Modelling and Simulation of Reactor

Pressure Vessel Failure during

Severe Accidents

PENG YU

Doctoral Thesis

Division of Nuclear Power Safety Department of Physics

School of Engineering Science KTH Royal Institute of Technology

Stockholm Sweden

AlbaNova University Center

Roslagstullsbacken 21

TRITA-SCI-FOU 202013 10691 Stockholm

ISBN 978-91-7873-540-2 Sweden

Akademisk avhandling som med tillstaringnd av Kungliga Tekniska Houmlgskolan

framlaumlgges till offentlig granskning foumlr avlaumlggande av teknologie doktorsexamen

i fysik den juni 12 2020 i AlbaNova Universitetscentrum Stockholm

copy Peng Yu June 2020

Tryck Universitetsservice US AB

To my family

and

all the people that I love

I

Abstract

This thesis aims at the development of new coupling approaches and new models for the thermo-fluid-structure coupling problem of reactor pressure vessel (RPV) failure during severe accidents and related physical phenomena The thesis work consists of five parts (i) development of a three-stage creep model for RPV steel 16MND5 (ii) development of a thermo-fluid-structure coupling approach for RPV failure analysis (iii) performance comparison of the new approach that uses volume loads mapping (VLM) for data transfer with the previous approach that uses surface loads mapping (SLM) (iv) development of a lumped-parameter code for quick estimate of transient melt pool heat transfer and (v) development of a hybrid coupling approach for efficient analysis of RPV failure

A creep model called lsquomodified theta projection modelrsquo was developed for the 16MND5 steel so that it covers three-stage creep process Creep curves are

expressed as a function of time with five parameters 120579119894 (i=14 and m) in the new creep model A dataset for the model parameters was constructed based on the available experimental creep curves given the monotonicity assumption of creep strain vs temperature and stress New creep curves can be predicted by interpolating model parameters from this dataset in contrast to the previous method that employs an extra fitting process The new treatment better accommodates all the experimental curves over the wide ranges of temperature and stress loads The model was implemented into the ANSYS Mechanical code and its predictions successfully captured all three creep stages and a good agreement was achieved between the experimental and predicted creep curves For dynamic loads that change with time the widely used time hardening and strain hardening models were implemented with a reasonable performance These properties fulfil the requirements of a creep model for structural analysis

A thermo-fluid-structure coupling approach was developed by coupling the ANSYS Fluent for the fluid dynamics of melt pool heat transfer and ANSYS Structural for structural mechanics of RPV An extension tool was introduced to realize transient load transfer from ANSYS Fluent to Structural and minimize the user effort Both CFD with turbulence models and the effective model PECM can be employed for predicting melt pool heat transfer The modified theta projection model was used for creep analysis of the RPV The coupling approach does not only capture the transient thermo-fluid-structure interaction feature but also support the advanced models in both melt pool convection and structural mechanics to improve fidelity and facilitate implementation The coupling approach performs well in the validation against the FOREVER-EC2 experiment and can be applied complex geometries such as a BWR lower head with forest of penetrations (control rod guide tubes and instrument guide tubes)

In the comparative analysis the VLM and SLM coupling approaches generally have the similar performance in terms of their predictability of the FOREVER-EC2 experiment and applicability to the reactor case Though the SLM approach predicted slightly earlier failure times than VLM in both cases the difference was negligible compared to the large scale of vessel failure time (~104 s) The VLM approach showed higher computational efficiency than the SLM

II

The idea of the hybrid coupling is to employ a lumped-parameter code for quick estimate of thermal load which can be employed in detailed structural analysis Such a coupling approach can significantly increase the calculation efficiency which is important to the case of a prototypical RPV where mechanistic simulation of melt pool convection is computationally expensive and unnecessary The transIVR code was developed for this purpose which is not only capable of quick estimate of transient heat transfer of one- and two- layer melt pool but also solving heat conduction problem in the RPV wall with 2D finite difference method to provide spatial thermal details for RPV structural analysis The capabilities of transIVR in modelling two-layer pool heat transfer and transient pool heat transfer were demonstrated by calculations against the UCSB FIBS benchmark case and the LIVE-7V experiment respectively The transIVR code was then coupled to the mechanical solver ANSYS Mechanical for detailed RPV failure analysis Validation against the FOREVER-EC2 experiment indicates the coupling framework successfully captured the vessel creep failure characteristics

Keywords RPV failure thermo-fluid-structure coupling creep modelling melt pool heat transfer severe accident CFD FEA transIVR

III

Sammanfattning

Syftet med denna avhandling aumlr att utveckla nya metoder och nya modeller foumlr att kunna koppla ihop termohydrauliska foumlrlopp och uppkommen strukturrespons i botten paring reaktortanken foumlr foumlrbaumlttrad analys av reaktortankbrott under ett svaringrt haveri Avhandlingen bestaringr av fem delar (i) utveckling av en trestegsmodell foumlr krypning i reaktortankens 16MND5-staringl (ii) utveckling av metoder foumlr kopp ling av termohydrauliska foumlrlopp och strukturrespons foumlr studier av reaktortankbrott (iii) jaumlmfoumlrelse av prestanda av den nya metoden som anvaumlnder volume loads mapping (VLM) foumlr dataoumlverfoumlring med den tidigare metoden som anvaumlnder surface loads mapping (SLM) (iv) utveckling av en lumped-parameter beraumlkningskod foumlr snabb uppskattning av transient vaumlrmeoumlverfoumlring i en smaumlltpoumll i botten paring reaktortanken och (v) utveckling av en hybrid kopplingsmetod foumlr effektiv analys av reaktortankbrott

En krypmodell kallad lsquomodified theta projection modelrsquo som behandlar krypprocessen i tre steg utvecklades foumlr 16MND5-staringlet Krypkurvor i den nya

krypmodellen ges som en funktion av tid med fem parametrar 120579119894 (i=14 och m) Ett dataset foumlr modellparametrar togs fram baserat paring tillgaumlngliga experimentalla krypkurvor under anatagande om att kryptoumljning aumlr en monoton funktion av temperatur och spaumlnning Nya krypkurvor kan genereras genom att interpolera modellparametrar fraringn detta dataset i kontrast till den tidigare metoden som anvaumlnder en speciell anpassningsprocess Foumlrdelen med det nya tillvaumlgagaringngssaumlttet aumlr att den hanterar alla experimentella krypkurvor foumlr ett brett spektrum av temperatur- och spaumlnningsbelastningar Modellen implementerades i ANSYS-koden och den korrekt behandlar alla tre krypsteg samt visar bra oumlverenstaumlmmelse mellan experimentella och beraumlknade krypkurvor Foumlr analys av tidsberoende dynamiska belastningar implementerades vaumllkaumlnda modeller av tids- och deformationshaumlrdning med rimliga resultat Dessa egenskaper uppfyller kravet paring en krypmodell foumlr strukturmekaniska analyser

En metod utvecklades foumlr koppling mellan termohydrauliska foumlrlopp och strukturmekanisk respons genom att koppla ihop ANSYS Fluent som analyserar fluiddynamiska aspekter av vaumlrmeoumlverfoumlring i en smaumlltpoumll med ANSYS Structural som analyserar strukturmekanisk respons av reaktortanksvaumlggen Ett speciellt verktyg togs fram foumlr att moumljliggoumlra oumlverfoumlring av transienta belastningar fraringn ANSYS Fluent till ANSYS Structural och foumlr att minimera anvaumlndarinsatsen Baringde CFD med turbulensmodeller och den effektiva PECM-modellen kan anvaumlndas foumlr att analysera vaumlrmeoumlverfoumlring i en smaumlltpoumll Foumlr krypanalysen av reaktortanken har lsquomodified theta projection modelrsquo anvaumlnts Den framtagna kopplingsmetoden har foumlrdelen att den inte bara faringngar de transienta aspekterna av interaktionen mellan termohydraulik och strukturmekanik men ocksaring stoumldjer avancerade modeller av konvektion i en smaumlltpoumll och strukturrespons och daumlrigenom bidrar till foumlrbaumlttrad precision och underlaumlttar implementeringen av modeller Kopplingsmetoden fungerar bra naumlr den valideras mot FOREVER-EC2 experimentet och den kan anvaumlndas i

IV

komplexa geometrier som till exempel i botten paring reaktortanken i en kokvattenreaktor (BWR) med en skog av genomfoumlringar (ledroumlr foumlr styrstavar och instrumentering)

Jaumlmfoumlrande beraumlkningar visar att de tvaring kopplingsmetoderna VLM-baserad och SLM-baserad presterar lika bra naumlr det gaumlller hur de predikterar resultat av FOREVER-EC2 experimentet och deras laumlmplighet foumlr reaktoranalyser Aumlven om SLM-metoden predikterar att tankbrott i baringda fallen intraumlffar lite tidigare saring aumlr skillnaden mellan metoderna foumlrsumbar i jaumlmfoumlrelse med tiden till tankbrott (~ 104 s) VLM-metoden var baumlttre aumln SLM-metoden naumlr det gaumlller beraumlkningseffektiviteten

Ideacuten om hybridkoppling garingr ut paring att anvaumlnda en lumped-parameter kod foumlr snabb uppskattning av termiska belastningar som senare kan anvaumlndas i detaljerad strukturmekanisk analys En saringdan kopplingsmetod kan paring ett signifikant saumltt oumlka effektiviteten av beraumlkningar vilket aumlr viktig i reaktorapplikationer daumlr mekanistisk simulering av konvektion i en smaumlltpoumll aumlr beraumlkningsintensiv och onoumldig I detta syfte utvecklades koden transIVR Koden kan dels ge en snabb uppskattning av transient vaumlrmeoumlverfoumlring i en smaumlltpoumll som bestaringr av en eller tvaring materialskikt och dels loumlsa vaumlrmeledningsproblem i reaktortanksvaumlggen med 2D finita differensmetoden vilket genererar detaljerad information om termiska foumlrharingllanden som behoumlvs foumlr strukturmekaniska analyser Foumlr att demonstrera att transIVR kan analysera dels vaumlrmeoumlverfoumlring i en smaumlltpoumll bestaringende av tvaring materialskikt och dels transient vaumlrmeoumlverfoumlring i en smaumlltpoumll jaumlmfoumlrdes transIVR prediktioner med UCSB FIBS-benchmarkfallet respektive LIVE-7V-experimentet I naumlsta steg kopplades transIVR-koden till ANSYS Structural foumlr att moumljliggoumlra mer detaljerad analys av reaktortankbrott Validering mot FOREVER-EC2 experimentet tyder paring att den utvecklade kopplingsmetodiken korrekt aringterger tankbrott orsakat av krypning

Nyckelord reaktortankbrott koppling termohydraulik-strukturmekanik modellering av krypning vaumlrmeoumlverfoumlring i smaumlltpoumll svaringra haverier CFD FEA transIVR

V

List of Publications

Papers included in this thesis

I P Yu and W Ma ldquoA modified theta projection model for creep behavior of RPV steel 16MND5rdquo Journal of Materials Science amp Technology vol 47 pp 231ndash242 2020

II P Yu W Ma W Villanueva A Karbojian and S Bechta ldquoValidation of a thermo-fluid-structure coupling approach for RPV creep failure analysis against FOREVER-EC2 experimentrdquo Annals of Nuclear Energy vol 133 pp 637ndash648 2019

III P Yu and W Ma ldquoComparative analysis of reactor pressure vessel failure using two thermo-fluid-structure coupling approachesrdquo Nuclear Engineering and Design submitted

IV P Yu and W Ma ldquoDevelopment of a lumped-parameter code for efficient assessment of in-vessel melt retention strategy of LWRsrdquo Annals of Nuclear Energy submitted

Respondentrsquos contribution to the papers included in the thesis

Peng Yu made major contributions to the publications of all the above four articles under the guidance of supervisors and with the collaboration of other co-authors including formulation of research questions development of coupling approaches and numerical methods analysis and generalization of simulation results as well as writing the first drafts and addressing review comments

Papers not included in this thesis

1 P Yu and W Ma ldquoRPV creep failure analysis using a Norton creep modelrdquo International Journal of Mechanical Sciences to be submitted

2 P Yu W Villanueva S Galushin W Ma and S Bechta ldquoCoupled thermo-mechanical creep analysis for a Nordic BWR lower head using non-homogeneous debris bed configuration from MELCORrdquo 12th International Topical Meeting on Nuclear Reactor Thermal-Hydraulics Operation and Safety (NUTHOS-12) Qingdao China October 14-18 2018

3 P Yu A Komlev Y Li W Villanueva W Ma and S Bechta ldquoPre-test simulation of SIMECO HT experiments for stratified melt pool heat transferrdquo 11th International Topical Meeting on Nuclear Reactor Thermal Hydraulics Operation and Safety (NUTHOS-11) Gyeongju Korea October 9-13 2016

VI

VII

Acknowledgements

First of all I would like to express my sincere gratitude to my supervisor Prof Weimin Ma for offering this opportunity to work in the severe accident research field and for the supports patience guidance discussions and encouragements throughout the entire period of my graduate study

I am very thankful to my co-supervisor Dr Walter Villanueva for the valuable help and discussions I learnt a lot during the times working with him I am also thankful to co-supervisor Prof Sevostian Bechta who is supportive and always has inspiring ideas that keep you thinking for a long time

Thank you also to all my colleagues at the Division of Nuclear Power Safety for the supports discussions and the very nice working environment during these years

I have taken many interesting courses during my PhD study for which I want to express my thanks to all the teachers With this regard a special thanks goes to Prof Johan Hoffman for his SF2561 Finite Element Method given in 2015 fall as well as the discussions and advices on numerical analysis

I am grateful to Ignacio Gallego-Marcos and Wenyuan Fan for their enthusiastic help on CFD and turbulence modelling during the time working on the turbulent

heat transfer I thank all my 师兄 for sharing their knowledge on nuclear engineering and giving advices when I was confused since 2012

I thank all my friends both in Sweden and in China for the company and the sharing of many wonderful moments in daily life

Finally I would like to express my deepest thank to my parents my girlfriend Siwen my sisters and all my other relatives for their unconditional supports and love

Financial supports from the China Scholarship Council and APRI make my study in KTH possible and are acknowledged

VIII

IX

Table of Contents

AbstracthelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphellipI

SammanfattninghelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphellipIII

List of PublicationshelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphellipV

AcknowledgementshelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphellipVII

NomenclaturehelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphellipXI

1 Introductionhelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphellip1

11 Nuclear Power Safety 1

12 RPV Failure 2

13 Literature Review Creep Behaviour and RPV Failure Analysis 4

131 Experiments on creep behaviour and RPV failure 4

132 Numerical studies on RPV failure 5

14 Motivation Objectives and Main Achievements 7

2 Numerical Solvers and Modelshelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphellip9

21 ANSYS Fluent 9

22 ANSYS Mechanical 9

23 PECM 11

3 Creep Modelling of 16MND5 Steelhelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphellip13

31 Creep and Creep Modelling 13

32 A Modified Theta Projection Model 14

321 Original theta projection model and its variations 14

322 A modified theta projection model 16

4 Coupled Thermo-Fluid-Structure Analysis of RPV Failurehelliphelliphelliphellip23

41 Thermo-Fluid-Structure Coupling Approach 23

42 Numerical Modelling 25

421 Conjugate heat transfer 25

422 Mechanical analysis 25

43 Validation against the FOREVER-EC2 Experiment 26

431 FOREVER-EC2 experiment 26

X

432 Conjugate heat transfer 27

433 Mechanical analysis 29

44 Application to BWR Case 32

45 Computational Efficiency 34

5 Hybrid Coupling Approach for Efficient RPV Failure Analysishelliphellip35

51 A Lumped-parameter Code for Melt Pool Heat Transfer 35

511 Heat transfer correlations 36

512 Energy balance in the pools 37

513 Energy balance in the RPV wall 39

52 Hybrid Coupling Approach 40

53 Verification and Validation 41

531 UCSB FIBS benchmark case 41

532 LIVE-7V experiment 43

533 FOREVER-EC2 experiment 45

6 Summary and Outlookhelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphellip51

Bibliographyhelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphellip55

XI

Nomenclature

Latin

119860 area of a surface

119862119901 heat capacity

119863 damage factor

119892 gravity

ℎ heat transfer coefficient

H Melt pool height

119896 thermal conductivity

119871 characteristic length

119873119906 Nusselt number

119875119903 Prandtl number

119902 heat flux

119902119894119899 heat flux through inner vessel surface

119876119881 internal volumetric heat source

119903 radial distance in a cylindrical or spherical coordinate

119877119886 external Rayleigh number

119877119886119894119899 internal Rayleigh number

119877119886119910 local Rayleigh number

119905 time

119879 temperature

119881119900119897 total volume

119909 thickness (of crust or RPV wall)

119911 height in a cylindrical coordinate

Greek

120572 thermal diffusivity

120573 thermal expansion

Δ119879 temperature difference

ε emissivity (in thermal analysis) strain (in structural analysis)

휀119888119903 creep strain

휀119903 creep strain rate

120579 polar angle

120579119894 parameters of a creep model 119894 = 1234119898

120584 kinematic viscosity

σ stress

XII

Super and subscripts

119886119898119887 ambient

119888119903 creep (in Section 3) crust (in Section 5)

119889119899 downward

119890119899119907 environment

119894 inner surface of RPV

119894119899119905 interface

119897119900119888119886119897 local

119898119886119909 maximum value

119898119890119897119905119894119899119892 melting point

119898119905 metal layer (bulk)

119898119905119889119899 downward surface of metal pool

119898119905119904119889 sideward surface of metal pool

119898119905119905119900119901 top surface of metal layer

119898119905119906119901 upward surface of metal pool

119900 outer surface of RPV

119900119909 oxide layer (bulk)

119900119909119889119899 downward surface of oxide layer

119900119909119906119901 upward surface of oxide layer

119903119890119891 reference

119904 internal structures inside the RPV

119904119889 sideward

119906119901 upward

119907119890119904 vessel

Acronyms

BWR boiling water reactor

CBD creep database

CFD computational fluid dynamics

CRGT control rod guide tube

ECCM effective convectivity conductivity model

FDM finite difference method

FE finite element

FEA finite element analysis

FEM finite element method

FVM finite volume method

IGT instrument guide tube

IVR in-vessel retention

XIII

IVMR in-vessel melt retention EU project

LES large eddy simulation

MPS moving particle semi-implicit method

NPP nuclear power plant

PECM phase-change effective convectivity model

RANS Reynolds-averaged Navier-Stokes

RPV reactor pressure vessel

SBO station blackout

SLM surface loads mapping

SPH smoothed-particle hydrodynamics

UDF user-defined function

UPF user programmable features

VLM volume loads mapping

XIV

1

1 Introduction

11 Nuclear Power Safety

Greenhouse gas emissions contribute to the global warming problem Phasing out fossil fuels by switching to low-carbon energy sources is considered as a mitigation strategy With this regard clean energy should be increased in the world energy consumption structure Being clean and stable nuclear energy is an attractive alternative to fossil fuels After the development over decades it also proves to be a mature and safe technology According to IAEA 2018 annual report [1] there were 450 nuclear power reactors under operation worldwide at the end of 2018 which had a record generating capacity of 3964 GW(e) 9 reactors were connected to the grid and 55 reactors under construction around the world

Due to the potential risk of the release of radioactivity nuclear power safety is emphasized throughout the stages of design construction operation and decommission of a nuclear power plant (NPP) The principal technical requirements [2] specify that the three fundamental safety functions should be fulfilled for all plant states of an NPP (i) control of reactivity (ii) removal of heat from the reactor and from the fuel store and (iii) confinement of radioactive material shielding against radiation and control of planned radioactive releases as well as limitation of accidental radioactive releases The defence-in-depth concept is also applied to provide several levels of defence to prevent consequences of accidents The fuel claddings reactor pressure vessel (RPV) and primary coolant pressure boundaries and the containment are the three physical barriers to prevent the release of radioactive materials to the environment

Severe accidents are events that cause significant damage to reactor fuel and result in core meltdown It may be caused by insufficient cooling of the reactor core due to various reasons or by the rapid introduction of huge reactivity which causes sharp increases in the power and fuel fragmentation eg Chernobyl accident During a severe accident the failure of the fuel claddings will result in the radioactive materials previously enclosed inside the cladding released to the primary loop If cooling is not sufficient to remove the decay heat the core materials will melt migrate down and relocate into the RPV lower head after which the melt will react with water and partially form fragments quench solidify and form a debris bed The debris bed formed by the melt may continue to melt to form a melt pool due to the decay heat As the corium contacts the RPV it would continuously heat the RPV and threaten its integrity Once the RPV fails containment becomes the last barrier to prevent the release of radioactivity The released melt would result in an increase of the pressure and temperature due to the high temperature fractions oxidation and hydrogen combustion which is called direct containment heating The melt would fall onto the reactor cavity Without sufficient cooling the molten core would interact with basemat concrete (MCCI) This process can last for few days It can result in containment failure due to either containment overpressure or the basemat melt-through of containment In addition containment is also subjected to isolation failure and early containment failure due to missiles and dynamic pressure [3] More

2

detailed severe accident progression or phenomena could also be found in eg [4] [5]

Though severe accidents are unlikely to occur the consequences can be severe if no proper measure is taken Related researches have been continuously going on aiming at mitigating the accident consequences and improving safety After the Fukushima Accident safety inspections and re-evaluation actions have been carried out worldwide especially on safety measurements regarding severe accidents The concept lsquoPractical Eliminationrsquo applied to new NPP designs demands that new nuclear installations be designed with the objective of mitigating the consequences of accidents and avoiding early radioactive releases and large radioactive releases [6] Severe accident mitigation strategies are applied to the existing and new NPPs (both Generation II and III) The strategies can generally be classified into two categories [3] the in-vessel retention (IVR) and ex-vessel retention (EVR) In simple words in-vessel retention tries to retain the core melt inside the RPV (second barrier) by applying external water cooling on the vessel as shown in Figure 1(a) Such method minimizes the dispersal of the melt within the containment and hence simplifies the later recovery work Examples of the application are AP1000 HPR1000 APR 1400 The idea of the EVR is to let the RPV fail but to retain the core melt inside the containment Two principle approaches of EVR are the retention within a dedicated lsquocore catcherrsquo and melt fragmentation Examples of core catcher would be EPR that directs the melt to a spreading compartment (Figure 1(b)) and VVER1000 that uses metallic crucible below the RPV An example of the melt fragmentation is the Nordic BWRs that the corium released from RPV would relocate in the cavity with a deep-water pool (7-12 m) and is expected to fragment quench and form a coolable debris bed (Figure 1(c))

Figure 1 Examples of IVR [7] EVR with core catcher [8] and melt fragmentation [9]

12 RPV Failure

RPV which contains the reactor core core shroud and coolant is a key component to the reactor system RPV and the primary loop constitute the primary coolant system which conveys the fission heat from the reactor core to the secondary loop through steam generator (PWR) or to the turbine (BWR) The pressure under

(a) IVR (b) Core catcher (c) Melt fragmentation

3

operation can be ~155 MPa for a 1000 MW PWR [10] and ~70 MPa for a 1000 MW BWR [11] It also has to withstand the neutron radiation damage The thickness of RPV [11] can be ~20 cm for PWR and ~16 cm for BWR

As the transition of in-vessel progression to ex-vessel progression RPV failure is important for analysis of accident progression and assessment of the effectiveness of severe accident mitigation strategies In the melt fragmentation strategy of Nordic BWR the melt-coolant interaction and debris bed coolability after the corium poured into the deep-water pool are essentially dependent on the vessel failure mode including the characteristic time of melt release rupture size and location and amount and superheat of melt available for release [12] New concern also arose [13] on the necessity of performing structural analysis for the RPV in the IVR strategy In the initial methodology and further development of the IVR the success of the strategy ie the integrity of the RPV was demonstrated (eg in [14]) with the following two conclusions (i) even for a local heat flux on the vessel as high as critical heat flux (CHF) the resulted very thin external ldquocold layerrdquo of the vessel would be sufficient to withstand the internal stress load and hold the vessel integrity (ii) as CHF is the critical value for the thermal load the main specific goal for assessing the effectiveness of IVR becomes to demonstrate that the local heat flux along RPV be lower than the CHF The new concern doubts that local lsquohot partrsquo of the vessel would have negative effects on the overall resistance due to creep and dilatation which could result in a vessel failure and hence violate the first conclusion in particular for ldquolong termrdquo retention of corium in RPV If so a structural integrity analysis of RPV would be needed in addition to assessing local heat flux especially for high-power reactors Therefore accurate prediction of vessel failure could be meaningful for both strategies

Figure 2 Illustration of RPV under thermal loads from corium

As the structural performance is highly affected by the thermal loads from the corium RPV failure is essentially a transient thermo-fluid-structure coupled problem Figure 2 illustrates the situation that a corium debris bedmelt pool formed in the RPV lower head The decay heat of nuclear fission products could continuously be generated inside the corium leading to the formation of melt pools as the debris melts The heated debris bedmelt pool imposes thermal and

4

mechanical loads on the RPV Such loads can be transient (change with time) and spatially non-uniform Complex phenomena are also involved in this process Strong turbulent natural convection induced by the decay heat may be expected in the melt pool under IVR The corresponding Rayleigh number 119877119886 can be 1016minus17 for prototypic reactor cases Material structural properties can become non-linear under high temperature and stress loads The creep deformation also becomes significant at high temperature Detailed modelling of turbulent natural convection and non-linear properties could be computationally expensive In addition large-scale geometry (hemisphere with a radius of 20-30 m) long-term transient deformation process (~10000 s) and required small time steps (eg 001 s) for numerical convergence consideration further increase the computational costs

13 Literature Review Creep Behaviour and RPV Failure Analysis

In this section the most relevant studies concerning the experiments modelling and analysis of creep behaviour and RPV failure are presented

131 Experiments on creep behaviour and RPV failure

Various experiments have been performed on the RPV failure problem some of which were to generate data of mechanical properties (especially the creep properties) and some to study the overall vessel failure with scaled or simplified geometries

The REVISA experiment [15] generated creep curves and stress strain curves of French steel 16MND5 ranging from 600 to 1300 degC This data is accessible in [16] as well as [17] which also documents other material properties eg density thermal expansion coefficient conductivity heat capacity Youngrsquos Modulus and plasticity Similar experiments were carried out to generate high-temperature creep and tensile test data for American RPV steels and Inconel [18]

The CORVIS experiments [19] investigated the lower head failure modes with and without penetrations Each experiment was carried out on a steel plate 100 mm thick representing a pressure vessel lower head The penetration case represented a boiling water reactor lower head carrying a tube penetration in the form of a drain line

The LHF (lower head failure) experiments [20] and the succeeding OLHF tests [21] investigated the vessel failure with a 15 scaled vessel heated by an electrical radiator allowing to locate the hot focus at different position of the vessel with a maximum wall temperature of around 900 degC Internal pressure ranging from 53 to 123 bar was also imposed to initiate the vessel failure

The FOREVER experiments [22] investigated the phenomena of melt pool convection vessel creep possible failure processes and failure modes occurred during the late phase of the in-vessel progression by heating up a melt pool in a vessel and observing possible vessel failures The study covers cases with and without penetration different melt level existence of gap cooling and in-vessel melt coolability

5

132 Numerical studies on RPV failure

Various numerical approaches have been employed on RPV failure Since a two-way thermo-fluid-structure coupling approach is challenging these approaches take simplifications and can generally be classified into two categories (i) RPV failure decoupled from melt pool heat transfer and (ii) RPV failure coupled with melt pool heat transfer in simplified manners

1) Decoupled approaches In the decoupled approaches only RPV is modelled Simplified constant thermal loads are imposed on RPV considering the melt pool heat transfer

In a post-test analysis of FOREVE-C2 experiment [23] it was treated as a thermo-structural coupled analysis of the vessel with the thermal load estimated by the outer thermocouples and the normal heat flux on the external vessel wall The analysis was done using finite element (FE) code SYSTUS A coupled creepdamage law based on a viscosity-hardening creep law describing primary and secondary creep was employed for creep analysis

Mao et al conducted various studies on reactor cases assessing the IVR strategy using FE code ABAQUS Thermal boundary conditions were imposed on the RPV which was then solved as a thermal-structural coupled problem The strain hardening creep model (Eq (1)) was mainly used Their studies cover the general RPV failure [24] as well as the effects of cooling water levels under IVR [25] and geometric discontinuity (ablated RPV) [26]

ε119888119903 = 1198891σ1198892ε1198893119890119909119901 [minus

1198894119879] (1)

In the EU IVMR project [13] the RPV integrity was also studied through a benchmark calculation which deals with the RPV of a PWR with a prescribed ablated profile with 2D axisymmetric geometry Prescribed temperature gradient was set as thermal load The objective was to calculate deformations for various pressure loads up to vessel rupture A smoother profile was planned in further research

The advantage of such approaches is once thermal loads are reasonably assumed the focus only needs to be put on the RPV The disadvantage is a reasonable thermal load should be consistent with the melt pool heat transfer while sometimes it is not well-noted eg constant uniform temperature boundary conditions were assumed for inner and outer RPV (with uniform thickness in lower head) surfaces [25] which ignored the non-uniform heat flux distribution caused by the turbulent convective corium pool As thermal loads strongly affect the structural performance they should be quantified carefully

2) Coupled approaches In coupled approaches both melt pool heat transfer and RPV failure are modelled and coupled in one-way or simplified manner With melt pool heat transfer modelled the coupled approaches are more flexible to provide reasonable thermal loads on the RPV In existing approaches CFD and the correlation-based distributed-parameter models eg the Effective Convectivity Conductivity Model

6

(ECCM) [27] and the Phase-change Effective Convectivity Model (PECM) [28] [29] are general methods for melt pool heat transfer

Sehgal et al [22] presented an approach that employed the MVITA code to calculate the thermal processes and ANSYS for structural analysis MVITA code employed the ECCM Bailey-Norton creep law was used as shown in Eq (2)

ε119888119903 = 6055 times 10minus26 sdot σ472119905085119890minus

37200

119879 (2)

This approach was used for the pre-test simulation of FOREVER experiment to determine the thermal and mechanical loadings Parametric investigations also helped in the identification of important factors that could affect the test procedure and test conditions

Willschutz et al [30] employed the CFD-module Flotran for the melt pool heat transfer and ANSYS for structural analysis of RPV Standard k-ε turbulence model was selected for the turbulent natural convection in the melt pool A user-defined three-stage creep model [31] was employed In this model experimental creep curves were discretized into a creep database (CDB) under given temperature 119879 and stress 120590 the creep curve was discretized into finite point pairs

(휀 ε) indicating under such thermal and mechanical loads the creep strain rate ε is subjected to creep strain 휀 The creep strain rate for any other case can be interpolated over known temperature stress and strain from the database This method was applied to the post-test calculation of the FOREVE-C2 experiment It was concluded that the unexpected deformation behaviour during the experiment was caused by the combination of the transient temperature field in the vessel wall and the three creep stages

They later replaced the Flotran solver with the ECCM for melt pool heat transfer in a following study [32] By implementing it into FEM-solver a recursively coupled analysis was constructed Moreover a modified creep model [33] was used with damage measures the CDB itself did not contain tertiary creep instead a factor 1(1 minus 119863) was multiplied to the original creep strain rate to get the modified creep strain rate where 119863 is a damage factor (0 lt 119863 lt 1) defined with creep strain information and plastic strain information As 119863 grows close to 1 the modified creep strain rate would be enlarged significantly by this factor 1(1 minus 119863) which then behaves like the accelerating tertiary stage This method was applied to pre- and post- test calculations for the FOREVER experiments proving to be closer to reality than the previous method

Madokoro et al [34] developed a thermal structural analysis tool based on open source solver OpenFOAM and numerically investigated the coupled RPV failure and melt pool convection The PECM was used in the melt pool heat transfer In structural part the plasticity and damage parameters were similar to Willschutz et al [32] except for that strain hardening creep model (Eq (3)) was used instead of a CDB Simulations of FOREVER-EC2 and -EC4 were investigated It was concluded from the simulations that although the emissivity and mesh type (updated or fixed) affect the results significantly the approach generally has the capability of predicting the vessel deformation and failure

7

ε119888119903 = 1198891 sdot σ1198892 sdot ε1198893 (3)

Villanueva et al [35] performed a coupled thermo-mechanical creep analysis of the reactor scale RPV (ABB-Atom design BWR) to evaluate efficacy of the cooling measures for the in-vessel melt coolability and retention The PECM was used for melt pool heat transfer and ANSYS for RPV integrity analysis The modified time hardening model (Eq (4)) of the American RPV steel SA533B was selected as creep model The transient thermal load from the melt pool was applied as a thermal boundary condition to the vessel wall Further studies were done with this approach on investigations of instrumentation guide tube (IGT) failure [36] and control rod guide tube (CRGT) cooling [37] of a Nordic BWR

ε119888119903 =1198881σ

11988821199051198883+1

1198883 + 1 1198881 gt 0 (4)

It is noted that only RPV analysis with finite element analysis (FEA) are reviewed here Engineering methods like the Larson-Miller method [38] are not covered in this thesis One can refer to theory manuals of severe accident codes like MELCOR ASTEC for more details Comprehensive description on modelling of melt pool heat transfer is not covered either One can refer to either [39] for a general view or to any severe accident code manuals for engineering methods

14 Motivation Objectives and Main Achievements

The existing experimental and numerical studies provide us a good understanding of RPV failure involved phenomena and valuable simulation experiences These numerical methods may vary in solvers (which may be related to code accessibility) and models (which may be related to model accessibility understanding of phenomena computational power) However improvements of modelling approaches are still possible as the improvements in any of the related aspects eg

bull Simulation tools are continuously being upgraded and providing new possibilities just like that PECM appeared and ECCM was no longer used

bull Better understanding of the phenomena and more simulation experiences from in-vessel retention and vessel failures researches benefit the further modelling and simulations

bull Increasing computational power allows more detailed modelling eg turbulence modelling of melt pool convection

bull There is still room for the improvement of creep modelling

In addition implementation difficulty of a coupling approach is another important issue to consider Since it is a TRANSIENT thermo-fluid-structure coupled problem it is almost impossible to replicate an approach simply by referring to a paper with limited details For BWRs it would be even better for the coupling approaches to support complex geometries considering forest of penetrations (CRGTs and IGTs) on the RPV lower head without significant increase of user effort in the coupling process

8

The objectives of this thesis are to investigate new approaches and develop new models for the thermo-fluid-structure coupled problem of RPV failure and involved phenomena These objectives are accomplished with the following achievements

1) A new creep model for the 16MND5 steel was developed which covers all three creep stages Good agreement was achieved between model predictions and experimental curves Both time and strain hardening models were implemented (Paper I)

2) A new thermo-fluid-structure coupling approach was developed for RPV failure analysis It supports the advanced models in both melt pool convection and structural mechanics to improve fidelity and facilitates implementation Validation of the approach with the new creep model was performed against FOREVER-EC2 experiment This approach is flexible for complex geometries and can be used in further analysis eg lower head with detailed modelling of penetrations in BWRs (Paper II)

3) The performance of this new approach which uses volume loads mapping for data transfer was compared with the previous approach using surface loads mapping showing good agreement with small discrepancy (Paper III)

4) A lumped-parameter code transIVR for melt pool heat transfer was developed which supports one- and two-layer pool configuration transient analysis and well-resolved RPV wall (Paper IV)

5) A hybrid coupling approach was developed for efficient analysis of RPV failure where transIVR is coupled with ANSYS Mechanical (Paper IV)

In the following general descriptions on the codes ANSYS Fluent Mechanical and the PECM are given in Section 2 the new creep model in Section 3 the new coupling approach as well as its comparison with the previous approach in Section 4 the efficient hybrid coupling approach in Section 5 and summary amp outlook in Section 6

9

2 Numerical Solvers and Models

In this section we briefly introduce the ANSYS Fluent as the CFD solver for melt pool heat transfer the ANSYS Mechanical as the mechanical solver for RPV failure and PECM as an efficient alternative for melt pool heat transfer that are heavily used in the following sections

21 ANSYS Fluent

In the fluid mechanics the fluid is treated as a continuum which is generally described by the conservative mass momentum and energy equations [40]

120597120588

120597119905+ 120571 sdot (120588) = 119878119898 (5)

120597(120588)

120597119905+ 120571 sdot (120588) = minus120571119901 + 120571 sdot (120591) + 120588 + (6)

120597120588119864

120597119905+ 120571 sdot ((120588119864 + 119901)) = 120571 sdot (119896120571119879 minussumℎ119895119869119895

119895

+ (120591 sdot )) + 119878ℎ

(7)

Computational Fluid Dynamics (CFD) studies topics on the numerical solutions of above equations Typical methods are the Finite Volume Method (FVM) Finite Difference Method (FDM) and Finite Element Method (FEM) as well as particle methods that are mesh-free eg SPH and MPS

ANSYS Fluent is a general purpose CFD code which solves above equations with FVM Turbulence modelling and multiphase flows are supported Turbulence can be modelled with RANS LES or hybrid LESRANS Customizations eg adding sourcesink time or spatial dependent boundary conditions are realized through the user-defined functions (UDF) which is programmed in C language

22 ANSYS Mechanical

In the continuum mechanics of solids three sets of variables are considered to describe the structure performance namely element displacement field strain field and stress field (119906 ε σ) A general 3-dimensional problem is described by 3 sets of equations the equilibrium equations (Eq (8)) strain-displacement relationship (Eq (9)) and constitute equations (Eq (10)) [41] For Eq (8) the reciprocal theorem of shear stress also gives τ119894119895 = τ119895119894 where 119894 119895 = 119909 119910 119911

indicating the stress tensor being symmetric For very simple applications eg 1D beam it can be solved analytically For complex problems where analytical solutions are not available approximation methods eg Galerkin weighted residual method principle of virtual work and principle of minimum potential energy would be used

10

partσ119909119909part119909

+partτ119909119910

part119910+partτ119909119911part119911

+ 119909 = 0

partτ119910119909

part119909+partσ119910119910

part119910+partτ119910119911

part119911+ 119910 = 0

partτ119911119909part119909

+partτ119911119910

part119910+partσ119911119911part119911

+ 119911 = 0

(8)

ε119909119909 =part119906

part119909 휀119910119910 =

120597119907

120597119910 휀119911119911 =

120597119908

120597119911

120574119909119910 =120597119906

120597119910+120597119907

120597119909 120574119910119911 =

120597119908

120597119910+120597119907

120597119911 120574119911119909 =

120597119908

120597119909+120597119906

120597119911

(9)

ε119909119909 =1

119864[σ119909119909 minus μ(σ119910119910 + σ119911119911)]

ε119910119910 =1

119864[σ119910119910 minus μ(σ119909119909 + σ119911119911)]

ε119911119911 =1

119864[σ119911119911 minus μ(σ119909119909 + σ119910119910)]

γ119909119910 =1

119866τ119909119910 γ119910119911 =

1

119866τ119910119911 γ119911119909 =

1

119866τ119911119909

(10)

ANSYS Mechanical [42] is a mechanical engineering software that uses finite element analysis (FEA) for structural analysis The principle of virtual work is adopted in the solver [43] The idea of this principle is that a virtual (very small) change of the internal strain energy must be offset by an identical change in external work due to the applied loads which reads

δ119880 = δ119881 (11) where 119880 is the strain energy or the internal work 119881 the external work and δ the virtual operator The virtual internal work can be expressed as

δ119880 = int δεσd(vol)T

vol

+int 119886119903119890119886119891

δ119908119899119879σ119889(119886119903119890119886119891) (12)

where 휀 is the strain vector 120590 stress vector vol volume of element 119908119899 motion normal to the surface 119886119903119890119886119891 area of the distributed resistance The virtual

external work is contributed from three parts the inertial effects the pressure force on the surface and nodal forces applied to the element which then can be expressed as

δ119881 = minusint 119907119900119897

δ119908119879119865119886

119907119900119897119889(119907119900119897) + int

119886119903119890119886119901

δ119908119899119879119875119889(119886119903119890119886119901)

+ δ119906119879119865119890119899119889

(13)

11

where 119908 is the vector of displacements 119865119886 acceleration force vector 119875 applied pressure vector 119886119903119890119886119901 area over which pressure acts and 119865119890

119899119889 nodal

forces applied to the element

Plugging Eq (12) and (13) into (11) would result in the equilibrium equation Since the virtual displacement can be arbitrary the final equations used to describe the equilibrium of an element is reduced to Eq (14) with multiple operations

([119870119890] + [119870119890119891])119906 minus 119865119890

119905ℎ = [119872119890] + 119865119890119901119903 + 119865119890

119899119889

(14)

where [119870119890] [119870119890119891] 119865119890

119905ℎ [119872119890] and 119865119890119901119903 are the element stiffness matrix

foundation stiffness matrix thermal load vector mass matrix acceleration vector and pressure vector respectively It represents the equilibrium equation on a one element basis Alternatively using a minimum total potential energy principle could result in a consistent expression to the principle of virtual work [41]

Customizations eg on material properties are realized through the user programmable features (UPF) which is programmed in FORTRAN language

23 PECM

The PECM [28] [29] developed in KTH is a correlation-based method to model the melt pool heat transfer It is implemented in ANSYS Fluent through UDF and has been validated against many experiments such as ACOPO COPO LIVE etc In the PECM the flow velocities 119906119909 119906119910 and 119906119911 are replaced by characteristic

velocities 119880119909 119880119910 and 119880119911 respectively in order to eliminate the necessity of solving

the Navier-Stokes equation

The characteristic velocities are calculated using heat transfer correlations

119880119906119901 =α

ℎ119901119900119900119897times (119873119906119906119901 minus

ℎ119901119900119900119897

ℎ119906119901) (15)

119880119889119899 =α

ℎ119901119900119900119897times (119873119906119889119899 minus

ℎ119901119900119900119897

ℎ119889119899) (16)

119880119904119889 =α

ℎ119901119900119900119897times (119873119906119904119889 minus

ℎ119901119900119900119897

119882119901119900119900119897) (17)

where ℎ119901119900119900119897 is the melt pool depth ℎ119906119901 the thickness of well mixed layer of the

pool ℎ119889119899 the thickness of lower stratified region of melt pool 119882119901119900119900119897 the width of

pool 120572 the thermal diffusivity The PECM employs reduced characteristic velocities to describe mushy zone convection heat transfer For internally heated pool natural convection the Steinberner-Reineke correlations (Eq (18)~(19)) [44] are employed The profile of sideward 119873119906 is described using Eckertrsquos type correlation (Eq (20)) for a vertical boundary layer where the local Rayleigh

number at pool height 119910 is defined as 119877119886119910 =119892βΔ1198791199103

αν

12

119873119906119906119901 = 03451198771198861198941198990233

(18)

119873119906119889119899 = 13891198771198861198941198990095

(19)

119873119906119904119889 = 050811987511990314 (

20

21+ 119875119903)

minus14

11987711988611991014

(20)

As the characteristic velocities are calculated the energy equation is the only transport equation that needs to be solved and can be expressed as follows

partρ119862119901119879

part119905+ nabla(ρ119862119901119932119879) = 119896nabla2119879 + 119876119907 (21)

where from left to right are the transient term convection term diffusion term and volumetric heat source respectively The convection term then can be calculated explicitly with use of values in the last time step and then can be treated together with the volumetric source term Therefore the PECM finally simplifies the energy equation to a conduction equation which can be easily solved and as a result will significantly increase the computational efficiency in simulations

In addition solidificationmelting process can also be modelled and the temperature-dependent physical properties are supported

13

3 Creep Modelling of 16MND5 Steel

Creep deformation becomes significant when a component is subjected to high temperature and stress Modelling of creep behaviour of the RPV is paramount since it is considered as a main mechanism leading to the RPV failure In this section we present the development of a three-stage creep model (called lsquomodified theta projection modelrsquo ) for the French RPV steel 16MND5 This model was successfully implemented into the ANSYS Mechanical code

31 Creep and Creep Modelling

Creep is an irreversible time-dependent inelastic deformation process Diffusion and dislocation are the two molecular mechanisms of creep The former normally occurs due to the presence of vacancies within the crystal lattice (or the grain) while the latter occurs if there is a slip by the passage of crystal dislocations [45] There are two types of diffusion creep [46] Coble creep (favoured at lower temperatures) if the diffusion paths are predominantly through the grain boundaries and Nabarro-Herring creep (favoured at higher temperatures) if through the grains themselves Dislocation creep tends to dominate at high stresses and relatively low temperatures Dislocations can move by gliding in a slip plane which is a process that requires little thermal activation But the rate-determining step for their motion is often a climb process which also requires diffusion and is thus a time-dependent process and favoured by higher temperatures Creep behaviour is highly affected by temperature and stress generally creep is more significant under higher temperature and stress

Creep curves describe the creep strain development with time Figure 3 shows a creep curve under constant temperature and stress which typically experiences three creep stages primary stage where the creep strain rate ε119888119903 (the change of creep strain ε119888119903 in unit time) decreases with time secondary stage where the creep strain rate almost keeps constant and tertiary stage where the creep strain rate drastically increases with time

Figure 3 Creep curve under constant temperature and stress

14

The built-in creep models in commercial software eg ANSYS [47] and ABAQUS [48] in default only cover the primary andor the secondary stage(s) Table 1 [47] gives the default creep models supported in ANSYS for example Possible reason why the tertiary stage is not modelled might be that for most industrial applications the failure analysis is beyond interest and therefore the built-in creep laws in the commercial software was supposed to be sufficient [49] Though the two codes offer the interfaces through either ANSYS UPF or ABAQUS user subroutines to support more general user-defined creep laws eg complex models that also include the tertiary creep stage

Table 1 Default creep models in ANSYS Mechanical

Creep model Equations Type

Strain hardening

ε119888119903 = 1198621σ1198622ε119888119903

11986231198901198624119879 1198621 gt 0 Primary

Time hardening ε119888119903 = 1198621σ11986221199051198623119890minus1198624119879 1198621 gt 0 Primary

Generalized exponential

ε119888119903 = 1198621σ1198622119903119890minus119903119905 r = 1198625σ

11986231199051198623119890minus1198624119879 1198621 gt 0

1198625 gt 0

Primary

Generalized Graham

ε119888119903 = 1198621σ1198622(1199051198623 + 1198624119905

1198625 + 11986261199051198627)119890minus1198628119879 1198621 gt 0 Primary

Generalized Blackburn

ε119888119903 = 119891(1 minus 119890minus119903119905) + 119892119905 119891 = 1198621119890

1198622σ 119903 = 1198623(σ1198624)1198625 119892 = 1198626119890

1198627σ

1198621 gt 0

1198623 gt 0

1198626 gt 0

Primary

Modified time hardening

ε119888119903 = 1198881σ11988821199051198883+1(1198883 + 1) 1198621 gt 0 Primary

Modified strain hardening

ε119888119903 = 1198621σ1198622[(1198623 + 1)ε119888119903]

1198623119890minus1198624119879 1198621 gt 0 Primary

Generalized Garofalo

ε119888119903 = 1198621[119904119894119899ℎ(1198622σ)]1198623119890minus1198624119879 1198621 gt 0 Secondary

Exponential form

ε119888119903 = 1198621119890σ1198622119890minus1198623119879 1198621 gt 0 Secondary

Norton ε119888119903 = 1198621σ1198622119890minus1198623119879 1198621 gt 0 Secondary

Combined time hardening

ε119888119903 = 1198621σ11986221199051198623+1119890minus1198624119879(1198883 + 1)

+ 1198625σ1198626119905119890minus1198627119879

1198621 gt 0

1198625 gt 0

Primary+ Secondary

Rational polynomial ε119888119903 = 1198621

partε119888part119905 ε119888 =

119888119901119905

1 + 119901119905+ ε119898119905

ε119898 = 1198622101198623σσ1198624 119888 = 1198627ε

1198628σ1198629

119901 = 11986210ε11986211σ11986212

1198622 gt 0 Primary+ Secondary

Generalized time hardening

ε119888119903 = 119891119905119903119890minus1198626119879 119891 = 1198621σ + 1198622σ2 + 1198623σ

3 119903 = 1198624 + 1198625

Primary

32 A Modified Theta Projection Model

321 Original theta projection model and its variations

The original theta projection model [50] was first proposed to accurate predict long-term creep and creep-fracture properties It assumes that a creep curve under constant temperature and stress can be described as

15

휀119888119903 = 1205791(1 minus 119890minus1205792119905) + 1205793(119890

1205794119905 minus 1) (22)

where 휀119888119903 is the creep strain 119905 the time and 1205791~1205794 the model parameters Figure 4 illustrates how the first term and second term on right hand side develop with time and constitute a creep curve which has obvious primary secondary and tertiary stages Model parameters 120579119894(119894 = 1234) are temperature and stress dependent and are described with the following expression

119897119899thinsp120579119894 = 119886119894 + 119887119894120590 + 119888119894119879 + 119889119894120590119879 (23)

where 119886119894 119887119894 119888119894 and 119889119894 ( 119894 = 1234 ) are constants It is noted [45] that this equation is suitable when the temperature range of experiment data sets is very narrow such that minus1119879 is almost proportional to 119879 These constants (119886119894 119887119894 119888119894 and 119889119894 (119894 = 1234)) are the final model parameters which are generated with two fitting procedures

(i) Fitting the experimental creep curves to Eq (22) generates several groups of 1205791~1205794 for each creep curve under certain temperature and stress We denote (119879119895 120590119895 1205791119895 1205792119895 1205793119895 1205794119895 ) as parameters for the 119895119905ℎ

experimental curve under temperature 119879119895 and stress 120590119895

(ii) Fitting 1205791 to Eq (23) as a function of temperature and stress with the obtained data ( 119879119895 120590119895 1205791119895 ) from ( 119879119895 120590119895 1205791119895 1205792119895 1205793119895 1205794119895 ) generates

parameters 1198861 1198871 1198881 and 1198891 Repeat this fitting procedure for 1205792 1205793 and 1205794 to get parameters 119886119894 119887119894 119888119894 and 119889119894 (119894 = 234)

Variations of the model were developed and applied to other different materials to better accommodate the experimental data which are all then called a lsquomodified theta projection methodrsquo independently Examples are models in [51] for lowhigh alloy ferritic austenitic steels Al-alloys Al-matrix composites [52] for Nickel-based Super-alloy [53] for pure metals and class M alloys with intermediate range of stress (40-70 of the yield stress) and temperature (40-60 of the melting temperature)

Figure 4 Illustration of theta projection method [54]

16

322 A modified theta projection model

The new modified theta projection model we proposed [54] was mainly adapted from the original theta projection method [50] and the modified version [51] The new model describes the creep curve with the following equation

휀119888119903 = 1205791(1 minus 119890minus1205792119905) + 120579119898119905 + 1205793(119890

1205794119905 minus 1) (24)

An additional term 120579119898119905 is seen compared with Eq (22) in the original method The reason for adding this term is that the 16MND5 steel shows fairly long secondary stages in many experimental creep curves and the creep curves can be better fitted by Eq (24) than Eq (22) As the structure undergoes different creep process under different temperature and stress the model parameters θ119894 (119894 =1~4 and 119898) are also temperature and stress dependent

The main difference compared with the original and all other modified projection models is that numerical interpolation is used to replace the second numerical fitting in previous models In other words instead of fitting the model parameters as functions of stress and temperature with Eq (23) the distributions of the model parameters are assumed to be piecewise in the temperature-stress space and they will be interpolated directly from known parameters obtained from experiments The consideration is that the experimental data ([16] [17]) for 16MND5 covers a wide temperature range varying from 600 degC to 1300 degC and a wide stress load range varying from 08 MPa to 248 MPa while Eq (23) in the existing methods is based on a narrow temperature span Poor parameter fitting results would be expected if we directly fit the parameters 1205791~1205794 with Eq (23) as used in the original methods

The creep model was developed using the following steps

(i) Fitting Each creep curve can be represented by a group of model parameters [1205791 1205792 120579119898 1205793 1205794] with respect to Eq (24) by fitting the creep curves to Eq (24) All parameters of the 32 experimental curves form an

initial dataset of the model parameters [119879119895 120590119895 1205791119895 1205792119895 120579119898119895 1205793119895 1205794119895] where j (0ltjle32) is the jth creep curve

(ii) Interpolation Parameters of a creep curve under any [119879 σ] are directly interpolated from the dataset

(iii) Parameter adjustment The dataset was further updated to satisfy the monotonicity assumption we have taken that creep strain monotonically increases with temperature and stress

(iv) Different hardening models for dynamic loads

Step (i) was done using MATLAB In the following steps (ii)-(iv) are described in detail

Model parameters for new creep curves are interpolated from the dataset over the 2D 119879 minus 120590 space The interpolation is divided into two sub-interpolation steps interpolate over stress with fixed temperature and then interpolate over temperature with fixed stress We take the interpolation of model parameters under (119879prime σprime) for example In the first interpolation we first find out the two

17

temperatures 119879119896119899119900119908119899119904119898119886119897119897 and 119879119896119899119900119908119899119897119886119903119892119890 most close to 119879prime from the experimental

temperature set 600 700 800900 1000 1100 1200 1300 Under each temperature there are four sets of model parameters in the dataset corresponding to four different stress loads Then the first interpolation is done over stress load from the known dataset of model parameters under each

temperature to output the parameters for (119879119896119899119900119908119899119904119898119886119897119897 120590rsquo) and (119879119896119899119900119908119899119897119886119903119892119890 120590rsquo)

In the second interpolation the model outputs parameters for (119879rsquo 120590rsquo) from

(119879119896119899119900119908119899119904119898119886119897119897 120590rsquo) and (119879119896119899119900119908119899119897119886119903119892119890 120590rsquo) For 120579i(119894 = 12 3 4) we assume a logarithm

relation between 120579119894 temperature 119879 and stress 120590 which is similar to Eq (23)

119897119899120579119894 = (119886 + 119887120590)(119888 + 119889119879) (25)

It reduces to 120579119894 = 11986011198901198871σ with fixed temperature in the first interpolation and to

θ119894 = 11986021198901198872T with fixed stress in the second interpolation The interpolation

expression for θ119894 = 11986011198901198871σwith two known points (1205901 1205791198941) and (1205902 1205791198942) is given

by Eq (26) The interpolation expression for θ119894 = 11986021198901198872T with two known points

(1198791 1205791198941) and (1198792 1205791198942) is given by Eq (27)

120579119894 = 120579i1

120590minus12059021205901minus1205902120579i2

1205901minus1205901205901minus1205902 (26)

120579119894 = 120579i1

119879minus11987921198791minus1198792120579i2

1198791minus1198791198791minus1198792 (27)

Similarly we assume the following relation between 120579119898 temperature 119879 and stress 120590

120579119898 = 119860120590119899119890minus119887 119879frasl (28)

The interpolation is done over the stress with 120579119898 = 1198603120590n by fixing temperature

and then over temperature with 120579119898 = 1198604119890minus1198874119879 by fixing stress The interpolation

expression over stress for 120579119898 = 1198603120590119899 with two known points (1205901 1205791198981) and

(1205902 1205791198982) would be given by Eq (29) The interpolation expression over

temperature for 120579119898 = 1198604119890minus1198874119879 with two known points (1198791 1205791198981) and (1198792 1205791198982)

would be given by Eq (30)

120579119898 = 119890119909119901(119897119899120590 minus 11989711989912059011198971198991205902 minus 1198971198991205901

1198971198991205791198982 +1198971198991205902 minus 119897119899120590

1198971198991205902 minus 11989711989912059011198971198991205791198981) (29)

120579119898 = 120579m2(11198791minus1119879) (

11198791minus11198792)fraslsdot 120579m1

(1119879minus11198792) (

11198791minus11198792)frasl (30)

In this model we also take the assumption that the creep strain monotonically increases with temperature and stress loads1 Or specifically with the same creep time the structural component under higher temperature or stress experiences

1 Intuitively materials deform more under higher stress and temperature This is also mostly observed in tensile experiments Though some exceptions exist one instance of which is the effect of phase change some materials under certain temperature around the phase change temperature may show much weaker structural performance Here we just simply assumed that the creep strain monotonically increases with temperature and stress

18

higher creep strain than that with lower temperature or stress (the creep curve under higher temperature or stress should be above the curve with lower temperature or stress) When generating the parameter dataset (Step (i)) we already kept the trend that under the same temperature cases under higher stress loads would have larger creep model parameters and give higher creep strains This trend would also hold for the interpolated temperatures which thus guarantees that creep strain monotonically increases with stress load under same temperature Then the question becomes if creep strain also monotonically increases with temperature given fixed stress However the answer could not always be a yes especially in the extrapolation range of the stress load Figure 5(a) illustrates an example of the interpolation of model parameters under (119879119872 σ119872) The red dots and blue dots are known parameters from the parameter dataset for corresponding high temperature 119879119867 and low temperature 119879119871 respectively The green dots (119879119867 σ119872) and (119879119871 σ119872) are interpolated from the red and blue dots in the first sub-interpolation respectively The black dot (119879119872 σ119872) is interpolated from the green dots in the second sub-interpolation It is seen that when σ119872 lt σ119862 the value of (119879119871 σ119872) is higher than that of (119879119867 σ119872) It means the higher 119879119872 is the smaller the model parameter will be ie the model would predict larger creep strain for lower temperatures which violates our monotonicity assumption One simple solution used here as shown in Figure 5(b) is to minorly adjust the values of the four red and blue dots such that the intersection point C can be sufficiently small and the situation σ119872 lt σ119862 would barely occur in the extrapolation range

Figure 5 Illustration of model parameter interpolation (a) before and (b) after parameter adjustment [54]

A corresponding parameter adjustment process was introduced to verify that the model predicts higher creep strain under higher temperature under fixed stress Figure 6 illustrates the general verification process for any (119879 120590) in the interested load range the creep strain of (119879 + 120575119879 120590) should be larger than that of (119879 120590) Otherwise parameters would be minorly adjusted and the verification process be repeated This process would be iterated several times until the requirement is reached The achieved verification shows

Stress

σC

High temperature TH

MPa

a

0

Model

par

amet

er θ

Low temperature TL

MPa

a

0

Model

par

amet

er θ

High temperature TH

Low temperature TL

σC

Stress

C C

(a) (b)

σM

σM

19

(i) For any (119879 120590) in the temperature range 273~ 1073 K with an interval of 1 K and the stress range 1 ~260 MPa with an interval of 1 MPa its creep strain is higher than that of (119879 minus 1119870 120590) for the first 12000000 s

(ii) For any (119879 120590) in the temperature range 1073~1673 K with an interval of 1 K and the stress range 1 ~80 MPa with an interval of 1MPa its creep strain is higher than that of (119879 minus 1119870 120590) for the first 12000000 s

(iii) For any (119879 120590) in the stress range 01~26 MPa with a smaller interval of 01 MPa for the full temperature range 273~1673 K with an interval of 1 K its creep strain is higher than that of (119879 minus 1 120590) for the first 12000000 s

Above verification covers a wide range of temperature and stress especially where the stress and temperature are low and extrapolation is used Further verification to cover even higher stress or temperature ranges is not considered as creep increases drastically under such high stress and temperature and the results would become meaningless With these regards we generally think the monotonicity requirement is reached The model parameter dataset was updated along with the adjustment The final dataset was given as Table 2 in Paper I

Figure 6 Parameter adjustment process [54]

Figure 7(a-d) present the model predictions of selected creep curves in ANSYS Mechanical (labelled as lsquoANSYSrsquo) compared with experimental curves (labelled as lsquoREVISArsquo) at temperature 600 800 1000 and 1200 degC respectively The predicted curves proved to clearly show the three-stage creep process and agreed well with the experimental creep curves

Initial parametersdataset

θ1 θ2 θm θ3 θ4 from creep

curve-fitting

Parameters interpolation

Is the monotonicity

assumption satisfied

IsMon= =1

Final model parameters

Adjust the

parametersdataset

θ1 θ2 θm θ3 θ4

No

Yes

IsMon=1 a flag is monotonic achieved

For σ in the stress range in MPa

For T in range Tstart Tend in K

For t in range 112000000 in s

If ε(σ T+1t)lt ε(σTt) IsMon=0

Iteration loop of

parameter adjustment

20

Figure 7 Selected model predictions compared with experimental creep curves [54]

Instead of being constant temperature and stress loads also dynamically change with time in real cases Hardening models determine how creep strain rate reacts to the change of loads during a creep process The time hardening model and strain hardening model2 are supported in our model implementation in ANSYS Mechanical which are two mostly used hardening models

Time hardening assumes that the following creep strain rate after a load change depends only on the current creep time from the beginning of the creep process (and also temperature and stress if used in the creep model) and ignores the current creep strain Strain hardening assumes that the creep strain rate depends only upon the current creep strain (and also temperature and stress if used in the creep model) [55]

2 The time hardening and strain hardening here are more general terms than the specific time hardening creep model and the strain hardening creep model mentioned in Table 1

(a) 600 degC (b) 800 degC

(c) 1000 degC (d) 1200 degC

21

Eq (24) in default is a time hardening model as creep strain is directly described by the time 119905 Eq (31) gives the creep strain increment 119889휀119888119903 at time 119905 during time interval 119889119905

119889휀119888119903 = (12057911205792119890minus1205792119905 + 120579119898 + 12057931205794119890

1205794119905)119889119905 (31)

The strain hardening model is derived from Eq (24) in an implicit form Specifically when temperature and stress load change to (119879 120590) at the time 119905 and current creep strain 휀1198881199030 the model will get the creep curve of constant (119879 120590) and

output how this curve will react under the current strain 휀1198881199030 Since the creep curve shows creep strain monotonically increases with time given the creep strain 휀1198881199030 under creep curve of (119879 120590) we can uniquely find this reference time

119905119903119890119891 satisfying Eq (32) One should notice that the reference time 119905119903119890119891 can be

different from the current physical time 119905

휀1198881199030 = 1205791(1 minus 119890minus1205792119905119903119890119891) + 120579119898119905119903119890119891 + 1205793(119890

1205794119905119903119890119891 minus 1) (32)

where all 120579119894(119894 = 1~4119898) are model parameters of the new creep curve (119879 120590) It can be solved for 119905119903119890119891 with methods eg Newton-Ramphson [56] Then the creep

strain increment at 119905119903119890119891 during time interval 119889119905 is given as

119889휀119888119903 = (12057911205792119890minus1205792119905119903119890119891 + 120579119898 + 12057931205794119890

1205794119905119903119890119891)119889119905 (33)

Therefore combining Eq (32) and (33) gives the creep strain rate at creep strain 휀1198881199030 In summary when loads are changed to (119879 120590) with current creep strain 휀1198881199030

the model traces the creep curve of (119879 120590) and calculates the creep strain rate of this curve at strain 휀1198881199030

One calculation was done to demonstrate the performance of the hardening models The stress load was set to 151 MPa and kept unchanged A step temperature load was applied as the orange dash curve shows in Figure 8 it was 900 degC during 0-36000 s and suddenly jumped to 1000 degC at 36000 s which was then kept till the end of the simulation The blue solid curve (the part before the temperature change was identical to and therefore covered by the green solid curve in the figure) and blue dash curve are the creep curves under constant temperature T=900 degC and T=1000 degC respectively The green solid curve and green dash curve represent the model predictions considering this temperature change with time hardening model and strain hardening model respectively Before this change the creep curves with both hardening models were identical to the creep curve of the T=900 degC till point A After that the creep strain rate then shifted from Point A to Point B (which had the same time with A) in the time hardening model and behaved the same as the curve T=1000 degC as shown in the rest part of the green solid curve In the strain hardening model the creep strain rate shifted from Point A to Point C (which had the same creep strain with A) and behaved the same as the curve T=1000 degC as shown in the rest part of the green dash curve The large difference between two models was partially caused by the drastic temperature jump In real cases the difference would be less significant as the temperature changes more mildly Overall this prediction is exactly the expected behaviors of the time hardening model and strain hardening model

22

Figure 8 Time hardening and strain hardening in ANSYS Mechanical [54]

23

4 Coupled Thermo-Fluid-Structure Analysis of RPV Failure

As described in Section 13 RPV failure is coupled with melt pool heat transfer leading it into a transient thermo-fluid-structure coupled problem The goal of this section is to develop and validate a reliable coupling approach which supports transient coupled thermo-fluid-structure analysis as well as advanced models and computational platforms so as to improve fidelity and facilitate implementation The approach was validated against the FOREVER-EC2 experiment and then applied to an RPV failure analysis of a Nordic BWR As the approach adopts the volume loads mapping for load transfer comparative analysis was also performed against the coupling approach using surface loads mapping Future work is considered in analysis of penetration failures (IGTs and CRGTs) of a BWR RPV since the approach is flexible to complex geometries Results of Paper II ampIII are jointly presented in this section

41 Thermo-Fluid-Structure Coupling Approach

Considering the involved phenomena a coupling approach should at least address three issues (i) melt pool heat transfer (ii) RPV deformation and (iii) the interaction between melt pool heat transfer and vessel deformation A two-way strong coupling considering both the transient thermal load from melt pool on vessel and the influence of vessel deformation on melt pool heat transfer involves complex data exchange and is still full of challenges Therefore the new coupling approach is realized in a one-way manner with the assumptions that the influence of vessel deformation on melt pool heat transfer can be ignored and thus only the thermal loads from melt pool to vessel are considered It is based on the considerations that on one hand the deformation is a slow process regarding the relatively long failure time and on the other hand it was pointed out [57] that the physics of natural convection heat transfer in internally heated liquid pools is not sensitive to the geometrical factors based on their calculation results and experimental data from [58] [59] as well as previous experiments in semicircular and square cavities eg [44] [60] In addition one-way coupling is also computationally more efficient especially when heat transfer turbulence structural non-linearity and large-scale geometries are involved The developed approach is based on the ANSYS platform which generally consists of three parts with respect to the three issues respectively

(i) conjugate heat transfer of the melt pool and vessel using ANSYS Fluent (ii) transient thermal loads transfer from ANSYS Fluent to ANSYS

Structural (the ANSYS Mechanical in ANSYS Workbench) (iii) structural analysis of the vessel in ANSYS Structural

ANSYS Structural in default only supports direct transfer of constantsteady-state spatial loads from ANSYS Fluent Transfer of transient loads is realized with the extension tool FSI Transient [61] It is a pre-processing tool (with good documentation) developed by a third-party CAE vendor EDRTM MEDESO [62] to map transient CFD results (heat transfer coefficient temperature and surface pressure) to thermal or structural analysis including the definition of load steps

24

This new procedure proves to be straightforward and reduces efforts in the load transfer

Since structural performance of RPV is highly dependent on temperature the transfer of transient temperature loads from conjugate heat transfer to mechanical solver is a core step of the coupling framework There are two general approaches to realize the transfer as shown in Figure 9 Both approaches contain three parts a conjugate heat transfer analysis for fluid and structure where flow and heat transfer are coupled a mechanical analysis for the structure and a load mapping process that transfers the transient loads The conjugate heat transfer part can be solved with any CFD solver and is treated in the same manner in both approaches In the first approach (Figure 9(a)) transient volume thermal loads are transferred and mapped into the mechanical simulation domain The structure domain reads the essential thermal loads from the mapping after which a structure analysis of this domain is performed In the second approach (Figure 9(b)) transient surface thermal loads are mapped onto the boundary surfaces of structure domain as thermal boundary conditions Then a thermo-structure coupled analysis of the structure domain in the mechanical part is performed To distinguish them we call the former approach volume loads mapping (VLM) and the latter surface loads mapping (SLM) In the RPV failure analysis the fluid and structure domains correspond to the melt pool and RPV respectively Examples of these methods are the studies of [63] and [30] using VLM and [35] using SLM The VLM is adopted in current approach

Figure 9 Frameworks of the thermo-fluid-structure coupling approaches (a) volume loads mapping (b) surface loads mapping [64]

fluid structure structure

fluid structure structure

Conjugate heat transfer problem Mechanical problem

(structure)

Conjugate heat transfer problem Mechanical problem

(thermo-structure coupled)

Transient volume

loads mapping

(VLM)

Transient surface

loads mapping

(SLM)

(a)

(b)

25

42 Numerical Modelling

421 Conjugate heat transfer

In the conjugate heat transfer analysis main focus is put on the modelling of melt pool heat transfer Within the ANSYS Fluent solver two options are supported CFD with turbulence modelling and PECM CFD is the choice when the model goes to detail and more mechanistic analysis are needed while PECM provides an alternative that is faster and possibly more suitable for large-scale problems

Within the scope of RANS models the SST (Shear Stress Transport) model [65] is selected as the turbulence model in present study This model is a combination of k-ε model and k-ω model The k-ω formulation is used in the inner parts of the boundary layer which makes the model directly usable all the way down to the wall through the viscous sub-layer It also switches to k-ε model in the free-stream and thereby avoids the common k-ω problem that the model is too sensitive to the inlet free-stream turbulence properties The considerations on choosing this model are

bull Several two-equation low-Re k-ε models have been proved in [66] not good enough for melt pool heat transfer Therefore these mentioned two-equation models as well as other one-equation models are not considered

bull SST is highly recommended [67] for simulating most industrial heat transfer problems

bull Convincing results ([68] [69]) using SST have been obtained in capturing the thermal behaviour of the natural convection in an internally heated fluid Especially good agreements were achieved between the predicted heat flux distribution along the vessel and experimental data which would be one of the most important factors to estimate model performance in this problem

The description of PECM as an effective model for melt pool heat transfer is given in Section 23

422 Mechanical analysis

The material for the RPV failure analysis considered in this study is the French RPV steel 16MND5 The three-stage modified theta projection model in Section 3 is used as the creep model Since the definition of an RPV failure criterion is not universal (it can be the time when damage variable reaching 1 or either specified value of creep strain creep strain rate or total deformation is reached) we do not explicitly define a failure criterion Instead the simulation is kept running until the creep converge criterion cannot be satisfied even with the defined minimum time step (indicating a fairly large creep strain rate) and the simulation will be automatically terminated The time the simulation stops is taken as the failure time This treatment implicitly implies a criterion of a large creep strain rate We refer to [17] for other material properties Youngrsquos modulus and thermal conductivity are temperature dependent Multilinear Isotropic Hardening (MISO) is used as the plasticity model

26

43 Validation against the FOREVER-EC2 Experiment

In this section the validation against the FOREVER-EC2 experiment is presented (indicated as lsquoVLMrsquo in the following) In addition the simulation results using the SLM approach (indicated as lsquoSLMrsquo in the following) is also included for comparison purpose

431 FOREVER-EC2 experiment

The FOREVER-EC2 experiment [70] is one of the FOREVER experiments that achieved a vessel failure Figure 10 shows the experimental setup A 110 scale pressure vessel with a wall thickness of 15 mm and an outer diameter of ~400 mm was employed The lower head was made of 16MND5 steel Since it was difficult to use prototypical corium materials in the experiment the binary salt 30 CaO-70 B2O3 was employed as simulant To generate the internal heat which corresponds to the nuclear decay heat a specially designed heater was used The power input was maintained ~38 kW during the experiment such that maximum melt pool temperature would be ~1300 degC The pressure inside the vessel was maintained at ~25 MPa Thermocouples were used to measure the melt pool and vessel wall temperatures and linear displacement transducers used to measure the vessel wall displacements The designed experimental conditions were reached at experimental time 11670 s Vessel failure was observed about 12510 s later (at experiment time 24180 s)

Figure 10 Scheme of the experimental setup [22]

27

Figure 11 Computational domains and boundary conditions (a) conjugate heat transfer (b) mechanical analysis [71]

Computational domains and boundary conditions are indicated in Figure 11 SST turbulence modelling was used for pool heat transfer The simulations started at the moment when experimental conditions were reached (experiment time 11670 s) So simulation time 0 s corresponds to experiment time 11670 s

432 Conjugate heat transfer

Figure 12 shows the temperature fields at simulation time 0 s 1225 s 2475 s 4975 s and 9975 s All the time points in this part refer to simulation time The field at time 0 s (Figure 12(a)) was the initial temperature that was reconstructed based on the thermocouple readings The temperature increased with time and reached a steady state when the maximum temperature reached ~1607 K Figure 12(b-f) show clear thermal stratification which is an expected phenomenon of natural convection

Internal pressure

25 bar

Internal pressure 25e6-2500times981timesy Pa

External pressure

1 bar

Zero displacement

y=0 m

Gravity 400

mm

R=203 mm

15 m

m

Tamb

=323 K radiation with ε = 08

+ convection with h=10Wm2K

Tamb

=800 K radiation with ε=02

(a) (b)

Tamb

=800 K radiation with ε=08

+ convection with h=10 Wm2K

28

Figure 12 Temperature fields at simulation time (a) 0 s (b)1225 s (c) 2475s (d) 4975 s (e) 7475 s and (f) 9975 s [64]

Figure 13 shows the temperature profiles of the external vessel along the polar angle in comparison with experimental data Regions with a polar angle larger than 90deg are those in the cylindrical part of the vessel It also shows the temperature profile increased with time until reaching a steady state The profile almost did not change after 4975 s which indicates that a steady state of the temperature was reached As time increased the temperature in the lower head region 0~90deg (where the vessel was contacted with the melt) evolved from a relatively flat distribution to a distribution that temperature at high angle was higher than that at low angle by more than 200 K Generally the steady state results agreed well with the experimental data except for the low angle part of vessel lower head and the high angle part of the cylindrical vessel In addition to modelling errors the discrepancy in the low angle part of the lower head may be caused by the crust and gap possibly formed at vessel bottom in the experiment which can reduce the downward heat transfer and result in lower temperature while they cannot be modelled in simulations As to the high angle cylindrical vessel part simplified boundary conditions were applied to the inner and outer vessel wall since it was far from the melt and not of interest which may result in some discrepancy

(a) (b)

(c) (d) (e) (f)

29

Figure 13 Temperature profiles of external vessel surface along the vessel polar angle [64]

433 Mechanical analysis

In the mechanical analysis the VLM predicted a failure time of 23959 s and the SLM predicted a failure time of 22445 s both are close to the experimental failure time 24180 s All the time points in this part refer to the experiment time Figure 14 shows the creep strain fields at failure time Both approaches predicted maximum creep strains at same location where the temperature was high (cf Figure 12) This is related to the fact that creep behavior is highly influenced by temperature Maximum values for both cases were also close to each other 0393 in VLM and 0366 in SLM

Figure 15 presents the total deformation of the vessel bottom point (also called ldquosouth polerdquo in some literature) in the vertical direction Since there was an initial displacement reading of ~355 mm at time 0 s which almost did not change during 0~11670 s (the whole period before the experimental conditions were reached) an initial value of 355 mm was added to the simulation result such that experimental and simulation results had the same total displacement at t=11670 s Similar trend was seen in the VLM SLM and experimental data that the deformation increased with time and experienced a drastic acceleration at the end Generally both simulation approaches agreed well with the experimental data though they both slightly under predicted the value of total deformation The discrepancy may possibly be related to the accuracy of the thermal predictions modelling errors of plasticity and creep and simplified boundary conditions

400

600

800

1000

1200

1400

1600

0 20 40 60 80 100 120 140 160

Tem

per

ature

(K

)

Polar angle (deg)

Experiment

25 s

1225 s

2475 s

4975 s

7475 s

9975 s

Weilding line

30

Figure 14 Creep strain fields at failure time (a) VLM (b) SLM [64]

Figure 15 Total deformation in vertical direction of the vessel bottom point [64] [71]

The wall thickness changes are plotted together with experimental results in Figure 16 The wall thickness change δ119909 is defined as

δ119909 = 119909119890119899119889 minus 119909119900119903119894119892119894119899119886119897 (34)

000

500

1000

1500

2000

2500

11670 13670 15670 17670 19670 21670 23670

Tota

l def

orm

atio

n (

mm

)

Time (s)

ExperimentVLMSLM

(a) (b)

31

where 119909119890119899119889 and 119909119900119903119894119892119894119899119886119897 are the wall thickness at failure time and the original wall

thickness respectively Since 119909119890119899119889 cannot directly be post-processed it was estimated in the following way (cf the subfigure of Figure 16) for the same polar angle θ find the corresponding points Pin (1199091 1199101) and Pout (1199092 1199102) on the inner and outer surfaces of the deformed vessel respectively the distance of Pin and

Pout radic(1199091 minus 1199092)2 + (1199101 minus 1199102)2 is calculated as 119909119890119899119889 at polar angle 120579 Such estimation of the wall thickness is more accurate when the line PinPout is closer to be orthogonal to the deformed local surface Considering the final deformed vessel as shown in Figure 14 the estimation should be fine at regions close to 0deg and 90deg and can be slightly overestimated at medium angle region The VLM curve was compared with the SLM curve and 4 experimental curves measured at vessel left right front and back sides Generally VLM curve agreed well with the experimental and SLM curves wall thickness gradually decreases with angle till around 75deg and then increases till 90deg the maximum thickness change occurred at around 75deg would be around -5 mm The positive values at low angle may be caused by the thermal expansion It also shows that the thickness transitions in experimental curves were smoother than those in both simulation curves while simulation curves showed narrower regions with large thickness changes (at angle 70deg~80deg) Possible reasons might be (i) sharper temperature profile predicted in simulation (ii) input uncertainty and (iii) modelling errors The SLM predicted slightly thinner vessel than the VLM except for the necking region The discrepancy might be caused by various reasons eg the interior vessel temperature can be slightly different between two approaches which would then result in different thermal stress and different deformation the final deformed vessel shapes cannot be exactly the same which also results in different errors in estimating PinPout

Figure 16 Wall thickness changes along the vessel polar angle [64] [71]

-6

-5

-4

-3

-2

-1

0

1

0 10 20 30 40 50 60 70 80 90

Wal

l th

icknes

s ch

ange

(mm

)

Polar angle (deg)

VLMSLMExperiment (left)Experiment (right)Experiment (front)Experiment (back)

32

44 Application to BWR Case

In this section the coupling approach (indicated as lsquoVLMrsquo in the following) was applied to an RPV failure analysis of a BWR A simulation using the SLM coupling approach (indicated as lsquoSLMrsquo in the following) was also performed for comparison purpose

A hypothetical severe accident of a Nordic BWR initiated by an SBO was considered The core materials would melt and migrate downward to the RPV lower head The total amount of corium relocated in the lower head was considered to form a 19 m corium melt pooldebris bed assuming the corium distribution was homogeneous

Figure 17 Computational domains and boundary conditions (a) conjugate heat transfer (b) mechanical analysis [64]

Figure 17(a) shows the computational domain of the conjugate heat transfer PECM was used in this calculation A general mesh with a pool height of 23 m was used to accommodate most of the relocation situations with the same mesh Since the corium only had a height of 19 m the top 04 m region was treated as injected water covering the corium The top water was expected to be boiling and we assumed a uniform temperature of 383 K Under the implementation of PECM this region was damped with artificially high heat capacity and high conductivity such that it would efficiently remove heat from the corium and suppress its own temperature increase For the corium part below water a volumetric heat source 11 MWm3 was implemented as the nuclear decay heat For the in-vessel surfaces a constant temperature of 383K was applied as they are covered by water which can be boiling Without water cooling on the external vessel wall a very small heat loss of 20 Wm2 was applied Melting and solidification of the corium were also considered with a solidus temperature of 2750 K and a liquidus temperature of 2770 K The initial temperature was set as

q=20 Wm2

T=383 K

Qv=11

MWm3

315 bar

315 bar+ hydrostatic

pressure of corium

Gravity

248 bar

Zero

displacement

Water

Corium

(a) (b)

33

1100 K for the corium 383 K for the water and 450 K for the vessel For the mechanical analysis (Figure 17(b)) an inner pressure of 315 bar and an outer pressure of 248 bar were considered For the region covered by corium an extra hydrostatic pressure was added on the vessel wall A 2 m long vessel extension was considered as well as the standard earth gravity

Figure 18 presents the wall temperature profile along the inner vessel surface at 50 s 2500 s 7500 s and 10000 s The temperature increased with time for the vessel surface contacted with corium and the distribution was flat which was due to that large part of the corium was still in solidus status It also shows that for the surface region that contacted the water the temperature was basically unchanged which indicates that the damping treatment indeed suppressed the temperature increment and guaranteed the heat removal upward by water

Figure 18 Temperature profiles of inner vessel wall along the vessel polar angle [64]

The predicted RPV failure time was 9869 s by VLM and 9650 s by SLM Compared with the FOREVER-EC2 case in Section 43 the VLM predicted a slightly later failure time than the SLM in this BWR case too Figure 19(a) compares the maximum creep strain developments with time of both approaches Before 9000 s the predicted values by both approaches are close to each other Afterward the creep acceleration occurred and the strain increased drastically The acceleration of the VLM occurred 200 s later than that of the SLM which agreed with the predicted later failure time Similar trend was observed for the total deformation in vertical direction of vessel bottom point as shown in Figure 19(b) The deformations of both approaches are close except that the acceleration of VLM occurred later than that of SLM Overall results of VLM agreed well with those of SLM

0

500

1000

1500

2000

2500

0 10 20 30 40 50 60 70 80 90 100

Tem

per

ature

(K

)

Polar angle (deg)

50 s

2500 s

5000 s

7000 s

10000 s

34

Figure 19 Results of structural analysis (a) creep strain and (b) total deformation in vertical direction of vessel bottom point [64]

45 Computational Efficiency

Computational efficiency is also an important consideration regarding the performance of numerical approaches The computational efficiencies of both VLM and SLM are analysed and compared in this section As parallel computing technique was used to speed up the simulations the number of parallel cores and corresponding costed computational time for each simulation are given together in Table 2 as a generally reflection of the computational efficiency All the simulations were done in a Lenovo P700 workstation equipped with 2 Intel Xeon E5-2630 v3 processors (8 cores per processor) and 32 G RAM For the simulations of FOREVER-EC2 SLM employed twice the parallel cores and costed 286 more time compared with VLM For the simulations of prototypic BWR case both VLM and SLM employed the same number of parallel cores but VLM only took 13 of the computational time compared with SLM Overall it can be concluded that VLM has higher computational efficiency than the SLM The reason is a structural analysis is performed in the VLM approach while a coupled thermo-structural analysis is performed in the SLM approach (cf Figure 9) a standalone structural analysis is computed faster than a coupled thermo-structural analysis

Table 2 Comparison of computational efficiency [64]

Case FOREVER-EC2 BWR VLM SLM VLM SLM

Number of cores 4 8 4 4 Computational time (hours) 21 27 18 72

35

5 Hybrid Coupling Approach for Efficient RPV Failure Analysis

In this section we propose a hybrid coupling approach for the efficient RPV failure analysis The idea is to replace the ANSYS Fluent in previous coupling approach with a lumped-parameter code for quick estimate of the melt pool heat transfer With the least modelling details (compared with CFD methods and distributed-parameter methods eg PECM) lumped-parameter codes prove to be the most efficient for melt pool heat transfer The new hybrid approach characterizes 1) utilization of efficient models for melt pool heat transfer and 2) the support of detailed structural modelling of RPV deformation allowing studies with different RPV geometric shapes under effects like local thinning due to ablation For studies of which the focus is mainly on structural behaviour of RPV eg [13] [26] while mechanistic simulation of melt pool convection is computationally expensive and unnecessary the lumped-parameter code could help providing reasonable thermal loads

The development of the approach consists of two tasks (i) development of a lumped-parameter code and (ii) its coupling with ANSYS Mechanical to form this hybrid coupling framework In the following we describe both tasks and the corresponding verification and validation processes respectively This section is mainly based on Paper IV

51 A Lumped-parameter Code for Melt Pool Heat Transfer

A lumped-parameter code transIVR was developed It features (i) quick estimate of transient melt pool heat transfer with reasonable accuracies and (ii) two-dimensional representation of heat conduction in the RPV wall facilitating the coupled analysis of the RPV failure under thermal loads The following assumptions are taken in the transIVR modelling

bull The crust between the oxide pool and metal pool is thin

bull The heat flux has uniform distributions on the bottom top and side surfaces of metal pool

bull Heat source only exists in oxide layer

bull Latent heat of fusion is ignored

Figure 20 illustrates the modelled three domains oxide layer metal layer and the RPV wall In addition the internal structure inside the vessel can also be considered as a radiative body to the top metal surface as in [72] An internal volumetric heat source generated by nuclear decay heat is considered in the oxide layer Heat in oxide layer transfers downward to the RPV wall and upward to the top metal layer Heat in metal layer can be transferred sideward to the RPV wall and upward to the surrounding internal structure through radiation (or to water if inside water flooding is applied) The heat transferred to RPV will be removed from outer RPV surface Possible formations of oxide crust in the top and bottom surface are illustrated as the black regions Possible formation of metal crust on the top surface is shown as the grey region The code automatically detects the

36

existence of the crust on oxide top and metal top surfaces during the simulation and solves the problem correspondingly withwithout crust Boundary conditions are applied to the external and internal surfaces of RPV and (if needed) top surface of metal layer Arbitrary boundary conditions are supported If a boundary condition is nonlinear eg mixed heat convection (linear dependent on 119879 ) and heat radiation (proportional to 1198794) the Picard method [73] would be used for solving this problem

This model consists of three main parts

(i) empirical heat transfer correlations (mostly from experiments) to determine global and local heat transfer coefficients

(ii) energy balances in oxide layer and metal layer where the heat transfer coefficients would be used and

(iii) energy balance in RPV

Figure 20 Two-layer melt pool configuration and RPV wall [74]

511 Heat transfer correlations

For internally heated natural convection (oxide layer) in transIVR the mini-ACOPO correlations [75] are used for the averaged heat transfer as expressed in Eq (35) and (36) and for the angular variation of the heat flux as expressed in Eq (37) and (38)

119873119906119906119901 = 03451198771198861198941198990233 (35)

119873119906119889119899 = 00038119877119886119894119899035(119867119877)025 (36)

119873119906119889119899(θ)

119873119906119889119899= 01 + 108(

θ

θ119898119886119909) minus 45 (

θ

θ119898119886119909)2

+ 86 (θ

θ119898119886119909)3

01 leθ

θ119898119886119909le 06 (37)

RPV

Oxide pool

Metal pool

37

119873119906119889119899(θ)

119873119906119889119899= 041 + 035 (

θ

θ119898119886119909) + (

θ

θ119898119886119909)2

06 ltθ

θ119898119886119909le 1 (38)

The 119877119886119894119899 that appears in Eq (35)~(36) is the internal Rayleigh number which is defined as

119877119886119894119899 =119892β119876119881119867

5

α119896ν (39)

For Rayleigh-Bernard convection in the metal layer heat transfer occurs in all the top bottom and side surfaces To side surface the reduced form [72] of the Churchill-Chu correlation [76] for 119875119903 = 013 is employed as shown in Eq (40)

119873119906119904119889 = 007611987711988613 (40)

To top and bottom surfaces the reduced form [72] of Globe-Dropkin correlation [77] for

119875119903 = 013 is used as expressed in Eq (41)

119873119906119906119901 = 005911987711988613 (41)

The 119877119886 that appears in Eq (40) and (41) is the external Rayleigh number which is defined

as

119877119886 =119892βΔ1198791198673

να (42)

Heat transfer coefficients can be calculated from Nusselt number by Eq (43)

ℎ =119873119906119896

119871 (43)

where 119896 and 119871 are the heat conductivity and characteristic length respectively 119873119906 can be the Nusselt numbers expressed in Eq (35)~(38) and (40)~(41)

512 Energy balance in the pools

The energy balance of oxide layer can be described as Eq (44) The total generated heat of oxide pool equals the sum of the increase of enthalpy (transient temperature change) upward heat flow to metal layer and downward heat flow to RPV

120588119862119901119881119900119897119889119879119900119909119889119905

+ ℎ119906119901119860119900119909119906119901(119879119900119909 minus 119879119900119909119906119901) + ℎ119889119899119860119900119909119889119899(119879119900119909 minus 119879119900119909119889119899) minus 119876119881119881119900119897 = 0 (44)

where ρ is the oxide density 119862119901 the heat capacity 119881119900119897 the total volume 119879119900119909 the bulk temperature 119905 the time ℎ119906119901 the upward heat transfer coefficient 119860119900119909119906119901

upward heat transfer area 119879119900119909119906119901 upward surface temperature ℎ119889119899 downward

heat transfer coefficient 119860119900119909119889119899 downward heat transfer area 119879119900119909119889119899 downward surface temperature and 119876119881 volumetric heat source

38

On the interface between oxide layer and metal layer energy balance requires that the heat transferred from oxide layer should equal the heat to the metal layer which can be described as Eq (45)

ℎ119906119901119860119900119909119906119901(119879119900119909 minus 119879119900119909119906119901) = ℎ119898119905119889119899119860119898119905119889119899(119879119894119899119905 minus 119879119898119905) (45)

where ℎ119898119905119889119899 is the downward heat transfer coefficient of metal layer 119860119898119905119889119899 the downward heat transfer area 119879119894119899119905 the interface temperature and 119879119898119905 the bulk temperature of metal layer

The energy balance of metal pool can be described as Eq (46) The increase of enthalpy (transient temperature change) equals the sum of upward downward and sideward heat flows to surrounding surfaces

ρ119898119905119862119901119898119905119881119900119897119898119905119889119879119898119905119889119905

+ ℎ119898119905119906119901119860119898119905119906119901(119879119898119905 minus 119879119898119905119906119901) + ℎ119898119905119904119889119860119898119905119904119889(119879119898119905 minus 119879119898119905119904119889)

+ ℎ119898119905119889119899119860119898119905119889119899(119879119898119905 minus 119879119894119899119905) = 0

(46)

where ρmt is the metal density 119862119901mt the heat capacity 119881119900119897mt the total volume 119879119898119905 the bulk metal temperature 119905 the time ℎ119898119905119906119901 the upward heat transfer

coefficient 119860119898119905119906119901 upward heat transfer area 119879119898119905119906119901 upward surface

temperature ℎ119898119905119904119889 sideward heat transfer coefficient 119860119898119905119904119889 sideward heat transfer area 119879119898119905119904119889 sideward surface temperature ℎ119898119905119889119899 downward heat transfer coefficient 119860119898119905119889119899 downward heat transfer area and 119879119894119899119905 oxide-metal interface temperature

The heat transferred from metal layer to the top surface should equal the heat radiated from this surface to the environment Considering the radiation and reflections of internal structures to this surface following energy balance equation [72] is achieved

ℎ119898119905119906119901(119879119898119905 minus 119879119898119905119906119901) = σ(1198791198981199051199051199001199014 minus 119879119904

4) (1

ε+1 minus ε119904ε119904

119860119898119905119906119901

119860119904)minus1

(47)

where 119879119898119905119905119900119901 is the temperature of metal top free surface ε the emissivity of

metal ε119904 the emissivity of internal structures 119860119898119905119906119901 the area of metal top surface

and 119860119904 the total area of internal structures Sometimes the internal water cooling may be available above metal layer In this case Eq (47) is unnecessary and 119879119898119905119905119900119901 can be set to a constant value

For the oxide layer top surface Eq (48) is used if not crust is formed in the interface between two layers ie 119879119894119899119905 gt 119879119898119890119897119905119894119899119892119900119909 Otherwise Eq (49) is used

119879119900119909119906119901 = 119879119894119899119905 (48)

119879119900119909119906119901 = 119879119898119890119897119905119894119899119892119900119909 (49)

For the metal layer top surface Eq (50) is used if not crust is formed on top of the metal ie 119879119906119901119904119906119903119891 gt 119879119898119890119897119905119894119899119892119898119905 Otherwise Eq (51) is used

119879119898119905119906119901 = 119879119898119905119905119900119901 (50)

39

119879119898119905119906119901 = 119879119898119890119897119905119894119899119892119898119905 (51)

Overall this part involves 6 equations regarding 6 unknows 119879119900119909 119879119900119909119906119901 119879119894119899119905 119879119898119905

119879119898119905119906119901 and 119879119898119905119905119900119901

513 Energy balance in the RPV wall

The heat transfer in the PRV wall can be modelled as a simple 1D heat conduction problem that heat only transfers in the wall thickness direction or 2D axisymmetric heat conduction problem

For 1D model the treatment in [72] was adopted As illustrated in Figure 21 the heat from the oxide pool transfers to inner vessel surface (119879119894) then to the interior vessel (119879119907119890119904) then to outer vessel (119879119900) and finally to environment

Figure 21 1D heat transfer model in RPV [74]

Unknown variables are the heat flux through inner vessel surface 119902119894119899 the crust thickness 119909119888119903 inner vessel temperature 119879119894 vessel center temperature 119879119907119890119904 and outer vessel temperature 119879119900 Eq (52)~(56) give a closed equation set for these variables considering a radiation boundary condition on the external vessel surface Eq (52)~(53) describe the heat transfer through the crust where an internal heat source 119876119888119903 is also considered Eq (52) can be easily derived from the one-dimensional heat transfer equation with heat source Eq (54)~(55) describe the energy balance on the inner vessel surface and vessel interior respectively Eq (56) describes the energy balance on the outer vessel surface regarding radiation boundary condition For Dirichlet boundary condition Eq (56) is replaced with Eq (57) For other boundary conditions we only need to correspondingly modify the terms on the right-hand side of Eq (56) For situations that no crust is formed on the surface of the vessel Eq (52)~(53) are eliminated and 119902119894119899 in Eq (54) can directly be determined from calculations in Section 512 and Eq (54)~(56) are solved for the unknows 119879119894 119879119907119890119904 and 119879119900 Once 119879119894 exceeds the vessel melting temperature (with or without crust formation) wall ablation is considered to occur Then 119879119894 is fixed to the vessel melting temperature and vessel thickness 119909119907119890119904 becomes the new unknown variable

119879119900119909119889119899 119879119894 119879119907119890119904 119879119900 119879119890119899119907

119909119888119903 119909119907119890119904

crust RPV

Qcr

119902119894119899 119902119897119900119888119886119897

40

119902119894119899 = 119896119888119903(119879119900119909119889119899 minus 119879119894)

119909119888119903+ 1198761198881199031199091198881199032

(52)

119902119894119899 = 119902119897119900119888119886119897 + 119876119888119903119909119888119903

(53)

119902119894119899 = 119896119907119890119904(119879119894 minus 119879119907119890119904)

05119909119907119890119904

(54)

120588119907119890119904119862119901119907119890119904119889119879119907119890119904119889119905

= 119896119907119890119904(minus2 lowast 119879119907119890119904 + 119879119894 + 119879119900)

(05119909119907119890119904)2

(55)

119896119907119890119904(119879119907119890119904 minus 119879119900)

05119909119907119890119904= 휀120590(119879119900

4 minus 1198791198901198991199074 )

(56)

119879119900 = 119879119900119888119900119899119904119905

(57)

For the 2D representation the heat conduction in the RPV is solved using FDM considering the heat transfer in both the thickness and the angular directions in the RPV wall The RPV lower head is treated as a spherical shell The corresponding heat conduction equation in the 2D spherical coordinate is given as Eq (58) The cylindrical part of the vessel is treated as a cylindrical shell The corresponding heat conduction equation in the 2D cylindrical coordinate is given as Eq (59) Eq (58) and (59) are coupled and solved together In default the mesh size is 5deg in angular direction and 0025119909119888119903 in radius direction This mesh discretization can be changed up to user demands

ρ119862119901part119879

part119905=

1

1199032part

part119903(1198961199032

part119879

part119903) +

1

r2sinθ

part

partθ(119896119904119894119899θ

part119879

partθ)

(58)

ρ119862119901part119879

part119905=1

119903

part

part119903(119896119903

part119879

part119903) +

part

part119911(119896part119879

part119911) (59)

52 Hybrid Coupling Approach

The hybrid coupling approach to assess the RPV integrity is illustrated in Figure 22 transIVR and ANSYS Mechanical are coupled in a one-way manner transIVR conducts the conjugate heat transfer of melt pool and RPV and provides transient thermal loads to the ANSYS Mechanical where structural analysis of RPV deformation is performed The 2D heat transfer of RPV is solved in transIVR to better capture the temperature distribution In RPV structural analysis same considerations are taken as in Section 422 for 16MND5 steel the modified theta projection model for creep and MISO for plasticity and temperature-dependent Young modulus etc The VLM is adopted for the transfer of transient thermal loads

41

Figure 22 Framework of the hybrid coupling approach [74]

53 Verification and Validation

The transIVR code was first benchmarked against the UCSB FIBS case of two-layer pool configuration and then validated against the LIVE-7V experiment showing the codersquos capability in predicting two-layer and transient melt pool heat transfers respectively Finally the hybrid coupling approach encompassing transIVR and ANSYS Mechanical was validated against the RPV failure experiment FOREVER-EC2 and the results showed acceptable agreements with those of both the experiment and the previous approach using CFD in Section 43 Hence the transIVR code is an efficient tool for melt pool heat transfer and the hybrid approach an efficient way for RPV failure analysis

531 UCSB FIBS benchmark case

The UCSB FIBS benchmark case [78] was calculated to demonstrate the capability of the transIVR code in simulating two-layer pool heat transfer problems The considered pool configuration was similar to that shown in Figure 20 with the following specific dimensions a two-layer melt pool with a 16 m oxide layer and a 08 m metallic pool on top and an RPV lower head with a thickness of 015 m The internal volumetric source in the oxide layer was 13 MWm3 The calculation results were compared with those of the DOE study [78] and the INEEL study [79]

The predicted results were compared with the digitalized results of the DOE and INEEL studies in Figure 23 All three studies show similar profiles of the heat flux distribution along the polar angle (Figure 23(a)) it increased with the polar angle in the oxide layer to the maximum value at around 77deg and then dropped to a significantly lower value in the metal layer The transIVR result agreed well with the other results with small discrepancy in the metal layer the DOE study gave a

heat flux of 500 MWm2 the INEEL study of 400 MWm2 and the transIVR of somewhere in between Figure 23(b) shows the predicted corium crust thickness distribution along the RPV polar angle also agreed well the DOE and INEEL

transIVR ANSYS Mechanical

Transient

thermal loads

42

studies The crust thickness decreased with polar angle in the oxide layer and no crust was formed in the metal layer As the crust formation is closely related to the heat flux distribution the crust thickness profile was reasonable regarding the heat flux distribution shown in Figure 23(a) Vessel wall ablation would occur due to the high thermal loads Figure 23(c) gives the profile of the remaining vessel wall thickness along the RPV polar angle considering the wall ablation It shows that no ablation occurred in low angle region (0deg~36deg) where heat flux was low and the wall thickness changed from 15 cm to around 6 cm as angle increased in high angle region of the oxide layer Because of the discrepancy in the heat flux of metal layer the predicted wall thicknesses also differ from those of the DOE and INEEL studies the thinnest wall was 73 cm predicted by the DOE study corresponding to the maximum heat flux followed by 88 cm of the present transIVR calculation and 94 cm of the INEEL study Generally speaking the prediction of transIVR is in an acceptable level compared with the other studies for the same benchmark case

Figure 23 Simulation results of the UCSB FIBS benchmark case [74]

(a) heat flux distribution (b) crust thickness distribution

(c) wall thickness distribution

43

532 LIVE-7V experiment

LIVE-7V is an experiment under the LIVE program conducted in KIT to study the core melt phenomena As Figure 24(a) shows it employed the LIVE-3D hemispherical facility with an inner diameter of 1 m and a wall thickness of ~25 mm The non-eutectic binary salt mixture of 20 mol NaNO3- 80 mol KNO3 was used as the simulant material The decay heat in the melt was simulated by cable-type heating with 8 planes of electrical resistance heating wires Water cooling was applied to the top and vessel surfaces Top surface cooling was realized by a cooling lid mounted at height 413 mm from the vessel bottom and vessel cooling by the flow channel between the test vessel and the outer cooling vessel The inlet and outlet water temperature for both top and vessel cooling surfaces were kept ~293 K and ~303 K throughout the whole experiment respectively A 413 mm height pool corresponds to a total volume of ~0194 m3 simulant material The experiment procedure can be divided into four stages with the heating power sequentially shifted from 29 kW to 24 kW 18 kW and 9 kW Figure 24(b) indicates the thermocouple positions in experiment

The simulation results of LIVE-7V using transIVR were compared with experimental data [80] and numerical study using REALPSCDAPSIM code [81] For simplicity the RELAPSCDAPSIM results in the following are labelled as lsquoCOUPLErsquo which is the name of the module in RELAPSCDAPSIM for melt pool heat transfer

Figure 24 LIVE test facility [80] and thermocouple positions [82]

Figure 25 compares the predicted heat removal upward to the top lid and downward to the bottom vessel surface with those of the experiment and COUPLE The black curve indicates the total input power at the four stages The experimental results (red and blue dots) were calculated from the Table 3 of [80] assuming no extra heat losses For the first three stages the transIVR predictions agreed well with the COUPLE results while the predicted upward heat flow was slightly lower and the downward heat flow higher than the experimental data For the last stage the transIVR agreed well with the experimental result while the COUPLE predicted lower upward and higher downward heat flows The short

(a) (b)

44

transition periods between two stages with different total inputs taken in transIVR also agreed with those in the COUPLE predictions

Figure 25 Comparison of heat removal upward to the top melt surface cooling lid and downward to bottom vessel wall [74]

Figure 26 shows the comparison of heat flux distribution along the vessel wall All curves show similar trend that a higher total input power would result in a higher heat flux profile As the COUPLE predictions had relatively flatter profiles transIVR predictions generally agreed better with the experimental results especially when total input was high A discrepancy is seen that the minimum value of a curve occurred at angle 0degfor both transIVR and COUPLE and at angle 30deg for the experimental data It is noted that the models in default assume heat flux monotonically increases with angle (cf Eq (37) and (38)) resulting it impossible to predict higher heat flux at angle 0deg than angle 30deg as in experimental results

45

Figure 26 Heat flux profiles along the vessel polar angle [74]

533 FOREVER-EC2 experiment

In this section we performed calculations against the FOREVER-EC2 experiment to validate the performance of the coupled transIVR and ANSYS Mechanical in assessing the RPV integrity The applied boundary conditions were identical to the settings of previous coupling approach in Section 4 (ANSYS Fluent+ Mechanical) as indicated in Figure 11 The initial bulk temperature of the melt pool was set to be 1261 K in transIVR which corresponds to the maximum value of the initial temperature field in CFD calculation of previous coupling approach Both the experimental data and the simulation results using previous coupling approach are included for comparison In the following results from current hybrid approach are labelled lsquotransIVRrsquo (and referred as transIVR results) results from previous approach labelled lsquoCFDrsquo (and referred as CFD results) and results from experiment labelled lsquoexperimentrsquo

Figure 27 compares the bulk temperature predicted by transIVR with the averaged and maximum temperatures of the CFD results All curves show the same trend that the temperature gradually increased with time till reaching a steady state and then kept unchanged till the end The times needed for reaching the steady state were ~5000 s for the CFD and ~4000 s for the transIVR which are close to each other It indicates again a good performance of the transIVR in predicting the transient behavior of melt pool Compared with the steady-state averaged temperature 1478 K and maximum temperature 1609 K of the CFD results the steady-state temperature of transIVR 1599 K is a reasonable prediction

0

3000

6000

9000

12000

15000

18000

0 20 40 60 80 100

Hea

t fl

ux

(W

m2)

Polar angle (deg)

29 kW-EXP24 kW-EXP18 kW-EXP9 kW-EXP29 kW-COUPLE24 kW-COUPLE18 kW-COUPLE9 kW-COUPLE29 kW-transIVR24 kW-transIVR18 kW-transIVR9 kW-transIVR

46

Figure 27 Temperature evolution with time [74]

Figure 28 Temperature fields at time (a) 01 s (b) 3038 s (c) 6038 s and (d) 13289 s [74]

Figure 28 shows the temperature fields of the vessel at time 01 s 3038 s 6038 s and 13289 s The temperature fields were visualized in the ANSYS Mechanical It also indicates that the transient temperature loads were properly transferred from transIVR to ANSYS Mechanical Figure 28(a-d) show both the minimum and maximum temperatures increased with time In the lower head region the temperature gradually increased with the height The maximum temperature occurs at the location slightly lower than the melt pool top surface This is because thermal loadheat flux imposed on the vessel generally increases with polar angle in the pool natural convection so the code transIVR inherently also predicted higher heat flux at higher angle positions The higher flux at higher position then correspondingly resulted in higher temperature In the cylindrical region the

(a) (b) (c) (d)

[K] [K] [K] [K]

47

temperature decreased as the height since the bottom part would be more significantly heated by the high-temperature lower head region

Figure 29 shows the steady-state temperature profiles of the external vessel surface along the polar angle The transIVR result agreed well with other two results in trend that the temperature increased with the angle and reach the peak values at angle ~80deg then gradually decreased with angle But slightly lower temperature was predicted in the low angle region Possible reason may be that the local heat flux distribution determined by Eq (37) and (38) is steeper than that in the experiment and CFD study which then resulted in a steeper increase of temperature along the angle and a lower temperature at low angle region In the cylindrical part (gt90deg) the transIVR result generally agreed with the experimental and CFD results The discrepancy with the experimental data may be caused by the simplified boundary conditions the ambient temperature inside the vessel was set to uniform constant value while it could decrease as height Therefore predicted heat loss in the higher position of cylindrical part could be smaller than that with height-dependent ambient temperature resulting in higher vessel temperature Overall the prediction was reasonable for a lumped-parameter model

Figure 29 Temperature profiles of the external vessel surface along the vessel polar angle [74]

The transIVR predicted a failure time of 24959 s which is close to 24180 s in experiment and 23959 s in the CFD results Figure 30 shows the total deformation of vessel bottom point with time All three curves increased gradually with time and experienced drastic accelerations at the end The

400

600

800

1000

1200

1400

0 40 80 120 160

Tem

per

ature

(K

)

Polar angle (deg)

transIVRExperimentCFDWeilding Line

48

transIVR curve agreed well with the CFD curve The final total deformation at failure time is also close to that of the CFD result but lower than the experiment data The discrepancy might be related to the simplifications in the boundary conditions accuracy of the thermal load prediction and the modelling errors in plasticity and creep

Figure 30 Total deformation in vertical direction of the vessel bottom point [74]

Figure 31 shows the profile of wall thickness change The transIVR curve was compared with the CFD curve and four experimental curves measured at vessel left right front and back sides The simulation result fitted the experimental and CFD results well in the region 0deg~50deg In the region 50deg~90deg wall thickness gradually decreased with angle till around 75deg and then increased till 90deg The maximum thickness change was -3 mm for the simulation which is close to the experiment left side curve and slightly larger than the -6~ -45 mm of the other curves from experiment and CFD results Concerning possible errors and uncertainties in the modelling and numerical settings the new approach generally gives reasonable predictions

0

5

10

15

20

25

11670 15170 18670 22170 25670

Tota

l def

orm

atio

n (

mm

)

Time (s)

Experiment

CFD

transIVR

49

Figure 31 Wall thickness changes along the vessel polar angle [74]

-6

-5

-4

-3

-2

-1

0

1

0 20 40 60 80 100

Wal

l th

icknes

s ch

ange

(mm

)

Polar angle (deg)

CFD

transIVR

Experiment (left)

Experiment (right)

Experiment (front)

Experiment (back)

50

51

6 Summary and Outlook

Reactor pressure vessel (RPV) is one of the key physical barriers in the defence-in-depth approach of nuclear power safety and its failure is also the transition point of core melt (corium) progression from in-vessel to ex-vessel phase Quantification of RPV failure during a hypothetical severe accident is therefore paramount to the specific qualification of severe accident mitigation strategies as well as the general assessment of corium risk in light water reactors The RPV failure is a problem of structural performance of RPV under the influence of thermal loads imposed by corium melt pool ie it is a transient thermo-fluid-structure coupled problem per se whose solution requires simultaneous modelling of both highly turbulent melt pool heat transfer and large-scale nonlinear RPV creep Due to the complexity of the problem only limited capabilities have been developed in the previous studies under different simplifications and assumptions in models and coupling methods The present study was motivated to advance modelling and coupling approaches for RPV performance analysis considering new understanding of physical phenomena and latest coupling platform of simulation tools

For the new development of coupling approaches and models for physical phenomena related to RPV failure analysis during severe accidents this thesis work was conducted in the following order (i) a new creep model for the RPV steel 16MND5 was developed which features the capability to cover all three creep stages (ii) a new thermo-fluid-structure coupling approach was developed for RPV failure analysis and validated against FOREVER-EC2 experiment (iii) the performance of the new coupling approach using volume loads mapping (VLM) for data transfer was further compared with another approach using surface loads mapping (SLM) (iv) a hybrid coupling approach was proposed for efficient analysis of RPV failure which employs a lumped-parameter code for quick estimate of thermal load and a mechanistic FE code for detailed structural analysis and (v) for the hybrid coupling approach the code transIVR was developed for calculation of transient melt pool heat transfer

The developed creep model for 16MND5 is called lsquomodified theta projection modelrsquo and describes all three stages of a creep process Creep curves are expressed as a function of time with five model parameters 120579119894(119894 = 14 119886119899119889 119898) A dataset for the model parameters was constructed by fitting experimental creep curves into this function and an iterative parameter adjustment process to satisfy the monotonicity assumption of creep strain vs temperature and stress Model parameters for new curves under different temperature and stress loads are directly interpolated from this dataset in contrast to the conventional ldquotheta projection modelrdquo that employs an extra fitting process The new treatment was proven to better accommodate all experimental curves over the wide ranges of temperature and stress loads The model was implemented into the ANSYS Mechanical code The simulation results indicate that the model successfully captured all three creep stages and a good agreement was achieved between each experimental creep curve and corresponding modelrsquos prediction Regarding the model response to transient changes of temperature and stress loads the widely used time hardening and strain hardening models were implemented and

52

performing reasonably well in the new method These properties fulfil the requirements of a creep model for structural analysis

The transient thermo-fluid-structure coupling approach was developed using the ANSYS Fluent code for thermofluid dynamics of the melt pool and ANSYS Structural code for structural mechanics of RPV An extension tool was introduced to realize transient load transfer from ANSYS Fluent to Structural and minimize the user effort Both turbulence models and the effective model PECM can be used for melt pool heat transfer The developed creep model was used for RPV failure analysis of the RPV Since it is insensitive to geometry the coupling approach allows more complex geometries such as a BWR lower head with a forest of penetrations (CRGTs and IGTs) modelled in detail Moreover the coupling approach does not only reflect the transient response of thermo-fluid-structure interaction but also facilitates the implementation of advanced (high-fidelity) models in both fluid mechanics and structural mechanics The validation of the coupling approach against the FOREVER-EC2 experiment showed that the temperature profile vessel failure time total displacement of vessel bottom point and wall thickness profile in the experiment were well predicted

Following the variation of the coupling approach from VLM to SLM for data transfer between ANSYS Fluent and ANSYS Structural the respective coupling approaches were investigated by comparative calculations against the FOREVER-EC2 experiment and a postulated severe accident of a reference BWR For the FOREVER-EC2 experiment both VLM and SLM approaches predicted generally well the total deformation in vertical direction of bottom point and the wall thickness change though the former predicted a slightly later failure time For the reference BWR case good agreements were achieved in predicting the maximum creep strain and the total deformation in vertical direction of bottom point For both experiment and reactor cases the VLM approach predicted slightly later failure times than SLM but the differences were negligible compared to the long time-scale of vessel failure (~104 s) The deformation rate of the vessel wall was slow at the early stage and accelerated with time till a drastic deformation and final vessel failure Generally speaking the VLM and SLM coupling approaches had the similar performance in terms of their predictability of experiment and applicability to reactor case but the VLM was computationally more efficient

In the development of the hybrid coupling approach the transIVR code was developed for quick estimate of transient melt pool heat transfer and determination of thermal loads on RPV transIVR does not only support transient analysis of one- and two- layer melt pool but also the heat transfer in RPV being resolved with 2D finite difference method which captures more spatial thermal details for RPV structural analysis in addition to 1D heat transfer model The capabilities of transIVR in modelling two-layer pool heat transfer and transient pool heat transfer were demonstrated by calculations against the UCSB FIBS benchmark case and the LIVE-7V experiment respectively It was then coupled to the mechanical solver ANSYS Mechanical which together form the hybrid coupling framework for detailed RPV failure analysis The calculation against the FOREVER-EC2 experiment did not only reproduce the major results of the experiment but also had a good agreement with the thermo-fluid-structure

53

coupling approach using CFD This indicated the new coupling approach is capable of capturing the vessel creep failure characteristics and can be employed as an efficient tool addressing the complex vessel failure problem

As a prospective more research topics can be addressed in future either as a continuation of the current work toward an effective tool for vessel failure analysis or from the general view of scientific interest in melt pool heat transfer and RPV failure The following topics are recommended

bull Extended applications to the reactor cases For instance the thermo-fluid-structure coupling approach can be used to study penetration failures in the RPV of an BWR with a detailed representation of penetrations (IGTs and CRGTs)

bull Synthesis or update of heat transfer correlations for melt pool Heat transfer correlations directly benefit the parametric codes such as transIVR in accurate quantification of heat flux distribution around a melt pool So far many empirical correlations have been proposed based on experiments which had a variety of geometries and melt simulants with a wide range of Rayleigh number (Ra) Sensitivity studies have also been performed on the influences of Ra Pa and other material properties Therefore a comprehensive synthesis or update of the existing heat transfer correlations will be a meaningful step for making reasonable extrapolations to the reactor applications with prototypic materials

bull Modelling of pool stratification The motivation starts with the improvement of the PECMrsquos modelling capability such that the dynamic stratification and resulting pool heat transfer can be modelled altogether Simulation of melt stratification has been supported in some system codes such as ASTEC using a predefined parametric approach The improvement of PECM can be considered in the similar approach or more mechanistic modelling

bull Development of a structural module for RPV The idea is to develop a simplified module for structural analysis so it can be coupled with transIVR ultimately forming a standalone code for quick estimate of vessel failure of LWRs during postulated severe accidents

bull Extension of vessel analysis capability The current version of tranIVR is limited to melt pool Although the precursory debris bed and melt ejection after vessel failure were not the focus of the present study they are equally important to assessment of corium risk especially for Swedish BWRs where the melt release conditions and history are crucial to qualification of their severe accident mitigation strategies

54

55

Bibliography

[1] IAEA ldquoIAEA Annual Report 2018rdquo IAEA 2019 [Online] Available httpswwwiaeaorgsitesdefaultfilesgcgc63-5pdf

[2] IAEA Safety of Nuclear Power Plants Design Vienna INTERNATIONAL ATOMIC ENERGY AGENCY 2012

[3] M Fischer ldquoGen II and III In-Vessel and Ex-Vessel Melt Retentionrdquo The Short Course on Severe Accident Phenomenology Oct 2017

[4] B R Sehgal Nuclear Safety in Light Water Reactors Severe Accident Phenomenology 1st ed Academic Press 2011

[5] IAEA Approaches and Tools for Severe Accident Analysis for Nuclear Power Plants Vienna INTERNATIONAL ATOMIC ENERGY AGENCY 2008

[6] WENRA ldquoPractical Elimination Applied to New NPP Designs - Key Elements and Expectationsrdquo Sep 2019 [Online] Available httpwwwwenraorgmediafiler_public20191111practical_elimination_applied_to_new_npp_designs_-_key_elements_and_expectations_-_for_issuepdf

[7] P Matejovic M Barnak M Bachraty and L Vranka ldquoASTEC applications to VVER-440V213 reactorsrdquo Nuclear Engineering and Design vol 272 pp 245ndash260 2014 doi 101016jnucengdes201308077

[8] AREVA ldquoUS EPR Severe Accident Evaluationrdquo Topical Report ANP-10268NP Oct 2006 [Online] Available httpswwwnrcgovdocsML0631ML063100157pdf

[9] C T Martin ldquoCoupled 3D Thermo-mechanical Analysis of Nordic BWR Lower Head Failure in case of Core Melt Severe Accidentrdquo master thesis KTH Stockholm 2013

[10] IAEA ldquoAdvanced Large Water Cooled Reactors A supplement to the IAEArsquos Advanced Reactor Information System (ARIS)rdquo Sep 2015 [Online] Available httpsarisiaeaorgPublicationsIAEA_WRC_Bookletpdf

[11] B Pershagen Light Water Reactor Safety Pergamon Press 1989 [12] Ninos Garis M Agrell H Glanneskog and L Agrenius ldquoAPRI 7 ndash Accident

Phenomena of Risk Importancerdquo SSM SSM 201212 2012 [Online] Available httpswwwstralsakerhetsmyndighetensecontentassets2a9f299f46b949b3b209e780cf71436a201212-apri-7-accident-phenomena-of-risk-importance-en-lagesrapport-om-forskningen-inom-omradet-svara-haverier-under-aren-2009-2011-

[13] F Fichot et al ldquoSome considerations to improve the methodology to assess In-Vessel Retention strategy for high-power reactorsrdquo Annals of Nuclear Energy vol 119 pp 36ndash45 2018 doi 101016janucene201803040

[14] T G Theofanous C Liu S Additon S Angelini O Kymaumllaumlinen and T Salmassi ldquoIn-vessel coolability and retention of a core meltrdquo Nuclear Engineering and Design vol 169 no 1 pp 1ndash48 1997 doi 101016S0029-5493(97)00009-5

[15] C Sainte Catherine ldquoTensile and creep tests material characterization of pressure vessel steel(16MND5) at high temperatures (20 up to 1300 degC)rdquo CEA France SEMTLISNRT98-009A 1998

56

[16] E Altstadt H-G Willschuetz B R Sehgal and F-P Weiss ldquoModelling of in-vessel retention after relocation of corium into the lower plenumrdquo Forschungszentrum Rossendorf FZR-437 2005

[17] H-G Willschuumltz and E Altstadt ldquoGeneration of a High Temperature Material DataBase and its Application to Creep Tests withFrench or German RPV-steelrdquo FORSCHUNGSZENTRUM ROSSENDORF FZR-353 ISSN 1437-322X Aug 2002

[18] J L Rempe S A Chavez and G L Thinnes ldquoLight water reactor lower head failure analysisrdquo Nuclear Regulatory Commission NUREGCR--5642 1993 Accessed Aug 23 2019 [Online] Available httpinisiaeaorgSearchsearchaspxorig_q=RN25025905

[19] S Brosi et al ldquoCORVIS Investigation of light water reactor lower head failure modesrdquo Nuclear Engineering and Design vol 168 no 1 pp 77ndash104 1997 doi 101016S0029-5493(96)00008-8

[20] T Y Chu M M Pilch J H Bentz J S Ludwigsen W Y Lu and L L Humphries ldquoLower Head Failure Experiments and Analysesrdquo Sandia National Laboratories Albuquerque NM USA NUREGCR-5582 SAND98-2047 1999

[21] L L Humphries et al ldquoOECD Lower Head Failure Project Final Reportrdquo Sandia National Laboratories Albuquerque NM 87185-1139 2002

[22] B R Sehgal R R Nourgaliev T N Dihn A Karbojian J A Green and V A Bui ldquoFOREVER Experiments on Thermal and Mechanical Behavior of a Reactor Pressure Vessel during a Severe Accidentrdquo in Proceedings of the Workshop on in-vessel core debris retention and coolability Garching near Munich Germany Mar 3-6 1998

[23] S Bhandari M Fischer B R Sehgal and A Theerthan ldquoPost Test Analysis of the FOREVER C2 Experimentrdquo The 9th International conference on Nuclear Engineering (ICONE-9) Nice France Apr 8-12 2001

[24] J F Mao J W Zhu S Y Bao L J Luo and Z L Gao ldquoCreep deformation and damage behavior of reactor pressure vessel under core meltdown scenariordquo International Journal of Pressure Vessels and Piping vol 139ndash140 pp 107ndash116 2016 doi 101016jijpvp201603009

[25] J Mao L Hu S Bao L Luo and Z Gao ldquoInvestigation on the RPV structural behaviors caused by various cooling water levels under severe accidentrdquo Engineering Failure Analysis vol 79 pp 274ndash284 2017 doi 101016jengfailanal201704029

[26] J F Mao J W Zhu S Y Bao L J Luo and Z L Gao ldquoStudy on structural failure of RPV with geometric discontinuity under severe accidentrdquo Nuclear Engineering and Design vol 307 pp 354ndash363 2016 doi 101016jnucengdes201607027

[27] B R Sehgal V A Bui T N Dinh and R R Nourgaliev ldquoHeat Transfer Processes in Reactor Vessel Lower Plenum During Late Phase of In-Vessel Core Melt Progressionrdquo in Advances in Nuclear Science and Technology J Lewins M Becker R W Albrecht E J Henley J D McKean K Oshima A Sesonske H B Smets and C P L Zaleski Eds Boston MA Springer US 1999 pp 103ndash135

[28] C-T Tran and T-N Dinh ldquoThe effective convectivity model for simulation of melt pool heat transfer in a light water reactor pressure vessel lower head Part I Physical processes modeling and model implementationrdquo Progress

57

in Nuclear Energy vol 51 no 8 pp 849ndash859 2009 doi 101016jpnucene200906007

[29] C-T Tran and T-N Dinh ldquoThe effective convectivity model for simulation of melt pool heat transfer in a light water reactor pressure vessel lower head Part II Model assessment and applicationrdquo Progress in Nuclear Energy vol 51 no 8 pp 860ndash871 2009 doi 101016jpnucene200906001

[30] H-G Willschuumltz E Altstadt B R Sehgal and F-P Weiss ldquoCoupled thermal structural analysis of LWR vessel creep failure experimentsrdquo Nuclear Engineering and Design vol 208 no 3 pp 265ndash282 2001 doi 101016S0029-5493(01)00364-8

[31] E Altstadt and T Moessner ldquoExtension of the ANSYS creep and damage simulation capabilitiesrdquo Forschungszentrum Rossendorf Technical report FZR-296 2000

[32] H-G Willschuumltz E Altstadt B R Sehgal and F-P Weiss ldquoRecursively coupled thermal and mechanical FEM-analysis of lower plenum creep failure experimentsrdquo Annals of Nuclear Energy vol 33 no 2 pp 126ndash148 2006 doi 101016janucene200508006

[33] H-G Willschuumltz E Altstadt B R Sehgal and F-P Weiss ldquoSimulation of creep tests with French or German RPV-steel and investigation of a RPV-support against failurerdquo Annals of Nuclear Energy vol 30 no 10 pp 1033ndash1063 2003 doi 101016S0306-4549(03)00036-7

[34] H Madokoro A Miassoedov and T Schulenberg ldquoA thermal structural analysis tool for RPV lower head behavior during severe accidents with core meltrdquo Mechanical Engineering Letters vol 4 pp 18-00038-18ndash00038 2018 doi 101299mel18-00038

[35] W Villanueva C-T Tran and P Kudinov ldquoCoupled thermo-mechanical creep analysis for boiling water reactor pressure vessel lower headrdquo Nuclear Engineering and Design vol 249 pp 146ndash153 2012 doi 101016jnucengdes201107048

[36] W Villanueva C T Tran and P Kudinov ldquoA Computational Study on Instrumentation Guide Tube Failure During a Severe Accident in Boiling Water Reactorsrdquo in The 14th International Topical Meeting on Nuclear Reactor Thermal hydraulics NURETH-14 Toronto Ontario Canada Sep25-30 2011

[37] W Villanueva C T Tran and P Kudinov ldquoAssessment with Coupled Thermo-Mechanical Creep Analysis of Combined CRGT and External Vessel Cooling Efficiency for A BWRrdquo The 14th International Topical Meeting on Nuclear Reactor Thermal hydraulics NURETH-14 Toronto Ontario Canada Sep 25-30 2011

[38] F R Larson and J Miller ldquoA Time-Temperature Relationship for Rupture and Creep Stressesrdquo Trans ASME vol 74 p 765minus775 1952

[39] C T Tran ldquoDevelopment validation and application of an effective convectivity model for simulation of melt pool heat transfer in a light water reactor lower headrdquo Licentiate thesis KTH Stockholm 2007

[40] ANSYS ldquoANSYS Fluent Theory Guide Release 162rdquo ANSYS Inc Jul 2015 [Online] Available httpwwwansyscom

[41] P Zeng Finite element analysis and application (in Chinese) Tsinghua University Press amp Springer 2004

58

[42] ANSYS ldquoANSYS Mechanical | Finite Element Analysis Softwarerdquo httpswwwansyscomproductsstructuresansys-mechanical-enterprise (accessed Sep 26 2019)

[43] ANSYS ldquoANSYS Mechanical APDL Theory Reference Release 162rdquo ANSYS Inc Jul 2015 [Online] Available httpwwwansyscom

[44] U Steinberner and H H Reineke ldquoTurbulent Buoyancy Convection Heat Transfer with Internal Heat Sourcesrdquo in Proceedings of the 6th Int Heat Transfer Conference Toronto Canada 1978 vol Vol2 pp 305ndash310 doi 101615IHTC63350

[45] R W Evans and B Wilshire Introduction to Creep London Institute of Materials 1993

[46] DoITPoMS ldquoCreep Deformation of Metalsrdquo Creep Deformation of Metals University of Cambridge 2006 httpswwwdoitpomsacuktlplibcreepindexphp (accessed Jul 09 2019)

[47] ANSYS ANSYS Mechanical APDL Material Reference Release 162 2015 [48] SIMULIA Abaqus 612 Analysis Userrsquos Manualvol 3 Materials Chapter

2324 2012 [49] S Imaoka ldquoUser Creep Subroutine STI0704A ANSYS Release110rdquo 2007 [50] R W Evans J D Parker and B Wilshire ldquoThe θ projection conceptmdashA

model-based approach to design and life extension of engineering plantrdquo International Journal of Pressure Vessels and Piping vol 50 no 1 pp 147ndash160 1992 doi 1010160308-0161(92)90035-E

[51] ECCC ldquoRecommendations and Guidance for the Assessment of Creep Strain and Creep Strength Data ECCC Recommendations Volume 5 Part 1 b[Issue 2]rdquo 2013

[52] W D Day and A P Gordon ldquoLife Fraction Hardening Applied to a Modified Theta Projection Creep Model for a Nickel-Based Super-Alloyrdquo The ASME Turbo Expo 2014 Turbine Technical Conference and Exposition Duumlsseldorf Germany Jun16-20 2014 vol 7A-Structures and Dynamics p V07AT29A010 doi 101115GT2014-25881

[53] M Kumar I V Singh B K Mishra S Ahmad A Venugopal Rao and V Kumar ldquoA Modified Theta Projection Model for Creep Behavior of Metals and Alloysrdquo J of Materi Eng and Perform vol 25 no 9 pp 3985ndash3992 2016 doi 101007s11665-016-2197-y

[54] P Yu and W Ma ldquoA modified theta projection model for creep behavior of RPV steel 16MND5rdquo Journal of Materials Science amp Technology vol 47 pp 231ndash242 2020 doi 101016jjmst202002016

[55] ANSYS ANSYS Mechanical Advanced Nonlinear Materials v160 Lecture 2 Rate Dependent Creep ANSYS training materials 2015

[56] Q Y Li N C Wang and D Y Yi Numerical Analysis (in Chinese) 5th ed Tsinghua University Press 2008

[57] T N Dinh R R Nourgaliev and B R Sehgal ldquoOn heat transfer characteristics of real and simulant melt pool experimentsrdquo Nuclear Engineering and Design vol 169 no 1 pp 151ndash164 1997 doi 101016S0029-5493(96)01283-6

[58] O Kymaumllaumlinen H Tuomisto O Hongisto and T G Theofanous ldquoHeat flux distribution from a volumetrically heated pool with high Rayleigh numberrdquo

59

Nuclear Engineering and Design vol 149 no 1ndash3 pp 401ndash408 1994 doi 1010160029-5493(94)90305-0

[59] F J Asfia and V K Dhir ldquoNatural convection heat transfer in volumetrically heated spherical poolsrdquo 1994 Accessed Aug 23 2019 [Online] Available httpinisiaeaorgSearchsearchaspxorig_q=RN26037727

[60] M Jahn and H H Reineke ldquoFree Convection Heat Transfer with Internal Heat Sources Calculations and Measurementsrdquo in Proceedings of the 5th Int Heat Transfer Conference Tokyo Japan 1974 vol Vol3 paper NC-28

[61] M Gustafsson ldquoFSI Transient extension V1704rdquo EDR MEDESO 2016 [62] EDR MEDESO httpsedrmedesocom [63] H Kim and I Namgung ldquoThermo-mechanical Analysis of RVLH for

APR1400 during a Severe Accidentrdquo in Transactions SMiRT-23 Manchester United Kingdom Aug10-14 2015 [Online] Available httpwwwlibncsueduresolver18402034197

[64] P Yu and W Ma ldquoComparative Analysis of Reactor Pressure Vessel Failure using Two Thermo-Fluid-Structure Coupling Approachesrdquo submitted to journal

[65] F R Menter ldquoTwo-equation eddy-viscosity turbulence models for engineering applicationsrdquo AIAA Journal vol 32 no 8 pp 1598ndash1605 1994 doi 102514312149

[66] T N Dinh and R R Nourgaliev ldquoTurbulence modelling for large volumetrically heated liquid poolsrdquo Nuclear Engineering and Design vol 169 no 1 pp 131ndash150 1997 doi 101016S0029-5493(96)01281-2

[67] ANSYS Inc ldquoIntroduction to ANSYS Fluent Lecture 7 Turbulence Modelling Fluent Tutorial Materialrdquo 2016

[68] A Horvat and M Borut ldquoNumerical investigation of natural convection heat transfer in volumetrically heated spherical segmentsrdquo in Proceedings of the International Conference Nuclear Energy for New Europe 2004

[69] Y Li CFD Pre-test Analysis of SIMECO-2 Experiment 2016 [70] A Theerthan A Karbojian and B R Sehgal ldquoEC-FOREVER experiments

on thermal and mechanical behavior of a reactor pressure vessel during a severe accident Technical report-1 EC-FOREVER-2 testrdquo SAM-ARVI-D008 Mar 2001

[71] P Yu W Ma W Villanueva A Karbojian and S Bechta ldquoValidation of a thermo-fluid-structure coupling approach for RPV creep failure analysis against FOREVER-EC2 experimentrdquo Annals of Nuclear Energy vol 133 pp 637ndash648 2019 doi 101016janucene201906067

[72] T G Theofanous C Liu S Additon S Angelini O Kymaumllaumlinen and T Salmassi ldquoIn-vessel coolability and retention of a core meltrdquo Nuclear Engineering and Design vol 169 no 1ndash3 pp 1ndash48 1997 doi 101016S0029-5493(97)00009-5

[73] H P Langtangen ldquoSolving nonlinear ODE and PDE problemsrdquo Simula Research Laboratory 2016 [Online] Available httpshplgitgithubionum-methods-for-PDEsdocpubnonlinpdfnonlin-4printpdf

[74] P Yu and W Ma ldquoDevelopment of a Lumped-Parameter Code for Efficient Assessment of In-Vessel Melt Retention Strategy of LWRsrdquo submitted to journal

60

[75] T G Theofanous M Maguire S Angelini and T Salmassi ldquoThe first results from the ACOPO experimentrdquo Nuclear Engineering and Design vol 169 no 1 pp 49ndash57 1997 doi 101016S0029-5493(97)00023-X

[76] S W Churchill and H H S Chu ldquoCorrelating equations for laminar and turbulent free convection from a vertical platerdquo International Journal of Heat and Mass Transfer vol 18 no 11 pp 1323ndash1329 1975 doi 1010160017-9310(75)90243-4

[77] S Globe and D Dropkin ldquoNatural-Convection Heat Transfer in Liquids Confined by Two Horizontal Plates and Heated from Belowrdquo J Heat Transfer vol 81 no 1 pp 24ndash28 1959 doi 10111514008124

[78] T G Theofanous C Liu S Additon S Angelini O Kymaelaeinen and T Salmassi ldquoIn-vessel coolability and retention of a core melt Volume 2rdquo Argonne National Lab DOEID--10460-VOL2 1996 Accessed Apr 18 2019 [Online] Available httpinisiaeaorgSearchsearchaspxorig_q=RN30000459

[79] J L Rempe D L Knudson C M Allison G L Thinnes and C L Atwood ldquoPotential for AP600 in-vessel retention through ex-vessel floodingrdquo United States INEELEXT--97-00779 1997 Accessed Apr 18 2019 [Online] Available httpinisiaeaorgSearchsearchaspxorig_q=RN30025872

[80] A Miassoedov X Gaus-Liu T Cron and B Fluhrer ldquoLive experiments on melt pool heat transfer in the reactor pressure vessel lower headrdquo The 10th International Conference on Heat Transfer Fluid Mechanics and Thermodynamics (HEFAT 2014) Orlando FL Jul 14-16 2014

[81] H Madokoro A Miassoedov and T Schulenberg ldquoAssessment of a lower head molten pool analysis module using LIVE experimentrdquo Nuclear Engineering and Design vol 330 pp 51ndash58 2018 doi 101016jnucengdes201801036

[82] B Fluhrer et al ldquoThe LIVE-L1 and LIVE-L3 experiments on melt behaviour in RPV lower headrdquo Forschungszentrum Karlsruhe GmbH Technik und Umwelt (Germany) Inst fuer Kern- und Energietechnik FZKA--7419 2008 Accessed Nov 12 2019 [Online] Available httpinisiaeaorgSearchsearchaspxorig_q=RN40036867

AlbaNova University Center

Roslagstullsbacken 21

TRITA-SCI-FOU 202013 10691 Stockholm

ISBN 978-91-7873-540-2 Sweden

Akademisk avhandling som med tillstaringnd av Kungliga Tekniska Houmlgskolan

framlaumlgges till offentlig granskning foumlr avlaumlggande av teknologie doktorsexamen

i fysik den juni 12 2020 i AlbaNova Universitetscentrum Stockholm

copy Peng Yu June 2020

Tryck Universitetsservice US AB

To my family

and

all the people that I love

I

Abstract

This thesis aims at the development of new coupling approaches and new models for the thermo-fluid-structure coupling problem of reactor pressure vessel (RPV) failure during severe accidents and related physical phenomena The thesis work consists of five parts (i) development of a three-stage creep model for RPV steel 16MND5 (ii) development of a thermo-fluid-structure coupling approach for RPV failure analysis (iii) performance comparison of the new approach that uses volume loads mapping (VLM) for data transfer with the previous approach that uses surface loads mapping (SLM) (iv) development of a lumped-parameter code for quick estimate of transient melt pool heat transfer and (v) development of a hybrid coupling approach for efficient analysis of RPV failure

A creep model called lsquomodified theta projection modelrsquo was developed for the 16MND5 steel so that it covers three-stage creep process Creep curves are

expressed as a function of time with five parameters 120579119894 (i=14 and m) in the new creep model A dataset for the model parameters was constructed based on the available experimental creep curves given the monotonicity assumption of creep strain vs temperature and stress New creep curves can be predicted by interpolating model parameters from this dataset in contrast to the previous method that employs an extra fitting process The new treatment better accommodates all the experimental curves over the wide ranges of temperature and stress loads The model was implemented into the ANSYS Mechanical code and its predictions successfully captured all three creep stages and a good agreement was achieved between the experimental and predicted creep curves For dynamic loads that change with time the widely used time hardening and strain hardening models were implemented with a reasonable performance These properties fulfil the requirements of a creep model for structural analysis

A thermo-fluid-structure coupling approach was developed by coupling the ANSYS Fluent for the fluid dynamics of melt pool heat transfer and ANSYS Structural for structural mechanics of RPV An extension tool was introduced to realize transient load transfer from ANSYS Fluent to Structural and minimize the user effort Both CFD with turbulence models and the effective model PECM can be employed for predicting melt pool heat transfer The modified theta projection model was used for creep analysis of the RPV The coupling approach does not only capture the transient thermo-fluid-structure interaction feature but also support the advanced models in both melt pool convection and structural mechanics to improve fidelity and facilitate implementation The coupling approach performs well in the validation against the FOREVER-EC2 experiment and can be applied complex geometries such as a BWR lower head with forest of penetrations (control rod guide tubes and instrument guide tubes)

In the comparative analysis the VLM and SLM coupling approaches generally have the similar performance in terms of their predictability of the FOREVER-EC2 experiment and applicability to the reactor case Though the SLM approach predicted slightly earlier failure times than VLM in both cases the difference was negligible compared to the large scale of vessel failure time (~104 s) The VLM approach showed higher computational efficiency than the SLM

II

The idea of the hybrid coupling is to employ a lumped-parameter code for quick estimate of thermal load which can be employed in detailed structural analysis Such a coupling approach can significantly increase the calculation efficiency which is important to the case of a prototypical RPV where mechanistic simulation of melt pool convection is computationally expensive and unnecessary The transIVR code was developed for this purpose which is not only capable of quick estimate of transient heat transfer of one- and two- layer melt pool but also solving heat conduction problem in the RPV wall with 2D finite difference method to provide spatial thermal details for RPV structural analysis The capabilities of transIVR in modelling two-layer pool heat transfer and transient pool heat transfer were demonstrated by calculations against the UCSB FIBS benchmark case and the LIVE-7V experiment respectively The transIVR code was then coupled to the mechanical solver ANSYS Mechanical for detailed RPV failure analysis Validation against the FOREVER-EC2 experiment indicates the coupling framework successfully captured the vessel creep failure characteristics

Keywords RPV failure thermo-fluid-structure coupling creep modelling melt pool heat transfer severe accident CFD FEA transIVR

III

Sammanfattning

Syftet med denna avhandling aumlr att utveckla nya metoder och nya modeller foumlr att kunna koppla ihop termohydrauliska foumlrlopp och uppkommen strukturrespons i botten paring reaktortanken foumlr foumlrbaumlttrad analys av reaktortankbrott under ett svaringrt haveri Avhandlingen bestaringr av fem delar (i) utveckling av en trestegsmodell foumlr krypning i reaktortankens 16MND5-staringl (ii) utveckling av metoder foumlr kopp ling av termohydrauliska foumlrlopp och strukturrespons foumlr studier av reaktortankbrott (iii) jaumlmfoumlrelse av prestanda av den nya metoden som anvaumlnder volume loads mapping (VLM) foumlr dataoumlverfoumlring med den tidigare metoden som anvaumlnder surface loads mapping (SLM) (iv) utveckling av en lumped-parameter beraumlkningskod foumlr snabb uppskattning av transient vaumlrmeoumlverfoumlring i en smaumlltpoumll i botten paring reaktortanken och (v) utveckling av en hybrid kopplingsmetod foumlr effektiv analys av reaktortankbrott

En krypmodell kallad lsquomodified theta projection modelrsquo som behandlar krypprocessen i tre steg utvecklades foumlr 16MND5-staringlet Krypkurvor i den nya

krypmodellen ges som en funktion av tid med fem parametrar 120579119894 (i=14 och m) Ett dataset foumlr modellparametrar togs fram baserat paring tillgaumlngliga experimentalla krypkurvor under anatagande om att kryptoumljning aumlr en monoton funktion av temperatur och spaumlnning Nya krypkurvor kan genereras genom att interpolera modellparametrar fraringn detta dataset i kontrast till den tidigare metoden som anvaumlnder en speciell anpassningsprocess Foumlrdelen med det nya tillvaumlgagaringngssaumlttet aumlr att den hanterar alla experimentella krypkurvor foumlr ett brett spektrum av temperatur- och spaumlnningsbelastningar Modellen implementerades i ANSYS-koden och den korrekt behandlar alla tre krypsteg samt visar bra oumlverenstaumlmmelse mellan experimentella och beraumlknade krypkurvor Foumlr analys av tidsberoende dynamiska belastningar implementerades vaumllkaumlnda modeller av tids- och deformationshaumlrdning med rimliga resultat Dessa egenskaper uppfyller kravet paring en krypmodell foumlr strukturmekaniska analyser

En metod utvecklades foumlr koppling mellan termohydrauliska foumlrlopp och strukturmekanisk respons genom att koppla ihop ANSYS Fluent som analyserar fluiddynamiska aspekter av vaumlrmeoumlverfoumlring i en smaumlltpoumll med ANSYS Structural som analyserar strukturmekanisk respons av reaktortanksvaumlggen Ett speciellt verktyg togs fram foumlr att moumljliggoumlra oumlverfoumlring av transienta belastningar fraringn ANSYS Fluent till ANSYS Structural och foumlr att minimera anvaumlndarinsatsen Baringde CFD med turbulensmodeller och den effektiva PECM-modellen kan anvaumlndas foumlr att analysera vaumlrmeoumlverfoumlring i en smaumlltpoumll Foumlr krypanalysen av reaktortanken har lsquomodified theta projection modelrsquo anvaumlnts Den framtagna kopplingsmetoden har foumlrdelen att den inte bara faringngar de transienta aspekterna av interaktionen mellan termohydraulik och strukturmekanik men ocksaring stoumldjer avancerade modeller av konvektion i en smaumlltpoumll och strukturrespons och daumlrigenom bidrar till foumlrbaumlttrad precision och underlaumlttar implementeringen av modeller Kopplingsmetoden fungerar bra naumlr den valideras mot FOREVER-EC2 experimentet och den kan anvaumlndas i

IV

komplexa geometrier som till exempel i botten paring reaktortanken i en kokvattenreaktor (BWR) med en skog av genomfoumlringar (ledroumlr foumlr styrstavar och instrumentering)

Jaumlmfoumlrande beraumlkningar visar att de tvaring kopplingsmetoderna VLM-baserad och SLM-baserad presterar lika bra naumlr det gaumlller hur de predikterar resultat av FOREVER-EC2 experimentet och deras laumlmplighet foumlr reaktoranalyser Aumlven om SLM-metoden predikterar att tankbrott i baringda fallen intraumlffar lite tidigare saring aumlr skillnaden mellan metoderna foumlrsumbar i jaumlmfoumlrelse med tiden till tankbrott (~ 104 s) VLM-metoden var baumlttre aumln SLM-metoden naumlr det gaumlller beraumlkningseffektiviteten

Ideacuten om hybridkoppling garingr ut paring att anvaumlnda en lumped-parameter kod foumlr snabb uppskattning av termiska belastningar som senare kan anvaumlndas i detaljerad strukturmekanisk analys En saringdan kopplingsmetod kan paring ett signifikant saumltt oumlka effektiviteten av beraumlkningar vilket aumlr viktig i reaktorapplikationer daumlr mekanistisk simulering av konvektion i en smaumlltpoumll aumlr beraumlkningsintensiv och onoumldig I detta syfte utvecklades koden transIVR Koden kan dels ge en snabb uppskattning av transient vaumlrmeoumlverfoumlring i en smaumlltpoumll som bestaringr av en eller tvaring materialskikt och dels loumlsa vaumlrmeledningsproblem i reaktortanksvaumlggen med 2D finita differensmetoden vilket genererar detaljerad information om termiska foumlrharingllanden som behoumlvs foumlr strukturmekaniska analyser Foumlr att demonstrera att transIVR kan analysera dels vaumlrmeoumlverfoumlring i en smaumlltpoumll bestaringende av tvaring materialskikt och dels transient vaumlrmeoumlverfoumlring i en smaumlltpoumll jaumlmfoumlrdes transIVR prediktioner med UCSB FIBS-benchmarkfallet respektive LIVE-7V-experimentet I naumlsta steg kopplades transIVR-koden till ANSYS Structural foumlr att moumljliggoumlra mer detaljerad analys av reaktortankbrott Validering mot FOREVER-EC2 experimentet tyder paring att den utvecklade kopplingsmetodiken korrekt aringterger tankbrott orsakat av krypning

Nyckelord reaktortankbrott koppling termohydraulik-strukturmekanik modellering av krypning vaumlrmeoumlverfoumlring i smaumlltpoumll svaringra haverier CFD FEA transIVR

V

List of Publications

Papers included in this thesis

I P Yu and W Ma ldquoA modified theta projection model for creep behavior of RPV steel 16MND5rdquo Journal of Materials Science amp Technology vol 47 pp 231ndash242 2020

II P Yu W Ma W Villanueva A Karbojian and S Bechta ldquoValidation of a thermo-fluid-structure coupling approach for RPV creep failure analysis against FOREVER-EC2 experimentrdquo Annals of Nuclear Energy vol 133 pp 637ndash648 2019

III P Yu and W Ma ldquoComparative analysis of reactor pressure vessel failure using two thermo-fluid-structure coupling approachesrdquo Nuclear Engineering and Design submitted

IV P Yu and W Ma ldquoDevelopment of a lumped-parameter code for efficient assessment of in-vessel melt retention strategy of LWRsrdquo Annals of Nuclear Energy submitted

Respondentrsquos contribution to the papers included in the thesis

Peng Yu made major contributions to the publications of all the above four articles under the guidance of supervisors and with the collaboration of other co-authors including formulation of research questions development of coupling approaches and numerical methods analysis and generalization of simulation results as well as writing the first drafts and addressing review comments

Papers not included in this thesis

1 P Yu and W Ma ldquoRPV creep failure analysis using a Norton creep modelrdquo International Journal of Mechanical Sciences to be submitted

2 P Yu W Villanueva S Galushin W Ma and S Bechta ldquoCoupled thermo-mechanical creep analysis for a Nordic BWR lower head using non-homogeneous debris bed configuration from MELCORrdquo 12th International Topical Meeting on Nuclear Reactor Thermal-Hydraulics Operation and Safety (NUTHOS-12) Qingdao China October 14-18 2018

3 P Yu A Komlev Y Li W Villanueva W Ma and S Bechta ldquoPre-test simulation of SIMECO HT experiments for stratified melt pool heat transferrdquo 11th International Topical Meeting on Nuclear Reactor Thermal Hydraulics Operation and Safety (NUTHOS-11) Gyeongju Korea October 9-13 2016

VI

VII

Acknowledgements

First of all I would like to express my sincere gratitude to my supervisor Prof Weimin Ma for offering this opportunity to work in the severe accident research field and for the supports patience guidance discussions and encouragements throughout the entire period of my graduate study

I am very thankful to my co-supervisor Dr Walter Villanueva for the valuable help and discussions I learnt a lot during the times working with him I am also thankful to co-supervisor Prof Sevostian Bechta who is supportive and always has inspiring ideas that keep you thinking for a long time

Thank you also to all my colleagues at the Division of Nuclear Power Safety for the supports discussions and the very nice working environment during these years

I have taken many interesting courses during my PhD study for which I want to express my thanks to all the teachers With this regard a special thanks goes to Prof Johan Hoffman for his SF2561 Finite Element Method given in 2015 fall as well as the discussions and advices on numerical analysis

I am grateful to Ignacio Gallego-Marcos and Wenyuan Fan for their enthusiastic help on CFD and turbulence modelling during the time working on the turbulent

heat transfer I thank all my 师兄 for sharing their knowledge on nuclear engineering and giving advices when I was confused since 2012

I thank all my friends both in Sweden and in China for the company and the sharing of many wonderful moments in daily life

Finally I would like to express my deepest thank to my parents my girlfriend Siwen my sisters and all my other relatives for their unconditional supports and love

Financial supports from the China Scholarship Council and APRI make my study in KTH possible and are acknowledged

VIII

IX

Table of Contents

AbstracthelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphellipI

SammanfattninghelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphellipIII

List of PublicationshelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphellipV

AcknowledgementshelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphellipVII

NomenclaturehelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphellipXI

1 Introductionhelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphellip1

11 Nuclear Power Safety 1

12 RPV Failure 2

13 Literature Review Creep Behaviour and RPV Failure Analysis 4

131 Experiments on creep behaviour and RPV failure 4

132 Numerical studies on RPV failure 5

14 Motivation Objectives and Main Achievements 7

2 Numerical Solvers and Modelshelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphellip9

21 ANSYS Fluent 9

22 ANSYS Mechanical 9

23 PECM 11

3 Creep Modelling of 16MND5 Steelhelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphellip13

31 Creep and Creep Modelling 13

32 A Modified Theta Projection Model 14

321 Original theta projection model and its variations 14

322 A modified theta projection model 16

4 Coupled Thermo-Fluid-Structure Analysis of RPV Failurehelliphelliphelliphellip23

41 Thermo-Fluid-Structure Coupling Approach 23

42 Numerical Modelling 25

421 Conjugate heat transfer 25

422 Mechanical analysis 25

43 Validation against the FOREVER-EC2 Experiment 26

431 FOREVER-EC2 experiment 26

X

432 Conjugate heat transfer 27

433 Mechanical analysis 29

44 Application to BWR Case 32

45 Computational Efficiency 34

5 Hybrid Coupling Approach for Efficient RPV Failure Analysishelliphellip35

51 A Lumped-parameter Code for Melt Pool Heat Transfer 35

511 Heat transfer correlations 36

512 Energy balance in the pools 37

513 Energy balance in the RPV wall 39

52 Hybrid Coupling Approach 40

53 Verification and Validation 41

531 UCSB FIBS benchmark case 41

532 LIVE-7V experiment 43

533 FOREVER-EC2 experiment 45

6 Summary and Outlookhelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphellip51

Bibliographyhelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphellip55

XI

Nomenclature

Latin

119860 area of a surface

119862119901 heat capacity

119863 damage factor

119892 gravity

ℎ heat transfer coefficient

H Melt pool height

119896 thermal conductivity

119871 characteristic length

119873119906 Nusselt number

119875119903 Prandtl number

119902 heat flux

119902119894119899 heat flux through inner vessel surface

119876119881 internal volumetric heat source

119903 radial distance in a cylindrical or spherical coordinate

119877119886 external Rayleigh number

119877119886119894119899 internal Rayleigh number

119877119886119910 local Rayleigh number

119905 time

119879 temperature

119881119900119897 total volume

119909 thickness (of crust or RPV wall)

119911 height in a cylindrical coordinate

Greek

120572 thermal diffusivity

120573 thermal expansion

Δ119879 temperature difference

ε emissivity (in thermal analysis) strain (in structural analysis)

휀119888119903 creep strain

휀119903 creep strain rate

120579 polar angle

120579119894 parameters of a creep model 119894 = 1234119898

120584 kinematic viscosity

σ stress

XII

Super and subscripts

119886119898119887 ambient

119888119903 creep (in Section 3) crust (in Section 5)

119889119899 downward

119890119899119907 environment

119894 inner surface of RPV

119894119899119905 interface

119897119900119888119886119897 local

119898119886119909 maximum value

119898119890119897119905119894119899119892 melting point

119898119905 metal layer (bulk)

119898119905119889119899 downward surface of metal pool

119898119905119904119889 sideward surface of metal pool

119898119905119905119900119901 top surface of metal layer

119898119905119906119901 upward surface of metal pool

119900 outer surface of RPV

119900119909 oxide layer (bulk)

119900119909119889119899 downward surface of oxide layer

119900119909119906119901 upward surface of oxide layer

119903119890119891 reference

119904 internal structures inside the RPV

119904119889 sideward

119906119901 upward

119907119890119904 vessel

Acronyms

BWR boiling water reactor

CBD creep database

CFD computational fluid dynamics

CRGT control rod guide tube

ECCM effective convectivity conductivity model

FDM finite difference method

FE finite element

FEA finite element analysis

FEM finite element method

FVM finite volume method

IGT instrument guide tube

IVR in-vessel retention

XIII

IVMR in-vessel melt retention EU project

LES large eddy simulation

MPS moving particle semi-implicit method

NPP nuclear power plant

PECM phase-change effective convectivity model

RANS Reynolds-averaged Navier-Stokes

RPV reactor pressure vessel

SBO station blackout

SLM surface loads mapping

SPH smoothed-particle hydrodynamics

UDF user-defined function

UPF user programmable features

VLM volume loads mapping

XIV

1

1 Introduction

11 Nuclear Power Safety

Greenhouse gas emissions contribute to the global warming problem Phasing out fossil fuels by switching to low-carbon energy sources is considered as a mitigation strategy With this regard clean energy should be increased in the world energy consumption structure Being clean and stable nuclear energy is an attractive alternative to fossil fuels After the development over decades it also proves to be a mature and safe technology According to IAEA 2018 annual report [1] there were 450 nuclear power reactors under operation worldwide at the end of 2018 which had a record generating capacity of 3964 GW(e) 9 reactors were connected to the grid and 55 reactors under construction around the world

Due to the potential risk of the release of radioactivity nuclear power safety is emphasized throughout the stages of design construction operation and decommission of a nuclear power plant (NPP) The principal technical requirements [2] specify that the three fundamental safety functions should be fulfilled for all plant states of an NPP (i) control of reactivity (ii) removal of heat from the reactor and from the fuel store and (iii) confinement of radioactive material shielding against radiation and control of planned radioactive releases as well as limitation of accidental radioactive releases The defence-in-depth concept is also applied to provide several levels of defence to prevent consequences of accidents The fuel claddings reactor pressure vessel (RPV) and primary coolant pressure boundaries and the containment are the three physical barriers to prevent the release of radioactive materials to the environment

Severe accidents are events that cause significant damage to reactor fuel and result in core meltdown It may be caused by insufficient cooling of the reactor core due to various reasons or by the rapid introduction of huge reactivity which causes sharp increases in the power and fuel fragmentation eg Chernobyl accident During a severe accident the failure of the fuel claddings will result in the radioactive materials previously enclosed inside the cladding released to the primary loop If cooling is not sufficient to remove the decay heat the core materials will melt migrate down and relocate into the RPV lower head after which the melt will react with water and partially form fragments quench solidify and form a debris bed The debris bed formed by the melt may continue to melt to form a melt pool due to the decay heat As the corium contacts the RPV it would continuously heat the RPV and threaten its integrity Once the RPV fails containment becomes the last barrier to prevent the release of radioactivity The released melt would result in an increase of the pressure and temperature due to the high temperature fractions oxidation and hydrogen combustion which is called direct containment heating The melt would fall onto the reactor cavity Without sufficient cooling the molten core would interact with basemat concrete (MCCI) This process can last for few days It can result in containment failure due to either containment overpressure or the basemat melt-through of containment In addition containment is also subjected to isolation failure and early containment failure due to missiles and dynamic pressure [3] More

2

detailed severe accident progression or phenomena could also be found in eg [4] [5]

Though severe accidents are unlikely to occur the consequences can be severe if no proper measure is taken Related researches have been continuously going on aiming at mitigating the accident consequences and improving safety After the Fukushima Accident safety inspections and re-evaluation actions have been carried out worldwide especially on safety measurements regarding severe accidents The concept lsquoPractical Eliminationrsquo applied to new NPP designs demands that new nuclear installations be designed with the objective of mitigating the consequences of accidents and avoiding early radioactive releases and large radioactive releases [6] Severe accident mitigation strategies are applied to the existing and new NPPs (both Generation II and III) The strategies can generally be classified into two categories [3] the in-vessel retention (IVR) and ex-vessel retention (EVR) In simple words in-vessel retention tries to retain the core melt inside the RPV (second barrier) by applying external water cooling on the vessel as shown in Figure 1(a) Such method minimizes the dispersal of the melt within the containment and hence simplifies the later recovery work Examples of the application are AP1000 HPR1000 APR 1400 The idea of the EVR is to let the RPV fail but to retain the core melt inside the containment Two principle approaches of EVR are the retention within a dedicated lsquocore catcherrsquo and melt fragmentation Examples of core catcher would be EPR that directs the melt to a spreading compartment (Figure 1(b)) and VVER1000 that uses metallic crucible below the RPV An example of the melt fragmentation is the Nordic BWRs that the corium released from RPV would relocate in the cavity with a deep-water pool (7-12 m) and is expected to fragment quench and form a coolable debris bed (Figure 1(c))

Figure 1 Examples of IVR [7] EVR with core catcher [8] and melt fragmentation [9]

12 RPV Failure

RPV which contains the reactor core core shroud and coolant is a key component to the reactor system RPV and the primary loop constitute the primary coolant system which conveys the fission heat from the reactor core to the secondary loop through steam generator (PWR) or to the turbine (BWR) The pressure under

(a) IVR (b) Core catcher (c) Melt fragmentation

3

operation can be ~155 MPa for a 1000 MW PWR [10] and ~70 MPa for a 1000 MW BWR [11] It also has to withstand the neutron radiation damage The thickness of RPV [11] can be ~20 cm for PWR and ~16 cm for BWR

As the transition of in-vessel progression to ex-vessel progression RPV failure is important for analysis of accident progression and assessment of the effectiveness of severe accident mitigation strategies In the melt fragmentation strategy of Nordic BWR the melt-coolant interaction and debris bed coolability after the corium poured into the deep-water pool are essentially dependent on the vessel failure mode including the characteristic time of melt release rupture size and location and amount and superheat of melt available for release [12] New concern also arose [13] on the necessity of performing structural analysis for the RPV in the IVR strategy In the initial methodology and further development of the IVR the success of the strategy ie the integrity of the RPV was demonstrated (eg in [14]) with the following two conclusions (i) even for a local heat flux on the vessel as high as critical heat flux (CHF) the resulted very thin external ldquocold layerrdquo of the vessel would be sufficient to withstand the internal stress load and hold the vessel integrity (ii) as CHF is the critical value for the thermal load the main specific goal for assessing the effectiveness of IVR becomes to demonstrate that the local heat flux along RPV be lower than the CHF The new concern doubts that local lsquohot partrsquo of the vessel would have negative effects on the overall resistance due to creep and dilatation which could result in a vessel failure and hence violate the first conclusion in particular for ldquolong termrdquo retention of corium in RPV If so a structural integrity analysis of RPV would be needed in addition to assessing local heat flux especially for high-power reactors Therefore accurate prediction of vessel failure could be meaningful for both strategies

Figure 2 Illustration of RPV under thermal loads from corium

As the structural performance is highly affected by the thermal loads from the corium RPV failure is essentially a transient thermo-fluid-structure coupled problem Figure 2 illustrates the situation that a corium debris bedmelt pool formed in the RPV lower head The decay heat of nuclear fission products could continuously be generated inside the corium leading to the formation of melt pools as the debris melts The heated debris bedmelt pool imposes thermal and

4

mechanical loads on the RPV Such loads can be transient (change with time) and spatially non-uniform Complex phenomena are also involved in this process Strong turbulent natural convection induced by the decay heat may be expected in the melt pool under IVR The corresponding Rayleigh number 119877119886 can be 1016minus17 for prototypic reactor cases Material structural properties can become non-linear under high temperature and stress loads The creep deformation also becomes significant at high temperature Detailed modelling of turbulent natural convection and non-linear properties could be computationally expensive In addition large-scale geometry (hemisphere with a radius of 20-30 m) long-term transient deformation process (~10000 s) and required small time steps (eg 001 s) for numerical convergence consideration further increase the computational costs

13 Literature Review Creep Behaviour and RPV Failure Analysis

In this section the most relevant studies concerning the experiments modelling and analysis of creep behaviour and RPV failure are presented

131 Experiments on creep behaviour and RPV failure

Various experiments have been performed on the RPV failure problem some of which were to generate data of mechanical properties (especially the creep properties) and some to study the overall vessel failure with scaled or simplified geometries

The REVISA experiment [15] generated creep curves and stress strain curves of French steel 16MND5 ranging from 600 to 1300 degC This data is accessible in [16] as well as [17] which also documents other material properties eg density thermal expansion coefficient conductivity heat capacity Youngrsquos Modulus and plasticity Similar experiments were carried out to generate high-temperature creep and tensile test data for American RPV steels and Inconel [18]

The CORVIS experiments [19] investigated the lower head failure modes with and without penetrations Each experiment was carried out on a steel plate 100 mm thick representing a pressure vessel lower head The penetration case represented a boiling water reactor lower head carrying a tube penetration in the form of a drain line

The LHF (lower head failure) experiments [20] and the succeeding OLHF tests [21] investigated the vessel failure with a 15 scaled vessel heated by an electrical radiator allowing to locate the hot focus at different position of the vessel with a maximum wall temperature of around 900 degC Internal pressure ranging from 53 to 123 bar was also imposed to initiate the vessel failure

The FOREVER experiments [22] investigated the phenomena of melt pool convection vessel creep possible failure processes and failure modes occurred during the late phase of the in-vessel progression by heating up a melt pool in a vessel and observing possible vessel failures The study covers cases with and without penetration different melt level existence of gap cooling and in-vessel melt coolability

5

132 Numerical studies on RPV failure

Various numerical approaches have been employed on RPV failure Since a two-way thermo-fluid-structure coupling approach is challenging these approaches take simplifications and can generally be classified into two categories (i) RPV failure decoupled from melt pool heat transfer and (ii) RPV failure coupled with melt pool heat transfer in simplified manners

1) Decoupled approaches In the decoupled approaches only RPV is modelled Simplified constant thermal loads are imposed on RPV considering the melt pool heat transfer

In a post-test analysis of FOREVE-C2 experiment [23] it was treated as a thermo-structural coupled analysis of the vessel with the thermal load estimated by the outer thermocouples and the normal heat flux on the external vessel wall The analysis was done using finite element (FE) code SYSTUS A coupled creepdamage law based on a viscosity-hardening creep law describing primary and secondary creep was employed for creep analysis

Mao et al conducted various studies on reactor cases assessing the IVR strategy using FE code ABAQUS Thermal boundary conditions were imposed on the RPV which was then solved as a thermal-structural coupled problem The strain hardening creep model (Eq (1)) was mainly used Their studies cover the general RPV failure [24] as well as the effects of cooling water levels under IVR [25] and geometric discontinuity (ablated RPV) [26]

ε119888119903 = 1198891σ1198892ε1198893119890119909119901 [minus

1198894119879] (1)

In the EU IVMR project [13] the RPV integrity was also studied through a benchmark calculation which deals with the RPV of a PWR with a prescribed ablated profile with 2D axisymmetric geometry Prescribed temperature gradient was set as thermal load The objective was to calculate deformations for various pressure loads up to vessel rupture A smoother profile was planned in further research

The advantage of such approaches is once thermal loads are reasonably assumed the focus only needs to be put on the RPV The disadvantage is a reasonable thermal load should be consistent with the melt pool heat transfer while sometimes it is not well-noted eg constant uniform temperature boundary conditions were assumed for inner and outer RPV (with uniform thickness in lower head) surfaces [25] which ignored the non-uniform heat flux distribution caused by the turbulent convective corium pool As thermal loads strongly affect the structural performance they should be quantified carefully

2) Coupled approaches In coupled approaches both melt pool heat transfer and RPV failure are modelled and coupled in one-way or simplified manner With melt pool heat transfer modelled the coupled approaches are more flexible to provide reasonable thermal loads on the RPV In existing approaches CFD and the correlation-based distributed-parameter models eg the Effective Convectivity Conductivity Model

6

(ECCM) [27] and the Phase-change Effective Convectivity Model (PECM) [28] [29] are general methods for melt pool heat transfer

Sehgal et al [22] presented an approach that employed the MVITA code to calculate the thermal processes and ANSYS for structural analysis MVITA code employed the ECCM Bailey-Norton creep law was used as shown in Eq (2)

ε119888119903 = 6055 times 10minus26 sdot σ472119905085119890minus

37200

119879 (2)

This approach was used for the pre-test simulation of FOREVER experiment to determine the thermal and mechanical loadings Parametric investigations also helped in the identification of important factors that could affect the test procedure and test conditions

Willschutz et al [30] employed the CFD-module Flotran for the melt pool heat transfer and ANSYS for structural analysis of RPV Standard k-ε turbulence model was selected for the turbulent natural convection in the melt pool A user-defined three-stage creep model [31] was employed In this model experimental creep curves were discretized into a creep database (CDB) under given temperature 119879 and stress 120590 the creep curve was discretized into finite point pairs

(휀 ε) indicating under such thermal and mechanical loads the creep strain rate ε is subjected to creep strain 휀 The creep strain rate for any other case can be interpolated over known temperature stress and strain from the database This method was applied to the post-test calculation of the FOREVE-C2 experiment It was concluded that the unexpected deformation behaviour during the experiment was caused by the combination of the transient temperature field in the vessel wall and the three creep stages

They later replaced the Flotran solver with the ECCM for melt pool heat transfer in a following study [32] By implementing it into FEM-solver a recursively coupled analysis was constructed Moreover a modified creep model [33] was used with damage measures the CDB itself did not contain tertiary creep instead a factor 1(1 minus 119863) was multiplied to the original creep strain rate to get the modified creep strain rate where 119863 is a damage factor (0 lt 119863 lt 1) defined with creep strain information and plastic strain information As 119863 grows close to 1 the modified creep strain rate would be enlarged significantly by this factor 1(1 minus 119863) which then behaves like the accelerating tertiary stage This method was applied to pre- and post- test calculations for the FOREVER experiments proving to be closer to reality than the previous method

Madokoro et al [34] developed a thermal structural analysis tool based on open source solver OpenFOAM and numerically investigated the coupled RPV failure and melt pool convection The PECM was used in the melt pool heat transfer In structural part the plasticity and damage parameters were similar to Willschutz et al [32] except for that strain hardening creep model (Eq (3)) was used instead of a CDB Simulations of FOREVER-EC2 and -EC4 were investigated It was concluded from the simulations that although the emissivity and mesh type (updated or fixed) affect the results significantly the approach generally has the capability of predicting the vessel deformation and failure

7

ε119888119903 = 1198891 sdot σ1198892 sdot ε1198893 (3)

Villanueva et al [35] performed a coupled thermo-mechanical creep analysis of the reactor scale RPV (ABB-Atom design BWR) to evaluate efficacy of the cooling measures for the in-vessel melt coolability and retention The PECM was used for melt pool heat transfer and ANSYS for RPV integrity analysis The modified time hardening model (Eq (4)) of the American RPV steel SA533B was selected as creep model The transient thermal load from the melt pool was applied as a thermal boundary condition to the vessel wall Further studies were done with this approach on investigations of instrumentation guide tube (IGT) failure [36] and control rod guide tube (CRGT) cooling [37] of a Nordic BWR

ε119888119903 =1198881σ

11988821199051198883+1

1198883 + 1 1198881 gt 0 (4)

It is noted that only RPV analysis with finite element analysis (FEA) are reviewed here Engineering methods like the Larson-Miller method [38] are not covered in this thesis One can refer to theory manuals of severe accident codes like MELCOR ASTEC for more details Comprehensive description on modelling of melt pool heat transfer is not covered either One can refer to either [39] for a general view or to any severe accident code manuals for engineering methods

14 Motivation Objectives and Main Achievements

The existing experimental and numerical studies provide us a good understanding of RPV failure involved phenomena and valuable simulation experiences These numerical methods may vary in solvers (which may be related to code accessibility) and models (which may be related to model accessibility understanding of phenomena computational power) However improvements of modelling approaches are still possible as the improvements in any of the related aspects eg

bull Simulation tools are continuously being upgraded and providing new possibilities just like that PECM appeared and ECCM was no longer used

bull Better understanding of the phenomena and more simulation experiences from in-vessel retention and vessel failures researches benefit the further modelling and simulations

bull Increasing computational power allows more detailed modelling eg turbulence modelling of melt pool convection

bull There is still room for the improvement of creep modelling

In addition implementation difficulty of a coupling approach is another important issue to consider Since it is a TRANSIENT thermo-fluid-structure coupled problem it is almost impossible to replicate an approach simply by referring to a paper with limited details For BWRs it would be even better for the coupling approaches to support complex geometries considering forest of penetrations (CRGTs and IGTs) on the RPV lower head without significant increase of user effort in the coupling process

8

The objectives of this thesis are to investigate new approaches and develop new models for the thermo-fluid-structure coupled problem of RPV failure and involved phenomena These objectives are accomplished with the following achievements

1) A new creep model for the 16MND5 steel was developed which covers all three creep stages Good agreement was achieved between model predictions and experimental curves Both time and strain hardening models were implemented (Paper I)

2) A new thermo-fluid-structure coupling approach was developed for RPV failure analysis It supports the advanced models in both melt pool convection and structural mechanics to improve fidelity and facilitates implementation Validation of the approach with the new creep model was performed against FOREVER-EC2 experiment This approach is flexible for complex geometries and can be used in further analysis eg lower head with detailed modelling of penetrations in BWRs (Paper II)

3) The performance of this new approach which uses volume loads mapping for data transfer was compared with the previous approach using surface loads mapping showing good agreement with small discrepancy (Paper III)

4) A lumped-parameter code transIVR for melt pool heat transfer was developed which supports one- and two-layer pool configuration transient analysis and well-resolved RPV wall (Paper IV)

5) A hybrid coupling approach was developed for efficient analysis of RPV failure where transIVR is coupled with ANSYS Mechanical (Paper IV)

In the following general descriptions on the codes ANSYS Fluent Mechanical and the PECM are given in Section 2 the new creep model in Section 3 the new coupling approach as well as its comparison with the previous approach in Section 4 the efficient hybrid coupling approach in Section 5 and summary amp outlook in Section 6

9

2 Numerical Solvers and Models

In this section we briefly introduce the ANSYS Fluent as the CFD solver for melt pool heat transfer the ANSYS Mechanical as the mechanical solver for RPV failure and PECM as an efficient alternative for melt pool heat transfer that are heavily used in the following sections

21 ANSYS Fluent

In the fluid mechanics the fluid is treated as a continuum which is generally described by the conservative mass momentum and energy equations [40]

120597120588

120597119905+ 120571 sdot (120588) = 119878119898 (5)

120597(120588)

120597119905+ 120571 sdot (120588) = minus120571119901 + 120571 sdot (120591) + 120588 + (6)

120597120588119864

120597119905+ 120571 sdot ((120588119864 + 119901)) = 120571 sdot (119896120571119879 minussumℎ119895119869119895

119895

+ (120591 sdot )) + 119878ℎ

(7)

Computational Fluid Dynamics (CFD) studies topics on the numerical solutions of above equations Typical methods are the Finite Volume Method (FVM) Finite Difference Method (FDM) and Finite Element Method (FEM) as well as particle methods that are mesh-free eg SPH and MPS

ANSYS Fluent is a general purpose CFD code which solves above equations with FVM Turbulence modelling and multiphase flows are supported Turbulence can be modelled with RANS LES or hybrid LESRANS Customizations eg adding sourcesink time or spatial dependent boundary conditions are realized through the user-defined functions (UDF) which is programmed in C language

22 ANSYS Mechanical

In the continuum mechanics of solids three sets of variables are considered to describe the structure performance namely element displacement field strain field and stress field (119906 ε σ) A general 3-dimensional problem is described by 3 sets of equations the equilibrium equations (Eq (8)) strain-displacement relationship (Eq (9)) and constitute equations (Eq (10)) [41] For Eq (8) the reciprocal theorem of shear stress also gives τ119894119895 = τ119895119894 where 119894 119895 = 119909 119910 119911

indicating the stress tensor being symmetric For very simple applications eg 1D beam it can be solved analytically For complex problems where analytical solutions are not available approximation methods eg Galerkin weighted residual method principle of virtual work and principle of minimum potential energy would be used

10

partσ119909119909part119909

+partτ119909119910

part119910+partτ119909119911part119911

+ 119909 = 0

partτ119910119909

part119909+partσ119910119910

part119910+partτ119910119911

part119911+ 119910 = 0

partτ119911119909part119909

+partτ119911119910

part119910+partσ119911119911part119911

+ 119911 = 0

(8)

ε119909119909 =part119906

part119909 휀119910119910 =

120597119907

120597119910 휀119911119911 =

120597119908

120597119911

120574119909119910 =120597119906

120597119910+120597119907

120597119909 120574119910119911 =

120597119908

120597119910+120597119907

120597119911 120574119911119909 =

120597119908

120597119909+120597119906

120597119911

(9)

ε119909119909 =1

119864[σ119909119909 minus μ(σ119910119910 + σ119911119911)]

ε119910119910 =1

119864[σ119910119910 minus μ(σ119909119909 + σ119911119911)]

ε119911119911 =1

119864[σ119911119911 minus μ(σ119909119909 + σ119910119910)]

γ119909119910 =1

119866τ119909119910 γ119910119911 =

1

119866τ119910119911 γ119911119909 =

1

119866τ119911119909

(10)

ANSYS Mechanical [42] is a mechanical engineering software that uses finite element analysis (FEA) for structural analysis The principle of virtual work is adopted in the solver [43] The idea of this principle is that a virtual (very small) change of the internal strain energy must be offset by an identical change in external work due to the applied loads which reads

δ119880 = δ119881 (11) where 119880 is the strain energy or the internal work 119881 the external work and δ the virtual operator The virtual internal work can be expressed as

δ119880 = int δεσd(vol)T

vol

+int 119886119903119890119886119891

δ119908119899119879σ119889(119886119903119890119886119891) (12)

where 휀 is the strain vector 120590 stress vector vol volume of element 119908119899 motion normal to the surface 119886119903119890119886119891 area of the distributed resistance The virtual

external work is contributed from three parts the inertial effects the pressure force on the surface and nodal forces applied to the element which then can be expressed as

δ119881 = minusint 119907119900119897

δ119908119879119865119886

119907119900119897119889(119907119900119897) + int

119886119903119890119886119901

δ119908119899119879119875119889(119886119903119890119886119901)

+ δ119906119879119865119890119899119889

(13)

11

where 119908 is the vector of displacements 119865119886 acceleration force vector 119875 applied pressure vector 119886119903119890119886119901 area over which pressure acts and 119865119890

119899119889 nodal

forces applied to the element

Plugging Eq (12) and (13) into (11) would result in the equilibrium equation Since the virtual displacement can be arbitrary the final equations used to describe the equilibrium of an element is reduced to Eq (14) with multiple operations

([119870119890] + [119870119890119891])119906 minus 119865119890

119905ℎ = [119872119890] + 119865119890119901119903 + 119865119890

119899119889

(14)

where [119870119890] [119870119890119891] 119865119890

119905ℎ [119872119890] and 119865119890119901119903 are the element stiffness matrix

foundation stiffness matrix thermal load vector mass matrix acceleration vector and pressure vector respectively It represents the equilibrium equation on a one element basis Alternatively using a minimum total potential energy principle could result in a consistent expression to the principle of virtual work [41]

Customizations eg on material properties are realized through the user programmable features (UPF) which is programmed in FORTRAN language

23 PECM

The PECM [28] [29] developed in KTH is a correlation-based method to model the melt pool heat transfer It is implemented in ANSYS Fluent through UDF and has been validated against many experiments such as ACOPO COPO LIVE etc In the PECM the flow velocities 119906119909 119906119910 and 119906119911 are replaced by characteristic

velocities 119880119909 119880119910 and 119880119911 respectively in order to eliminate the necessity of solving

the Navier-Stokes equation

The characteristic velocities are calculated using heat transfer correlations

119880119906119901 =α

ℎ119901119900119900119897times (119873119906119906119901 minus

ℎ119901119900119900119897

ℎ119906119901) (15)

119880119889119899 =α

ℎ119901119900119900119897times (119873119906119889119899 minus

ℎ119901119900119900119897

ℎ119889119899) (16)

119880119904119889 =α

ℎ119901119900119900119897times (119873119906119904119889 minus

ℎ119901119900119900119897

119882119901119900119900119897) (17)

where ℎ119901119900119900119897 is the melt pool depth ℎ119906119901 the thickness of well mixed layer of the

pool ℎ119889119899 the thickness of lower stratified region of melt pool 119882119901119900119900119897 the width of

pool 120572 the thermal diffusivity The PECM employs reduced characteristic velocities to describe mushy zone convection heat transfer For internally heated pool natural convection the Steinberner-Reineke correlations (Eq (18)~(19)) [44] are employed The profile of sideward 119873119906 is described using Eckertrsquos type correlation (Eq (20)) for a vertical boundary layer where the local Rayleigh

number at pool height 119910 is defined as 119877119886119910 =119892βΔ1198791199103

αν

12

119873119906119906119901 = 03451198771198861198941198990233

(18)

119873119906119889119899 = 13891198771198861198941198990095

(19)

119873119906119904119889 = 050811987511990314 (

20

21+ 119875119903)

minus14

11987711988611991014

(20)

As the characteristic velocities are calculated the energy equation is the only transport equation that needs to be solved and can be expressed as follows

partρ119862119901119879

part119905+ nabla(ρ119862119901119932119879) = 119896nabla2119879 + 119876119907 (21)

where from left to right are the transient term convection term diffusion term and volumetric heat source respectively The convection term then can be calculated explicitly with use of values in the last time step and then can be treated together with the volumetric source term Therefore the PECM finally simplifies the energy equation to a conduction equation which can be easily solved and as a result will significantly increase the computational efficiency in simulations

In addition solidificationmelting process can also be modelled and the temperature-dependent physical properties are supported

13

3 Creep Modelling of 16MND5 Steel

Creep deformation becomes significant when a component is subjected to high temperature and stress Modelling of creep behaviour of the RPV is paramount since it is considered as a main mechanism leading to the RPV failure In this section we present the development of a three-stage creep model (called lsquomodified theta projection modelrsquo ) for the French RPV steel 16MND5 This model was successfully implemented into the ANSYS Mechanical code

31 Creep and Creep Modelling

Creep is an irreversible time-dependent inelastic deformation process Diffusion and dislocation are the two molecular mechanisms of creep The former normally occurs due to the presence of vacancies within the crystal lattice (or the grain) while the latter occurs if there is a slip by the passage of crystal dislocations [45] There are two types of diffusion creep [46] Coble creep (favoured at lower temperatures) if the diffusion paths are predominantly through the grain boundaries and Nabarro-Herring creep (favoured at higher temperatures) if through the grains themselves Dislocation creep tends to dominate at high stresses and relatively low temperatures Dislocations can move by gliding in a slip plane which is a process that requires little thermal activation But the rate-determining step for their motion is often a climb process which also requires diffusion and is thus a time-dependent process and favoured by higher temperatures Creep behaviour is highly affected by temperature and stress generally creep is more significant under higher temperature and stress

Creep curves describe the creep strain development with time Figure 3 shows a creep curve under constant temperature and stress which typically experiences three creep stages primary stage where the creep strain rate ε119888119903 (the change of creep strain ε119888119903 in unit time) decreases with time secondary stage where the creep strain rate almost keeps constant and tertiary stage where the creep strain rate drastically increases with time

Figure 3 Creep curve under constant temperature and stress

14

The built-in creep models in commercial software eg ANSYS [47] and ABAQUS [48] in default only cover the primary andor the secondary stage(s) Table 1 [47] gives the default creep models supported in ANSYS for example Possible reason why the tertiary stage is not modelled might be that for most industrial applications the failure analysis is beyond interest and therefore the built-in creep laws in the commercial software was supposed to be sufficient [49] Though the two codes offer the interfaces through either ANSYS UPF or ABAQUS user subroutines to support more general user-defined creep laws eg complex models that also include the tertiary creep stage

Table 1 Default creep models in ANSYS Mechanical

Creep model Equations Type

Strain hardening

ε119888119903 = 1198621σ1198622ε119888119903

11986231198901198624119879 1198621 gt 0 Primary

Time hardening ε119888119903 = 1198621σ11986221199051198623119890minus1198624119879 1198621 gt 0 Primary

Generalized exponential

ε119888119903 = 1198621σ1198622119903119890minus119903119905 r = 1198625σ

11986231199051198623119890minus1198624119879 1198621 gt 0

1198625 gt 0

Primary

Generalized Graham

ε119888119903 = 1198621σ1198622(1199051198623 + 1198624119905

1198625 + 11986261199051198627)119890minus1198628119879 1198621 gt 0 Primary

Generalized Blackburn

ε119888119903 = 119891(1 minus 119890minus119903119905) + 119892119905 119891 = 1198621119890

1198622σ 119903 = 1198623(σ1198624)1198625 119892 = 1198626119890

1198627σ

1198621 gt 0

1198623 gt 0

1198626 gt 0

Primary

Modified time hardening

ε119888119903 = 1198881σ11988821199051198883+1(1198883 + 1) 1198621 gt 0 Primary

Modified strain hardening

ε119888119903 = 1198621σ1198622[(1198623 + 1)ε119888119903]

1198623119890minus1198624119879 1198621 gt 0 Primary

Generalized Garofalo

ε119888119903 = 1198621[119904119894119899ℎ(1198622σ)]1198623119890minus1198624119879 1198621 gt 0 Secondary

Exponential form

ε119888119903 = 1198621119890σ1198622119890minus1198623119879 1198621 gt 0 Secondary

Norton ε119888119903 = 1198621σ1198622119890minus1198623119879 1198621 gt 0 Secondary

Combined time hardening

ε119888119903 = 1198621σ11986221199051198623+1119890minus1198624119879(1198883 + 1)

+ 1198625σ1198626119905119890minus1198627119879

1198621 gt 0

1198625 gt 0

Primary+ Secondary

Rational polynomial ε119888119903 = 1198621

partε119888part119905 ε119888 =

119888119901119905

1 + 119901119905+ ε119898119905

ε119898 = 1198622101198623σσ1198624 119888 = 1198627ε

1198628σ1198629

119901 = 11986210ε11986211σ11986212

1198622 gt 0 Primary+ Secondary

Generalized time hardening

ε119888119903 = 119891119905119903119890minus1198626119879 119891 = 1198621σ + 1198622σ2 + 1198623σ

3 119903 = 1198624 + 1198625

Primary

32 A Modified Theta Projection Model

321 Original theta projection model and its variations

The original theta projection model [50] was first proposed to accurate predict long-term creep and creep-fracture properties It assumes that a creep curve under constant temperature and stress can be described as

15

휀119888119903 = 1205791(1 minus 119890minus1205792119905) + 1205793(119890

1205794119905 minus 1) (22)

where 휀119888119903 is the creep strain 119905 the time and 1205791~1205794 the model parameters Figure 4 illustrates how the first term and second term on right hand side develop with time and constitute a creep curve which has obvious primary secondary and tertiary stages Model parameters 120579119894(119894 = 1234) are temperature and stress dependent and are described with the following expression

119897119899thinsp120579119894 = 119886119894 + 119887119894120590 + 119888119894119879 + 119889119894120590119879 (23)

where 119886119894 119887119894 119888119894 and 119889119894 ( 119894 = 1234 ) are constants It is noted [45] that this equation is suitable when the temperature range of experiment data sets is very narrow such that minus1119879 is almost proportional to 119879 These constants (119886119894 119887119894 119888119894 and 119889119894 (119894 = 1234)) are the final model parameters which are generated with two fitting procedures

(i) Fitting the experimental creep curves to Eq (22) generates several groups of 1205791~1205794 for each creep curve under certain temperature and stress We denote (119879119895 120590119895 1205791119895 1205792119895 1205793119895 1205794119895 ) as parameters for the 119895119905ℎ

experimental curve under temperature 119879119895 and stress 120590119895

(ii) Fitting 1205791 to Eq (23) as a function of temperature and stress with the obtained data ( 119879119895 120590119895 1205791119895 ) from ( 119879119895 120590119895 1205791119895 1205792119895 1205793119895 1205794119895 ) generates

parameters 1198861 1198871 1198881 and 1198891 Repeat this fitting procedure for 1205792 1205793 and 1205794 to get parameters 119886119894 119887119894 119888119894 and 119889119894 (119894 = 234)

Variations of the model were developed and applied to other different materials to better accommodate the experimental data which are all then called a lsquomodified theta projection methodrsquo independently Examples are models in [51] for lowhigh alloy ferritic austenitic steels Al-alloys Al-matrix composites [52] for Nickel-based Super-alloy [53] for pure metals and class M alloys with intermediate range of stress (40-70 of the yield stress) and temperature (40-60 of the melting temperature)

Figure 4 Illustration of theta projection method [54]

16

322 A modified theta projection model

The new modified theta projection model we proposed [54] was mainly adapted from the original theta projection method [50] and the modified version [51] The new model describes the creep curve with the following equation

휀119888119903 = 1205791(1 minus 119890minus1205792119905) + 120579119898119905 + 1205793(119890

1205794119905 minus 1) (24)

An additional term 120579119898119905 is seen compared with Eq (22) in the original method The reason for adding this term is that the 16MND5 steel shows fairly long secondary stages in many experimental creep curves and the creep curves can be better fitted by Eq (24) than Eq (22) As the structure undergoes different creep process under different temperature and stress the model parameters θ119894 (119894 =1~4 and 119898) are also temperature and stress dependent

The main difference compared with the original and all other modified projection models is that numerical interpolation is used to replace the second numerical fitting in previous models In other words instead of fitting the model parameters as functions of stress and temperature with Eq (23) the distributions of the model parameters are assumed to be piecewise in the temperature-stress space and they will be interpolated directly from known parameters obtained from experiments The consideration is that the experimental data ([16] [17]) for 16MND5 covers a wide temperature range varying from 600 degC to 1300 degC and a wide stress load range varying from 08 MPa to 248 MPa while Eq (23) in the existing methods is based on a narrow temperature span Poor parameter fitting results would be expected if we directly fit the parameters 1205791~1205794 with Eq (23) as used in the original methods

The creep model was developed using the following steps

(i) Fitting Each creep curve can be represented by a group of model parameters [1205791 1205792 120579119898 1205793 1205794] with respect to Eq (24) by fitting the creep curves to Eq (24) All parameters of the 32 experimental curves form an

initial dataset of the model parameters [119879119895 120590119895 1205791119895 1205792119895 120579119898119895 1205793119895 1205794119895] where j (0ltjle32) is the jth creep curve

(ii) Interpolation Parameters of a creep curve under any [119879 σ] are directly interpolated from the dataset

(iii) Parameter adjustment The dataset was further updated to satisfy the monotonicity assumption we have taken that creep strain monotonically increases with temperature and stress

(iv) Different hardening models for dynamic loads

Step (i) was done using MATLAB In the following steps (ii)-(iv) are described in detail

Model parameters for new creep curves are interpolated from the dataset over the 2D 119879 minus 120590 space The interpolation is divided into two sub-interpolation steps interpolate over stress with fixed temperature and then interpolate over temperature with fixed stress We take the interpolation of model parameters under (119879prime σprime) for example In the first interpolation we first find out the two

17

temperatures 119879119896119899119900119908119899119904119898119886119897119897 and 119879119896119899119900119908119899119897119886119903119892119890 most close to 119879prime from the experimental

temperature set 600 700 800900 1000 1100 1200 1300 Under each temperature there are four sets of model parameters in the dataset corresponding to four different stress loads Then the first interpolation is done over stress load from the known dataset of model parameters under each

temperature to output the parameters for (119879119896119899119900119908119899119904119898119886119897119897 120590rsquo) and (119879119896119899119900119908119899119897119886119903119892119890 120590rsquo)

In the second interpolation the model outputs parameters for (119879rsquo 120590rsquo) from

(119879119896119899119900119908119899119904119898119886119897119897 120590rsquo) and (119879119896119899119900119908119899119897119886119903119892119890 120590rsquo) For 120579i(119894 = 12 3 4) we assume a logarithm

relation between 120579119894 temperature 119879 and stress 120590 which is similar to Eq (23)

119897119899120579119894 = (119886 + 119887120590)(119888 + 119889119879) (25)

It reduces to 120579119894 = 11986011198901198871σ with fixed temperature in the first interpolation and to

θ119894 = 11986021198901198872T with fixed stress in the second interpolation The interpolation

expression for θ119894 = 11986011198901198871σwith two known points (1205901 1205791198941) and (1205902 1205791198942) is given

by Eq (26) The interpolation expression for θ119894 = 11986021198901198872T with two known points

(1198791 1205791198941) and (1198792 1205791198942) is given by Eq (27)

120579119894 = 120579i1

120590minus12059021205901minus1205902120579i2

1205901minus1205901205901minus1205902 (26)

120579119894 = 120579i1

119879minus11987921198791minus1198792120579i2

1198791minus1198791198791minus1198792 (27)

Similarly we assume the following relation between 120579119898 temperature 119879 and stress 120590

120579119898 = 119860120590119899119890minus119887 119879frasl (28)

The interpolation is done over the stress with 120579119898 = 1198603120590n by fixing temperature

and then over temperature with 120579119898 = 1198604119890minus1198874119879 by fixing stress The interpolation

expression over stress for 120579119898 = 1198603120590119899 with two known points (1205901 1205791198981) and

(1205902 1205791198982) would be given by Eq (29) The interpolation expression over

temperature for 120579119898 = 1198604119890minus1198874119879 with two known points (1198791 1205791198981) and (1198792 1205791198982)

would be given by Eq (30)

120579119898 = 119890119909119901(119897119899120590 minus 11989711989912059011198971198991205902 minus 1198971198991205901

1198971198991205791198982 +1198971198991205902 minus 119897119899120590

1198971198991205902 minus 11989711989912059011198971198991205791198981) (29)

120579119898 = 120579m2(11198791minus1119879) (

11198791minus11198792)fraslsdot 120579m1

(1119879minus11198792) (

11198791minus11198792)frasl (30)

In this model we also take the assumption that the creep strain monotonically increases with temperature and stress loads1 Or specifically with the same creep time the structural component under higher temperature or stress experiences

1 Intuitively materials deform more under higher stress and temperature This is also mostly observed in tensile experiments Though some exceptions exist one instance of which is the effect of phase change some materials under certain temperature around the phase change temperature may show much weaker structural performance Here we just simply assumed that the creep strain monotonically increases with temperature and stress

18

higher creep strain than that with lower temperature or stress (the creep curve under higher temperature or stress should be above the curve with lower temperature or stress) When generating the parameter dataset (Step (i)) we already kept the trend that under the same temperature cases under higher stress loads would have larger creep model parameters and give higher creep strains This trend would also hold for the interpolated temperatures which thus guarantees that creep strain monotonically increases with stress load under same temperature Then the question becomes if creep strain also monotonically increases with temperature given fixed stress However the answer could not always be a yes especially in the extrapolation range of the stress load Figure 5(a) illustrates an example of the interpolation of model parameters under (119879119872 σ119872) The red dots and blue dots are known parameters from the parameter dataset for corresponding high temperature 119879119867 and low temperature 119879119871 respectively The green dots (119879119867 σ119872) and (119879119871 σ119872) are interpolated from the red and blue dots in the first sub-interpolation respectively The black dot (119879119872 σ119872) is interpolated from the green dots in the second sub-interpolation It is seen that when σ119872 lt σ119862 the value of (119879119871 σ119872) is higher than that of (119879119867 σ119872) It means the higher 119879119872 is the smaller the model parameter will be ie the model would predict larger creep strain for lower temperatures which violates our monotonicity assumption One simple solution used here as shown in Figure 5(b) is to minorly adjust the values of the four red and blue dots such that the intersection point C can be sufficiently small and the situation σ119872 lt σ119862 would barely occur in the extrapolation range

Figure 5 Illustration of model parameter interpolation (a) before and (b) after parameter adjustment [54]

A corresponding parameter adjustment process was introduced to verify that the model predicts higher creep strain under higher temperature under fixed stress Figure 6 illustrates the general verification process for any (119879 120590) in the interested load range the creep strain of (119879 + 120575119879 120590) should be larger than that of (119879 120590) Otherwise parameters would be minorly adjusted and the verification process be repeated This process would be iterated several times until the requirement is reached The achieved verification shows

Stress

σC

High temperature TH

MPa

a

0

Model

par

amet

er θ

Low temperature TL

MPa

a

0

Model

par

amet

er θ

High temperature TH

Low temperature TL

σC

Stress

C C

(a) (b)

σM

σM

19

(i) For any (119879 120590) in the temperature range 273~ 1073 K with an interval of 1 K and the stress range 1 ~260 MPa with an interval of 1 MPa its creep strain is higher than that of (119879 minus 1119870 120590) for the first 12000000 s

(ii) For any (119879 120590) in the temperature range 1073~1673 K with an interval of 1 K and the stress range 1 ~80 MPa with an interval of 1MPa its creep strain is higher than that of (119879 minus 1119870 120590) for the first 12000000 s

(iii) For any (119879 120590) in the stress range 01~26 MPa with a smaller interval of 01 MPa for the full temperature range 273~1673 K with an interval of 1 K its creep strain is higher than that of (119879 minus 1 120590) for the first 12000000 s

Above verification covers a wide range of temperature and stress especially where the stress and temperature are low and extrapolation is used Further verification to cover even higher stress or temperature ranges is not considered as creep increases drastically under such high stress and temperature and the results would become meaningless With these regards we generally think the monotonicity requirement is reached The model parameter dataset was updated along with the adjustment The final dataset was given as Table 2 in Paper I

Figure 6 Parameter adjustment process [54]

Figure 7(a-d) present the model predictions of selected creep curves in ANSYS Mechanical (labelled as lsquoANSYSrsquo) compared with experimental curves (labelled as lsquoREVISArsquo) at temperature 600 800 1000 and 1200 degC respectively The predicted curves proved to clearly show the three-stage creep process and agreed well with the experimental creep curves

Initial parametersdataset

θ1 θ2 θm θ3 θ4 from creep

curve-fitting

Parameters interpolation

Is the monotonicity

assumption satisfied

IsMon= =1

Final model parameters

Adjust the

parametersdataset

θ1 θ2 θm θ3 θ4

No

Yes

IsMon=1 a flag is monotonic achieved

For σ in the stress range in MPa

For T in range Tstart Tend in K

For t in range 112000000 in s

If ε(σ T+1t)lt ε(σTt) IsMon=0

Iteration loop of

parameter adjustment

20

Figure 7 Selected model predictions compared with experimental creep curves [54]

Instead of being constant temperature and stress loads also dynamically change with time in real cases Hardening models determine how creep strain rate reacts to the change of loads during a creep process The time hardening model and strain hardening model2 are supported in our model implementation in ANSYS Mechanical which are two mostly used hardening models

Time hardening assumes that the following creep strain rate after a load change depends only on the current creep time from the beginning of the creep process (and also temperature and stress if used in the creep model) and ignores the current creep strain Strain hardening assumes that the creep strain rate depends only upon the current creep strain (and also temperature and stress if used in the creep model) [55]

2 The time hardening and strain hardening here are more general terms than the specific time hardening creep model and the strain hardening creep model mentioned in Table 1

(a) 600 degC (b) 800 degC

(c) 1000 degC (d) 1200 degC

21

Eq (24) in default is a time hardening model as creep strain is directly described by the time 119905 Eq (31) gives the creep strain increment 119889휀119888119903 at time 119905 during time interval 119889119905

119889휀119888119903 = (12057911205792119890minus1205792119905 + 120579119898 + 12057931205794119890

1205794119905)119889119905 (31)

The strain hardening model is derived from Eq (24) in an implicit form Specifically when temperature and stress load change to (119879 120590) at the time 119905 and current creep strain 휀1198881199030 the model will get the creep curve of constant (119879 120590) and

output how this curve will react under the current strain 휀1198881199030 Since the creep curve shows creep strain monotonically increases with time given the creep strain 휀1198881199030 under creep curve of (119879 120590) we can uniquely find this reference time

119905119903119890119891 satisfying Eq (32) One should notice that the reference time 119905119903119890119891 can be

different from the current physical time 119905

휀1198881199030 = 1205791(1 minus 119890minus1205792119905119903119890119891) + 120579119898119905119903119890119891 + 1205793(119890

1205794119905119903119890119891 minus 1) (32)

where all 120579119894(119894 = 1~4119898) are model parameters of the new creep curve (119879 120590) It can be solved for 119905119903119890119891 with methods eg Newton-Ramphson [56] Then the creep

strain increment at 119905119903119890119891 during time interval 119889119905 is given as

119889휀119888119903 = (12057911205792119890minus1205792119905119903119890119891 + 120579119898 + 12057931205794119890

1205794119905119903119890119891)119889119905 (33)

Therefore combining Eq (32) and (33) gives the creep strain rate at creep strain 휀1198881199030 In summary when loads are changed to (119879 120590) with current creep strain 휀1198881199030

the model traces the creep curve of (119879 120590) and calculates the creep strain rate of this curve at strain 휀1198881199030

One calculation was done to demonstrate the performance of the hardening models The stress load was set to 151 MPa and kept unchanged A step temperature load was applied as the orange dash curve shows in Figure 8 it was 900 degC during 0-36000 s and suddenly jumped to 1000 degC at 36000 s which was then kept till the end of the simulation The blue solid curve (the part before the temperature change was identical to and therefore covered by the green solid curve in the figure) and blue dash curve are the creep curves under constant temperature T=900 degC and T=1000 degC respectively The green solid curve and green dash curve represent the model predictions considering this temperature change with time hardening model and strain hardening model respectively Before this change the creep curves with both hardening models were identical to the creep curve of the T=900 degC till point A After that the creep strain rate then shifted from Point A to Point B (which had the same time with A) in the time hardening model and behaved the same as the curve T=1000 degC as shown in the rest part of the green solid curve In the strain hardening model the creep strain rate shifted from Point A to Point C (which had the same creep strain with A) and behaved the same as the curve T=1000 degC as shown in the rest part of the green dash curve The large difference between two models was partially caused by the drastic temperature jump In real cases the difference would be less significant as the temperature changes more mildly Overall this prediction is exactly the expected behaviors of the time hardening model and strain hardening model

22

Figure 8 Time hardening and strain hardening in ANSYS Mechanical [54]

23

4 Coupled Thermo-Fluid-Structure Analysis of RPV Failure

As described in Section 13 RPV failure is coupled with melt pool heat transfer leading it into a transient thermo-fluid-structure coupled problem The goal of this section is to develop and validate a reliable coupling approach which supports transient coupled thermo-fluid-structure analysis as well as advanced models and computational platforms so as to improve fidelity and facilitate implementation The approach was validated against the FOREVER-EC2 experiment and then applied to an RPV failure analysis of a Nordic BWR As the approach adopts the volume loads mapping for load transfer comparative analysis was also performed against the coupling approach using surface loads mapping Future work is considered in analysis of penetration failures (IGTs and CRGTs) of a BWR RPV since the approach is flexible to complex geometries Results of Paper II ampIII are jointly presented in this section

41 Thermo-Fluid-Structure Coupling Approach

Considering the involved phenomena a coupling approach should at least address three issues (i) melt pool heat transfer (ii) RPV deformation and (iii) the interaction between melt pool heat transfer and vessel deformation A two-way strong coupling considering both the transient thermal load from melt pool on vessel and the influence of vessel deformation on melt pool heat transfer involves complex data exchange and is still full of challenges Therefore the new coupling approach is realized in a one-way manner with the assumptions that the influence of vessel deformation on melt pool heat transfer can be ignored and thus only the thermal loads from melt pool to vessel are considered It is based on the considerations that on one hand the deformation is a slow process regarding the relatively long failure time and on the other hand it was pointed out [57] that the physics of natural convection heat transfer in internally heated liquid pools is not sensitive to the geometrical factors based on their calculation results and experimental data from [58] [59] as well as previous experiments in semicircular and square cavities eg [44] [60] In addition one-way coupling is also computationally more efficient especially when heat transfer turbulence structural non-linearity and large-scale geometries are involved The developed approach is based on the ANSYS platform which generally consists of three parts with respect to the three issues respectively

(i) conjugate heat transfer of the melt pool and vessel using ANSYS Fluent (ii) transient thermal loads transfer from ANSYS Fluent to ANSYS

Structural (the ANSYS Mechanical in ANSYS Workbench) (iii) structural analysis of the vessel in ANSYS Structural

ANSYS Structural in default only supports direct transfer of constantsteady-state spatial loads from ANSYS Fluent Transfer of transient loads is realized with the extension tool FSI Transient [61] It is a pre-processing tool (with good documentation) developed by a third-party CAE vendor EDRTM MEDESO [62] to map transient CFD results (heat transfer coefficient temperature and surface pressure) to thermal or structural analysis including the definition of load steps

24

This new procedure proves to be straightforward and reduces efforts in the load transfer

Since structural performance of RPV is highly dependent on temperature the transfer of transient temperature loads from conjugate heat transfer to mechanical solver is a core step of the coupling framework There are two general approaches to realize the transfer as shown in Figure 9 Both approaches contain three parts a conjugate heat transfer analysis for fluid and structure where flow and heat transfer are coupled a mechanical analysis for the structure and a load mapping process that transfers the transient loads The conjugate heat transfer part can be solved with any CFD solver and is treated in the same manner in both approaches In the first approach (Figure 9(a)) transient volume thermal loads are transferred and mapped into the mechanical simulation domain The structure domain reads the essential thermal loads from the mapping after which a structure analysis of this domain is performed In the second approach (Figure 9(b)) transient surface thermal loads are mapped onto the boundary surfaces of structure domain as thermal boundary conditions Then a thermo-structure coupled analysis of the structure domain in the mechanical part is performed To distinguish them we call the former approach volume loads mapping (VLM) and the latter surface loads mapping (SLM) In the RPV failure analysis the fluid and structure domains correspond to the melt pool and RPV respectively Examples of these methods are the studies of [63] and [30] using VLM and [35] using SLM The VLM is adopted in current approach

Figure 9 Frameworks of the thermo-fluid-structure coupling approaches (a) volume loads mapping (b) surface loads mapping [64]

fluid structure structure

fluid structure structure

Conjugate heat transfer problem Mechanical problem

(structure)

Conjugate heat transfer problem Mechanical problem

(thermo-structure coupled)

Transient volume

loads mapping

(VLM)

Transient surface

loads mapping

(SLM)

(a)

(b)

25

42 Numerical Modelling

421 Conjugate heat transfer

In the conjugate heat transfer analysis main focus is put on the modelling of melt pool heat transfer Within the ANSYS Fluent solver two options are supported CFD with turbulence modelling and PECM CFD is the choice when the model goes to detail and more mechanistic analysis are needed while PECM provides an alternative that is faster and possibly more suitable for large-scale problems

Within the scope of RANS models the SST (Shear Stress Transport) model [65] is selected as the turbulence model in present study This model is a combination of k-ε model and k-ω model The k-ω formulation is used in the inner parts of the boundary layer which makes the model directly usable all the way down to the wall through the viscous sub-layer It also switches to k-ε model in the free-stream and thereby avoids the common k-ω problem that the model is too sensitive to the inlet free-stream turbulence properties The considerations on choosing this model are

bull Several two-equation low-Re k-ε models have been proved in [66] not good enough for melt pool heat transfer Therefore these mentioned two-equation models as well as other one-equation models are not considered

bull SST is highly recommended [67] for simulating most industrial heat transfer problems

bull Convincing results ([68] [69]) using SST have been obtained in capturing the thermal behaviour of the natural convection in an internally heated fluid Especially good agreements were achieved between the predicted heat flux distribution along the vessel and experimental data which would be one of the most important factors to estimate model performance in this problem

The description of PECM as an effective model for melt pool heat transfer is given in Section 23

422 Mechanical analysis

The material for the RPV failure analysis considered in this study is the French RPV steel 16MND5 The three-stage modified theta projection model in Section 3 is used as the creep model Since the definition of an RPV failure criterion is not universal (it can be the time when damage variable reaching 1 or either specified value of creep strain creep strain rate or total deformation is reached) we do not explicitly define a failure criterion Instead the simulation is kept running until the creep converge criterion cannot be satisfied even with the defined minimum time step (indicating a fairly large creep strain rate) and the simulation will be automatically terminated The time the simulation stops is taken as the failure time This treatment implicitly implies a criterion of a large creep strain rate We refer to [17] for other material properties Youngrsquos modulus and thermal conductivity are temperature dependent Multilinear Isotropic Hardening (MISO) is used as the plasticity model

26

43 Validation against the FOREVER-EC2 Experiment

In this section the validation against the FOREVER-EC2 experiment is presented (indicated as lsquoVLMrsquo in the following) In addition the simulation results using the SLM approach (indicated as lsquoSLMrsquo in the following) is also included for comparison purpose

431 FOREVER-EC2 experiment

The FOREVER-EC2 experiment [70] is one of the FOREVER experiments that achieved a vessel failure Figure 10 shows the experimental setup A 110 scale pressure vessel with a wall thickness of 15 mm and an outer diameter of ~400 mm was employed The lower head was made of 16MND5 steel Since it was difficult to use prototypical corium materials in the experiment the binary salt 30 CaO-70 B2O3 was employed as simulant To generate the internal heat which corresponds to the nuclear decay heat a specially designed heater was used The power input was maintained ~38 kW during the experiment such that maximum melt pool temperature would be ~1300 degC The pressure inside the vessel was maintained at ~25 MPa Thermocouples were used to measure the melt pool and vessel wall temperatures and linear displacement transducers used to measure the vessel wall displacements The designed experimental conditions were reached at experimental time 11670 s Vessel failure was observed about 12510 s later (at experiment time 24180 s)

Figure 10 Scheme of the experimental setup [22]

27

Figure 11 Computational domains and boundary conditions (a) conjugate heat transfer (b) mechanical analysis [71]

Computational domains and boundary conditions are indicated in Figure 11 SST turbulence modelling was used for pool heat transfer The simulations started at the moment when experimental conditions were reached (experiment time 11670 s) So simulation time 0 s corresponds to experiment time 11670 s

432 Conjugate heat transfer

Figure 12 shows the temperature fields at simulation time 0 s 1225 s 2475 s 4975 s and 9975 s All the time points in this part refer to simulation time The field at time 0 s (Figure 12(a)) was the initial temperature that was reconstructed based on the thermocouple readings The temperature increased with time and reached a steady state when the maximum temperature reached ~1607 K Figure 12(b-f) show clear thermal stratification which is an expected phenomenon of natural convection

Internal pressure

25 bar

Internal pressure 25e6-2500times981timesy Pa

External pressure

1 bar

Zero displacement

y=0 m

Gravity 400

mm

R=203 mm

15 m

m

Tamb

=323 K radiation with ε = 08

+ convection with h=10Wm2K

Tamb

=800 K radiation with ε=02

(a) (b)

Tamb

=800 K radiation with ε=08

+ convection with h=10 Wm2K

28

Figure 12 Temperature fields at simulation time (a) 0 s (b)1225 s (c) 2475s (d) 4975 s (e) 7475 s and (f) 9975 s [64]

Figure 13 shows the temperature profiles of the external vessel along the polar angle in comparison with experimental data Regions with a polar angle larger than 90deg are those in the cylindrical part of the vessel It also shows the temperature profile increased with time until reaching a steady state The profile almost did not change after 4975 s which indicates that a steady state of the temperature was reached As time increased the temperature in the lower head region 0~90deg (where the vessel was contacted with the melt) evolved from a relatively flat distribution to a distribution that temperature at high angle was higher than that at low angle by more than 200 K Generally the steady state results agreed well with the experimental data except for the low angle part of vessel lower head and the high angle part of the cylindrical vessel In addition to modelling errors the discrepancy in the low angle part of the lower head may be caused by the crust and gap possibly formed at vessel bottom in the experiment which can reduce the downward heat transfer and result in lower temperature while they cannot be modelled in simulations As to the high angle cylindrical vessel part simplified boundary conditions were applied to the inner and outer vessel wall since it was far from the melt and not of interest which may result in some discrepancy

(a) (b)

(c) (d) (e) (f)

29

Figure 13 Temperature profiles of external vessel surface along the vessel polar angle [64]

433 Mechanical analysis

In the mechanical analysis the VLM predicted a failure time of 23959 s and the SLM predicted a failure time of 22445 s both are close to the experimental failure time 24180 s All the time points in this part refer to the experiment time Figure 14 shows the creep strain fields at failure time Both approaches predicted maximum creep strains at same location where the temperature was high (cf Figure 12) This is related to the fact that creep behavior is highly influenced by temperature Maximum values for both cases were also close to each other 0393 in VLM and 0366 in SLM

Figure 15 presents the total deformation of the vessel bottom point (also called ldquosouth polerdquo in some literature) in the vertical direction Since there was an initial displacement reading of ~355 mm at time 0 s which almost did not change during 0~11670 s (the whole period before the experimental conditions were reached) an initial value of 355 mm was added to the simulation result such that experimental and simulation results had the same total displacement at t=11670 s Similar trend was seen in the VLM SLM and experimental data that the deformation increased with time and experienced a drastic acceleration at the end Generally both simulation approaches agreed well with the experimental data though they both slightly under predicted the value of total deformation The discrepancy may possibly be related to the accuracy of the thermal predictions modelling errors of plasticity and creep and simplified boundary conditions

400

600

800

1000

1200

1400

1600

0 20 40 60 80 100 120 140 160

Tem

per

ature

(K

)

Polar angle (deg)

Experiment

25 s

1225 s

2475 s

4975 s

7475 s

9975 s

Weilding line

30

Figure 14 Creep strain fields at failure time (a) VLM (b) SLM [64]

Figure 15 Total deformation in vertical direction of the vessel bottom point [64] [71]

The wall thickness changes are plotted together with experimental results in Figure 16 The wall thickness change δ119909 is defined as

δ119909 = 119909119890119899119889 minus 119909119900119903119894119892119894119899119886119897 (34)

000

500

1000

1500

2000

2500

11670 13670 15670 17670 19670 21670 23670

Tota

l def

orm

atio

n (

mm

)

Time (s)

ExperimentVLMSLM

(a) (b)

31

where 119909119890119899119889 and 119909119900119903119894119892119894119899119886119897 are the wall thickness at failure time and the original wall

thickness respectively Since 119909119890119899119889 cannot directly be post-processed it was estimated in the following way (cf the subfigure of Figure 16) for the same polar angle θ find the corresponding points Pin (1199091 1199101) and Pout (1199092 1199102) on the inner and outer surfaces of the deformed vessel respectively the distance of Pin and

Pout radic(1199091 minus 1199092)2 + (1199101 minus 1199102)2 is calculated as 119909119890119899119889 at polar angle 120579 Such estimation of the wall thickness is more accurate when the line PinPout is closer to be orthogonal to the deformed local surface Considering the final deformed vessel as shown in Figure 14 the estimation should be fine at regions close to 0deg and 90deg and can be slightly overestimated at medium angle region The VLM curve was compared with the SLM curve and 4 experimental curves measured at vessel left right front and back sides Generally VLM curve agreed well with the experimental and SLM curves wall thickness gradually decreases with angle till around 75deg and then increases till 90deg the maximum thickness change occurred at around 75deg would be around -5 mm The positive values at low angle may be caused by the thermal expansion It also shows that the thickness transitions in experimental curves were smoother than those in both simulation curves while simulation curves showed narrower regions with large thickness changes (at angle 70deg~80deg) Possible reasons might be (i) sharper temperature profile predicted in simulation (ii) input uncertainty and (iii) modelling errors The SLM predicted slightly thinner vessel than the VLM except for the necking region The discrepancy might be caused by various reasons eg the interior vessel temperature can be slightly different between two approaches which would then result in different thermal stress and different deformation the final deformed vessel shapes cannot be exactly the same which also results in different errors in estimating PinPout

Figure 16 Wall thickness changes along the vessel polar angle [64] [71]

-6

-5

-4

-3

-2

-1

0

1

0 10 20 30 40 50 60 70 80 90

Wal

l th

icknes

s ch

ange

(mm

)

Polar angle (deg)

VLMSLMExperiment (left)Experiment (right)Experiment (front)Experiment (back)

32

44 Application to BWR Case

In this section the coupling approach (indicated as lsquoVLMrsquo in the following) was applied to an RPV failure analysis of a BWR A simulation using the SLM coupling approach (indicated as lsquoSLMrsquo in the following) was also performed for comparison purpose

A hypothetical severe accident of a Nordic BWR initiated by an SBO was considered The core materials would melt and migrate downward to the RPV lower head The total amount of corium relocated in the lower head was considered to form a 19 m corium melt pooldebris bed assuming the corium distribution was homogeneous

Figure 17 Computational domains and boundary conditions (a) conjugate heat transfer (b) mechanical analysis [64]

Figure 17(a) shows the computational domain of the conjugate heat transfer PECM was used in this calculation A general mesh with a pool height of 23 m was used to accommodate most of the relocation situations with the same mesh Since the corium only had a height of 19 m the top 04 m region was treated as injected water covering the corium The top water was expected to be boiling and we assumed a uniform temperature of 383 K Under the implementation of PECM this region was damped with artificially high heat capacity and high conductivity such that it would efficiently remove heat from the corium and suppress its own temperature increase For the corium part below water a volumetric heat source 11 MWm3 was implemented as the nuclear decay heat For the in-vessel surfaces a constant temperature of 383K was applied as they are covered by water which can be boiling Without water cooling on the external vessel wall a very small heat loss of 20 Wm2 was applied Melting and solidification of the corium were also considered with a solidus temperature of 2750 K and a liquidus temperature of 2770 K The initial temperature was set as

q=20 Wm2

T=383 K

Qv=11

MWm3

315 bar

315 bar+ hydrostatic

pressure of corium

Gravity

248 bar

Zero

displacement

Water

Corium

(a) (b)

33

1100 K for the corium 383 K for the water and 450 K for the vessel For the mechanical analysis (Figure 17(b)) an inner pressure of 315 bar and an outer pressure of 248 bar were considered For the region covered by corium an extra hydrostatic pressure was added on the vessel wall A 2 m long vessel extension was considered as well as the standard earth gravity

Figure 18 presents the wall temperature profile along the inner vessel surface at 50 s 2500 s 7500 s and 10000 s The temperature increased with time for the vessel surface contacted with corium and the distribution was flat which was due to that large part of the corium was still in solidus status It also shows that for the surface region that contacted the water the temperature was basically unchanged which indicates that the damping treatment indeed suppressed the temperature increment and guaranteed the heat removal upward by water

Figure 18 Temperature profiles of inner vessel wall along the vessel polar angle [64]

The predicted RPV failure time was 9869 s by VLM and 9650 s by SLM Compared with the FOREVER-EC2 case in Section 43 the VLM predicted a slightly later failure time than the SLM in this BWR case too Figure 19(a) compares the maximum creep strain developments with time of both approaches Before 9000 s the predicted values by both approaches are close to each other Afterward the creep acceleration occurred and the strain increased drastically The acceleration of the VLM occurred 200 s later than that of the SLM which agreed with the predicted later failure time Similar trend was observed for the total deformation in vertical direction of vessel bottom point as shown in Figure 19(b) The deformations of both approaches are close except that the acceleration of VLM occurred later than that of SLM Overall results of VLM agreed well with those of SLM

0

500

1000

1500

2000

2500

0 10 20 30 40 50 60 70 80 90 100

Tem

per

ature

(K

)

Polar angle (deg)

50 s

2500 s

5000 s

7000 s

10000 s

34

Figure 19 Results of structural analysis (a) creep strain and (b) total deformation in vertical direction of vessel bottom point [64]

45 Computational Efficiency

Computational efficiency is also an important consideration regarding the performance of numerical approaches The computational efficiencies of both VLM and SLM are analysed and compared in this section As parallel computing technique was used to speed up the simulations the number of parallel cores and corresponding costed computational time for each simulation are given together in Table 2 as a generally reflection of the computational efficiency All the simulations were done in a Lenovo P700 workstation equipped with 2 Intel Xeon E5-2630 v3 processors (8 cores per processor) and 32 G RAM For the simulations of FOREVER-EC2 SLM employed twice the parallel cores and costed 286 more time compared with VLM For the simulations of prototypic BWR case both VLM and SLM employed the same number of parallel cores but VLM only took 13 of the computational time compared with SLM Overall it can be concluded that VLM has higher computational efficiency than the SLM The reason is a structural analysis is performed in the VLM approach while a coupled thermo-structural analysis is performed in the SLM approach (cf Figure 9) a standalone structural analysis is computed faster than a coupled thermo-structural analysis

Table 2 Comparison of computational efficiency [64]

Case FOREVER-EC2 BWR VLM SLM VLM SLM

Number of cores 4 8 4 4 Computational time (hours) 21 27 18 72

35

5 Hybrid Coupling Approach for Efficient RPV Failure Analysis

In this section we propose a hybrid coupling approach for the efficient RPV failure analysis The idea is to replace the ANSYS Fluent in previous coupling approach with a lumped-parameter code for quick estimate of the melt pool heat transfer With the least modelling details (compared with CFD methods and distributed-parameter methods eg PECM) lumped-parameter codes prove to be the most efficient for melt pool heat transfer The new hybrid approach characterizes 1) utilization of efficient models for melt pool heat transfer and 2) the support of detailed structural modelling of RPV deformation allowing studies with different RPV geometric shapes under effects like local thinning due to ablation For studies of which the focus is mainly on structural behaviour of RPV eg [13] [26] while mechanistic simulation of melt pool convection is computationally expensive and unnecessary the lumped-parameter code could help providing reasonable thermal loads

The development of the approach consists of two tasks (i) development of a lumped-parameter code and (ii) its coupling with ANSYS Mechanical to form this hybrid coupling framework In the following we describe both tasks and the corresponding verification and validation processes respectively This section is mainly based on Paper IV

51 A Lumped-parameter Code for Melt Pool Heat Transfer

A lumped-parameter code transIVR was developed It features (i) quick estimate of transient melt pool heat transfer with reasonable accuracies and (ii) two-dimensional representation of heat conduction in the RPV wall facilitating the coupled analysis of the RPV failure under thermal loads The following assumptions are taken in the transIVR modelling

bull The crust between the oxide pool and metal pool is thin

bull The heat flux has uniform distributions on the bottom top and side surfaces of metal pool

bull Heat source only exists in oxide layer

bull Latent heat of fusion is ignored

Figure 20 illustrates the modelled three domains oxide layer metal layer and the RPV wall In addition the internal structure inside the vessel can also be considered as a radiative body to the top metal surface as in [72] An internal volumetric heat source generated by nuclear decay heat is considered in the oxide layer Heat in oxide layer transfers downward to the RPV wall and upward to the top metal layer Heat in metal layer can be transferred sideward to the RPV wall and upward to the surrounding internal structure through radiation (or to water if inside water flooding is applied) The heat transferred to RPV will be removed from outer RPV surface Possible formations of oxide crust in the top and bottom surface are illustrated as the black regions Possible formation of metal crust on the top surface is shown as the grey region The code automatically detects the

36

existence of the crust on oxide top and metal top surfaces during the simulation and solves the problem correspondingly withwithout crust Boundary conditions are applied to the external and internal surfaces of RPV and (if needed) top surface of metal layer Arbitrary boundary conditions are supported If a boundary condition is nonlinear eg mixed heat convection (linear dependent on 119879 ) and heat radiation (proportional to 1198794) the Picard method [73] would be used for solving this problem

This model consists of three main parts

(i) empirical heat transfer correlations (mostly from experiments) to determine global and local heat transfer coefficients

(ii) energy balances in oxide layer and metal layer where the heat transfer coefficients would be used and

(iii) energy balance in RPV

Figure 20 Two-layer melt pool configuration and RPV wall [74]

511 Heat transfer correlations

For internally heated natural convection (oxide layer) in transIVR the mini-ACOPO correlations [75] are used for the averaged heat transfer as expressed in Eq (35) and (36) and for the angular variation of the heat flux as expressed in Eq (37) and (38)

119873119906119906119901 = 03451198771198861198941198990233 (35)

119873119906119889119899 = 00038119877119886119894119899035(119867119877)025 (36)

119873119906119889119899(θ)

119873119906119889119899= 01 + 108(

θ

θ119898119886119909) minus 45 (

θ

θ119898119886119909)2

+ 86 (θ

θ119898119886119909)3

01 leθ

θ119898119886119909le 06 (37)

RPV

Oxide pool

Metal pool

37

119873119906119889119899(θ)

119873119906119889119899= 041 + 035 (

θ

θ119898119886119909) + (

θ

θ119898119886119909)2

06 ltθ

θ119898119886119909le 1 (38)

The 119877119886119894119899 that appears in Eq (35)~(36) is the internal Rayleigh number which is defined as

119877119886119894119899 =119892β119876119881119867

5

α119896ν (39)

For Rayleigh-Bernard convection in the metal layer heat transfer occurs in all the top bottom and side surfaces To side surface the reduced form [72] of the Churchill-Chu correlation [76] for 119875119903 = 013 is employed as shown in Eq (40)

119873119906119904119889 = 007611987711988613 (40)

To top and bottom surfaces the reduced form [72] of Globe-Dropkin correlation [77] for

119875119903 = 013 is used as expressed in Eq (41)

119873119906119906119901 = 005911987711988613 (41)

The 119877119886 that appears in Eq (40) and (41) is the external Rayleigh number which is defined

as

119877119886 =119892βΔ1198791198673

να (42)

Heat transfer coefficients can be calculated from Nusselt number by Eq (43)

ℎ =119873119906119896

119871 (43)

where 119896 and 119871 are the heat conductivity and characteristic length respectively 119873119906 can be the Nusselt numbers expressed in Eq (35)~(38) and (40)~(41)

512 Energy balance in the pools

The energy balance of oxide layer can be described as Eq (44) The total generated heat of oxide pool equals the sum of the increase of enthalpy (transient temperature change) upward heat flow to metal layer and downward heat flow to RPV

120588119862119901119881119900119897119889119879119900119909119889119905

+ ℎ119906119901119860119900119909119906119901(119879119900119909 minus 119879119900119909119906119901) + ℎ119889119899119860119900119909119889119899(119879119900119909 minus 119879119900119909119889119899) minus 119876119881119881119900119897 = 0 (44)

where ρ is the oxide density 119862119901 the heat capacity 119881119900119897 the total volume 119879119900119909 the bulk temperature 119905 the time ℎ119906119901 the upward heat transfer coefficient 119860119900119909119906119901

upward heat transfer area 119879119900119909119906119901 upward surface temperature ℎ119889119899 downward

heat transfer coefficient 119860119900119909119889119899 downward heat transfer area 119879119900119909119889119899 downward surface temperature and 119876119881 volumetric heat source

38

On the interface between oxide layer and metal layer energy balance requires that the heat transferred from oxide layer should equal the heat to the metal layer which can be described as Eq (45)

ℎ119906119901119860119900119909119906119901(119879119900119909 minus 119879119900119909119906119901) = ℎ119898119905119889119899119860119898119905119889119899(119879119894119899119905 minus 119879119898119905) (45)

where ℎ119898119905119889119899 is the downward heat transfer coefficient of metal layer 119860119898119905119889119899 the downward heat transfer area 119879119894119899119905 the interface temperature and 119879119898119905 the bulk temperature of metal layer

The energy balance of metal pool can be described as Eq (46) The increase of enthalpy (transient temperature change) equals the sum of upward downward and sideward heat flows to surrounding surfaces

ρ119898119905119862119901119898119905119881119900119897119898119905119889119879119898119905119889119905

+ ℎ119898119905119906119901119860119898119905119906119901(119879119898119905 minus 119879119898119905119906119901) + ℎ119898119905119904119889119860119898119905119904119889(119879119898119905 minus 119879119898119905119904119889)

+ ℎ119898119905119889119899119860119898119905119889119899(119879119898119905 minus 119879119894119899119905) = 0

(46)

where ρmt is the metal density 119862119901mt the heat capacity 119881119900119897mt the total volume 119879119898119905 the bulk metal temperature 119905 the time ℎ119898119905119906119901 the upward heat transfer

coefficient 119860119898119905119906119901 upward heat transfer area 119879119898119905119906119901 upward surface

temperature ℎ119898119905119904119889 sideward heat transfer coefficient 119860119898119905119904119889 sideward heat transfer area 119879119898119905119904119889 sideward surface temperature ℎ119898119905119889119899 downward heat transfer coefficient 119860119898119905119889119899 downward heat transfer area and 119879119894119899119905 oxide-metal interface temperature

The heat transferred from metal layer to the top surface should equal the heat radiated from this surface to the environment Considering the radiation and reflections of internal structures to this surface following energy balance equation [72] is achieved

ℎ119898119905119906119901(119879119898119905 minus 119879119898119905119906119901) = σ(1198791198981199051199051199001199014 minus 119879119904

4) (1

ε+1 minus ε119904ε119904

119860119898119905119906119901

119860119904)minus1

(47)

where 119879119898119905119905119900119901 is the temperature of metal top free surface ε the emissivity of

metal ε119904 the emissivity of internal structures 119860119898119905119906119901 the area of metal top surface

and 119860119904 the total area of internal structures Sometimes the internal water cooling may be available above metal layer In this case Eq (47) is unnecessary and 119879119898119905119905119900119901 can be set to a constant value

For the oxide layer top surface Eq (48) is used if not crust is formed in the interface between two layers ie 119879119894119899119905 gt 119879119898119890119897119905119894119899119892119900119909 Otherwise Eq (49) is used

119879119900119909119906119901 = 119879119894119899119905 (48)

119879119900119909119906119901 = 119879119898119890119897119905119894119899119892119900119909 (49)

For the metal layer top surface Eq (50) is used if not crust is formed on top of the metal ie 119879119906119901119904119906119903119891 gt 119879119898119890119897119905119894119899119892119898119905 Otherwise Eq (51) is used

119879119898119905119906119901 = 119879119898119905119905119900119901 (50)

39

119879119898119905119906119901 = 119879119898119890119897119905119894119899119892119898119905 (51)

Overall this part involves 6 equations regarding 6 unknows 119879119900119909 119879119900119909119906119901 119879119894119899119905 119879119898119905

119879119898119905119906119901 and 119879119898119905119905119900119901

513 Energy balance in the RPV wall

The heat transfer in the PRV wall can be modelled as a simple 1D heat conduction problem that heat only transfers in the wall thickness direction or 2D axisymmetric heat conduction problem

For 1D model the treatment in [72] was adopted As illustrated in Figure 21 the heat from the oxide pool transfers to inner vessel surface (119879119894) then to the interior vessel (119879119907119890119904) then to outer vessel (119879119900) and finally to environment

Figure 21 1D heat transfer model in RPV [74]

Unknown variables are the heat flux through inner vessel surface 119902119894119899 the crust thickness 119909119888119903 inner vessel temperature 119879119894 vessel center temperature 119879119907119890119904 and outer vessel temperature 119879119900 Eq (52)~(56) give a closed equation set for these variables considering a radiation boundary condition on the external vessel surface Eq (52)~(53) describe the heat transfer through the crust where an internal heat source 119876119888119903 is also considered Eq (52) can be easily derived from the one-dimensional heat transfer equation with heat source Eq (54)~(55) describe the energy balance on the inner vessel surface and vessel interior respectively Eq (56) describes the energy balance on the outer vessel surface regarding radiation boundary condition For Dirichlet boundary condition Eq (56) is replaced with Eq (57) For other boundary conditions we only need to correspondingly modify the terms on the right-hand side of Eq (56) For situations that no crust is formed on the surface of the vessel Eq (52)~(53) are eliminated and 119902119894119899 in Eq (54) can directly be determined from calculations in Section 512 and Eq (54)~(56) are solved for the unknows 119879119894 119879119907119890119904 and 119879119900 Once 119879119894 exceeds the vessel melting temperature (with or without crust formation) wall ablation is considered to occur Then 119879119894 is fixed to the vessel melting temperature and vessel thickness 119909119907119890119904 becomes the new unknown variable

119879119900119909119889119899 119879119894 119879119907119890119904 119879119900 119879119890119899119907

119909119888119903 119909119907119890119904

crust RPV

Qcr

119902119894119899 119902119897119900119888119886119897

40

119902119894119899 = 119896119888119903(119879119900119909119889119899 minus 119879119894)

119909119888119903+ 1198761198881199031199091198881199032

(52)

119902119894119899 = 119902119897119900119888119886119897 + 119876119888119903119909119888119903

(53)

119902119894119899 = 119896119907119890119904(119879119894 minus 119879119907119890119904)

05119909119907119890119904

(54)

120588119907119890119904119862119901119907119890119904119889119879119907119890119904119889119905

= 119896119907119890119904(minus2 lowast 119879119907119890119904 + 119879119894 + 119879119900)

(05119909119907119890119904)2

(55)

119896119907119890119904(119879119907119890119904 minus 119879119900)

05119909119907119890119904= 휀120590(119879119900

4 minus 1198791198901198991199074 )

(56)

119879119900 = 119879119900119888119900119899119904119905

(57)

For the 2D representation the heat conduction in the RPV is solved using FDM considering the heat transfer in both the thickness and the angular directions in the RPV wall The RPV lower head is treated as a spherical shell The corresponding heat conduction equation in the 2D spherical coordinate is given as Eq (58) The cylindrical part of the vessel is treated as a cylindrical shell The corresponding heat conduction equation in the 2D cylindrical coordinate is given as Eq (59) Eq (58) and (59) are coupled and solved together In default the mesh size is 5deg in angular direction and 0025119909119888119903 in radius direction This mesh discretization can be changed up to user demands

ρ119862119901part119879

part119905=

1

1199032part

part119903(1198961199032

part119879

part119903) +

1

r2sinθ

part

partθ(119896119904119894119899θ

part119879

partθ)

(58)

ρ119862119901part119879

part119905=1

119903

part

part119903(119896119903

part119879

part119903) +

part

part119911(119896part119879

part119911) (59)

52 Hybrid Coupling Approach

The hybrid coupling approach to assess the RPV integrity is illustrated in Figure 22 transIVR and ANSYS Mechanical are coupled in a one-way manner transIVR conducts the conjugate heat transfer of melt pool and RPV and provides transient thermal loads to the ANSYS Mechanical where structural analysis of RPV deformation is performed The 2D heat transfer of RPV is solved in transIVR to better capture the temperature distribution In RPV structural analysis same considerations are taken as in Section 422 for 16MND5 steel the modified theta projection model for creep and MISO for plasticity and temperature-dependent Young modulus etc The VLM is adopted for the transfer of transient thermal loads

41

Figure 22 Framework of the hybrid coupling approach [74]

53 Verification and Validation

The transIVR code was first benchmarked against the UCSB FIBS case of two-layer pool configuration and then validated against the LIVE-7V experiment showing the codersquos capability in predicting two-layer and transient melt pool heat transfers respectively Finally the hybrid coupling approach encompassing transIVR and ANSYS Mechanical was validated against the RPV failure experiment FOREVER-EC2 and the results showed acceptable agreements with those of both the experiment and the previous approach using CFD in Section 43 Hence the transIVR code is an efficient tool for melt pool heat transfer and the hybrid approach an efficient way for RPV failure analysis

531 UCSB FIBS benchmark case

The UCSB FIBS benchmark case [78] was calculated to demonstrate the capability of the transIVR code in simulating two-layer pool heat transfer problems The considered pool configuration was similar to that shown in Figure 20 with the following specific dimensions a two-layer melt pool with a 16 m oxide layer and a 08 m metallic pool on top and an RPV lower head with a thickness of 015 m The internal volumetric source in the oxide layer was 13 MWm3 The calculation results were compared with those of the DOE study [78] and the INEEL study [79]

The predicted results were compared with the digitalized results of the DOE and INEEL studies in Figure 23 All three studies show similar profiles of the heat flux distribution along the polar angle (Figure 23(a)) it increased with the polar angle in the oxide layer to the maximum value at around 77deg and then dropped to a significantly lower value in the metal layer The transIVR result agreed well with the other results with small discrepancy in the metal layer the DOE study gave a

heat flux of 500 MWm2 the INEEL study of 400 MWm2 and the transIVR of somewhere in between Figure 23(b) shows the predicted corium crust thickness distribution along the RPV polar angle also agreed well the DOE and INEEL

transIVR ANSYS Mechanical

Transient

thermal loads

42

studies The crust thickness decreased with polar angle in the oxide layer and no crust was formed in the metal layer As the crust formation is closely related to the heat flux distribution the crust thickness profile was reasonable regarding the heat flux distribution shown in Figure 23(a) Vessel wall ablation would occur due to the high thermal loads Figure 23(c) gives the profile of the remaining vessel wall thickness along the RPV polar angle considering the wall ablation It shows that no ablation occurred in low angle region (0deg~36deg) where heat flux was low and the wall thickness changed from 15 cm to around 6 cm as angle increased in high angle region of the oxide layer Because of the discrepancy in the heat flux of metal layer the predicted wall thicknesses also differ from those of the DOE and INEEL studies the thinnest wall was 73 cm predicted by the DOE study corresponding to the maximum heat flux followed by 88 cm of the present transIVR calculation and 94 cm of the INEEL study Generally speaking the prediction of transIVR is in an acceptable level compared with the other studies for the same benchmark case

Figure 23 Simulation results of the UCSB FIBS benchmark case [74]

(a) heat flux distribution (b) crust thickness distribution

(c) wall thickness distribution

43

532 LIVE-7V experiment

LIVE-7V is an experiment under the LIVE program conducted in KIT to study the core melt phenomena As Figure 24(a) shows it employed the LIVE-3D hemispherical facility with an inner diameter of 1 m and a wall thickness of ~25 mm The non-eutectic binary salt mixture of 20 mol NaNO3- 80 mol KNO3 was used as the simulant material The decay heat in the melt was simulated by cable-type heating with 8 planes of electrical resistance heating wires Water cooling was applied to the top and vessel surfaces Top surface cooling was realized by a cooling lid mounted at height 413 mm from the vessel bottom and vessel cooling by the flow channel between the test vessel and the outer cooling vessel The inlet and outlet water temperature for both top and vessel cooling surfaces were kept ~293 K and ~303 K throughout the whole experiment respectively A 413 mm height pool corresponds to a total volume of ~0194 m3 simulant material The experiment procedure can be divided into four stages with the heating power sequentially shifted from 29 kW to 24 kW 18 kW and 9 kW Figure 24(b) indicates the thermocouple positions in experiment

The simulation results of LIVE-7V using transIVR were compared with experimental data [80] and numerical study using REALPSCDAPSIM code [81] For simplicity the RELAPSCDAPSIM results in the following are labelled as lsquoCOUPLErsquo which is the name of the module in RELAPSCDAPSIM for melt pool heat transfer

Figure 24 LIVE test facility [80] and thermocouple positions [82]

Figure 25 compares the predicted heat removal upward to the top lid and downward to the bottom vessel surface with those of the experiment and COUPLE The black curve indicates the total input power at the four stages The experimental results (red and blue dots) were calculated from the Table 3 of [80] assuming no extra heat losses For the first three stages the transIVR predictions agreed well with the COUPLE results while the predicted upward heat flow was slightly lower and the downward heat flow higher than the experimental data For the last stage the transIVR agreed well with the experimental result while the COUPLE predicted lower upward and higher downward heat flows The short

(a) (b)

44

transition periods between two stages with different total inputs taken in transIVR also agreed with those in the COUPLE predictions

Figure 25 Comparison of heat removal upward to the top melt surface cooling lid and downward to bottom vessel wall [74]

Figure 26 shows the comparison of heat flux distribution along the vessel wall All curves show similar trend that a higher total input power would result in a higher heat flux profile As the COUPLE predictions had relatively flatter profiles transIVR predictions generally agreed better with the experimental results especially when total input was high A discrepancy is seen that the minimum value of a curve occurred at angle 0degfor both transIVR and COUPLE and at angle 30deg for the experimental data It is noted that the models in default assume heat flux monotonically increases with angle (cf Eq (37) and (38)) resulting it impossible to predict higher heat flux at angle 0deg than angle 30deg as in experimental results

45

Figure 26 Heat flux profiles along the vessel polar angle [74]

533 FOREVER-EC2 experiment

In this section we performed calculations against the FOREVER-EC2 experiment to validate the performance of the coupled transIVR and ANSYS Mechanical in assessing the RPV integrity The applied boundary conditions were identical to the settings of previous coupling approach in Section 4 (ANSYS Fluent+ Mechanical) as indicated in Figure 11 The initial bulk temperature of the melt pool was set to be 1261 K in transIVR which corresponds to the maximum value of the initial temperature field in CFD calculation of previous coupling approach Both the experimental data and the simulation results using previous coupling approach are included for comparison In the following results from current hybrid approach are labelled lsquotransIVRrsquo (and referred as transIVR results) results from previous approach labelled lsquoCFDrsquo (and referred as CFD results) and results from experiment labelled lsquoexperimentrsquo

Figure 27 compares the bulk temperature predicted by transIVR with the averaged and maximum temperatures of the CFD results All curves show the same trend that the temperature gradually increased with time till reaching a steady state and then kept unchanged till the end The times needed for reaching the steady state were ~5000 s for the CFD and ~4000 s for the transIVR which are close to each other It indicates again a good performance of the transIVR in predicting the transient behavior of melt pool Compared with the steady-state averaged temperature 1478 K and maximum temperature 1609 K of the CFD results the steady-state temperature of transIVR 1599 K is a reasonable prediction

0

3000

6000

9000

12000

15000

18000

0 20 40 60 80 100

Hea

t fl

ux

(W

m2)

Polar angle (deg)

29 kW-EXP24 kW-EXP18 kW-EXP9 kW-EXP29 kW-COUPLE24 kW-COUPLE18 kW-COUPLE9 kW-COUPLE29 kW-transIVR24 kW-transIVR18 kW-transIVR9 kW-transIVR

46

Figure 27 Temperature evolution with time [74]

Figure 28 Temperature fields at time (a) 01 s (b) 3038 s (c) 6038 s and (d) 13289 s [74]

Figure 28 shows the temperature fields of the vessel at time 01 s 3038 s 6038 s and 13289 s The temperature fields were visualized in the ANSYS Mechanical It also indicates that the transient temperature loads were properly transferred from transIVR to ANSYS Mechanical Figure 28(a-d) show both the minimum and maximum temperatures increased with time In the lower head region the temperature gradually increased with the height The maximum temperature occurs at the location slightly lower than the melt pool top surface This is because thermal loadheat flux imposed on the vessel generally increases with polar angle in the pool natural convection so the code transIVR inherently also predicted higher heat flux at higher angle positions The higher flux at higher position then correspondingly resulted in higher temperature In the cylindrical region the

(a) (b) (c) (d)

[K] [K] [K] [K]

47

temperature decreased as the height since the bottom part would be more significantly heated by the high-temperature lower head region

Figure 29 shows the steady-state temperature profiles of the external vessel surface along the polar angle The transIVR result agreed well with other two results in trend that the temperature increased with the angle and reach the peak values at angle ~80deg then gradually decreased with angle But slightly lower temperature was predicted in the low angle region Possible reason may be that the local heat flux distribution determined by Eq (37) and (38) is steeper than that in the experiment and CFD study which then resulted in a steeper increase of temperature along the angle and a lower temperature at low angle region In the cylindrical part (gt90deg) the transIVR result generally agreed with the experimental and CFD results The discrepancy with the experimental data may be caused by the simplified boundary conditions the ambient temperature inside the vessel was set to uniform constant value while it could decrease as height Therefore predicted heat loss in the higher position of cylindrical part could be smaller than that with height-dependent ambient temperature resulting in higher vessel temperature Overall the prediction was reasonable for a lumped-parameter model

Figure 29 Temperature profiles of the external vessel surface along the vessel polar angle [74]

The transIVR predicted a failure time of 24959 s which is close to 24180 s in experiment and 23959 s in the CFD results Figure 30 shows the total deformation of vessel bottom point with time All three curves increased gradually with time and experienced drastic accelerations at the end The

400

600

800

1000

1200

1400

0 40 80 120 160

Tem

per

ature

(K

)

Polar angle (deg)

transIVRExperimentCFDWeilding Line

48

transIVR curve agreed well with the CFD curve The final total deformation at failure time is also close to that of the CFD result but lower than the experiment data The discrepancy might be related to the simplifications in the boundary conditions accuracy of the thermal load prediction and the modelling errors in plasticity and creep

Figure 30 Total deformation in vertical direction of the vessel bottom point [74]

Figure 31 shows the profile of wall thickness change The transIVR curve was compared with the CFD curve and four experimental curves measured at vessel left right front and back sides The simulation result fitted the experimental and CFD results well in the region 0deg~50deg In the region 50deg~90deg wall thickness gradually decreased with angle till around 75deg and then increased till 90deg The maximum thickness change was -3 mm for the simulation which is close to the experiment left side curve and slightly larger than the -6~ -45 mm of the other curves from experiment and CFD results Concerning possible errors and uncertainties in the modelling and numerical settings the new approach generally gives reasonable predictions

0

5

10

15

20

25

11670 15170 18670 22170 25670

Tota

l def

orm

atio

n (

mm

)

Time (s)

Experiment

CFD

transIVR

49

Figure 31 Wall thickness changes along the vessel polar angle [74]

-6

-5

-4

-3

-2

-1

0

1

0 20 40 60 80 100

Wal

l th

icknes

s ch

ange

(mm

)

Polar angle (deg)

CFD

transIVR

Experiment (left)

Experiment (right)

Experiment (front)

Experiment (back)

50

51

6 Summary and Outlook

Reactor pressure vessel (RPV) is one of the key physical barriers in the defence-in-depth approach of nuclear power safety and its failure is also the transition point of core melt (corium) progression from in-vessel to ex-vessel phase Quantification of RPV failure during a hypothetical severe accident is therefore paramount to the specific qualification of severe accident mitigation strategies as well as the general assessment of corium risk in light water reactors The RPV failure is a problem of structural performance of RPV under the influence of thermal loads imposed by corium melt pool ie it is a transient thermo-fluid-structure coupled problem per se whose solution requires simultaneous modelling of both highly turbulent melt pool heat transfer and large-scale nonlinear RPV creep Due to the complexity of the problem only limited capabilities have been developed in the previous studies under different simplifications and assumptions in models and coupling methods The present study was motivated to advance modelling and coupling approaches for RPV performance analysis considering new understanding of physical phenomena and latest coupling platform of simulation tools

For the new development of coupling approaches and models for physical phenomena related to RPV failure analysis during severe accidents this thesis work was conducted in the following order (i) a new creep model for the RPV steel 16MND5 was developed which features the capability to cover all three creep stages (ii) a new thermo-fluid-structure coupling approach was developed for RPV failure analysis and validated against FOREVER-EC2 experiment (iii) the performance of the new coupling approach using volume loads mapping (VLM) for data transfer was further compared with another approach using surface loads mapping (SLM) (iv) a hybrid coupling approach was proposed for efficient analysis of RPV failure which employs a lumped-parameter code for quick estimate of thermal load and a mechanistic FE code for detailed structural analysis and (v) for the hybrid coupling approach the code transIVR was developed for calculation of transient melt pool heat transfer

The developed creep model for 16MND5 is called lsquomodified theta projection modelrsquo and describes all three stages of a creep process Creep curves are expressed as a function of time with five model parameters 120579119894(119894 = 14 119886119899119889 119898) A dataset for the model parameters was constructed by fitting experimental creep curves into this function and an iterative parameter adjustment process to satisfy the monotonicity assumption of creep strain vs temperature and stress Model parameters for new curves under different temperature and stress loads are directly interpolated from this dataset in contrast to the conventional ldquotheta projection modelrdquo that employs an extra fitting process The new treatment was proven to better accommodate all experimental curves over the wide ranges of temperature and stress loads The model was implemented into the ANSYS Mechanical code The simulation results indicate that the model successfully captured all three creep stages and a good agreement was achieved between each experimental creep curve and corresponding modelrsquos prediction Regarding the model response to transient changes of temperature and stress loads the widely used time hardening and strain hardening models were implemented and

52

performing reasonably well in the new method These properties fulfil the requirements of a creep model for structural analysis

The transient thermo-fluid-structure coupling approach was developed using the ANSYS Fluent code for thermofluid dynamics of the melt pool and ANSYS Structural code for structural mechanics of RPV An extension tool was introduced to realize transient load transfer from ANSYS Fluent to Structural and minimize the user effort Both turbulence models and the effective model PECM can be used for melt pool heat transfer The developed creep model was used for RPV failure analysis of the RPV Since it is insensitive to geometry the coupling approach allows more complex geometries such as a BWR lower head with a forest of penetrations (CRGTs and IGTs) modelled in detail Moreover the coupling approach does not only reflect the transient response of thermo-fluid-structure interaction but also facilitates the implementation of advanced (high-fidelity) models in both fluid mechanics and structural mechanics The validation of the coupling approach against the FOREVER-EC2 experiment showed that the temperature profile vessel failure time total displacement of vessel bottom point and wall thickness profile in the experiment were well predicted

Following the variation of the coupling approach from VLM to SLM for data transfer between ANSYS Fluent and ANSYS Structural the respective coupling approaches were investigated by comparative calculations against the FOREVER-EC2 experiment and a postulated severe accident of a reference BWR For the FOREVER-EC2 experiment both VLM and SLM approaches predicted generally well the total deformation in vertical direction of bottom point and the wall thickness change though the former predicted a slightly later failure time For the reference BWR case good agreements were achieved in predicting the maximum creep strain and the total deformation in vertical direction of bottom point For both experiment and reactor cases the VLM approach predicted slightly later failure times than SLM but the differences were negligible compared to the long time-scale of vessel failure (~104 s) The deformation rate of the vessel wall was slow at the early stage and accelerated with time till a drastic deformation and final vessel failure Generally speaking the VLM and SLM coupling approaches had the similar performance in terms of their predictability of experiment and applicability to reactor case but the VLM was computationally more efficient

In the development of the hybrid coupling approach the transIVR code was developed for quick estimate of transient melt pool heat transfer and determination of thermal loads on RPV transIVR does not only support transient analysis of one- and two- layer melt pool but also the heat transfer in RPV being resolved with 2D finite difference method which captures more spatial thermal details for RPV structural analysis in addition to 1D heat transfer model The capabilities of transIVR in modelling two-layer pool heat transfer and transient pool heat transfer were demonstrated by calculations against the UCSB FIBS benchmark case and the LIVE-7V experiment respectively It was then coupled to the mechanical solver ANSYS Mechanical which together form the hybrid coupling framework for detailed RPV failure analysis The calculation against the FOREVER-EC2 experiment did not only reproduce the major results of the experiment but also had a good agreement with the thermo-fluid-structure

53

coupling approach using CFD This indicated the new coupling approach is capable of capturing the vessel creep failure characteristics and can be employed as an efficient tool addressing the complex vessel failure problem

As a prospective more research topics can be addressed in future either as a continuation of the current work toward an effective tool for vessel failure analysis or from the general view of scientific interest in melt pool heat transfer and RPV failure The following topics are recommended

bull Extended applications to the reactor cases For instance the thermo-fluid-structure coupling approach can be used to study penetration failures in the RPV of an BWR with a detailed representation of penetrations (IGTs and CRGTs)

bull Synthesis or update of heat transfer correlations for melt pool Heat transfer correlations directly benefit the parametric codes such as transIVR in accurate quantification of heat flux distribution around a melt pool So far many empirical correlations have been proposed based on experiments which had a variety of geometries and melt simulants with a wide range of Rayleigh number (Ra) Sensitivity studies have also been performed on the influences of Ra Pa and other material properties Therefore a comprehensive synthesis or update of the existing heat transfer correlations will be a meaningful step for making reasonable extrapolations to the reactor applications with prototypic materials

bull Modelling of pool stratification The motivation starts with the improvement of the PECMrsquos modelling capability such that the dynamic stratification and resulting pool heat transfer can be modelled altogether Simulation of melt stratification has been supported in some system codes such as ASTEC using a predefined parametric approach The improvement of PECM can be considered in the similar approach or more mechanistic modelling

bull Development of a structural module for RPV The idea is to develop a simplified module for structural analysis so it can be coupled with transIVR ultimately forming a standalone code for quick estimate of vessel failure of LWRs during postulated severe accidents

bull Extension of vessel analysis capability The current version of tranIVR is limited to melt pool Although the precursory debris bed and melt ejection after vessel failure were not the focus of the present study they are equally important to assessment of corium risk especially for Swedish BWRs where the melt release conditions and history are crucial to qualification of their severe accident mitigation strategies

54

55

Bibliography

[1] IAEA ldquoIAEA Annual Report 2018rdquo IAEA 2019 [Online] Available httpswwwiaeaorgsitesdefaultfilesgcgc63-5pdf

[2] IAEA Safety of Nuclear Power Plants Design Vienna INTERNATIONAL ATOMIC ENERGY AGENCY 2012

[3] M Fischer ldquoGen II and III In-Vessel and Ex-Vessel Melt Retentionrdquo The Short Course on Severe Accident Phenomenology Oct 2017

[4] B R Sehgal Nuclear Safety in Light Water Reactors Severe Accident Phenomenology 1st ed Academic Press 2011

[5] IAEA Approaches and Tools for Severe Accident Analysis for Nuclear Power Plants Vienna INTERNATIONAL ATOMIC ENERGY AGENCY 2008

[6] WENRA ldquoPractical Elimination Applied to New NPP Designs - Key Elements and Expectationsrdquo Sep 2019 [Online] Available httpwwwwenraorgmediafiler_public20191111practical_elimination_applied_to_new_npp_designs_-_key_elements_and_expectations_-_for_issuepdf

[7] P Matejovic M Barnak M Bachraty and L Vranka ldquoASTEC applications to VVER-440V213 reactorsrdquo Nuclear Engineering and Design vol 272 pp 245ndash260 2014 doi 101016jnucengdes201308077

[8] AREVA ldquoUS EPR Severe Accident Evaluationrdquo Topical Report ANP-10268NP Oct 2006 [Online] Available httpswwwnrcgovdocsML0631ML063100157pdf

[9] C T Martin ldquoCoupled 3D Thermo-mechanical Analysis of Nordic BWR Lower Head Failure in case of Core Melt Severe Accidentrdquo master thesis KTH Stockholm 2013

[10] IAEA ldquoAdvanced Large Water Cooled Reactors A supplement to the IAEArsquos Advanced Reactor Information System (ARIS)rdquo Sep 2015 [Online] Available httpsarisiaeaorgPublicationsIAEA_WRC_Bookletpdf

[11] B Pershagen Light Water Reactor Safety Pergamon Press 1989 [12] Ninos Garis M Agrell H Glanneskog and L Agrenius ldquoAPRI 7 ndash Accident

Phenomena of Risk Importancerdquo SSM SSM 201212 2012 [Online] Available httpswwwstralsakerhetsmyndighetensecontentassets2a9f299f46b949b3b209e780cf71436a201212-apri-7-accident-phenomena-of-risk-importance-en-lagesrapport-om-forskningen-inom-omradet-svara-haverier-under-aren-2009-2011-

[13] F Fichot et al ldquoSome considerations to improve the methodology to assess In-Vessel Retention strategy for high-power reactorsrdquo Annals of Nuclear Energy vol 119 pp 36ndash45 2018 doi 101016janucene201803040

[14] T G Theofanous C Liu S Additon S Angelini O Kymaumllaumlinen and T Salmassi ldquoIn-vessel coolability and retention of a core meltrdquo Nuclear Engineering and Design vol 169 no 1 pp 1ndash48 1997 doi 101016S0029-5493(97)00009-5

[15] C Sainte Catherine ldquoTensile and creep tests material characterization of pressure vessel steel(16MND5) at high temperatures (20 up to 1300 degC)rdquo CEA France SEMTLISNRT98-009A 1998

56

[16] E Altstadt H-G Willschuetz B R Sehgal and F-P Weiss ldquoModelling of in-vessel retention after relocation of corium into the lower plenumrdquo Forschungszentrum Rossendorf FZR-437 2005

[17] H-G Willschuumltz and E Altstadt ldquoGeneration of a High Temperature Material DataBase and its Application to Creep Tests withFrench or German RPV-steelrdquo FORSCHUNGSZENTRUM ROSSENDORF FZR-353 ISSN 1437-322X Aug 2002

[18] J L Rempe S A Chavez and G L Thinnes ldquoLight water reactor lower head failure analysisrdquo Nuclear Regulatory Commission NUREGCR--5642 1993 Accessed Aug 23 2019 [Online] Available httpinisiaeaorgSearchsearchaspxorig_q=RN25025905

[19] S Brosi et al ldquoCORVIS Investigation of light water reactor lower head failure modesrdquo Nuclear Engineering and Design vol 168 no 1 pp 77ndash104 1997 doi 101016S0029-5493(96)00008-8

[20] T Y Chu M M Pilch J H Bentz J S Ludwigsen W Y Lu and L L Humphries ldquoLower Head Failure Experiments and Analysesrdquo Sandia National Laboratories Albuquerque NM USA NUREGCR-5582 SAND98-2047 1999

[21] L L Humphries et al ldquoOECD Lower Head Failure Project Final Reportrdquo Sandia National Laboratories Albuquerque NM 87185-1139 2002

[22] B R Sehgal R R Nourgaliev T N Dihn A Karbojian J A Green and V A Bui ldquoFOREVER Experiments on Thermal and Mechanical Behavior of a Reactor Pressure Vessel during a Severe Accidentrdquo in Proceedings of the Workshop on in-vessel core debris retention and coolability Garching near Munich Germany Mar 3-6 1998

[23] S Bhandari M Fischer B R Sehgal and A Theerthan ldquoPost Test Analysis of the FOREVER C2 Experimentrdquo The 9th International conference on Nuclear Engineering (ICONE-9) Nice France Apr 8-12 2001

[24] J F Mao J W Zhu S Y Bao L J Luo and Z L Gao ldquoCreep deformation and damage behavior of reactor pressure vessel under core meltdown scenariordquo International Journal of Pressure Vessels and Piping vol 139ndash140 pp 107ndash116 2016 doi 101016jijpvp201603009

[25] J Mao L Hu S Bao L Luo and Z Gao ldquoInvestigation on the RPV structural behaviors caused by various cooling water levels under severe accidentrdquo Engineering Failure Analysis vol 79 pp 274ndash284 2017 doi 101016jengfailanal201704029

[26] J F Mao J W Zhu S Y Bao L J Luo and Z L Gao ldquoStudy on structural failure of RPV with geometric discontinuity under severe accidentrdquo Nuclear Engineering and Design vol 307 pp 354ndash363 2016 doi 101016jnucengdes201607027

[27] B R Sehgal V A Bui T N Dinh and R R Nourgaliev ldquoHeat Transfer Processes in Reactor Vessel Lower Plenum During Late Phase of In-Vessel Core Melt Progressionrdquo in Advances in Nuclear Science and Technology J Lewins M Becker R W Albrecht E J Henley J D McKean K Oshima A Sesonske H B Smets and C P L Zaleski Eds Boston MA Springer US 1999 pp 103ndash135

[28] C-T Tran and T-N Dinh ldquoThe effective convectivity model for simulation of melt pool heat transfer in a light water reactor pressure vessel lower head Part I Physical processes modeling and model implementationrdquo Progress

57

in Nuclear Energy vol 51 no 8 pp 849ndash859 2009 doi 101016jpnucene200906007

[29] C-T Tran and T-N Dinh ldquoThe effective convectivity model for simulation of melt pool heat transfer in a light water reactor pressure vessel lower head Part II Model assessment and applicationrdquo Progress in Nuclear Energy vol 51 no 8 pp 860ndash871 2009 doi 101016jpnucene200906001

[30] H-G Willschuumltz E Altstadt B R Sehgal and F-P Weiss ldquoCoupled thermal structural analysis of LWR vessel creep failure experimentsrdquo Nuclear Engineering and Design vol 208 no 3 pp 265ndash282 2001 doi 101016S0029-5493(01)00364-8

[31] E Altstadt and T Moessner ldquoExtension of the ANSYS creep and damage simulation capabilitiesrdquo Forschungszentrum Rossendorf Technical report FZR-296 2000

[32] H-G Willschuumltz E Altstadt B R Sehgal and F-P Weiss ldquoRecursively coupled thermal and mechanical FEM-analysis of lower plenum creep failure experimentsrdquo Annals of Nuclear Energy vol 33 no 2 pp 126ndash148 2006 doi 101016janucene200508006

[33] H-G Willschuumltz E Altstadt B R Sehgal and F-P Weiss ldquoSimulation of creep tests with French or German RPV-steel and investigation of a RPV-support against failurerdquo Annals of Nuclear Energy vol 30 no 10 pp 1033ndash1063 2003 doi 101016S0306-4549(03)00036-7

[34] H Madokoro A Miassoedov and T Schulenberg ldquoA thermal structural analysis tool for RPV lower head behavior during severe accidents with core meltrdquo Mechanical Engineering Letters vol 4 pp 18-00038-18ndash00038 2018 doi 101299mel18-00038

[35] W Villanueva C-T Tran and P Kudinov ldquoCoupled thermo-mechanical creep analysis for boiling water reactor pressure vessel lower headrdquo Nuclear Engineering and Design vol 249 pp 146ndash153 2012 doi 101016jnucengdes201107048

[36] W Villanueva C T Tran and P Kudinov ldquoA Computational Study on Instrumentation Guide Tube Failure During a Severe Accident in Boiling Water Reactorsrdquo in The 14th International Topical Meeting on Nuclear Reactor Thermal hydraulics NURETH-14 Toronto Ontario Canada Sep25-30 2011

[37] W Villanueva C T Tran and P Kudinov ldquoAssessment with Coupled Thermo-Mechanical Creep Analysis of Combined CRGT and External Vessel Cooling Efficiency for A BWRrdquo The 14th International Topical Meeting on Nuclear Reactor Thermal hydraulics NURETH-14 Toronto Ontario Canada Sep 25-30 2011

[38] F R Larson and J Miller ldquoA Time-Temperature Relationship for Rupture and Creep Stressesrdquo Trans ASME vol 74 p 765minus775 1952

[39] C T Tran ldquoDevelopment validation and application of an effective convectivity model for simulation of melt pool heat transfer in a light water reactor lower headrdquo Licentiate thesis KTH Stockholm 2007

[40] ANSYS ldquoANSYS Fluent Theory Guide Release 162rdquo ANSYS Inc Jul 2015 [Online] Available httpwwwansyscom

[41] P Zeng Finite element analysis and application (in Chinese) Tsinghua University Press amp Springer 2004

58

[42] ANSYS ldquoANSYS Mechanical | Finite Element Analysis Softwarerdquo httpswwwansyscomproductsstructuresansys-mechanical-enterprise (accessed Sep 26 2019)

[43] ANSYS ldquoANSYS Mechanical APDL Theory Reference Release 162rdquo ANSYS Inc Jul 2015 [Online] Available httpwwwansyscom

[44] U Steinberner and H H Reineke ldquoTurbulent Buoyancy Convection Heat Transfer with Internal Heat Sourcesrdquo in Proceedings of the 6th Int Heat Transfer Conference Toronto Canada 1978 vol Vol2 pp 305ndash310 doi 101615IHTC63350

[45] R W Evans and B Wilshire Introduction to Creep London Institute of Materials 1993

[46] DoITPoMS ldquoCreep Deformation of Metalsrdquo Creep Deformation of Metals University of Cambridge 2006 httpswwwdoitpomsacuktlplibcreepindexphp (accessed Jul 09 2019)

[47] ANSYS ANSYS Mechanical APDL Material Reference Release 162 2015 [48] SIMULIA Abaqus 612 Analysis Userrsquos Manualvol 3 Materials Chapter

2324 2012 [49] S Imaoka ldquoUser Creep Subroutine STI0704A ANSYS Release110rdquo 2007 [50] R W Evans J D Parker and B Wilshire ldquoThe θ projection conceptmdashA

model-based approach to design and life extension of engineering plantrdquo International Journal of Pressure Vessels and Piping vol 50 no 1 pp 147ndash160 1992 doi 1010160308-0161(92)90035-E

[51] ECCC ldquoRecommendations and Guidance for the Assessment of Creep Strain and Creep Strength Data ECCC Recommendations Volume 5 Part 1 b[Issue 2]rdquo 2013

[52] W D Day and A P Gordon ldquoLife Fraction Hardening Applied to a Modified Theta Projection Creep Model for a Nickel-Based Super-Alloyrdquo The ASME Turbo Expo 2014 Turbine Technical Conference and Exposition Duumlsseldorf Germany Jun16-20 2014 vol 7A-Structures and Dynamics p V07AT29A010 doi 101115GT2014-25881

[53] M Kumar I V Singh B K Mishra S Ahmad A Venugopal Rao and V Kumar ldquoA Modified Theta Projection Model for Creep Behavior of Metals and Alloysrdquo J of Materi Eng and Perform vol 25 no 9 pp 3985ndash3992 2016 doi 101007s11665-016-2197-y

[54] P Yu and W Ma ldquoA modified theta projection model for creep behavior of RPV steel 16MND5rdquo Journal of Materials Science amp Technology vol 47 pp 231ndash242 2020 doi 101016jjmst202002016

[55] ANSYS ANSYS Mechanical Advanced Nonlinear Materials v160 Lecture 2 Rate Dependent Creep ANSYS training materials 2015

[56] Q Y Li N C Wang and D Y Yi Numerical Analysis (in Chinese) 5th ed Tsinghua University Press 2008

[57] T N Dinh R R Nourgaliev and B R Sehgal ldquoOn heat transfer characteristics of real and simulant melt pool experimentsrdquo Nuclear Engineering and Design vol 169 no 1 pp 151ndash164 1997 doi 101016S0029-5493(96)01283-6

[58] O Kymaumllaumlinen H Tuomisto O Hongisto and T G Theofanous ldquoHeat flux distribution from a volumetrically heated pool with high Rayleigh numberrdquo

59

Nuclear Engineering and Design vol 149 no 1ndash3 pp 401ndash408 1994 doi 1010160029-5493(94)90305-0

[59] F J Asfia and V K Dhir ldquoNatural convection heat transfer in volumetrically heated spherical poolsrdquo 1994 Accessed Aug 23 2019 [Online] Available httpinisiaeaorgSearchsearchaspxorig_q=RN26037727

[60] M Jahn and H H Reineke ldquoFree Convection Heat Transfer with Internal Heat Sources Calculations and Measurementsrdquo in Proceedings of the 5th Int Heat Transfer Conference Tokyo Japan 1974 vol Vol3 paper NC-28

[61] M Gustafsson ldquoFSI Transient extension V1704rdquo EDR MEDESO 2016 [62] EDR MEDESO httpsedrmedesocom [63] H Kim and I Namgung ldquoThermo-mechanical Analysis of RVLH for

APR1400 during a Severe Accidentrdquo in Transactions SMiRT-23 Manchester United Kingdom Aug10-14 2015 [Online] Available httpwwwlibncsueduresolver18402034197

[64] P Yu and W Ma ldquoComparative Analysis of Reactor Pressure Vessel Failure using Two Thermo-Fluid-Structure Coupling Approachesrdquo submitted to journal

[65] F R Menter ldquoTwo-equation eddy-viscosity turbulence models for engineering applicationsrdquo AIAA Journal vol 32 no 8 pp 1598ndash1605 1994 doi 102514312149

[66] T N Dinh and R R Nourgaliev ldquoTurbulence modelling for large volumetrically heated liquid poolsrdquo Nuclear Engineering and Design vol 169 no 1 pp 131ndash150 1997 doi 101016S0029-5493(96)01281-2

[67] ANSYS Inc ldquoIntroduction to ANSYS Fluent Lecture 7 Turbulence Modelling Fluent Tutorial Materialrdquo 2016

[68] A Horvat and M Borut ldquoNumerical investigation of natural convection heat transfer in volumetrically heated spherical segmentsrdquo in Proceedings of the International Conference Nuclear Energy for New Europe 2004

[69] Y Li CFD Pre-test Analysis of SIMECO-2 Experiment 2016 [70] A Theerthan A Karbojian and B R Sehgal ldquoEC-FOREVER experiments

on thermal and mechanical behavior of a reactor pressure vessel during a severe accident Technical report-1 EC-FOREVER-2 testrdquo SAM-ARVI-D008 Mar 2001

[71] P Yu W Ma W Villanueva A Karbojian and S Bechta ldquoValidation of a thermo-fluid-structure coupling approach for RPV creep failure analysis against FOREVER-EC2 experimentrdquo Annals of Nuclear Energy vol 133 pp 637ndash648 2019 doi 101016janucene201906067

[72] T G Theofanous C Liu S Additon S Angelini O Kymaumllaumlinen and T Salmassi ldquoIn-vessel coolability and retention of a core meltrdquo Nuclear Engineering and Design vol 169 no 1ndash3 pp 1ndash48 1997 doi 101016S0029-5493(97)00009-5

[73] H P Langtangen ldquoSolving nonlinear ODE and PDE problemsrdquo Simula Research Laboratory 2016 [Online] Available httpshplgitgithubionum-methods-for-PDEsdocpubnonlinpdfnonlin-4printpdf

[74] P Yu and W Ma ldquoDevelopment of a Lumped-Parameter Code for Efficient Assessment of In-Vessel Melt Retention Strategy of LWRsrdquo submitted to journal

60

[75] T G Theofanous M Maguire S Angelini and T Salmassi ldquoThe first results from the ACOPO experimentrdquo Nuclear Engineering and Design vol 169 no 1 pp 49ndash57 1997 doi 101016S0029-5493(97)00023-X

[76] S W Churchill and H H S Chu ldquoCorrelating equations for laminar and turbulent free convection from a vertical platerdquo International Journal of Heat and Mass Transfer vol 18 no 11 pp 1323ndash1329 1975 doi 1010160017-9310(75)90243-4

[77] S Globe and D Dropkin ldquoNatural-Convection Heat Transfer in Liquids Confined by Two Horizontal Plates and Heated from Belowrdquo J Heat Transfer vol 81 no 1 pp 24ndash28 1959 doi 10111514008124

[78] T G Theofanous C Liu S Additon S Angelini O Kymaelaeinen and T Salmassi ldquoIn-vessel coolability and retention of a core melt Volume 2rdquo Argonne National Lab DOEID--10460-VOL2 1996 Accessed Apr 18 2019 [Online] Available httpinisiaeaorgSearchsearchaspxorig_q=RN30000459

[79] J L Rempe D L Knudson C M Allison G L Thinnes and C L Atwood ldquoPotential for AP600 in-vessel retention through ex-vessel floodingrdquo United States INEELEXT--97-00779 1997 Accessed Apr 18 2019 [Online] Available httpinisiaeaorgSearchsearchaspxorig_q=RN30025872

[80] A Miassoedov X Gaus-Liu T Cron and B Fluhrer ldquoLive experiments on melt pool heat transfer in the reactor pressure vessel lower headrdquo The 10th International Conference on Heat Transfer Fluid Mechanics and Thermodynamics (HEFAT 2014) Orlando FL Jul 14-16 2014

[81] H Madokoro A Miassoedov and T Schulenberg ldquoAssessment of a lower head molten pool analysis module using LIVE experimentrdquo Nuclear Engineering and Design vol 330 pp 51ndash58 2018 doi 101016jnucengdes201801036

[82] B Fluhrer et al ldquoThe LIVE-L1 and LIVE-L3 experiments on melt behaviour in RPV lower headrdquo Forschungszentrum Karlsruhe GmbH Technik und Umwelt (Germany) Inst fuer Kern- und Energietechnik FZKA--7419 2008 Accessed Nov 12 2019 [Online] Available httpinisiaeaorgSearchsearchaspxorig_q=RN40036867

To my family

and

all the people that I love

I

Abstract

This thesis aims at the development of new coupling approaches and new models for the thermo-fluid-structure coupling problem of reactor pressure vessel (RPV) failure during severe accidents and related physical phenomena The thesis work consists of five parts (i) development of a three-stage creep model for RPV steel 16MND5 (ii) development of a thermo-fluid-structure coupling approach for RPV failure analysis (iii) performance comparison of the new approach that uses volume loads mapping (VLM) for data transfer with the previous approach that uses surface loads mapping (SLM) (iv) development of a lumped-parameter code for quick estimate of transient melt pool heat transfer and (v) development of a hybrid coupling approach for efficient analysis of RPV failure

A creep model called lsquomodified theta projection modelrsquo was developed for the 16MND5 steel so that it covers three-stage creep process Creep curves are

expressed as a function of time with five parameters 120579119894 (i=14 and m) in the new creep model A dataset for the model parameters was constructed based on the available experimental creep curves given the monotonicity assumption of creep strain vs temperature and stress New creep curves can be predicted by interpolating model parameters from this dataset in contrast to the previous method that employs an extra fitting process The new treatment better accommodates all the experimental curves over the wide ranges of temperature and stress loads The model was implemented into the ANSYS Mechanical code and its predictions successfully captured all three creep stages and a good agreement was achieved between the experimental and predicted creep curves For dynamic loads that change with time the widely used time hardening and strain hardening models were implemented with a reasonable performance These properties fulfil the requirements of a creep model for structural analysis

A thermo-fluid-structure coupling approach was developed by coupling the ANSYS Fluent for the fluid dynamics of melt pool heat transfer and ANSYS Structural for structural mechanics of RPV An extension tool was introduced to realize transient load transfer from ANSYS Fluent to Structural and minimize the user effort Both CFD with turbulence models and the effective model PECM can be employed for predicting melt pool heat transfer The modified theta projection model was used for creep analysis of the RPV The coupling approach does not only capture the transient thermo-fluid-structure interaction feature but also support the advanced models in both melt pool convection and structural mechanics to improve fidelity and facilitate implementation The coupling approach performs well in the validation against the FOREVER-EC2 experiment and can be applied complex geometries such as a BWR lower head with forest of penetrations (control rod guide tubes and instrument guide tubes)

In the comparative analysis the VLM and SLM coupling approaches generally have the similar performance in terms of their predictability of the FOREVER-EC2 experiment and applicability to the reactor case Though the SLM approach predicted slightly earlier failure times than VLM in both cases the difference was negligible compared to the large scale of vessel failure time (~104 s) The VLM approach showed higher computational efficiency than the SLM

II

The idea of the hybrid coupling is to employ a lumped-parameter code for quick estimate of thermal load which can be employed in detailed structural analysis Such a coupling approach can significantly increase the calculation efficiency which is important to the case of a prototypical RPV where mechanistic simulation of melt pool convection is computationally expensive and unnecessary The transIVR code was developed for this purpose which is not only capable of quick estimate of transient heat transfer of one- and two- layer melt pool but also solving heat conduction problem in the RPV wall with 2D finite difference method to provide spatial thermal details for RPV structural analysis The capabilities of transIVR in modelling two-layer pool heat transfer and transient pool heat transfer were demonstrated by calculations against the UCSB FIBS benchmark case and the LIVE-7V experiment respectively The transIVR code was then coupled to the mechanical solver ANSYS Mechanical for detailed RPV failure analysis Validation against the FOREVER-EC2 experiment indicates the coupling framework successfully captured the vessel creep failure characteristics

Keywords RPV failure thermo-fluid-structure coupling creep modelling melt pool heat transfer severe accident CFD FEA transIVR

III

Sammanfattning

Syftet med denna avhandling aumlr att utveckla nya metoder och nya modeller foumlr att kunna koppla ihop termohydrauliska foumlrlopp och uppkommen strukturrespons i botten paring reaktortanken foumlr foumlrbaumlttrad analys av reaktortankbrott under ett svaringrt haveri Avhandlingen bestaringr av fem delar (i) utveckling av en trestegsmodell foumlr krypning i reaktortankens 16MND5-staringl (ii) utveckling av metoder foumlr kopp ling av termohydrauliska foumlrlopp och strukturrespons foumlr studier av reaktortankbrott (iii) jaumlmfoumlrelse av prestanda av den nya metoden som anvaumlnder volume loads mapping (VLM) foumlr dataoumlverfoumlring med den tidigare metoden som anvaumlnder surface loads mapping (SLM) (iv) utveckling av en lumped-parameter beraumlkningskod foumlr snabb uppskattning av transient vaumlrmeoumlverfoumlring i en smaumlltpoumll i botten paring reaktortanken och (v) utveckling av en hybrid kopplingsmetod foumlr effektiv analys av reaktortankbrott

En krypmodell kallad lsquomodified theta projection modelrsquo som behandlar krypprocessen i tre steg utvecklades foumlr 16MND5-staringlet Krypkurvor i den nya

krypmodellen ges som en funktion av tid med fem parametrar 120579119894 (i=14 och m) Ett dataset foumlr modellparametrar togs fram baserat paring tillgaumlngliga experimentalla krypkurvor under anatagande om att kryptoumljning aumlr en monoton funktion av temperatur och spaumlnning Nya krypkurvor kan genereras genom att interpolera modellparametrar fraringn detta dataset i kontrast till den tidigare metoden som anvaumlnder en speciell anpassningsprocess Foumlrdelen med det nya tillvaumlgagaringngssaumlttet aumlr att den hanterar alla experimentella krypkurvor foumlr ett brett spektrum av temperatur- och spaumlnningsbelastningar Modellen implementerades i ANSYS-koden och den korrekt behandlar alla tre krypsteg samt visar bra oumlverenstaumlmmelse mellan experimentella och beraumlknade krypkurvor Foumlr analys av tidsberoende dynamiska belastningar implementerades vaumllkaumlnda modeller av tids- och deformationshaumlrdning med rimliga resultat Dessa egenskaper uppfyller kravet paring en krypmodell foumlr strukturmekaniska analyser

En metod utvecklades foumlr koppling mellan termohydrauliska foumlrlopp och strukturmekanisk respons genom att koppla ihop ANSYS Fluent som analyserar fluiddynamiska aspekter av vaumlrmeoumlverfoumlring i en smaumlltpoumll med ANSYS Structural som analyserar strukturmekanisk respons av reaktortanksvaumlggen Ett speciellt verktyg togs fram foumlr att moumljliggoumlra oumlverfoumlring av transienta belastningar fraringn ANSYS Fluent till ANSYS Structural och foumlr att minimera anvaumlndarinsatsen Baringde CFD med turbulensmodeller och den effektiva PECM-modellen kan anvaumlndas foumlr att analysera vaumlrmeoumlverfoumlring i en smaumlltpoumll Foumlr krypanalysen av reaktortanken har lsquomodified theta projection modelrsquo anvaumlnts Den framtagna kopplingsmetoden har foumlrdelen att den inte bara faringngar de transienta aspekterna av interaktionen mellan termohydraulik och strukturmekanik men ocksaring stoumldjer avancerade modeller av konvektion i en smaumlltpoumll och strukturrespons och daumlrigenom bidrar till foumlrbaumlttrad precision och underlaumlttar implementeringen av modeller Kopplingsmetoden fungerar bra naumlr den valideras mot FOREVER-EC2 experimentet och den kan anvaumlndas i

IV

komplexa geometrier som till exempel i botten paring reaktortanken i en kokvattenreaktor (BWR) med en skog av genomfoumlringar (ledroumlr foumlr styrstavar och instrumentering)

Jaumlmfoumlrande beraumlkningar visar att de tvaring kopplingsmetoderna VLM-baserad och SLM-baserad presterar lika bra naumlr det gaumlller hur de predikterar resultat av FOREVER-EC2 experimentet och deras laumlmplighet foumlr reaktoranalyser Aumlven om SLM-metoden predikterar att tankbrott i baringda fallen intraumlffar lite tidigare saring aumlr skillnaden mellan metoderna foumlrsumbar i jaumlmfoumlrelse med tiden till tankbrott (~ 104 s) VLM-metoden var baumlttre aumln SLM-metoden naumlr det gaumlller beraumlkningseffektiviteten

Ideacuten om hybridkoppling garingr ut paring att anvaumlnda en lumped-parameter kod foumlr snabb uppskattning av termiska belastningar som senare kan anvaumlndas i detaljerad strukturmekanisk analys En saringdan kopplingsmetod kan paring ett signifikant saumltt oumlka effektiviteten av beraumlkningar vilket aumlr viktig i reaktorapplikationer daumlr mekanistisk simulering av konvektion i en smaumlltpoumll aumlr beraumlkningsintensiv och onoumldig I detta syfte utvecklades koden transIVR Koden kan dels ge en snabb uppskattning av transient vaumlrmeoumlverfoumlring i en smaumlltpoumll som bestaringr av en eller tvaring materialskikt och dels loumlsa vaumlrmeledningsproblem i reaktortanksvaumlggen med 2D finita differensmetoden vilket genererar detaljerad information om termiska foumlrharingllanden som behoumlvs foumlr strukturmekaniska analyser Foumlr att demonstrera att transIVR kan analysera dels vaumlrmeoumlverfoumlring i en smaumlltpoumll bestaringende av tvaring materialskikt och dels transient vaumlrmeoumlverfoumlring i en smaumlltpoumll jaumlmfoumlrdes transIVR prediktioner med UCSB FIBS-benchmarkfallet respektive LIVE-7V-experimentet I naumlsta steg kopplades transIVR-koden till ANSYS Structural foumlr att moumljliggoumlra mer detaljerad analys av reaktortankbrott Validering mot FOREVER-EC2 experimentet tyder paring att den utvecklade kopplingsmetodiken korrekt aringterger tankbrott orsakat av krypning

Nyckelord reaktortankbrott koppling termohydraulik-strukturmekanik modellering av krypning vaumlrmeoumlverfoumlring i smaumlltpoumll svaringra haverier CFD FEA transIVR

V

List of Publications

Papers included in this thesis

I P Yu and W Ma ldquoA modified theta projection model for creep behavior of RPV steel 16MND5rdquo Journal of Materials Science amp Technology vol 47 pp 231ndash242 2020

II P Yu W Ma W Villanueva A Karbojian and S Bechta ldquoValidation of a thermo-fluid-structure coupling approach for RPV creep failure analysis against FOREVER-EC2 experimentrdquo Annals of Nuclear Energy vol 133 pp 637ndash648 2019

III P Yu and W Ma ldquoComparative analysis of reactor pressure vessel failure using two thermo-fluid-structure coupling approachesrdquo Nuclear Engineering and Design submitted

IV P Yu and W Ma ldquoDevelopment of a lumped-parameter code for efficient assessment of in-vessel melt retention strategy of LWRsrdquo Annals of Nuclear Energy submitted

Respondentrsquos contribution to the papers included in the thesis

Peng Yu made major contributions to the publications of all the above four articles under the guidance of supervisors and with the collaboration of other co-authors including formulation of research questions development of coupling approaches and numerical methods analysis and generalization of simulation results as well as writing the first drafts and addressing review comments

Papers not included in this thesis

1 P Yu and W Ma ldquoRPV creep failure analysis using a Norton creep modelrdquo International Journal of Mechanical Sciences to be submitted

2 P Yu W Villanueva S Galushin W Ma and S Bechta ldquoCoupled thermo-mechanical creep analysis for a Nordic BWR lower head using non-homogeneous debris bed configuration from MELCORrdquo 12th International Topical Meeting on Nuclear Reactor Thermal-Hydraulics Operation and Safety (NUTHOS-12) Qingdao China October 14-18 2018

3 P Yu A Komlev Y Li W Villanueva W Ma and S Bechta ldquoPre-test simulation of SIMECO HT experiments for stratified melt pool heat transferrdquo 11th International Topical Meeting on Nuclear Reactor Thermal Hydraulics Operation and Safety (NUTHOS-11) Gyeongju Korea October 9-13 2016

VI

VII

Acknowledgements

First of all I would like to express my sincere gratitude to my supervisor Prof Weimin Ma for offering this opportunity to work in the severe accident research field and for the supports patience guidance discussions and encouragements throughout the entire period of my graduate study

I am very thankful to my co-supervisor Dr Walter Villanueva for the valuable help and discussions I learnt a lot during the times working with him I am also thankful to co-supervisor Prof Sevostian Bechta who is supportive and always has inspiring ideas that keep you thinking for a long time

Thank you also to all my colleagues at the Division of Nuclear Power Safety for the supports discussions and the very nice working environment during these years

I have taken many interesting courses during my PhD study for which I want to express my thanks to all the teachers With this regard a special thanks goes to Prof Johan Hoffman for his SF2561 Finite Element Method given in 2015 fall as well as the discussions and advices on numerical analysis

I am grateful to Ignacio Gallego-Marcos and Wenyuan Fan for their enthusiastic help on CFD and turbulence modelling during the time working on the turbulent

heat transfer I thank all my 师兄 for sharing their knowledge on nuclear engineering and giving advices when I was confused since 2012

I thank all my friends both in Sweden and in China for the company and the sharing of many wonderful moments in daily life

Finally I would like to express my deepest thank to my parents my girlfriend Siwen my sisters and all my other relatives for their unconditional supports and love

Financial supports from the China Scholarship Council and APRI make my study in KTH possible and are acknowledged

VIII

IX

Table of Contents

AbstracthelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphellipI

SammanfattninghelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphellipIII

List of PublicationshelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphellipV

AcknowledgementshelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphellipVII

NomenclaturehelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphellipXI

1 Introductionhelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphellip1

11 Nuclear Power Safety 1

12 RPV Failure 2

13 Literature Review Creep Behaviour and RPV Failure Analysis 4

131 Experiments on creep behaviour and RPV failure 4

132 Numerical studies on RPV failure 5

14 Motivation Objectives and Main Achievements 7

2 Numerical Solvers and Modelshelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphellip9

21 ANSYS Fluent 9

22 ANSYS Mechanical 9

23 PECM 11

3 Creep Modelling of 16MND5 Steelhelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphellip13

31 Creep and Creep Modelling 13

32 A Modified Theta Projection Model 14

321 Original theta projection model and its variations 14

322 A modified theta projection model 16

4 Coupled Thermo-Fluid-Structure Analysis of RPV Failurehelliphelliphelliphellip23

41 Thermo-Fluid-Structure Coupling Approach 23

42 Numerical Modelling 25

421 Conjugate heat transfer 25

422 Mechanical analysis 25

43 Validation against the FOREVER-EC2 Experiment 26

431 FOREVER-EC2 experiment 26

X

432 Conjugate heat transfer 27

433 Mechanical analysis 29

44 Application to BWR Case 32

45 Computational Efficiency 34

5 Hybrid Coupling Approach for Efficient RPV Failure Analysishelliphellip35

51 A Lumped-parameter Code for Melt Pool Heat Transfer 35

511 Heat transfer correlations 36

512 Energy balance in the pools 37

513 Energy balance in the RPV wall 39

52 Hybrid Coupling Approach 40

53 Verification and Validation 41

531 UCSB FIBS benchmark case 41

532 LIVE-7V experiment 43

533 FOREVER-EC2 experiment 45

6 Summary and Outlookhelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphellip51

Bibliographyhelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphellip55

XI

Nomenclature

Latin

119860 area of a surface

119862119901 heat capacity

119863 damage factor

119892 gravity

ℎ heat transfer coefficient

H Melt pool height

119896 thermal conductivity

119871 characteristic length

119873119906 Nusselt number

119875119903 Prandtl number

119902 heat flux

119902119894119899 heat flux through inner vessel surface

119876119881 internal volumetric heat source

119903 radial distance in a cylindrical or spherical coordinate

119877119886 external Rayleigh number

119877119886119894119899 internal Rayleigh number

119877119886119910 local Rayleigh number

119905 time

119879 temperature

119881119900119897 total volume

119909 thickness (of crust or RPV wall)

119911 height in a cylindrical coordinate

Greek

120572 thermal diffusivity

120573 thermal expansion

Δ119879 temperature difference

ε emissivity (in thermal analysis) strain (in structural analysis)

휀119888119903 creep strain

휀119903 creep strain rate

120579 polar angle

120579119894 parameters of a creep model 119894 = 1234119898

120584 kinematic viscosity

σ stress

XII

Super and subscripts

119886119898119887 ambient

119888119903 creep (in Section 3) crust (in Section 5)

119889119899 downward

119890119899119907 environment

119894 inner surface of RPV

119894119899119905 interface

119897119900119888119886119897 local

119898119886119909 maximum value

119898119890119897119905119894119899119892 melting point

119898119905 metal layer (bulk)

119898119905119889119899 downward surface of metal pool

119898119905119904119889 sideward surface of metal pool

119898119905119905119900119901 top surface of metal layer

119898119905119906119901 upward surface of metal pool

119900 outer surface of RPV

119900119909 oxide layer (bulk)

119900119909119889119899 downward surface of oxide layer

119900119909119906119901 upward surface of oxide layer

119903119890119891 reference

119904 internal structures inside the RPV

119904119889 sideward

119906119901 upward

119907119890119904 vessel

Acronyms

BWR boiling water reactor

CBD creep database

CFD computational fluid dynamics

CRGT control rod guide tube

ECCM effective convectivity conductivity model

FDM finite difference method

FE finite element

FEA finite element analysis

FEM finite element method

FVM finite volume method

IGT instrument guide tube

IVR in-vessel retention

XIII

IVMR in-vessel melt retention EU project

LES large eddy simulation

MPS moving particle semi-implicit method

NPP nuclear power plant

PECM phase-change effective convectivity model

RANS Reynolds-averaged Navier-Stokes

RPV reactor pressure vessel

SBO station blackout

SLM surface loads mapping

SPH smoothed-particle hydrodynamics

UDF user-defined function

UPF user programmable features

VLM volume loads mapping

XIV

1

1 Introduction

11 Nuclear Power Safety

Greenhouse gas emissions contribute to the global warming problem Phasing out fossil fuels by switching to low-carbon energy sources is considered as a mitigation strategy With this regard clean energy should be increased in the world energy consumption structure Being clean and stable nuclear energy is an attractive alternative to fossil fuels After the development over decades it also proves to be a mature and safe technology According to IAEA 2018 annual report [1] there were 450 nuclear power reactors under operation worldwide at the end of 2018 which had a record generating capacity of 3964 GW(e) 9 reactors were connected to the grid and 55 reactors under construction around the world

Due to the potential risk of the release of radioactivity nuclear power safety is emphasized throughout the stages of design construction operation and decommission of a nuclear power plant (NPP) The principal technical requirements [2] specify that the three fundamental safety functions should be fulfilled for all plant states of an NPP (i) control of reactivity (ii) removal of heat from the reactor and from the fuel store and (iii) confinement of radioactive material shielding against radiation and control of planned radioactive releases as well as limitation of accidental radioactive releases The defence-in-depth concept is also applied to provide several levels of defence to prevent consequences of accidents The fuel claddings reactor pressure vessel (RPV) and primary coolant pressure boundaries and the containment are the three physical barriers to prevent the release of radioactive materials to the environment

Severe accidents are events that cause significant damage to reactor fuel and result in core meltdown It may be caused by insufficient cooling of the reactor core due to various reasons or by the rapid introduction of huge reactivity which causes sharp increases in the power and fuel fragmentation eg Chernobyl accident During a severe accident the failure of the fuel claddings will result in the radioactive materials previously enclosed inside the cladding released to the primary loop If cooling is not sufficient to remove the decay heat the core materials will melt migrate down and relocate into the RPV lower head after which the melt will react with water and partially form fragments quench solidify and form a debris bed The debris bed formed by the melt may continue to melt to form a melt pool due to the decay heat As the corium contacts the RPV it would continuously heat the RPV and threaten its integrity Once the RPV fails containment becomes the last barrier to prevent the release of radioactivity The released melt would result in an increase of the pressure and temperature due to the high temperature fractions oxidation and hydrogen combustion which is called direct containment heating The melt would fall onto the reactor cavity Without sufficient cooling the molten core would interact with basemat concrete (MCCI) This process can last for few days It can result in containment failure due to either containment overpressure or the basemat melt-through of containment In addition containment is also subjected to isolation failure and early containment failure due to missiles and dynamic pressure [3] More

2

detailed severe accident progression or phenomena could also be found in eg [4] [5]

Though severe accidents are unlikely to occur the consequences can be severe if no proper measure is taken Related researches have been continuously going on aiming at mitigating the accident consequences and improving safety After the Fukushima Accident safety inspections and re-evaluation actions have been carried out worldwide especially on safety measurements regarding severe accidents The concept lsquoPractical Eliminationrsquo applied to new NPP designs demands that new nuclear installations be designed with the objective of mitigating the consequences of accidents and avoiding early radioactive releases and large radioactive releases [6] Severe accident mitigation strategies are applied to the existing and new NPPs (both Generation II and III) The strategies can generally be classified into two categories [3] the in-vessel retention (IVR) and ex-vessel retention (EVR) In simple words in-vessel retention tries to retain the core melt inside the RPV (second barrier) by applying external water cooling on the vessel as shown in Figure 1(a) Such method minimizes the dispersal of the melt within the containment and hence simplifies the later recovery work Examples of the application are AP1000 HPR1000 APR 1400 The idea of the EVR is to let the RPV fail but to retain the core melt inside the containment Two principle approaches of EVR are the retention within a dedicated lsquocore catcherrsquo and melt fragmentation Examples of core catcher would be EPR that directs the melt to a spreading compartment (Figure 1(b)) and VVER1000 that uses metallic crucible below the RPV An example of the melt fragmentation is the Nordic BWRs that the corium released from RPV would relocate in the cavity with a deep-water pool (7-12 m) and is expected to fragment quench and form a coolable debris bed (Figure 1(c))

Figure 1 Examples of IVR [7] EVR with core catcher [8] and melt fragmentation [9]

12 RPV Failure

RPV which contains the reactor core core shroud and coolant is a key component to the reactor system RPV and the primary loop constitute the primary coolant system which conveys the fission heat from the reactor core to the secondary loop through steam generator (PWR) or to the turbine (BWR) The pressure under

(a) IVR (b) Core catcher (c) Melt fragmentation

3

operation can be ~155 MPa for a 1000 MW PWR [10] and ~70 MPa for a 1000 MW BWR [11] It also has to withstand the neutron radiation damage The thickness of RPV [11] can be ~20 cm for PWR and ~16 cm for BWR

As the transition of in-vessel progression to ex-vessel progression RPV failure is important for analysis of accident progression and assessment of the effectiveness of severe accident mitigation strategies In the melt fragmentation strategy of Nordic BWR the melt-coolant interaction and debris bed coolability after the corium poured into the deep-water pool are essentially dependent on the vessel failure mode including the characteristic time of melt release rupture size and location and amount and superheat of melt available for release [12] New concern also arose [13] on the necessity of performing structural analysis for the RPV in the IVR strategy In the initial methodology and further development of the IVR the success of the strategy ie the integrity of the RPV was demonstrated (eg in [14]) with the following two conclusions (i) even for a local heat flux on the vessel as high as critical heat flux (CHF) the resulted very thin external ldquocold layerrdquo of the vessel would be sufficient to withstand the internal stress load and hold the vessel integrity (ii) as CHF is the critical value for the thermal load the main specific goal for assessing the effectiveness of IVR becomes to demonstrate that the local heat flux along RPV be lower than the CHF The new concern doubts that local lsquohot partrsquo of the vessel would have negative effects on the overall resistance due to creep and dilatation which could result in a vessel failure and hence violate the first conclusion in particular for ldquolong termrdquo retention of corium in RPV If so a structural integrity analysis of RPV would be needed in addition to assessing local heat flux especially for high-power reactors Therefore accurate prediction of vessel failure could be meaningful for both strategies

Figure 2 Illustration of RPV under thermal loads from corium

As the structural performance is highly affected by the thermal loads from the corium RPV failure is essentially a transient thermo-fluid-structure coupled problem Figure 2 illustrates the situation that a corium debris bedmelt pool formed in the RPV lower head The decay heat of nuclear fission products could continuously be generated inside the corium leading to the formation of melt pools as the debris melts The heated debris bedmelt pool imposes thermal and

4

mechanical loads on the RPV Such loads can be transient (change with time) and spatially non-uniform Complex phenomena are also involved in this process Strong turbulent natural convection induced by the decay heat may be expected in the melt pool under IVR The corresponding Rayleigh number 119877119886 can be 1016minus17 for prototypic reactor cases Material structural properties can become non-linear under high temperature and stress loads The creep deformation also becomes significant at high temperature Detailed modelling of turbulent natural convection and non-linear properties could be computationally expensive In addition large-scale geometry (hemisphere with a radius of 20-30 m) long-term transient deformation process (~10000 s) and required small time steps (eg 001 s) for numerical convergence consideration further increase the computational costs

13 Literature Review Creep Behaviour and RPV Failure Analysis

In this section the most relevant studies concerning the experiments modelling and analysis of creep behaviour and RPV failure are presented

131 Experiments on creep behaviour and RPV failure

Various experiments have been performed on the RPV failure problem some of which were to generate data of mechanical properties (especially the creep properties) and some to study the overall vessel failure with scaled or simplified geometries

The REVISA experiment [15] generated creep curves and stress strain curves of French steel 16MND5 ranging from 600 to 1300 degC This data is accessible in [16] as well as [17] which also documents other material properties eg density thermal expansion coefficient conductivity heat capacity Youngrsquos Modulus and plasticity Similar experiments were carried out to generate high-temperature creep and tensile test data for American RPV steels and Inconel [18]

The CORVIS experiments [19] investigated the lower head failure modes with and without penetrations Each experiment was carried out on a steel plate 100 mm thick representing a pressure vessel lower head The penetration case represented a boiling water reactor lower head carrying a tube penetration in the form of a drain line

The LHF (lower head failure) experiments [20] and the succeeding OLHF tests [21] investigated the vessel failure with a 15 scaled vessel heated by an electrical radiator allowing to locate the hot focus at different position of the vessel with a maximum wall temperature of around 900 degC Internal pressure ranging from 53 to 123 bar was also imposed to initiate the vessel failure

The FOREVER experiments [22] investigated the phenomena of melt pool convection vessel creep possible failure processes and failure modes occurred during the late phase of the in-vessel progression by heating up a melt pool in a vessel and observing possible vessel failures The study covers cases with and without penetration different melt level existence of gap cooling and in-vessel melt coolability

5

132 Numerical studies on RPV failure

Various numerical approaches have been employed on RPV failure Since a two-way thermo-fluid-structure coupling approach is challenging these approaches take simplifications and can generally be classified into two categories (i) RPV failure decoupled from melt pool heat transfer and (ii) RPV failure coupled with melt pool heat transfer in simplified manners

1) Decoupled approaches In the decoupled approaches only RPV is modelled Simplified constant thermal loads are imposed on RPV considering the melt pool heat transfer

In a post-test analysis of FOREVE-C2 experiment [23] it was treated as a thermo-structural coupled analysis of the vessel with the thermal load estimated by the outer thermocouples and the normal heat flux on the external vessel wall The analysis was done using finite element (FE) code SYSTUS A coupled creepdamage law based on a viscosity-hardening creep law describing primary and secondary creep was employed for creep analysis

Mao et al conducted various studies on reactor cases assessing the IVR strategy using FE code ABAQUS Thermal boundary conditions were imposed on the RPV which was then solved as a thermal-structural coupled problem The strain hardening creep model (Eq (1)) was mainly used Their studies cover the general RPV failure [24] as well as the effects of cooling water levels under IVR [25] and geometric discontinuity (ablated RPV) [26]

ε119888119903 = 1198891σ1198892ε1198893119890119909119901 [minus

1198894119879] (1)

In the EU IVMR project [13] the RPV integrity was also studied through a benchmark calculation which deals with the RPV of a PWR with a prescribed ablated profile with 2D axisymmetric geometry Prescribed temperature gradient was set as thermal load The objective was to calculate deformations for various pressure loads up to vessel rupture A smoother profile was planned in further research

The advantage of such approaches is once thermal loads are reasonably assumed the focus only needs to be put on the RPV The disadvantage is a reasonable thermal load should be consistent with the melt pool heat transfer while sometimes it is not well-noted eg constant uniform temperature boundary conditions were assumed for inner and outer RPV (with uniform thickness in lower head) surfaces [25] which ignored the non-uniform heat flux distribution caused by the turbulent convective corium pool As thermal loads strongly affect the structural performance they should be quantified carefully

2) Coupled approaches In coupled approaches both melt pool heat transfer and RPV failure are modelled and coupled in one-way or simplified manner With melt pool heat transfer modelled the coupled approaches are more flexible to provide reasonable thermal loads on the RPV In existing approaches CFD and the correlation-based distributed-parameter models eg the Effective Convectivity Conductivity Model

6

(ECCM) [27] and the Phase-change Effective Convectivity Model (PECM) [28] [29] are general methods for melt pool heat transfer

Sehgal et al [22] presented an approach that employed the MVITA code to calculate the thermal processes and ANSYS for structural analysis MVITA code employed the ECCM Bailey-Norton creep law was used as shown in Eq (2)

ε119888119903 = 6055 times 10minus26 sdot σ472119905085119890minus

37200

119879 (2)

This approach was used for the pre-test simulation of FOREVER experiment to determine the thermal and mechanical loadings Parametric investigations also helped in the identification of important factors that could affect the test procedure and test conditions

Willschutz et al [30] employed the CFD-module Flotran for the melt pool heat transfer and ANSYS for structural analysis of RPV Standard k-ε turbulence model was selected for the turbulent natural convection in the melt pool A user-defined three-stage creep model [31] was employed In this model experimental creep curves were discretized into a creep database (CDB) under given temperature 119879 and stress 120590 the creep curve was discretized into finite point pairs

(휀 ε) indicating under such thermal and mechanical loads the creep strain rate ε is subjected to creep strain 휀 The creep strain rate for any other case can be interpolated over known temperature stress and strain from the database This method was applied to the post-test calculation of the FOREVE-C2 experiment It was concluded that the unexpected deformation behaviour during the experiment was caused by the combination of the transient temperature field in the vessel wall and the three creep stages

They later replaced the Flotran solver with the ECCM for melt pool heat transfer in a following study [32] By implementing it into FEM-solver a recursively coupled analysis was constructed Moreover a modified creep model [33] was used with damage measures the CDB itself did not contain tertiary creep instead a factor 1(1 minus 119863) was multiplied to the original creep strain rate to get the modified creep strain rate where 119863 is a damage factor (0 lt 119863 lt 1) defined with creep strain information and plastic strain information As 119863 grows close to 1 the modified creep strain rate would be enlarged significantly by this factor 1(1 minus 119863) which then behaves like the accelerating tertiary stage This method was applied to pre- and post- test calculations for the FOREVER experiments proving to be closer to reality than the previous method

Madokoro et al [34] developed a thermal structural analysis tool based on open source solver OpenFOAM and numerically investigated the coupled RPV failure and melt pool convection The PECM was used in the melt pool heat transfer In structural part the plasticity and damage parameters were similar to Willschutz et al [32] except for that strain hardening creep model (Eq (3)) was used instead of a CDB Simulations of FOREVER-EC2 and -EC4 were investigated It was concluded from the simulations that although the emissivity and mesh type (updated or fixed) affect the results significantly the approach generally has the capability of predicting the vessel deformation and failure

7

ε119888119903 = 1198891 sdot σ1198892 sdot ε1198893 (3)

Villanueva et al [35] performed a coupled thermo-mechanical creep analysis of the reactor scale RPV (ABB-Atom design BWR) to evaluate efficacy of the cooling measures for the in-vessel melt coolability and retention The PECM was used for melt pool heat transfer and ANSYS for RPV integrity analysis The modified time hardening model (Eq (4)) of the American RPV steel SA533B was selected as creep model The transient thermal load from the melt pool was applied as a thermal boundary condition to the vessel wall Further studies were done with this approach on investigations of instrumentation guide tube (IGT) failure [36] and control rod guide tube (CRGT) cooling [37] of a Nordic BWR

ε119888119903 =1198881σ

11988821199051198883+1

1198883 + 1 1198881 gt 0 (4)

It is noted that only RPV analysis with finite element analysis (FEA) are reviewed here Engineering methods like the Larson-Miller method [38] are not covered in this thesis One can refer to theory manuals of severe accident codes like MELCOR ASTEC for more details Comprehensive description on modelling of melt pool heat transfer is not covered either One can refer to either [39] for a general view or to any severe accident code manuals for engineering methods

14 Motivation Objectives and Main Achievements

The existing experimental and numerical studies provide us a good understanding of RPV failure involved phenomena and valuable simulation experiences These numerical methods may vary in solvers (which may be related to code accessibility) and models (which may be related to model accessibility understanding of phenomena computational power) However improvements of modelling approaches are still possible as the improvements in any of the related aspects eg

bull Simulation tools are continuously being upgraded and providing new possibilities just like that PECM appeared and ECCM was no longer used

bull Better understanding of the phenomena and more simulation experiences from in-vessel retention and vessel failures researches benefit the further modelling and simulations

bull Increasing computational power allows more detailed modelling eg turbulence modelling of melt pool convection

bull There is still room for the improvement of creep modelling

In addition implementation difficulty of a coupling approach is another important issue to consider Since it is a TRANSIENT thermo-fluid-structure coupled problem it is almost impossible to replicate an approach simply by referring to a paper with limited details For BWRs it would be even better for the coupling approaches to support complex geometries considering forest of penetrations (CRGTs and IGTs) on the RPV lower head without significant increase of user effort in the coupling process

8

The objectives of this thesis are to investigate new approaches and develop new models for the thermo-fluid-structure coupled problem of RPV failure and involved phenomena These objectives are accomplished with the following achievements

1) A new creep model for the 16MND5 steel was developed which covers all three creep stages Good agreement was achieved between model predictions and experimental curves Both time and strain hardening models were implemented (Paper I)

2) A new thermo-fluid-structure coupling approach was developed for RPV failure analysis It supports the advanced models in both melt pool convection and structural mechanics to improve fidelity and facilitates implementation Validation of the approach with the new creep model was performed against FOREVER-EC2 experiment This approach is flexible for complex geometries and can be used in further analysis eg lower head with detailed modelling of penetrations in BWRs (Paper II)

3) The performance of this new approach which uses volume loads mapping for data transfer was compared with the previous approach using surface loads mapping showing good agreement with small discrepancy (Paper III)

4) A lumped-parameter code transIVR for melt pool heat transfer was developed which supports one- and two-layer pool configuration transient analysis and well-resolved RPV wall (Paper IV)

5) A hybrid coupling approach was developed for efficient analysis of RPV failure where transIVR is coupled with ANSYS Mechanical (Paper IV)

In the following general descriptions on the codes ANSYS Fluent Mechanical and the PECM are given in Section 2 the new creep model in Section 3 the new coupling approach as well as its comparison with the previous approach in Section 4 the efficient hybrid coupling approach in Section 5 and summary amp outlook in Section 6

9

2 Numerical Solvers and Models

In this section we briefly introduce the ANSYS Fluent as the CFD solver for melt pool heat transfer the ANSYS Mechanical as the mechanical solver for RPV failure and PECM as an efficient alternative for melt pool heat transfer that are heavily used in the following sections

21 ANSYS Fluent

In the fluid mechanics the fluid is treated as a continuum which is generally described by the conservative mass momentum and energy equations [40]

120597120588

120597119905+ 120571 sdot (120588) = 119878119898 (5)

120597(120588)

120597119905+ 120571 sdot (120588) = minus120571119901 + 120571 sdot (120591) + 120588 + (6)

120597120588119864

120597119905+ 120571 sdot ((120588119864 + 119901)) = 120571 sdot (119896120571119879 minussumℎ119895119869119895

119895

+ (120591 sdot )) + 119878ℎ

(7)

Computational Fluid Dynamics (CFD) studies topics on the numerical solutions of above equations Typical methods are the Finite Volume Method (FVM) Finite Difference Method (FDM) and Finite Element Method (FEM) as well as particle methods that are mesh-free eg SPH and MPS

ANSYS Fluent is a general purpose CFD code which solves above equations with FVM Turbulence modelling and multiphase flows are supported Turbulence can be modelled with RANS LES or hybrid LESRANS Customizations eg adding sourcesink time or spatial dependent boundary conditions are realized through the user-defined functions (UDF) which is programmed in C language

22 ANSYS Mechanical

In the continuum mechanics of solids three sets of variables are considered to describe the structure performance namely element displacement field strain field and stress field (119906 ε σ) A general 3-dimensional problem is described by 3 sets of equations the equilibrium equations (Eq (8)) strain-displacement relationship (Eq (9)) and constitute equations (Eq (10)) [41] For Eq (8) the reciprocal theorem of shear stress also gives τ119894119895 = τ119895119894 where 119894 119895 = 119909 119910 119911

indicating the stress tensor being symmetric For very simple applications eg 1D beam it can be solved analytically For complex problems where analytical solutions are not available approximation methods eg Galerkin weighted residual method principle of virtual work and principle of minimum potential energy would be used

10

partσ119909119909part119909

+partτ119909119910

part119910+partτ119909119911part119911

+ 119909 = 0

partτ119910119909

part119909+partσ119910119910

part119910+partτ119910119911

part119911+ 119910 = 0

partτ119911119909part119909

+partτ119911119910

part119910+partσ119911119911part119911

+ 119911 = 0

(8)

ε119909119909 =part119906

part119909 휀119910119910 =

120597119907

120597119910 휀119911119911 =

120597119908

120597119911

120574119909119910 =120597119906

120597119910+120597119907

120597119909 120574119910119911 =

120597119908

120597119910+120597119907

120597119911 120574119911119909 =

120597119908

120597119909+120597119906

120597119911

(9)

ε119909119909 =1

119864[σ119909119909 minus μ(σ119910119910 + σ119911119911)]

ε119910119910 =1

119864[σ119910119910 minus μ(σ119909119909 + σ119911119911)]

ε119911119911 =1

119864[σ119911119911 minus μ(σ119909119909 + σ119910119910)]

γ119909119910 =1

119866τ119909119910 γ119910119911 =

1

119866τ119910119911 γ119911119909 =

1

119866τ119911119909

(10)

ANSYS Mechanical [42] is a mechanical engineering software that uses finite element analysis (FEA) for structural analysis The principle of virtual work is adopted in the solver [43] The idea of this principle is that a virtual (very small) change of the internal strain energy must be offset by an identical change in external work due to the applied loads which reads

δ119880 = δ119881 (11) where 119880 is the strain energy or the internal work 119881 the external work and δ the virtual operator The virtual internal work can be expressed as

δ119880 = int δεσd(vol)T

vol

+int 119886119903119890119886119891

δ119908119899119879σ119889(119886119903119890119886119891) (12)

where 휀 is the strain vector 120590 stress vector vol volume of element 119908119899 motion normal to the surface 119886119903119890119886119891 area of the distributed resistance The virtual

external work is contributed from three parts the inertial effects the pressure force on the surface and nodal forces applied to the element which then can be expressed as

δ119881 = minusint 119907119900119897

δ119908119879119865119886

119907119900119897119889(119907119900119897) + int

119886119903119890119886119901

δ119908119899119879119875119889(119886119903119890119886119901)

+ δ119906119879119865119890119899119889

(13)

11

where 119908 is the vector of displacements 119865119886 acceleration force vector 119875 applied pressure vector 119886119903119890119886119901 area over which pressure acts and 119865119890

119899119889 nodal

forces applied to the element

Plugging Eq (12) and (13) into (11) would result in the equilibrium equation Since the virtual displacement can be arbitrary the final equations used to describe the equilibrium of an element is reduced to Eq (14) with multiple operations

([119870119890] + [119870119890119891])119906 minus 119865119890

119905ℎ = [119872119890] + 119865119890119901119903 + 119865119890

119899119889

(14)

where [119870119890] [119870119890119891] 119865119890

119905ℎ [119872119890] and 119865119890119901119903 are the element stiffness matrix

foundation stiffness matrix thermal load vector mass matrix acceleration vector and pressure vector respectively It represents the equilibrium equation on a one element basis Alternatively using a minimum total potential energy principle could result in a consistent expression to the principle of virtual work [41]

Customizations eg on material properties are realized through the user programmable features (UPF) which is programmed in FORTRAN language

23 PECM

The PECM [28] [29] developed in KTH is a correlation-based method to model the melt pool heat transfer It is implemented in ANSYS Fluent through UDF and has been validated against many experiments such as ACOPO COPO LIVE etc In the PECM the flow velocities 119906119909 119906119910 and 119906119911 are replaced by characteristic

velocities 119880119909 119880119910 and 119880119911 respectively in order to eliminate the necessity of solving

the Navier-Stokes equation

The characteristic velocities are calculated using heat transfer correlations

119880119906119901 =α

ℎ119901119900119900119897times (119873119906119906119901 minus

ℎ119901119900119900119897

ℎ119906119901) (15)

119880119889119899 =α

ℎ119901119900119900119897times (119873119906119889119899 minus

ℎ119901119900119900119897

ℎ119889119899) (16)

119880119904119889 =α

ℎ119901119900119900119897times (119873119906119904119889 minus

ℎ119901119900119900119897

119882119901119900119900119897) (17)

where ℎ119901119900119900119897 is the melt pool depth ℎ119906119901 the thickness of well mixed layer of the

pool ℎ119889119899 the thickness of lower stratified region of melt pool 119882119901119900119900119897 the width of

pool 120572 the thermal diffusivity The PECM employs reduced characteristic velocities to describe mushy zone convection heat transfer For internally heated pool natural convection the Steinberner-Reineke correlations (Eq (18)~(19)) [44] are employed The profile of sideward 119873119906 is described using Eckertrsquos type correlation (Eq (20)) for a vertical boundary layer where the local Rayleigh

number at pool height 119910 is defined as 119877119886119910 =119892βΔ1198791199103

αν

12

119873119906119906119901 = 03451198771198861198941198990233

(18)

119873119906119889119899 = 13891198771198861198941198990095

(19)

119873119906119904119889 = 050811987511990314 (

20

21+ 119875119903)

minus14

11987711988611991014

(20)

As the characteristic velocities are calculated the energy equation is the only transport equation that needs to be solved and can be expressed as follows

partρ119862119901119879

part119905+ nabla(ρ119862119901119932119879) = 119896nabla2119879 + 119876119907 (21)

where from left to right are the transient term convection term diffusion term and volumetric heat source respectively The convection term then can be calculated explicitly with use of values in the last time step and then can be treated together with the volumetric source term Therefore the PECM finally simplifies the energy equation to a conduction equation which can be easily solved and as a result will significantly increase the computational efficiency in simulations

In addition solidificationmelting process can also be modelled and the temperature-dependent physical properties are supported

13

3 Creep Modelling of 16MND5 Steel

Creep deformation becomes significant when a component is subjected to high temperature and stress Modelling of creep behaviour of the RPV is paramount since it is considered as a main mechanism leading to the RPV failure In this section we present the development of a three-stage creep model (called lsquomodified theta projection modelrsquo ) for the French RPV steel 16MND5 This model was successfully implemented into the ANSYS Mechanical code

31 Creep and Creep Modelling

Creep is an irreversible time-dependent inelastic deformation process Diffusion and dislocation are the two molecular mechanisms of creep The former normally occurs due to the presence of vacancies within the crystal lattice (or the grain) while the latter occurs if there is a slip by the passage of crystal dislocations [45] There are two types of diffusion creep [46] Coble creep (favoured at lower temperatures) if the diffusion paths are predominantly through the grain boundaries and Nabarro-Herring creep (favoured at higher temperatures) if through the grains themselves Dislocation creep tends to dominate at high stresses and relatively low temperatures Dislocations can move by gliding in a slip plane which is a process that requires little thermal activation But the rate-determining step for their motion is often a climb process which also requires diffusion and is thus a time-dependent process and favoured by higher temperatures Creep behaviour is highly affected by temperature and stress generally creep is more significant under higher temperature and stress

Creep curves describe the creep strain development with time Figure 3 shows a creep curve under constant temperature and stress which typically experiences three creep stages primary stage where the creep strain rate ε119888119903 (the change of creep strain ε119888119903 in unit time) decreases with time secondary stage where the creep strain rate almost keeps constant and tertiary stage where the creep strain rate drastically increases with time

Figure 3 Creep curve under constant temperature and stress

14

The built-in creep models in commercial software eg ANSYS [47] and ABAQUS [48] in default only cover the primary andor the secondary stage(s) Table 1 [47] gives the default creep models supported in ANSYS for example Possible reason why the tertiary stage is not modelled might be that for most industrial applications the failure analysis is beyond interest and therefore the built-in creep laws in the commercial software was supposed to be sufficient [49] Though the two codes offer the interfaces through either ANSYS UPF or ABAQUS user subroutines to support more general user-defined creep laws eg complex models that also include the tertiary creep stage

Table 1 Default creep models in ANSYS Mechanical

Creep model Equations Type

Strain hardening

ε119888119903 = 1198621σ1198622ε119888119903

11986231198901198624119879 1198621 gt 0 Primary

Time hardening ε119888119903 = 1198621σ11986221199051198623119890minus1198624119879 1198621 gt 0 Primary

Generalized exponential

ε119888119903 = 1198621σ1198622119903119890minus119903119905 r = 1198625σ

11986231199051198623119890minus1198624119879 1198621 gt 0

1198625 gt 0

Primary

Generalized Graham

ε119888119903 = 1198621σ1198622(1199051198623 + 1198624119905

1198625 + 11986261199051198627)119890minus1198628119879 1198621 gt 0 Primary

Generalized Blackburn

ε119888119903 = 119891(1 minus 119890minus119903119905) + 119892119905 119891 = 1198621119890

1198622σ 119903 = 1198623(σ1198624)1198625 119892 = 1198626119890

1198627σ

1198621 gt 0

1198623 gt 0

1198626 gt 0

Primary

Modified time hardening

ε119888119903 = 1198881σ11988821199051198883+1(1198883 + 1) 1198621 gt 0 Primary

Modified strain hardening

ε119888119903 = 1198621σ1198622[(1198623 + 1)ε119888119903]

1198623119890minus1198624119879 1198621 gt 0 Primary

Generalized Garofalo

ε119888119903 = 1198621[119904119894119899ℎ(1198622σ)]1198623119890minus1198624119879 1198621 gt 0 Secondary

Exponential form

ε119888119903 = 1198621119890σ1198622119890minus1198623119879 1198621 gt 0 Secondary

Norton ε119888119903 = 1198621σ1198622119890minus1198623119879 1198621 gt 0 Secondary

Combined time hardening

ε119888119903 = 1198621σ11986221199051198623+1119890minus1198624119879(1198883 + 1)

+ 1198625σ1198626119905119890minus1198627119879

1198621 gt 0

1198625 gt 0

Primary+ Secondary

Rational polynomial ε119888119903 = 1198621

partε119888part119905 ε119888 =

119888119901119905

1 + 119901119905+ ε119898119905

ε119898 = 1198622101198623σσ1198624 119888 = 1198627ε

1198628σ1198629

119901 = 11986210ε11986211σ11986212

1198622 gt 0 Primary+ Secondary

Generalized time hardening

ε119888119903 = 119891119905119903119890minus1198626119879 119891 = 1198621σ + 1198622σ2 + 1198623σ

3 119903 = 1198624 + 1198625

Primary

32 A Modified Theta Projection Model

321 Original theta projection model and its variations

The original theta projection model [50] was first proposed to accurate predict long-term creep and creep-fracture properties It assumes that a creep curve under constant temperature and stress can be described as

15

휀119888119903 = 1205791(1 minus 119890minus1205792119905) + 1205793(119890

1205794119905 minus 1) (22)

where 휀119888119903 is the creep strain 119905 the time and 1205791~1205794 the model parameters Figure 4 illustrates how the first term and second term on right hand side develop with time and constitute a creep curve which has obvious primary secondary and tertiary stages Model parameters 120579119894(119894 = 1234) are temperature and stress dependent and are described with the following expression

119897119899thinsp120579119894 = 119886119894 + 119887119894120590 + 119888119894119879 + 119889119894120590119879 (23)

where 119886119894 119887119894 119888119894 and 119889119894 ( 119894 = 1234 ) are constants It is noted [45] that this equation is suitable when the temperature range of experiment data sets is very narrow such that minus1119879 is almost proportional to 119879 These constants (119886119894 119887119894 119888119894 and 119889119894 (119894 = 1234)) are the final model parameters which are generated with two fitting procedures

(i) Fitting the experimental creep curves to Eq (22) generates several groups of 1205791~1205794 for each creep curve under certain temperature and stress We denote (119879119895 120590119895 1205791119895 1205792119895 1205793119895 1205794119895 ) as parameters for the 119895119905ℎ

experimental curve under temperature 119879119895 and stress 120590119895

(ii) Fitting 1205791 to Eq (23) as a function of temperature and stress with the obtained data ( 119879119895 120590119895 1205791119895 ) from ( 119879119895 120590119895 1205791119895 1205792119895 1205793119895 1205794119895 ) generates

parameters 1198861 1198871 1198881 and 1198891 Repeat this fitting procedure for 1205792 1205793 and 1205794 to get parameters 119886119894 119887119894 119888119894 and 119889119894 (119894 = 234)

Variations of the model were developed and applied to other different materials to better accommodate the experimental data which are all then called a lsquomodified theta projection methodrsquo independently Examples are models in [51] for lowhigh alloy ferritic austenitic steels Al-alloys Al-matrix composites [52] for Nickel-based Super-alloy [53] for pure metals and class M alloys with intermediate range of stress (40-70 of the yield stress) and temperature (40-60 of the melting temperature)

Figure 4 Illustration of theta projection method [54]

16

322 A modified theta projection model

The new modified theta projection model we proposed [54] was mainly adapted from the original theta projection method [50] and the modified version [51] The new model describes the creep curve with the following equation

휀119888119903 = 1205791(1 minus 119890minus1205792119905) + 120579119898119905 + 1205793(119890

1205794119905 minus 1) (24)

An additional term 120579119898119905 is seen compared with Eq (22) in the original method The reason for adding this term is that the 16MND5 steel shows fairly long secondary stages in many experimental creep curves and the creep curves can be better fitted by Eq (24) than Eq (22) As the structure undergoes different creep process under different temperature and stress the model parameters θ119894 (119894 =1~4 and 119898) are also temperature and stress dependent

The main difference compared with the original and all other modified projection models is that numerical interpolation is used to replace the second numerical fitting in previous models In other words instead of fitting the model parameters as functions of stress and temperature with Eq (23) the distributions of the model parameters are assumed to be piecewise in the temperature-stress space and they will be interpolated directly from known parameters obtained from experiments The consideration is that the experimental data ([16] [17]) for 16MND5 covers a wide temperature range varying from 600 degC to 1300 degC and a wide stress load range varying from 08 MPa to 248 MPa while Eq (23) in the existing methods is based on a narrow temperature span Poor parameter fitting results would be expected if we directly fit the parameters 1205791~1205794 with Eq (23) as used in the original methods

The creep model was developed using the following steps

(i) Fitting Each creep curve can be represented by a group of model parameters [1205791 1205792 120579119898 1205793 1205794] with respect to Eq (24) by fitting the creep curves to Eq (24) All parameters of the 32 experimental curves form an

initial dataset of the model parameters [119879119895 120590119895 1205791119895 1205792119895 120579119898119895 1205793119895 1205794119895] where j (0ltjle32) is the jth creep curve

(ii) Interpolation Parameters of a creep curve under any [119879 σ] are directly interpolated from the dataset

(iii) Parameter adjustment The dataset was further updated to satisfy the monotonicity assumption we have taken that creep strain monotonically increases with temperature and stress

(iv) Different hardening models for dynamic loads

Step (i) was done using MATLAB In the following steps (ii)-(iv) are described in detail

Model parameters for new creep curves are interpolated from the dataset over the 2D 119879 minus 120590 space The interpolation is divided into two sub-interpolation steps interpolate over stress with fixed temperature and then interpolate over temperature with fixed stress We take the interpolation of model parameters under (119879prime σprime) for example In the first interpolation we first find out the two

17

temperatures 119879119896119899119900119908119899119904119898119886119897119897 and 119879119896119899119900119908119899119897119886119903119892119890 most close to 119879prime from the experimental

temperature set 600 700 800900 1000 1100 1200 1300 Under each temperature there are four sets of model parameters in the dataset corresponding to four different stress loads Then the first interpolation is done over stress load from the known dataset of model parameters under each

temperature to output the parameters for (119879119896119899119900119908119899119904119898119886119897119897 120590rsquo) and (119879119896119899119900119908119899119897119886119903119892119890 120590rsquo)

In the second interpolation the model outputs parameters for (119879rsquo 120590rsquo) from

(119879119896119899119900119908119899119904119898119886119897119897 120590rsquo) and (119879119896119899119900119908119899119897119886119903119892119890 120590rsquo) For 120579i(119894 = 12 3 4) we assume a logarithm

relation between 120579119894 temperature 119879 and stress 120590 which is similar to Eq (23)

119897119899120579119894 = (119886 + 119887120590)(119888 + 119889119879) (25)

It reduces to 120579119894 = 11986011198901198871σ with fixed temperature in the first interpolation and to

θ119894 = 11986021198901198872T with fixed stress in the second interpolation The interpolation

expression for θ119894 = 11986011198901198871σwith two known points (1205901 1205791198941) and (1205902 1205791198942) is given

by Eq (26) The interpolation expression for θ119894 = 11986021198901198872T with two known points

(1198791 1205791198941) and (1198792 1205791198942) is given by Eq (27)

120579119894 = 120579i1

120590minus12059021205901minus1205902120579i2

1205901minus1205901205901minus1205902 (26)

120579119894 = 120579i1

119879minus11987921198791minus1198792120579i2

1198791minus1198791198791minus1198792 (27)

Similarly we assume the following relation between 120579119898 temperature 119879 and stress 120590

120579119898 = 119860120590119899119890minus119887 119879frasl (28)

The interpolation is done over the stress with 120579119898 = 1198603120590n by fixing temperature

and then over temperature with 120579119898 = 1198604119890minus1198874119879 by fixing stress The interpolation

expression over stress for 120579119898 = 1198603120590119899 with two known points (1205901 1205791198981) and

(1205902 1205791198982) would be given by Eq (29) The interpolation expression over

temperature for 120579119898 = 1198604119890minus1198874119879 with two known points (1198791 1205791198981) and (1198792 1205791198982)

would be given by Eq (30)

120579119898 = 119890119909119901(119897119899120590 minus 11989711989912059011198971198991205902 minus 1198971198991205901

1198971198991205791198982 +1198971198991205902 minus 119897119899120590

1198971198991205902 minus 11989711989912059011198971198991205791198981) (29)

120579119898 = 120579m2(11198791minus1119879) (

11198791minus11198792)fraslsdot 120579m1

(1119879minus11198792) (

11198791minus11198792)frasl (30)

In this model we also take the assumption that the creep strain monotonically increases with temperature and stress loads1 Or specifically with the same creep time the structural component under higher temperature or stress experiences

1 Intuitively materials deform more under higher stress and temperature This is also mostly observed in tensile experiments Though some exceptions exist one instance of which is the effect of phase change some materials under certain temperature around the phase change temperature may show much weaker structural performance Here we just simply assumed that the creep strain monotonically increases with temperature and stress

18

higher creep strain than that with lower temperature or stress (the creep curve under higher temperature or stress should be above the curve with lower temperature or stress) When generating the parameter dataset (Step (i)) we already kept the trend that under the same temperature cases under higher stress loads would have larger creep model parameters and give higher creep strains This trend would also hold for the interpolated temperatures which thus guarantees that creep strain monotonically increases with stress load under same temperature Then the question becomes if creep strain also monotonically increases with temperature given fixed stress However the answer could not always be a yes especially in the extrapolation range of the stress load Figure 5(a) illustrates an example of the interpolation of model parameters under (119879119872 σ119872) The red dots and blue dots are known parameters from the parameter dataset for corresponding high temperature 119879119867 and low temperature 119879119871 respectively The green dots (119879119867 σ119872) and (119879119871 σ119872) are interpolated from the red and blue dots in the first sub-interpolation respectively The black dot (119879119872 σ119872) is interpolated from the green dots in the second sub-interpolation It is seen that when σ119872 lt σ119862 the value of (119879119871 σ119872) is higher than that of (119879119867 σ119872) It means the higher 119879119872 is the smaller the model parameter will be ie the model would predict larger creep strain for lower temperatures which violates our monotonicity assumption One simple solution used here as shown in Figure 5(b) is to minorly adjust the values of the four red and blue dots such that the intersection point C can be sufficiently small and the situation σ119872 lt σ119862 would barely occur in the extrapolation range

Figure 5 Illustration of model parameter interpolation (a) before and (b) after parameter adjustment [54]

A corresponding parameter adjustment process was introduced to verify that the model predicts higher creep strain under higher temperature under fixed stress Figure 6 illustrates the general verification process for any (119879 120590) in the interested load range the creep strain of (119879 + 120575119879 120590) should be larger than that of (119879 120590) Otherwise parameters would be minorly adjusted and the verification process be repeated This process would be iterated several times until the requirement is reached The achieved verification shows

Stress

σC

High temperature TH

MPa

a

0

Model

par

amet

er θ

Low temperature TL

MPa

a

0

Model

par

amet

er θ

High temperature TH

Low temperature TL

σC

Stress

C C

(a) (b)

σM

σM

19

(i) For any (119879 120590) in the temperature range 273~ 1073 K with an interval of 1 K and the stress range 1 ~260 MPa with an interval of 1 MPa its creep strain is higher than that of (119879 minus 1119870 120590) for the first 12000000 s

(ii) For any (119879 120590) in the temperature range 1073~1673 K with an interval of 1 K and the stress range 1 ~80 MPa with an interval of 1MPa its creep strain is higher than that of (119879 minus 1119870 120590) for the first 12000000 s

(iii) For any (119879 120590) in the stress range 01~26 MPa with a smaller interval of 01 MPa for the full temperature range 273~1673 K with an interval of 1 K its creep strain is higher than that of (119879 minus 1 120590) for the first 12000000 s

Above verification covers a wide range of temperature and stress especially where the stress and temperature are low and extrapolation is used Further verification to cover even higher stress or temperature ranges is not considered as creep increases drastically under such high stress and temperature and the results would become meaningless With these regards we generally think the monotonicity requirement is reached The model parameter dataset was updated along with the adjustment The final dataset was given as Table 2 in Paper I

Figure 6 Parameter adjustment process [54]

Figure 7(a-d) present the model predictions of selected creep curves in ANSYS Mechanical (labelled as lsquoANSYSrsquo) compared with experimental curves (labelled as lsquoREVISArsquo) at temperature 600 800 1000 and 1200 degC respectively The predicted curves proved to clearly show the three-stage creep process and agreed well with the experimental creep curves

Initial parametersdataset

θ1 θ2 θm θ3 θ4 from creep

curve-fitting

Parameters interpolation

Is the monotonicity

assumption satisfied

IsMon= =1

Final model parameters

Adjust the

parametersdataset

θ1 θ2 θm θ3 θ4

No

Yes

IsMon=1 a flag is monotonic achieved

For σ in the stress range in MPa

For T in range Tstart Tend in K

For t in range 112000000 in s

If ε(σ T+1t)lt ε(σTt) IsMon=0

Iteration loop of

parameter adjustment

20

Figure 7 Selected model predictions compared with experimental creep curves [54]

Instead of being constant temperature and stress loads also dynamically change with time in real cases Hardening models determine how creep strain rate reacts to the change of loads during a creep process The time hardening model and strain hardening model2 are supported in our model implementation in ANSYS Mechanical which are two mostly used hardening models

Time hardening assumes that the following creep strain rate after a load change depends only on the current creep time from the beginning of the creep process (and also temperature and stress if used in the creep model) and ignores the current creep strain Strain hardening assumes that the creep strain rate depends only upon the current creep strain (and also temperature and stress if used in the creep model) [55]

2 The time hardening and strain hardening here are more general terms than the specific time hardening creep model and the strain hardening creep model mentioned in Table 1

(a) 600 degC (b) 800 degC

(c) 1000 degC (d) 1200 degC

21

Eq (24) in default is a time hardening model as creep strain is directly described by the time 119905 Eq (31) gives the creep strain increment 119889휀119888119903 at time 119905 during time interval 119889119905

119889휀119888119903 = (12057911205792119890minus1205792119905 + 120579119898 + 12057931205794119890

1205794119905)119889119905 (31)

The strain hardening model is derived from Eq (24) in an implicit form Specifically when temperature and stress load change to (119879 120590) at the time 119905 and current creep strain 휀1198881199030 the model will get the creep curve of constant (119879 120590) and

output how this curve will react under the current strain 휀1198881199030 Since the creep curve shows creep strain monotonically increases with time given the creep strain 휀1198881199030 under creep curve of (119879 120590) we can uniquely find this reference time

119905119903119890119891 satisfying Eq (32) One should notice that the reference time 119905119903119890119891 can be

different from the current physical time 119905

휀1198881199030 = 1205791(1 minus 119890minus1205792119905119903119890119891) + 120579119898119905119903119890119891 + 1205793(119890

1205794119905119903119890119891 minus 1) (32)

where all 120579119894(119894 = 1~4119898) are model parameters of the new creep curve (119879 120590) It can be solved for 119905119903119890119891 with methods eg Newton-Ramphson [56] Then the creep

strain increment at 119905119903119890119891 during time interval 119889119905 is given as

119889휀119888119903 = (12057911205792119890minus1205792119905119903119890119891 + 120579119898 + 12057931205794119890

1205794119905119903119890119891)119889119905 (33)

Therefore combining Eq (32) and (33) gives the creep strain rate at creep strain 휀1198881199030 In summary when loads are changed to (119879 120590) with current creep strain 휀1198881199030

the model traces the creep curve of (119879 120590) and calculates the creep strain rate of this curve at strain 휀1198881199030

One calculation was done to demonstrate the performance of the hardening models The stress load was set to 151 MPa and kept unchanged A step temperature load was applied as the orange dash curve shows in Figure 8 it was 900 degC during 0-36000 s and suddenly jumped to 1000 degC at 36000 s which was then kept till the end of the simulation The blue solid curve (the part before the temperature change was identical to and therefore covered by the green solid curve in the figure) and blue dash curve are the creep curves under constant temperature T=900 degC and T=1000 degC respectively The green solid curve and green dash curve represent the model predictions considering this temperature change with time hardening model and strain hardening model respectively Before this change the creep curves with both hardening models were identical to the creep curve of the T=900 degC till point A After that the creep strain rate then shifted from Point A to Point B (which had the same time with A) in the time hardening model and behaved the same as the curve T=1000 degC as shown in the rest part of the green solid curve In the strain hardening model the creep strain rate shifted from Point A to Point C (which had the same creep strain with A) and behaved the same as the curve T=1000 degC as shown in the rest part of the green dash curve The large difference between two models was partially caused by the drastic temperature jump In real cases the difference would be less significant as the temperature changes more mildly Overall this prediction is exactly the expected behaviors of the time hardening model and strain hardening model

22

Figure 8 Time hardening and strain hardening in ANSYS Mechanical [54]

23

4 Coupled Thermo-Fluid-Structure Analysis of RPV Failure

As described in Section 13 RPV failure is coupled with melt pool heat transfer leading it into a transient thermo-fluid-structure coupled problem The goal of this section is to develop and validate a reliable coupling approach which supports transient coupled thermo-fluid-structure analysis as well as advanced models and computational platforms so as to improve fidelity and facilitate implementation The approach was validated against the FOREVER-EC2 experiment and then applied to an RPV failure analysis of a Nordic BWR As the approach adopts the volume loads mapping for load transfer comparative analysis was also performed against the coupling approach using surface loads mapping Future work is considered in analysis of penetration failures (IGTs and CRGTs) of a BWR RPV since the approach is flexible to complex geometries Results of Paper II ampIII are jointly presented in this section

41 Thermo-Fluid-Structure Coupling Approach

Considering the involved phenomena a coupling approach should at least address three issues (i) melt pool heat transfer (ii) RPV deformation and (iii) the interaction between melt pool heat transfer and vessel deformation A two-way strong coupling considering both the transient thermal load from melt pool on vessel and the influence of vessel deformation on melt pool heat transfer involves complex data exchange and is still full of challenges Therefore the new coupling approach is realized in a one-way manner with the assumptions that the influence of vessel deformation on melt pool heat transfer can be ignored and thus only the thermal loads from melt pool to vessel are considered It is based on the considerations that on one hand the deformation is a slow process regarding the relatively long failure time and on the other hand it was pointed out [57] that the physics of natural convection heat transfer in internally heated liquid pools is not sensitive to the geometrical factors based on their calculation results and experimental data from [58] [59] as well as previous experiments in semicircular and square cavities eg [44] [60] In addition one-way coupling is also computationally more efficient especially when heat transfer turbulence structural non-linearity and large-scale geometries are involved The developed approach is based on the ANSYS platform which generally consists of three parts with respect to the three issues respectively

(i) conjugate heat transfer of the melt pool and vessel using ANSYS Fluent (ii) transient thermal loads transfer from ANSYS Fluent to ANSYS

Structural (the ANSYS Mechanical in ANSYS Workbench) (iii) structural analysis of the vessel in ANSYS Structural

ANSYS Structural in default only supports direct transfer of constantsteady-state spatial loads from ANSYS Fluent Transfer of transient loads is realized with the extension tool FSI Transient [61] It is a pre-processing tool (with good documentation) developed by a third-party CAE vendor EDRTM MEDESO [62] to map transient CFD results (heat transfer coefficient temperature and surface pressure) to thermal or structural analysis including the definition of load steps

24

This new procedure proves to be straightforward and reduces efforts in the load transfer

Since structural performance of RPV is highly dependent on temperature the transfer of transient temperature loads from conjugate heat transfer to mechanical solver is a core step of the coupling framework There are two general approaches to realize the transfer as shown in Figure 9 Both approaches contain three parts a conjugate heat transfer analysis for fluid and structure where flow and heat transfer are coupled a mechanical analysis for the structure and a load mapping process that transfers the transient loads The conjugate heat transfer part can be solved with any CFD solver and is treated in the same manner in both approaches In the first approach (Figure 9(a)) transient volume thermal loads are transferred and mapped into the mechanical simulation domain The structure domain reads the essential thermal loads from the mapping after which a structure analysis of this domain is performed In the second approach (Figure 9(b)) transient surface thermal loads are mapped onto the boundary surfaces of structure domain as thermal boundary conditions Then a thermo-structure coupled analysis of the structure domain in the mechanical part is performed To distinguish them we call the former approach volume loads mapping (VLM) and the latter surface loads mapping (SLM) In the RPV failure analysis the fluid and structure domains correspond to the melt pool and RPV respectively Examples of these methods are the studies of [63] and [30] using VLM and [35] using SLM The VLM is adopted in current approach

Figure 9 Frameworks of the thermo-fluid-structure coupling approaches (a) volume loads mapping (b) surface loads mapping [64]

fluid structure structure

fluid structure structure

Conjugate heat transfer problem Mechanical problem

(structure)

Conjugate heat transfer problem Mechanical problem

(thermo-structure coupled)

Transient volume

loads mapping

(VLM)

Transient surface

loads mapping

(SLM)

(a)

(b)

25

42 Numerical Modelling

421 Conjugate heat transfer

In the conjugate heat transfer analysis main focus is put on the modelling of melt pool heat transfer Within the ANSYS Fluent solver two options are supported CFD with turbulence modelling and PECM CFD is the choice when the model goes to detail and more mechanistic analysis are needed while PECM provides an alternative that is faster and possibly more suitable for large-scale problems

Within the scope of RANS models the SST (Shear Stress Transport) model [65] is selected as the turbulence model in present study This model is a combination of k-ε model and k-ω model The k-ω formulation is used in the inner parts of the boundary layer which makes the model directly usable all the way down to the wall through the viscous sub-layer It also switches to k-ε model in the free-stream and thereby avoids the common k-ω problem that the model is too sensitive to the inlet free-stream turbulence properties The considerations on choosing this model are

bull Several two-equation low-Re k-ε models have been proved in [66] not good enough for melt pool heat transfer Therefore these mentioned two-equation models as well as other one-equation models are not considered

bull SST is highly recommended [67] for simulating most industrial heat transfer problems

bull Convincing results ([68] [69]) using SST have been obtained in capturing the thermal behaviour of the natural convection in an internally heated fluid Especially good agreements were achieved between the predicted heat flux distribution along the vessel and experimental data which would be one of the most important factors to estimate model performance in this problem

The description of PECM as an effective model for melt pool heat transfer is given in Section 23

422 Mechanical analysis

The material for the RPV failure analysis considered in this study is the French RPV steel 16MND5 The three-stage modified theta projection model in Section 3 is used as the creep model Since the definition of an RPV failure criterion is not universal (it can be the time when damage variable reaching 1 or either specified value of creep strain creep strain rate or total deformation is reached) we do not explicitly define a failure criterion Instead the simulation is kept running until the creep converge criterion cannot be satisfied even with the defined minimum time step (indicating a fairly large creep strain rate) and the simulation will be automatically terminated The time the simulation stops is taken as the failure time This treatment implicitly implies a criterion of a large creep strain rate We refer to [17] for other material properties Youngrsquos modulus and thermal conductivity are temperature dependent Multilinear Isotropic Hardening (MISO) is used as the plasticity model

26

43 Validation against the FOREVER-EC2 Experiment

In this section the validation against the FOREVER-EC2 experiment is presented (indicated as lsquoVLMrsquo in the following) In addition the simulation results using the SLM approach (indicated as lsquoSLMrsquo in the following) is also included for comparison purpose

431 FOREVER-EC2 experiment

The FOREVER-EC2 experiment [70] is one of the FOREVER experiments that achieved a vessel failure Figure 10 shows the experimental setup A 110 scale pressure vessel with a wall thickness of 15 mm and an outer diameter of ~400 mm was employed The lower head was made of 16MND5 steel Since it was difficult to use prototypical corium materials in the experiment the binary salt 30 CaO-70 B2O3 was employed as simulant To generate the internal heat which corresponds to the nuclear decay heat a specially designed heater was used The power input was maintained ~38 kW during the experiment such that maximum melt pool temperature would be ~1300 degC The pressure inside the vessel was maintained at ~25 MPa Thermocouples were used to measure the melt pool and vessel wall temperatures and linear displacement transducers used to measure the vessel wall displacements The designed experimental conditions were reached at experimental time 11670 s Vessel failure was observed about 12510 s later (at experiment time 24180 s)

Figure 10 Scheme of the experimental setup [22]

27

Figure 11 Computational domains and boundary conditions (a) conjugate heat transfer (b) mechanical analysis [71]

Computational domains and boundary conditions are indicated in Figure 11 SST turbulence modelling was used for pool heat transfer The simulations started at the moment when experimental conditions were reached (experiment time 11670 s) So simulation time 0 s corresponds to experiment time 11670 s

432 Conjugate heat transfer

Figure 12 shows the temperature fields at simulation time 0 s 1225 s 2475 s 4975 s and 9975 s All the time points in this part refer to simulation time The field at time 0 s (Figure 12(a)) was the initial temperature that was reconstructed based on the thermocouple readings The temperature increased with time and reached a steady state when the maximum temperature reached ~1607 K Figure 12(b-f) show clear thermal stratification which is an expected phenomenon of natural convection

Internal pressure

25 bar

Internal pressure 25e6-2500times981timesy Pa

External pressure

1 bar

Zero displacement

y=0 m

Gravity 400

mm

R=203 mm

15 m

m

Tamb

=323 K radiation with ε = 08

+ convection with h=10Wm2K

Tamb

=800 K radiation with ε=02

(a) (b)

Tamb

=800 K radiation with ε=08

+ convection with h=10 Wm2K

28

Figure 12 Temperature fields at simulation time (a) 0 s (b)1225 s (c) 2475s (d) 4975 s (e) 7475 s and (f) 9975 s [64]

Figure 13 shows the temperature profiles of the external vessel along the polar angle in comparison with experimental data Regions with a polar angle larger than 90deg are those in the cylindrical part of the vessel It also shows the temperature profile increased with time until reaching a steady state The profile almost did not change after 4975 s which indicates that a steady state of the temperature was reached As time increased the temperature in the lower head region 0~90deg (where the vessel was contacted with the melt) evolved from a relatively flat distribution to a distribution that temperature at high angle was higher than that at low angle by more than 200 K Generally the steady state results agreed well with the experimental data except for the low angle part of vessel lower head and the high angle part of the cylindrical vessel In addition to modelling errors the discrepancy in the low angle part of the lower head may be caused by the crust and gap possibly formed at vessel bottom in the experiment which can reduce the downward heat transfer and result in lower temperature while they cannot be modelled in simulations As to the high angle cylindrical vessel part simplified boundary conditions were applied to the inner and outer vessel wall since it was far from the melt and not of interest which may result in some discrepancy

(a) (b)

(c) (d) (e) (f)

29

Figure 13 Temperature profiles of external vessel surface along the vessel polar angle [64]

433 Mechanical analysis

In the mechanical analysis the VLM predicted a failure time of 23959 s and the SLM predicted a failure time of 22445 s both are close to the experimental failure time 24180 s All the time points in this part refer to the experiment time Figure 14 shows the creep strain fields at failure time Both approaches predicted maximum creep strains at same location where the temperature was high (cf Figure 12) This is related to the fact that creep behavior is highly influenced by temperature Maximum values for both cases were also close to each other 0393 in VLM and 0366 in SLM

Figure 15 presents the total deformation of the vessel bottom point (also called ldquosouth polerdquo in some literature) in the vertical direction Since there was an initial displacement reading of ~355 mm at time 0 s which almost did not change during 0~11670 s (the whole period before the experimental conditions were reached) an initial value of 355 mm was added to the simulation result such that experimental and simulation results had the same total displacement at t=11670 s Similar trend was seen in the VLM SLM and experimental data that the deformation increased with time and experienced a drastic acceleration at the end Generally both simulation approaches agreed well with the experimental data though they both slightly under predicted the value of total deformation The discrepancy may possibly be related to the accuracy of the thermal predictions modelling errors of plasticity and creep and simplified boundary conditions

400

600

800

1000

1200

1400

1600

0 20 40 60 80 100 120 140 160

Tem

per

ature

(K

)

Polar angle (deg)

Experiment

25 s

1225 s

2475 s

4975 s

7475 s

9975 s

Weilding line

30

Figure 14 Creep strain fields at failure time (a) VLM (b) SLM [64]

Figure 15 Total deformation in vertical direction of the vessel bottom point [64] [71]

The wall thickness changes are plotted together with experimental results in Figure 16 The wall thickness change δ119909 is defined as

δ119909 = 119909119890119899119889 minus 119909119900119903119894119892119894119899119886119897 (34)

000

500

1000

1500

2000

2500

11670 13670 15670 17670 19670 21670 23670

Tota

l def

orm

atio

n (

mm

)

Time (s)

ExperimentVLMSLM

(a) (b)

31

where 119909119890119899119889 and 119909119900119903119894119892119894119899119886119897 are the wall thickness at failure time and the original wall

thickness respectively Since 119909119890119899119889 cannot directly be post-processed it was estimated in the following way (cf the subfigure of Figure 16) for the same polar angle θ find the corresponding points Pin (1199091 1199101) and Pout (1199092 1199102) on the inner and outer surfaces of the deformed vessel respectively the distance of Pin and

Pout radic(1199091 minus 1199092)2 + (1199101 minus 1199102)2 is calculated as 119909119890119899119889 at polar angle 120579 Such estimation of the wall thickness is more accurate when the line PinPout is closer to be orthogonal to the deformed local surface Considering the final deformed vessel as shown in Figure 14 the estimation should be fine at regions close to 0deg and 90deg and can be slightly overestimated at medium angle region The VLM curve was compared with the SLM curve and 4 experimental curves measured at vessel left right front and back sides Generally VLM curve agreed well with the experimental and SLM curves wall thickness gradually decreases with angle till around 75deg and then increases till 90deg the maximum thickness change occurred at around 75deg would be around -5 mm The positive values at low angle may be caused by the thermal expansion It also shows that the thickness transitions in experimental curves were smoother than those in both simulation curves while simulation curves showed narrower regions with large thickness changes (at angle 70deg~80deg) Possible reasons might be (i) sharper temperature profile predicted in simulation (ii) input uncertainty and (iii) modelling errors The SLM predicted slightly thinner vessel than the VLM except for the necking region The discrepancy might be caused by various reasons eg the interior vessel temperature can be slightly different between two approaches which would then result in different thermal stress and different deformation the final deformed vessel shapes cannot be exactly the same which also results in different errors in estimating PinPout

Figure 16 Wall thickness changes along the vessel polar angle [64] [71]

-6

-5

-4

-3

-2

-1

0

1

0 10 20 30 40 50 60 70 80 90

Wal

l th

icknes

s ch

ange

(mm

)

Polar angle (deg)

VLMSLMExperiment (left)Experiment (right)Experiment (front)Experiment (back)

32

44 Application to BWR Case

In this section the coupling approach (indicated as lsquoVLMrsquo in the following) was applied to an RPV failure analysis of a BWR A simulation using the SLM coupling approach (indicated as lsquoSLMrsquo in the following) was also performed for comparison purpose

A hypothetical severe accident of a Nordic BWR initiated by an SBO was considered The core materials would melt and migrate downward to the RPV lower head The total amount of corium relocated in the lower head was considered to form a 19 m corium melt pooldebris bed assuming the corium distribution was homogeneous

Figure 17 Computational domains and boundary conditions (a) conjugate heat transfer (b) mechanical analysis [64]

Figure 17(a) shows the computational domain of the conjugate heat transfer PECM was used in this calculation A general mesh with a pool height of 23 m was used to accommodate most of the relocation situations with the same mesh Since the corium only had a height of 19 m the top 04 m region was treated as injected water covering the corium The top water was expected to be boiling and we assumed a uniform temperature of 383 K Under the implementation of PECM this region was damped with artificially high heat capacity and high conductivity such that it would efficiently remove heat from the corium and suppress its own temperature increase For the corium part below water a volumetric heat source 11 MWm3 was implemented as the nuclear decay heat For the in-vessel surfaces a constant temperature of 383K was applied as they are covered by water which can be boiling Without water cooling on the external vessel wall a very small heat loss of 20 Wm2 was applied Melting and solidification of the corium were also considered with a solidus temperature of 2750 K and a liquidus temperature of 2770 K The initial temperature was set as

q=20 Wm2

T=383 K

Qv=11

MWm3

315 bar

315 bar+ hydrostatic

pressure of corium

Gravity

248 bar

Zero

displacement

Water

Corium

(a) (b)

33

1100 K for the corium 383 K for the water and 450 K for the vessel For the mechanical analysis (Figure 17(b)) an inner pressure of 315 bar and an outer pressure of 248 bar were considered For the region covered by corium an extra hydrostatic pressure was added on the vessel wall A 2 m long vessel extension was considered as well as the standard earth gravity

Figure 18 presents the wall temperature profile along the inner vessel surface at 50 s 2500 s 7500 s and 10000 s The temperature increased with time for the vessel surface contacted with corium and the distribution was flat which was due to that large part of the corium was still in solidus status It also shows that for the surface region that contacted the water the temperature was basically unchanged which indicates that the damping treatment indeed suppressed the temperature increment and guaranteed the heat removal upward by water

Figure 18 Temperature profiles of inner vessel wall along the vessel polar angle [64]

The predicted RPV failure time was 9869 s by VLM and 9650 s by SLM Compared with the FOREVER-EC2 case in Section 43 the VLM predicted a slightly later failure time than the SLM in this BWR case too Figure 19(a) compares the maximum creep strain developments with time of both approaches Before 9000 s the predicted values by both approaches are close to each other Afterward the creep acceleration occurred and the strain increased drastically The acceleration of the VLM occurred 200 s later than that of the SLM which agreed with the predicted later failure time Similar trend was observed for the total deformation in vertical direction of vessel bottom point as shown in Figure 19(b) The deformations of both approaches are close except that the acceleration of VLM occurred later than that of SLM Overall results of VLM agreed well with those of SLM

0

500

1000

1500

2000

2500

0 10 20 30 40 50 60 70 80 90 100

Tem

per

ature

(K

)

Polar angle (deg)

50 s

2500 s

5000 s

7000 s

10000 s

34

Figure 19 Results of structural analysis (a) creep strain and (b) total deformation in vertical direction of vessel bottom point [64]

45 Computational Efficiency

Computational efficiency is also an important consideration regarding the performance of numerical approaches The computational efficiencies of both VLM and SLM are analysed and compared in this section As parallel computing technique was used to speed up the simulations the number of parallel cores and corresponding costed computational time for each simulation are given together in Table 2 as a generally reflection of the computational efficiency All the simulations were done in a Lenovo P700 workstation equipped with 2 Intel Xeon E5-2630 v3 processors (8 cores per processor) and 32 G RAM For the simulations of FOREVER-EC2 SLM employed twice the parallel cores and costed 286 more time compared with VLM For the simulations of prototypic BWR case both VLM and SLM employed the same number of parallel cores but VLM only took 13 of the computational time compared with SLM Overall it can be concluded that VLM has higher computational efficiency than the SLM The reason is a structural analysis is performed in the VLM approach while a coupled thermo-structural analysis is performed in the SLM approach (cf Figure 9) a standalone structural analysis is computed faster than a coupled thermo-structural analysis

Table 2 Comparison of computational efficiency [64]

Case FOREVER-EC2 BWR VLM SLM VLM SLM

Number of cores 4 8 4 4 Computational time (hours) 21 27 18 72

35

5 Hybrid Coupling Approach for Efficient RPV Failure Analysis

In this section we propose a hybrid coupling approach for the efficient RPV failure analysis The idea is to replace the ANSYS Fluent in previous coupling approach with a lumped-parameter code for quick estimate of the melt pool heat transfer With the least modelling details (compared with CFD methods and distributed-parameter methods eg PECM) lumped-parameter codes prove to be the most efficient for melt pool heat transfer The new hybrid approach characterizes 1) utilization of efficient models for melt pool heat transfer and 2) the support of detailed structural modelling of RPV deformation allowing studies with different RPV geometric shapes under effects like local thinning due to ablation For studies of which the focus is mainly on structural behaviour of RPV eg [13] [26] while mechanistic simulation of melt pool convection is computationally expensive and unnecessary the lumped-parameter code could help providing reasonable thermal loads

The development of the approach consists of two tasks (i) development of a lumped-parameter code and (ii) its coupling with ANSYS Mechanical to form this hybrid coupling framework In the following we describe both tasks and the corresponding verification and validation processes respectively This section is mainly based on Paper IV

51 A Lumped-parameter Code for Melt Pool Heat Transfer

A lumped-parameter code transIVR was developed It features (i) quick estimate of transient melt pool heat transfer with reasonable accuracies and (ii) two-dimensional representation of heat conduction in the RPV wall facilitating the coupled analysis of the RPV failure under thermal loads The following assumptions are taken in the transIVR modelling

bull The crust between the oxide pool and metal pool is thin

bull The heat flux has uniform distributions on the bottom top and side surfaces of metal pool

bull Heat source only exists in oxide layer

bull Latent heat of fusion is ignored

Figure 20 illustrates the modelled three domains oxide layer metal layer and the RPV wall In addition the internal structure inside the vessel can also be considered as a radiative body to the top metal surface as in [72] An internal volumetric heat source generated by nuclear decay heat is considered in the oxide layer Heat in oxide layer transfers downward to the RPV wall and upward to the top metal layer Heat in metal layer can be transferred sideward to the RPV wall and upward to the surrounding internal structure through radiation (or to water if inside water flooding is applied) The heat transferred to RPV will be removed from outer RPV surface Possible formations of oxide crust in the top and bottom surface are illustrated as the black regions Possible formation of metal crust on the top surface is shown as the grey region The code automatically detects the

36

existence of the crust on oxide top and metal top surfaces during the simulation and solves the problem correspondingly withwithout crust Boundary conditions are applied to the external and internal surfaces of RPV and (if needed) top surface of metal layer Arbitrary boundary conditions are supported If a boundary condition is nonlinear eg mixed heat convection (linear dependent on 119879 ) and heat radiation (proportional to 1198794) the Picard method [73] would be used for solving this problem

This model consists of three main parts

(i) empirical heat transfer correlations (mostly from experiments) to determine global and local heat transfer coefficients

(ii) energy balances in oxide layer and metal layer where the heat transfer coefficients would be used and

(iii) energy balance in RPV

Figure 20 Two-layer melt pool configuration and RPV wall [74]

511 Heat transfer correlations

For internally heated natural convection (oxide layer) in transIVR the mini-ACOPO correlations [75] are used for the averaged heat transfer as expressed in Eq (35) and (36) and for the angular variation of the heat flux as expressed in Eq (37) and (38)

119873119906119906119901 = 03451198771198861198941198990233 (35)

119873119906119889119899 = 00038119877119886119894119899035(119867119877)025 (36)

119873119906119889119899(θ)

119873119906119889119899= 01 + 108(

θ

θ119898119886119909) minus 45 (

θ

θ119898119886119909)2

+ 86 (θ

θ119898119886119909)3

01 leθ

θ119898119886119909le 06 (37)

RPV

Oxide pool

Metal pool

37

119873119906119889119899(θ)

119873119906119889119899= 041 + 035 (

θ

θ119898119886119909) + (

θ

θ119898119886119909)2

06 ltθ

θ119898119886119909le 1 (38)

The 119877119886119894119899 that appears in Eq (35)~(36) is the internal Rayleigh number which is defined as

119877119886119894119899 =119892β119876119881119867

5

α119896ν (39)

For Rayleigh-Bernard convection in the metal layer heat transfer occurs in all the top bottom and side surfaces To side surface the reduced form [72] of the Churchill-Chu correlation [76] for 119875119903 = 013 is employed as shown in Eq (40)

119873119906119904119889 = 007611987711988613 (40)

To top and bottom surfaces the reduced form [72] of Globe-Dropkin correlation [77] for

119875119903 = 013 is used as expressed in Eq (41)

119873119906119906119901 = 005911987711988613 (41)

The 119877119886 that appears in Eq (40) and (41) is the external Rayleigh number which is defined

as

119877119886 =119892βΔ1198791198673

να (42)

Heat transfer coefficients can be calculated from Nusselt number by Eq (43)

ℎ =119873119906119896

119871 (43)

where 119896 and 119871 are the heat conductivity and characteristic length respectively 119873119906 can be the Nusselt numbers expressed in Eq (35)~(38) and (40)~(41)

512 Energy balance in the pools

The energy balance of oxide layer can be described as Eq (44) The total generated heat of oxide pool equals the sum of the increase of enthalpy (transient temperature change) upward heat flow to metal layer and downward heat flow to RPV

120588119862119901119881119900119897119889119879119900119909119889119905

+ ℎ119906119901119860119900119909119906119901(119879119900119909 minus 119879119900119909119906119901) + ℎ119889119899119860119900119909119889119899(119879119900119909 minus 119879119900119909119889119899) minus 119876119881119881119900119897 = 0 (44)

where ρ is the oxide density 119862119901 the heat capacity 119881119900119897 the total volume 119879119900119909 the bulk temperature 119905 the time ℎ119906119901 the upward heat transfer coefficient 119860119900119909119906119901

upward heat transfer area 119879119900119909119906119901 upward surface temperature ℎ119889119899 downward

heat transfer coefficient 119860119900119909119889119899 downward heat transfer area 119879119900119909119889119899 downward surface temperature and 119876119881 volumetric heat source

38

On the interface between oxide layer and metal layer energy balance requires that the heat transferred from oxide layer should equal the heat to the metal layer which can be described as Eq (45)

ℎ119906119901119860119900119909119906119901(119879119900119909 minus 119879119900119909119906119901) = ℎ119898119905119889119899119860119898119905119889119899(119879119894119899119905 minus 119879119898119905) (45)

where ℎ119898119905119889119899 is the downward heat transfer coefficient of metal layer 119860119898119905119889119899 the downward heat transfer area 119879119894119899119905 the interface temperature and 119879119898119905 the bulk temperature of metal layer

The energy balance of metal pool can be described as Eq (46) The increase of enthalpy (transient temperature change) equals the sum of upward downward and sideward heat flows to surrounding surfaces

ρ119898119905119862119901119898119905119881119900119897119898119905119889119879119898119905119889119905

+ ℎ119898119905119906119901119860119898119905119906119901(119879119898119905 minus 119879119898119905119906119901) + ℎ119898119905119904119889119860119898119905119904119889(119879119898119905 minus 119879119898119905119904119889)

+ ℎ119898119905119889119899119860119898119905119889119899(119879119898119905 minus 119879119894119899119905) = 0

(46)

where ρmt is the metal density 119862119901mt the heat capacity 119881119900119897mt the total volume 119879119898119905 the bulk metal temperature 119905 the time ℎ119898119905119906119901 the upward heat transfer

coefficient 119860119898119905119906119901 upward heat transfer area 119879119898119905119906119901 upward surface

temperature ℎ119898119905119904119889 sideward heat transfer coefficient 119860119898119905119904119889 sideward heat transfer area 119879119898119905119904119889 sideward surface temperature ℎ119898119905119889119899 downward heat transfer coefficient 119860119898119905119889119899 downward heat transfer area and 119879119894119899119905 oxide-metal interface temperature

The heat transferred from metal layer to the top surface should equal the heat radiated from this surface to the environment Considering the radiation and reflections of internal structures to this surface following energy balance equation [72] is achieved

ℎ119898119905119906119901(119879119898119905 minus 119879119898119905119906119901) = σ(1198791198981199051199051199001199014 minus 119879119904

4) (1

ε+1 minus ε119904ε119904

119860119898119905119906119901

119860119904)minus1

(47)

where 119879119898119905119905119900119901 is the temperature of metal top free surface ε the emissivity of

metal ε119904 the emissivity of internal structures 119860119898119905119906119901 the area of metal top surface

and 119860119904 the total area of internal structures Sometimes the internal water cooling may be available above metal layer In this case Eq (47) is unnecessary and 119879119898119905119905119900119901 can be set to a constant value

For the oxide layer top surface Eq (48) is used if not crust is formed in the interface between two layers ie 119879119894119899119905 gt 119879119898119890119897119905119894119899119892119900119909 Otherwise Eq (49) is used

119879119900119909119906119901 = 119879119894119899119905 (48)

119879119900119909119906119901 = 119879119898119890119897119905119894119899119892119900119909 (49)

For the metal layer top surface Eq (50) is used if not crust is formed on top of the metal ie 119879119906119901119904119906119903119891 gt 119879119898119890119897119905119894119899119892119898119905 Otherwise Eq (51) is used

119879119898119905119906119901 = 119879119898119905119905119900119901 (50)

39

119879119898119905119906119901 = 119879119898119890119897119905119894119899119892119898119905 (51)

Overall this part involves 6 equations regarding 6 unknows 119879119900119909 119879119900119909119906119901 119879119894119899119905 119879119898119905

119879119898119905119906119901 and 119879119898119905119905119900119901

513 Energy balance in the RPV wall

The heat transfer in the PRV wall can be modelled as a simple 1D heat conduction problem that heat only transfers in the wall thickness direction or 2D axisymmetric heat conduction problem

For 1D model the treatment in [72] was adopted As illustrated in Figure 21 the heat from the oxide pool transfers to inner vessel surface (119879119894) then to the interior vessel (119879119907119890119904) then to outer vessel (119879119900) and finally to environment

Figure 21 1D heat transfer model in RPV [74]

Unknown variables are the heat flux through inner vessel surface 119902119894119899 the crust thickness 119909119888119903 inner vessel temperature 119879119894 vessel center temperature 119879119907119890119904 and outer vessel temperature 119879119900 Eq (52)~(56) give a closed equation set for these variables considering a radiation boundary condition on the external vessel surface Eq (52)~(53) describe the heat transfer through the crust where an internal heat source 119876119888119903 is also considered Eq (52) can be easily derived from the one-dimensional heat transfer equation with heat source Eq (54)~(55) describe the energy balance on the inner vessel surface and vessel interior respectively Eq (56) describes the energy balance on the outer vessel surface regarding radiation boundary condition For Dirichlet boundary condition Eq (56) is replaced with Eq (57) For other boundary conditions we only need to correspondingly modify the terms on the right-hand side of Eq (56) For situations that no crust is formed on the surface of the vessel Eq (52)~(53) are eliminated and 119902119894119899 in Eq (54) can directly be determined from calculations in Section 512 and Eq (54)~(56) are solved for the unknows 119879119894 119879119907119890119904 and 119879119900 Once 119879119894 exceeds the vessel melting temperature (with or without crust formation) wall ablation is considered to occur Then 119879119894 is fixed to the vessel melting temperature and vessel thickness 119909119907119890119904 becomes the new unknown variable

119879119900119909119889119899 119879119894 119879119907119890119904 119879119900 119879119890119899119907

119909119888119903 119909119907119890119904

crust RPV

Qcr

119902119894119899 119902119897119900119888119886119897

40

119902119894119899 = 119896119888119903(119879119900119909119889119899 minus 119879119894)

119909119888119903+ 1198761198881199031199091198881199032

(52)

119902119894119899 = 119902119897119900119888119886119897 + 119876119888119903119909119888119903

(53)

119902119894119899 = 119896119907119890119904(119879119894 minus 119879119907119890119904)

05119909119907119890119904

(54)

120588119907119890119904119862119901119907119890119904119889119879119907119890119904119889119905

= 119896119907119890119904(minus2 lowast 119879119907119890119904 + 119879119894 + 119879119900)

(05119909119907119890119904)2

(55)

119896119907119890119904(119879119907119890119904 minus 119879119900)

05119909119907119890119904= 휀120590(119879119900

4 minus 1198791198901198991199074 )

(56)

119879119900 = 119879119900119888119900119899119904119905

(57)

For the 2D representation the heat conduction in the RPV is solved using FDM considering the heat transfer in both the thickness and the angular directions in the RPV wall The RPV lower head is treated as a spherical shell The corresponding heat conduction equation in the 2D spherical coordinate is given as Eq (58) The cylindrical part of the vessel is treated as a cylindrical shell The corresponding heat conduction equation in the 2D cylindrical coordinate is given as Eq (59) Eq (58) and (59) are coupled and solved together In default the mesh size is 5deg in angular direction and 0025119909119888119903 in radius direction This mesh discretization can be changed up to user demands

ρ119862119901part119879

part119905=

1

1199032part

part119903(1198961199032

part119879

part119903) +

1

r2sinθ

part

partθ(119896119904119894119899θ

part119879

partθ)

(58)

ρ119862119901part119879

part119905=1

119903

part

part119903(119896119903

part119879

part119903) +

part

part119911(119896part119879

part119911) (59)

52 Hybrid Coupling Approach

The hybrid coupling approach to assess the RPV integrity is illustrated in Figure 22 transIVR and ANSYS Mechanical are coupled in a one-way manner transIVR conducts the conjugate heat transfer of melt pool and RPV and provides transient thermal loads to the ANSYS Mechanical where structural analysis of RPV deformation is performed The 2D heat transfer of RPV is solved in transIVR to better capture the temperature distribution In RPV structural analysis same considerations are taken as in Section 422 for 16MND5 steel the modified theta projection model for creep and MISO for plasticity and temperature-dependent Young modulus etc The VLM is adopted for the transfer of transient thermal loads

41

Figure 22 Framework of the hybrid coupling approach [74]

53 Verification and Validation

The transIVR code was first benchmarked against the UCSB FIBS case of two-layer pool configuration and then validated against the LIVE-7V experiment showing the codersquos capability in predicting two-layer and transient melt pool heat transfers respectively Finally the hybrid coupling approach encompassing transIVR and ANSYS Mechanical was validated against the RPV failure experiment FOREVER-EC2 and the results showed acceptable agreements with those of both the experiment and the previous approach using CFD in Section 43 Hence the transIVR code is an efficient tool for melt pool heat transfer and the hybrid approach an efficient way for RPV failure analysis

531 UCSB FIBS benchmark case

The UCSB FIBS benchmark case [78] was calculated to demonstrate the capability of the transIVR code in simulating two-layer pool heat transfer problems The considered pool configuration was similar to that shown in Figure 20 with the following specific dimensions a two-layer melt pool with a 16 m oxide layer and a 08 m metallic pool on top and an RPV lower head with a thickness of 015 m The internal volumetric source in the oxide layer was 13 MWm3 The calculation results were compared with those of the DOE study [78] and the INEEL study [79]

The predicted results were compared with the digitalized results of the DOE and INEEL studies in Figure 23 All three studies show similar profiles of the heat flux distribution along the polar angle (Figure 23(a)) it increased with the polar angle in the oxide layer to the maximum value at around 77deg and then dropped to a significantly lower value in the metal layer The transIVR result agreed well with the other results with small discrepancy in the metal layer the DOE study gave a

heat flux of 500 MWm2 the INEEL study of 400 MWm2 and the transIVR of somewhere in between Figure 23(b) shows the predicted corium crust thickness distribution along the RPV polar angle also agreed well the DOE and INEEL

transIVR ANSYS Mechanical

Transient

thermal loads

42

studies The crust thickness decreased with polar angle in the oxide layer and no crust was formed in the metal layer As the crust formation is closely related to the heat flux distribution the crust thickness profile was reasonable regarding the heat flux distribution shown in Figure 23(a) Vessel wall ablation would occur due to the high thermal loads Figure 23(c) gives the profile of the remaining vessel wall thickness along the RPV polar angle considering the wall ablation It shows that no ablation occurred in low angle region (0deg~36deg) where heat flux was low and the wall thickness changed from 15 cm to around 6 cm as angle increased in high angle region of the oxide layer Because of the discrepancy in the heat flux of metal layer the predicted wall thicknesses also differ from those of the DOE and INEEL studies the thinnest wall was 73 cm predicted by the DOE study corresponding to the maximum heat flux followed by 88 cm of the present transIVR calculation and 94 cm of the INEEL study Generally speaking the prediction of transIVR is in an acceptable level compared with the other studies for the same benchmark case

Figure 23 Simulation results of the UCSB FIBS benchmark case [74]

(a) heat flux distribution (b) crust thickness distribution

(c) wall thickness distribution

43

532 LIVE-7V experiment

LIVE-7V is an experiment under the LIVE program conducted in KIT to study the core melt phenomena As Figure 24(a) shows it employed the LIVE-3D hemispherical facility with an inner diameter of 1 m and a wall thickness of ~25 mm The non-eutectic binary salt mixture of 20 mol NaNO3- 80 mol KNO3 was used as the simulant material The decay heat in the melt was simulated by cable-type heating with 8 planes of electrical resistance heating wires Water cooling was applied to the top and vessel surfaces Top surface cooling was realized by a cooling lid mounted at height 413 mm from the vessel bottom and vessel cooling by the flow channel between the test vessel and the outer cooling vessel The inlet and outlet water temperature for both top and vessel cooling surfaces were kept ~293 K and ~303 K throughout the whole experiment respectively A 413 mm height pool corresponds to a total volume of ~0194 m3 simulant material The experiment procedure can be divided into four stages with the heating power sequentially shifted from 29 kW to 24 kW 18 kW and 9 kW Figure 24(b) indicates the thermocouple positions in experiment

The simulation results of LIVE-7V using transIVR were compared with experimental data [80] and numerical study using REALPSCDAPSIM code [81] For simplicity the RELAPSCDAPSIM results in the following are labelled as lsquoCOUPLErsquo which is the name of the module in RELAPSCDAPSIM for melt pool heat transfer

Figure 24 LIVE test facility [80] and thermocouple positions [82]

Figure 25 compares the predicted heat removal upward to the top lid and downward to the bottom vessel surface with those of the experiment and COUPLE The black curve indicates the total input power at the four stages The experimental results (red and blue dots) were calculated from the Table 3 of [80] assuming no extra heat losses For the first three stages the transIVR predictions agreed well with the COUPLE results while the predicted upward heat flow was slightly lower and the downward heat flow higher than the experimental data For the last stage the transIVR agreed well with the experimental result while the COUPLE predicted lower upward and higher downward heat flows The short

(a) (b)

44

transition periods between two stages with different total inputs taken in transIVR also agreed with those in the COUPLE predictions

Figure 25 Comparison of heat removal upward to the top melt surface cooling lid and downward to bottom vessel wall [74]

Figure 26 shows the comparison of heat flux distribution along the vessel wall All curves show similar trend that a higher total input power would result in a higher heat flux profile As the COUPLE predictions had relatively flatter profiles transIVR predictions generally agreed better with the experimental results especially when total input was high A discrepancy is seen that the minimum value of a curve occurred at angle 0degfor both transIVR and COUPLE and at angle 30deg for the experimental data It is noted that the models in default assume heat flux monotonically increases with angle (cf Eq (37) and (38)) resulting it impossible to predict higher heat flux at angle 0deg than angle 30deg as in experimental results

45

Figure 26 Heat flux profiles along the vessel polar angle [74]

533 FOREVER-EC2 experiment

In this section we performed calculations against the FOREVER-EC2 experiment to validate the performance of the coupled transIVR and ANSYS Mechanical in assessing the RPV integrity The applied boundary conditions were identical to the settings of previous coupling approach in Section 4 (ANSYS Fluent+ Mechanical) as indicated in Figure 11 The initial bulk temperature of the melt pool was set to be 1261 K in transIVR which corresponds to the maximum value of the initial temperature field in CFD calculation of previous coupling approach Both the experimental data and the simulation results using previous coupling approach are included for comparison In the following results from current hybrid approach are labelled lsquotransIVRrsquo (and referred as transIVR results) results from previous approach labelled lsquoCFDrsquo (and referred as CFD results) and results from experiment labelled lsquoexperimentrsquo

Figure 27 compares the bulk temperature predicted by transIVR with the averaged and maximum temperatures of the CFD results All curves show the same trend that the temperature gradually increased with time till reaching a steady state and then kept unchanged till the end The times needed for reaching the steady state were ~5000 s for the CFD and ~4000 s for the transIVR which are close to each other It indicates again a good performance of the transIVR in predicting the transient behavior of melt pool Compared with the steady-state averaged temperature 1478 K and maximum temperature 1609 K of the CFD results the steady-state temperature of transIVR 1599 K is a reasonable prediction

0

3000

6000

9000

12000

15000

18000

0 20 40 60 80 100

Hea

t fl

ux

(W

m2)

Polar angle (deg)

29 kW-EXP24 kW-EXP18 kW-EXP9 kW-EXP29 kW-COUPLE24 kW-COUPLE18 kW-COUPLE9 kW-COUPLE29 kW-transIVR24 kW-transIVR18 kW-transIVR9 kW-transIVR

46

Figure 27 Temperature evolution with time [74]

Figure 28 Temperature fields at time (a) 01 s (b) 3038 s (c) 6038 s and (d) 13289 s [74]

Figure 28 shows the temperature fields of the vessel at time 01 s 3038 s 6038 s and 13289 s The temperature fields were visualized in the ANSYS Mechanical It also indicates that the transient temperature loads were properly transferred from transIVR to ANSYS Mechanical Figure 28(a-d) show both the minimum and maximum temperatures increased with time In the lower head region the temperature gradually increased with the height The maximum temperature occurs at the location slightly lower than the melt pool top surface This is because thermal loadheat flux imposed on the vessel generally increases with polar angle in the pool natural convection so the code transIVR inherently also predicted higher heat flux at higher angle positions The higher flux at higher position then correspondingly resulted in higher temperature In the cylindrical region the

(a) (b) (c) (d)

[K] [K] [K] [K]

47

temperature decreased as the height since the bottom part would be more significantly heated by the high-temperature lower head region

Figure 29 shows the steady-state temperature profiles of the external vessel surface along the polar angle The transIVR result agreed well with other two results in trend that the temperature increased with the angle and reach the peak values at angle ~80deg then gradually decreased with angle But slightly lower temperature was predicted in the low angle region Possible reason may be that the local heat flux distribution determined by Eq (37) and (38) is steeper than that in the experiment and CFD study which then resulted in a steeper increase of temperature along the angle and a lower temperature at low angle region In the cylindrical part (gt90deg) the transIVR result generally agreed with the experimental and CFD results The discrepancy with the experimental data may be caused by the simplified boundary conditions the ambient temperature inside the vessel was set to uniform constant value while it could decrease as height Therefore predicted heat loss in the higher position of cylindrical part could be smaller than that with height-dependent ambient temperature resulting in higher vessel temperature Overall the prediction was reasonable for a lumped-parameter model

Figure 29 Temperature profiles of the external vessel surface along the vessel polar angle [74]

The transIVR predicted a failure time of 24959 s which is close to 24180 s in experiment and 23959 s in the CFD results Figure 30 shows the total deformation of vessel bottom point with time All three curves increased gradually with time and experienced drastic accelerations at the end The

400

600

800

1000

1200

1400

0 40 80 120 160

Tem

per

ature

(K

)

Polar angle (deg)

transIVRExperimentCFDWeilding Line

48

transIVR curve agreed well with the CFD curve The final total deformation at failure time is also close to that of the CFD result but lower than the experiment data The discrepancy might be related to the simplifications in the boundary conditions accuracy of the thermal load prediction and the modelling errors in plasticity and creep

Figure 30 Total deformation in vertical direction of the vessel bottom point [74]

Figure 31 shows the profile of wall thickness change The transIVR curve was compared with the CFD curve and four experimental curves measured at vessel left right front and back sides The simulation result fitted the experimental and CFD results well in the region 0deg~50deg In the region 50deg~90deg wall thickness gradually decreased with angle till around 75deg and then increased till 90deg The maximum thickness change was -3 mm for the simulation which is close to the experiment left side curve and slightly larger than the -6~ -45 mm of the other curves from experiment and CFD results Concerning possible errors and uncertainties in the modelling and numerical settings the new approach generally gives reasonable predictions

0

5

10

15

20

25

11670 15170 18670 22170 25670

Tota

l def

orm

atio

n (

mm

)

Time (s)

Experiment

CFD

transIVR

49

Figure 31 Wall thickness changes along the vessel polar angle [74]

-6

-5

-4

-3

-2

-1

0

1

0 20 40 60 80 100

Wal

l th

icknes

s ch

ange

(mm

)

Polar angle (deg)

CFD

transIVR

Experiment (left)

Experiment (right)

Experiment (front)

Experiment (back)

50

51

6 Summary and Outlook

Reactor pressure vessel (RPV) is one of the key physical barriers in the defence-in-depth approach of nuclear power safety and its failure is also the transition point of core melt (corium) progression from in-vessel to ex-vessel phase Quantification of RPV failure during a hypothetical severe accident is therefore paramount to the specific qualification of severe accident mitigation strategies as well as the general assessment of corium risk in light water reactors The RPV failure is a problem of structural performance of RPV under the influence of thermal loads imposed by corium melt pool ie it is a transient thermo-fluid-structure coupled problem per se whose solution requires simultaneous modelling of both highly turbulent melt pool heat transfer and large-scale nonlinear RPV creep Due to the complexity of the problem only limited capabilities have been developed in the previous studies under different simplifications and assumptions in models and coupling methods The present study was motivated to advance modelling and coupling approaches for RPV performance analysis considering new understanding of physical phenomena and latest coupling platform of simulation tools

For the new development of coupling approaches and models for physical phenomena related to RPV failure analysis during severe accidents this thesis work was conducted in the following order (i) a new creep model for the RPV steel 16MND5 was developed which features the capability to cover all three creep stages (ii) a new thermo-fluid-structure coupling approach was developed for RPV failure analysis and validated against FOREVER-EC2 experiment (iii) the performance of the new coupling approach using volume loads mapping (VLM) for data transfer was further compared with another approach using surface loads mapping (SLM) (iv) a hybrid coupling approach was proposed for efficient analysis of RPV failure which employs a lumped-parameter code for quick estimate of thermal load and a mechanistic FE code for detailed structural analysis and (v) for the hybrid coupling approach the code transIVR was developed for calculation of transient melt pool heat transfer

The developed creep model for 16MND5 is called lsquomodified theta projection modelrsquo and describes all three stages of a creep process Creep curves are expressed as a function of time with five model parameters 120579119894(119894 = 14 119886119899119889 119898) A dataset for the model parameters was constructed by fitting experimental creep curves into this function and an iterative parameter adjustment process to satisfy the monotonicity assumption of creep strain vs temperature and stress Model parameters for new curves under different temperature and stress loads are directly interpolated from this dataset in contrast to the conventional ldquotheta projection modelrdquo that employs an extra fitting process The new treatment was proven to better accommodate all experimental curves over the wide ranges of temperature and stress loads The model was implemented into the ANSYS Mechanical code The simulation results indicate that the model successfully captured all three creep stages and a good agreement was achieved between each experimental creep curve and corresponding modelrsquos prediction Regarding the model response to transient changes of temperature and stress loads the widely used time hardening and strain hardening models were implemented and

52

performing reasonably well in the new method These properties fulfil the requirements of a creep model for structural analysis

The transient thermo-fluid-structure coupling approach was developed using the ANSYS Fluent code for thermofluid dynamics of the melt pool and ANSYS Structural code for structural mechanics of RPV An extension tool was introduced to realize transient load transfer from ANSYS Fluent to Structural and minimize the user effort Both turbulence models and the effective model PECM can be used for melt pool heat transfer The developed creep model was used for RPV failure analysis of the RPV Since it is insensitive to geometry the coupling approach allows more complex geometries such as a BWR lower head with a forest of penetrations (CRGTs and IGTs) modelled in detail Moreover the coupling approach does not only reflect the transient response of thermo-fluid-structure interaction but also facilitates the implementation of advanced (high-fidelity) models in both fluid mechanics and structural mechanics The validation of the coupling approach against the FOREVER-EC2 experiment showed that the temperature profile vessel failure time total displacement of vessel bottom point and wall thickness profile in the experiment were well predicted

Following the variation of the coupling approach from VLM to SLM for data transfer between ANSYS Fluent and ANSYS Structural the respective coupling approaches were investigated by comparative calculations against the FOREVER-EC2 experiment and a postulated severe accident of a reference BWR For the FOREVER-EC2 experiment both VLM and SLM approaches predicted generally well the total deformation in vertical direction of bottom point and the wall thickness change though the former predicted a slightly later failure time For the reference BWR case good agreements were achieved in predicting the maximum creep strain and the total deformation in vertical direction of bottom point For both experiment and reactor cases the VLM approach predicted slightly later failure times than SLM but the differences were negligible compared to the long time-scale of vessel failure (~104 s) The deformation rate of the vessel wall was slow at the early stage and accelerated with time till a drastic deformation and final vessel failure Generally speaking the VLM and SLM coupling approaches had the similar performance in terms of their predictability of experiment and applicability to reactor case but the VLM was computationally more efficient

In the development of the hybrid coupling approach the transIVR code was developed for quick estimate of transient melt pool heat transfer and determination of thermal loads on RPV transIVR does not only support transient analysis of one- and two- layer melt pool but also the heat transfer in RPV being resolved with 2D finite difference method which captures more spatial thermal details for RPV structural analysis in addition to 1D heat transfer model The capabilities of transIVR in modelling two-layer pool heat transfer and transient pool heat transfer were demonstrated by calculations against the UCSB FIBS benchmark case and the LIVE-7V experiment respectively It was then coupled to the mechanical solver ANSYS Mechanical which together form the hybrid coupling framework for detailed RPV failure analysis The calculation against the FOREVER-EC2 experiment did not only reproduce the major results of the experiment but also had a good agreement with the thermo-fluid-structure

53

coupling approach using CFD This indicated the new coupling approach is capable of capturing the vessel creep failure characteristics and can be employed as an efficient tool addressing the complex vessel failure problem

As a prospective more research topics can be addressed in future either as a continuation of the current work toward an effective tool for vessel failure analysis or from the general view of scientific interest in melt pool heat transfer and RPV failure The following topics are recommended

bull Extended applications to the reactor cases For instance the thermo-fluid-structure coupling approach can be used to study penetration failures in the RPV of an BWR with a detailed representation of penetrations (IGTs and CRGTs)

bull Synthesis or update of heat transfer correlations for melt pool Heat transfer correlations directly benefit the parametric codes such as transIVR in accurate quantification of heat flux distribution around a melt pool So far many empirical correlations have been proposed based on experiments which had a variety of geometries and melt simulants with a wide range of Rayleigh number (Ra) Sensitivity studies have also been performed on the influences of Ra Pa and other material properties Therefore a comprehensive synthesis or update of the existing heat transfer correlations will be a meaningful step for making reasonable extrapolations to the reactor applications with prototypic materials

bull Modelling of pool stratification The motivation starts with the improvement of the PECMrsquos modelling capability such that the dynamic stratification and resulting pool heat transfer can be modelled altogether Simulation of melt stratification has been supported in some system codes such as ASTEC using a predefined parametric approach The improvement of PECM can be considered in the similar approach or more mechanistic modelling

bull Development of a structural module for RPV The idea is to develop a simplified module for structural analysis so it can be coupled with transIVR ultimately forming a standalone code for quick estimate of vessel failure of LWRs during postulated severe accidents

bull Extension of vessel analysis capability The current version of tranIVR is limited to melt pool Although the precursory debris bed and melt ejection after vessel failure were not the focus of the present study they are equally important to assessment of corium risk especially for Swedish BWRs where the melt release conditions and history are crucial to qualification of their severe accident mitigation strategies

54

55

Bibliography

[1] IAEA ldquoIAEA Annual Report 2018rdquo IAEA 2019 [Online] Available httpswwwiaeaorgsitesdefaultfilesgcgc63-5pdf

[2] IAEA Safety of Nuclear Power Plants Design Vienna INTERNATIONAL ATOMIC ENERGY AGENCY 2012

[3] M Fischer ldquoGen II and III In-Vessel and Ex-Vessel Melt Retentionrdquo The Short Course on Severe Accident Phenomenology Oct 2017

[4] B R Sehgal Nuclear Safety in Light Water Reactors Severe Accident Phenomenology 1st ed Academic Press 2011

[5] IAEA Approaches and Tools for Severe Accident Analysis for Nuclear Power Plants Vienna INTERNATIONAL ATOMIC ENERGY AGENCY 2008

[6] WENRA ldquoPractical Elimination Applied to New NPP Designs - Key Elements and Expectationsrdquo Sep 2019 [Online] Available httpwwwwenraorgmediafiler_public20191111practical_elimination_applied_to_new_npp_designs_-_key_elements_and_expectations_-_for_issuepdf

[7] P Matejovic M Barnak M Bachraty and L Vranka ldquoASTEC applications to VVER-440V213 reactorsrdquo Nuclear Engineering and Design vol 272 pp 245ndash260 2014 doi 101016jnucengdes201308077

[8] AREVA ldquoUS EPR Severe Accident Evaluationrdquo Topical Report ANP-10268NP Oct 2006 [Online] Available httpswwwnrcgovdocsML0631ML063100157pdf

[9] C T Martin ldquoCoupled 3D Thermo-mechanical Analysis of Nordic BWR Lower Head Failure in case of Core Melt Severe Accidentrdquo master thesis KTH Stockholm 2013

[10] IAEA ldquoAdvanced Large Water Cooled Reactors A supplement to the IAEArsquos Advanced Reactor Information System (ARIS)rdquo Sep 2015 [Online] Available httpsarisiaeaorgPublicationsIAEA_WRC_Bookletpdf

[11] B Pershagen Light Water Reactor Safety Pergamon Press 1989 [12] Ninos Garis M Agrell H Glanneskog and L Agrenius ldquoAPRI 7 ndash Accident

Phenomena of Risk Importancerdquo SSM SSM 201212 2012 [Online] Available httpswwwstralsakerhetsmyndighetensecontentassets2a9f299f46b949b3b209e780cf71436a201212-apri-7-accident-phenomena-of-risk-importance-en-lagesrapport-om-forskningen-inom-omradet-svara-haverier-under-aren-2009-2011-

[13] F Fichot et al ldquoSome considerations to improve the methodology to assess In-Vessel Retention strategy for high-power reactorsrdquo Annals of Nuclear Energy vol 119 pp 36ndash45 2018 doi 101016janucene201803040

[14] T G Theofanous C Liu S Additon S Angelini O Kymaumllaumlinen and T Salmassi ldquoIn-vessel coolability and retention of a core meltrdquo Nuclear Engineering and Design vol 169 no 1 pp 1ndash48 1997 doi 101016S0029-5493(97)00009-5

[15] C Sainte Catherine ldquoTensile and creep tests material characterization of pressure vessel steel(16MND5) at high temperatures (20 up to 1300 degC)rdquo CEA France SEMTLISNRT98-009A 1998

56

[16] E Altstadt H-G Willschuetz B R Sehgal and F-P Weiss ldquoModelling of in-vessel retention after relocation of corium into the lower plenumrdquo Forschungszentrum Rossendorf FZR-437 2005

[17] H-G Willschuumltz and E Altstadt ldquoGeneration of a High Temperature Material DataBase and its Application to Creep Tests withFrench or German RPV-steelrdquo FORSCHUNGSZENTRUM ROSSENDORF FZR-353 ISSN 1437-322X Aug 2002

[18] J L Rempe S A Chavez and G L Thinnes ldquoLight water reactor lower head failure analysisrdquo Nuclear Regulatory Commission NUREGCR--5642 1993 Accessed Aug 23 2019 [Online] Available httpinisiaeaorgSearchsearchaspxorig_q=RN25025905

[19] S Brosi et al ldquoCORVIS Investigation of light water reactor lower head failure modesrdquo Nuclear Engineering and Design vol 168 no 1 pp 77ndash104 1997 doi 101016S0029-5493(96)00008-8

[20] T Y Chu M M Pilch J H Bentz J S Ludwigsen W Y Lu and L L Humphries ldquoLower Head Failure Experiments and Analysesrdquo Sandia National Laboratories Albuquerque NM USA NUREGCR-5582 SAND98-2047 1999

[21] L L Humphries et al ldquoOECD Lower Head Failure Project Final Reportrdquo Sandia National Laboratories Albuquerque NM 87185-1139 2002

[22] B R Sehgal R R Nourgaliev T N Dihn A Karbojian J A Green and V A Bui ldquoFOREVER Experiments on Thermal and Mechanical Behavior of a Reactor Pressure Vessel during a Severe Accidentrdquo in Proceedings of the Workshop on in-vessel core debris retention and coolability Garching near Munich Germany Mar 3-6 1998

[23] S Bhandari M Fischer B R Sehgal and A Theerthan ldquoPost Test Analysis of the FOREVER C2 Experimentrdquo The 9th International conference on Nuclear Engineering (ICONE-9) Nice France Apr 8-12 2001

[24] J F Mao J W Zhu S Y Bao L J Luo and Z L Gao ldquoCreep deformation and damage behavior of reactor pressure vessel under core meltdown scenariordquo International Journal of Pressure Vessels and Piping vol 139ndash140 pp 107ndash116 2016 doi 101016jijpvp201603009

[25] J Mao L Hu S Bao L Luo and Z Gao ldquoInvestigation on the RPV structural behaviors caused by various cooling water levels under severe accidentrdquo Engineering Failure Analysis vol 79 pp 274ndash284 2017 doi 101016jengfailanal201704029

[26] J F Mao J W Zhu S Y Bao L J Luo and Z L Gao ldquoStudy on structural failure of RPV with geometric discontinuity under severe accidentrdquo Nuclear Engineering and Design vol 307 pp 354ndash363 2016 doi 101016jnucengdes201607027

[27] B R Sehgal V A Bui T N Dinh and R R Nourgaliev ldquoHeat Transfer Processes in Reactor Vessel Lower Plenum During Late Phase of In-Vessel Core Melt Progressionrdquo in Advances in Nuclear Science and Technology J Lewins M Becker R W Albrecht E J Henley J D McKean K Oshima A Sesonske H B Smets and C P L Zaleski Eds Boston MA Springer US 1999 pp 103ndash135

[28] C-T Tran and T-N Dinh ldquoThe effective convectivity model for simulation of melt pool heat transfer in a light water reactor pressure vessel lower head Part I Physical processes modeling and model implementationrdquo Progress

57

in Nuclear Energy vol 51 no 8 pp 849ndash859 2009 doi 101016jpnucene200906007

[29] C-T Tran and T-N Dinh ldquoThe effective convectivity model for simulation of melt pool heat transfer in a light water reactor pressure vessel lower head Part II Model assessment and applicationrdquo Progress in Nuclear Energy vol 51 no 8 pp 860ndash871 2009 doi 101016jpnucene200906001

[30] H-G Willschuumltz E Altstadt B R Sehgal and F-P Weiss ldquoCoupled thermal structural analysis of LWR vessel creep failure experimentsrdquo Nuclear Engineering and Design vol 208 no 3 pp 265ndash282 2001 doi 101016S0029-5493(01)00364-8

[31] E Altstadt and T Moessner ldquoExtension of the ANSYS creep and damage simulation capabilitiesrdquo Forschungszentrum Rossendorf Technical report FZR-296 2000

[32] H-G Willschuumltz E Altstadt B R Sehgal and F-P Weiss ldquoRecursively coupled thermal and mechanical FEM-analysis of lower plenum creep failure experimentsrdquo Annals of Nuclear Energy vol 33 no 2 pp 126ndash148 2006 doi 101016janucene200508006

[33] H-G Willschuumltz E Altstadt B R Sehgal and F-P Weiss ldquoSimulation of creep tests with French or German RPV-steel and investigation of a RPV-support against failurerdquo Annals of Nuclear Energy vol 30 no 10 pp 1033ndash1063 2003 doi 101016S0306-4549(03)00036-7

[34] H Madokoro A Miassoedov and T Schulenberg ldquoA thermal structural analysis tool for RPV lower head behavior during severe accidents with core meltrdquo Mechanical Engineering Letters vol 4 pp 18-00038-18ndash00038 2018 doi 101299mel18-00038

[35] W Villanueva C-T Tran and P Kudinov ldquoCoupled thermo-mechanical creep analysis for boiling water reactor pressure vessel lower headrdquo Nuclear Engineering and Design vol 249 pp 146ndash153 2012 doi 101016jnucengdes201107048

[36] W Villanueva C T Tran and P Kudinov ldquoA Computational Study on Instrumentation Guide Tube Failure During a Severe Accident in Boiling Water Reactorsrdquo in The 14th International Topical Meeting on Nuclear Reactor Thermal hydraulics NURETH-14 Toronto Ontario Canada Sep25-30 2011

[37] W Villanueva C T Tran and P Kudinov ldquoAssessment with Coupled Thermo-Mechanical Creep Analysis of Combined CRGT and External Vessel Cooling Efficiency for A BWRrdquo The 14th International Topical Meeting on Nuclear Reactor Thermal hydraulics NURETH-14 Toronto Ontario Canada Sep 25-30 2011

[38] F R Larson and J Miller ldquoA Time-Temperature Relationship for Rupture and Creep Stressesrdquo Trans ASME vol 74 p 765minus775 1952

[39] C T Tran ldquoDevelopment validation and application of an effective convectivity model for simulation of melt pool heat transfer in a light water reactor lower headrdquo Licentiate thesis KTH Stockholm 2007

[40] ANSYS ldquoANSYS Fluent Theory Guide Release 162rdquo ANSYS Inc Jul 2015 [Online] Available httpwwwansyscom

[41] P Zeng Finite element analysis and application (in Chinese) Tsinghua University Press amp Springer 2004

58

[42] ANSYS ldquoANSYS Mechanical | Finite Element Analysis Softwarerdquo httpswwwansyscomproductsstructuresansys-mechanical-enterprise (accessed Sep 26 2019)

[43] ANSYS ldquoANSYS Mechanical APDL Theory Reference Release 162rdquo ANSYS Inc Jul 2015 [Online] Available httpwwwansyscom

[44] U Steinberner and H H Reineke ldquoTurbulent Buoyancy Convection Heat Transfer with Internal Heat Sourcesrdquo in Proceedings of the 6th Int Heat Transfer Conference Toronto Canada 1978 vol Vol2 pp 305ndash310 doi 101615IHTC63350

[45] R W Evans and B Wilshire Introduction to Creep London Institute of Materials 1993

[46] DoITPoMS ldquoCreep Deformation of Metalsrdquo Creep Deformation of Metals University of Cambridge 2006 httpswwwdoitpomsacuktlplibcreepindexphp (accessed Jul 09 2019)

[47] ANSYS ANSYS Mechanical APDL Material Reference Release 162 2015 [48] SIMULIA Abaqus 612 Analysis Userrsquos Manualvol 3 Materials Chapter

2324 2012 [49] S Imaoka ldquoUser Creep Subroutine STI0704A ANSYS Release110rdquo 2007 [50] R W Evans J D Parker and B Wilshire ldquoThe θ projection conceptmdashA

model-based approach to design and life extension of engineering plantrdquo International Journal of Pressure Vessels and Piping vol 50 no 1 pp 147ndash160 1992 doi 1010160308-0161(92)90035-E

[51] ECCC ldquoRecommendations and Guidance for the Assessment of Creep Strain and Creep Strength Data ECCC Recommendations Volume 5 Part 1 b[Issue 2]rdquo 2013

[52] W D Day and A P Gordon ldquoLife Fraction Hardening Applied to a Modified Theta Projection Creep Model for a Nickel-Based Super-Alloyrdquo The ASME Turbo Expo 2014 Turbine Technical Conference and Exposition Duumlsseldorf Germany Jun16-20 2014 vol 7A-Structures and Dynamics p V07AT29A010 doi 101115GT2014-25881

[53] M Kumar I V Singh B K Mishra S Ahmad A Venugopal Rao and V Kumar ldquoA Modified Theta Projection Model for Creep Behavior of Metals and Alloysrdquo J of Materi Eng and Perform vol 25 no 9 pp 3985ndash3992 2016 doi 101007s11665-016-2197-y

[54] P Yu and W Ma ldquoA modified theta projection model for creep behavior of RPV steel 16MND5rdquo Journal of Materials Science amp Technology vol 47 pp 231ndash242 2020 doi 101016jjmst202002016

[55] ANSYS ANSYS Mechanical Advanced Nonlinear Materials v160 Lecture 2 Rate Dependent Creep ANSYS training materials 2015

[56] Q Y Li N C Wang and D Y Yi Numerical Analysis (in Chinese) 5th ed Tsinghua University Press 2008

[57] T N Dinh R R Nourgaliev and B R Sehgal ldquoOn heat transfer characteristics of real and simulant melt pool experimentsrdquo Nuclear Engineering and Design vol 169 no 1 pp 151ndash164 1997 doi 101016S0029-5493(96)01283-6

[58] O Kymaumllaumlinen H Tuomisto O Hongisto and T G Theofanous ldquoHeat flux distribution from a volumetrically heated pool with high Rayleigh numberrdquo

59

Nuclear Engineering and Design vol 149 no 1ndash3 pp 401ndash408 1994 doi 1010160029-5493(94)90305-0

[59] F J Asfia and V K Dhir ldquoNatural convection heat transfer in volumetrically heated spherical poolsrdquo 1994 Accessed Aug 23 2019 [Online] Available httpinisiaeaorgSearchsearchaspxorig_q=RN26037727

[60] M Jahn and H H Reineke ldquoFree Convection Heat Transfer with Internal Heat Sources Calculations and Measurementsrdquo in Proceedings of the 5th Int Heat Transfer Conference Tokyo Japan 1974 vol Vol3 paper NC-28

[61] M Gustafsson ldquoFSI Transient extension V1704rdquo EDR MEDESO 2016 [62] EDR MEDESO httpsedrmedesocom [63] H Kim and I Namgung ldquoThermo-mechanical Analysis of RVLH for

APR1400 during a Severe Accidentrdquo in Transactions SMiRT-23 Manchester United Kingdom Aug10-14 2015 [Online] Available httpwwwlibncsueduresolver18402034197

[64] P Yu and W Ma ldquoComparative Analysis of Reactor Pressure Vessel Failure using Two Thermo-Fluid-Structure Coupling Approachesrdquo submitted to journal

[65] F R Menter ldquoTwo-equation eddy-viscosity turbulence models for engineering applicationsrdquo AIAA Journal vol 32 no 8 pp 1598ndash1605 1994 doi 102514312149

[66] T N Dinh and R R Nourgaliev ldquoTurbulence modelling for large volumetrically heated liquid poolsrdquo Nuclear Engineering and Design vol 169 no 1 pp 131ndash150 1997 doi 101016S0029-5493(96)01281-2

[67] ANSYS Inc ldquoIntroduction to ANSYS Fluent Lecture 7 Turbulence Modelling Fluent Tutorial Materialrdquo 2016

[68] A Horvat and M Borut ldquoNumerical investigation of natural convection heat transfer in volumetrically heated spherical segmentsrdquo in Proceedings of the International Conference Nuclear Energy for New Europe 2004

[69] Y Li CFD Pre-test Analysis of SIMECO-2 Experiment 2016 [70] A Theerthan A Karbojian and B R Sehgal ldquoEC-FOREVER experiments

on thermal and mechanical behavior of a reactor pressure vessel during a severe accident Technical report-1 EC-FOREVER-2 testrdquo SAM-ARVI-D008 Mar 2001

[71] P Yu W Ma W Villanueva A Karbojian and S Bechta ldquoValidation of a thermo-fluid-structure coupling approach for RPV creep failure analysis against FOREVER-EC2 experimentrdquo Annals of Nuclear Energy vol 133 pp 637ndash648 2019 doi 101016janucene201906067

[72] T G Theofanous C Liu S Additon S Angelini O Kymaumllaumlinen and T Salmassi ldquoIn-vessel coolability and retention of a core meltrdquo Nuclear Engineering and Design vol 169 no 1ndash3 pp 1ndash48 1997 doi 101016S0029-5493(97)00009-5

[73] H P Langtangen ldquoSolving nonlinear ODE and PDE problemsrdquo Simula Research Laboratory 2016 [Online] Available httpshplgitgithubionum-methods-for-PDEsdocpubnonlinpdfnonlin-4printpdf

[74] P Yu and W Ma ldquoDevelopment of a Lumped-Parameter Code for Efficient Assessment of In-Vessel Melt Retention Strategy of LWRsrdquo submitted to journal

60

[75] T G Theofanous M Maguire S Angelini and T Salmassi ldquoThe first results from the ACOPO experimentrdquo Nuclear Engineering and Design vol 169 no 1 pp 49ndash57 1997 doi 101016S0029-5493(97)00023-X

[76] S W Churchill and H H S Chu ldquoCorrelating equations for laminar and turbulent free convection from a vertical platerdquo International Journal of Heat and Mass Transfer vol 18 no 11 pp 1323ndash1329 1975 doi 1010160017-9310(75)90243-4

[77] S Globe and D Dropkin ldquoNatural-Convection Heat Transfer in Liquids Confined by Two Horizontal Plates and Heated from Belowrdquo J Heat Transfer vol 81 no 1 pp 24ndash28 1959 doi 10111514008124

[78] T G Theofanous C Liu S Additon S Angelini O Kymaelaeinen and T Salmassi ldquoIn-vessel coolability and retention of a core melt Volume 2rdquo Argonne National Lab DOEID--10460-VOL2 1996 Accessed Apr 18 2019 [Online] Available httpinisiaeaorgSearchsearchaspxorig_q=RN30000459

[79] J L Rempe D L Knudson C M Allison G L Thinnes and C L Atwood ldquoPotential for AP600 in-vessel retention through ex-vessel floodingrdquo United States INEELEXT--97-00779 1997 Accessed Apr 18 2019 [Online] Available httpinisiaeaorgSearchsearchaspxorig_q=RN30025872

[80] A Miassoedov X Gaus-Liu T Cron and B Fluhrer ldquoLive experiments on melt pool heat transfer in the reactor pressure vessel lower headrdquo The 10th International Conference on Heat Transfer Fluid Mechanics and Thermodynamics (HEFAT 2014) Orlando FL Jul 14-16 2014

[81] H Madokoro A Miassoedov and T Schulenberg ldquoAssessment of a lower head molten pool analysis module using LIVE experimentrdquo Nuclear Engineering and Design vol 330 pp 51ndash58 2018 doi 101016jnucengdes201801036

[82] B Fluhrer et al ldquoThe LIVE-L1 and LIVE-L3 experiments on melt behaviour in RPV lower headrdquo Forschungszentrum Karlsruhe GmbH Technik und Umwelt (Germany) Inst fuer Kern- und Energietechnik FZKA--7419 2008 Accessed Nov 12 2019 [Online] Available httpinisiaeaorgSearchsearchaspxorig_q=RN40036867

I

Abstract

This thesis aims at the development of new coupling approaches and new models for the thermo-fluid-structure coupling problem of reactor pressure vessel (RPV) failure during severe accidents and related physical phenomena The thesis work consists of five parts (i) development of a three-stage creep model for RPV steel 16MND5 (ii) development of a thermo-fluid-structure coupling approach for RPV failure analysis (iii) performance comparison of the new approach that uses volume loads mapping (VLM) for data transfer with the previous approach that uses surface loads mapping (SLM) (iv) development of a lumped-parameter code for quick estimate of transient melt pool heat transfer and (v) development of a hybrid coupling approach for efficient analysis of RPV failure

A creep model called lsquomodified theta projection modelrsquo was developed for the 16MND5 steel so that it covers three-stage creep process Creep curves are

expressed as a function of time with five parameters 120579119894 (i=14 and m) in the new creep model A dataset for the model parameters was constructed based on the available experimental creep curves given the monotonicity assumption of creep strain vs temperature and stress New creep curves can be predicted by interpolating model parameters from this dataset in contrast to the previous method that employs an extra fitting process The new treatment better accommodates all the experimental curves over the wide ranges of temperature and stress loads The model was implemented into the ANSYS Mechanical code and its predictions successfully captured all three creep stages and a good agreement was achieved between the experimental and predicted creep curves For dynamic loads that change with time the widely used time hardening and strain hardening models were implemented with a reasonable performance These properties fulfil the requirements of a creep model for structural analysis

A thermo-fluid-structure coupling approach was developed by coupling the ANSYS Fluent for the fluid dynamics of melt pool heat transfer and ANSYS Structural for structural mechanics of RPV An extension tool was introduced to realize transient load transfer from ANSYS Fluent to Structural and minimize the user effort Both CFD with turbulence models and the effective model PECM can be employed for predicting melt pool heat transfer The modified theta projection model was used for creep analysis of the RPV The coupling approach does not only capture the transient thermo-fluid-structure interaction feature but also support the advanced models in both melt pool convection and structural mechanics to improve fidelity and facilitate implementation The coupling approach performs well in the validation against the FOREVER-EC2 experiment and can be applied complex geometries such as a BWR lower head with forest of penetrations (control rod guide tubes and instrument guide tubes)

In the comparative analysis the VLM and SLM coupling approaches generally have the similar performance in terms of their predictability of the FOREVER-EC2 experiment and applicability to the reactor case Though the SLM approach predicted slightly earlier failure times than VLM in both cases the difference was negligible compared to the large scale of vessel failure time (~104 s) The VLM approach showed higher computational efficiency than the SLM

II

The idea of the hybrid coupling is to employ a lumped-parameter code for quick estimate of thermal load which can be employed in detailed structural analysis Such a coupling approach can significantly increase the calculation efficiency which is important to the case of a prototypical RPV where mechanistic simulation of melt pool convection is computationally expensive and unnecessary The transIVR code was developed for this purpose which is not only capable of quick estimate of transient heat transfer of one- and two- layer melt pool but also solving heat conduction problem in the RPV wall with 2D finite difference method to provide spatial thermal details for RPV structural analysis The capabilities of transIVR in modelling two-layer pool heat transfer and transient pool heat transfer were demonstrated by calculations against the UCSB FIBS benchmark case and the LIVE-7V experiment respectively The transIVR code was then coupled to the mechanical solver ANSYS Mechanical for detailed RPV failure analysis Validation against the FOREVER-EC2 experiment indicates the coupling framework successfully captured the vessel creep failure characteristics

Keywords RPV failure thermo-fluid-structure coupling creep modelling melt pool heat transfer severe accident CFD FEA transIVR

III

Sammanfattning

Syftet med denna avhandling aumlr att utveckla nya metoder och nya modeller foumlr att kunna koppla ihop termohydrauliska foumlrlopp och uppkommen strukturrespons i botten paring reaktortanken foumlr foumlrbaumlttrad analys av reaktortankbrott under ett svaringrt haveri Avhandlingen bestaringr av fem delar (i) utveckling av en trestegsmodell foumlr krypning i reaktortankens 16MND5-staringl (ii) utveckling av metoder foumlr kopp ling av termohydrauliska foumlrlopp och strukturrespons foumlr studier av reaktortankbrott (iii) jaumlmfoumlrelse av prestanda av den nya metoden som anvaumlnder volume loads mapping (VLM) foumlr dataoumlverfoumlring med den tidigare metoden som anvaumlnder surface loads mapping (SLM) (iv) utveckling av en lumped-parameter beraumlkningskod foumlr snabb uppskattning av transient vaumlrmeoumlverfoumlring i en smaumlltpoumll i botten paring reaktortanken och (v) utveckling av en hybrid kopplingsmetod foumlr effektiv analys av reaktortankbrott

En krypmodell kallad lsquomodified theta projection modelrsquo som behandlar krypprocessen i tre steg utvecklades foumlr 16MND5-staringlet Krypkurvor i den nya

krypmodellen ges som en funktion av tid med fem parametrar 120579119894 (i=14 och m) Ett dataset foumlr modellparametrar togs fram baserat paring tillgaumlngliga experimentalla krypkurvor under anatagande om att kryptoumljning aumlr en monoton funktion av temperatur och spaumlnning Nya krypkurvor kan genereras genom att interpolera modellparametrar fraringn detta dataset i kontrast till den tidigare metoden som anvaumlnder en speciell anpassningsprocess Foumlrdelen med det nya tillvaumlgagaringngssaumlttet aumlr att den hanterar alla experimentella krypkurvor foumlr ett brett spektrum av temperatur- och spaumlnningsbelastningar Modellen implementerades i ANSYS-koden och den korrekt behandlar alla tre krypsteg samt visar bra oumlverenstaumlmmelse mellan experimentella och beraumlknade krypkurvor Foumlr analys av tidsberoende dynamiska belastningar implementerades vaumllkaumlnda modeller av tids- och deformationshaumlrdning med rimliga resultat Dessa egenskaper uppfyller kravet paring en krypmodell foumlr strukturmekaniska analyser

En metod utvecklades foumlr koppling mellan termohydrauliska foumlrlopp och strukturmekanisk respons genom att koppla ihop ANSYS Fluent som analyserar fluiddynamiska aspekter av vaumlrmeoumlverfoumlring i en smaumlltpoumll med ANSYS Structural som analyserar strukturmekanisk respons av reaktortanksvaumlggen Ett speciellt verktyg togs fram foumlr att moumljliggoumlra oumlverfoumlring av transienta belastningar fraringn ANSYS Fluent till ANSYS Structural och foumlr att minimera anvaumlndarinsatsen Baringde CFD med turbulensmodeller och den effektiva PECM-modellen kan anvaumlndas foumlr att analysera vaumlrmeoumlverfoumlring i en smaumlltpoumll Foumlr krypanalysen av reaktortanken har lsquomodified theta projection modelrsquo anvaumlnts Den framtagna kopplingsmetoden har foumlrdelen att den inte bara faringngar de transienta aspekterna av interaktionen mellan termohydraulik och strukturmekanik men ocksaring stoumldjer avancerade modeller av konvektion i en smaumlltpoumll och strukturrespons och daumlrigenom bidrar till foumlrbaumlttrad precision och underlaumlttar implementeringen av modeller Kopplingsmetoden fungerar bra naumlr den valideras mot FOREVER-EC2 experimentet och den kan anvaumlndas i

IV

komplexa geometrier som till exempel i botten paring reaktortanken i en kokvattenreaktor (BWR) med en skog av genomfoumlringar (ledroumlr foumlr styrstavar och instrumentering)

Jaumlmfoumlrande beraumlkningar visar att de tvaring kopplingsmetoderna VLM-baserad och SLM-baserad presterar lika bra naumlr det gaumlller hur de predikterar resultat av FOREVER-EC2 experimentet och deras laumlmplighet foumlr reaktoranalyser Aumlven om SLM-metoden predikterar att tankbrott i baringda fallen intraumlffar lite tidigare saring aumlr skillnaden mellan metoderna foumlrsumbar i jaumlmfoumlrelse med tiden till tankbrott (~ 104 s) VLM-metoden var baumlttre aumln SLM-metoden naumlr det gaumlller beraumlkningseffektiviteten

Ideacuten om hybridkoppling garingr ut paring att anvaumlnda en lumped-parameter kod foumlr snabb uppskattning av termiska belastningar som senare kan anvaumlndas i detaljerad strukturmekanisk analys En saringdan kopplingsmetod kan paring ett signifikant saumltt oumlka effektiviteten av beraumlkningar vilket aumlr viktig i reaktorapplikationer daumlr mekanistisk simulering av konvektion i en smaumlltpoumll aumlr beraumlkningsintensiv och onoumldig I detta syfte utvecklades koden transIVR Koden kan dels ge en snabb uppskattning av transient vaumlrmeoumlverfoumlring i en smaumlltpoumll som bestaringr av en eller tvaring materialskikt och dels loumlsa vaumlrmeledningsproblem i reaktortanksvaumlggen med 2D finita differensmetoden vilket genererar detaljerad information om termiska foumlrharingllanden som behoumlvs foumlr strukturmekaniska analyser Foumlr att demonstrera att transIVR kan analysera dels vaumlrmeoumlverfoumlring i en smaumlltpoumll bestaringende av tvaring materialskikt och dels transient vaumlrmeoumlverfoumlring i en smaumlltpoumll jaumlmfoumlrdes transIVR prediktioner med UCSB FIBS-benchmarkfallet respektive LIVE-7V-experimentet I naumlsta steg kopplades transIVR-koden till ANSYS Structural foumlr att moumljliggoumlra mer detaljerad analys av reaktortankbrott Validering mot FOREVER-EC2 experimentet tyder paring att den utvecklade kopplingsmetodiken korrekt aringterger tankbrott orsakat av krypning

Nyckelord reaktortankbrott koppling termohydraulik-strukturmekanik modellering av krypning vaumlrmeoumlverfoumlring i smaumlltpoumll svaringra haverier CFD FEA transIVR

V

List of Publications

Papers included in this thesis

I P Yu and W Ma ldquoA modified theta projection model for creep behavior of RPV steel 16MND5rdquo Journal of Materials Science amp Technology vol 47 pp 231ndash242 2020

II P Yu W Ma W Villanueva A Karbojian and S Bechta ldquoValidation of a thermo-fluid-structure coupling approach for RPV creep failure analysis against FOREVER-EC2 experimentrdquo Annals of Nuclear Energy vol 133 pp 637ndash648 2019

III P Yu and W Ma ldquoComparative analysis of reactor pressure vessel failure using two thermo-fluid-structure coupling approachesrdquo Nuclear Engineering and Design submitted

IV P Yu and W Ma ldquoDevelopment of a lumped-parameter code for efficient assessment of in-vessel melt retention strategy of LWRsrdquo Annals of Nuclear Energy submitted

Respondentrsquos contribution to the papers included in the thesis

Peng Yu made major contributions to the publications of all the above four articles under the guidance of supervisors and with the collaboration of other co-authors including formulation of research questions development of coupling approaches and numerical methods analysis and generalization of simulation results as well as writing the first drafts and addressing review comments

Papers not included in this thesis

1 P Yu and W Ma ldquoRPV creep failure analysis using a Norton creep modelrdquo International Journal of Mechanical Sciences to be submitted

2 P Yu W Villanueva S Galushin W Ma and S Bechta ldquoCoupled thermo-mechanical creep analysis for a Nordic BWR lower head using non-homogeneous debris bed configuration from MELCORrdquo 12th International Topical Meeting on Nuclear Reactor Thermal-Hydraulics Operation and Safety (NUTHOS-12) Qingdao China October 14-18 2018

3 P Yu A Komlev Y Li W Villanueva W Ma and S Bechta ldquoPre-test simulation of SIMECO HT experiments for stratified melt pool heat transferrdquo 11th International Topical Meeting on Nuclear Reactor Thermal Hydraulics Operation and Safety (NUTHOS-11) Gyeongju Korea October 9-13 2016

VI

VII

Acknowledgements

First of all I would like to express my sincere gratitude to my supervisor Prof Weimin Ma for offering this opportunity to work in the severe accident research field and for the supports patience guidance discussions and encouragements throughout the entire period of my graduate study

I am very thankful to my co-supervisor Dr Walter Villanueva for the valuable help and discussions I learnt a lot during the times working with him I am also thankful to co-supervisor Prof Sevostian Bechta who is supportive and always has inspiring ideas that keep you thinking for a long time

Thank you also to all my colleagues at the Division of Nuclear Power Safety for the supports discussions and the very nice working environment during these years

I have taken many interesting courses during my PhD study for which I want to express my thanks to all the teachers With this regard a special thanks goes to Prof Johan Hoffman for his SF2561 Finite Element Method given in 2015 fall as well as the discussions and advices on numerical analysis

I am grateful to Ignacio Gallego-Marcos and Wenyuan Fan for their enthusiastic help on CFD and turbulence modelling during the time working on the turbulent

heat transfer I thank all my 师兄 for sharing their knowledge on nuclear engineering and giving advices when I was confused since 2012

I thank all my friends both in Sweden and in China for the company and the sharing of many wonderful moments in daily life

Finally I would like to express my deepest thank to my parents my girlfriend Siwen my sisters and all my other relatives for their unconditional supports and love

Financial supports from the China Scholarship Council and APRI make my study in KTH possible and are acknowledged

VIII

IX

Table of Contents

AbstracthelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphellipI

SammanfattninghelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphellipIII

List of PublicationshelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphellipV

AcknowledgementshelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphellipVII

NomenclaturehelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphellipXI

1 Introductionhelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphellip1

11 Nuclear Power Safety 1

12 RPV Failure 2

13 Literature Review Creep Behaviour and RPV Failure Analysis 4

131 Experiments on creep behaviour and RPV failure 4

132 Numerical studies on RPV failure 5

14 Motivation Objectives and Main Achievements 7

2 Numerical Solvers and Modelshelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphellip9

21 ANSYS Fluent 9

22 ANSYS Mechanical 9

23 PECM 11

3 Creep Modelling of 16MND5 Steelhelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphellip13

31 Creep and Creep Modelling 13

32 A Modified Theta Projection Model 14

321 Original theta projection model and its variations 14

322 A modified theta projection model 16

4 Coupled Thermo-Fluid-Structure Analysis of RPV Failurehelliphelliphelliphellip23

41 Thermo-Fluid-Structure Coupling Approach 23

42 Numerical Modelling 25

421 Conjugate heat transfer 25

422 Mechanical analysis 25

43 Validation against the FOREVER-EC2 Experiment 26

431 FOREVER-EC2 experiment 26

X

432 Conjugate heat transfer 27

433 Mechanical analysis 29

44 Application to BWR Case 32

45 Computational Efficiency 34

5 Hybrid Coupling Approach for Efficient RPV Failure Analysishelliphellip35

51 A Lumped-parameter Code for Melt Pool Heat Transfer 35

511 Heat transfer correlations 36

512 Energy balance in the pools 37

513 Energy balance in the RPV wall 39

52 Hybrid Coupling Approach 40

53 Verification and Validation 41

531 UCSB FIBS benchmark case 41

532 LIVE-7V experiment 43

533 FOREVER-EC2 experiment 45

6 Summary and Outlookhelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphellip51

Bibliographyhelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphellip55

XI

Nomenclature

Latin

119860 area of a surface

119862119901 heat capacity

119863 damage factor

119892 gravity

ℎ heat transfer coefficient

H Melt pool height

119896 thermal conductivity

119871 characteristic length

119873119906 Nusselt number

119875119903 Prandtl number

119902 heat flux

119902119894119899 heat flux through inner vessel surface

119876119881 internal volumetric heat source

119903 radial distance in a cylindrical or spherical coordinate

119877119886 external Rayleigh number

119877119886119894119899 internal Rayleigh number

119877119886119910 local Rayleigh number

119905 time

119879 temperature

119881119900119897 total volume

119909 thickness (of crust or RPV wall)

119911 height in a cylindrical coordinate

Greek

120572 thermal diffusivity

120573 thermal expansion

Δ119879 temperature difference

ε emissivity (in thermal analysis) strain (in structural analysis)

휀119888119903 creep strain

휀119903 creep strain rate

120579 polar angle

120579119894 parameters of a creep model 119894 = 1234119898

120584 kinematic viscosity

σ stress

XII

Super and subscripts

119886119898119887 ambient

119888119903 creep (in Section 3) crust (in Section 5)

119889119899 downward

119890119899119907 environment

119894 inner surface of RPV

119894119899119905 interface

119897119900119888119886119897 local

119898119886119909 maximum value

119898119890119897119905119894119899119892 melting point

119898119905 metal layer (bulk)

119898119905119889119899 downward surface of metal pool

119898119905119904119889 sideward surface of metal pool

119898119905119905119900119901 top surface of metal layer

119898119905119906119901 upward surface of metal pool

119900 outer surface of RPV

119900119909 oxide layer (bulk)

119900119909119889119899 downward surface of oxide layer

119900119909119906119901 upward surface of oxide layer

119903119890119891 reference

119904 internal structures inside the RPV

119904119889 sideward

119906119901 upward

119907119890119904 vessel

Acronyms

BWR boiling water reactor

CBD creep database

CFD computational fluid dynamics

CRGT control rod guide tube

ECCM effective convectivity conductivity model

FDM finite difference method

FE finite element

FEA finite element analysis

FEM finite element method

FVM finite volume method

IGT instrument guide tube

IVR in-vessel retention

XIII

IVMR in-vessel melt retention EU project

LES large eddy simulation

MPS moving particle semi-implicit method

NPP nuclear power plant

PECM phase-change effective convectivity model

RANS Reynolds-averaged Navier-Stokes

RPV reactor pressure vessel

SBO station blackout

SLM surface loads mapping

SPH smoothed-particle hydrodynamics

UDF user-defined function

UPF user programmable features

VLM volume loads mapping

XIV

1

1 Introduction

11 Nuclear Power Safety

Greenhouse gas emissions contribute to the global warming problem Phasing out fossil fuels by switching to low-carbon energy sources is considered as a mitigation strategy With this regard clean energy should be increased in the world energy consumption structure Being clean and stable nuclear energy is an attractive alternative to fossil fuels After the development over decades it also proves to be a mature and safe technology According to IAEA 2018 annual report [1] there were 450 nuclear power reactors under operation worldwide at the end of 2018 which had a record generating capacity of 3964 GW(e) 9 reactors were connected to the grid and 55 reactors under construction around the world

Due to the potential risk of the release of radioactivity nuclear power safety is emphasized throughout the stages of design construction operation and decommission of a nuclear power plant (NPP) The principal technical requirements [2] specify that the three fundamental safety functions should be fulfilled for all plant states of an NPP (i) control of reactivity (ii) removal of heat from the reactor and from the fuel store and (iii) confinement of radioactive material shielding against radiation and control of planned radioactive releases as well as limitation of accidental radioactive releases The defence-in-depth concept is also applied to provide several levels of defence to prevent consequences of accidents The fuel claddings reactor pressure vessel (RPV) and primary coolant pressure boundaries and the containment are the three physical barriers to prevent the release of radioactive materials to the environment

Severe accidents are events that cause significant damage to reactor fuel and result in core meltdown It may be caused by insufficient cooling of the reactor core due to various reasons or by the rapid introduction of huge reactivity which causes sharp increases in the power and fuel fragmentation eg Chernobyl accident During a severe accident the failure of the fuel claddings will result in the radioactive materials previously enclosed inside the cladding released to the primary loop If cooling is not sufficient to remove the decay heat the core materials will melt migrate down and relocate into the RPV lower head after which the melt will react with water and partially form fragments quench solidify and form a debris bed The debris bed formed by the melt may continue to melt to form a melt pool due to the decay heat As the corium contacts the RPV it would continuously heat the RPV and threaten its integrity Once the RPV fails containment becomes the last barrier to prevent the release of radioactivity The released melt would result in an increase of the pressure and temperature due to the high temperature fractions oxidation and hydrogen combustion which is called direct containment heating The melt would fall onto the reactor cavity Without sufficient cooling the molten core would interact with basemat concrete (MCCI) This process can last for few days It can result in containment failure due to either containment overpressure or the basemat melt-through of containment In addition containment is also subjected to isolation failure and early containment failure due to missiles and dynamic pressure [3] More

2

detailed severe accident progression or phenomena could also be found in eg [4] [5]

Though severe accidents are unlikely to occur the consequences can be severe if no proper measure is taken Related researches have been continuously going on aiming at mitigating the accident consequences and improving safety After the Fukushima Accident safety inspections and re-evaluation actions have been carried out worldwide especially on safety measurements regarding severe accidents The concept lsquoPractical Eliminationrsquo applied to new NPP designs demands that new nuclear installations be designed with the objective of mitigating the consequences of accidents and avoiding early radioactive releases and large radioactive releases [6] Severe accident mitigation strategies are applied to the existing and new NPPs (both Generation II and III) The strategies can generally be classified into two categories [3] the in-vessel retention (IVR) and ex-vessel retention (EVR) In simple words in-vessel retention tries to retain the core melt inside the RPV (second barrier) by applying external water cooling on the vessel as shown in Figure 1(a) Such method minimizes the dispersal of the melt within the containment and hence simplifies the later recovery work Examples of the application are AP1000 HPR1000 APR 1400 The idea of the EVR is to let the RPV fail but to retain the core melt inside the containment Two principle approaches of EVR are the retention within a dedicated lsquocore catcherrsquo and melt fragmentation Examples of core catcher would be EPR that directs the melt to a spreading compartment (Figure 1(b)) and VVER1000 that uses metallic crucible below the RPV An example of the melt fragmentation is the Nordic BWRs that the corium released from RPV would relocate in the cavity with a deep-water pool (7-12 m) and is expected to fragment quench and form a coolable debris bed (Figure 1(c))

Figure 1 Examples of IVR [7] EVR with core catcher [8] and melt fragmentation [9]

12 RPV Failure

RPV which contains the reactor core core shroud and coolant is a key component to the reactor system RPV and the primary loop constitute the primary coolant system which conveys the fission heat from the reactor core to the secondary loop through steam generator (PWR) or to the turbine (BWR) The pressure under

(a) IVR (b) Core catcher (c) Melt fragmentation

3

operation can be ~155 MPa for a 1000 MW PWR [10] and ~70 MPa for a 1000 MW BWR [11] It also has to withstand the neutron radiation damage The thickness of RPV [11] can be ~20 cm for PWR and ~16 cm for BWR

As the transition of in-vessel progression to ex-vessel progression RPV failure is important for analysis of accident progression and assessment of the effectiveness of severe accident mitigation strategies In the melt fragmentation strategy of Nordic BWR the melt-coolant interaction and debris bed coolability after the corium poured into the deep-water pool are essentially dependent on the vessel failure mode including the characteristic time of melt release rupture size and location and amount and superheat of melt available for release [12] New concern also arose [13] on the necessity of performing structural analysis for the RPV in the IVR strategy In the initial methodology and further development of the IVR the success of the strategy ie the integrity of the RPV was demonstrated (eg in [14]) with the following two conclusions (i) even for a local heat flux on the vessel as high as critical heat flux (CHF) the resulted very thin external ldquocold layerrdquo of the vessel would be sufficient to withstand the internal stress load and hold the vessel integrity (ii) as CHF is the critical value for the thermal load the main specific goal for assessing the effectiveness of IVR becomes to demonstrate that the local heat flux along RPV be lower than the CHF The new concern doubts that local lsquohot partrsquo of the vessel would have negative effects on the overall resistance due to creep and dilatation which could result in a vessel failure and hence violate the first conclusion in particular for ldquolong termrdquo retention of corium in RPV If so a structural integrity analysis of RPV would be needed in addition to assessing local heat flux especially for high-power reactors Therefore accurate prediction of vessel failure could be meaningful for both strategies

Figure 2 Illustration of RPV under thermal loads from corium

As the structural performance is highly affected by the thermal loads from the corium RPV failure is essentially a transient thermo-fluid-structure coupled problem Figure 2 illustrates the situation that a corium debris bedmelt pool formed in the RPV lower head The decay heat of nuclear fission products could continuously be generated inside the corium leading to the formation of melt pools as the debris melts The heated debris bedmelt pool imposes thermal and

4

mechanical loads on the RPV Such loads can be transient (change with time) and spatially non-uniform Complex phenomena are also involved in this process Strong turbulent natural convection induced by the decay heat may be expected in the melt pool under IVR The corresponding Rayleigh number 119877119886 can be 1016minus17 for prototypic reactor cases Material structural properties can become non-linear under high temperature and stress loads The creep deformation also becomes significant at high temperature Detailed modelling of turbulent natural convection and non-linear properties could be computationally expensive In addition large-scale geometry (hemisphere with a radius of 20-30 m) long-term transient deformation process (~10000 s) and required small time steps (eg 001 s) for numerical convergence consideration further increase the computational costs

13 Literature Review Creep Behaviour and RPV Failure Analysis

In this section the most relevant studies concerning the experiments modelling and analysis of creep behaviour and RPV failure are presented

131 Experiments on creep behaviour and RPV failure

Various experiments have been performed on the RPV failure problem some of which were to generate data of mechanical properties (especially the creep properties) and some to study the overall vessel failure with scaled or simplified geometries

The REVISA experiment [15] generated creep curves and stress strain curves of French steel 16MND5 ranging from 600 to 1300 degC This data is accessible in [16] as well as [17] which also documents other material properties eg density thermal expansion coefficient conductivity heat capacity Youngrsquos Modulus and plasticity Similar experiments were carried out to generate high-temperature creep and tensile test data for American RPV steels and Inconel [18]

The CORVIS experiments [19] investigated the lower head failure modes with and without penetrations Each experiment was carried out on a steel plate 100 mm thick representing a pressure vessel lower head The penetration case represented a boiling water reactor lower head carrying a tube penetration in the form of a drain line

The LHF (lower head failure) experiments [20] and the succeeding OLHF tests [21] investigated the vessel failure with a 15 scaled vessel heated by an electrical radiator allowing to locate the hot focus at different position of the vessel with a maximum wall temperature of around 900 degC Internal pressure ranging from 53 to 123 bar was also imposed to initiate the vessel failure

The FOREVER experiments [22] investigated the phenomena of melt pool convection vessel creep possible failure processes and failure modes occurred during the late phase of the in-vessel progression by heating up a melt pool in a vessel and observing possible vessel failures The study covers cases with and without penetration different melt level existence of gap cooling and in-vessel melt coolability

5

132 Numerical studies on RPV failure

Various numerical approaches have been employed on RPV failure Since a two-way thermo-fluid-structure coupling approach is challenging these approaches take simplifications and can generally be classified into two categories (i) RPV failure decoupled from melt pool heat transfer and (ii) RPV failure coupled with melt pool heat transfer in simplified manners

1) Decoupled approaches In the decoupled approaches only RPV is modelled Simplified constant thermal loads are imposed on RPV considering the melt pool heat transfer

In a post-test analysis of FOREVE-C2 experiment [23] it was treated as a thermo-structural coupled analysis of the vessel with the thermal load estimated by the outer thermocouples and the normal heat flux on the external vessel wall The analysis was done using finite element (FE) code SYSTUS A coupled creepdamage law based on a viscosity-hardening creep law describing primary and secondary creep was employed for creep analysis

Mao et al conducted various studies on reactor cases assessing the IVR strategy using FE code ABAQUS Thermal boundary conditions were imposed on the RPV which was then solved as a thermal-structural coupled problem The strain hardening creep model (Eq (1)) was mainly used Their studies cover the general RPV failure [24] as well as the effects of cooling water levels under IVR [25] and geometric discontinuity (ablated RPV) [26]

ε119888119903 = 1198891σ1198892ε1198893119890119909119901 [minus

1198894119879] (1)

In the EU IVMR project [13] the RPV integrity was also studied through a benchmark calculation which deals with the RPV of a PWR with a prescribed ablated profile with 2D axisymmetric geometry Prescribed temperature gradient was set as thermal load The objective was to calculate deformations for various pressure loads up to vessel rupture A smoother profile was planned in further research

The advantage of such approaches is once thermal loads are reasonably assumed the focus only needs to be put on the RPV The disadvantage is a reasonable thermal load should be consistent with the melt pool heat transfer while sometimes it is not well-noted eg constant uniform temperature boundary conditions were assumed for inner and outer RPV (with uniform thickness in lower head) surfaces [25] which ignored the non-uniform heat flux distribution caused by the turbulent convective corium pool As thermal loads strongly affect the structural performance they should be quantified carefully

2) Coupled approaches In coupled approaches both melt pool heat transfer and RPV failure are modelled and coupled in one-way or simplified manner With melt pool heat transfer modelled the coupled approaches are more flexible to provide reasonable thermal loads on the RPV In existing approaches CFD and the correlation-based distributed-parameter models eg the Effective Convectivity Conductivity Model

6

(ECCM) [27] and the Phase-change Effective Convectivity Model (PECM) [28] [29] are general methods for melt pool heat transfer

Sehgal et al [22] presented an approach that employed the MVITA code to calculate the thermal processes and ANSYS for structural analysis MVITA code employed the ECCM Bailey-Norton creep law was used as shown in Eq (2)

ε119888119903 = 6055 times 10minus26 sdot σ472119905085119890minus

37200

119879 (2)

This approach was used for the pre-test simulation of FOREVER experiment to determine the thermal and mechanical loadings Parametric investigations also helped in the identification of important factors that could affect the test procedure and test conditions

Willschutz et al [30] employed the CFD-module Flotran for the melt pool heat transfer and ANSYS for structural analysis of RPV Standard k-ε turbulence model was selected for the turbulent natural convection in the melt pool A user-defined three-stage creep model [31] was employed In this model experimental creep curves were discretized into a creep database (CDB) under given temperature 119879 and stress 120590 the creep curve was discretized into finite point pairs

(휀 ε) indicating under such thermal and mechanical loads the creep strain rate ε is subjected to creep strain 휀 The creep strain rate for any other case can be interpolated over known temperature stress and strain from the database This method was applied to the post-test calculation of the FOREVE-C2 experiment It was concluded that the unexpected deformation behaviour during the experiment was caused by the combination of the transient temperature field in the vessel wall and the three creep stages

They later replaced the Flotran solver with the ECCM for melt pool heat transfer in a following study [32] By implementing it into FEM-solver a recursively coupled analysis was constructed Moreover a modified creep model [33] was used with damage measures the CDB itself did not contain tertiary creep instead a factor 1(1 minus 119863) was multiplied to the original creep strain rate to get the modified creep strain rate where 119863 is a damage factor (0 lt 119863 lt 1) defined with creep strain information and plastic strain information As 119863 grows close to 1 the modified creep strain rate would be enlarged significantly by this factor 1(1 minus 119863) which then behaves like the accelerating tertiary stage This method was applied to pre- and post- test calculations for the FOREVER experiments proving to be closer to reality than the previous method

Madokoro et al [34] developed a thermal structural analysis tool based on open source solver OpenFOAM and numerically investigated the coupled RPV failure and melt pool convection The PECM was used in the melt pool heat transfer In structural part the plasticity and damage parameters were similar to Willschutz et al [32] except for that strain hardening creep model (Eq (3)) was used instead of a CDB Simulations of FOREVER-EC2 and -EC4 were investigated It was concluded from the simulations that although the emissivity and mesh type (updated or fixed) affect the results significantly the approach generally has the capability of predicting the vessel deformation and failure

7

ε119888119903 = 1198891 sdot σ1198892 sdot ε1198893 (3)

Villanueva et al [35] performed a coupled thermo-mechanical creep analysis of the reactor scale RPV (ABB-Atom design BWR) to evaluate efficacy of the cooling measures for the in-vessel melt coolability and retention The PECM was used for melt pool heat transfer and ANSYS for RPV integrity analysis The modified time hardening model (Eq (4)) of the American RPV steel SA533B was selected as creep model The transient thermal load from the melt pool was applied as a thermal boundary condition to the vessel wall Further studies were done with this approach on investigations of instrumentation guide tube (IGT) failure [36] and control rod guide tube (CRGT) cooling [37] of a Nordic BWR

ε119888119903 =1198881σ

11988821199051198883+1

1198883 + 1 1198881 gt 0 (4)

It is noted that only RPV analysis with finite element analysis (FEA) are reviewed here Engineering methods like the Larson-Miller method [38] are not covered in this thesis One can refer to theory manuals of severe accident codes like MELCOR ASTEC for more details Comprehensive description on modelling of melt pool heat transfer is not covered either One can refer to either [39] for a general view or to any severe accident code manuals for engineering methods

14 Motivation Objectives and Main Achievements

The existing experimental and numerical studies provide us a good understanding of RPV failure involved phenomena and valuable simulation experiences These numerical methods may vary in solvers (which may be related to code accessibility) and models (which may be related to model accessibility understanding of phenomena computational power) However improvements of modelling approaches are still possible as the improvements in any of the related aspects eg

bull Simulation tools are continuously being upgraded and providing new possibilities just like that PECM appeared and ECCM was no longer used

bull Better understanding of the phenomena and more simulation experiences from in-vessel retention and vessel failures researches benefit the further modelling and simulations

bull Increasing computational power allows more detailed modelling eg turbulence modelling of melt pool convection

bull There is still room for the improvement of creep modelling

In addition implementation difficulty of a coupling approach is another important issue to consider Since it is a TRANSIENT thermo-fluid-structure coupled problem it is almost impossible to replicate an approach simply by referring to a paper with limited details For BWRs it would be even better for the coupling approaches to support complex geometries considering forest of penetrations (CRGTs and IGTs) on the RPV lower head without significant increase of user effort in the coupling process

8

The objectives of this thesis are to investigate new approaches and develop new models for the thermo-fluid-structure coupled problem of RPV failure and involved phenomena These objectives are accomplished with the following achievements

1) A new creep model for the 16MND5 steel was developed which covers all three creep stages Good agreement was achieved between model predictions and experimental curves Both time and strain hardening models were implemented (Paper I)

2) A new thermo-fluid-structure coupling approach was developed for RPV failure analysis It supports the advanced models in both melt pool convection and structural mechanics to improve fidelity and facilitates implementation Validation of the approach with the new creep model was performed against FOREVER-EC2 experiment This approach is flexible for complex geometries and can be used in further analysis eg lower head with detailed modelling of penetrations in BWRs (Paper II)

3) The performance of this new approach which uses volume loads mapping for data transfer was compared with the previous approach using surface loads mapping showing good agreement with small discrepancy (Paper III)

4) A lumped-parameter code transIVR for melt pool heat transfer was developed which supports one- and two-layer pool configuration transient analysis and well-resolved RPV wall (Paper IV)

5) A hybrid coupling approach was developed for efficient analysis of RPV failure where transIVR is coupled with ANSYS Mechanical (Paper IV)

In the following general descriptions on the codes ANSYS Fluent Mechanical and the PECM are given in Section 2 the new creep model in Section 3 the new coupling approach as well as its comparison with the previous approach in Section 4 the efficient hybrid coupling approach in Section 5 and summary amp outlook in Section 6

9

2 Numerical Solvers and Models

In this section we briefly introduce the ANSYS Fluent as the CFD solver for melt pool heat transfer the ANSYS Mechanical as the mechanical solver for RPV failure and PECM as an efficient alternative for melt pool heat transfer that are heavily used in the following sections

21 ANSYS Fluent

In the fluid mechanics the fluid is treated as a continuum which is generally described by the conservative mass momentum and energy equations [40]

120597120588

120597119905+ 120571 sdot (120588) = 119878119898 (5)

120597(120588)

120597119905+ 120571 sdot (120588) = minus120571119901 + 120571 sdot (120591) + 120588 + (6)

120597120588119864

120597119905+ 120571 sdot ((120588119864 + 119901)) = 120571 sdot (119896120571119879 minussumℎ119895119869119895

119895

+ (120591 sdot )) + 119878ℎ

(7)

Computational Fluid Dynamics (CFD) studies topics on the numerical solutions of above equations Typical methods are the Finite Volume Method (FVM) Finite Difference Method (FDM) and Finite Element Method (FEM) as well as particle methods that are mesh-free eg SPH and MPS

ANSYS Fluent is a general purpose CFD code which solves above equations with FVM Turbulence modelling and multiphase flows are supported Turbulence can be modelled with RANS LES or hybrid LESRANS Customizations eg adding sourcesink time or spatial dependent boundary conditions are realized through the user-defined functions (UDF) which is programmed in C language

22 ANSYS Mechanical

In the continuum mechanics of solids three sets of variables are considered to describe the structure performance namely element displacement field strain field and stress field (119906 ε σ) A general 3-dimensional problem is described by 3 sets of equations the equilibrium equations (Eq (8)) strain-displacement relationship (Eq (9)) and constitute equations (Eq (10)) [41] For Eq (8) the reciprocal theorem of shear stress also gives τ119894119895 = τ119895119894 where 119894 119895 = 119909 119910 119911

indicating the stress tensor being symmetric For very simple applications eg 1D beam it can be solved analytically For complex problems where analytical solutions are not available approximation methods eg Galerkin weighted residual method principle of virtual work and principle of minimum potential energy would be used

10

partσ119909119909part119909

+partτ119909119910

part119910+partτ119909119911part119911

+ 119909 = 0

partτ119910119909

part119909+partσ119910119910

part119910+partτ119910119911

part119911+ 119910 = 0

partτ119911119909part119909

+partτ119911119910

part119910+partσ119911119911part119911

+ 119911 = 0

(8)

ε119909119909 =part119906

part119909 휀119910119910 =

120597119907

120597119910 휀119911119911 =

120597119908

120597119911

120574119909119910 =120597119906

120597119910+120597119907

120597119909 120574119910119911 =

120597119908

120597119910+120597119907

120597119911 120574119911119909 =

120597119908

120597119909+120597119906

120597119911

(9)

ε119909119909 =1

119864[σ119909119909 minus μ(σ119910119910 + σ119911119911)]

ε119910119910 =1

119864[σ119910119910 minus μ(σ119909119909 + σ119911119911)]

ε119911119911 =1

119864[σ119911119911 minus μ(σ119909119909 + σ119910119910)]

γ119909119910 =1

119866τ119909119910 γ119910119911 =

1

119866τ119910119911 γ119911119909 =

1

119866τ119911119909

(10)

ANSYS Mechanical [42] is a mechanical engineering software that uses finite element analysis (FEA) for structural analysis The principle of virtual work is adopted in the solver [43] The idea of this principle is that a virtual (very small) change of the internal strain energy must be offset by an identical change in external work due to the applied loads which reads

δ119880 = δ119881 (11) where 119880 is the strain energy or the internal work 119881 the external work and δ the virtual operator The virtual internal work can be expressed as

δ119880 = int δεσd(vol)T

vol

+int 119886119903119890119886119891

δ119908119899119879σ119889(119886119903119890119886119891) (12)

where 휀 is the strain vector 120590 stress vector vol volume of element 119908119899 motion normal to the surface 119886119903119890119886119891 area of the distributed resistance The virtual

external work is contributed from three parts the inertial effects the pressure force on the surface and nodal forces applied to the element which then can be expressed as

δ119881 = minusint 119907119900119897

δ119908119879119865119886

119907119900119897119889(119907119900119897) + int

119886119903119890119886119901

δ119908119899119879119875119889(119886119903119890119886119901)

+ δ119906119879119865119890119899119889

(13)

11

where 119908 is the vector of displacements 119865119886 acceleration force vector 119875 applied pressure vector 119886119903119890119886119901 area over which pressure acts and 119865119890

119899119889 nodal

forces applied to the element

Plugging Eq (12) and (13) into (11) would result in the equilibrium equation Since the virtual displacement can be arbitrary the final equations used to describe the equilibrium of an element is reduced to Eq (14) with multiple operations

([119870119890] + [119870119890119891])119906 minus 119865119890

119905ℎ = [119872119890] + 119865119890119901119903 + 119865119890

119899119889

(14)

where [119870119890] [119870119890119891] 119865119890

119905ℎ [119872119890] and 119865119890119901119903 are the element stiffness matrix

foundation stiffness matrix thermal load vector mass matrix acceleration vector and pressure vector respectively It represents the equilibrium equation on a one element basis Alternatively using a minimum total potential energy principle could result in a consistent expression to the principle of virtual work [41]

Customizations eg on material properties are realized through the user programmable features (UPF) which is programmed in FORTRAN language

23 PECM

The PECM [28] [29] developed in KTH is a correlation-based method to model the melt pool heat transfer It is implemented in ANSYS Fluent through UDF and has been validated against many experiments such as ACOPO COPO LIVE etc In the PECM the flow velocities 119906119909 119906119910 and 119906119911 are replaced by characteristic

velocities 119880119909 119880119910 and 119880119911 respectively in order to eliminate the necessity of solving

the Navier-Stokes equation

The characteristic velocities are calculated using heat transfer correlations

119880119906119901 =α

ℎ119901119900119900119897times (119873119906119906119901 minus

ℎ119901119900119900119897

ℎ119906119901) (15)

119880119889119899 =α

ℎ119901119900119900119897times (119873119906119889119899 minus

ℎ119901119900119900119897

ℎ119889119899) (16)

119880119904119889 =α

ℎ119901119900119900119897times (119873119906119904119889 minus

ℎ119901119900119900119897

119882119901119900119900119897) (17)

where ℎ119901119900119900119897 is the melt pool depth ℎ119906119901 the thickness of well mixed layer of the

pool ℎ119889119899 the thickness of lower stratified region of melt pool 119882119901119900119900119897 the width of

pool 120572 the thermal diffusivity The PECM employs reduced characteristic velocities to describe mushy zone convection heat transfer For internally heated pool natural convection the Steinberner-Reineke correlations (Eq (18)~(19)) [44] are employed The profile of sideward 119873119906 is described using Eckertrsquos type correlation (Eq (20)) for a vertical boundary layer where the local Rayleigh

number at pool height 119910 is defined as 119877119886119910 =119892βΔ1198791199103

αν

12

119873119906119906119901 = 03451198771198861198941198990233

(18)

119873119906119889119899 = 13891198771198861198941198990095

(19)

119873119906119904119889 = 050811987511990314 (

20

21+ 119875119903)

minus14

11987711988611991014

(20)

As the characteristic velocities are calculated the energy equation is the only transport equation that needs to be solved and can be expressed as follows

partρ119862119901119879

part119905+ nabla(ρ119862119901119932119879) = 119896nabla2119879 + 119876119907 (21)

where from left to right are the transient term convection term diffusion term and volumetric heat source respectively The convection term then can be calculated explicitly with use of values in the last time step and then can be treated together with the volumetric source term Therefore the PECM finally simplifies the energy equation to a conduction equation which can be easily solved and as a result will significantly increase the computational efficiency in simulations

In addition solidificationmelting process can also be modelled and the temperature-dependent physical properties are supported

13

3 Creep Modelling of 16MND5 Steel

Creep deformation becomes significant when a component is subjected to high temperature and stress Modelling of creep behaviour of the RPV is paramount since it is considered as a main mechanism leading to the RPV failure In this section we present the development of a three-stage creep model (called lsquomodified theta projection modelrsquo ) for the French RPV steel 16MND5 This model was successfully implemented into the ANSYS Mechanical code

31 Creep and Creep Modelling

Creep is an irreversible time-dependent inelastic deformation process Diffusion and dislocation are the two molecular mechanisms of creep The former normally occurs due to the presence of vacancies within the crystal lattice (or the grain) while the latter occurs if there is a slip by the passage of crystal dislocations [45] There are two types of diffusion creep [46] Coble creep (favoured at lower temperatures) if the diffusion paths are predominantly through the grain boundaries and Nabarro-Herring creep (favoured at higher temperatures) if through the grains themselves Dislocation creep tends to dominate at high stresses and relatively low temperatures Dislocations can move by gliding in a slip plane which is a process that requires little thermal activation But the rate-determining step for their motion is often a climb process which also requires diffusion and is thus a time-dependent process and favoured by higher temperatures Creep behaviour is highly affected by temperature and stress generally creep is more significant under higher temperature and stress

Creep curves describe the creep strain development with time Figure 3 shows a creep curve under constant temperature and stress which typically experiences three creep stages primary stage where the creep strain rate ε119888119903 (the change of creep strain ε119888119903 in unit time) decreases with time secondary stage where the creep strain rate almost keeps constant and tertiary stage where the creep strain rate drastically increases with time

Figure 3 Creep curve under constant temperature and stress

14

The built-in creep models in commercial software eg ANSYS [47] and ABAQUS [48] in default only cover the primary andor the secondary stage(s) Table 1 [47] gives the default creep models supported in ANSYS for example Possible reason why the tertiary stage is not modelled might be that for most industrial applications the failure analysis is beyond interest and therefore the built-in creep laws in the commercial software was supposed to be sufficient [49] Though the two codes offer the interfaces through either ANSYS UPF or ABAQUS user subroutines to support more general user-defined creep laws eg complex models that also include the tertiary creep stage

Table 1 Default creep models in ANSYS Mechanical

Creep model Equations Type

Strain hardening

ε119888119903 = 1198621σ1198622ε119888119903

11986231198901198624119879 1198621 gt 0 Primary

Time hardening ε119888119903 = 1198621σ11986221199051198623119890minus1198624119879 1198621 gt 0 Primary

Generalized exponential

ε119888119903 = 1198621σ1198622119903119890minus119903119905 r = 1198625σ

11986231199051198623119890minus1198624119879 1198621 gt 0

1198625 gt 0

Primary

Generalized Graham

ε119888119903 = 1198621σ1198622(1199051198623 + 1198624119905

1198625 + 11986261199051198627)119890minus1198628119879 1198621 gt 0 Primary

Generalized Blackburn

ε119888119903 = 119891(1 minus 119890minus119903119905) + 119892119905 119891 = 1198621119890

1198622σ 119903 = 1198623(σ1198624)1198625 119892 = 1198626119890

1198627σ

1198621 gt 0

1198623 gt 0

1198626 gt 0

Primary

Modified time hardening

ε119888119903 = 1198881σ11988821199051198883+1(1198883 + 1) 1198621 gt 0 Primary

Modified strain hardening

ε119888119903 = 1198621σ1198622[(1198623 + 1)ε119888119903]

1198623119890minus1198624119879 1198621 gt 0 Primary

Generalized Garofalo

ε119888119903 = 1198621[119904119894119899ℎ(1198622σ)]1198623119890minus1198624119879 1198621 gt 0 Secondary

Exponential form

ε119888119903 = 1198621119890σ1198622119890minus1198623119879 1198621 gt 0 Secondary

Norton ε119888119903 = 1198621σ1198622119890minus1198623119879 1198621 gt 0 Secondary

Combined time hardening

ε119888119903 = 1198621σ11986221199051198623+1119890minus1198624119879(1198883 + 1)

+ 1198625σ1198626119905119890minus1198627119879

1198621 gt 0

1198625 gt 0

Primary+ Secondary

Rational polynomial ε119888119903 = 1198621

partε119888part119905 ε119888 =

119888119901119905

1 + 119901119905+ ε119898119905

ε119898 = 1198622101198623σσ1198624 119888 = 1198627ε

1198628σ1198629

119901 = 11986210ε11986211σ11986212

1198622 gt 0 Primary+ Secondary

Generalized time hardening

ε119888119903 = 119891119905119903119890minus1198626119879 119891 = 1198621σ + 1198622σ2 + 1198623σ

3 119903 = 1198624 + 1198625

Primary

32 A Modified Theta Projection Model

321 Original theta projection model and its variations

The original theta projection model [50] was first proposed to accurate predict long-term creep and creep-fracture properties It assumes that a creep curve under constant temperature and stress can be described as

15

휀119888119903 = 1205791(1 minus 119890minus1205792119905) + 1205793(119890

1205794119905 minus 1) (22)

where 휀119888119903 is the creep strain 119905 the time and 1205791~1205794 the model parameters Figure 4 illustrates how the first term and second term on right hand side develop with time and constitute a creep curve which has obvious primary secondary and tertiary stages Model parameters 120579119894(119894 = 1234) are temperature and stress dependent and are described with the following expression

119897119899thinsp120579119894 = 119886119894 + 119887119894120590 + 119888119894119879 + 119889119894120590119879 (23)

where 119886119894 119887119894 119888119894 and 119889119894 ( 119894 = 1234 ) are constants It is noted [45] that this equation is suitable when the temperature range of experiment data sets is very narrow such that minus1119879 is almost proportional to 119879 These constants (119886119894 119887119894 119888119894 and 119889119894 (119894 = 1234)) are the final model parameters which are generated with two fitting procedures

(i) Fitting the experimental creep curves to Eq (22) generates several groups of 1205791~1205794 for each creep curve under certain temperature and stress We denote (119879119895 120590119895 1205791119895 1205792119895 1205793119895 1205794119895 ) as parameters for the 119895119905ℎ

experimental curve under temperature 119879119895 and stress 120590119895

(ii) Fitting 1205791 to Eq (23) as a function of temperature and stress with the obtained data ( 119879119895 120590119895 1205791119895 ) from ( 119879119895 120590119895 1205791119895 1205792119895 1205793119895 1205794119895 ) generates

parameters 1198861 1198871 1198881 and 1198891 Repeat this fitting procedure for 1205792 1205793 and 1205794 to get parameters 119886119894 119887119894 119888119894 and 119889119894 (119894 = 234)

Variations of the model were developed and applied to other different materials to better accommodate the experimental data which are all then called a lsquomodified theta projection methodrsquo independently Examples are models in [51] for lowhigh alloy ferritic austenitic steels Al-alloys Al-matrix composites [52] for Nickel-based Super-alloy [53] for pure metals and class M alloys with intermediate range of stress (40-70 of the yield stress) and temperature (40-60 of the melting temperature)

Figure 4 Illustration of theta projection method [54]

16

322 A modified theta projection model

The new modified theta projection model we proposed [54] was mainly adapted from the original theta projection method [50] and the modified version [51] The new model describes the creep curve with the following equation

휀119888119903 = 1205791(1 minus 119890minus1205792119905) + 120579119898119905 + 1205793(119890

1205794119905 minus 1) (24)

An additional term 120579119898119905 is seen compared with Eq (22) in the original method The reason for adding this term is that the 16MND5 steel shows fairly long secondary stages in many experimental creep curves and the creep curves can be better fitted by Eq (24) than Eq (22) As the structure undergoes different creep process under different temperature and stress the model parameters θ119894 (119894 =1~4 and 119898) are also temperature and stress dependent

The main difference compared with the original and all other modified projection models is that numerical interpolation is used to replace the second numerical fitting in previous models In other words instead of fitting the model parameters as functions of stress and temperature with Eq (23) the distributions of the model parameters are assumed to be piecewise in the temperature-stress space and they will be interpolated directly from known parameters obtained from experiments The consideration is that the experimental data ([16] [17]) for 16MND5 covers a wide temperature range varying from 600 degC to 1300 degC and a wide stress load range varying from 08 MPa to 248 MPa while Eq (23) in the existing methods is based on a narrow temperature span Poor parameter fitting results would be expected if we directly fit the parameters 1205791~1205794 with Eq (23) as used in the original methods

The creep model was developed using the following steps

(i) Fitting Each creep curve can be represented by a group of model parameters [1205791 1205792 120579119898 1205793 1205794] with respect to Eq (24) by fitting the creep curves to Eq (24) All parameters of the 32 experimental curves form an

initial dataset of the model parameters [119879119895 120590119895 1205791119895 1205792119895 120579119898119895 1205793119895 1205794119895] where j (0ltjle32) is the jth creep curve

(ii) Interpolation Parameters of a creep curve under any [119879 σ] are directly interpolated from the dataset

(iii) Parameter adjustment The dataset was further updated to satisfy the monotonicity assumption we have taken that creep strain monotonically increases with temperature and stress

(iv) Different hardening models for dynamic loads

Step (i) was done using MATLAB In the following steps (ii)-(iv) are described in detail

Model parameters for new creep curves are interpolated from the dataset over the 2D 119879 minus 120590 space The interpolation is divided into two sub-interpolation steps interpolate over stress with fixed temperature and then interpolate over temperature with fixed stress We take the interpolation of model parameters under (119879prime σprime) for example In the first interpolation we first find out the two

17

temperatures 119879119896119899119900119908119899119904119898119886119897119897 and 119879119896119899119900119908119899119897119886119903119892119890 most close to 119879prime from the experimental

temperature set 600 700 800900 1000 1100 1200 1300 Under each temperature there are four sets of model parameters in the dataset corresponding to four different stress loads Then the first interpolation is done over stress load from the known dataset of model parameters under each

temperature to output the parameters for (119879119896119899119900119908119899119904119898119886119897119897 120590rsquo) and (119879119896119899119900119908119899119897119886119903119892119890 120590rsquo)

In the second interpolation the model outputs parameters for (119879rsquo 120590rsquo) from

(119879119896119899119900119908119899119904119898119886119897119897 120590rsquo) and (119879119896119899119900119908119899119897119886119903119892119890 120590rsquo) For 120579i(119894 = 12 3 4) we assume a logarithm

relation between 120579119894 temperature 119879 and stress 120590 which is similar to Eq (23)

119897119899120579119894 = (119886 + 119887120590)(119888 + 119889119879) (25)

It reduces to 120579119894 = 11986011198901198871σ with fixed temperature in the first interpolation and to

θ119894 = 11986021198901198872T with fixed stress in the second interpolation The interpolation

expression for θ119894 = 11986011198901198871σwith two known points (1205901 1205791198941) and (1205902 1205791198942) is given

by Eq (26) The interpolation expression for θ119894 = 11986021198901198872T with two known points

(1198791 1205791198941) and (1198792 1205791198942) is given by Eq (27)

120579119894 = 120579i1

120590minus12059021205901minus1205902120579i2

1205901minus1205901205901minus1205902 (26)

120579119894 = 120579i1

119879minus11987921198791minus1198792120579i2

1198791minus1198791198791minus1198792 (27)

Similarly we assume the following relation between 120579119898 temperature 119879 and stress 120590

120579119898 = 119860120590119899119890minus119887 119879frasl (28)

The interpolation is done over the stress with 120579119898 = 1198603120590n by fixing temperature

and then over temperature with 120579119898 = 1198604119890minus1198874119879 by fixing stress The interpolation

expression over stress for 120579119898 = 1198603120590119899 with two known points (1205901 1205791198981) and

(1205902 1205791198982) would be given by Eq (29) The interpolation expression over

temperature for 120579119898 = 1198604119890minus1198874119879 with two known points (1198791 1205791198981) and (1198792 1205791198982)

would be given by Eq (30)

120579119898 = 119890119909119901(119897119899120590 minus 11989711989912059011198971198991205902 minus 1198971198991205901

1198971198991205791198982 +1198971198991205902 minus 119897119899120590

1198971198991205902 minus 11989711989912059011198971198991205791198981) (29)

120579119898 = 120579m2(11198791minus1119879) (

11198791minus11198792)fraslsdot 120579m1

(1119879minus11198792) (

11198791minus11198792)frasl (30)

In this model we also take the assumption that the creep strain monotonically increases with temperature and stress loads1 Or specifically with the same creep time the structural component under higher temperature or stress experiences

1 Intuitively materials deform more under higher stress and temperature This is also mostly observed in tensile experiments Though some exceptions exist one instance of which is the effect of phase change some materials under certain temperature around the phase change temperature may show much weaker structural performance Here we just simply assumed that the creep strain monotonically increases with temperature and stress

18

higher creep strain than that with lower temperature or stress (the creep curve under higher temperature or stress should be above the curve with lower temperature or stress) When generating the parameter dataset (Step (i)) we already kept the trend that under the same temperature cases under higher stress loads would have larger creep model parameters and give higher creep strains This trend would also hold for the interpolated temperatures which thus guarantees that creep strain monotonically increases with stress load under same temperature Then the question becomes if creep strain also monotonically increases with temperature given fixed stress However the answer could not always be a yes especially in the extrapolation range of the stress load Figure 5(a) illustrates an example of the interpolation of model parameters under (119879119872 σ119872) The red dots and blue dots are known parameters from the parameter dataset for corresponding high temperature 119879119867 and low temperature 119879119871 respectively The green dots (119879119867 σ119872) and (119879119871 σ119872) are interpolated from the red and blue dots in the first sub-interpolation respectively The black dot (119879119872 σ119872) is interpolated from the green dots in the second sub-interpolation It is seen that when σ119872 lt σ119862 the value of (119879119871 σ119872) is higher than that of (119879119867 σ119872) It means the higher 119879119872 is the smaller the model parameter will be ie the model would predict larger creep strain for lower temperatures which violates our monotonicity assumption One simple solution used here as shown in Figure 5(b) is to minorly adjust the values of the four red and blue dots such that the intersection point C can be sufficiently small and the situation σ119872 lt σ119862 would barely occur in the extrapolation range

Figure 5 Illustration of model parameter interpolation (a) before and (b) after parameter adjustment [54]

A corresponding parameter adjustment process was introduced to verify that the model predicts higher creep strain under higher temperature under fixed stress Figure 6 illustrates the general verification process for any (119879 120590) in the interested load range the creep strain of (119879 + 120575119879 120590) should be larger than that of (119879 120590) Otherwise parameters would be minorly adjusted and the verification process be repeated This process would be iterated several times until the requirement is reached The achieved verification shows

Stress

σC

High temperature TH

MPa

a

0

Model

par

amet

er θ

Low temperature TL

MPa

a

0

Model

par

amet

er θ

High temperature TH

Low temperature TL

σC

Stress

C C

(a) (b)

σM

σM

19

(i) For any (119879 120590) in the temperature range 273~ 1073 K with an interval of 1 K and the stress range 1 ~260 MPa with an interval of 1 MPa its creep strain is higher than that of (119879 minus 1119870 120590) for the first 12000000 s

(ii) For any (119879 120590) in the temperature range 1073~1673 K with an interval of 1 K and the stress range 1 ~80 MPa with an interval of 1MPa its creep strain is higher than that of (119879 minus 1119870 120590) for the first 12000000 s

(iii) For any (119879 120590) in the stress range 01~26 MPa with a smaller interval of 01 MPa for the full temperature range 273~1673 K with an interval of 1 K its creep strain is higher than that of (119879 minus 1 120590) for the first 12000000 s

Above verification covers a wide range of temperature and stress especially where the stress and temperature are low and extrapolation is used Further verification to cover even higher stress or temperature ranges is not considered as creep increases drastically under such high stress and temperature and the results would become meaningless With these regards we generally think the monotonicity requirement is reached The model parameter dataset was updated along with the adjustment The final dataset was given as Table 2 in Paper I

Figure 6 Parameter adjustment process [54]

Figure 7(a-d) present the model predictions of selected creep curves in ANSYS Mechanical (labelled as lsquoANSYSrsquo) compared with experimental curves (labelled as lsquoREVISArsquo) at temperature 600 800 1000 and 1200 degC respectively The predicted curves proved to clearly show the three-stage creep process and agreed well with the experimental creep curves

Initial parametersdataset

θ1 θ2 θm θ3 θ4 from creep

curve-fitting

Parameters interpolation

Is the monotonicity

assumption satisfied

IsMon= =1

Final model parameters

Adjust the

parametersdataset

θ1 θ2 θm θ3 θ4

No

Yes

IsMon=1 a flag is monotonic achieved

For σ in the stress range in MPa

For T in range Tstart Tend in K

For t in range 112000000 in s

If ε(σ T+1t)lt ε(σTt) IsMon=0

Iteration loop of

parameter adjustment

20

Figure 7 Selected model predictions compared with experimental creep curves [54]

Instead of being constant temperature and stress loads also dynamically change with time in real cases Hardening models determine how creep strain rate reacts to the change of loads during a creep process The time hardening model and strain hardening model2 are supported in our model implementation in ANSYS Mechanical which are two mostly used hardening models

Time hardening assumes that the following creep strain rate after a load change depends only on the current creep time from the beginning of the creep process (and also temperature and stress if used in the creep model) and ignores the current creep strain Strain hardening assumes that the creep strain rate depends only upon the current creep strain (and also temperature and stress if used in the creep model) [55]

2 The time hardening and strain hardening here are more general terms than the specific time hardening creep model and the strain hardening creep model mentioned in Table 1

(a) 600 degC (b) 800 degC

(c) 1000 degC (d) 1200 degC

21

Eq (24) in default is a time hardening model as creep strain is directly described by the time 119905 Eq (31) gives the creep strain increment 119889휀119888119903 at time 119905 during time interval 119889119905

119889휀119888119903 = (12057911205792119890minus1205792119905 + 120579119898 + 12057931205794119890

1205794119905)119889119905 (31)

The strain hardening model is derived from Eq (24) in an implicit form Specifically when temperature and stress load change to (119879 120590) at the time 119905 and current creep strain 휀1198881199030 the model will get the creep curve of constant (119879 120590) and

output how this curve will react under the current strain 휀1198881199030 Since the creep curve shows creep strain monotonically increases with time given the creep strain 휀1198881199030 under creep curve of (119879 120590) we can uniquely find this reference time

119905119903119890119891 satisfying Eq (32) One should notice that the reference time 119905119903119890119891 can be

different from the current physical time 119905

휀1198881199030 = 1205791(1 minus 119890minus1205792119905119903119890119891) + 120579119898119905119903119890119891 + 1205793(119890

1205794119905119903119890119891 minus 1) (32)

where all 120579119894(119894 = 1~4119898) are model parameters of the new creep curve (119879 120590) It can be solved for 119905119903119890119891 with methods eg Newton-Ramphson [56] Then the creep

strain increment at 119905119903119890119891 during time interval 119889119905 is given as

119889휀119888119903 = (12057911205792119890minus1205792119905119903119890119891 + 120579119898 + 12057931205794119890

1205794119905119903119890119891)119889119905 (33)

Therefore combining Eq (32) and (33) gives the creep strain rate at creep strain 휀1198881199030 In summary when loads are changed to (119879 120590) with current creep strain 휀1198881199030

the model traces the creep curve of (119879 120590) and calculates the creep strain rate of this curve at strain 휀1198881199030

One calculation was done to demonstrate the performance of the hardening models The stress load was set to 151 MPa and kept unchanged A step temperature load was applied as the orange dash curve shows in Figure 8 it was 900 degC during 0-36000 s and suddenly jumped to 1000 degC at 36000 s which was then kept till the end of the simulation The blue solid curve (the part before the temperature change was identical to and therefore covered by the green solid curve in the figure) and blue dash curve are the creep curves under constant temperature T=900 degC and T=1000 degC respectively The green solid curve and green dash curve represent the model predictions considering this temperature change with time hardening model and strain hardening model respectively Before this change the creep curves with both hardening models were identical to the creep curve of the T=900 degC till point A After that the creep strain rate then shifted from Point A to Point B (which had the same time with A) in the time hardening model and behaved the same as the curve T=1000 degC as shown in the rest part of the green solid curve In the strain hardening model the creep strain rate shifted from Point A to Point C (which had the same creep strain with A) and behaved the same as the curve T=1000 degC as shown in the rest part of the green dash curve The large difference between two models was partially caused by the drastic temperature jump In real cases the difference would be less significant as the temperature changes more mildly Overall this prediction is exactly the expected behaviors of the time hardening model and strain hardening model

22

Figure 8 Time hardening and strain hardening in ANSYS Mechanical [54]

23

4 Coupled Thermo-Fluid-Structure Analysis of RPV Failure

As described in Section 13 RPV failure is coupled with melt pool heat transfer leading it into a transient thermo-fluid-structure coupled problem The goal of this section is to develop and validate a reliable coupling approach which supports transient coupled thermo-fluid-structure analysis as well as advanced models and computational platforms so as to improve fidelity and facilitate implementation The approach was validated against the FOREVER-EC2 experiment and then applied to an RPV failure analysis of a Nordic BWR As the approach adopts the volume loads mapping for load transfer comparative analysis was also performed against the coupling approach using surface loads mapping Future work is considered in analysis of penetration failures (IGTs and CRGTs) of a BWR RPV since the approach is flexible to complex geometries Results of Paper II ampIII are jointly presented in this section

41 Thermo-Fluid-Structure Coupling Approach

Considering the involved phenomena a coupling approach should at least address three issues (i) melt pool heat transfer (ii) RPV deformation and (iii) the interaction between melt pool heat transfer and vessel deformation A two-way strong coupling considering both the transient thermal load from melt pool on vessel and the influence of vessel deformation on melt pool heat transfer involves complex data exchange and is still full of challenges Therefore the new coupling approach is realized in a one-way manner with the assumptions that the influence of vessel deformation on melt pool heat transfer can be ignored and thus only the thermal loads from melt pool to vessel are considered It is based on the considerations that on one hand the deformation is a slow process regarding the relatively long failure time and on the other hand it was pointed out [57] that the physics of natural convection heat transfer in internally heated liquid pools is not sensitive to the geometrical factors based on their calculation results and experimental data from [58] [59] as well as previous experiments in semicircular and square cavities eg [44] [60] In addition one-way coupling is also computationally more efficient especially when heat transfer turbulence structural non-linearity and large-scale geometries are involved The developed approach is based on the ANSYS platform which generally consists of three parts with respect to the three issues respectively

(i) conjugate heat transfer of the melt pool and vessel using ANSYS Fluent (ii) transient thermal loads transfer from ANSYS Fluent to ANSYS

Structural (the ANSYS Mechanical in ANSYS Workbench) (iii) structural analysis of the vessel in ANSYS Structural

ANSYS Structural in default only supports direct transfer of constantsteady-state spatial loads from ANSYS Fluent Transfer of transient loads is realized with the extension tool FSI Transient [61] It is a pre-processing tool (with good documentation) developed by a third-party CAE vendor EDRTM MEDESO [62] to map transient CFD results (heat transfer coefficient temperature and surface pressure) to thermal or structural analysis including the definition of load steps

24

This new procedure proves to be straightforward and reduces efforts in the load transfer

Since structural performance of RPV is highly dependent on temperature the transfer of transient temperature loads from conjugate heat transfer to mechanical solver is a core step of the coupling framework There are two general approaches to realize the transfer as shown in Figure 9 Both approaches contain three parts a conjugate heat transfer analysis for fluid and structure where flow and heat transfer are coupled a mechanical analysis for the structure and a load mapping process that transfers the transient loads The conjugate heat transfer part can be solved with any CFD solver and is treated in the same manner in both approaches In the first approach (Figure 9(a)) transient volume thermal loads are transferred and mapped into the mechanical simulation domain The structure domain reads the essential thermal loads from the mapping after which a structure analysis of this domain is performed In the second approach (Figure 9(b)) transient surface thermal loads are mapped onto the boundary surfaces of structure domain as thermal boundary conditions Then a thermo-structure coupled analysis of the structure domain in the mechanical part is performed To distinguish them we call the former approach volume loads mapping (VLM) and the latter surface loads mapping (SLM) In the RPV failure analysis the fluid and structure domains correspond to the melt pool and RPV respectively Examples of these methods are the studies of [63] and [30] using VLM and [35] using SLM The VLM is adopted in current approach

Figure 9 Frameworks of the thermo-fluid-structure coupling approaches (a) volume loads mapping (b) surface loads mapping [64]

fluid structure structure

fluid structure structure

Conjugate heat transfer problem Mechanical problem

(structure)

Conjugate heat transfer problem Mechanical problem

(thermo-structure coupled)

Transient volume

loads mapping

(VLM)

Transient surface

loads mapping

(SLM)

(a)

(b)

25

42 Numerical Modelling

421 Conjugate heat transfer

In the conjugate heat transfer analysis main focus is put on the modelling of melt pool heat transfer Within the ANSYS Fluent solver two options are supported CFD with turbulence modelling and PECM CFD is the choice when the model goes to detail and more mechanistic analysis are needed while PECM provides an alternative that is faster and possibly more suitable for large-scale problems

Within the scope of RANS models the SST (Shear Stress Transport) model [65] is selected as the turbulence model in present study This model is a combination of k-ε model and k-ω model The k-ω formulation is used in the inner parts of the boundary layer which makes the model directly usable all the way down to the wall through the viscous sub-layer It also switches to k-ε model in the free-stream and thereby avoids the common k-ω problem that the model is too sensitive to the inlet free-stream turbulence properties The considerations on choosing this model are

bull Several two-equation low-Re k-ε models have been proved in [66] not good enough for melt pool heat transfer Therefore these mentioned two-equation models as well as other one-equation models are not considered

bull SST is highly recommended [67] for simulating most industrial heat transfer problems

bull Convincing results ([68] [69]) using SST have been obtained in capturing the thermal behaviour of the natural convection in an internally heated fluid Especially good agreements were achieved between the predicted heat flux distribution along the vessel and experimental data which would be one of the most important factors to estimate model performance in this problem

The description of PECM as an effective model for melt pool heat transfer is given in Section 23

422 Mechanical analysis

The material for the RPV failure analysis considered in this study is the French RPV steel 16MND5 The three-stage modified theta projection model in Section 3 is used as the creep model Since the definition of an RPV failure criterion is not universal (it can be the time when damage variable reaching 1 or either specified value of creep strain creep strain rate or total deformation is reached) we do not explicitly define a failure criterion Instead the simulation is kept running until the creep converge criterion cannot be satisfied even with the defined minimum time step (indicating a fairly large creep strain rate) and the simulation will be automatically terminated The time the simulation stops is taken as the failure time This treatment implicitly implies a criterion of a large creep strain rate We refer to [17] for other material properties Youngrsquos modulus and thermal conductivity are temperature dependent Multilinear Isotropic Hardening (MISO) is used as the plasticity model

26

43 Validation against the FOREVER-EC2 Experiment

In this section the validation against the FOREVER-EC2 experiment is presented (indicated as lsquoVLMrsquo in the following) In addition the simulation results using the SLM approach (indicated as lsquoSLMrsquo in the following) is also included for comparison purpose

431 FOREVER-EC2 experiment

The FOREVER-EC2 experiment [70] is one of the FOREVER experiments that achieved a vessel failure Figure 10 shows the experimental setup A 110 scale pressure vessel with a wall thickness of 15 mm and an outer diameter of ~400 mm was employed The lower head was made of 16MND5 steel Since it was difficult to use prototypical corium materials in the experiment the binary salt 30 CaO-70 B2O3 was employed as simulant To generate the internal heat which corresponds to the nuclear decay heat a specially designed heater was used The power input was maintained ~38 kW during the experiment such that maximum melt pool temperature would be ~1300 degC The pressure inside the vessel was maintained at ~25 MPa Thermocouples were used to measure the melt pool and vessel wall temperatures and linear displacement transducers used to measure the vessel wall displacements The designed experimental conditions were reached at experimental time 11670 s Vessel failure was observed about 12510 s later (at experiment time 24180 s)

Figure 10 Scheme of the experimental setup [22]

27

Figure 11 Computational domains and boundary conditions (a) conjugate heat transfer (b) mechanical analysis [71]

Computational domains and boundary conditions are indicated in Figure 11 SST turbulence modelling was used for pool heat transfer The simulations started at the moment when experimental conditions were reached (experiment time 11670 s) So simulation time 0 s corresponds to experiment time 11670 s

432 Conjugate heat transfer

Figure 12 shows the temperature fields at simulation time 0 s 1225 s 2475 s 4975 s and 9975 s All the time points in this part refer to simulation time The field at time 0 s (Figure 12(a)) was the initial temperature that was reconstructed based on the thermocouple readings The temperature increased with time and reached a steady state when the maximum temperature reached ~1607 K Figure 12(b-f) show clear thermal stratification which is an expected phenomenon of natural convection

Internal pressure

25 bar

Internal pressure 25e6-2500times981timesy Pa

External pressure

1 bar

Zero displacement

y=0 m

Gravity 400

mm

R=203 mm

15 m

m

Tamb

=323 K radiation with ε = 08

+ convection with h=10Wm2K

Tamb

=800 K radiation with ε=02

(a) (b)

Tamb

=800 K radiation with ε=08

+ convection with h=10 Wm2K

28

Figure 12 Temperature fields at simulation time (a) 0 s (b)1225 s (c) 2475s (d) 4975 s (e) 7475 s and (f) 9975 s [64]

Figure 13 shows the temperature profiles of the external vessel along the polar angle in comparison with experimental data Regions with a polar angle larger than 90deg are those in the cylindrical part of the vessel It also shows the temperature profile increased with time until reaching a steady state The profile almost did not change after 4975 s which indicates that a steady state of the temperature was reached As time increased the temperature in the lower head region 0~90deg (where the vessel was contacted with the melt) evolved from a relatively flat distribution to a distribution that temperature at high angle was higher than that at low angle by more than 200 K Generally the steady state results agreed well with the experimental data except for the low angle part of vessel lower head and the high angle part of the cylindrical vessel In addition to modelling errors the discrepancy in the low angle part of the lower head may be caused by the crust and gap possibly formed at vessel bottom in the experiment which can reduce the downward heat transfer and result in lower temperature while they cannot be modelled in simulations As to the high angle cylindrical vessel part simplified boundary conditions were applied to the inner and outer vessel wall since it was far from the melt and not of interest which may result in some discrepancy

(a) (b)

(c) (d) (e) (f)

29

Figure 13 Temperature profiles of external vessel surface along the vessel polar angle [64]

433 Mechanical analysis

In the mechanical analysis the VLM predicted a failure time of 23959 s and the SLM predicted a failure time of 22445 s both are close to the experimental failure time 24180 s All the time points in this part refer to the experiment time Figure 14 shows the creep strain fields at failure time Both approaches predicted maximum creep strains at same location where the temperature was high (cf Figure 12) This is related to the fact that creep behavior is highly influenced by temperature Maximum values for both cases were also close to each other 0393 in VLM and 0366 in SLM

Figure 15 presents the total deformation of the vessel bottom point (also called ldquosouth polerdquo in some literature) in the vertical direction Since there was an initial displacement reading of ~355 mm at time 0 s which almost did not change during 0~11670 s (the whole period before the experimental conditions were reached) an initial value of 355 mm was added to the simulation result such that experimental and simulation results had the same total displacement at t=11670 s Similar trend was seen in the VLM SLM and experimental data that the deformation increased with time and experienced a drastic acceleration at the end Generally both simulation approaches agreed well with the experimental data though they both slightly under predicted the value of total deformation The discrepancy may possibly be related to the accuracy of the thermal predictions modelling errors of plasticity and creep and simplified boundary conditions

400

600

800

1000

1200

1400

1600

0 20 40 60 80 100 120 140 160

Tem

per

ature

(K

)

Polar angle (deg)

Experiment

25 s

1225 s

2475 s

4975 s

7475 s

9975 s

Weilding line

30

Figure 14 Creep strain fields at failure time (a) VLM (b) SLM [64]

Figure 15 Total deformation in vertical direction of the vessel bottom point [64] [71]

The wall thickness changes are plotted together with experimental results in Figure 16 The wall thickness change δ119909 is defined as

δ119909 = 119909119890119899119889 minus 119909119900119903119894119892119894119899119886119897 (34)

000

500

1000

1500

2000

2500

11670 13670 15670 17670 19670 21670 23670

Tota

l def

orm

atio

n (

mm

)

Time (s)

ExperimentVLMSLM

(a) (b)

31

where 119909119890119899119889 and 119909119900119903119894119892119894119899119886119897 are the wall thickness at failure time and the original wall

thickness respectively Since 119909119890119899119889 cannot directly be post-processed it was estimated in the following way (cf the subfigure of Figure 16) for the same polar angle θ find the corresponding points Pin (1199091 1199101) and Pout (1199092 1199102) on the inner and outer surfaces of the deformed vessel respectively the distance of Pin and

Pout radic(1199091 minus 1199092)2 + (1199101 minus 1199102)2 is calculated as 119909119890119899119889 at polar angle 120579 Such estimation of the wall thickness is more accurate when the line PinPout is closer to be orthogonal to the deformed local surface Considering the final deformed vessel as shown in Figure 14 the estimation should be fine at regions close to 0deg and 90deg and can be slightly overestimated at medium angle region The VLM curve was compared with the SLM curve and 4 experimental curves measured at vessel left right front and back sides Generally VLM curve agreed well with the experimental and SLM curves wall thickness gradually decreases with angle till around 75deg and then increases till 90deg the maximum thickness change occurred at around 75deg would be around -5 mm The positive values at low angle may be caused by the thermal expansion It also shows that the thickness transitions in experimental curves were smoother than those in both simulation curves while simulation curves showed narrower regions with large thickness changes (at angle 70deg~80deg) Possible reasons might be (i) sharper temperature profile predicted in simulation (ii) input uncertainty and (iii) modelling errors The SLM predicted slightly thinner vessel than the VLM except for the necking region The discrepancy might be caused by various reasons eg the interior vessel temperature can be slightly different between two approaches which would then result in different thermal stress and different deformation the final deformed vessel shapes cannot be exactly the same which also results in different errors in estimating PinPout

Figure 16 Wall thickness changes along the vessel polar angle [64] [71]

-6

-5

-4

-3

-2

-1

0

1

0 10 20 30 40 50 60 70 80 90

Wal

l th

icknes

s ch

ange

(mm

)

Polar angle (deg)

VLMSLMExperiment (left)Experiment (right)Experiment (front)Experiment (back)

32

44 Application to BWR Case

In this section the coupling approach (indicated as lsquoVLMrsquo in the following) was applied to an RPV failure analysis of a BWR A simulation using the SLM coupling approach (indicated as lsquoSLMrsquo in the following) was also performed for comparison purpose

A hypothetical severe accident of a Nordic BWR initiated by an SBO was considered The core materials would melt and migrate downward to the RPV lower head The total amount of corium relocated in the lower head was considered to form a 19 m corium melt pooldebris bed assuming the corium distribution was homogeneous

Figure 17 Computational domains and boundary conditions (a) conjugate heat transfer (b) mechanical analysis [64]

Figure 17(a) shows the computational domain of the conjugate heat transfer PECM was used in this calculation A general mesh with a pool height of 23 m was used to accommodate most of the relocation situations with the same mesh Since the corium only had a height of 19 m the top 04 m region was treated as injected water covering the corium The top water was expected to be boiling and we assumed a uniform temperature of 383 K Under the implementation of PECM this region was damped with artificially high heat capacity and high conductivity such that it would efficiently remove heat from the corium and suppress its own temperature increase For the corium part below water a volumetric heat source 11 MWm3 was implemented as the nuclear decay heat For the in-vessel surfaces a constant temperature of 383K was applied as they are covered by water which can be boiling Without water cooling on the external vessel wall a very small heat loss of 20 Wm2 was applied Melting and solidification of the corium were also considered with a solidus temperature of 2750 K and a liquidus temperature of 2770 K The initial temperature was set as

q=20 Wm2

T=383 K

Qv=11

MWm3

315 bar

315 bar+ hydrostatic

pressure of corium

Gravity

248 bar

Zero

displacement

Water

Corium

(a) (b)

33

1100 K for the corium 383 K for the water and 450 K for the vessel For the mechanical analysis (Figure 17(b)) an inner pressure of 315 bar and an outer pressure of 248 bar were considered For the region covered by corium an extra hydrostatic pressure was added on the vessel wall A 2 m long vessel extension was considered as well as the standard earth gravity

Figure 18 presents the wall temperature profile along the inner vessel surface at 50 s 2500 s 7500 s and 10000 s The temperature increased with time for the vessel surface contacted with corium and the distribution was flat which was due to that large part of the corium was still in solidus status It also shows that for the surface region that contacted the water the temperature was basically unchanged which indicates that the damping treatment indeed suppressed the temperature increment and guaranteed the heat removal upward by water

Figure 18 Temperature profiles of inner vessel wall along the vessel polar angle [64]

The predicted RPV failure time was 9869 s by VLM and 9650 s by SLM Compared with the FOREVER-EC2 case in Section 43 the VLM predicted a slightly later failure time than the SLM in this BWR case too Figure 19(a) compares the maximum creep strain developments with time of both approaches Before 9000 s the predicted values by both approaches are close to each other Afterward the creep acceleration occurred and the strain increased drastically The acceleration of the VLM occurred 200 s later than that of the SLM which agreed with the predicted later failure time Similar trend was observed for the total deformation in vertical direction of vessel bottom point as shown in Figure 19(b) The deformations of both approaches are close except that the acceleration of VLM occurred later than that of SLM Overall results of VLM agreed well with those of SLM

0

500

1000

1500

2000

2500

0 10 20 30 40 50 60 70 80 90 100

Tem

per

ature

(K

)

Polar angle (deg)

50 s

2500 s

5000 s

7000 s

10000 s

34

Figure 19 Results of structural analysis (a) creep strain and (b) total deformation in vertical direction of vessel bottom point [64]

45 Computational Efficiency

Computational efficiency is also an important consideration regarding the performance of numerical approaches The computational efficiencies of both VLM and SLM are analysed and compared in this section As parallel computing technique was used to speed up the simulations the number of parallel cores and corresponding costed computational time for each simulation are given together in Table 2 as a generally reflection of the computational efficiency All the simulations were done in a Lenovo P700 workstation equipped with 2 Intel Xeon E5-2630 v3 processors (8 cores per processor) and 32 G RAM For the simulations of FOREVER-EC2 SLM employed twice the parallel cores and costed 286 more time compared with VLM For the simulations of prototypic BWR case both VLM and SLM employed the same number of parallel cores but VLM only took 13 of the computational time compared with SLM Overall it can be concluded that VLM has higher computational efficiency than the SLM The reason is a structural analysis is performed in the VLM approach while a coupled thermo-structural analysis is performed in the SLM approach (cf Figure 9) a standalone structural analysis is computed faster than a coupled thermo-structural analysis

Table 2 Comparison of computational efficiency [64]

Case FOREVER-EC2 BWR VLM SLM VLM SLM

Number of cores 4 8 4 4 Computational time (hours) 21 27 18 72

35

5 Hybrid Coupling Approach for Efficient RPV Failure Analysis

In this section we propose a hybrid coupling approach for the efficient RPV failure analysis The idea is to replace the ANSYS Fluent in previous coupling approach with a lumped-parameter code for quick estimate of the melt pool heat transfer With the least modelling details (compared with CFD methods and distributed-parameter methods eg PECM) lumped-parameter codes prove to be the most efficient for melt pool heat transfer The new hybrid approach characterizes 1) utilization of efficient models for melt pool heat transfer and 2) the support of detailed structural modelling of RPV deformation allowing studies with different RPV geometric shapes under effects like local thinning due to ablation For studies of which the focus is mainly on structural behaviour of RPV eg [13] [26] while mechanistic simulation of melt pool convection is computationally expensive and unnecessary the lumped-parameter code could help providing reasonable thermal loads

The development of the approach consists of two tasks (i) development of a lumped-parameter code and (ii) its coupling with ANSYS Mechanical to form this hybrid coupling framework In the following we describe both tasks and the corresponding verification and validation processes respectively This section is mainly based on Paper IV

51 A Lumped-parameter Code for Melt Pool Heat Transfer

A lumped-parameter code transIVR was developed It features (i) quick estimate of transient melt pool heat transfer with reasonable accuracies and (ii) two-dimensional representation of heat conduction in the RPV wall facilitating the coupled analysis of the RPV failure under thermal loads The following assumptions are taken in the transIVR modelling

bull The crust between the oxide pool and metal pool is thin

bull The heat flux has uniform distributions on the bottom top and side surfaces of metal pool

bull Heat source only exists in oxide layer

bull Latent heat of fusion is ignored

Figure 20 illustrates the modelled three domains oxide layer metal layer and the RPV wall In addition the internal structure inside the vessel can also be considered as a radiative body to the top metal surface as in [72] An internal volumetric heat source generated by nuclear decay heat is considered in the oxide layer Heat in oxide layer transfers downward to the RPV wall and upward to the top metal layer Heat in metal layer can be transferred sideward to the RPV wall and upward to the surrounding internal structure through radiation (or to water if inside water flooding is applied) The heat transferred to RPV will be removed from outer RPV surface Possible formations of oxide crust in the top and bottom surface are illustrated as the black regions Possible formation of metal crust on the top surface is shown as the grey region The code automatically detects the

36

existence of the crust on oxide top and metal top surfaces during the simulation and solves the problem correspondingly withwithout crust Boundary conditions are applied to the external and internal surfaces of RPV and (if needed) top surface of metal layer Arbitrary boundary conditions are supported If a boundary condition is nonlinear eg mixed heat convection (linear dependent on 119879 ) and heat radiation (proportional to 1198794) the Picard method [73] would be used for solving this problem

This model consists of three main parts

(i) empirical heat transfer correlations (mostly from experiments) to determine global and local heat transfer coefficients

(ii) energy balances in oxide layer and metal layer where the heat transfer coefficients would be used and

(iii) energy balance in RPV

Figure 20 Two-layer melt pool configuration and RPV wall [74]

511 Heat transfer correlations

For internally heated natural convection (oxide layer) in transIVR the mini-ACOPO correlations [75] are used for the averaged heat transfer as expressed in Eq (35) and (36) and for the angular variation of the heat flux as expressed in Eq (37) and (38)

119873119906119906119901 = 03451198771198861198941198990233 (35)

119873119906119889119899 = 00038119877119886119894119899035(119867119877)025 (36)

119873119906119889119899(θ)

119873119906119889119899= 01 + 108(

θ

θ119898119886119909) minus 45 (

θ

θ119898119886119909)2

+ 86 (θ

θ119898119886119909)3

01 leθ

θ119898119886119909le 06 (37)

RPV

Oxide pool

Metal pool

37

119873119906119889119899(θ)

119873119906119889119899= 041 + 035 (

θ

θ119898119886119909) + (

θ

θ119898119886119909)2

06 ltθ

θ119898119886119909le 1 (38)

The 119877119886119894119899 that appears in Eq (35)~(36) is the internal Rayleigh number which is defined as

119877119886119894119899 =119892β119876119881119867

5

α119896ν (39)

For Rayleigh-Bernard convection in the metal layer heat transfer occurs in all the top bottom and side surfaces To side surface the reduced form [72] of the Churchill-Chu correlation [76] for 119875119903 = 013 is employed as shown in Eq (40)

119873119906119904119889 = 007611987711988613 (40)

To top and bottom surfaces the reduced form [72] of Globe-Dropkin correlation [77] for

119875119903 = 013 is used as expressed in Eq (41)

119873119906119906119901 = 005911987711988613 (41)

The 119877119886 that appears in Eq (40) and (41) is the external Rayleigh number which is defined

as

119877119886 =119892βΔ1198791198673

να (42)

Heat transfer coefficients can be calculated from Nusselt number by Eq (43)

ℎ =119873119906119896

119871 (43)

where 119896 and 119871 are the heat conductivity and characteristic length respectively 119873119906 can be the Nusselt numbers expressed in Eq (35)~(38) and (40)~(41)

512 Energy balance in the pools

The energy balance of oxide layer can be described as Eq (44) The total generated heat of oxide pool equals the sum of the increase of enthalpy (transient temperature change) upward heat flow to metal layer and downward heat flow to RPV

120588119862119901119881119900119897119889119879119900119909119889119905

+ ℎ119906119901119860119900119909119906119901(119879119900119909 minus 119879119900119909119906119901) + ℎ119889119899119860119900119909119889119899(119879119900119909 minus 119879119900119909119889119899) minus 119876119881119881119900119897 = 0 (44)

where ρ is the oxide density 119862119901 the heat capacity 119881119900119897 the total volume 119879119900119909 the bulk temperature 119905 the time ℎ119906119901 the upward heat transfer coefficient 119860119900119909119906119901

upward heat transfer area 119879119900119909119906119901 upward surface temperature ℎ119889119899 downward

heat transfer coefficient 119860119900119909119889119899 downward heat transfer area 119879119900119909119889119899 downward surface temperature and 119876119881 volumetric heat source

38

On the interface between oxide layer and metal layer energy balance requires that the heat transferred from oxide layer should equal the heat to the metal layer which can be described as Eq (45)

ℎ119906119901119860119900119909119906119901(119879119900119909 minus 119879119900119909119906119901) = ℎ119898119905119889119899119860119898119905119889119899(119879119894119899119905 minus 119879119898119905) (45)

where ℎ119898119905119889119899 is the downward heat transfer coefficient of metal layer 119860119898119905119889119899 the downward heat transfer area 119879119894119899119905 the interface temperature and 119879119898119905 the bulk temperature of metal layer

The energy balance of metal pool can be described as Eq (46) The increase of enthalpy (transient temperature change) equals the sum of upward downward and sideward heat flows to surrounding surfaces

ρ119898119905119862119901119898119905119881119900119897119898119905119889119879119898119905119889119905

+ ℎ119898119905119906119901119860119898119905119906119901(119879119898119905 minus 119879119898119905119906119901) + ℎ119898119905119904119889119860119898119905119904119889(119879119898119905 minus 119879119898119905119904119889)

+ ℎ119898119905119889119899119860119898119905119889119899(119879119898119905 minus 119879119894119899119905) = 0

(46)

where ρmt is the metal density 119862119901mt the heat capacity 119881119900119897mt the total volume 119879119898119905 the bulk metal temperature 119905 the time ℎ119898119905119906119901 the upward heat transfer

coefficient 119860119898119905119906119901 upward heat transfer area 119879119898119905119906119901 upward surface

temperature ℎ119898119905119904119889 sideward heat transfer coefficient 119860119898119905119904119889 sideward heat transfer area 119879119898119905119904119889 sideward surface temperature ℎ119898119905119889119899 downward heat transfer coefficient 119860119898119905119889119899 downward heat transfer area and 119879119894119899119905 oxide-metal interface temperature

The heat transferred from metal layer to the top surface should equal the heat radiated from this surface to the environment Considering the radiation and reflections of internal structures to this surface following energy balance equation [72] is achieved

ℎ119898119905119906119901(119879119898119905 minus 119879119898119905119906119901) = σ(1198791198981199051199051199001199014 minus 119879119904

4) (1

ε+1 minus ε119904ε119904

119860119898119905119906119901

119860119904)minus1

(47)

where 119879119898119905119905119900119901 is the temperature of metal top free surface ε the emissivity of

metal ε119904 the emissivity of internal structures 119860119898119905119906119901 the area of metal top surface

and 119860119904 the total area of internal structures Sometimes the internal water cooling may be available above metal layer In this case Eq (47) is unnecessary and 119879119898119905119905119900119901 can be set to a constant value

For the oxide layer top surface Eq (48) is used if not crust is formed in the interface between two layers ie 119879119894119899119905 gt 119879119898119890119897119905119894119899119892119900119909 Otherwise Eq (49) is used

119879119900119909119906119901 = 119879119894119899119905 (48)

119879119900119909119906119901 = 119879119898119890119897119905119894119899119892119900119909 (49)

For the metal layer top surface Eq (50) is used if not crust is formed on top of the metal ie 119879119906119901119904119906119903119891 gt 119879119898119890119897119905119894119899119892119898119905 Otherwise Eq (51) is used

119879119898119905119906119901 = 119879119898119905119905119900119901 (50)

39

119879119898119905119906119901 = 119879119898119890119897119905119894119899119892119898119905 (51)

Overall this part involves 6 equations regarding 6 unknows 119879119900119909 119879119900119909119906119901 119879119894119899119905 119879119898119905

119879119898119905119906119901 and 119879119898119905119905119900119901

513 Energy balance in the RPV wall

The heat transfer in the PRV wall can be modelled as a simple 1D heat conduction problem that heat only transfers in the wall thickness direction or 2D axisymmetric heat conduction problem

For 1D model the treatment in [72] was adopted As illustrated in Figure 21 the heat from the oxide pool transfers to inner vessel surface (119879119894) then to the interior vessel (119879119907119890119904) then to outer vessel (119879119900) and finally to environment

Figure 21 1D heat transfer model in RPV [74]

Unknown variables are the heat flux through inner vessel surface 119902119894119899 the crust thickness 119909119888119903 inner vessel temperature 119879119894 vessel center temperature 119879119907119890119904 and outer vessel temperature 119879119900 Eq (52)~(56) give a closed equation set for these variables considering a radiation boundary condition on the external vessel surface Eq (52)~(53) describe the heat transfer through the crust where an internal heat source 119876119888119903 is also considered Eq (52) can be easily derived from the one-dimensional heat transfer equation with heat source Eq (54)~(55) describe the energy balance on the inner vessel surface and vessel interior respectively Eq (56) describes the energy balance on the outer vessel surface regarding radiation boundary condition For Dirichlet boundary condition Eq (56) is replaced with Eq (57) For other boundary conditions we only need to correspondingly modify the terms on the right-hand side of Eq (56) For situations that no crust is formed on the surface of the vessel Eq (52)~(53) are eliminated and 119902119894119899 in Eq (54) can directly be determined from calculations in Section 512 and Eq (54)~(56) are solved for the unknows 119879119894 119879119907119890119904 and 119879119900 Once 119879119894 exceeds the vessel melting temperature (with or without crust formation) wall ablation is considered to occur Then 119879119894 is fixed to the vessel melting temperature and vessel thickness 119909119907119890119904 becomes the new unknown variable

119879119900119909119889119899 119879119894 119879119907119890119904 119879119900 119879119890119899119907

119909119888119903 119909119907119890119904

crust RPV

Qcr

119902119894119899 119902119897119900119888119886119897

40

119902119894119899 = 119896119888119903(119879119900119909119889119899 minus 119879119894)

119909119888119903+ 1198761198881199031199091198881199032

(52)

119902119894119899 = 119902119897119900119888119886119897 + 119876119888119903119909119888119903

(53)

119902119894119899 = 119896119907119890119904(119879119894 minus 119879119907119890119904)

05119909119907119890119904

(54)

120588119907119890119904119862119901119907119890119904119889119879119907119890119904119889119905

= 119896119907119890119904(minus2 lowast 119879119907119890119904 + 119879119894 + 119879119900)

(05119909119907119890119904)2

(55)

119896119907119890119904(119879119907119890119904 minus 119879119900)

05119909119907119890119904= 휀120590(119879119900

4 minus 1198791198901198991199074 )

(56)

119879119900 = 119879119900119888119900119899119904119905

(57)

For the 2D representation the heat conduction in the RPV is solved using FDM considering the heat transfer in both the thickness and the angular directions in the RPV wall The RPV lower head is treated as a spherical shell The corresponding heat conduction equation in the 2D spherical coordinate is given as Eq (58) The cylindrical part of the vessel is treated as a cylindrical shell The corresponding heat conduction equation in the 2D cylindrical coordinate is given as Eq (59) Eq (58) and (59) are coupled and solved together In default the mesh size is 5deg in angular direction and 0025119909119888119903 in radius direction This mesh discretization can be changed up to user demands

ρ119862119901part119879

part119905=

1

1199032part

part119903(1198961199032

part119879

part119903) +

1

r2sinθ

part

partθ(119896119904119894119899θ

part119879

partθ)

(58)

ρ119862119901part119879

part119905=1

119903

part

part119903(119896119903

part119879

part119903) +

part

part119911(119896part119879

part119911) (59)

52 Hybrid Coupling Approach

The hybrid coupling approach to assess the RPV integrity is illustrated in Figure 22 transIVR and ANSYS Mechanical are coupled in a one-way manner transIVR conducts the conjugate heat transfer of melt pool and RPV and provides transient thermal loads to the ANSYS Mechanical where structural analysis of RPV deformation is performed The 2D heat transfer of RPV is solved in transIVR to better capture the temperature distribution In RPV structural analysis same considerations are taken as in Section 422 for 16MND5 steel the modified theta projection model for creep and MISO for plasticity and temperature-dependent Young modulus etc The VLM is adopted for the transfer of transient thermal loads

41

Figure 22 Framework of the hybrid coupling approach [74]

53 Verification and Validation

The transIVR code was first benchmarked against the UCSB FIBS case of two-layer pool configuration and then validated against the LIVE-7V experiment showing the codersquos capability in predicting two-layer and transient melt pool heat transfers respectively Finally the hybrid coupling approach encompassing transIVR and ANSYS Mechanical was validated against the RPV failure experiment FOREVER-EC2 and the results showed acceptable agreements with those of both the experiment and the previous approach using CFD in Section 43 Hence the transIVR code is an efficient tool for melt pool heat transfer and the hybrid approach an efficient way for RPV failure analysis

531 UCSB FIBS benchmark case

The UCSB FIBS benchmark case [78] was calculated to demonstrate the capability of the transIVR code in simulating two-layer pool heat transfer problems The considered pool configuration was similar to that shown in Figure 20 with the following specific dimensions a two-layer melt pool with a 16 m oxide layer and a 08 m metallic pool on top and an RPV lower head with a thickness of 015 m The internal volumetric source in the oxide layer was 13 MWm3 The calculation results were compared with those of the DOE study [78] and the INEEL study [79]

The predicted results were compared with the digitalized results of the DOE and INEEL studies in Figure 23 All three studies show similar profiles of the heat flux distribution along the polar angle (Figure 23(a)) it increased with the polar angle in the oxide layer to the maximum value at around 77deg and then dropped to a significantly lower value in the metal layer The transIVR result agreed well with the other results with small discrepancy in the metal layer the DOE study gave a

heat flux of 500 MWm2 the INEEL study of 400 MWm2 and the transIVR of somewhere in between Figure 23(b) shows the predicted corium crust thickness distribution along the RPV polar angle also agreed well the DOE and INEEL

transIVR ANSYS Mechanical

Transient

thermal loads

42

studies The crust thickness decreased with polar angle in the oxide layer and no crust was formed in the metal layer As the crust formation is closely related to the heat flux distribution the crust thickness profile was reasonable regarding the heat flux distribution shown in Figure 23(a) Vessel wall ablation would occur due to the high thermal loads Figure 23(c) gives the profile of the remaining vessel wall thickness along the RPV polar angle considering the wall ablation It shows that no ablation occurred in low angle region (0deg~36deg) where heat flux was low and the wall thickness changed from 15 cm to around 6 cm as angle increased in high angle region of the oxide layer Because of the discrepancy in the heat flux of metal layer the predicted wall thicknesses also differ from those of the DOE and INEEL studies the thinnest wall was 73 cm predicted by the DOE study corresponding to the maximum heat flux followed by 88 cm of the present transIVR calculation and 94 cm of the INEEL study Generally speaking the prediction of transIVR is in an acceptable level compared with the other studies for the same benchmark case

Figure 23 Simulation results of the UCSB FIBS benchmark case [74]

(a) heat flux distribution (b) crust thickness distribution

(c) wall thickness distribution

43

532 LIVE-7V experiment

LIVE-7V is an experiment under the LIVE program conducted in KIT to study the core melt phenomena As Figure 24(a) shows it employed the LIVE-3D hemispherical facility with an inner diameter of 1 m and a wall thickness of ~25 mm The non-eutectic binary salt mixture of 20 mol NaNO3- 80 mol KNO3 was used as the simulant material The decay heat in the melt was simulated by cable-type heating with 8 planes of electrical resistance heating wires Water cooling was applied to the top and vessel surfaces Top surface cooling was realized by a cooling lid mounted at height 413 mm from the vessel bottom and vessel cooling by the flow channel between the test vessel and the outer cooling vessel The inlet and outlet water temperature for both top and vessel cooling surfaces were kept ~293 K and ~303 K throughout the whole experiment respectively A 413 mm height pool corresponds to a total volume of ~0194 m3 simulant material The experiment procedure can be divided into four stages with the heating power sequentially shifted from 29 kW to 24 kW 18 kW and 9 kW Figure 24(b) indicates the thermocouple positions in experiment

The simulation results of LIVE-7V using transIVR were compared with experimental data [80] and numerical study using REALPSCDAPSIM code [81] For simplicity the RELAPSCDAPSIM results in the following are labelled as lsquoCOUPLErsquo which is the name of the module in RELAPSCDAPSIM for melt pool heat transfer

Figure 24 LIVE test facility [80] and thermocouple positions [82]

Figure 25 compares the predicted heat removal upward to the top lid and downward to the bottom vessel surface with those of the experiment and COUPLE The black curve indicates the total input power at the four stages The experimental results (red and blue dots) were calculated from the Table 3 of [80] assuming no extra heat losses For the first three stages the transIVR predictions agreed well with the COUPLE results while the predicted upward heat flow was slightly lower and the downward heat flow higher than the experimental data For the last stage the transIVR agreed well with the experimental result while the COUPLE predicted lower upward and higher downward heat flows The short

(a) (b)

44

transition periods between two stages with different total inputs taken in transIVR also agreed with those in the COUPLE predictions

Figure 25 Comparison of heat removal upward to the top melt surface cooling lid and downward to bottom vessel wall [74]

Figure 26 shows the comparison of heat flux distribution along the vessel wall All curves show similar trend that a higher total input power would result in a higher heat flux profile As the COUPLE predictions had relatively flatter profiles transIVR predictions generally agreed better with the experimental results especially when total input was high A discrepancy is seen that the minimum value of a curve occurred at angle 0degfor both transIVR and COUPLE and at angle 30deg for the experimental data It is noted that the models in default assume heat flux monotonically increases with angle (cf Eq (37) and (38)) resulting it impossible to predict higher heat flux at angle 0deg than angle 30deg as in experimental results

45

Figure 26 Heat flux profiles along the vessel polar angle [74]

533 FOREVER-EC2 experiment

In this section we performed calculations against the FOREVER-EC2 experiment to validate the performance of the coupled transIVR and ANSYS Mechanical in assessing the RPV integrity The applied boundary conditions were identical to the settings of previous coupling approach in Section 4 (ANSYS Fluent+ Mechanical) as indicated in Figure 11 The initial bulk temperature of the melt pool was set to be 1261 K in transIVR which corresponds to the maximum value of the initial temperature field in CFD calculation of previous coupling approach Both the experimental data and the simulation results using previous coupling approach are included for comparison In the following results from current hybrid approach are labelled lsquotransIVRrsquo (and referred as transIVR results) results from previous approach labelled lsquoCFDrsquo (and referred as CFD results) and results from experiment labelled lsquoexperimentrsquo

Figure 27 compares the bulk temperature predicted by transIVR with the averaged and maximum temperatures of the CFD results All curves show the same trend that the temperature gradually increased with time till reaching a steady state and then kept unchanged till the end The times needed for reaching the steady state were ~5000 s for the CFD and ~4000 s for the transIVR which are close to each other It indicates again a good performance of the transIVR in predicting the transient behavior of melt pool Compared with the steady-state averaged temperature 1478 K and maximum temperature 1609 K of the CFD results the steady-state temperature of transIVR 1599 K is a reasonable prediction

0

3000

6000

9000

12000

15000

18000

0 20 40 60 80 100

Hea

t fl

ux

(W

m2)

Polar angle (deg)

29 kW-EXP24 kW-EXP18 kW-EXP9 kW-EXP29 kW-COUPLE24 kW-COUPLE18 kW-COUPLE9 kW-COUPLE29 kW-transIVR24 kW-transIVR18 kW-transIVR9 kW-transIVR

46

Figure 27 Temperature evolution with time [74]

Figure 28 Temperature fields at time (a) 01 s (b) 3038 s (c) 6038 s and (d) 13289 s [74]

Figure 28 shows the temperature fields of the vessel at time 01 s 3038 s 6038 s and 13289 s The temperature fields were visualized in the ANSYS Mechanical It also indicates that the transient temperature loads were properly transferred from transIVR to ANSYS Mechanical Figure 28(a-d) show both the minimum and maximum temperatures increased with time In the lower head region the temperature gradually increased with the height The maximum temperature occurs at the location slightly lower than the melt pool top surface This is because thermal loadheat flux imposed on the vessel generally increases with polar angle in the pool natural convection so the code transIVR inherently also predicted higher heat flux at higher angle positions The higher flux at higher position then correspondingly resulted in higher temperature In the cylindrical region the

(a) (b) (c) (d)

[K] [K] [K] [K]

47

temperature decreased as the height since the bottom part would be more significantly heated by the high-temperature lower head region

Figure 29 shows the steady-state temperature profiles of the external vessel surface along the polar angle The transIVR result agreed well with other two results in trend that the temperature increased with the angle and reach the peak values at angle ~80deg then gradually decreased with angle But slightly lower temperature was predicted in the low angle region Possible reason may be that the local heat flux distribution determined by Eq (37) and (38) is steeper than that in the experiment and CFD study which then resulted in a steeper increase of temperature along the angle and a lower temperature at low angle region In the cylindrical part (gt90deg) the transIVR result generally agreed with the experimental and CFD results The discrepancy with the experimental data may be caused by the simplified boundary conditions the ambient temperature inside the vessel was set to uniform constant value while it could decrease as height Therefore predicted heat loss in the higher position of cylindrical part could be smaller than that with height-dependent ambient temperature resulting in higher vessel temperature Overall the prediction was reasonable for a lumped-parameter model

Figure 29 Temperature profiles of the external vessel surface along the vessel polar angle [74]

The transIVR predicted a failure time of 24959 s which is close to 24180 s in experiment and 23959 s in the CFD results Figure 30 shows the total deformation of vessel bottom point with time All three curves increased gradually with time and experienced drastic accelerations at the end The

400

600

800

1000

1200

1400

0 40 80 120 160

Tem

per

ature

(K

)

Polar angle (deg)

transIVRExperimentCFDWeilding Line

48

transIVR curve agreed well with the CFD curve The final total deformation at failure time is also close to that of the CFD result but lower than the experiment data The discrepancy might be related to the simplifications in the boundary conditions accuracy of the thermal load prediction and the modelling errors in plasticity and creep

Figure 30 Total deformation in vertical direction of the vessel bottom point [74]

Figure 31 shows the profile of wall thickness change The transIVR curve was compared with the CFD curve and four experimental curves measured at vessel left right front and back sides The simulation result fitted the experimental and CFD results well in the region 0deg~50deg In the region 50deg~90deg wall thickness gradually decreased with angle till around 75deg and then increased till 90deg The maximum thickness change was -3 mm for the simulation which is close to the experiment left side curve and slightly larger than the -6~ -45 mm of the other curves from experiment and CFD results Concerning possible errors and uncertainties in the modelling and numerical settings the new approach generally gives reasonable predictions

0

5

10

15

20

25

11670 15170 18670 22170 25670

Tota

l def

orm

atio

n (

mm

)

Time (s)

Experiment

CFD

transIVR

49

Figure 31 Wall thickness changes along the vessel polar angle [74]

-6

-5

-4

-3

-2

-1

0

1

0 20 40 60 80 100

Wal

l th

icknes

s ch

ange

(mm

)

Polar angle (deg)

CFD

transIVR

Experiment (left)

Experiment (right)

Experiment (front)

Experiment (back)

50

51

6 Summary and Outlook

Reactor pressure vessel (RPV) is one of the key physical barriers in the defence-in-depth approach of nuclear power safety and its failure is also the transition point of core melt (corium) progression from in-vessel to ex-vessel phase Quantification of RPV failure during a hypothetical severe accident is therefore paramount to the specific qualification of severe accident mitigation strategies as well as the general assessment of corium risk in light water reactors The RPV failure is a problem of structural performance of RPV under the influence of thermal loads imposed by corium melt pool ie it is a transient thermo-fluid-structure coupled problem per se whose solution requires simultaneous modelling of both highly turbulent melt pool heat transfer and large-scale nonlinear RPV creep Due to the complexity of the problem only limited capabilities have been developed in the previous studies under different simplifications and assumptions in models and coupling methods The present study was motivated to advance modelling and coupling approaches for RPV performance analysis considering new understanding of physical phenomena and latest coupling platform of simulation tools

For the new development of coupling approaches and models for physical phenomena related to RPV failure analysis during severe accidents this thesis work was conducted in the following order (i) a new creep model for the RPV steel 16MND5 was developed which features the capability to cover all three creep stages (ii) a new thermo-fluid-structure coupling approach was developed for RPV failure analysis and validated against FOREVER-EC2 experiment (iii) the performance of the new coupling approach using volume loads mapping (VLM) for data transfer was further compared with another approach using surface loads mapping (SLM) (iv) a hybrid coupling approach was proposed for efficient analysis of RPV failure which employs a lumped-parameter code for quick estimate of thermal load and a mechanistic FE code for detailed structural analysis and (v) for the hybrid coupling approach the code transIVR was developed for calculation of transient melt pool heat transfer

The developed creep model for 16MND5 is called lsquomodified theta projection modelrsquo and describes all three stages of a creep process Creep curves are expressed as a function of time with five model parameters 120579119894(119894 = 14 119886119899119889 119898) A dataset for the model parameters was constructed by fitting experimental creep curves into this function and an iterative parameter adjustment process to satisfy the monotonicity assumption of creep strain vs temperature and stress Model parameters for new curves under different temperature and stress loads are directly interpolated from this dataset in contrast to the conventional ldquotheta projection modelrdquo that employs an extra fitting process The new treatment was proven to better accommodate all experimental curves over the wide ranges of temperature and stress loads The model was implemented into the ANSYS Mechanical code The simulation results indicate that the model successfully captured all three creep stages and a good agreement was achieved between each experimental creep curve and corresponding modelrsquos prediction Regarding the model response to transient changes of temperature and stress loads the widely used time hardening and strain hardening models were implemented and

52

performing reasonably well in the new method These properties fulfil the requirements of a creep model for structural analysis

The transient thermo-fluid-structure coupling approach was developed using the ANSYS Fluent code for thermofluid dynamics of the melt pool and ANSYS Structural code for structural mechanics of RPV An extension tool was introduced to realize transient load transfer from ANSYS Fluent to Structural and minimize the user effort Both turbulence models and the effective model PECM can be used for melt pool heat transfer The developed creep model was used for RPV failure analysis of the RPV Since it is insensitive to geometry the coupling approach allows more complex geometries such as a BWR lower head with a forest of penetrations (CRGTs and IGTs) modelled in detail Moreover the coupling approach does not only reflect the transient response of thermo-fluid-structure interaction but also facilitates the implementation of advanced (high-fidelity) models in both fluid mechanics and structural mechanics The validation of the coupling approach against the FOREVER-EC2 experiment showed that the temperature profile vessel failure time total displacement of vessel bottom point and wall thickness profile in the experiment were well predicted

Following the variation of the coupling approach from VLM to SLM for data transfer between ANSYS Fluent and ANSYS Structural the respective coupling approaches were investigated by comparative calculations against the FOREVER-EC2 experiment and a postulated severe accident of a reference BWR For the FOREVER-EC2 experiment both VLM and SLM approaches predicted generally well the total deformation in vertical direction of bottom point and the wall thickness change though the former predicted a slightly later failure time For the reference BWR case good agreements were achieved in predicting the maximum creep strain and the total deformation in vertical direction of bottom point For both experiment and reactor cases the VLM approach predicted slightly later failure times than SLM but the differences were negligible compared to the long time-scale of vessel failure (~104 s) The deformation rate of the vessel wall was slow at the early stage and accelerated with time till a drastic deformation and final vessel failure Generally speaking the VLM and SLM coupling approaches had the similar performance in terms of their predictability of experiment and applicability to reactor case but the VLM was computationally more efficient

In the development of the hybrid coupling approach the transIVR code was developed for quick estimate of transient melt pool heat transfer and determination of thermal loads on RPV transIVR does not only support transient analysis of one- and two- layer melt pool but also the heat transfer in RPV being resolved with 2D finite difference method which captures more spatial thermal details for RPV structural analysis in addition to 1D heat transfer model The capabilities of transIVR in modelling two-layer pool heat transfer and transient pool heat transfer were demonstrated by calculations against the UCSB FIBS benchmark case and the LIVE-7V experiment respectively It was then coupled to the mechanical solver ANSYS Mechanical which together form the hybrid coupling framework for detailed RPV failure analysis The calculation against the FOREVER-EC2 experiment did not only reproduce the major results of the experiment but also had a good agreement with the thermo-fluid-structure

53

coupling approach using CFD This indicated the new coupling approach is capable of capturing the vessel creep failure characteristics and can be employed as an efficient tool addressing the complex vessel failure problem

As a prospective more research topics can be addressed in future either as a continuation of the current work toward an effective tool for vessel failure analysis or from the general view of scientific interest in melt pool heat transfer and RPV failure The following topics are recommended

bull Extended applications to the reactor cases For instance the thermo-fluid-structure coupling approach can be used to study penetration failures in the RPV of an BWR with a detailed representation of penetrations (IGTs and CRGTs)

bull Synthesis or update of heat transfer correlations for melt pool Heat transfer correlations directly benefit the parametric codes such as transIVR in accurate quantification of heat flux distribution around a melt pool So far many empirical correlations have been proposed based on experiments which had a variety of geometries and melt simulants with a wide range of Rayleigh number (Ra) Sensitivity studies have also been performed on the influences of Ra Pa and other material properties Therefore a comprehensive synthesis or update of the existing heat transfer correlations will be a meaningful step for making reasonable extrapolations to the reactor applications with prototypic materials

bull Modelling of pool stratification The motivation starts with the improvement of the PECMrsquos modelling capability such that the dynamic stratification and resulting pool heat transfer can be modelled altogether Simulation of melt stratification has been supported in some system codes such as ASTEC using a predefined parametric approach The improvement of PECM can be considered in the similar approach or more mechanistic modelling

bull Development of a structural module for RPV The idea is to develop a simplified module for structural analysis so it can be coupled with transIVR ultimately forming a standalone code for quick estimate of vessel failure of LWRs during postulated severe accidents

bull Extension of vessel analysis capability The current version of tranIVR is limited to melt pool Although the precursory debris bed and melt ejection after vessel failure were not the focus of the present study they are equally important to assessment of corium risk especially for Swedish BWRs where the melt release conditions and history are crucial to qualification of their severe accident mitigation strategies

54

55

Bibliography

[1] IAEA ldquoIAEA Annual Report 2018rdquo IAEA 2019 [Online] Available httpswwwiaeaorgsitesdefaultfilesgcgc63-5pdf

[2] IAEA Safety of Nuclear Power Plants Design Vienna INTERNATIONAL ATOMIC ENERGY AGENCY 2012

[3] M Fischer ldquoGen II and III In-Vessel and Ex-Vessel Melt Retentionrdquo The Short Course on Severe Accident Phenomenology Oct 2017

[4] B R Sehgal Nuclear Safety in Light Water Reactors Severe Accident Phenomenology 1st ed Academic Press 2011

[5] IAEA Approaches and Tools for Severe Accident Analysis for Nuclear Power Plants Vienna INTERNATIONAL ATOMIC ENERGY AGENCY 2008

[6] WENRA ldquoPractical Elimination Applied to New NPP Designs - Key Elements and Expectationsrdquo Sep 2019 [Online] Available httpwwwwenraorgmediafiler_public20191111practical_elimination_applied_to_new_npp_designs_-_key_elements_and_expectations_-_for_issuepdf

[7] P Matejovic M Barnak M Bachraty and L Vranka ldquoASTEC applications to VVER-440V213 reactorsrdquo Nuclear Engineering and Design vol 272 pp 245ndash260 2014 doi 101016jnucengdes201308077

[8] AREVA ldquoUS EPR Severe Accident Evaluationrdquo Topical Report ANP-10268NP Oct 2006 [Online] Available httpswwwnrcgovdocsML0631ML063100157pdf

[9] C T Martin ldquoCoupled 3D Thermo-mechanical Analysis of Nordic BWR Lower Head Failure in case of Core Melt Severe Accidentrdquo master thesis KTH Stockholm 2013

[10] IAEA ldquoAdvanced Large Water Cooled Reactors A supplement to the IAEArsquos Advanced Reactor Information System (ARIS)rdquo Sep 2015 [Online] Available httpsarisiaeaorgPublicationsIAEA_WRC_Bookletpdf

[11] B Pershagen Light Water Reactor Safety Pergamon Press 1989 [12] Ninos Garis M Agrell H Glanneskog and L Agrenius ldquoAPRI 7 ndash Accident

Phenomena of Risk Importancerdquo SSM SSM 201212 2012 [Online] Available httpswwwstralsakerhetsmyndighetensecontentassets2a9f299f46b949b3b209e780cf71436a201212-apri-7-accident-phenomena-of-risk-importance-en-lagesrapport-om-forskningen-inom-omradet-svara-haverier-under-aren-2009-2011-

[13] F Fichot et al ldquoSome considerations to improve the methodology to assess In-Vessel Retention strategy for high-power reactorsrdquo Annals of Nuclear Energy vol 119 pp 36ndash45 2018 doi 101016janucene201803040

[14] T G Theofanous C Liu S Additon S Angelini O Kymaumllaumlinen and T Salmassi ldquoIn-vessel coolability and retention of a core meltrdquo Nuclear Engineering and Design vol 169 no 1 pp 1ndash48 1997 doi 101016S0029-5493(97)00009-5

[15] C Sainte Catherine ldquoTensile and creep tests material characterization of pressure vessel steel(16MND5) at high temperatures (20 up to 1300 degC)rdquo CEA France SEMTLISNRT98-009A 1998

56

[16] E Altstadt H-G Willschuetz B R Sehgal and F-P Weiss ldquoModelling of in-vessel retention after relocation of corium into the lower plenumrdquo Forschungszentrum Rossendorf FZR-437 2005

[17] H-G Willschuumltz and E Altstadt ldquoGeneration of a High Temperature Material DataBase and its Application to Creep Tests withFrench or German RPV-steelrdquo FORSCHUNGSZENTRUM ROSSENDORF FZR-353 ISSN 1437-322X Aug 2002

[18] J L Rempe S A Chavez and G L Thinnes ldquoLight water reactor lower head failure analysisrdquo Nuclear Regulatory Commission NUREGCR--5642 1993 Accessed Aug 23 2019 [Online] Available httpinisiaeaorgSearchsearchaspxorig_q=RN25025905

[19] S Brosi et al ldquoCORVIS Investigation of light water reactor lower head failure modesrdquo Nuclear Engineering and Design vol 168 no 1 pp 77ndash104 1997 doi 101016S0029-5493(96)00008-8

[20] T Y Chu M M Pilch J H Bentz J S Ludwigsen W Y Lu and L L Humphries ldquoLower Head Failure Experiments and Analysesrdquo Sandia National Laboratories Albuquerque NM USA NUREGCR-5582 SAND98-2047 1999

[21] L L Humphries et al ldquoOECD Lower Head Failure Project Final Reportrdquo Sandia National Laboratories Albuquerque NM 87185-1139 2002

[22] B R Sehgal R R Nourgaliev T N Dihn A Karbojian J A Green and V A Bui ldquoFOREVER Experiments on Thermal and Mechanical Behavior of a Reactor Pressure Vessel during a Severe Accidentrdquo in Proceedings of the Workshop on in-vessel core debris retention and coolability Garching near Munich Germany Mar 3-6 1998

[23] S Bhandari M Fischer B R Sehgal and A Theerthan ldquoPost Test Analysis of the FOREVER C2 Experimentrdquo The 9th International conference on Nuclear Engineering (ICONE-9) Nice France Apr 8-12 2001

[24] J F Mao J W Zhu S Y Bao L J Luo and Z L Gao ldquoCreep deformation and damage behavior of reactor pressure vessel under core meltdown scenariordquo International Journal of Pressure Vessels and Piping vol 139ndash140 pp 107ndash116 2016 doi 101016jijpvp201603009

[25] J Mao L Hu S Bao L Luo and Z Gao ldquoInvestigation on the RPV structural behaviors caused by various cooling water levels under severe accidentrdquo Engineering Failure Analysis vol 79 pp 274ndash284 2017 doi 101016jengfailanal201704029

[26] J F Mao J W Zhu S Y Bao L J Luo and Z L Gao ldquoStudy on structural failure of RPV with geometric discontinuity under severe accidentrdquo Nuclear Engineering and Design vol 307 pp 354ndash363 2016 doi 101016jnucengdes201607027

[27] B R Sehgal V A Bui T N Dinh and R R Nourgaliev ldquoHeat Transfer Processes in Reactor Vessel Lower Plenum During Late Phase of In-Vessel Core Melt Progressionrdquo in Advances in Nuclear Science and Technology J Lewins M Becker R W Albrecht E J Henley J D McKean K Oshima A Sesonske H B Smets and C P L Zaleski Eds Boston MA Springer US 1999 pp 103ndash135

[28] C-T Tran and T-N Dinh ldquoThe effective convectivity model for simulation of melt pool heat transfer in a light water reactor pressure vessel lower head Part I Physical processes modeling and model implementationrdquo Progress

57

in Nuclear Energy vol 51 no 8 pp 849ndash859 2009 doi 101016jpnucene200906007

[29] C-T Tran and T-N Dinh ldquoThe effective convectivity model for simulation of melt pool heat transfer in a light water reactor pressure vessel lower head Part II Model assessment and applicationrdquo Progress in Nuclear Energy vol 51 no 8 pp 860ndash871 2009 doi 101016jpnucene200906001

[30] H-G Willschuumltz E Altstadt B R Sehgal and F-P Weiss ldquoCoupled thermal structural analysis of LWR vessel creep failure experimentsrdquo Nuclear Engineering and Design vol 208 no 3 pp 265ndash282 2001 doi 101016S0029-5493(01)00364-8

[31] E Altstadt and T Moessner ldquoExtension of the ANSYS creep and damage simulation capabilitiesrdquo Forschungszentrum Rossendorf Technical report FZR-296 2000

[32] H-G Willschuumltz E Altstadt B R Sehgal and F-P Weiss ldquoRecursively coupled thermal and mechanical FEM-analysis of lower plenum creep failure experimentsrdquo Annals of Nuclear Energy vol 33 no 2 pp 126ndash148 2006 doi 101016janucene200508006

[33] H-G Willschuumltz E Altstadt B R Sehgal and F-P Weiss ldquoSimulation of creep tests with French or German RPV-steel and investigation of a RPV-support against failurerdquo Annals of Nuclear Energy vol 30 no 10 pp 1033ndash1063 2003 doi 101016S0306-4549(03)00036-7

[34] H Madokoro A Miassoedov and T Schulenberg ldquoA thermal structural analysis tool for RPV lower head behavior during severe accidents with core meltrdquo Mechanical Engineering Letters vol 4 pp 18-00038-18ndash00038 2018 doi 101299mel18-00038

[35] W Villanueva C-T Tran and P Kudinov ldquoCoupled thermo-mechanical creep analysis for boiling water reactor pressure vessel lower headrdquo Nuclear Engineering and Design vol 249 pp 146ndash153 2012 doi 101016jnucengdes201107048

[36] W Villanueva C T Tran and P Kudinov ldquoA Computational Study on Instrumentation Guide Tube Failure During a Severe Accident in Boiling Water Reactorsrdquo in The 14th International Topical Meeting on Nuclear Reactor Thermal hydraulics NURETH-14 Toronto Ontario Canada Sep25-30 2011

[37] W Villanueva C T Tran and P Kudinov ldquoAssessment with Coupled Thermo-Mechanical Creep Analysis of Combined CRGT and External Vessel Cooling Efficiency for A BWRrdquo The 14th International Topical Meeting on Nuclear Reactor Thermal hydraulics NURETH-14 Toronto Ontario Canada Sep 25-30 2011

[38] F R Larson and J Miller ldquoA Time-Temperature Relationship for Rupture and Creep Stressesrdquo Trans ASME vol 74 p 765minus775 1952

[39] C T Tran ldquoDevelopment validation and application of an effective convectivity model for simulation of melt pool heat transfer in a light water reactor lower headrdquo Licentiate thesis KTH Stockholm 2007

[40] ANSYS ldquoANSYS Fluent Theory Guide Release 162rdquo ANSYS Inc Jul 2015 [Online] Available httpwwwansyscom

[41] P Zeng Finite element analysis and application (in Chinese) Tsinghua University Press amp Springer 2004

58

[42] ANSYS ldquoANSYS Mechanical | Finite Element Analysis Softwarerdquo httpswwwansyscomproductsstructuresansys-mechanical-enterprise (accessed Sep 26 2019)

[43] ANSYS ldquoANSYS Mechanical APDL Theory Reference Release 162rdquo ANSYS Inc Jul 2015 [Online] Available httpwwwansyscom

[44] U Steinberner and H H Reineke ldquoTurbulent Buoyancy Convection Heat Transfer with Internal Heat Sourcesrdquo in Proceedings of the 6th Int Heat Transfer Conference Toronto Canada 1978 vol Vol2 pp 305ndash310 doi 101615IHTC63350

[45] R W Evans and B Wilshire Introduction to Creep London Institute of Materials 1993

[46] DoITPoMS ldquoCreep Deformation of Metalsrdquo Creep Deformation of Metals University of Cambridge 2006 httpswwwdoitpomsacuktlplibcreepindexphp (accessed Jul 09 2019)

[47] ANSYS ANSYS Mechanical APDL Material Reference Release 162 2015 [48] SIMULIA Abaqus 612 Analysis Userrsquos Manualvol 3 Materials Chapter

2324 2012 [49] S Imaoka ldquoUser Creep Subroutine STI0704A ANSYS Release110rdquo 2007 [50] R W Evans J D Parker and B Wilshire ldquoThe θ projection conceptmdashA

model-based approach to design and life extension of engineering plantrdquo International Journal of Pressure Vessels and Piping vol 50 no 1 pp 147ndash160 1992 doi 1010160308-0161(92)90035-E

[51] ECCC ldquoRecommendations and Guidance for the Assessment of Creep Strain and Creep Strength Data ECCC Recommendations Volume 5 Part 1 b[Issue 2]rdquo 2013

[52] W D Day and A P Gordon ldquoLife Fraction Hardening Applied to a Modified Theta Projection Creep Model for a Nickel-Based Super-Alloyrdquo The ASME Turbo Expo 2014 Turbine Technical Conference and Exposition Duumlsseldorf Germany Jun16-20 2014 vol 7A-Structures and Dynamics p V07AT29A010 doi 101115GT2014-25881

[53] M Kumar I V Singh B K Mishra S Ahmad A Venugopal Rao and V Kumar ldquoA Modified Theta Projection Model for Creep Behavior of Metals and Alloysrdquo J of Materi Eng and Perform vol 25 no 9 pp 3985ndash3992 2016 doi 101007s11665-016-2197-y

[54] P Yu and W Ma ldquoA modified theta projection model for creep behavior of RPV steel 16MND5rdquo Journal of Materials Science amp Technology vol 47 pp 231ndash242 2020 doi 101016jjmst202002016

[55] ANSYS ANSYS Mechanical Advanced Nonlinear Materials v160 Lecture 2 Rate Dependent Creep ANSYS training materials 2015

[56] Q Y Li N C Wang and D Y Yi Numerical Analysis (in Chinese) 5th ed Tsinghua University Press 2008

[57] T N Dinh R R Nourgaliev and B R Sehgal ldquoOn heat transfer characteristics of real and simulant melt pool experimentsrdquo Nuclear Engineering and Design vol 169 no 1 pp 151ndash164 1997 doi 101016S0029-5493(96)01283-6

[58] O Kymaumllaumlinen H Tuomisto O Hongisto and T G Theofanous ldquoHeat flux distribution from a volumetrically heated pool with high Rayleigh numberrdquo

59

Nuclear Engineering and Design vol 149 no 1ndash3 pp 401ndash408 1994 doi 1010160029-5493(94)90305-0

[59] F J Asfia and V K Dhir ldquoNatural convection heat transfer in volumetrically heated spherical poolsrdquo 1994 Accessed Aug 23 2019 [Online] Available httpinisiaeaorgSearchsearchaspxorig_q=RN26037727

[60] M Jahn and H H Reineke ldquoFree Convection Heat Transfer with Internal Heat Sources Calculations and Measurementsrdquo in Proceedings of the 5th Int Heat Transfer Conference Tokyo Japan 1974 vol Vol3 paper NC-28

[61] M Gustafsson ldquoFSI Transient extension V1704rdquo EDR MEDESO 2016 [62] EDR MEDESO httpsedrmedesocom [63] H Kim and I Namgung ldquoThermo-mechanical Analysis of RVLH for

APR1400 during a Severe Accidentrdquo in Transactions SMiRT-23 Manchester United Kingdom Aug10-14 2015 [Online] Available httpwwwlibncsueduresolver18402034197

[64] P Yu and W Ma ldquoComparative Analysis of Reactor Pressure Vessel Failure using Two Thermo-Fluid-Structure Coupling Approachesrdquo submitted to journal

[65] F R Menter ldquoTwo-equation eddy-viscosity turbulence models for engineering applicationsrdquo AIAA Journal vol 32 no 8 pp 1598ndash1605 1994 doi 102514312149

[66] T N Dinh and R R Nourgaliev ldquoTurbulence modelling for large volumetrically heated liquid poolsrdquo Nuclear Engineering and Design vol 169 no 1 pp 131ndash150 1997 doi 101016S0029-5493(96)01281-2

[67] ANSYS Inc ldquoIntroduction to ANSYS Fluent Lecture 7 Turbulence Modelling Fluent Tutorial Materialrdquo 2016

[68] A Horvat and M Borut ldquoNumerical investigation of natural convection heat transfer in volumetrically heated spherical segmentsrdquo in Proceedings of the International Conference Nuclear Energy for New Europe 2004

[69] Y Li CFD Pre-test Analysis of SIMECO-2 Experiment 2016 [70] A Theerthan A Karbojian and B R Sehgal ldquoEC-FOREVER experiments

on thermal and mechanical behavior of a reactor pressure vessel during a severe accident Technical report-1 EC-FOREVER-2 testrdquo SAM-ARVI-D008 Mar 2001

[71] P Yu W Ma W Villanueva A Karbojian and S Bechta ldquoValidation of a thermo-fluid-structure coupling approach for RPV creep failure analysis against FOREVER-EC2 experimentrdquo Annals of Nuclear Energy vol 133 pp 637ndash648 2019 doi 101016janucene201906067

[72] T G Theofanous C Liu S Additon S Angelini O Kymaumllaumlinen and T Salmassi ldquoIn-vessel coolability and retention of a core meltrdquo Nuclear Engineering and Design vol 169 no 1ndash3 pp 1ndash48 1997 doi 101016S0029-5493(97)00009-5

[73] H P Langtangen ldquoSolving nonlinear ODE and PDE problemsrdquo Simula Research Laboratory 2016 [Online] Available httpshplgitgithubionum-methods-for-PDEsdocpubnonlinpdfnonlin-4printpdf

[74] P Yu and W Ma ldquoDevelopment of a Lumped-Parameter Code for Efficient Assessment of In-Vessel Melt Retention Strategy of LWRsrdquo submitted to journal

60

[75] T G Theofanous M Maguire S Angelini and T Salmassi ldquoThe first results from the ACOPO experimentrdquo Nuclear Engineering and Design vol 169 no 1 pp 49ndash57 1997 doi 101016S0029-5493(97)00023-X

[76] S W Churchill and H H S Chu ldquoCorrelating equations for laminar and turbulent free convection from a vertical platerdquo International Journal of Heat and Mass Transfer vol 18 no 11 pp 1323ndash1329 1975 doi 1010160017-9310(75)90243-4

[77] S Globe and D Dropkin ldquoNatural-Convection Heat Transfer in Liquids Confined by Two Horizontal Plates and Heated from Belowrdquo J Heat Transfer vol 81 no 1 pp 24ndash28 1959 doi 10111514008124

[78] T G Theofanous C Liu S Additon S Angelini O Kymaelaeinen and T Salmassi ldquoIn-vessel coolability and retention of a core melt Volume 2rdquo Argonne National Lab DOEID--10460-VOL2 1996 Accessed Apr 18 2019 [Online] Available httpinisiaeaorgSearchsearchaspxorig_q=RN30000459

[79] J L Rempe D L Knudson C M Allison G L Thinnes and C L Atwood ldquoPotential for AP600 in-vessel retention through ex-vessel floodingrdquo United States INEELEXT--97-00779 1997 Accessed Apr 18 2019 [Online] Available httpinisiaeaorgSearchsearchaspxorig_q=RN30025872

[80] A Miassoedov X Gaus-Liu T Cron and B Fluhrer ldquoLive experiments on melt pool heat transfer in the reactor pressure vessel lower headrdquo The 10th International Conference on Heat Transfer Fluid Mechanics and Thermodynamics (HEFAT 2014) Orlando FL Jul 14-16 2014

[81] H Madokoro A Miassoedov and T Schulenberg ldquoAssessment of a lower head molten pool analysis module using LIVE experimentrdquo Nuclear Engineering and Design vol 330 pp 51ndash58 2018 doi 101016jnucengdes201801036

[82] B Fluhrer et al ldquoThe LIVE-L1 and LIVE-L3 experiments on melt behaviour in RPV lower headrdquo Forschungszentrum Karlsruhe GmbH Technik und Umwelt (Germany) Inst fuer Kern- und Energietechnik FZKA--7419 2008 Accessed Nov 12 2019 [Online] Available httpinisiaeaorgSearchsearchaspxorig_q=RN40036867

II

The idea of the hybrid coupling is to employ a lumped-parameter code for quick estimate of thermal load which can be employed in detailed structural analysis Such a coupling approach can significantly increase the calculation efficiency which is important to the case of a prototypical RPV where mechanistic simulation of melt pool convection is computationally expensive and unnecessary The transIVR code was developed for this purpose which is not only capable of quick estimate of transient heat transfer of one- and two- layer melt pool but also solving heat conduction problem in the RPV wall with 2D finite difference method to provide spatial thermal details for RPV structural analysis The capabilities of transIVR in modelling two-layer pool heat transfer and transient pool heat transfer were demonstrated by calculations against the UCSB FIBS benchmark case and the LIVE-7V experiment respectively The transIVR code was then coupled to the mechanical solver ANSYS Mechanical for detailed RPV failure analysis Validation against the FOREVER-EC2 experiment indicates the coupling framework successfully captured the vessel creep failure characteristics

Keywords RPV failure thermo-fluid-structure coupling creep modelling melt pool heat transfer severe accident CFD FEA transIVR

III

Sammanfattning

Syftet med denna avhandling aumlr att utveckla nya metoder och nya modeller foumlr att kunna koppla ihop termohydrauliska foumlrlopp och uppkommen strukturrespons i botten paring reaktortanken foumlr foumlrbaumlttrad analys av reaktortankbrott under ett svaringrt haveri Avhandlingen bestaringr av fem delar (i) utveckling av en trestegsmodell foumlr krypning i reaktortankens 16MND5-staringl (ii) utveckling av metoder foumlr kopp ling av termohydrauliska foumlrlopp och strukturrespons foumlr studier av reaktortankbrott (iii) jaumlmfoumlrelse av prestanda av den nya metoden som anvaumlnder volume loads mapping (VLM) foumlr dataoumlverfoumlring med den tidigare metoden som anvaumlnder surface loads mapping (SLM) (iv) utveckling av en lumped-parameter beraumlkningskod foumlr snabb uppskattning av transient vaumlrmeoumlverfoumlring i en smaumlltpoumll i botten paring reaktortanken och (v) utveckling av en hybrid kopplingsmetod foumlr effektiv analys av reaktortankbrott

En krypmodell kallad lsquomodified theta projection modelrsquo som behandlar krypprocessen i tre steg utvecklades foumlr 16MND5-staringlet Krypkurvor i den nya

krypmodellen ges som en funktion av tid med fem parametrar 120579119894 (i=14 och m) Ett dataset foumlr modellparametrar togs fram baserat paring tillgaumlngliga experimentalla krypkurvor under anatagande om att kryptoumljning aumlr en monoton funktion av temperatur och spaumlnning Nya krypkurvor kan genereras genom att interpolera modellparametrar fraringn detta dataset i kontrast till den tidigare metoden som anvaumlnder en speciell anpassningsprocess Foumlrdelen med det nya tillvaumlgagaringngssaumlttet aumlr att den hanterar alla experimentella krypkurvor foumlr ett brett spektrum av temperatur- och spaumlnningsbelastningar Modellen implementerades i ANSYS-koden och den korrekt behandlar alla tre krypsteg samt visar bra oumlverenstaumlmmelse mellan experimentella och beraumlknade krypkurvor Foumlr analys av tidsberoende dynamiska belastningar implementerades vaumllkaumlnda modeller av tids- och deformationshaumlrdning med rimliga resultat Dessa egenskaper uppfyller kravet paring en krypmodell foumlr strukturmekaniska analyser

En metod utvecklades foumlr koppling mellan termohydrauliska foumlrlopp och strukturmekanisk respons genom att koppla ihop ANSYS Fluent som analyserar fluiddynamiska aspekter av vaumlrmeoumlverfoumlring i en smaumlltpoumll med ANSYS Structural som analyserar strukturmekanisk respons av reaktortanksvaumlggen Ett speciellt verktyg togs fram foumlr att moumljliggoumlra oumlverfoumlring av transienta belastningar fraringn ANSYS Fluent till ANSYS Structural och foumlr att minimera anvaumlndarinsatsen Baringde CFD med turbulensmodeller och den effektiva PECM-modellen kan anvaumlndas foumlr att analysera vaumlrmeoumlverfoumlring i en smaumlltpoumll Foumlr krypanalysen av reaktortanken har lsquomodified theta projection modelrsquo anvaumlnts Den framtagna kopplingsmetoden har foumlrdelen att den inte bara faringngar de transienta aspekterna av interaktionen mellan termohydraulik och strukturmekanik men ocksaring stoumldjer avancerade modeller av konvektion i en smaumlltpoumll och strukturrespons och daumlrigenom bidrar till foumlrbaumlttrad precision och underlaumlttar implementeringen av modeller Kopplingsmetoden fungerar bra naumlr den valideras mot FOREVER-EC2 experimentet och den kan anvaumlndas i

IV

komplexa geometrier som till exempel i botten paring reaktortanken i en kokvattenreaktor (BWR) med en skog av genomfoumlringar (ledroumlr foumlr styrstavar och instrumentering)

Jaumlmfoumlrande beraumlkningar visar att de tvaring kopplingsmetoderna VLM-baserad och SLM-baserad presterar lika bra naumlr det gaumlller hur de predikterar resultat av FOREVER-EC2 experimentet och deras laumlmplighet foumlr reaktoranalyser Aumlven om SLM-metoden predikterar att tankbrott i baringda fallen intraumlffar lite tidigare saring aumlr skillnaden mellan metoderna foumlrsumbar i jaumlmfoumlrelse med tiden till tankbrott (~ 104 s) VLM-metoden var baumlttre aumln SLM-metoden naumlr det gaumlller beraumlkningseffektiviteten

Ideacuten om hybridkoppling garingr ut paring att anvaumlnda en lumped-parameter kod foumlr snabb uppskattning av termiska belastningar som senare kan anvaumlndas i detaljerad strukturmekanisk analys En saringdan kopplingsmetod kan paring ett signifikant saumltt oumlka effektiviteten av beraumlkningar vilket aumlr viktig i reaktorapplikationer daumlr mekanistisk simulering av konvektion i en smaumlltpoumll aumlr beraumlkningsintensiv och onoumldig I detta syfte utvecklades koden transIVR Koden kan dels ge en snabb uppskattning av transient vaumlrmeoumlverfoumlring i en smaumlltpoumll som bestaringr av en eller tvaring materialskikt och dels loumlsa vaumlrmeledningsproblem i reaktortanksvaumlggen med 2D finita differensmetoden vilket genererar detaljerad information om termiska foumlrharingllanden som behoumlvs foumlr strukturmekaniska analyser Foumlr att demonstrera att transIVR kan analysera dels vaumlrmeoumlverfoumlring i en smaumlltpoumll bestaringende av tvaring materialskikt och dels transient vaumlrmeoumlverfoumlring i en smaumlltpoumll jaumlmfoumlrdes transIVR prediktioner med UCSB FIBS-benchmarkfallet respektive LIVE-7V-experimentet I naumlsta steg kopplades transIVR-koden till ANSYS Structural foumlr att moumljliggoumlra mer detaljerad analys av reaktortankbrott Validering mot FOREVER-EC2 experimentet tyder paring att den utvecklade kopplingsmetodiken korrekt aringterger tankbrott orsakat av krypning

Nyckelord reaktortankbrott koppling termohydraulik-strukturmekanik modellering av krypning vaumlrmeoumlverfoumlring i smaumlltpoumll svaringra haverier CFD FEA transIVR

V

List of Publications

Papers included in this thesis

I P Yu and W Ma ldquoA modified theta projection model for creep behavior of RPV steel 16MND5rdquo Journal of Materials Science amp Technology vol 47 pp 231ndash242 2020

II P Yu W Ma W Villanueva A Karbojian and S Bechta ldquoValidation of a thermo-fluid-structure coupling approach for RPV creep failure analysis against FOREVER-EC2 experimentrdquo Annals of Nuclear Energy vol 133 pp 637ndash648 2019

III P Yu and W Ma ldquoComparative analysis of reactor pressure vessel failure using two thermo-fluid-structure coupling approachesrdquo Nuclear Engineering and Design submitted

IV P Yu and W Ma ldquoDevelopment of a lumped-parameter code for efficient assessment of in-vessel melt retention strategy of LWRsrdquo Annals of Nuclear Energy submitted

Respondentrsquos contribution to the papers included in the thesis

Peng Yu made major contributions to the publications of all the above four articles under the guidance of supervisors and with the collaboration of other co-authors including formulation of research questions development of coupling approaches and numerical methods analysis and generalization of simulation results as well as writing the first drafts and addressing review comments

Papers not included in this thesis

1 P Yu and W Ma ldquoRPV creep failure analysis using a Norton creep modelrdquo International Journal of Mechanical Sciences to be submitted

2 P Yu W Villanueva S Galushin W Ma and S Bechta ldquoCoupled thermo-mechanical creep analysis for a Nordic BWR lower head using non-homogeneous debris bed configuration from MELCORrdquo 12th International Topical Meeting on Nuclear Reactor Thermal-Hydraulics Operation and Safety (NUTHOS-12) Qingdao China October 14-18 2018

3 P Yu A Komlev Y Li W Villanueva W Ma and S Bechta ldquoPre-test simulation of SIMECO HT experiments for stratified melt pool heat transferrdquo 11th International Topical Meeting on Nuclear Reactor Thermal Hydraulics Operation and Safety (NUTHOS-11) Gyeongju Korea October 9-13 2016

VI

VII

Acknowledgements

First of all I would like to express my sincere gratitude to my supervisor Prof Weimin Ma for offering this opportunity to work in the severe accident research field and for the supports patience guidance discussions and encouragements throughout the entire period of my graduate study

I am very thankful to my co-supervisor Dr Walter Villanueva for the valuable help and discussions I learnt a lot during the times working with him I am also thankful to co-supervisor Prof Sevostian Bechta who is supportive and always has inspiring ideas that keep you thinking for a long time

Thank you also to all my colleagues at the Division of Nuclear Power Safety for the supports discussions and the very nice working environment during these years

I have taken many interesting courses during my PhD study for which I want to express my thanks to all the teachers With this regard a special thanks goes to Prof Johan Hoffman for his SF2561 Finite Element Method given in 2015 fall as well as the discussions and advices on numerical analysis

I am grateful to Ignacio Gallego-Marcos and Wenyuan Fan for their enthusiastic help on CFD and turbulence modelling during the time working on the turbulent

heat transfer I thank all my 师兄 for sharing their knowledge on nuclear engineering and giving advices when I was confused since 2012

I thank all my friends both in Sweden and in China for the company and the sharing of many wonderful moments in daily life

Finally I would like to express my deepest thank to my parents my girlfriend Siwen my sisters and all my other relatives for their unconditional supports and love

Financial supports from the China Scholarship Council and APRI make my study in KTH possible and are acknowledged

VIII

IX

Table of Contents

AbstracthelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphellipI

SammanfattninghelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphellipIII

List of PublicationshelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphellipV

AcknowledgementshelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphellipVII

NomenclaturehelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphellipXI

1 Introductionhelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphellip1

11 Nuclear Power Safety 1

12 RPV Failure 2

13 Literature Review Creep Behaviour and RPV Failure Analysis 4

131 Experiments on creep behaviour and RPV failure 4

132 Numerical studies on RPV failure 5

14 Motivation Objectives and Main Achievements 7

2 Numerical Solvers and Modelshelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphellip9

21 ANSYS Fluent 9

22 ANSYS Mechanical 9

23 PECM 11

3 Creep Modelling of 16MND5 Steelhelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphellip13

31 Creep and Creep Modelling 13

32 A Modified Theta Projection Model 14

321 Original theta projection model and its variations 14

322 A modified theta projection model 16

4 Coupled Thermo-Fluid-Structure Analysis of RPV Failurehelliphelliphelliphellip23

41 Thermo-Fluid-Structure Coupling Approach 23

42 Numerical Modelling 25

421 Conjugate heat transfer 25

422 Mechanical analysis 25

43 Validation against the FOREVER-EC2 Experiment 26

431 FOREVER-EC2 experiment 26

X

432 Conjugate heat transfer 27

433 Mechanical analysis 29

44 Application to BWR Case 32

45 Computational Efficiency 34

5 Hybrid Coupling Approach for Efficient RPV Failure Analysishelliphellip35

51 A Lumped-parameter Code for Melt Pool Heat Transfer 35

511 Heat transfer correlations 36

512 Energy balance in the pools 37

513 Energy balance in the RPV wall 39

52 Hybrid Coupling Approach 40

53 Verification and Validation 41

531 UCSB FIBS benchmark case 41

532 LIVE-7V experiment 43

533 FOREVER-EC2 experiment 45

6 Summary and Outlookhelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphellip51

Bibliographyhelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphellip55

XI

Nomenclature

Latin

119860 area of a surface

119862119901 heat capacity

119863 damage factor

119892 gravity

ℎ heat transfer coefficient

H Melt pool height

119896 thermal conductivity

119871 characteristic length

119873119906 Nusselt number

119875119903 Prandtl number

119902 heat flux

119902119894119899 heat flux through inner vessel surface

119876119881 internal volumetric heat source

119903 radial distance in a cylindrical or spherical coordinate

119877119886 external Rayleigh number

119877119886119894119899 internal Rayleigh number

119877119886119910 local Rayleigh number

119905 time

119879 temperature

119881119900119897 total volume

119909 thickness (of crust or RPV wall)

119911 height in a cylindrical coordinate

Greek

120572 thermal diffusivity

120573 thermal expansion

Δ119879 temperature difference

ε emissivity (in thermal analysis) strain (in structural analysis)

휀119888119903 creep strain

휀119903 creep strain rate

120579 polar angle

120579119894 parameters of a creep model 119894 = 1234119898

120584 kinematic viscosity

σ stress

XII

Super and subscripts

119886119898119887 ambient

119888119903 creep (in Section 3) crust (in Section 5)

119889119899 downward

119890119899119907 environment

119894 inner surface of RPV

119894119899119905 interface

119897119900119888119886119897 local

119898119886119909 maximum value

119898119890119897119905119894119899119892 melting point

119898119905 metal layer (bulk)

119898119905119889119899 downward surface of metal pool

119898119905119904119889 sideward surface of metal pool

119898119905119905119900119901 top surface of metal layer

119898119905119906119901 upward surface of metal pool

119900 outer surface of RPV

119900119909 oxide layer (bulk)

119900119909119889119899 downward surface of oxide layer

119900119909119906119901 upward surface of oxide layer

119903119890119891 reference

119904 internal structures inside the RPV

119904119889 sideward

119906119901 upward

119907119890119904 vessel

Acronyms

BWR boiling water reactor

CBD creep database

CFD computational fluid dynamics

CRGT control rod guide tube

ECCM effective convectivity conductivity model

FDM finite difference method

FE finite element

FEA finite element analysis

FEM finite element method

FVM finite volume method

IGT instrument guide tube

IVR in-vessel retention

XIII

IVMR in-vessel melt retention EU project

LES large eddy simulation

MPS moving particle semi-implicit method

NPP nuclear power plant

PECM phase-change effective convectivity model

RANS Reynolds-averaged Navier-Stokes

RPV reactor pressure vessel

SBO station blackout

SLM surface loads mapping

SPH smoothed-particle hydrodynamics

UDF user-defined function

UPF user programmable features

VLM volume loads mapping

XIV

1

1 Introduction

11 Nuclear Power Safety

Greenhouse gas emissions contribute to the global warming problem Phasing out fossil fuels by switching to low-carbon energy sources is considered as a mitigation strategy With this regard clean energy should be increased in the world energy consumption structure Being clean and stable nuclear energy is an attractive alternative to fossil fuels After the development over decades it also proves to be a mature and safe technology According to IAEA 2018 annual report [1] there were 450 nuclear power reactors under operation worldwide at the end of 2018 which had a record generating capacity of 3964 GW(e) 9 reactors were connected to the grid and 55 reactors under construction around the world

Due to the potential risk of the release of radioactivity nuclear power safety is emphasized throughout the stages of design construction operation and decommission of a nuclear power plant (NPP) The principal technical requirements [2] specify that the three fundamental safety functions should be fulfilled for all plant states of an NPP (i) control of reactivity (ii) removal of heat from the reactor and from the fuel store and (iii) confinement of radioactive material shielding against radiation and control of planned radioactive releases as well as limitation of accidental radioactive releases The defence-in-depth concept is also applied to provide several levels of defence to prevent consequences of accidents The fuel claddings reactor pressure vessel (RPV) and primary coolant pressure boundaries and the containment are the three physical barriers to prevent the release of radioactive materials to the environment

Severe accidents are events that cause significant damage to reactor fuel and result in core meltdown It may be caused by insufficient cooling of the reactor core due to various reasons or by the rapid introduction of huge reactivity which causes sharp increases in the power and fuel fragmentation eg Chernobyl accident During a severe accident the failure of the fuel claddings will result in the radioactive materials previously enclosed inside the cladding released to the primary loop If cooling is not sufficient to remove the decay heat the core materials will melt migrate down and relocate into the RPV lower head after which the melt will react with water and partially form fragments quench solidify and form a debris bed The debris bed formed by the melt may continue to melt to form a melt pool due to the decay heat As the corium contacts the RPV it would continuously heat the RPV and threaten its integrity Once the RPV fails containment becomes the last barrier to prevent the release of radioactivity The released melt would result in an increase of the pressure and temperature due to the high temperature fractions oxidation and hydrogen combustion which is called direct containment heating The melt would fall onto the reactor cavity Without sufficient cooling the molten core would interact with basemat concrete (MCCI) This process can last for few days It can result in containment failure due to either containment overpressure or the basemat melt-through of containment In addition containment is also subjected to isolation failure and early containment failure due to missiles and dynamic pressure [3] More

2

detailed severe accident progression or phenomena could also be found in eg [4] [5]

Though severe accidents are unlikely to occur the consequences can be severe if no proper measure is taken Related researches have been continuously going on aiming at mitigating the accident consequences and improving safety After the Fukushima Accident safety inspections and re-evaluation actions have been carried out worldwide especially on safety measurements regarding severe accidents The concept lsquoPractical Eliminationrsquo applied to new NPP designs demands that new nuclear installations be designed with the objective of mitigating the consequences of accidents and avoiding early radioactive releases and large radioactive releases [6] Severe accident mitigation strategies are applied to the existing and new NPPs (both Generation II and III) The strategies can generally be classified into two categories [3] the in-vessel retention (IVR) and ex-vessel retention (EVR) In simple words in-vessel retention tries to retain the core melt inside the RPV (second barrier) by applying external water cooling on the vessel as shown in Figure 1(a) Such method minimizes the dispersal of the melt within the containment and hence simplifies the later recovery work Examples of the application are AP1000 HPR1000 APR 1400 The idea of the EVR is to let the RPV fail but to retain the core melt inside the containment Two principle approaches of EVR are the retention within a dedicated lsquocore catcherrsquo and melt fragmentation Examples of core catcher would be EPR that directs the melt to a spreading compartment (Figure 1(b)) and VVER1000 that uses metallic crucible below the RPV An example of the melt fragmentation is the Nordic BWRs that the corium released from RPV would relocate in the cavity with a deep-water pool (7-12 m) and is expected to fragment quench and form a coolable debris bed (Figure 1(c))

Figure 1 Examples of IVR [7] EVR with core catcher [8] and melt fragmentation [9]

12 RPV Failure

RPV which contains the reactor core core shroud and coolant is a key component to the reactor system RPV and the primary loop constitute the primary coolant system which conveys the fission heat from the reactor core to the secondary loop through steam generator (PWR) or to the turbine (BWR) The pressure under

(a) IVR (b) Core catcher (c) Melt fragmentation

3

operation can be ~155 MPa for a 1000 MW PWR [10] and ~70 MPa for a 1000 MW BWR [11] It also has to withstand the neutron radiation damage The thickness of RPV [11] can be ~20 cm for PWR and ~16 cm for BWR

As the transition of in-vessel progression to ex-vessel progression RPV failure is important for analysis of accident progression and assessment of the effectiveness of severe accident mitigation strategies In the melt fragmentation strategy of Nordic BWR the melt-coolant interaction and debris bed coolability after the corium poured into the deep-water pool are essentially dependent on the vessel failure mode including the characteristic time of melt release rupture size and location and amount and superheat of melt available for release [12] New concern also arose [13] on the necessity of performing structural analysis for the RPV in the IVR strategy In the initial methodology and further development of the IVR the success of the strategy ie the integrity of the RPV was demonstrated (eg in [14]) with the following two conclusions (i) even for a local heat flux on the vessel as high as critical heat flux (CHF) the resulted very thin external ldquocold layerrdquo of the vessel would be sufficient to withstand the internal stress load and hold the vessel integrity (ii) as CHF is the critical value for the thermal load the main specific goal for assessing the effectiveness of IVR becomes to demonstrate that the local heat flux along RPV be lower than the CHF The new concern doubts that local lsquohot partrsquo of the vessel would have negative effects on the overall resistance due to creep and dilatation which could result in a vessel failure and hence violate the first conclusion in particular for ldquolong termrdquo retention of corium in RPV If so a structural integrity analysis of RPV would be needed in addition to assessing local heat flux especially for high-power reactors Therefore accurate prediction of vessel failure could be meaningful for both strategies

Figure 2 Illustration of RPV under thermal loads from corium

As the structural performance is highly affected by the thermal loads from the corium RPV failure is essentially a transient thermo-fluid-structure coupled problem Figure 2 illustrates the situation that a corium debris bedmelt pool formed in the RPV lower head The decay heat of nuclear fission products could continuously be generated inside the corium leading to the formation of melt pools as the debris melts The heated debris bedmelt pool imposes thermal and

4

mechanical loads on the RPV Such loads can be transient (change with time) and spatially non-uniform Complex phenomena are also involved in this process Strong turbulent natural convection induced by the decay heat may be expected in the melt pool under IVR The corresponding Rayleigh number 119877119886 can be 1016minus17 for prototypic reactor cases Material structural properties can become non-linear under high temperature and stress loads The creep deformation also becomes significant at high temperature Detailed modelling of turbulent natural convection and non-linear properties could be computationally expensive In addition large-scale geometry (hemisphere with a radius of 20-30 m) long-term transient deformation process (~10000 s) and required small time steps (eg 001 s) for numerical convergence consideration further increase the computational costs

13 Literature Review Creep Behaviour and RPV Failure Analysis

In this section the most relevant studies concerning the experiments modelling and analysis of creep behaviour and RPV failure are presented

131 Experiments on creep behaviour and RPV failure

Various experiments have been performed on the RPV failure problem some of which were to generate data of mechanical properties (especially the creep properties) and some to study the overall vessel failure with scaled or simplified geometries

The REVISA experiment [15] generated creep curves and stress strain curves of French steel 16MND5 ranging from 600 to 1300 degC This data is accessible in [16] as well as [17] which also documents other material properties eg density thermal expansion coefficient conductivity heat capacity Youngrsquos Modulus and plasticity Similar experiments were carried out to generate high-temperature creep and tensile test data for American RPV steels and Inconel [18]

The CORVIS experiments [19] investigated the lower head failure modes with and without penetrations Each experiment was carried out on a steel plate 100 mm thick representing a pressure vessel lower head The penetration case represented a boiling water reactor lower head carrying a tube penetration in the form of a drain line

The LHF (lower head failure) experiments [20] and the succeeding OLHF tests [21] investigated the vessel failure with a 15 scaled vessel heated by an electrical radiator allowing to locate the hot focus at different position of the vessel with a maximum wall temperature of around 900 degC Internal pressure ranging from 53 to 123 bar was also imposed to initiate the vessel failure

The FOREVER experiments [22] investigated the phenomena of melt pool convection vessel creep possible failure processes and failure modes occurred during the late phase of the in-vessel progression by heating up a melt pool in a vessel and observing possible vessel failures The study covers cases with and without penetration different melt level existence of gap cooling and in-vessel melt coolability

5

132 Numerical studies on RPV failure

Various numerical approaches have been employed on RPV failure Since a two-way thermo-fluid-structure coupling approach is challenging these approaches take simplifications and can generally be classified into two categories (i) RPV failure decoupled from melt pool heat transfer and (ii) RPV failure coupled with melt pool heat transfer in simplified manners

1) Decoupled approaches In the decoupled approaches only RPV is modelled Simplified constant thermal loads are imposed on RPV considering the melt pool heat transfer

In a post-test analysis of FOREVE-C2 experiment [23] it was treated as a thermo-structural coupled analysis of the vessel with the thermal load estimated by the outer thermocouples and the normal heat flux on the external vessel wall The analysis was done using finite element (FE) code SYSTUS A coupled creepdamage law based on a viscosity-hardening creep law describing primary and secondary creep was employed for creep analysis

Mao et al conducted various studies on reactor cases assessing the IVR strategy using FE code ABAQUS Thermal boundary conditions were imposed on the RPV which was then solved as a thermal-structural coupled problem The strain hardening creep model (Eq (1)) was mainly used Their studies cover the general RPV failure [24] as well as the effects of cooling water levels under IVR [25] and geometric discontinuity (ablated RPV) [26]

ε119888119903 = 1198891σ1198892ε1198893119890119909119901 [minus

1198894119879] (1)

In the EU IVMR project [13] the RPV integrity was also studied through a benchmark calculation which deals with the RPV of a PWR with a prescribed ablated profile with 2D axisymmetric geometry Prescribed temperature gradient was set as thermal load The objective was to calculate deformations for various pressure loads up to vessel rupture A smoother profile was planned in further research

The advantage of such approaches is once thermal loads are reasonably assumed the focus only needs to be put on the RPV The disadvantage is a reasonable thermal load should be consistent with the melt pool heat transfer while sometimes it is not well-noted eg constant uniform temperature boundary conditions were assumed for inner and outer RPV (with uniform thickness in lower head) surfaces [25] which ignored the non-uniform heat flux distribution caused by the turbulent convective corium pool As thermal loads strongly affect the structural performance they should be quantified carefully

2) Coupled approaches In coupled approaches both melt pool heat transfer and RPV failure are modelled and coupled in one-way or simplified manner With melt pool heat transfer modelled the coupled approaches are more flexible to provide reasonable thermal loads on the RPV In existing approaches CFD and the correlation-based distributed-parameter models eg the Effective Convectivity Conductivity Model

6

(ECCM) [27] and the Phase-change Effective Convectivity Model (PECM) [28] [29] are general methods for melt pool heat transfer

Sehgal et al [22] presented an approach that employed the MVITA code to calculate the thermal processes and ANSYS for structural analysis MVITA code employed the ECCM Bailey-Norton creep law was used as shown in Eq (2)

ε119888119903 = 6055 times 10minus26 sdot σ472119905085119890minus

37200

119879 (2)

This approach was used for the pre-test simulation of FOREVER experiment to determine the thermal and mechanical loadings Parametric investigations also helped in the identification of important factors that could affect the test procedure and test conditions

Willschutz et al [30] employed the CFD-module Flotran for the melt pool heat transfer and ANSYS for structural analysis of RPV Standard k-ε turbulence model was selected for the turbulent natural convection in the melt pool A user-defined three-stage creep model [31] was employed In this model experimental creep curves were discretized into a creep database (CDB) under given temperature 119879 and stress 120590 the creep curve was discretized into finite point pairs

(휀 ε) indicating under such thermal and mechanical loads the creep strain rate ε is subjected to creep strain 휀 The creep strain rate for any other case can be interpolated over known temperature stress and strain from the database This method was applied to the post-test calculation of the FOREVE-C2 experiment It was concluded that the unexpected deformation behaviour during the experiment was caused by the combination of the transient temperature field in the vessel wall and the three creep stages

They later replaced the Flotran solver with the ECCM for melt pool heat transfer in a following study [32] By implementing it into FEM-solver a recursively coupled analysis was constructed Moreover a modified creep model [33] was used with damage measures the CDB itself did not contain tertiary creep instead a factor 1(1 minus 119863) was multiplied to the original creep strain rate to get the modified creep strain rate where 119863 is a damage factor (0 lt 119863 lt 1) defined with creep strain information and plastic strain information As 119863 grows close to 1 the modified creep strain rate would be enlarged significantly by this factor 1(1 minus 119863) which then behaves like the accelerating tertiary stage This method was applied to pre- and post- test calculations for the FOREVER experiments proving to be closer to reality than the previous method

Madokoro et al [34] developed a thermal structural analysis tool based on open source solver OpenFOAM and numerically investigated the coupled RPV failure and melt pool convection The PECM was used in the melt pool heat transfer In structural part the plasticity and damage parameters were similar to Willschutz et al [32] except for that strain hardening creep model (Eq (3)) was used instead of a CDB Simulations of FOREVER-EC2 and -EC4 were investigated It was concluded from the simulations that although the emissivity and mesh type (updated or fixed) affect the results significantly the approach generally has the capability of predicting the vessel deformation and failure

7

ε119888119903 = 1198891 sdot σ1198892 sdot ε1198893 (3)

Villanueva et al [35] performed a coupled thermo-mechanical creep analysis of the reactor scale RPV (ABB-Atom design BWR) to evaluate efficacy of the cooling measures for the in-vessel melt coolability and retention The PECM was used for melt pool heat transfer and ANSYS for RPV integrity analysis The modified time hardening model (Eq (4)) of the American RPV steel SA533B was selected as creep model The transient thermal load from the melt pool was applied as a thermal boundary condition to the vessel wall Further studies were done with this approach on investigations of instrumentation guide tube (IGT) failure [36] and control rod guide tube (CRGT) cooling [37] of a Nordic BWR

ε119888119903 =1198881σ

11988821199051198883+1

1198883 + 1 1198881 gt 0 (4)

It is noted that only RPV analysis with finite element analysis (FEA) are reviewed here Engineering methods like the Larson-Miller method [38] are not covered in this thesis One can refer to theory manuals of severe accident codes like MELCOR ASTEC for more details Comprehensive description on modelling of melt pool heat transfer is not covered either One can refer to either [39] for a general view or to any severe accident code manuals for engineering methods

14 Motivation Objectives and Main Achievements

The existing experimental and numerical studies provide us a good understanding of RPV failure involved phenomena and valuable simulation experiences These numerical methods may vary in solvers (which may be related to code accessibility) and models (which may be related to model accessibility understanding of phenomena computational power) However improvements of modelling approaches are still possible as the improvements in any of the related aspects eg

bull Simulation tools are continuously being upgraded and providing new possibilities just like that PECM appeared and ECCM was no longer used

bull Better understanding of the phenomena and more simulation experiences from in-vessel retention and vessel failures researches benefit the further modelling and simulations

bull Increasing computational power allows more detailed modelling eg turbulence modelling of melt pool convection

bull There is still room for the improvement of creep modelling

In addition implementation difficulty of a coupling approach is another important issue to consider Since it is a TRANSIENT thermo-fluid-structure coupled problem it is almost impossible to replicate an approach simply by referring to a paper with limited details For BWRs it would be even better for the coupling approaches to support complex geometries considering forest of penetrations (CRGTs and IGTs) on the RPV lower head without significant increase of user effort in the coupling process

8

The objectives of this thesis are to investigate new approaches and develop new models for the thermo-fluid-structure coupled problem of RPV failure and involved phenomena These objectives are accomplished with the following achievements

1) A new creep model for the 16MND5 steel was developed which covers all three creep stages Good agreement was achieved between model predictions and experimental curves Both time and strain hardening models were implemented (Paper I)

2) A new thermo-fluid-structure coupling approach was developed for RPV failure analysis It supports the advanced models in both melt pool convection and structural mechanics to improve fidelity and facilitates implementation Validation of the approach with the new creep model was performed against FOREVER-EC2 experiment This approach is flexible for complex geometries and can be used in further analysis eg lower head with detailed modelling of penetrations in BWRs (Paper II)

3) The performance of this new approach which uses volume loads mapping for data transfer was compared with the previous approach using surface loads mapping showing good agreement with small discrepancy (Paper III)

4) A lumped-parameter code transIVR for melt pool heat transfer was developed which supports one- and two-layer pool configuration transient analysis and well-resolved RPV wall (Paper IV)

5) A hybrid coupling approach was developed for efficient analysis of RPV failure where transIVR is coupled with ANSYS Mechanical (Paper IV)

In the following general descriptions on the codes ANSYS Fluent Mechanical and the PECM are given in Section 2 the new creep model in Section 3 the new coupling approach as well as its comparison with the previous approach in Section 4 the efficient hybrid coupling approach in Section 5 and summary amp outlook in Section 6

9

2 Numerical Solvers and Models

In this section we briefly introduce the ANSYS Fluent as the CFD solver for melt pool heat transfer the ANSYS Mechanical as the mechanical solver for RPV failure and PECM as an efficient alternative for melt pool heat transfer that are heavily used in the following sections

21 ANSYS Fluent

In the fluid mechanics the fluid is treated as a continuum which is generally described by the conservative mass momentum and energy equations [40]

120597120588

120597119905+ 120571 sdot (120588) = 119878119898 (5)

120597(120588)

120597119905+ 120571 sdot (120588) = minus120571119901 + 120571 sdot (120591) + 120588 + (6)

120597120588119864

120597119905+ 120571 sdot ((120588119864 + 119901)) = 120571 sdot (119896120571119879 minussumℎ119895119869119895

119895

+ (120591 sdot )) + 119878ℎ

(7)

Computational Fluid Dynamics (CFD) studies topics on the numerical solutions of above equations Typical methods are the Finite Volume Method (FVM) Finite Difference Method (FDM) and Finite Element Method (FEM) as well as particle methods that are mesh-free eg SPH and MPS

ANSYS Fluent is a general purpose CFD code which solves above equations with FVM Turbulence modelling and multiphase flows are supported Turbulence can be modelled with RANS LES or hybrid LESRANS Customizations eg adding sourcesink time or spatial dependent boundary conditions are realized through the user-defined functions (UDF) which is programmed in C language

22 ANSYS Mechanical

In the continuum mechanics of solids three sets of variables are considered to describe the structure performance namely element displacement field strain field and stress field (119906 ε σ) A general 3-dimensional problem is described by 3 sets of equations the equilibrium equations (Eq (8)) strain-displacement relationship (Eq (9)) and constitute equations (Eq (10)) [41] For Eq (8) the reciprocal theorem of shear stress also gives τ119894119895 = τ119895119894 where 119894 119895 = 119909 119910 119911

indicating the stress tensor being symmetric For very simple applications eg 1D beam it can be solved analytically For complex problems where analytical solutions are not available approximation methods eg Galerkin weighted residual method principle of virtual work and principle of minimum potential energy would be used

10

partσ119909119909part119909

+partτ119909119910

part119910+partτ119909119911part119911

+ 119909 = 0

partτ119910119909

part119909+partσ119910119910

part119910+partτ119910119911

part119911+ 119910 = 0

partτ119911119909part119909

+partτ119911119910

part119910+partσ119911119911part119911

+ 119911 = 0

(8)

ε119909119909 =part119906

part119909 휀119910119910 =

120597119907

120597119910 휀119911119911 =

120597119908

120597119911

120574119909119910 =120597119906

120597119910+120597119907

120597119909 120574119910119911 =

120597119908

120597119910+120597119907

120597119911 120574119911119909 =

120597119908

120597119909+120597119906

120597119911

(9)

ε119909119909 =1

119864[σ119909119909 minus μ(σ119910119910 + σ119911119911)]

ε119910119910 =1

119864[σ119910119910 minus μ(σ119909119909 + σ119911119911)]

ε119911119911 =1

119864[σ119911119911 minus μ(σ119909119909 + σ119910119910)]

γ119909119910 =1

119866τ119909119910 γ119910119911 =

1

119866τ119910119911 γ119911119909 =

1

119866τ119911119909

(10)

ANSYS Mechanical [42] is a mechanical engineering software that uses finite element analysis (FEA) for structural analysis The principle of virtual work is adopted in the solver [43] The idea of this principle is that a virtual (very small) change of the internal strain energy must be offset by an identical change in external work due to the applied loads which reads

δ119880 = δ119881 (11) where 119880 is the strain energy or the internal work 119881 the external work and δ the virtual operator The virtual internal work can be expressed as

δ119880 = int δεσd(vol)T

vol

+int 119886119903119890119886119891

δ119908119899119879σ119889(119886119903119890119886119891) (12)

where 휀 is the strain vector 120590 stress vector vol volume of element 119908119899 motion normal to the surface 119886119903119890119886119891 area of the distributed resistance The virtual

external work is contributed from three parts the inertial effects the pressure force on the surface and nodal forces applied to the element which then can be expressed as

δ119881 = minusint 119907119900119897

δ119908119879119865119886

119907119900119897119889(119907119900119897) + int

119886119903119890119886119901

δ119908119899119879119875119889(119886119903119890119886119901)

+ δ119906119879119865119890119899119889

(13)

11

where 119908 is the vector of displacements 119865119886 acceleration force vector 119875 applied pressure vector 119886119903119890119886119901 area over which pressure acts and 119865119890

119899119889 nodal

forces applied to the element

Plugging Eq (12) and (13) into (11) would result in the equilibrium equation Since the virtual displacement can be arbitrary the final equations used to describe the equilibrium of an element is reduced to Eq (14) with multiple operations

([119870119890] + [119870119890119891])119906 minus 119865119890

119905ℎ = [119872119890] + 119865119890119901119903 + 119865119890

119899119889

(14)

where [119870119890] [119870119890119891] 119865119890

119905ℎ [119872119890] and 119865119890119901119903 are the element stiffness matrix

foundation stiffness matrix thermal load vector mass matrix acceleration vector and pressure vector respectively It represents the equilibrium equation on a one element basis Alternatively using a minimum total potential energy principle could result in a consistent expression to the principle of virtual work [41]

Customizations eg on material properties are realized through the user programmable features (UPF) which is programmed in FORTRAN language

23 PECM

The PECM [28] [29] developed in KTH is a correlation-based method to model the melt pool heat transfer It is implemented in ANSYS Fluent through UDF and has been validated against many experiments such as ACOPO COPO LIVE etc In the PECM the flow velocities 119906119909 119906119910 and 119906119911 are replaced by characteristic

velocities 119880119909 119880119910 and 119880119911 respectively in order to eliminate the necessity of solving

the Navier-Stokes equation

The characteristic velocities are calculated using heat transfer correlations

119880119906119901 =α

ℎ119901119900119900119897times (119873119906119906119901 minus

ℎ119901119900119900119897

ℎ119906119901) (15)

119880119889119899 =α

ℎ119901119900119900119897times (119873119906119889119899 minus

ℎ119901119900119900119897

ℎ119889119899) (16)

119880119904119889 =α

ℎ119901119900119900119897times (119873119906119904119889 minus

ℎ119901119900119900119897

119882119901119900119900119897) (17)

where ℎ119901119900119900119897 is the melt pool depth ℎ119906119901 the thickness of well mixed layer of the

pool ℎ119889119899 the thickness of lower stratified region of melt pool 119882119901119900119900119897 the width of

pool 120572 the thermal diffusivity The PECM employs reduced characteristic velocities to describe mushy zone convection heat transfer For internally heated pool natural convection the Steinberner-Reineke correlations (Eq (18)~(19)) [44] are employed The profile of sideward 119873119906 is described using Eckertrsquos type correlation (Eq (20)) for a vertical boundary layer where the local Rayleigh

number at pool height 119910 is defined as 119877119886119910 =119892βΔ1198791199103

αν

12

119873119906119906119901 = 03451198771198861198941198990233

(18)

119873119906119889119899 = 13891198771198861198941198990095

(19)

119873119906119904119889 = 050811987511990314 (

20

21+ 119875119903)

minus14

11987711988611991014

(20)

As the characteristic velocities are calculated the energy equation is the only transport equation that needs to be solved and can be expressed as follows

partρ119862119901119879

part119905+ nabla(ρ119862119901119932119879) = 119896nabla2119879 + 119876119907 (21)

where from left to right are the transient term convection term diffusion term and volumetric heat source respectively The convection term then can be calculated explicitly with use of values in the last time step and then can be treated together with the volumetric source term Therefore the PECM finally simplifies the energy equation to a conduction equation which can be easily solved and as a result will significantly increase the computational efficiency in simulations

In addition solidificationmelting process can also be modelled and the temperature-dependent physical properties are supported

13

3 Creep Modelling of 16MND5 Steel

Creep deformation becomes significant when a component is subjected to high temperature and stress Modelling of creep behaviour of the RPV is paramount since it is considered as a main mechanism leading to the RPV failure In this section we present the development of a three-stage creep model (called lsquomodified theta projection modelrsquo ) for the French RPV steel 16MND5 This model was successfully implemented into the ANSYS Mechanical code

31 Creep and Creep Modelling

Creep is an irreversible time-dependent inelastic deformation process Diffusion and dislocation are the two molecular mechanisms of creep The former normally occurs due to the presence of vacancies within the crystal lattice (or the grain) while the latter occurs if there is a slip by the passage of crystal dislocations [45] There are two types of diffusion creep [46] Coble creep (favoured at lower temperatures) if the diffusion paths are predominantly through the grain boundaries and Nabarro-Herring creep (favoured at higher temperatures) if through the grains themselves Dislocation creep tends to dominate at high stresses and relatively low temperatures Dislocations can move by gliding in a slip plane which is a process that requires little thermal activation But the rate-determining step for their motion is often a climb process which also requires diffusion and is thus a time-dependent process and favoured by higher temperatures Creep behaviour is highly affected by temperature and stress generally creep is more significant under higher temperature and stress

Creep curves describe the creep strain development with time Figure 3 shows a creep curve under constant temperature and stress which typically experiences three creep stages primary stage where the creep strain rate ε119888119903 (the change of creep strain ε119888119903 in unit time) decreases with time secondary stage where the creep strain rate almost keeps constant and tertiary stage where the creep strain rate drastically increases with time

Figure 3 Creep curve under constant temperature and stress

14

The built-in creep models in commercial software eg ANSYS [47] and ABAQUS [48] in default only cover the primary andor the secondary stage(s) Table 1 [47] gives the default creep models supported in ANSYS for example Possible reason why the tertiary stage is not modelled might be that for most industrial applications the failure analysis is beyond interest and therefore the built-in creep laws in the commercial software was supposed to be sufficient [49] Though the two codes offer the interfaces through either ANSYS UPF or ABAQUS user subroutines to support more general user-defined creep laws eg complex models that also include the tertiary creep stage

Table 1 Default creep models in ANSYS Mechanical

Creep model Equations Type

Strain hardening

ε119888119903 = 1198621σ1198622ε119888119903

11986231198901198624119879 1198621 gt 0 Primary

Time hardening ε119888119903 = 1198621σ11986221199051198623119890minus1198624119879 1198621 gt 0 Primary

Generalized exponential

ε119888119903 = 1198621σ1198622119903119890minus119903119905 r = 1198625σ

11986231199051198623119890minus1198624119879 1198621 gt 0

1198625 gt 0

Primary

Generalized Graham

ε119888119903 = 1198621σ1198622(1199051198623 + 1198624119905

1198625 + 11986261199051198627)119890minus1198628119879 1198621 gt 0 Primary

Generalized Blackburn

ε119888119903 = 119891(1 minus 119890minus119903119905) + 119892119905 119891 = 1198621119890

1198622σ 119903 = 1198623(σ1198624)1198625 119892 = 1198626119890

1198627σ

1198621 gt 0

1198623 gt 0

1198626 gt 0

Primary

Modified time hardening

ε119888119903 = 1198881σ11988821199051198883+1(1198883 + 1) 1198621 gt 0 Primary

Modified strain hardening

ε119888119903 = 1198621σ1198622[(1198623 + 1)ε119888119903]

1198623119890minus1198624119879 1198621 gt 0 Primary

Generalized Garofalo

ε119888119903 = 1198621[119904119894119899ℎ(1198622σ)]1198623119890minus1198624119879 1198621 gt 0 Secondary

Exponential form

ε119888119903 = 1198621119890σ1198622119890minus1198623119879 1198621 gt 0 Secondary

Norton ε119888119903 = 1198621σ1198622119890minus1198623119879 1198621 gt 0 Secondary

Combined time hardening

ε119888119903 = 1198621σ11986221199051198623+1119890minus1198624119879(1198883 + 1)

+ 1198625σ1198626119905119890minus1198627119879

1198621 gt 0

1198625 gt 0

Primary+ Secondary

Rational polynomial ε119888119903 = 1198621

partε119888part119905 ε119888 =

119888119901119905

1 + 119901119905+ ε119898119905

ε119898 = 1198622101198623σσ1198624 119888 = 1198627ε

1198628σ1198629

119901 = 11986210ε11986211σ11986212

1198622 gt 0 Primary+ Secondary

Generalized time hardening

ε119888119903 = 119891119905119903119890minus1198626119879 119891 = 1198621σ + 1198622σ2 + 1198623σ

3 119903 = 1198624 + 1198625

Primary

32 A Modified Theta Projection Model

321 Original theta projection model and its variations

The original theta projection model [50] was first proposed to accurate predict long-term creep and creep-fracture properties It assumes that a creep curve under constant temperature and stress can be described as

15

휀119888119903 = 1205791(1 minus 119890minus1205792119905) + 1205793(119890

1205794119905 minus 1) (22)

where 휀119888119903 is the creep strain 119905 the time and 1205791~1205794 the model parameters Figure 4 illustrates how the first term and second term on right hand side develop with time and constitute a creep curve which has obvious primary secondary and tertiary stages Model parameters 120579119894(119894 = 1234) are temperature and stress dependent and are described with the following expression

119897119899thinsp120579119894 = 119886119894 + 119887119894120590 + 119888119894119879 + 119889119894120590119879 (23)

where 119886119894 119887119894 119888119894 and 119889119894 ( 119894 = 1234 ) are constants It is noted [45] that this equation is suitable when the temperature range of experiment data sets is very narrow such that minus1119879 is almost proportional to 119879 These constants (119886119894 119887119894 119888119894 and 119889119894 (119894 = 1234)) are the final model parameters which are generated with two fitting procedures

(i) Fitting the experimental creep curves to Eq (22) generates several groups of 1205791~1205794 for each creep curve under certain temperature and stress We denote (119879119895 120590119895 1205791119895 1205792119895 1205793119895 1205794119895 ) as parameters for the 119895119905ℎ

experimental curve under temperature 119879119895 and stress 120590119895

(ii) Fitting 1205791 to Eq (23) as a function of temperature and stress with the obtained data ( 119879119895 120590119895 1205791119895 ) from ( 119879119895 120590119895 1205791119895 1205792119895 1205793119895 1205794119895 ) generates

parameters 1198861 1198871 1198881 and 1198891 Repeat this fitting procedure for 1205792 1205793 and 1205794 to get parameters 119886119894 119887119894 119888119894 and 119889119894 (119894 = 234)

Variations of the model were developed and applied to other different materials to better accommodate the experimental data which are all then called a lsquomodified theta projection methodrsquo independently Examples are models in [51] for lowhigh alloy ferritic austenitic steels Al-alloys Al-matrix composites [52] for Nickel-based Super-alloy [53] for pure metals and class M alloys with intermediate range of stress (40-70 of the yield stress) and temperature (40-60 of the melting temperature)

Figure 4 Illustration of theta projection method [54]

16

322 A modified theta projection model

The new modified theta projection model we proposed [54] was mainly adapted from the original theta projection method [50] and the modified version [51] The new model describes the creep curve with the following equation

휀119888119903 = 1205791(1 minus 119890minus1205792119905) + 120579119898119905 + 1205793(119890

1205794119905 minus 1) (24)

An additional term 120579119898119905 is seen compared with Eq (22) in the original method The reason for adding this term is that the 16MND5 steel shows fairly long secondary stages in many experimental creep curves and the creep curves can be better fitted by Eq (24) than Eq (22) As the structure undergoes different creep process under different temperature and stress the model parameters θ119894 (119894 =1~4 and 119898) are also temperature and stress dependent

The main difference compared with the original and all other modified projection models is that numerical interpolation is used to replace the second numerical fitting in previous models In other words instead of fitting the model parameters as functions of stress and temperature with Eq (23) the distributions of the model parameters are assumed to be piecewise in the temperature-stress space and they will be interpolated directly from known parameters obtained from experiments The consideration is that the experimental data ([16] [17]) for 16MND5 covers a wide temperature range varying from 600 degC to 1300 degC and a wide stress load range varying from 08 MPa to 248 MPa while Eq (23) in the existing methods is based on a narrow temperature span Poor parameter fitting results would be expected if we directly fit the parameters 1205791~1205794 with Eq (23) as used in the original methods

The creep model was developed using the following steps

(i) Fitting Each creep curve can be represented by a group of model parameters [1205791 1205792 120579119898 1205793 1205794] with respect to Eq (24) by fitting the creep curves to Eq (24) All parameters of the 32 experimental curves form an

initial dataset of the model parameters [119879119895 120590119895 1205791119895 1205792119895 120579119898119895 1205793119895 1205794119895] where j (0ltjle32) is the jth creep curve

(ii) Interpolation Parameters of a creep curve under any [119879 σ] are directly interpolated from the dataset

(iii) Parameter adjustment The dataset was further updated to satisfy the monotonicity assumption we have taken that creep strain monotonically increases with temperature and stress

(iv) Different hardening models for dynamic loads

Step (i) was done using MATLAB In the following steps (ii)-(iv) are described in detail

Model parameters for new creep curves are interpolated from the dataset over the 2D 119879 minus 120590 space The interpolation is divided into two sub-interpolation steps interpolate over stress with fixed temperature and then interpolate over temperature with fixed stress We take the interpolation of model parameters under (119879prime σprime) for example In the first interpolation we first find out the two

17

temperatures 119879119896119899119900119908119899119904119898119886119897119897 and 119879119896119899119900119908119899119897119886119903119892119890 most close to 119879prime from the experimental

temperature set 600 700 800900 1000 1100 1200 1300 Under each temperature there are four sets of model parameters in the dataset corresponding to four different stress loads Then the first interpolation is done over stress load from the known dataset of model parameters under each

temperature to output the parameters for (119879119896119899119900119908119899119904119898119886119897119897 120590rsquo) and (119879119896119899119900119908119899119897119886119903119892119890 120590rsquo)

In the second interpolation the model outputs parameters for (119879rsquo 120590rsquo) from

(119879119896119899119900119908119899119904119898119886119897119897 120590rsquo) and (119879119896119899119900119908119899119897119886119903119892119890 120590rsquo) For 120579i(119894 = 12 3 4) we assume a logarithm

relation between 120579119894 temperature 119879 and stress 120590 which is similar to Eq (23)

119897119899120579119894 = (119886 + 119887120590)(119888 + 119889119879) (25)

It reduces to 120579119894 = 11986011198901198871σ with fixed temperature in the first interpolation and to

θ119894 = 11986021198901198872T with fixed stress in the second interpolation The interpolation

expression for θ119894 = 11986011198901198871σwith two known points (1205901 1205791198941) and (1205902 1205791198942) is given

by Eq (26) The interpolation expression for θ119894 = 11986021198901198872T with two known points

(1198791 1205791198941) and (1198792 1205791198942) is given by Eq (27)

120579119894 = 120579i1

120590minus12059021205901minus1205902120579i2

1205901minus1205901205901minus1205902 (26)

120579119894 = 120579i1

119879minus11987921198791minus1198792120579i2

1198791minus1198791198791minus1198792 (27)

Similarly we assume the following relation between 120579119898 temperature 119879 and stress 120590

120579119898 = 119860120590119899119890minus119887 119879frasl (28)

The interpolation is done over the stress with 120579119898 = 1198603120590n by fixing temperature

and then over temperature with 120579119898 = 1198604119890minus1198874119879 by fixing stress The interpolation

expression over stress for 120579119898 = 1198603120590119899 with two known points (1205901 1205791198981) and

(1205902 1205791198982) would be given by Eq (29) The interpolation expression over

temperature for 120579119898 = 1198604119890minus1198874119879 with two known points (1198791 1205791198981) and (1198792 1205791198982)

would be given by Eq (30)

120579119898 = 119890119909119901(119897119899120590 minus 11989711989912059011198971198991205902 minus 1198971198991205901

1198971198991205791198982 +1198971198991205902 minus 119897119899120590

1198971198991205902 minus 11989711989912059011198971198991205791198981) (29)

120579119898 = 120579m2(11198791minus1119879) (

11198791minus11198792)fraslsdot 120579m1

(1119879minus11198792) (

11198791minus11198792)frasl (30)

In this model we also take the assumption that the creep strain monotonically increases with temperature and stress loads1 Or specifically with the same creep time the structural component under higher temperature or stress experiences

1 Intuitively materials deform more under higher stress and temperature This is also mostly observed in tensile experiments Though some exceptions exist one instance of which is the effect of phase change some materials under certain temperature around the phase change temperature may show much weaker structural performance Here we just simply assumed that the creep strain monotonically increases with temperature and stress

18

higher creep strain than that with lower temperature or stress (the creep curve under higher temperature or stress should be above the curve with lower temperature or stress) When generating the parameter dataset (Step (i)) we already kept the trend that under the same temperature cases under higher stress loads would have larger creep model parameters and give higher creep strains This trend would also hold for the interpolated temperatures which thus guarantees that creep strain monotonically increases with stress load under same temperature Then the question becomes if creep strain also monotonically increases with temperature given fixed stress However the answer could not always be a yes especially in the extrapolation range of the stress load Figure 5(a) illustrates an example of the interpolation of model parameters under (119879119872 σ119872) The red dots and blue dots are known parameters from the parameter dataset for corresponding high temperature 119879119867 and low temperature 119879119871 respectively The green dots (119879119867 σ119872) and (119879119871 σ119872) are interpolated from the red and blue dots in the first sub-interpolation respectively The black dot (119879119872 σ119872) is interpolated from the green dots in the second sub-interpolation It is seen that when σ119872 lt σ119862 the value of (119879119871 σ119872) is higher than that of (119879119867 σ119872) It means the higher 119879119872 is the smaller the model parameter will be ie the model would predict larger creep strain for lower temperatures which violates our monotonicity assumption One simple solution used here as shown in Figure 5(b) is to minorly adjust the values of the four red and blue dots such that the intersection point C can be sufficiently small and the situation σ119872 lt σ119862 would barely occur in the extrapolation range

Figure 5 Illustration of model parameter interpolation (a) before and (b) after parameter adjustment [54]

A corresponding parameter adjustment process was introduced to verify that the model predicts higher creep strain under higher temperature under fixed stress Figure 6 illustrates the general verification process for any (119879 120590) in the interested load range the creep strain of (119879 + 120575119879 120590) should be larger than that of (119879 120590) Otherwise parameters would be minorly adjusted and the verification process be repeated This process would be iterated several times until the requirement is reached The achieved verification shows

Stress

σC

High temperature TH

MPa

a

0

Model

par

amet

er θ

Low temperature TL

MPa

a

0

Model

par

amet

er θ

High temperature TH

Low temperature TL

σC

Stress

C C

(a) (b)

σM

σM

19

(i) For any (119879 120590) in the temperature range 273~ 1073 K with an interval of 1 K and the stress range 1 ~260 MPa with an interval of 1 MPa its creep strain is higher than that of (119879 minus 1119870 120590) for the first 12000000 s

(ii) For any (119879 120590) in the temperature range 1073~1673 K with an interval of 1 K and the stress range 1 ~80 MPa with an interval of 1MPa its creep strain is higher than that of (119879 minus 1119870 120590) for the first 12000000 s

(iii) For any (119879 120590) in the stress range 01~26 MPa with a smaller interval of 01 MPa for the full temperature range 273~1673 K with an interval of 1 K its creep strain is higher than that of (119879 minus 1 120590) for the first 12000000 s

Above verification covers a wide range of temperature and stress especially where the stress and temperature are low and extrapolation is used Further verification to cover even higher stress or temperature ranges is not considered as creep increases drastically under such high stress and temperature and the results would become meaningless With these regards we generally think the monotonicity requirement is reached The model parameter dataset was updated along with the adjustment The final dataset was given as Table 2 in Paper I

Figure 6 Parameter adjustment process [54]

Figure 7(a-d) present the model predictions of selected creep curves in ANSYS Mechanical (labelled as lsquoANSYSrsquo) compared with experimental curves (labelled as lsquoREVISArsquo) at temperature 600 800 1000 and 1200 degC respectively The predicted curves proved to clearly show the three-stage creep process and agreed well with the experimental creep curves

Initial parametersdataset

θ1 θ2 θm θ3 θ4 from creep

curve-fitting

Parameters interpolation

Is the monotonicity

assumption satisfied

IsMon= =1

Final model parameters

Adjust the

parametersdataset

θ1 θ2 θm θ3 θ4

No

Yes

IsMon=1 a flag is monotonic achieved

For σ in the stress range in MPa

For T in range Tstart Tend in K

For t in range 112000000 in s

If ε(σ T+1t)lt ε(σTt) IsMon=0

Iteration loop of

parameter adjustment

20

Figure 7 Selected model predictions compared with experimental creep curves [54]

Instead of being constant temperature and stress loads also dynamically change with time in real cases Hardening models determine how creep strain rate reacts to the change of loads during a creep process The time hardening model and strain hardening model2 are supported in our model implementation in ANSYS Mechanical which are two mostly used hardening models

Time hardening assumes that the following creep strain rate after a load change depends only on the current creep time from the beginning of the creep process (and also temperature and stress if used in the creep model) and ignores the current creep strain Strain hardening assumes that the creep strain rate depends only upon the current creep strain (and also temperature and stress if used in the creep model) [55]

2 The time hardening and strain hardening here are more general terms than the specific time hardening creep model and the strain hardening creep model mentioned in Table 1

(a) 600 degC (b) 800 degC

(c) 1000 degC (d) 1200 degC

21

Eq (24) in default is a time hardening model as creep strain is directly described by the time 119905 Eq (31) gives the creep strain increment 119889휀119888119903 at time 119905 during time interval 119889119905

119889휀119888119903 = (12057911205792119890minus1205792119905 + 120579119898 + 12057931205794119890

1205794119905)119889119905 (31)

The strain hardening model is derived from Eq (24) in an implicit form Specifically when temperature and stress load change to (119879 120590) at the time 119905 and current creep strain 휀1198881199030 the model will get the creep curve of constant (119879 120590) and

output how this curve will react under the current strain 휀1198881199030 Since the creep curve shows creep strain monotonically increases with time given the creep strain 휀1198881199030 under creep curve of (119879 120590) we can uniquely find this reference time

119905119903119890119891 satisfying Eq (32) One should notice that the reference time 119905119903119890119891 can be

different from the current physical time 119905

휀1198881199030 = 1205791(1 minus 119890minus1205792119905119903119890119891) + 120579119898119905119903119890119891 + 1205793(119890

1205794119905119903119890119891 minus 1) (32)

where all 120579119894(119894 = 1~4119898) are model parameters of the new creep curve (119879 120590) It can be solved for 119905119903119890119891 with methods eg Newton-Ramphson [56] Then the creep

strain increment at 119905119903119890119891 during time interval 119889119905 is given as

119889휀119888119903 = (12057911205792119890minus1205792119905119903119890119891 + 120579119898 + 12057931205794119890

1205794119905119903119890119891)119889119905 (33)

Therefore combining Eq (32) and (33) gives the creep strain rate at creep strain 휀1198881199030 In summary when loads are changed to (119879 120590) with current creep strain 휀1198881199030

the model traces the creep curve of (119879 120590) and calculates the creep strain rate of this curve at strain 휀1198881199030

One calculation was done to demonstrate the performance of the hardening models The stress load was set to 151 MPa and kept unchanged A step temperature load was applied as the orange dash curve shows in Figure 8 it was 900 degC during 0-36000 s and suddenly jumped to 1000 degC at 36000 s which was then kept till the end of the simulation The blue solid curve (the part before the temperature change was identical to and therefore covered by the green solid curve in the figure) and blue dash curve are the creep curves under constant temperature T=900 degC and T=1000 degC respectively The green solid curve and green dash curve represent the model predictions considering this temperature change with time hardening model and strain hardening model respectively Before this change the creep curves with both hardening models were identical to the creep curve of the T=900 degC till point A After that the creep strain rate then shifted from Point A to Point B (which had the same time with A) in the time hardening model and behaved the same as the curve T=1000 degC as shown in the rest part of the green solid curve In the strain hardening model the creep strain rate shifted from Point A to Point C (which had the same creep strain with A) and behaved the same as the curve T=1000 degC as shown in the rest part of the green dash curve The large difference between two models was partially caused by the drastic temperature jump In real cases the difference would be less significant as the temperature changes more mildly Overall this prediction is exactly the expected behaviors of the time hardening model and strain hardening model

22

Figure 8 Time hardening and strain hardening in ANSYS Mechanical [54]

23

4 Coupled Thermo-Fluid-Structure Analysis of RPV Failure

As described in Section 13 RPV failure is coupled with melt pool heat transfer leading it into a transient thermo-fluid-structure coupled problem The goal of this section is to develop and validate a reliable coupling approach which supports transient coupled thermo-fluid-structure analysis as well as advanced models and computational platforms so as to improve fidelity and facilitate implementation The approach was validated against the FOREVER-EC2 experiment and then applied to an RPV failure analysis of a Nordic BWR As the approach adopts the volume loads mapping for load transfer comparative analysis was also performed against the coupling approach using surface loads mapping Future work is considered in analysis of penetration failures (IGTs and CRGTs) of a BWR RPV since the approach is flexible to complex geometries Results of Paper II ampIII are jointly presented in this section

41 Thermo-Fluid-Structure Coupling Approach

Considering the involved phenomena a coupling approach should at least address three issues (i) melt pool heat transfer (ii) RPV deformation and (iii) the interaction between melt pool heat transfer and vessel deformation A two-way strong coupling considering both the transient thermal load from melt pool on vessel and the influence of vessel deformation on melt pool heat transfer involves complex data exchange and is still full of challenges Therefore the new coupling approach is realized in a one-way manner with the assumptions that the influence of vessel deformation on melt pool heat transfer can be ignored and thus only the thermal loads from melt pool to vessel are considered It is based on the considerations that on one hand the deformation is a slow process regarding the relatively long failure time and on the other hand it was pointed out [57] that the physics of natural convection heat transfer in internally heated liquid pools is not sensitive to the geometrical factors based on their calculation results and experimental data from [58] [59] as well as previous experiments in semicircular and square cavities eg [44] [60] In addition one-way coupling is also computationally more efficient especially when heat transfer turbulence structural non-linearity and large-scale geometries are involved The developed approach is based on the ANSYS platform which generally consists of three parts with respect to the three issues respectively

(i) conjugate heat transfer of the melt pool and vessel using ANSYS Fluent (ii) transient thermal loads transfer from ANSYS Fluent to ANSYS

Structural (the ANSYS Mechanical in ANSYS Workbench) (iii) structural analysis of the vessel in ANSYS Structural

ANSYS Structural in default only supports direct transfer of constantsteady-state spatial loads from ANSYS Fluent Transfer of transient loads is realized with the extension tool FSI Transient [61] It is a pre-processing tool (with good documentation) developed by a third-party CAE vendor EDRTM MEDESO [62] to map transient CFD results (heat transfer coefficient temperature and surface pressure) to thermal or structural analysis including the definition of load steps

24

This new procedure proves to be straightforward and reduces efforts in the load transfer

Since structural performance of RPV is highly dependent on temperature the transfer of transient temperature loads from conjugate heat transfer to mechanical solver is a core step of the coupling framework There are two general approaches to realize the transfer as shown in Figure 9 Both approaches contain three parts a conjugate heat transfer analysis for fluid and structure where flow and heat transfer are coupled a mechanical analysis for the structure and a load mapping process that transfers the transient loads The conjugate heat transfer part can be solved with any CFD solver and is treated in the same manner in both approaches In the first approach (Figure 9(a)) transient volume thermal loads are transferred and mapped into the mechanical simulation domain The structure domain reads the essential thermal loads from the mapping after which a structure analysis of this domain is performed In the second approach (Figure 9(b)) transient surface thermal loads are mapped onto the boundary surfaces of structure domain as thermal boundary conditions Then a thermo-structure coupled analysis of the structure domain in the mechanical part is performed To distinguish them we call the former approach volume loads mapping (VLM) and the latter surface loads mapping (SLM) In the RPV failure analysis the fluid and structure domains correspond to the melt pool and RPV respectively Examples of these methods are the studies of [63] and [30] using VLM and [35] using SLM The VLM is adopted in current approach

Figure 9 Frameworks of the thermo-fluid-structure coupling approaches (a) volume loads mapping (b) surface loads mapping [64]

fluid structure structure

fluid structure structure

Conjugate heat transfer problem Mechanical problem

(structure)

Conjugate heat transfer problem Mechanical problem

(thermo-structure coupled)

Transient volume

loads mapping

(VLM)

Transient surface

loads mapping

(SLM)

(a)

(b)

25

42 Numerical Modelling

421 Conjugate heat transfer

In the conjugate heat transfer analysis main focus is put on the modelling of melt pool heat transfer Within the ANSYS Fluent solver two options are supported CFD with turbulence modelling and PECM CFD is the choice when the model goes to detail and more mechanistic analysis are needed while PECM provides an alternative that is faster and possibly more suitable for large-scale problems

Within the scope of RANS models the SST (Shear Stress Transport) model [65] is selected as the turbulence model in present study This model is a combination of k-ε model and k-ω model The k-ω formulation is used in the inner parts of the boundary layer which makes the model directly usable all the way down to the wall through the viscous sub-layer It also switches to k-ε model in the free-stream and thereby avoids the common k-ω problem that the model is too sensitive to the inlet free-stream turbulence properties The considerations on choosing this model are

bull Several two-equation low-Re k-ε models have been proved in [66] not good enough for melt pool heat transfer Therefore these mentioned two-equation models as well as other one-equation models are not considered

bull SST is highly recommended [67] for simulating most industrial heat transfer problems

bull Convincing results ([68] [69]) using SST have been obtained in capturing the thermal behaviour of the natural convection in an internally heated fluid Especially good agreements were achieved between the predicted heat flux distribution along the vessel and experimental data which would be one of the most important factors to estimate model performance in this problem

The description of PECM as an effective model for melt pool heat transfer is given in Section 23

422 Mechanical analysis

The material for the RPV failure analysis considered in this study is the French RPV steel 16MND5 The three-stage modified theta projection model in Section 3 is used as the creep model Since the definition of an RPV failure criterion is not universal (it can be the time when damage variable reaching 1 or either specified value of creep strain creep strain rate or total deformation is reached) we do not explicitly define a failure criterion Instead the simulation is kept running until the creep converge criterion cannot be satisfied even with the defined minimum time step (indicating a fairly large creep strain rate) and the simulation will be automatically terminated The time the simulation stops is taken as the failure time This treatment implicitly implies a criterion of a large creep strain rate We refer to [17] for other material properties Youngrsquos modulus and thermal conductivity are temperature dependent Multilinear Isotropic Hardening (MISO) is used as the plasticity model

26

43 Validation against the FOREVER-EC2 Experiment

In this section the validation against the FOREVER-EC2 experiment is presented (indicated as lsquoVLMrsquo in the following) In addition the simulation results using the SLM approach (indicated as lsquoSLMrsquo in the following) is also included for comparison purpose

431 FOREVER-EC2 experiment

The FOREVER-EC2 experiment [70] is one of the FOREVER experiments that achieved a vessel failure Figure 10 shows the experimental setup A 110 scale pressure vessel with a wall thickness of 15 mm and an outer diameter of ~400 mm was employed The lower head was made of 16MND5 steel Since it was difficult to use prototypical corium materials in the experiment the binary salt 30 CaO-70 B2O3 was employed as simulant To generate the internal heat which corresponds to the nuclear decay heat a specially designed heater was used The power input was maintained ~38 kW during the experiment such that maximum melt pool temperature would be ~1300 degC The pressure inside the vessel was maintained at ~25 MPa Thermocouples were used to measure the melt pool and vessel wall temperatures and linear displacement transducers used to measure the vessel wall displacements The designed experimental conditions were reached at experimental time 11670 s Vessel failure was observed about 12510 s later (at experiment time 24180 s)

Figure 10 Scheme of the experimental setup [22]

27

Figure 11 Computational domains and boundary conditions (a) conjugate heat transfer (b) mechanical analysis [71]

Computational domains and boundary conditions are indicated in Figure 11 SST turbulence modelling was used for pool heat transfer The simulations started at the moment when experimental conditions were reached (experiment time 11670 s) So simulation time 0 s corresponds to experiment time 11670 s

432 Conjugate heat transfer

Figure 12 shows the temperature fields at simulation time 0 s 1225 s 2475 s 4975 s and 9975 s All the time points in this part refer to simulation time The field at time 0 s (Figure 12(a)) was the initial temperature that was reconstructed based on the thermocouple readings The temperature increased with time and reached a steady state when the maximum temperature reached ~1607 K Figure 12(b-f) show clear thermal stratification which is an expected phenomenon of natural convection

Internal pressure

25 bar

Internal pressure 25e6-2500times981timesy Pa

External pressure

1 bar

Zero displacement

y=0 m

Gravity 400

mm

R=203 mm

15 m

m

Tamb

=323 K radiation with ε = 08

+ convection with h=10Wm2K

Tamb

=800 K radiation with ε=02

(a) (b)

Tamb

=800 K radiation with ε=08

+ convection with h=10 Wm2K

28

Figure 12 Temperature fields at simulation time (a) 0 s (b)1225 s (c) 2475s (d) 4975 s (e) 7475 s and (f) 9975 s [64]

Figure 13 shows the temperature profiles of the external vessel along the polar angle in comparison with experimental data Regions with a polar angle larger than 90deg are those in the cylindrical part of the vessel It also shows the temperature profile increased with time until reaching a steady state The profile almost did not change after 4975 s which indicates that a steady state of the temperature was reached As time increased the temperature in the lower head region 0~90deg (where the vessel was contacted with the melt) evolved from a relatively flat distribution to a distribution that temperature at high angle was higher than that at low angle by more than 200 K Generally the steady state results agreed well with the experimental data except for the low angle part of vessel lower head and the high angle part of the cylindrical vessel In addition to modelling errors the discrepancy in the low angle part of the lower head may be caused by the crust and gap possibly formed at vessel bottom in the experiment which can reduce the downward heat transfer and result in lower temperature while they cannot be modelled in simulations As to the high angle cylindrical vessel part simplified boundary conditions were applied to the inner and outer vessel wall since it was far from the melt and not of interest which may result in some discrepancy

(a) (b)

(c) (d) (e) (f)

29

Figure 13 Temperature profiles of external vessel surface along the vessel polar angle [64]

433 Mechanical analysis

In the mechanical analysis the VLM predicted a failure time of 23959 s and the SLM predicted a failure time of 22445 s both are close to the experimental failure time 24180 s All the time points in this part refer to the experiment time Figure 14 shows the creep strain fields at failure time Both approaches predicted maximum creep strains at same location where the temperature was high (cf Figure 12) This is related to the fact that creep behavior is highly influenced by temperature Maximum values for both cases were also close to each other 0393 in VLM and 0366 in SLM

Figure 15 presents the total deformation of the vessel bottom point (also called ldquosouth polerdquo in some literature) in the vertical direction Since there was an initial displacement reading of ~355 mm at time 0 s which almost did not change during 0~11670 s (the whole period before the experimental conditions were reached) an initial value of 355 mm was added to the simulation result such that experimental and simulation results had the same total displacement at t=11670 s Similar trend was seen in the VLM SLM and experimental data that the deformation increased with time and experienced a drastic acceleration at the end Generally both simulation approaches agreed well with the experimental data though they both slightly under predicted the value of total deformation The discrepancy may possibly be related to the accuracy of the thermal predictions modelling errors of plasticity and creep and simplified boundary conditions

400

600

800

1000

1200

1400

1600

0 20 40 60 80 100 120 140 160

Tem

per

ature

(K

)

Polar angle (deg)

Experiment

25 s

1225 s

2475 s

4975 s

7475 s

9975 s

Weilding line

30

Figure 14 Creep strain fields at failure time (a) VLM (b) SLM [64]

Figure 15 Total deformation in vertical direction of the vessel bottom point [64] [71]

The wall thickness changes are plotted together with experimental results in Figure 16 The wall thickness change δ119909 is defined as

δ119909 = 119909119890119899119889 minus 119909119900119903119894119892119894119899119886119897 (34)

000

500

1000

1500

2000

2500

11670 13670 15670 17670 19670 21670 23670

Tota

l def

orm

atio

n (

mm

)

Time (s)

ExperimentVLMSLM

(a) (b)

31

where 119909119890119899119889 and 119909119900119903119894119892119894119899119886119897 are the wall thickness at failure time and the original wall

thickness respectively Since 119909119890119899119889 cannot directly be post-processed it was estimated in the following way (cf the subfigure of Figure 16) for the same polar angle θ find the corresponding points Pin (1199091 1199101) and Pout (1199092 1199102) on the inner and outer surfaces of the deformed vessel respectively the distance of Pin and

Pout radic(1199091 minus 1199092)2 + (1199101 minus 1199102)2 is calculated as 119909119890119899119889 at polar angle 120579 Such estimation of the wall thickness is more accurate when the line PinPout is closer to be orthogonal to the deformed local surface Considering the final deformed vessel as shown in Figure 14 the estimation should be fine at regions close to 0deg and 90deg and can be slightly overestimated at medium angle region The VLM curve was compared with the SLM curve and 4 experimental curves measured at vessel left right front and back sides Generally VLM curve agreed well with the experimental and SLM curves wall thickness gradually decreases with angle till around 75deg and then increases till 90deg the maximum thickness change occurred at around 75deg would be around -5 mm The positive values at low angle may be caused by the thermal expansion It also shows that the thickness transitions in experimental curves were smoother than those in both simulation curves while simulation curves showed narrower regions with large thickness changes (at angle 70deg~80deg) Possible reasons might be (i) sharper temperature profile predicted in simulation (ii) input uncertainty and (iii) modelling errors The SLM predicted slightly thinner vessel than the VLM except for the necking region The discrepancy might be caused by various reasons eg the interior vessel temperature can be slightly different between two approaches which would then result in different thermal stress and different deformation the final deformed vessel shapes cannot be exactly the same which also results in different errors in estimating PinPout

Figure 16 Wall thickness changes along the vessel polar angle [64] [71]

-6

-5

-4

-3

-2

-1

0

1

0 10 20 30 40 50 60 70 80 90

Wal

l th

icknes

s ch

ange

(mm

)

Polar angle (deg)

VLMSLMExperiment (left)Experiment (right)Experiment (front)Experiment (back)

32

44 Application to BWR Case

In this section the coupling approach (indicated as lsquoVLMrsquo in the following) was applied to an RPV failure analysis of a BWR A simulation using the SLM coupling approach (indicated as lsquoSLMrsquo in the following) was also performed for comparison purpose

A hypothetical severe accident of a Nordic BWR initiated by an SBO was considered The core materials would melt and migrate downward to the RPV lower head The total amount of corium relocated in the lower head was considered to form a 19 m corium melt pooldebris bed assuming the corium distribution was homogeneous

Figure 17 Computational domains and boundary conditions (a) conjugate heat transfer (b) mechanical analysis [64]

Figure 17(a) shows the computational domain of the conjugate heat transfer PECM was used in this calculation A general mesh with a pool height of 23 m was used to accommodate most of the relocation situations with the same mesh Since the corium only had a height of 19 m the top 04 m region was treated as injected water covering the corium The top water was expected to be boiling and we assumed a uniform temperature of 383 K Under the implementation of PECM this region was damped with artificially high heat capacity and high conductivity such that it would efficiently remove heat from the corium and suppress its own temperature increase For the corium part below water a volumetric heat source 11 MWm3 was implemented as the nuclear decay heat For the in-vessel surfaces a constant temperature of 383K was applied as they are covered by water which can be boiling Without water cooling on the external vessel wall a very small heat loss of 20 Wm2 was applied Melting and solidification of the corium were also considered with a solidus temperature of 2750 K and a liquidus temperature of 2770 K The initial temperature was set as

q=20 Wm2

T=383 K

Qv=11

MWm3

315 bar

315 bar+ hydrostatic

pressure of corium

Gravity

248 bar

Zero

displacement

Water

Corium

(a) (b)

33

1100 K for the corium 383 K for the water and 450 K for the vessel For the mechanical analysis (Figure 17(b)) an inner pressure of 315 bar and an outer pressure of 248 bar were considered For the region covered by corium an extra hydrostatic pressure was added on the vessel wall A 2 m long vessel extension was considered as well as the standard earth gravity

Figure 18 presents the wall temperature profile along the inner vessel surface at 50 s 2500 s 7500 s and 10000 s The temperature increased with time for the vessel surface contacted with corium and the distribution was flat which was due to that large part of the corium was still in solidus status It also shows that for the surface region that contacted the water the temperature was basically unchanged which indicates that the damping treatment indeed suppressed the temperature increment and guaranteed the heat removal upward by water

Figure 18 Temperature profiles of inner vessel wall along the vessel polar angle [64]

The predicted RPV failure time was 9869 s by VLM and 9650 s by SLM Compared with the FOREVER-EC2 case in Section 43 the VLM predicted a slightly later failure time than the SLM in this BWR case too Figure 19(a) compares the maximum creep strain developments with time of both approaches Before 9000 s the predicted values by both approaches are close to each other Afterward the creep acceleration occurred and the strain increased drastically The acceleration of the VLM occurred 200 s later than that of the SLM which agreed with the predicted later failure time Similar trend was observed for the total deformation in vertical direction of vessel bottom point as shown in Figure 19(b) The deformations of both approaches are close except that the acceleration of VLM occurred later than that of SLM Overall results of VLM agreed well with those of SLM

0

500

1000

1500

2000

2500

0 10 20 30 40 50 60 70 80 90 100

Tem

per

ature

(K

)

Polar angle (deg)

50 s

2500 s

5000 s

7000 s

10000 s

34

Figure 19 Results of structural analysis (a) creep strain and (b) total deformation in vertical direction of vessel bottom point [64]

45 Computational Efficiency

Computational efficiency is also an important consideration regarding the performance of numerical approaches The computational efficiencies of both VLM and SLM are analysed and compared in this section As parallel computing technique was used to speed up the simulations the number of parallel cores and corresponding costed computational time for each simulation are given together in Table 2 as a generally reflection of the computational efficiency All the simulations were done in a Lenovo P700 workstation equipped with 2 Intel Xeon E5-2630 v3 processors (8 cores per processor) and 32 G RAM For the simulations of FOREVER-EC2 SLM employed twice the parallel cores and costed 286 more time compared with VLM For the simulations of prototypic BWR case both VLM and SLM employed the same number of parallel cores but VLM only took 13 of the computational time compared with SLM Overall it can be concluded that VLM has higher computational efficiency than the SLM The reason is a structural analysis is performed in the VLM approach while a coupled thermo-structural analysis is performed in the SLM approach (cf Figure 9) a standalone structural analysis is computed faster than a coupled thermo-structural analysis

Table 2 Comparison of computational efficiency [64]

Case FOREVER-EC2 BWR VLM SLM VLM SLM

Number of cores 4 8 4 4 Computational time (hours) 21 27 18 72

35

5 Hybrid Coupling Approach for Efficient RPV Failure Analysis

In this section we propose a hybrid coupling approach for the efficient RPV failure analysis The idea is to replace the ANSYS Fluent in previous coupling approach with a lumped-parameter code for quick estimate of the melt pool heat transfer With the least modelling details (compared with CFD methods and distributed-parameter methods eg PECM) lumped-parameter codes prove to be the most efficient for melt pool heat transfer The new hybrid approach characterizes 1) utilization of efficient models for melt pool heat transfer and 2) the support of detailed structural modelling of RPV deformation allowing studies with different RPV geometric shapes under effects like local thinning due to ablation For studies of which the focus is mainly on structural behaviour of RPV eg [13] [26] while mechanistic simulation of melt pool convection is computationally expensive and unnecessary the lumped-parameter code could help providing reasonable thermal loads

The development of the approach consists of two tasks (i) development of a lumped-parameter code and (ii) its coupling with ANSYS Mechanical to form this hybrid coupling framework In the following we describe both tasks and the corresponding verification and validation processes respectively This section is mainly based on Paper IV

51 A Lumped-parameter Code for Melt Pool Heat Transfer

A lumped-parameter code transIVR was developed It features (i) quick estimate of transient melt pool heat transfer with reasonable accuracies and (ii) two-dimensional representation of heat conduction in the RPV wall facilitating the coupled analysis of the RPV failure under thermal loads The following assumptions are taken in the transIVR modelling

bull The crust between the oxide pool and metal pool is thin

bull The heat flux has uniform distributions on the bottom top and side surfaces of metal pool

bull Heat source only exists in oxide layer

bull Latent heat of fusion is ignored

Figure 20 illustrates the modelled three domains oxide layer metal layer and the RPV wall In addition the internal structure inside the vessel can also be considered as a radiative body to the top metal surface as in [72] An internal volumetric heat source generated by nuclear decay heat is considered in the oxide layer Heat in oxide layer transfers downward to the RPV wall and upward to the top metal layer Heat in metal layer can be transferred sideward to the RPV wall and upward to the surrounding internal structure through radiation (or to water if inside water flooding is applied) The heat transferred to RPV will be removed from outer RPV surface Possible formations of oxide crust in the top and bottom surface are illustrated as the black regions Possible formation of metal crust on the top surface is shown as the grey region The code automatically detects the

36

existence of the crust on oxide top and metal top surfaces during the simulation and solves the problem correspondingly withwithout crust Boundary conditions are applied to the external and internal surfaces of RPV and (if needed) top surface of metal layer Arbitrary boundary conditions are supported If a boundary condition is nonlinear eg mixed heat convection (linear dependent on 119879 ) and heat radiation (proportional to 1198794) the Picard method [73] would be used for solving this problem

This model consists of three main parts

(i) empirical heat transfer correlations (mostly from experiments) to determine global and local heat transfer coefficients

(ii) energy balances in oxide layer and metal layer where the heat transfer coefficients would be used and

(iii) energy balance in RPV

Figure 20 Two-layer melt pool configuration and RPV wall [74]

511 Heat transfer correlations

For internally heated natural convection (oxide layer) in transIVR the mini-ACOPO correlations [75] are used for the averaged heat transfer as expressed in Eq (35) and (36) and for the angular variation of the heat flux as expressed in Eq (37) and (38)

119873119906119906119901 = 03451198771198861198941198990233 (35)

119873119906119889119899 = 00038119877119886119894119899035(119867119877)025 (36)

119873119906119889119899(θ)

119873119906119889119899= 01 + 108(

θ

θ119898119886119909) minus 45 (

θ

θ119898119886119909)2

+ 86 (θ

θ119898119886119909)3

01 leθ

θ119898119886119909le 06 (37)

RPV

Oxide pool

Metal pool

37

119873119906119889119899(θ)

119873119906119889119899= 041 + 035 (

θ

θ119898119886119909) + (

θ

θ119898119886119909)2

06 ltθ

θ119898119886119909le 1 (38)

The 119877119886119894119899 that appears in Eq (35)~(36) is the internal Rayleigh number which is defined as

119877119886119894119899 =119892β119876119881119867

5

α119896ν (39)

For Rayleigh-Bernard convection in the metal layer heat transfer occurs in all the top bottom and side surfaces To side surface the reduced form [72] of the Churchill-Chu correlation [76] for 119875119903 = 013 is employed as shown in Eq (40)

119873119906119904119889 = 007611987711988613 (40)

To top and bottom surfaces the reduced form [72] of Globe-Dropkin correlation [77] for

119875119903 = 013 is used as expressed in Eq (41)

119873119906119906119901 = 005911987711988613 (41)

The 119877119886 that appears in Eq (40) and (41) is the external Rayleigh number which is defined

as

119877119886 =119892βΔ1198791198673

να (42)

Heat transfer coefficients can be calculated from Nusselt number by Eq (43)

ℎ =119873119906119896

119871 (43)

where 119896 and 119871 are the heat conductivity and characteristic length respectively 119873119906 can be the Nusselt numbers expressed in Eq (35)~(38) and (40)~(41)

512 Energy balance in the pools

The energy balance of oxide layer can be described as Eq (44) The total generated heat of oxide pool equals the sum of the increase of enthalpy (transient temperature change) upward heat flow to metal layer and downward heat flow to RPV

120588119862119901119881119900119897119889119879119900119909119889119905

+ ℎ119906119901119860119900119909119906119901(119879119900119909 minus 119879119900119909119906119901) + ℎ119889119899119860119900119909119889119899(119879119900119909 minus 119879119900119909119889119899) minus 119876119881119881119900119897 = 0 (44)

where ρ is the oxide density 119862119901 the heat capacity 119881119900119897 the total volume 119879119900119909 the bulk temperature 119905 the time ℎ119906119901 the upward heat transfer coefficient 119860119900119909119906119901

upward heat transfer area 119879119900119909119906119901 upward surface temperature ℎ119889119899 downward

heat transfer coefficient 119860119900119909119889119899 downward heat transfer area 119879119900119909119889119899 downward surface temperature and 119876119881 volumetric heat source

38

On the interface between oxide layer and metal layer energy balance requires that the heat transferred from oxide layer should equal the heat to the metal layer which can be described as Eq (45)

ℎ119906119901119860119900119909119906119901(119879119900119909 minus 119879119900119909119906119901) = ℎ119898119905119889119899119860119898119905119889119899(119879119894119899119905 minus 119879119898119905) (45)

where ℎ119898119905119889119899 is the downward heat transfer coefficient of metal layer 119860119898119905119889119899 the downward heat transfer area 119879119894119899119905 the interface temperature and 119879119898119905 the bulk temperature of metal layer

The energy balance of metal pool can be described as Eq (46) The increase of enthalpy (transient temperature change) equals the sum of upward downward and sideward heat flows to surrounding surfaces

ρ119898119905119862119901119898119905119881119900119897119898119905119889119879119898119905119889119905

+ ℎ119898119905119906119901119860119898119905119906119901(119879119898119905 minus 119879119898119905119906119901) + ℎ119898119905119904119889119860119898119905119904119889(119879119898119905 minus 119879119898119905119904119889)

+ ℎ119898119905119889119899119860119898119905119889119899(119879119898119905 minus 119879119894119899119905) = 0

(46)

where ρmt is the metal density 119862119901mt the heat capacity 119881119900119897mt the total volume 119879119898119905 the bulk metal temperature 119905 the time ℎ119898119905119906119901 the upward heat transfer

coefficient 119860119898119905119906119901 upward heat transfer area 119879119898119905119906119901 upward surface

temperature ℎ119898119905119904119889 sideward heat transfer coefficient 119860119898119905119904119889 sideward heat transfer area 119879119898119905119904119889 sideward surface temperature ℎ119898119905119889119899 downward heat transfer coefficient 119860119898119905119889119899 downward heat transfer area and 119879119894119899119905 oxide-metal interface temperature

The heat transferred from metal layer to the top surface should equal the heat radiated from this surface to the environment Considering the radiation and reflections of internal structures to this surface following energy balance equation [72] is achieved

ℎ119898119905119906119901(119879119898119905 minus 119879119898119905119906119901) = σ(1198791198981199051199051199001199014 minus 119879119904

4) (1

ε+1 minus ε119904ε119904

119860119898119905119906119901

119860119904)minus1

(47)

where 119879119898119905119905119900119901 is the temperature of metal top free surface ε the emissivity of

metal ε119904 the emissivity of internal structures 119860119898119905119906119901 the area of metal top surface

and 119860119904 the total area of internal structures Sometimes the internal water cooling may be available above metal layer In this case Eq (47) is unnecessary and 119879119898119905119905119900119901 can be set to a constant value

For the oxide layer top surface Eq (48) is used if not crust is formed in the interface between two layers ie 119879119894119899119905 gt 119879119898119890119897119905119894119899119892119900119909 Otherwise Eq (49) is used

119879119900119909119906119901 = 119879119894119899119905 (48)

119879119900119909119906119901 = 119879119898119890119897119905119894119899119892119900119909 (49)

For the metal layer top surface Eq (50) is used if not crust is formed on top of the metal ie 119879119906119901119904119906119903119891 gt 119879119898119890119897119905119894119899119892119898119905 Otherwise Eq (51) is used

119879119898119905119906119901 = 119879119898119905119905119900119901 (50)

39

119879119898119905119906119901 = 119879119898119890119897119905119894119899119892119898119905 (51)

Overall this part involves 6 equations regarding 6 unknows 119879119900119909 119879119900119909119906119901 119879119894119899119905 119879119898119905

119879119898119905119906119901 and 119879119898119905119905119900119901

513 Energy balance in the RPV wall

The heat transfer in the PRV wall can be modelled as a simple 1D heat conduction problem that heat only transfers in the wall thickness direction or 2D axisymmetric heat conduction problem

For 1D model the treatment in [72] was adopted As illustrated in Figure 21 the heat from the oxide pool transfers to inner vessel surface (119879119894) then to the interior vessel (119879119907119890119904) then to outer vessel (119879119900) and finally to environment

Figure 21 1D heat transfer model in RPV [74]

Unknown variables are the heat flux through inner vessel surface 119902119894119899 the crust thickness 119909119888119903 inner vessel temperature 119879119894 vessel center temperature 119879119907119890119904 and outer vessel temperature 119879119900 Eq (52)~(56) give a closed equation set for these variables considering a radiation boundary condition on the external vessel surface Eq (52)~(53) describe the heat transfer through the crust where an internal heat source 119876119888119903 is also considered Eq (52) can be easily derived from the one-dimensional heat transfer equation with heat source Eq (54)~(55) describe the energy balance on the inner vessel surface and vessel interior respectively Eq (56) describes the energy balance on the outer vessel surface regarding radiation boundary condition For Dirichlet boundary condition Eq (56) is replaced with Eq (57) For other boundary conditions we only need to correspondingly modify the terms on the right-hand side of Eq (56) For situations that no crust is formed on the surface of the vessel Eq (52)~(53) are eliminated and 119902119894119899 in Eq (54) can directly be determined from calculations in Section 512 and Eq (54)~(56) are solved for the unknows 119879119894 119879119907119890119904 and 119879119900 Once 119879119894 exceeds the vessel melting temperature (with or without crust formation) wall ablation is considered to occur Then 119879119894 is fixed to the vessel melting temperature and vessel thickness 119909119907119890119904 becomes the new unknown variable

119879119900119909119889119899 119879119894 119879119907119890119904 119879119900 119879119890119899119907

119909119888119903 119909119907119890119904

crust RPV

Qcr

119902119894119899 119902119897119900119888119886119897

40

119902119894119899 = 119896119888119903(119879119900119909119889119899 minus 119879119894)

119909119888119903+ 1198761198881199031199091198881199032

(52)

119902119894119899 = 119902119897119900119888119886119897 + 119876119888119903119909119888119903

(53)

119902119894119899 = 119896119907119890119904(119879119894 minus 119879119907119890119904)

05119909119907119890119904

(54)

120588119907119890119904119862119901119907119890119904119889119879119907119890119904119889119905

= 119896119907119890119904(minus2 lowast 119879119907119890119904 + 119879119894 + 119879119900)

(05119909119907119890119904)2

(55)

119896119907119890119904(119879119907119890119904 minus 119879119900)

05119909119907119890119904= 휀120590(119879119900

4 minus 1198791198901198991199074 )

(56)

119879119900 = 119879119900119888119900119899119904119905

(57)

For the 2D representation the heat conduction in the RPV is solved using FDM considering the heat transfer in both the thickness and the angular directions in the RPV wall The RPV lower head is treated as a spherical shell The corresponding heat conduction equation in the 2D spherical coordinate is given as Eq (58) The cylindrical part of the vessel is treated as a cylindrical shell The corresponding heat conduction equation in the 2D cylindrical coordinate is given as Eq (59) Eq (58) and (59) are coupled and solved together In default the mesh size is 5deg in angular direction and 0025119909119888119903 in radius direction This mesh discretization can be changed up to user demands

ρ119862119901part119879

part119905=

1

1199032part

part119903(1198961199032

part119879

part119903) +

1

r2sinθ

part

partθ(119896119904119894119899θ

part119879

partθ)

(58)

ρ119862119901part119879

part119905=1

119903

part

part119903(119896119903

part119879

part119903) +

part

part119911(119896part119879

part119911) (59)

52 Hybrid Coupling Approach

The hybrid coupling approach to assess the RPV integrity is illustrated in Figure 22 transIVR and ANSYS Mechanical are coupled in a one-way manner transIVR conducts the conjugate heat transfer of melt pool and RPV and provides transient thermal loads to the ANSYS Mechanical where structural analysis of RPV deformation is performed The 2D heat transfer of RPV is solved in transIVR to better capture the temperature distribution In RPV structural analysis same considerations are taken as in Section 422 for 16MND5 steel the modified theta projection model for creep and MISO for plasticity and temperature-dependent Young modulus etc The VLM is adopted for the transfer of transient thermal loads

41

Figure 22 Framework of the hybrid coupling approach [74]

53 Verification and Validation

The transIVR code was first benchmarked against the UCSB FIBS case of two-layer pool configuration and then validated against the LIVE-7V experiment showing the codersquos capability in predicting two-layer and transient melt pool heat transfers respectively Finally the hybrid coupling approach encompassing transIVR and ANSYS Mechanical was validated against the RPV failure experiment FOREVER-EC2 and the results showed acceptable agreements with those of both the experiment and the previous approach using CFD in Section 43 Hence the transIVR code is an efficient tool for melt pool heat transfer and the hybrid approach an efficient way for RPV failure analysis

531 UCSB FIBS benchmark case

The UCSB FIBS benchmark case [78] was calculated to demonstrate the capability of the transIVR code in simulating two-layer pool heat transfer problems The considered pool configuration was similar to that shown in Figure 20 with the following specific dimensions a two-layer melt pool with a 16 m oxide layer and a 08 m metallic pool on top and an RPV lower head with a thickness of 015 m The internal volumetric source in the oxide layer was 13 MWm3 The calculation results were compared with those of the DOE study [78] and the INEEL study [79]

The predicted results were compared with the digitalized results of the DOE and INEEL studies in Figure 23 All three studies show similar profiles of the heat flux distribution along the polar angle (Figure 23(a)) it increased with the polar angle in the oxide layer to the maximum value at around 77deg and then dropped to a significantly lower value in the metal layer The transIVR result agreed well with the other results with small discrepancy in the metal layer the DOE study gave a

heat flux of 500 MWm2 the INEEL study of 400 MWm2 and the transIVR of somewhere in between Figure 23(b) shows the predicted corium crust thickness distribution along the RPV polar angle also agreed well the DOE and INEEL

transIVR ANSYS Mechanical

Transient

thermal loads

42

studies The crust thickness decreased with polar angle in the oxide layer and no crust was formed in the metal layer As the crust formation is closely related to the heat flux distribution the crust thickness profile was reasonable regarding the heat flux distribution shown in Figure 23(a) Vessel wall ablation would occur due to the high thermal loads Figure 23(c) gives the profile of the remaining vessel wall thickness along the RPV polar angle considering the wall ablation It shows that no ablation occurred in low angle region (0deg~36deg) where heat flux was low and the wall thickness changed from 15 cm to around 6 cm as angle increased in high angle region of the oxide layer Because of the discrepancy in the heat flux of metal layer the predicted wall thicknesses also differ from those of the DOE and INEEL studies the thinnest wall was 73 cm predicted by the DOE study corresponding to the maximum heat flux followed by 88 cm of the present transIVR calculation and 94 cm of the INEEL study Generally speaking the prediction of transIVR is in an acceptable level compared with the other studies for the same benchmark case

Figure 23 Simulation results of the UCSB FIBS benchmark case [74]

(a) heat flux distribution (b) crust thickness distribution

(c) wall thickness distribution

43

532 LIVE-7V experiment

LIVE-7V is an experiment under the LIVE program conducted in KIT to study the core melt phenomena As Figure 24(a) shows it employed the LIVE-3D hemispherical facility with an inner diameter of 1 m and a wall thickness of ~25 mm The non-eutectic binary salt mixture of 20 mol NaNO3- 80 mol KNO3 was used as the simulant material The decay heat in the melt was simulated by cable-type heating with 8 planes of electrical resistance heating wires Water cooling was applied to the top and vessel surfaces Top surface cooling was realized by a cooling lid mounted at height 413 mm from the vessel bottom and vessel cooling by the flow channel between the test vessel and the outer cooling vessel The inlet and outlet water temperature for both top and vessel cooling surfaces were kept ~293 K and ~303 K throughout the whole experiment respectively A 413 mm height pool corresponds to a total volume of ~0194 m3 simulant material The experiment procedure can be divided into four stages with the heating power sequentially shifted from 29 kW to 24 kW 18 kW and 9 kW Figure 24(b) indicates the thermocouple positions in experiment

The simulation results of LIVE-7V using transIVR were compared with experimental data [80] and numerical study using REALPSCDAPSIM code [81] For simplicity the RELAPSCDAPSIM results in the following are labelled as lsquoCOUPLErsquo which is the name of the module in RELAPSCDAPSIM for melt pool heat transfer

Figure 24 LIVE test facility [80] and thermocouple positions [82]

Figure 25 compares the predicted heat removal upward to the top lid and downward to the bottom vessel surface with those of the experiment and COUPLE The black curve indicates the total input power at the four stages The experimental results (red and blue dots) were calculated from the Table 3 of [80] assuming no extra heat losses For the first three stages the transIVR predictions agreed well with the COUPLE results while the predicted upward heat flow was slightly lower and the downward heat flow higher than the experimental data For the last stage the transIVR agreed well with the experimental result while the COUPLE predicted lower upward and higher downward heat flows The short

(a) (b)

44

transition periods between two stages with different total inputs taken in transIVR also agreed with those in the COUPLE predictions

Figure 25 Comparison of heat removal upward to the top melt surface cooling lid and downward to bottom vessel wall [74]

Figure 26 shows the comparison of heat flux distribution along the vessel wall All curves show similar trend that a higher total input power would result in a higher heat flux profile As the COUPLE predictions had relatively flatter profiles transIVR predictions generally agreed better with the experimental results especially when total input was high A discrepancy is seen that the minimum value of a curve occurred at angle 0degfor both transIVR and COUPLE and at angle 30deg for the experimental data It is noted that the models in default assume heat flux monotonically increases with angle (cf Eq (37) and (38)) resulting it impossible to predict higher heat flux at angle 0deg than angle 30deg as in experimental results

45

Figure 26 Heat flux profiles along the vessel polar angle [74]

533 FOREVER-EC2 experiment

In this section we performed calculations against the FOREVER-EC2 experiment to validate the performance of the coupled transIVR and ANSYS Mechanical in assessing the RPV integrity The applied boundary conditions were identical to the settings of previous coupling approach in Section 4 (ANSYS Fluent+ Mechanical) as indicated in Figure 11 The initial bulk temperature of the melt pool was set to be 1261 K in transIVR which corresponds to the maximum value of the initial temperature field in CFD calculation of previous coupling approach Both the experimental data and the simulation results using previous coupling approach are included for comparison In the following results from current hybrid approach are labelled lsquotransIVRrsquo (and referred as transIVR results) results from previous approach labelled lsquoCFDrsquo (and referred as CFD results) and results from experiment labelled lsquoexperimentrsquo

Figure 27 compares the bulk temperature predicted by transIVR with the averaged and maximum temperatures of the CFD results All curves show the same trend that the temperature gradually increased with time till reaching a steady state and then kept unchanged till the end The times needed for reaching the steady state were ~5000 s for the CFD and ~4000 s for the transIVR which are close to each other It indicates again a good performance of the transIVR in predicting the transient behavior of melt pool Compared with the steady-state averaged temperature 1478 K and maximum temperature 1609 K of the CFD results the steady-state temperature of transIVR 1599 K is a reasonable prediction

0

3000

6000

9000

12000

15000

18000

0 20 40 60 80 100

Hea

t fl

ux

(W

m2)

Polar angle (deg)

29 kW-EXP24 kW-EXP18 kW-EXP9 kW-EXP29 kW-COUPLE24 kW-COUPLE18 kW-COUPLE9 kW-COUPLE29 kW-transIVR24 kW-transIVR18 kW-transIVR9 kW-transIVR

46

Figure 27 Temperature evolution with time [74]

Figure 28 Temperature fields at time (a) 01 s (b) 3038 s (c) 6038 s and (d) 13289 s [74]

Figure 28 shows the temperature fields of the vessel at time 01 s 3038 s 6038 s and 13289 s The temperature fields were visualized in the ANSYS Mechanical It also indicates that the transient temperature loads were properly transferred from transIVR to ANSYS Mechanical Figure 28(a-d) show both the minimum and maximum temperatures increased with time In the lower head region the temperature gradually increased with the height The maximum temperature occurs at the location slightly lower than the melt pool top surface This is because thermal loadheat flux imposed on the vessel generally increases with polar angle in the pool natural convection so the code transIVR inherently also predicted higher heat flux at higher angle positions The higher flux at higher position then correspondingly resulted in higher temperature In the cylindrical region the

(a) (b) (c) (d)

[K] [K] [K] [K]

47

temperature decreased as the height since the bottom part would be more significantly heated by the high-temperature lower head region

Figure 29 shows the steady-state temperature profiles of the external vessel surface along the polar angle The transIVR result agreed well with other two results in trend that the temperature increased with the angle and reach the peak values at angle ~80deg then gradually decreased with angle But slightly lower temperature was predicted in the low angle region Possible reason may be that the local heat flux distribution determined by Eq (37) and (38) is steeper than that in the experiment and CFD study which then resulted in a steeper increase of temperature along the angle and a lower temperature at low angle region In the cylindrical part (gt90deg) the transIVR result generally agreed with the experimental and CFD results The discrepancy with the experimental data may be caused by the simplified boundary conditions the ambient temperature inside the vessel was set to uniform constant value while it could decrease as height Therefore predicted heat loss in the higher position of cylindrical part could be smaller than that with height-dependent ambient temperature resulting in higher vessel temperature Overall the prediction was reasonable for a lumped-parameter model

Figure 29 Temperature profiles of the external vessel surface along the vessel polar angle [74]

The transIVR predicted a failure time of 24959 s which is close to 24180 s in experiment and 23959 s in the CFD results Figure 30 shows the total deformation of vessel bottom point with time All three curves increased gradually with time and experienced drastic accelerations at the end The

400

600

800

1000

1200

1400

0 40 80 120 160

Tem

per

ature

(K

)

Polar angle (deg)

transIVRExperimentCFDWeilding Line

48

transIVR curve agreed well with the CFD curve The final total deformation at failure time is also close to that of the CFD result but lower than the experiment data The discrepancy might be related to the simplifications in the boundary conditions accuracy of the thermal load prediction and the modelling errors in plasticity and creep

Figure 30 Total deformation in vertical direction of the vessel bottom point [74]

Figure 31 shows the profile of wall thickness change The transIVR curve was compared with the CFD curve and four experimental curves measured at vessel left right front and back sides The simulation result fitted the experimental and CFD results well in the region 0deg~50deg In the region 50deg~90deg wall thickness gradually decreased with angle till around 75deg and then increased till 90deg The maximum thickness change was -3 mm for the simulation which is close to the experiment left side curve and slightly larger than the -6~ -45 mm of the other curves from experiment and CFD results Concerning possible errors and uncertainties in the modelling and numerical settings the new approach generally gives reasonable predictions

0

5

10

15

20

25

11670 15170 18670 22170 25670

Tota

l def

orm

atio

n (

mm

)

Time (s)

Experiment

CFD

transIVR

49

Figure 31 Wall thickness changes along the vessel polar angle [74]

-6

-5

-4

-3

-2

-1

0

1

0 20 40 60 80 100

Wal

l th

icknes

s ch

ange

(mm

)

Polar angle (deg)

CFD

transIVR

Experiment (left)

Experiment (right)

Experiment (front)

Experiment (back)

50

51

6 Summary and Outlook

Reactor pressure vessel (RPV) is one of the key physical barriers in the defence-in-depth approach of nuclear power safety and its failure is also the transition point of core melt (corium) progression from in-vessel to ex-vessel phase Quantification of RPV failure during a hypothetical severe accident is therefore paramount to the specific qualification of severe accident mitigation strategies as well as the general assessment of corium risk in light water reactors The RPV failure is a problem of structural performance of RPV under the influence of thermal loads imposed by corium melt pool ie it is a transient thermo-fluid-structure coupled problem per se whose solution requires simultaneous modelling of both highly turbulent melt pool heat transfer and large-scale nonlinear RPV creep Due to the complexity of the problem only limited capabilities have been developed in the previous studies under different simplifications and assumptions in models and coupling methods The present study was motivated to advance modelling and coupling approaches for RPV performance analysis considering new understanding of physical phenomena and latest coupling platform of simulation tools

For the new development of coupling approaches and models for physical phenomena related to RPV failure analysis during severe accidents this thesis work was conducted in the following order (i) a new creep model for the RPV steel 16MND5 was developed which features the capability to cover all three creep stages (ii) a new thermo-fluid-structure coupling approach was developed for RPV failure analysis and validated against FOREVER-EC2 experiment (iii) the performance of the new coupling approach using volume loads mapping (VLM) for data transfer was further compared with another approach using surface loads mapping (SLM) (iv) a hybrid coupling approach was proposed for efficient analysis of RPV failure which employs a lumped-parameter code for quick estimate of thermal load and a mechanistic FE code for detailed structural analysis and (v) for the hybrid coupling approach the code transIVR was developed for calculation of transient melt pool heat transfer

The developed creep model for 16MND5 is called lsquomodified theta projection modelrsquo and describes all three stages of a creep process Creep curves are expressed as a function of time with five model parameters 120579119894(119894 = 14 119886119899119889 119898) A dataset for the model parameters was constructed by fitting experimental creep curves into this function and an iterative parameter adjustment process to satisfy the monotonicity assumption of creep strain vs temperature and stress Model parameters for new curves under different temperature and stress loads are directly interpolated from this dataset in contrast to the conventional ldquotheta projection modelrdquo that employs an extra fitting process The new treatment was proven to better accommodate all experimental curves over the wide ranges of temperature and stress loads The model was implemented into the ANSYS Mechanical code The simulation results indicate that the model successfully captured all three creep stages and a good agreement was achieved between each experimental creep curve and corresponding modelrsquos prediction Regarding the model response to transient changes of temperature and stress loads the widely used time hardening and strain hardening models were implemented and

52

performing reasonably well in the new method These properties fulfil the requirements of a creep model for structural analysis

The transient thermo-fluid-structure coupling approach was developed using the ANSYS Fluent code for thermofluid dynamics of the melt pool and ANSYS Structural code for structural mechanics of RPV An extension tool was introduced to realize transient load transfer from ANSYS Fluent to Structural and minimize the user effort Both turbulence models and the effective model PECM can be used for melt pool heat transfer The developed creep model was used for RPV failure analysis of the RPV Since it is insensitive to geometry the coupling approach allows more complex geometries such as a BWR lower head with a forest of penetrations (CRGTs and IGTs) modelled in detail Moreover the coupling approach does not only reflect the transient response of thermo-fluid-structure interaction but also facilitates the implementation of advanced (high-fidelity) models in both fluid mechanics and structural mechanics The validation of the coupling approach against the FOREVER-EC2 experiment showed that the temperature profile vessel failure time total displacement of vessel bottom point and wall thickness profile in the experiment were well predicted

Following the variation of the coupling approach from VLM to SLM for data transfer between ANSYS Fluent and ANSYS Structural the respective coupling approaches were investigated by comparative calculations against the FOREVER-EC2 experiment and a postulated severe accident of a reference BWR For the FOREVER-EC2 experiment both VLM and SLM approaches predicted generally well the total deformation in vertical direction of bottom point and the wall thickness change though the former predicted a slightly later failure time For the reference BWR case good agreements were achieved in predicting the maximum creep strain and the total deformation in vertical direction of bottom point For both experiment and reactor cases the VLM approach predicted slightly later failure times than SLM but the differences were negligible compared to the long time-scale of vessel failure (~104 s) The deformation rate of the vessel wall was slow at the early stage and accelerated with time till a drastic deformation and final vessel failure Generally speaking the VLM and SLM coupling approaches had the similar performance in terms of their predictability of experiment and applicability to reactor case but the VLM was computationally more efficient

In the development of the hybrid coupling approach the transIVR code was developed for quick estimate of transient melt pool heat transfer and determination of thermal loads on RPV transIVR does not only support transient analysis of one- and two- layer melt pool but also the heat transfer in RPV being resolved with 2D finite difference method which captures more spatial thermal details for RPV structural analysis in addition to 1D heat transfer model The capabilities of transIVR in modelling two-layer pool heat transfer and transient pool heat transfer were demonstrated by calculations against the UCSB FIBS benchmark case and the LIVE-7V experiment respectively It was then coupled to the mechanical solver ANSYS Mechanical which together form the hybrid coupling framework for detailed RPV failure analysis The calculation against the FOREVER-EC2 experiment did not only reproduce the major results of the experiment but also had a good agreement with the thermo-fluid-structure

53

coupling approach using CFD This indicated the new coupling approach is capable of capturing the vessel creep failure characteristics and can be employed as an efficient tool addressing the complex vessel failure problem

As a prospective more research topics can be addressed in future either as a continuation of the current work toward an effective tool for vessel failure analysis or from the general view of scientific interest in melt pool heat transfer and RPV failure The following topics are recommended

bull Extended applications to the reactor cases For instance the thermo-fluid-structure coupling approach can be used to study penetration failures in the RPV of an BWR with a detailed representation of penetrations (IGTs and CRGTs)

bull Synthesis or update of heat transfer correlations for melt pool Heat transfer correlations directly benefit the parametric codes such as transIVR in accurate quantification of heat flux distribution around a melt pool So far many empirical correlations have been proposed based on experiments which had a variety of geometries and melt simulants with a wide range of Rayleigh number (Ra) Sensitivity studies have also been performed on the influences of Ra Pa and other material properties Therefore a comprehensive synthesis or update of the existing heat transfer correlations will be a meaningful step for making reasonable extrapolations to the reactor applications with prototypic materials

bull Modelling of pool stratification The motivation starts with the improvement of the PECMrsquos modelling capability such that the dynamic stratification and resulting pool heat transfer can be modelled altogether Simulation of melt stratification has been supported in some system codes such as ASTEC using a predefined parametric approach The improvement of PECM can be considered in the similar approach or more mechanistic modelling

bull Development of a structural module for RPV The idea is to develop a simplified module for structural analysis so it can be coupled with transIVR ultimately forming a standalone code for quick estimate of vessel failure of LWRs during postulated severe accidents

bull Extension of vessel analysis capability The current version of tranIVR is limited to melt pool Although the precursory debris bed and melt ejection after vessel failure were not the focus of the present study they are equally important to assessment of corium risk especially for Swedish BWRs where the melt release conditions and history are crucial to qualification of their severe accident mitigation strategies

54

55

Bibliography

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[2] IAEA Safety of Nuclear Power Plants Design Vienna INTERNATIONAL ATOMIC ENERGY AGENCY 2012

[3] M Fischer ldquoGen II and III In-Vessel and Ex-Vessel Melt Retentionrdquo The Short Course on Severe Accident Phenomenology Oct 2017

[4] B R Sehgal Nuclear Safety in Light Water Reactors Severe Accident Phenomenology 1st ed Academic Press 2011

[5] IAEA Approaches and Tools for Severe Accident Analysis for Nuclear Power Plants Vienna INTERNATIONAL ATOMIC ENERGY AGENCY 2008

[6] WENRA ldquoPractical Elimination Applied to New NPP Designs - Key Elements and Expectationsrdquo Sep 2019 [Online] Available httpwwwwenraorgmediafiler_public20191111practical_elimination_applied_to_new_npp_designs_-_key_elements_and_expectations_-_for_issuepdf

[7] P Matejovic M Barnak M Bachraty and L Vranka ldquoASTEC applications to VVER-440V213 reactorsrdquo Nuclear Engineering and Design vol 272 pp 245ndash260 2014 doi 101016jnucengdes201308077

[8] AREVA ldquoUS EPR Severe Accident Evaluationrdquo Topical Report ANP-10268NP Oct 2006 [Online] Available httpswwwnrcgovdocsML0631ML063100157pdf

[9] C T Martin ldquoCoupled 3D Thermo-mechanical Analysis of Nordic BWR Lower Head Failure in case of Core Melt Severe Accidentrdquo master thesis KTH Stockholm 2013

[10] IAEA ldquoAdvanced Large Water Cooled Reactors A supplement to the IAEArsquos Advanced Reactor Information System (ARIS)rdquo Sep 2015 [Online] Available httpsarisiaeaorgPublicationsIAEA_WRC_Bookletpdf

[11] B Pershagen Light Water Reactor Safety Pergamon Press 1989 [12] Ninos Garis M Agrell H Glanneskog and L Agrenius ldquoAPRI 7 ndash Accident

Phenomena of Risk Importancerdquo SSM SSM 201212 2012 [Online] Available httpswwwstralsakerhetsmyndighetensecontentassets2a9f299f46b949b3b209e780cf71436a201212-apri-7-accident-phenomena-of-risk-importance-en-lagesrapport-om-forskningen-inom-omradet-svara-haverier-under-aren-2009-2011-

[13] F Fichot et al ldquoSome considerations to improve the methodology to assess In-Vessel Retention strategy for high-power reactorsrdquo Annals of Nuclear Energy vol 119 pp 36ndash45 2018 doi 101016janucene201803040

[14] T G Theofanous C Liu S Additon S Angelini O Kymaumllaumlinen and T Salmassi ldquoIn-vessel coolability and retention of a core meltrdquo Nuclear Engineering and Design vol 169 no 1 pp 1ndash48 1997 doi 101016S0029-5493(97)00009-5

[15] C Sainte Catherine ldquoTensile and creep tests material characterization of pressure vessel steel(16MND5) at high temperatures (20 up to 1300 degC)rdquo CEA France SEMTLISNRT98-009A 1998

56

[16] E Altstadt H-G Willschuetz B R Sehgal and F-P Weiss ldquoModelling of in-vessel retention after relocation of corium into the lower plenumrdquo Forschungszentrum Rossendorf FZR-437 2005

[17] H-G Willschuumltz and E Altstadt ldquoGeneration of a High Temperature Material DataBase and its Application to Creep Tests withFrench or German RPV-steelrdquo FORSCHUNGSZENTRUM ROSSENDORF FZR-353 ISSN 1437-322X Aug 2002

[18] J L Rempe S A Chavez and G L Thinnes ldquoLight water reactor lower head failure analysisrdquo Nuclear Regulatory Commission NUREGCR--5642 1993 Accessed Aug 23 2019 [Online] Available httpinisiaeaorgSearchsearchaspxorig_q=RN25025905

[19] S Brosi et al ldquoCORVIS Investigation of light water reactor lower head failure modesrdquo Nuclear Engineering and Design vol 168 no 1 pp 77ndash104 1997 doi 101016S0029-5493(96)00008-8

[20] T Y Chu M M Pilch J H Bentz J S Ludwigsen W Y Lu and L L Humphries ldquoLower Head Failure Experiments and Analysesrdquo Sandia National Laboratories Albuquerque NM USA NUREGCR-5582 SAND98-2047 1999

[21] L L Humphries et al ldquoOECD Lower Head Failure Project Final Reportrdquo Sandia National Laboratories Albuquerque NM 87185-1139 2002

[22] B R Sehgal R R Nourgaliev T N Dihn A Karbojian J A Green and V A Bui ldquoFOREVER Experiments on Thermal and Mechanical Behavior of a Reactor Pressure Vessel during a Severe Accidentrdquo in Proceedings of the Workshop on in-vessel core debris retention and coolability Garching near Munich Germany Mar 3-6 1998

[23] S Bhandari M Fischer B R Sehgal and A Theerthan ldquoPost Test Analysis of the FOREVER C2 Experimentrdquo The 9th International conference on Nuclear Engineering (ICONE-9) Nice France Apr 8-12 2001

[24] J F Mao J W Zhu S Y Bao L J Luo and Z L Gao ldquoCreep deformation and damage behavior of reactor pressure vessel under core meltdown scenariordquo International Journal of Pressure Vessels and Piping vol 139ndash140 pp 107ndash116 2016 doi 101016jijpvp201603009

[25] J Mao L Hu S Bao L Luo and Z Gao ldquoInvestigation on the RPV structural behaviors caused by various cooling water levels under severe accidentrdquo Engineering Failure Analysis vol 79 pp 274ndash284 2017 doi 101016jengfailanal201704029

[26] J F Mao J W Zhu S Y Bao L J Luo and Z L Gao ldquoStudy on structural failure of RPV with geometric discontinuity under severe accidentrdquo Nuclear Engineering and Design vol 307 pp 354ndash363 2016 doi 101016jnucengdes201607027

[27] B R Sehgal V A Bui T N Dinh and R R Nourgaliev ldquoHeat Transfer Processes in Reactor Vessel Lower Plenum During Late Phase of In-Vessel Core Melt Progressionrdquo in Advances in Nuclear Science and Technology J Lewins M Becker R W Albrecht E J Henley J D McKean K Oshima A Sesonske H B Smets and C P L Zaleski Eds Boston MA Springer US 1999 pp 103ndash135

[28] C-T Tran and T-N Dinh ldquoThe effective convectivity model for simulation of melt pool heat transfer in a light water reactor pressure vessel lower head Part I Physical processes modeling and model implementationrdquo Progress

57

in Nuclear Energy vol 51 no 8 pp 849ndash859 2009 doi 101016jpnucene200906007

[29] C-T Tran and T-N Dinh ldquoThe effective convectivity model for simulation of melt pool heat transfer in a light water reactor pressure vessel lower head Part II Model assessment and applicationrdquo Progress in Nuclear Energy vol 51 no 8 pp 860ndash871 2009 doi 101016jpnucene200906001

[30] H-G Willschuumltz E Altstadt B R Sehgal and F-P Weiss ldquoCoupled thermal structural analysis of LWR vessel creep failure experimentsrdquo Nuclear Engineering and Design vol 208 no 3 pp 265ndash282 2001 doi 101016S0029-5493(01)00364-8

[31] E Altstadt and T Moessner ldquoExtension of the ANSYS creep and damage simulation capabilitiesrdquo Forschungszentrum Rossendorf Technical report FZR-296 2000

[32] H-G Willschuumltz E Altstadt B R Sehgal and F-P Weiss ldquoRecursively coupled thermal and mechanical FEM-analysis of lower plenum creep failure experimentsrdquo Annals of Nuclear Energy vol 33 no 2 pp 126ndash148 2006 doi 101016janucene200508006

[33] H-G Willschuumltz E Altstadt B R Sehgal and F-P Weiss ldquoSimulation of creep tests with French or German RPV-steel and investigation of a RPV-support against failurerdquo Annals of Nuclear Energy vol 30 no 10 pp 1033ndash1063 2003 doi 101016S0306-4549(03)00036-7

[34] H Madokoro A Miassoedov and T Schulenberg ldquoA thermal structural analysis tool for RPV lower head behavior during severe accidents with core meltrdquo Mechanical Engineering Letters vol 4 pp 18-00038-18ndash00038 2018 doi 101299mel18-00038

[35] W Villanueva C-T Tran and P Kudinov ldquoCoupled thermo-mechanical creep analysis for boiling water reactor pressure vessel lower headrdquo Nuclear Engineering and Design vol 249 pp 146ndash153 2012 doi 101016jnucengdes201107048

[36] W Villanueva C T Tran and P Kudinov ldquoA Computational Study on Instrumentation Guide Tube Failure During a Severe Accident in Boiling Water Reactorsrdquo in The 14th International Topical Meeting on Nuclear Reactor Thermal hydraulics NURETH-14 Toronto Ontario Canada Sep25-30 2011

[37] W Villanueva C T Tran and P Kudinov ldquoAssessment with Coupled Thermo-Mechanical Creep Analysis of Combined CRGT and External Vessel Cooling Efficiency for A BWRrdquo The 14th International Topical Meeting on Nuclear Reactor Thermal hydraulics NURETH-14 Toronto Ontario Canada Sep 25-30 2011

[38] F R Larson and J Miller ldquoA Time-Temperature Relationship for Rupture and Creep Stressesrdquo Trans ASME vol 74 p 765minus775 1952

[39] C T Tran ldquoDevelopment validation and application of an effective convectivity model for simulation of melt pool heat transfer in a light water reactor lower headrdquo Licentiate thesis KTH Stockholm 2007

[40] ANSYS ldquoANSYS Fluent Theory Guide Release 162rdquo ANSYS Inc Jul 2015 [Online] Available httpwwwansyscom

[41] P Zeng Finite element analysis and application (in Chinese) Tsinghua University Press amp Springer 2004

58

[42] ANSYS ldquoANSYS Mechanical | Finite Element Analysis Softwarerdquo httpswwwansyscomproductsstructuresansys-mechanical-enterprise (accessed Sep 26 2019)

[43] ANSYS ldquoANSYS Mechanical APDL Theory Reference Release 162rdquo ANSYS Inc Jul 2015 [Online] Available httpwwwansyscom

[44] U Steinberner and H H Reineke ldquoTurbulent Buoyancy Convection Heat Transfer with Internal Heat Sourcesrdquo in Proceedings of the 6th Int Heat Transfer Conference Toronto Canada 1978 vol Vol2 pp 305ndash310 doi 101615IHTC63350

[45] R W Evans and B Wilshire Introduction to Creep London Institute of Materials 1993

[46] DoITPoMS ldquoCreep Deformation of Metalsrdquo Creep Deformation of Metals University of Cambridge 2006 httpswwwdoitpomsacuktlplibcreepindexphp (accessed Jul 09 2019)

[47] ANSYS ANSYS Mechanical APDL Material Reference Release 162 2015 [48] SIMULIA Abaqus 612 Analysis Userrsquos Manualvol 3 Materials Chapter

2324 2012 [49] S Imaoka ldquoUser Creep Subroutine STI0704A ANSYS Release110rdquo 2007 [50] R W Evans J D Parker and B Wilshire ldquoThe θ projection conceptmdashA

model-based approach to design and life extension of engineering plantrdquo International Journal of Pressure Vessels and Piping vol 50 no 1 pp 147ndash160 1992 doi 1010160308-0161(92)90035-E

[51] ECCC ldquoRecommendations and Guidance for the Assessment of Creep Strain and Creep Strength Data ECCC Recommendations Volume 5 Part 1 b[Issue 2]rdquo 2013

[52] W D Day and A P Gordon ldquoLife Fraction Hardening Applied to a Modified Theta Projection Creep Model for a Nickel-Based Super-Alloyrdquo The ASME Turbo Expo 2014 Turbine Technical Conference and Exposition Duumlsseldorf Germany Jun16-20 2014 vol 7A-Structures and Dynamics p V07AT29A010 doi 101115GT2014-25881

[53] M Kumar I V Singh B K Mishra S Ahmad A Venugopal Rao and V Kumar ldquoA Modified Theta Projection Model for Creep Behavior of Metals and Alloysrdquo J of Materi Eng and Perform vol 25 no 9 pp 3985ndash3992 2016 doi 101007s11665-016-2197-y

[54] P Yu and W Ma ldquoA modified theta projection model for creep behavior of RPV steel 16MND5rdquo Journal of Materials Science amp Technology vol 47 pp 231ndash242 2020 doi 101016jjmst202002016

[55] ANSYS ANSYS Mechanical Advanced Nonlinear Materials v160 Lecture 2 Rate Dependent Creep ANSYS training materials 2015

[56] Q Y Li N C Wang and D Y Yi Numerical Analysis (in Chinese) 5th ed Tsinghua University Press 2008

[57] T N Dinh R R Nourgaliev and B R Sehgal ldquoOn heat transfer characteristics of real and simulant melt pool experimentsrdquo Nuclear Engineering and Design vol 169 no 1 pp 151ndash164 1997 doi 101016S0029-5493(96)01283-6

[58] O Kymaumllaumlinen H Tuomisto O Hongisto and T G Theofanous ldquoHeat flux distribution from a volumetrically heated pool with high Rayleigh numberrdquo

59

Nuclear Engineering and Design vol 149 no 1ndash3 pp 401ndash408 1994 doi 1010160029-5493(94)90305-0

[59] F J Asfia and V K Dhir ldquoNatural convection heat transfer in volumetrically heated spherical poolsrdquo 1994 Accessed Aug 23 2019 [Online] Available httpinisiaeaorgSearchsearchaspxorig_q=RN26037727

[60] M Jahn and H H Reineke ldquoFree Convection Heat Transfer with Internal Heat Sources Calculations and Measurementsrdquo in Proceedings of the 5th Int Heat Transfer Conference Tokyo Japan 1974 vol Vol3 paper NC-28

[61] M Gustafsson ldquoFSI Transient extension V1704rdquo EDR MEDESO 2016 [62] EDR MEDESO httpsedrmedesocom [63] H Kim and I Namgung ldquoThermo-mechanical Analysis of RVLH for

APR1400 during a Severe Accidentrdquo in Transactions SMiRT-23 Manchester United Kingdom Aug10-14 2015 [Online] Available httpwwwlibncsueduresolver18402034197

[64] P Yu and W Ma ldquoComparative Analysis of Reactor Pressure Vessel Failure using Two Thermo-Fluid-Structure Coupling Approachesrdquo submitted to journal

[65] F R Menter ldquoTwo-equation eddy-viscosity turbulence models for engineering applicationsrdquo AIAA Journal vol 32 no 8 pp 1598ndash1605 1994 doi 102514312149

[66] T N Dinh and R R Nourgaliev ldquoTurbulence modelling for large volumetrically heated liquid poolsrdquo Nuclear Engineering and Design vol 169 no 1 pp 131ndash150 1997 doi 101016S0029-5493(96)01281-2

[67] ANSYS Inc ldquoIntroduction to ANSYS Fluent Lecture 7 Turbulence Modelling Fluent Tutorial Materialrdquo 2016

[68] A Horvat and M Borut ldquoNumerical investigation of natural convection heat transfer in volumetrically heated spherical segmentsrdquo in Proceedings of the International Conference Nuclear Energy for New Europe 2004

[69] Y Li CFD Pre-test Analysis of SIMECO-2 Experiment 2016 [70] A Theerthan A Karbojian and B R Sehgal ldquoEC-FOREVER experiments

on thermal and mechanical behavior of a reactor pressure vessel during a severe accident Technical report-1 EC-FOREVER-2 testrdquo SAM-ARVI-D008 Mar 2001

[71] P Yu W Ma W Villanueva A Karbojian and S Bechta ldquoValidation of a thermo-fluid-structure coupling approach for RPV creep failure analysis against FOREVER-EC2 experimentrdquo Annals of Nuclear Energy vol 133 pp 637ndash648 2019 doi 101016janucene201906067

[72] T G Theofanous C Liu S Additon S Angelini O Kymaumllaumlinen and T Salmassi ldquoIn-vessel coolability and retention of a core meltrdquo Nuclear Engineering and Design vol 169 no 1ndash3 pp 1ndash48 1997 doi 101016S0029-5493(97)00009-5

[73] H P Langtangen ldquoSolving nonlinear ODE and PDE problemsrdquo Simula Research Laboratory 2016 [Online] Available httpshplgitgithubionum-methods-for-PDEsdocpubnonlinpdfnonlin-4printpdf

[74] P Yu and W Ma ldquoDevelopment of a Lumped-Parameter Code for Efficient Assessment of In-Vessel Melt Retention Strategy of LWRsrdquo submitted to journal

60

[75] T G Theofanous M Maguire S Angelini and T Salmassi ldquoThe first results from the ACOPO experimentrdquo Nuclear Engineering and Design vol 169 no 1 pp 49ndash57 1997 doi 101016S0029-5493(97)00023-X

[76] S W Churchill and H H S Chu ldquoCorrelating equations for laminar and turbulent free convection from a vertical platerdquo International Journal of Heat and Mass Transfer vol 18 no 11 pp 1323ndash1329 1975 doi 1010160017-9310(75)90243-4

[77] S Globe and D Dropkin ldquoNatural-Convection Heat Transfer in Liquids Confined by Two Horizontal Plates and Heated from Belowrdquo J Heat Transfer vol 81 no 1 pp 24ndash28 1959 doi 10111514008124

[78] T G Theofanous C Liu S Additon S Angelini O Kymaelaeinen and T Salmassi ldquoIn-vessel coolability and retention of a core melt Volume 2rdquo Argonne National Lab DOEID--10460-VOL2 1996 Accessed Apr 18 2019 [Online] Available httpinisiaeaorgSearchsearchaspxorig_q=RN30000459

[79] J L Rempe D L Knudson C M Allison G L Thinnes and C L Atwood ldquoPotential for AP600 in-vessel retention through ex-vessel floodingrdquo United States INEELEXT--97-00779 1997 Accessed Apr 18 2019 [Online] Available httpinisiaeaorgSearchsearchaspxorig_q=RN30025872

[80] A Miassoedov X Gaus-Liu T Cron and B Fluhrer ldquoLive experiments on melt pool heat transfer in the reactor pressure vessel lower headrdquo The 10th International Conference on Heat Transfer Fluid Mechanics and Thermodynamics (HEFAT 2014) Orlando FL Jul 14-16 2014

[81] H Madokoro A Miassoedov and T Schulenberg ldquoAssessment of a lower head molten pool analysis module using LIVE experimentrdquo Nuclear Engineering and Design vol 330 pp 51ndash58 2018 doi 101016jnucengdes201801036

[82] B Fluhrer et al ldquoThe LIVE-L1 and LIVE-L3 experiments on melt behaviour in RPV lower headrdquo Forschungszentrum Karlsruhe GmbH Technik und Umwelt (Germany) Inst fuer Kern- und Energietechnik FZKA--7419 2008 Accessed Nov 12 2019 [Online] Available httpinisiaeaorgSearchsearchaspxorig_q=RN40036867

III

Sammanfattning

Syftet med denna avhandling aumlr att utveckla nya metoder och nya modeller foumlr att kunna koppla ihop termohydrauliska foumlrlopp och uppkommen strukturrespons i botten paring reaktortanken foumlr foumlrbaumlttrad analys av reaktortankbrott under ett svaringrt haveri Avhandlingen bestaringr av fem delar (i) utveckling av en trestegsmodell foumlr krypning i reaktortankens 16MND5-staringl (ii) utveckling av metoder foumlr kopp ling av termohydrauliska foumlrlopp och strukturrespons foumlr studier av reaktortankbrott (iii) jaumlmfoumlrelse av prestanda av den nya metoden som anvaumlnder volume loads mapping (VLM) foumlr dataoumlverfoumlring med den tidigare metoden som anvaumlnder surface loads mapping (SLM) (iv) utveckling av en lumped-parameter beraumlkningskod foumlr snabb uppskattning av transient vaumlrmeoumlverfoumlring i en smaumlltpoumll i botten paring reaktortanken och (v) utveckling av en hybrid kopplingsmetod foumlr effektiv analys av reaktortankbrott

En krypmodell kallad lsquomodified theta projection modelrsquo som behandlar krypprocessen i tre steg utvecklades foumlr 16MND5-staringlet Krypkurvor i den nya

krypmodellen ges som en funktion av tid med fem parametrar 120579119894 (i=14 och m) Ett dataset foumlr modellparametrar togs fram baserat paring tillgaumlngliga experimentalla krypkurvor under anatagande om att kryptoumljning aumlr en monoton funktion av temperatur och spaumlnning Nya krypkurvor kan genereras genom att interpolera modellparametrar fraringn detta dataset i kontrast till den tidigare metoden som anvaumlnder en speciell anpassningsprocess Foumlrdelen med det nya tillvaumlgagaringngssaumlttet aumlr att den hanterar alla experimentella krypkurvor foumlr ett brett spektrum av temperatur- och spaumlnningsbelastningar Modellen implementerades i ANSYS-koden och den korrekt behandlar alla tre krypsteg samt visar bra oumlverenstaumlmmelse mellan experimentella och beraumlknade krypkurvor Foumlr analys av tidsberoende dynamiska belastningar implementerades vaumllkaumlnda modeller av tids- och deformationshaumlrdning med rimliga resultat Dessa egenskaper uppfyller kravet paring en krypmodell foumlr strukturmekaniska analyser

En metod utvecklades foumlr koppling mellan termohydrauliska foumlrlopp och strukturmekanisk respons genom att koppla ihop ANSYS Fluent som analyserar fluiddynamiska aspekter av vaumlrmeoumlverfoumlring i en smaumlltpoumll med ANSYS Structural som analyserar strukturmekanisk respons av reaktortanksvaumlggen Ett speciellt verktyg togs fram foumlr att moumljliggoumlra oumlverfoumlring av transienta belastningar fraringn ANSYS Fluent till ANSYS Structural och foumlr att minimera anvaumlndarinsatsen Baringde CFD med turbulensmodeller och den effektiva PECM-modellen kan anvaumlndas foumlr att analysera vaumlrmeoumlverfoumlring i en smaumlltpoumll Foumlr krypanalysen av reaktortanken har lsquomodified theta projection modelrsquo anvaumlnts Den framtagna kopplingsmetoden har foumlrdelen att den inte bara faringngar de transienta aspekterna av interaktionen mellan termohydraulik och strukturmekanik men ocksaring stoumldjer avancerade modeller av konvektion i en smaumlltpoumll och strukturrespons och daumlrigenom bidrar till foumlrbaumlttrad precision och underlaumlttar implementeringen av modeller Kopplingsmetoden fungerar bra naumlr den valideras mot FOREVER-EC2 experimentet och den kan anvaumlndas i

IV

komplexa geometrier som till exempel i botten paring reaktortanken i en kokvattenreaktor (BWR) med en skog av genomfoumlringar (ledroumlr foumlr styrstavar och instrumentering)

Jaumlmfoumlrande beraumlkningar visar att de tvaring kopplingsmetoderna VLM-baserad och SLM-baserad presterar lika bra naumlr det gaumlller hur de predikterar resultat av FOREVER-EC2 experimentet och deras laumlmplighet foumlr reaktoranalyser Aumlven om SLM-metoden predikterar att tankbrott i baringda fallen intraumlffar lite tidigare saring aumlr skillnaden mellan metoderna foumlrsumbar i jaumlmfoumlrelse med tiden till tankbrott (~ 104 s) VLM-metoden var baumlttre aumln SLM-metoden naumlr det gaumlller beraumlkningseffektiviteten

Ideacuten om hybridkoppling garingr ut paring att anvaumlnda en lumped-parameter kod foumlr snabb uppskattning av termiska belastningar som senare kan anvaumlndas i detaljerad strukturmekanisk analys En saringdan kopplingsmetod kan paring ett signifikant saumltt oumlka effektiviteten av beraumlkningar vilket aumlr viktig i reaktorapplikationer daumlr mekanistisk simulering av konvektion i en smaumlltpoumll aumlr beraumlkningsintensiv och onoumldig I detta syfte utvecklades koden transIVR Koden kan dels ge en snabb uppskattning av transient vaumlrmeoumlverfoumlring i en smaumlltpoumll som bestaringr av en eller tvaring materialskikt och dels loumlsa vaumlrmeledningsproblem i reaktortanksvaumlggen med 2D finita differensmetoden vilket genererar detaljerad information om termiska foumlrharingllanden som behoumlvs foumlr strukturmekaniska analyser Foumlr att demonstrera att transIVR kan analysera dels vaumlrmeoumlverfoumlring i en smaumlltpoumll bestaringende av tvaring materialskikt och dels transient vaumlrmeoumlverfoumlring i en smaumlltpoumll jaumlmfoumlrdes transIVR prediktioner med UCSB FIBS-benchmarkfallet respektive LIVE-7V-experimentet I naumlsta steg kopplades transIVR-koden till ANSYS Structural foumlr att moumljliggoumlra mer detaljerad analys av reaktortankbrott Validering mot FOREVER-EC2 experimentet tyder paring att den utvecklade kopplingsmetodiken korrekt aringterger tankbrott orsakat av krypning

Nyckelord reaktortankbrott koppling termohydraulik-strukturmekanik modellering av krypning vaumlrmeoumlverfoumlring i smaumlltpoumll svaringra haverier CFD FEA transIVR

V

List of Publications

Papers included in this thesis

I P Yu and W Ma ldquoA modified theta projection model for creep behavior of RPV steel 16MND5rdquo Journal of Materials Science amp Technology vol 47 pp 231ndash242 2020

II P Yu W Ma W Villanueva A Karbojian and S Bechta ldquoValidation of a thermo-fluid-structure coupling approach for RPV creep failure analysis against FOREVER-EC2 experimentrdquo Annals of Nuclear Energy vol 133 pp 637ndash648 2019

III P Yu and W Ma ldquoComparative analysis of reactor pressure vessel failure using two thermo-fluid-structure coupling approachesrdquo Nuclear Engineering and Design submitted

IV P Yu and W Ma ldquoDevelopment of a lumped-parameter code for efficient assessment of in-vessel melt retention strategy of LWRsrdquo Annals of Nuclear Energy submitted

Respondentrsquos contribution to the papers included in the thesis

Peng Yu made major contributions to the publications of all the above four articles under the guidance of supervisors and with the collaboration of other co-authors including formulation of research questions development of coupling approaches and numerical methods analysis and generalization of simulation results as well as writing the first drafts and addressing review comments

Papers not included in this thesis

1 P Yu and W Ma ldquoRPV creep failure analysis using a Norton creep modelrdquo International Journal of Mechanical Sciences to be submitted

2 P Yu W Villanueva S Galushin W Ma and S Bechta ldquoCoupled thermo-mechanical creep analysis for a Nordic BWR lower head using non-homogeneous debris bed configuration from MELCORrdquo 12th International Topical Meeting on Nuclear Reactor Thermal-Hydraulics Operation and Safety (NUTHOS-12) Qingdao China October 14-18 2018

3 P Yu A Komlev Y Li W Villanueva W Ma and S Bechta ldquoPre-test simulation of SIMECO HT experiments for stratified melt pool heat transferrdquo 11th International Topical Meeting on Nuclear Reactor Thermal Hydraulics Operation and Safety (NUTHOS-11) Gyeongju Korea October 9-13 2016

VI

VII

Acknowledgements

First of all I would like to express my sincere gratitude to my supervisor Prof Weimin Ma for offering this opportunity to work in the severe accident research field and for the supports patience guidance discussions and encouragements throughout the entire period of my graduate study

I am very thankful to my co-supervisor Dr Walter Villanueva for the valuable help and discussions I learnt a lot during the times working with him I am also thankful to co-supervisor Prof Sevostian Bechta who is supportive and always has inspiring ideas that keep you thinking for a long time

Thank you also to all my colleagues at the Division of Nuclear Power Safety for the supports discussions and the very nice working environment during these years

I have taken many interesting courses during my PhD study for which I want to express my thanks to all the teachers With this regard a special thanks goes to Prof Johan Hoffman for his SF2561 Finite Element Method given in 2015 fall as well as the discussions and advices on numerical analysis

I am grateful to Ignacio Gallego-Marcos and Wenyuan Fan for their enthusiastic help on CFD and turbulence modelling during the time working on the turbulent

heat transfer I thank all my 师兄 for sharing their knowledge on nuclear engineering and giving advices when I was confused since 2012

I thank all my friends both in Sweden and in China for the company and the sharing of many wonderful moments in daily life

Finally I would like to express my deepest thank to my parents my girlfriend Siwen my sisters and all my other relatives for their unconditional supports and love

Financial supports from the China Scholarship Council and APRI make my study in KTH possible and are acknowledged

VIII

IX

Table of Contents

AbstracthelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphellipI

SammanfattninghelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphellipIII

List of PublicationshelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphellipV

AcknowledgementshelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphellipVII

NomenclaturehelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphellipXI

1 Introductionhelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphellip1

11 Nuclear Power Safety 1

12 RPV Failure 2

13 Literature Review Creep Behaviour and RPV Failure Analysis 4

131 Experiments on creep behaviour and RPV failure 4

132 Numerical studies on RPV failure 5

14 Motivation Objectives and Main Achievements 7

2 Numerical Solvers and Modelshelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphellip9

21 ANSYS Fluent 9

22 ANSYS Mechanical 9

23 PECM 11

3 Creep Modelling of 16MND5 Steelhelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphellip13

31 Creep and Creep Modelling 13

32 A Modified Theta Projection Model 14

321 Original theta projection model and its variations 14

322 A modified theta projection model 16

4 Coupled Thermo-Fluid-Structure Analysis of RPV Failurehelliphelliphelliphellip23

41 Thermo-Fluid-Structure Coupling Approach 23

42 Numerical Modelling 25

421 Conjugate heat transfer 25

422 Mechanical analysis 25

43 Validation against the FOREVER-EC2 Experiment 26

431 FOREVER-EC2 experiment 26

X

432 Conjugate heat transfer 27

433 Mechanical analysis 29

44 Application to BWR Case 32

45 Computational Efficiency 34

5 Hybrid Coupling Approach for Efficient RPV Failure Analysishelliphellip35

51 A Lumped-parameter Code for Melt Pool Heat Transfer 35

511 Heat transfer correlations 36

512 Energy balance in the pools 37

513 Energy balance in the RPV wall 39

52 Hybrid Coupling Approach 40

53 Verification and Validation 41

531 UCSB FIBS benchmark case 41

532 LIVE-7V experiment 43

533 FOREVER-EC2 experiment 45

6 Summary and Outlookhelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphellip51

Bibliographyhelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphellip55

XI

Nomenclature

Latin

119860 area of a surface

119862119901 heat capacity

119863 damage factor

119892 gravity

ℎ heat transfer coefficient

H Melt pool height

119896 thermal conductivity

119871 characteristic length

119873119906 Nusselt number

119875119903 Prandtl number

119902 heat flux

119902119894119899 heat flux through inner vessel surface

119876119881 internal volumetric heat source

119903 radial distance in a cylindrical or spherical coordinate

119877119886 external Rayleigh number

119877119886119894119899 internal Rayleigh number

119877119886119910 local Rayleigh number

119905 time

119879 temperature

119881119900119897 total volume

119909 thickness (of crust or RPV wall)

119911 height in a cylindrical coordinate

Greek

120572 thermal diffusivity

120573 thermal expansion

Δ119879 temperature difference

ε emissivity (in thermal analysis) strain (in structural analysis)

휀119888119903 creep strain

휀119903 creep strain rate

120579 polar angle

120579119894 parameters of a creep model 119894 = 1234119898

120584 kinematic viscosity

σ stress

XII

Super and subscripts

119886119898119887 ambient

119888119903 creep (in Section 3) crust (in Section 5)

119889119899 downward

119890119899119907 environment

119894 inner surface of RPV

119894119899119905 interface

119897119900119888119886119897 local

119898119886119909 maximum value

119898119890119897119905119894119899119892 melting point

119898119905 metal layer (bulk)

119898119905119889119899 downward surface of metal pool

119898119905119904119889 sideward surface of metal pool

119898119905119905119900119901 top surface of metal layer

119898119905119906119901 upward surface of metal pool

119900 outer surface of RPV

119900119909 oxide layer (bulk)

119900119909119889119899 downward surface of oxide layer

119900119909119906119901 upward surface of oxide layer

119903119890119891 reference

119904 internal structures inside the RPV

119904119889 sideward

119906119901 upward

119907119890119904 vessel

Acronyms

BWR boiling water reactor

CBD creep database

CFD computational fluid dynamics

CRGT control rod guide tube

ECCM effective convectivity conductivity model

FDM finite difference method

FE finite element

FEA finite element analysis

FEM finite element method

FVM finite volume method

IGT instrument guide tube

IVR in-vessel retention

XIII

IVMR in-vessel melt retention EU project

LES large eddy simulation

MPS moving particle semi-implicit method

NPP nuclear power plant

PECM phase-change effective convectivity model

RANS Reynolds-averaged Navier-Stokes

RPV reactor pressure vessel

SBO station blackout

SLM surface loads mapping

SPH smoothed-particle hydrodynamics

UDF user-defined function

UPF user programmable features

VLM volume loads mapping

XIV

1

1 Introduction

11 Nuclear Power Safety

Greenhouse gas emissions contribute to the global warming problem Phasing out fossil fuels by switching to low-carbon energy sources is considered as a mitigation strategy With this regard clean energy should be increased in the world energy consumption structure Being clean and stable nuclear energy is an attractive alternative to fossil fuels After the development over decades it also proves to be a mature and safe technology According to IAEA 2018 annual report [1] there were 450 nuclear power reactors under operation worldwide at the end of 2018 which had a record generating capacity of 3964 GW(e) 9 reactors were connected to the grid and 55 reactors under construction around the world

Due to the potential risk of the release of radioactivity nuclear power safety is emphasized throughout the stages of design construction operation and decommission of a nuclear power plant (NPP) The principal technical requirements [2] specify that the three fundamental safety functions should be fulfilled for all plant states of an NPP (i) control of reactivity (ii) removal of heat from the reactor and from the fuel store and (iii) confinement of radioactive material shielding against radiation and control of planned radioactive releases as well as limitation of accidental radioactive releases The defence-in-depth concept is also applied to provide several levels of defence to prevent consequences of accidents The fuel claddings reactor pressure vessel (RPV) and primary coolant pressure boundaries and the containment are the three physical barriers to prevent the release of radioactive materials to the environment

Severe accidents are events that cause significant damage to reactor fuel and result in core meltdown It may be caused by insufficient cooling of the reactor core due to various reasons or by the rapid introduction of huge reactivity which causes sharp increases in the power and fuel fragmentation eg Chernobyl accident During a severe accident the failure of the fuel claddings will result in the radioactive materials previously enclosed inside the cladding released to the primary loop If cooling is not sufficient to remove the decay heat the core materials will melt migrate down and relocate into the RPV lower head after which the melt will react with water and partially form fragments quench solidify and form a debris bed The debris bed formed by the melt may continue to melt to form a melt pool due to the decay heat As the corium contacts the RPV it would continuously heat the RPV and threaten its integrity Once the RPV fails containment becomes the last barrier to prevent the release of radioactivity The released melt would result in an increase of the pressure and temperature due to the high temperature fractions oxidation and hydrogen combustion which is called direct containment heating The melt would fall onto the reactor cavity Without sufficient cooling the molten core would interact with basemat concrete (MCCI) This process can last for few days It can result in containment failure due to either containment overpressure or the basemat melt-through of containment In addition containment is also subjected to isolation failure and early containment failure due to missiles and dynamic pressure [3] More

2

detailed severe accident progression or phenomena could also be found in eg [4] [5]

Though severe accidents are unlikely to occur the consequences can be severe if no proper measure is taken Related researches have been continuously going on aiming at mitigating the accident consequences and improving safety After the Fukushima Accident safety inspections and re-evaluation actions have been carried out worldwide especially on safety measurements regarding severe accidents The concept lsquoPractical Eliminationrsquo applied to new NPP designs demands that new nuclear installations be designed with the objective of mitigating the consequences of accidents and avoiding early radioactive releases and large radioactive releases [6] Severe accident mitigation strategies are applied to the existing and new NPPs (both Generation II and III) The strategies can generally be classified into two categories [3] the in-vessel retention (IVR) and ex-vessel retention (EVR) In simple words in-vessel retention tries to retain the core melt inside the RPV (second barrier) by applying external water cooling on the vessel as shown in Figure 1(a) Such method minimizes the dispersal of the melt within the containment and hence simplifies the later recovery work Examples of the application are AP1000 HPR1000 APR 1400 The idea of the EVR is to let the RPV fail but to retain the core melt inside the containment Two principle approaches of EVR are the retention within a dedicated lsquocore catcherrsquo and melt fragmentation Examples of core catcher would be EPR that directs the melt to a spreading compartment (Figure 1(b)) and VVER1000 that uses metallic crucible below the RPV An example of the melt fragmentation is the Nordic BWRs that the corium released from RPV would relocate in the cavity with a deep-water pool (7-12 m) and is expected to fragment quench and form a coolable debris bed (Figure 1(c))

Figure 1 Examples of IVR [7] EVR with core catcher [8] and melt fragmentation [9]

12 RPV Failure

RPV which contains the reactor core core shroud and coolant is a key component to the reactor system RPV and the primary loop constitute the primary coolant system which conveys the fission heat from the reactor core to the secondary loop through steam generator (PWR) or to the turbine (BWR) The pressure under

(a) IVR (b) Core catcher (c) Melt fragmentation

3

operation can be ~155 MPa for a 1000 MW PWR [10] and ~70 MPa for a 1000 MW BWR [11] It also has to withstand the neutron radiation damage The thickness of RPV [11] can be ~20 cm for PWR and ~16 cm for BWR

As the transition of in-vessel progression to ex-vessel progression RPV failure is important for analysis of accident progression and assessment of the effectiveness of severe accident mitigation strategies In the melt fragmentation strategy of Nordic BWR the melt-coolant interaction and debris bed coolability after the corium poured into the deep-water pool are essentially dependent on the vessel failure mode including the characteristic time of melt release rupture size and location and amount and superheat of melt available for release [12] New concern also arose [13] on the necessity of performing structural analysis for the RPV in the IVR strategy In the initial methodology and further development of the IVR the success of the strategy ie the integrity of the RPV was demonstrated (eg in [14]) with the following two conclusions (i) even for a local heat flux on the vessel as high as critical heat flux (CHF) the resulted very thin external ldquocold layerrdquo of the vessel would be sufficient to withstand the internal stress load and hold the vessel integrity (ii) as CHF is the critical value for the thermal load the main specific goal for assessing the effectiveness of IVR becomes to demonstrate that the local heat flux along RPV be lower than the CHF The new concern doubts that local lsquohot partrsquo of the vessel would have negative effects on the overall resistance due to creep and dilatation which could result in a vessel failure and hence violate the first conclusion in particular for ldquolong termrdquo retention of corium in RPV If so a structural integrity analysis of RPV would be needed in addition to assessing local heat flux especially for high-power reactors Therefore accurate prediction of vessel failure could be meaningful for both strategies

Figure 2 Illustration of RPV under thermal loads from corium

As the structural performance is highly affected by the thermal loads from the corium RPV failure is essentially a transient thermo-fluid-structure coupled problem Figure 2 illustrates the situation that a corium debris bedmelt pool formed in the RPV lower head The decay heat of nuclear fission products could continuously be generated inside the corium leading to the formation of melt pools as the debris melts The heated debris bedmelt pool imposes thermal and

4

mechanical loads on the RPV Such loads can be transient (change with time) and spatially non-uniform Complex phenomena are also involved in this process Strong turbulent natural convection induced by the decay heat may be expected in the melt pool under IVR The corresponding Rayleigh number 119877119886 can be 1016minus17 for prototypic reactor cases Material structural properties can become non-linear under high temperature and stress loads The creep deformation also becomes significant at high temperature Detailed modelling of turbulent natural convection and non-linear properties could be computationally expensive In addition large-scale geometry (hemisphere with a radius of 20-30 m) long-term transient deformation process (~10000 s) and required small time steps (eg 001 s) for numerical convergence consideration further increase the computational costs

13 Literature Review Creep Behaviour and RPV Failure Analysis

In this section the most relevant studies concerning the experiments modelling and analysis of creep behaviour and RPV failure are presented

131 Experiments on creep behaviour and RPV failure

Various experiments have been performed on the RPV failure problem some of which were to generate data of mechanical properties (especially the creep properties) and some to study the overall vessel failure with scaled or simplified geometries

The REVISA experiment [15] generated creep curves and stress strain curves of French steel 16MND5 ranging from 600 to 1300 degC This data is accessible in [16] as well as [17] which also documents other material properties eg density thermal expansion coefficient conductivity heat capacity Youngrsquos Modulus and plasticity Similar experiments were carried out to generate high-temperature creep and tensile test data for American RPV steels and Inconel [18]

The CORVIS experiments [19] investigated the lower head failure modes with and without penetrations Each experiment was carried out on a steel plate 100 mm thick representing a pressure vessel lower head The penetration case represented a boiling water reactor lower head carrying a tube penetration in the form of a drain line

The LHF (lower head failure) experiments [20] and the succeeding OLHF tests [21] investigated the vessel failure with a 15 scaled vessel heated by an electrical radiator allowing to locate the hot focus at different position of the vessel with a maximum wall temperature of around 900 degC Internal pressure ranging from 53 to 123 bar was also imposed to initiate the vessel failure

The FOREVER experiments [22] investigated the phenomena of melt pool convection vessel creep possible failure processes and failure modes occurred during the late phase of the in-vessel progression by heating up a melt pool in a vessel and observing possible vessel failures The study covers cases with and without penetration different melt level existence of gap cooling and in-vessel melt coolability

5

132 Numerical studies on RPV failure

Various numerical approaches have been employed on RPV failure Since a two-way thermo-fluid-structure coupling approach is challenging these approaches take simplifications and can generally be classified into two categories (i) RPV failure decoupled from melt pool heat transfer and (ii) RPV failure coupled with melt pool heat transfer in simplified manners

1) Decoupled approaches In the decoupled approaches only RPV is modelled Simplified constant thermal loads are imposed on RPV considering the melt pool heat transfer

In a post-test analysis of FOREVE-C2 experiment [23] it was treated as a thermo-structural coupled analysis of the vessel with the thermal load estimated by the outer thermocouples and the normal heat flux on the external vessel wall The analysis was done using finite element (FE) code SYSTUS A coupled creepdamage law based on a viscosity-hardening creep law describing primary and secondary creep was employed for creep analysis

Mao et al conducted various studies on reactor cases assessing the IVR strategy using FE code ABAQUS Thermal boundary conditions were imposed on the RPV which was then solved as a thermal-structural coupled problem The strain hardening creep model (Eq (1)) was mainly used Their studies cover the general RPV failure [24] as well as the effects of cooling water levels under IVR [25] and geometric discontinuity (ablated RPV) [26]

ε119888119903 = 1198891σ1198892ε1198893119890119909119901 [minus

1198894119879] (1)

In the EU IVMR project [13] the RPV integrity was also studied through a benchmark calculation which deals with the RPV of a PWR with a prescribed ablated profile with 2D axisymmetric geometry Prescribed temperature gradient was set as thermal load The objective was to calculate deformations for various pressure loads up to vessel rupture A smoother profile was planned in further research

The advantage of such approaches is once thermal loads are reasonably assumed the focus only needs to be put on the RPV The disadvantage is a reasonable thermal load should be consistent with the melt pool heat transfer while sometimes it is not well-noted eg constant uniform temperature boundary conditions were assumed for inner and outer RPV (with uniform thickness in lower head) surfaces [25] which ignored the non-uniform heat flux distribution caused by the turbulent convective corium pool As thermal loads strongly affect the structural performance they should be quantified carefully

2) Coupled approaches In coupled approaches both melt pool heat transfer and RPV failure are modelled and coupled in one-way or simplified manner With melt pool heat transfer modelled the coupled approaches are more flexible to provide reasonable thermal loads on the RPV In existing approaches CFD and the correlation-based distributed-parameter models eg the Effective Convectivity Conductivity Model

6

(ECCM) [27] and the Phase-change Effective Convectivity Model (PECM) [28] [29] are general methods for melt pool heat transfer

Sehgal et al [22] presented an approach that employed the MVITA code to calculate the thermal processes and ANSYS for structural analysis MVITA code employed the ECCM Bailey-Norton creep law was used as shown in Eq (2)

ε119888119903 = 6055 times 10minus26 sdot σ472119905085119890minus

37200

119879 (2)

This approach was used for the pre-test simulation of FOREVER experiment to determine the thermal and mechanical loadings Parametric investigations also helped in the identification of important factors that could affect the test procedure and test conditions

Willschutz et al [30] employed the CFD-module Flotran for the melt pool heat transfer and ANSYS for structural analysis of RPV Standard k-ε turbulence model was selected for the turbulent natural convection in the melt pool A user-defined three-stage creep model [31] was employed In this model experimental creep curves were discretized into a creep database (CDB) under given temperature 119879 and stress 120590 the creep curve was discretized into finite point pairs

(휀 ε) indicating under such thermal and mechanical loads the creep strain rate ε is subjected to creep strain 휀 The creep strain rate for any other case can be interpolated over known temperature stress and strain from the database This method was applied to the post-test calculation of the FOREVE-C2 experiment It was concluded that the unexpected deformation behaviour during the experiment was caused by the combination of the transient temperature field in the vessel wall and the three creep stages

They later replaced the Flotran solver with the ECCM for melt pool heat transfer in a following study [32] By implementing it into FEM-solver a recursively coupled analysis was constructed Moreover a modified creep model [33] was used with damage measures the CDB itself did not contain tertiary creep instead a factor 1(1 minus 119863) was multiplied to the original creep strain rate to get the modified creep strain rate where 119863 is a damage factor (0 lt 119863 lt 1) defined with creep strain information and plastic strain information As 119863 grows close to 1 the modified creep strain rate would be enlarged significantly by this factor 1(1 minus 119863) which then behaves like the accelerating tertiary stage This method was applied to pre- and post- test calculations for the FOREVER experiments proving to be closer to reality than the previous method

Madokoro et al [34] developed a thermal structural analysis tool based on open source solver OpenFOAM and numerically investigated the coupled RPV failure and melt pool convection The PECM was used in the melt pool heat transfer In structural part the plasticity and damage parameters were similar to Willschutz et al [32] except for that strain hardening creep model (Eq (3)) was used instead of a CDB Simulations of FOREVER-EC2 and -EC4 were investigated It was concluded from the simulations that although the emissivity and mesh type (updated or fixed) affect the results significantly the approach generally has the capability of predicting the vessel deformation and failure

7

ε119888119903 = 1198891 sdot σ1198892 sdot ε1198893 (3)

Villanueva et al [35] performed a coupled thermo-mechanical creep analysis of the reactor scale RPV (ABB-Atom design BWR) to evaluate efficacy of the cooling measures for the in-vessel melt coolability and retention The PECM was used for melt pool heat transfer and ANSYS for RPV integrity analysis The modified time hardening model (Eq (4)) of the American RPV steel SA533B was selected as creep model The transient thermal load from the melt pool was applied as a thermal boundary condition to the vessel wall Further studies were done with this approach on investigations of instrumentation guide tube (IGT) failure [36] and control rod guide tube (CRGT) cooling [37] of a Nordic BWR

ε119888119903 =1198881σ

11988821199051198883+1

1198883 + 1 1198881 gt 0 (4)

It is noted that only RPV analysis with finite element analysis (FEA) are reviewed here Engineering methods like the Larson-Miller method [38] are not covered in this thesis One can refer to theory manuals of severe accident codes like MELCOR ASTEC for more details Comprehensive description on modelling of melt pool heat transfer is not covered either One can refer to either [39] for a general view or to any severe accident code manuals for engineering methods

14 Motivation Objectives and Main Achievements

The existing experimental and numerical studies provide us a good understanding of RPV failure involved phenomena and valuable simulation experiences These numerical methods may vary in solvers (which may be related to code accessibility) and models (which may be related to model accessibility understanding of phenomena computational power) However improvements of modelling approaches are still possible as the improvements in any of the related aspects eg

bull Simulation tools are continuously being upgraded and providing new possibilities just like that PECM appeared and ECCM was no longer used

bull Better understanding of the phenomena and more simulation experiences from in-vessel retention and vessel failures researches benefit the further modelling and simulations

bull Increasing computational power allows more detailed modelling eg turbulence modelling of melt pool convection

bull There is still room for the improvement of creep modelling

In addition implementation difficulty of a coupling approach is another important issue to consider Since it is a TRANSIENT thermo-fluid-structure coupled problem it is almost impossible to replicate an approach simply by referring to a paper with limited details For BWRs it would be even better for the coupling approaches to support complex geometries considering forest of penetrations (CRGTs and IGTs) on the RPV lower head without significant increase of user effort in the coupling process

8

The objectives of this thesis are to investigate new approaches and develop new models for the thermo-fluid-structure coupled problem of RPV failure and involved phenomena These objectives are accomplished with the following achievements

1) A new creep model for the 16MND5 steel was developed which covers all three creep stages Good agreement was achieved between model predictions and experimental curves Both time and strain hardening models were implemented (Paper I)

2) A new thermo-fluid-structure coupling approach was developed for RPV failure analysis It supports the advanced models in both melt pool convection and structural mechanics to improve fidelity and facilitates implementation Validation of the approach with the new creep model was performed against FOREVER-EC2 experiment This approach is flexible for complex geometries and can be used in further analysis eg lower head with detailed modelling of penetrations in BWRs (Paper II)

3) The performance of this new approach which uses volume loads mapping for data transfer was compared with the previous approach using surface loads mapping showing good agreement with small discrepancy (Paper III)

4) A lumped-parameter code transIVR for melt pool heat transfer was developed which supports one- and two-layer pool configuration transient analysis and well-resolved RPV wall (Paper IV)

5) A hybrid coupling approach was developed for efficient analysis of RPV failure where transIVR is coupled with ANSYS Mechanical (Paper IV)

In the following general descriptions on the codes ANSYS Fluent Mechanical and the PECM are given in Section 2 the new creep model in Section 3 the new coupling approach as well as its comparison with the previous approach in Section 4 the efficient hybrid coupling approach in Section 5 and summary amp outlook in Section 6

9

2 Numerical Solvers and Models

In this section we briefly introduce the ANSYS Fluent as the CFD solver for melt pool heat transfer the ANSYS Mechanical as the mechanical solver for RPV failure and PECM as an efficient alternative for melt pool heat transfer that are heavily used in the following sections

21 ANSYS Fluent

In the fluid mechanics the fluid is treated as a continuum which is generally described by the conservative mass momentum and energy equations [40]

120597120588

120597119905+ 120571 sdot (120588) = 119878119898 (5)

120597(120588)

120597119905+ 120571 sdot (120588) = minus120571119901 + 120571 sdot (120591) + 120588 + (6)

120597120588119864

120597119905+ 120571 sdot ((120588119864 + 119901)) = 120571 sdot (119896120571119879 minussumℎ119895119869119895

119895

+ (120591 sdot )) + 119878ℎ

(7)

Computational Fluid Dynamics (CFD) studies topics on the numerical solutions of above equations Typical methods are the Finite Volume Method (FVM) Finite Difference Method (FDM) and Finite Element Method (FEM) as well as particle methods that are mesh-free eg SPH and MPS

ANSYS Fluent is a general purpose CFD code which solves above equations with FVM Turbulence modelling and multiphase flows are supported Turbulence can be modelled with RANS LES or hybrid LESRANS Customizations eg adding sourcesink time or spatial dependent boundary conditions are realized through the user-defined functions (UDF) which is programmed in C language

22 ANSYS Mechanical

In the continuum mechanics of solids three sets of variables are considered to describe the structure performance namely element displacement field strain field and stress field (119906 ε σ) A general 3-dimensional problem is described by 3 sets of equations the equilibrium equations (Eq (8)) strain-displacement relationship (Eq (9)) and constitute equations (Eq (10)) [41] For Eq (8) the reciprocal theorem of shear stress also gives τ119894119895 = τ119895119894 where 119894 119895 = 119909 119910 119911

indicating the stress tensor being symmetric For very simple applications eg 1D beam it can be solved analytically For complex problems where analytical solutions are not available approximation methods eg Galerkin weighted residual method principle of virtual work and principle of minimum potential energy would be used

10

partσ119909119909part119909

+partτ119909119910

part119910+partτ119909119911part119911

+ 119909 = 0

partτ119910119909

part119909+partσ119910119910

part119910+partτ119910119911

part119911+ 119910 = 0

partτ119911119909part119909

+partτ119911119910

part119910+partσ119911119911part119911

+ 119911 = 0

(8)

ε119909119909 =part119906

part119909 휀119910119910 =

120597119907

120597119910 휀119911119911 =

120597119908

120597119911

120574119909119910 =120597119906

120597119910+120597119907

120597119909 120574119910119911 =

120597119908

120597119910+120597119907

120597119911 120574119911119909 =

120597119908

120597119909+120597119906

120597119911

(9)

ε119909119909 =1

119864[σ119909119909 minus μ(σ119910119910 + σ119911119911)]

ε119910119910 =1

119864[σ119910119910 minus μ(σ119909119909 + σ119911119911)]

ε119911119911 =1

119864[σ119911119911 minus μ(σ119909119909 + σ119910119910)]

γ119909119910 =1

119866τ119909119910 γ119910119911 =

1

119866τ119910119911 γ119911119909 =

1

119866τ119911119909

(10)

ANSYS Mechanical [42] is a mechanical engineering software that uses finite element analysis (FEA) for structural analysis The principle of virtual work is adopted in the solver [43] The idea of this principle is that a virtual (very small) change of the internal strain energy must be offset by an identical change in external work due to the applied loads which reads

δ119880 = δ119881 (11) where 119880 is the strain energy or the internal work 119881 the external work and δ the virtual operator The virtual internal work can be expressed as

δ119880 = int δεσd(vol)T

vol

+int 119886119903119890119886119891

δ119908119899119879σ119889(119886119903119890119886119891) (12)

where 휀 is the strain vector 120590 stress vector vol volume of element 119908119899 motion normal to the surface 119886119903119890119886119891 area of the distributed resistance The virtual

external work is contributed from three parts the inertial effects the pressure force on the surface and nodal forces applied to the element which then can be expressed as

δ119881 = minusint 119907119900119897

δ119908119879119865119886

119907119900119897119889(119907119900119897) + int

119886119903119890119886119901

δ119908119899119879119875119889(119886119903119890119886119901)

+ δ119906119879119865119890119899119889

(13)

11

where 119908 is the vector of displacements 119865119886 acceleration force vector 119875 applied pressure vector 119886119903119890119886119901 area over which pressure acts and 119865119890

119899119889 nodal

forces applied to the element

Plugging Eq (12) and (13) into (11) would result in the equilibrium equation Since the virtual displacement can be arbitrary the final equations used to describe the equilibrium of an element is reduced to Eq (14) with multiple operations

([119870119890] + [119870119890119891])119906 minus 119865119890

119905ℎ = [119872119890] + 119865119890119901119903 + 119865119890

119899119889

(14)

where [119870119890] [119870119890119891] 119865119890

119905ℎ [119872119890] and 119865119890119901119903 are the element stiffness matrix

foundation stiffness matrix thermal load vector mass matrix acceleration vector and pressure vector respectively It represents the equilibrium equation on a one element basis Alternatively using a minimum total potential energy principle could result in a consistent expression to the principle of virtual work [41]

Customizations eg on material properties are realized through the user programmable features (UPF) which is programmed in FORTRAN language

23 PECM

The PECM [28] [29] developed in KTH is a correlation-based method to model the melt pool heat transfer It is implemented in ANSYS Fluent through UDF and has been validated against many experiments such as ACOPO COPO LIVE etc In the PECM the flow velocities 119906119909 119906119910 and 119906119911 are replaced by characteristic

velocities 119880119909 119880119910 and 119880119911 respectively in order to eliminate the necessity of solving

the Navier-Stokes equation

The characteristic velocities are calculated using heat transfer correlations

119880119906119901 =α

ℎ119901119900119900119897times (119873119906119906119901 minus

ℎ119901119900119900119897

ℎ119906119901) (15)

119880119889119899 =α

ℎ119901119900119900119897times (119873119906119889119899 minus

ℎ119901119900119900119897

ℎ119889119899) (16)

119880119904119889 =α

ℎ119901119900119900119897times (119873119906119904119889 minus

ℎ119901119900119900119897

119882119901119900119900119897) (17)

where ℎ119901119900119900119897 is the melt pool depth ℎ119906119901 the thickness of well mixed layer of the

pool ℎ119889119899 the thickness of lower stratified region of melt pool 119882119901119900119900119897 the width of

pool 120572 the thermal diffusivity The PECM employs reduced characteristic velocities to describe mushy zone convection heat transfer For internally heated pool natural convection the Steinberner-Reineke correlations (Eq (18)~(19)) [44] are employed The profile of sideward 119873119906 is described using Eckertrsquos type correlation (Eq (20)) for a vertical boundary layer where the local Rayleigh

number at pool height 119910 is defined as 119877119886119910 =119892βΔ1198791199103

αν

12

119873119906119906119901 = 03451198771198861198941198990233

(18)

119873119906119889119899 = 13891198771198861198941198990095

(19)

119873119906119904119889 = 050811987511990314 (

20

21+ 119875119903)

minus14

11987711988611991014

(20)

As the characteristic velocities are calculated the energy equation is the only transport equation that needs to be solved and can be expressed as follows

partρ119862119901119879

part119905+ nabla(ρ119862119901119932119879) = 119896nabla2119879 + 119876119907 (21)

where from left to right are the transient term convection term diffusion term and volumetric heat source respectively The convection term then can be calculated explicitly with use of values in the last time step and then can be treated together with the volumetric source term Therefore the PECM finally simplifies the energy equation to a conduction equation which can be easily solved and as a result will significantly increase the computational efficiency in simulations

In addition solidificationmelting process can also be modelled and the temperature-dependent physical properties are supported

13

3 Creep Modelling of 16MND5 Steel

Creep deformation becomes significant when a component is subjected to high temperature and stress Modelling of creep behaviour of the RPV is paramount since it is considered as a main mechanism leading to the RPV failure In this section we present the development of a three-stage creep model (called lsquomodified theta projection modelrsquo ) for the French RPV steel 16MND5 This model was successfully implemented into the ANSYS Mechanical code

31 Creep and Creep Modelling

Creep is an irreversible time-dependent inelastic deformation process Diffusion and dislocation are the two molecular mechanisms of creep The former normally occurs due to the presence of vacancies within the crystal lattice (or the grain) while the latter occurs if there is a slip by the passage of crystal dislocations [45] There are two types of diffusion creep [46] Coble creep (favoured at lower temperatures) if the diffusion paths are predominantly through the grain boundaries and Nabarro-Herring creep (favoured at higher temperatures) if through the grains themselves Dislocation creep tends to dominate at high stresses and relatively low temperatures Dislocations can move by gliding in a slip plane which is a process that requires little thermal activation But the rate-determining step for their motion is often a climb process which also requires diffusion and is thus a time-dependent process and favoured by higher temperatures Creep behaviour is highly affected by temperature and stress generally creep is more significant under higher temperature and stress

Creep curves describe the creep strain development with time Figure 3 shows a creep curve under constant temperature and stress which typically experiences three creep stages primary stage where the creep strain rate ε119888119903 (the change of creep strain ε119888119903 in unit time) decreases with time secondary stage where the creep strain rate almost keeps constant and tertiary stage where the creep strain rate drastically increases with time

Figure 3 Creep curve under constant temperature and stress

14

The built-in creep models in commercial software eg ANSYS [47] and ABAQUS [48] in default only cover the primary andor the secondary stage(s) Table 1 [47] gives the default creep models supported in ANSYS for example Possible reason why the tertiary stage is not modelled might be that for most industrial applications the failure analysis is beyond interest and therefore the built-in creep laws in the commercial software was supposed to be sufficient [49] Though the two codes offer the interfaces through either ANSYS UPF or ABAQUS user subroutines to support more general user-defined creep laws eg complex models that also include the tertiary creep stage

Table 1 Default creep models in ANSYS Mechanical

Creep model Equations Type

Strain hardening

ε119888119903 = 1198621σ1198622ε119888119903

11986231198901198624119879 1198621 gt 0 Primary

Time hardening ε119888119903 = 1198621σ11986221199051198623119890minus1198624119879 1198621 gt 0 Primary

Generalized exponential

ε119888119903 = 1198621σ1198622119903119890minus119903119905 r = 1198625σ

11986231199051198623119890minus1198624119879 1198621 gt 0

1198625 gt 0

Primary

Generalized Graham

ε119888119903 = 1198621σ1198622(1199051198623 + 1198624119905

1198625 + 11986261199051198627)119890minus1198628119879 1198621 gt 0 Primary

Generalized Blackburn

ε119888119903 = 119891(1 minus 119890minus119903119905) + 119892119905 119891 = 1198621119890

1198622σ 119903 = 1198623(σ1198624)1198625 119892 = 1198626119890

1198627σ

1198621 gt 0

1198623 gt 0

1198626 gt 0

Primary

Modified time hardening

ε119888119903 = 1198881σ11988821199051198883+1(1198883 + 1) 1198621 gt 0 Primary

Modified strain hardening

ε119888119903 = 1198621σ1198622[(1198623 + 1)ε119888119903]

1198623119890minus1198624119879 1198621 gt 0 Primary

Generalized Garofalo

ε119888119903 = 1198621[119904119894119899ℎ(1198622σ)]1198623119890minus1198624119879 1198621 gt 0 Secondary

Exponential form

ε119888119903 = 1198621119890σ1198622119890minus1198623119879 1198621 gt 0 Secondary

Norton ε119888119903 = 1198621σ1198622119890minus1198623119879 1198621 gt 0 Secondary

Combined time hardening

ε119888119903 = 1198621σ11986221199051198623+1119890minus1198624119879(1198883 + 1)

+ 1198625σ1198626119905119890minus1198627119879

1198621 gt 0

1198625 gt 0

Primary+ Secondary

Rational polynomial ε119888119903 = 1198621

partε119888part119905 ε119888 =

119888119901119905

1 + 119901119905+ ε119898119905

ε119898 = 1198622101198623σσ1198624 119888 = 1198627ε

1198628σ1198629

119901 = 11986210ε11986211σ11986212

1198622 gt 0 Primary+ Secondary

Generalized time hardening

ε119888119903 = 119891119905119903119890minus1198626119879 119891 = 1198621σ + 1198622σ2 + 1198623σ

3 119903 = 1198624 + 1198625

Primary

32 A Modified Theta Projection Model

321 Original theta projection model and its variations

The original theta projection model [50] was first proposed to accurate predict long-term creep and creep-fracture properties It assumes that a creep curve under constant temperature and stress can be described as

15

휀119888119903 = 1205791(1 minus 119890minus1205792119905) + 1205793(119890

1205794119905 minus 1) (22)

where 휀119888119903 is the creep strain 119905 the time and 1205791~1205794 the model parameters Figure 4 illustrates how the first term and second term on right hand side develop with time and constitute a creep curve which has obvious primary secondary and tertiary stages Model parameters 120579119894(119894 = 1234) are temperature and stress dependent and are described with the following expression

119897119899thinsp120579119894 = 119886119894 + 119887119894120590 + 119888119894119879 + 119889119894120590119879 (23)

where 119886119894 119887119894 119888119894 and 119889119894 ( 119894 = 1234 ) are constants It is noted [45] that this equation is suitable when the temperature range of experiment data sets is very narrow such that minus1119879 is almost proportional to 119879 These constants (119886119894 119887119894 119888119894 and 119889119894 (119894 = 1234)) are the final model parameters which are generated with two fitting procedures

(i) Fitting the experimental creep curves to Eq (22) generates several groups of 1205791~1205794 for each creep curve under certain temperature and stress We denote (119879119895 120590119895 1205791119895 1205792119895 1205793119895 1205794119895 ) as parameters for the 119895119905ℎ

experimental curve under temperature 119879119895 and stress 120590119895

(ii) Fitting 1205791 to Eq (23) as a function of temperature and stress with the obtained data ( 119879119895 120590119895 1205791119895 ) from ( 119879119895 120590119895 1205791119895 1205792119895 1205793119895 1205794119895 ) generates

parameters 1198861 1198871 1198881 and 1198891 Repeat this fitting procedure for 1205792 1205793 and 1205794 to get parameters 119886119894 119887119894 119888119894 and 119889119894 (119894 = 234)

Variations of the model were developed and applied to other different materials to better accommodate the experimental data which are all then called a lsquomodified theta projection methodrsquo independently Examples are models in [51] for lowhigh alloy ferritic austenitic steels Al-alloys Al-matrix composites [52] for Nickel-based Super-alloy [53] for pure metals and class M alloys with intermediate range of stress (40-70 of the yield stress) and temperature (40-60 of the melting temperature)

Figure 4 Illustration of theta projection method [54]

16

322 A modified theta projection model

The new modified theta projection model we proposed [54] was mainly adapted from the original theta projection method [50] and the modified version [51] The new model describes the creep curve with the following equation

휀119888119903 = 1205791(1 minus 119890minus1205792119905) + 120579119898119905 + 1205793(119890

1205794119905 minus 1) (24)

An additional term 120579119898119905 is seen compared with Eq (22) in the original method The reason for adding this term is that the 16MND5 steel shows fairly long secondary stages in many experimental creep curves and the creep curves can be better fitted by Eq (24) than Eq (22) As the structure undergoes different creep process under different temperature and stress the model parameters θ119894 (119894 =1~4 and 119898) are also temperature and stress dependent

The main difference compared with the original and all other modified projection models is that numerical interpolation is used to replace the second numerical fitting in previous models In other words instead of fitting the model parameters as functions of stress and temperature with Eq (23) the distributions of the model parameters are assumed to be piecewise in the temperature-stress space and they will be interpolated directly from known parameters obtained from experiments The consideration is that the experimental data ([16] [17]) for 16MND5 covers a wide temperature range varying from 600 degC to 1300 degC and a wide stress load range varying from 08 MPa to 248 MPa while Eq (23) in the existing methods is based on a narrow temperature span Poor parameter fitting results would be expected if we directly fit the parameters 1205791~1205794 with Eq (23) as used in the original methods

The creep model was developed using the following steps

(i) Fitting Each creep curve can be represented by a group of model parameters [1205791 1205792 120579119898 1205793 1205794] with respect to Eq (24) by fitting the creep curves to Eq (24) All parameters of the 32 experimental curves form an

initial dataset of the model parameters [119879119895 120590119895 1205791119895 1205792119895 120579119898119895 1205793119895 1205794119895] where j (0ltjle32) is the jth creep curve

(ii) Interpolation Parameters of a creep curve under any [119879 σ] are directly interpolated from the dataset

(iii) Parameter adjustment The dataset was further updated to satisfy the monotonicity assumption we have taken that creep strain monotonically increases with temperature and stress

(iv) Different hardening models for dynamic loads

Step (i) was done using MATLAB In the following steps (ii)-(iv) are described in detail

Model parameters for new creep curves are interpolated from the dataset over the 2D 119879 minus 120590 space The interpolation is divided into two sub-interpolation steps interpolate over stress with fixed temperature and then interpolate over temperature with fixed stress We take the interpolation of model parameters under (119879prime σprime) for example In the first interpolation we first find out the two

17

temperatures 119879119896119899119900119908119899119904119898119886119897119897 and 119879119896119899119900119908119899119897119886119903119892119890 most close to 119879prime from the experimental

temperature set 600 700 800900 1000 1100 1200 1300 Under each temperature there are four sets of model parameters in the dataset corresponding to four different stress loads Then the first interpolation is done over stress load from the known dataset of model parameters under each

temperature to output the parameters for (119879119896119899119900119908119899119904119898119886119897119897 120590rsquo) and (119879119896119899119900119908119899119897119886119903119892119890 120590rsquo)

In the second interpolation the model outputs parameters for (119879rsquo 120590rsquo) from

(119879119896119899119900119908119899119904119898119886119897119897 120590rsquo) and (119879119896119899119900119908119899119897119886119903119892119890 120590rsquo) For 120579i(119894 = 12 3 4) we assume a logarithm

relation between 120579119894 temperature 119879 and stress 120590 which is similar to Eq (23)

119897119899120579119894 = (119886 + 119887120590)(119888 + 119889119879) (25)

It reduces to 120579119894 = 11986011198901198871σ with fixed temperature in the first interpolation and to

θ119894 = 11986021198901198872T with fixed stress in the second interpolation The interpolation

expression for θ119894 = 11986011198901198871σwith two known points (1205901 1205791198941) and (1205902 1205791198942) is given

by Eq (26) The interpolation expression for θ119894 = 11986021198901198872T with two known points

(1198791 1205791198941) and (1198792 1205791198942) is given by Eq (27)

120579119894 = 120579i1

120590minus12059021205901minus1205902120579i2

1205901minus1205901205901minus1205902 (26)

120579119894 = 120579i1

119879minus11987921198791minus1198792120579i2

1198791minus1198791198791minus1198792 (27)

Similarly we assume the following relation between 120579119898 temperature 119879 and stress 120590

120579119898 = 119860120590119899119890minus119887 119879frasl (28)

The interpolation is done over the stress with 120579119898 = 1198603120590n by fixing temperature

and then over temperature with 120579119898 = 1198604119890minus1198874119879 by fixing stress The interpolation

expression over stress for 120579119898 = 1198603120590119899 with two known points (1205901 1205791198981) and

(1205902 1205791198982) would be given by Eq (29) The interpolation expression over

temperature for 120579119898 = 1198604119890minus1198874119879 with two known points (1198791 1205791198981) and (1198792 1205791198982)

would be given by Eq (30)

120579119898 = 119890119909119901(119897119899120590 minus 11989711989912059011198971198991205902 minus 1198971198991205901

1198971198991205791198982 +1198971198991205902 minus 119897119899120590

1198971198991205902 minus 11989711989912059011198971198991205791198981) (29)

120579119898 = 120579m2(11198791minus1119879) (

11198791minus11198792)fraslsdot 120579m1

(1119879minus11198792) (

11198791minus11198792)frasl (30)

In this model we also take the assumption that the creep strain monotonically increases with temperature and stress loads1 Or specifically with the same creep time the structural component under higher temperature or stress experiences

1 Intuitively materials deform more under higher stress and temperature This is also mostly observed in tensile experiments Though some exceptions exist one instance of which is the effect of phase change some materials under certain temperature around the phase change temperature may show much weaker structural performance Here we just simply assumed that the creep strain monotonically increases with temperature and stress

18

higher creep strain than that with lower temperature or stress (the creep curve under higher temperature or stress should be above the curve with lower temperature or stress) When generating the parameter dataset (Step (i)) we already kept the trend that under the same temperature cases under higher stress loads would have larger creep model parameters and give higher creep strains This trend would also hold for the interpolated temperatures which thus guarantees that creep strain monotonically increases with stress load under same temperature Then the question becomes if creep strain also monotonically increases with temperature given fixed stress However the answer could not always be a yes especially in the extrapolation range of the stress load Figure 5(a) illustrates an example of the interpolation of model parameters under (119879119872 σ119872) The red dots and blue dots are known parameters from the parameter dataset for corresponding high temperature 119879119867 and low temperature 119879119871 respectively The green dots (119879119867 σ119872) and (119879119871 σ119872) are interpolated from the red and blue dots in the first sub-interpolation respectively The black dot (119879119872 σ119872) is interpolated from the green dots in the second sub-interpolation It is seen that when σ119872 lt σ119862 the value of (119879119871 σ119872) is higher than that of (119879119867 σ119872) It means the higher 119879119872 is the smaller the model parameter will be ie the model would predict larger creep strain for lower temperatures which violates our monotonicity assumption One simple solution used here as shown in Figure 5(b) is to minorly adjust the values of the four red and blue dots such that the intersection point C can be sufficiently small and the situation σ119872 lt σ119862 would barely occur in the extrapolation range

Figure 5 Illustration of model parameter interpolation (a) before and (b) after parameter adjustment [54]

A corresponding parameter adjustment process was introduced to verify that the model predicts higher creep strain under higher temperature under fixed stress Figure 6 illustrates the general verification process for any (119879 120590) in the interested load range the creep strain of (119879 + 120575119879 120590) should be larger than that of (119879 120590) Otherwise parameters would be minorly adjusted and the verification process be repeated This process would be iterated several times until the requirement is reached The achieved verification shows

Stress

σC

High temperature TH

MPa

a

0

Model

par

amet

er θ

Low temperature TL

MPa

a

0

Model

par

amet

er θ

High temperature TH

Low temperature TL

σC

Stress

C C

(a) (b)

σM

σM

19

(i) For any (119879 120590) in the temperature range 273~ 1073 K with an interval of 1 K and the stress range 1 ~260 MPa with an interval of 1 MPa its creep strain is higher than that of (119879 minus 1119870 120590) for the first 12000000 s

(ii) For any (119879 120590) in the temperature range 1073~1673 K with an interval of 1 K and the stress range 1 ~80 MPa with an interval of 1MPa its creep strain is higher than that of (119879 minus 1119870 120590) for the first 12000000 s

(iii) For any (119879 120590) in the stress range 01~26 MPa with a smaller interval of 01 MPa for the full temperature range 273~1673 K with an interval of 1 K its creep strain is higher than that of (119879 minus 1 120590) for the first 12000000 s

Above verification covers a wide range of temperature and stress especially where the stress and temperature are low and extrapolation is used Further verification to cover even higher stress or temperature ranges is not considered as creep increases drastically under such high stress and temperature and the results would become meaningless With these regards we generally think the monotonicity requirement is reached The model parameter dataset was updated along with the adjustment The final dataset was given as Table 2 in Paper I

Figure 6 Parameter adjustment process [54]

Figure 7(a-d) present the model predictions of selected creep curves in ANSYS Mechanical (labelled as lsquoANSYSrsquo) compared with experimental curves (labelled as lsquoREVISArsquo) at temperature 600 800 1000 and 1200 degC respectively The predicted curves proved to clearly show the three-stage creep process and agreed well with the experimental creep curves

Initial parametersdataset

θ1 θ2 θm θ3 θ4 from creep

curve-fitting

Parameters interpolation

Is the monotonicity

assumption satisfied

IsMon= =1

Final model parameters

Adjust the

parametersdataset

θ1 θ2 θm θ3 θ4

No

Yes

IsMon=1 a flag is monotonic achieved

For σ in the stress range in MPa

For T in range Tstart Tend in K

For t in range 112000000 in s

If ε(σ T+1t)lt ε(σTt) IsMon=0

Iteration loop of

parameter adjustment

20

Figure 7 Selected model predictions compared with experimental creep curves [54]

Instead of being constant temperature and stress loads also dynamically change with time in real cases Hardening models determine how creep strain rate reacts to the change of loads during a creep process The time hardening model and strain hardening model2 are supported in our model implementation in ANSYS Mechanical which are two mostly used hardening models

Time hardening assumes that the following creep strain rate after a load change depends only on the current creep time from the beginning of the creep process (and also temperature and stress if used in the creep model) and ignores the current creep strain Strain hardening assumes that the creep strain rate depends only upon the current creep strain (and also temperature and stress if used in the creep model) [55]

2 The time hardening and strain hardening here are more general terms than the specific time hardening creep model and the strain hardening creep model mentioned in Table 1

(a) 600 degC (b) 800 degC

(c) 1000 degC (d) 1200 degC

21

Eq (24) in default is a time hardening model as creep strain is directly described by the time 119905 Eq (31) gives the creep strain increment 119889휀119888119903 at time 119905 during time interval 119889119905

119889휀119888119903 = (12057911205792119890minus1205792119905 + 120579119898 + 12057931205794119890

1205794119905)119889119905 (31)

The strain hardening model is derived from Eq (24) in an implicit form Specifically when temperature and stress load change to (119879 120590) at the time 119905 and current creep strain 휀1198881199030 the model will get the creep curve of constant (119879 120590) and

output how this curve will react under the current strain 휀1198881199030 Since the creep curve shows creep strain monotonically increases with time given the creep strain 휀1198881199030 under creep curve of (119879 120590) we can uniquely find this reference time

119905119903119890119891 satisfying Eq (32) One should notice that the reference time 119905119903119890119891 can be

different from the current physical time 119905

휀1198881199030 = 1205791(1 minus 119890minus1205792119905119903119890119891) + 120579119898119905119903119890119891 + 1205793(119890

1205794119905119903119890119891 minus 1) (32)

where all 120579119894(119894 = 1~4119898) are model parameters of the new creep curve (119879 120590) It can be solved for 119905119903119890119891 with methods eg Newton-Ramphson [56] Then the creep

strain increment at 119905119903119890119891 during time interval 119889119905 is given as

119889휀119888119903 = (12057911205792119890minus1205792119905119903119890119891 + 120579119898 + 12057931205794119890

1205794119905119903119890119891)119889119905 (33)

Therefore combining Eq (32) and (33) gives the creep strain rate at creep strain 휀1198881199030 In summary when loads are changed to (119879 120590) with current creep strain 휀1198881199030

the model traces the creep curve of (119879 120590) and calculates the creep strain rate of this curve at strain 휀1198881199030

One calculation was done to demonstrate the performance of the hardening models The stress load was set to 151 MPa and kept unchanged A step temperature load was applied as the orange dash curve shows in Figure 8 it was 900 degC during 0-36000 s and suddenly jumped to 1000 degC at 36000 s which was then kept till the end of the simulation The blue solid curve (the part before the temperature change was identical to and therefore covered by the green solid curve in the figure) and blue dash curve are the creep curves under constant temperature T=900 degC and T=1000 degC respectively The green solid curve and green dash curve represent the model predictions considering this temperature change with time hardening model and strain hardening model respectively Before this change the creep curves with both hardening models were identical to the creep curve of the T=900 degC till point A After that the creep strain rate then shifted from Point A to Point B (which had the same time with A) in the time hardening model and behaved the same as the curve T=1000 degC as shown in the rest part of the green solid curve In the strain hardening model the creep strain rate shifted from Point A to Point C (which had the same creep strain with A) and behaved the same as the curve T=1000 degC as shown in the rest part of the green dash curve The large difference between two models was partially caused by the drastic temperature jump In real cases the difference would be less significant as the temperature changes more mildly Overall this prediction is exactly the expected behaviors of the time hardening model and strain hardening model

22

Figure 8 Time hardening and strain hardening in ANSYS Mechanical [54]

23

4 Coupled Thermo-Fluid-Structure Analysis of RPV Failure

As described in Section 13 RPV failure is coupled with melt pool heat transfer leading it into a transient thermo-fluid-structure coupled problem The goal of this section is to develop and validate a reliable coupling approach which supports transient coupled thermo-fluid-structure analysis as well as advanced models and computational platforms so as to improve fidelity and facilitate implementation The approach was validated against the FOREVER-EC2 experiment and then applied to an RPV failure analysis of a Nordic BWR As the approach adopts the volume loads mapping for load transfer comparative analysis was also performed against the coupling approach using surface loads mapping Future work is considered in analysis of penetration failures (IGTs and CRGTs) of a BWR RPV since the approach is flexible to complex geometries Results of Paper II ampIII are jointly presented in this section

41 Thermo-Fluid-Structure Coupling Approach

Considering the involved phenomena a coupling approach should at least address three issues (i) melt pool heat transfer (ii) RPV deformation and (iii) the interaction between melt pool heat transfer and vessel deformation A two-way strong coupling considering both the transient thermal load from melt pool on vessel and the influence of vessel deformation on melt pool heat transfer involves complex data exchange and is still full of challenges Therefore the new coupling approach is realized in a one-way manner with the assumptions that the influence of vessel deformation on melt pool heat transfer can be ignored and thus only the thermal loads from melt pool to vessel are considered It is based on the considerations that on one hand the deformation is a slow process regarding the relatively long failure time and on the other hand it was pointed out [57] that the physics of natural convection heat transfer in internally heated liquid pools is not sensitive to the geometrical factors based on their calculation results and experimental data from [58] [59] as well as previous experiments in semicircular and square cavities eg [44] [60] In addition one-way coupling is also computationally more efficient especially when heat transfer turbulence structural non-linearity and large-scale geometries are involved The developed approach is based on the ANSYS platform which generally consists of three parts with respect to the three issues respectively

(i) conjugate heat transfer of the melt pool and vessel using ANSYS Fluent (ii) transient thermal loads transfer from ANSYS Fluent to ANSYS

Structural (the ANSYS Mechanical in ANSYS Workbench) (iii) structural analysis of the vessel in ANSYS Structural

ANSYS Structural in default only supports direct transfer of constantsteady-state spatial loads from ANSYS Fluent Transfer of transient loads is realized with the extension tool FSI Transient [61] It is a pre-processing tool (with good documentation) developed by a third-party CAE vendor EDRTM MEDESO [62] to map transient CFD results (heat transfer coefficient temperature and surface pressure) to thermal or structural analysis including the definition of load steps

24

This new procedure proves to be straightforward and reduces efforts in the load transfer

Since structural performance of RPV is highly dependent on temperature the transfer of transient temperature loads from conjugate heat transfer to mechanical solver is a core step of the coupling framework There are two general approaches to realize the transfer as shown in Figure 9 Both approaches contain three parts a conjugate heat transfer analysis for fluid and structure where flow and heat transfer are coupled a mechanical analysis for the structure and a load mapping process that transfers the transient loads The conjugate heat transfer part can be solved with any CFD solver and is treated in the same manner in both approaches In the first approach (Figure 9(a)) transient volume thermal loads are transferred and mapped into the mechanical simulation domain The structure domain reads the essential thermal loads from the mapping after which a structure analysis of this domain is performed In the second approach (Figure 9(b)) transient surface thermal loads are mapped onto the boundary surfaces of structure domain as thermal boundary conditions Then a thermo-structure coupled analysis of the structure domain in the mechanical part is performed To distinguish them we call the former approach volume loads mapping (VLM) and the latter surface loads mapping (SLM) In the RPV failure analysis the fluid and structure domains correspond to the melt pool and RPV respectively Examples of these methods are the studies of [63] and [30] using VLM and [35] using SLM The VLM is adopted in current approach

Figure 9 Frameworks of the thermo-fluid-structure coupling approaches (a) volume loads mapping (b) surface loads mapping [64]

fluid structure structure

fluid structure structure

Conjugate heat transfer problem Mechanical problem

(structure)

Conjugate heat transfer problem Mechanical problem

(thermo-structure coupled)

Transient volume

loads mapping

(VLM)

Transient surface

loads mapping

(SLM)

(a)

(b)

25

42 Numerical Modelling

421 Conjugate heat transfer

In the conjugate heat transfer analysis main focus is put on the modelling of melt pool heat transfer Within the ANSYS Fluent solver two options are supported CFD with turbulence modelling and PECM CFD is the choice when the model goes to detail and more mechanistic analysis are needed while PECM provides an alternative that is faster and possibly more suitable for large-scale problems

Within the scope of RANS models the SST (Shear Stress Transport) model [65] is selected as the turbulence model in present study This model is a combination of k-ε model and k-ω model The k-ω formulation is used in the inner parts of the boundary layer which makes the model directly usable all the way down to the wall through the viscous sub-layer It also switches to k-ε model in the free-stream and thereby avoids the common k-ω problem that the model is too sensitive to the inlet free-stream turbulence properties The considerations on choosing this model are

bull Several two-equation low-Re k-ε models have been proved in [66] not good enough for melt pool heat transfer Therefore these mentioned two-equation models as well as other one-equation models are not considered

bull SST is highly recommended [67] for simulating most industrial heat transfer problems

bull Convincing results ([68] [69]) using SST have been obtained in capturing the thermal behaviour of the natural convection in an internally heated fluid Especially good agreements were achieved between the predicted heat flux distribution along the vessel and experimental data which would be one of the most important factors to estimate model performance in this problem

The description of PECM as an effective model for melt pool heat transfer is given in Section 23

422 Mechanical analysis

The material for the RPV failure analysis considered in this study is the French RPV steel 16MND5 The three-stage modified theta projection model in Section 3 is used as the creep model Since the definition of an RPV failure criterion is not universal (it can be the time when damage variable reaching 1 or either specified value of creep strain creep strain rate or total deformation is reached) we do not explicitly define a failure criterion Instead the simulation is kept running until the creep converge criterion cannot be satisfied even with the defined minimum time step (indicating a fairly large creep strain rate) and the simulation will be automatically terminated The time the simulation stops is taken as the failure time This treatment implicitly implies a criterion of a large creep strain rate We refer to [17] for other material properties Youngrsquos modulus and thermal conductivity are temperature dependent Multilinear Isotropic Hardening (MISO) is used as the plasticity model

26

43 Validation against the FOREVER-EC2 Experiment

In this section the validation against the FOREVER-EC2 experiment is presented (indicated as lsquoVLMrsquo in the following) In addition the simulation results using the SLM approach (indicated as lsquoSLMrsquo in the following) is also included for comparison purpose

431 FOREVER-EC2 experiment

The FOREVER-EC2 experiment [70] is one of the FOREVER experiments that achieved a vessel failure Figure 10 shows the experimental setup A 110 scale pressure vessel with a wall thickness of 15 mm and an outer diameter of ~400 mm was employed The lower head was made of 16MND5 steel Since it was difficult to use prototypical corium materials in the experiment the binary salt 30 CaO-70 B2O3 was employed as simulant To generate the internal heat which corresponds to the nuclear decay heat a specially designed heater was used The power input was maintained ~38 kW during the experiment such that maximum melt pool temperature would be ~1300 degC The pressure inside the vessel was maintained at ~25 MPa Thermocouples were used to measure the melt pool and vessel wall temperatures and linear displacement transducers used to measure the vessel wall displacements The designed experimental conditions were reached at experimental time 11670 s Vessel failure was observed about 12510 s later (at experiment time 24180 s)

Figure 10 Scheme of the experimental setup [22]

27

Figure 11 Computational domains and boundary conditions (a) conjugate heat transfer (b) mechanical analysis [71]

Computational domains and boundary conditions are indicated in Figure 11 SST turbulence modelling was used for pool heat transfer The simulations started at the moment when experimental conditions were reached (experiment time 11670 s) So simulation time 0 s corresponds to experiment time 11670 s

432 Conjugate heat transfer

Figure 12 shows the temperature fields at simulation time 0 s 1225 s 2475 s 4975 s and 9975 s All the time points in this part refer to simulation time The field at time 0 s (Figure 12(a)) was the initial temperature that was reconstructed based on the thermocouple readings The temperature increased with time and reached a steady state when the maximum temperature reached ~1607 K Figure 12(b-f) show clear thermal stratification which is an expected phenomenon of natural convection

Internal pressure

25 bar

Internal pressure 25e6-2500times981timesy Pa

External pressure

1 bar

Zero displacement

y=0 m

Gravity 400

mm

R=203 mm

15 m

m

Tamb

=323 K radiation with ε = 08

+ convection with h=10Wm2K

Tamb

=800 K radiation with ε=02

(a) (b)

Tamb

=800 K radiation with ε=08

+ convection with h=10 Wm2K

28

Figure 12 Temperature fields at simulation time (a) 0 s (b)1225 s (c) 2475s (d) 4975 s (e) 7475 s and (f) 9975 s [64]

Figure 13 shows the temperature profiles of the external vessel along the polar angle in comparison with experimental data Regions with a polar angle larger than 90deg are those in the cylindrical part of the vessel It also shows the temperature profile increased with time until reaching a steady state The profile almost did not change after 4975 s which indicates that a steady state of the temperature was reached As time increased the temperature in the lower head region 0~90deg (where the vessel was contacted with the melt) evolved from a relatively flat distribution to a distribution that temperature at high angle was higher than that at low angle by more than 200 K Generally the steady state results agreed well with the experimental data except for the low angle part of vessel lower head and the high angle part of the cylindrical vessel In addition to modelling errors the discrepancy in the low angle part of the lower head may be caused by the crust and gap possibly formed at vessel bottom in the experiment which can reduce the downward heat transfer and result in lower temperature while they cannot be modelled in simulations As to the high angle cylindrical vessel part simplified boundary conditions were applied to the inner and outer vessel wall since it was far from the melt and not of interest which may result in some discrepancy

(a) (b)

(c) (d) (e) (f)

29

Figure 13 Temperature profiles of external vessel surface along the vessel polar angle [64]

433 Mechanical analysis

In the mechanical analysis the VLM predicted a failure time of 23959 s and the SLM predicted a failure time of 22445 s both are close to the experimental failure time 24180 s All the time points in this part refer to the experiment time Figure 14 shows the creep strain fields at failure time Both approaches predicted maximum creep strains at same location where the temperature was high (cf Figure 12) This is related to the fact that creep behavior is highly influenced by temperature Maximum values for both cases were also close to each other 0393 in VLM and 0366 in SLM

Figure 15 presents the total deformation of the vessel bottom point (also called ldquosouth polerdquo in some literature) in the vertical direction Since there was an initial displacement reading of ~355 mm at time 0 s which almost did not change during 0~11670 s (the whole period before the experimental conditions were reached) an initial value of 355 mm was added to the simulation result such that experimental and simulation results had the same total displacement at t=11670 s Similar trend was seen in the VLM SLM and experimental data that the deformation increased with time and experienced a drastic acceleration at the end Generally both simulation approaches agreed well with the experimental data though they both slightly under predicted the value of total deformation The discrepancy may possibly be related to the accuracy of the thermal predictions modelling errors of plasticity and creep and simplified boundary conditions

400

600

800

1000

1200

1400

1600

0 20 40 60 80 100 120 140 160

Tem

per

ature

(K

)

Polar angle (deg)

Experiment

25 s

1225 s

2475 s

4975 s

7475 s

9975 s

Weilding line

30

Figure 14 Creep strain fields at failure time (a) VLM (b) SLM [64]

Figure 15 Total deformation in vertical direction of the vessel bottom point [64] [71]

The wall thickness changes are plotted together with experimental results in Figure 16 The wall thickness change δ119909 is defined as

δ119909 = 119909119890119899119889 minus 119909119900119903119894119892119894119899119886119897 (34)

000

500

1000

1500

2000

2500

11670 13670 15670 17670 19670 21670 23670

Tota

l def

orm

atio

n (

mm

)

Time (s)

ExperimentVLMSLM

(a) (b)

31

where 119909119890119899119889 and 119909119900119903119894119892119894119899119886119897 are the wall thickness at failure time and the original wall

thickness respectively Since 119909119890119899119889 cannot directly be post-processed it was estimated in the following way (cf the subfigure of Figure 16) for the same polar angle θ find the corresponding points Pin (1199091 1199101) and Pout (1199092 1199102) on the inner and outer surfaces of the deformed vessel respectively the distance of Pin and

Pout radic(1199091 minus 1199092)2 + (1199101 minus 1199102)2 is calculated as 119909119890119899119889 at polar angle 120579 Such estimation of the wall thickness is more accurate when the line PinPout is closer to be orthogonal to the deformed local surface Considering the final deformed vessel as shown in Figure 14 the estimation should be fine at regions close to 0deg and 90deg and can be slightly overestimated at medium angle region The VLM curve was compared with the SLM curve and 4 experimental curves measured at vessel left right front and back sides Generally VLM curve agreed well with the experimental and SLM curves wall thickness gradually decreases with angle till around 75deg and then increases till 90deg the maximum thickness change occurred at around 75deg would be around -5 mm The positive values at low angle may be caused by the thermal expansion It also shows that the thickness transitions in experimental curves were smoother than those in both simulation curves while simulation curves showed narrower regions with large thickness changes (at angle 70deg~80deg) Possible reasons might be (i) sharper temperature profile predicted in simulation (ii) input uncertainty and (iii) modelling errors The SLM predicted slightly thinner vessel than the VLM except for the necking region The discrepancy might be caused by various reasons eg the interior vessel temperature can be slightly different between two approaches which would then result in different thermal stress and different deformation the final deformed vessel shapes cannot be exactly the same which also results in different errors in estimating PinPout

Figure 16 Wall thickness changes along the vessel polar angle [64] [71]

-6

-5

-4

-3

-2

-1

0

1

0 10 20 30 40 50 60 70 80 90

Wal

l th

icknes

s ch

ange

(mm

)

Polar angle (deg)

VLMSLMExperiment (left)Experiment (right)Experiment (front)Experiment (back)

32

44 Application to BWR Case

In this section the coupling approach (indicated as lsquoVLMrsquo in the following) was applied to an RPV failure analysis of a BWR A simulation using the SLM coupling approach (indicated as lsquoSLMrsquo in the following) was also performed for comparison purpose

A hypothetical severe accident of a Nordic BWR initiated by an SBO was considered The core materials would melt and migrate downward to the RPV lower head The total amount of corium relocated in the lower head was considered to form a 19 m corium melt pooldebris bed assuming the corium distribution was homogeneous

Figure 17 Computational domains and boundary conditions (a) conjugate heat transfer (b) mechanical analysis [64]

Figure 17(a) shows the computational domain of the conjugate heat transfer PECM was used in this calculation A general mesh with a pool height of 23 m was used to accommodate most of the relocation situations with the same mesh Since the corium only had a height of 19 m the top 04 m region was treated as injected water covering the corium The top water was expected to be boiling and we assumed a uniform temperature of 383 K Under the implementation of PECM this region was damped with artificially high heat capacity and high conductivity such that it would efficiently remove heat from the corium and suppress its own temperature increase For the corium part below water a volumetric heat source 11 MWm3 was implemented as the nuclear decay heat For the in-vessel surfaces a constant temperature of 383K was applied as they are covered by water which can be boiling Without water cooling on the external vessel wall a very small heat loss of 20 Wm2 was applied Melting and solidification of the corium were also considered with a solidus temperature of 2750 K and a liquidus temperature of 2770 K The initial temperature was set as

q=20 Wm2

T=383 K

Qv=11

MWm3

315 bar

315 bar+ hydrostatic

pressure of corium

Gravity

248 bar

Zero

displacement

Water

Corium

(a) (b)

33

1100 K for the corium 383 K for the water and 450 K for the vessel For the mechanical analysis (Figure 17(b)) an inner pressure of 315 bar and an outer pressure of 248 bar were considered For the region covered by corium an extra hydrostatic pressure was added on the vessel wall A 2 m long vessel extension was considered as well as the standard earth gravity

Figure 18 presents the wall temperature profile along the inner vessel surface at 50 s 2500 s 7500 s and 10000 s The temperature increased with time for the vessel surface contacted with corium and the distribution was flat which was due to that large part of the corium was still in solidus status It also shows that for the surface region that contacted the water the temperature was basically unchanged which indicates that the damping treatment indeed suppressed the temperature increment and guaranteed the heat removal upward by water

Figure 18 Temperature profiles of inner vessel wall along the vessel polar angle [64]

The predicted RPV failure time was 9869 s by VLM and 9650 s by SLM Compared with the FOREVER-EC2 case in Section 43 the VLM predicted a slightly later failure time than the SLM in this BWR case too Figure 19(a) compares the maximum creep strain developments with time of both approaches Before 9000 s the predicted values by both approaches are close to each other Afterward the creep acceleration occurred and the strain increased drastically The acceleration of the VLM occurred 200 s later than that of the SLM which agreed with the predicted later failure time Similar trend was observed for the total deformation in vertical direction of vessel bottom point as shown in Figure 19(b) The deformations of both approaches are close except that the acceleration of VLM occurred later than that of SLM Overall results of VLM agreed well with those of SLM

0

500

1000

1500

2000

2500

0 10 20 30 40 50 60 70 80 90 100

Tem

per

ature

(K

)

Polar angle (deg)

50 s

2500 s

5000 s

7000 s

10000 s

34

Figure 19 Results of structural analysis (a) creep strain and (b) total deformation in vertical direction of vessel bottom point [64]

45 Computational Efficiency

Computational efficiency is also an important consideration regarding the performance of numerical approaches The computational efficiencies of both VLM and SLM are analysed and compared in this section As parallel computing technique was used to speed up the simulations the number of parallel cores and corresponding costed computational time for each simulation are given together in Table 2 as a generally reflection of the computational efficiency All the simulations were done in a Lenovo P700 workstation equipped with 2 Intel Xeon E5-2630 v3 processors (8 cores per processor) and 32 G RAM For the simulations of FOREVER-EC2 SLM employed twice the parallel cores and costed 286 more time compared with VLM For the simulations of prototypic BWR case both VLM and SLM employed the same number of parallel cores but VLM only took 13 of the computational time compared with SLM Overall it can be concluded that VLM has higher computational efficiency than the SLM The reason is a structural analysis is performed in the VLM approach while a coupled thermo-structural analysis is performed in the SLM approach (cf Figure 9) a standalone structural analysis is computed faster than a coupled thermo-structural analysis

Table 2 Comparison of computational efficiency [64]

Case FOREVER-EC2 BWR VLM SLM VLM SLM

Number of cores 4 8 4 4 Computational time (hours) 21 27 18 72

35

5 Hybrid Coupling Approach for Efficient RPV Failure Analysis

In this section we propose a hybrid coupling approach for the efficient RPV failure analysis The idea is to replace the ANSYS Fluent in previous coupling approach with a lumped-parameter code for quick estimate of the melt pool heat transfer With the least modelling details (compared with CFD methods and distributed-parameter methods eg PECM) lumped-parameter codes prove to be the most efficient for melt pool heat transfer The new hybrid approach characterizes 1) utilization of efficient models for melt pool heat transfer and 2) the support of detailed structural modelling of RPV deformation allowing studies with different RPV geometric shapes under effects like local thinning due to ablation For studies of which the focus is mainly on structural behaviour of RPV eg [13] [26] while mechanistic simulation of melt pool convection is computationally expensive and unnecessary the lumped-parameter code could help providing reasonable thermal loads

The development of the approach consists of two tasks (i) development of a lumped-parameter code and (ii) its coupling with ANSYS Mechanical to form this hybrid coupling framework In the following we describe both tasks and the corresponding verification and validation processes respectively This section is mainly based on Paper IV

51 A Lumped-parameter Code for Melt Pool Heat Transfer

A lumped-parameter code transIVR was developed It features (i) quick estimate of transient melt pool heat transfer with reasonable accuracies and (ii) two-dimensional representation of heat conduction in the RPV wall facilitating the coupled analysis of the RPV failure under thermal loads The following assumptions are taken in the transIVR modelling

bull The crust between the oxide pool and metal pool is thin

bull The heat flux has uniform distributions on the bottom top and side surfaces of metal pool

bull Heat source only exists in oxide layer

bull Latent heat of fusion is ignored

Figure 20 illustrates the modelled three domains oxide layer metal layer and the RPV wall In addition the internal structure inside the vessel can also be considered as a radiative body to the top metal surface as in [72] An internal volumetric heat source generated by nuclear decay heat is considered in the oxide layer Heat in oxide layer transfers downward to the RPV wall and upward to the top metal layer Heat in metal layer can be transferred sideward to the RPV wall and upward to the surrounding internal structure through radiation (or to water if inside water flooding is applied) The heat transferred to RPV will be removed from outer RPV surface Possible formations of oxide crust in the top and bottom surface are illustrated as the black regions Possible formation of metal crust on the top surface is shown as the grey region The code automatically detects the

36

existence of the crust on oxide top and metal top surfaces during the simulation and solves the problem correspondingly withwithout crust Boundary conditions are applied to the external and internal surfaces of RPV and (if needed) top surface of metal layer Arbitrary boundary conditions are supported If a boundary condition is nonlinear eg mixed heat convection (linear dependent on 119879 ) and heat radiation (proportional to 1198794) the Picard method [73] would be used for solving this problem

This model consists of three main parts

(i) empirical heat transfer correlations (mostly from experiments) to determine global and local heat transfer coefficients

(ii) energy balances in oxide layer and metal layer where the heat transfer coefficients would be used and

(iii) energy balance in RPV

Figure 20 Two-layer melt pool configuration and RPV wall [74]

511 Heat transfer correlations

For internally heated natural convection (oxide layer) in transIVR the mini-ACOPO correlations [75] are used for the averaged heat transfer as expressed in Eq (35) and (36) and for the angular variation of the heat flux as expressed in Eq (37) and (38)

119873119906119906119901 = 03451198771198861198941198990233 (35)

119873119906119889119899 = 00038119877119886119894119899035(119867119877)025 (36)

119873119906119889119899(θ)

119873119906119889119899= 01 + 108(

θ

θ119898119886119909) minus 45 (

θ

θ119898119886119909)2

+ 86 (θ

θ119898119886119909)3

01 leθ

θ119898119886119909le 06 (37)

RPV

Oxide pool

Metal pool

37

119873119906119889119899(θ)

119873119906119889119899= 041 + 035 (

θ

θ119898119886119909) + (

θ

θ119898119886119909)2

06 ltθ

θ119898119886119909le 1 (38)

The 119877119886119894119899 that appears in Eq (35)~(36) is the internal Rayleigh number which is defined as

119877119886119894119899 =119892β119876119881119867

5

α119896ν (39)

For Rayleigh-Bernard convection in the metal layer heat transfer occurs in all the top bottom and side surfaces To side surface the reduced form [72] of the Churchill-Chu correlation [76] for 119875119903 = 013 is employed as shown in Eq (40)

119873119906119904119889 = 007611987711988613 (40)

To top and bottom surfaces the reduced form [72] of Globe-Dropkin correlation [77] for

119875119903 = 013 is used as expressed in Eq (41)

119873119906119906119901 = 005911987711988613 (41)

The 119877119886 that appears in Eq (40) and (41) is the external Rayleigh number which is defined

as

119877119886 =119892βΔ1198791198673

να (42)

Heat transfer coefficients can be calculated from Nusselt number by Eq (43)

ℎ =119873119906119896

119871 (43)

where 119896 and 119871 are the heat conductivity and characteristic length respectively 119873119906 can be the Nusselt numbers expressed in Eq (35)~(38) and (40)~(41)

512 Energy balance in the pools

The energy balance of oxide layer can be described as Eq (44) The total generated heat of oxide pool equals the sum of the increase of enthalpy (transient temperature change) upward heat flow to metal layer and downward heat flow to RPV

120588119862119901119881119900119897119889119879119900119909119889119905

+ ℎ119906119901119860119900119909119906119901(119879119900119909 minus 119879119900119909119906119901) + ℎ119889119899119860119900119909119889119899(119879119900119909 minus 119879119900119909119889119899) minus 119876119881119881119900119897 = 0 (44)

where ρ is the oxide density 119862119901 the heat capacity 119881119900119897 the total volume 119879119900119909 the bulk temperature 119905 the time ℎ119906119901 the upward heat transfer coefficient 119860119900119909119906119901

upward heat transfer area 119879119900119909119906119901 upward surface temperature ℎ119889119899 downward

heat transfer coefficient 119860119900119909119889119899 downward heat transfer area 119879119900119909119889119899 downward surface temperature and 119876119881 volumetric heat source

38

On the interface between oxide layer and metal layer energy balance requires that the heat transferred from oxide layer should equal the heat to the metal layer which can be described as Eq (45)

ℎ119906119901119860119900119909119906119901(119879119900119909 minus 119879119900119909119906119901) = ℎ119898119905119889119899119860119898119905119889119899(119879119894119899119905 minus 119879119898119905) (45)

where ℎ119898119905119889119899 is the downward heat transfer coefficient of metal layer 119860119898119905119889119899 the downward heat transfer area 119879119894119899119905 the interface temperature and 119879119898119905 the bulk temperature of metal layer

The energy balance of metal pool can be described as Eq (46) The increase of enthalpy (transient temperature change) equals the sum of upward downward and sideward heat flows to surrounding surfaces

ρ119898119905119862119901119898119905119881119900119897119898119905119889119879119898119905119889119905

+ ℎ119898119905119906119901119860119898119905119906119901(119879119898119905 minus 119879119898119905119906119901) + ℎ119898119905119904119889119860119898119905119904119889(119879119898119905 minus 119879119898119905119904119889)

+ ℎ119898119905119889119899119860119898119905119889119899(119879119898119905 minus 119879119894119899119905) = 0

(46)

where ρmt is the metal density 119862119901mt the heat capacity 119881119900119897mt the total volume 119879119898119905 the bulk metal temperature 119905 the time ℎ119898119905119906119901 the upward heat transfer

coefficient 119860119898119905119906119901 upward heat transfer area 119879119898119905119906119901 upward surface

temperature ℎ119898119905119904119889 sideward heat transfer coefficient 119860119898119905119904119889 sideward heat transfer area 119879119898119905119904119889 sideward surface temperature ℎ119898119905119889119899 downward heat transfer coefficient 119860119898119905119889119899 downward heat transfer area and 119879119894119899119905 oxide-metal interface temperature

The heat transferred from metal layer to the top surface should equal the heat radiated from this surface to the environment Considering the radiation and reflections of internal structures to this surface following energy balance equation [72] is achieved

ℎ119898119905119906119901(119879119898119905 minus 119879119898119905119906119901) = σ(1198791198981199051199051199001199014 minus 119879119904

4) (1

ε+1 minus ε119904ε119904

119860119898119905119906119901

119860119904)minus1

(47)

where 119879119898119905119905119900119901 is the temperature of metal top free surface ε the emissivity of

metal ε119904 the emissivity of internal structures 119860119898119905119906119901 the area of metal top surface

and 119860119904 the total area of internal structures Sometimes the internal water cooling may be available above metal layer In this case Eq (47) is unnecessary and 119879119898119905119905119900119901 can be set to a constant value

For the oxide layer top surface Eq (48) is used if not crust is formed in the interface between two layers ie 119879119894119899119905 gt 119879119898119890119897119905119894119899119892119900119909 Otherwise Eq (49) is used

119879119900119909119906119901 = 119879119894119899119905 (48)

119879119900119909119906119901 = 119879119898119890119897119905119894119899119892119900119909 (49)

For the metal layer top surface Eq (50) is used if not crust is formed on top of the metal ie 119879119906119901119904119906119903119891 gt 119879119898119890119897119905119894119899119892119898119905 Otherwise Eq (51) is used

119879119898119905119906119901 = 119879119898119905119905119900119901 (50)

39

119879119898119905119906119901 = 119879119898119890119897119905119894119899119892119898119905 (51)

Overall this part involves 6 equations regarding 6 unknows 119879119900119909 119879119900119909119906119901 119879119894119899119905 119879119898119905

119879119898119905119906119901 and 119879119898119905119905119900119901

513 Energy balance in the RPV wall

The heat transfer in the PRV wall can be modelled as a simple 1D heat conduction problem that heat only transfers in the wall thickness direction or 2D axisymmetric heat conduction problem

For 1D model the treatment in [72] was adopted As illustrated in Figure 21 the heat from the oxide pool transfers to inner vessel surface (119879119894) then to the interior vessel (119879119907119890119904) then to outer vessel (119879119900) and finally to environment

Figure 21 1D heat transfer model in RPV [74]

Unknown variables are the heat flux through inner vessel surface 119902119894119899 the crust thickness 119909119888119903 inner vessel temperature 119879119894 vessel center temperature 119879119907119890119904 and outer vessel temperature 119879119900 Eq (52)~(56) give a closed equation set for these variables considering a radiation boundary condition on the external vessel surface Eq (52)~(53) describe the heat transfer through the crust where an internal heat source 119876119888119903 is also considered Eq (52) can be easily derived from the one-dimensional heat transfer equation with heat source Eq (54)~(55) describe the energy balance on the inner vessel surface and vessel interior respectively Eq (56) describes the energy balance on the outer vessel surface regarding radiation boundary condition For Dirichlet boundary condition Eq (56) is replaced with Eq (57) For other boundary conditions we only need to correspondingly modify the terms on the right-hand side of Eq (56) For situations that no crust is formed on the surface of the vessel Eq (52)~(53) are eliminated and 119902119894119899 in Eq (54) can directly be determined from calculations in Section 512 and Eq (54)~(56) are solved for the unknows 119879119894 119879119907119890119904 and 119879119900 Once 119879119894 exceeds the vessel melting temperature (with or without crust formation) wall ablation is considered to occur Then 119879119894 is fixed to the vessel melting temperature and vessel thickness 119909119907119890119904 becomes the new unknown variable

119879119900119909119889119899 119879119894 119879119907119890119904 119879119900 119879119890119899119907

119909119888119903 119909119907119890119904

crust RPV

Qcr

119902119894119899 119902119897119900119888119886119897

40

119902119894119899 = 119896119888119903(119879119900119909119889119899 minus 119879119894)

119909119888119903+ 1198761198881199031199091198881199032

(52)

119902119894119899 = 119902119897119900119888119886119897 + 119876119888119903119909119888119903

(53)

119902119894119899 = 119896119907119890119904(119879119894 minus 119879119907119890119904)

05119909119907119890119904

(54)

120588119907119890119904119862119901119907119890119904119889119879119907119890119904119889119905

= 119896119907119890119904(minus2 lowast 119879119907119890119904 + 119879119894 + 119879119900)

(05119909119907119890119904)2

(55)

119896119907119890119904(119879119907119890119904 minus 119879119900)

05119909119907119890119904= 휀120590(119879119900

4 minus 1198791198901198991199074 )

(56)

119879119900 = 119879119900119888119900119899119904119905

(57)

For the 2D representation the heat conduction in the RPV is solved using FDM considering the heat transfer in both the thickness and the angular directions in the RPV wall The RPV lower head is treated as a spherical shell The corresponding heat conduction equation in the 2D spherical coordinate is given as Eq (58) The cylindrical part of the vessel is treated as a cylindrical shell The corresponding heat conduction equation in the 2D cylindrical coordinate is given as Eq (59) Eq (58) and (59) are coupled and solved together In default the mesh size is 5deg in angular direction and 0025119909119888119903 in radius direction This mesh discretization can be changed up to user demands

ρ119862119901part119879

part119905=

1

1199032part

part119903(1198961199032

part119879

part119903) +

1

r2sinθ

part

partθ(119896119904119894119899θ

part119879

partθ)

(58)

ρ119862119901part119879

part119905=1

119903

part

part119903(119896119903

part119879

part119903) +

part

part119911(119896part119879

part119911) (59)

52 Hybrid Coupling Approach

The hybrid coupling approach to assess the RPV integrity is illustrated in Figure 22 transIVR and ANSYS Mechanical are coupled in a one-way manner transIVR conducts the conjugate heat transfer of melt pool and RPV and provides transient thermal loads to the ANSYS Mechanical where structural analysis of RPV deformation is performed The 2D heat transfer of RPV is solved in transIVR to better capture the temperature distribution In RPV structural analysis same considerations are taken as in Section 422 for 16MND5 steel the modified theta projection model for creep and MISO for plasticity and temperature-dependent Young modulus etc The VLM is adopted for the transfer of transient thermal loads

41

Figure 22 Framework of the hybrid coupling approach [74]

53 Verification and Validation

The transIVR code was first benchmarked against the UCSB FIBS case of two-layer pool configuration and then validated against the LIVE-7V experiment showing the codersquos capability in predicting two-layer and transient melt pool heat transfers respectively Finally the hybrid coupling approach encompassing transIVR and ANSYS Mechanical was validated against the RPV failure experiment FOREVER-EC2 and the results showed acceptable agreements with those of both the experiment and the previous approach using CFD in Section 43 Hence the transIVR code is an efficient tool for melt pool heat transfer and the hybrid approach an efficient way for RPV failure analysis

531 UCSB FIBS benchmark case

The UCSB FIBS benchmark case [78] was calculated to demonstrate the capability of the transIVR code in simulating two-layer pool heat transfer problems The considered pool configuration was similar to that shown in Figure 20 with the following specific dimensions a two-layer melt pool with a 16 m oxide layer and a 08 m metallic pool on top and an RPV lower head with a thickness of 015 m The internal volumetric source in the oxide layer was 13 MWm3 The calculation results were compared with those of the DOE study [78] and the INEEL study [79]

The predicted results were compared with the digitalized results of the DOE and INEEL studies in Figure 23 All three studies show similar profiles of the heat flux distribution along the polar angle (Figure 23(a)) it increased with the polar angle in the oxide layer to the maximum value at around 77deg and then dropped to a significantly lower value in the metal layer The transIVR result agreed well with the other results with small discrepancy in the metal layer the DOE study gave a

heat flux of 500 MWm2 the INEEL study of 400 MWm2 and the transIVR of somewhere in between Figure 23(b) shows the predicted corium crust thickness distribution along the RPV polar angle also agreed well the DOE and INEEL

transIVR ANSYS Mechanical

Transient

thermal loads

42

studies The crust thickness decreased with polar angle in the oxide layer and no crust was formed in the metal layer As the crust formation is closely related to the heat flux distribution the crust thickness profile was reasonable regarding the heat flux distribution shown in Figure 23(a) Vessel wall ablation would occur due to the high thermal loads Figure 23(c) gives the profile of the remaining vessel wall thickness along the RPV polar angle considering the wall ablation It shows that no ablation occurred in low angle region (0deg~36deg) where heat flux was low and the wall thickness changed from 15 cm to around 6 cm as angle increased in high angle region of the oxide layer Because of the discrepancy in the heat flux of metal layer the predicted wall thicknesses also differ from those of the DOE and INEEL studies the thinnest wall was 73 cm predicted by the DOE study corresponding to the maximum heat flux followed by 88 cm of the present transIVR calculation and 94 cm of the INEEL study Generally speaking the prediction of transIVR is in an acceptable level compared with the other studies for the same benchmark case

Figure 23 Simulation results of the UCSB FIBS benchmark case [74]

(a) heat flux distribution (b) crust thickness distribution

(c) wall thickness distribution

43

532 LIVE-7V experiment

LIVE-7V is an experiment under the LIVE program conducted in KIT to study the core melt phenomena As Figure 24(a) shows it employed the LIVE-3D hemispherical facility with an inner diameter of 1 m and a wall thickness of ~25 mm The non-eutectic binary salt mixture of 20 mol NaNO3- 80 mol KNO3 was used as the simulant material The decay heat in the melt was simulated by cable-type heating with 8 planes of electrical resistance heating wires Water cooling was applied to the top and vessel surfaces Top surface cooling was realized by a cooling lid mounted at height 413 mm from the vessel bottom and vessel cooling by the flow channel between the test vessel and the outer cooling vessel The inlet and outlet water temperature for both top and vessel cooling surfaces were kept ~293 K and ~303 K throughout the whole experiment respectively A 413 mm height pool corresponds to a total volume of ~0194 m3 simulant material The experiment procedure can be divided into four stages with the heating power sequentially shifted from 29 kW to 24 kW 18 kW and 9 kW Figure 24(b) indicates the thermocouple positions in experiment

The simulation results of LIVE-7V using transIVR were compared with experimental data [80] and numerical study using REALPSCDAPSIM code [81] For simplicity the RELAPSCDAPSIM results in the following are labelled as lsquoCOUPLErsquo which is the name of the module in RELAPSCDAPSIM for melt pool heat transfer

Figure 24 LIVE test facility [80] and thermocouple positions [82]

Figure 25 compares the predicted heat removal upward to the top lid and downward to the bottom vessel surface with those of the experiment and COUPLE The black curve indicates the total input power at the four stages The experimental results (red and blue dots) were calculated from the Table 3 of [80] assuming no extra heat losses For the first three stages the transIVR predictions agreed well with the COUPLE results while the predicted upward heat flow was slightly lower and the downward heat flow higher than the experimental data For the last stage the transIVR agreed well with the experimental result while the COUPLE predicted lower upward and higher downward heat flows The short

(a) (b)

44

transition periods between two stages with different total inputs taken in transIVR also agreed with those in the COUPLE predictions

Figure 25 Comparison of heat removal upward to the top melt surface cooling lid and downward to bottom vessel wall [74]

Figure 26 shows the comparison of heat flux distribution along the vessel wall All curves show similar trend that a higher total input power would result in a higher heat flux profile As the COUPLE predictions had relatively flatter profiles transIVR predictions generally agreed better with the experimental results especially when total input was high A discrepancy is seen that the minimum value of a curve occurred at angle 0degfor both transIVR and COUPLE and at angle 30deg for the experimental data It is noted that the models in default assume heat flux monotonically increases with angle (cf Eq (37) and (38)) resulting it impossible to predict higher heat flux at angle 0deg than angle 30deg as in experimental results

45

Figure 26 Heat flux profiles along the vessel polar angle [74]

533 FOREVER-EC2 experiment

In this section we performed calculations against the FOREVER-EC2 experiment to validate the performance of the coupled transIVR and ANSYS Mechanical in assessing the RPV integrity The applied boundary conditions were identical to the settings of previous coupling approach in Section 4 (ANSYS Fluent+ Mechanical) as indicated in Figure 11 The initial bulk temperature of the melt pool was set to be 1261 K in transIVR which corresponds to the maximum value of the initial temperature field in CFD calculation of previous coupling approach Both the experimental data and the simulation results using previous coupling approach are included for comparison In the following results from current hybrid approach are labelled lsquotransIVRrsquo (and referred as transIVR results) results from previous approach labelled lsquoCFDrsquo (and referred as CFD results) and results from experiment labelled lsquoexperimentrsquo

Figure 27 compares the bulk temperature predicted by transIVR with the averaged and maximum temperatures of the CFD results All curves show the same trend that the temperature gradually increased with time till reaching a steady state and then kept unchanged till the end The times needed for reaching the steady state were ~5000 s for the CFD and ~4000 s for the transIVR which are close to each other It indicates again a good performance of the transIVR in predicting the transient behavior of melt pool Compared with the steady-state averaged temperature 1478 K and maximum temperature 1609 K of the CFD results the steady-state temperature of transIVR 1599 K is a reasonable prediction

0

3000

6000

9000

12000

15000

18000

0 20 40 60 80 100

Hea

t fl

ux

(W

m2)

Polar angle (deg)

29 kW-EXP24 kW-EXP18 kW-EXP9 kW-EXP29 kW-COUPLE24 kW-COUPLE18 kW-COUPLE9 kW-COUPLE29 kW-transIVR24 kW-transIVR18 kW-transIVR9 kW-transIVR

46

Figure 27 Temperature evolution with time [74]

Figure 28 Temperature fields at time (a) 01 s (b) 3038 s (c) 6038 s and (d) 13289 s [74]

Figure 28 shows the temperature fields of the vessel at time 01 s 3038 s 6038 s and 13289 s The temperature fields were visualized in the ANSYS Mechanical It also indicates that the transient temperature loads were properly transferred from transIVR to ANSYS Mechanical Figure 28(a-d) show both the minimum and maximum temperatures increased with time In the lower head region the temperature gradually increased with the height The maximum temperature occurs at the location slightly lower than the melt pool top surface This is because thermal loadheat flux imposed on the vessel generally increases with polar angle in the pool natural convection so the code transIVR inherently also predicted higher heat flux at higher angle positions The higher flux at higher position then correspondingly resulted in higher temperature In the cylindrical region the

(a) (b) (c) (d)

[K] [K] [K] [K]

47

temperature decreased as the height since the bottom part would be more significantly heated by the high-temperature lower head region

Figure 29 shows the steady-state temperature profiles of the external vessel surface along the polar angle The transIVR result agreed well with other two results in trend that the temperature increased with the angle and reach the peak values at angle ~80deg then gradually decreased with angle But slightly lower temperature was predicted in the low angle region Possible reason may be that the local heat flux distribution determined by Eq (37) and (38) is steeper than that in the experiment and CFD study which then resulted in a steeper increase of temperature along the angle and a lower temperature at low angle region In the cylindrical part (gt90deg) the transIVR result generally agreed with the experimental and CFD results The discrepancy with the experimental data may be caused by the simplified boundary conditions the ambient temperature inside the vessel was set to uniform constant value while it could decrease as height Therefore predicted heat loss in the higher position of cylindrical part could be smaller than that with height-dependent ambient temperature resulting in higher vessel temperature Overall the prediction was reasonable for a lumped-parameter model

Figure 29 Temperature profiles of the external vessel surface along the vessel polar angle [74]

The transIVR predicted a failure time of 24959 s which is close to 24180 s in experiment and 23959 s in the CFD results Figure 30 shows the total deformation of vessel bottom point with time All three curves increased gradually with time and experienced drastic accelerations at the end The

400

600

800

1000

1200

1400

0 40 80 120 160

Tem

per

ature

(K

)

Polar angle (deg)

transIVRExperimentCFDWeilding Line

48

transIVR curve agreed well with the CFD curve The final total deformation at failure time is also close to that of the CFD result but lower than the experiment data The discrepancy might be related to the simplifications in the boundary conditions accuracy of the thermal load prediction and the modelling errors in plasticity and creep

Figure 30 Total deformation in vertical direction of the vessel bottom point [74]

Figure 31 shows the profile of wall thickness change The transIVR curve was compared with the CFD curve and four experimental curves measured at vessel left right front and back sides The simulation result fitted the experimental and CFD results well in the region 0deg~50deg In the region 50deg~90deg wall thickness gradually decreased with angle till around 75deg and then increased till 90deg The maximum thickness change was -3 mm for the simulation which is close to the experiment left side curve and slightly larger than the -6~ -45 mm of the other curves from experiment and CFD results Concerning possible errors and uncertainties in the modelling and numerical settings the new approach generally gives reasonable predictions

0

5

10

15

20

25

11670 15170 18670 22170 25670

Tota

l def

orm

atio

n (

mm

)

Time (s)

Experiment

CFD

transIVR

49

Figure 31 Wall thickness changes along the vessel polar angle [74]

-6

-5

-4

-3

-2

-1

0

1

0 20 40 60 80 100

Wal

l th

icknes

s ch

ange

(mm

)

Polar angle (deg)

CFD

transIVR

Experiment (left)

Experiment (right)

Experiment (front)

Experiment (back)

50

51

6 Summary and Outlook

Reactor pressure vessel (RPV) is one of the key physical barriers in the defence-in-depth approach of nuclear power safety and its failure is also the transition point of core melt (corium) progression from in-vessel to ex-vessel phase Quantification of RPV failure during a hypothetical severe accident is therefore paramount to the specific qualification of severe accident mitigation strategies as well as the general assessment of corium risk in light water reactors The RPV failure is a problem of structural performance of RPV under the influence of thermal loads imposed by corium melt pool ie it is a transient thermo-fluid-structure coupled problem per se whose solution requires simultaneous modelling of both highly turbulent melt pool heat transfer and large-scale nonlinear RPV creep Due to the complexity of the problem only limited capabilities have been developed in the previous studies under different simplifications and assumptions in models and coupling methods The present study was motivated to advance modelling and coupling approaches for RPV performance analysis considering new understanding of physical phenomena and latest coupling platform of simulation tools

For the new development of coupling approaches and models for physical phenomena related to RPV failure analysis during severe accidents this thesis work was conducted in the following order (i) a new creep model for the RPV steel 16MND5 was developed which features the capability to cover all three creep stages (ii) a new thermo-fluid-structure coupling approach was developed for RPV failure analysis and validated against FOREVER-EC2 experiment (iii) the performance of the new coupling approach using volume loads mapping (VLM) for data transfer was further compared with another approach using surface loads mapping (SLM) (iv) a hybrid coupling approach was proposed for efficient analysis of RPV failure which employs a lumped-parameter code for quick estimate of thermal load and a mechanistic FE code for detailed structural analysis and (v) for the hybrid coupling approach the code transIVR was developed for calculation of transient melt pool heat transfer

The developed creep model for 16MND5 is called lsquomodified theta projection modelrsquo and describes all three stages of a creep process Creep curves are expressed as a function of time with five model parameters 120579119894(119894 = 14 119886119899119889 119898) A dataset for the model parameters was constructed by fitting experimental creep curves into this function and an iterative parameter adjustment process to satisfy the monotonicity assumption of creep strain vs temperature and stress Model parameters for new curves under different temperature and stress loads are directly interpolated from this dataset in contrast to the conventional ldquotheta projection modelrdquo that employs an extra fitting process The new treatment was proven to better accommodate all experimental curves over the wide ranges of temperature and stress loads The model was implemented into the ANSYS Mechanical code The simulation results indicate that the model successfully captured all three creep stages and a good agreement was achieved between each experimental creep curve and corresponding modelrsquos prediction Regarding the model response to transient changes of temperature and stress loads the widely used time hardening and strain hardening models were implemented and

52

performing reasonably well in the new method These properties fulfil the requirements of a creep model for structural analysis

The transient thermo-fluid-structure coupling approach was developed using the ANSYS Fluent code for thermofluid dynamics of the melt pool and ANSYS Structural code for structural mechanics of RPV An extension tool was introduced to realize transient load transfer from ANSYS Fluent to Structural and minimize the user effort Both turbulence models and the effective model PECM can be used for melt pool heat transfer The developed creep model was used for RPV failure analysis of the RPV Since it is insensitive to geometry the coupling approach allows more complex geometries such as a BWR lower head with a forest of penetrations (CRGTs and IGTs) modelled in detail Moreover the coupling approach does not only reflect the transient response of thermo-fluid-structure interaction but also facilitates the implementation of advanced (high-fidelity) models in both fluid mechanics and structural mechanics The validation of the coupling approach against the FOREVER-EC2 experiment showed that the temperature profile vessel failure time total displacement of vessel bottom point and wall thickness profile in the experiment were well predicted

Following the variation of the coupling approach from VLM to SLM for data transfer between ANSYS Fluent and ANSYS Structural the respective coupling approaches were investigated by comparative calculations against the FOREVER-EC2 experiment and a postulated severe accident of a reference BWR For the FOREVER-EC2 experiment both VLM and SLM approaches predicted generally well the total deformation in vertical direction of bottom point and the wall thickness change though the former predicted a slightly later failure time For the reference BWR case good agreements were achieved in predicting the maximum creep strain and the total deformation in vertical direction of bottom point For both experiment and reactor cases the VLM approach predicted slightly later failure times than SLM but the differences were negligible compared to the long time-scale of vessel failure (~104 s) The deformation rate of the vessel wall was slow at the early stage and accelerated with time till a drastic deformation and final vessel failure Generally speaking the VLM and SLM coupling approaches had the similar performance in terms of their predictability of experiment and applicability to reactor case but the VLM was computationally more efficient

In the development of the hybrid coupling approach the transIVR code was developed for quick estimate of transient melt pool heat transfer and determination of thermal loads on RPV transIVR does not only support transient analysis of one- and two- layer melt pool but also the heat transfer in RPV being resolved with 2D finite difference method which captures more spatial thermal details for RPV structural analysis in addition to 1D heat transfer model The capabilities of transIVR in modelling two-layer pool heat transfer and transient pool heat transfer were demonstrated by calculations against the UCSB FIBS benchmark case and the LIVE-7V experiment respectively It was then coupled to the mechanical solver ANSYS Mechanical which together form the hybrid coupling framework for detailed RPV failure analysis The calculation against the FOREVER-EC2 experiment did not only reproduce the major results of the experiment but also had a good agreement with the thermo-fluid-structure

53

coupling approach using CFD This indicated the new coupling approach is capable of capturing the vessel creep failure characteristics and can be employed as an efficient tool addressing the complex vessel failure problem

As a prospective more research topics can be addressed in future either as a continuation of the current work toward an effective tool for vessel failure analysis or from the general view of scientific interest in melt pool heat transfer and RPV failure The following topics are recommended

bull Extended applications to the reactor cases For instance the thermo-fluid-structure coupling approach can be used to study penetration failures in the RPV of an BWR with a detailed representation of penetrations (IGTs and CRGTs)

bull Synthesis or update of heat transfer correlations for melt pool Heat transfer correlations directly benefit the parametric codes such as transIVR in accurate quantification of heat flux distribution around a melt pool So far many empirical correlations have been proposed based on experiments which had a variety of geometries and melt simulants with a wide range of Rayleigh number (Ra) Sensitivity studies have also been performed on the influences of Ra Pa and other material properties Therefore a comprehensive synthesis or update of the existing heat transfer correlations will be a meaningful step for making reasonable extrapolations to the reactor applications with prototypic materials

bull Modelling of pool stratification The motivation starts with the improvement of the PECMrsquos modelling capability such that the dynamic stratification and resulting pool heat transfer can be modelled altogether Simulation of melt stratification has been supported in some system codes such as ASTEC using a predefined parametric approach The improvement of PECM can be considered in the similar approach or more mechanistic modelling

bull Development of a structural module for RPV The idea is to develop a simplified module for structural analysis so it can be coupled with transIVR ultimately forming a standalone code for quick estimate of vessel failure of LWRs during postulated severe accidents

bull Extension of vessel analysis capability The current version of tranIVR is limited to melt pool Although the precursory debris bed and melt ejection after vessel failure were not the focus of the present study they are equally important to assessment of corium risk especially for Swedish BWRs where the melt release conditions and history are crucial to qualification of their severe accident mitigation strategies

54

55

Bibliography

[1] IAEA ldquoIAEA Annual Report 2018rdquo IAEA 2019 [Online] Available httpswwwiaeaorgsitesdefaultfilesgcgc63-5pdf

[2] IAEA Safety of Nuclear Power Plants Design Vienna INTERNATIONAL ATOMIC ENERGY AGENCY 2012

[3] M Fischer ldquoGen II and III In-Vessel and Ex-Vessel Melt Retentionrdquo The Short Course on Severe Accident Phenomenology Oct 2017

[4] B R Sehgal Nuclear Safety in Light Water Reactors Severe Accident Phenomenology 1st ed Academic Press 2011

[5] IAEA Approaches and Tools for Severe Accident Analysis for Nuclear Power Plants Vienna INTERNATIONAL ATOMIC ENERGY AGENCY 2008

[6] WENRA ldquoPractical Elimination Applied to New NPP Designs - Key Elements and Expectationsrdquo Sep 2019 [Online] Available httpwwwwenraorgmediafiler_public20191111practical_elimination_applied_to_new_npp_designs_-_key_elements_and_expectations_-_for_issuepdf

[7] P Matejovic M Barnak M Bachraty and L Vranka ldquoASTEC applications to VVER-440V213 reactorsrdquo Nuclear Engineering and Design vol 272 pp 245ndash260 2014 doi 101016jnucengdes201308077

[8] AREVA ldquoUS EPR Severe Accident Evaluationrdquo Topical Report ANP-10268NP Oct 2006 [Online] Available httpswwwnrcgovdocsML0631ML063100157pdf

[9] C T Martin ldquoCoupled 3D Thermo-mechanical Analysis of Nordic BWR Lower Head Failure in case of Core Melt Severe Accidentrdquo master thesis KTH Stockholm 2013

[10] IAEA ldquoAdvanced Large Water Cooled Reactors A supplement to the IAEArsquos Advanced Reactor Information System (ARIS)rdquo Sep 2015 [Online] Available httpsarisiaeaorgPublicationsIAEA_WRC_Bookletpdf

[11] B Pershagen Light Water Reactor Safety Pergamon Press 1989 [12] Ninos Garis M Agrell H Glanneskog and L Agrenius ldquoAPRI 7 ndash Accident

Phenomena of Risk Importancerdquo SSM SSM 201212 2012 [Online] Available httpswwwstralsakerhetsmyndighetensecontentassets2a9f299f46b949b3b209e780cf71436a201212-apri-7-accident-phenomena-of-risk-importance-en-lagesrapport-om-forskningen-inom-omradet-svara-haverier-under-aren-2009-2011-

[13] F Fichot et al ldquoSome considerations to improve the methodology to assess In-Vessel Retention strategy for high-power reactorsrdquo Annals of Nuclear Energy vol 119 pp 36ndash45 2018 doi 101016janucene201803040

[14] T G Theofanous C Liu S Additon S Angelini O Kymaumllaumlinen and T Salmassi ldquoIn-vessel coolability and retention of a core meltrdquo Nuclear Engineering and Design vol 169 no 1 pp 1ndash48 1997 doi 101016S0029-5493(97)00009-5

[15] C Sainte Catherine ldquoTensile and creep tests material characterization of pressure vessel steel(16MND5) at high temperatures (20 up to 1300 degC)rdquo CEA France SEMTLISNRT98-009A 1998

56

[16] E Altstadt H-G Willschuetz B R Sehgal and F-P Weiss ldquoModelling of in-vessel retention after relocation of corium into the lower plenumrdquo Forschungszentrum Rossendorf FZR-437 2005

[17] H-G Willschuumltz and E Altstadt ldquoGeneration of a High Temperature Material DataBase and its Application to Creep Tests withFrench or German RPV-steelrdquo FORSCHUNGSZENTRUM ROSSENDORF FZR-353 ISSN 1437-322X Aug 2002

[18] J L Rempe S A Chavez and G L Thinnes ldquoLight water reactor lower head failure analysisrdquo Nuclear Regulatory Commission NUREGCR--5642 1993 Accessed Aug 23 2019 [Online] Available httpinisiaeaorgSearchsearchaspxorig_q=RN25025905

[19] S Brosi et al ldquoCORVIS Investigation of light water reactor lower head failure modesrdquo Nuclear Engineering and Design vol 168 no 1 pp 77ndash104 1997 doi 101016S0029-5493(96)00008-8

[20] T Y Chu M M Pilch J H Bentz J S Ludwigsen W Y Lu and L L Humphries ldquoLower Head Failure Experiments and Analysesrdquo Sandia National Laboratories Albuquerque NM USA NUREGCR-5582 SAND98-2047 1999

[21] L L Humphries et al ldquoOECD Lower Head Failure Project Final Reportrdquo Sandia National Laboratories Albuquerque NM 87185-1139 2002

[22] B R Sehgal R R Nourgaliev T N Dihn A Karbojian J A Green and V A Bui ldquoFOREVER Experiments on Thermal and Mechanical Behavior of a Reactor Pressure Vessel during a Severe Accidentrdquo in Proceedings of the Workshop on in-vessel core debris retention and coolability Garching near Munich Germany Mar 3-6 1998

[23] S Bhandari M Fischer B R Sehgal and A Theerthan ldquoPost Test Analysis of the FOREVER C2 Experimentrdquo The 9th International conference on Nuclear Engineering (ICONE-9) Nice France Apr 8-12 2001

[24] J F Mao J W Zhu S Y Bao L J Luo and Z L Gao ldquoCreep deformation and damage behavior of reactor pressure vessel under core meltdown scenariordquo International Journal of Pressure Vessels and Piping vol 139ndash140 pp 107ndash116 2016 doi 101016jijpvp201603009

[25] J Mao L Hu S Bao L Luo and Z Gao ldquoInvestigation on the RPV structural behaviors caused by various cooling water levels under severe accidentrdquo Engineering Failure Analysis vol 79 pp 274ndash284 2017 doi 101016jengfailanal201704029

[26] J F Mao J W Zhu S Y Bao L J Luo and Z L Gao ldquoStudy on structural failure of RPV with geometric discontinuity under severe accidentrdquo Nuclear Engineering and Design vol 307 pp 354ndash363 2016 doi 101016jnucengdes201607027

[27] B R Sehgal V A Bui T N Dinh and R R Nourgaliev ldquoHeat Transfer Processes in Reactor Vessel Lower Plenum During Late Phase of In-Vessel Core Melt Progressionrdquo in Advances in Nuclear Science and Technology J Lewins M Becker R W Albrecht E J Henley J D McKean K Oshima A Sesonske H B Smets and C P L Zaleski Eds Boston MA Springer US 1999 pp 103ndash135

[28] C-T Tran and T-N Dinh ldquoThe effective convectivity model for simulation of melt pool heat transfer in a light water reactor pressure vessel lower head Part I Physical processes modeling and model implementationrdquo Progress

57

in Nuclear Energy vol 51 no 8 pp 849ndash859 2009 doi 101016jpnucene200906007

[29] C-T Tran and T-N Dinh ldquoThe effective convectivity model for simulation of melt pool heat transfer in a light water reactor pressure vessel lower head Part II Model assessment and applicationrdquo Progress in Nuclear Energy vol 51 no 8 pp 860ndash871 2009 doi 101016jpnucene200906001

[30] H-G Willschuumltz E Altstadt B R Sehgal and F-P Weiss ldquoCoupled thermal structural analysis of LWR vessel creep failure experimentsrdquo Nuclear Engineering and Design vol 208 no 3 pp 265ndash282 2001 doi 101016S0029-5493(01)00364-8

[31] E Altstadt and T Moessner ldquoExtension of the ANSYS creep and damage simulation capabilitiesrdquo Forschungszentrum Rossendorf Technical report FZR-296 2000

[32] H-G Willschuumltz E Altstadt B R Sehgal and F-P Weiss ldquoRecursively coupled thermal and mechanical FEM-analysis of lower plenum creep failure experimentsrdquo Annals of Nuclear Energy vol 33 no 2 pp 126ndash148 2006 doi 101016janucene200508006

[33] H-G Willschuumltz E Altstadt B R Sehgal and F-P Weiss ldquoSimulation of creep tests with French or German RPV-steel and investigation of a RPV-support against failurerdquo Annals of Nuclear Energy vol 30 no 10 pp 1033ndash1063 2003 doi 101016S0306-4549(03)00036-7

[34] H Madokoro A Miassoedov and T Schulenberg ldquoA thermal structural analysis tool for RPV lower head behavior during severe accidents with core meltrdquo Mechanical Engineering Letters vol 4 pp 18-00038-18ndash00038 2018 doi 101299mel18-00038

[35] W Villanueva C-T Tran and P Kudinov ldquoCoupled thermo-mechanical creep analysis for boiling water reactor pressure vessel lower headrdquo Nuclear Engineering and Design vol 249 pp 146ndash153 2012 doi 101016jnucengdes201107048

[36] W Villanueva C T Tran and P Kudinov ldquoA Computational Study on Instrumentation Guide Tube Failure During a Severe Accident in Boiling Water Reactorsrdquo in The 14th International Topical Meeting on Nuclear Reactor Thermal hydraulics NURETH-14 Toronto Ontario Canada Sep25-30 2011

[37] W Villanueva C T Tran and P Kudinov ldquoAssessment with Coupled Thermo-Mechanical Creep Analysis of Combined CRGT and External Vessel Cooling Efficiency for A BWRrdquo The 14th International Topical Meeting on Nuclear Reactor Thermal hydraulics NURETH-14 Toronto Ontario Canada Sep 25-30 2011

[38] F R Larson and J Miller ldquoA Time-Temperature Relationship for Rupture and Creep Stressesrdquo Trans ASME vol 74 p 765minus775 1952

[39] C T Tran ldquoDevelopment validation and application of an effective convectivity model for simulation of melt pool heat transfer in a light water reactor lower headrdquo Licentiate thesis KTH Stockholm 2007

[40] ANSYS ldquoANSYS Fluent Theory Guide Release 162rdquo ANSYS Inc Jul 2015 [Online] Available httpwwwansyscom

[41] P Zeng Finite element analysis and application (in Chinese) Tsinghua University Press amp Springer 2004

58

[42] ANSYS ldquoANSYS Mechanical | Finite Element Analysis Softwarerdquo httpswwwansyscomproductsstructuresansys-mechanical-enterprise (accessed Sep 26 2019)

[43] ANSYS ldquoANSYS Mechanical APDL Theory Reference Release 162rdquo ANSYS Inc Jul 2015 [Online] Available httpwwwansyscom

[44] U Steinberner and H H Reineke ldquoTurbulent Buoyancy Convection Heat Transfer with Internal Heat Sourcesrdquo in Proceedings of the 6th Int Heat Transfer Conference Toronto Canada 1978 vol Vol2 pp 305ndash310 doi 101615IHTC63350

[45] R W Evans and B Wilshire Introduction to Creep London Institute of Materials 1993

[46] DoITPoMS ldquoCreep Deformation of Metalsrdquo Creep Deformation of Metals University of Cambridge 2006 httpswwwdoitpomsacuktlplibcreepindexphp (accessed Jul 09 2019)

[47] ANSYS ANSYS Mechanical APDL Material Reference Release 162 2015 [48] SIMULIA Abaqus 612 Analysis Userrsquos Manualvol 3 Materials Chapter

2324 2012 [49] S Imaoka ldquoUser Creep Subroutine STI0704A ANSYS Release110rdquo 2007 [50] R W Evans J D Parker and B Wilshire ldquoThe θ projection conceptmdashA

model-based approach to design and life extension of engineering plantrdquo International Journal of Pressure Vessels and Piping vol 50 no 1 pp 147ndash160 1992 doi 1010160308-0161(92)90035-E

[51] ECCC ldquoRecommendations and Guidance for the Assessment of Creep Strain and Creep Strength Data ECCC Recommendations Volume 5 Part 1 b[Issue 2]rdquo 2013

[52] W D Day and A P Gordon ldquoLife Fraction Hardening Applied to a Modified Theta Projection Creep Model for a Nickel-Based Super-Alloyrdquo The ASME Turbo Expo 2014 Turbine Technical Conference and Exposition Duumlsseldorf Germany Jun16-20 2014 vol 7A-Structures and Dynamics p V07AT29A010 doi 101115GT2014-25881

[53] M Kumar I V Singh B K Mishra S Ahmad A Venugopal Rao and V Kumar ldquoA Modified Theta Projection Model for Creep Behavior of Metals and Alloysrdquo J of Materi Eng and Perform vol 25 no 9 pp 3985ndash3992 2016 doi 101007s11665-016-2197-y

[54] P Yu and W Ma ldquoA modified theta projection model for creep behavior of RPV steel 16MND5rdquo Journal of Materials Science amp Technology vol 47 pp 231ndash242 2020 doi 101016jjmst202002016

[55] ANSYS ANSYS Mechanical Advanced Nonlinear Materials v160 Lecture 2 Rate Dependent Creep ANSYS training materials 2015

[56] Q Y Li N C Wang and D Y Yi Numerical Analysis (in Chinese) 5th ed Tsinghua University Press 2008

[57] T N Dinh R R Nourgaliev and B R Sehgal ldquoOn heat transfer characteristics of real and simulant melt pool experimentsrdquo Nuclear Engineering and Design vol 169 no 1 pp 151ndash164 1997 doi 101016S0029-5493(96)01283-6

[58] O Kymaumllaumlinen H Tuomisto O Hongisto and T G Theofanous ldquoHeat flux distribution from a volumetrically heated pool with high Rayleigh numberrdquo

59

Nuclear Engineering and Design vol 149 no 1ndash3 pp 401ndash408 1994 doi 1010160029-5493(94)90305-0

[59] F J Asfia and V K Dhir ldquoNatural convection heat transfer in volumetrically heated spherical poolsrdquo 1994 Accessed Aug 23 2019 [Online] Available httpinisiaeaorgSearchsearchaspxorig_q=RN26037727

[60] M Jahn and H H Reineke ldquoFree Convection Heat Transfer with Internal Heat Sources Calculations and Measurementsrdquo in Proceedings of the 5th Int Heat Transfer Conference Tokyo Japan 1974 vol Vol3 paper NC-28

[61] M Gustafsson ldquoFSI Transient extension V1704rdquo EDR MEDESO 2016 [62] EDR MEDESO httpsedrmedesocom [63] H Kim and I Namgung ldquoThermo-mechanical Analysis of RVLH for

APR1400 during a Severe Accidentrdquo in Transactions SMiRT-23 Manchester United Kingdom Aug10-14 2015 [Online] Available httpwwwlibncsueduresolver18402034197

[64] P Yu and W Ma ldquoComparative Analysis of Reactor Pressure Vessel Failure using Two Thermo-Fluid-Structure Coupling Approachesrdquo submitted to journal

[65] F R Menter ldquoTwo-equation eddy-viscosity turbulence models for engineering applicationsrdquo AIAA Journal vol 32 no 8 pp 1598ndash1605 1994 doi 102514312149

[66] T N Dinh and R R Nourgaliev ldquoTurbulence modelling for large volumetrically heated liquid poolsrdquo Nuclear Engineering and Design vol 169 no 1 pp 131ndash150 1997 doi 101016S0029-5493(96)01281-2

[67] ANSYS Inc ldquoIntroduction to ANSYS Fluent Lecture 7 Turbulence Modelling Fluent Tutorial Materialrdquo 2016

[68] A Horvat and M Borut ldquoNumerical investigation of natural convection heat transfer in volumetrically heated spherical segmentsrdquo in Proceedings of the International Conference Nuclear Energy for New Europe 2004

[69] Y Li CFD Pre-test Analysis of SIMECO-2 Experiment 2016 [70] A Theerthan A Karbojian and B R Sehgal ldquoEC-FOREVER experiments

on thermal and mechanical behavior of a reactor pressure vessel during a severe accident Technical report-1 EC-FOREVER-2 testrdquo SAM-ARVI-D008 Mar 2001

[71] P Yu W Ma W Villanueva A Karbojian and S Bechta ldquoValidation of a thermo-fluid-structure coupling approach for RPV creep failure analysis against FOREVER-EC2 experimentrdquo Annals of Nuclear Energy vol 133 pp 637ndash648 2019 doi 101016janucene201906067

[72] T G Theofanous C Liu S Additon S Angelini O Kymaumllaumlinen and T Salmassi ldquoIn-vessel coolability and retention of a core meltrdquo Nuclear Engineering and Design vol 169 no 1ndash3 pp 1ndash48 1997 doi 101016S0029-5493(97)00009-5

[73] H P Langtangen ldquoSolving nonlinear ODE and PDE problemsrdquo Simula Research Laboratory 2016 [Online] Available httpshplgitgithubionum-methods-for-PDEsdocpubnonlinpdfnonlin-4printpdf

[74] P Yu and W Ma ldquoDevelopment of a Lumped-Parameter Code for Efficient Assessment of In-Vessel Melt Retention Strategy of LWRsrdquo submitted to journal

60

[75] T G Theofanous M Maguire S Angelini and T Salmassi ldquoThe first results from the ACOPO experimentrdquo Nuclear Engineering and Design vol 169 no 1 pp 49ndash57 1997 doi 101016S0029-5493(97)00023-X

[76] S W Churchill and H H S Chu ldquoCorrelating equations for laminar and turbulent free convection from a vertical platerdquo International Journal of Heat and Mass Transfer vol 18 no 11 pp 1323ndash1329 1975 doi 1010160017-9310(75)90243-4

[77] S Globe and D Dropkin ldquoNatural-Convection Heat Transfer in Liquids Confined by Two Horizontal Plates and Heated from Belowrdquo J Heat Transfer vol 81 no 1 pp 24ndash28 1959 doi 10111514008124

[78] T G Theofanous C Liu S Additon S Angelini O Kymaelaeinen and T Salmassi ldquoIn-vessel coolability and retention of a core melt Volume 2rdquo Argonne National Lab DOEID--10460-VOL2 1996 Accessed Apr 18 2019 [Online] Available httpinisiaeaorgSearchsearchaspxorig_q=RN30000459

[79] J L Rempe D L Knudson C M Allison G L Thinnes and C L Atwood ldquoPotential for AP600 in-vessel retention through ex-vessel floodingrdquo United States INEELEXT--97-00779 1997 Accessed Apr 18 2019 [Online] Available httpinisiaeaorgSearchsearchaspxorig_q=RN30025872

[80] A Miassoedov X Gaus-Liu T Cron and B Fluhrer ldquoLive experiments on melt pool heat transfer in the reactor pressure vessel lower headrdquo The 10th International Conference on Heat Transfer Fluid Mechanics and Thermodynamics (HEFAT 2014) Orlando FL Jul 14-16 2014

[81] H Madokoro A Miassoedov and T Schulenberg ldquoAssessment of a lower head molten pool analysis module using LIVE experimentrdquo Nuclear Engineering and Design vol 330 pp 51ndash58 2018 doi 101016jnucengdes201801036

[82] B Fluhrer et al ldquoThe LIVE-L1 and LIVE-L3 experiments on melt behaviour in RPV lower headrdquo Forschungszentrum Karlsruhe GmbH Technik und Umwelt (Germany) Inst fuer Kern- und Energietechnik FZKA--7419 2008 Accessed Nov 12 2019 [Online] Available httpinisiaeaorgSearchsearchaspxorig_q=RN40036867

IV

komplexa geometrier som till exempel i botten paring reaktortanken i en kokvattenreaktor (BWR) med en skog av genomfoumlringar (ledroumlr foumlr styrstavar och instrumentering)

Jaumlmfoumlrande beraumlkningar visar att de tvaring kopplingsmetoderna VLM-baserad och SLM-baserad presterar lika bra naumlr det gaumlller hur de predikterar resultat av FOREVER-EC2 experimentet och deras laumlmplighet foumlr reaktoranalyser Aumlven om SLM-metoden predikterar att tankbrott i baringda fallen intraumlffar lite tidigare saring aumlr skillnaden mellan metoderna foumlrsumbar i jaumlmfoumlrelse med tiden till tankbrott (~ 104 s) VLM-metoden var baumlttre aumln SLM-metoden naumlr det gaumlller beraumlkningseffektiviteten

Ideacuten om hybridkoppling garingr ut paring att anvaumlnda en lumped-parameter kod foumlr snabb uppskattning av termiska belastningar som senare kan anvaumlndas i detaljerad strukturmekanisk analys En saringdan kopplingsmetod kan paring ett signifikant saumltt oumlka effektiviteten av beraumlkningar vilket aumlr viktig i reaktorapplikationer daumlr mekanistisk simulering av konvektion i en smaumlltpoumll aumlr beraumlkningsintensiv och onoumldig I detta syfte utvecklades koden transIVR Koden kan dels ge en snabb uppskattning av transient vaumlrmeoumlverfoumlring i en smaumlltpoumll som bestaringr av en eller tvaring materialskikt och dels loumlsa vaumlrmeledningsproblem i reaktortanksvaumlggen med 2D finita differensmetoden vilket genererar detaljerad information om termiska foumlrharingllanden som behoumlvs foumlr strukturmekaniska analyser Foumlr att demonstrera att transIVR kan analysera dels vaumlrmeoumlverfoumlring i en smaumlltpoumll bestaringende av tvaring materialskikt och dels transient vaumlrmeoumlverfoumlring i en smaumlltpoumll jaumlmfoumlrdes transIVR prediktioner med UCSB FIBS-benchmarkfallet respektive LIVE-7V-experimentet I naumlsta steg kopplades transIVR-koden till ANSYS Structural foumlr att moumljliggoumlra mer detaljerad analys av reaktortankbrott Validering mot FOREVER-EC2 experimentet tyder paring att den utvecklade kopplingsmetodiken korrekt aringterger tankbrott orsakat av krypning

Nyckelord reaktortankbrott koppling termohydraulik-strukturmekanik modellering av krypning vaumlrmeoumlverfoumlring i smaumlltpoumll svaringra haverier CFD FEA transIVR

V

List of Publications

Papers included in this thesis

I P Yu and W Ma ldquoA modified theta projection model for creep behavior of RPV steel 16MND5rdquo Journal of Materials Science amp Technology vol 47 pp 231ndash242 2020

II P Yu W Ma W Villanueva A Karbojian and S Bechta ldquoValidation of a thermo-fluid-structure coupling approach for RPV creep failure analysis against FOREVER-EC2 experimentrdquo Annals of Nuclear Energy vol 133 pp 637ndash648 2019

III P Yu and W Ma ldquoComparative analysis of reactor pressure vessel failure using two thermo-fluid-structure coupling approachesrdquo Nuclear Engineering and Design submitted

IV P Yu and W Ma ldquoDevelopment of a lumped-parameter code for efficient assessment of in-vessel melt retention strategy of LWRsrdquo Annals of Nuclear Energy submitted

Respondentrsquos contribution to the papers included in the thesis

Peng Yu made major contributions to the publications of all the above four articles under the guidance of supervisors and with the collaboration of other co-authors including formulation of research questions development of coupling approaches and numerical methods analysis and generalization of simulation results as well as writing the first drafts and addressing review comments

Papers not included in this thesis

1 P Yu and W Ma ldquoRPV creep failure analysis using a Norton creep modelrdquo International Journal of Mechanical Sciences to be submitted

2 P Yu W Villanueva S Galushin W Ma and S Bechta ldquoCoupled thermo-mechanical creep analysis for a Nordic BWR lower head using non-homogeneous debris bed configuration from MELCORrdquo 12th International Topical Meeting on Nuclear Reactor Thermal-Hydraulics Operation and Safety (NUTHOS-12) Qingdao China October 14-18 2018

3 P Yu A Komlev Y Li W Villanueva W Ma and S Bechta ldquoPre-test simulation of SIMECO HT experiments for stratified melt pool heat transferrdquo 11th International Topical Meeting on Nuclear Reactor Thermal Hydraulics Operation and Safety (NUTHOS-11) Gyeongju Korea October 9-13 2016

VI

VII

Acknowledgements

First of all I would like to express my sincere gratitude to my supervisor Prof Weimin Ma for offering this opportunity to work in the severe accident research field and for the supports patience guidance discussions and encouragements throughout the entire period of my graduate study

I am very thankful to my co-supervisor Dr Walter Villanueva for the valuable help and discussions I learnt a lot during the times working with him I am also thankful to co-supervisor Prof Sevostian Bechta who is supportive and always has inspiring ideas that keep you thinking for a long time

Thank you also to all my colleagues at the Division of Nuclear Power Safety for the supports discussions and the very nice working environment during these years

I have taken many interesting courses during my PhD study for which I want to express my thanks to all the teachers With this regard a special thanks goes to Prof Johan Hoffman for his SF2561 Finite Element Method given in 2015 fall as well as the discussions and advices on numerical analysis

I am grateful to Ignacio Gallego-Marcos and Wenyuan Fan for their enthusiastic help on CFD and turbulence modelling during the time working on the turbulent

heat transfer I thank all my 师兄 for sharing their knowledge on nuclear engineering and giving advices when I was confused since 2012

I thank all my friends both in Sweden and in China for the company and the sharing of many wonderful moments in daily life

Finally I would like to express my deepest thank to my parents my girlfriend Siwen my sisters and all my other relatives for their unconditional supports and love

Financial supports from the China Scholarship Council and APRI make my study in KTH possible and are acknowledged

VIII

IX

Table of Contents

AbstracthelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphellipI

SammanfattninghelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphellipIII

List of PublicationshelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphellipV

AcknowledgementshelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphellipVII

NomenclaturehelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphellipXI

1 Introductionhelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphellip1

11 Nuclear Power Safety 1

12 RPV Failure 2

13 Literature Review Creep Behaviour and RPV Failure Analysis 4

131 Experiments on creep behaviour and RPV failure 4

132 Numerical studies on RPV failure 5

14 Motivation Objectives and Main Achievements 7

2 Numerical Solvers and Modelshelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphellip9

21 ANSYS Fluent 9

22 ANSYS Mechanical 9

23 PECM 11

3 Creep Modelling of 16MND5 Steelhelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphellip13

31 Creep and Creep Modelling 13

32 A Modified Theta Projection Model 14

321 Original theta projection model and its variations 14

322 A modified theta projection model 16

4 Coupled Thermo-Fluid-Structure Analysis of RPV Failurehelliphelliphelliphellip23

41 Thermo-Fluid-Structure Coupling Approach 23

42 Numerical Modelling 25

421 Conjugate heat transfer 25

422 Mechanical analysis 25

43 Validation against the FOREVER-EC2 Experiment 26

431 FOREVER-EC2 experiment 26

X

432 Conjugate heat transfer 27

433 Mechanical analysis 29

44 Application to BWR Case 32

45 Computational Efficiency 34

5 Hybrid Coupling Approach for Efficient RPV Failure Analysishelliphellip35

51 A Lumped-parameter Code for Melt Pool Heat Transfer 35

511 Heat transfer correlations 36

512 Energy balance in the pools 37

513 Energy balance in the RPV wall 39

52 Hybrid Coupling Approach 40

53 Verification and Validation 41

531 UCSB FIBS benchmark case 41

532 LIVE-7V experiment 43

533 FOREVER-EC2 experiment 45

6 Summary and Outlookhelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphellip51

Bibliographyhelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphellip55

XI

Nomenclature

Latin

119860 area of a surface

119862119901 heat capacity

119863 damage factor

119892 gravity

ℎ heat transfer coefficient

H Melt pool height

119896 thermal conductivity

119871 characteristic length

119873119906 Nusselt number

119875119903 Prandtl number

119902 heat flux

119902119894119899 heat flux through inner vessel surface

119876119881 internal volumetric heat source

119903 radial distance in a cylindrical or spherical coordinate

119877119886 external Rayleigh number

119877119886119894119899 internal Rayleigh number

119877119886119910 local Rayleigh number

119905 time

119879 temperature

119881119900119897 total volume

119909 thickness (of crust or RPV wall)

119911 height in a cylindrical coordinate

Greek

120572 thermal diffusivity

120573 thermal expansion

Δ119879 temperature difference

ε emissivity (in thermal analysis) strain (in structural analysis)

휀119888119903 creep strain

휀119903 creep strain rate

120579 polar angle

120579119894 parameters of a creep model 119894 = 1234119898

120584 kinematic viscosity

σ stress

XII

Super and subscripts

119886119898119887 ambient

119888119903 creep (in Section 3) crust (in Section 5)

119889119899 downward

119890119899119907 environment

119894 inner surface of RPV

119894119899119905 interface

119897119900119888119886119897 local

119898119886119909 maximum value

119898119890119897119905119894119899119892 melting point

119898119905 metal layer (bulk)

119898119905119889119899 downward surface of metal pool

119898119905119904119889 sideward surface of metal pool

119898119905119905119900119901 top surface of metal layer

119898119905119906119901 upward surface of metal pool

119900 outer surface of RPV

119900119909 oxide layer (bulk)

119900119909119889119899 downward surface of oxide layer

119900119909119906119901 upward surface of oxide layer

119903119890119891 reference

119904 internal structures inside the RPV

119904119889 sideward

119906119901 upward

119907119890119904 vessel

Acronyms

BWR boiling water reactor

CBD creep database

CFD computational fluid dynamics

CRGT control rod guide tube

ECCM effective convectivity conductivity model

FDM finite difference method

FE finite element

FEA finite element analysis

FEM finite element method

FVM finite volume method

IGT instrument guide tube

IVR in-vessel retention

XIII

IVMR in-vessel melt retention EU project

LES large eddy simulation

MPS moving particle semi-implicit method

NPP nuclear power plant

PECM phase-change effective convectivity model

RANS Reynolds-averaged Navier-Stokes

RPV reactor pressure vessel

SBO station blackout

SLM surface loads mapping

SPH smoothed-particle hydrodynamics

UDF user-defined function

UPF user programmable features

VLM volume loads mapping

XIV

1

1 Introduction

11 Nuclear Power Safety

Greenhouse gas emissions contribute to the global warming problem Phasing out fossil fuels by switching to low-carbon energy sources is considered as a mitigation strategy With this regard clean energy should be increased in the world energy consumption structure Being clean and stable nuclear energy is an attractive alternative to fossil fuels After the development over decades it also proves to be a mature and safe technology According to IAEA 2018 annual report [1] there were 450 nuclear power reactors under operation worldwide at the end of 2018 which had a record generating capacity of 3964 GW(e) 9 reactors were connected to the grid and 55 reactors under construction around the world

Due to the potential risk of the release of radioactivity nuclear power safety is emphasized throughout the stages of design construction operation and decommission of a nuclear power plant (NPP) The principal technical requirements [2] specify that the three fundamental safety functions should be fulfilled for all plant states of an NPP (i) control of reactivity (ii) removal of heat from the reactor and from the fuel store and (iii) confinement of radioactive material shielding against radiation and control of planned radioactive releases as well as limitation of accidental radioactive releases The defence-in-depth concept is also applied to provide several levels of defence to prevent consequences of accidents The fuel claddings reactor pressure vessel (RPV) and primary coolant pressure boundaries and the containment are the three physical barriers to prevent the release of radioactive materials to the environment

Severe accidents are events that cause significant damage to reactor fuel and result in core meltdown It may be caused by insufficient cooling of the reactor core due to various reasons or by the rapid introduction of huge reactivity which causes sharp increases in the power and fuel fragmentation eg Chernobyl accident During a severe accident the failure of the fuel claddings will result in the radioactive materials previously enclosed inside the cladding released to the primary loop If cooling is not sufficient to remove the decay heat the core materials will melt migrate down and relocate into the RPV lower head after which the melt will react with water and partially form fragments quench solidify and form a debris bed The debris bed formed by the melt may continue to melt to form a melt pool due to the decay heat As the corium contacts the RPV it would continuously heat the RPV and threaten its integrity Once the RPV fails containment becomes the last barrier to prevent the release of radioactivity The released melt would result in an increase of the pressure and temperature due to the high temperature fractions oxidation and hydrogen combustion which is called direct containment heating The melt would fall onto the reactor cavity Without sufficient cooling the molten core would interact with basemat concrete (MCCI) This process can last for few days It can result in containment failure due to either containment overpressure or the basemat melt-through of containment In addition containment is also subjected to isolation failure and early containment failure due to missiles and dynamic pressure [3] More

2

detailed severe accident progression or phenomena could also be found in eg [4] [5]

Though severe accidents are unlikely to occur the consequences can be severe if no proper measure is taken Related researches have been continuously going on aiming at mitigating the accident consequences and improving safety After the Fukushima Accident safety inspections and re-evaluation actions have been carried out worldwide especially on safety measurements regarding severe accidents The concept lsquoPractical Eliminationrsquo applied to new NPP designs demands that new nuclear installations be designed with the objective of mitigating the consequences of accidents and avoiding early radioactive releases and large radioactive releases [6] Severe accident mitigation strategies are applied to the existing and new NPPs (both Generation II and III) The strategies can generally be classified into two categories [3] the in-vessel retention (IVR) and ex-vessel retention (EVR) In simple words in-vessel retention tries to retain the core melt inside the RPV (second barrier) by applying external water cooling on the vessel as shown in Figure 1(a) Such method minimizes the dispersal of the melt within the containment and hence simplifies the later recovery work Examples of the application are AP1000 HPR1000 APR 1400 The idea of the EVR is to let the RPV fail but to retain the core melt inside the containment Two principle approaches of EVR are the retention within a dedicated lsquocore catcherrsquo and melt fragmentation Examples of core catcher would be EPR that directs the melt to a spreading compartment (Figure 1(b)) and VVER1000 that uses metallic crucible below the RPV An example of the melt fragmentation is the Nordic BWRs that the corium released from RPV would relocate in the cavity with a deep-water pool (7-12 m) and is expected to fragment quench and form a coolable debris bed (Figure 1(c))

Figure 1 Examples of IVR [7] EVR with core catcher [8] and melt fragmentation [9]

12 RPV Failure

RPV which contains the reactor core core shroud and coolant is a key component to the reactor system RPV and the primary loop constitute the primary coolant system which conveys the fission heat from the reactor core to the secondary loop through steam generator (PWR) or to the turbine (BWR) The pressure under

(a) IVR (b) Core catcher (c) Melt fragmentation

3

operation can be ~155 MPa for a 1000 MW PWR [10] and ~70 MPa for a 1000 MW BWR [11] It also has to withstand the neutron radiation damage The thickness of RPV [11] can be ~20 cm for PWR and ~16 cm for BWR

As the transition of in-vessel progression to ex-vessel progression RPV failure is important for analysis of accident progression and assessment of the effectiveness of severe accident mitigation strategies In the melt fragmentation strategy of Nordic BWR the melt-coolant interaction and debris bed coolability after the corium poured into the deep-water pool are essentially dependent on the vessel failure mode including the characteristic time of melt release rupture size and location and amount and superheat of melt available for release [12] New concern also arose [13] on the necessity of performing structural analysis for the RPV in the IVR strategy In the initial methodology and further development of the IVR the success of the strategy ie the integrity of the RPV was demonstrated (eg in [14]) with the following two conclusions (i) even for a local heat flux on the vessel as high as critical heat flux (CHF) the resulted very thin external ldquocold layerrdquo of the vessel would be sufficient to withstand the internal stress load and hold the vessel integrity (ii) as CHF is the critical value for the thermal load the main specific goal for assessing the effectiveness of IVR becomes to demonstrate that the local heat flux along RPV be lower than the CHF The new concern doubts that local lsquohot partrsquo of the vessel would have negative effects on the overall resistance due to creep and dilatation which could result in a vessel failure and hence violate the first conclusion in particular for ldquolong termrdquo retention of corium in RPV If so a structural integrity analysis of RPV would be needed in addition to assessing local heat flux especially for high-power reactors Therefore accurate prediction of vessel failure could be meaningful for both strategies

Figure 2 Illustration of RPV under thermal loads from corium

As the structural performance is highly affected by the thermal loads from the corium RPV failure is essentially a transient thermo-fluid-structure coupled problem Figure 2 illustrates the situation that a corium debris bedmelt pool formed in the RPV lower head The decay heat of nuclear fission products could continuously be generated inside the corium leading to the formation of melt pools as the debris melts The heated debris bedmelt pool imposes thermal and

4

mechanical loads on the RPV Such loads can be transient (change with time) and spatially non-uniform Complex phenomena are also involved in this process Strong turbulent natural convection induced by the decay heat may be expected in the melt pool under IVR The corresponding Rayleigh number 119877119886 can be 1016minus17 for prototypic reactor cases Material structural properties can become non-linear under high temperature and stress loads The creep deformation also becomes significant at high temperature Detailed modelling of turbulent natural convection and non-linear properties could be computationally expensive In addition large-scale geometry (hemisphere with a radius of 20-30 m) long-term transient deformation process (~10000 s) and required small time steps (eg 001 s) for numerical convergence consideration further increase the computational costs

13 Literature Review Creep Behaviour and RPV Failure Analysis

In this section the most relevant studies concerning the experiments modelling and analysis of creep behaviour and RPV failure are presented

131 Experiments on creep behaviour and RPV failure

Various experiments have been performed on the RPV failure problem some of which were to generate data of mechanical properties (especially the creep properties) and some to study the overall vessel failure with scaled or simplified geometries

The REVISA experiment [15] generated creep curves and stress strain curves of French steel 16MND5 ranging from 600 to 1300 degC This data is accessible in [16] as well as [17] which also documents other material properties eg density thermal expansion coefficient conductivity heat capacity Youngrsquos Modulus and plasticity Similar experiments were carried out to generate high-temperature creep and tensile test data for American RPV steels and Inconel [18]

The CORVIS experiments [19] investigated the lower head failure modes with and without penetrations Each experiment was carried out on a steel plate 100 mm thick representing a pressure vessel lower head The penetration case represented a boiling water reactor lower head carrying a tube penetration in the form of a drain line

The LHF (lower head failure) experiments [20] and the succeeding OLHF tests [21] investigated the vessel failure with a 15 scaled vessel heated by an electrical radiator allowing to locate the hot focus at different position of the vessel with a maximum wall temperature of around 900 degC Internal pressure ranging from 53 to 123 bar was also imposed to initiate the vessel failure

The FOREVER experiments [22] investigated the phenomena of melt pool convection vessel creep possible failure processes and failure modes occurred during the late phase of the in-vessel progression by heating up a melt pool in a vessel and observing possible vessel failures The study covers cases with and without penetration different melt level existence of gap cooling and in-vessel melt coolability

5

132 Numerical studies on RPV failure

Various numerical approaches have been employed on RPV failure Since a two-way thermo-fluid-structure coupling approach is challenging these approaches take simplifications and can generally be classified into two categories (i) RPV failure decoupled from melt pool heat transfer and (ii) RPV failure coupled with melt pool heat transfer in simplified manners

1) Decoupled approaches In the decoupled approaches only RPV is modelled Simplified constant thermal loads are imposed on RPV considering the melt pool heat transfer

In a post-test analysis of FOREVE-C2 experiment [23] it was treated as a thermo-structural coupled analysis of the vessel with the thermal load estimated by the outer thermocouples and the normal heat flux on the external vessel wall The analysis was done using finite element (FE) code SYSTUS A coupled creepdamage law based on a viscosity-hardening creep law describing primary and secondary creep was employed for creep analysis

Mao et al conducted various studies on reactor cases assessing the IVR strategy using FE code ABAQUS Thermal boundary conditions were imposed on the RPV which was then solved as a thermal-structural coupled problem The strain hardening creep model (Eq (1)) was mainly used Their studies cover the general RPV failure [24] as well as the effects of cooling water levels under IVR [25] and geometric discontinuity (ablated RPV) [26]

ε119888119903 = 1198891σ1198892ε1198893119890119909119901 [minus

1198894119879] (1)

In the EU IVMR project [13] the RPV integrity was also studied through a benchmark calculation which deals with the RPV of a PWR with a prescribed ablated profile with 2D axisymmetric geometry Prescribed temperature gradient was set as thermal load The objective was to calculate deformations for various pressure loads up to vessel rupture A smoother profile was planned in further research

The advantage of such approaches is once thermal loads are reasonably assumed the focus only needs to be put on the RPV The disadvantage is a reasonable thermal load should be consistent with the melt pool heat transfer while sometimes it is not well-noted eg constant uniform temperature boundary conditions were assumed for inner and outer RPV (with uniform thickness in lower head) surfaces [25] which ignored the non-uniform heat flux distribution caused by the turbulent convective corium pool As thermal loads strongly affect the structural performance they should be quantified carefully

2) Coupled approaches In coupled approaches both melt pool heat transfer and RPV failure are modelled and coupled in one-way or simplified manner With melt pool heat transfer modelled the coupled approaches are more flexible to provide reasonable thermal loads on the RPV In existing approaches CFD and the correlation-based distributed-parameter models eg the Effective Convectivity Conductivity Model

6

(ECCM) [27] and the Phase-change Effective Convectivity Model (PECM) [28] [29] are general methods for melt pool heat transfer

Sehgal et al [22] presented an approach that employed the MVITA code to calculate the thermal processes and ANSYS for structural analysis MVITA code employed the ECCM Bailey-Norton creep law was used as shown in Eq (2)

ε119888119903 = 6055 times 10minus26 sdot σ472119905085119890minus

37200

119879 (2)

This approach was used for the pre-test simulation of FOREVER experiment to determine the thermal and mechanical loadings Parametric investigations also helped in the identification of important factors that could affect the test procedure and test conditions

Willschutz et al [30] employed the CFD-module Flotran for the melt pool heat transfer and ANSYS for structural analysis of RPV Standard k-ε turbulence model was selected for the turbulent natural convection in the melt pool A user-defined three-stage creep model [31] was employed In this model experimental creep curves were discretized into a creep database (CDB) under given temperature 119879 and stress 120590 the creep curve was discretized into finite point pairs

(휀 ε) indicating under such thermal and mechanical loads the creep strain rate ε is subjected to creep strain 휀 The creep strain rate for any other case can be interpolated over known temperature stress and strain from the database This method was applied to the post-test calculation of the FOREVE-C2 experiment It was concluded that the unexpected deformation behaviour during the experiment was caused by the combination of the transient temperature field in the vessel wall and the three creep stages

They later replaced the Flotran solver with the ECCM for melt pool heat transfer in a following study [32] By implementing it into FEM-solver a recursively coupled analysis was constructed Moreover a modified creep model [33] was used with damage measures the CDB itself did not contain tertiary creep instead a factor 1(1 minus 119863) was multiplied to the original creep strain rate to get the modified creep strain rate where 119863 is a damage factor (0 lt 119863 lt 1) defined with creep strain information and plastic strain information As 119863 grows close to 1 the modified creep strain rate would be enlarged significantly by this factor 1(1 minus 119863) which then behaves like the accelerating tertiary stage This method was applied to pre- and post- test calculations for the FOREVER experiments proving to be closer to reality than the previous method

Madokoro et al [34] developed a thermal structural analysis tool based on open source solver OpenFOAM and numerically investigated the coupled RPV failure and melt pool convection The PECM was used in the melt pool heat transfer In structural part the plasticity and damage parameters were similar to Willschutz et al [32] except for that strain hardening creep model (Eq (3)) was used instead of a CDB Simulations of FOREVER-EC2 and -EC4 were investigated It was concluded from the simulations that although the emissivity and mesh type (updated or fixed) affect the results significantly the approach generally has the capability of predicting the vessel deformation and failure

7

ε119888119903 = 1198891 sdot σ1198892 sdot ε1198893 (3)

Villanueva et al [35] performed a coupled thermo-mechanical creep analysis of the reactor scale RPV (ABB-Atom design BWR) to evaluate efficacy of the cooling measures for the in-vessel melt coolability and retention The PECM was used for melt pool heat transfer and ANSYS for RPV integrity analysis The modified time hardening model (Eq (4)) of the American RPV steel SA533B was selected as creep model The transient thermal load from the melt pool was applied as a thermal boundary condition to the vessel wall Further studies were done with this approach on investigations of instrumentation guide tube (IGT) failure [36] and control rod guide tube (CRGT) cooling [37] of a Nordic BWR

ε119888119903 =1198881σ

11988821199051198883+1

1198883 + 1 1198881 gt 0 (4)

It is noted that only RPV analysis with finite element analysis (FEA) are reviewed here Engineering methods like the Larson-Miller method [38] are not covered in this thesis One can refer to theory manuals of severe accident codes like MELCOR ASTEC for more details Comprehensive description on modelling of melt pool heat transfer is not covered either One can refer to either [39] for a general view or to any severe accident code manuals for engineering methods

14 Motivation Objectives and Main Achievements

The existing experimental and numerical studies provide us a good understanding of RPV failure involved phenomena and valuable simulation experiences These numerical methods may vary in solvers (which may be related to code accessibility) and models (which may be related to model accessibility understanding of phenomena computational power) However improvements of modelling approaches are still possible as the improvements in any of the related aspects eg

bull Simulation tools are continuously being upgraded and providing new possibilities just like that PECM appeared and ECCM was no longer used

bull Better understanding of the phenomena and more simulation experiences from in-vessel retention and vessel failures researches benefit the further modelling and simulations

bull Increasing computational power allows more detailed modelling eg turbulence modelling of melt pool convection

bull There is still room for the improvement of creep modelling

In addition implementation difficulty of a coupling approach is another important issue to consider Since it is a TRANSIENT thermo-fluid-structure coupled problem it is almost impossible to replicate an approach simply by referring to a paper with limited details For BWRs it would be even better for the coupling approaches to support complex geometries considering forest of penetrations (CRGTs and IGTs) on the RPV lower head without significant increase of user effort in the coupling process

8

The objectives of this thesis are to investigate new approaches and develop new models for the thermo-fluid-structure coupled problem of RPV failure and involved phenomena These objectives are accomplished with the following achievements

1) A new creep model for the 16MND5 steel was developed which covers all three creep stages Good agreement was achieved between model predictions and experimental curves Both time and strain hardening models were implemented (Paper I)

2) A new thermo-fluid-structure coupling approach was developed for RPV failure analysis It supports the advanced models in both melt pool convection and structural mechanics to improve fidelity and facilitates implementation Validation of the approach with the new creep model was performed against FOREVER-EC2 experiment This approach is flexible for complex geometries and can be used in further analysis eg lower head with detailed modelling of penetrations in BWRs (Paper II)

3) The performance of this new approach which uses volume loads mapping for data transfer was compared with the previous approach using surface loads mapping showing good agreement with small discrepancy (Paper III)

4) A lumped-parameter code transIVR for melt pool heat transfer was developed which supports one- and two-layer pool configuration transient analysis and well-resolved RPV wall (Paper IV)

5) A hybrid coupling approach was developed for efficient analysis of RPV failure where transIVR is coupled with ANSYS Mechanical (Paper IV)

In the following general descriptions on the codes ANSYS Fluent Mechanical and the PECM are given in Section 2 the new creep model in Section 3 the new coupling approach as well as its comparison with the previous approach in Section 4 the efficient hybrid coupling approach in Section 5 and summary amp outlook in Section 6

9

2 Numerical Solvers and Models

In this section we briefly introduce the ANSYS Fluent as the CFD solver for melt pool heat transfer the ANSYS Mechanical as the mechanical solver for RPV failure and PECM as an efficient alternative for melt pool heat transfer that are heavily used in the following sections

21 ANSYS Fluent

In the fluid mechanics the fluid is treated as a continuum which is generally described by the conservative mass momentum and energy equations [40]

120597120588

120597119905+ 120571 sdot (120588) = 119878119898 (5)

120597(120588)

120597119905+ 120571 sdot (120588) = minus120571119901 + 120571 sdot (120591) + 120588 + (6)

120597120588119864

120597119905+ 120571 sdot ((120588119864 + 119901)) = 120571 sdot (119896120571119879 minussumℎ119895119869119895

119895

+ (120591 sdot )) + 119878ℎ

(7)

Computational Fluid Dynamics (CFD) studies topics on the numerical solutions of above equations Typical methods are the Finite Volume Method (FVM) Finite Difference Method (FDM) and Finite Element Method (FEM) as well as particle methods that are mesh-free eg SPH and MPS

ANSYS Fluent is a general purpose CFD code which solves above equations with FVM Turbulence modelling and multiphase flows are supported Turbulence can be modelled with RANS LES or hybrid LESRANS Customizations eg adding sourcesink time or spatial dependent boundary conditions are realized through the user-defined functions (UDF) which is programmed in C language

22 ANSYS Mechanical

In the continuum mechanics of solids three sets of variables are considered to describe the structure performance namely element displacement field strain field and stress field (119906 ε σ) A general 3-dimensional problem is described by 3 sets of equations the equilibrium equations (Eq (8)) strain-displacement relationship (Eq (9)) and constitute equations (Eq (10)) [41] For Eq (8) the reciprocal theorem of shear stress also gives τ119894119895 = τ119895119894 where 119894 119895 = 119909 119910 119911

indicating the stress tensor being symmetric For very simple applications eg 1D beam it can be solved analytically For complex problems where analytical solutions are not available approximation methods eg Galerkin weighted residual method principle of virtual work and principle of minimum potential energy would be used

10

partσ119909119909part119909

+partτ119909119910

part119910+partτ119909119911part119911

+ 119909 = 0

partτ119910119909

part119909+partσ119910119910

part119910+partτ119910119911

part119911+ 119910 = 0

partτ119911119909part119909

+partτ119911119910

part119910+partσ119911119911part119911

+ 119911 = 0

(8)

ε119909119909 =part119906

part119909 휀119910119910 =

120597119907

120597119910 휀119911119911 =

120597119908

120597119911

120574119909119910 =120597119906

120597119910+120597119907

120597119909 120574119910119911 =

120597119908

120597119910+120597119907

120597119911 120574119911119909 =

120597119908

120597119909+120597119906

120597119911

(9)

ε119909119909 =1

119864[σ119909119909 minus μ(σ119910119910 + σ119911119911)]

ε119910119910 =1

119864[σ119910119910 minus μ(σ119909119909 + σ119911119911)]

ε119911119911 =1

119864[σ119911119911 minus μ(σ119909119909 + σ119910119910)]

γ119909119910 =1

119866τ119909119910 γ119910119911 =

1

119866τ119910119911 γ119911119909 =

1

119866τ119911119909

(10)

ANSYS Mechanical [42] is a mechanical engineering software that uses finite element analysis (FEA) for structural analysis The principle of virtual work is adopted in the solver [43] The idea of this principle is that a virtual (very small) change of the internal strain energy must be offset by an identical change in external work due to the applied loads which reads

δ119880 = δ119881 (11) where 119880 is the strain energy or the internal work 119881 the external work and δ the virtual operator The virtual internal work can be expressed as

δ119880 = int δεσd(vol)T

vol

+int 119886119903119890119886119891

δ119908119899119879σ119889(119886119903119890119886119891) (12)

where 휀 is the strain vector 120590 stress vector vol volume of element 119908119899 motion normal to the surface 119886119903119890119886119891 area of the distributed resistance The virtual

external work is contributed from three parts the inertial effects the pressure force on the surface and nodal forces applied to the element which then can be expressed as

δ119881 = minusint 119907119900119897

δ119908119879119865119886

119907119900119897119889(119907119900119897) + int

119886119903119890119886119901

δ119908119899119879119875119889(119886119903119890119886119901)

+ δ119906119879119865119890119899119889

(13)

11

where 119908 is the vector of displacements 119865119886 acceleration force vector 119875 applied pressure vector 119886119903119890119886119901 area over which pressure acts and 119865119890

119899119889 nodal

forces applied to the element

Plugging Eq (12) and (13) into (11) would result in the equilibrium equation Since the virtual displacement can be arbitrary the final equations used to describe the equilibrium of an element is reduced to Eq (14) with multiple operations

([119870119890] + [119870119890119891])119906 minus 119865119890

119905ℎ = [119872119890] + 119865119890119901119903 + 119865119890

119899119889

(14)

where [119870119890] [119870119890119891] 119865119890

119905ℎ [119872119890] and 119865119890119901119903 are the element stiffness matrix

foundation stiffness matrix thermal load vector mass matrix acceleration vector and pressure vector respectively It represents the equilibrium equation on a one element basis Alternatively using a minimum total potential energy principle could result in a consistent expression to the principle of virtual work [41]

Customizations eg on material properties are realized through the user programmable features (UPF) which is programmed in FORTRAN language

23 PECM

The PECM [28] [29] developed in KTH is a correlation-based method to model the melt pool heat transfer It is implemented in ANSYS Fluent through UDF and has been validated against many experiments such as ACOPO COPO LIVE etc In the PECM the flow velocities 119906119909 119906119910 and 119906119911 are replaced by characteristic

velocities 119880119909 119880119910 and 119880119911 respectively in order to eliminate the necessity of solving

the Navier-Stokes equation

The characteristic velocities are calculated using heat transfer correlations

119880119906119901 =α

ℎ119901119900119900119897times (119873119906119906119901 minus

ℎ119901119900119900119897

ℎ119906119901) (15)

119880119889119899 =α

ℎ119901119900119900119897times (119873119906119889119899 minus

ℎ119901119900119900119897

ℎ119889119899) (16)

119880119904119889 =α

ℎ119901119900119900119897times (119873119906119904119889 minus

ℎ119901119900119900119897

119882119901119900119900119897) (17)

where ℎ119901119900119900119897 is the melt pool depth ℎ119906119901 the thickness of well mixed layer of the

pool ℎ119889119899 the thickness of lower stratified region of melt pool 119882119901119900119900119897 the width of

pool 120572 the thermal diffusivity The PECM employs reduced characteristic velocities to describe mushy zone convection heat transfer For internally heated pool natural convection the Steinberner-Reineke correlations (Eq (18)~(19)) [44] are employed The profile of sideward 119873119906 is described using Eckertrsquos type correlation (Eq (20)) for a vertical boundary layer where the local Rayleigh

number at pool height 119910 is defined as 119877119886119910 =119892βΔ1198791199103

αν

12

119873119906119906119901 = 03451198771198861198941198990233

(18)

119873119906119889119899 = 13891198771198861198941198990095

(19)

119873119906119904119889 = 050811987511990314 (

20

21+ 119875119903)

minus14

11987711988611991014

(20)

As the characteristic velocities are calculated the energy equation is the only transport equation that needs to be solved and can be expressed as follows

partρ119862119901119879

part119905+ nabla(ρ119862119901119932119879) = 119896nabla2119879 + 119876119907 (21)

where from left to right are the transient term convection term diffusion term and volumetric heat source respectively The convection term then can be calculated explicitly with use of values in the last time step and then can be treated together with the volumetric source term Therefore the PECM finally simplifies the energy equation to a conduction equation which can be easily solved and as a result will significantly increase the computational efficiency in simulations

In addition solidificationmelting process can also be modelled and the temperature-dependent physical properties are supported

13

3 Creep Modelling of 16MND5 Steel

Creep deformation becomes significant when a component is subjected to high temperature and stress Modelling of creep behaviour of the RPV is paramount since it is considered as a main mechanism leading to the RPV failure In this section we present the development of a three-stage creep model (called lsquomodified theta projection modelrsquo ) for the French RPV steel 16MND5 This model was successfully implemented into the ANSYS Mechanical code

31 Creep and Creep Modelling

Creep is an irreversible time-dependent inelastic deformation process Diffusion and dislocation are the two molecular mechanisms of creep The former normally occurs due to the presence of vacancies within the crystal lattice (or the grain) while the latter occurs if there is a slip by the passage of crystal dislocations [45] There are two types of diffusion creep [46] Coble creep (favoured at lower temperatures) if the diffusion paths are predominantly through the grain boundaries and Nabarro-Herring creep (favoured at higher temperatures) if through the grains themselves Dislocation creep tends to dominate at high stresses and relatively low temperatures Dislocations can move by gliding in a slip plane which is a process that requires little thermal activation But the rate-determining step for their motion is often a climb process which also requires diffusion and is thus a time-dependent process and favoured by higher temperatures Creep behaviour is highly affected by temperature and stress generally creep is more significant under higher temperature and stress

Creep curves describe the creep strain development with time Figure 3 shows a creep curve under constant temperature and stress which typically experiences three creep stages primary stage where the creep strain rate ε119888119903 (the change of creep strain ε119888119903 in unit time) decreases with time secondary stage where the creep strain rate almost keeps constant and tertiary stage where the creep strain rate drastically increases with time

Figure 3 Creep curve under constant temperature and stress

14

The built-in creep models in commercial software eg ANSYS [47] and ABAQUS [48] in default only cover the primary andor the secondary stage(s) Table 1 [47] gives the default creep models supported in ANSYS for example Possible reason why the tertiary stage is not modelled might be that for most industrial applications the failure analysis is beyond interest and therefore the built-in creep laws in the commercial software was supposed to be sufficient [49] Though the two codes offer the interfaces through either ANSYS UPF or ABAQUS user subroutines to support more general user-defined creep laws eg complex models that also include the tertiary creep stage

Table 1 Default creep models in ANSYS Mechanical

Creep model Equations Type

Strain hardening

ε119888119903 = 1198621σ1198622ε119888119903

11986231198901198624119879 1198621 gt 0 Primary

Time hardening ε119888119903 = 1198621σ11986221199051198623119890minus1198624119879 1198621 gt 0 Primary

Generalized exponential

ε119888119903 = 1198621σ1198622119903119890minus119903119905 r = 1198625σ

11986231199051198623119890minus1198624119879 1198621 gt 0

1198625 gt 0

Primary

Generalized Graham

ε119888119903 = 1198621σ1198622(1199051198623 + 1198624119905

1198625 + 11986261199051198627)119890minus1198628119879 1198621 gt 0 Primary

Generalized Blackburn

ε119888119903 = 119891(1 minus 119890minus119903119905) + 119892119905 119891 = 1198621119890

1198622σ 119903 = 1198623(σ1198624)1198625 119892 = 1198626119890

1198627σ

1198621 gt 0

1198623 gt 0

1198626 gt 0

Primary

Modified time hardening

ε119888119903 = 1198881σ11988821199051198883+1(1198883 + 1) 1198621 gt 0 Primary

Modified strain hardening

ε119888119903 = 1198621σ1198622[(1198623 + 1)ε119888119903]

1198623119890minus1198624119879 1198621 gt 0 Primary

Generalized Garofalo

ε119888119903 = 1198621[119904119894119899ℎ(1198622σ)]1198623119890minus1198624119879 1198621 gt 0 Secondary

Exponential form

ε119888119903 = 1198621119890σ1198622119890minus1198623119879 1198621 gt 0 Secondary

Norton ε119888119903 = 1198621σ1198622119890minus1198623119879 1198621 gt 0 Secondary

Combined time hardening

ε119888119903 = 1198621σ11986221199051198623+1119890minus1198624119879(1198883 + 1)

+ 1198625σ1198626119905119890minus1198627119879

1198621 gt 0

1198625 gt 0

Primary+ Secondary

Rational polynomial ε119888119903 = 1198621

partε119888part119905 ε119888 =

119888119901119905

1 + 119901119905+ ε119898119905

ε119898 = 1198622101198623σσ1198624 119888 = 1198627ε

1198628σ1198629

119901 = 11986210ε11986211σ11986212

1198622 gt 0 Primary+ Secondary

Generalized time hardening

ε119888119903 = 119891119905119903119890minus1198626119879 119891 = 1198621σ + 1198622σ2 + 1198623σ

3 119903 = 1198624 + 1198625

Primary

32 A Modified Theta Projection Model

321 Original theta projection model and its variations

The original theta projection model [50] was first proposed to accurate predict long-term creep and creep-fracture properties It assumes that a creep curve under constant temperature and stress can be described as

15

휀119888119903 = 1205791(1 minus 119890minus1205792119905) + 1205793(119890

1205794119905 minus 1) (22)

where 휀119888119903 is the creep strain 119905 the time and 1205791~1205794 the model parameters Figure 4 illustrates how the first term and second term on right hand side develop with time and constitute a creep curve which has obvious primary secondary and tertiary stages Model parameters 120579119894(119894 = 1234) are temperature and stress dependent and are described with the following expression

119897119899thinsp120579119894 = 119886119894 + 119887119894120590 + 119888119894119879 + 119889119894120590119879 (23)

where 119886119894 119887119894 119888119894 and 119889119894 ( 119894 = 1234 ) are constants It is noted [45] that this equation is suitable when the temperature range of experiment data sets is very narrow such that minus1119879 is almost proportional to 119879 These constants (119886119894 119887119894 119888119894 and 119889119894 (119894 = 1234)) are the final model parameters which are generated with two fitting procedures

(i) Fitting the experimental creep curves to Eq (22) generates several groups of 1205791~1205794 for each creep curve under certain temperature and stress We denote (119879119895 120590119895 1205791119895 1205792119895 1205793119895 1205794119895 ) as parameters for the 119895119905ℎ

experimental curve under temperature 119879119895 and stress 120590119895

(ii) Fitting 1205791 to Eq (23) as a function of temperature and stress with the obtained data ( 119879119895 120590119895 1205791119895 ) from ( 119879119895 120590119895 1205791119895 1205792119895 1205793119895 1205794119895 ) generates

parameters 1198861 1198871 1198881 and 1198891 Repeat this fitting procedure for 1205792 1205793 and 1205794 to get parameters 119886119894 119887119894 119888119894 and 119889119894 (119894 = 234)

Variations of the model were developed and applied to other different materials to better accommodate the experimental data which are all then called a lsquomodified theta projection methodrsquo independently Examples are models in [51] for lowhigh alloy ferritic austenitic steels Al-alloys Al-matrix composites [52] for Nickel-based Super-alloy [53] for pure metals and class M alloys with intermediate range of stress (40-70 of the yield stress) and temperature (40-60 of the melting temperature)

Figure 4 Illustration of theta projection method [54]

16

322 A modified theta projection model

The new modified theta projection model we proposed [54] was mainly adapted from the original theta projection method [50] and the modified version [51] The new model describes the creep curve with the following equation

휀119888119903 = 1205791(1 minus 119890minus1205792119905) + 120579119898119905 + 1205793(119890

1205794119905 minus 1) (24)

An additional term 120579119898119905 is seen compared with Eq (22) in the original method The reason for adding this term is that the 16MND5 steel shows fairly long secondary stages in many experimental creep curves and the creep curves can be better fitted by Eq (24) than Eq (22) As the structure undergoes different creep process under different temperature and stress the model parameters θ119894 (119894 =1~4 and 119898) are also temperature and stress dependent

The main difference compared with the original and all other modified projection models is that numerical interpolation is used to replace the second numerical fitting in previous models In other words instead of fitting the model parameters as functions of stress and temperature with Eq (23) the distributions of the model parameters are assumed to be piecewise in the temperature-stress space and they will be interpolated directly from known parameters obtained from experiments The consideration is that the experimental data ([16] [17]) for 16MND5 covers a wide temperature range varying from 600 degC to 1300 degC and a wide stress load range varying from 08 MPa to 248 MPa while Eq (23) in the existing methods is based on a narrow temperature span Poor parameter fitting results would be expected if we directly fit the parameters 1205791~1205794 with Eq (23) as used in the original methods

The creep model was developed using the following steps

(i) Fitting Each creep curve can be represented by a group of model parameters [1205791 1205792 120579119898 1205793 1205794] with respect to Eq (24) by fitting the creep curves to Eq (24) All parameters of the 32 experimental curves form an

initial dataset of the model parameters [119879119895 120590119895 1205791119895 1205792119895 120579119898119895 1205793119895 1205794119895] where j (0ltjle32) is the jth creep curve

(ii) Interpolation Parameters of a creep curve under any [119879 σ] are directly interpolated from the dataset

(iii) Parameter adjustment The dataset was further updated to satisfy the monotonicity assumption we have taken that creep strain monotonically increases with temperature and stress

(iv) Different hardening models for dynamic loads

Step (i) was done using MATLAB In the following steps (ii)-(iv) are described in detail

Model parameters for new creep curves are interpolated from the dataset over the 2D 119879 minus 120590 space The interpolation is divided into two sub-interpolation steps interpolate over stress with fixed temperature and then interpolate over temperature with fixed stress We take the interpolation of model parameters under (119879prime σprime) for example In the first interpolation we first find out the two

17

temperatures 119879119896119899119900119908119899119904119898119886119897119897 and 119879119896119899119900119908119899119897119886119903119892119890 most close to 119879prime from the experimental

temperature set 600 700 800900 1000 1100 1200 1300 Under each temperature there are four sets of model parameters in the dataset corresponding to four different stress loads Then the first interpolation is done over stress load from the known dataset of model parameters under each

temperature to output the parameters for (119879119896119899119900119908119899119904119898119886119897119897 120590rsquo) and (119879119896119899119900119908119899119897119886119903119892119890 120590rsquo)

In the second interpolation the model outputs parameters for (119879rsquo 120590rsquo) from

(119879119896119899119900119908119899119904119898119886119897119897 120590rsquo) and (119879119896119899119900119908119899119897119886119903119892119890 120590rsquo) For 120579i(119894 = 12 3 4) we assume a logarithm

relation between 120579119894 temperature 119879 and stress 120590 which is similar to Eq (23)

119897119899120579119894 = (119886 + 119887120590)(119888 + 119889119879) (25)

It reduces to 120579119894 = 11986011198901198871σ with fixed temperature in the first interpolation and to

θ119894 = 11986021198901198872T with fixed stress in the second interpolation The interpolation

expression for θ119894 = 11986011198901198871σwith two known points (1205901 1205791198941) and (1205902 1205791198942) is given

by Eq (26) The interpolation expression for θ119894 = 11986021198901198872T with two known points

(1198791 1205791198941) and (1198792 1205791198942) is given by Eq (27)

120579119894 = 120579i1

120590minus12059021205901minus1205902120579i2

1205901minus1205901205901minus1205902 (26)

120579119894 = 120579i1

119879minus11987921198791minus1198792120579i2

1198791minus1198791198791minus1198792 (27)

Similarly we assume the following relation between 120579119898 temperature 119879 and stress 120590

120579119898 = 119860120590119899119890minus119887 119879frasl (28)

The interpolation is done over the stress with 120579119898 = 1198603120590n by fixing temperature

and then over temperature with 120579119898 = 1198604119890minus1198874119879 by fixing stress The interpolation

expression over stress for 120579119898 = 1198603120590119899 with two known points (1205901 1205791198981) and

(1205902 1205791198982) would be given by Eq (29) The interpolation expression over

temperature for 120579119898 = 1198604119890minus1198874119879 with two known points (1198791 1205791198981) and (1198792 1205791198982)

would be given by Eq (30)

120579119898 = 119890119909119901(119897119899120590 minus 11989711989912059011198971198991205902 minus 1198971198991205901

1198971198991205791198982 +1198971198991205902 minus 119897119899120590

1198971198991205902 minus 11989711989912059011198971198991205791198981) (29)

120579119898 = 120579m2(11198791minus1119879) (

11198791minus11198792)fraslsdot 120579m1

(1119879minus11198792) (

11198791minus11198792)frasl (30)

In this model we also take the assumption that the creep strain monotonically increases with temperature and stress loads1 Or specifically with the same creep time the structural component under higher temperature or stress experiences

1 Intuitively materials deform more under higher stress and temperature This is also mostly observed in tensile experiments Though some exceptions exist one instance of which is the effect of phase change some materials under certain temperature around the phase change temperature may show much weaker structural performance Here we just simply assumed that the creep strain monotonically increases with temperature and stress

18

higher creep strain than that with lower temperature or stress (the creep curve under higher temperature or stress should be above the curve with lower temperature or stress) When generating the parameter dataset (Step (i)) we already kept the trend that under the same temperature cases under higher stress loads would have larger creep model parameters and give higher creep strains This trend would also hold for the interpolated temperatures which thus guarantees that creep strain monotonically increases with stress load under same temperature Then the question becomes if creep strain also monotonically increases with temperature given fixed stress However the answer could not always be a yes especially in the extrapolation range of the stress load Figure 5(a) illustrates an example of the interpolation of model parameters under (119879119872 σ119872) The red dots and blue dots are known parameters from the parameter dataset for corresponding high temperature 119879119867 and low temperature 119879119871 respectively The green dots (119879119867 σ119872) and (119879119871 σ119872) are interpolated from the red and blue dots in the first sub-interpolation respectively The black dot (119879119872 σ119872) is interpolated from the green dots in the second sub-interpolation It is seen that when σ119872 lt σ119862 the value of (119879119871 σ119872) is higher than that of (119879119867 σ119872) It means the higher 119879119872 is the smaller the model parameter will be ie the model would predict larger creep strain for lower temperatures which violates our monotonicity assumption One simple solution used here as shown in Figure 5(b) is to minorly adjust the values of the four red and blue dots such that the intersection point C can be sufficiently small and the situation σ119872 lt σ119862 would barely occur in the extrapolation range

Figure 5 Illustration of model parameter interpolation (a) before and (b) after parameter adjustment [54]

A corresponding parameter adjustment process was introduced to verify that the model predicts higher creep strain under higher temperature under fixed stress Figure 6 illustrates the general verification process for any (119879 120590) in the interested load range the creep strain of (119879 + 120575119879 120590) should be larger than that of (119879 120590) Otherwise parameters would be minorly adjusted and the verification process be repeated This process would be iterated several times until the requirement is reached The achieved verification shows

Stress

σC

High temperature TH

MPa

a

0

Model

par

amet

er θ

Low temperature TL

MPa

a

0

Model

par

amet

er θ

High temperature TH

Low temperature TL

σC

Stress

C C

(a) (b)

σM

σM

19

(i) For any (119879 120590) in the temperature range 273~ 1073 K with an interval of 1 K and the stress range 1 ~260 MPa with an interval of 1 MPa its creep strain is higher than that of (119879 minus 1119870 120590) for the first 12000000 s

(ii) For any (119879 120590) in the temperature range 1073~1673 K with an interval of 1 K and the stress range 1 ~80 MPa with an interval of 1MPa its creep strain is higher than that of (119879 minus 1119870 120590) for the first 12000000 s

(iii) For any (119879 120590) in the stress range 01~26 MPa with a smaller interval of 01 MPa for the full temperature range 273~1673 K with an interval of 1 K its creep strain is higher than that of (119879 minus 1 120590) for the first 12000000 s

Above verification covers a wide range of temperature and stress especially where the stress and temperature are low and extrapolation is used Further verification to cover even higher stress or temperature ranges is not considered as creep increases drastically under such high stress and temperature and the results would become meaningless With these regards we generally think the monotonicity requirement is reached The model parameter dataset was updated along with the adjustment The final dataset was given as Table 2 in Paper I

Figure 6 Parameter adjustment process [54]

Figure 7(a-d) present the model predictions of selected creep curves in ANSYS Mechanical (labelled as lsquoANSYSrsquo) compared with experimental curves (labelled as lsquoREVISArsquo) at temperature 600 800 1000 and 1200 degC respectively The predicted curves proved to clearly show the three-stage creep process and agreed well with the experimental creep curves

Initial parametersdataset

θ1 θ2 θm θ3 θ4 from creep

curve-fitting

Parameters interpolation

Is the monotonicity

assumption satisfied

IsMon= =1

Final model parameters

Adjust the

parametersdataset

θ1 θ2 θm θ3 θ4

No

Yes

IsMon=1 a flag is monotonic achieved

For σ in the stress range in MPa

For T in range Tstart Tend in K

For t in range 112000000 in s

If ε(σ T+1t)lt ε(σTt) IsMon=0

Iteration loop of

parameter adjustment

20

Figure 7 Selected model predictions compared with experimental creep curves [54]

Instead of being constant temperature and stress loads also dynamically change with time in real cases Hardening models determine how creep strain rate reacts to the change of loads during a creep process The time hardening model and strain hardening model2 are supported in our model implementation in ANSYS Mechanical which are two mostly used hardening models

Time hardening assumes that the following creep strain rate after a load change depends only on the current creep time from the beginning of the creep process (and also temperature and stress if used in the creep model) and ignores the current creep strain Strain hardening assumes that the creep strain rate depends only upon the current creep strain (and also temperature and stress if used in the creep model) [55]

2 The time hardening and strain hardening here are more general terms than the specific time hardening creep model and the strain hardening creep model mentioned in Table 1

(a) 600 degC (b) 800 degC

(c) 1000 degC (d) 1200 degC

21

Eq (24) in default is a time hardening model as creep strain is directly described by the time 119905 Eq (31) gives the creep strain increment 119889휀119888119903 at time 119905 during time interval 119889119905

119889휀119888119903 = (12057911205792119890minus1205792119905 + 120579119898 + 12057931205794119890

1205794119905)119889119905 (31)

The strain hardening model is derived from Eq (24) in an implicit form Specifically when temperature and stress load change to (119879 120590) at the time 119905 and current creep strain 휀1198881199030 the model will get the creep curve of constant (119879 120590) and

output how this curve will react under the current strain 휀1198881199030 Since the creep curve shows creep strain monotonically increases with time given the creep strain 휀1198881199030 under creep curve of (119879 120590) we can uniquely find this reference time

119905119903119890119891 satisfying Eq (32) One should notice that the reference time 119905119903119890119891 can be

different from the current physical time 119905

휀1198881199030 = 1205791(1 minus 119890minus1205792119905119903119890119891) + 120579119898119905119903119890119891 + 1205793(119890

1205794119905119903119890119891 minus 1) (32)

where all 120579119894(119894 = 1~4119898) are model parameters of the new creep curve (119879 120590) It can be solved for 119905119903119890119891 with methods eg Newton-Ramphson [56] Then the creep

strain increment at 119905119903119890119891 during time interval 119889119905 is given as

119889휀119888119903 = (12057911205792119890minus1205792119905119903119890119891 + 120579119898 + 12057931205794119890

1205794119905119903119890119891)119889119905 (33)

Therefore combining Eq (32) and (33) gives the creep strain rate at creep strain 휀1198881199030 In summary when loads are changed to (119879 120590) with current creep strain 휀1198881199030

the model traces the creep curve of (119879 120590) and calculates the creep strain rate of this curve at strain 휀1198881199030

One calculation was done to demonstrate the performance of the hardening models The stress load was set to 151 MPa and kept unchanged A step temperature load was applied as the orange dash curve shows in Figure 8 it was 900 degC during 0-36000 s and suddenly jumped to 1000 degC at 36000 s which was then kept till the end of the simulation The blue solid curve (the part before the temperature change was identical to and therefore covered by the green solid curve in the figure) and blue dash curve are the creep curves under constant temperature T=900 degC and T=1000 degC respectively The green solid curve and green dash curve represent the model predictions considering this temperature change with time hardening model and strain hardening model respectively Before this change the creep curves with both hardening models were identical to the creep curve of the T=900 degC till point A After that the creep strain rate then shifted from Point A to Point B (which had the same time with A) in the time hardening model and behaved the same as the curve T=1000 degC as shown in the rest part of the green solid curve In the strain hardening model the creep strain rate shifted from Point A to Point C (which had the same creep strain with A) and behaved the same as the curve T=1000 degC as shown in the rest part of the green dash curve The large difference between two models was partially caused by the drastic temperature jump In real cases the difference would be less significant as the temperature changes more mildly Overall this prediction is exactly the expected behaviors of the time hardening model and strain hardening model

22

Figure 8 Time hardening and strain hardening in ANSYS Mechanical [54]

23

4 Coupled Thermo-Fluid-Structure Analysis of RPV Failure

As described in Section 13 RPV failure is coupled with melt pool heat transfer leading it into a transient thermo-fluid-structure coupled problem The goal of this section is to develop and validate a reliable coupling approach which supports transient coupled thermo-fluid-structure analysis as well as advanced models and computational platforms so as to improve fidelity and facilitate implementation The approach was validated against the FOREVER-EC2 experiment and then applied to an RPV failure analysis of a Nordic BWR As the approach adopts the volume loads mapping for load transfer comparative analysis was also performed against the coupling approach using surface loads mapping Future work is considered in analysis of penetration failures (IGTs and CRGTs) of a BWR RPV since the approach is flexible to complex geometries Results of Paper II ampIII are jointly presented in this section

41 Thermo-Fluid-Structure Coupling Approach

Considering the involved phenomena a coupling approach should at least address three issues (i) melt pool heat transfer (ii) RPV deformation and (iii) the interaction between melt pool heat transfer and vessel deformation A two-way strong coupling considering both the transient thermal load from melt pool on vessel and the influence of vessel deformation on melt pool heat transfer involves complex data exchange and is still full of challenges Therefore the new coupling approach is realized in a one-way manner with the assumptions that the influence of vessel deformation on melt pool heat transfer can be ignored and thus only the thermal loads from melt pool to vessel are considered It is based on the considerations that on one hand the deformation is a slow process regarding the relatively long failure time and on the other hand it was pointed out [57] that the physics of natural convection heat transfer in internally heated liquid pools is not sensitive to the geometrical factors based on their calculation results and experimental data from [58] [59] as well as previous experiments in semicircular and square cavities eg [44] [60] In addition one-way coupling is also computationally more efficient especially when heat transfer turbulence structural non-linearity and large-scale geometries are involved The developed approach is based on the ANSYS platform which generally consists of three parts with respect to the three issues respectively

(i) conjugate heat transfer of the melt pool and vessel using ANSYS Fluent (ii) transient thermal loads transfer from ANSYS Fluent to ANSYS

Structural (the ANSYS Mechanical in ANSYS Workbench) (iii) structural analysis of the vessel in ANSYS Structural

ANSYS Structural in default only supports direct transfer of constantsteady-state spatial loads from ANSYS Fluent Transfer of transient loads is realized with the extension tool FSI Transient [61] It is a pre-processing tool (with good documentation) developed by a third-party CAE vendor EDRTM MEDESO [62] to map transient CFD results (heat transfer coefficient temperature and surface pressure) to thermal or structural analysis including the definition of load steps

24

This new procedure proves to be straightforward and reduces efforts in the load transfer

Since structural performance of RPV is highly dependent on temperature the transfer of transient temperature loads from conjugate heat transfer to mechanical solver is a core step of the coupling framework There are two general approaches to realize the transfer as shown in Figure 9 Both approaches contain three parts a conjugate heat transfer analysis for fluid and structure where flow and heat transfer are coupled a mechanical analysis for the structure and a load mapping process that transfers the transient loads The conjugate heat transfer part can be solved with any CFD solver and is treated in the same manner in both approaches In the first approach (Figure 9(a)) transient volume thermal loads are transferred and mapped into the mechanical simulation domain The structure domain reads the essential thermal loads from the mapping after which a structure analysis of this domain is performed In the second approach (Figure 9(b)) transient surface thermal loads are mapped onto the boundary surfaces of structure domain as thermal boundary conditions Then a thermo-structure coupled analysis of the structure domain in the mechanical part is performed To distinguish them we call the former approach volume loads mapping (VLM) and the latter surface loads mapping (SLM) In the RPV failure analysis the fluid and structure domains correspond to the melt pool and RPV respectively Examples of these methods are the studies of [63] and [30] using VLM and [35] using SLM The VLM is adopted in current approach

Figure 9 Frameworks of the thermo-fluid-structure coupling approaches (a) volume loads mapping (b) surface loads mapping [64]

fluid structure structure

fluid structure structure

Conjugate heat transfer problem Mechanical problem

(structure)

Conjugate heat transfer problem Mechanical problem

(thermo-structure coupled)

Transient volume

loads mapping

(VLM)

Transient surface

loads mapping

(SLM)

(a)

(b)

25

42 Numerical Modelling

421 Conjugate heat transfer

In the conjugate heat transfer analysis main focus is put on the modelling of melt pool heat transfer Within the ANSYS Fluent solver two options are supported CFD with turbulence modelling and PECM CFD is the choice when the model goes to detail and more mechanistic analysis are needed while PECM provides an alternative that is faster and possibly more suitable for large-scale problems

Within the scope of RANS models the SST (Shear Stress Transport) model [65] is selected as the turbulence model in present study This model is a combination of k-ε model and k-ω model The k-ω formulation is used in the inner parts of the boundary layer which makes the model directly usable all the way down to the wall through the viscous sub-layer It also switches to k-ε model in the free-stream and thereby avoids the common k-ω problem that the model is too sensitive to the inlet free-stream turbulence properties The considerations on choosing this model are

bull Several two-equation low-Re k-ε models have been proved in [66] not good enough for melt pool heat transfer Therefore these mentioned two-equation models as well as other one-equation models are not considered

bull SST is highly recommended [67] for simulating most industrial heat transfer problems

bull Convincing results ([68] [69]) using SST have been obtained in capturing the thermal behaviour of the natural convection in an internally heated fluid Especially good agreements were achieved between the predicted heat flux distribution along the vessel and experimental data which would be one of the most important factors to estimate model performance in this problem

The description of PECM as an effective model for melt pool heat transfer is given in Section 23

422 Mechanical analysis

The material for the RPV failure analysis considered in this study is the French RPV steel 16MND5 The three-stage modified theta projection model in Section 3 is used as the creep model Since the definition of an RPV failure criterion is not universal (it can be the time when damage variable reaching 1 or either specified value of creep strain creep strain rate or total deformation is reached) we do not explicitly define a failure criterion Instead the simulation is kept running until the creep converge criterion cannot be satisfied even with the defined minimum time step (indicating a fairly large creep strain rate) and the simulation will be automatically terminated The time the simulation stops is taken as the failure time This treatment implicitly implies a criterion of a large creep strain rate We refer to [17] for other material properties Youngrsquos modulus and thermal conductivity are temperature dependent Multilinear Isotropic Hardening (MISO) is used as the plasticity model

26

43 Validation against the FOREVER-EC2 Experiment

In this section the validation against the FOREVER-EC2 experiment is presented (indicated as lsquoVLMrsquo in the following) In addition the simulation results using the SLM approach (indicated as lsquoSLMrsquo in the following) is also included for comparison purpose

431 FOREVER-EC2 experiment

The FOREVER-EC2 experiment [70] is one of the FOREVER experiments that achieved a vessel failure Figure 10 shows the experimental setup A 110 scale pressure vessel with a wall thickness of 15 mm and an outer diameter of ~400 mm was employed The lower head was made of 16MND5 steel Since it was difficult to use prototypical corium materials in the experiment the binary salt 30 CaO-70 B2O3 was employed as simulant To generate the internal heat which corresponds to the nuclear decay heat a specially designed heater was used The power input was maintained ~38 kW during the experiment such that maximum melt pool temperature would be ~1300 degC The pressure inside the vessel was maintained at ~25 MPa Thermocouples were used to measure the melt pool and vessel wall temperatures and linear displacement transducers used to measure the vessel wall displacements The designed experimental conditions were reached at experimental time 11670 s Vessel failure was observed about 12510 s later (at experiment time 24180 s)

Figure 10 Scheme of the experimental setup [22]

27

Figure 11 Computational domains and boundary conditions (a) conjugate heat transfer (b) mechanical analysis [71]

Computational domains and boundary conditions are indicated in Figure 11 SST turbulence modelling was used for pool heat transfer The simulations started at the moment when experimental conditions were reached (experiment time 11670 s) So simulation time 0 s corresponds to experiment time 11670 s

432 Conjugate heat transfer

Figure 12 shows the temperature fields at simulation time 0 s 1225 s 2475 s 4975 s and 9975 s All the time points in this part refer to simulation time The field at time 0 s (Figure 12(a)) was the initial temperature that was reconstructed based on the thermocouple readings The temperature increased with time and reached a steady state when the maximum temperature reached ~1607 K Figure 12(b-f) show clear thermal stratification which is an expected phenomenon of natural convection

Internal pressure

25 bar

Internal pressure 25e6-2500times981timesy Pa

External pressure

1 bar

Zero displacement

y=0 m

Gravity 400

mm

R=203 mm

15 m

m

Tamb

=323 K radiation with ε = 08

+ convection with h=10Wm2K

Tamb

=800 K radiation with ε=02

(a) (b)

Tamb

=800 K radiation with ε=08

+ convection with h=10 Wm2K

28

Figure 12 Temperature fields at simulation time (a) 0 s (b)1225 s (c) 2475s (d) 4975 s (e) 7475 s and (f) 9975 s [64]

Figure 13 shows the temperature profiles of the external vessel along the polar angle in comparison with experimental data Regions with a polar angle larger than 90deg are those in the cylindrical part of the vessel It also shows the temperature profile increased with time until reaching a steady state The profile almost did not change after 4975 s which indicates that a steady state of the temperature was reached As time increased the temperature in the lower head region 0~90deg (where the vessel was contacted with the melt) evolved from a relatively flat distribution to a distribution that temperature at high angle was higher than that at low angle by more than 200 K Generally the steady state results agreed well with the experimental data except for the low angle part of vessel lower head and the high angle part of the cylindrical vessel In addition to modelling errors the discrepancy in the low angle part of the lower head may be caused by the crust and gap possibly formed at vessel bottom in the experiment which can reduce the downward heat transfer and result in lower temperature while they cannot be modelled in simulations As to the high angle cylindrical vessel part simplified boundary conditions were applied to the inner and outer vessel wall since it was far from the melt and not of interest which may result in some discrepancy

(a) (b)

(c) (d) (e) (f)

29

Figure 13 Temperature profiles of external vessel surface along the vessel polar angle [64]

433 Mechanical analysis

In the mechanical analysis the VLM predicted a failure time of 23959 s and the SLM predicted a failure time of 22445 s both are close to the experimental failure time 24180 s All the time points in this part refer to the experiment time Figure 14 shows the creep strain fields at failure time Both approaches predicted maximum creep strains at same location where the temperature was high (cf Figure 12) This is related to the fact that creep behavior is highly influenced by temperature Maximum values for both cases were also close to each other 0393 in VLM and 0366 in SLM

Figure 15 presents the total deformation of the vessel bottom point (also called ldquosouth polerdquo in some literature) in the vertical direction Since there was an initial displacement reading of ~355 mm at time 0 s which almost did not change during 0~11670 s (the whole period before the experimental conditions were reached) an initial value of 355 mm was added to the simulation result such that experimental and simulation results had the same total displacement at t=11670 s Similar trend was seen in the VLM SLM and experimental data that the deformation increased with time and experienced a drastic acceleration at the end Generally both simulation approaches agreed well with the experimental data though they both slightly under predicted the value of total deformation The discrepancy may possibly be related to the accuracy of the thermal predictions modelling errors of plasticity and creep and simplified boundary conditions

400

600

800

1000

1200

1400

1600

0 20 40 60 80 100 120 140 160

Tem

per

ature

(K

)

Polar angle (deg)

Experiment

25 s

1225 s

2475 s

4975 s

7475 s

9975 s

Weilding line

30

Figure 14 Creep strain fields at failure time (a) VLM (b) SLM [64]

Figure 15 Total deformation in vertical direction of the vessel bottom point [64] [71]

The wall thickness changes are plotted together with experimental results in Figure 16 The wall thickness change δ119909 is defined as

δ119909 = 119909119890119899119889 minus 119909119900119903119894119892119894119899119886119897 (34)

000

500

1000

1500

2000

2500

11670 13670 15670 17670 19670 21670 23670

Tota

l def

orm

atio

n (

mm

)

Time (s)

ExperimentVLMSLM

(a) (b)

31

where 119909119890119899119889 and 119909119900119903119894119892119894119899119886119897 are the wall thickness at failure time and the original wall

thickness respectively Since 119909119890119899119889 cannot directly be post-processed it was estimated in the following way (cf the subfigure of Figure 16) for the same polar angle θ find the corresponding points Pin (1199091 1199101) and Pout (1199092 1199102) on the inner and outer surfaces of the deformed vessel respectively the distance of Pin and

Pout radic(1199091 minus 1199092)2 + (1199101 minus 1199102)2 is calculated as 119909119890119899119889 at polar angle 120579 Such estimation of the wall thickness is more accurate when the line PinPout is closer to be orthogonal to the deformed local surface Considering the final deformed vessel as shown in Figure 14 the estimation should be fine at regions close to 0deg and 90deg and can be slightly overestimated at medium angle region The VLM curve was compared with the SLM curve and 4 experimental curves measured at vessel left right front and back sides Generally VLM curve agreed well with the experimental and SLM curves wall thickness gradually decreases with angle till around 75deg and then increases till 90deg the maximum thickness change occurred at around 75deg would be around -5 mm The positive values at low angle may be caused by the thermal expansion It also shows that the thickness transitions in experimental curves were smoother than those in both simulation curves while simulation curves showed narrower regions with large thickness changes (at angle 70deg~80deg) Possible reasons might be (i) sharper temperature profile predicted in simulation (ii) input uncertainty and (iii) modelling errors The SLM predicted slightly thinner vessel than the VLM except for the necking region The discrepancy might be caused by various reasons eg the interior vessel temperature can be slightly different between two approaches which would then result in different thermal stress and different deformation the final deformed vessel shapes cannot be exactly the same which also results in different errors in estimating PinPout

Figure 16 Wall thickness changes along the vessel polar angle [64] [71]

-6

-5

-4

-3

-2

-1

0

1

0 10 20 30 40 50 60 70 80 90

Wal

l th

icknes

s ch

ange

(mm

)

Polar angle (deg)

VLMSLMExperiment (left)Experiment (right)Experiment (front)Experiment (back)

32

44 Application to BWR Case

In this section the coupling approach (indicated as lsquoVLMrsquo in the following) was applied to an RPV failure analysis of a BWR A simulation using the SLM coupling approach (indicated as lsquoSLMrsquo in the following) was also performed for comparison purpose

A hypothetical severe accident of a Nordic BWR initiated by an SBO was considered The core materials would melt and migrate downward to the RPV lower head The total amount of corium relocated in the lower head was considered to form a 19 m corium melt pooldebris bed assuming the corium distribution was homogeneous

Figure 17 Computational domains and boundary conditions (a) conjugate heat transfer (b) mechanical analysis [64]

Figure 17(a) shows the computational domain of the conjugate heat transfer PECM was used in this calculation A general mesh with a pool height of 23 m was used to accommodate most of the relocation situations with the same mesh Since the corium only had a height of 19 m the top 04 m region was treated as injected water covering the corium The top water was expected to be boiling and we assumed a uniform temperature of 383 K Under the implementation of PECM this region was damped with artificially high heat capacity and high conductivity such that it would efficiently remove heat from the corium and suppress its own temperature increase For the corium part below water a volumetric heat source 11 MWm3 was implemented as the nuclear decay heat For the in-vessel surfaces a constant temperature of 383K was applied as they are covered by water which can be boiling Without water cooling on the external vessel wall a very small heat loss of 20 Wm2 was applied Melting and solidification of the corium were also considered with a solidus temperature of 2750 K and a liquidus temperature of 2770 K The initial temperature was set as

q=20 Wm2

T=383 K

Qv=11

MWm3

315 bar

315 bar+ hydrostatic

pressure of corium

Gravity

248 bar

Zero

displacement

Water

Corium

(a) (b)

33

1100 K for the corium 383 K for the water and 450 K for the vessel For the mechanical analysis (Figure 17(b)) an inner pressure of 315 bar and an outer pressure of 248 bar were considered For the region covered by corium an extra hydrostatic pressure was added on the vessel wall A 2 m long vessel extension was considered as well as the standard earth gravity

Figure 18 presents the wall temperature profile along the inner vessel surface at 50 s 2500 s 7500 s and 10000 s The temperature increased with time for the vessel surface contacted with corium and the distribution was flat which was due to that large part of the corium was still in solidus status It also shows that for the surface region that contacted the water the temperature was basically unchanged which indicates that the damping treatment indeed suppressed the temperature increment and guaranteed the heat removal upward by water

Figure 18 Temperature profiles of inner vessel wall along the vessel polar angle [64]

The predicted RPV failure time was 9869 s by VLM and 9650 s by SLM Compared with the FOREVER-EC2 case in Section 43 the VLM predicted a slightly later failure time than the SLM in this BWR case too Figure 19(a) compares the maximum creep strain developments with time of both approaches Before 9000 s the predicted values by both approaches are close to each other Afterward the creep acceleration occurred and the strain increased drastically The acceleration of the VLM occurred 200 s later than that of the SLM which agreed with the predicted later failure time Similar trend was observed for the total deformation in vertical direction of vessel bottom point as shown in Figure 19(b) The deformations of both approaches are close except that the acceleration of VLM occurred later than that of SLM Overall results of VLM agreed well with those of SLM

0

500

1000

1500

2000

2500

0 10 20 30 40 50 60 70 80 90 100

Tem

per

ature

(K

)

Polar angle (deg)

50 s

2500 s

5000 s

7000 s

10000 s

34

Figure 19 Results of structural analysis (a) creep strain and (b) total deformation in vertical direction of vessel bottom point [64]

45 Computational Efficiency

Computational efficiency is also an important consideration regarding the performance of numerical approaches The computational efficiencies of both VLM and SLM are analysed and compared in this section As parallel computing technique was used to speed up the simulations the number of parallel cores and corresponding costed computational time for each simulation are given together in Table 2 as a generally reflection of the computational efficiency All the simulations were done in a Lenovo P700 workstation equipped with 2 Intel Xeon E5-2630 v3 processors (8 cores per processor) and 32 G RAM For the simulations of FOREVER-EC2 SLM employed twice the parallel cores and costed 286 more time compared with VLM For the simulations of prototypic BWR case both VLM and SLM employed the same number of parallel cores but VLM only took 13 of the computational time compared with SLM Overall it can be concluded that VLM has higher computational efficiency than the SLM The reason is a structural analysis is performed in the VLM approach while a coupled thermo-structural analysis is performed in the SLM approach (cf Figure 9) a standalone structural analysis is computed faster than a coupled thermo-structural analysis

Table 2 Comparison of computational efficiency [64]

Case FOREVER-EC2 BWR VLM SLM VLM SLM

Number of cores 4 8 4 4 Computational time (hours) 21 27 18 72

35

5 Hybrid Coupling Approach for Efficient RPV Failure Analysis

In this section we propose a hybrid coupling approach for the efficient RPV failure analysis The idea is to replace the ANSYS Fluent in previous coupling approach with a lumped-parameter code for quick estimate of the melt pool heat transfer With the least modelling details (compared with CFD methods and distributed-parameter methods eg PECM) lumped-parameter codes prove to be the most efficient for melt pool heat transfer The new hybrid approach characterizes 1) utilization of efficient models for melt pool heat transfer and 2) the support of detailed structural modelling of RPV deformation allowing studies with different RPV geometric shapes under effects like local thinning due to ablation For studies of which the focus is mainly on structural behaviour of RPV eg [13] [26] while mechanistic simulation of melt pool convection is computationally expensive and unnecessary the lumped-parameter code could help providing reasonable thermal loads

The development of the approach consists of two tasks (i) development of a lumped-parameter code and (ii) its coupling with ANSYS Mechanical to form this hybrid coupling framework In the following we describe both tasks and the corresponding verification and validation processes respectively This section is mainly based on Paper IV

51 A Lumped-parameter Code for Melt Pool Heat Transfer

A lumped-parameter code transIVR was developed It features (i) quick estimate of transient melt pool heat transfer with reasonable accuracies and (ii) two-dimensional representation of heat conduction in the RPV wall facilitating the coupled analysis of the RPV failure under thermal loads The following assumptions are taken in the transIVR modelling

bull The crust between the oxide pool and metal pool is thin

bull The heat flux has uniform distributions on the bottom top and side surfaces of metal pool

bull Heat source only exists in oxide layer

bull Latent heat of fusion is ignored

Figure 20 illustrates the modelled three domains oxide layer metal layer and the RPV wall In addition the internal structure inside the vessel can also be considered as a radiative body to the top metal surface as in [72] An internal volumetric heat source generated by nuclear decay heat is considered in the oxide layer Heat in oxide layer transfers downward to the RPV wall and upward to the top metal layer Heat in metal layer can be transferred sideward to the RPV wall and upward to the surrounding internal structure through radiation (or to water if inside water flooding is applied) The heat transferred to RPV will be removed from outer RPV surface Possible formations of oxide crust in the top and bottom surface are illustrated as the black regions Possible formation of metal crust on the top surface is shown as the grey region The code automatically detects the

36

existence of the crust on oxide top and metal top surfaces during the simulation and solves the problem correspondingly withwithout crust Boundary conditions are applied to the external and internal surfaces of RPV and (if needed) top surface of metal layer Arbitrary boundary conditions are supported If a boundary condition is nonlinear eg mixed heat convection (linear dependent on 119879 ) and heat radiation (proportional to 1198794) the Picard method [73] would be used for solving this problem

This model consists of three main parts

(i) empirical heat transfer correlations (mostly from experiments) to determine global and local heat transfer coefficients

(ii) energy balances in oxide layer and metal layer where the heat transfer coefficients would be used and

(iii) energy balance in RPV

Figure 20 Two-layer melt pool configuration and RPV wall [74]

511 Heat transfer correlations

For internally heated natural convection (oxide layer) in transIVR the mini-ACOPO correlations [75] are used for the averaged heat transfer as expressed in Eq (35) and (36) and for the angular variation of the heat flux as expressed in Eq (37) and (38)

119873119906119906119901 = 03451198771198861198941198990233 (35)

119873119906119889119899 = 00038119877119886119894119899035(119867119877)025 (36)

119873119906119889119899(θ)

119873119906119889119899= 01 + 108(

θ

θ119898119886119909) minus 45 (

θ

θ119898119886119909)2

+ 86 (θ

θ119898119886119909)3

01 leθ

θ119898119886119909le 06 (37)

RPV

Oxide pool

Metal pool

37

119873119906119889119899(θ)

119873119906119889119899= 041 + 035 (

θ

θ119898119886119909) + (

θ

θ119898119886119909)2

06 ltθ

θ119898119886119909le 1 (38)

The 119877119886119894119899 that appears in Eq (35)~(36) is the internal Rayleigh number which is defined as

119877119886119894119899 =119892β119876119881119867

5

α119896ν (39)

For Rayleigh-Bernard convection in the metal layer heat transfer occurs in all the top bottom and side surfaces To side surface the reduced form [72] of the Churchill-Chu correlation [76] for 119875119903 = 013 is employed as shown in Eq (40)

119873119906119904119889 = 007611987711988613 (40)

To top and bottom surfaces the reduced form [72] of Globe-Dropkin correlation [77] for

119875119903 = 013 is used as expressed in Eq (41)

119873119906119906119901 = 005911987711988613 (41)

The 119877119886 that appears in Eq (40) and (41) is the external Rayleigh number which is defined

as

119877119886 =119892βΔ1198791198673

να (42)

Heat transfer coefficients can be calculated from Nusselt number by Eq (43)

ℎ =119873119906119896

119871 (43)

where 119896 and 119871 are the heat conductivity and characteristic length respectively 119873119906 can be the Nusselt numbers expressed in Eq (35)~(38) and (40)~(41)

512 Energy balance in the pools

The energy balance of oxide layer can be described as Eq (44) The total generated heat of oxide pool equals the sum of the increase of enthalpy (transient temperature change) upward heat flow to metal layer and downward heat flow to RPV

120588119862119901119881119900119897119889119879119900119909119889119905

+ ℎ119906119901119860119900119909119906119901(119879119900119909 minus 119879119900119909119906119901) + ℎ119889119899119860119900119909119889119899(119879119900119909 minus 119879119900119909119889119899) minus 119876119881119881119900119897 = 0 (44)

where ρ is the oxide density 119862119901 the heat capacity 119881119900119897 the total volume 119879119900119909 the bulk temperature 119905 the time ℎ119906119901 the upward heat transfer coefficient 119860119900119909119906119901

upward heat transfer area 119879119900119909119906119901 upward surface temperature ℎ119889119899 downward

heat transfer coefficient 119860119900119909119889119899 downward heat transfer area 119879119900119909119889119899 downward surface temperature and 119876119881 volumetric heat source

38

On the interface between oxide layer and metal layer energy balance requires that the heat transferred from oxide layer should equal the heat to the metal layer which can be described as Eq (45)

ℎ119906119901119860119900119909119906119901(119879119900119909 minus 119879119900119909119906119901) = ℎ119898119905119889119899119860119898119905119889119899(119879119894119899119905 minus 119879119898119905) (45)

where ℎ119898119905119889119899 is the downward heat transfer coefficient of metal layer 119860119898119905119889119899 the downward heat transfer area 119879119894119899119905 the interface temperature and 119879119898119905 the bulk temperature of metal layer

The energy balance of metal pool can be described as Eq (46) The increase of enthalpy (transient temperature change) equals the sum of upward downward and sideward heat flows to surrounding surfaces

ρ119898119905119862119901119898119905119881119900119897119898119905119889119879119898119905119889119905

+ ℎ119898119905119906119901119860119898119905119906119901(119879119898119905 minus 119879119898119905119906119901) + ℎ119898119905119904119889119860119898119905119904119889(119879119898119905 minus 119879119898119905119904119889)

+ ℎ119898119905119889119899119860119898119905119889119899(119879119898119905 minus 119879119894119899119905) = 0

(46)

where ρmt is the metal density 119862119901mt the heat capacity 119881119900119897mt the total volume 119879119898119905 the bulk metal temperature 119905 the time ℎ119898119905119906119901 the upward heat transfer

coefficient 119860119898119905119906119901 upward heat transfer area 119879119898119905119906119901 upward surface

temperature ℎ119898119905119904119889 sideward heat transfer coefficient 119860119898119905119904119889 sideward heat transfer area 119879119898119905119904119889 sideward surface temperature ℎ119898119905119889119899 downward heat transfer coefficient 119860119898119905119889119899 downward heat transfer area and 119879119894119899119905 oxide-metal interface temperature

The heat transferred from metal layer to the top surface should equal the heat radiated from this surface to the environment Considering the radiation and reflections of internal structures to this surface following energy balance equation [72] is achieved

ℎ119898119905119906119901(119879119898119905 minus 119879119898119905119906119901) = σ(1198791198981199051199051199001199014 minus 119879119904

4) (1

ε+1 minus ε119904ε119904

119860119898119905119906119901

119860119904)minus1

(47)

where 119879119898119905119905119900119901 is the temperature of metal top free surface ε the emissivity of

metal ε119904 the emissivity of internal structures 119860119898119905119906119901 the area of metal top surface

and 119860119904 the total area of internal structures Sometimes the internal water cooling may be available above metal layer In this case Eq (47) is unnecessary and 119879119898119905119905119900119901 can be set to a constant value

For the oxide layer top surface Eq (48) is used if not crust is formed in the interface between two layers ie 119879119894119899119905 gt 119879119898119890119897119905119894119899119892119900119909 Otherwise Eq (49) is used

119879119900119909119906119901 = 119879119894119899119905 (48)

119879119900119909119906119901 = 119879119898119890119897119905119894119899119892119900119909 (49)

For the metal layer top surface Eq (50) is used if not crust is formed on top of the metal ie 119879119906119901119904119906119903119891 gt 119879119898119890119897119905119894119899119892119898119905 Otherwise Eq (51) is used

119879119898119905119906119901 = 119879119898119905119905119900119901 (50)

39

119879119898119905119906119901 = 119879119898119890119897119905119894119899119892119898119905 (51)

Overall this part involves 6 equations regarding 6 unknows 119879119900119909 119879119900119909119906119901 119879119894119899119905 119879119898119905

119879119898119905119906119901 and 119879119898119905119905119900119901

513 Energy balance in the RPV wall

The heat transfer in the PRV wall can be modelled as a simple 1D heat conduction problem that heat only transfers in the wall thickness direction or 2D axisymmetric heat conduction problem

For 1D model the treatment in [72] was adopted As illustrated in Figure 21 the heat from the oxide pool transfers to inner vessel surface (119879119894) then to the interior vessel (119879119907119890119904) then to outer vessel (119879119900) and finally to environment

Figure 21 1D heat transfer model in RPV [74]

Unknown variables are the heat flux through inner vessel surface 119902119894119899 the crust thickness 119909119888119903 inner vessel temperature 119879119894 vessel center temperature 119879119907119890119904 and outer vessel temperature 119879119900 Eq (52)~(56) give a closed equation set for these variables considering a radiation boundary condition on the external vessel surface Eq (52)~(53) describe the heat transfer through the crust where an internal heat source 119876119888119903 is also considered Eq (52) can be easily derived from the one-dimensional heat transfer equation with heat source Eq (54)~(55) describe the energy balance on the inner vessel surface and vessel interior respectively Eq (56) describes the energy balance on the outer vessel surface regarding radiation boundary condition For Dirichlet boundary condition Eq (56) is replaced with Eq (57) For other boundary conditions we only need to correspondingly modify the terms on the right-hand side of Eq (56) For situations that no crust is formed on the surface of the vessel Eq (52)~(53) are eliminated and 119902119894119899 in Eq (54) can directly be determined from calculations in Section 512 and Eq (54)~(56) are solved for the unknows 119879119894 119879119907119890119904 and 119879119900 Once 119879119894 exceeds the vessel melting temperature (with or without crust formation) wall ablation is considered to occur Then 119879119894 is fixed to the vessel melting temperature and vessel thickness 119909119907119890119904 becomes the new unknown variable

119879119900119909119889119899 119879119894 119879119907119890119904 119879119900 119879119890119899119907

119909119888119903 119909119907119890119904

crust RPV

Qcr

119902119894119899 119902119897119900119888119886119897

40

119902119894119899 = 119896119888119903(119879119900119909119889119899 minus 119879119894)

119909119888119903+ 1198761198881199031199091198881199032

(52)

119902119894119899 = 119902119897119900119888119886119897 + 119876119888119903119909119888119903

(53)

119902119894119899 = 119896119907119890119904(119879119894 minus 119879119907119890119904)

05119909119907119890119904

(54)

120588119907119890119904119862119901119907119890119904119889119879119907119890119904119889119905

= 119896119907119890119904(minus2 lowast 119879119907119890119904 + 119879119894 + 119879119900)

(05119909119907119890119904)2

(55)

119896119907119890119904(119879119907119890119904 minus 119879119900)

05119909119907119890119904= 휀120590(119879119900

4 minus 1198791198901198991199074 )

(56)

119879119900 = 119879119900119888119900119899119904119905

(57)

For the 2D representation the heat conduction in the RPV is solved using FDM considering the heat transfer in both the thickness and the angular directions in the RPV wall The RPV lower head is treated as a spherical shell The corresponding heat conduction equation in the 2D spherical coordinate is given as Eq (58) The cylindrical part of the vessel is treated as a cylindrical shell The corresponding heat conduction equation in the 2D cylindrical coordinate is given as Eq (59) Eq (58) and (59) are coupled and solved together In default the mesh size is 5deg in angular direction and 0025119909119888119903 in radius direction This mesh discretization can be changed up to user demands

ρ119862119901part119879

part119905=

1

1199032part

part119903(1198961199032

part119879

part119903) +

1

r2sinθ

part

partθ(119896119904119894119899θ

part119879

partθ)

(58)

ρ119862119901part119879

part119905=1

119903

part

part119903(119896119903

part119879

part119903) +

part

part119911(119896part119879

part119911) (59)

52 Hybrid Coupling Approach

The hybrid coupling approach to assess the RPV integrity is illustrated in Figure 22 transIVR and ANSYS Mechanical are coupled in a one-way manner transIVR conducts the conjugate heat transfer of melt pool and RPV and provides transient thermal loads to the ANSYS Mechanical where structural analysis of RPV deformation is performed The 2D heat transfer of RPV is solved in transIVR to better capture the temperature distribution In RPV structural analysis same considerations are taken as in Section 422 for 16MND5 steel the modified theta projection model for creep and MISO for plasticity and temperature-dependent Young modulus etc The VLM is adopted for the transfer of transient thermal loads

41

Figure 22 Framework of the hybrid coupling approach [74]

53 Verification and Validation

The transIVR code was first benchmarked against the UCSB FIBS case of two-layer pool configuration and then validated against the LIVE-7V experiment showing the codersquos capability in predicting two-layer and transient melt pool heat transfers respectively Finally the hybrid coupling approach encompassing transIVR and ANSYS Mechanical was validated against the RPV failure experiment FOREVER-EC2 and the results showed acceptable agreements with those of both the experiment and the previous approach using CFD in Section 43 Hence the transIVR code is an efficient tool for melt pool heat transfer and the hybrid approach an efficient way for RPV failure analysis

531 UCSB FIBS benchmark case

The UCSB FIBS benchmark case [78] was calculated to demonstrate the capability of the transIVR code in simulating two-layer pool heat transfer problems The considered pool configuration was similar to that shown in Figure 20 with the following specific dimensions a two-layer melt pool with a 16 m oxide layer and a 08 m metallic pool on top and an RPV lower head with a thickness of 015 m The internal volumetric source in the oxide layer was 13 MWm3 The calculation results were compared with those of the DOE study [78] and the INEEL study [79]

The predicted results were compared with the digitalized results of the DOE and INEEL studies in Figure 23 All three studies show similar profiles of the heat flux distribution along the polar angle (Figure 23(a)) it increased with the polar angle in the oxide layer to the maximum value at around 77deg and then dropped to a significantly lower value in the metal layer The transIVR result agreed well with the other results with small discrepancy in the metal layer the DOE study gave a

heat flux of 500 MWm2 the INEEL study of 400 MWm2 and the transIVR of somewhere in between Figure 23(b) shows the predicted corium crust thickness distribution along the RPV polar angle also agreed well the DOE and INEEL

transIVR ANSYS Mechanical

Transient

thermal loads

42

studies The crust thickness decreased with polar angle in the oxide layer and no crust was formed in the metal layer As the crust formation is closely related to the heat flux distribution the crust thickness profile was reasonable regarding the heat flux distribution shown in Figure 23(a) Vessel wall ablation would occur due to the high thermal loads Figure 23(c) gives the profile of the remaining vessel wall thickness along the RPV polar angle considering the wall ablation It shows that no ablation occurred in low angle region (0deg~36deg) where heat flux was low and the wall thickness changed from 15 cm to around 6 cm as angle increased in high angle region of the oxide layer Because of the discrepancy in the heat flux of metal layer the predicted wall thicknesses also differ from those of the DOE and INEEL studies the thinnest wall was 73 cm predicted by the DOE study corresponding to the maximum heat flux followed by 88 cm of the present transIVR calculation and 94 cm of the INEEL study Generally speaking the prediction of transIVR is in an acceptable level compared with the other studies for the same benchmark case

Figure 23 Simulation results of the UCSB FIBS benchmark case [74]

(a) heat flux distribution (b) crust thickness distribution

(c) wall thickness distribution

43

532 LIVE-7V experiment

LIVE-7V is an experiment under the LIVE program conducted in KIT to study the core melt phenomena As Figure 24(a) shows it employed the LIVE-3D hemispherical facility with an inner diameter of 1 m and a wall thickness of ~25 mm The non-eutectic binary salt mixture of 20 mol NaNO3- 80 mol KNO3 was used as the simulant material The decay heat in the melt was simulated by cable-type heating with 8 planes of electrical resistance heating wires Water cooling was applied to the top and vessel surfaces Top surface cooling was realized by a cooling lid mounted at height 413 mm from the vessel bottom and vessel cooling by the flow channel between the test vessel and the outer cooling vessel The inlet and outlet water temperature for both top and vessel cooling surfaces were kept ~293 K and ~303 K throughout the whole experiment respectively A 413 mm height pool corresponds to a total volume of ~0194 m3 simulant material The experiment procedure can be divided into four stages with the heating power sequentially shifted from 29 kW to 24 kW 18 kW and 9 kW Figure 24(b) indicates the thermocouple positions in experiment

The simulation results of LIVE-7V using transIVR were compared with experimental data [80] and numerical study using REALPSCDAPSIM code [81] For simplicity the RELAPSCDAPSIM results in the following are labelled as lsquoCOUPLErsquo which is the name of the module in RELAPSCDAPSIM for melt pool heat transfer

Figure 24 LIVE test facility [80] and thermocouple positions [82]

Figure 25 compares the predicted heat removal upward to the top lid and downward to the bottom vessel surface with those of the experiment and COUPLE The black curve indicates the total input power at the four stages The experimental results (red and blue dots) were calculated from the Table 3 of [80] assuming no extra heat losses For the first three stages the transIVR predictions agreed well with the COUPLE results while the predicted upward heat flow was slightly lower and the downward heat flow higher than the experimental data For the last stage the transIVR agreed well with the experimental result while the COUPLE predicted lower upward and higher downward heat flows The short

(a) (b)

44

transition periods between two stages with different total inputs taken in transIVR also agreed with those in the COUPLE predictions

Figure 25 Comparison of heat removal upward to the top melt surface cooling lid and downward to bottom vessel wall [74]

Figure 26 shows the comparison of heat flux distribution along the vessel wall All curves show similar trend that a higher total input power would result in a higher heat flux profile As the COUPLE predictions had relatively flatter profiles transIVR predictions generally agreed better with the experimental results especially when total input was high A discrepancy is seen that the minimum value of a curve occurred at angle 0degfor both transIVR and COUPLE and at angle 30deg for the experimental data It is noted that the models in default assume heat flux monotonically increases with angle (cf Eq (37) and (38)) resulting it impossible to predict higher heat flux at angle 0deg than angle 30deg as in experimental results

45

Figure 26 Heat flux profiles along the vessel polar angle [74]

533 FOREVER-EC2 experiment

In this section we performed calculations against the FOREVER-EC2 experiment to validate the performance of the coupled transIVR and ANSYS Mechanical in assessing the RPV integrity The applied boundary conditions were identical to the settings of previous coupling approach in Section 4 (ANSYS Fluent+ Mechanical) as indicated in Figure 11 The initial bulk temperature of the melt pool was set to be 1261 K in transIVR which corresponds to the maximum value of the initial temperature field in CFD calculation of previous coupling approach Both the experimental data and the simulation results using previous coupling approach are included for comparison In the following results from current hybrid approach are labelled lsquotransIVRrsquo (and referred as transIVR results) results from previous approach labelled lsquoCFDrsquo (and referred as CFD results) and results from experiment labelled lsquoexperimentrsquo

Figure 27 compares the bulk temperature predicted by transIVR with the averaged and maximum temperatures of the CFD results All curves show the same trend that the temperature gradually increased with time till reaching a steady state and then kept unchanged till the end The times needed for reaching the steady state were ~5000 s for the CFD and ~4000 s for the transIVR which are close to each other It indicates again a good performance of the transIVR in predicting the transient behavior of melt pool Compared with the steady-state averaged temperature 1478 K and maximum temperature 1609 K of the CFD results the steady-state temperature of transIVR 1599 K is a reasonable prediction

0

3000

6000

9000

12000

15000

18000

0 20 40 60 80 100

Hea

t fl

ux

(W

m2)

Polar angle (deg)

29 kW-EXP24 kW-EXP18 kW-EXP9 kW-EXP29 kW-COUPLE24 kW-COUPLE18 kW-COUPLE9 kW-COUPLE29 kW-transIVR24 kW-transIVR18 kW-transIVR9 kW-transIVR

46

Figure 27 Temperature evolution with time [74]

Figure 28 Temperature fields at time (a) 01 s (b) 3038 s (c) 6038 s and (d) 13289 s [74]

Figure 28 shows the temperature fields of the vessel at time 01 s 3038 s 6038 s and 13289 s The temperature fields were visualized in the ANSYS Mechanical It also indicates that the transient temperature loads were properly transferred from transIVR to ANSYS Mechanical Figure 28(a-d) show both the minimum and maximum temperatures increased with time In the lower head region the temperature gradually increased with the height The maximum temperature occurs at the location slightly lower than the melt pool top surface This is because thermal loadheat flux imposed on the vessel generally increases with polar angle in the pool natural convection so the code transIVR inherently also predicted higher heat flux at higher angle positions The higher flux at higher position then correspondingly resulted in higher temperature In the cylindrical region the

(a) (b) (c) (d)

[K] [K] [K] [K]

47

temperature decreased as the height since the bottom part would be more significantly heated by the high-temperature lower head region

Figure 29 shows the steady-state temperature profiles of the external vessel surface along the polar angle The transIVR result agreed well with other two results in trend that the temperature increased with the angle and reach the peak values at angle ~80deg then gradually decreased with angle But slightly lower temperature was predicted in the low angle region Possible reason may be that the local heat flux distribution determined by Eq (37) and (38) is steeper than that in the experiment and CFD study which then resulted in a steeper increase of temperature along the angle and a lower temperature at low angle region In the cylindrical part (gt90deg) the transIVR result generally agreed with the experimental and CFD results The discrepancy with the experimental data may be caused by the simplified boundary conditions the ambient temperature inside the vessel was set to uniform constant value while it could decrease as height Therefore predicted heat loss in the higher position of cylindrical part could be smaller than that with height-dependent ambient temperature resulting in higher vessel temperature Overall the prediction was reasonable for a lumped-parameter model

Figure 29 Temperature profiles of the external vessel surface along the vessel polar angle [74]

The transIVR predicted a failure time of 24959 s which is close to 24180 s in experiment and 23959 s in the CFD results Figure 30 shows the total deformation of vessel bottom point with time All three curves increased gradually with time and experienced drastic accelerations at the end The

400

600

800

1000

1200

1400

0 40 80 120 160

Tem

per

ature

(K

)

Polar angle (deg)

transIVRExperimentCFDWeilding Line

48

transIVR curve agreed well with the CFD curve The final total deformation at failure time is also close to that of the CFD result but lower than the experiment data The discrepancy might be related to the simplifications in the boundary conditions accuracy of the thermal load prediction and the modelling errors in plasticity and creep

Figure 30 Total deformation in vertical direction of the vessel bottom point [74]

Figure 31 shows the profile of wall thickness change The transIVR curve was compared with the CFD curve and four experimental curves measured at vessel left right front and back sides The simulation result fitted the experimental and CFD results well in the region 0deg~50deg In the region 50deg~90deg wall thickness gradually decreased with angle till around 75deg and then increased till 90deg The maximum thickness change was -3 mm for the simulation which is close to the experiment left side curve and slightly larger than the -6~ -45 mm of the other curves from experiment and CFD results Concerning possible errors and uncertainties in the modelling and numerical settings the new approach generally gives reasonable predictions

0

5

10

15

20

25

11670 15170 18670 22170 25670

Tota

l def

orm

atio

n (

mm

)

Time (s)

Experiment

CFD

transIVR

49

Figure 31 Wall thickness changes along the vessel polar angle [74]

-6

-5

-4

-3

-2

-1

0

1

0 20 40 60 80 100

Wal

l th

icknes

s ch

ange

(mm

)

Polar angle (deg)

CFD

transIVR

Experiment (left)

Experiment (right)

Experiment (front)

Experiment (back)

50

51

6 Summary and Outlook

Reactor pressure vessel (RPV) is one of the key physical barriers in the defence-in-depth approach of nuclear power safety and its failure is also the transition point of core melt (corium) progression from in-vessel to ex-vessel phase Quantification of RPV failure during a hypothetical severe accident is therefore paramount to the specific qualification of severe accident mitigation strategies as well as the general assessment of corium risk in light water reactors The RPV failure is a problem of structural performance of RPV under the influence of thermal loads imposed by corium melt pool ie it is a transient thermo-fluid-structure coupled problem per se whose solution requires simultaneous modelling of both highly turbulent melt pool heat transfer and large-scale nonlinear RPV creep Due to the complexity of the problem only limited capabilities have been developed in the previous studies under different simplifications and assumptions in models and coupling methods The present study was motivated to advance modelling and coupling approaches for RPV performance analysis considering new understanding of physical phenomena and latest coupling platform of simulation tools

For the new development of coupling approaches and models for physical phenomena related to RPV failure analysis during severe accidents this thesis work was conducted in the following order (i) a new creep model for the RPV steel 16MND5 was developed which features the capability to cover all three creep stages (ii) a new thermo-fluid-structure coupling approach was developed for RPV failure analysis and validated against FOREVER-EC2 experiment (iii) the performance of the new coupling approach using volume loads mapping (VLM) for data transfer was further compared with another approach using surface loads mapping (SLM) (iv) a hybrid coupling approach was proposed for efficient analysis of RPV failure which employs a lumped-parameter code for quick estimate of thermal load and a mechanistic FE code for detailed structural analysis and (v) for the hybrid coupling approach the code transIVR was developed for calculation of transient melt pool heat transfer

The developed creep model for 16MND5 is called lsquomodified theta projection modelrsquo and describes all three stages of a creep process Creep curves are expressed as a function of time with five model parameters 120579119894(119894 = 14 119886119899119889 119898) A dataset for the model parameters was constructed by fitting experimental creep curves into this function and an iterative parameter adjustment process to satisfy the monotonicity assumption of creep strain vs temperature and stress Model parameters for new curves under different temperature and stress loads are directly interpolated from this dataset in contrast to the conventional ldquotheta projection modelrdquo that employs an extra fitting process The new treatment was proven to better accommodate all experimental curves over the wide ranges of temperature and stress loads The model was implemented into the ANSYS Mechanical code The simulation results indicate that the model successfully captured all three creep stages and a good agreement was achieved between each experimental creep curve and corresponding modelrsquos prediction Regarding the model response to transient changes of temperature and stress loads the widely used time hardening and strain hardening models were implemented and

52

performing reasonably well in the new method These properties fulfil the requirements of a creep model for structural analysis

The transient thermo-fluid-structure coupling approach was developed using the ANSYS Fluent code for thermofluid dynamics of the melt pool and ANSYS Structural code for structural mechanics of RPV An extension tool was introduced to realize transient load transfer from ANSYS Fluent to Structural and minimize the user effort Both turbulence models and the effective model PECM can be used for melt pool heat transfer The developed creep model was used for RPV failure analysis of the RPV Since it is insensitive to geometry the coupling approach allows more complex geometries such as a BWR lower head with a forest of penetrations (CRGTs and IGTs) modelled in detail Moreover the coupling approach does not only reflect the transient response of thermo-fluid-structure interaction but also facilitates the implementation of advanced (high-fidelity) models in both fluid mechanics and structural mechanics The validation of the coupling approach against the FOREVER-EC2 experiment showed that the temperature profile vessel failure time total displacement of vessel bottom point and wall thickness profile in the experiment were well predicted

Following the variation of the coupling approach from VLM to SLM for data transfer between ANSYS Fluent and ANSYS Structural the respective coupling approaches were investigated by comparative calculations against the FOREVER-EC2 experiment and a postulated severe accident of a reference BWR For the FOREVER-EC2 experiment both VLM and SLM approaches predicted generally well the total deformation in vertical direction of bottom point and the wall thickness change though the former predicted a slightly later failure time For the reference BWR case good agreements were achieved in predicting the maximum creep strain and the total deformation in vertical direction of bottom point For both experiment and reactor cases the VLM approach predicted slightly later failure times than SLM but the differences were negligible compared to the long time-scale of vessel failure (~104 s) The deformation rate of the vessel wall was slow at the early stage and accelerated with time till a drastic deformation and final vessel failure Generally speaking the VLM and SLM coupling approaches had the similar performance in terms of their predictability of experiment and applicability to reactor case but the VLM was computationally more efficient

In the development of the hybrid coupling approach the transIVR code was developed for quick estimate of transient melt pool heat transfer and determination of thermal loads on RPV transIVR does not only support transient analysis of one- and two- layer melt pool but also the heat transfer in RPV being resolved with 2D finite difference method which captures more spatial thermal details for RPV structural analysis in addition to 1D heat transfer model The capabilities of transIVR in modelling two-layer pool heat transfer and transient pool heat transfer were demonstrated by calculations against the UCSB FIBS benchmark case and the LIVE-7V experiment respectively It was then coupled to the mechanical solver ANSYS Mechanical which together form the hybrid coupling framework for detailed RPV failure analysis The calculation against the FOREVER-EC2 experiment did not only reproduce the major results of the experiment but also had a good agreement with the thermo-fluid-structure

53

coupling approach using CFD This indicated the new coupling approach is capable of capturing the vessel creep failure characteristics and can be employed as an efficient tool addressing the complex vessel failure problem

As a prospective more research topics can be addressed in future either as a continuation of the current work toward an effective tool for vessel failure analysis or from the general view of scientific interest in melt pool heat transfer and RPV failure The following topics are recommended

bull Extended applications to the reactor cases For instance the thermo-fluid-structure coupling approach can be used to study penetration failures in the RPV of an BWR with a detailed representation of penetrations (IGTs and CRGTs)

bull Synthesis or update of heat transfer correlations for melt pool Heat transfer correlations directly benefit the parametric codes such as transIVR in accurate quantification of heat flux distribution around a melt pool So far many empirical correlations have been proposed based on experiments which had a variety of geometries and melt simulants with a wide range of Rayleigh number (Ra) Sensitivity studies have also been performed on the influences of Ra Pa and other material properties Therefore a comprehensive synthesis or update of the existing heat transfer correlations will be a meaningful step for making reasonable extrapolations to the reactor applications with prototypic materials

bull Modelling of pool stratification The motivation starts with the improvement of the PECMrsquos modelling capability such that the dynamic stratification and resulting pool heat transfer can be modelled altogether Simulation of melt stratification has been supported in some system codes such as ASTEC using a predefined parametric approach The improvement of PECM can be considered in the similar approach or more mechanistic modelling

bull Development of a structural module for RPV The idea is to develop a simplified module for structural analysis so it can be coupled with transIVR ultimately forming a standalone code for quick estimate of vessel failure of LWRs during postulated severe accidents

bull Extension of vessel analysis capability The current version of tranIVR is limited to melt pool Although the precursory debris bed and melt ejection after vessel failure were not the focus of the present study they are equally important to assessment of corium risk especially for Swedish BWRs where the melt release conditions and history are crucial to qualification of their severe accident mitigation strategies

54

55

Bibliography

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[2] IAEA Safety of Nuclear Power Plants Design Vienna INTERNATIONAL ATOMIC ENERGY AGENCY 2012

[3] M Fischer ldquoGen II and III In-Vessel and Ex-Vessel Melt Retentionrdquo The Short Course on Severe Accident Phenomenology Oct 2017

[4] B R Sehgal Nuclear Safety in Light Water Reactors Severe Accident Phenomenology 1st ed Academic Press 2011

[5] IAEA Approaches and Tools for Severe Accident Analysis for Nuclear Power Plants Vienna INTERNATIONAL ATOMIC ENERGY AGENCY 2008

[6] WENRA ldquoPractical Elimination Applied to New NPP Designs - Key Elements and Expectationsrdquo Sep 2019 [Online] Available httpwwwwenraorgmediafiler_public20191111practical_elimination_applied_to_new_npp_designs_-_key_elements_and_expectations_-_for_issuepdf

[7] P Matejovic M Barnak M Bachraty and L Vranka ldquoASTEC applications to VVER-440V213 reactorsrdquo Nuclear Engineering and Design vol 272 pp 245ndash260 2014 doi 101016jnucengdes201308077

[8] AREVA ldquoUS EPR Severe Accident Evaluationrdquo Topical Report ANP-10268NP Oct 2006 [Online] Available httpswwwnrcgovdocsML0631ML063100157pdf

[9] C T Martin ldquoCoupled 3D Thermo-mechanical Analysis of Nordic BWR Lower Head Failure in case of Core Melt Severe Accidentrdquo master thesis KTH Stockholm 2013

[10] IAEA ldquoAdvanced Large Water Cooled Reactors A supplement to the IAEArsquos Advanced Reactor Information System (ARIS)rdquo Sep 2015 [Online] Available httpsarisiaeaorgPublicationsIAEA_WRC_Bookletpdf

[11] B Pershagen Light Water Reactor Safety Pergamon Press 1989 [12] Ninos Garis M Agrell H Glanneskog and L Agrenius ldquoAPRI 7 ndash Accident

Phenomena of Risk Importancerdquo SSM SSM 201212 2012 [Online] Available httpswwwstralsakerhetsmyndighetensecontentassets2a9f299f46b949b3b209e780cf71436a201212-apri-7-accident-phenomena-of-risk-importance-en-lagesrapport-om-forskningen-inom-omradet-svara-haverier-under-aren-2009-2011-

[13] F Fichot et al ldquoSome considerations to improve the methodology to assess In-Vessel Retention strategy for high-power reactorsrdquo Annals of Nuclear Energy vol 119 pp 36ndash45 2018 doi 101016janucene201803040

[14] T G Theofanous C Liu S Additon S Angelini O Kymaumllaumlinen and T Salmassi ldquoIn-vessel coolability and retention of a core meltrdquo Nuclear Engineering and Design vol 169 no 1 pp 1ndash48 1997 doi 101016S0029-5493(97)00009-5

[15] C Sainte Catherine ldquoTensile and creep tests material characterization of pressure vessel steel(16MND5) at high temperatures (20 up to 1300 degC)rdquo CEA France SEMTLISNRT98-009A 1998

56

[16] E Altstadt H-G Willschuetz B R Sehgal and F-P Weiss ldquoModelling of in-vessel retention after relocation of corium into the lower plenumrdquo Forschungszentrum Rossendorf FZR-437 2005

[17] H-G Willschuumltz and E Altstadt ldquoGeneration of a High Temperature Material DataBase and its Application to Creep Tests withFrench or German RPV-steelrdquo FORSCHUNGSZENTRUM ROSSENDORF FZR-353 ISSN 1437-322X Aug 2002

[18] J L Rempe S A Chavez and G L Thinnes ldquoLight water reactor lower head failure analysisrdquo Nuclear Regulatory Commission NUREGCR--5642 1993 Accessed Aug 23 2019 [Online] Available httpinisiaeaorgSearchsearchaspxorig_q=RN25025905

[19] S Brosi et al ldquoCORVIS Investigation of light water reactor lower head failure modesrdquo Nuclear Engineering and Design vol 168 no 1 pp 77ndash104 1997 doi 101016S0029-5493(96)00008-8

[20] T Y Chu M M Pilch J H Bentz J S Ludwigsen W Y Lu and L L Humphries ldquoLower Head Failure Experiments and Analysesrdquo Sandia National Laboratories Albuquerque NM USA NUREGCR-5582 SAND98-2047 1999

[21] L L Humphries et al ldquoOECD Lower Head Failure Project Final Reportrdquo Sandia National Laboratories Albuquerque NM 87185-1139 2002

[22] B R Sehgal R R Nourgaliev T N Dihn A Karbojian J A Green and V A Bui ldquoFOREVER Experiments on Thermal and Mechanical Behavior of a Reactor Pressure Vessel during a Severe Accidentrdquo in Proceedings of the Workshop on in-vessel core debris retention and coolability Garching near Munich Germany Mar 3-6 1998

[23] S Bhandari M Fischer B R Sehgal and A Theerthan ldquoPost Test Analysis of the FOREVER C2 Experimentrdquo The 9th International conference on Nuclear Engineering (ICONE-9) Nice France Apr 8-12 2001

[24] J F Mao J W Zhu S Y Bao L J Luo and Z L Gao ldquoCreep deformation and damage behavior of reactor pressure vessel under core meltdown scenariordquo International Journal of Pressure Vessels and Piping vol 139ndash140 pp 107ndash116 2016 doi 101016jijpvp201603009

[25] J Mao L Hu S Bao L Luo and Z Gao ldquoInvestigation on the RPV structural behaviors caused by various cooling water levels under severe accidentrdquo Engineering Failure Analysis vol 79 pp 274ndash284 2017 doi 101016jengfailanal201704029

[26] J F Mao J W Zhu S Y Bao L J Luo and Z L Gao ldquoStudy on structural failure of RPV with geometric discontinuity under severe accidentrdquo Nuclear Engineering and Design vol 307 pp 354ndash363 2016 doi 101016jnucengdes201607027

[27] B R Sehgal V A Bui T N Dinh and R R Nourgaliev ldquoHeat Transfer Processes in Reactor Vessel Lower Plenum During Late Phase of In-Vessel Core Melt Progressionrdquo in Advances in Nuclear Science and Technology J Lewins M Becker R W Albrecht E J Henley J D McKean K Oshima A Sesonske H B Smets and C P L Zaleski Eds Boston MA Springer US 1999 pp 103ndash135

[28] C-T Tran and T-N Dinh ldquoThe effective convectivity model for simulation of melt pool heat transfer in a light water reactor pressure vessel lower head Part I Physical processes modeling and model implementationrdquo Progress

57

in Nuclear Energy vol 51 no 8 pp 849ndash859 2009 doi 101016jpnucene200906007

[29] C-T Tran and T-N Dinh ldquoThe effective convectivity model for simulation of melt pool heat transfer in a light water reactor pressure vessel lower head Part II Model assessment and applicationrdquo Progress in Nuclear Energy vol 51 no 8 pp 860ndash871 2009 doi 101016jpnucene200906001

[30] H-G Willschuumltz E Altstadt B R Sehgal and F-P Weiss ldquoCoupled thermal structural analysis of LWR vessel creep failure experimentsrdquo Nuclear Engineering and Design vol 208 no 3 pp 265ndash282 2001 doi 101016S0029-5493(01)00364-8

[31] E Altstadt and T Moessner ldquoExtension of the ANSYS creep and damage simulation capabilitiesrdquo Forschungszentrum Rossendorf Technical report FZR-296 2000

[32] H-G Willschuumltz E Altstadt B R Sehgal and F-P Weiss ldquoRecursively coupled thermal and mechanical FEM-analysis of lower plenum creep failure experimentsrdquo Annals of Nuclear Energy vol 33 no 2 pp 126ndash148 2006 doi 101016janucene200508006

[33] H-G Willschuumltz E Altstadt B R Sehgal and F-P Weiss ldquoSimulation of creep tests with French or German RPV-steel and investigation of a RPV-support against failurerdquo Annals of Nuclear Energy vol 30 no 10 pp 1033ndash1063 2003 doi 101016S0306-4549(03)00036-7

[34] H Madokoro A Miassoedov and T Schulenberg ldquoA thermal structural analysis tool for RPV lower head behavior during severe accidents with core meltrdquo Mechanical Engineering Letters vol 4 pp 18-00038-18ndash00038 2018 doi 101299mel18-00038

[35] W Villanueva C-T Tran and P Kudinov ldquoCoupled thermo-mechanical creep analysis for boiling water reactor pressure vessel lower headrdquo Nuclear Engineering and Design vol 249 pp 146ndash153 2012 doi 101016jnucengdes201107048

[36] W Villanueva C T Tran and P Kudinov ldquoA Computational Study on Instrumentation Guide Tube Failure During a Severe Accident in Boiling Water Reactorsrdquo in The 14th International Topical Meeting on Nuclear Reactor Thermal hydraulics NURETH-14 Toronto Ontario Canada Sep25-30 2011

[37] W Villanueva C T Tran and P Kudinov ldquoAssessment with Coupled Thermo-Mechanical Creep Analysis of Combined CRGT and External Vessel Cooling Efficiency for A BWRrdquo The 14th International Topical Meeting on Nuclear Reactor Thermal hydraulics NURETH-14 Toronto Ontario Canada Sep 25-30 2011

[38] F R Larson and J Miller ldquoA Time-Temperature Relationship for Rupture and Creep Stressesrdquo Trans ASME vol 74 p 765minus775 1952

[39] C T Tran ldquoDevelopment validation and application of an effective convectivity model for simulation of melt pool heat transfer in a light water reactor lower headrdquo Licentiate thesis KTH Stockholm 2007

[40] ANSYS ldquoANSYS Fluent Theory Guide Release 162rdquo ANSYS Inc Jul 2015 [Online] Available httpwwwansyscom

[41] P Zeng Finite element analysis and application (in Chinese) Tsinghua University Press amp Springer 2004

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[42] ANSYS ldquoANSYS Mechanical | Finite Element Analysis Softwarerdquo httpswwwansyscomproductsstructuresansys-mechanical-enterprise (accessed Sep 26 2019)

[43] ANSYS ldquoANSYS Mechanical APDL Theory Reference Release 162rdquo ANSYS Inc Jul 2015 [Online] Available httpwwwansyscom

[44] U Steinberner and H H Reineke ldquoTurbulent Buoyancy Convection Heat Transfer with Internal Heat Sourcesrdquo in Proceedings of the 6th Int Heat Transfer Conference Toronto Canada 1978 vol Vol2 pp 305ndash310 doi 101615IHTC63350

[45] R W Evans and B Wilshire Introduction to Creep London Institute of Materials 1993

[46] DoITPoMS ldquoCreep Deformation of Metalsrdquo Creep Deformation of Metals University of Cambridge 2006 httpswwwdoitpomsacuktlplibcreepindexphp (accessed Jul 09 2019)

[47] ANSYS ANSYS Mechanical APDL Material Reference Release 162 2015 [48] SIMULIA Abaqus 612 Analysis Userrsquos Manualvol 3 Materials Chapter

2324 2012 [49] S Imaoka ldquoUser Creep Subroutine STI0704A ANSYS Release110rdquo 2007 [50] R W Evans J D Parker and B Wilshire ldquoThe θ projection conceptmdashA

model-based approach to design and life extension of engineering plantrdquo International Journal of Pressure Vessels and Piping vol 50 no 1 pp 147ndash160 1992 doi 1010160308-0161(92)90035-E

[51] ECCC ldquoRecommendations and Guidance for the Assessment of Creep Strain and Creep Strength Data ECCC Recommendations Volume 5 Part 1 b[Issue 2]rdquo 2013

[52] W D Day and A P Gordon ldquoLife Fraction Hardening Applied to a Modified Theta Projection Creep Model for a Nickel-Based Super-Alloyrdquo The ASME Turbo Expo 2014 Turbine Technical Conference and Exposition Duumlsseldorf Germany Jun16-20 2014 vol 7A-Structures and Dynamics p V07AT29A010 doi 101115GT2014-25881

[53] M Kumar I V Singh B K Mishra S Ahmad A Venugopal Rao and V Kumar ldquoA Modified Theta Projection Model for Creep Behavior of Metals and Alloysrdquo J of Materi Eng and Perform vol 25 no 9 pp 3985ndash3992 2016 doi 101007s11665-016-2197-y

[54] P Yu and W Ma ldquoA modified theta projection model for creep behavior of RPV steel 16MND5rdquo Journal of Materials Science amp Technology vol 47 pp 231ndash242 2020 doi 101016jjmst202002016

[55] ANSYS ANSYS Mechanical Advanced Nonlinear Materials v160 Lecture 2 Rate Dependent Creep ANSYS training materials 2015

[56] Q Y Li N C Wang and D Y Yi Numerical Analysis (in Chinese) 5th ed Tsinghua University Press 2008

[57] T N Dinh R R Nourgaliev and B R Sehgal ldquoOn heat transfer characteristics of real and simulant melt pool experimentsrdquo Nuclear Engineering and Design vol 169 no 1 pp 151ndash164 1997 doi 101016S0029-5493(96)01283-6

[58] O Kymaumllaumlinen H Tuomisto O Hongisto and T G Theofanous ldquoHeat flux distribution from a volumetrically heated pool with high Rayleigh numberrdquo

59

Nuclear Engineering and Design vol 149 no 1ndash3 pp 401ndash408 1994 doi 1010160029-5493(94)90305-0

[59] F J Asfia and V K Dhir ldquoNatural convection heat transfer in volumetrically heated spherical poolsrdquo 1994 Accessed Aug 23 2019 [Online] Available httpinisiaeaorgSearchsearchaspxorig_q=RN26037727

[60] M Jahn and H H Reineke ldquoFree Convection Heat Transfer with Internal Heat Sources Calculations and Measurementsrdquo in Proceedings of the 5th Int Heat Transfer Conference Tokyo Japan 1974 vol Vol3 paper NC-28

[61] M Gustafsson ldquoFSI Transient extension V1704rdquo EDR MEDESO 2016 [62] EDR MEDESO httpsedrmedesocom [63] H Kim and I Namgung ldquoThermo-mechanical Analysis of RVLH for

APR1400 during a Severe Accidentrdquo in Transactions SMiRT-23 Manchester United Kingdom Aug10-14 2015 [Online] Available httpwwwlibncsueduresolver18402034197

[64] P Yu and W Ma ldquoComparative Analysis of Reactor Pressure Vessel Failure using Two Thermo-Fluid-Structure Coupling Approachesrdquo submitted to journal

[65] F R Menter ldquoTwo-equation eddy-viscosity turbulence models for engineering applicationsrdquo AIAA Journal vol 32 no 8 pp 1598ndash1605 1994 doi 102514312149

[66] T N Dinh and R R Nourgaliev ldquoTurbulence modelling for large volumetrically heated liquid poolsrdquo Nuclear Engineering and Design vol 169 no 1 pp 131ndash150 1997 doi 101016S0029-5493(96)01281-2

[67] ANSYS Inc ldquoIntroduction to ANSYS Fluent Lecture 7 Turbulence Modelling Fluent Tutorial Materialrdquo 2016

[68] A Horvat and M Borut ldquoNumerical investigation of natural convection heat transfer in volumetrically heated spherical segmentsrdquo in Proceedings of the International Conference Nuclear Energy for New Europe 2004

[69] Y Li CFD Pre-test Analysis of SIMECO-2 Experiment 2016 [70] A Theerthan A Karbojian and B R Sehgal ldquoEC-FOREVER experiments

on thermal and mechanical behavior of a reactor pressure vessel during a severe accident Technical report-1 EC-FOREVER-2 testrdquo SAM-ARVI-D008 Mar 2001

[71] P Yu W Ma W Villanueva A Karbojian and S Bechta ldquoValidation of a thermo-fluid-structure coupling approach for RPV creep failure analysis against FOREVER-EC2 experimentrdquo Annals of Nuclear Energy vol 133 pp 637ndash648 2019 doi 101016janucene201906067

[72] T G Theofanous C Liu S Additon S Angelini O Kymaumllaumlinen and T Salmassi ldquoIn-vessel coolability and retention of a core meltrdquo Nuclear Engineering and Design vol 169 no 1ndash3 pp 1ndash48 1997 doi 101016S0029-5493(97)00009-5

[73] H P Langtangen ldquoSolving nonlinear ODE and PDE problemsrdquo Simula Research Laboratory 2016 [Online] Available httpshplgitgithubionum-methods-for-PDEsdocpubnonlinpdfnonlin-4printpdf

[74] P Yu and W Ma ldquoDevelopment of a Lumped-Parameter Code for Efficient Assessment of In-Vessel Melt Retention Strategy of LWRsrdquo submitted to journal

60

[75] T G Theofanous M Maguire S Angelini and T Salmassi ldquoThe first results from the ACOPO experimentrdquo Nuclear Engineering and Design vol 169 no 1 pp 49ndash57 1997 doi 101016S0029-5493(97)00023-X

[76] S W Churchill and H H S Chu ldquoCorrelating equations for laminar and turbulent free convection from a vertical platerdquo International Journal of Heat and Mass Transfer vol 18 no 11 pp 1323ndash1329 1975 doi 1010160017-9310(75)90243-4

[77] S Globe and D Dropkin ldquoNatural-Convection Heat Transfer in Liquids Confined by Two Horizontal Plates and Heated from Belowrdquo J Heat Transfer vol 81 no 1 pp 24ndash28 1959 doi 10111514008124

[78] T G Theofanous C Liu S Additon S Angelini O Kymaelaeinen and T Salmassi ldquoIn-vessel coolability and retention of a core melt Volume 2rdquo Argonne National Lab DOEID--10460-VOL2 1996 Accessed Apr 18 2019 [Online] Available httpinisiaeaorgSearchsearchaspxorig_q=RN30000459

[79] J L Rempe D L Knudson C M Allison G L Thinnes and C L Atwood ldquoPotential for AP600 in-vessel retention through ex-vessel floodingrdquo United States INEELEXT--97-00779 1997 Accessed Apr 18 2019 [Online] Available httpinisiaeaorgSearchsearchaspxorig_q=RN30025872

[80] A Miassoedov X Gaus-Liu T Cron and B Fluhrer ldquoLive experiments on melt pool heat transfer in the reactor pressure vessel lower headrdquo The 10th International Conference on Heat Transfer Fluid Mechanics and Thermodynamics (HEFAT 2014) Orlando FL Jul 14-16 2014

[81] H Madokoro A Miassoedov and T Schulenberg ldquoAssessment of a lower head molten pool analysis module using LIVE experimentrdquo Nuclear Engineering and Design vol 330 pp 51ndash58 2018 doi 101016jnucengdes201801036

[82] B Fluhrer et al ldquoThe LIVE-L1 and LIVE-L3 experiments on melt behaviour in RPV lower headrdquo Forschungszentrum Karlsruhe GmbH Technik und Umwelt (Germany) Inst fuer Kern- und Energietechnik FZKA--7419 2008 Accessed Nov 12 2019 [Online] Available httpinisiaeaorgSearchsearchaspxorig_q=RN40036867

V

List of Publications

Papers included in this thesis

I P Yu and W Ma ldquoA modified theta projection model for creep behavior of RPV steel 16MND5rdquo Journal of Materials Science amp Technology vol 47 pp 231ndash242 2020

II P Yu W Ma W Villanueva A Karbojian and S Bechta ldquoValidation of a thermo-fluid-structure coupling approach for RPV creep failure analysis against FOREVER-EC2 experimentrdquo Annals of Nuclear Energy vol 133 pp 637ndash648 2019

III P Yu and W Ma ldquoComparative analysis of reactor pressure vessel failure using two thermo-fluid-structure coupling approachesrdquo Nuclear Engineering and Design submitted

IV P Yu and W Ma ldquoDevelopment of a lumped-parameter code for efficient assessment of in-vessel melt retention strategy of LWRsrdquo Annals of Nuclear Energy submitted

Respondentrsquos contribution to the papers included in the thesis

Peng Yu made major contributions to the publications of all the above four articles under the guidance of supervisors and with the collaboration of other co-authors including formulation of research questions development of coupling approaches and numerical methods analysis and generalization of simulation results as well as writing the first drafts and addressing review comments

Papers not included in this thesis

1 P Yu and W Ma ldquoRPV creep failure analysis using a Norton creep modelrdquo International Journal of Mechanical Sciences to be submitted

2 P Yu W Villanueva S Galushin W Ma and S Bechta ldquoCoupled thermo-mechanical creep analysis for a Nordic BWR lower head using non-homogeneous debris bed configuration from MELCORrdquo 12th International Topical Meeting on Nuclear Reactor Thermal-Hydraulics Operation and Safety (NUTHOS-12) Qingdao China October 14-18 2018

3 P Yu A Komlev Y Li W Villanueva W Ma and S Bechta ldquoPre-test simulation of SIMECO HT experiments for stratified melt pool heat transferrdquo 11th International Topical Meeting on Nuclear Reactor Thermal Hydraulics Operation and Safety (NUTHOS-11) Gyeongju Korea October 9-13 2016

VI

VII

Acknowledgements

First of all I would like to express my sincere gratitude to my supervisor Prof Weimin Ma for offering this opportunity to work in the severe accident research field and for the supports patience guidance discussions and encouragements throughout the entire period of my graduate study

I am very thankful to my co-supervisor Dr Walter Villanueva for the valuable help and discussions I learnt a lot during the times working with him I am also thankful to co-supervisor Prof Sevostian Bechta who is supportive and always has inspiring ideas that keep you thinking for a long time

Thank you also to all my colleagues at the Division of Nuclear Power Safety for the supports discussions and the very nice working environment during these years

I have taken many interesting courses during my PhD study for which I want to express my thanks to all the teachers With this regard a special thanks goes to Prof Johan Hoffman for his SF2561 Finite Element Method given in 2015 fall as well as the discussions and advices on numerical analysis

I am grateful to Ignacio Gallego-Marcos and Wenyuan Fan for their enthusiastic help on CFD and turbulence modelling during the time working on the turbulent

heat transfer I thank all my 师兄 for sharing their knowledge on nuclear engineering and giving advices when I was confused since 2012

I thank all my friends both in Sweden and in China for the company and the sharing of many wonderful moments in daily life

Finally I would like to express my deepest thank to my parents my girlfriend Siwen my sisters and all my other relatives for their unconditional supports and love

Financial supports from the China Scholarship Council and APRI make my study in KTH possible and are acknowledged

VIII

IX

Table of Contents

AbstracthelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphellipI

SammanfattninghelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphellipIII

List of PublicationshelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphellipV

AcknowledgementshelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphellipVII

NomenclaturehelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphellipXI

1 Introductionhelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphellip1

11 Nuclear Power Safety 1

12 RPV Failure 2

13 Literature Review Creep Behaviour and RPV Failure Analysis 4

131 Experiments on creep behaviour and RPV failure 4

132 Numerical studies on RPV failure 5

14 Motivation Objectives and Main Achievements 7

2 Numerical Solvers and Modelshelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphellip9

21 ANSYS Fluent 9

22 ANSYS Mechanical 9

23 PECM 11

3 Creep Modelling of 16MND5 Steelhelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphellip13

31 Creep and Creep Modelling 13

32 A Modified Theta Projection Model 14

321 Original theta projection model and its variations 14

322 A modified theta projection model 16

4 Coupled Thermo-Fluid-Structure Analysis of RPV Failurehelliphelliphelliphellip23

41 Thermo-Fluid-Structure Coupling Approach 23

42 Numerical Modelling 25

421 Conjugate heat transfer 25

422 Mechanical analysis 25

43 Validation against the FOREVER-EC2 Experiment 26

431 FOREVER-EC2 experiment 26

X

432 Conjugate heat transfer 27

433 Mechanical analysis 29

44 Application to BWR Case 32

45 Computational Efficiency 34

5 Hybrid Coupling Approach for Efficient RPV Failure Analysishelliphellip35

51 A Lumped-parameter Code for Melt Pool Heat Transfer 35

511 Heat transfer correlations 36

512 Energy balance in the pools 37

513 Energy balance in the RPV wall 39

52 Hybrid Coupling Approach 40

53 Verification and Validation 41

531 UCSB FIBS benchmark case 41

532 LIVE-7V experiment 43

533 FOREVER-EC2 experiment 45

6 Summary and Outlookhelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphellip51

Bibliographyhelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphellip55

XI

Nomenclature

Latin

119860 area of a surface

119862119901 heat capacity

119863 damage factor

119892 gravity

ℎ heat transfer coefficient

H Melt pool height

119896 thermal conductivity

119871 characteristic length

119873119906 Nusselt number

119875119903 Prandtl number

119902 heat flux

119902119894119899 heat flux through inner vessel surface

119876119881 internal volumetric heat source

119903 radial distance in a cylindrical or spherical coordinate

119877119886 external Rayleigh number

119877119886119894119899 internal Rayleigh number

119877119886119910 local Rayleigh number

119905 time

119879 temperature

119881119900119897 total volume

119909 thickness (of crust or RPV wall)

119911 height in a cylindrical coordinate

Greek

120572 thermal diffusivity

120573 thermal expansion

Δ119879 temperature difference

ε emissivity (in thermal analysis) strain (in structural analysis)

휀119888119903 creep strain

휀119903 creep strain rate

120579 polar angle

120579119894 parameters of a creep model 119894 = 1234119898

120584 kinematic viscosity

σ stress

XII

Super and subscripts

119886119898119887 ambient

119888119903 creep (in Section 3) crust (in Section 5)

119889119899 downward

119890119899119907 environment

119894 inner surface of RPV

119894119899119905 interface

119897119900119888119886119897 local

119898119886119909 maximum value

119898119890119897119905119894119899119892 melting point

119898119905 metal layer (bulk)

119898119905119889119899 downward surface of metal pool

119898119905119904119889 sideward surface of metal pool

119898119905119905119900119901 top surface of metal layer

119898119905119906119901 upward surface of metal pool

119900 outer surface of RPV

119900119909 oxide layer (bulk)

119900119909119889119899 downward surface of oxide layer

119900119909119906119901 upward surface of oxide layer

119903119890119891 reference

119904 internal structures inside the RPV

119904119889 sideward

119906119901 upward

119907119890119904 vessel

Acronyms

BWR boiling water reactor

CBD creep database

CFD computational fluid dynamics

CRGT control rod guide tube

ECCM effective convectivity conductivity model

FDM finite difference method

FE finite element

FEA finite element analysis

FEM finite element method

FVM finite volume method

IGT instrument guide tube

IVR in-vessel retention

XIII

IVMR in-vessel melt retention EU project

LES large eddy simulation

MPS moving particle semi-implicit method

NPP nuclear power plant

PECM phase-change effective convectivity model

RANS Reynolds-averaged Navier-Stokes

RPV reactor pressure vessel

SBO station blackout

SLM surface loads mapping

SPH smoothed-particle hydrodynamics

UDF user-defined function

UPF user programmable features

VLM volume loads mapping

XIV

1

1 Introduction

11 Nuclear Power Safety

Greenhouse gas emissions contribute to the global warming problem Phasing out fossil fuels by switching to low-carbon energy sources is considered as a mitigation strategy With this regard clean energy should be increased in the world energy consumption structure Being clean and stable nuclear energy is an attractive alternative to fossil fuels After the development over decades it also proves to be a mature and safe technology According to IAEA 2018 annual report [1] there were 450 nuclear power reactors under operation worldwide at the end of 2018 which had a record generating capacity of 3964 GW(e) 9 reactors were connected to the grid and 55 reactors under construction around the world

Due to the potential risk of the release of radioactivity nuclear power safety is emphasized throughout the stages of design construction operation and decommission of a nuclear power plant (NPP) The principal technical requirements [2] specify that the three fundamental safety functions should be fulfilled for all plant states of an NPP (i) control of reactivity (ii) removal of heat from the reactor and from the fuel store and (iii) confinement of radioactive material shielding against radiation and control of planned radioactive releases as well as limitation of accidental radioactive releases The defence-in-depth concept is also applied to provide several levels of defence to prevent consequences of accidents The fuel claddings reactor pressure vessel (RPV) and primary coolant pressure boundaries and the containment are the three physical barriers to prevent the release of radioactive materials to the environment

Severe accidents are events that cause significant damage to reactor fuel and result in core meltdown It may be caused by insufficient cooling of the reactor core due to various reasons or by the rapid introduction of huge reactivity which causes sharp increases in the power and fuel fragmentation eg Chernobyl accident During a severe accident the failure of the fuel claddings will result in the radioactive materials previously enclosed inside the cladding released to the primary loop If cooling is not sufficient to remove the decay heat the core materials will melt migrate down and relocate into the RPV lower head after which the melt will react with water and partially form fragments quench solidify and form a debris bed The debris bed formed by the melt may continue to melt to form a melt pool due to the decay heat As the corium contacts the RPV it would continuously heat the RPV and threaten its integrity Once the RPV fails containment becomes the last barrier to prevent the release of radioactivity The released melt would result in an increase of the pressure and temperature due to the high temperature fractions oxidation and hydrogen combustion which is called direct containment heating The melt would fall onto the reactor cavity Without sufficient cooling the molten core would interact with basemat concrete (MCCI) This process can last for few days It can result in containment failure due to either containment overpressure or the basemat melt-through of containment In addition containment is also subjected to isolation failure and early containment failure due to missiles and dynamic pressure [3] More

2

detailed severe accident progression or phenomena could also be found in eg [4] [5]

Though severe accidents are unlikely to occur the consequences can be severe if no proper measure is taken Related researches have been continuously going on aiming at mitigating the accident consequences and improving safety After the Fukushima Accident safety inspections and re-evaluation actions have been carried out worldwide especially on safety measurements regarding severe accidents The concept lsquoPractical Eliminationrsquo applied to new NPP designs demands that new nuclear installations be designed with the objective of mitigating the consequences of accidents and avoiding early radioactive releases and large radioactive releases [6] Severe accident mitigation strategies are applied to the existing and new NPPs (both Generation II and III) The strategies can generally be classified into two categories [3] the in-vessel retention (IVR) and ex-vessel retention (EVR) In simple words in-vessel retention tries to retain the core melt inside the RPV (second barrier) by applying external water cooling on the vessel as shown in Figure 1(a) Such method minimizes the dispersal of the melt within the containment and hence simplifies the later recovery work Examples of the application are AP1000 HPR1000 APR 1400 The idea of the EVR is to let the RPV fail but to retain the core melt inside the containment Two principle approaches of EVR are the retention within a dedicated lsquocore catcherrsquo and melt fragmentation Examples of core catcher would be EPR that directs the melt to a spreading compartment (Figure 1(b)) and VVER1000 that uses metallic crucible below the RPV An example of the melt fragmentation is the Nordic BWRs that the corium released from RPV would relocate in the cavity with a deep-water pool (7-12 m) and is expected to fragment quench and form a coolable debris bed (Figure 1(c))

Figure 1 Examples of IVR [7] EVR with core catcher [8] and melt fragmentation [9]

12 RPV Failure

RPV which contains the reactor core core shroud and coolant is a key component to the reactor system RPV and the primary loop constitute the primary coolant system which conveys the fission heat from the reactor core to the secondary loop through steam generator (PWR) or to the turbine (BWR) The pressure under

(a) IVR (b) Core catcher (c) Melt fragmentation

3

operation can be ~155 MPa for a 1000 MW PWR [10] and ~70 MPa for a 1000 MW BWR [11] It also has to withstand the neutron radiation damage The thickness of RPV [11] can be ~20 cm for PWR and ~16 cm for BWR

As the transition of in-vessel progression to ex-vessel progression RPV failure is important for analysis of accident progression and assessment of the effectiveness of severe accident mitigation strategies In the melt fragmentation strategy of Nordic BWR the melt-coolant interaction and debris bed coolability after the corium poured into the deep-water pool are essentially dependent on the vessel failure mode including the characteristic time of melt release rupture size and location and amount and superheat of melt available for release [12] New concern also arose [13] on the necessity of performing structural analysis for the RPV in the IVR strategy In the initial methodology and further development of the IVR the success of the strategy ie the integrity of the RPV was demonstrated (eg in [14]) with the following two conclusions (i) even for a local heat flux on the vessel as high as critical heat flux (CHF) the resulted very thin external ldquocold layerrdquo of the vessel would be sufficient to withstand the internal stress load and hold the vessel integrity (ii) as CHF is the critical value for the thermal load the main specific goal for assessing the effectiveness of IVR becomes to demonstrate that the local heat flux along RPV be lower than the CHF The new concern doubts that local lsquohot partrsquo of the vessel would have negative effects on the overall resistance due to creep and dilatation which could result in a vessel failure and hence violate the first conclusion in particular for ldquolong termrdquo retention of corium in RPV If so a structural integrity analysis of RPV would be needed in addition to assessing local heat flux especially for high-power reactors Therefore accurate prediction of vessel failure could be meaningful for both strategies

Figure 2 Illustration of RPV under thermal loads from corium

As the structural performance is highly affected by the thermal loads from the corium RPV failure is essentially a transient thermo-fluid-structure coupled problem Figure 2 illustrates the situation that a corium debris bedmelt pool formed in the RPV lower head The decay heat of nuclear fission products could continuously be generated inside the corium leading to the formation of melt pools as the debris melts The heated debris bedmelt pool imposes thermal and

4

mechanical loads on the RPV Such loads can be transient (change with time) and spatially non-uniform Complex phenomena are also involved in this process Strong turbulent natural convection induced by the decay heat may be expected in the melt pool under IVR The corresponding Rayleigh number 119877119886 can be 1016minus17 for prototypic reactor cases Material structural properties can become non-linear under high temperature and stress loads The creep deformation also becomes significant at high temperature Detailed modelling of turbulent natural convection and non-linear properties could be computationally expensive In addition large-scale geometry (hemisphere with a radius of 20-30 m) long-term transient deformation process (~10000 s) and required small time steps (eg 001 s) for numerical convergence consideration further increase the computational costs

13 Literature Review Creep Behaviour and RPV Failure Analysis

In this section the most relevant studies concerning the experiments modelling and analysis of creep behaviour and RPV failure are presented

131 Experiments on creep behaviour and RPV failure

Various experiments have been performed on the RPV failure problem some of which were to generate data of mechanical properties (especially the creep properties) and some to study the overall vessel failure with scaled or simplified geometries

The REVISA experiment [15] generated creep curves and stress strain curves of French steel 16MND5 ranging from 600 to 1300 degC This data is accessible in [16] as well as [17] which also documents other material properties eg density thermal expansion coefficient conductivity heat capacity Youngrsquos Modulus and plasticity Similar experiments were carried out to generate high-temperature creep and tensile test data for American RPV steels and Inconel [18]

The CORVIS experiments [19] investigated the lower head failure modes with and without penetrations Each experiment was carried out on a steel plate 100 mm thick representing a pressure vessel lower head The penetration case represented a boiling water reactor lower head carrying a tube penetration in the form of a drain line

The LHF (lower head failure) experiments [20] and the succeeding OLHF tests [21] investigated the vessel failure with a 15 scaled vessel heated by an electrical radiator allowing to locate the hot focus at different position of the vessel with a maximum wall temperature of around 900 degC Internal pressure ranging from 53 to 123 bar was also imposed to initiate the vessel failure

The FOREVER experiments [22] investigated the phenomena of melt pool convection vessel creep possible failure processes and failure modes occurred during the late phase of the in-vessel progression by heating up a melt pool in a vessel and observing possible vessel failures The study covers cases with and without penetration different melt level existence of gap cooling and in-vessel melt coolability

5

132 Numerical studies on RPV failure

Various numerical approaches have been employed on RPV failure Since a two-way thermo-fluid-structure coupling approach is challenging these approaches take simplifications and can generally be classified into two categories (i) RPV failure decoupled from melt pool heat transfer and (ii) RPV failure coupled with melt pool heat transfer in simplified manners

1) Decoupled approaches In the decoupled approaches only RPV is modelled Simplified constant thermal loads are imposed on RPV considering the melt pool heat transfer

In a post-test analysis of FOREVE-C2 experiment [23] it was treated as a thermo-structural coupled analysis of the vessel with the thermal load estimated by the outer thermocouples and the normal heat flux on the external vessel wall The analysis was done using finite element (FE) code SYSTUS A coupled creepdamage law based on a viscosity-hardening creep law describing primary and secondary creep was employed for creep analysis

Mao et al conducted various studies on reactor cases assessing the IVR strategy using FE code ABAQUS Thermal boundary conditions were imposed on the RPV which was then solved as a thermal-structural coupled problem The strain hardening creep model (Eq (1)) was mainly used Their studies cover the general RPV failure [24] as well as the effects of cooling water levels under IVR [25] and geometric discontinuity (ablated RPV) [26]

ε119888119903 = 1198891σ1198892ε1198893119890119909119901 [minus

1198894119879] (1)

In the EU IVMR project [13] the RPV integrity was also studied through a benchmark calculation which deals with the RPV of a PWR with a prescribed ablated profile with 2D axisymmetric geometry Prescribed temperature gradient was set as thermal load The objective was to calculate deformations for various pressure loads up to vessel rupture A smoother profile was planned in further research

The advantage of such approaches is once thermal loads are reasonably assumed the focus only needs to be put on the RPV The disadvantage is a reasonable thermal load should be consistent with the melt pool heat transfer while sometimes it is not well-noted eg constant uniform temperature boundary conditions were assumed for inner and outer RPV (with uniform thickness in lower head) surfaces [25] which ignored the non-uniform heat flux distribution caused by the turbulent convective corium pool As thermal loads strongly affect the structural performance they should be quantified carefully

2) Coupled approaches In coupled approaches both melt pool heat transfer and RPV failure are modelled and coupled in one-way or simplified manner With melt pool heat transfer modelled the coupled approaches are more flexible to provide reasonable thermal loads on the RPV In existing approaches CFD and the correlation-based distributed-parameter models eg the Effective Convectivity Conductivity Model

6

(ECCM) [27] and the Phase-change Effective Convectivity Model (PECM) [28] [29] are general methods for melt pool heat transfer

Sehgal et al [22] presented an approach that employed the MVITA code to calculate the thermal processes and ANSYS for structural analysis MVITA code employed the ECCM Bailey-Norton creep law was used as shown in Eq (2)

ε119888119903 = 6055 times 10minus26 sdot σ472119905085119890minus

37200

119879 (2)

This approach was used for the pre-test simulation of FOREVER experiment to determine the thermal and mechanical loadings Parametric investigations also helped in the identification of important factors that could affect the test procedure and test conditions

Willschutz et al [30] employed the CFD-module Flotran for the melt pool heat transfer and ANSYS for structural analysis of RPV Standard k-ε turbulence model was selected for the turbulent natural convection in the melt pool A user-defined three-stage creep model [31] was employed In this model experimental creep curves were discretized into a creep database (CDB) under given temperature 119879 and stress 120590 the creep curve was discretized into finite point pairs

(휀 ε) indicating under such thermal and mechanical loads the creep strain rate ε is subjected to creep strain 휀 The creep strain rate for any other case can be interpolated over known temperature stress and strain from the database This method was applied to the post-test calculation of the FOREVE-C2 experiment It was concluded that the unexpected deformation behaviour during the experiment was caused by the combination of the transient temperature field in the vessel wall and the three creep stages

They later replaced the Flotran solver with the ECCM for melt pool heat transfer in a following study [32] By implementing it into FEM-solver a recursively coupled analysis was constructed Moreover a modified creep model [33] was used with damage measures the CDB itself did not contain tertiary creep instead a factor 1(1 minus 119863) was multiplied to the original creep strain rate to get the modified creep strain rate where 119863 is a damage factor (0 lt 119863 lt 1) defined with creep strain information and plastic strain information As 119863 grows close to 1 the modified creep strain rate would be enlarged significantly by this factor 1(1 minus 119863) which then behaves like the accelerating tertiary stage This method was applied to pre- and post- test calculations for the FOREVER experiments proving to be closer to reality than the previous method

Madokoro et al [34] developed a thermal structural analysis tool based on open source solver OpenFOAM and numerically investigated the coupled RPV failure and melt pool convection The PECM was used in the melt pool heat transfer In structural part the plasticity and damage parameters were similar to Willschutz et al [32] except for that strain hardening creep model (Eq (3)) was used instead of a CDB Simulations of FOREVER-EC2 and -EC4 were investigated It was concluded from the simulations that although the emissivity and mesh type (updated or fixed) affect the results significantly the approach generally has the capability of predicting the vessel deformation and failure

7

ε119888119903 = 1198891 sdot σ1198892 sdot ε1198893 (3)

Villanueva et al [35] performed a coupled thermo-mechanical creep analysis of the reactor scale RPV (ABB-Atom design BWR) to evaluate efficacy of the cooling measures for the in-vessel melt coolability and retention The PECM was used for melt pool heat transfer and ANSYS for RPV integrity analysis The modified time hardening model (Eq (4)) of the American RPV steel SA533B was selected as creep model The transient thermal load from the melt pool was applied as a thermal boundary condition to the vessel wall Further studies were done with this approach on investigations of instrumentation guide tube (IGT) failure [36] and control rod guide tube (CRGT) cooling [37] of a Nordic BWR

ε119888119903 =1198881σ

11988821199051198883+1

1198883 + 1 1198881 gt 0 (4)

It is noted that only RPV analysis with finite element analysis (FEA) are reviewed here Engineering methods like the Larson-Miller method [38] are not covered in this thesis One can refer to theory manuals of severe accident codes like MELCOR ASTEC for more details Comprehensive description on modelling of melt pool heat transfer is not covered either One can refer to either [39] for a general view or to any severe accident code manuals for engineering methods

14 Motivation Objectives and Main Achievements

The existing experimental and numerical studies provide us a good understanding of RPV failure involved phenomena and valuable simulation experiences These numerical methods may vary in solvers (which may be related to code accessibility) and models (which may be related to model accessibility understanding of phenomena computational power) However improvements of modelling approaches are still possible as the improvements in any of the related aspects eg

bull Simulation tools are continuously being upgraded and providing new possibilities just like that PECM appeared and ECCM was no longer used

bull Better understanding of the phenomena and more simulation experiences from in-vessel retention and vessel failures researches benefit the further modelling and simulations

bull Increasing computational power allows more detailed modelling eg turbulence modelling of melt pool convection

bull There is still room for the improvement of creep modelling

In addition implementation difficulty of a coupling approach is another important issue to consider Since it is a TRANSIENT thermo-fluid-structure coupled problem it is almost impossible to replicate an approach simply by referring to a paper with limited details For BWRs it would be even better for the coupling approaches to support complex geometries considering forest of penetrations (CRGTs and IGTs) on the RPV lower head without significant increase of user effort in the coupling process

8

The objectives of this thesis are to investigate new approaches and develop new models for the thermo-fluid-structure coupled problem of RPV failure and involved phenomena These objectives are accomplished with the following achievements

1) A new creep model for the 16MND5 steel was developed which covers all three creep stages Good agreement was achieved between model predictions and experimental curves Both time and strain hardening models were implemented (Paper I)

2) A new thermo-fluid-structure coupling approach was developed for RPV failure analysis It supports the advanced models in both melt pool convection and structural mechanics to improve fidelity and facilitates implementation Validation of the approach with the new creep model was performed against FOREVER-EC2 experiment This approach is flexible for complex geometries and can be used in further analysis eg lower head with detailed modelling of penetrations in BWRs (Paper II)

3) The performance of this new approach which uses volume loads mapping for data transfer was compared with the previous approach using surface loads mapping showing good agreement with small discrepancy (Paper III)

4) A lumped-parameter code transIVR for melt pool heat transfer was developed which supports one- and two-layer pool configuration transient analysis and well-resolved RPV wall (Paper IV)

5) A hybrid coupling approach was developed for efficient analysis of RPV failure where transIVR is coupled with ANSYS Mechanical (Paper IV)

In the following general descriptions on the codes ANSYS Fluent Mechanical and the PECM are given in Section 2 the new creep model in Section 3 the new coupling approach as well as its comparison with the previous approach in Section 4 the efficient hybrid coupling approach in Section 5 and summary amp outlook in Section 6

9

2 Numerical Solvers and Models

In this section we briefly introduce the ANSYS Fluent as the CFD solver for melt pool heat transfer the ANSYS Mechanical as the mechanical solver for RPV failure and PECM as an efficient alternative for melt pool heat transfer that are heavily used in the following sections

21 ANSYS Fluent

In the fluid mechanics the fluid is treated as a continuum which is generally described by the conservative mass momentum and energy equations [40]

120597120588

120597119905+ 120571 sdot (120588) = 119878119898 (5)

120597(120588)

120597119905+ 120571 sdot (120588) = minus120571119901 + 120571 sdot (120591) + 120588 + (6)

120597120588119864

120597119905+ 120571 sdot ((120588119864 + 119901)) = 120571 sdot (119896120571119879 minussumℎ119895119869119895

119895

+ (120591 sdot )) + 119878ℎ

(7)

Computational Fluid Dynamics (CFD) studies topics on the numerical solutions of above equations Typical methods are the Finite Volume Method (FVM) Finite Difference Method (FDM) and Finite Element Method (FEM) as well as particle methods that are mesh-free eg SPH and MPS

ANSYS Fluent is a general purpose CFD code which solves above equations with FVM Turbulence modelling and multiphase flows are supported Turbulence can be modelled with RANS LES or hybrid LESRANS Customizations eg adding sourcesink time or spatial dependent boundary conditions are realized through the user-defined functions (UDF) which is programmed in C language

22 ANSYS Mechanical

In the continuum mechanics of solids three sets of variables are considered to describe the structure performance namely element displacement field strain field and stress field (119906 ε σ) A general 3-dimensional problem is described by 3 sets of equations the equilibrium equations (Eq (8)) strain-displacement relationship (Eq (9)) and constitute equations (Eq (10)) [41] For Eq (8) the reciprocal theorem of shear stress also gives τ119894119895 = τ119895119894 where 119894 119895 = 119909 119910 119911

indicating the stress tensor being symmetric For very simple applications eg 1D beam it can be solved analytically For complex problems where analytical solutions are not available approximation methods eg Galerkin weighted residual method principle of virtual work and principle of minimum potential energy would be used

10

partσ119909119909part119909

+partτ119909119910

part119910+partτ119909119911part119911

+ 119909 = 0

partτ119910119909

part119909+partσ119910119910

part119910+partτ119910119911

part119911+ 119910 = 0

partτ119911119909part119909

+partτ119911119910

part119910+partσ119911119911part119911

+ 119911 = 0

(8)

ε119909119909 =part119906

part119909 휀119910119910 =

120597119907

120597119910 휀119911119911 =

120597119908

120597119911

120574119909119910 =120597119906

120597119910+120597119907

120597119909 120574119910119911 =

120597119908

120597119910+120597119907

120597119911 120574119911119909 =

120597119908

120597119909+120597119906

120597119911

(9)

ε119909119909 =1

119864[σ119909119909 minus μ(σ119910119910 + σ119911119911)]

ε119910119910 =1

119864[σ119910119910 minus μ(σ119909119909 + σ119911119911)]

ε119911119911 =1

119864[σ119911119911 minus μ(σ119909119909 + σ119910119910)]

γ119909119910 =1

119866τ119909119910 γ119910119911 =

1

119866τ119910119911 γ119911119909 =

1

119866τ119911119909

(10)

ANSYS Mechanical [42] is a mechanical engineering software that uses finite element analysis (FEA) for structural analysis The principle of virtual work is adopted in the solver [43] The idea of this principle is that a virtual (very small) change of the internal strain energy must be offset by an identical change in external work due to the applied loads which reads

δ119880 = δ119881 (11) where 119880 is the strain energy or the internal work 119881 the external work and δ the virtual operator The virtual internal work can be expressed as

δ119880 = int δεσd(vol)T

vol

+int 119886119903119890119886119891

δ119908119899119879σ119889(119886119903119890119886119891) (12)

where 휀 is the strain vector 120590 stress vector vol volume of element 119908119899 motion normal to the surface 119886119903119890119886119891 area of the distributed resistance The virtual

external work is contributed from three parts the inertial effects the pressure force on the surface and nodal forces applied to the element which then can be expressed as

δ119881 = minusint 119907119900119897

δ119908119879119865119886

119907119900119897119889(119907119900119897) + int

119886119903119890119886119901

δ119908119899119879119875119889(119886119903119890119886119901)

+ δ119906119879119865119890119899119889

(13)

11

where 119908 is the vector of displacements 119865119886 acceleration force vector 119875 applied pressure vector 119886119903119890119886119901 area over which pressure acts and 119865119890

119899119889 nodal

forces applied to the element

Plugging Eq (12) and (13) into (11) would result in the equilibrium equation Since the virtual displacement can be arbitrary the final equations used to describe the equilibrium of an element is reduced to Eq (14) with multiple operations

([119870119890] + [119870119890119891])119906 minus 119865119890

119905ℎ = [119872119890] + 119865119890119901119903 + 119865119890

119899119889

(14)

where [119870119890] [119870119890119891] 119865119890

119905ℎ [119872119890] and 119865119890119901119903 are the element stiffness matrix

foundation stiffness matrix thermal load vector mass matrix acceleration vector and pressure vector respectively It represents the equilibrium equation on a one element basis Alternatively using a minimum total potential energy principle could result in a consistent expression to the principle of virtual work [41]

Customizations eg on material properties are realized through the user programmable features (UPF) which is programmed in FORTRAN language

23 PECM

The PECM [28] [29] developed in KTH is a correlation-based method to model the melt pool heat transfer It is implemented in ANSYS Fluent through UDF and has been validated against many experiments such as ACOPO COPO LIVE etc In the PECM the flow velocities 119906119909 119906119910 and 119906119911 are replaced by characteristic

velocities 119880119909 119880119910 and 119880119911 respectively in order to eliminate the necessity of solving

the Navier-Stokes equation

The characteristic velocities are calculated using heat transfer correlations

119880119906119901 =α

ℎ119901119900119900119897times (119873119906119906119901 minus

ℎ119901119900119900119897

ℎ119906119901) (15)

119880119889119899 =α

ℎ119901119900119900119897times (119873119906119889119899 minus

ℎ119901119900119900119897

ℎ119889119899) (16)

119880119904119889 =α

ℎ119901119900119900119897times (119873119906119904119889 minus

ℎ119901119900119900119897

119882119901119900119900119897) (17)

where ℎ119901119900119900119897 is the melt pool depth ℎ119906119901 the thickness of well mixed layer of the

pool ℎ119889119899 the thickness of lower stratified region of melt pool 119882119901119900119900119897 the width of

pool 120572 the thermal diffusivity The PECM employs reduced characteristic velocities to describe mushy zone convection heat transfer For internally heated pool natural convection the Steinberner-Reineke correlations (Eq (18)~(19)) [44] are employed The profile of sideward 119873119906 is described using Eckertrsquos type correlation (Eq (20)) for a vertical boundary layer where the local Rayleigh

number at pool height 119910 is defined as 119877119886119910 =119892βΔ1198791199103

αν

12

119873119906119906119901 = 03451198771198861198941198990233

(18)

119873119906119889119899 = 13891198771198861198941198990095

(19)

119873119906119904119889 = 050811987511990314 (

20

21+ 119875119903)

minus14

11987711988611991014

(20)

As the characteristic velocities are calculated the energy equation is the only transport equation that needs to be solved and can be expressed as follows

partρ119862119901119879

part119905+ nabla(ρ119862119901119932119879) = 119896nabla2119879 + 119876119907 (21)

where from left to right are the transient term convection term diffusion term and volumetric heat source respectively The convection term then can be calculated explicitly with use of values in the last time step and then can be treated together with the volumetric source term Therefore the PECM finally simplifies the energy equation to a conduction equation which can be easily solved and as a result will significantly increase the computational efficiency in simulations

In addition solidificationmelting process can also be modelled and the temperature-dependent physical properties are supported

13

3 Creep Modelling of 16MND5 Steel

Creep deformation becomes significant when a component is subjected to high temperature and stress Modelling of creep behaviour of the RPV is paramount since it is considered as a main mechanism leading to the RPV failure In this section we present the development of a three-stage creep model (called lsquomodified theta projection modelrsquo ) for the French RPV steel 16MND5 This model was successfully implemented into the ANSYS Mechanical code

31 Creep and Creep Modelling

Creep is an irreversible time-dependent inelastic deformation process Diffusion and dislocation are the two molecular mechanisms of creep The former normally occurs due to the presence of vacancies within the crystal lattice (or the grain) while the latter occurs if there is a slip by the passage of crystal dislocations [45] There are two types of diffusion creep [46] Coble creep (favoured at lower temperatures) if the diffusion paths are predominantly through the grain boundaries and Nabarro-Herring creep (favoured at higher temperatures) if through the grains themselves Dislocation creep tends to dominate at high stresses and relatively low temperatures Dislocations can move by gliding in a slip plane which is a process that requires little thermal activation But the rate-determining step for their motion is often a climb process which also requires diffusion and is thus a time-dependent process and favoured by higher temperatures Creep behaviour is highly affected by temperature and stress generally creep is more significant under higher temperature and stress

Creep curves describe the creep strain development with time Figure 3 shows a creep curve under constant temperature and stress which typically experiences three creep stages primary stage where the creep strain rate ε119888119903 (the change of creep strain ε119888119903 in unit time) decreases with time secondary stage where the creep strain rate almost keeps constant and tertiary stage where the creep strain rate drastically increases with time

Figure 3 Creep curve under constant temperature and stress

14

The built-in creep models in commercial software eg ANSYS [47] and ABAQUS [48] in default only cover the primary andor the secondary stage(s) Table 1 [47] gives the default creep models supported in ANSYS for example Possible reason why the tertiary stage is not modelled might be that for most industrial applications the failure analysis is beyond interest and therefore the built-in creep laws in the commercial software was supposed to be sufficient [49] Though the two codes offer the interfaces through either ANSYS UPF or ABAQUS user subroutines to support more general user-defined creep laws eg complex models that also include the tertiary creep stage

Table 1 Default creep models in ANSYS Mechanical

Creep model Equations Type

Strain hardening

ε119888119903 = 1198621σ1198622ε119888119903

11986231198901198624119879 1198621 gt 0 Primary

Time hardening ε119888119903 = 1198621σ11986221199051198623119890minus1198624119879 1198621 gt 0 Primary

Generalized exponential

ε119888119903 = 1198621σ1198622119903119890minus119903119905 r = 1198625σ

11986231199051198623119890minus1198624119879 1198621 gt 0

1198625 gt 0

Primary

Generalized Graham

ε119888119903 = 1198621σ1198622(1199051198623 + 1198624119905

1198625 + 11986261199051198627)119890minus1198628119879 1198621 gt 0 Primary

Generalized Blackburn

ε119888119903 = 119891(1 minus 119890minus119903119905) + 119892119905 119891 = 1198621119890

1198622σ 119903 = 1198623(σ1198624)1198625 119892 = 1198626119890

1198627σ

1198621 gt 0

1198623 gt 0

1198626 gt 0

Primary

Modified time hardening

ε119888119903 = 1198881σ11988821199051198883+1(1198883 + 1) 1198621 gt 0 Primary

Modified strain hardening

ε119888119903 = 1198621σ1198622[(1198623 + 1)ε119888119903]

1198623119890minus1198624119879 1198621 gt 0 Primary

Generalized Garofalo

ε119888119903 = 1198621[119904119894119899ℎ(1198622σ)]1198623119890minus1198624119879 1198621 gt 0 Secondary

Exponential form

ε119888119903 = 1198621119890σ1198622119890minus1198623119879 1198621 gt 0 Secondary

Norton ε119888119903 = 1198621σ1198622119890minus1198623119879 1198621 gt 0 Secondary

Combined time hardening

ε119888119903 = 1198621σ11986221199051198623+1119890minus1198624119879(1198883 + 1)

+ 1198625σ1198626119905119890minus1198627119879

1198621 gt 0

1198625 gt 0

Primary+ Secondary

Rational polynomial ε119888119903 = 1198621

partε119888part119905 ε119888 =

119888119901119905

1 + 119901119905+ ε119898119905

ε119898 = 1198622101198623σσ1198624 119888 = 1198627ε

1198628σ1198629

119901 = 11986210ε11986211σ11986212

1198622 gt 0 Primary+ Secondary

Generalized time hardening

ε119888119903 = 119891119905119903119890minus1198626119879 119891 = 1198621σ + 1198622σ2 + 1198623σ

3 119903 = 1198624 + 1198625

Primary

32 A Modified Theta Projection Model

321 Original theta projection model and its variations

The original theta projection model [50] was first proposed to accurate predict long-term creep and creep-fracture properties It assumes that a creep curve under constant temperature and stress can be described as

15

휀119888119903 = 1205791(1 minus 119890minus1205792119905) + 1205793(119890

1205794119905 minus 1) (22)

where 휀119888119903 is the creep strain 119905 the time and 1205791~1205794 the model parameters Figure 4 illustrates how the first term and second term on right hand side develop with time and constitute a creep curve which has obvious primary secondary and tertiary stages Model parameters 120579119894(119894 = 1234) are temperature and stress dependent and are described with the following expression

119897119899thinsp120579119894 = 119886119894 + 119887119894120590 + 119888119894119879 + 119889119894120590119879 (23)

where 119886119894 119887119894 119888119894 and 119889119894 ( 119894 = 1234 ) are constants It is noted [45] that this equation is suitable when the temperature range of experiment data sets is very narrow such that minus1119879 is almost proportional to 119879 These constants (119886119894 119887119894 119888119894 and 119889119894 (119894 = 1234)) are the final model parameters which are generated with two fitting procedures

(i) Fitting the experimental creep curves to Eq (22) generates several groups of 1205791~1205794 for each creep curve under certain temperature and stress We denote (119879119895 120590119895 1205791119895 1205792119895 1205793119895 1205794119895 ) as parameters for the 119895119905ℎ

experimental curve under temperature 119879119895 and stress 120590119895

(ii) Fitting 1205791 to Eq (23) as a function of temperature and stress with the obtained data ( 119879119895 120590119895 1205791119895 ) from ( 119879119895 120590119895 1205791119895 1205792119895 1205793119895 1205794119895 ) generates

parameters 1198861 1198871 1198881 and 1198891 Repeat this fitting procedure for 1205792 1205793 and 1205794 to get parameters 119886119894 119887119894 119888119894 and 119889119894 (119894 = 234)

Variations of the model were developed and applied to other different materials to better accommodate the experimental data which are all then called a lsquomodified theta projection methodrsquo independently Examples are models in [51] for lowhigh alloy ferritic austenitic steels Al-alloys Al-matrix composites [52] for Nickel-based Super-alloy [53] for pure metals and class M alloys with intermediate range of stress (40-70 of the yield stress) and temperature (40-60 of the melting temperature)

Figure 4 Illustration of theta projection method [54]

16

322 A modified theta projection model

The new modified theta projection model we proposed [54] was mainly adapted from the original theta projection method [50] and the modified version [51] The new model describes the creep curve with the following equation

휀119888119903 = 1205791(1 minus 119890minus1205792119905) + 120579119898119905 + 1205793(119890

1205794119905 minus 1) (24)

An additional term 120579119898119905 is seen compared with Eq (22) in the original method The reason for adding this term is that the 16MND5 steel shows fairly long secondary stages in many experimental creep curves and the creep curves can be better fitted by Eq (24) than Eq (22) As the structure undergoes different creep process under different temperature and stress the model parameters θ119894 (119894 =1~4 and 119898) are also temperature and stress dependent

The main difference compared with the original and all other modified projection models is that numerical interpolation is used to replace the second numerical fitting in previous models In other words instead of fitting the model parameters as functions of stress and temperature with Eq (23) the distributions of the model parameters are assumed to be piecewise in the temperature-stress space and they will be interpolated directly from known parameters obtained from experiments The consideration is that the experimental data ([16] [17]) for 16MND5 covers a wide temperature range varying from 600 degC to 1300 degC and a wide stress load range varying from 08 MPa to 248 MPa while Eq (23) in the existing methods is based on a narrow temperature span Poor parameter fitting results would be expected if we directly fit the parameters 1205791~1205794 with Eq (23) as used in the original methods

The creep model was developed using the following steps

(i) Fitting Each creep curve can be represented by a group of model parameters [1205791 1205792 120579119898 1205793 1205794] with respect to Eq (24) by fitting the creep curves to Eq (24) All parameters of the 32 experimental curves form an

initial dataset of the model parameters [119879119895 120590119895 1205791119895 1205792119895 120579119898119895 1205793119895 1205794119895] where j (0ltjle32) is the jth creep curve

(ii) Interpolation Parameters of a creep curve under any [119879 σ] are directly interpolated from the dataset

(iii) Parameter adjustment The dataset was further updated to satisfy the monotonicity assumption we have taken that creep strain monotonically increases with temperature and stress

(iv) Different hardening models for dynamic loads

Step (i) was done using MATLAB In the following steps (ii)-(iv) are described in detail

Model parameters for new creep curves are interpolated from the dataset over the 2D 119879 minus 120590 space The interpolation is divided into two sub-interpolation steps interpolate over stress with fixed temperature and then interpolate over temperature with fixed stress We take the interpolation of model parameters under (119879prime σprime) for example In the first interpolation we first find out the two

17

temperatures 119879119896119899119900119908119899119904119898119886119897119897 and 119879119896119899119900119908119899119897119886119903119892119890 most close to 119879prime from the experimental

temperature set 600 700 800900 1000 1100 1200 1300 Under each temperature there are four sets of model parameters in the dataset corresponding to four different stress loads Then the first interpolation is done over stress load from the known dataset of model parameters under each

temperature to output the parameters for (119879119896119899119900119908119899119904119898119886119897119897 120590rsquo) and (119879119896119899119900119908119899119897119886119903119892119890 120590rsquo)

In the second interpolation the model outputs parameters for (119879rsquo 120590rsquo) from

(119879119896119899119900119908119899119904119898119886119897119897 120590rsquo) and (119879119896119899119900119908119899119897119886119903119892119890 120590rsquo) For 120579i(119894 = 12 3 4) we assume a logarithm

relation between 120579119894 temperature 119879 and stress 120590 which is similar to Eq (23)

119897119899120579119894 = (119886 + 119887120590)(119888 + 119889119879) (25)

It reduces to 120579119894 = 11986011198901198871σ with fixed temperature in the first interpolation and to

θ119894 = 11986021198901198872T with fixed stress in the second interpolation The interpolation

expression for θ119894 = 11986011198901198871σwith two known points (1205901 1205791198941) and (1205902 1205791198942) is given

by Eq (26) The interpolation expression for θ119894 = 11986021198901198872T with two known points

(1198791 1205791198941) and (1198792 1205791198942) is given by Eq (27)

120579119894 = 120579i1

120590minus12059021205901minus1205902120579i2

1205901minus1205901205901minus1205902 (26)

120579119894 = 120579i1

119879minus11987921198791minus1198792120579i2

1198791minus1198791198791minus1198792 (27)

Similarly we assume the following relation between 120579119898 temperature 119879 and stress 120590

120579119898 = 119860120590119899119890minus119887 119879frasl (28)

The interpolation is done over the stress with 120579119898 = 1198603120590n by fixing temperature

and then over temperature with 120579119898 = 1198604119890minus1198874119879 by fixing stress The interpolation

expression over stress for 120579119898 = 1198603120590119899 with two known points (1205901 1205791198981) and

(1205902 1205791198982) would be given by Eq (29) The interpolation expression over

temperature for 120579119898 = 1198604119890minus1198874119879 with two known points (1198791 1205791198981) and (1198792 1205791198982)

would be given by Eq (30)

120579119898 = 119890119909119901(119897119899120590 minus 11989711989912059011198971198991205902 minus 1198971198991205901

1198971198991205791198982 +1198971198991205902 minus 119897119899120590

1198971198991205902 minus 11989711989912059011198971198991205791198981) (29)

120579119898 = 120579m2(11198791minus1119879) (

11198791minus11198792)fraslsdot 120579m1

(1119879minus11198792) (

11198791minus11198792)frasl (30)

In this model we also take the assumption that the creep strain monotonically increases with temperature and stress loads1 Or specifically with the same creep time the structural component under higher temperature or stress experiences

1 Intuitively materials deform more under higher stress and temperature This is also mostly observed in tensile experiments Though some exceptions exist one instance of which is the effect of phase change some materials under certain temperature around the phase change temperature may show much weaker structural performance Here we just simply assumed that the creep strain monotonically increases with temperature and stress

18

higher creep strain than that with lower temperature or stress (the creep curve under higher temperature or stress should be above the curve with lower temperature or stress) When generating the parameter dataset (Step (i)) we already kept the trend that under the same temperature cases under higher stress loads would have larger creep model parameters and give higher creep strains This trend would also hold for the interpolated temperatures which thus guarantees that creep strain monotonically increases with stress load under same temperature Then the question becomes if creep strain also monotonically increases with temperature given fixed stress However the answer could not always be a yes especially in the extrapolation range of the stress load Figure 5(a) illustrates an example of the interpolation of model parameters under (119879119872 σ119872) The red dots and blue dots are known parameters from the parameter dataset for corresponding high temperature 119879119867 and low temperature 119879119871 respectively The green dots (119879119867 σ119872) and (119879119871 σ119872) are interpolated from the red and blue dots in the first sub-interpolation respectively The black dot (119879119872 σ119872) is interpolated from the green dots in the second sub-interpolation It is seen that when σ119872 lt σ119862 the value of (119879119871 σ119872) is higher than that of (119879119867 σ119872) It means the higher 119879119872 is the smaller the model parameter will be ie the model would predict larger creep strain for lower temperatures which violates our monotonicity assumption One simple solution used here as shown in Figure 5(b) is to minorly adjust the values of the four red and blue dots such that the intersection point C can be sufficiently small and the situation σ119872 lt σ119862 would barely occur in the extrapolation range

Figure 5 Illustration of model parameter interpolation (a) before and (b) after parameter adjustment [54]

A corresponding parameter adjustment process was introduced to verify that the model predicts higher creep strain under higher temperature under fixed stress Figure 6 illustrates the general verification process for any (119879 120590) in the interested load range the creep strain of (119879 + 120575119879 120590) should be larger than that of (119879 120590) Otherwise parameters would be minorly adjusted and the verification process be repeated This process would be iterated several times until the requirement is reached The achieved verification shows

Stress

σC

High temperature TH

MPa

a

0

Model

par

amet

er θ

Low temperature TL

MPa

a

0

Model

par

amet

er θ

High temperature TH

Low temperature TL

σC

Stress

C C

(a) (b)

σM

σM

19

(i) For any (119879 120590) in the temperature range 273~ 1073 K with an interval of 1 K and the stress range 1 ~260 MPa with an interval of 1 MPa its creep strain is higher than that of (119879 minus 1119870 120590) for the first 12000000 s

(ii) For any (119879 120590) in the temperature range 1073~1673 K with an interval of 1 K and the stress range 1 ~80 MPa with an interval of 1MPa its creep strain is higher than that of (119879 minus 1119870 120590) for the first 12000000 s

(iii) For any (119879 120590) in the stress range 01~26 MPa with a smaller interval of 01 MPa for the full temperature range 273~1673 K with an interval of 1 K its creep strain is higher than that of (119879 minus 1 120590) for the first 12000000 s

Above verification covers a wide range of temperature and stress especially where the stress and temperature are low and extrapolation is used Further verification to cover even higher stress or temperature ranges is not considered as creep increases drastically under such high stress and temperature and the results would become meaningless With these regards we generally think the monotonicity requirement is reached The model parameter dataset was updated along with the adjustment The final dataset was given as Table 2 in Paper I

Figure 6 Parameter adjustment process [54]

Figure 7(a-d) present the model predictions of selected creep curves in ANSYS Mechanical (labelled as lsquoANSYSrsquo) compared with experimental curves (labelled as lsquoREVISArsquo) at temperature 600 800 1000 and 1200 degC respectively The predicted curves proved to clearly show the three-stage creep process and agreed well with the experimental creep curves

Initial parametersdataset

θ1 θ2 θm θ3 θ4 from creep

curve-fitting

Parameters interpolation

Is the monotonicity

assumption satisfied

IsMon= =1

Final model parameters

Adjust the

parametersdataset

θ1 θ2 θm θ3 θ4

No

Yes

IsMon=1 a flag is monotonic achieved

For σ in the stress range in MPa

For T in range Tstart Tend in K

For t in range 112000000 in s

If ε(σ T+1t)lt ε(σTt) IsMon=0

Iteration loop of

parameter adjustment

20

Figure 7 Selected model predictions compared with experimental creep curves [54]

Instead of being constant temperature and stress loads also dynamically change with time in real cases Hardening models determine how creep strain rate reacts to the change of loads during a creep process The time hardening model and strain hardening model2 are supported in our model implementation in ANSYS Mechanical which are two mostly used hardening models

Time hardening assumes that the following creep strain rate after a load change depends only on the current creep time from the beginning of the creep process (and also temperature and stress if used in the creep model) and ignores the current creep strain Strain hardening assumes that the creep strain rate depends only upon the current creep strain (and also temperature and stress if used in the creep model) [55]

2 The time hardening and strain hardening here are more general terms than the specific time hardening creep model and the strain hardening creep model mentioned in Table 1

(a) 600 degC (b) 800 degC

(c) 1000 degC (d) 1200 degC

21

Eq (24) in default is a time hardening model as creep strain is directly described by the time 119905 Eq (31) gives the creep strain increment 119889휀119888119903 at time 119905 during time interval 119889119905

119889휀119888119903 = (12057911205792119890minus1205792119905 + 120579119898 + 12057931205794119890

1205794119905)119889119905 (31)

The strain hardening model is derived from Eq (24) in an implicit form Specifically when temperature and stress load change to (119879 120590) at the time 119905 and current creep strain 휀1198881199030 the model will get the creep curve of constant (119879 120590) and

output how this curve will react under the current strain 휀1198881199030 Since the creep curve shows creep strain monotonically increases with time given the creep strain 휀1198881199030 under creep curve of (119879 120590) we can uniquely find this reference time

119905119903119890119891 satisfying Eq (32) One should notice that the reference time 119905119903119890119891 can be

different from the current physical time 119905

휀1198881199030 = 1205791(1 minus 119890minus1205792119905119903119890119891) + 120579119898119905119903119890119891 + 1205793(119890

1205794119905119903119890119891 minus 1) (32)

where all 120579119894(119894 = 1~4119898) are model parameters of the new creep curve (119879 120590) It can be solved for 119905119903119890119891 with methods eg Newton-Ramphson [56] Then the creep

strain increment at 119905119903119890119891 during time interval 119889119905 is given as

119889휀119888119903 = (12057911205792119890minus1205792119905119903119890119891 + 120579119898 + 12057931205794119890

1205794119905119903119890119891)119889119905 (33)

Therefore combining Eq (32) and (33) gives the creep strain rate at creep strain 휀1198881199030 In summary when loads are changed to (119879 120590) with current creep strain 휀1198881199030

the model traces the creep curve of (119879 120590) and calculates the creep strain rate of this curve at strain 휀1198881199030

One calculation was done to demonstrate the performance of the hardening models The stress load was set to 151 MPa and kept unchanged A step temperature load was applied as the orange dash curve shows in Figure 8 it was 900 degC during 0-36000 s and suddenly jumped to 1000 degC at 36000 s which was then kept till the end of the simulation The blue solid curve (the part before the temperature change was identical to and therefore covered by the green solid curve in the figure) and blue dash curve are the creep curves under constant temperature T=900 degC and T=1000 degC respectively The green solid curve and green dash curve represent the model predictions considering this temperature change with time hardening model and strain hardening model respectively Before this change the creep curves with both hardening models were identical to the creep curve of the T=900 degC till point A After that the creep strain rate then shifted from Point A to Point B (which had the same time with A) in the time hardening model and behaved the same as the curve T=1000 degC as shown in the rest part of the green solid curve In the strain hardening model the creep strain rate shifted from Point A to Point C (which had the same creep strain with A) and behaved the same as the curve T=1000 degC as shown in the rest part of the green dash curve The large difference between two models was partially caused by the drastic temperature jump In real cases the difference would be less significant as the temperature changes more mildly Overall this prediction is exactly the expected behaviors of the time hardening model and strain hardening model

22

Figure 8 Time hardening and strain hardening in ANSYS Mechanical [54]

23

4 Coupled Thermo-Fluid-Structure Analysis of RPV Failure

As described in Section 13 RPV failure is coupled with melt pool heat transfer leading it into a transient thermo-fluid-structure coupled problem The goal of this section is to develop and validate a reliable coupling approach which supports transient coupled thermo-fluid-structure analysis as well as advanced models and computational platforms so as to improve fidelity and facilitate implementation The approach was validated against the FOREVER-EC2 experiment and then applied to an RPV failure analysis of a Nordic BWR As the approach adopts the volume loads mapping for load transfer comparative analysis was also performed against the coupling approach using surface loads mapping Future work is considered in analysis of penetration failures (IGTs and CRGTs) of a BWR RPV since the approach is flexible to complex geometries Results of Paper II ampIII are jointly presented in this section

41 Thermo-Fluid-Structure Coupling Approach

Considering the involved phenomena a coupling approach should at least address three issues (i) melt pool heat transfer (ii) RPV deformation and (iii) the interaction between melt pool heat transfer and vessel deformation A two-way strong coupling considering both the transient thermal load from melt pool on vessel and the influence of vessel deformation on melt pool heat transfer involves complex data exchange and is still full of challenges Therefore the new coupling approach is realized in a one-way manner with the assumptions that the influence of vessel deformation on melt pool heat transfer can be ignored and thus only the thermal loads from melt pool to vessel are considered It is based on the considerations that on one hand the deformation is a slow process regarding the relatively long failure time and on the other hand it was pointed out [57] that the physics of natural convection heat transfer in internally heated liquid pools is not sensitive to the geometrical factors based on their calculation results and experimental data from [58] [59] as well as previous experiments in semicircular and square cavities eg [44] [60] In addition one-way coupling is also computationally more efficient especially when heat transfer turbulence structural non-linearity and large-scale geometries are involved The developed approach is based on the ANSYS platform which generally consists of three parts with respect to the three issues respectively

(i) conjugate heat transfer of the melt pool and vessel using ANSYS Fluent (ii) transient thermal loads transfer from ANSYS Fluent to ANSYS

Structural (the ANSYS Mechanical in ANSYS Workbench) (iii) structural analysis of the vessel in ANSYS Structural

ANSYS Structural in default only supports direct transfer of constantsteady-state spatial loads from ANSYS Fluent Transfer of transient loads is realized with the extension tool FSI Transient [61] It is a pre-processing tool (with good documentation) developed by a third-party CAE vendor EDRTM MEDESO [62] to map transient CFD results (heat transfer coefficient temperature and surface pressure) to thermal or structural analysis including the definition of load steps

24

This new procedure proves to be straightforward and reduces efforts in the load transfer

Since structural performance of RPV is highly dependent on temperature the transfer of transient temperature loads from conjugate heat transfer to mechanical solver is a core step of the coupling framework There are two general approaches to realize the transfer as shown in Figure 9 Both approaches contain three parts a conjugate heat transfer analysis for fluid and structure where flow and heat transfer are coupled a mechanical analysis for the structure and a load mapping process that transfers the transient loads The conjugate heat transfer part can be solved with any CFD solver and is treated in the same manner in both approaches In the first approach (Figure 9(a)) transient volume thermal loads are transferred and mapped into the mechanical simulation domain The structure domain reads the essential thermal loads from the mapping after which a structure analysis of this domain is performed In the second approach (Figure 9(b)) transient surface thermal loads are mapped onto the boundary surfaces of structure domain as thermal boundary conditions Then a thermo-structure coupled analysis of the structure domain in the mechanical part is performed To distinguish them we call the former approach volume loads mapping (VLM) and the latter surface loads mapping (SLM) In the RPV failure analysis the fluid and structure domains correspond to the melt pool and RPV respectively Examples of these methods are the studies of [63] and [30] using VLM and [35] using SLM The VLM is adopted in current approach

Figure 9 Frameworks of the thermo-fluid-structure coupling approaches (a) volume loads mapping (b) surface loads mapping [64]

fluid structure structure

fluid structure structure

Conjugate heat transfer problem Mechanical problem

(structure)

Conjugate heat transfer problem Mechanical problem

(thermo-structure coupled)

Transient volume

loads mapping

(VLM)

Transient surface

loads mapping

(SLM)

(a)

(b)

25

42 Numerical Modelling

421 Conjugate heat transfer

In the conjugate heat transfer analysis main focus is put on the modelling of melt pool heat transfer Within the ANSYS Fluent solver two options are supported CFD with turbulence modelling and PECM CFD is the choice when the model goes to detail and more mechanistic analysis are needed while PECM provides an alternative that is faster and possibly more suitable for large-scale problems

Within the scope of RANS models the SST (Shear Stress Transport) model [65] is selected as the turbulence model in present study This model is a combination of k-ε model and k-ω model The k-ω formulation is used in the inner parts of the boundary layer which makes the model directly usable all the way down to the wall through the viscous sub-layer It also switches to k-ε model in the free-stream and thereby avoids the common k-ω problem that the model is too sensitive to the inlet free-stream turbulence properties The considerations on choosing this model are

bull Several two-equation low-Re k-ε models have been proved in [66] not good enough for melt pool heat transfer Therefore these mentioned two-equation models as well as other one-equation models are not considered

bull SST is highly recommended [67] for simulating most industrial heat transfer problems

bull Convincing results ([68] [69]) using SST have been obtained in capturing the thermal behaviour of the natural convection in an internally heated fluid Especially good agreements were achieved between the predicted heat flux distribution along the vessel and experimental data which would be one of the most important factors to estimate model performance in this problem

The description of PECM as an effective model for melt pool heat transfer is given in Section 23

422 Mechanical analysis

The material for the RPV failure analysis considered in this study is the French RPV steel 16MND5 The three-stage modified theta projection model in Section 3 is used as the creep model Since the definition of an RPV failure criterion is not universal (it can be the time when damage variable reaching 1 or either specified value of creep strain creep strain rate or total deformation is reached) we do not explicitly define a failure criterion Instead the simulation is kept running until the creep converge criterion cannot be satisfied even with the defined minimum time step (indicating a fairly large creep strain rate) and the simulation will be automatically terminated The time the simulation stops is taken as the failure time This treatment implicitly implies a criterion of a large creep strain rate We refer to [17] for other material properties Youngrsquos modulus and thermal conductivity are temperature dependent Multilinear Isotropic Hardening (MISO) is used as the plasticity model

26

43 Validation against the FOREVER-EC2 Experiment

In this section the validation against the FOREVER-EC2 experiment is presented (indicated as lsquoVLMrsquo in the following) In addition the simulation results using the SLM approach (indicated as lsquoSLMrsquo in the following) is also included for comparison purpose

431 FOREVER-EC2 experiment

The FOREVER-EC2 experiment [70] is one of the FOREVER experiments that achieved a vessel failure Figure 10 shows the experimental setup A 110 scale pressure vessel with a wall thickness of 15 mm and an outer diameter of ~400 mm was employed The lower head was made of 16MND5 steel Since it was difficult to use prototypical corium materials in the experiment the binary salt 30 CaO-70 B2O3 was employed as simulant To generate the internal heat which corresponds to the nuclear decay heat a specially designed heater was used The power input was maintained ~38 kW during the experiment such that maximum melt pool temperature would be ~1300 degC The pressure inside the vessel was maintained at ~25 MPa Thermocouples were used to measure the melt pool and vessel wall temperatures and linear displacement transducers used to measure the vessel wall displacements The designed experimental conditions were reached at experimental time 11670 s Vessel failure was observed about 12510 s later (at experiment time 24180 s)

Figure 10 Scheme of the experimental setup [22]

27

Figure 11 Computational domains and boundary conditions (a) conjugate heat transfer (b) mechanical analysis [71]

Computational domains and boundary conditions are indicated in Figure 11 SST turbulence modelling was used for pool heat transfer The simulations started at the moment when experimental conditions were reached (experiment time 11670 s) So simulation time 0 s corresponds to experiment time 11670 s

432 Conjugate heat transfer

Figure 12 shows the temperature fields at simulation time 0 s 1225 s 2475 s 4975 s and 9975 s All the time points in this part refer to simulation time The field at time 0 s (Figure 12(a)) was the initial temperature that was reconstructed based on the thermocouple readings The temperature increased with time and reached a steady state when the maximum temperature reached ~1607 K Figure 12(b-f) show clear thermal stratification which is an expected phenomenon of natural convection

Internal pressure

25 bar

Internal pressure 25e6-2500times981timesy Pa

External pressure

1 bar

Zero displacement

y=0 m

Gravity 400

mm

R=203 mm

15 m

m

Tamb

=323 K radiation with ε = 08

+ convection with h=10Wm2K

Tamb

=800 K radiation with ε=02

(a) (b)

Tamb

=800 K radiation with ε=08

+ convection with h=10 Wm2K

28

Figure 12 Temperature fields at simulation time (a) 0 s (b)1225 s (c) 2475s (d) 4975 s (e) 7475 s and (f) 9975 s [64]

Figure 13 shows the temperature profiles of the external vessel along the polar angle in comparison with experimental data Regions with a polar angle larger than 90deg are those in the cylindrical part of the vessel It also shows the temperature profile increased with time until reaching a steady state The profile almost did not change after 4975 s which indicates that a steady state of the temperature was reached As time increased the temperature in the lower head region 0~90deg (where the vessel was contacted with the melt) evolved from a relatively flat distribution to a distribution that temperature at high angle was higher than that at low angle by more than 200 K Generally the steady state results agreed well with the experimental data except for the low angle part of vessel lower head and the high angle part of the cylindrical vessel In addition to modelling errors the discrepancy in the low angle part of the lower head may be caused by the crust and gap possibly formed at vessel bottom in the experiment which can reduce the downward heat transfer and result in lower temperature while they cannot be modelled in simulations As to the high angle cylindrical vessel part simplified boundary conditions were applied to the inner and outer vessel wall since it was far from the melt and not of interest which may result in some discrepancy

(a) (b)

(c) (d) (e) (f)

29

Figure 13 Temperature profiles of external vessel surface along the vessel polar angle [64]

433 Mechanical analysis

In the mechanical analysis the VLM predicted a failure time of 23959 s and the SLM predicted a failure time of 22445 s both are close to the experimental failure time 24180 s All the time points in this part refer to the experiment time Figure 14 shows the creep strain fields at failure time Both approaches predicted maximum creep strains at same location where the temperature was high (cf Figure 12) This is related to the fact that creep behavior is highly influenced by temperature Maximum values for both cases were also close to each other 0393 in VLM and 0366 in SLM

Figure 15 presents the total deformation of the vessel bottom point (also called ldquosouth polerdquo in some literature) in the vertical direction Since there was an initial displacement reading of ~355 mm at time 0 s which almost did not change during 0~11670 s (the whole period before the experimental conditions were reached) an initial value of 355 mm was added to the simulation result such that experimental and simulation results had the same total displacement at t=11670 s Similar trend was seen in the VLM SLM and experimental data that the deformation increased with time and experienced a drastic acceleration at the end Generally both simulation approaches agreed well with the experimental data though they both slightly under predicted the value of total deformation The discrepancy may possibly be related to the accuracy of the thermal predictions modelling errors of plasticity and creep and simplified boundary conditions

400

600

800

1000

1200

1400

1600

0 20 40 60 80 100 120 140 160

Tem

per

ature

(K

)

Polar angle (deg)

Experiment

25 s

1225 s

2475 s

4975 s

7475 s

9975 s

Weilding line

30

Figure 14 Creep strain fields at failure time (a) VLM (b) SLM [64]

Figure 15 Total deformation in vertical direction of the vessel bottom point [64] [71]

The wall thickness changes are plotted together with experimental results in Figure 16 The wall thickness change δ119909 is defined as

δ119909 = 119909119890119899119889 minus 119909119900119903119894119892119894119899119886119897 (34)

000

500

1000

1500

2000

2500

11670 13670 15670 17670 19670 21670 23670

Tota

l def

orm

atio

n (

mm

)

Time (s)

ExperimentVLMSLM

(a) (b)

31

where 119909119890119899119889 and 119909119900119903119894119892119894119899119886119897 are the wall thickness at failure time and the original wall

thickness respectively Since 119909119890119899119889 cannot directly be post-processed it was estimated in the following way (cf the subfigure of Figure 16) for the same polar angle θ find the corresponding points Pin (1199091 1199101) and Pout (1199092 1199102) on the inner and outer surfaces of the deformed vessel respectively the distance of Pin and

Pout radic(1199091 minus 1199092)2 + (1199101 minus 1199102)2 is calculated as 119909119890119899119889 at polar angle 120579 Such estimation of the wall thickness is more accurate when the line PinPout is closer to be orthogonal to the deformed local surface Considering the final deformed vessel as shown in Figure 14 the estimation should be fine at regions close to 0deg and 90deg and can be slightly overestimated at medium angle region The VLM curve was compared with the SLM curve and 4 experimental curves measured at vessel left right front and back sides Generally VLM curve agreed well with the experimental and SLM curves wall thickness gradually decreases with angle till around 75deg and then increases till 90deg the maximum thickness change occurred at around 75deg would be around -5 mm The positive values at low angle may be caused by the thermal expansion It also shows that the thickness transitions in experimental curves were smoother than those in both simulation curves while simulation curves showed narrower regions with large thickness changes (at angle 70deg~80deg) Possible reasons might be (i) sharper temperature profile predicted in simulation (ii) input uncertainty and (iii) modelling errors The SLM predicted slightly thinner vessel than the VLM except for the necking region The discrepancy might be caused by various reasons eg the interior vessel temperature can be slightly different between two approaches which would then result in different thermal stress and different deformation the final deformed vessel shapes cannot be exactly the same which also results in different errors in estimating PinPout

Figure 16 Wall thickness changes along the vessel polar angle [64] [71]

-6

-5

-4

-3

-2

-1

0

1

0 10 20 30 40 50 60 70 80 90

Wal

l th

icknes

s ch

ange

(mm

)

Polar angle (deg)

VLMSLMExperiment (left)Experiment (right)Experiment (front)Experiment (back)

32

44 Application to BWR Case

In this section the coupling approach (indicated as lsquoVLMrsquo in the following) was applied to an RPV failure analysis of a BWR A simulation using the SLM coupling approach (indicated as lsquoSLMrsquo in the following) was also performed for comparison purpose

A hypothetical severe accident of a Nordic BWR initiated by an SBO was considered The core materials would melt and migrate downward to the RPV lower head The total amount of corium relocated in the lower head was considered to form a 19 m corium melt pooldebris bed assuming the corium distribution was homogeneous

Figure 17 Computational domains and boundary conditions (a) conjugate heat transfer (b) mechanical analysis [64]

Figure 17(a) shows the computational domain of the conjugate heat transfer PECM was used in this calculation A general mesh with a pool height of 23 m was used to accommodate most of the relocation situations with the same mesh Since the corium only had a height of 19 m the top 04 m region was treated as injected water covering the corium The top water was expected to be boiling and we assumed a uniform temperature of 383 K Under the implementation of PECM this region was damped with artificially high heat capacity and high conductivity such that it would efficiently remove heat from the corium and suppress its own temperature increase For the corium part below water a volumetric heat source 11 MWm3 was implemented as the nuclear decay heat For the in-vessel surfaces a constant temperature of 383K was applied as they are covered by water which can be boiling Without water cooling on the external vessel wall a very small heat loss of 20 Wm2 was applied Melting and solidification of the corium were also considered with a solidus temperature of 2750 K and a liquidus temperature of 2770 K The initial temperature was set as

q=20 Wm2

T=383 K

Qv=11

MWm3

315 bar

315 bar+ hydrostatic

pressure of corium

Gravity

248 bar

Zero

displacement

Water

Corium

(a) (b)

33

1100 K for the corium 383 K for the water and 450 K for the vessel For the mechanical analysis (Figure 17(b)) an inner pressure of 315 bar and an outer pressure of 248 bar were considered For the region covered by corium an extra hydrostatic pressure was added on the vessel wall A 2 m long vessel extension was considered as well as the standard earth gravity

Figure 18 presents the wall temperature profile along the inner vessel surface at 50 s 2500 s 7500 s and 10000 s The temperature increased with time for the vessel surface contacted with corium and the distribution was flat which was due to that large part of the corium was still in solidus status It also shows that for the surface region that contacted the water the temperature was basically unchanged which indicates that the damping treatment indeed suppressed the temperature increment and guaranteed the heat removal upward by water

Figure 18 Temperature profiles of inner vessel wall along the vessel polar angle [64]

The predicted RPV failure time was 9869 s by VLM and 9650 s by SLM Compared with the FOREVER-EC2 case in Section 43 the VLM predicted a slightly later failure time than the SLM in this BWR case too Figure 19(a) compares the maximum creep strain developments with time of both approaches Before 9000 s the predicted values by both approaches are close to each other Afterward the creep acceleration occurred and the strain increased drastically The acceleration of the VLM occurred 200 s later than that of the SLM which agreed with the predicted later failure time Similar trend was observed for the total deformation in vertical direction of vessel bottom point as shown in Figure 19(b) The deformations of both approaches are close except that the acceleration of VLM occurred later than that of SLM Overall results of VLM agreed well with those of SLM

0

500

1000

1500

2000

2500

0 10 20 30 40 50 60 70 80 90 100

Tem

per

ature

(K

)

Polar angle (deg)

50 s

2500 s

5000 s

7000 s

10000 s

34

Figure 19 Results of structural analysis (a) creep strain and (b) total deformation in vertical direction of vessel bottom point [64]

45 Computational Efficiency

Computational efficiency is also an important consideration regarding the performance of numerical approaches The computational efficiencies of both VLM and SLM are analysed and compared in this section As parallel computing technique was used to speed up the simulations the number of parallel cores and corresponding costed computational time for each simulation are given together in Table 2 as a generally reflection of the computational efficiency All the simulations were done in a Lenovo P700 workstation equipped with 2 Intel Xeon E5-2630 v3 processors (8 cores per processor) and 32 G RAM For the simulations of FOREVER-EC2 SLM employed twice the parallel cores and costed 286 more time compared with VLM For the simulations of prototypic BWR case both VLM and SLM employed the same number of parallel cores but VLM only took 13 of the computational time compared with SLM Overall it can be concluded that VLM has higher computational efficiency than the SLM The reason is a structural analysis is performed in the VLM approach while a coupled thermo-structural analysis is performed in the SLM approach (cf Figure 9) a standalone structural analysis is computed faster than a coupled thermo-structural analysis

Table 2 Comparison of computational efficiency [64]

Case FOREVER-EC2 BWR VLM SLM VLM SLM

Number of cores 4 8 4 4 Computational time (hours) 21 27 18 72

35

5 Hybrid Coupling Approach for Efficient RPV Failure Analysis

In this section we propose a hybrid coupling approach for the efficient RPV failure analysis The idea is to replace the ANSYS Fluent in previous coupling approach with a lumped-parameter code for quick estimate of the melt pool heat transfer With the least modelling details (compared with CFD methods and distributed-parameter methods eg PECM) lumped-parameter codes prove to be the most efficient for melt pool heat transfer The new hybrid approach characterizes 1) utilization of efficient models for melt pool heat transfer and 2) the support of detailed structural modelling of RPV deformation allowing studies with different RPV geometric shapes under effects like local thinning due to ablation For studies of which the focus is mainly on structural behaviour of RPV eg [13] [26] while mechanistic simulation of melt pool convection is computationally expensive and unnecessary the lumped-parameter code could help providing reasonable thermal loads

The development of the approach consists of two tasks (i) development of a lumped-parameter code and (ii) its coupling with ANSYS Mechanical to form this hybrid coupling framework In the following we describe both tasks and the corresponding verification and validation processes respectively This section is mainly based on Paper IV

51 A Lumped-parameter Code for Melt Pool Heat Transfer

A lumped-parameter code transIVR was developed It features (i) quick estimate of transient melt pool heat transfer with reasonable accuracies and (ii) two-dimensional representation of heat conduction in the RPV wall facilitating the coupled analysis of the RPV failure under thermal loads The following assumptions are taken in the transIVR modelling

bull The crust between the oxide pool and metal pool is thin

bull The heat flux has uniform distributions on the bottom top and side surfaces of metal pool

bull Heat source only exists in oxide layer

bull Latent heat of fusion is ignored

Figure 20 illustrates the modelled three domains oxide layer metal layer and the RPV wall In addition the internal structure inside the vessel can also be considered as a radiative body to the top metal surface as in [72] An internal volumetric heat source generated by nuclear decay heat is considered in the oxide layer Heat in oxide layer transfers downward to the RPV wall and upward to the top metal layer Heat in metal layer can be transferred sideward to the RPV wall and upward to the surrounding internal structure through radiation (or to water if inside water flooding is applied) The heat transferred to RPV will be removed from outer RPV surface Possible formations of oxide crust in the top and bottom surface are illustrated as the black regions Possible formation of metal crust on the top surface is shown as the grey region The code automatically detects the

36

existence of the crust on oxide top and metal top surfaces during the simulation and solves the problem correspondingly withwithout crust Boundary conditions are applied to the external and internal surfaces of RPV and (if needed) top surface of metal layer Arbitrary boundary conditions are supported If a boundary condition is nonlinear eg mixed heat convection (linear dependent on 119879 ) and heat radiation (proportional to 1198794) the Picard method [73] would be used for solving this problem

This model consists of three main parts

(i) empirical heat transfer correlations (mostly from experiments) to determine global and local heat transfer coefficients

(ii) energy balances in oxide layer and metal layer where the heat transfer coefficients would be used and

(iii) energy balance in RPV

Figure 20 Two-layer melt pool configuration and RPV wall [74]

511 Heat transfer correlations

For internally heated natural convection (oxide layer) in transIVR the mini-ACOPO correlations [75] are used for the averaged heat transfer as expressed in Eq (35) and (36) and for the angular variation of the heat flux as expressed in Eq (37) and (38)

119873119906119906119901 = 03451198771198861198941198990233 (35)

119873119906119889119899 = 00038119877119886119894119899035(119867119877)025 (36)

119873119906119889119899(θ)

119873119906119889119899= 01 + 108(

θ

θ119898119886119909) minus 45 (

θ

θ119898119886119909)2

+ 86 (θ

θ119898119886119909)3

01 leθ

θ119898119886119909le 06 (37)

RPV

Oxide pool

Metal pool

37

119873119906119889119899(θ)

119873119906119889119899= 041 + 035 (

θ

θ119898119886119909) + (

θ

θ119898119886119909)2

06 ltθ

θ119898119886119909le 1 (38)

The 119877119886119894119899 that appears in Eq (35)~(36) is the internal Rayleigh number which is defined as

119877119886119894119899 =119892β119876119881119867

5

α119896ν (39)

For Rayleigh-Bernard convection in the metal layer heat transfer occurs in all the top bottom and side surfaces To side surface the reduced form [72] of the Churchill-Chu correlation [76] for 119875119903 = 013 is employed as shown in Eq (40)

119873119906119904119889 = 007611987711988613 (40)

To top and bottom surfaces the reduced form [72] of Globe-Dropkin correlation [77] for

119875119903 = 013 is used as expressed in Eq (41)

119873119906119906119901 = 005911987711988613 (41)

The 119877119886 that appears in Eq (40) and (41) is the external Rayleigh number which is defined

as

119877119886 =119892βΔ1198791198673

να (42)

Heat transfer coefficients can be calculated from Nusselt number by Eq (43)

ℎ =119873119906119896

119871 (43)

where 119896 and 119871 are the heat conductivity and characteristic length respectively 119873119906 can be the Nusselt numbers expressed in Eq (35)~(38) and (40)~(41)

512 Energy balance in the pools

The energy balance of oxide layer can be described as Eq (44) The total generated heat of oxide pool equals the sum of the increase of enthalpy (transient temperature change) upward heat flow to metal layer and downward heat flow to RPV

120588119862119901119881119900119897119889119879119900119909119889119905

+ ℎ119906119901119860119900119909119906119901(119879119900119909 minus 119879119900119909119906119901) + ℎ119889119899119860119900119909119889119899(119879119900119909 minus 119879119900119909119889119899) minus 119876119881119881119900119897 = 0 (44)

where ρ is the oxide density 119862119901 the heat capacity 119881119900119897 the total volume 119879119900119909 the bulk temperature 119905 the time ℎ119906119901 the upward heat transfer coefficient 119860119900119909119906119901

upward heat transfer area 119879119900119909119906119901 upward surface temperature ℎ119889119899 downward

heat transfer coefficient 119860119900119909119889119899 downward heat transfer area 119879119900119909119889119899 downward surface temperature and 119876119881 volumetric heat source

38

On the interface between oxide layer and metal layer energy balance requires that the heat transferred from oxide layer should equal the heat to the metal layer which can be described as Eq (45)

ℎ119906119901119860119900119909119906119901(119879119900119909 minus 119879119900119909119906119901) = ℎ119898119905119889119899119860119898119905119889119899(119879119894119899119905 minus 119879119898119905) (45)

where ℎ119898119905119889119899 is the downward heat transfer coefficient of metal layer 119860119898119905119889119899 the downward heat transfer area 119879119894119899119905 the interface temperature and 119879119898119905 the bulk temperature of metal layer

The energy balance of metal pool can be described as Eq (46) The increase of enthalpy (transient temperature change) equals the sum of upward downward and sideward heat flows to surrounding surfaces

ρ119898119905119862119901119898119905119881119900119897119898119905119889119879119898119905119889119905

+ ℎ119898119905119906119901119860119898119905119906119901(119879119898119905 minus 119879119898119905119906119901) + ℎ119898119905119904119889119860119898119905119904119889(119879119898119905 minus 119879119898119905119904119889)

+ ℎ119898119905119889119899119860119898119905119889119899(119879119898119905 minus 119879119894119899119905) = 0

(46)

where ρmt is the metal density 119862119901mt the heat capacity 119881119900119897mt the total volume 119879119898119905 the bulk metal temperature 119905 the time ℎ119898119905119906119901 the upward heat transfer

coefficient 119860119898119905119906119901 upward heat transfer area 119879119898119905119906119901 upward surface

temperature ℎ119898119905119904119889 sideward heat transfer coefficient 119860119898119905119904119889 sideward heat transfer area 119879119898119905119904119889 sideward surface temperature ℎ119898119905119889119899 downward heat transfer coefficient 119860119898119905119889119899 downward heat transfer area and 119879119894119899119905 oxide-metal interface temperature

The heat transferred from metal layer to the top surface should equal the heat radiated from this surface to the environment Considering the radiation and reflections of internal structures to this surface following energy balance equation [72] is achieved

ℎ119898119905119906119901(119879119898119905 minus 119879119898119905119906119901) = σ(1198791198981199051199051199001199014 minus 119879119904

4) (1

ε+1 minus ε119904ε119904

119860119898119905119906119901

119860119904)minus1

(47)

where 119879119898119905119905119900119901 is the temperature of metal top free surface ε the emissivity of

metal ε119904 the emissivity of internal structures 119860119898119905119906119901 the area of metal top surface

and 119860119904 the total area of internal structures Sometimes the internal water cooling may be available above metal layer In this case Eq (47) is unnecessary and 119879119898119905119905119900119901 can be set to a constant value

For the oxide layer top surface Eq (48) is used if not crust is formed in the interface between two layers ie 119879119894119899119905 gt 119879119898119890119897119905119894119899119892119900119909 Otherwise Eq (49) is used

119879119900119909119906119901 = 119879119894119899119905 (48)

119879119900119909119906119901 = 119879119898119890119897119905119894119899119892119900119909 (49)

For the metal layer top surface Eq (50) is used if not crust is formed on top of the metal ie 119879119906119901119904119906119903119891 gt 119879119898119890119897119905119894119899119892119898119905 Otherwise Eq (51) is used

119879119898119905119906119901 = 119879119898119905119905119900119901 (50)

39

119879119898119905119906119901 = 119879119898119890119897119905119894119899119892119898119905 (51)

Overall this part involves 6 equations regarding 6 unknows 119879119900119909 119879119900119909119906119901 119879119894119899119905 119879119898119905

119879119898119905119906119901 and 119879119898119905119905119900119901

513 Energy balance in the RPV wall

The heat transfer in the PRV wall can be modelled as a simple 1D heat conduction problem that heat only transfers in the wall thickness direction or 2D axisymmetric heat conduction problem

For 1D model the treatment in [72] was adopted As illustrated in Figure 21 the heat from the oxide pool transfers to inner vessel surface (119879119894) then to the interior vessel (119879119907119890119904) then to outer vessel (119879119900) and finally to environment

Figure 21 1D heat transfer model in RPV [74]

Unknown variables are the heat flux through inner vessel surface 119902119894119899 the crust thickness 119909119888119903 inner vessel temperature 119879119894 vessel center temperature 119879119907119890119904 and outer vessel temperature 119879119900 Eq (52)~(56) give a closed equation set for these variables considering a radiation boundary condition on the external vessel surface Eq (52)~(53) describe the heat transfer through the crust where an internal heat source 119876119888119903 is also considered Eq (52) can be easily derived from the one-dimensional heat transfer equation with heat source Eq (54)~(55) describe the energy balance on the inner vessel surface and vessel interior respectively Eq (56) describes the energy balance on the outer vessel surface regarding radiation boundary condition For Dirichlet boundary condition Eq (56) is replaced with Eq (57) For other boundary conditions we only need to correspondingly modify the terms on the right-hand side of Eq (56) For situations that no crust is formed on the surface of the vessel Eq (52)~(53) are eliminated and 119902119894119899 in Eq (54) can directly be determined from calculations in Section 512 and Eq (54)~(56) are solved for the unknows 119879119894 119879119907119890119904 and 119879119900 Once 119879119894 exceeds the vessel melting temperature (with or without crust formation) wall ablation is considered to occur Then 119879119894 is fixed to the vessel melting temperature and vessel thickness 119909119907119890119904 becomes the new unknown variable

119879119900119909119889119899 119879119894 119879119907119890119904 119879119900 119879119890119899119907

119909119888119903 119909119907119890119904

crust RPV

Qcr

119902119894119899 119902119897119900119888119886119897

40

119902119894119899 = 119896119888119903(119879119900119909119889119899 minus 119879119894)

119909119888119903+ 1198761198881199031199091198881199032

(52)

119902119894119899 = 119902119897119900119888119886119897 + 119876119888119903119909119888119903

(53)

119902119894119899 = 119896119907119890119904(119879119894 minus 119879119907119890119904)

05119909119907119890119904

(54)

120588119907119890119904119862119901119907119890119904119889119879119907119890119904119889119905

= 119896119907119890119904(minus2 lowast 119879119907119890119904 + 119879119894 + 119879119900)

(05119909119907119890119904)2

(55)

119896119907119890119904(119879119907119890119904 minus 119879119900)

05119909119907119890119904= 휀120590(119879119900

4 minus 1198791198901198991199074 )

(56)

119879119900 = 119879119900119888119900119899119904119905

(57)

For the 2D representation the heat conduction in the RPV is solved using FDM considering the heat transfer in both the thickness and the angular directions in the RPV wall The RPV lower head is treated as a spherical shell The corresponding heat conduction equation in the 2D spherical coordinate is given as Eq (58) The cylindrical part of the vessel is treated as a cylindrical shell The corresponding heat conduction equation in the 2D cylindrical coordinate is given as Eq (59) Eq (58) and (59) are coupled and solved together In default the mesh size is 5deg in angular direction and 0025119909119888119903 in radius direction This mesh discretization can be changed up to user demands

ρ119862119901part119879

part119905=

1

1199032part

part119903(1198961199032

part119879

part119903) +

1

r2sinθ

part

partθ(119896119904119894119899θ

part119879

partθ)

(58)

ρ119862119901part119879

part119905=1

119903

part

part119903(119896119903

part119879

part119903) +

part

part119911(119896part119879

part119911) (59)

52 Hybrid Coupling Approach

The hybrid coupling approach to assess the RPV integrity is illustrated in Figure 22 transIVR and ANSYS Mechanical are coupled in a one-way manner transIVR conducts the conjugate heat transfer of melt pool and RPV and provides transient thermal loads to the ANSYS Mechanical where structural analysis of RPV deformation is performed The 2D heat transfer of RPV is solved in transIVR to better capture the temperature distribution In RPV structural analysis same considerations are taken as in Section 422 for 16MND5 steel the modified theta projection model for creep and MISO for plasticity and temperature-dependent Young modulus etc The VLM is adopted for the transfer of transient thermal loads

41

Figure 22 Framework of the hybrid coupling approach [74]

53 Verification and Validation

The transIVR code was first benchmarked against the UCSB FIBS case of two-layer pool configuration and then validated against the LIVE-7V experiment showing the codersquos capability in predicting two-layer and transient melt pool heat transfers respectively Finally the hybrid coupling approach encompassing transIVR and ANSYS Mechanical was validated against the RPV failure experiment FOREVER-EC2 and the results showed acceptable agreements with those of both the experiment and the previous approach using CFD in Section 43 Hence the transIVR code is an efficient tool for melt pool heat transfer and the hybrid approach an efficient way for RPV failure analysis

531 UCSB FIBS benchmark case

The UCSB FIBS benchmark case [78] was calculated to demonstrate the capability of the transIVR code in simulating two-layer pool heat transfer problems The considered pool configuration was similar to that shown in Figure 20 with the following specific dimensions a two-layer melt pool with a 16 m oxide layer and a 08 m metallic pool on top and an RPV lower head with a thickness of 015 m The internal volumetric source in the oxide layer was 13 MWm3 The calculation results were compared with those of the DOE study [78] and the INEEL study [79]

The predicted results were compared with the digitalized results of the DOE and INEEL studies in Figure 23 All three studies show similar profiles of the heat flux distribution along the polar angle (Figure 23(a)) it increased with the polar angle in the oxide layer to the maximum value at around 77deg and then dropped to a significantly lower value in the metal layer The transIVR result agreed well with the other results with small discrepancy in the metal layer the DOE study gave a

heat flux of 500 MWm2 the INEEL study of 400 MWm2 and the transIVR of somewhere in between Figure 23(b) shows the predicted corium crust thickness distribution along the RPV polar angle also agreed well the DOE and INEEL

transIVR ANSYS Mechanical

Transient

thermal loads

42

studies The crust thickness decreased with polar angle in the oxide layer and no crust was formed in the metal layer As the crust formation is closely related to the heat flux distribution the crust thickness profile was reasonable regarding the heat flux distribution shown in Figure 23(a) Vessel wall ablation would occur due to the high thermal loads Figure 23(c) gives the profile of the remaining vessel wall thickness along the RPV polar angle considering the wall ablation It shows that no ablation occurred in low angle region (0deg~36deg) where heat flux was low and the wall thickness changed from 15 cm to around 6 cm as angle increased in high angle region of the oxide layer Because of the discrepancy in the heat flux of metal layer the predicted wall thicknesses also differ from those of the DOE and INEEL studies the thinnest wall was 73 cm predicted by the DOE study corresponding to the maximum heat flux followed by 88 cm of the present transIVR calculation and 94 cm of the INEEL study Generally speaking the prediction of transIVR is in an acceptable level compared with the other studies for the same benchmark case

Figure 23 Simulation results of the UCSB FIBS benchmark case [74]

(a) heat flux distribution (b) crust thickness distribution

(c) wall thickness distribution

43

532 LIVE-7V experiment

LIVE-7V is an experiment under the LIVE program conducted in KIT to study the core melt phenomena As Figure 24(a) shows it employed the LIVE-3D hemispherical facility with an inner diameter of 1 m and a wall thickness of ~25 mm The non-eutectic binary salt mixture of 20 mol NaNO3- 80 mol KNO3 was used as the simulant material The decay heat in the melt was simulated by cable-type heating with 8 planes of electrical resistance heating wires Water cooling was applied to the top and vessel surfaces Top surface cooling was realized by a cooling lid mounted at height 413 mm from the vessel bottom and vessel cooling by the flow channel between the test vessel and the outer cooling vessel The inlet and outlet water temperature for both top and vessel cooling surfaces were kept ~293 K and ~303 K throughout the whole experiment respectively A 413 mm height pool corresponds to a total volume of ~0194 m3 simulant material The experiment procedure can be divided into four stages with the heating power sequentially shifted from 29 kW to 24 kW 18 kW and 9 kW Figure 24(b) indicates the thermocouple positions in experiment

The simulation results of LIVE-7V using transIVR were compared with experimental data [80] and numerical study using REALPSCDAPSIM code [81] For simplicity the RELAPSCDAPSIM results in the following are labelled as lsquoCOUPLErsquo which is the name of the module in RELAPSCDAPSIM for melt pool heat transfer

Figure 24 LIVE test facility [80] and thermocouple positions [82]

Figure 25 compares the predicted heat removal upward to the top lid and downward to the bottom vessel surface with those of the experiment and COUPLE The black curve indicates the total input power at the four stages The experimental results (red and blue dots) were calculated from the Table 3 of [80] assuming no extra heat losses For the first three stages the transIVR predictions agreed well with the COUPLE results while the predicted upward heat flow was slightly lower and the downward heat flow higher than the experimental data For the last stage the transIVR agreed well with the experimental result while the COUPLE predicted lower upward and higher downward heat flows The short

(a) (b)

44

transition periods between two stages with different total inputs taken in transIVR also agreed with those in the COUPLE predictions

Figure 25 Comparison of heat removal upward to the top melt surface cooling lid and downward to bottom vessel wall [74]

Figure 26 shows the comparison of heat flux distribution along the vessel wall All curves show similar trend that a higher total input power would result in a higher heat flux profile As the COUPLE predictions had relatively flatter profiles transIVR predictions generally agreed better with the experimental results especially when total input was high A discrepancy is seen that the minimum value of a curve occurred at angle 0degfor both transIVR and COUPLE and at angle 30deg for the experimental data It is noted that the models in default assume heat flux monotonically increases with angle (cf Eq (37) and (38)) resulting it impossible to predict higher heat flux at angle 0deg than angle 30deg as in experimental results

45

Figure 26 Heat flux profiles along the vessel polar angle [74]

533 FOREVER-EC2 experiment

In this section we performed calculations against the FOREVER-EC2 experiment to validate the performance of the coupled transIVR and ANSYS Mechanical in assessing the RPV integrity The applied boundary conditions were identical to the settings of previous coupling approach in Section 4 (ANSYS Fluent+ Mechanical) as indicated in Figure 11 The initial bulk temperature of the melt pool was set to be 1261 K in transIVR which corresponds to the maximum value of the initial temperature field in CFD calculation of previous coupling approach Both the experimental data and the simulation results using previous coupling approach are included for comparison In the following results from current hybrid approach are labelled lsquotransIVRrsquo (and referred as transIVR results) results from previous approach labelled lsquoCFDrsquo (and referred as CFD results) and results from experiment labelled lsquoexperimentrsquo

Figure 27 compares the bulk temperature predicted by transIVR with the averaged and maximum temperatures of the CFD results All curves show the same trend that the temperature gradually increased with time till reaching a steady state and then kept unchanged till the end The times needed for reaching the steady state were ~5000 s for the CFD and ~4000 s for the transIVR which are close to each other It indicates again a good performance of the transIVR in predicting the transient behavior of melt pool Compared with the steady-state averaged temperature 1478 K and maximum temperature 1609 K of the CFD results the steady-state temperature of transIVR 1599 K is a reasonable prediction

0

3000

6000

9000

12000

15000

18000

0 20 40 60 80 100

Hea

t fl

ux

(W

m2)

Polar angle (deg)

29 kW-EXP24 kW-EXP18 kW-EXP9 kW-EXP29 kW-COUPLE24 kW-COUPLE18 kW-COUPLE9 kW-COUPLE29 kW-transIVR24 kW-transIVR18 kW-transIVR9 kW-transIVR

46

Figure 27 Temperature evolution with time [74]

Figure 28 Temperature fields at time (a) 01 s (b) 3038 s (c) 6038 s and (d) 13289 s [74]

Figure 28 shows the temperature fields of the vessel at time 01 s 3038 s 6038 s and 13289 s The temperature fields were visualized in the ANSYS Mechanical It also indicates that the transient temperature loads were properly transferred from transIVR to ANSYS Mechanical Figure 28(a-d) show both the minimum and maximum temperatures increased with time In the lower head region the temperature gradually increased with the height The maximum temperature occurs at the location slightly lower than the melt pool top surface This is because thermal loadheat flux imposed on the vessel generally increases with polar angle in the pool natural convection so the code transIVR inherently also predicted higher heat flux at higher angle positions The higher flux at higher position then correspondingly resulted in higher temperature In the cylindrical region the

(a) (b) (c) (d)

[K] [K] [K] [K]

47

temperature decreased as the height since the bottom part would be more significantly heated by the high-temperature lower head region

Figure 29 shows the steady-state temperature profiles of the external vessel surface along the polar angle The transIVR result agreed well with other two results in trend that the temperature increased with the angle and reach the peak values at angle ~80deg then gradually decreased with angle But slightly lower temperature was predicted in the low angle region Possible reason may be that the local heat flux distribution determined by Eq (37) and (38) is steeper than that in the experiment and CFD study which then resulted in a steeper increase of temperature along the angle and a lower temperature at low angle region In the cylindrical part (gt90deg) the transIVR result generally agreed with the experimental and CFD results The discrepancy with the experimental data may be caused by the simplified boundary conditions the ambient temperature inside the vessel was set to uniform constant value while it could decrease as height Therefore predicted heat loss in the higher position of cylindrical part could be smaller than that with height-dependent ambient temperature resulting in higher vessel temperature Overall the prediction was reasonable for a lumped-parameter model

Figure 29 Temperature profiles of the external vessel surface along the vessel polar angle [74]

The transIVR predicted a failure time of 24959 s which is close to 24180 s in experiment and 23959 s in the CFD results Figure 30 shows the total deformation of vessel bottom point with time All three curves increased gradually with time and experienced drastic accelerations at the end The

400

600

800

1000

1200

1400

0 40 80 120 160

Tem

per

ature

(K

)

Polar angle (deg)

transIVRExperimentCFDWeilding Line

48

transIVR curve agreed well with the CFD curve The final total deformation at failure time is also close to that of the CFD result but lower than the experiment data The discrepancy might be related to the simplifications in the boundary conditions accuracy of the thermal load prediction and the modelling errors in plasticity and creep

Figure 30 Total deformation in vertical direction of the vessel bottom point [74]

Figure 31 shows the profile of wall thickness change The transIVR curve was compared with the CFD curve and four experimental curves measured at vessel left right front and back sides The simulation result fitted the experimental and CFD results well in the region 0deg~50deg In the region 50deg~90deg wall thickness gradually decreased with angle till around 75deg and then increased till 90deg The maximum thickness change was -3 mm for the simulation which is close to the experiment left side curve and slightly larger than the -6~ -45 mm of the other curves from experiment and CFD results Concerning possible errors and uncertainties in the modelling and numerical settings the new approach generally gives reasonable predictions

0

5

10

15

20

25

11670 15170 18670 22170 25670

Tota

l def

orm

atio

n (

mm

)

Time (s)

Experiment

CFD

transIVR

49

Figure 31 Wall thickness changes along the vessel polar angle [74]

-6

-5

-4

-3

-2

-1

0

1

0 20 40 60 80 100

Wal

l th

icknes

s ch

ange

(mm

)

Polar angle (deg)

CFD

transIVR

Experiment (left)

Experiment (right)

Experiment (front)

Experiment (back)

50

51

6 Summary and Outlook

Reactor pressure vessel (RPV) is one of the key physical barriers in the defence-in-depth approach of nuclear power safety and its failure is also the transition point of core melt (corium) progression from in-vessel to ex-vessel phase Quantification of RPV failure during a hypothetical severe accident is therefore paramount to the specific qualification of severe accident mitigation strategies as well as the general assessment of corium risk in light water reactors The RPV failure is a problem of structural performance of RPV under the influence of thermal loads imposed by corium melt pool ie it is a transient thermo-fluid-structure coupled problem per se whose solution requires simultaneous modelling of both highly turbulent melt pool heat transfer and large-scale nonlinear RPV creep Due to the complexity of the problem only limited capabilities have been developed in the previous studies under different simplifications and assumptions in models and coupling methods The present study was motivated to advance modelling and coupling approaches for RPV performance analysis considering new understanding of physical phenomena and latest coupling platform of simulation tools

For the new development of coupling approaches and models for physical phenomena related to RPV failure analysis during severe accidents this thesis work was conducted in the following order (i) a new creep model for the RPV steel 16MND5 was developed which features the capability to cover all three creep stages (ii) a new thermo-fluid-structure coupling approach was developed for RPV failure analysis and validated against FOREVER-EC2 experiment (iii) the performance of the new coupling approach using volume loads mapping (VLM) for data transfer was further compared with another approach using surface loads mapping (SLM) (iv) a hybrid coupling approach was proposed for efficient analysis of RPV failure which employs a lumped-parameter code for quick estimate of thermal load and a mechanistic FE code for detailed structural analysis and (v) for the hybrid coupling approach the code transIVR was developed for calculation of transient melt pool heat transfer

The developed creep model for 16MND5 is called lsquomodified theta projection modelrsquo and describes all three stages of a creep process Creep curves are expressed as a function of time with five model parameters 120579119894(119894 = 14 119886119899119889 119898) A dataset for the model parameters was constructed by fitting experimental creep curves into this function and an iterative parameter adjustment process to satisfy the monotonicity assumption of creep strain vs temperature and stress Model parameters for new curves under different temperature and stress loads are directly interpolated from this dataset in contrast to the conventional ldquotheta projection modelrdquo that employs an extra fitting process The new treatment was proven to better accommodate all experimental curves over the wide ranges of temperature and stress loads The model was implemented into the ANSYS Mechanical code The simulation results indicate that the model successfully captured all three creep stages and a good agreement was achieved between each experimental creep curve and corresponding modelrsquos prediction Regarding the model response to transient changes of temperature and stress loads the widely used time hardening and strain hardening models were implemented and

52

performing reasonably well in the new method These properties fulfil the requirements of a creep model for structural analysis

The transient thermo-fluid-structure coupling approach was developed using the ANSYS Fluent code for thermofluid dynamics of the melt pool and ANSYS Structural code for structural mechanics of RPV An extension tool was introduced to realize transient load transfer from ANSYS Fluent to Structural and minimize the user effort Both turbulence models and the effective model PECM can be used for melt pool heat transfer The developed creep model was used for RPV failure analysis of the RPV Since it is insensitive to geometry the coupling approach allows more complex geometries such as a BWR lower head with a forest of penetrations (CRGTs and IGTs) modelled in detail Moreover the coupling approach does not only reflect the transient response of thermo-fluid-structure interaction but also facilitates the implementation of advanced (high-fidelity) models in both fluid mechanics and structural mechanics The validation of the coupling approach against the FOREVER-EC2 experiment showed that the temperature profile vessel failure time total displacement of vessel bottom point and wall thickness profile in the experiment were well predicted

Following the variation of the coupling approach from VLM to SLM for data transfer between ANSYS Fluent and ANSYS Structural the respective coupling approaches were investigated by comparative calculations against the FOREVER-EC2 experiment and a postulated severe accident of a reference BWR For the FOREVER-EC2 experiment both VLM and SLM approaches predicted generally well the total deformation in vertical direction of bottom point and the wall thickness change though the former predicted a slightly later failure time For the reference BWR case good agreements were achieved in predicting the maximum creep strain and the total deformation in vertical direction of bottom point For both experiment and reactor cases the VLM approach predicted slightly later failure times than SLM but the differences were negligible compared to the long time-scale of vessel failure (~104 s) The deformation rate of the vessel wall was slow at the early stage and accelerated with time till a drastic deformation and final vessel failure Generally speaking the VLM and SLM coupling approaches had the similar performance in terms of their predictability of experiment and applicability to reactor case but the VLM was computationally more efficient

In the development of the hybrid coupling approach the transIVR code was developed for quick estimate of transient melt pool heat transfer and determination of thermal loads on RPV transIVR does not only support transient analysis of one- and two- layer melt pool but also the heat transfer in RPV being resolved with 2D finite difference method which captures more spatial thermal details for RPV structural analysis in addition to 1D heat transfer model The capabilities of transIVR in modelling two-layer pool heat transfer and transient pool heat transfer were demonstrated by calculations against the UCSB FIBS benchmark case and the LIVE-7V experiment respectively It was then coupled to the mechanical solver ANSYS Mechanical which together form the hybrid coupling framework for detailed RPV failure analysis The calculation against the FOREVER-EC2 experiment did not only reproduce the major results of the experiment but also had a good agreement with the thermo-fluid-structure

53

coupling approach using CFD This indicated the new coupling approach is capable of capturing the vessel creep failure characteristics and can be employed as an efficient tool addressing the complex vessel failure problem

As a prospective more research topics can be addressed in future either as a continuation of the current work toward an effective tool for vessel failure analysis or from the general view of scientific interest in melt pool heat transfer and RPV failure The following topics are recommended

bull Extended applications to the reactor cases For instance the thermo-fluid-structure coupling approach can be used to study penetration failures in the RPV of an BWR with a detailed representation of penetrations (IGTs and CRGTs)

bull Synthesis or update of heat transfer correlations for melt pool Heat transfer correlations directly benefit the parametric codes such as transIVR in accurate quantification of heat flux distribution around a melt pool So far many empirical correlations have been proposed based on experiments which had a variety of geometries and melt simulants with a wide range of Rayleigh number (Ra) Sensitivity studies have also been performed on the influences of Ra Pa and other material properties Therefore a comprehensive synthesis or update of the existing heat transfer correlations will be a meaningful step for making reasonable extrapolations to the reactor applications with prototypic materials

bull Modelling of pool stratification The motivation starts with the improvement of the PECMrsquos modelling capability such that the dynamic stratification and resulting pool heat transfer can be modelled altogether Simulation of melt stratification has been supported in some system codes such as ASTEC using a predefined parametric approach The improvement of PECM can be considered in the similar approach or more mechanistic modelling

bull Development of a structural module for RPV The idea is to develop a simplified module for structural analysis so it can be coupled with transIVR ultimately forming a standalone code for quick estimate of vessel failure of LWRs during postulated severe accidents

bull Extension of vessel analysis capability The current version of tranIVR is limited to melt pool Although the precursory debris bed and melt ejection after vessel failure were not the focus of the present study they are equally important to assessment of corium risk especially for Swedish BWRs where the melt release conditions and history are crucial to qualification of their severe accident mitigation strategies

54

55

Bibliography

[1] IAEA ldquoIAEA Annual Report 2018rdquo IAEA 2019 [Online] Available httpswwwiaeaorgsitesdefaultfilesgcgc63-5pdf

[2] IAEA Safety of Nuclear Power Plants Design Vienna INTERNATIONAL ATOMIC ENERGY AGENCY 2012

[3] M Fischer ldquoGen II and III In-Vessel and Ex-Vessel Melt Retentionrdquo The Short Course on Severe Accident Phenomenology Oct 2017

[4] B R Sehgal Nuclear Safety in Light Water Reactors Severe Accident Phenomenology 1st ed Academic Press 2011

[5] IAEA Approaches and Tools for Severe Accident Analysis for Nuclear Power Plants Vienna INTERNATIONAL ATOMIC ENERGY AGENCY 2008

[6] WENRA ldquoPractical Elimination Applied to New NPP Designs - Key Elements and Expectationsrdquo Sep 2019 [Online] Available httpwwwwenraorgmediafiler_public20191111practical_elimination_applied_to_new_npp_designs_-_key_elements_and_expectations_-_for_issuepdf

[7] P Matejovic M Barnak M Bachraty and L Vranka ldquoASTEC applications to VVER-440V213 reactorsrdquo Nuclear Engineering and Design vol 272 pp 245ndash260 2014 doi 101016jnucengdes201308077

[8] AREVA ldquoUS EPR Severe Accident Evaluationrdquo Topical Report ANP-10268NP Oct 2006 [Online] Available httpswwwnrcgovdocsML0631ML063100157pdf

[9] C T Martin ldquoCoupled 3D Thermo-mechanical Analysis of Nordic BWR Lower Head Failure in case of Core Melt Severe Accidentrdquo master thesis KTH Stockholm 2013

[10] IAEA ldquoAdvanced Large Water Cooled Reactors A supplement to the IAEArsquos Advanced Reactor Information System (ARIS)rdquo Sep 2015 [Online] Available httpsarisiaeaorgPublicationsIAEA_WRC_Bookletpdf

[11] B Pershagen Light Water Reactor Safety Pergamon Press 1989 [12] Ninos Garis M Agrell H Glanneskog and L Agrenius ldquoAPRI 7 ndash Accident

Phenomena of Risk Importancerdquo SSM SSM 201212 2012 [Online] Available httpswwwstralsakerhetsmyndighetensecontentassets2a9f299f46b949b3b209e780cf71436a201212-apri-7-accident-phenomena-of-risk-importance-en-lagesrapport-om-forskningen-inom-omradet-svara-haverier-under-aren-2009-2011-

[13] F Fichot et al ldquoSome considerations to improve the methodology to assess In-Vessel Retention strategy for high-power reactorsrdquo Annals of Nuclear Energy vol 119 pp 36ndash45 2018 doi 101016janucene201803040

[14] T G Theofanous C Liu S Additon S Angelini O Kymaumllaumlinen and T Salmassi ldquoIn-vessel coolability and retention of a core meltrdquo Nuclear Engineering and Design vol 169 no 1 pp 1ndash48 1997 doi 101016S0029-5493(97)00009-5

[15] C Sainte Catherine ldquoTensile and creep tests material characterization of pressure vessel steel(16MND5) at high temperatures (20 up to 1300 degC)rdquo CEA France SEMTLISNRT98-009A 1998

56

[16] E Altstadt H-G Willschuetz B R Sehgal and F-P Weiss ldquoModelling of in-vessel retention after relocation of corium into the lower plenumrdquo Forschungszentrum Rossendorf FZR-437 2005

[17] H-G Willschuumltz and E Altstadt ldquoGeneration of a High Temperature Material DataBase and its Application to Creep Tests withFrench or German RPV-steelrdquo FORSCHUNGSZENTRUM ROSSENDORF FZR-353 ISSN 1437-322X Aug 2002

[18] J L Rempe S A Chavez and G L Thinnes ldquoLight water reactor lower head failure analysisrdquo Nuclear Regulatory Commission NUREGCR--5642 1993 Accessed Aug 23 2019 [Online] Available httpinisiaeaorgSearchsearchaspxorig_q=RN25025905

[19] S Brosi et al ldquoCORVIS Investigation of light water reactor lower head failure modesrdquo Nuclear Engineering and Design vol 168 no 1 pp 77ndash104 1997 doi 101016S0029-5493(96)00008-8

[20] T Y Chu M M Pilch J H Bentz J S Ludwigsen W Y Lu and L L Humphries ldquoLower Head Failure Experiments and Analysesrdquo Sandia National Laboratories Albuquerque NM USA NUREGCR-5582 SAND98-2047 1999

[21] L L Humphries et al ldquoOECD Lower Head Failure Project Final Reportrdquo Sandia National Laboratories Albuquerque NM 87185-1139 2002

[22] B R Sehgal R R Nourgaliev T N Dihn A Karbojian J A Green and V A Bui ldquoFOREVER Experiments on Thermal and Mechanical Behavior of a Reactor Pressure Vessel during a Severe Accidentrdquo in Proceedings of the Workshop on in-vessel core debris retention and coolability Garching near Munich Germany Mar 3-6 1998

[23] S Bhandari M Fischer B R Sehgal and A Theerthan ldquoPost Test Analysis of the FOREVER C2 Experimentrdquo The 9th International conference on Nuclear Engineering (ICONE-9) Nice France Apr 8-12 2001

[24] J F Mao J W Zhu S Y Bao L J Luo and Z L Gao ldquoCreep deformation and damage behavior of reactor pressure vessel under core meltdown scenariordquo International Journal of Pressure Vessels and Piping vol 139ndash140 pp 107ndash116 2016 doi 101016jijpvp201603009

[25] J Mao L Hu S Bao L Luo and Z Gao ldquoInvestigation on the RPV structural behaviors caused by various cooling water levels under severe accidentrdquo Engineering Failure Analysis vol 79 pp 274ndash284 2017 doi 101016jengfailanal201704029

[26] J F Mao J W Zhu S Y Bao L J Luo and Z L Gao ldquoStudy on structural failure of RPV with geometric discontinuity under severe accidentrdquo Nuclear Engineering and Design vol 307 pp 354ndash363 2016 doi 101016jnucengdes201607027

[27] B R Sehgal V A Bui T N Dinh and R R Nourgaliev ldquoHeat Transfer Processes in Reactor Vessel Lower Plenum During Late Phase of In-Vessel Core Melt Progressionrdquo in Advances in Nuclear Science and Technology J Lewins M Becker R W Albrecht E J Henley J D McKean K Oshima A Sesonske H B Smets and C P L Zaleski Eds Boston MA Springer US 1999 pp 103ndash135

[28] C-T Tran and T-N Dinh ldquoThe effective convectivity model for simulation of melt pool heat transfer in a light water reactor pressure vessel lower head Part I Physical processes modeling and model implementationrdquo Progress

57

in Nuclear Energy vol 51 no 8 pp 849ndash859 2009 doi 101016jpnucene200906007

[29] C-T Tran and T-N Dinh ldquoThe effective convectivity model for simulation of melt pool heat transfer in a light water reactor pressure vessel lower head Part II Model assessment and applicationrdquo Progress in Nuclear Energy vol 51 no 8 pp 860ndash871 2009 doi 101016jpnucene200906001

[30] H-G Willschuumltz E Altstadt B R Sehgal and F-P Weiss ldquoCoupled thermal structural analysis of LWR vessel creep failure experimentsrdquo Nuclear Engineering and Design vol 208 no 3 pp 265ndash282 2001 doi 101016S0029-5493(01)00364-8

[31] E Altstadt and T Moessner ldquoExtension of the ANSYS creep and damage simulation capabilitiesrdquo Forschungszentrum Rossendorf Technical report FZR-296 2000

[32] H-G Willschuumltz E Altstadt B R Sehgal and F-P Weiss ldquoRecursively coupled thermal and mechanical FEM-analysis of lower plenum creep failure experimentsrdquo Annals of Nuclear Energy vol 33 no 2 pp 126ndash148 2006 doi 101016janucene200508006

[33] H-G Willschuumltz E Altstadt B R Sehgal and F-P Weiss ldquoSimulation of creep tests with French or German RPV-steel and investigation of a RPV-support against failurerdquo Annals of Nuclear Energy vol 30 no 10 pp 1033ndash1063 2003 doi 101016S0306-4549(03)00036-7

[34] H Madokoro A Miassoedov and T Schulenberg ldquoA thermal structural analysis tool for RPV lower head behavior during severe accidents with core meltrdquo Mechanical Engineering Letters vol 4 pp 18-00038-18ndash00038 2018 doi 101299mel18-00038

[35] W Villanueva C-T Tran and P Kudinov ldquoCoupled thermo-mechanical creep analysis for boiling water reactor pressure vessel lower headrdquo Nuclear Engineering and Design vol 249 pp 146ndash153 2012 doi 101016jnucengdes201107048

[36] W Villanueva C T Tran and P Kudinov ldquoA Computational Study on Instrumentation Guide Tube Failure During a Severe Accident in Boiling Water Reactorsrdquo in The 14th International Topical Meeting on Nuclear Reactor Thermal hydraulics NURETH-14 Toronto Ontario Canada Sep25-30 2011

[37] W Villanueva C T Tran and P Kudinov ldquoAssessment with Coupled Thermo-Mechanical Creep Analysis of Combined CRGT and External Vessel Cooling Efficiency for A BWRrdquo The 14th International Topical Meeting on Nuclear Reactor Thermal hydraulics NURETH-14 Toronto Ontario Canada Sep 25-30 2011

[38] F R Larson and J Miller ldquoA Time-Temperature Relationship for Rupture and Creep Stressesrdquo Trans ASME vol 74 p 765minus775 1952

[39] C T Tran ldquoDevelopment validation and application of an effective convectivity model for simulation of melt pool heat transfer in a light water reactor lower headrdquo Licentiate thesis KTH Stockholm 2007

[40] ANSYS ldquoANSYS Fluent Theory Guide Release 162rdquo ANSYS Inc Jul 2015 [Online] Available httpwwwansyscom

[41] P Zeng Finite element analysis and application (in Chinese) Tsinghua University Press amp Springer 2004

58

[42] ANSYS ldquoANSYS Mechanical | Finite Element Analysis Softwarerdquo httpswwwansyscomproductsstructuresansys-mechanical-enterprise (accessed Sep 26 2019)

[43] ANSYS ldquoANSYS Mechanical APDL Theory Reference Release 162rdquo ANSYS Inc Jul 2015 [Online] Available httpwwwansyscom

[44] U Steinberner and H H Reineke ldquoTurbulent Buoyancy Convection Heat Transfer with Internal Heat Sourcesrdquo in Proceedings of the 6th Int Heat Transfer Conference Toronto Canada 1978 vol Vol2 pp 305ndash310 doi 101615IHTC63350

[45] R W Evans and B Wilshire Introduction to Creep London Institute of Materials 1993

[46] DoITPoMS ldquoCreep Deformation of Metalsrdquo Creep Deformation of Metals University of Cambridge 2006 httpswwwdoitpomsacuktlplibcreepindexphp (accessed Jul 09 2019)

[47] ANSYS ANSYS Mechanical APDL Material Reference Release 162 2015 [48] SIMULIA Abaqus 612 Analysis Userrsquos Manualvol 3 Materials Chapter

2324 2012 [49] S Imaoka ldquoUser Creep Subroutine STI0704A ANSYS Release110rdquo 2007 [50] R W Evans J D Parker and B Wilshire ldquoThe θ projection conceptmdashA

model-based approach to design and life extension of engineering plantrdquo International Journal of Pressure Vessels and Piping vol 50 no 1 pp 147ndash160 1992 doi 1010160308-0161(92)90035-E

[51] ECCC ldquoRecommendations and Guidance for the Assessment of Creep Strain and Creep Strength Data ECCC Recommendations Volume 5 Part 1 b[Issue 2]rdquo 2013

[52] W D Day and A P Gordon ldquoLife Fraction Hardening Applied to a Modified Theta Projection Creep Model for a Nickel-Based Super-Alloyrdquo The ASME Turbo Expo 2014 Turbine Technical Conference and Exposition Duumlsseldorf Germany Jun16-20 2014 vol 7A-Structures and Dynamics p V07AT29A010 doi 101115GT2014-25881

[53] M Kumar I V Singh B K Mishra S Ahmad A Venugopal Rao and V Kumar ldquoA Modified Theta Projection Model for Creep Behavior of Metals and Alloysrdquo J of Materi Eng and Perform vol 25 no 9 pp 3985ndash3992 2016 doi 101007s11665-016-2197-y

[54] P Yu and W Ma ldquoA modified theta projection model for creep behavior of RPV steel 16MND5rdquo Journal of Materials Science amp Technology vol 47 pp 231ndash242 2020 doi 101016jjmst202002016

[55] ANSYS ANSYS Mechanical Advanced Nonlinear Materials v160 Lecture 2 Rate Dependent Creep ANSYS training materials 2015

[56] Q Y Li N C Wang and D Y Yi Numerical Analysis (in Chinese) 5th ed Tsinghua University Press 2008

[57] T N Dinh R R Nourgaliev and B R Sehgal ldquoOn heat transfer characteristics of real and simulant melt pool experimentsrdquo Nuclear Engineering and Design vol 169 no 1 pp 151ndash164 1997 doi 101016S0029-5493(96)01283-6

[58] O Kymaumllaumlinen H Tuomisto O Hongisto and T G Theofanous ldquoHeat flux distribution from a volumetrically heated pool with high Rayleigh numberrdquo

59

Nuclear Engineering and Design vol 149 no 1ndash3 pp 401ndash408 1994 doi 1010160029-5493(94)90305-0

[59] F J Asfia and V K Dhir ldquoNatural convection heat transfer in volumetrically heated spherical poolsrdquo 1994 Accessed Aug 23 2019 [Online] Available httpinisiaeaorgSearchsearchaspxorig_q=RN26037727

[60] M Jahn and H H Reineke ldquoFree Convection Heat Transfer with Internal Heat Sources Calculations and Measurementsrdquo in Proceedings of the 5th Int Heat Transfer Conference Tokyo Japan 1974 vol Vol3 paper NC-28

[61] M Gustafsson ldquoFSI Transient extension V1704rdquo EDR MEDESO 2016 [62] EDR MEDESO httpsedrmedesocom [63] H Kim and I Namgung ldquoThermo-mechanical Analysis of RVLH for

APR1400 during a Severe Accidentrdquo in Transactions SMiRT-23 Manchester United Kingdom Aug10-14 2015 [Online] Available httpwwwlibncsueduresolver18402034197

[64] P Yu and W Ma ldquoComparative Analysis of Reactor Pressure Vessel Failure using Two Thermo-Fluid-Structure Coupling Approachesrdquo submitted to journal

[65] F R Menter ldquoTwo-equation eddy-viscosity turbulence models for engineering applicationsrdquo AIAA Journal vol 32 no 8 pp 1598ndash1605 1994 doi 102514312149

[66] T N Dinh and R R Nourgaliev ldquoTurbulence modelling for large volumetrically heated liquid poolsrdquo Nuclear Engineering and Design vol 169 no 1 pp 131ndash150 1997 doi 101016S0029-5493(96)01281-2

[67] ANSYS Inc ldquoIntroduction to ANSYS Fluent Lecture 7 Turbulence Modelling Fluent Tutorial Materialrdquo 2016

[68] A Horvat and M Borut ldquoNumerical investigation of natural convection heat transfer in volumetrically heated spherical segmentsrdquo in Proceedings of the International Conference Nuclear Energy for New Europe 2004

[69] Y Li CFD Pre-test Analysis of SIMECO-2 Experiment 2016 [70] A Theerthan A Karbojian and B R Sehgal ldquoEC-FOREVER experiments

on thermal and mechanical behavior of a reactor pressure vessel during a severe accident Technical report-1 EC-FOREVER-2 testrdquo SAM-ARVI-D008 Mar 2001

[71] P Yu W Ma W Villanueva A Karbojian and S Bechta ldquoValidation of a thermo-fluid-structure coupling approach for RPV creep failure analysis against FOREVER-EC2 experimentrdquo Annals of Nuclear Energy vol 133 pp 637ndash648 2019 doi 101016janucene201906067

[72] T G Theofanous C Liu S Additon S Angelini O Kymaumllaumlinen and T Salmassi ldquoIn-vessel coolability and retention of a core meltrdquo Nuclear Engineering and Design vol 169 no 1ndash3 pp 1ndash48 1997 doi 101016S0029-5493(97)00009-5

[73] H P Langtangen ldquoSolving nonlinear ODE and PDE problemsrdquo Simula Research Laboratory 2016 [Online] Available httpshplgitgithubionum-methods-for-PDEsdocpubnonlinpdfnonlin-4printpdf

[74] P Yu and W Ma ldquoDevelopment of a Lumped-Parameter Code for Efficient Assessment of In-Vessel Melt Retention Strategy of LWRsrdquo submitted to journal

60

[75] T G Theofanous M Maguire S Angelini and T Salmassi ldquoThe first results from the ACOPO experimentrdquo Nuclear Engineering and Design vol 169 no 1 pp 49ndash57 1997 doi 101016S0029-5493(97)00023-X

[76] S W Churchill and H H S Chu ldquoCorrelating equations for laminar and turbulent free convection from a vertical platerdquo International Journal of Heat and Mass Transfer vol 18 no 11 pp 1323ndash1329 1975 doi 1010160017-9310(75)90243-4

[77] S Globe and D Dropkin ldquoNatural-Convection Heat Transfer in Liquids Confined by Two Horizontal Plates and Heated from Belowrdquo J Heat Transfer vol 81 no 1 pp 24ndash28 1959 doi 10111514008124

[78] T G Theofanous C Liu S Additon S Angelini O Kymaelaeinen and T Salmassi ldquoIn-vessel coolability and retention of a core melt Volume 2rdquo Argonne National Lab DOEID--10460-VOL2 1996 Accessed Apr 18 2019 [Online] Available httpinisiaeaorgSearchsearchaspxorig_q=RN30000459

[79] J L Rempe D L Knudson C M Allison G L Thinnes and C L Atwood ldquoPotential for AP600 in-vessel retention through ex-vessel floodingrdquo United States INEELEXT--97-00779 1997 Accessed Apr 18 2019 [Online] Available httpinisiaeaorgSearchsearchaspxorig_q=RN30025872

[80] A Miassoedov X Gaus-Liu T Cron and B Fluhrer ldquoLive experiments on melt pool heat transfer in the reactor pressure vessel lower headrdquo The 10th International Conference on Heat Transfer Fluid Mechanics and Thermodynamics (HEFAT 2014) Orlando FL Jul 14-16 2014

[81] H Madokoro A Miassoedov and T Schulenberg ldquoAssessment of a lower head molten pool analysis module using LIVE experimentrdquo Nuclear Engineering and Design vol 330 pp 51ndash58 2018 doi 101016jnucengdes201801036

[82] B Fluhrer et al ldquoThe LIVE-L1 and LIVE-L3 experiments on melt behaviour in RPV lower headrdquo Forschungszentrum Karlsruhe GmbH Technik und Umwelt (Germany) Inst fuer Kern- und Energietechnik FZKA--7419 2008 Accessed Nov 12 2019 [Online] Available httpinisiaeaorgSearchsearchaspxorig_q=RN40036867

VI

VII

Acknowledgements

First of all I would like to express my sincere gratitude to my supervisor Prof Weimin Ma for offering this opportunity to work in the severe accident research field and for the supports patience guidance discussions and encouragements throughout the entire period of my graduate study

I am very thankful to my co-supervisor Dr Walter Villanueva for the valuable help and discussions I learnt a lot during the times working with him I am also thankful to co-supervisor Prof Sevostian Bechta who is supportive and always has inspiring ideas that keep you thinking for a long time

Thank you also to all my colleagues at the Division of Nuclear Power Safety for the supports discussions and the very nice working environment during these years

I have taken many interesting courses during my PhD study for which I want to express my thanks to all the teachers With this regard a special thanks goes to Prof Johan Hoffman for his SF2561 Finite Element Method given in 2015 fall as well as the discussions and advices on numerical analysis

I am grateful to Ignacio Gallego-Marcos and Wenyuan Fan for their enthusiastic help on CFD and turbulence modelling during the time working on the turbulent

heat transfer I thank all my 师兄 for sharing their knowledge on nuclear engineering and giving advices when I was confused since 2012

I thank all my friends both in Sweden and in China for the company and the sharing of many wonderful moments in daily life

Finally I would like to express my deepest thank to my parents my girlfriend Siwen my sisters and all my other relatives for their unconditional supports and love

Financial supports from the China Scholarship Council and APRI make my study in KTH possible and are acknowledged

VIII

IX

Table of Contents

AbstracthelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphellipI

SammanfattninghelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphellipIII

List of PublicationshelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphellipV

AcknowledgementshelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphellipVII

NomenclaturehelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphellipXI

1 Introductionhelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphellip1

11 Nuclear Power Safety 1

12 RPV Failure 2

13 Literature Review Creep Behaviour and RPV Failure Analysis 4

131 Experiments on creep behaviour and RPV failure 4

132 Numerical studies on RPV failure 5

14 Motivation Objectives and Main Achievements 7

2 Numerical Solvers and Modelshelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphellip9

21 ANSYS Fluent 9

22 ANSYS Mechanical 9

23 PECM 11

3 Creep Modelling of 16MND5 Steelhelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphellip13

31 Creep and Creep Modelling 13

32 A Modified Theta Projection Model 14

321 Original theta projection model and its variations 14

322 A modified theta projection model 16

4 Coupled Thermo-Fluid-Structure Analysis of RPV Failurehelliphelliphelliphellip23

41 Thermo-Fluid-Structure Coupling Approach 23

42 Numerical Modelling 25

421 Conjugate heat transfer 25

422 Mechanical analysis 25

43 Validation against the FOREVER-EC2 Experiment 26

431 FOREVER-EC2 experiment 26

X

432 Conjugate heat transfer 27

433 Mechanical analysis 29

44 Application to BWR Case 32

45 Computational Efficiency 34

5 Hybrid Coupling Approach for Efficient RPV Failure Analysishelliphellip35

51 A Lumped-parameter Code for Melt Pool Heat Transfer 35

511 Heat transfer correlations 36

512 Energy balance in the pools 37

513 Energy balance in the RPV wall 39

52 Hybrid Coupling Approach 40

53 Verification and Validation 41

531 UCSB FIBS benchmark case 41

532 LIVE-7V experiment 43

533 FOREVER-EC2 experiment 45

6 Summary and Outlookhelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphellip51

Bibliographyhelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphellip55

XI

Nomenclature

Latin

119860 area of a surface

119862119901 heat capacity

119863 damage factor

119892 gravity

ℎ heat transfer coefficient

H Melt pool height

119896 thermal conductivity

119871 characteristic length

119873119906 Nusselt number

119875119903 Prandtl number

119902 heat flux

119902119894119899 heat flux through inner vessel surface

119876119881 internal volumetric heat source

119903 radial distance in a cylindrical or spherical coordinate

119877119886 external Rayleigh number

119877119886119894119899 internal Rayleigh number

119877119886119910 local Rayleigh number

119905 time

119879 temperature

119881119900119897 total volume

119909 thickness (of crust or RPV wall)

119911 height in a cylindrical coordinate

Greek

120572 thermal diffusivity

120573 thermal expansion

Δ119879 temperature difference

ε emissivity (in thermal analysis) strain (in structural analysis)

휀119888119903 creep strain

휀119903 creep strain rate

120579 polar angle

120579119894 parameters of a creep model 119894 = 1234119898

120584 kinematic viscosity

σ stress

XII

Super and subscripts

119886119898119887 ambient

119888119903 creep (in Section 3) crust (in Section 5)

119889119899 downward

119890119899119907 environment

119894 inner surface of RPV

119894119899119905 interface

119897119900119888119886119897 local

119898119886119909 maximum value

119898119890119897119905119894119899119892 melting point

119898119905 metal layer (bulk)

119898119905119889119899 downward surface of metal pool

119898119905119904119889 sideward surface of metal pool

119898119905119905119900119901 top surface of metal layer

119898119905119906119901 upward surface of metal pool

119900 outer surface of RPV

119900119909 oxide layer (bulk)

119900119909119889119899 downward surface of oxide layer

119900119909119906119901 upward surface of oxide layer

119903119890119891 reference

119904 internal structures inside the RPV

119904119889 sideward

119906119901 upward

119907119890119904 vessel

Acronyms

BWR boiling water reactor

CBD creep database

CFD computational fluid dynamics

CRGT control rod guide tube

ECCM effective convectivity conductivity model

FDM finite difference method

FE finite element

FEA finite element analysis

FEM finite element method

FVM finite volume method

IGT instrument guide tube

IVR in-vessel retention

XIII

IVMR in-vessel melt retention EU project

LES large eddy simulation

MPS moving particle semi-implicit method

NPP nuclear power plant

PECM phase-change effective convectivity model

RANS Reynolds-averaged Navier-Stokes

RPV reactor pressure vessel

SBO station blackout

SLM surface loads mapping

SPH smoothed-particle hydrodynamics

UDF user-defined function

UPF user programmable features

VLM volume loads mapping

XIV

1

1 Introduction

11 Nuclear Power Safety

Greenhouse gas emissions contribute to the global warming problem Phasing out fossil fuels by switching to low-carbon energy sources is considered as a mitigation strategy With this regard clean energy should be increased in the world energy consumption structure Being clean and stable nuclear energy is an attractive alternative to fossil fuels After the development over decades it also proves to be a mature and safe technology According to IAEA 2018 annual report [1] there were 450 nuclear power reactors under operation worldwide at the end of 2018 which had a record generating capacity of 3964 GW(e) 9 reactors were connected to the grid and 55 reactors under construction around the world

Due to the potential risk of the release of radioactivity nuclear power safety is emphasized throughout the stages of design construction operation and decommission of a nuclear power plant (NPP) The principal technical requirements [2] specify that the three fundamental safety functions should be fulfilled for all plant states of an NPP (i) control of reactivity (ii) removal of heat from the reactor and from the fuel store and (iii) confinement of radioactive material shielding against radiation and control of planned radioactive releases as well as limitation of accidental radioactive releases The defence-in-depth concept is also applied to provide several levels of defence to prevent consequences of accidents The fuel claddings reactor pressure vessel (RPV) and primary coolant pressure boundaries and the containment are the three physical barriers to prevent the release of radioactive materials to the environment

Severe accidents are events that cause significant damage to reactor fuel and result in core meltdown It may be caused by insufficient cooling of the reactor core due to various reasons or by the rapid introduction of huge reactivity which causes sharp increases in the power and fuel fragmentation eg Chernobyl accident During a severe accident the failure of the fuel claddings will result in the radioactive materials previously enclosed inside the cladding released to the primary loop If cooling is not sufficient to remove the decay heat the core materials will melt migrate down and relocate into the RPV lower head after which the melt will react with water and partially form fragments quench solidify and form a debris bed The debris bed formed by the melt may continue to melt to form a melt pool due to the decay heat As the corium contacts the RPV it would continuously heat the RPV and threaten its integrity Once the RPV fails containment becomes the last barrier to prevent the release of radioactivity The released melt would result in an increase of the pressure and temperature due to the high temperature fractions oxidation and hydrogen combustion which is called direct containment heating The melt would fall onto the reactor cavity Without sufficient cooling the molten core would interact with basemat concrete (MCCI) This process can last for few days It can result in containment failure due to either containment overpressure or the basemat melt-through of containment In addition containment is also subjected to isolation failure and early containment failure due to missiles and dynamic pressure [3] More

2

detailed severe accident progression or phenomena could also be found in eg [4] [5]

Though severe accidents are unlikely to occur the consequences can be severe if no proper measure is taken Related researches have been continuously going on aiming at mitigating the accident consequences and improving safety After the Fukushima Accident safety inspections and re-evaluation actions have been carried out worldwide especially on safety measurements regarding severe accidents The concept lsquoPractical Eliminationrsquo applied to new NPP designs demands that new nuclear installations be designed with the objective of mitigating the consequences of accidents and avoiding early radioactive releases and large radioactive releases [6] Severe accident mitigation strategies are applied to the existing and new NPPs (both Generation II and III) The strategies can generally be classified into two categories [3] the in-vessel retention (IVR) and ex-vessel retention (EVR) In simple words in-vessel retention tries to retain the core melt inside the RPV (second barrier) by applying external water cooling on the vessel as shown in Figure 1(a) Such method minimizes the dispersal of the melt within the containment and hence simplifies the later recovery work Examples of the application are AP1000 HPR1000 APR 1400 The idea of the EVR is to let the RPV fail but to retain the core melt inside the containment Two principle approaches of EVR are the retention within a dedicated lsquocore catcherrsquo and melt fragmentation Examples of core catcher would be EPR that directs the melt to a spreading compartment (Figure 1(b)) and VVER1000 that uses metallic crucible below the RPV An example of the melt fragmentation is the Nordic BWRs that the corium released from RPV would relocate in the cavity with a deep-water pool (7-12 m) and is expected to fragment quench and form a coolable debris bed (Figure 1(c))

Figure 1 Examples of IVR [7] EVR with core catcher [8] and melt fragmentation [9]

12 RPV Failure

RPV which contains the reactor core core shroud and coolant is a key component to the reactor system RPV and the primary loop constitute the primary coolant system which conveys the fission heat from the reactor core to the secondary loop through steam generator (PWR) or to the turbine (BWR) The pressure under

(a) IVR (b) Core catcher (c) Melt fragmentation

3

operation can be ~155 MPa for a 1000 MW PWR [10] and ~70 MPa for a 1000 MW BWR [11] It also has to withstand the neutron radiation damage The thickness of RPV [11] can be ~20 cm for PWR and ~16 cm for BWR

As the transition of in-vessel progression to ex-vessel progression RPV failure is important for analysis of accident progression and assessment of the effectiveness of severe accident mitigation strategies In the melt fragmentation strategy of Nordic BWR the melt-coolant interaction and debris bed coolability after the corium poured into the deep-water pool are essentially dependent on the vessel failure mode including the characteristic time of melt release rupture size and location and amount and superheat of melt available for release [12] New concern also arose [13] on the necessity of performing structural analysis for the RPV in the IVR strategy In the initial methodology and further development of the IVR the success of the strategy ie the integrity of the RPV was demonstrated (eg in [14]) with the following two conclusions (i) even for a local heat flux on the vessel as high as critical heat flux (CHF) the resulted very thin external ldquocold layerrdquo of the vessel would be sufficient to withstand the internal stress load and hold the vessel integrity (ii) as CHF is the critical value for the thermal load the main specific goal for assessing the effectiveness of IVR becomes to demonstrate that the local heat flux along RPV be lower than the CHF The new concern doubts that local lsquohot partrsquo of the vessel would have negative effects on the overall resistance due to creep and dilatation which could result in a vessel failure and hence violate the first conclusion in particular for ldquolong termrdquo retention of corium in RPV If so a structural integrity analysis of RPV would be needed in addition to assessing local heat flux especially for high-power reactors Therefore accurate prediction of vessel failure could be meaningful for both strategies

Figure 2 Illustration of RPV under thermal loads from corium

As the structural performance is highly affected by the thermal loads from the corium RPV failure is essentially a transient thermo-fluid-structure coupled problem Figure 2 illustrates the situation that a corium debris bedmelt pool formed in the RPV lower head The decay heat of nuclear fission products could continuously be generated inside the corium leading to the formation of melt pools as the debris melts The heated debris bedmelt pool imposes thermal and

4

mechanical loads on the RPV Such loads can be transient (change with time) and spatially non-uniform Complex phenomena are also involved in this process Strong turbulent natural convection induced by the decay heat may be expected in the melt pool under IVR The corresponding Rayleigh number 119877119886 can be 1016minus17 for prototypic reactor cases Material structural properties can become non-linear under high temperature and stress loads The creep deformation also becomes significant at high temperature Detailed modelling of turbulent natural convection and non-linear properties could be computationally expensive In addition large-scale geometry (hemisphere with a radius of 20-30 m) long-term transient deformation process (~10000 s) and required small time steps (eg 001 s) for numerical convergence consideration further increase the computational costs

13 Literature Review Creep Behaviour and RPV Failure Analysis

In this section the most relevant studies concerning the experiments modelling and analysis of creep behaviour and RPV failure are presented

131 Experiments on creep behaviour and RPV failure

Various experiments have been performed on the RPV failure problem some of which were to generate data of mechanical properties (especially the creep properties) and some to study the overall vessel failure with scaled or simplified geometries

The REVISA experiment [15] generated creep curves and stress strain curves of French steel 16MND5 ranging from 600 to 1300 degC This data is accessible in [16] as well as [17] which also documents other material properties eg density thermal expansion coefficient conductivity heat capacity Youngrsquos Modulus and plasticity Similar experiments were carried out to generate high-temperature creep and tensile test data for American RPV steels and Inconel [18]

The CORVIS experiments [19] investigated the lower head failure modes with and without penetrations Each experiment was carried out on a steel plate 100 mm thick representing a pressure vessel lower head The penetration case represented a boiling water reactor lower head carrying a tube penetration in the form of a drain line

The LHF (lower head failure) experiments [20] and the succeeding OLHF tests [21] investigated the vessel failure with a 15 scaled vessel heated by an electrical radiator allowing to locate the hot focus at different position of the vessel with a maximum wall temperature of around 900 degC Internal pressure ranging from 53 to 123 bar was also imposed to initiate the vessel failure

The FOREVER experiments [22] investigated the phenomena of melt pool convection vessel creep possible failure processes and failure modes occurred during the late phase of the in-vessel progression by heating up a melt pool in a vessel and observing possible vessel failures The study covers cases with and without penetration different melt level existence of gap cooling and in-vessel melt coolability

5

132 Numerical studies on RPV failure

Various numerical approaches have been employed on RPV failure Since a two-way thermo-fluid-structure coupling approach is challenging these approaches take simplifications and can generally be classified into two categories (i) RPV failure decoupled from melt pool heat transfer and (ii) RPV failure coupled with melt pool heat transfer in simplified manners

1) Decoupled approaches In the decoupled approaches only RPV is modelled Simplified constant thermal loads are imposed on RPV considering the melt pool heat transfer

In a post-test analysis of FOREVE-C2 experiment [23] it was treated as a thermo-structural coupled analysis of the vessel with the thermal load estimated by the outer thermocouples and the normal heat flux on the external vessel wall The analysis was done using finite element (FE) code SYSTUS A coupled creepdamage law based on a viscosity-hardening creep law describing primary and secondary creep was employed for creep analysis

Mao et al conducted various studies on reactor cases assessing the IVR strategy using FE code ABAQUS Thermal boundary conditions were imposed on the RPV which was then solved as a thermal-structural coupled problem The strain hardening creep model (Eq (1)) was mainly used Their studies cover the general RPV failure [24] as well as the effects of cooling water levels under IVR [25] and geometric discontinuity (ablated RPV) [26]

ε119888119903 = 1198891σ1198892ε1198893119890119909119901 [minus

1198894119879] (1)

In the EU IVMR project [13] the RPV integrity was also studied through a benchmark calculation which deals with the RPV of a PWR with a prescribed ablated profile with 2D axisymmetric geometry Prescribed temperature gradient was set as thermal load The objective was to calculate deformations for various pressure loads up to vessel rupture A smoother profile was planned in further research

The advantage of such approaches is once thermal loads are reasonably assumed the focus only needs to be put on the RPV The disadvantage is a reasonable thermal load should be consistent with the melt pool heat transfer while sometimes it is not well-noted eg constant uniform temperature boundary conditions were assumed for inner and outer RPV (with uniform thickness in lower head) surfaces [25] which ignored the non-uniform heat flux distribution caused by the turbulent convective corium pool As thermal loads strongly affect the structural performance they should be quantified carefully

2) Coupled approaches In coupled approaches both melt pool heat transfer and RPV failure are modelled and coupled in one-way or simplified manner With melt pool heat transfer modelled the coupled approaches are more flexible to provide reasonable thermal loads on the RPV In existing approaches CFD and the correlation-based distributed-parameter models eg the Effective Convectivity Conductivity Model

6

(ECCM) [27] and the Phase-change Effective Convectivity Model (PECM) [28] [29] are general methods for melt pool heat transfer

Sehgal et al [22] presented an approach that employed the MVITA code to calculate the thermal processes and ANSYS for structural analysis MVITA code employed the ECCM Bailey-Norton creep law was used as shown in Eq (2)

ε119888119903 = 6055 times 10minus26 sdot σ472119905085119890minus

37200

119879 (2)

This approach was used for the pre-test simulation of FOREVER experiment to determine the thermal and mechanical loadings Parametric investigations also helped in the identification of important factors that could affect the test procedure and test conditions

Willschutz et al [30] employed the CFD-module Flotran for the melt pool heat transfer and ANSYS for structural analysis of RPV Standard k-ε turbulence model was selected for the turbulent natural convection in the melt pool A user-defined three-stage creep model [31] was employed In this model experimental creep curves were discretized into a creep database (CDB) under given temperature 119879 and stress 120590 the creep curve was discretized into finite point pairs

(휀 ε) indicating under such thermal and mechanical loads the creep strain rate ε is subjected to creep strain 휀 The creep strain rate for any other case can be interpolated over known temperature stress and strain from the database This method was applied to the post-test calculation of the FOREVE-C2 experiment It was concluded that the unexpected deformation behaviour during the experiment was caused by the combination of the transient temperature field in the vessel wall and the three creep stages

They later replaced the Flotran solver with the ECCM for melt pool heat transfer in a following study [32] By implementing it into FEM-solver a recursively coupled analysis was constructed Moreover a modified creep model [33] was used with damage measures the CDB itself did not contain tertiary creep instead a factor 1(1 minus 119863) was multiplied to the original creep strain rate to get the modified creep strain rate where 119863 is a damage factor (0 lt 119863 lt 1) defined with creep strain information and plastic strain information As 119863 grows close to 1 the modified creep strain rate would be enlarged significantly by this factor 1(1 minus 119863) which then behaves like the accelerating tertiary stage This method was applied to pre- and post- test calculations for the FOREVER experiments proving to be closer to reality than the previous method

Madokoro et al [34] developed a thermal structural analysis tool based on open source solver OpenFOAM and numerically investigated the coupled RPV failure and melt pool convection The PECM was used in the melt pool heat transfer In structural part the plasticity and damage parameters were similar to Willschutz et al [32] except for that strain hardening creep model (Eq (3)) was used instead of a CDB Simulations of FOREVER-EC2 and -EC4 were investigated It was concluded from the simulations that although the emissivity and mesh type (updated or fixed) affect the results significantly the approach generally has the capability of predicting the vessel deformation and failure

7

ε119888119903 = 1198891 sdot σ1198892 sdot ε1198893 (3)

Villanueva et al [35] performed a coupled thermo-mechanical creep analysis of the reactor scale RPV (ABB-Atom design BWR) to evaluate efficacy of the cooling measures for the in-vessel melt coolability and retention The PECM was used for melt pool heat transfer and ANSYS for RPV integrity analysis The modified time hardening model (Eq (4)) of the American RPV steel SA533B was selected as creep model The transient thermal load from the melt pool was applied as a thermal boundary condition to the vessel wall Further studies were done with this approach on investigations of instrumentation guide tube (IGT) failure [36] and control rod guide tube (CRGT) cooling [37] of a Nordic BWR

ε119888119903 =1198881σ

11988821199051198883+1

1198883 + 1 1198881 gt 0 (4)

It is noted that only RPV analysis with finite element analysis (FEA) are reviewed here Engineering methods like the Larson-Miller method [38] are not covered in this thesis One can refer to theory manuals of severe accident codes like MELCOR ASTEC for more details Comprehensive description on modelling of melt pool heat transfer is not covered either One can refer to either [39] for a general view or to any severe accident code manuals for engineering methods

14 Motivation Objectives and Main Achievements

The existing experimental and numerical studies provide us a good understanding of RPV failure involved phenomena and valuable simulation experiences These numerical methods may vary in solvers (which may be related to code accessibility) and models (which may be related to model accessibility understanding of phenomena computational power) However improvements of modelling approaches are still possible as the improvements in any of the related aspects eg

bull Simulation tools are continuously being upgraded and providing new possibilities just like that PECM appeared and ECCM was no longer used

bull Better understanding of the phenomena and more simulation experiences from in-vessel retention and vessel failures researches benefit the further modelling and simulations

bull Increasing computational power allows more detailed modelling eg turbulence modelling of melt pool convection

bull There is still room for the improvement of creep modelling

In addition implementation difficulty of a coupling approach is another important issue to consider Since it is a TRANSIENT thermo-fluid-structure coupled problem it is almost impossible to replicate an approach simply by referring to a paper with limited details For BWRs it would be even better for the coupling approaches to support complex geometries considering forest of penetrations (CRGTs and IGTs) on the RPV lower head without significant increase of user effort in the coupling process

8

The objectives of this thesis are to investigate new approaches and develop new models for the thermo-fluid-structure coupled problem of RPV failure and involved phenomena These objectives are accomplished with the following achievements

1) A new creep model for the 16MND5 steel was developed which covers all three creep stages Good agreement was achieved between model predictions and experimental curves Both time and strain hardening models were implemented (Paper I)

2) A new thermo-fluid-structure coupling approach was developed for RPV failure analysis It supports the advanced models in both melt pool convection and structural mechanics to improve fidelity and facilitates implementation Validation of the approach with the new creep model was performed against FOREVER-EC2 experiment This approach is flexible for complex geometries and can be used in further analysis eg lower head with detailed modelling of penetrations in BWRs (Paper II)

3) The performance of this new approach which uses volume loads mapping for data transfer was compared with the previous approach using surface loads mapping showing good agreement with small discrepancy (Paper III)

4) A lumped-parameter code transIVR for melt pool heat transfer was developed which supports one- and two-layer pool configuration transient analysis and well-resolved RPV wall (Paper IV)

5) A hybrid coupling approach was developed for efficient analysis of RPV failure where transIVR is coupled with ANSYS Mechanical (Paper IV)

In the following general descriptions on the codes ANSYS Fluent Mechanical and the PECM are given in Section 2 the new creep model in Section 3 the new coupling approach as well as its comparison with the previous approach in Section 4 the efficient hybrid coupling approach in Section 5 and summary amp outlook in Section 6

9

2 Numerical Solvers and Models

In this section we briefly introduce the ANSYS Fluent as the CFD solver for melt pool heat transfer the ANSYS Mechanical as the mechanical solver for RPV failure and PECM as an efficient alternative for melt pool heat transfer that are heavily used in the following sections

21 ANSYS Fluent

In the fluid mechanics the fluid is treated as a continuum which is generally described by the conservative mass momentum and energy equations [40]

120597120588

120597119905+ 120571 sdot (120588) = 119878119898 (5)

120597(120588)

120597119905+ 120571 sdot (120588) = minus120571119901 + 120571 sdot (120591) + 120588 + (6)

120597120588119864

120597119905+ 120571 sdot ((120588119864 + 119901)) = 120571 sdot (119896120571119879 minussumℎ119895119869119895

119895

+ (120591 sdot )) + 119878ℎ

(7)

Computational Fluid Dynamics (CFD) studies topics on the numerical solutions of above equations Typical methods are the Finite Volume Method (FVM) Finite Difference Method (FDM) and Finite Element Method (FEM) as well as particle methods that are mesh-free eg SPH and MPS

ANSYS Fluent is a general purpose CFD code which solves above equations with FVM Turbulence modelling and multiphase flows are supported Turbulence can be modelled with RANS LES or hybrid LESRANS Customizations eg adding sourcesink time or spatial dependent boundary conditions are realized through the user-defined functions (UDF) which is programmed in C language

22 ANSYS Mechanical

In the continuum mechanics of solids three sets of variables are considered to describe the structure performance namely element displacement field strain field and stress field (119906 ε σ) A general 3-dimensional problem is described by 3 sets of equations the equilibrium equations (Eq (8)) strain-displacement relationship (Eq (9)) and constitute equations (Eq (10)) [41] For Eq (8) the reciprocal theorem of shear stress also gives τ119894119895 = τ119895119894 where 119894 119895 = 119909 119910 119911

indicating the stress tensor being symmetric For very simple applications eg 1D beam it can be solved analytically For complex problems where analytical solutions are not available approximation methods eg Galerkin weighted residual method principle of virtual work and principle of minimum potential energy would be used

10

partσ119909119909part119909

+partτ119909119910

part119910+partτ119909119911part119911

+ 119909 = 0

partτ119910119909

part119909+partσ119910119910

part119910+partτ119910119911

part119911+ 119910 = 0

partτ119911119909part119909

+partτ119911119910

part119910+partσ119911119911part119911

+ 119911 = 0

(8)

ε119909119909 =part119906

part119909 휀119910119910 =

120597119907

120597119910 휀119911119911 =

120597119908

120597119911

120574119909119910 =120597119906

120597119910+120597119907

120597119909 120574119910119911 =

120597119908

120597119910+120597119907

120597119911 120574119911119909 =

120597119908

120597119909+120597119906

120597119911

(9)

ε119909119909 =1

119864[σ119909119909 minus μ(σ119910119910 + σ119911119911)]

ε119910119910 =1

119864[σ119910119910 minus μ(σ119909119909 + σ119911119911)]

ε119911119911 =1

119864[σ119911119911 minus μ(σ119909119909 + σ119910119910)]

γ119909119910 =1

119866τ119909119910 γ119910119911 =

1

119866τ119910119911 γ119911119909 =

1

119866τ119911119909

(10)

ANSYS Mechanical [42] is a mechanical engineering software that uses finite element analysis (FEA) for structural analysis The principle of virtual work is adopted in the solver [43] The idea of this principle is that a virtual (very small) change of the internal strain energy must be offset by an identical change in external work due to the applied loads which reads

δ119880 = δ119881 (11) where 119880 is the strain energy or the internal work 119881 the external work and δ the virtual operator The virtual internal work can be expressed as

δ119880 = int δεσd(vol)T

vol

+int 119886119903119890119886119891

δ119908119899119879σ119889(119886119903119890119886119891) (12)

where 휀 is the strain vector 120590 stress vector vol volume of element 119908119899 motion normal to the surface 119886119903119890119886119891 area of the distributed resistance The virtual

external work is contributed from three parts the inertial effects the pressure force on the surface and nodal forces applied to the element which then can be expressed as

δ119881 = minusint 119907119900119897

δ119908119879119865119886

119907119900119897119889(119907119900119897) + int

119886119903119890119886119901

δ119908119899119879119875119889(119886119903119890119886119901)

+ δ119906119879119865119890119899119889

(13)

11

where 119908 is the vector of displacements 119865119886 acceleration force vector 119875 applied pressure vector 119886119903119890119886119901 area over which pressure acts and 119865119890

119899119889 nodal

forces applied to the element

Plugging Eq (12) and (13) into (11) would result in the equilibrium equation Since the virtual displacement can be arbitrary the final equations used to describe the equilibrium of an element is reduced to Eq (14) with multiple operations

([119870119890] + [119870119890119891])119906 minus 119865119890

119905ℎ = [119872119890] + 119865119890119901119903 + 119865119890

119899119889

(14)

where [119870119890] [119870119890119891] 119865119890

119905ℎ [119872119890] and 119865119890119901119903 are the element stiffness matrix

foundation stiffness matrix thermal load vector mass matrix acceleration vector and pressure vector respectively It represents the equilibrium equation on a one element basis Alternatively using a minimum total potential energy principle could result in a consistent expression to the principle of virtual work [41]

Customizations eg on material properties are realized through the user programmable features (UPF) which is programmed in FORTRAN language

23 PECM

The PECM [28] [29] developed in KTH is a correlation-based method to model the melt pool heat transfer It is implemented in ANSYS Fluent through UDF and has been validated against many experiments such as ACOPO COPO LIVE etc In the PECM the flow velocities 119906119909 119906119910 and 119906119911 are replaced by characteristic

velocities 119880119909 119880119910 and 119880119911 respectively in order to eliminate the necessity of solving

the Navier-Stokes equation

The characteristic velocities are calculated using heat transfer correlations

119880119906119901 =α

ℎ119901119900119900119897times (119873119906119906119901 minus

ℎ119901119900119900119897

ℎ119906119901) (15)

119880119889119899 =α

ℎ119901119900119900119897times (119873119906119889119899 minus

ℎ119901119900119900119897

ℎ119889119899) (16)

119880119904119889 =α

ℎ119901119900119900119897times (119873119906119904119889 minus

ℎ119901119900119900119897

119882119901119900119900119897) (17)

where ℎ119901119900119900119897 is the melt pool depth ℎ119906119901 the thickness of well mixed layer of the

pool ℎ119889119899 the thickness of lower stratified region of melt pool 119882119901119900119900119897 the width of

pool 120572 the thermal diffusivity The PECM employs reduced characteristic velocities to describe mushy zone convection heat transfer For internally heated pool natural convection the Steinberner-Reineke correlations (Eq (18)~(19)) [44] are employed The profile of sideward 119873119906 is described using Eckertrsquos type correlation (Eq (20)) for a vertical boundary layer where the local Rayleigh

number at pool height 119910 is defined as 119877119886119910 =119892βΔ1198791199103

αν

12

119873119906119906119901 = 03451198771198861198941198990233

(18)

119873119906119889119899 = 13891198771198861198941198990095

(19)

119873119906119904119889 = 050811987511990314 (

20

21+ 119875119903)

minus14

11987711988611991014

(20)

As the characteristic velocities are calculated the energy equation is the only transport equation that needs to be solved and can be expressed as follows

partρ119862119901119879

part119905+ nabla(ρ119862119901119932119879) = 119896nabla2119879 + 119876119907 (21)

where from left to right are the transient term convection term diffusion term and volumetric heat source respectively The convection term then can be calculated explicitly with use of values in the last time step and then can be treated together with the volumetric source term Therefore the PECM finally simplifies the energy equation to a conduction equation which can be easily solved and as a result will significantly increase the computational efficiency in simulations

In addition solidificationmelting process can also be modelled and the temperature-dependent physical properties are supported

13

3 Creep Modelling of 16MND5 Steel

Creep deformation becomes significant when a component is subjected to high temperature and stress Modelling of creep behaviour of the RPV is paramount since it is considered as a main mechanism leading to the RPV failure In this section we present the development of a three-stage creep model (called lsquomodified theta projection modelrsquo ) for the French RPV steel 16MND5 This model was successfully implemented into the ANSYS Mechanical code

31 Creep and Creep Modelling

Creep is an irreversible time-dependent inelastic deformation process Diffusion and dislocation are the two molecular mechanisms of creep The former normally occurs due to the presence of vacancies within the crystal lattice (or the grain) while the latter occurs if there is a slip by the passage of crystal dislocations [45] There are two types of diffusion creep [46] Coble creep (favoured at lower temperatures) if the diffusion paths are predominantly through the grain boundaries and Nabarro-Herring creep (favoured at higher temperatures) if through the grains themselves Dislocation creep tends to dominate at high stresses and relatively low temperatures Dislocations can move by gliding in a slip plane which is a process that requires little thermal activation But the rate-determining step for their motion is often a climb process which also requires diffusion and is thus a time-dependent process and favoured by higher temperatures Creep behaviour is highly affected by temperature and stress generally creep is more significant under higher temperature and stress

Creep curves describe the creep strain development with time Figure 3 shows a creep curve under constant temperature and stress which typically experiences three creep stages primary stage where the creep strain rate ε119888119903 (the change of creep strain ε119888119903 in unit time) decreases with time secondary stage where the creep strain rate almost keeps constant and tertiary stage where the creep strain rate drastically increases with time

Figure 3 Creep curve under constant temperature and stress

14

The built-in creep models in commercial software eg ANSYS [47] and ABAQUS [48] in default only cover the primary andor the secondary stage(s) Table 1 [47] gives the default creep models supported in ANSYS for example Possible reason why the tertiary stage is not modelled might be that for most industrial applications the failure analysis is beyond interest and therefore the built-in creep laws in the commercial software was supposed to be sufficient [49] Though the two codes offer the interfaces through either ANSYS UPF or ABAQUS user subroutines to support more general user-defined creep laws eg complex models that also include the tertiary creep stage

Table 1 Default creep models in ANSYS Mechanical

Creep model Equations Type

Strain hardening

ε119888119903 = 1198621σ1198622ε119888119903

11986231198901198624119879 1198621 gt 0 Primary

Time hardening ε119888119903 = 1198621σ11986221199051198623119890minus1198624119879 1198621 gt 0 Primary

Generalized exponential

ε119888119903 = 1198621σ1198622119903119890minus119903119905 r = 1198625σ

11986231199051198623119890minus1198624119879 1198621 gt 0

1198625 gt 0

Primary

Generalized Graham

ε119888119903 = 1198621σ1198622(1199051198623 + 1198624119905

1198625 + 11986261199051198627)119890minus1198628119879 1198621 gt 0 Primary

Generalized Blackburn

ε119888119903 = 119891(1 minus 119890minus119903119905) + 119892119905 119891 = 1198621119890

1198622σ 119903 = 1198623(σ1198624)1198625 119892 = 1198626119890

1198627σ

1198621 gt 0

1198623 gt 0

1198626 gt 0

Primary

Modified time hardening

ε119888119903 = 1198881σ11988821199051198883+1(1198883 + 1) 1198621 gt 0 Primary

Modified strain hardening

ε119888119903 = 1198621σ1198622[(1198623 + 1)ε119888119903]

1198623119890minus1198624119879 1198621 gt 0 Primary

Generalized Garofalo

ε119888119903 = 1198621[119904119894119899ℎ(1198622σ)]1198623119890minus1198624119879 1198621 gt 0 Secondary

Exponential form

ε119888119903 = 1198621119890σ1198622119890minus1198623119879 1198621 gt 0 Secondary

Norton ε119888119903 = 1198621σ1198622119890minus1198623119879 1198621 gt 0 Secondary

Combined time hardening

ε119888119903 = 1198621σ11986221199051198623+1119890minus1198624119879(1198883 + 1)

+ 1198625σ1198626119905119890minus1198627119879

1198621 gt 0

1198625 gt 0

Primary+ Secondary

Rational polynomial ε119888119903 = 1198621

partε119888part119905 ε119888 =

119888119901119905

1 + 119901119905+ ε119898119905

ε119898 = 1198622101198623σσ1198624 119888 = 1198627ε

1198628σ1198629

119901 = 11986210ε11986211σ11986212

1198622 gt 0 Primary+ Secondary

Generalized time hardening

ε119888119903 = 119891119905119903119890minus1198626119879 119891 = 1198621σ + 1198622σ2 + 1198623σ

3 119903 = 1198624 + 1198625

Primary

32 A Modified Theta Projection Model

321 Original theta projection model and its variations

The original theta projection model [50] was first proposed to accurate predict long-term creep and creep-fracture properties It assumes that a creep curve under constant temperature and stress can be described as

15

휀119888119903 = 1205791(1 minus 119890minus1205792119905) + 1205793(119890

1205794119905 minus 1) (22)

where 휀119888119903 is the creep strain 119905 the time and 1205791~1205794 the model parameters Figure 4 illustrates how the first term and second term on right hand side develop with time and constitute a creep curve which has obvious primary secondary and tertiary stages Model parameters 120579119894(119894 = 1234) are temperature and stress dependent and are described with the following expression

119897119899thinsp120579119894 = 119886119894 + 119887119894120590 + 119888119894119879 + 119889119894120590119879 (23)

where 119886119894 119887119894 119888119894 and 119889119894 ( 119894 = 1234 ) are constants It is noted [45] that this equation is suitable when the temperature range of experiment data sets is very narrow such that minus1119879 is almost proportional to 119879 These constants (119886119894 119887119894 119888119894 and 119889119894 (119894 = 1234)) are the final model parameters which are generated with two fitting procedures

(i) Fitting the experimental creep curves to Eq (22) generates several groups of 1205791~1205794 for each creep curve under certain temperature and stress We denote (119879119895 120590119895 1205791119895 1205792119895 1205793119895 1205794119895 ) as parameters for the 119895119905ℎ

experimental curve under temperature 119879119895 and stress 120590119895

(ii) Fitting 1205791 to Eq (23) as a function of temperature and stress with the obtained data ( 119879119895 120590119895 1205791119895 ) from ( 119879119895 120590119895 1205791119895 1205792119895 1205793119895 1205794119895 ) generates

parameters 1198861 1198871 1198881 and 1198891 Repeat this fitting procedure for 1205792 1205793 and 1205794 to get parameters 119886119894 119887119894 119888119894 and 119889119894 (119894 = 234)

Variations of the model were developed and applied to other different materials to better accommodate the experimental data which are all then called a lsquomodified theta projection methodrsquo independently Examples are models in [51] for lowhigh alloy ferritic austenitic steels Al-alloys Al-matrix composites [52] for Nickel-based Super-alloy [53] for pure metals and class M alloys with intermediate range of stress (40-70 of the yield stress) and temperature (40-60 of the melting temperature)

Figure 4 Illustration of theta projection method [54]

16

322 A modified theta projection model

The new modified theta projection model we proposed [54] was mainly adapted from the original theta projection method [50] and the modified version [51] The new model describes the creep curve with the following equation

휀119888119903 = 1205791(1 minus 119890minus1205792119905) + 120579119898119905 + 1205793(119890

1205794119905 minus 1) (24)

An additional term 120579119898119905 is seen compared with Eq (22) in the original method The reason for adding this term is that the 16MND5 steel shows fairly long secondary stages in many experimental creep curves and the creep curves can be better fitted by Eq (24) than Eq (22) As the structure undergoes different creep process under different temperature and stress the model parameters θ119894 (119894 =1~4 and 119898) are also temperature and stress dependent

The main difference compared with the original and all other modified projection models is that numerical interpolation is used to replace the second numerical fitting in previous models In other words instead of fitting the model parameters as functions of stress and temperature with Eq (23) the distributions of the model parameters are assumed to be piecewise in the temperature-stress space and they will be interpolated directly from known parameters obtained from experiments The consideration is that the experimental data ([16] [17]) for 16MND5 covers a wide temperature range varying from 600 degC to 1300 degC and a wide stress load range varying from 08 MPa to 248 MPa while Eq (23) in the existing methods is based on a narrow temperature span Poor parameter fitting results would be expected if we directly fit the parameters 1205791~1205794 with Eq (23) as used in the original methods

The creep model was developed using the following steps

(i) Fitting Each creep curve can be represented by a group of model parameters [1205791 1205792 120579119898 1205793 1205794] with respect to Eq (24) by fitting the creep curves to Eq (24) All parameters of the 32 experimental curves form an

initial dataset of the model parameters [119879119895 120590119895 1205791119895 1205792119895 120579119898119895 1205793119895 1205794119895] where j (0ltjle32) is the jth creep curve

(ii) Interpolation Parameters of a creep curve under any [119879 σ] are directly interpolated from the dataset

(iii) Parameter adjustment The dataset was further updated to satisfy the monotonicity assumption we have taken that creep strain monotonically increases with temperature and stress

(iv) Different hardening models for dynamic loads

Step (i) was done using MATLAB In the following steps (ii)-(iv) are described in detail

Model parameters for new creep curves are interpolated from the dataset over the 2D 119879 minus 120590 space The interpolation is divided into two sub-interpolation steps interpolate over stress with fixed temperature and then interpolate over temperature with fixed stress We take the interpolation of model parameters under (119879prime σprime) for example In the first interpolation we first find out the two

17

temperatures 119879119896119899119900119908119899119904119898119886119897119897 and 119879119896119899119900119908119899119897119886119903119892119890 most close to 119879prime from the experimental

temperature set 600 700 800900 1000 1100 1200 1300 Under each temperature there are four sets of model parameters in the dataset corresponding to four different stress loads Then the first interpolation is done over stress load from the known dataset of model parameters under each

temperature to output the parameters for (119879119896119899119900119908119899119904119898119886119897119897 120590rsquo) and (119879119896119899119900119908119899119897119886119903119892119890 120590rsquo)

In the second interpolation the model outputs parameters for (119879rsquo 120590rsquo) from

(119879119896119899119900119908119899119904119898119886119897119897 120590rsquo) and (119879119896119899119900119908119899119897119886119903119892119890 120590rsquo) For 120579i(119894 = 12 3 4) we assume a logarithm

relation between 120579119894 temperature 119879 and stress 120590 which is similar to Eq (23)

119897119899120579119894 = (119886 + 119887120590)(119888 + 119889119879) (25)

It reduces to 120579119894 = 11986011198901198871σ with fixed temperature in the first interpolation and to

θ119894 = 11986021198901198872T with fixed stress in the second interpolation The interpolation

expression for θ119894 = 11986011198901198871σwith two known points (1205901 1205791198941) and (1205902 1205791198942) is given

by Eq (26) The interpolation expression for θ119894 = 11986021198901198872T with two known points

(1198791 1205791198941) and (1198792 1205791198942) is given by Eq (27)

120579119894 = 120579i1

120590minus12059021205901minus1205902120579i2

1205901minus1205901205901minus1205902 (26)

120579119894 = 120579i1

119879minus11987921198791minus1198792120579i2

1198791minus1198791198791minus1198792 (27)

Similarly we assume the following relation between 120579119898 temperature 119879 and stress 120590

120579119898 = 119860120590119899119890minus119887 119879frasl (28)

The interpolation is done over the stress with 120579119898 = 1198603120590n by fixing temperature

and then over temperature with 120579119898 = 1198604119890minus1198874119879 by fixing stress The interpolation

expression over stress for 120579119898 = 1198603120590119899 with two known points (1205901 1205791198981) and

(1205902 1205791198982) would be given by Eq (29) The interpolation expression over

temperature for 120579119898 = 1198604119890minus1198874119879 with two known points (1198791 1205791198981) and (1198792 1205791198982)

would be given by Eq (30)

120579119898 = 119890119909119901(119897119899120590 minus 11989711989912059011198971198991205902 minus 1198971198991205901

1198971198991205791198982 +1198971198991205902 minus 119897119899120590

1198971198991205902 minus 11989711989912059011198971198991205791198981) (29)

120579119898 = 120579m2(11198791minus1119879) (

11198791minus11198792)fraslsdot 120579m1

(1119879minus11198792) (

11198791minus11198792)frasl (30)

In this model we also take the assumption that the creep strain monotonically increases with temperature and stress loads1 Or specifically with the same creep time the structural component under higher temperature or stress experiences

1 Intuitively materials deform more under higher stress and temperature This is also mostly observed in tensile experiments Though some exceptions exist one instance of which is the effect of phase change some materials under certain temperature around the phase change temperature may show much weaker structural performance Here we just simply assumed that the creep strain monotonically increases with temperature and stress

18

higher creep strain than that with lower temperature or stress (the creep curve under higher temperature or stress should be above the curve with lower temperature or stress) When generating the parameter dataset (Step (i)) we already kept the trend that under the same temperature cases under higher stress loads would have larger creep model parameters and give higher creep strains This trend would also hold for the interpolated temperatures which thus guarantees that creep strain monotonically increases with stress load under same temperature Then the question becomes if creep strain also monotonically increases with temperature given fixed stress However the answer could not always be a yes especially in the extrapolation range of the stress load Figure 5(a) illustrates an example of the interpolation of model parameters under (119879119872 σ119872) The red dots and blue dots are known parameters from the parameter dataset for corresponding high temperature 119879119867 and low temperature 119879119871 respectively The green dots (119879119867 σ119872) and (119879119871 σ119872) are interpolated from the red and blue dots in the first sub-interpolation respectively The black dot (119879119872 σ119872) is interpolated from the green dots in the second sub-interpolation It is seen that when σ119872 lt σ119862 the value of (119879119871 σ119872) is higher than that of (119879119867 σ119872) It means the higher 119879119872 is the smaller the model parameter will be ie the model would predict larger creep strain for lower temperatures which violates our monotonicity assumption One simple solution used here as shown in Figure 5(b) is to minorly adjust the values of the four red and blue dots such that the intersection point C can be sufficiently small and the situation σ119872 lt σ119862 would barely occur in the extrapolation range

Figure 5 Illustration of model parameter interpolation (a) before and (b) after parameter adjustment [54]

A corresponding parameter adjustment process was introduced to verify that the model predicts higher creep strain under higher temperature under fixed stress Figure 6 illustrates the general verification process for any (119879 120590) in the interested load range the creep strain of (119879 + 120575119879 120590) should be larger than that of (119879 120590) Otherwise parameters would be minorly adjusted and the verification process be repeated This process would be iterated several times until the requirement is reached The achieved verification shows

Stress

σC

High temperature TH

MPa

a

0

Model

par

amet

er θ

Low temperature TL

MPa

a

0

Model

par

amet

er θ

High temperature TH

Low temperature TL

σC

Stress

C C

(a) (b)

σM

σM

19

(i) For any (119879 120590) in the temperature range 273~ 1073 K with an interval of 1 K and the stress range 1 ~260 MPa with an interval of 1 MPa its creep strain is higher than that of (119879 minus 1119870 120590) for the first 12000000 s

(ii) For any (119879 120590) in the temperature range 1073~1673 K with an interval of 1 K and the stress range 1 ~80 MPa with an interval of 1MPa its creep strain is higher than that of (119879 minus 1119870 120590) for the first 12000000 s

(iii) For any (119879 120590) in the stress range 01~26 MPa with a smaller interval of 01 MPa for the full temperature range 273~1673 K with an interval of 1 K its creep strain is higher than that of (119879 minus 1 120590) for the first 12000000 s

Above verification covers a wide range of temperature and stress especially where the stress and temperature are low and extrapolation is used Further verification to cover even higher stress or temperature ranges is not considered as creep increases drastically under such high stress and temperature and the results would become meaningless With these regards we generally think the monotonicity requirement is reached The model parameter dataset was updated along with the adjustment The final dataset was given as Table 2 in Paper I

Figure 6 Parameter adjustment process [54]

Figure 7(a-d) present the model predictions of selected creep curves in ANSYS Mechanical (labelled as lsquoANSYSrsquo) compared with experimental curves (labelled as lsquoREVISArsquo) at temperature 600 800 1000 and 1200 degC respectively The predicted curves proved to clearly show the three-stage creep process and agreed well with the experimental creep curves

Initial parametersdataset

θ1 θ2 θm θ3 θ4 from creep

curve-fitting

Parameters interpolation

Is the monotonicity

assumption satisfied

IsMon= =1

Final model parameters

Adjust the

parametersdataset

θ1 θ2 θm θ3 θ4

No

Yes

IsMon=1 a flag is monotonic achieved

For σ in the stress range in MPa

For T in range Tstart Tend in K

For t in range 112000000 in s

If ε(σ T+1t)lt ε(σTt) IsMon=0

Iteration loop of

parameter adjustment

20

Figure 7 Selected model predictions compared with experimental creep curves [54]

Instead of being constant temperature and stress loads also dynamically change with time in real cases Hardening models determine how creep strain rate reacts to the change of loads during a creep process The time hardening model and strain hardening model2 are supported in our model implementation in ANSYS Mechanical which are two mostly used hardening models

Time hardening assumes that the following creep strain rate after a load change depends only on the current creep time from the beginning of the creep process (and also temperature and stress if used in the creep model) and ignores the current creep strain Strain hardening assumes that the creep strain rate depends only upon the current creep strain (and also temperature and stress if used in the creep model) [55]

2 The time hardening and strain hardening here are more general terms than the specific time hardening creep model and the strain hardening creep model mentioned in Table 1

(a) 600 degC (b) 800 degC

(c) 1000 degC (d) 1200 degC

21

Eq (24) in default is a time hardening model as creep strain is directly described by the time 119905 Eq (31) gives the creep strain increment 119889휀119888119903 at time 119905 during time interval 119889119905

119889휀119888119903 = (12057911205792119890minus1205792119905 + 120579119898 + 12057931205794119890

1205794119905)119889119905 (31)

The strain hardening model is derived from Eq (24) in an implicit form Specifically when temperature and stress load change to (119879 120590) at the time 119905 and current creep strain 휀1198881199030 the model will get the creep curve of constant (119879 120590) and

output how this curve will react under the current strain 휀1198881199030 Since the creep curve shows creep strain monotonically increases with time given the creep strain 휀1198881199030 under creep curve of (119879 120590) we can uniquely find this reference time

119905119903119890119891 satisfying Eq (32) One should notice that the reference time 119905119903119890119891 can be

different from the current physical time 119905

휀1198881199030 = 1205791(1 minus 119890minus1205792119905119903119890119891) + 120579119898119905119903119890119891 + 1205793(119890

1205794119905119903119890119891 minus 1) (32)

where all 120579119894(119894 = 1~4119898) are model parameters of the new creep curve (119879 120590) It can be solved for 119905119903119890119891 with methods eg Newton-Ramphson [56] Then the creep

strain increment at 119905119903119890119891 during time interval 119889119905 is given as

119889휀119888119903 = (12057911205792119890minus1205792119905119903119890119891 + 120579119898 + 12057931205794119890

1205794119905119903119890119891)119889119905 (33)

Therefore combining Eq (32) and (33) gives the creep strain rate at creep strain 휀1198881199030 In summary when loads are changed to (119879 120590) with current creep strain 휀1198881199030

the model traces the creep curve of (119879 120590) and calculates the creep strain rate of this curve at strain 휀1198881199030

One calculation was done to demonstrate the performance of the hardening models The stress load was set to 151 MPa and kept unchanged A step temperature load was applied as the orange dash curve shows in Figure 8 it was 900 degC during 0-36000 s and suddenly jumped to 1000 degC at 36000 s which was then kept till the end of the simulation The blue solid curve (the part before the temperature change was identical to and therefore covered by the green solid curve in the figure) and blue dash curve are the creep curves under constant temperature T=900 degC and T=1000 degC respectively The green solid curve and green dash curve represent the model predictions considering this temperature change with time hardening model and strain hardening model respectively Before this change the creep curves with both hardening models were identical to the creep curve of the T=900 degC till point A After that the creep strain rate then shifted from Point A to Point B (which had the same time with A) in the time hardening model and behaved the same as the curve T=1000 degC as shown in the rest part of the green solid curve In the strain hardening model the creep strain rate shifted from Point A to Point C (which had the same creep strain with A) and behaved the same as the curve T=1000 degC as shown in the rest part of the green dash curve The large difference between two models was partially caused by the drastic temperature jump In real cases the difference would be less significant as the temperature changes more mildly Overall this prediction is exactly the expected behaviors of the time hardening model and strain hardening model

22

Figure 8 Time hardening and strain hardening in ANSYS Mechanical [54]

23

4 Coupled Thermo-Fluid-Structure Analysis of RPV Failure

As described in Section 13 RPV failure is coupled with melt pool heat transfer leading it into a transient thermo-fluid-structure coupled problem The goal of this section is to develop and validate a reliable coupling approach which supports transient coupled thermo-fluid-structure analysis as well as advanced models and computational platforms so as to improve fidelity and facilitate implementation The approach was validated against the FOREVER-EC2 experiment and then applied to an RPV failure analysis of a Nordic BWR As the approach adopts the volume loads mapping for load transfer comparative analysis was also performed against the coupling approach using surface loads mapping Future work is considered in analysis of penetration failures (IGTs and CRGTs) of a BWR RPV since the approach is flexible to complex geometries Results of Paper II ampIII are jointly presented in this section

41 Thermo-Fluid-Structure Coupling Approach

Considering the involved phenomena a coupling approach should at least address three issues (i) melt pool heat transfer (ii) RPV deformation and (iii) the interaction between melt pool heat transfer and vessel deformation A two-way strong coupling considering both the transient thermal load from melt pool on vessel and the influence of vessel deformation on melt pool heat transfer involves complex data exchange and is still full of challenges Therefore the new coupling approach is realized in a one-way manner with the assumptions that the influence of vessel deformation on melt pool heat transfer can be ignored and thus only the thermal loads from melt pool to vessel are considered It is based on the considerations that on one hand the deformation is a slow process regarding the relatively long failure time and on the other hand it was pointed out [57] that the physics of natural convection heat transfer in internally heated liquid pools is not sensitive to the geometrical factors based on their calculation results and experimental data from [58] [59] as well as previous experiments in semicircular and square cavities eg [44] [60] In addition one-way coupling is also computationally more efficient especially when heat transfer turbulence structural non-linearity and large-scale geometries are involved The developed approach is based on the ANSYS platform which generally consists of three parts with respect to the three issues respectively

(i) conjugate heat transfer of the melt pool and vessel using ANSYS Fluent (ii) transient thermal loads transfer from ANSYS Fluent to ANSYS

Structural (the ANSYS Mechanical in ANSYS Workbench) (iii) structural analysis of the vessel in ANSYS Structural

ANSYS Structural in default only supports direct transfer of constantsteady-state spatial loads from ANSYS Fluent Transfer of transient loads is realized with the extension tool FSI Transient [61] It is a pre-processing tool (with good documentation) developed by a third-party CAE vendor EDRTM MEDESO [62] to map transient CFD results (heat transfer coefficient temperature and surface pressure) to thermal or structural analysis including the definition of load steps

24

This new procedure proves to be straightforward and reduces efforts in the load transfer

Since structural performance of RPV is highly dependent on temperature the transfer of transient temperature loads from conjugate heat transfer to mechanical solver is a core step of the coupling framework There are two general approaches to realize the transfer as shown in Figure 9 Both approaches contain three parts a conjugate heat transfer analysis for fluid and structure where flow and heat transfer are coupled a mechanical analysis for the structure and a load mapping process that transfers the transient loads The conjugate heat transfer part can be solved with any CFD solver and is treated in the same manner in both approaches In the first approach (Figure 9(a)) transient volume thermal loads are transferred and mapped into the mechanical simulation domain The structure domain reads the essential thermal loads from the mapping after which a structure analysis of this domain is performed In the second approach (Figure 9(b)) transient surface thermal loads are mapped onto the boundary surfaces of structure domain as thermal boundary conditions Then a thermo-structure coupled analysis of the structure domain in the mechanical part is performed To distinguish them we call the former approach volume loads mapping (VLM) and the latter surface loads mapping (SLM) In the RPV failure analysis the fluid and structure domains correspond to the melt pool and RPV respectively Examples of these methods are the studies of [63] and [30] using VLM and [35] using SLM The VLM is adopted in current approach

Figure 9 Frameworks of the thermo-fluid-structure coupling approaches (a) volume loads mapping (b) surface loads mapping [64]

fluid structure structure

fluid structure structure

Conjugate heat transfer problem Mechanical problem

(structure)

Conjugate heat transfer problem Mechanical problem

(thermo-structure coupled)

Transient volume

loads mapping

(VLM)

Transient surface

loads mapping

(SLM)

(a)

(b)

25

42 Numerical Modelling

421 Conjugate heat transfer

In the conjugate heat transfer analysis main focus is put on the modelling of melt pool heat transfer Within the ANSYS Fluent solver two options are supported CFD with turbulence modelling and PECM CFD is the choice when the model goes to detail and more mechanistic analysis are needed while PECM provides an alternative that is faster and possibly more suitable for large-scale problems

Within the scope of RANS models the SST (Shear Stress Transport) model [65] is selected as the turbulence model in present study This model is a combination of k-ε model and k-ω model The k-ω formulation is used in the inner parts of the boundary layer which makes the model directly usable all the way down to the wall through the viscous sub-layer It also switches to k-ε model in the free-stream and thereby avoids the common k-ω problem that the model is too sensitive to the inlet free-stream turbulence properties The considerations on choosing this model are

bull Several two-equation low-Re k-ε models have been proved in [66] not good enough for melt pool heat transfer Therefore these mentioned two-equation models as well as other one-equation models are not considered

bull SST is highly recommended [67] for simulating most industrial heat transfer problems

bull Convincing results ([68] [69]) using SST have been obtained in capturing the thermal behaviour of the natural convection in an internally heated fluid Especially good agreements were achieved between the predicted heat flux distribution along the vessel and experimental data which would be one of the most important factors to estimate model performance in this problem

The description of PECM as an effective model for melt pool heat transfer is given in Section 23

422 Mechanical analysis

The material for the RPV failure analysis considered in this study is the French RPV steel 16MND5 The three-stage modified theta projection model in Section 3 is used as the creep model Since the definition of an RPV failure criterion is not universal (it can be the time when damage variable reaching 1 or either specified value of creep strain creep strain rate or total deformation is reached) we do not explicitly define a failure criterion Instead the simulation is kept running until the creep converge criterion cannot be satisfied even with the defined minimum time step (indicating a fairly large creep strain rate) and the simulation will be automatically terminated The time the simulation stops is taken as the failure time This treatment implicitly implies a criterion of a large creep strain rate We refer to [17] for other material properties Youngrsquos modulus and thermal conductivity are temperature dependent Multilinear Isotropic Hardening (MISO) is used as the plasticity model

26

43 Validation against the FOREVER-EC2 Experiment

In this section the validation against the FOREVER-EC2 experiment is presented (indicated as lsquoVLMrsquo in the following) In addition the simulation results using the SLM approach (indicated as lsquoSLMrsquo in the following) is also included for comparison purpose

431 FOREVER-EC2 experiment

The FOREVER-EC2 experiment [70] is one of the FOREVER experiments that achieved a vessel failure Figure 10 shows the experimental setup A 110 scale pressure vessel with a wall thickness of 15 mm and an outer diameter of ~400 mm was employed The lower head was made of 16MND5 steel Since it was difficult to use prototypical corium materials in the experiment the binary salt 30 CaO-70 B2O3 was employed as simulant To generate the internal heat which corresponds to the nuclear decay heat a specially designed heater was used The power input was maintained ~38 kW during the experiment such that maximum melt pool temperature would be ~1300 degC The pressure inside the vessel was maintained at ~25 MPa Thermocouples were used to measure the melt pool and vessel wall temperatures and linear displacement transducers used to measure the vessel wall displacements The designed experimental conditions were reached at experimental time 11670 s Vessel failure was observed about 12510 s later (at experiment time 24180 s)

Figure 10 Scheme of the experimental setup [22]

27

Figure 11 Computational domains and boundary conditions (a) conjugate heat transfer (b) mechanical analysis [71]

Computational domains and boundary conditions are indicated in Figure 11 SST turbulence modelling was used for pool heat transfer The simulations started at the moment when experimental conditions were reached (experiment time 11670 s) So simulation time 0 s corresponds to experiment time 11670 s

432 Conjugate heat transfer

Figure 12 shows the temperature fields at simulation time 0 s 1225 s 2475 s 4975 s and 9975 s All the time points in this part refer to simulation time The field at time 0 s (Figure 12(a)) was the initial temperature that was reconstructed based on the thermocouple readings The temperature increased with time and reached a steady state when the maximum temperature reached ~1607 K Figure 12(b-f) show clear thermal stratification which is an expected phenomenon of natural convection

Internal pressure

25 bar

Internal pressure 25e6-2500times981timesy Pa

External pressure

1 bar

Zero displacement

y=0 m

Gravity 400

mm

R=203 mm

15 m

m

Tamb

=323 K radiation with ε = 08

+ convection with h=10Wm2K

Tamb

=800 K radiation with ε=02

(a) (b)

Tamb

=800 K radiation with ε=08

+ convection with h=10 Wm2K

28

Figure 12 Temperature fields at simulation time (a) 0 s (b)1225 s (c) 2475s (d) 4975 s (e) 7475 s and (f) 9975 s [64]

Figure 13 shows the temperature profiles of the external vessel along the polar angle in comparison with experimental data Regions with a polar angle larger than 90deg are those in the cylindrical part of the vessel It also shows the temperature profile increased with time until reaching a steady state The profile almost did not change after 4975 s which indicates that a steady state of the temperature was reached As time increased the temperature in the lower head region 0~90deg (where the vessel was contacted with the melt) evolved from a relatively flat distribution to a distribution that temperature at high angle was higher than that at low angle by more than 200 K Generally the steady state results agreed well with the experimental data except for the low angle part of vessel lower head and the high angle part of the cylindrical vessel In addition to modelling errors the discrepancy in the low angle part of the lower head may be caused by the crust and gap possibly formed at vessel bottom in the experiment which can reduce the downward heat transfer and result in lower temperature while they cannot be modelled in simulations As to the high angle cylindrical vessel part simplified boundary conditions were applied to the inner and outer vessel wall since it was far from the melt and not of interest which may result in some discrepancy

(a) (b)

(c) (d) (e) (f)

29

Figure 13 Temperature profiles of external vessel surface along the vessel polar angle [64]

433 Mechanical analysis

In the mechanical analysis the VLM predicted a failure time of 23959 s and the SLM predicted a failure time of 22445 s both are close to the experimental failure time 24180 s All the time points in this part refer to the experiment time Figure 14 shows the creep strain fields at failure time Both approaches predicted maximum creep strains at same location where the temperature was high (cf Figure 12) This is related to the fact that creep behavior is highly influenced by temperature Maximum values for both cases were also close to each other 0393 in VLM and 0366 in SLM

Figure 15 presents the total deformation of the vessel bottom point (also called ldquosouth polerdquo in some literature) in the vertical direction Since there was an initial displacement reading of ~355 mm at time 0 s which almost did not change during 0~11670 s (the whole period before the experimental conditions were reached) an initial value of 355 mm was added to the simulation result such that experimental and simulation results had the same total displacement at t=11670 s Similar trend was seen in the VLM SLM and experimental data that the deformation increased with time and experienced a drastic acceleration at the end Generally both simulation approaches agreed well with the experimental data though they both slightly under predicted the value of total deformation The discrepancy may possibly be related to the accuracy of the thermal predictions modelling errors of plasticity and creep and simplified boundary conditions

400

600

800

1000

1200

1400

1600

0 20 40 60 80 100 120 140 160

Tem

per

ature

(K

)

Polar angle (deg)

Experiment

25 s

1225 s

2475 s

4975 s

7475 s

9975 s

Weilding line

30

Figure 14 Creep strain fields at failure time (a) VLM (b) SLM [64]

Figure 15 Total deformation in vertical direction of the vessel bottom point [64] [71]

The wall thickness changes are plotted together with experimental results in Figure 16 The wall thickness change δ119909 is defined as

δ119909 = 119909119890119899119889 minus 119909119900119903119894119892119894119899119886119897 (34)

000

500

1000

1500

2000

2500

11670 13670 15670 17670 19670 21670 23670

Tota

l def

orm

atio

n (

mm

)

Time (s)

ExperimentVLMSLM

(a) (b)

31

where 119909119890119899119889 and 119909119900119903119894119892119894119899119886119897 are the wall thickness at failure time and the original wall

thickness respectively Since 119909119890119899119889 cannot directly be post-processed it was estimated in the following way (cf the subfigure of Figure 16) for the same polar angle θ find the corresponding points Pin (1199091 1199101) and Pout (1199092 1199102) on the inner and outer surfaces of the deformed vessel respectively the distance of Pin and

Pout radic(1199091 minus 1199092)2 + (1199101 minus 1199102)2 is calculated as 119909119890119899119889 at polar angle 120579 Such estimation of the wall thickness is more accurate when the line PinPout is closer to be orthogonal to the deformed local surface Considering the final deformed vessel as shown in Figure 14 the estimation should be fine at regions close to 0deg and 90deg and can be slightly overestimated at medium angle region The VLM curve was compared with the SLM curve and 4 experimental curves measured at vessel left right front and back sides Generally VLM curve agreed well with the experimental and SLM curves wall thickness gradually decreases with angle till around 75deg and then increases till 90deg the maximum thickness change occurred at around 75deg would be around -5 mm The positive values at low angle may be caused by the thermal expansion It also shows that the thickness transitions in experimental curves were smoother than those in both simulation curves while simulation curves showed narrower regions with large thickness changes (at angle 70deg~80deg) Possible reasons might be (i) sharper temperature profile predicted in simulation (ii) input uncertainty and (iii) modelling errors The SLM predicted slightly thinner vessel than the VLM except for the necking region The discrepancy might be caused by various reasons eg the interior vessel temperature can be slightly different between two approaches which would then result in different thermal stress and different deformation the final deformed vessel shapes cannot be exactly the same which also results in different errors in estimating PinPout

Figure 16 Wall thickness changes along the vessel polar angle [64] [71]

-6

-5

-4

-3

-2

-1

0

1

0 10 20 30 40 50 60 70 80 90

Wal

l th

icknes

s ch

ange

(mm

)

Polar angle (deg)

VLMSLMExperiment (left)Experiment (right)Experiment (front)Experiment (back)

32

44 Application to BWR Case

In this section the coupling approach (indicated as lsquoVLMrsquo in the following) was applied to an RPV failure analysis of a BWR A simulation using the SLM coupling approach (indicated as lsquoSLMrsquo in the following) was also performed for comparison purpose

A hypothetical severe accident of a Nordic BWR initiated by an SBO was considered The core materials would melt and migrate downward to the RPV lower head The total amount of corium relocated in the lower head was considered to form a 19 m corium melt pooldebris bed assuming the corium distribution was homogeneous

Figure 17 Computational domains and boundary conditions (a) conjugate heat transfer (b) mechanical analysis [64]

Figure 17(a) shows the computational domain of the conjugate heat transfer PECM was used in this calculation A general mesh with a pool height of 23 m was used to accommodate most of the relocation situations with the same mesh Since the corium only had a height of 19 m the top 04 m region was treated as injected water covering the corium The top water was expected to be boiling and we assumed a uniform temperature of 383 K Under the implementation of PECM this region was damped with artificially high heat capacity and high conductivity such that it would efficiently remove heat from the corium and suppress its own temperature increase For the corium part below water a volumetric heat source 11 MWm3 was implemented as the nuclear decay heat For the in-vessel surfaces a constant temperature of 383K was applied as they are covered by water which can be boiling Without water cooling on the external vessel wall a very small heat loss of 20 Wm2 was applied Melting and solidification of the corium were also considered with a solidus temperature of 2750 K and a liquidus temperature of 2770 K The initial temperature was set as

q=20 Wm2

T=383 K

Qv=11

MWm3

315 bar

315 bar+ hydrostatic

pressure of corium

Gravity

248 bar

Zero

displacement

Water

Corium

(a) (b)

33

1100 K for the corium 383 K for the water and 450 K for the vessel For the mechanical analysis (Figure 17(b)) an inner pressure of 315 bar and an outer pressure of 248 bar were considered For the region covered by corium an extra hydrostatic pressure was added on the vessel wall A 2 m long vessel extension was considered as well as the standard earth gravity

Figure 18 presents the wall temperature profile along the inner vessel surface at 50 s 2500 s 7500 s and 10000 s The temperature increased with time for the vessel surface contacted with corium and the distribution was flat which was due to that large part of the corium was still in solidus status It also shows that for the surface region that contacted the water the temperature was basically unchanged which indicates that the damping treatment indeed suppressed the temperature increment and guaranteed the heat removal upward by water

Figure 18 Temperature profiles of inner vessel wall along the vessel polar angle [64]

The predicted RPV failure time was 9869 s by VLM and 9650 s by SLM Compared with the FOREVER-EC2 case in Section 43 the VLM predicted a slightly later failure time than the SLM in this BWR case too Figure 19(a) compares the maximum creep strain developments with time of both approaches Before 9000 s the predicted values by both approaches are close to each other Afterward the creep acceleration occurred and the strain increased drastically The acceleration of the VLM occurred 200 s later than that of the SLM which agreed with the predicted later failure time Similar trend was observed for the total deformation in vertical direction of vessel bottom point as shown in Figure 19(b) The deformations of both approaches are close except that the acceleration of VLM occurred later than that of SLM Overall results of VLM agreed well with those of SLM

0

500

1000

1500

2000

2500

0 10 20 30 40 50 60 70 80 90 100

Tem

per

ature

(K

)

Polar angle (deg)

50 s

2500 s

5000 s

7000 s

10000 s

34

Figure 19 Results of structural analysis (a) creep strain and (b) total deformation in vertical direction of vessel bottom point [64]

45 Computational Efficiency

Computational efficiency is also an important consideration regarding the performance of numerical approaches The computational efficiencies of both VLM and SLM are analysed and compared in this section As parallel computing technique was used to speed up the simulations the number of parallel cores and corresponding costed computational time for each simulation are given together in Table 2 as a generally reflection of the computational efficiency All the simulations were done in a Lenovo P700 workstation equipped with 2 Intel Xeon E5-2630 v3 processors (8 cores per processor) and 32 G RAM For the simulations of FOREVER-EC2 SLM employed twice the parallel cores and costed 286 more time compared with VLM For the simulations of prototypic BWR case both VLM and SLM employed the same number of parallel cores but VLM only took 13 of the computational time compared with SLM Overall it can be concluded that VLM has higher computational efficiency than the SLM The reason is a structural analysis is performed in the VLM approach while a coupled thermo-structural analysis is performed in the SLM approach (cf Figure 9) a standalone structural analysis is computed faster than a coupled thermo-structural analysis

Table 2 Comparison of computational efficiency [64]

Case FOREVER-EC2 BWR VLM SLM VLM SLM

Number of cores 4 8 4 4 Computational time (hours) 21 27 18 72

35

5 Hybrid Coupling Approach for Efficient RPV Failure Analysis

In this section we propose a hybrid coupling approach for the efficient RPV failure analysis The idea is to replace the ANSYS Fluent in previous coupling approach with a lumped-parameter code for quick estimate of the melt pool heat transfer With the least modelling details (compared with CFD methods and distributed-parameter methods eg PECM) lumped-parameter codes prove to be the most efficient for melt pool heat transfer The new hybrid approach characterizes 1) utilization of efficient models for melt pool heat transfer and 2) the support of detailed structural modelling of RPV deformation allowing studies with different RPV geometric shapes under effects like local thinning due to ablation For studies of which the focus is mainly on structural behaviour of RPV eg [13] [26] while mechanistic simulation of melt pool convection is computationally expensive and unnecessary the lumped-parameter code could help providing reasonable thermal loads

The development of the approach consists of two tasks (i) development of a lumped-parameter code and (ii) its coupling with ANSYS Mechanical to form this hybrid coupling framework In the following we describe both tasks and the corresponding verification and validation processes respectively This section is mainly based on Paper IV

51 A Lumped-parameter Code for Melt Pool Heat Transfer

A lumped-parameter code transIVR was developed It features (i) quick estimate of transient melt pool heat transfer with reasonable accuracies and (ii) two-dimensional representation of heat conduction in the RPV wall facilitating the coupled analysis of the RPV failure under thermal loads The following assumptions are taken in the transIVR modelling

bull The crust between the oxide pool and metal pool is thin

bull The heat flux has uniform distributions on the bottom top and side surfaces of metal pool

bull Heat source only exists in oxide layer

bull Latent heat of fusion is ignored

Figure 20 illustrates the modelled three domains oxide layer metal layer and the RPV wall In addition the internal structure inside the vessel can also be considered as a radiative body to the top metal surface as in [72] An internal volumetric heat source generated by nuclear decay heat is considered in the oxide layer Heat in oxide layer transfers downward to the RPV wall and upward to the top metal layer Heat in metal layer can be transferred sideward to the RPV wall and upward to the surrounding internal structure through radiation (or to water if inside water flooding is applied) The heat transferred to RPV will be removed from outer RPV surface Possible formations of oxide crust in the top and bottom surface are illustrated as the black regions Possible formation of metal crust on the top surface is shown as the grey region The code automatically detects the

36

existence of the crust on oxide top and metal top surfaces during the simulation and solves the problem correspondingly withwithout crust Boundary conditions are applied to the external and internal surfaces of RPV and (if needed) top surface of metal layer Arbitrary boundary conditions are supported If a boundary condition is nonlinear eg mixed heat convection (linear dependent on 119879 ) and heat radiation (proportional to 1198794) the Picard method [73] would be used for solving this problem

This model consists of three main parts

(i) empirical heat transfer correlations (mostly from experiments) to determine global and local heat transfer coefficients

(ii) energy balances in oxide layer and metal layer where the heat transfer coefficients would be used and

(iii) energy balance in RPV

Figure 20 Two-layer melt pool configuration and RPV wall [74]

511 Heat transfer correlations

For internally heated natural convection (oxide layer) in transIVR the mini-ACOPO correlations [75] are used for the averaged heat transfer as expressed in Eq (35) and (36) and for the angular variation of the heat flux as expressed in Eq (37) and (38)

119873119906119906119901 = 03451198771198861198941198990233 (35)

119873119906119889119899 = 00038119877119886119894119899035(119867119877)025 (36)

119873119906119889119899(θ)

119873119906119889119899= 01 + 108(

θ

θ119898119886119909) minus 45 (

θ

θ119898119886119909)2

+ 86 (θ

θ119898119886119909)3

01 leθ

θ119898119886119909le 06 (37)

RPV

Oxide pool

Metal pool

37

119873119906119889119899(θ)

119873119906119889119899= 041 + 035 (

θ

θ119898119886119909) + (

θ

θ119898119886119909)2

06 ltθ

θ119898119886119909le 1 (38)

The 119877119886119894119899 that appears in Eq (35)~(36) is the internal Rayleigh number which is defined as

119877119886119894119899 =119892β119876119881119867

5

α119896ν (39)

For Rayleigh-Bernard convection in the metal layer heat transfer occurs in all the top bottom and side surfaces To side surface the reduced form [72] of the Churchill-Chu correlation [76] for 119875119903 = 013 is employed as shown in Eq (40)

119873119906119904119889 = 007611987711988613 (40)

To top and bottom surfaces the reduced form [72] of Globe-Dropkin correlation [77] for

119875119903 = 013 is used as expressed in Eq (41)

119873119906119906119901 = 005911987711988613 (41)

The 119877119886 that appears in Eq (40) and (41) is the external Rayleigh number which is defined

as

119877119886 =119892βΔ1198791198673

να (42)

Heat transfer coefficients can be calculated from Nusselt number by Eq (43)

ℎ =119873119906119896

119871 (43)

where 119896 and 119871 are the heat conductivity and characteristic length respectively 119873119906 can be the Nusselt numbers expressed in Eq (35)~(38) and (40)~(41)

512 Energy balance in the pools

The energy balance of oxide layer can be described as Eq (44) The total generated heat of oxide pool equals the sum of the increase of enthalpy (transient temperature change) upward heat flow to metal layer and downward heat flow to RPV

120588119862119901119881119900119897119889119879119900119909119889119905

+ ℎ119906119901119860119900119909119906119901(119879119900119909 minus 119879119900119909119906119901) + ℎ119889119899119860119900119909119889119899(119879119900119909 minus 119879119900119909119889119899) minus 119876119881119881119900119897 = 0 (44)

where ρ is the oxide density 119862119901 the heat capacity 119881119900119897 the total volume 119879119900119909 the bulk temperature 119905 the time ℎ119906119901 the upward heat transfer coefficient 119860119900119909119906119901

upward heat transfer area 119879119900119909119906119901 upward surface temperature ℎ119889119899 downward

heat transfer coefficient 119860119900119909119889119899 downward heat transfer area 119879119900119909119889119899 downward surface temperature and 119876119881 volumetric heat source

38

On the interface between oxide layer and metal layer energy balance requires that the heat transferred from oxide layer should equal the heat to the metal layer which can be described as Eq (45)

ℎ119906119901119860119900119909119906119901(119879119900119909 minus 119879119900119909119906119901) = ℎ119898119905119889119899119860119898119905119889119899(119879119894119899119905 minus 119879119898119905) (45)

where ℎ119898119905119889119899 is the downward heat transfer coefficient of metal layer 119860119898119905119889119899 the downward heat transfer area 119879119894119899119905 the interface temperature and 119879119898119905 the bulk temperature of metal layer

The energy balance of metal pool can be described as Eq (46) The increase of enthalpy (transient temperature change) equals the sum of upward downward and sideward heat flows to surrounding surfaces

ρ119898119905119862119901119898119905119881119900119897119898119905119889119879119898119905119889119905

+ ℎ119898119905119906119901119860119898119905119906119901(119879119898119905 minus 119879119898119905119906119901) + ℎ119898119905119904119889119860119898119905119904119889(119879119898119905 minus 119879119898119905119904119889)

+ ℎ119898119905119889119899119860119898119905119889119899(119879119898119905 minus 119879119894119899119905) = 0

(46)

where ρmt is the metal density 119862119901mt the heat capacity 119881119900119897mt the total volume 119879119898119905 the bulk metal temperature 119905 the time ℎ119898119905119906119901 the upward heat transfer

coefficient 119860119898119905119906119901 upward heat transfer area 119879119898119905119906119901 upward surface

temperature ℎ119898119905119904119889 sideward heat transfer coefficient 119860119898119905119904119889 sideward heat transfer area 119879119898119905119904119889 sideward surface temperature ℎ119898119905119889119899 downward heat transfer coefficient 119860119898119905119889119899 downward heat transfer area and 119879119894119899119905 oxide-metal interface temperature

The heat transferred from metal layer to the top surface should equal the heat radiated from this surface to the environment Considering the radiation and reflections of internal structures to this surface following energy balance equation [72] is achieved

ℎ119898119905119906119901(119879119898119905 minus 119879119898119905119906119901) = σ(1198791198981199051199051199001199014 minus 119879119904

4) (1

ε+1 minus ε119904ε119904

119860119898119905119906119901

119860119904)minus1

(47)

where 119879119898119905119905119900119901 is the temperature of metal top free surface ε the emissivity of

metal ε119904 the emissivity of internal structures 119860119898119905119906119901 the area of metal top surface

and 119860119904 the total area of internal structures Sometimes the internal water cooling may be available above metal layer In this case Eq (47) is unnecessary and 119879119898119905119905119900119901 can be set to a constant value

For the oxide layer top surface Eq (48) is used if not crust is formed in the interface between two layers ie 119879119894119899119905 gt 119879119898119890119897119905119894119899119892119900119909 Otherwise Eq (49) is used

119879119900119909119906119901 = 119879119894119899119905 (48)

119879119900119909119906119901 = 119879119898119890119897119905119894119899119892119900119909 (49)

For the metal layer top surface Eq (50) is used if not crust is formed on top of the metal ie 119879119906119901119904119906119903119891 gt 119879119898119890119897119905119894119899119892119898119905 Otherwise Eq (51) is used

119879119898119905119906119901 = 119879119898119905119905119900119901 (50)

39

119879119898119905119906119901 = 119879119898119890119897119905119894119899119892119898119905 (51)

Overall this part involves 6 equations regarding 6 unknows 119879119900119909 119879119900119909119906119901 119879119894119899119905 119879119898119905

119879119898119905119906119901 and 119879119898119905119905119900119901

513 Energy balance in the RPV wall

The heat transfer in the PRV wall can be modelled as a simple 1D heat conduction problem that heat only transfers in the wall thickness direction or 2D axisymmetric heat conduction problem

For 1D model the treatment in [72] was adopted As illustrated in Figure 21 the heat from the oxide pool transfers to inner vessel surface (119879119894) then to the interior vessel (119879119907119890119904) then to outer vessel (119879119900) and finally to environment

Figure 21 1D heat transfer model in RPV [74]

Unknown variables are the heat flux through inner vessel surface 119902119894119899 the crust thickness 119909119888119903 inner vessel temperature 119879119894 vessel center temperature 119879119907119890119904 and outer vessel temperature 119879119900 Eq (52)~(56) give a closed equation set for these variables considering a radiation boundary condition on the external vessel surface Eq (52)~(53) describe the heat transfer through the crust where an internal heat source 119876119888119903 is also considered Eq (52) can be easily derived from the one-dimensional heat transfer equation with heat source Eq (54)~(55) describe the energy balance on the inner vessel surface and vessel interior respectively Eq (56) describes the energy balance on the outer vessel surface regarding radiation boundary condition For Dirichlet boundary condition Eq (56) is replaced with Eq (57) For other boundary conditions we only need to correspondingly modify the terms on the right-hand side of Eq (56) For situations that no crust is formed on the surface of the vessel Eq (52)~(53) are eliminated and 119902119894119899 in Eq (54) can directly be determined from calculations in Section 512 and Eq (54)~(56) are solved for the unknows 119879119894 119879119907119890119904 and 119879119900 Once 119879119894 exceeds the vessel melting temperature (with or without crust formation) wall ablation is considered to occur Then 119879119894 is fixed to the vessel melting temperature and vessel thickness 119909119907119890119904 becomes the new unknown variable

119879119900119909119889119899 119879119894 119879119907119890119904 119879119900 119879119890119899119907

119909119888119903 119909119907119890119904

crust RPV

Qcr

119902119894119899 119902119897119900119888119886119897

40

119902119894119899 = 119896119888119903(119879119900119909119889119899 minus 119879119894)

119909119888119903+ 1198761198881199031199091198881199032

(52)

119902119894119899 = 119902119897119900119888119886119897 + 119876119888119903119909119888119903

(53)

119902119894119899 = 119896119907119890119904(119879119894 minus 119879119907119890119904)

05119909119907119890119904

(54)

120588119907119890119904119862119901119907119890119904119889119879119907119890119904119889119905

= 119896119907119890119904(minus2 lowast 119879119907119890119904 + 119879119894 + 119879119900)

(05119909119907119890119904)2

(55)

119896119907119890119904(119879119907119890119904 minus 119879119900)

05119909119907119890119904= 휀120590(119879119900

4 minus 1198791198901198991199074 )

(56)

119879119900 = 119879119900119888119900119899119904119905

(57)

For the 2D representation the heat conduction in the RPV is solved using FDM considering the heat transfer in both the thickness and the angular directions in the RPV wall The RPV lower head is treated as a spherical shell The corresponding heat conduction equation in the 2D spherical coordinate is given as Eq (58) The cylindrical part of the vessel is treated as a cylindrical shell The corresponding heat conduction equation in the 2D cylindrical coordinate is given as Eq (59) Eq (58) and (59) are coupled and solved together In default the mesh size is 5deg in angular direction and 0025119909119888119903 in radius direction This mesh discretization can be changed up to user demands

ρ119862119901part119879

part119905=

1

1199032part

part119903(1198961199032

part119879

part119903) +

1

r2sinθ

part

partθ(119896119904119894119899θ

part119879

partθ)

(58)

ρ119862119901part119879

part119905=1

119903

part

part119903(119896119903

part119879

part119903) +

part

part119911(119896part119879

part119911) (59)

52 Hybrid Coupling Approach

The hybrid coupling approach to assess the RPV integrity is illustrated in Figure 22 transIVR and ANSYS Mechanical are coupled in a one-way manner transIVR conducts the conjugate heat transfer of melt pool and RPV and provides transient thermal loads to the ANSYS Mechanical where structural analysis of RPV deformation is performed The 2D heat transfer of RPV is solved in transIVR to better capture the temperature distribution In RPV structural analysis same considerations are taken as in Section 422 for 16MND5 steel the modified theta projection model for creep and MISO for plasticity and temperature-dependent Young modulus etc The VLM is adopted for the transfer of transient thermal loads

41

Figure 22 Framework of the hybrid coupling approach [74]

53 Verification and Validation

The transIVR code was first benchmarked against the UCSB FIBS case of two-layer pool configuration and then validated against the LIVE-7V experiment showing the codersquos capability in predicting two-layer and transient melt pool heat transfers respectively Finally the hybrid coupling approach encompassing transIVR and ANSYS Mechanical was validated against the RPV failure experiment FOREVER-EC2 and the results showed acceptable agreements with those of both the experiment and the previous approach using CFD in Section 43 Hence the transIVR code is an efficient tool for melt pool heat transfer and the hybrid approach an efficient way for RPV failure analysis

531 UCSB FIBS benchmark case

The UCSB FIBS benchmark case [78] was calculated to demonstrate the capability of the transIVR code in simulating two-layer pool heat transfer problems The considered pool configuration was similar to that shown in Figure 20 with the following specific dimensions a two-layer melt pool with a 16 m oxide layer and a 08 m metallic pool on top and an RPV lower head with a thickness of 015 m The internal volumetric source in the oxide layer was 13 MWm3 The calculation results were compared with those of the DOE study [78] and the INEEL study [79]

The predicted results were compared with the digitalized results of the DOE and INEEL studies in Figure 23 All three studies show similar profiles of the heat flux distribution along the polar angle (Figure 23(a)) it increased with the polar angle in the oxide layer to the maximum value at around 77deg and then dropped to a significantly lower value in the metal layer The transIVR result agreed well with the other results with small discrepancy in the metal layer the DOE study gave a

heat flux of 500 MWm2 the INEEL study of 400 MWm2 and the transIVR of somewhere in between Figure 23(b) shows the predicted corium crust thickness distribution along the RPV polar angle also agreed well the DOE and INEEL

transIVR ANSYS Mechanical

Transient

thermal loads

42

studies The crust thickness decreased with polar angle in the oxide layer and no crust was formed in the metal layer As the crust formation is closely related to the heat flux distribution the crust thickness profile was reasonable regarding the heat flux distribution shown in Figure 23(a) Vessel wall ablation would occur due to the high thermal loads Figure 23(c) gives the profile of the remaining vessel wall thickness along the RPV polar angle considering the wall ablation It shows that no ablation occurred in low angle region (0deg~36deg) where heat flux was low and the wall thickness changed from 15 cm to around 6 cm as angle increased in high angle region of the oxide layer Because of the discrepancy in the heat flux of metal layer the predicted wall thicknesses also differ from those of the DOE and INEEL studies the thinnest wall was 73 cm predicted by the DOE study corresponding to the maximum heat flux followed by 88 cm of the present transIVR calculation and 94 cm of the INEEL study Generally speaking the prediction of transIVR is in an acceptable level compared with the other studies for the same benchmark case

Figure 23 Simulation results of the UCSB FIBS benchmark case [74]

(a) heat flux distribution (b) crust thickness distribution

(c) wall thickness distribution

43

532 LIVE-7V experiment

LIVE-7V is an experiment under the LIVE program conducted in KIT to study the core melt phenomena As Figure 24(a) shows it employed the LIVE-3D hemispherical facility with an inner diameter of 1 m and a wall thickness of ~25 mm The non-eutectic binary salt mixture of 20 mol NaNO3- 80 mol KNO3 was used as the simulant material The decay heat in the melt was simulated by cable-type heating with 8 planes of electrical resistance heating wires Water cooling was applied to the top and vessel surfaces Top surface cooling was realized by a cooling lid mounted at height 413 mm from the vessel bottom and vessel cooling by the flow channel between the test vessel and the outer cooling vessel The inlet and outlet water temperature for both top and vessel cooling surfaces were kept ~293 K and ~303 K throughout the whole experiment respectively A 413 mm height pool corresponds to a total volume of ~0194 m3 simulant material The experiment procedure can be divided into four stages with the heating power sequentially shifted from 29 kW to 24 kW 18 kW and 9 kW Figure 24(b) indicates the thermocouple positions in experiment

The simulation results of LIVE-7V using transIVR were compared with experimental data [80] and numerical study using REALPSCDAPSIM code [81] For simplicity the RELAPSCDAPSIM results in the following are labelled as lsquoCOUPLErsquo which is the name of the module in RELAPSCDAPSIM for melt pool heat transfer

Figure 24 LIVE test facility [80] and thermocouple positions [82]

Figure 25 compares the predicted heat removal upward to the top lid and downward to the bottom vessel surface with those of the experiment and COUPLE The black curve indicates the total input power at the four stages The experimental results (red and blue dots) were calculated from the Table 3 of [80] assuming no extra heat losses For the first three stages the transIVR predictions agreed well with the COUPLE results while the predicted upward heat flow was slightly lower and the downward heat flow higher than the experimental data For the last stage the transIVR agreed well with the experimental result while the COUPLE predicted lower upward and higher downward heat flows The short

(a) (b)

44

transition periods between two stages with different total inputs taken in transIVR also agreed with those in the COUPLE predictions

Figure 25 Comparison of heat removal upward to the top melt surface cooling lid and downward to bottom vessel wall [74]

Figure 26 shows the comparison of heat flux distribution along the vessel wall All curves show similar trend that a higher total input power would result in a higher heat flux profile As the COUPLE predictions had relatively flatter profiles transIVR predictions generally agreed better with the experimental results especially when total input was high A discrepancy is seen that the minimum value of a curve occurred at angle 0degfor both transIVR and COUPLE and at angle 30deg for the experimental data It is noted that the models in default assume heat flux monotonically increases with angle (cf Eq (37) and (38)) resulting it impossible to predict higher heat flux at angle 0deg than angle 30deg as in experimental results

45

Figure 26 Heat flux profiles along the vessel polar angle [74]

533 FOREVER-EC2 experiment

In this section we performed calculations against the FOREVER-EC2 experiment to validate the performance of the coupled transIVR and ANSYS Mechanical in assessing the RPV integrity The applied boundary conditions were identical to the settings of previous coupling approach in Section 4 (ANSYS Fluent+ Mechanical) as indicated in Figure 11 The initial bulk temperature of the melt pool was set to be 1261 K in transIVR which corresponds to the maximum value of the initial temperature field in CFD calculation of previous coupling approach Both the experimental data and the simulation results using previous coupling approach are included for comparison In the following results from current hybrid approach are labelled lsquotransIVRrsquo (and referred as transIVR results) results from previous approach labelled lsquoCFDrsquo (and referred as CFD results) and results from experiment labelled lsquoexperimentrsquo

Figure 27 compares the bulk temperature predicted by transIVR with the averaged and maximum temperatures of the CFD results All curves show the same trend that the temperature gradually increased with time till reaching a steady state and then kept unchanged till the end The times needed for reaching the steady state were ~5000 s for the CFD and ~4000 s for the transIVR which are close to each other It indicates again a good performance of the transIVR in predicting the transient behavior of melt pool Compared with the steady-state averaged temperature 1478 K and maximum temperature 1609 K of the CFD results the steady-state temperature of transIVR 1599 K is a reasonable prediction

0

3000

6000

9000

12000

15000

18000

0 20 40 60 80 100

Hea

t fl

ux

(W

m2)

Polar angle (deg)

29 kW-EXP24 kW-EXP18 kW-EXP9 kW-EXP29 kW-COUPLE24 kW-COUPLE18 kW-COUPLE9 kW-COUPLE29 kW-transIVR24 kW-transIVR18 kW-transIVR9 kW-transIVR

46

Figure 27 Temperature evolution with time [74]

Figure 28 Temperature fields at time (a) 01 s (b) 3038 s (c) 6038 s and (d) 13289 s [74]

Figure 28 shows the temperature fields of the vessel at time 01 s 3038 s 6038 s and 13289 s The temperature fields were visualized in the ANSYS Mechanical It also indicates that the transient temperature loads were properly transferred from transIVR to ANSYS Mechanical Figure 28(a-d) show both the minimum and maximum temperatures increased with time In the lower head region the temperature gradually increased with the height The maximum temperature occurs at the location slightly lower than the melt pool top surface This is because thermal loadheat flux imposed on the vessel generally increases with polar angle in the pool natural convection so the code transIVR inherently also predicted higher heat flux at higher angle positions The higher flux at higher position then correspondingly resulted in higher temperature In the cylindrical region the

(a) (b) (c) (d)

[K] [K] [K] [K]

47

temperature decreased as the height since the bottom part would be more significantly heated by the high-temperature lower head region

Figure 29 shows the steady-state temperature profiles of the external vessel surface along the polar angle The transIVR result agreed well with other two results in trend that the temperature increased with the angle and reach the peak values at angle ~80deg then gradually decreased with angle But slightly lower temperature was predicted in the low angle region Possible reason may be that the local heat flux distribution determined by Eq (37) and (38) is steeper than that in the experiment and CFD study which then resulted in a steeper increase of temperature along the angle and a lower temperature at low angle region In the cylindrical part (gt90deg) the transIVR result generally agreed with the experimental and CFD results The discrepancy with the experimental data may be caused by the simplified boundary conditions the ambient temperature inside the vessel was set to uniform constant value while it could decrease as height Therefore predicted heat loss in the higher position of cylindrical part could be smaller than that with height-dependent ambient temperature resulting in higher vessel temperature Overall the prediction was reasonable for a lumped-parameter model

Figure 29 Temperature profiles of the external vessel surface along the vessel polar angle [74]

The transIVR predicted a failure time of 24959 s which is close to 24180 s in experiment and 23959 s in the CFD results Figure 30 shows the total deformation of vessel bottom point with time All three curves increased gradually with time and experienced drastic accelerations at the end The

400

600

800

1000

1200

1400

0 40 80 120 160

Tem

per

ature

(K

)

Polar angle (deg)

transIVRExperimentCFDWeilding Line

48

transIVR curve agreed well with the CFD curve The final total deformation at failure time is also close to that of the CFD result but lower than the experiment data The discrepancy might be related to the simplifications in the boundary conditions accuracy of the thermal load prediction and the modelling errors in plasticity and creep

Figure 30 Total deformation in vertical direction of the vessel bottom point [74]

Figure 31 shows the profile of wall thickness change The transIVR curve was compared with the CFD curve and four experimental curves measured at vessel left right front and back sides The simulation result fitted the experimental and CFD results well in the region 0deg~50deg In the region 50deg~90deg wall thickness gradually decreased with angle till around 75deg and then increased till 90deg The maximum thickness change was -3 mm for the simulation which is close to the experiment left side curve and slightly larger than the -6~ -45 mm of the other curves from experiment and CFD results Concerning possible errors and uncertainties in the modelling and numerical settings the new approach generally gives reasonable predictions

0

5

10

15

20

25

11670 15170 18670 22170 25670

Tota

l def

orm

atio

n (

mm

)

Time (s)

Experiment

CFD

transIVR

49

Figure 31 Wall thickness changes along the vessel polar angle [74]

-6

-5

-4

-3

-2

-1

0

1

0 20 40 60 80 100

Wal

l th

icknes

s ch

ange

(mm

)

Polar angle (deg)

CFD

transIVR

Experiment (left)

Experiment (right)

Experiment (front)

Experiment (back)

50

51

6 Summary and Outlook

Reactor pressure vessel (RPV) is one of the key physical barriers in the defence-in-depth approach of nuclear power safety and its failure is also the transition point of core melt (corium) progression from in-vessel to ex-vessel phase Quantification of RPV failure during a hypothetical severe accident is therefore paramount to the specific qualification of severe accident mitigation strategies as well as the general assessment of corium risk in light water reactors The RPV failure is a problem of structural performance of RPV under the influence of thermal loads imposed by corium melt pool ie it is a transient thermo-fluid-structure coupled problem per se whose solution requires simultaneous modelling of both highly turbulent melt pool heat transfer and large-scale nonlinear RPV creep Due to the complexity of the problem only limited capabilities have been developed in the previous studies under different simplifications and assumptions in models and coupling methods The present study was motivated to advance modelling and coupling approaches for RPV performance analysis considering new understanding of physical phenomena and latest coupling platform of simulation tools

For the new development of coupling approaches and models for physical phenomena related to RPV failure analysis during severe accidents this thesis work was conducted in the following order (i) a new creep model for the RPV steel 16MND5 was developed which features the capability to cover all three creep stages (ii) a new thermo-fluid-structure coupling approach was developed for RPV failure analysis and validated against FOREVER-EC2 experiment (iii) the performance of the new coupling approach using volume loads mapping (VLM) for data transfer was further compared with another approach using surface loads mapping (SLM) (iv) a hybrid coupling approach was proposed for efficient analysis of RPV failure which employs a lumped-parameter code for quick estimate of thermal load and a mechanistic FE code for detailed structural analysis and (v) for the hybrid coupling approach the code transIVR was developed for calculation of transient melt pool heat transfer

The developed creep model for 16MND5 is called lsquomodified theta projection modelrsquo and describes all three stages of a creep process Creep curves are expressed as a function of time with five model parameters 120579119894(119894 = 14 119886119899119889 119898) A dataset for the model parameters was constructed by fitting experimental creep curves into this function and an iterative parameter adjustment process to satisfy the monotonicity assumption of creep strain vs temperature and stress Model parameters for new curves under different temperature and stress loads are directly interpolated from this dataset in contrast to the conventional ldquotheta projection modelrdquo that employs an extra fitting process The new treatment was proven to better accommodate all experimental curves over the wide ranges of temperature and stress loads The model was implemented into the ANSYS Mechanical code The simulation results indicate that the model successfully captured all three creep stages and a good agreement was achieved between each experimental creep curve and corresponding modelrsquos prediction Regarding the model response to transient changes of temperature and stress loads the widely used time hardening and strain hardening models were implemented and

52

performing reasonably well in the new method These properties fulfil the requirements of a creep model for structural analysis

The transient thermo-fluid-structure coupling approach was developed using the ANSYS Fluent code for thermofluid dynamics of the melt pool and ANSYS Structural code for structural mechanics of RPV An extension tool was introduced to realize transient load transfer from ANSYS Fluent to Structural and minimize the user effort Both turbulence models and the effective model PECM can be used for melt pool heat transfer The developed creep model was used for RPV failure analysis of the RPV Since it is insensitive to geometry the coupling approach allows more complex geometries such as a BWR lower head with a forest of penetrations (CRGTs and IGTs) modelled in detail Moreover the coupling approach does not only reflect the transient response of thermo-fluid-structure interaction but also facilitates the implementation of advanced (high-fidelity) models in both fluid mechanics and structural mechanics The validation of the coupling approach against the FOREVER-EC2 experiment showed that the temperature profile vessel failure time total displacement of vessel bottom point and wall thickness profile in the experiment were well predicted

Following the variation of the coupling approach from VLM to SLM for data transfer between ANSYS Fluent and ANSYS Structural the respective coupling approaches were investigated by comparative calculations against the FOREVER-EC2 experiment and a postulated severe accident of a reference BWR For the FOREVER-EC2 experiment both VLM and SLM approaches predicted generally well the total deformation in vertical direction of bottom point and the wall thickness change though the former predicted a slightly later failure time For the reference BWR case good agreements were achieved in predicting the maximum creep strain and the total deformation in vertical direction of bottom point For both experiment and reactor cases the VLM approach predicted slightly later failure times than SLM but the differences were negligible compared to the long time-scale of vessel failure (~104 s) The deformation rate of the vessel wall was slow at the early stage and accelerated with time till a drastic deformation and final vessel failure Generally speaking the VLM and SLM coupling approaches had the similar performance in terms of their predictability of experiment and applicability to reactor case but the VLM was computationally more efficient

In the development of the hybrid coupling approach the transIVR code was developed for quick estimate of transient melt pool heat transfer and determination of thermal loads on RPV transIVR does not only support transient analysis of one- and two- layer melt pool but also the heat transfer in RPV being resolved with 2D finite difference method which captures more spatial thermal details for RPV structural analysis in addition to 1D heat transfer model The capabilities of transIVR in modelling two-layer pool heat transfer and transient pool heat transfer were demonstrated by calculations against the UCSB FIBS benchmark case and the LIVE-7V experiment respectively It was then coupled to the mechanical solver ANSYS Mechanical which together form the hybrid coupling framework for detailed RPV failure analysis The calculation against the FOREVER-EC2 experiment did not only reproduce the major results of the experiment but also had a good agreement with the thermo-fluid-structure

53

coupling approach using CFD This indicated the new coupling approach is capable of capturing the vessel creep failure characteristics and can be employed as an efficient tool addressing the complex vessel failure problem

As a prospective more research topics can be addressed in future either as a continuation of the current work toward an effective tool for vessel failure analysis or from the general view of scientific interest in melt pool heat transfer and RPV failure The following topics are recommended

bull Extended applications to the reactor cases For instance the thermo-fluid-structure coupling approach can be used to study penetration failures in the RPV of an BWR with a detailed representation of penetrations (IGTs and CRGTs)

bull Synthesis or update of heat transfer correlations for melt pool Heat transfer correlations directly benefit the parametric codes such as transIVR in accurate quantification of heat flux distribution around a melt pool So far many empirical correlations have been proposed based on experiments which had a variety of geometries and melt simulants with a wide range of Rayleigh number (Ra) Sensitivity studies have also been performed on the influences of Ra Pa and other material properties Therefore a comprehensive synthesis or update of the existing heat transfer correlations will be a meaningful step for making reasonable extrapolations to the reactor applications with prototypic materials

bull Modelling of pool stratification The motivation starts with the improvement of the PECMrsquos modelling capability such that the dynamic stratification and resulting pool heat transfer can be modelled altogether Simulation of melt stratification has been supported in some system codes such as ASTEC using a predefined parametric approach The improvement of PECM can be considered in the similar approach or more mechanistic modelling

bull Development of a structural module for RPV The idea is to develop a simplified module for structural analysis so it can be coupled with transIVR ultimately forming a standalone code for quick estimate of vessel failure of LWRs during postulated severe accidents

bull Extension of vessel analysis capability The current version of tranIVR is limited to melt pool Although the precursory debris bed and melt ejection after vessel failure were not the focus of the present study they are equally important to assessment of corium risk especially for Swedish BWRs where the melt release conditions and history are crucial to qualification of their severe accident mitigation strategies

54

55

Bibliography

[1] IAEA ldquoIAEA Annual Report 2018rdquo IAEA 2019 [Online] Available httpswwwiaeaorgsitesdefaultfilesgcgc63-5pdf

[2] IAEA Safety of Nuclear Power Plants Design Vienna INTERNATIONAL ATOMIC ENERGY AGENCY 2012

[3] M Fischer ldquoGen II and III In-Vessel and Ex-Vessel Melt Retentionrdquo The Short Course on Severe Accident Phenomenology Oct 2017

[4] B R Sehgal Nuclear Safety in Light Water Reactors Severe Accident Phenomenology 1st ed Academic Press 2011

[5] IAEA Approaches and Tools for Severe Accident Analysis for Nuclear Power Plants Vienna INTERNATIONAL ATOMIC ENERGY AGENCY 2008

[6] WENRA ldquoPractical Elimination Applied to New NPP Designs - Key Elements and Expectationsrdquo Sep 2019 [Online] Available httpwwwwenraorgmediafiler_public20191111practical_elimination_applied_to_new_npp_designs_-_key_elements_and_expectations_-_for_issuepdf

[7] P Matejovic M Barnak M Bachraty and L Vranka ldquoASTEC applications to VVER-440V213 reactorsrdquo Nuclear Engineering and Design vol 272 pp 245ndash260 2014 doi 101016jnucengdes201308077

[8] AREVA ldquoUS EPR Severe Accident Evaluationrdquo Topical Report ANP-10268NP Oct 2006 [Online] Available httpswwwnrcgovdocsML0631ML063100157pdf

[9] C T Martin ldquoCoupled 3D Thermo-mechanical Analysis of Nordic BWR Lower Head Failure in case of Core Melt Severe Accidentrdquo master thesis KTH Stockholm 2013

[10] IAEA ldquoAdvanced Large Water Cooled Reactors A supplement to the IAEArsquos Advanced Reactor Information System (ARIS)rdquo Sep 2015 [Online] Available httpsarisiaeaorgPublicationsIAEA_WRC_Bookletpdf

[11] B Pershagen Light Water Reactor Safety Pergamon Press 1989 [12] Ninos Garis M Agrell H Glanneskog and L Agrenius ldquoAPRI 7 ndash Accident

Phenomena of Risk Importancerdquo SSM SSM 201212 2012 [Online] Available httpswwwstralsakerhetsmyndighetensecontentassets2a9f299f46b949b3b209e780cf71436a201212-apri-7-accident-phenomena-of-risk-importance-en-lagesrapport-om-forskningen-inom-omradet-svara-haverier-under-aren-2009-2011-

[13] F Fichot et al ldquoSome considerations to improve the methodology to assess In-Vessel Retention strategy for high-power reactorsrdquo Annals of Nuclear Energy vol 119 pp 36ndash45 2018 doi 101016janucene201803040

[14] T G Theofanous C Liu S Additon S Angelini O Kymaumllaumlinen and T Salmassi ldquoIn-vessel coolability and retention of a core meltrdquo Nuclear Engineering and Design vol 169 no 1 pp 1ndash48 1997 doi 101016S0029-5493(97)00009-5

[15] C Sainte Catherine ldquoTensile and creep tests material characterization of pressure vessel steel(16MND5) at high temperatures (20 up to 1300 degC)rdquo CEA France SEMTLISNRT98-009A 1998

56

[16] E Altstadt H-G Willschuetz B R Sehgal and F-P Weiss ldquoModelling of in-vessel retention after relocation of corium into the lower plenumrdquo Forschungszentrum Rossendorf FZR-437 2005

[17] H-G Willschuumltz and E Altstadt ldquoGeneration of a High Temperature Material DataBase and its Application to Creep Tests withFrench or German RPV-steelrdquo FORSCHUNGSZENTRUM ROSSENDORF FZR-353 ISSN 1437-322X Aug 2002

[18] J L Rempe S A Chavez and G L Thinnes ldquoLight water reactor lower head failure analysisrdquo Nuclear Regulatory Commission NUREGCR--5642 1993 Accessed Aug 23 2019 [Online] Available httpinisiaeaorgSearchsearchaspxorig_q=RN25025905

[19] S Brosi et al ldquoCORVIS Investigation of light water reactor lower head failure modesrdquo Nuclear Engineering and Design vol 168 no 1 pp 77ndash104 1997 doi 101016S0029-5493(96)00008-8

[20] T Y Chu M M Pilch J H Bentz J S Ludwigsen W Y Lu and L L Humphries ldquoLower Head Failure Experiments and Analysesrdquo Sandia National Laboratories Albuquerque NM USA NUREGCR-5582 SAND98-2047 1999

[21] L L Humphries et al ldquoOECD Lower Head Failure Project Final Reportrdquo Sandia National Laboratories Albuquerque NM 87185-1139 2002

[22] B R Sehgal R R Nourgaliev T N Dihn A Karbojian J A Green and V A Bui ldquoFOREVER Experiments on Thermal and Mechanical Behavior of a Reactor Pressure Vessel during a Severe Accidentrdquo in Proceedings of the Workshop on in-vessel core debris retention and coolability Garching near Munich Germany Mar 3-6 1998

[23] S Bhandari M Fischer B R Sehgal and A Theerthan ldquoPost Test Analysis of the FOREVER C2 Experimentrdquo The 9th International conference on Nuclear Engineering (ICONE-9) Nice France Apr 8-12 2001

[24] J F Mao J W Zhu S Y Bao L J Luo and Z L Gao ldquoCreep deformation and damage behavior of reactor pressure vessel under core meltdown scenariordquo International Journal of Pressure Vessels and Piping vol 139ndash140 pp 107ndash116 2016 doi 101016jijpvp201603009

[25] J Mao L Hu S Bao L Luo and Z Gao ldquoInvestigation on the RPV structural behaviors caused by various cooling water levels under severe accidentrdquo Engineering Failure Analysis vol 79 pp 274ndash284 2017 doi 101016jengfailanal201704029

[26] J F Mao J W Zhu S Y Bao L J Luo and Z L Gao ldquoStudy on structural failure of RPV with geometric discontinuity under severe accidentrdquo Nuclear Engineering and Design vol 307 pp 354ndash363 2016 doi 101016jnucengdes201607027

[27] B R Sehgal V A Bui T N Dinh and R R Nourgaliev ldquoHeat Transfer Processes in Reactor Vessel Lower Plenum During Late Phase of In-Vessel Core Melt Progressionrdquo in Advances in Nuclear Science and Technology J Lewins M Becker R W Albrecht E J Henley J D McKean K Oshima A Sesonske H B Smets and C P L Zaleski Eds Boston MA Springer US 1999 pp 103ndash135

[28] C-T Tran and T-N Dinh ldquoThe effective convectivity model for simulation of melt pool heat transfer in a light water reactor pressure vessel lower head Part I Physical processes modeling and model implementationrdquo Progress

57

in Nuclear Energy vol 51 no 8 pp 849ndash859 2009 doi 101016jpnucene200906007

[29] C-T Tran and T-N Dinh ldquoThe effective convectivity model for simulation of melt pool heat transfer in a light water reactor pressure vessel lower head Part II Model assessment and applicationrdquo Progress in Nuclear Energy vol 51 no 8 pp 860ndash871 2009 doi 101016jpnucene200906001

[30] H-G Willschuumltz E Altstadt B R Sehgal and F-P Weiss ldquoCoupled thermal structural analysis of LWR vessel creep failure experimentsrdquo Nuclear Engineering and Design vol 208 no 3 pp 265ndash282 2001 doi 101016S0029-5493(01)00364-8

[31] E Altstadt and T Moessner ldquoExtension of the ANSYS creep and damage simulation capabilitiesrdquo Forschungszentrum Rossendorf Technical report FZR-296 2000

[32] H-G Willschuumltz E Altstadt B R Sehgal and F-P Weiss ldquoRecursively coupled thermal and mechanical FEM-analysis of lower plenum creep failure experimentsrdquo Annals of Nuclear Energy vol 33 no 2 pp 126ndash148 2006 doi 101016janucene200508006

[33] H-G Willschuumltz E Altstadt B R Sehgal and F-P Weiss ldquoSimulation of creep tests with French or German RPV-steel and investigation of a RPV-support against failurerdquo Annals of Nuclear Energy vol 30 no 10 pp 1033ndash1063 2003 doi 101016S0306-4549(03)00036-7

[34] H Madokoro A Miassoedov and T Schulenberg ldquoA thermal structural analysis tool for RPV lower head behavior during severe accidents with core meltrdquo Mechanical Engineering Letters vol 4 pp 18-00038-18ndash00038 2018 doi 101299mel18-00038

[35] W Villanueva C-T Tran and P Kudinov ldquoCoupled thermo-mechanical creep analysis for boiling water reactor pressure vessel lower headrdquo Nuclear Engineering and Design vol 249 pp 146ndash153 2012 doi 101016jnucengdes201107048

[36] W Villanueva C T Tran and P Kudinov ldquoA Computational Study on Instrumentation Guide Tube Failure During a Severe Accident in Boiling Water Reactorsrdquo in The 14th International Topical Meeting on Nuclear Reactor Thermal hydraulics NURETH-14 Toronto Ontario Canada Sep25-30 2011

[37] W Villanueva C T Tran and P Kudinov ldquoAssessment with Coupled Thermo-Mechanical Creep Analysis of Combined CRGT and External Vessel Cooling Efficiency for A BWRrdquo The 14th International Topical Meeting on Nuclear Reactor Thermal hydraulics NURETH-14 Toronto Ontario Canada Sep 25-30 2011

[38] F R Larson and J Miller ldquoA Time-Temperature Relationship for Rupture and Creep Stressesrdquo Trans ASME vol 74 p 765minus775 1952

[39] C T Tran ldquoDevelopment validation and application of an effective convectivity model for simulation of melt pool heat transfer in a light water reactor lower headrdquo Licentiate thesis KTH Stockholm 2007

[40] ANSYS ldquoANSYS Fluent Theory Guide Release 162rdquo ANSYS Inc Jul 2015 [Online] Available httpwwwansyscom

[41] P Zeng Finite element analysis and application (in Chinese) Tsinghua University Press amp Springer 2004

58

[42] ANSYS ldquoANSYS Mechanical | Finite Element Analysis Softwarerdquo httpswwwansyscomproductsstructuresansys-mechanical-enterprise (accessed Sep 26 2019)

[43] ANSYS ldquoANSYS Mechanical APDL Theory Reference Release 162rdquo ANSYS Inc Jul 2015 [Online] Available httpwwwansyscom

[44] U Steinberner and H H Reineke ldquoTurbulent Buoyancy Convection Heat Transfer with Internal Heat Sourcesrdquo in Proceedings of the 6th Int Heat Transfer Conference Toronto Canada 1978 vol Vol2 pp 305ndash310 doi 101615IHTC63350

[45] R W Evans and B Wilshire Introduction to Creep London Institute of Materials 1993

[46] DoITPoMS ldquoCreep Deformation of Metalsrdquo Creep Deformation of Metals University of Cambridge 2006 httpswwwdoitpomsacuktlplibcreepindexphp (accessed Jul 09 2019)

[47] ANSYS ANSYS Mechanical APDL Material Reference Release 162 2015 [48] SIMULIA Abaqus 612 Analysis Userrsquos Manualvol 3 Materials Chapter

2324 2012 [49] S Imaoka ldquoUser Creep Subroutine STI0704A ANSYS Release110rdquo 2007 [50] R W Evans J D Parker and B Wilshire ldquoThe θ projection conceptmdashA

model-based approach to design and life extension of engineering plantrdquo International Journal of Pressure Vessels and Piping vol 50 no 1 pp 147ndash160 1992 doi 1010160308-0161(92)90035-E

[51] ECCC ldquoRecommendations and Guidance for the Assessment of Creep Strain and Creep Strength Data ECCC Recommendations Volume 5 Part 1 b[Issue 2]rdquo 2013

[52] W D Day and A P Gordon ldquoLife Fraction Hardening Applied to a Modified Theta Projection Creep Model for a Nickel-Based Super-Alloyrdquo The ASME Turbo Expo 2014 Turbine Technical Conference and Exposition Duumlsseldorf Germany Jun16-20 2014 vol 7A-Structures and Dynamics p V07AT29A010 doi 101115GT2014-25881

[53] M Kumar I V Singh B K Mishra S Ahmad A Venugopal Rao and V Kumar ldquoA Modified Theta Projection Model for Creep Behavior of Metals and Alloysrdquo J of Materi Eng and Perform vol 25 no 9 pp 3985ndash3992 2016 doi 101007s11665-016-2197-y

[54] P Yu and W Ma ldquoA modified theta projection model for creep behavior of RPV steel 16MND5rdquo Journal of Materials Science amp Technology vol 47 pp 231ndash242 2020 doi 101016jjmst202002016

[55] ANSYS ANSYS Mechanical Advanced Nonlinear Materials v160 Lecture 2 Rate Dependent Creep ANSYS training materials 2015

[56] Q Y Li N C Wang and D Y Yi Numerical Analysis (in Chinese) 5th ed Tsinghua University Press 2008

[57] T N Dinh R R Nourgaliev and B R Sehgal ldquoOn heat transfer characteristics of real and simulant melt pool experimentsrdquo Nuclear Engineering and Design vol 169 no 1 pp 151ndash164 1997 doi 101016S0029-5493(96)01283-6

[58] O Kymaumllaumlinen H Tuomisto O Hongisto and T G Theofanous ldquoHeat flux distribution from a volumetrically heated pool with high Rayleigh numberrdquo

59

Nuclear Engineering and Design vol 149 no 1ndash3 pp 401ndash408 1994 doi 1010160029-5493(94)90305-0

[59] F J Asfia and V K Dhir ldquoNatural convection heat transfer in volumetrically heated spherical poolsrdquo 1994 Accessed Aug 23 2019 [Online] Available httpinisiaeaorgSearchsearchaspxorig_q=RN26037727

[60] M Jahn and H H Reineke ldquoFree Convection Heat Transfer with Internal Heat Sources Calculations and Measurementsrdquo in Proceedings of the 5th Int Heat Transfer Conference Tokyo Japan 1974 vol Vol3 paper NC-28

[61] M Gustafsson ldquoFSI Transient extension V1704rdquo EDR MEDESO 2016 [62] EDR MEDESO httpsedrmedesocom [63] H Kim and I Namgung ldquoThermo-mechanical Analysis of RVLH for

APR1400 during a Severe Accidentrdquo in Transactions SMiRT-23 Manchester United Kingdom Aug10-14 2015 [Online] Available httpwwwlibncsueduresolver18402034197

[64] P Yu and W Ma ldquoComparative Analysis of Reactor Pressure Vessel Failure using Two Thermo-Fluid-Structure Coupling Approachesrdquo submitted to journal

[65] F R Menter ldquoTwo-equation eddy-viscosity turbulence models for engineering applicationsrdquo AIAA Journal vol 32 no 8 pp 1598ndash1605 1994 doi 102514312149

[66] T N Dinh and R R Nourgaliev ldquoTurbulence modelling for large volumetrically heated liquid poolsrdquo Nuclear Engineering and Design vol 169 no 1 pp 131ndash150 1997 doi 101016S0029-5493(96)01281-2

[67] ANSYS Inc ldquoIntroduction to ANSYS Fluent Lecture 7 Turbulence Modelling Fluent Tutorial Materialrdquo 2016

[68] A Horvat and M Borut ldquoNumerical investigation of natural convection heat transfer in volumetrically heated spherical segmentsrdquo in Proceedings of the International Conference Nuclear Energy for New Europe 2004

[69] Y Li CFD Pre-test Analysis of SIMECO-2 Experiment 2016 [70] A Theerthan A Karbojian and B R Sehgal ldquoEC-FOREVER experiments

on thermal and mechanical behavior of a reactor pressure vessel during a severe accident Technical report-1 EC-FOREVER-2 testrdquo SAM-ARVI-D008 Mar 2001

[71] P Yu W Ma W Villanueva A Karbojian and S Bechta ldquoValidation of a thermo-fluid-structure coupling approach for RPV creep failure analysis against FOREVER-EC2 experimentrdquo Annals of Nuclear Energy vol 133 pp 637ndash648 2019 doi 101016janucene201906067

[72] T G Theofanous C Liu S Additon S Angelini O Kymaumllaumlinen and T Salmassi ldquoIn-vessel coolability and retention of a core meltrdquo Nuclear Engineering and Design vol 169 no 1ndash3 pp 1ndash48 1997 doi 101016S0029-5493(97)00009-5

[73] H P Langtangen ldquoSolving nonlinear ODE and PDE problemsrdquo Simula Research Laboratory 2016 [Online] Available httpshplgitgithubionum-methods-for-PDEsdocpubnonlinpdfnonlin-4printpdf

[74] P Yu and W Ma ldquoDevelopment of a Lumped-Parameter Code for Efficient Assessment of In-Vessel Melt Retention Strategy of LWRsrdquo submitted to journal

60

[75] T G Theofanous M Maguire S Angelini and T Salmassi ldquoThe first results from the ACOPO experimentrdquo Nuclear Engineering and Design vol 169 no 1 pp 49ndash57 1997 doi 101016S0029-5493(97)00023-X

[76] S W Churchill and H H S Chu ldquoCorrelating equations for laminar and turbulent free convection from a vertical platerdquo International Journal of Heat and Mass Transfer vol 18 no 11 pp 1323ndash1329 1975 doi 1010160017-9310(75)90243-4

[77] S Globe and D Dropkin ldquoNatural-Convection Heat Transfer in Liquids Confined by Two Horizontal Plates and Heated from Belowrdquo J Heat Transfer vol 81 no 1 pp 24ndash28 1959 doi 10111514008124

[78] T G Theofanous C Liu S Additon S Angelini O Kymaelaeinen and T Salmassi ldquoIn-vessel coolability and retention of a core melt Volume 2rdquo Argonne National Lab DOEID--10460-VOL2 1996 Accessed Apr 18 2019 [Online] Available httpinisiaeaorgSearchsearchaspxorig_q=RN30000459

[79] J L Rempe D L Knudson C M Allison G L Thinnes and C L Atwood ldquoPotential for AP600 in-vessel retention through ex-vessel floodingrdquo United States INEELEXT--97-00779 1997 Accessed Apr 18 2019 [Online] Available httpinisiaeaorgSearchsearchaspxorig_q=RN30025872

[80] A Miassoedov X Gaus-Liu T Cron and B Fluhrer ldquoLive experiments on melt pool heat transfer in the reactor pressure vessel lower headrdquo The 10th International Conference on Heat Transfer Fluid Mechanics and Thermodynamics (HEFAT 2014) Orlando FL Jul 14-16 2014

[81] H Madokoro A Miassoedov and T Schulenberg ldquoAssessment of a lower head molten pool analysis module using LIVE experimentrdquo Nuclear Engineering and Design vol 330 pp 51ndash58 2018 doi 101016jnucengdes201801036

[82] B Fluhrer et al ldquoThe LIVE-L1 and LIVE-L3 experiments on melt behaviour in RPV lower headrdquo Forschungszentrum Karlsruhe GmbH Technik und Umwelt (Germany) Inst fuer Kern- und Energietechnik FZKA--7419 2008 Accessed Nov 12 2019 [Online] Available httpinisiaeaorgSearchsearchaspxorig_q=RN40036867

VII

Acknowledgements

First of all I would like to express my sincere gratitude to my supervisor Prof Weimin Ma for offering this opportunity to work in the severe accident research field and for the supports patience guidance discussions and encouragements throughout the entire period of my graduate study

I am very thankful to my co-supervisor Dr Walter Villanueva for the valuable help and discussions I learnt a lot during the times working with him I am also thankful to co-supervisor Prof Sevostian Bechta who is supportive and always has inspiring ideas that keep you thinking for a long time

Thank you also to all my colleagues at the Division of Nuclear Power Safety for the supports discussions and the very nice working environment during these years

I have taken many interesting courses during my PhD study for which I want to express my thanks to all the teachers With this regard a special thanks goes to Prof Johan Hoffman for his SF2561 Finite Element Method given in 2015 fall as well as the discussions and advices on numerical analysis

I am grateful to Ignacio Gallego-Marcos and Wenyuan Fan for their enthusiastic help on CFD and turbulence modelling during the time working on the turbulent

heat transfer I thank all my 师兄 for sharing their knowledge on nuclear engineering and giving advices when I was confused since 2012

I thank all my friends both in Sweden and in China for the company and the sharing of many wonderful moments in daily life

Finally I would like to express my deepest thank to my parents my girlfriend Siwen my sisters and all my other relatives for their unconditional supports and love

Financial supports from the China Scholarship Council and APRI make my study in KTH possible and are acknowledged

VIII

IX

Table of Contents

AbstracthelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphellipI

SammanfattninghelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphellipIII

List of PublicationshelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphellipV

AcknowledgementshelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphellipVII

NomenclaturehelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphellipXI

1 Introductionhelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphellip1

11 Nuclear Power Safety 1

12 RPV Failure 2

13 Literature Review Creep Behaviour and RPV Failure Analysis 4

131 Experiments on creep behaviour and RPV failure 4

132 Numerical studies on RPV failure 5

14 Motivation Objectives and Main Achievements 7

2 Numerical Solvers and Modelshelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphellip9

21 ANSYS Fluent 9

22 ANSYS Mechanical 9

23 PECM 11

3 Creep Modelling of 16MND5 Steelhelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphellip13

31 Creep and Creep Modelling 13

32 A Modified Theta Projection Model 14

321 Original theta projection model and its variations 14

322 A modified theta projection model 16

4 Coupled Thermo-Fluid-Structure Analysis of RPV Failurehelliphelliphelliphellip23

41 Thermo-Fluid-Structure Coupling Approach 23

42 Numerical Modelling 25

421 Conjugate heat transfer 25

422 Mechanical analysis 25

43 Validation against the FOREVER-EC2 Experiment 26

431 FOREVER-EC2 experiment 26

X

432 Conjugate heat transfer 27

433 Mechanical analysis 29

44 Application to BWR Case 32

45 Computational Efficiency 34

5 Hybrid Coupling Approach for Efficient RPV Failure Analysishelliphellip35

51 A Lumped-parameter Code for Melt Pool Heat Transfer 35

511 Heat transfer correlations 36

512 Energy balance in the pools 37

513 Energy balance in the RPV wall 39

52 Hybrid Coupling Approach 40

53 Verification and Validation 41

531 UCSB FIBS benchmark case 41

532 LIVE-7V experiment 43

533 FOREVER-EC2 experiment 45

6 Summary and Outlookhelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphellip51

Bibliographyhelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphellip55

XI

Nomenclature

Latin

119860 area of a surface

119862119901 heat capacity

119863 damage factor

119892 gravity

ℎ heat transfer coefficient

H Melt pool height

119896 thermal conductivity

119871 characteristic length

119873119906 Nusselt number

119875119903 Prandtl number

119902 heat flux

119902119894119899 heat flux through inner vessel surface

119876119881 internal volumetric heat source

119903 radial distance in a cylindrical or spherical coordinate

119877119886 external Rayleigh number

119877119886119894119899 internal Rayleigh number

119877119886119910 local Rayleigh number

119905 time

119879 temperature

119881119900119897 total volume

119909 thickness (of crust or RPV wall)

119911 height in a cylindrical coordinate

Greek

120572 thermal diffusivity

120573 thermal expansion

Δ119879 temperature difference

ε emissivity (in thermal analysis) strain (in structural analysis)

휀119888119903 creep strain

휀119903 creep strain rate

120579 polar angle

120579119894 parameters of a creep model 119894 = 1234119898

120584 kinematic viscosity

σ stress

XII

Super and subscripts

119886119898119887 ambient

119888119903 creep (in Section 3) crust (in Section 5)

119889119899 downward

119890119899119907 environment

119894 inner surface of RPV

119894119899119905 interface

119897119900119888119886119897 local

119898119886119909 maximum value

119898119890119897119905119894119899119892 melting point

119898119905 metal layer (bulk)

119898119905119889119899 downward surface of metal pool

119898119905119904119889 sideward surface of metal pool

119898119905119905119900119901 top surface of metal layer

119898119905119906119901 upward surface of metal pool

119900 outer surface of RPV

119900119909 oxide layer (bulk)

119900119909119889119899 downward surface of oxide layer

119900119909119906119901 upward surface of oxide layer

119903119890119891 reference

119904 internal structures inside the RPV

119904119889 sideward

119906119901 upward

119907119890119904 vessel

Acronyms

BWR boiling water reactor

CBD creep database

CFD computational fluid dynamics

CRGT control rod guide tube

ECCM effective convectivity conductivity model

FDM finite difference method

FE finite element

FEA finite element analysis

FEM finite element method

FVM finite volume method

IGT instrument guide tube

IVR in-vessel retention

XIII

IVMR in-vessel melt retention EU project

LES large eddy simulation

MPS moving particle semi-implicit method

NPP nuclear power plant

PECM phase-change effective convectivity model

RANS Reynolds-averaged Navier-Stokes

RPV reactor pressure vessel

SBO station blackout

SLM surface loads mapping

SPH smoothed-particle hydrodynamics

UDF user-defined function

UPF user programmable features

VLM volume loads mapping

XIV

1

1 Introduction

11 Nuclear Power Safety

Greenhouse gas emissions contribute to the global warming problem Phasing out fossil fuels by switching to low-carbon energy sources is considered as a mitigation strategy With this regard clean energy should be increased in the world energy consumption structure Being clean and stable nuclear energy is an attractive alternative to fossil fuels After the development over decades it also proves to be a mature and safe technology According to IAEA 2018 annual report [1] there were 450 nuclear power reactors under operation worldwide at the end of 2018 which had a record generating capacity of 3964 GW(e) 9 reactors were connected to the grid and 55 reactors under construction around the world

Due to the potential risk of the release of radioactivity nuclear power safety is emphasized throughout the stages of design construction operation and decommission of a nuclear power plant (NPP) The principal technical requirements [2] specify that the three fundamental safety functions should be fulfilled for all plant states of an NPP (i) control of reactivity (ii) removal of heat from the reactor and from the fuel store and (iii) confinement of radioactive material shielding against radiation and control of planned radioactive releases as well as limitation of accidental radioactive releases The defence-in-depth concept is also applied to provide several levels of defence to prevent consequences of accidents The fuel claddings reactor pressure vessel (RPV) and primary coolant pressure boundaries and the containment are the three physical barriers to prevent the release of radioactive materials to the environment

Severe accidents are events that cause significant damage to reactor fuel and result in core meltdown It may be caused by insufficient cooling of the reactor core due to various reasons or by the rapid introduction of huge reactivity which causes sharp increases in the power and fuel fragmentation eg Chernobyl accident During a severe accident the failure of the fuel claddings will result in the radioactive materials previously enclosed inside the cladding released to the primary loop If cooling is not sufficient to remove the decay heat the core materials will melt migrate down and relocate into the RPV lower head after which the melt will react with water and partially form fragments quench solidify and form a debris bed The debris bed formed by the melt may continue to melt to form a melt pool due to the decay heat As the corium contacts the RPV it would continuously heat the RPV and threaten its integrity Once the RPV fails containment becomes the last barrier to prevent the release of radioactivity The released melt would result in an increase of the pressure and temperature due to the high temperature fractions oxidation and hydrogen combustion which is called direct containment heating The melt would fall onto the reactor cavity Without sufficient cooling the molten core would interact with basemat concrete (MCCI) This process can last for few days It can result in containment failure due to either containment overpressure or the basemat melt-through of containment In addition containment is also subjected to isolation failure and early containment failure due to missiles and dynamic pressure [3] More

2

detailed severe accident progression or phenomena could also be found in eg [4] [5]

Though severe accidents are unlikely to occur the consequences can be severe if no proper measure is taken Related researches have been continuously going on aiming at mitigating the accident consequences and improving safety After the Fukushima Accident safety inspections and re-evaluation actions have been carried out worldwide especially on safety measurements regarding severe accidents The concept lsquoPractical Eliminationrsquo applied to new NPP designs demands that new nuclear installations be designed with the objective of mitigating the consequences of accidents and avoiding early radioactive releases and large radioactive releases [6] Severe accident mitigation strategies are applied to the existing and new NPPs (both Generation II and III) The strategies can generally be classified into two categories [3] the in-vessel retention (IVR) and ex-vessel retention (EVR) In simple words in-vessel retention tries to retain the core melt inside the RPV (second barrier) by applying external water cooling on the vessel as shown in Figure 1(a) Such method minimizes the dispersal of the melt within the containment and hence simplifies the later recovery work Examples of the application are AP1000 HPR1000 APR 1400 The idea of the EVR is to let the RPV fail but to retain the core melt inside the containment Two principle approaches of EVR are the retention within a dedicated lsquocore catcherrsquo and melt fragmentation Examples of core catcher would be EPR that directs the melt to a spreading compartment (Figure 1(b)) and VVER1000 that uses metallic crucible below the RPV An example of the melt fragmentation is the Nordic BWRs that the corium released from RPV would relocate in the cavity with a deep-water pool (7-12 m) and is expected to fragment quench and form a coolable debris bed (Figure 1(c))

Figure 1 Examples of IVR [7] EVR with core catcher [8] and melt fragmentation [9]

12 RPV Failure

RPV which contains the reactor core core shroud and coolant is a key component to the reactor system RPV and the primary loop constitute the primary coolant system which conveys the fission heat from the reactor core to the secondary loop through steam generator (PWR) or to the turbine (BWR) The pressure under

(a) IVR (b) Core catcher (c) Melt fragmentation

3

operation can be ~155 MPa for a 1000 MW PWR [10] and ~70 MPa for a 1000 MW BWR [11] It also has to withstand the neutron radiation damage The thickness of RPV [11] can be ~20 cm for PWR and ~16 cm for BWR

As the transition of in-vessel progression to ex-vessel progression RPV failure is important for analysis of accident progression and assessment of the effectiveness of severe accident mitigation strategies In the melt fragmentation strategy of Nordic BWR the melt-coolant interaction and debris bed coolability after the corium poured into the deep-water pool are essentially dependent on the vessel failure mode including the characteristic time of melt release rupture size and location and amount and superheat of melt available for release [12] New concern also arose [13] on the necessity of performing structural analysis for the RPV in the IVR strategy In the initial methodology and further development of the IVR the success of the strategy ie the integrity of the RPV was demonstrated (eg in [14]) with the following two conclusions (i) even for a local heat flux on the vessel as high as critical heat flux (CHF) the resulted very thin external ldquocold layerrdquo of the vessel would be sufficient to withstand the internal stress load and hold the vessel integrity (ii) as CHF is the critical value for the thermal load the main specific goal for assessing the effectiveness of IVR becomes to demonstrate that the local heat flux along RPV be lower than the CHF The new concern doubts that local lsquohot partrsquo of the vessel would have negative effects on the overall resistance due to creep and dilatation which could result in a vessel failure and hence violate the first conclusion in particular for ldquolong termrdquo retention of corium in RPV If so a structural integrity analysis of RPV would be needed in addition to assessing local heat flux especially for high-power reactors Therefore accurate prediction of vessel failure could be meaningful for both strategies

Figure 2 Illustration of RPV under thermal loads from corium

As the structural performance is highly affected by the thermal loads from the corium RPV failure is essentially a transient thermo-fluid-structure coupled problem Figure 2 illustrates the situation that a corium debris bedmelt pool formed in the RPV lower head The decay heat of nuclear fission products could continuously be generated inside the corium leading to the formation of melt pools as the debris melts The heated debris bedmelt pool imposes thermal and

4

mechanical loads on the RPV Such loads can be transient (change with time) and spatially non-uniform Complex phenomena are also involved in this process Strong turbulent natural convection induced by the decay heat may be expected in the melt pool under IVR The corresponding Rayleigh number 119877119886 can be 1016minus17 for prototypic reactor cases Material structural properties can become non-linear under high temperature and stress loads The creep deformation also becomes significant at high temperature Detailed modelling of turbulent natural convection and non-linear properties could be computationally expensive In addition large-scale geometry (hemisphere with a radius of 20-30 m) long-term transient deformation process (~10000 s) and required small time steps (eg 001 s) for numerical convergence consideration further increase the computational costs

13 Literature Review Creep Behaviour and RPV Failure Analysis

In this section the most relevant studies concerning the experiments modelling and analysis of creep behaviour and RPV failure are presented

131 Experiments on creep behaviour and RPV failure

Various experiments have been performed on the RPV failure problem some of which were to generate data of mechanical properties (especially the creep properties) and some to study the overall vessel failure with scaled or simplified geometries

The REVISA experiment [15] generated creep curves and stress strain curves of French steel 16MND5 ranging from 600 to 1300 degC This data is accessible in [16] as well as [17] which also documents other material properties eg density thermal expansion coefficient conductivity heat capacity Youngrsquos Modulus and plasticity Similar experiments were carried out to generate high-temperature creep and tensile test data for American RPV steels and Inconel [18]

The CORVIS experiments [19] investigated the lower head failure modes with and without penetrations Each experiment was carried out on a steel plate 100 mm thick representing a pressure vessel lower head The penetration case represented a boiling water reactor lower head carrying a tube penetration in the form of a drain line

The LHF (lower head failure) experiments [20] and the succeeding OLHF tests [21] investigated the vessel failure with a 15 scaled vessel heated by an electrical radiator allowing to locate the hot focus at different position of the vessel with a maximum wall temperature of around 900 degC Internal pressure ranging from 53 to 123 bar was also imposed to initiate the vessel failure

The FOREVER experiments [22] investigated the phenomena of melt pool convection vessel creep possible failure processes and failure modes occurred during the late phase of the in-vessel progression by heating up a melt pool in a vessel and observing possible vessel failures The study covers cases with and without penetration different melt level existence of gap cooling and in-vessel melt coolability

5

132 Numerical studies on RPV failure

Various numerical approaches have been employed on RPV failure Since a two-way thermo-fluid-structure coupling approach is challenging these approaches take simplifications and can generally be classified into two categories (i) RPV failure decoupled from melt pool heat transfer and (ii) RPV failure coupled with melt pool heat transfer in simplified manners

1) Decoupled approaches In the decoupled approaches only RPV is modelled Simplified constant thermal loads are imposed on RPV considering the melt pool heat transfer

In a post-test analysis of FOREVE-C2 experiment [23] it was treated as a thermo-structural coupled analysis of the vessel with the thermal load estimated by the outer thermocouples and the normal heat flux on the external vessel wall The analysis was done using finite element (FE) code SYSTUS A coupled creepdamage law based on a viscosity-hardening creep law describing primary and secondary creep was employed for creep analysis

Mao et al conducted various studies on reactor cases assessing the IVR strategy using FE code ABAQUS Thermal boundary conditions were imposed on the RPV which was then solved as a thermal-structural coupled problem The strain hardening creep model (Eq (1)) was mainly used Their studies cover the general RPV failure [24] as well as the effects of cooling water levels under IVR [25] and geometric discontinuity (ablated RPV) [26]

ε119888119903 = 1198891σ1198892ε1198893119890119909119901 [minus

1198894119879] (1)

In the EU IVMR project [13] the RPV integrity was also studied through a benchmark calculation which deals with the RPV of a PWR with a prescribed ablated profile with 2D axisymmetric geometry Prescribed temperature gradient was set as thermal load The objective was to calculate deformations for various pressure loads up to vessel rupture A smoother profile was planned in further research

The advantage of such approaches is once thermal loads are reasonably assumed the focus only needs to be put on the RPV The disadvantage is a reasonable thermal load should be consistent with the melt pool heat transfer while sometimes it is not well-noted eg constant uniform temperature boundary conditions were assumed for inner and outer RPV (with uniform thickness in lower head) surfaces [25] which ignored the non-uniform heat flux distribution caused by the turbulent convective corium pool As thermal loads strongly affect the structural performance they should be quantified carefully

2) Coupled approaches In coupled approaches both melt pool heat transfer and RPV failure are modelled and coupled in one-way or simplified manner With melt pool heat transfer modelled the coupled approaches are more flexible to provide reasonable thermal loads on the RPV In existing approaches CFD and the correlation-based distributed-parameter models eg the Effective Convectivity Conductivity Model

6

(ECCM) [27] and the Phase-change Effective Convectivity Model (PECM) [28] [29] are general methods for melt pool heat transfer

Sehgal et al [22] presented an approach that employed the MVITA code to calculate the thermal processes and ANSYS for structural analysis MVITA code employed the ECCM Bailey-Norton creep law was used as shown in Eq (2)

ε119888119903 = 6055 times 10minus26 sdot σ472119905085119890minus

37200

119879 (2)

This approach was used for the pre-test simulation of FOREVER experiment to determine the thermal and mechanical loadings Parametric investigations also helped in the identification of important factors that could affect the test procedure and test conditions

Willschutz et al [30] employed the CFD-module Flotran for the melt pool heat transfer and ANSYS for structural analysis of RPV Standard k-ε turbulence model was selected for the turbulent natural convection in the melt pool A user-defined three-stage creep model [31] was employed In this model experimental creep curves were discretized into a creep database (CDB) under given temperature 119879 and stress 120590 the creep curve was discretized into finite point pairs

(휀 ε) indicating under such thermal and mechanical loads the creep strain rate ε is subjected to creep strain 휀 The creep strain rate for any other case can be interpolated over known temperature stress and strain from the database This method was applied to the post-test calculation of the FOREVE-C2 experiment It was concluded that the unexpected deformation behaviour during the experiment was caused by the combination of the transient temperature field in the vessel wall and the three creep stages

They later replaced the Flotran solver with the ECCM for melt pool heat transfer in a following study [32] By implementing it into FEM-solver a recursively coupled analysis was constructed Moreover a modified creep model [33] was used with damage measures the CDB itself did not contain tertiary creep instead a factor 1(1 minus 119863) was multiplied to the original creep strain rate to get the modified creep strain rate where 119863 is a damage factor (0 lt 119863 lt 1) defined with creep strain information and plastic strain information As 119863 grows close to 1 the modified creep strain rate would be enlarged significantly by this factor 1(1 minus 119863) which then behaves like the accelerating tertiary stage This method was applied to pre- and post- test calculations for the FOREVER experiments proving to be closer to reality than the previous method

Madokoro et al [34] developed a thermal structural analysis tool based on open source solver OpenFOAM and numerically investigated the coupled RPV failure and melt pool convection The PECM was used in the melt pool heat transfer In structural part the plasticity and damage parameters were similar to Willschutz et al [32] except for that strain hardening creep model (Eq (3)) was used instead of a CDB Simulations of FOREVER-EC2 and -EC4 were investigated It was concluded from the simulations that although the emissivity and mesh type (updated or fixed) affect the results significantly the approach generally has the capability of predicting the vessel deformation and failure

7

ε119888119903 = 1198891 sdot σ1198892 sdot ε1198893 (3)

Villanueva et al [35] performed a coupled thermo-mechanical creep analysis of the reactor scale RPV (ABB-Atom design BWR) to evaluate efficacy of the cooling measures for the in-vessel melt coolability and retention The PECM was used for melt pool heat transfer and ANSYS for RPV integrity analysis The modified time hardening model (Eq (4)) of the American RPV steel SA533B was selected as creep model The transient thermal load from the melt pool was applied as a thermal boundary condition to the vessel wall Further studies were done with this approach on investigations of instrumentation guide tube (IGT) failure [36] and control rod guide tube (CRGT) cooling [37] of a Nordic BWR

ε119888119903 =1198881σ

11988821199051198883+1

1198883 + 1 1198881 gt 0 (4)

It is noted that only RPV analysis with finite element analysis (FEA) are reviewed here Engineering methods like the Larson-Miller method [38] are not covered in this thesis One can refer to theory manuals of severe accident codes like MELCOR ASTEC for more details Comprehensive description on modelling of melt pool heat transfer is not covered either One can refer to either [39] for a general view or to any severe accident code manuals for engineering methods

14 Motivation Objectives and Main Achievements

The existing experimental and numerical studies provide us a good understanding of RPV failure involved phenomena and valuable simulation experiences These numerical methods may vary in solvers (which may be related to code accessibility) and models (which may be related to model accessibility understanding of phenomena computational power) However improvements of modelling approaches are still possible as the improvements in any of the related aspects eg

bull Simulation tools are continuously being upgraded and providing new possibilities just like that PECM appeared and ECCM was no longer used

bull Better understanding of the phenomena and more simulation experiences from in-vessel retention and vessel failures researches benefit the further modelling and simulations

bull Increasing computational power allows more detailed modelling eg turbulence modelling of melt pool convection

bull There is still room for the improvement of creep modelling

In addition implementation difficulty of a coupling approach is another important issue to consider Since it is a TRANSIENT thermo-fluid-structure coupled problem it is almost impossible to replicate an approach simply by referring to a paper with limited details For BWRs it would be even better for the coupling approaches to support complex geometries considering forest of penetrations (CRGTs and IGTs) on the RPV lower head without significant increase of user effort in the coupling process

8

The objectives of this thesis are to investigate new approaches and develop new models for the thermo-fluid-structure coupled problem of RPV failure and involved phenomena These objectives are accomplished with the following achievements

1) A new creep model for the 16MND5 steel was developed which covers all three creep stages Good agreement was achieved between model predictions and experimental curves Both time and strain hardening models were implemented (Paper I)

2) A new thermo-fluid-structure coupling approach was developed for RPV failure analysis It supports the advanced models in both melt pool convection and structural mechanics to improve fidelity and facilitates implementation Validation of the approach with the new creep model was performed against FOREVER-EC2 experiment This approach is flexible for complex geometries and can be used in further analysis eg lower head with detailed modelling of penetrations in BWRs (Paper II)

3) The performance of this new approach which uses volume loads mapping for data transfer was compared with the previous approach using surface loads mapping showing good agreement with small discrepancy (Paper III)

4) A lumped-parameter code transIVR for melt pool heat transfer was developed which supports one- and two-layer pool configuration transient analysis and well-resolved RPV wall (Paper IV)

5) A hybrid coupling approach was developed for efficient analysis of RPV failure where transIVR is coupled with ANSYS Mechanical (Paper IV)

In the following general descriptions on the codes ANSYS Fluent Mechanical and the PECM are given in Section 2 the new creep model in Section 3 the new coupling approach as well as its comparison with the previous approach in Section 4 the efficient hybrid coupling approach in Section 5 and summary amp outlook in Section 6

9

2 Numerical Solvers and Models

In this section we briefly introduce the ANSYS Fluent as the CFD solver for melt pool heat transfer the ANSYS Mechanical as the mechanical solver for RPV failure and PECM as an efficient alternative for melt pool heat transfer that are heavily used in the following sections

21 ANSYS Fluent

In the fluid mechanics the fluid is treated as a continuum which is generally described by the conservative mass momentum and energy equations [40]

120597120588

120597119905+ 120571 sdot (120588) = 119878119898 (5)

120597(120588)

120597119905+ 120571 sdot (120588) = minus120571119901 + 120571 sdot (120591) + 120588 + (6)

120597120588119864

120597119905+ 120571 sdot ((120588119864 + 119901)) = 120571 sdot (119896120571119879 minussumℎ119895119869119895

119895

+ (120591 sdot )) + 119878ℎ

(7)

Computational Fluid Dynamics (CFD) studies topics on the numerical solutions of above equations Typical methods are the Finite Volume Method (FVM) Finite Difference Method (FDM) and Finite Element Method (FEM) as well as particle methods that are mesh-free eg SPH and MPS

ANSYS Fluent is a general purpose CFD code which solves above equations with FVM Turbulence modelling and multiphase flows are supported Turbulence can be modelled with RANS LES or hybrid LESRANS Customizations eg adding sourcesink time or spatial dependent boundary conditions are realized through the user-defined functions (UDF) which is programmed in C language

22 ANSYS Mechanical

In the continuum mechanics of solids three sets of variables are considered to describe the structure performance namely element displacement field strain field and stress field (119906 ε σ) A general 3-dimensional problem is described by 3 sets of equations the equilibrium equations (Eq (8)) strain-displacement relationship (Eq (9)) and constitute equations (Eq (10)) [41] For Eq (8) the reciprocal theorem of shear stress also gives τ119894119895 = τ119895119894 where 119894 119895 = 119909 119910 119911

indicating the stress tensor being symmetric For very simple applications eg 1D beam it can be solved analytically For complex problems where analytical solutions are not available approximation methods eg Galerkin weighted residual method principle of virtual work and principle of minimum potential energy would be used

10

partσ119909119909part119909

+partτ119909119910

part119910+partτ119909119911part119911

+ 119909 = 0

partτ119910119909

part119909+partσ119910119910

part119910+partτ119910119911

part119911+ 119910 = 0

partτ119911119909part119909

+partτ119911119910

part119910+partσ119911119911part119911

+ 119911 = 0

(8)

ε119909119909 =part119906

part119909 휀119910119910 =

120597119907

120597119910 휀119911119911 =

120597119908

120597119911

120574119909119910 =120597119906

120597119910+120597119907

120597119909 120574119910119911 =

120597119908

120597119910+120597119907

120597119911 120574119911119909 =

120597119908

120597119909+120597119906

120597119911

(9)

ε119909119909 =1

119864[σ119909119909 minus μ(σ119910119910 + σ119911119911)]

ε119910119910 =1

119864[σ119910119910 minus μ(σ119909119909 + σ119911119911)]

ε119911119911 =1

119864[σ119911119911 minus μ(σ119909119909 + σ119910119910)]

γ119909119910 =1

119866τ119909119910 γ119910119911 =

1

119866τ119910119911 γ119911119909 =

1

119866τ119911119909

(10)

ANSYS Mechanical [42] is a mechanical engineering software that uses finite element analysis (FEA) for structural analysis The principle of virtual work is adopted in the solver [43] The idea of this principle is that a virtual (very small) change of the internal strain energy must be offset by an identical change in external work due to the applied loads which reads

δ119880 = δ119881 (11) where 119880 is the strain energy or the internal work 119881 the external work and δ the virtual operator The virtual internal work can be expressed as

δ119880 = int δεσd(vol)T

vol

+int 119886119903119890119886119891

δ119908119899119879σ119889(119886119903119890119886119891) (12)

where 휀 is the strain vector 120590 stress vector vol volume of element 119908119899 motion normal to the surface 119886119903119890119886119891 area of the distributed resistance The virtual

external work is contributed from three parts the inertial effects the pressure force on the surface and nodal forces applied to the element which then can be expressed as

δ119881 = minusint 119907119900119897

δ119908119879119865119886

119907119900119897119889(119907119900119897) + int

119886119903119890119886119901

δ119908119899119879119875119889(119886119903119890119886119901)

+ δ119906119879119865119890119899119889

(13)

11

where 119908 is the vector of displacements 119865119886 acceleration force vector 119875 applied pressure vector 119886119903119890119886119901 area over which pressure acts and 119865119890

119899119889 nodal

forces applied to the element

Plugging Eq (12) and (13) into (11) would result in the equilibrium equation Since the virtual displacement can be arbitrary the final equations used to describe the equilibrium of an element is reduced to Eq (14) with multiple operations

([119870119890] + [119870119890119891])119906 minus 119865119890

119905ℎ = [119872119890] + 119865119890119901119903 + 119865119890

119899119889

(14)

where [119870119890] [119870119890119891] 119865119890

119905ℎ [119872119890] and 119865119890119901119903 are the element stiffness matrix

foundation stiffness matrix thermal load vector mass matrix acceleration vector and pressure vector respectively It represents the equilibrium equation on a one element basis Alternatively using a minimum total potential energy principle could result in a consistent expression to the principle of virtual work [41]

Customizations eg on material properties are realized through the user programmable features (UPF) which is programmed in FORTRAN language

23 PECM

The PECM [28] [29] developed in KTH is a correlation-based method to model the melt pool heat transfer It is implemented in ANSYS Fluent through UDF and has been validated against many experiments such as ACOPO COPO LIVE etc In the PECM the flow velocities 119906119909 119906119910 and 119906119911 are replaced by characteristic

velocities 119880119909 119880119910 and 119880119911 respectively in order to eliminate the necessity of solving

the Navier-Stokes equation

The characteristic velocities are calculated using heat transfer correlations

119880119906119901 =α

ℎ119901119900119900119897times (119873119906119906119901 minus

ℎ119901119900119900119897

ℎ119906119901) (15)

119880119889119899 =α

ℎ119901119900119900119897times (119873119906119889119899 minus

ℎ119901119900119900119897

ℎ119889119899) (16)

119880119904119889 =α

ℎ119901119900119900119897times (119873119906119904119889 minus

ℎ119901119900119900119897

119882119901119900119900119897) (17)

where ℎ119901119900119900119897 is the melt pool depth ℎ119906119901 the thickness of well mixed layer of the

pool ℎ119889119899 the thickness of lower stratified region of melt pool 119882119901119900119900119897 the width of

pool 120572 the thermal diffusivity The PECM employs reduced characteristic velocities to describe mushy zone convection heat transfer For internally heated pool natural convection the Steinberner-Reineke correlations (Eq (18)~(19)) [44] are employed The profile of sideward 119873119906 is described using Eckertrsquos type correlation (Eq (20)) for a vertical boundary layer where the local Rayleigh

number at pool height 119910 is defined as 119877119886119910 =119892βΔ1198791199103

αν

12

119873119906119906119901 = 03451198771198861198941198990233

(18)

119873119906119889119899 = 13891198771198861198941198990095

(19)

119873119906119904119889 = 050811987511990314 (

20

21+ 119875119903)

minus14

11987711988611991014

(20)

As the characteristic velocities are calculated the energy equation is the only transport equation that needs to be solved and can be expressed as follows

partρ119862119901119879

part119905+ nabla(ρ119862119901119932119879) = 119896nabla2119879 + 119876119907 (21)

where from left to right are the transient term convection term diffusion term and volumetric heat source respectively The convection term then can be calculated explicitly with use of values in the last time step and then can be treated together with the volumetric source term Therefore the PECM finally simplifies the energy equation to a conduction equation which can be easily solved and as a result will significantly increase the computational efficiency in simulations

In addition solidificationmelting process can also be modelled and the temperature-dependent physical properties are supported

13

3 Creep Modelling of 16MND5 Steel

Creep deformation becomes significant when a component is subjected to high temperature and stress Modelling of creep behaviour of the RPV is paramount since it is considered as a main mechanism leading to the RPV failure In this section we present the development of a three-stage creep model (called lsquomodified theta projection modelrsquo ) for the French RPV steel 16MND5 This model was successfully implemented into the ANSYS Mechanical code

31 Creep and Creep Modelling

Creep is an irreversible time-dependent inelastic deformation process Diffusion and dislocation are the two molecular mechanisms of creep The former normally occurs due to the presence of vacancies within the crystal lattice (or the grain) while the latter occurs if there is a slip by the passage of crystal dislocations [45] There are two types of diffusion creep [46] Coble creep (favoured at lower temperatures) if the diffusion paths are predominantly through the grain boundaries and Nabarro-Herring creep (favoured at higher temperatures) if through the grains themselves Dislocation creep tends to dominate at high stresses and relatively low temperatures Dislocations can move by gliding in a slip plane which is a process that requires little thermal activation But the rate-determining step for their motion is often a climb process which also requires diffusion and is thus a time-dependent process and favoured by higher temperatures Creep behaviour is highly affected by temperature and stress generally creep is more significant under higher temperature and stress

Creep curves describe the creep strain development with time Figure 3 shows a creep curve under constant temperature and stress which typically experiences three creep stages primary stage where the creep strain rate ε119888119903 (the change of creep strain ε119888119903 in unit time) decreases with time secondary stage where the creep strain rate almost keeps constant and tertiary stage where the creep strain rate drastically increases with time

Figure 3 Creep curve under constant temperature and stress

14

The built-in creep models in commercial software eg ANSYS [47] and ABAQUS [48] in default only cover the primary andor the secondary stage(s) Table 1 [47] gives the default creep models supported in ANSYS for example Possible reason why the tertiary stage is not modelled might be that for most industrial applications the failure analysis is beyond interest and therefore the built-in creep laws in the commercial software was supposed to be sufficient [49] Though the two codes offer the interfaces through either ANSYS UPF or ABAQUS user subroutines to support more general user-defined creep laws eg complex models that also include the tertiary creep stage

Table 1 Default creep models in ANSYS Mechanical

Creep model Equations Type

Strain hardening

ε119888119903 = 1198621σ1198622ε119888119903

11986231198901198624119879 1198621 gt 0 Primary

Time hardening ε119888119903 = 1198621σ11986221199051198623119890minus1198624119879 1198621 gt 0 Primary

Generalized exponential

ε119888119903 = 1198621σ1198622119903119890minus119903119905 r = 1198625σ

11986231199051198623119890minus1198624119879 1198621 gt 0

1198625 gt 0

Primary

Generalized Graham

ε119888119903 = 1198621σ1198622(1199051198623 + 1198624119905

1198625 + 11986261199051198627)119890minus1198628119879 1198621 gt 0 Primary

Generalized Blackburn

ε119888119903 = 119891(1 minus 119890minus119903119905) + 119892119905 119891 = 1198621119890

1198622σ 119903 = 1198623(σ1198624)1198625 119892 = 1198626119890

1198627σ

1198621 gt 0

1198623 gt 0

1198626 gt 0

Primary

Modified time hardening

ε119888119903 = 1198881σ11988821199051198883+1(1198883 + 1) 1198621 gt 0 Primary

Modified strain hardening

ε119888119903 = 1198621σ1198622[(1198623 + 1)ε119888119903]

1198623119890minus1198624119879 1198621 gt 0 Primary

Generalized Garofalo

ε119888119903 = 1198621[119904119894119899ℎ(1198622σ)]1198623119890minus1198624119879 1198621 gt 0 Secondary

Exponential form

ε119888119903 = 1198621119890σ1198622119890minus1198623119879 1198621 gt 0 Secondary

Norton ε119888119903 = 1198621σ1198622119890minus1198623119879 1198621 gt 0 Secondary

Combined time hardening

ε119888119903 = 1198621σ11986221199051198623+1119890minus1198624119879(1198883 + 1)

+ 1198625σ1198626119905119890minus1198627119879

1198621 gt 0

1198625 gt 0

Primary+ Secondary

Rational polynomial ε119888119903 = 1198621

partε119888part119905 ε119888 =

119888119901119905

1 + 119901119905+ ε119898119905

ε119898 = 1198622101198623σσ1198624 119888 = 1198627ε

1198628σ1198629

119901 = 11986210ε11986211σ11986212

1198622 gt 0 Primary+ Secondary

Generalized time hardening

ε119888119903 = 119891119905119903119890minus1198626119879 119891 = 1198621σ + 1198622σ2 + 1198623σ

3 119903 = 1198624 + 1198625

Primary

32 A Modified Theta Projection Model

321 Original theta projection model and its variations

The original theta projection model [50] was first proposed to accurate predict long-term creep and creep-fracture properties It assumes that a creep curve under constant temperature and stress can be described as

15

휀119888119903 = 1205791(1 minus 119890minus1205792119905) + 1205793(119890

1205794119905 minus 1) (22)

where 휀119888119903 is the creep strain 119905 the time and 1205791~1205794 the model parameters Figure 4 illustrates how the first term and second term on right hand side develop with time and constitute a creep curve which has obvious primary secondary and tertiary stages Model parameters 120579119894(119894 = 1234) are temperature and stress dependent and are described with the following expression

119897119899thinsp120579119894 = 119886119894 + 119887119894120590 + 119888119894119879 + 119889119894120590119879 (23)

where 119886119894 119887119894 119888119894 and 119889119894 ( 119894 = 1234 ) are constants It is noted [45] that this equation is suitable when the temperature range of experiment data sets is very narrow such that minus1119879 is almost proportional to 119879 These constants (119886119894 119887119894 119888119894 and 119889119894 (119894 = 1234)) are the final model parameters which are generated with two fitting procedures

(i) Fitting the experimental creep curves to Eq (22) generates several groups of 1205791~1205794 for each creep curve under certain temperature and stress We denote (119879119895 120590119895 1205791119895 1205792119895 1205793119895 1205794119895 ) as parameters for the 119895119905ℎ

experimental curve under temperature 119879119895 and stress 120590119895

(ii) Fitting 1205791 to Eq (23) as a function of temperature and stress with the obtained data ( 119879119895 120590119895 1205791119895 ) from ( 119879119895 120590119895 1205791119895 1205792119895 1205793119895 1205794119895 ) generates

parameters 1198861 1198871 1198881 and 1198891 Repeat this fitting procedure for 1205792 1205793 and 1205794 to get parameters 119886119894 119887119894 119888119894 and 119889119894 (119894 = 234)

Variations of the model were developed and applied to other different materials to better accommodate the experimental data which are all then called a lsquomodified theta projection methodrsquo independently Examples are models in [51] for lowhigh alloy ferritic austenitic steels Al-alloys Al-matrix composites [52] for Nickel-based Super-alloy [53] for pure metals and class M alloys with intermediate range of stress (40-70 of the yield stress) and temperature (40-60 of the melting temperature)

Figure 4 Illustration of theta projection method [54]

16

322 A modified theta projection model

The new modified theta projection model we proposed [54] was mainly adapted from the original theta projection method [50] and the modified version [51] The new model describes the creep curve with the following equation

휀119888119903 = 1205791(1 minus 119890minus1205792119905) + 120579119898119905 + 1205793(119890

1205794119905 minus 1) (24)

An additional term 120579119898119905 is seen compared with Eq (22) in the original method The reason for adding this term is that the 16MND5 steel shows fairly long secondary stages in many experimental creep curves and the creep curves can be better fitted by Eq (24) than Eq (22) As the structure undergoes different creep process under different temperature and stress the model parameters θ119894 (119894 =1~4 and 119898) are also temperature and stress dependent

The main difference compared with the original and all other modified projection models is that numerical interpolation is used to replace the second numerical fitting in previous models In other words instead of fitting the model parameters as functions of stress and temperature with Eq (23) the distributions of the model parameters are assumed to be piecewise in the temperature-stress space and they will be interpolated directly from known parameters obtained from experiments The consideration is that the experimental data ([16] [17]) for 16MND5 covers a wide temperature range varying from 600 degC to 1300 degC and a wide stress load range varying from 08 MPa to 248 MPa while Eq (23) in the existing methods is based on a narrow temperature span Poor parameter fitting results would be expected if we directly fit the parameters 1205791~1205794 with Eq (23) as used in the original methods

The creep model was developed using the following steps

(i) Fitting Each creep curve can be represented by a group of model parameters [1205791 1205792 120579119898 1205793 1205794] with respect to Eq (24) by fitting the creep curves to Eq (24) All parameters of the 32 experimental curves form an

initial dataset of the model parameters [119879119895 120590119895 1205791119895 1205792119895 120579119898119895 1205793119895 1205794119895] where j (0ltjle32) is the jth creep curve

(ii) Interpolation Parameters of a creep curve under any [119879 σ] are directly interpolated from the dataset

(iii) Parameter adjustment The dataset was further updated to satisfy the monotonicity assumption we have taken that creep strain monotonically increases with temperature and stress

(iv) Different hardening models for dynamic loads

Step (i) was done using MATLAB In the following steps (ii)-(iv) are described in detail

Model parameters for new creep curves are interpolated from the dataset over the 2D 119879 minus 120590 space The interpolation is divided into two sub-interpolation steps interpolate over stress with fixed temperature and then interpolate over temperature with fixed stress We take the interpolation of model parameters under (119879prime σprime) for example In the first interpolation we first find out the two

17

temperatures 119879119896119899119900119908119899119904119898119886119897119897 and 119879119896119899119900119908119899119897119886119903119892119890 most close to 119879prime from the experimental

temperature set 600 700 800900 1000 1100 1200 1300 Under each temperature there are four sets of model parameters in the dataset corresponding to four different stress loads Then the first interpolation is done over stress load from the known dataset of model parameters under each

temperature to output the parameters for (119879119896119899119900119908119899119904119898119886119897119897 120590rsquo) and (119879119896119899119900119908119899119897119886119903119892119890 120590rsquo)

In the second interpolation the model outputs parameters for (119879rsquo 120590rsquo) from

(119879119896119899119900119908119899119904119898119886119897119897 120590rsquo) and (119879119896119899119900119908119899119897119886119903119892119890 120590rsquo) For 120579i(119894 = 12 3 4) we assume a logarithm

relation between 120579119894 temperature 119879 and stress 120590 which is similar to Eq (23)

119897119899120579119894 = (119886 + 119887120590)(119888 + 119889119879) (25)

It reduces to 120579119894 = 11986011198901198871σ with fixed temperature in the first interpolation and to

θ119894 = 11986021198901198872T with fixed stress in the second interpolation The interpolation

expression for θ119894 = 11986011198901198871σwith two known points (1205901 1205791198941) and (1205902 1205791198942) is given

by Eq (26) The interpolation expression for θ119894 = 11986021198901198872T with two known points

(1198791 1205791198941) and (1198792 1205791198942) is given by Eq (27)

120579119894 = 120579i1

120590minus12059021205901minus1205902120579i2

1205901minus1205901205901minus1205902 (26)

120579119894 = 120579i1

119879minus11987921198791minus1198792120579i2

1198791minus1198791198791minus1198792 (27)

Similarly we assume the following relation between 120579119898 temperature 119879 and stress 120590

120579119898 = 119860120590119899119890minus119887 119879frasl (28)

The interpolation is done over the stress with 120579119898 = 1198603120590n by fixing temperature

and then over temperature with 120579119898 = 1198604119890minus1198874119879 by fixing stress The interpolation

expression over stress for 120579119898 = 1198603120590119899 with two known points (1205901 1205791198981) and

(1205902 1205791198982) would be given by Eq (29) The interpolation expression over

temperature for 120579119898 = 1198604119890minus1198874119879 with two known points (1198791 1205791198981) and (1198792 1205791198982)

would be given by Eq (30)

120579119898 = 119890119909119901(119897119899120590 minus 11989711989912059011198971198991205902 minus 1198971198991205901

1198971198991205791198982 +1198971198991205902 minus 119897119899120590

1198971198991205902 minus 11989711989912059011198971198991205791198981) (29)

120579119898 = 120579m2(11198791minus1119879) (

11198791minus11198792)fraslsdot 120579m1

(1119879minus11198792) (

11198791minus11198792)frasl (30)

In this model we also take the assumption that the creep strain monotonically increases with temperature and stress loads1 Or specifically with the same creep time the structural component under higher temperature or stress experiences

1 Intuitively materials deform more under higher stress and temperature This is also mostly observed in tensile experiments Though some exceptions exist one instance of which is the effect of phase change some materials under certain temperature around the phase change temperature may show much weaker structural performance Here we just simply assumed that the creep strain monotonically increases with temperature and stress

18

higher creep strain than that with lower temperature or stress (the creep curve under higher temperature or stress should be above the curve with lower temperature or stress) When generating the parameter dataset (Step (i)) we already kept the trend that under the same temperature cases under higher stress loads would have larger creep model parameters and give higher creep strains This trend would also hold for the interpolated temperatures which thus guarantees that creep strain monotonically increases with stress load under same temperature Then the question becomes if creep strain also monotonically increases with temperature given fixed stress However the answer could not always be a yes especially in the extrapolation range of the stress load Figure 5(a) illustrates an example of the interpolation of model parameters under (119879119872 σ119872) The red dots and blue dots are known parameters from the parameter dataset for corresponding high temperature 119879119867 and low temperature 119879119871 respectively The green dots (119879119867 σ119872) and (119879119871 σ119872) are interpolated from the red and blue dots in the first sub-interpolation respectively The black dot (119879119872 σ119872) is interpolated from the green dots in the second sub-interpolation It is seen that when σ119872 lt σ119862 the value of (119879119871 σ119872) is higher than that of (119879119867 σ119872) It means the higher 119879119872 is the smaller the model parameter will be ie the model would predict larger creep strain for lower temperatures which violates our monotonicity assumption One simple solution used here as shown in Figure 5(b) is to minorly adjust the values of the four red and blue dots such that the intersection point C can be sufficiently small and the situation σ119872 lt σ119862 would barely occur in the extrapolation range

Figure 5 Illustration of model parameter interpolation (a) before and (b) after parameter adjustment [54]

A corresponding parameter adjustment process was introduced to verify that the model predicts higher creep strain under higher temperature under fixed stress Figure 6 illustrates the general verification process for any (119879 120590) in the interested load range the creep strain of (119879 + 120575119879 120590) should be larger than that of (119879 120590) Otherwise parameters would be minorly adjusted and the verification process be repeated This process would be iterated several times until the requirement is reached The achieved verification shows

Stress

σC

High temperature TH

MPa

a

0

Model

par

amet

er θ

Low temperature TL

MPa

a

0

Model

par

amet

er θ

High temperature TH

Low temperature TL

σC

Stress

C C

(a) (b)

σM

σM

19

(i) For any (119879 120590) in the temperature range 273~ 1073 K with an interval of 1 K and the stress range 1 ~260 MPa with an interval of 1 MPa its creep strain is higher than that of (119879 minus 1119870 120590) for the first 12000000 s

(ii) For any (119879 120590) in the temperature range 1073~1673 K with an interval of 1 K and the stress range 1 ~80 MPa with an interval of 1MPa its creep strain is higher than that of (119879 minus 1119870 120590) for the first 12000000 s

(iii) For any (119879 120590) in the stress range 01~26 MPa with a smaller interval of 01 MPa for the full temperature range 273~1673 K with an interval of 1 K its creep strain is higher than that of (119879 minus 1 120590) for the first 12000000 s

Above verification covers a wide range of temperature and stress especially where the stress and temperature are low and extrapolation is used Further verification to cover even higher stress or temperature ranges is not considered as creep increases drastically under such high stress and temperature and the results would become meaningless With these regards we generally think the monotonicity requirement is reached The model parameter dataset was updated along with the adjustment The final dataset was given as Table 2 in Paper I

Figure 6 Parameter adjustment process [54]

Figure 7(a-d) present the model predictions of selected creep curves in ANSYS Mechanical (labelled as lsquoANSYSrsquo) compared with experimental curves (labelled as lsquoREVISArsquo) at temperature 600 800 1000 and 1200 degC respectively The predicted curves proved to clearly show the three-stage creep process and agreed well with the experimental creep curves

Initial parametersdataset

θ1 θ2 θm θ3 θ4 from creep

curve-fitting

Parameters interpolation

Is the monotonicity

assumption satisfied

IsMon= =1

Final model parameters

Adjust the

parametersdataset

θ1 θ2 θm θ3 θ4

No

Yes

IsMon=1 a flag is monotonic achieved

For σ in the stress range in MPa

For T in range Tstart Tend in K

For t in range 112000000 in s

If ε(σ T+1t)lt ε(σTt) IsMon=0

Iteration loop of

parameter adjustment

20

Figure 7 Selected model predictions compared with experimental creep curves [54]

Instead of being constant temperature and stress loads also dynamically change with time in real cases Hardening models determine how creep strain rate reacts to the change of loads during a creep process The time hardening model and strain hardening model2 are supported in our model implementation in ANSYS Mechanical which are two mostly used hardening models

Time hardening assumes that the following creep strain rate after a load change depends only on the current creep time from the beginning of the creep process (and also temperature and stress if used in the creep model) and ignores the current creep strain Strain hardening assumes that the creep strain rate depends only upon the current creep strain (and also temperature and stress if used in the creep model) [55]

2 The time hardening and strain hardening here are more general terms than the specific time hardening creep model and the strain hardening creep model mentioned in Table 1

(a) 600 degC (b) 800 degC

(c) 1000 degC (d) 1200 degC

21

Eq (24) in default is a time hardening model as creep strain is directly described by the time 119905 Eq (31) gives the creep strain increment 119889휀119888119903 at time 119905 during time interval 119889119905

119889휀119888119903 = (12057911205792119890minus1205792119905 + 120579119898 + 12057931205794119890

1205794119905)119889119905 (31)

The strain hardening model is derived from Eq (24) in an implicit form Specifically when temperature and stress load change to (119879 120590) at the time 119905 and current creep strain 휀1198881199030 the model will get the creep curve of constant (119879 120590) and

output how this curve will react under the current strain 휀1198881199030 Since the creep curve shows creep strain monotonically increases with time given the creep strain 휀1198881199030 under creep curve of (119879 120590) we can uniquely find this reference time

119905119903119890119891 satisfying Eq (32) One should notice that the reference time 119905119903119890119891 can be

different from the current physical time 119905

휀1198881199030 = 1205791(1 minus 119890minus1205792119905119903119890119891) + 120579119898119905119903119890119891 + 1205793(119890

1205794119905119903119890119891 minus 1) (32)

where all 120579119894(119894 = 1~4119898) are model parameters of the new creep curve (119879 120590) It can be solved for 119905119903119890119891 with methods eg Newton-Ramphson [56] Then the creep

strain increment at 119905119903119890119891 during time interval 119889119905 is given as

119889휀119888119903 = (12057911205792119890minus1205792119905119903119890119891 + 120579119898 + 12057931205794119890

1205794119905119903119890119891)119889119905 (33)

Therefore combining Eq (32) and (33) gives the creep strain rate at creep strain 휀1198881199030 In summary when loads are changed to (119879 120590) with current creep strain 휀1198881199030

the model traces the creep curve of (119879 120590) and calculates the creep strain rate of this curve at strain 휀1198881199030

One calculation was done to demonstrate the performance of the hardening models The stress load was set to 151 MPa and kept unchanged A step temperature load was applied as the orange dash curve shows in Figure 8 it was 900 degC during 0-36000 s and suddenly jumped to 1000 degC at 36000 s which was then kept till the end of the simulation The blue solid curve (the part before the temperature change was identical to and therefore covered by the green solid curve in the figure) and blue dash curve are the creep curves under constant temperature T=900 degC and T=1000 degC respectively The green solid curve and green dash curve represent the model predictions considering this temperature change with time hardening model and strain hardening model respectively Before this change the creep curves with both hardening models were identical to the creep curve of the T=900 degC till point A After that the creep strain rate then shifted from Point A to Point B (which had the same time with A) in the time hardening model and behaved the same as the curve T=1000 degC as shown in the rest part of the green solid curve In the strain hardening model the creep strain rate shifted from Point A to Point C (which had the same creep strain with A) and behaved the same as the curve T=1000 degC as shown in the rest part of the green dash curve The large difference between two models was partially caused by the drastic temperature jump In real cases the difference would be less significant as the temperature changes more mildly Overall this prediction is exactly the expected behaviors of the time hardening model and strain hardening model

22

Figure 8 Time hardening and strain hardening in ANSYS Mechanical [54]

23

4 Coupled Thermo-Fluid-Structure Analysis of RPV Failure

As described in Section 13 RPV failure is coupled with melt pool heat transfer leading it into a transient thermo-fluid-structure coupled problem The goal of this section is to develop and validate a reliable coupling approach which supports transient coupled thermo-fluid-structure analysis as well as advanced models and computational platforms so as to improve fidelity and facilitate implementation The approach was validated against the FOREVER-EC2 experiment and then applied to an RPV failure analysis of a Nordic BWR As the approach adopts the volume loads mapping for load transfer comparative analysis was also performed against the coupling approach using surface loads mapping Future work is considered in analysis of penetration failures (IGTs and CRGTs) of a BWR RPV since the approach is flexible to complex geometries Results of Paper II ampIII are jointly presented in this section

41 Thermo-Fluid-Structure Coupling Approach

Considering the involved phenomena a coupling approach should at least address three issues (i) melt pool heat transfer (ii) RPV deformation and (iii) the interaction between melt pool heat transfer and vessel deformation A two-way strong coupling considering both the transient thermal load from melt pool on vessel and the influence of vessel deformation on melt pool heat transfer involves complex data exchange and is still full of challenges Therefore the new coupling approach is realized in a one-way manner with the assumptions that the influence of vessel deformation on melt pool heat transfer can be ignored and thus only the thermal loads from melt pool to vessel are considered It is based on the considerations that on one hand the deformation is a slow process regarding the relatively long failure time and on the other hand it was pointed out [57] that the physics of natural convection heat transfer in internally heated liquid pools is not sensitive to the geometrical factors based on their calculation results and experimental data from [58] [59] as well as previous experiments in semicircular and square cavities eg [44] [60] In addition one-way coupling is also computationally more efficient especially when heat transfer turbulence structural non-linearity and large-scale geometries are involved The developed approach is based on the ANSYS platform which generally consists of three parts with respect to the three issues respectively

(i) conjugate heat transfer of the melt pool and vessel using ANSYS Fluent (ii) transient thermal loads transfer from ANSYS Fluent to ANSYS

Structural (the ANSYS Mechanical in ANSYS Workbench) (iii) structural analysis of the vessel in ANSYS Structural

ANSYS Structural in default only supports direct transfer of constantsteady-state spatial loads from ANSYS Fluent Transfer of transient loads is realized with the extension tool FSI Transient [61] It is a pre-processing tool (with good documentation) developed by a third-party CAE vendor EDRTM MEDESO [62] to map transient CFD results (heat transfer coefficient temperature and surface pressure) to thermal or structural analysis including the definition of load steps

24

This new procedure proves to be straightforward and reduces efforts in the load transfer

Since structural performance of RPV is highly dependent on temperature the transfer of transient temperature loads from conjugate heat transfer to mechanical solver is a core step of the coupling framework There are two general approaches to realize the transfer as shown in Figure 9 Both approaches contain three parts a conjugate heat transfer analysis for fluid and structure where flow and heat transfer are coupled a mechanical analysis for the structure and a load mapping process that transfers the transient loads The conjugate heat transfer part can be solved with any CFD solver and is treated in the same manner in both approaches In the first approach (Figure 9(a)) transient volume thermal loads are transferred and mapped into the mechanical simulation domain The structure domain reads the essential thermal loads from the mapping after which a structure analysis of this domain is performed In the second approach (Figure 9(b)) transient surface thermal loads are mapped onto the boundary surfaces of structure domain as thermal boundary conditions Then a thermo-structure coupled analysis of the structure domain in the mechanical part is performed To distinguish them we call the former approach volume loads mapping (VLM) and the latter surface loads mapping (SLM) In the RPV failure analysis the fluid and structure domains correspond to the melt pool and RPV respectively Examples of these methods are the studies of [63] and [30] using VLM and [35] using SLM The VLM is adopted in current approach

Figure 9 Frameworks of the thermo-fluid-structure coupling approaches (a) volume loads mapping (b) surface loads mapping [64]

fluid structure structure

fluid structure structure

Conjugate heat transfer problem Mechanical problem

(structure)

Conjugate heat transfer problem Mechanical problem

(thermo-structure coupled)

Transient volume

loads mapping

(VLM)

Transient surface

loads mapping

(SLM)

(a)

(b)

25

42 Numerical Modelling

421 Conjugate heat transfer

In the conjugate heat transfer analysis main focus is put on the modelling of melt pool heat transfer Within the ANSYS Fluent solver two options are supported CFD with turbulence modelling and PECM CFD is the choice when the model goes to detail and more mechanistic analysis are needed while PECM provides an alternative that is faster and possibly more suitable for large-scale problems

Within the scope of RANS models the SST (Shear Stress Transport) model [65] is selected as the turbulence model in present study This model is a combination of k-ε model and k-ω model The k-ω formulation is used in the inner parts of the boundary layer which makes the model directly usable all the way down to the wall through the viscous sub-layer It also switches to k-ε model in the free-stream and thereby avoids the common k-ω problem that the model is too sensitive to the inlet free-stream turbulence properties The considerations on choosing this model are

bull Several two-equation low-Re k-ε models have been proved in [66] not good enough for melt pool heat transfer Therefore these mentioned two-equation models as well as other one-equation models are not considered

bull SST is highly recommended [67] for simulating most industrial heat transfer problems

bull Convincing results ([68] [69]) using SST have been obtained in capturing the thermal behaviour of the natural convection in an internally heated fluid Especially good agreements were achieved between the predicted heat flux distribution along the vessel and experimental data which would be one of the most important factors to estimate model performance in this problem

The description of PECM as an effective model for melt pool heat transfer is given in Section 23

422 Mechanical analysis

The material for the RPV failure analysis considered in this study is the French RPV steel 16MND5 The three-stage modified theta projection model in Section 3 is used as the creep model Since the definition of an RPV failure criterion is not universal (it can be the time when damage variable reaching 1 or either specified value of creep strain creep strain rate or total deformation is reached) we do not explicitly define a failure criterion Instead the simulation is kept running until the creep converge criterion cannot be satisfied even with the defined minimum time step (indicating a fairly large creep strain rate) and the simulation will be automatically terminated The time the simulation stops is taken as the failure time This treatment implicitly implies a criterion of a large creep strain rate We refer to [17] for other material properties Youngrsquos modulus and thermal conductivity are temperature dependent Multilinear Isotropic Hardening (MISO) is used as the plasticity model

26

43 Validation against the FOREVER-EC2 Experiment

In this section the validation against the FOREVER-EC2 experiment is presented (indicated as lsquoVLMrsquo in the following) In addition the simulation results using the SLM approach (indicated as lsquoSLMrsquo in the following) is also included for comparison purpose

431 FOREVER-EC2 experiment

The FOREVER-EC2 experiment [70] is one of the FOREVER experiments that achieved a vessel failure Figure 10 shows the experimental setup A 110 scale pressure vessel with a wall thickness of 15 mm and an outer diameter of ~400 mm was employed The lower head was made of 16MND5 steel Since it was difficult to use prototypical corium materials in the experiment the binary salt 30 CaO-70 B2O3 was employed as simulant To generate the internal heat which corresponds to the nuclear decay heat a specially designed heater was used The power input was maintained ~38 kW during the experiment such that maximum melt pool temperature would be ~1300 degC The pressure inside the vessel was maintained at ~25 MPa Thermocouples were used to measure the melt pool and vessel wall temperatures and linear displacement transducers used to measure the vessel wall displacements The designed experimental conditions were reached at experimental time 11670 s Vessel failure was observed about 12510 s later (at experiment time 24180 s)

Figure 10 Scheme of the experimental setup [22]

27

Figure 11 Computational domains and boundary conditions (a) conjugate heat transfer (b) mechanical analysis [71]

Computational domains and boundary conditions are indicated in Figure 11 SST turbulence modelling was used for pool heat transfer The simulations started at the moment when experimental conditions were reached (experiment time 11670 s) So simulation time 0 s corresponds to experiment time 11670 s

432 Conjugate heat transfer

Figure 12 shows the temperature fields at simulation time 0 s 1225 s 2475 s 4975 s and 9975 s All the time points in this part refer to simulation time The field at time 0 s (Figure 12(a)) was the initial temperature that was reconstructed based on the thermocouple readings The temperature increased with time and reached a steady state when the maximum temperature reached ~1607 K Figure 12(b-f) show clear thermal stratification which is an expected phenomenon of natural convection

Internal pressure

25 bar

Internal pressure 25e6-2500times981timesy Pa

External pressure

1 bar

Zero displacement

y=0 m

Gravity 400

mm

R=203 mm

15 m

m

Tamb

=323 K radiation with ε = 08

+ convection with h=10Wm2K

Tamb

=800 K radiation with ε=02

(a) (b)

Tamb

=800 K radiation with ε=08

+ convection with h=10 Wm2K

28

Figure 12 Temperature fields at simulation time (a) 0 s (b)1225 s (c) 2475s (d) 4975 s (e) 7475 s and (f) 9975 s [64]

Figure 13 shows the temperature profiles of the external vessel along the polar angle in comparison with experimental data Regions with a polar angle larger than 90deg are those in the cylindrical part of the vessel It also shows the temperature profile increased with time until reaching a steady state The profile almost did not change after 4975 s which indicates that a steady state of the temperature was reached As time increased the temperature in the lower head region 0~90deg (where the vessel was contacted with the melt) evolved from a relatively flat distribution to a distribution that temperature at high angle was higher than that at low angle by more than 200 K Generally the steady state results agreed well with the experimental data except for the low angle part of vessel lower head and the high angle part of the cylindrical vessel In addition to modelling errors the discrepancy in the low angle part of the lower head may be caused by the crust and gap possibly formed at vessel bottom in the experiment which can reduce the downward heat transfer and result in lower temperature while they cannot be modelled in simulations As to the high angle cylindrical vessel part simplified boundary conditions were applied to the inner and outer vessel wall since it was far from the melt and not of interest which may result in some discrepancy

(a) (b)

(c) (d) (e) (f)

29

Figure 13 Temperature profiles of external vessel surface along the vessel polar angle [64]

433 Mechanical analysis

In the mechanical analysis the VLM predicted a failure time of 23959 s and the SLM predicted a failure time of 22445 s both are close to the experimental failure time 24180 s All the time points in this part refer to the experiment time Figure 14 shows the creep strain fields at failure time Both approaches predicted maximum creep strains at same location where the temperature was high (cf Figure 12) This is related to the fact that creep behavior is highly influenced by temperature Maximum values for both cases were also close to each other 0393 in VLM and 0366 in SLM

Figure 15 presents the total deformation of the vessel bottom point (also called ldquosouth polerdquo in some literature) in the vertical direction Since there was an initial displacement reading of ~355 mm at time 0 s which almost did not change during 0~11670 s (the whole period before the experimental conditions were reached) an initial value of 355 mm was added to the simulation result such that experimental and simulation results had the same total displacement at t=11670 s Similar trend was seen in the VLM SLM and experimental data that the deformation increased with time and experienced a drastic acceleration at the end Generally both simulation approaches agreed well with the experimental data though they both slightly under predicted the value of total deformation The discrepancy may possibly be related to the accuracy of the thermal predictions modelling errors of plasticity and creep and simplified boundary conditions

400

600

800

1000

1200

1400

1600

0 20 40 60 80 100 120 140 160

Tem

per

ature

(K

)

Polar angle (deg)

Experiment

25 s

1225 s

2475 s

4975 s

7475 s

9975 s

Weilding line

30

Figure 14 Creep strain fields at failure time (a) VLM (b) SLM [64]

Figure 15 Total deformation in vertical direction of the vessel bottom point [64] [71]

The wall thickness changes are plotted together with experimental results in Figure 16 The wall thickness change δ119909 is defined as

δ119909 = 119909119890119899119889 minus 119909119900119903119894119892119894119899119886119897 (34)

000

500

1000

1500

2000

2500

11670 13670 15670 17670 19670 21670 23670

Tota

l def

orm

atio

n (

mm

)

Time (s)

ExperimentVLMSLM

(a) (b)

31

where 119909119890119899119889 and 119909119900119903119894119892119894119899119886119897 are the wall thickness at failure time and the original wall

thickness respectively Since 119909119890119899119889 cannot directly be post-processed it was estimated in the following way (cf the subfigure of Figure 16) for the same polar angle θ find the corresponding points Pin (1199091 1199101) and Pout (1199092 1199102) on the inner and outer surfaces of the deformed vessel respectively the distance of Pin and

Pout radic(1199091 minus 1199092)2 + (1199101 minus 1199102)2 is calculated as 119909119890119899119889 at polar angle 120579 Such estimation of the wall thickness is more accurate when the line PinPout is closer to be orthogonal to the deformed local surface Considering the final deformed vessel as shown in Figure 14 the estimation should be fine at regions close to 0deg and 90deg and can be slightly overestimated at medium angle region The VLM curve was compared with the SLM curve and 4 experimental curves measured at vessel left right front and back sides Generally VLM curve agreed well with the experimental and SLM curves wall thickness gradually decreases with angle till around 75deg and then increases till 90deg the maximum thickness change occurred at around 75deg would be around -5 mm The positive values at low angle may be caused by the thermal expansion It also shows that the thickness transitions in experimental curves were smoother than those in both simulation curves while simulation curves showed narrower regions with large thickness changes (at angle 70deg~80deg) Possible reasons might be (i) sharper temperature profile predicted in simulation (ii) input uncertainty and (iii) modelling errors The SLM predicted slightly thinner vessel than the VLM except for the necking region The discrepancy might be caused by various reasons eg the interior vessel temperature can be slightly different between two approaches which would then result in different thermal stress and different deformation the final deformed vessel shapes cannot be exactly the same which also results in different errors in estimating PinPout

Figure 16 Wall thickness changes along the vessel polar angle [64] [71]

-6

-5

-4

-3

-2

-1

0

1

0 10 20 30 40 50 60 70 80 90

Wal

l th

icknes

s ch

ange

(mm

)

Polar angle (deg)

VLMSLMExperiment (left)Experiment (right)Experiment (front)Experiment (back)

32

44 Application to BWR Case

In this section the coupling approach (indicated as lsquoVLMrsquo in the following) was applied to an RPV failure analysis of a BWR A simulation using the SLM coupling approach (indicated as lsquoSLMrsquo in the following) was also performed for comparison purpose

A hypothetical severe accident of a Nordic BWR initiated by an SBO was considered The core materials would melt and migrate downward to the RPV lower head The total amount of corium relocated in the lower head was considered to form a 19 m corium melt pooldebris bed assuming the corium distribution was homogeneous

Figure 17 Computational domains and boundary conditions (a) conjugate heat transfer (b) mechanical analysis [64]

Figure 17(a) shows the computational domain of the conjugate heat transfer PECM was used in this calculation A general mesh with a pool height of 23 m was used to accommodate most of the relocation situations with the same mesh Since the corium only had a height of 19 m the top 04 m region was treated as injected water covering the corium The top water was expected to be boiling and we assumed a uniform temperature of 383 K Under the implementation of PECM this region was damped with artificially high heat capacity and high conductivity such that it would efficiently remove heat from the corium and suppress its own temperature increase For the corium part below water a volumetric heat source 11 MWm3 was implemented as the nuclear decay heat For the in-vessel surfaces a constant temperature of 383K was applied as they are covered by water which can be boiling Without water cooling on the external vessel wall a very small heat loss of 20 Wm2 was applied Melting and solidification of the corium were also considered with a solidus temperature of 2750 K and a liquidus temperature of 2770 K The initial temperature was set as

q=20 Wm2

T=383 K

Qv=11

MWm3

315 bar

315 bar+ hydrostatic

pressure of corium

Gravity

248 bar

Zero

displacement

Water

Corium

(a) (b)

33

1100 K for the corium 383 K for the water and 450 K for the vessel For the mechanical analysis (Figure 17(b)) an inner pressure of 315 bar and an outer pressure of 248 bar were considered For the region covered by corium an extra hydrostatic pressure was added on the vessel wall A 2 m long vessel extension was considered as well as the standard earth gravity

Figure 18 presents the wall temperature profile along the inner vessel surface at 50 s 2500 s 7500 s and 10000 s The temperature increased with time for the vessel surface contacted with corium and the distribution was flat which was due to that large part of the corium was still in solidus status It also shows that for the surface region that contacted the water the temperature was basically unchanged which indicates that the damping treatment indeed suppressed the temperature increment and guaranteed the heat removal upward by water

Figure 18 Temperature profiles of inner vessel wall along the vessel polar angle [64]

The predicted RPV failure time was 9869 s by VLM and 9650 s by SLM Compared with the FOREVER-EC2 case in Section 43 the VLM predicted a slightly later failure time than the SLM in this BWR case too Figure 19(a) compares the maximum creep strain developments with time of both approaches Before 9000 s the predicted values by both approaches are close to each other Afterward the creep acceleration occurred and the strain increased drastically The acceleration of the VLM occurred 200 s later than that of the SLM which agreed with the predicted later failure time Similar trend was observed for the total deformation in vertical direction of vessel bottom point as shown in Figure 19(b) The deformations of both approaches are close except that the acceleration of VLM occurred later than that of SLM Overall results of VLM agreed well with those of SLM

0

500

1000

1500

2000

2500

0 10 20 30 40 50 60 70 80 90 100

Tem

per

ature

(K

)

Polar angle (deg)

50 s

2500 s

5000 s

7000 s

10000 s

34

Figure 19 Results of structural analysis (a) creep strain and (b) total deformation in vertical direction of vessel bottom point [64]

45 Computational Efficiency

Computational efficiency is also an important consideration regarding the performance of numerical approaches The computational efficiencies of both VLM and SLM are analysed and compared in this section As parallel computing technique was used to speed up the simulations the number of parallel cores and corresponding costed computational time for each simulation are given together in Table 2 as a generally reflection of the computational efficiency All the simulations were done in a Lenovo P700 workstation equipped with 2 Intel Xeon E5-2630 v3 processors (8 cores per processor) and 32 G RAM For the simulations of FOREVER-EC2 SLM employed twice the parallel cores and costed 286 more time compared with VLM For the simulations of prototypic BWR case both VLM and SLM employed the same number of parallel cores but VLM only took 13 of the computational time compared with SLM Overall it can be concluded that VLM has higher computational efficiency than the SLM The reason is a structural analysis is performed in the VLM approach while a coupled thermo-structural analysis is performed in the SLM approach (cf Figure 9) a standalone structural analysis is computed faster than a coupled thermo-structural analysis

Table 2 Comparison of computational efficiency [64]

Case FOREVER-EC2 BWR VLM SLM VLM SLM

Number of cores 4 8 4 4 Computational time (hours) 21 27 18 72

35

5 Hybrid Coupling Approach for Efficient RPV Failure Analysis

In this section we propose a hybrid coupling approach for the efficient RPV failure analysis The idea is to replace the ANSYS Fluent in previous coupling approach with a lumped-parameter code for quick estimate of the melt pool heat transfer With the least modelling details (compared with CFD methods and distributed-parameter methods eg PECM) lumped-parameter codes prove to be the most efficient for melt pool heat transfer The new hybrid approach characterizes 1) utilization of efficient models for melt pool heat transfer and 2) the support of detailed structural modelling of RPV deformation allowing studies with different RPV geometric shapes under effects like local thinning due to ablation For studies of which the focus is mainly on structural behaviour of RPV eg [13] [26] while mechanistic simulation of melt pool convection is computationally expensive and unnecessary the lumped-parameter code could help providing reasonable thermal loads

The development of the approach consists of two tasks (i) development of a lumped-parameter code and (ii) its coupling with ANSYS Mechanical to form this hybrid coupling framework In the following we describe both tasks and the corresponding verification and validation processes respectively This section is mainly based on Paper IV

51 A Lumped-parameter Code for Melt Pool Heat Transfer

A lumped-parameter code transIVR was developed It features (i) quick estimate of transient melt pool heat transfer with reasonable accuracies and (ii) two-dimensional representation of heat conduction in the RPV wall facilitating the coupled analysis of the RPV failure under thermal loads The following assumptions are taken in the transIVR modelling

bull The crust between the oxide pool and metal pool is thin

bull The heat flux has uniform distributions on the bottom top and side surfaces of metal pool

bull Heat source only exists in oxide layer

bull Latent heat of fusion is ignored

Figure 20 illustrates the modelled three domains oxide layer metal layer and the RPV wall In addition the internal structure inside the vessel can also be considered as a radiative body to the top metal surface as in [72] An internal volumetric heat source generated by nuclear decay heat is considered in the oxide layer Heat in oxide layer transfers downward to the RPV wall and upward to the top metal layer Heat in metal layer can be transferred sideward to the RPV wall and upward to the surrounding internal structure through radiation (or to water if inside water flooding is applied) The heat transferred to RPV will be removed from outer RPV surface Possible formations of oxide crust in the top and bottom surface are illustrated as the black regions Possible formation of metal crust on the top surface is shown as the grey region The code automatically detects the

36

existence of the crust on oxide top and metal top surfaces during the simulation and solves the problem correspondingly withwithout crust Boundary conditions are applied to the external and internal surfaces of RPV and (if needed) top surface of metal layer Arbitrary boundary conditions are supported If a boundary condition is nonlinear eg mixed heat convection (linear dependent on 119879 ) and heat radiation (proportional to 1198794) the Picard method [73] would be used for solving this problem

This model consists of three main parts

(i) empirical heat transfer correlations (mostly from experiments) to determine global and local heat transfer coefficients

(ii) energy balances in oxide layer and metal layer where the heat transfer coefficients would be used and

(iii) energy balance in RPV

Figure 20 Two-layer melt pool configuration and RPV wall [74]

511 Heat transfer correlations

For internally heated natural convection (oxide layer) in transIVR the mini-ACOPO correlations [75] are used for the averaged heat transfer as expressed in Eq (35) and (36) and for the angular variation of the heat flux as expressed in Eq (37) and (38)

119873119906119906119901 = 03451198771198861198941198990233 (35)

119873119906119889119899 = 00038119877119886119894119899035(119867119877)025 (36)

119873119906119889119899(θ)

119873119906119889119899= 01 + 108(

θ

θ119898119886119909) minus 45 (

θ

θ119898119886119909)2

+ 86 (θ

θ119898119886119909)3

01 leθ

θ119898119886119909le 06 (37)

RPV

Oxide pool

Metal pool

37

119873119906119889119899(θ)

119873119906119889119899= 041 + 035 (

θ

θ119898119886119909) + (

θ

θ119898119886119909)2

06 ltθ

θ119898119886119909le 1 (38)

The 119877119886119894119899 that appears in Eq (35)~(36) is the internal Rayleigh number which is defined as

119877119886119894119899 =119892β119876119881119867

5

α119896ν (39)

For Rayleigh-Bernard convection in the metal layer heat transfer occurs in all the top bottom and side surfaces To side surface the reduced form [72] of the Churchill-Chu correlation [76] for 119875119903 = 013 is employed as shown in Eq (40)

119873119906119904119889 = 007611987711988613 (40)

To top and bottom surfaces the reduced form [72] of Globe-Dropkin correlation [77] for

119875119903 = 013 is used as expressed in Eq (41)

119873119906119906119901 = 005911987711988613 (41)

The 119877119886 that appears in Eq (40) and (41) is the external Rayleigh number which is defined

as

119877119886 =119892βΔ1198791198673

να (42)

Heat transfer coefficients can be calculated from Nusselt number by Eq (43)

ℎ =119873119906119896

119871 (43)

where 119896 and 119871 are the heat conductivity and characteristic length respectively 119873119906 can be the Nusselt numbers expressed in Eq (35)~(38) and (40)~(41)

512 Energy balance in the pools

The energy balance of oxide layer can be described as Eq (44) The total generated heat of oxide pool equals the sum of the increase of enthalpy (transient temperature change) upward heat flow to metal layer and downward heat flow to RPV

120588119862119901119881119900119897119889119879119900119909119889119905

+ ℎ119906119901119860119900119909119906119901(119879119900119909 minus 119879119900119909119906119901) + ℎ119889119899119860119900119909119889119899(119879119900119909 minus 119879119900119909119889119899) minus 119876119881119881119900119897 = 0 (44)

where ρ is the oxide density 119862119901 the heat capacity 119881119900119897 the total volume 119879119900119909 the bulk temperature 119905 the time ℎ119906119901 the upward heat transfer coefficient 119860119900119909119906119901

upward heat transfer area 119879119900119909119906119901 upward surface temperature ℎ119889119899 downward

heat transfer coefficient 119860119900119909119889119899 downward heat transfer area 119879119900119909119889119899 downward surface temperature and 119876119881 volumetric heat source

38

On the interface between oxide layer and metal layer energy balance requires that the heat transferred from oxide layer should equal the heat to the metal layer which can be described as Eq (45)

ℎ119906119901119860119900119909119906119901(119879119900119909 minus 119879119900119909119906119901) = ℎ119898119905119889119899119860119898119905119889119899(119879119894119899119905 minus 119879119898119905) (45)

where ℎ119898119905119889119899 is the downward heat transfer coefficient of metal layer 119860119898119905119889119899 the downward heat transfer area 119879119894119899119905 the interface temperature and 119879119898119905 the bulk temperature of metal layer

The energy balance of metal pool can be described as Eq (46) The increase of enthalpy (transient temperature change) equals the sum of upward downward and sideward heat flows to surrounding surfaces

ρ119898119905119862119901119898119905119881119900119897119898119905119889119879119898119905119889119905

+ ℎ119898119905119906119901119860119898119905119906119901(119879119898119905 minus 119879119898119905119906119901) + ℎ119898119905119904119889119860119898119905119904119889(119879119898119905 minus 119879119898119905119904119889)

+ ℎ119898119905119889119899119860119898119905119889119899(119879119898119905 minus 119879119894119899119905) = 0

(46)

where ρmt is the metal density 119862119901mt the heat capacity 119881119900119897mt the total volume 119879119898119905 the bulk metal temperature 119905 the time ℎ119898119905119906119901 the upward heat transfer

coefficient 119860119898119905119906119901 upward heat transfer area 119879119898119905119906119901 upward surface

temperature ℎ119898119905119904119889 sideward heat transfer coefficient 119860119898119905119904119889 sideward heat transfer area 119879119898119905119904119889 sideward surface temperature ℎ119898119905119889119899 downward heat transfer coefficient 119860119898119905119889119899 downward heat transfer area and 119879119894119899119905 oxide-metal interface temperature

The heat transferred from metal layer to the top surface should equal the heat radiated from this surface to the environment Considering the radiation and reflections of internal structures to this surface following energy balance equation [72] is achieved

ℎ119898119905119906119901(119879119898119905 minus 119879119898119905119906119901) = σ(1198791198981199051199051199001199014 minus 119879119904

4) (1

ε+1 minus ε119904ε119904

119860119898119905119906119901

119860119904)minus1

(47)

where 119879119898119905119905119900119901 is the temperature of metal top free surface ε the emissivity of

metal ε119904 the emissivity of internal structures 119860119898119905119906119901 the area of metal top surface

and 119860119904 the total area of internal structures Sometimes the internal water cooling may be available above metal layer In this case Eq (47) is unnecessary and 119879119898119905119905119900119901 can be set to a constant value

For the oxide layer top surface Eq (48) is used if not crust is formed in the interface between two layers ie 119879119894119899119905 gt 119879119898119890119897119905119894119899119892119900119909 Otherwise Eq (49) is used

119879119900119909119906119901 = 119879119894119899119905 (48)

119879119900119909119906119901 = 119879119898119890119897119905119894119899119892119900119909 (49)

For the metal layer top surface Eq (50) is used if not crust is formed on top of the metal ie 119879119906119901119904119906119903119891 gt 119879119898119890119897119905119894119899119892119898119905 Otherwise Eq (51) is used

119879119898119905119906119901 = 119879119898119905119905119900119901 (50)

39

119879119898119905119906119901 = 119879119898119890119897119905119894119899119892119898119905 (51)

Overall this part involves 6 equations regarding 6 unknows 119879119900119909 119879119900119909119906119901 119879119894119899119905 119879119898119905

119879119898119905119906119901 and 119879119898119905119905119900119901

513 Energy balance in the RPV wall

The heat transfer in the PRV wall can be modelled as a simple 1D heat conduction problem that heat only transfers in the wall thickness direction or 2D axisymmetric heat conduction problem

For 1D model the treatment in [72] was adopted As illustrated in Figure 21 the heat from the oxide pool transfers to inner vessel surface (119879119894) then to the interior vessel (119879119907119890119904) then to outer vessel (119879119900) and finally to environment

Figure 21 1D heat transfer model in RPV [74]

Unknown variables are the heat flux through inner vessel surface 119902119894119899 the crust thickness 119909119888119903 inner vessel temperature 119879119894 vessel center temperature 119879119907119890119904 and outer vessel temperature 119879119900 Eq (52)~(56) give a closed equation set for these variables considering a radiation boundary condition on the external vessel surface Eq (52)~(53) describe the heat transfer through the crust where an internal heat source 119876119888119903 is also considered Eq (52) can be easily derived from the one-dimensional heat transfer equation with heat source Eq (54)~(55) describe the energy balance on the inner vessel surface and vessel interior respectively Eq (56) describes the energy balance on the outer vessel surface regarding radiation boundary condition For Dirichlet boundary condition Eq (56) is replaced with Eq (57) For other boundary conditions we only need to correspondingly modify the terms on the right-hand side of Eq (56) For situations that no crust is formed on the surface of the vessel Eq (52)~(53) are eliminated and 119902119894119899 in Eq (54) can directly be determined from calculations in Section 512 and Eq (54)~(56) are solved for the unknows 119879119894 119879119907119890119904 and 119879119900 Once 119879119894 exceeds the vessel melting temperature (with or without crust formation) wall ablation is considered to occur Then 119879119894 is fixed to the vessel melting temperature and vessel thickness 119909119907119890119904 becomes the new unknown variable

119879119900119909119889119899 119879119894 119879119907119890119904 119879119900 119879119890119899119907

119909119888119903 119909119907119890119904

crust RPV

Qcr

119902119894119899 119902119897119900119888119886119897

40

119902119894119899 = 119896119888119903(119879119900119909119889119899 minus 119879119894)

119909119888119903+ 1198761198881199031199091198881199032

(52)

119902119894119899 = 119902119897119900119888119886119897 + 119876119888119903119909119888119903

(53)

119902119894119899 = 119896119907119890119904(119879119894 minus 119879119907119890119904)

05119909119907119890119904

(54)

120588119907119890119904119862119901119907119890119904119889119879119907119890119904119889119905

= 119896119907119890119904(minus2 lowast 119879119907119890119904 + 119879119894 + 119879119900)

(05119909119907119890119904)2

(55)

119896119907119890119904(119879119907119890119904 minus 119879119900)

05119909119907119890119904= 휀120590(119879119900

4 minus 1198791198901198991199074 )

(56)

119879119900 = 119879119900119888119900119899119904119905

(57)

For the 2D representation the heat conduction in the RPV is solved using FDM considering the heat transfer in both the thickness and the angular directions in the RPV wall The RPV lower head is treated as a spherical shell The corresponding heat conduction equation in the 2D spherical coordinate is given as Eq (58) The cylindrical part of the vessel is treated as a cylindrical shell The corresponding heat conduction equation in the 2D cylindrical coordinate is given as Eq (59) Eq (58) and (59) are coupled and solved together In default the mesh size is 5deg in angular direction and 0025119909119888119903 in radius direction This mesh discretization can be changed up to user demands

ρ119862119901part119879

part119905=

1

1199032part

part119903(1198961199032

part119879

part119903) +

1

r2sinθ

part

partθ(119896119904119894119899θ

part119879

partθ)

(58)

ρ119862119901part119879

part119905=1

119903

part

part119903(119896119903

part119879

part119903) +

part

part119911(119896part119879

part119911) (59)

52 Hybrid Coupling Approach

The hybrid coupling approach to assess the RPV integrity is illustrated in Figure 22 transIVR and ANSYS Mechanical are coupled in a one-way manner transIVR conducts the conjugate heat transfer of melt pool and RPV and provides transient thermal loads to the ANSYS Mechanical where structural analysis of RPV deformation is performed The 2D heat transfer of RPV is solved in transIVR to better capture the temperature distribution In RPV structural analysis same considerations are taken as in Section 422 for 16MND5 steel the modified theta projection model for creep and MISO for plasticity and temperature-dependent Young modulus etc The VLM is adopted for the transfer of transient thermal loads

41

Figure 22 Framework of the hybrid coupling approach [74]

53 Verification and Validation

The transIVR code was first benchmarked against the UCSB FIBS case of two-layer pool configuration and then validated against the LIVE-7V experiment showing the codersquos capability in predicting two-layer and transient melt pool heat transfers respectively Finally the hybrid coupling approach encompassing transIVR and ANSYS Mechanical was validated against the RPV failure experiment FOREVER-EC2 and the results showed acceptable agreements with those of both the experiment and the previous approach using CFD in Section 43 Hence the transIVR code is an efficient tool for melt pool heat transfer and the hybrid approach an efficient way for RPV failure analysis

531 UCSB FIBS benchmark case

The UCSB FIBS benchmark case [78] was calculated to demonstrate the capability of the transIVR code in simulating two-layer pool heat transfer problems The considered pool configuration was similar to that shown in Figure 20 with the following specific dimensions a two-layer melt pool with a 16 m oxide layer and a 08 m metallic pool on top and an RPV lower head with a thickness of 015 m The internal volumetric source in the oxide layer was 13 MWm3 The calculation results were compared with those of the DOE study [78] and the INEEL study [79]

The predicted results were compared with the digitalized results of the DOE and INEEL studies in Figure 23 All three studies show similar profiles of the heat flux distribution along the polar angle (Figure 23(a)) it increased with the polar angle in the oxide layer to the maximum value at around 77deg and then dropped to a significantly lower value in the metal layer The transIVR result agreed well with the other results with small discrepancy in the metal layer the DOE study gave a

heat flux of 500 MWm2 the INEEL study of 400 MWm2 and the transIVR of somewhere in between Figure 23(b) shows the predicted corium crust thickness distribution along the RPV polar angle also agreed well the DOE and INEEL

transIVR ANSYS Mechanical

Transient

thermal loads

42

studies The crust thickness decreased with polar angle in the oxide layer and no crust was formed in the metal layer As the crust formation is closely related to the heat flux distribution the crust thickness profile was reasonable regarding the heat flux distribution shown in Figure 23(a) Vessel wall ablation would occur due to the high thermal loads Figure 23(c) gives the profile of the remaining vessel wall thickness along the RPV polar angle considering the wall ablation It shows that no ablation occurred in low angle region (0deg~36deg) where heat flux was low and the wall thickness changed from 15 cm to around 6 cm as angle increased in high angle region of the oxide layer Because of the discrepancy in the heat flux of metal layer the predicted wall thicknesses also differ from those of the DOE and INEEL studies the thinnest wall was 73 cm predicted by the DOE study corresponding to the maximum heat flux followed by 88 cm of the present transIVR calculation and 94 cm of the INEEL study Generally speaking the prediction of transIVR is in an acceptable level compared with the other studies for the same benchmark case

Figure 23 Simulation results of the UCSB FIBS benchmark case [74]

(a) heat flux distribution (b) crust thickness distribution

(c) wall thickness distribution

43

532 LIVE-7V experiment

LIVE-7V is an experiment under the LIVE program conducted in KIT to study the core melt phenomena As Figure 24(a) shows it employed the LIVE-3D hemispherical facility with an inner diameter of 1 m and a wall thickness of ~25 mm The non-eutectic binary salt mixture of 20 mol NaNO3- 80 mol KNO3 was used as the simulant material The decay heat in the melt was simulated by cable-type heating with 8 planes of electrical resistance heating wires Water cooling was applied to the top and vessel surfaces Top surface cooling was realized by a cooling lid mounted at height 413 mm from the vessel bottom and vessel cooling by the flow channel between the test vessel and the outer cooling vessel The inlet and outlet water temperature for both top and vessel cooling surfaces were kept ~293 K and ~303 K throughout the whole experiment respectively A 413 mm height pool corresponds to a total volume of ~0194 m3 simulant material The experiment procedure can be divided into four stages with the heating power sequentially shifted from 29 kW to 24 kW 18 kW and 9 kW Figure 24(b) indicates the thermocouple positions in experiment

The simulation results of LIVE-7V using transIVR were compared with experimental data [80] and numerical study using REALPSCDAPSIM code [81] For simplicity the RELAPSCDAPSIM results in the following are labelled as lsquoCOUPLErsquo which is the name of the module in RELAPSCDAPSIM for melt pool heat transfer

Figure 24 LIVE test facility [80] and thermocouple positions [82]

Figure 25 compares the predicted heat removal upward to the top lid and downward to the bottom vessel surface with those of the experiment and COUPLE The black curve indicates the total input power at the four stages The experimental results (red and blue dots) were calculated from the Table 3 of [80] assuming no extra heat losses For the first three stages the transIVR predictions agreed well with the COUPLE results while the predicted upward heat flow was slightly lower and the downward heat flow higher than the experimental data For the last stage the transIVR agreed well with the experimental result while the COUPLE predicted lower upward and higher downward heat flows The short

(a) (b)

44

transition periods between two stages with different total inputs taken in transIVR also agreed with those in the COUPLE predictions

Figure 25 Comparison of heat removal upward to the top melt surface cooling lid and downward to bottom vessel wall [74]

Figure 26 shows the comparison of heat flux distribution along the vessel wall All curves show similar trend that a higher total input power would result in a higher heat flux profile As the COUPLE predictions had relatively flatter profiles transIVR predictions generally agreed better with the experimental results especially when total input was high A discrepancy is seen that the minimum value of a curve occurred at angle 0degfor both transIVR and COUPLE and at angle 30deg for the experimental data It is noted that the models in default assume heat flux monotonically increases with angle (cf Eq (37) and (38)) resulting it impossible to predict higher heat flux at angle 0deg than angle 30deg as in experimental results

45

Figure 26 Heat flux profiles along the vessel polar angle [74]

533 FOREVER-EC2 experiment

In this section we performed calculations against the FOREVER-EC2 experiment to validate the performance of the coupled transIVR and ANSYS Mechanical in assessing the RPV integrity The applied boundary conditions were identical to the settings of previous coupling approach in Section 4 (ANSYS Fluent+ Mechanical) as indicated in Figure 11 The initial bulk temperature of the melt pool was set to be 1261 K in transIVR which corresponds to the maximum value of the initial temperature field in CFD calculation of previous coupling approach Both the experimental data and the simulation results using previous coupling approach are included for comparison In the following results from current hybrid approach are labelled lsquotransIVRrsquo (and referred as transIVR results) results from previous approach labelled lsquoCFDrsquo (and referred as CFD results) and results from experiment labelled lsquoexperimentrsquo

Figure 27 compares the bulk temperature predicted by transIVR with the averaged and maximum temperatures of the CFD results All curves show the same trend that the temperature gradually increased with time till reaching a steady state and then kept unchanged till the end The times needed for reaching the steady state were ~5000 s for the CFD and ~4000 s for the transIVR which are close to each other It indicates again a good performance of the transIVR in predicting the transient behavior of melt pool Compared with the steady-state averaged temperature 1478 K and maximum temperature 1609 K of the CFD results the steady-state temperature of transIVR 1599 K is a reasonable prediction

0

3000

6000

9000

12000

15000

18000

0 20 40 60 80 100

Hea

t fl

ux

(W

m2)

Polar angle (deg)

29 kW-EXP24 kW-EXP18 kW-EXP9 kW-EXP29 kW-COUPLE24 kW-COUPLE18 kW-COUPLE9 kW-COUPLE29 kW-transIVR24 kW-transIVR18 kW-transIVR9 kW-transIVR

46

Figure 27 Temperature evolution with time [74]

Figure 28 Temperature fields at time (a) 01 s (b) 3038 s (c) 6038 s and (d) 13289 s [74]

Figure 28 shows the temperature fields of the vessel at time 01 s 3038 s 6038 s and 13289 s The temperature fields were visualized in the ANSYS Mechanical It also indicates that the transient temperature loads were properly transferred from transIVR to ANSYS Mechanical Figure 28(a-d) show both the minimum and maximum temperatures increased with time In the lower head region the temperature gradually increased with the height The maximum temperature occurs at the location slightly lower than the melt pool top surface This is because thermal loadheat flux imposed on the vessel generally increases with polar angle in the pool natural convection so the code transIVR inherently also predicted higher heat flux at higher angle positions The higher flux at higher position then correspondingly resulted in higher temperature In the cylindrical region the

(a) (b) (c) (d)

[K] [K] [K] [K]

47

temperature decreased as the height since the bottom part would be more significantly heated by the high-temperature lower head region

Figure 29 shows the steady-state temperature profiles of the external vessel surface along the polar angle The transIVR result agreed well with other two results in trend that the temperature increased with the angle and reach the peak values at angle ~80deg then gradually decreased with angle But slightly lower temperature was predicted in the low angle region Possible reason may be that the local heat flux distribution determined by Eq (37) and (38) is steeper than that in the experiment and CFD study which then resulted in a steeper increase of temperature along the angle and a lower temperature at low angle region In the cylindrical part (gt90deg) the transIVR result generally agreed with the experimental and CFD results The discrepancy with the experimental data may be caused by the simplified boundary conditions the ambient temperature inside the vessel was set to uniform constant value while it could decrease as height Therefore predicted heat loss in the higher position of cylindrical part could be smaller than that with height-dependent ambient temperature resulting in higher vessel temperature Overall the prediction was reasonable for a lumped-parameter model

Figure 29 Temperature profiles of the external vessel surface along the vessel polar angle [74]

The transIVR predicted a failure time of 24959 s which is close to 24180 s in experiment and 23959 s in the CFD results Figure 30 shows the total deformation of vessel bottom point with time All three curves increased gradually with time and experienced drastic accelerations at the end The

400

600

800

1000

1200

1400

0 40 80 120 160

Tem

per

ature

(K

)

Polar angle (deg)

transIVRExperimentCFDWeilding Line

48

transIVR curve agreed well with the CFD curve The final total deformation at failure time is also close to that of the CFD result but lower than the experiment data The discrepancy might be related to the simplifications in the boundary conditions accuracy of the thermal load prediction and the modelling errors in plasticity and creep

Figure 30 Total deformation in vertical direction of the vessel bottom point [74]

Figure 31 shows the profile of wall thickness change The transIVR curve was compared with the CFD curve and four experimental curves measured at vessel left right front and back sides The simulation result fitted the experimental and CFD results well in the region 0deg~50deg In the region 50deg~90deg wall thickness gradually decreased with angle till around 75deg and then increased till 90deg The maximum thickness change was -3 mm for the simulation which is close to the experiment left side curve and slightly larger than the -6~ -45 mm of the other curves from experiment and CFD results Concerning possible errors and uncertainties in the modelling and numerical settings the new approach generally gives reasonable predictions

0

5

10

15

20

25

11670 15170 18670 22170 25670

Tota

l def

orm

atio

n (

mm

)

Time (s)

Experiment

CFD

transIVR

49

Figure 31 Wall thickness changes along the vessel polar angle [74]

-6

-5

-4

-3

-2

-1

0

1

0 20 40 60 80 100

Wal

l th

icknes

s ch

ange

(mm

)

Polar angle (deg)

CFD

transIVR

Experiment (left)

Experiment (right)

Experiment (front)

Experiment (back)

50

51

6 Summary and Outlook

Reactor pressure vessel (RPV) is one of the key physical barriers in the defence-in-depth approach of nuclear power safety and its failure is also the transition point of core melt (corium) progression from in-vessel to ex-vessel phase Quantification of RPV failure during a hypothetical severe accident is therefore paramount to the specific qualification of severe accident mitigation strategies as well as the general assessment of corium risk in light water reactors The RPV failure is a problem of structural performance of RPV under the influence of thermal loads imposed by corium melt pool ie it is a transient thermo-fluid-structure coupled problem per se whose solution requires simultaneous modelling of both highly turbulent melt pool heat transfer and large-scale nonlinear RPV creep Due to the complexity of the problem only limited capabilities have been developed in the previous studies under different simplifications and assumptions in models and coupling methods The present study was motivated to advance modelling and coupling approaches for RPV performance analysis considering new understanding of physical phenomena and latest coupling platform of simulation tools

For the new development of coupling approaches and models for physical phenomena related to RPV failure analysis during severe accidents this thesis work was conducted in the following order (i) a new creep model for the RPV steel 16MND5 was developed which features the capability to cover all three creep stages (ii) a new thermo-fluid-structure coupling approach was developed for RPV failure analysis and validated against FOREVER-EC2 experiment (iii) the performance of the new coupling approach using volume loads mapping (VLM) for data transfer was further compared with another approach using surface loads mapping (SLM) (iv) a hybrid coupling approach was proposed for efficient analysis of RPV failure which employs a lumped-parameter code for quick estimate of thermal load and a mechanistic FE code for detailed structural analysis and (v) for the hybrid coupling approach the code transIVR was developed for calculation of transient melt pool heat transfer

The developed creep model for 16MND5 is called lsquomodified theta projection modelrsquo and describes all three stages of a creep process Creep curves are expressed as a function of time with five model parameters 120579119894(119894 = 14 119886119899119889 119898) A dataset for the model parameters was constructed by fitting experimental creep curves into this function and an iterative parameter adjustment process to satisfy the monotonicity assumption of creep strain vs temperature and stress Model parameters for new curves under different temperature and stress loads are directly interpolated from this dataset in contrast to the conventional ldquotheta projection modelrdquo that employs an extra fitting process The new treatment was proven to better accommodate all experimental curves over the wide ranges of temperature and stress loads The model was implemented into the ANSYS Mechanical code The simulation results indicate that the model successfully captured all three creep stages and a good agreement was achieved between each experimental creep curve and corresponding modelrsquos prediction Regarding the model response to transient changes of temperature and stress loads the widely used time hardening and strain hardening models were implemented and

52

performing reasonably well in the new method These properties fulfil the requirements of a creep model for structural analysis

The transient thermo-fluid-structure coupling approach was developed using the ANSYS Fluent code for thermofluid dynamics of the melt pool and ANSYS Structural code for structural mechanics of RPV An extension tool was introduced to realize transient load transfer from ANSYS Fluent to Structural and minimize the user effort Both turbulence models and the effective model PECM can be used for melt pool heat transfer The developed creep model was used for RPV failure analysis of the RPV Since it is insensitive to geometry the coupling approach allows more complex geometries such as a BWR lower head with a forest of penetrations (CRGTs and IGTs) modelled in detail Moreover the coupling approach does not only reflect the transient response of thermo-fluid-structure interaction but also facilitates the implementation of advanced (high-fidelity) models in both fluid mechanics and structural mechanics The validation of the coupling approach against the FOREVER-EC2 experiment showed that the temperature profile vessel failure time total displacement of vessel bottom point and wall thickness profile in the experiment were well predicted

Following the variation of the coupling approach from VLM to SLM for data transfer between ANSYS Fluent and ANSYS Structural the respective coupling approaches were investigated by comparative calculations against the FOREVER-EC2 experiment and a postulated severe accident of a reference BWR For the FOREVER-EC2 experiment both VLM and SLM approaches predicted generally well the total deformation in vertical direction of bottom point and the wall thickness change though the former predicted a slightly later failure time For the reference BWR case good agreements were achieved in predicting the maximum creep strain and the total deformation in vertical direction of bottom point For both experiment and reactor cases the VLM approach predicted slightly later failure times than SLM but the differences were negligible compared to the long time-scale of vessel failure (~104 s) The deformation rate of the vessel wall was slow at the early stage and accelerated with time till a drastic deformation and final vessel failure Generally speaking the VLM and SLM coupling approaches had the similar performance in terms of their predictability of experiment and applicability to reactor case but the VLM was computationally more efficient

In the development of the hybrid coupling approach the transIVR code was developed for quick estimate of transient melt pool heat transfer and determination of thermal loads on RPV transIVR does not only support transient analysis of one- and two- layer melt pool but also the heat transfer in RPV being resolved with 2D finite difference method which captures more spatial thermal details for RPV structural analysis in addition to 1D heat transfer model The capabilities of transIVR in modelling two-layer pool heat transfer and transient pool heat transfer were demonstrated by calculations against the UCSB FIBS benchmark case and the LIVE-7V experiment respectively It was then coupled to the mechanical solver ANSYS Mechanical which together form the hybrid coupling framework for detailed RPV failure analysis The calculation against the FOREVER-EC2 experiment did not only reproduce the major results of the experiment but also had a good agreement with the thermo-fluid-structure

53

coupling approach using CFD This indicated the new coupling approach is capable of capturing the vessel creep failure characteristics and can be employed as an efficient tool addressing the complex vessel failure problem

As a prospective more research topics can be addressed in future either as a continuation of the current work toward an effective tool for vessel failure analysis or from the general view of scientific interest in melt pool heat transfer and RPV failure The following topics are recommended

bull Extended applications to the reactor cases For instance the thermo-fluid-structure coupling approach can be used to study penetration failures in the RPV of an BWR with a detailed representation of penetrations (IGTs and CRGTs)

bull Synthesis or update of heat transfer correlations for melt pool Heat transfer correlations directly benefit the parametric codes such as transIVR in accurate quantification of heat flux distribution around a melt pool So far many empirical correlations have been proposed based on experiments which had a variety of geometries and melt simulants with a wide range of Rayleigh number (Ra) Sensitivity studies have also been performed on the influences of Ra Pa and other material properties Therefore a comprehensive synthesis or update of the existing heat transfer correlations will be a meaningful step for making reasonable extrapolations to the reactor applications with prototypic materials

bull Modelling of pool stratification The motivation starts with the improvement of the PECMrsquos modelling capability such that the dynamic stratification and resulting pool heat transfer can be modelled altogether Simulation of melt stratification has been supported in some system codes such as ASTEC using a predefined parametric approach The improvement of PECM can be considered in the similar approach or more mechanistic modelling

bull Development of a structural module for RPV The idea is to develop a simplified module for structural analysis so it can be coupled with transIVR ultimately forming a standalone code for quick estimate of vessel failure of LWRs during postulated severe accidents

bull Extension of vessel analysis capability The current version of tranIVR is limited to melt pool Although the precursory debris bed and melt ejection after vessel failure were not the focus of the present study they are equally important to assessment of corium risk especially for Swedish BWRs where the melt release conditions and history are crucial to qualification of their severe accident mitigation strategies

54

55

Bibliography

[1] IAEA ldquoIAEA Annual Report 2018rdquo IAEA 2019 [Online] Available httpswwwiaeaorgsitesdefaultfilesgcgc63-5pdf

[2] IAEA Safety of Nuclear Power Plants Design Vienna INTERNATIONAL ATOMIC ENERGY AGENCY 2012

[3] M Fischer ldquoGen II and III In-Vessel and Ex-Vessel Melt Retentionrdquo The Short Course on Severe Accident Phenomenology Oct 2017

[4] B R Sehgal Nuclear Safety in Light Water Reactors Severe Accident Phenomenology 1st ed Academic Press 2011

[5] IAEA Approaches and Tools for Severe Accident Analysis for Nuclear Power Plants Vienna INTERNATIONAL ATOMIC ENERGY AGENCY 2008

[6] WENRA ldquoPractical Elimination Applied to New NPP Designs - Key Elements and Expectationsrdquo Sep 2019 [Online] Available httpwwwwenraorgmediafiler_public20191111practical_elimination_applied_to_new_npp_designs_-_key_elements_and_expectations_-_for_issuepdf

[7] P Matejovic M Barnak M Bachraty and L Vranka ldquoASTEC applications to VVER-440V213 reactorsrdquo Nuclear Engineering and Design vol 272 pp 245ndash260 2014 doi 101016jnucengdes201308077

[8] AREVA ldquoUS EPR Severe Accident Evaluationrdquo Topical Report ANP-10268NP Oct 2006 [Online] Available httpswwwnrcgovdocsML0631ML063100157pdf

[9] C T Martin ldquoCoupled 3D Thermo-mechanical Analysis of Nordic BWR Lower Head Failure in case of Core Melt Severe Accidentrdquo master thesis KTH Stockholm 2013

[10] IAEA ldquoAdvanced Large Water Cooled Reactors A supplement to the IAEArsquos Advanced Reactor Information System (ARIS)rdquo Sep 2015 [Online] Available httpsarisiaeaorgPublicationsIAEA_WRC_Bookletpdf

[11] B Pershagen Light Water Reactor Safety Pergamon Press 1989 [12] Ninos Garis M Agrell H Glanneskog and L Agrenius ldquoAPRI 7 ndash Accident

Phenomena of Risk Importancerdquo SSM SSM 201212 2012 [Online] Available httpswwwstralsakerhetsmyndighetensecontentassets2a9f299f46b949b3b209e780cf71436a201212-apri-7-accident-phenomena-of-risk-importance-en-lagesrapport-om-forskningen-inom-omradet-svara-haverier-under-aren-2009-2011-

[13] F Fichot et al ldquoSome considerations to improve the methodology to assess In-Vessel Retention strategy for high-power reactorsrdquo Annals of Nuclear Energy vol 119 pp 36ndash45 2018 doi 101016janucene201803040

[14] T G Theofanous C Liu S Additon S Angelini O Kymaumllaumlinen and T Salmassi ldquoIn-vessel coolability and retention of a core meltrdquo Nuclear Engineering and Design vol 169 no 1 pp 1ndash48 1997 doi 101016S0029-5493(97)00009-5

[15] C Sainte Catherine ldquoTensile and creep tests material characterization of pressure vessel steel(16MND5) at high temperatures (20 up to 1300 degC)rdquo CEA France SEMTLISNRT98-009A 1998

56

[16] E Altstadt H-G Willschuetz B R Sehgal and F-P Weiss ldquoModelling of in-vessel retention after relocation of corium into the lower plenumrdquo Forschungszentrum Rossendorf FZR-437 2005

[17] H-G Willschuumltz and E Altstadt ldquoGeneration of a High Temperature Material DataBase and its Application to Creep Tests withFrench or German RPV-steelrdquo FORSCHUNGSZENTRUM ROSSENDORF FZR-353 ISSN 1437-322X Aug 2002

[18] J L Rempe S A Chavez and G L Thinnes ldquoLight water reactor lower head failure analysisrdquo Nuclear Regulatory Commission NUREGCR--5642 1993 Accessed Aug 23 2019 [Online] Available httpinisiaeaorgSearchsearchaspxorig_q=RN25025905

[19] S Brosi et al ldquoCORVIS Investigation of light water reactor lower head failure modesrdquo Nuclear Engineering and Design vol 168 no 1 pp 77ndash104 1997 doi 101016S0029-5493(96)00008-8

[20] T Y Chu M M Pilch J H Bentz J S Ludwigsen W Y Lu and L L Humphries ldquoLower Head Failure Experiments and Analysesrdquo Sandia National Laboratories Albuquerque NM USA NUREGCR-5582 SAND98-2047 1999

[21] L L Humphries et al ldquoOECD Lower Head Failure Project Final Reportrdquo Sandia National Laboratories Albuquerque NM 87185-1139 2002

[22] B R Sehgal R R Nourgaliev T N Dihn A Karbojian J A Green and V A Bui ldquoFOREVER Experiments on Thermal and Mechanical Behavior of a Reactor Pressure Vessel during a Severe Accidentrdquo in Proceedings of the Workshop on in-vessel core debris retention and coolability Garching near Munich Germany Mar 3-6 1998

[23] S Bhandari M Fischer B R Sehgal and A Theerthan ldquoPost Test Analysis of the FOREVER C2 Experimentrdquo The 9th International conference on Nuclear Engineering (ICONE-9) Nice France Apr 8-12 2001

[24] J F Mao J W Zhu S Y Bao L J Luo and Z L Gao ldquoCreep deformation and damage behavior of reactor pressure vessel under core meltdown scenariordquo International Journal of Pressure Vessels and Piping vol 139ndash140 pp 107ndash116 2016 doi 101016jijpvp201603009

[25] J Mao L Hu S Bao L Luo and Z Gao ldquoInvestigation on the RPV structural behaviors caused by various cooling water levels under severe accidentrdquo Engineering Failure Analysis vol 79 pp 274ndash284 2017 doi 101016jengfailanal201704029

[26] J F Mao J W Zhu S Y Bao L J Luo and Z L Gao ldquoStudy on structural failure of RPV with geometric discontinuity under severe accidentrdquo Nuclear Engineering and Design vol 307 pp 354ndash363 2016 doi 101016jnucengdes201607027

[27] B R Sehgal V A Bui T N Dinh and R R Nourgaliev ldquoHeat Transfer Processes in Reactor Vessel Lower Plenum During Late Phase of In-Vessel Core Melt Progressionrdquo in Advances in Nuclear Science and Technology J Lewins M Becker R W Albrecht E J Henley J D McKean K Oshima A Sesonske H B Smets and C P L Zaleski Eds Boston MA Springer US 1999 pp 103ndash135

[28] C-T Tran and T-N Dinh ldquoThe effective convectivity model for simulation of melt pool heat transfer in a light water reactor pressure vessel lower head Part I Physical processes modeling and model implementationrdquo Progress

57

in Nuclear Energy vol 51 no 8 pp 849ndash859 2009 doi 101016jpnucene200906007

[29] C-T Tran and T-N Dinh ldquoThe effective convectivity model for simulation of melt pool heat transfer in a light water reactor pressure vessel lower head Part II Model assessment and applicationrdquo Progress in Nuclear Energy vol 51 no 8 pp 860ndash871 2009 doi 101016jpnucene200906001

[30] H-G Willschuumltz E Altstadt B R Sehgal and F-P Weiss ldquoCoupled thermal structural analysis of LWR vessel creep failure experimentsrdquo Nuclear Engineering and Design vol 208 no 3 pp 265ndash282 2001 doi 101016S0029-5493(01)00364-8

[31] E Altstadt and T Moessner ldquoExtension of the ANSYS creep and damage simulation capabilitiesrdquo Forschungszentrum Rossendorf Technical report FZR-296 2000

[32] H-G Willschuumltz E Altstadt B R Sehgal and F-P Weiss ldquoRecursively coupled thermal and mechanical FEM-analysis of lower plenum creep failure experimentsrdquo Annals of Nuclear Energy vol 33 no 2 pp 126ndash148 2006 doi 101016janucene200508006

[33] H-G Willschuumltz E Altstadt B R Sehgal and F-P Weiss ldquoSimulation of creep tests with French or German RPV-steel and investigation of a RPV-support against failurerdquo Annals of Nuclear Energy vol 30 no 10 pp 1033ndash1063 2003 doi 101016S0306-4549(03)00036-7

[34] H Madokoro A Miassoedov and T Schulenberg ldquoA thermal structural analysis tool for RPV lower head behavior during severe accidents with core meltrdquo Mechanical Engineering Letters vol 4 pp 18-00038-18ndash00038 2018 doi 101299mel18-00038

[35] W Villanueva C-T Tran and P Kudinov ldquoCoupled thermo-mechanical creep analysis for boiling water reactor pressure vessel lower headrdquo Nuclear Engineering and Design vol 249 pp 146ndash153 2012 doi 101016jnucengdes201107048

[36] W Villanueva C T Tran and P Kudinov ldquoA Computational Study on Instrumentation Guide Tube Failure During a Severe Accident in Boiling Water Reactorsrdquo in The 14th International Topical Meeting on Nuclear Reactor Thermal hydraulics NURETH-14 Toronto Ontario Canada Sep25-30 2011

[37] W Villanueva C T Tran and P Kudinov ldquoAssessment with Coupled Thermo-Mechanical Creep Analysis of Combined CRGT and External Vessel Cooling Efficiency for A BWRrdquo The 14th International Topical Meeting on Nuclear Reactor Thermal hydraulics NURETH-14 Toronto Ontario Canada Sep 25-30 2011

[38] F R Larson and J Miller ldquoA Time-Temperature Relationship for Rupture and Creep Stressesrdquo Trans ASME vol 74 p 765minus775 1952

[39] C T Tran ldquoDevelopment validation and application of an effective convectivity model for simulation of melt pool heat transfer in a light water reactor lower headrdquo Licentiate thesis KTH Stockholm 2007

[40] ANSYS ldquoANSYS Fluent Theory Guide Release 162rdquo ANSYS Inc Jul 2015 [Online] Available httpwwwansyscom

[41] P Zeng Finite element analysis and application (in Chinese) Tsinghua University Press amp Springer 2004

58

[42] ANSYS ldquoANSYS Mechanical | Finite Element Analysis Softwarerdquo httpswwwansyscomproductsstructuresansys-mechanical-enterprise (accessed Sep 26 2019)

[43] ANSYS ldquoANSYS Mechanical APDL Theory Reference Release 162rdquo ANSYS Inc Jul 2015 [Online] Available httpwwwansyscom

[44] U Steinberner and H H Reineke ldquoTurbulent Buoyancy Convection Heat Transfer with Internal Heat Sourcesrdquo in Proceedings of the 6th Int Heat Transfer Conference Toronto Canada 1978 vol Vol2 pp 305ndash310 doi 101615IHTC63350

[45] R W Evans and B Wilshire Introduction to Creep London Institute of Materials 1993

[46] DoITPoMS ldquoCreep Deformation of Metalsrdquo Creep Deformation of Metals University of Cambridge 2006 httpswwwdoitpomsacuktlplibcreepindexphp (accessed Jul 09 2019)

[47] ANSYS ANSYS Mechanical APDL Material Reference Release 162 2015 [48] SIMULIA Abaqus 612 Analysis Userrsquos Manualvol 3 Materials Chapter

2324 2012 [49] S Imaoka ldquoUser Creep Subroutine STI0704A ANSYS Release110rdquo 2007 [50] R W Evans J D Parker and B Wilshire ldquoThe θ projection conceptmdashA

model-based approach to design and life extension of engineering plantrdquo International Journal of Pressure Vessels and Piping vol 50 no 1 pp 147ndash160 1992 doi 1010160308-0161(92)90035-E

[51] ECCC ldquoRecommendations and Guidance for the Assessment of Creep Strain and Creep Strength Data ECCC Recommendations Volume 5 Part 1 b[Issue 2]rdquo 2013

[52] W D Day and A P Gordon ldquoLife Fraction Hardening Applied to a Modified Theta Projection Creep Model for a Nickel-Based Super-Alloyrdquo The ASME Turbo Expo 2014 Turbine Technical Conference and Exposition Duumlsseldorf Germany Jun16-20 2014 vol 7A-Structures and Dynamics p V07AT29A010 doi 101115GT2014-25881

[53] M Kumar I V Singh B K Mishra S Ahmad A Venugopal Rao and V Kumar ldquoA Modified Theta Projection Model for Creep Behavior of Metals and Alloysrdquo J of Materi Eng and Perform vol 25 no 9 pp 3985ndash3992 2016 doi 101007s11665-016-2197-y

[54] P Yu and W Ma ldquoA modified theta projection model for creep behavior of RPV steel 16MND5rdquo Journal of Materials Science amp Technology vol 47 pp 231ndash242 2020 doi 101016jjmst202002016

[55] ANSYS ANSYS Mechanical Advanced Nonlinear Materials v160 Lecture 2 Rate Dependent Creep ANSYS training materials 2015

[56] Q Y Li N C Wang and D Y Yi Numerical Analysis (in Chinese) 5th ed Tsinghua University Press 2008

[57] T N Dinh R R Nourgaliev and B R Sehgal ldquoOn heat transfer characteristics of real and simulant melt pool experimentsrdquo Nuclear Engineering and Design vol 169 no 1 pp 151ndash164 1997 doi 101016S0029-5493(96)01283-6

[58] O Kymaumllaumlinen H Tuomisto O Hongisto and T G Theofanous ldquoHeat flux distribution from a volumetrically heated pool with high Rayleigh numberrdquo

59

Nuclear Engineering and Design vol 149 no 1ndash3 pp 401ndash408 1994 doi 1010160029-5493(94)90305-0

[59] F J Asfia and V K Dhir ldquoNatural convection heat transfer in volumetrically heated spherical poolsrdquo 1994 Accessed Aug 23 2019 [Online] Available httpinisiaeaorgSearchsearchaspxorig_q=RN26037727

[60] M Jahn and H H Reineke ldquoFree Convection Heat Transfer with Internal Heat Sources Calculations and Measurementsrdquo in Proceedings of the 5th Int Heat Transfer Conference Tokyo Japan 1974 vol Vol3 paper NC-28

[61] M Gustafsson ldquoFSI Transient extension V1704rdquo EDR MEDESO 2016 [62] EDR MEDESO httpsedrmedesocom [63] H Kim and I Namgung ldquoThermo-mechanical Analysis of RVLH for

APR1400 during a Severe Accidentrdquo in Transactions SMiRT-23 Manchester United Kingdom Aug10-14 2015 [Online] Available httpwwwlibncsueduresolver18402034197

[64] P Yu and W Ma ldquoComparative Analysis of Reactor Pressure Vessel Failure using Two Thermo-Fluid-Structure Coupling Approachesrdquo submitted to journal

[65] F R Menter ldquoTwo-equation eddy-viscosity turbulence models for engineering applicationsrdquo AIAA Journal vol 32 no 8 pp 1598ndash1605 1994 doi 102514312149

[66] T N Dinh and R R Nourgaliev ldquoTurbulence modelling for large volumetrically heated liquid poolsrdquo Nuclear Engineering and Design vol 169 no 1 pp 131ndash150 1997 doi 101016S0029-5493(96)01281-2

[67] ANSYS Inc ldquoIntroduction to ANSYS Fluent Lecture 7 Turbulence Modelling Fluent Tutorial Materialrdquo 2016

[68] A Horvat and M Borut ldquoNumerical investigation of natural convection heat transfer in volumetrically heated spherical segmentsrdquo in Proceedings of the International Conference Nuclear Energy for New Europe 2004

[69] Y Li CFD Pre-test Analysis of SIMECO-2 Experiment 2016 [70] A Theerthan A Karbojian and B R Sehgal ldquoEC-FOREVER experiments

on thermal and mechanical behavior of a reactor pressure vessel during a severe accident Technical report-1 EC-FOREVER-2 testrdquo SAM-ARVI-D008 Mar 2001

[71] P Yu W Ma W Villanueva A Karbojian and S Bechta ldquoValidation of a thermo-fluid-structure coupling approach for RPV creep failure analysis against FOREVER-EC2 experimentrdquo Annals of Nuclear Energy vol 133 pp 637ndash648 2019 doi 101016janucene201906067

[72] T G Theofanous C Liu S Additon S Angelini O Kymaumllaumlinen and T Salmassi ldquoIn-vessel coolability and retention of a core meltrdquo Nuclear Engineering and Design vol 169 no 1ndash3 pp 1ndash48 1997 doi 101016S0029-5493(97)00009-5

[73] H P Langtangen ldquoSolving nonlinear ODE and PDE problemsrdquo Simula Research Laboratory 2016 [Online] Available httpshplgitgithubionum-methods-for-PDEsdocpubnonlinpdfnonlin-4printpdf

[74] P Yu and W Ma ldquoDevelopment of a Lumped-Parameter Code for Efficient Assessment of In-Vessel Melt Retention Strategy of LWRsrdquo submitted to journal

60

[75] T G Theofanous M Maguire S Angelini and T Salmassi ldquoThe first results from the ACOPO experimentrdquo Nuclear Engineering and Design vol 169 no 1 pp 49ndash57 1997 doi 101016S0029-5493(97)00023-X

[76] S W Churchill and H H S Chu ldquoCorrelating equations for laminar and turbulent free convection from a vertical platerdquo International Journal of Heat and Mass Transfer vol 18 no 11 pp 1323ndash1329 1975 doi 1010160017-9310(75)90243-4

[77] S Globe and D Dropkin ldquoNatural-Convection Heat Transfer in Liquids Confined by Two Horizontal Plates and Heated from Belowrdquo J Heat Transfer vol 81 no 1 pp 24ndash28 1959 doi 10111514008124

[78] T G Theofanous C Liu S Additon S Angelini O Kymaelaeinen and T Salmassi ldquoIn-vessel coolability and retention of a core melt Volume 2rdquo Argonne National Lab DOEID--10460-VOL2 1996 Accessed Apr 18 2019 [Online] Available httpinisiaeaorgSearchsearchaspxorig_q=RN30000459

[79] J L Rempe D L Knudson C M Allison G L Thinnes and C L Atwood ldquoPotential for AP600 in-vessel retention through ex-vessel floodingrdquo United States INEELEXT--97-00779 1997 Accessed Apr 18 2019 [Online] Available httpinisiaeaorgSearchsearchaspxorig_q=RN30025872

[80] A Miassoedov X Gaus-Liu T Cron and B Fluhrer ldquoLive experiments on melt pool heat transfer in the reactor pressure vessel lower headrdquo The 10th International Conference on Heat Transfer Fluid Mechanics and Thermodynamics (HEFAT 2014) Orlando FL Jul 14-16 2014

[81] H Madokoro A Miassoedov and T Schulenberg ldquoAssessment of a lower head molten pool analysis module using LIVE experimentrdquo Nuclear Engineering and Design vol 330 pp 51ndash58 2018 doi 101016jnucengdes201801036

[82] B Fluhrer et al ldquoThe LIVE-L1 and LIVE-L3 experiments on melt behaviour in RPV lower headrdquo Forschungszentrum Karlsruhe GmbH Technik und Umwelt (Germany) Inst fuer Kern- und Energietechnik FZKA--7419 2008 Accessed Nov 12 2019 [Online] Available httpinisiaeaorgSearchsearchaspxorig_q=RN40036867

VIII

IX

Table of Contents

AbstracthelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphellipI

SammanfattninghelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphellipIII

List of PublicationshelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphellipV

AcknowledgementshelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphellipVII

NomenclaturehelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphellipXI

1 Introductionhelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphellip1

11 Nuclear Power Safety 1

12 RPV Failure 2

13 Literature Review Creep Behaviour and RPV Failure Analysis 4

131 Experiments on creep behaviour and RPV failure 4

132 Numerical studies on RPV failure 5

14 Motivation Objectives and Main Achievements 7

2 Numerical Solvers and Modelshelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphellip9

21 ANSYS Fluent 9

22 ANSYS Mechanical 9

23 PECM 11

3 Creep Modelling of 16MND5 Steelhelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphellip13

31 Creep and Creep Modelling 13

32 A Modified Theta Projection Model 14

321 Original theta projection model and its variations 14

322 A modified theta projection model 16

4 Coupled Thermo-Fluid-Structure Analysis of RPV Failurehelliphelliphelliphellip23

41 Thermo-Fluid-Structure Coupling Approach 23

42 Numerical Modelling 25

421 Conjugate heat transfer 25

422 Mechanical analysis 25

43 Validation against the FOREVER-EC2 Experiment 26

431 FOREVER-EC2 experiment 26

X

432 Conjugate heat transfer 27

433 Mechanical analysis 29

44 Application to BWR Case 32

45 Computational Efficiency 34

5 Hybrid Coupling Approach for Efficient RPV Failure Analysishelliphellip35

51 A Lumped-parameter Code for Melt Pool Heat Transfer 35

511 Heat transfer correlations 36

512 Energy balance in the pools 37

513 Energy balance in the RPV wall 39

52 Hybrid Coupling Approach 40

53 Verification and Validation 41

531 UCSB FIBS benchmark case 41

532 LIVE-7V experiment 43

533 FOREVER-EC2 experiment 45

6 Summary and Outlookhelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphellip51

Bibliographyhelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphellip55

XI

Nomenclature

Latin

119860 area of a surface

119862119901 heat capacity

119863 damage factor

119892 gravity

ℎ heat transfer coefficient

H Melt pool height

119896 thermal conductivity

119871 characteristic length

119873119906 Nusselt number

119875119903 Prandtl number

119902 heat flux

119902119894119899 heat flux through inner vessel surface

119876119881 internal volumetric heat source

119903 radial distance in a cylindrical or spherical coordinate

119877119886 external Rayleigh number

119877119886119894119899 internal Rayleigh number

119877119886119910 local Rayleigh number

119905 time

119879 temperature

119881119900119897 total volume

119909 thickness (of crust or RPV wall)

119911 height in a cylindrical coordinate

Greek

120572 thermal diffusivity

120573 thermal expansion

Δ119879 temperature difference

ε emissivity (in thermal analysis) strain (in structural analysis)

휀119888119903 creep strain

휀119903 creep strain rate

120579 polar angle

120579119894 parameters of a creep model 119894 = 1234119898

120584 kinematic viscosity

σ stress

XII

Super and subscripts

119886119898119887 ambient

119888119903 creep (in Section 3) crust (in Section 5)

119889119899 downward

119890119899119907 environment

119894 inner surface of RPV

119894119899119905 interface

119897119900119888119886119897 local

119898119886119909 maximum value

119898119890119897119905119894119899119892 melting point

119898119905 metal layer (bulk)

119898119905119889119899 downward surface of metal pool

119898119905119904119889 sideward surface of metal pool

119898119905119905119900119901 top surface of metal layer

119898119905119906119901 upward surface of metal pool

119900 outer surface of RPV

119900119909 oxide layer (bulk)

119900119909119889119899 downward surface of oxide layer

119900119909119906119901 upward surface of oxide layer

119903119890119891 reference

119904 internal structures inside the RPV

119904119889 sideward

119906119901 upward

119907119890119904 vessel

Acronyms

BWR boiling water reactor

CBD creep database

CFD computational fluid dynamics

CRGT control rod guide tube

ECCM effective convectivity conductivity model

FDM finite difference method

FE finite element

FEA finite element analysis

FEM finite element method

FVM finite volume method

IGT instrument guide tube

IVR in-vessel retention

XIII

IVMR in-vessel melt retention EU project

LES large eddy simulation

MPS moving particle semi-implicit method

NPP nuclear power plant

PECM phase-change effective convectivity model

RANS Reynolds-averaged Navier-Stokes

RPV reactor pressure vessel

SBO station blackout

SLM surface loads mapping

SPH smoothed-particle hydrodynamics

UDF user-defined function

UPF user programmable features

VLM volume loads mapping

XIV

1

1 Introduction

11 Nuclear Power Safety

Greenhouse gas emissions contribute to the global warming problem Phasing out fossil fuels by switching to low-carbon energy sources is considered as a mitigation strategy With this regard clean energy should be increased in the world energy consumption structure Being clean and stable nuclear energy is an attractive alternative to fossil fuels After the development over decades it also proves to be a mature and safe technology According to IAEA 2018 annual report [1] there were 450 nuclear power reactors under operation worldwide at the end of 2018 which had a record generating capacity of 3964 GW(e) 9 reactors were connected to the grid and 55 reactors under construction around the world

Due to the potential risk of the release of radioactivity nuclear power safety is emphasized throughout the stages of design construction operation and decommission of a nuclear power plant (NPP) The principal technical requirements [2] specify that the three fundamental safety functions should be fulfilled for all plant states of an NPP (i) control of reactivity (ii) removal of heat from the reactor and from the fuel store and (iii) confinement of radioactive material shielding against radiation and control of planned radioactive releases as well as limitation of accidental radioactive releases The defence-in-depth concept is also applied to provide several levels of defence to prevent consequences of accidents The fuel claddings reactor pressure vessel (RPV) and primary coolant pressure boundaries and the containment are the three physical barriers to prevent the release of radioactive materials to the environment

Severe accidents are events that cause significant damage to reactor fuel and result in core meltdown It may be caused by insufficient cooling of the reactor core due to various reasons or by the rapid introduction of huge reactivity which causes sharp increases in the power and fuel fragmentation eg Chernobyl accident During a severe accident the failure of the fuel claddings will result in the radioactive materials previously enclosed inside the cladding released to the primary loop If cooling is not sufficient to remove the decay heat the core materials will melt migrate down and relocate into the RPV lower head after which the melt will react with water and partially form fragments quench solidify and form a debris bed The debris bed formed by the melt may continue to melt to form a melt pool due to the decay heat As the corium contacts the RPV it would continuously heat the RPV and threaten its integrity Once the RPV fails containment becomes the last barrier to prevent the release of radioactivity The released melt would result in an increase of the pressure and temperature due to the high temperature fractions oxidation and hydrogen combustion which is called direct containment heating The melt would fall onto the reactor cavity Without sufficient cooling the molten core would interact with basemat concrete (MCCI) This process can last for few days It can result in containment failure due to either containment overpressure or the basemat melt-through of containment In addition containment is also subjected to isolation failure and early containment failure due to missiles and dynamic pressure [3] More

2

detailed severe accident progression or phenomena could also be found in eg [4] [5]

Though severe accidents are unlikely to occur the consequences can be severe if no proper measure is taken Related researches have been continuously going on aiming at mitigating the accident consequences and improving safety After the Fukushima Accident safety inspections and re-evaluation actions have been carried out worldwide especially on safety measurements regarding severe accidents The concept lsquoPractical Eliminationrsquo applied to new NPP designs demands that new nuclear installations be designed with the objective of mitigating the consequences of accidents and avoiding early radioactive releases and large radioactive releases [6] Severe accident mitigation strategies are applied to the existing and new NPPs (both Generation II and III) The strategies can generally be classified into two categories [3] the in-vessel retention (IVR) and ex-vessel retention (EVR) In simple words in-vessel retention tries to retain the core melt inside the RPV (second barrier) by applying external water cooling on the vessel as shown in Figure 1(a) Such method minimizes the dispersal of the melt within the containment and hence simplifies the later recovery work Examples of the application are AP1000 HPR1000 APR 1400 The idea of the EVR is to let the RPV fail but to retain the core melt inside the containment Two principle approaches of EVR are the retention within a dedicated lsquocore catcherrsquo and melt fragmentation Examples of core catcher would be EPR that directs the melt to a spreading compartment (Figure 1(b)) and VVER1000 that uses metallic crucible below the RPV An example of the melt fragmentation is the Nordic BWRs that the corium released from RPV would relocate in the cavity with a deep-water pool (7-12 m) and is expected to fragment quench and form a coolable debris bed (Figure 1(c))

Figure 1 Examples of IVR [7] EVR with core catcher [8] and melt fragmentation [9]

12 RPV Failure

RPV which contains the reactor core core shroud and coolant is a key component to the reactor system RPV and the primary loop constitute the primary coolant system which conveys the fission heat from the reactor core to the secondary loop through steam generator (PWR) or to the turbine (BWR) The pressure under

(a) IVR (b) Core catcher (c) Melt fragmentation

3

operation can be ~155 MPa for a 1000 MW PWR [10] and ~70 MPa for a 1000 MW BWR [11] It also has to withstand the neutron radiation damage The thickness of RPV [11] can be ~20 cm for PWR and ~16 cm for BWR

As the transition of in-vessel progression to ex-vessel progression RPV failure is important for analysis of accident progression and assessment of the effectiveness of severe accident mitigation strategies In the melt fragmentation strategy of Nordic BWR the melt-coolant interaction and debris bed coolability after the corium poured into the deep-water pool are essentially dependent on the vessel failure mode including the characteristic time of melt release rupture size and location and amount and superheat of melt available for release [12] New concern also arose [13] on the necessity of performing structural analysis for the RPV in the IVR strategy In the initial methodology and further development of the IVR the success of the strategy ie the integrity of the RPV was demonstrated (eg in [14]) with the following two conclusions (i) even for a local heat flux on the vessel as high as critical heat flux (CHF) the resulted very thin external ldquocold layerrdquo of the vessel would be sufficient to withstand the internal stress load and hold the vessel integrity (ii) as CHF is the critical value for the thermal load the main specific goal for assessing the effectiveness of IVR becomes to demonstrate that the local heat flux along RPV be lower than the CHF The new concern doubts that local lsquohot partrsquo of the vessel would have negative effects on the overall resistance due to creep and dilatation which could result in a vessel failure and hence violate the first conclusion in particular for ldquolong termrdquo retention of corium in RPV If so a structural integrity analysis of RPV would be needed in addition to assessing local heat flux especially for high-power reactors Therefore accurate prediction of vessel failure could be meaningful for both strategies

Figure 2 Illustration of RPV under thermal loads from corium

As the structural performance is highly affected by the thermal loads from the corium RPV failure is essentially a transient thermo-fluid-structure coupled problem Figure 2 illustrates the situation that a corium debris bedmelt pool formed in the RPV lower head The decay heat of nuclear fission products could continuously be generated inside the corium leading to the formation of melt pools as the debris melts The heated debris bedmelt pool imposes thermal and

4

mechanical loads on the RPV Such loads can be transient (change with time) and spatially non-uniform Complex phenomena are also involved in this process Strong turbulent natural convection induced by the decay heat may be expected in the melt pool under IVR The corresponding Rayleigh number 119877119886 can be 1016minus17 for prototypic reactor cases Material structural properties can become non-linear under high temperature and stress loads The creep deformation also becomes significant at high temperature Detailed modelling of turbulent natural convection and non-linear properties could be computationally expensive In addition large-scale geometry (hemisphere with a radius of 20-30 m) long-term transient deformation process (~10000 s) and required small time steps (eg 001 s) for numerical convergence consideration further increase the computational costs

13 Literature Review Creep Behaviour and RPV Failure Analysis

In this section the most relevant studies concerning the experiments modelling and analysis of creep behaviour and RPV failure are presented

131 Experiments on creep behaviour and RPV failure

Various experiments have been performed on the RPV failure problem some of which were to generate data of mechanical properties (especially the creep properties) and some to study the overall vessel failure with scaled or simplified geometries

The REVISA experiment [15] generated creep curves and stress strain curves of French steel 16MND5 ranging from 600 to 1300 degC This data is accessible in [16] as well as [17] which also documents other material properties eg density thermal expansion coefficient conductivity heat capacity Youngrsquos Modulus and plasticity Similar experiments were carried out to generate high-temperature creep and tensile test data for American RPV steels and Inconel [18]

The CORVIS experiments [19] investigated the lower head failure modes with and without penetrations Each experiment was carried out on a steel plate 100 mm thick representing a pressure vessel lower head The penetration case represented a boiling water reactor lower head carrying a tube penetration in the form of a drain line

The LHF (lower head failure) experiments [20] and the succeeding OLHF tests [21] investigated the vessel failure with a 15 scaled vessel heated by an electrical radiator allowing to locate the hot focus at different position of the vessel with a maximum wall temperature of around 900 degC Internal pressure ranging from 53 to 123 bar was also imposed to initiate the vessel failure

The FOREVER experiments [22] investigated the phenomena of melt pool convection vessel creep possible failure processes and failure modes occurred during the late phase of the in-vessel progression by heating up a melt pool in a vessel and observing possible vessel failures The study covers cases with and without penetration different melt level existence of gap cooling and in-vessel melt coolability

5

132 Numerical studies on RPV failure

Various numerical approaches have been employed on RPV failure Since a two-way thermo-fluid-structure coupling approach is challenging these approaches take simplifications and can generally be classified into two categories (i) RPV failure decoupled from melt pool heat transfer and (ii) RPV failure coupled with melt pool heat transfer in simplified manners

1) Decoupled approaches In the decoupled approaches only RPV is modelled Simplified constant thermal loads are imposed on RPV considering the melt pool heat transfer

In a post-test analysis of FOREVE-C2 experiment [23] it was treated as a thermo-structural coupled analysis of the vessel with the thermal load estimated by the outer thermocouples and the normal heat flux on the external vessel wall The analysis was done using finite element (FE) code SYSTUS A coupled creepdamage law based on a viscosity-hardening creep law describing primary and secondary creep was employed for creep analysis

Mao et al conducted various studies on reactor cases assessing the IVR strategy using FE code ABAQUS Thermal boundary conditions were imposed on the RPV which was then solved as a thermal-structural coupled problem The strain hardening creep model (Eq (1)) was mainly used Their studies cover the general RPV failure [24] as well as the effects of cooling water levels under IVR [25] and geometric discontinuity (ablated RPV) [26]

ε119888119903 = 1198891σ1198892ε1198893119890119909119901 [minus

1198894119879] (1)

In the EU IVMR project [13] the RPV integrity was also studied through a benchmark calculation which deals with the RPV of a PWR with a prescribed ablated profile with 2D axisymmetric geometry Prescribed temperature gradient was set as thermal load The objective was to calculate deformations for various pressure loads up to vessel rupture A smoother profile was planned in further research

The advantage of such approaches is once thermal loads are reasonably assumed the focus only needs to be put on the RPV The disadvantage is a reasonable thermal load should be consistent with the melt pool heat transfer while sometimes it is not well-noted eg constant uniform temperature boundary conditions were assumed for inner and outer RPV (with uniform thickness in lower head) surfaces [25] which ignored the non-uniform heat flux distribution caused by the turbulent convective corium pool As thermal loads strongly affect the structural performance they should be quantified carefully

2) Coupled approaches In coupled approaches both melt pool heat transfer and RPV failure are modelled and coupled in one-way or simplified manner With melt pool heat transfer modelled the coupled approaches are more flexible to provide reasonable thermal loads on the RPV In existing approaches CFD and the correlation-based distributed-parameter models eg the Effective Convectivity Conductivity Model

6

(ECCM) [27] and the Phase-change Effective Convectivity Model (PECM) [28] [29] are general methods for melt pool heat transfer

Sehgal et al [22] presented an approach that employed the MVITA code to calculate the thermal processes and ANSYS for structural analysis MVITA code employed the ECCM Bailey-Norton creep law was used as shown in Eq (2)

ε119888119903 = 6055 times 10minus26 sdot σ472119905085119890minus

37200

119879 (2)

This approach was used for the pre-test simulation of FOREVER experiment to determine the thermal and mechanical loadings Parametric investigations also helped in the identification of important factors that could affect the test procedure and test conditions

Willschutz et al [30] employed the CFD-module Flotran for the melt pool heat transfer and ANSYS for structural analysis of RPV Standard k-ε turbulence model was selected for the turbulent natural convection in the melt pool A user-defined three-stage creep model [31] was employed In this model experimental creep curves were discretized into a creep database (CDB) under given temperature 119879 and stress 120590 the creep curve was discretized into finite point pairs

(휀 ε) indicating under such thermal and mechanical loads the creep strain rate ε is subjected to creep strain 휀 The creep strain rate for any other case can be interpolated over known temperature stress and strain from the database This method was applied to the post-test calculation of the FOREVE-C2 experiment It was concluded that the unexpected deformation behaviour during the experiment was caused by the combination of the transient temperature field in the vessel wall and the three creep stages

They later replaced the Flotran solver with the ECCM for melt pool heat transfer in a following study [32] By implementing it into FEM-solver a recursively coupled analysis was constructed Moreover a modified creep model [33] was used with damage measures the CDB itself did not contain tertiary creep instead a factor 1(1 minus 119863) was multiplied to the original creep strain rate to get the modified creep strain rate where 119863 is a damage factor (0 lt 119863 lt 1) defined with creep strain information and plastic strain information As 119863 grows close to 1 the modified creep strain rate would be enlarged significantly by this factor 1(1 minus 119863) which then behaves like the accelerating tertiary stage This method was applied to pre- and post- test calculations for the FOREVER experiments proving to be closer to reality than the previous method

Madokoro et al [34] developed a thermal structural analysis tool based on open source solver OpenFOAM and numerically investigated the coupled RPV failure and melt pool convection The PECM was used in the melt pool heat transfer In structural part the plasticity and damage parameters were similar to Willschutz et al [32] except for that strain hardening creep model (Eq (3)) was used instead of a CDB Simulations of FOREVER-EC2 and -EC4 were investigated It was concluded from the simulations that although the emissivity and mesh type (updated or fixed) affect the results significantly the approach generally has the capability of predicting the vessel deformation and failure

7

ε119888119903 = 1198891 sdot σ1198892 sdot ε1198893 (3)

Villanueva et al [35] performed a coupled thermo-mechanical creep analysis of the reactor scale RPV (ABB-Atom design BWR) to evaluate efficacy of the cooling measures for the in-vessel melt coolability and retention The PECM was used for melt pool heat transfer and ANSYS for RPV integrity analysis The modified time hardening model (Eq (4)) of the American RPV steel SA533B was selected as creep model The transient thermal load from the melt pool was applied as a thermal boundary condition to the vessel wall Further studies were done with this approach on investigations of instrumentation guide tube (IGT) failure [36] and control rod guide tube (CRGT) cooling [37] of a Nordic BWR

ε119888119903 =1198881σ

11988821199051198883+1

1198883 + 1 1198881 gt 0 (4)

It is noted that only RPV analysis with finite element analysis (FEA) are reviewed here Engineering methods like the Larson-Miller method [38] are not covered in this thesis One can refer to theory manuals of severe accident codes like MELCOR ASTEC for more details Comprehensive description on modelling of melt pool heat transfer is not covered either One can refer to either [39] for a general view or to any severe accident code manuals for engineering methods

14 Motivation Objectives and Main Achievements

The existing experimental and numerical studies provide us a good understanding of RPV failure involved phenomena and valuable simulation experiences These numerical methods may vary in solvers (which may be related to code accessibility) and models (which may be related to model accessibility understanding of phenomena computational power) However improvements of modelling approaches are still possible as the improvements in any of the related aspects eg

bull Simulation tools are continuously being upgraded and providing new possibilities just like that PECM appeared and ECCM was no longer used

bull Better understanding of the phenomena and more simulation experiences from in-vessel retention and vessel failures researches benefit the further modelling and simulations

bull Increasing computational power allows more detailed modelling eg turbulence modelling of melt pool convection

bull There is still room for the improvement of creep modelling

In addition implementation difficulty of a coupling approach is another important issue to consider Since it is a TRANSIENT thermo-fluid-structure coupled problem it is almost impossible to replicate an approach simply by referring to a paper with limited details For BWRs it would be even better for the coupling approaches to support complex geometries considering forest of penetrations (CRGTs and IGTs) on the RPV lower head without significant increase of user effort in the coupling process

8

The objectives of this thesis are to investigate new approaches and develop new models for the thermo-fluid-structure coupled problem of RPV failure and involved phenomena These objectives are accomplished with the following achievements

1) A new creep model for the 16MND5 steel was developed which covers all three creep stages Good agreement was achieved between model predictions and experimental curves Both time and strain hardening models were implemented (Paper I)

2) A new thermo-fluid-structure coupling approach was developed for RPV failure analysis It supports the advanced models in both melt pool convection and structural mechanics to improve fidelity and facilitates implementation Validation of the approach with the new creep model was performed against FOREVER-EC2 experiment This approach is flexible for complex geometries and can be used in further analysis eg lower head with detailed modelling of penetrations in BWRs (Paper II)

3) The performance of this new approach which uses volume loads mapping for data transfer was compared with the previous approach using surface loads mapping showing good agreement with small discrepancy (Paper III)

4) A lumped-parameter code transIVR for melt pool heat transfer was developed which supports one- and two-layer pool configuration transient analysis and well-resolved RPV wall (Paper IV)

5) A hybrid coupling approach was developed for efficient analysis of RPV failure where transIVR is coupled with ANSYS Mechanical (Paper IV)

In the following general descriptions on the codes ANSYS Fluent Mechanical and the PECM are given in Section 2 the new creep model in Section 3 the new coupling approach as well as its comparison with the previous approach in Section 4 the efficient hybrid coupling approach in Section 5 and summary amp outlook in Section 6

9

2 Numerical Solvers and Models

In this section we briefly introduce the ANSYS Fluent as the CFD solver for melt pool heat transfer the ANSYS Mechanical as the mechanical solver for RPV failure and PECM as an efficient alternative for melt pool heat transfer that are heavily used in the following sections

21 ANSYS Fluent

In the fluid mechanics the fluid is treated as a continuum which is generally described by the conservative mass momentum and energy equations [40]

120597120588

120597119905+ 120571 sdot (120588) = 119878119898 (5)

120597(120588)

120597119905+ 120571 sdot (120588) = minus120571119901 + 120571 sdot (120591) + 120588 + (6)

120597120588119864

120597119905+ 120571 sdot ((120588119864 + 119901)) = 120571 sdot (119896120571119879 minussumℎ119895119869119895

119895

+ (120591 sdot )) + 119878ℎ

(7)

Computational Fluid Dynamics (CFD) studies topics on the numerical solutions of above equations Typical methods are the Finite Volume Method (FVM) Finite Difference Method (FDM) and Finite Element Method (FEM) as well as particle methods that are mesh-free eg SPH and MPS

ANSYS Fluent is a general purpose CFD code which solves above equations with FVM Turbulence modelling and multiphase flows are supported Turbulence can be modelled with RANS LES or hybrid LESRANS Customizations eg adding sourcesink time or spatial dependent boundary conditions are realized through the user-defined functions (UDF) which is programmed in C language

22 ANSYS Mechanical

In the continuum mechanics of solids three sets of variables are considered to describe the structure performance namely element displacement field strain field and stress field (119906 ε σ) A general 3-dimensional problem is described by 3 sets of equations the equilibrium equations (Eq (8)) strain-displacement relationship (Eq (9)) and constitute equations (Eq (10)) [41] For Eq (8) the reciprocal theorem of shear stress also gives τ119894119895 = τ119895119894 where 119894 119895 = 119909 119910 119911

indicating the stress tensor being symmetric For very simple applications eg 1D beam it can be solved analytically For complex problems where analytical solutions are not available approximation methods eg Galerkin weighted residual method principle of virtual work and principle of minimum potential energy would be used

10

partσ119909119909part119909

+partτ119909119910

part119910+partτ119909119911part119911

+ 119909 = 0

partτ119910119909

part119909+partσ119910119910

part119910+partτ119910119911

part119911+ 119910 = 0

partτ119911119909part119909

+partτ119911119910

part119910+partσ119911119911part119911

+ 119911 = 0

(8)

ε119909119909 =part119906

part119909 휀119910119910 =

120597119907

120597119910 휀119911119911 =

120597119908

120597119911

120574119909119910 =120597119906

120597119910+120597119907

120597119909 120574119910119911 =

120597119908

120597119910+120597119907

120597119911 120574119911119909 =

120597119908

120597119909+120597119906

120597119911

(9)

ε119909119909 =1

119864[σ119909119909 minus μ(σ119910119910 + σ119911119911)]

ε119910119910 =1

119864[σ119910119910 minus μ(σ119909119909 + σ119911119911)]

ε119911119911 =1

119864[σ119911119911 minus μ(σ119909119909 + σ119910119910)]

γ119909119910 =1

119866τ119909119910 γ119910119911 =

1

119866τ119910119911 γ119911119909 =

1

119866τ119911119909

(10)

ANSYS Mechanical [42] is a mechanical engineering software that uses finite element analysis (FEA) for structural analysis The principle of virtual work is adopted in the solver [43] The idea of this principle is that a virtual (very small) change of the internal strain energy must be offset by an identical change in external work due to the applied loads which reads

δ119880 = δ119881 (11) where 119880 is the strain energy or the internal work 119881 the external work and δ the virtual operator The virtual internal work can be expressed as

δ119880 = int δεσd(vol)T

vol

+int 119886119903119890119886119891

δ119908119899119879σ119889(119886119903119890119886119891) (12)

where 휀 is the strain vector 120590 stress vector vol volume of element 119908119899 motion normal to the surface 119886119903119890119886119891 area of the distributed resistance The virtual

external work is contributed from three parts the inertial effects the pressure force on the surface and nodal forces applied to the element which then can be expressed as

δ119881 = minusint 119907119900119897

δ119908119879119865119886

119907119900119897119889(119907119900119897) + int

119886119903119890119886119901

δ119908119899119879119875119889(119886119903119890119886119901)

+ δ119906119879119865119890119899119889

(13)

11

where 119908 is the vector of displacements 119865119886 acceleration force vector 119875 applied pressure vector 119886119903119890119886119901 area over which pressure acts and 119865119890

119899119889 nodal

forces applied to the element

Plugging Eq (12) and (13) into (11) would result in the equilibrium equation Since the virtual displacement can be arbitrary the final equations used to describe the equilibrium of an element is reduced to Eq (14) with multiple operations

([119870119890] + [119870119890119891])119906 minus 119865119890

119905ℎ = [119872119890] + 119865119890119901119903 + 119865119890

119899119889

(14)

where [119870119890] [119870119890119891] 119865119890

119905ℎ [119872119890] and 119865119890119901119903 are the element stiffness matrix

foundation stiffness matrix thermal load vector mass matrix acceleration vector and pressure vector respectively It represents the equilibrium equation on a one element basis Alternatively using a minimum total potential energy principle could result in a consistent expression to the principle of virtual work [41]

Customizations eg on material properties are realized through the user programmable features (UPF) which is programmed in FORTRAN language

23 PECM

The PECM [28] [29] developed in KTH is a correlation-based method to model the melt pool heat transfer It is implemented in ANSYS Fluent through UDF and has been validated against many experiments such as ACOPO COPO LIVE etc In the PECM the flow velocities 119906119909 119906119910 and 119906119911 are replaced by characteristic

velocities 119880119909 119880119910 and 119880119911 respectively in order to eliminate the necessity of solving

the Navier-Stokes equation

The characteristic velocities are calculated using heat transfer correlations

119880119906119901 =α

ℎ119901119900119900119897times (119873119906119906119901 minus

ℎ119901119900119900119897

ℎ119906119901) (15)

119880119889119899 =α

ℎ119901119900119900119897times (119873119906119889119899 minus

ℎ119901119900119900119897

ℎ119889119899) (16)

119880119904119889 =α

ℎ119901119900119900119897times (119873119906119904119889 minus

ℎ119901119900119900119897

119882119901119900119900119897) (17)

where ℎ119901119900119900119897 is the melt pool depth ℎ119906119901 the thickness of well mixed layer of the

pool ℎ119889119899 the thickness of lower stratified region of melt pool 119882119901119900119900119897 the width of

pool 120572 the thermal diffusivity The PECM employs reduced characteristic velocities to describe mushy zone convection heat transfer For internally heated pool natural convection the Steinberner-Reineke correlations (Eq (18)~(19)) [44] are employed The profile of sideward 119873119906 is described using Eckertrsquos type correlation (Eq (20)) for a vertical boundary layer where the local Rayleigh

number at pool height 119910 is defined as 119877119886119910 =119892βΔ1198791199103

αν

12

119873119906119906119901 = 03451198771198861198941198990233

(18)

119873119906119889119899 = 13891198771198861198941198990095

(19)

119873119906119904119889 = 050811987511990314 (

20

21+ 119875119903)

minus14

11987711988611991014

(20)

As the characteristic velocities are calculated the energy equation is the only transport equation that needs to be solved and can be expressed as follows

partρ119862119901119879

part119905+ nabla(ρ119862119901119932119879) = 119896nabla2119879 + 119876119907 (21)

where from left to right are the transient term convection term diffusion term and volumetric heat source respectively The convection term then can be calculated explicitly with use of values in the last time step and then can be treated together with the volumetric source term Therefore the PECM finally simplifies the energy equation to a conduction equation which can be easily solved and as a result will significantly increase the computational efficiency in simulations

In addition solidificationmelting process can also be modelled and the temperature-dependent physical properties are supported

13

3 Creep Modelling of 16MND5 Steel

Creep deformation becomes significant when a component is subjected to high temperature and stress Modelling of creep behaviour of the RPV is paramount since it is considered as a main mechanism leading to the RPV failure In this section we present the development of a three-stage creep model (called lsquomodified theta projection modelrsquo ) for the French RPV steel 16MND5 This model was successfully implemented into the ANSYS Mechanical code

31 Creep and Creep Modelling

Creep is an irreversible time-dependent inelastic deformation process Diffusion and dislocation are the two molecular mechanisms of creep The former normally occurs due to the presence of vacancies within the crystal lattice (or the grain) while the latter occurs if there is a slip by the passage of crystal dislocations [45] There are two types of diffusion creep [46] Coble creep (favoured at lower temperatures) if the diffusion paths are predominantly through the grain boundaries and Nabarro-Herring creep (favoured at higher temperatures) if through the grains themselves Dislocation creep tends to dominate at high stresses and relatively low temperatures Dislocations can move by gliding in a slip plane which is a process that requires little thermal activation But the rate-determining step for their motion is often a climb process which also requires diffusion and is thus a time-dependent process and favoured by higher temperatures Creep behaviour is highly affected by temperature and stress generally creep is more significant under higher temperature and stress

Creep curves describe the creep strain development with time Figure 3 shows a creep curve under constant temperature and stress which typically experiences three creep stages primary stage where the creep strain rate ε119888119903 (the change of creep strain ε119888119903 in unit time) decreases with time secondary stage where the creep strain rate almost keeps constant and tertiary stage where the creep strain rate drastically increases with time

Figure 3 Creep curve under constant temperature and stress

14

The built-in creep models in commercial software eg ANSYS [47] and ABAQUS [48] in default only cover the primary andor the secondary stage(s) Table 1 [47] gives the default creep models supported in ANSYS for example Possible reason why the tertiary stage is not modelled might be that for most industrial applications the failure analysis is beyond interest and therefore the built-in creep laws in the commercial software was supposed to be sufficient [49] Though the two codes offer the interfaces through either ANSYS UPF or ABAQUS user subroutines to support more general user-defined creep laws eg complex models that also include the tertiary creep stage

Table 1 Default creep models in ANSYS Mechanical

Creep model Equations Type

Strain hardening

ε119888119903 = 1198621σ1198622ε119888119903

11986231198901198624119879 1198621 gt 0 Primary

Time hardening ε119888119903 = 1198621σ11986221199051198623119890minus1198624119879 1198621 gt 0 Primary

Generalized exponential

ε119888119903 = 1198621σ1198622119903119890minus119903119905 r = 1198625σ

11986231199051198623119890minus1198624119879 1198621 gt 0

1198625 gt 0

Primary

Generalized Graham

ε119888119903 = 1198621σ1198622(1199051198623 + 1198624119905

1198625 + 11986261199051198627)119890minus1198628119879 1198621 gt 0 Primary

Generalized Blackburn

ε119888119903 = 119891(1 minus 119890minus119903119905) + 119892119905 119891 = 1198621119890

1198622σ 119903 = 1198623(σ1198624)1198625 119892 = 1198626119890

1198627σ

1198621 gt 0

1198623 gt 0

1198626 gt 0

Primary

Modified time hardening

ε119888119903 = 1198881σ11988821199051198883+1(1198883 + 1) 1198621 gt 0 Primary

Modified strain hardening

ε119888119903 = 1198621σ1198622[(1198623 + 1)ε119888119903]

1198623119890minus1198624119879 1198621 gt 0 Primary

Generalized Garofalo

ε119888119903 = 1198621[119904119894119899ℎ(1198622σ)]1198623119890minus1198624119879 1198621 gt 0 Secondary

Exponential form

ε119888119903 = 1198621119890σ1198622119890minus1198623119879 1198621 gt 0 Secondary

Norton ε119888119903 = 1198621σ1198622119890minus1198623119879 1198621 gt 0 Secondary

Combined time hardening

ε119888119903 = 1198621σ11986221199051198623+1119890minus1198624119879(1198883 + 1)

+ 1198625σ1198626119905119890minus1198627119879

1198621 gt 0

1198625 gt 0

Primary+ Secondary

Rational polynomial ε119888119903 = 1198621

partε119888part119905 ε119888 =

119888119901119905

1 + 119901119905+ ε119898119905

ε119898 = 1198622101198623σσ1198624 119888 = 1198627ε

1198628σ1198629

119901 = 11986210ε11986211σ11986212

1198622 gt 0 Primary+ Secondary

Generalized time hardening

ε119888119903 = 119891119905119903119890minus1198626119879 119891 = 1198621σ + 1198622σ2 + 1198623σ

3 119903 = 1198624 + 1198625

Primary

32 A Modified Theta Projection Model

321 Original theta projection model and its variations

The original theta projection model [50] was first proposed to accurate predict long-term creep and creep-fracture properties It assumes that a creep curve under constant temperature and stress can be described as

15

휀119888119903 = 1205791(1 minus 119890minus1205792119905) + 1205793(119890

1205794119905 minus 1) (22)

where 휀119888119903 is the creep strain 119905 the time and 1205791~1205794 the model parameters Figure 4 illustrates how the first term and second term on right hand side develop with time and constitute a creep curve which has obvious primary secondary and tertiary stages Model parameters 120579119894(119894 = 1234) are temperature and stress dependent and are described with the following expression

119897119899thinsp120579119894 = 119886119894 + 119887119894120590 + 119888119894119879 + 119889119894120590119879 (23)

where 119886119894 119887119894 119888119894 and 119889119894 ( 119894 = 1234 ) are constants It is noted [45] that this equation is suitable when the temperature range of experiment data sets is very narrow such that minus1119879 is almost proportional to 119879 These constants (119886119894 119887119894 119888119894 and 119889119894 (119894 = 1234)) are the final model parameters which are generated with two fitting procedures

(i) Fitting the experimental creep curves to Eq (22) generates several groups of 1205791~1205794 for each creep curve under certain temperature and stress We denote (119879119895 120590119895 1205791119895 1205792119895 1205793119895 1205794119895 ) as parameters for the 119895119905ℎ

experimental curve under temperature 119879119895 and stress 120590119895

(ii) Fitting 1205791 to Eq (23) as a function of temperature and stress with the obtained data ( 119879119895 120590119895 1205791119895 ) from ( 119879119895 120590119895 1205791119895 1205792119895 1205793119895 1205794119895 ) generates

parameters 1198861 1198871 1198881 and 1198891 Repeat this fitting procedure for 1205792 1205793 and 1205794 to get parameters 119886119894 119887119894 119888119894 and 119889119894 (119894 = 234)

Variations of the model were developed and applied to other different materials to better accommodate the experimental data which are all then called a lsquomodified theta projection methodrsquo independently Examples are models in [51] for lowhigh alloy ferritic austenitic steels Al-alloys Al-matrix composites [52] for Nickel-based Super-alloy [53] for pure metals and class M alloys with intermediate range of stress (40-70 of the yield stress) and temperature (40-60 of the melting temperature)

Figure 4 Illustration of theta projection method [54]

16

322 A modified theta projection model

The new modified theta projection model we proposed [54] was mainly adapted from the original theta projection method [50] and the modified version [51] The new model describes the creep curve with the following equation

휀119888119903 = 1205791(1 minus 119890minus1205792119905) + 120579119898119905 + 1205793(119890

1205794119905 minus 1) (24)

An additional term 120579119898119905 is seen compared with Eq (22) in the original method The reason for adding this term is that the 16MND5 steel shows fairly long secondary stages in many experimental creep curves and the creep curves can be better fitted by Eq (24) than Eq (22) As the structure undergoes different creep process under different temperature and stress the model parameters θ119894 (119894 =1~4 and 119898) are also temperature and stress dependent

The main difference compared with the original and all other modified projection models is that numerical interpolation is used to replace the second numerical fitting in previous models In other words instead of fitting the model parameters as functions of stress and temperature with Eq (23) the distributions of the model parameters are assumed to be piecewise in the temperature-stress space and they will be interpolated directly from known parameters obtained from experiments The consideration is that the experimental data ([16] [17]) for 16MND5 covers a wide temperature range varying from 600 degC to 1300 degC and a wide stress load range varying from 08 MPa to 248 MPa while Eq (23) in the existing methods is based on a narrow temperature span Poor parameter fitting results would be expected if we directly fit the parameters 1205791~1205794 with Eq (23) as used in the original methods

The creep model was developed using the following steps

(i) Fitting Each creep curve can be represented by a group of model parameters [1205791 1205792 120579119898 1205793 1205794] with respect to Eq (24) by fitting the creep curves to Eq (24) All parameters of the 32 experimental curves form an

initial dataset of the model parameters [119879119895 120590119895 1205791119895 1205792119895 120579119898119895 1205793119895 1205794119895] where j (0ltjle32) is the jth creep curve

(ii) Interpolation Parameters of a creep curve under any [119879 σ] are directly interpolated from the dataset

(iii) Parameter adjustment The dataset was further updated to satisfy the monotonicity assumption we have taken that creep strain monotonically increases with temperature and stress

(iv) Different hardening models for dynamic loads

Step (i) was done using MATLAB In the following steps (ii)-(iv) are described in detail

Model parameters for new creep curves are interpolated from the dataset over the 2D 119879 minus 120590 space The interpolation is divided into two sub-interpolation steps interpolate over stress with fixed temperature and then interpolate over temperature with fixed stress We take the interpolation of model parameters under (119879prime σprime) for example In the first interpolation we first find out the two

17

temperatures 119879119896119899119900119908119899119904119898119886119897119897 and 119879119896119899119900119908119899119897119886119903119892119890 most close to 119879prime from the experimental

temperature set 600 700 800900 1000 1100 1200 1300 Under each temperature there are four sets of model parameters in the dataset corresponding to four different stress loads Then the first interpolation is done over stress load from the known dataset of model parameters under each

temperature to output the parameters for (119879119896119899119900119908119899119904119898119886119897119897 120590rsquo) and (119879119896119899119900119908119899119897119886119903119892119890 120590rsquo)

In the second interpolation the model outputs parameters for (119879rsquo 120590rsquo) from

(119879119896119899119900119908119899119904119898119886119897119897 120590rsquo) and (119879119896119899119900119908119899119897119886119903119892119890 120590rsquo) For 120579i(119894 = 12 3 4) we assume a logarithm

relation between 120579119894 temperature 119879 and stress 120590 which is similar to Eq (23)

119897119899120579119894 = (119886 + 119887120590)(119888 + 119889119879) (25)

It reduces to 120579119894 = 11986011198901198871σ with fixed temperature in the first interpolation and to

θ119894 = 11986021198901198872T with fixed stress in the second interpolation The interpolation

expression for θ119894 = 11986011198901198871σwith two known points (1205901 1205791198941) and (1205902 1205791198942) is given

by Eq (26) The interpolation expression for θ119894 = 11986021198901198872T with two known points

(1198791 1205791198941) and (1198792 1205791198942) is given by Eq (27)

120579119894 = 120579i1

120590minus12059021205901minus1205902120579i2

1205901minus1205901205901minus1205902 (26)

120579119894 = 120579i1

119879minus11987921198791minus1198792120579i2

1198791minus1198791198791minus1198792 (27)

Similarly we assume the following relation between 120579119898 temperature 119879 and stress 120590

120579119898 = 119860120590119899119890minus119887 119879frasl (28)

The interpolation is done over the stress with 120579119898 = 1198603120590n by fixing temperature

and then over temperature with 120579119898 = 1198604119890minus1198874119879 by fixing stress The interpolation

expression over stress for 120579119898 = 1198603120590119899 with two known points (1205901 1205791198981) and

(1205902 1205791198982) would be given by Eq (29) The interpolation expression over

temperature for 120579119898 = 1198604119890minus1198874119879 with two known points (1198791 1205791198981) and (1198792 1205791198982)

would be given by Eq (30)

120579119898 = 119890119909119901(119897119899120590 minus 11989711989912059011198971198991205902 minus 1198971198991205901

1198971198991205791198982 +1198971198991205902 minus 119897119899120590

1198971198991205902 minus 11989711989912059011198971198991205791198981) (29)

120579119898 = 120579m2(11198791minus1119879) (

11198791minus11198792)fraslsdot 120579m1

(1119879minus11198792) (

11198791minus11198792)frasl (30)

In this model we also take the assumption that the creep strain monotonically increases with temperature and stress loads1 Or specifically with the same creep time the structural component under higher temperature or stress experiences

1 Intuitively materials deform more under higher stress and temperature This is also mostly observed in tensile experiments Though some exceptions exist one instance of which is the effect of phase change some materials under certain temperature around the phase change temperature may show much weaker structural performance Here we just simply assumed that the creep strain monotonically increases with temperature and stress

18

higher creep strain than that with lower temperature or stress (the creep curve under higher temperature or stress should be above the curve with lower temperature or stress) When generating the parameter dataset (Step (i)) we already kept the trend that under the same temperature cases under higher stress loads would have larger creep model parameters and give higher creep strains This trend would also hold for the interpolated temperatures which thus guarantees that creep strain monotonically increases with stress load under same temperature Then the question becomes if creep strain also monotonically increases with temperature given fixed stress However the answer could not always be a yes especially in the extrapolation range of the stress load Figure 5(a) illustrates an example of the interpolation of model parameters under (119879119872 σ119872) The red dots and blue dots are known parameters from the parameter dataset for corresponding high temperature 119879119867 and low temperature 119879119871 respectively The green dots (119879119867 σ119872) and (119879119871 σ119872) are interpolated from the red and blue dots in the first sub-interpolation respectively The black dot (119879119872 σ119872) is interpolated from the green dots in the second sub-interpolation It is seen that when σ119872 lt σ119862 the value of (119879119871 σ119872) is higher than that of (119879119867 σ119872) It means the higher 119879119872 is the smaller the model parameter will be ie the model would predict larger creep strain for lower temperatures which violates our monotonicity assumption One simple solution used here as shown in Figure 5(b) is to minorly adjust the values of the four red and blue dots such that the intersection point C can be sufficiently small and the situation σ119872 lt σ119862 would barely occur in the extrapolation range

Figure 5 Illustration of model parameter interpolation (a) before and (b) after parameter adjustment [54]

A corresponding parameter adjustment process was introduced to verify that the model predicts higher creep strain under higher temperature under fixed stress Figure 6 illustrates the general verification process for any (119879 120590) in the interested load range the creep strain of (119879 + 120575119879 120590) should be larger than that of (119879 120590) Otherwise parameters would be minorly adjusted and the verification process be repeated This process would be iterated several times until the requirement is reached The achieved verification shows

Stress

σC

High temperature TH

MPa

a

0

Model

par

amet

er θ

Low temperature TL

MPa

a

0

Model

par

amet

er θ

High temperature TH

Low temperature TL

σC

Stress

C C

(a) (b)

σM

σM

19

(i) For any (119879 120590) in the temperature range 273~ 1073 K with an interval of 1 K and the stress range 1 ~260 MPa with an interval of 1 MPa its creep strain is higher than that of (119879 minus 1119870 120590) for the first 12000000 s

(ii) For any (119879 120590) in the temperature range 1073~1673 K with an interval of 1 K and the stress range 1 ~80 MPa with an interval of 1MPa its creep strain is higher than that of (119879 minus 1119870 120590) for the first 12000000 s

(iii) For any (119879 120590) in the stress range 01~26 MPa with a smaller interval of 01 MPa for the full temperature range 273~1673 K with an interval of 1 K its creep strain is higher than that of (119879 minus 1 120590) for the first 12000000 s

Above verification covers a wide range of temperature and stress especially where the stress and temperature are low and extrapolation is used Further verification to cover even higher stress or temperature ranges is not considered as creep increases drastically under such high stress and temperature and the results would become meaningless With these regards we generally think the monotonicity requirement is reached The model parameter dataset was updated along with the adjustment The final dataset was given as Table 2 in Paper I

Figure 6 Parameter adjustment process [54]

Figure 7(a-d) present the model predictions of selected creep curves in ANSYS Mechanical (labelled as lsquoANSYSrsquo) compared with experimental curves (labelled as lsquoREVISArsquo) at temperature 600 800 1000 and 1200 degC respectively The predicted curves proved to clearly show the three-stage creep process and agreed well with the experimental creep curves

Initial parametersdataset

θ1 θ2 θm θ3 θ4 from creep

curve-fitting

Parameters interpolation

Is the monotonicity

assumption satisfied

IsMon= =1

Final model parameters

Adjust the

parametersdataset

θ1 θ2 θm θ3 θ4

No

Yes

IsMon=1 a flag is monotonic achieved

For σ in the stress range in MPa

For T in range Tstart Tend in K

For t in range 112000000 in s

If ε(σ T+1t)lt ε(σTt) IsMon=0

Iteration loop of

parameter adjustment

20

Figure 7 Selected model predictions compared with experimental creep curves [54]

Instead of being constant temperature and stress loads also dynamically change with time in real cases Hardening models determine how creep strain rate reacts to the change of loads during a creep process The time hardening model and strain hardening model2 are supported in our model implementation in ANSYS Mechanical which are two mostly used hardening models

Time hardening assumes that the following creep strain rate after a load change depends only on the current creep time from the beginning of the creep process (and also temperature and stress if used in the creep model) and ignores the current creep strain Strain hardening assumes that the creep strain rate depends only upon the current creep strain (and also temperature and stress if used in the creep model) [55]

2 The time hardening and strain hardening here are more general terms than the specific time hardening creep model and the strain hardening creep model mentioned in Table 1

(a) 600 degC (b) 800 degC

(c) 1000 degC (d) 1200 degC

21

Eq (24) in default is a time hardening model as creep strain is directly described by the time 119905 Eq (31) gives the creep strain increment 119889휀119888119903 at time 119905 during time interval 119889119905

119889휀119888119903 = (12057911205792119890minus1205792119905 + 120579119898 + 12057931205794119890

1205794119905)119889119905 (31)

The strain hardening model is derived from Eq (24) in an implicit form Specifically when temperature and stress load change to (119879 120590) at the time 119905 and current creep strain 휀1198881199030 the model will get the creep curve of constant (119879 120590) and

output how this curve will react under the current strain 휀1198881199030 Since the creep curve shows creep strain monotonically increases with time given the creep strain 휀1198881199030 under creep curve of (119879 120590) we can uniquely find this reference time

119905119903119890119891 satisfying Eq (32) One should notice that the reference time 119905119903119890119891 can be

different from the current physical time 119905

휀1198881199030 = 1205791(1 minus 119890minus1205792119905119903119890119891) + 120579119898119905119903119890119891 + 1205793(119890

1205794119905119903119890119891 minus 1) (32)

where all 120579119894(119894 = 1~4119898) are model parameters of the new creep curve (119879 120590) It can be solved for 119905119903119890119891 with methods eg Newton-Ramphson [56] Then the creep

strain increment at 119905119903119890119891 during time interval 119889119905 is given as

119889휀119888119903 = (12057911205792119890minus1205792119905119903119890119891 + 120579119898 + 12057931205794119890

1205794119905119903119890119891)119889119905 (33)

Therefore combining Eq (32) and (33) gives the creep strain rate at creep strain 휀1198881199030 In summary when loads are changed to (119879 120590) with current creep strain 휀1198881199030

the model traces the creep curve of (119879 120590) and calculates the creep strain rate of this curve at strain 휀1198881199030

One calculation was done to demonstrate the performance of the hardening models The stress load was set to 151 MPa and kept unchanged A step temperature load was applied as the orange dash curve shows in Figure 8 it was 900 degC during 0-36000 s and suddenly jumped to 1000 degC at 36000 s which was then kept till the end of the simulation The blue solid curve (the part before the temperature change was identical to and therefore covered by the green solid curve in the figure) and blue dash curve are the creep curves under constant temperature T=900 degC and T=1000 degC respectively The green solid curve and green dash curve represent the model predictions considering this temperature change with time hardening model and strain hardening model respectively Before this change the creep curves with both hardening models were identical to the creep curve of the T=900 degC till point A After that the creep strain rate then shifted from Point A to Point B (which had the same time with A) in the time hardening model and behaved the same as the curve T=1000 degC as shown in the rest part of the green solid curve In the strain hardening model the creep strain rate shifted from Point A to Point C (which had the same creep strain with A) and behaved the same as the curve T=1000 degC as shown in the rest part of the green dash curve The large difference between two models was partially caused by the drastic temperature jump In real cases the difference would be less significant as the temperature changes more mildly Overall this prediction is exactly the expected behaviors of the time hardening model and strain hardening model

22

Figure 8 Time hardening and strain hardening in ANSYS Mechanical [54]

23

4 Coupled Thermo-Fluid-Structure Analysis of RPV Failure

As described in Section 13 RPV failure is coupled with melt pool heat transfer leading it into a transient thermo-fluid-structure coupled problem The goal of this section is to develop and validate a reliable coupling approach which supports transient coupled thermo-fluid-structure analysis as well as advanced models and computational platforms so as to improve fidelity and facilitate implementation The approach was validated against the FOREVER-EC2 experiment and then applied to an RPV failure analysis of a Nordic BWR As the approach adopts the volume loads mapping for load transfer comparative analysis was also performed against the coupling approach using surface loads mapping Future work is considered in analysis of penetration failures (IGTs and CRGTs) of a BWR RPV since the approach is flexible to complex geometries Results of Paper II ampIII are jointly presented in this section

41 Thermo-Fluid-Structure Coupling Approach

Considering the involved phenomena a coupling approach should at least address three issues (i) melt pool heat transfer (ii) RPV deformation and (iii) the interaction between melt pool heat transfer and vessel deformation A two-way strong coupling considering both the transient thermal load from melt pool on vessel and the influence of vessel deformation on melt pool heat transfer involves complex data exchange and is still full of challenges Therefore the new coupling approach is realized in a one-way manner with the assumptions that the influence of vessel deformation on melt pool heat transfer can be ignored and thus only the thermal loads from melt pool to vessel are considered It is based on the considerations that on one hand the deformation is a slow process regarding the relatively long failure time and on the other hand it was pointed out [57] that the physics of natural convection heat transfer in internally heated liquid pools is not sensitive to the geometrical factors based on their calculation results and experimental data from [58] [59] as well as previous experiments in semicircular and square cavities eg [44] [60] In addition one-way coupling is also computationally more efficient especially when heat transfer turbulence structural non-linearity and large-scale geometries are involved The developed approach is based on the ANSYS platform which generally consists of three parts with respect to the three issues respectively

(i) conjugate heat transfer of the melt pool and vessel using ANSYS Fluent (ii) transient thermal loads transfer from ANSYS Fluent to ANSYS

Structural (the ANSYS Mechanical in ANSYS Workbench) (iii) structural analysis of the vessel in ANSYS Structural

ANSYS Structural in default only supports direct transfer of constantsteady-state spatial loads from ANSYS Fluent Transfer of transient loads is realized with the extension tool FSI Transient [61] It is a pre-processing tool (with good documentation) developed by a third-party CAE vendor EDRTM MEDESO [62] to map transient CFD results (heat transfer coefficient temperature and surface pressure) to thermal or structural analysis including the definition of load steps

24

This new procedure proves to be straightforward and reduces efforts in the load transfer

Since structural performance of RPV is highly dependent on temperature the transfer of transient temperature loads from conjugate heat transfer to mechanical solver is a core step of the coupling framework There are two general approaches to realize the transfer as shown in Figure 9 Both approaches contain three parts a conjugate heat transfer analysis for fluid and structure where flow and heat transfer are coupled a mechanical analysis for the structure and a load mapping process that transfers the transient loads The conjugate heat transfer part can be solved with any CFD solver and is treated in the same manner in both approaches In the first approach (Figure 9(a)) transient volume thermal loads are transferred and mapped into the mechanical simulation domain The structure domain reads the essential thermal loads from the mapping after which a structure analysis of this domain is performed In the second approach (Figure 9(b)) transient surface thermal loads are mapped onto the boundary surfaces of structure domain as thermal boundary conditions Then a thermo-structure coupled analysis of the structure domain in the mechanical part is performed To distinguish them we call the former approach volume loads mapping (VLM) and the latter surface loads mapping (SLM) In the RPV failure analysis the fluid and structure domains correspond to the melt pool and RPV respectively Examples of these methods are the studies of [63] and [30] using VLM and [35] using SLM The VLM is adopted in current approach

Figure 9 Frameworks of the thermo-fluid-structure coupling approaches (a) volume loads mapping (b) surface loads mapping [64]

fluid structure structure

fluid structure structure

Conjugate heat transfer problem Mechanical problem

(structure)

Conjugate heat transfer problem Mechanical problem

(thermo-structure coupled)

Transient volume

loads mapping

(VLM)

Transient surface

loads mapping

(SLM)

(a)

(b)

25

42 Numerical Modelling

421 Conjugate heat transfer

In the conjugate heat transfer analysis main focus is put on the modelling of melt pool heat transfer Within the ANSYS Fluent solver two options are supported CFD with turbulence modelling and PECM CFD is the choice when the model goes to detail and more mechanistic analysis are needed while PECM provides an alternative that is faster and possibly more suitable for large-scale problems

Within the scope of RANS models the SST (Shear Stress Transport) model [65] is selected as the turbulence model in present study This model is a combination of k-ε model and k-ω model The k-ω formulation is used in the inner parts of the boundary layer which makes the model directly usable all the way down to the wall through the viscous sub-layer It also switches to k-ε model in the free-stream and thereby avoids the common k-ω problem that the model is too sensitive to the inlet free-stream turbulence properties The considerations on choosing this model are

bull Several two-equation low-Re k-ε models have been proved in [66] not good enough for melt pool heat transfer Therefore these mentioned two-equation models as well as other one-equation models are not considered

bull SST is highly recommended [67] for simulating most industrial heat transfer problems

bull Convincing results ([68] [69]) using SST have been obtained in capturing the thermal behaviour of the natural convection in an internally heated fluid Especially good agreements were achieved between the predicted heat flux distribution along the vessel and experimental data which would be one of the most important factors to estimate model performance in this problem

The description of PECM as an effective model for melt pool heat transfer is given in Section 23

422 Mechanical analysis

The material for the RPV failure analysis considered in this study is the French RPV steel 16MND5 The three-stage modified theta projection model in Section 3 is used as the creep model Since the definition of an RPV failure criterion is not universal (it can be the time when damage variable reaching 1 or either specified value of creep strain creep strain rate or total deformation is reached) we do not explicitly define a failure criterion Instead the simulation is kept running until the creep converge criterion cannot be satisfied even with the defined minimum time step (indicating a fairly large creep strain rate) and the simulation will be automatically terminated The time the simulation stops is taken as the failure time This treatment implicitly implies a criterion of a large creep strain rate We refer to [17] for other material properties Youngrsquos modulus and thermal conductivity are temperature dependent Multilinear Isotropic Hardening (MISO) is used as the plasticity model

26

43 Validation against the FOREVER-EC2 Experiment

In this section the validation against the FOREVER-EC2 experiment is presented (indicated as lsquoVLMrsquo in the following) In addition the simulation results using the SLM approach (indicated as lsquoSLMrsquo in the following) is also included for comparison purpose

431 FOREVER-EC2 experiment

The FOREVER-EC2 experiment [70] is one of the FOREVER experiments that achieved a vessel failure Figure 10 shows the experimental setup A 110 scale pressure vessel with a wall thickness of 15 mm and an outer diameter of ~400 mm was employed The lower head was made of 16MND5 steel Since it was difficult to use prototypical corium materials in the experiment the binary salt 30 CaO-70 B2O3 was employed as simulant To generate the internal heat which corresponds to the nuclear decay heat a specially designed heater was used The power input was maintained ~38 kW during the experiment such that maximum melt pool temperature would be ~1300 degC The pressure inside the vessel was maintained at ~25 MPa Thermocouples were used to measure the melt pool and vessel wall temperatures and linear displacement transducers used to measure the vessel wall displacements The designed experimental conditions were reached at experimental time 11670 s Vessel failure was observed about 12510 s later (at experiment time 24180 s)

Figure 10 Scheme of the experimental setup [22]

27

Figure 11 Computational domains and boundary conditions (a) conjugate heat transfer (b) mechanical analysis [71]

Computational domains and boundary conditions are indicated in Figure 11 SST turbulence modelling was used for pool heat transfer The simulations started at the moment when experimental conditions were reached (experiment time 11670 s) So simulation time 0 s corresponds to experiment time 11670 s

432 Conjugate heat transfer

Figure 12 shows the temperature fields at simulation time 0 s 1225 s 2475 s 4975 s and 9975 s All the time points in this part refer to simulation time The field at time 0 s (Figure 12(a)) was the initial temperature that was reconstructed based on the thermocouple readings The temperature increased with time and reached a steady state when the maximum temperature reached ~1607 K Figure 12(b-f) show clear thermal stratification which is an expected phenomenon of natural convection

Internal pressure

25 bar

Internal pressure 25e6-2500times981timesy Pa

External pressure

1 bar

Zero displacement

y=0 m

Gravity 400

mm

R=203 mm

15 m

m

Tamb

=323 K radiation with ε = 08

+ convection with h=10Wm2K

Tamb

=800 K radiation with ε=02

(a) (b)

Tamb

=800 K radiation with ε=08

+ convection with h=10 Wm2K

28

Figure 12 Temperature fields at simulation time (a) 0 s (b)1225 s (c) 2475s (d) 4975 s (e) 7475 s and (f) 9975 s [64]

Figure 13 shows the temperature profiles of the external vessel along the polar angle in comparison with experimental data Regions with a polar angle larger than 90deg are those in the cylindrical part of the vessel It also shows the temperature profile increased with time until reaching a steady state The profile almost did not change after 4975 s which indicates that a steady state of the temperature was reached As time increased the temperature in the lower head region 0~90deg (where the vessel was contacted with the melt) evolved from a relatively flat distribution to a distribution that temperature at high angle was higher than that at low angle by more than 200 K Generally the steady state results agreed well with the experimental data except for the low angle part of vessel lower head and the high angle part of the cylindrical vessel In addition to modelling errors the discrepancy in the low angle part of the lower head may be caused by the crust and gap possibly formed at vessel bottom in the experiment which can reduce the downward heat transfer and result in lower temperature while they cannot be modelled in simulations As to the high angle cylindrical vessel part simplified boundary conditions were applied to the inner and outer vessel wall since it was far from the melt and not of interest which may result in some discrepancy

(a) (b)

(c) (d) (e) (f)

29

Figure 13 Temperature profiles of external vessel surface along the vessel polar angle [64]

433 Mechanical analysis

In the mechanical analysis the VLM predicted a failure time of 23959 s and the SLM predicted a failure time of 22445 s both are close to the experimental failure time 24180 s All the time points in this part refer to the experiment time Figure 14 shows the creep strain fields at failure time Both approaches predicted maximum creep strains at same location where the temperature was high (cf Figure 12) This is related to the fact that creep behavior is highly influenced by temperature Maximum values for both cases were also close to each other 0393 in VLM and 0366 in SLM

Figure 15 presents the total deformation of the vessel bottom point (also called ldquosouth polerdquo in some literature) in the vertical direction Since there was an initial displacement reading of ~355 mm at time 0 s which almost did not change during 0~11670 s (the whole period before the experimental conditions were reached) an initial value of 355 mm was added to the simulation result such that experimental and simulation results had the same total displacement at t=11670 s Similar trend was seen in the VLM SLM and experimental data that the deformation increased with time and experienced a drastic acceleration at the end Generally both simulation approaches agreed well with the experimental data though they both slightly under predicted the value of total deformation The discrepancy may possibly be related to the accuracy of the thermal predictions modelling errors of plasticity and creep and simplified boundary conditions

400

600

800

1000

1200

1400

1600

0 20 40 60 80 100 120 140 160

Tem

per

ature

(K

)

Polar angle (deg)

Experiment

25 s

1225 s

2475 s

4975 s

7475 s

9975 s

Weilding line

30

Figure 14 Creep strain fields at failure time (a) VLM (b) SLM [64]

Figure 15 Total deformation in vertical direction of the vessel bottom point [64] [71]

The wall thickness changes are plotted together with experimental results in Figure 16 The wall thickness change δ119909 is defined as

δ119909 = 119909119890119899119889 minus 119909119900119903119894119892119894119899119886119897 (34)

000

500

1000

1500

2000

2500

11670 13670 15670 17670 19670 21670 23670

Tota

l def

orm

atio

n (

mm

)

Time (s)

ExperimentVLMSLM

(a) (b)

31

where 119909119890119899119889 and 119909119900119903119894119892119894119899119886119897 are the wall thickness at failure time and the original wall

thickness respectively Since 119909119890119899119889 cannot directly be post-processed it was estimated in the following way (cf the subfigure of Figure 16) for the same polar angle θ find the corresponding points Pin (1199091 1199101) and Pout (1199092 1199102) on the inner and outer surfaces of the deformed vessel respectively the distance of Pin and

Pout radic(1199091 minus 1199092)2 + (1199101 minus 1199102)2 is calculated as 119909119890119899119889 at polar angle 120579 Such estimation of the wall thickness is more accurate when the line PinPout is closer to be orthogonal to the deformed local surface Considering the final deformed vessel as shown in Figure 14 the estimation should be fine at regions close to 0deg and 90deg and can be slightly overestimated at medium angle region The VLM curve was compared with the SLM curve and 4 experimental curves measured at vessel left right front and back sides Generally VLM curve agreed well with the experimental and SLM curves wall thickness gradually decreases with angle till around 75deg and then increases till 90deg the maximum thickness change occurred at around 75deg would be around -5 mm The positive values at low angle may be caused by the thermal expansion It also shows that the thickness transitions in experimental curves were smoother than those in both simulation curves while simulation curves showed narrower regions with large thickness changes (at angle 70deg~80deg) Possible reasons might be (i) sharper temperature profile predicted in simulation (ii) input uncertainty and (iii) modelling errors The SLM predicted slightly thinner vessel than the VLM except for the necking region The discrepancy might be caused by various reasons eg the interior vessel temperature can be slightly different between two approaches which would then result in different thermal stress and different deformation the final deformed vessel shapes cannot be exactly the same which also results in different errors in estimating PinPout

Figure 16 Wall thickness changes along the vessel polar angle [64] [71]

-6

-5

-4

-3

-2

-1

0

1

0 10 20 30 40 50 60 70 80 90

Wal

l th

icknes

s ch

ange

(mm

)

Polar angle (deg)

VLMSLMExperiment (left)Experiment (right)Experiment (front)Experiment (back)

32

44 Application to BWR Case

In this section the coupling approach (indicated as lsquoVLMrsquo in the following) was applied to an RPV failure analysis of a BWR A simulation using the SLM coupling approach (indicated as lsquoSLMrsquo in the following) was also performed for comparison purpose

A hypothetical severe accident of a Nordic BWR initiated by an SBO was considered The core materials would melt and migrate downward to the RPV lower head The total amount of corium relocated in the lower head was considered to form a 19 m corium melt pooldebris bed assuming the corium distribution was homogeneous

Figure 17 Computational domains and boundary conditions (a) conjugate heat transfer (b) mechanical analysis [64]

Figure 17(a) shows the computational domain of the conjugate heat transfer PECM was used in this calculation A general mesh with a pool height of 23 m was used to accommodate most of the relocation situations with the same mesh Since the corium only had a height of 19 m the top 04 m region was treated as injected water covering the corium The top water was expected to be boiling and we assumed a uniform temperature of 383 K Under the implementation of PECM this region was damped with artificially high heat capacity and high conductivity such that it would efficiently remove heat from the corium and suppress its own temperature increase For the corium part below water a volumetric heat source 11 MWm3 was implemented as the nuclear decay heat For the in-vessel surfaces a constant temperature of 383K was applied as they are covered by water which can be boiling Without water cooling on the external vessel wall a very small heat loss of 20 Wm2 was applied Melting and solidification of the corium were also considered with a solidus temperature of 2750 K and a liquidus temperature of 2770 K The initial temperature was set as

q=20 Wm2

T=383 K

Qv=11

MWm3

315 bar

315 bar+ hydrostatic

pressure of corium

Gravity

248 bar

Zero

displacement

Water

Corium

(a) (b)

33

1100 K for the corium 383 K for the water and 450 K for the vessel For the mechanical analysis (Figure 17(b)) an inner pressure of 315 bar and an outer pressure of 248 bar were considered For the region covered by corium an extra hydrostatic pressure was added on the vessel wall A 2 m long vessel extension was considered as well as the standard earth gravity

Figure 18 presents the wall temperature profile along the inner vessel surface at 50 s 2500 s 7500 s and 10000 s The temperature increased with time for the vessel surface contacted with corium and the distribution was flat which was due to that large part of the corium was still in solidus status It also shows that for the surface region that contacted the water the temperature was basically unchanged which indicates that the damping treatment indeed suppressed the temperature increment and guaranteed the heat removal upward by water

Figure 18 Temperature profiles of inner vessel wall along the vessel polar angle [64]

The predicted RPV failure time was 9869 s by VLM and 9650 s by SLM Compared with the FOREVER-EC2 case in Section 43 the VLM predicted a slightly later failure time than the SLM in this BWR case too Figure 19(a) compares the maximum creep strain developments with time of both approaches Before 9000 s the predicted values by both approaches are close to each other Afterward the creep acceleration occurred and the strain increased drastically The acceleration of the VLM occurred 200 s later than that of the SLM which agreed with the predicted later failure time Similar trend was observed for the total deformation in vertical direction of vessel bottom point as shown in Figure 19(b) The deformations of both approaches are close except that the acceleration of VLM occurred later than that of SLM Overall results of VLM agreed well with those of SLM

0

500

1000

1500

2000

2500

0 10 20 30 40 50 60 70 80 90 100

Tem

per

ature

(K

)

Polar angle (deg)

50 s

2500 s

5000 s

7000 s

10000 s

34

Figure 19 Results of structural analysis (a) creep strain and (b) total deformation in vertical direction of vessel bottom point [64]

45 Computational Efficiency

Computational efficiency is also an important consideration regarding the performance of numerical approaches The computational efficiencies of both VLM and SLM are analysed and compared in this section As parallel computing technique was used to speed up the simulations the number of parallel cores and corresponding costed computational time for each simulation are given together in Table 2 as a generally reflection of the computational efficiency All the simulations were done in a Lenovo P700 workstation equipped with 2 Intel Xeon E5-2630 v3 processors (8 cores per processor) and 32 G RAM For the simulations of FOREVER-EC2 SLM employed twice the parallel cores and costed 286 more time compared with VLM For the simulations of prototypic BWR case both VLM and SLM employed the same number of parallel cores but VLM only took 13 of the computational time compared with SLM Overall it can be concluded that VLM has higher computational efficiency than the SLM The reason is a structural analysis is performed in the VLM approach while a coupled thermo-structural analysis is performed in the SLM approach (cf Figure 9) a standalone structural analysis is computed faster than a coupled thermo-structural analysis

Table 2 Comparison of computational efficiency [64]

Case FOREVER-EC2 BWR VLM SLM VLM SLM

Number of cores 4 8 4 4 Computational time (hours) 21 27 18 72

35

5 Hybrid Coupling Approach for Efficient RPV Failure Analysis

In this section we propose a hybrid coupling approach for the efficient RPV failure analysis The idea is to replace the ANSYS Fluent in previous coupling approach with a lumped-parameter code for quick estimate of the melt pool heat transfer With the least modelling details (compared with CFD methods and distributed-parameter methods eg PECM) lumped-parameter codes prove to be the most efficient for melt pool heat transfer The new hybrid approach characterizes 1) utilization of efficient models for melt pool heat transfer and 2) the support of detailed structural modelling of RPV deformation allowing studies with different RPV geometric shapes under effects like local thinning due to ablation For studies of which the focus is mainly on structural behaviour of RPV eg [13] [26] while mechanistic simulation of melt pool convection is computationally expensive and unnecessary the lumped-parameter code could help providing reasonable thermal loads

The development of the approach consists of two tasks (i) development of a lumped-parameter code and (ii) its coupling with ANSYS Mechanical to form this hybrid coupling framework In the following we describe both tasks and the corresponding verification and validation processes respectively This section is mainly based on Paper IV

51 A Lumped-parameter Code for Melt Pool Heat Transfer

A lumped-parameter code transIVR was developed It features (i) quick estimate of transient melt pool heat transfer with reasonable accuracies and (ii) two-dimensional representation of heat conduction in the RPV wall facilitating the coupled analysis of the RPV failure under thermal loads The following assumptions are taken in the transIVR modelling

bull The crust between the oxide pool and metal pool is thin

bull The heat flux has uniform distributions on the bottom top and side surfaces of metal pool

bull Heat source only exists in oxide layer

bull Latent heat of fusion is ignored

Figure 20 illustrates the modelled three domains oxide layer metal layer and the RPV wall In addition the internal structure inside the vessel can also be considered as a radiative body to the top metal surface as in [72] An internal volumetric heat source generated by nuclear decay heat is considered in the oxide layer Heat in oxide layer transfers downward to the RPV wall and upward to the top metal layer Heat in metal layer can be transferred sideward to the RPV wall and upward to the surrounding internal structure through radiation (or to water if inside water flooding is applied) The heat transferred to RPV will be removed from outer RPV surface Possible formations of oxide crust in the top and bottom surface are illustrated as the black regions Possible formation of metal crust on the top surface is shown as the grey region The code automatically detects the

36

existence of the crust on oxide top and metal top surfaces during the simulation and solves the problem correspondingly withwithout crust Boundary conditions are applied to the external and internal surfaces of RPV and (if needed) top surface of metal layer Arbitrary boundary conditions are supported If a boundary condition is nonlinear eg mixed heat convection (linear dependent on 119879 ) and heat radiation (proportional to 1198794) the Picard method [73] would be used for solving this problem

This model consists of three main parts

(i) empirical heat transfer correlations (mostly from experiments) to determine global and local heat transfer coefficients

(ii) energy balances in oxide layer and metal layer where the heat transfer coefficients would be used and

(iii) energy balance in RPV

Figure 20 Two-layer melt pool configuration and RPV wall [74]

511 Heat transfer correlations

For internally heated natural convection (oxide layer) in transIVR the mini-ACOPO correlations [75] are used for the averaged heat transfer as expressed in Eq (35) and (36) and for the angular variation of the heat flux as expressed in Eq (37) and (38)

119873119906119906119901 = 03451198771198861198941198990233 (35)

119873119906119889119899 = 00038119877119886119894119899035(119867119877)025 (36)

119873119906119889119899(θ)

119873119906119889119899= 01 + 108(

θ

θ119898119886119909) minus 45 (

θ

θ119898119886119909)2

+ 86 (θ

θ119898119886119909)3

01 leθ

θ119898119886119909le 06 (37)

RPV

Oxide pool

Metal pool

37

119873119906119889119899(θ)

119873119906119889119899= 041 + 035 (

θ

θ119898119886119909) + (

θ

θ119898119886119909)2

06 ltθ

θ119898119886119909le 1 (38)

The 119877119886119894119899 that appears in Eq (35)~(36) is the internal Rayleigh number which is defined as

119877119886119894119899 =119892β119876119881119867

5

α119896ν (39)

For Rayleigh-Bernard convection in the metal layer heat transfer occurs in all the top bottom and side surfaces To side surface the reduced form [72] of the Churchill-Chu correlation [76] for 119875119903 = 013 is employed as shown in Eq (40)

119873119906119904119889 = 007611987711988613 (40)

To top and bottom surfaces the reduced form [72] of Globe-Dropkin correlation [77] for

119875119903 = 013 is used as expressed in Eq (41)

119873119906119906119901 = 005911987711988613 (41)

The 119877119886 that appears in Eq (40) and (41) is the external Rayleigh number which is defined

as

119877119886 =119892βΔ1198791198673

να (42)

Heat transfer coefficients can be calculated from Nusselt number by Eq (43)

ℎ =119873119906119896

119871 (43)

where 119896 and 119871 are the heat conductivity and characteristic length respectively 119873119906 can be the Nusselt numbers expressed in Eq (35)~(38) and (40)~(41)

512 Energy balance in the pools

The energy balance of oxide layer can be described as Eq (44) The total generated heat of oxide pool equals the sum of the increase of enthalpy (transient temperature change) upward heat flow to metal layer and downward heat flow to RPV

120588119862119901119881119900119897119889119879119900119909119889119905

+ ℎ119906119901119860119900119909119906119901(119879119900119909 minus 119879119900119909119906119901) + ℎ119889119899119860119900119909119889119899(119879119900119909 minus 119879119900119909119889119899) minus 119876119881119881119900119897 = 0 (44)

where ρ is the oxide density 119862119901 the heat capacity 119881119900119897 the total volume 119879119900119909 the bulk temperature 119905 the time ℎ119906119901 the upward heat transfer coefficient 119860119900119909119906119901

upward heat transfer area 119879119900119909119906119901 upward surface temperature ℎ119889119899 downward

heat transfer coefficient 119860119900119909119889119899 downward heat transfer area 119879119900119909119889119899 downward surface temperature and 119876119881 volumetric heat source

38

On the interface between oxide layer and metal layer energy balance requires that the heat transferred from oxide layer should equal the heat to the metal layer which can be described as Eq (45)

ℎ119906119901119860119900119909119906119901(119879119900119909 minus 119879119900119909119906119901) = ℎ119898119905119889119899119860119898119905119889119899(119879119894119899119905 minus 119879119898119905) (45)

where ℎ119898119905119889119899 is the downward heat transfer coefficient of metal layer 119860119898119905119889119899 the downward heat transfer area 119879119894119899119905 the interface temperature and 119879119898119905 the bulk temperature of metal layer

The energy balance of metal pool can be described as Eq (46) The increase of enthalpy (transient temperature change) equals the sum of upward downward and sideward heat flows to surrounding surfaces

ρ119898119905119862119901119898119905119881119900119897119898119905119889119879119898119905119889119905

+ ℎ119898119905119906119901119860119898119905119906119901(119879119898119905 minus 119879119898119905119906119901) + ℎ119898119905119904119889119860119898119905119904119889(119879119898119905 minus 119879119898119905119904119889)

+ ℎ119898119905119889119899119860119898119905119889119899(119879119898119905 minus 119879119894119899119905) = 0

(46)

where ρmt is the metal density 119862119901mt the heat capacity 119881119900119897mt the total volume 119879119898119905 the bulk metal temperature 119905 the time ℎ119898119905119906119901 the upward heat transfer

coefficient 119860119898119905119906119901 upward heat transfer area 119879119898119905119906119901 upward surface

temperature ℎ119898119905119904119889 sideward heat transfer coefficient 119860119898119905119904119889 sideward heat transfer area 119879119898119905119904119889 sideward surface temperature ℎ119898119905119889119899 downward heat transfer coefficient 119860119898119905119889119899 downward heat transfer area and 119879119894119899119905 oxide-metal interface temperature

The heat transferred from metal layer to the top surface should equal the heat radiated from this surface to the environment Considering the radiation and reflections of internal structures to this surface following energy balance equation [72] is achieved

ℎ119898119905119906119901(119879119898119905 minus 119879119898119905119906119901) = σ(1198791198981199051199051199001199014 minus 119879119904

4) (1

ε+1 minus ε119904ε119904

119860119898119905119906119901

119860119904)minus1

(47)

where 119879119898119905119905119900119901 is the temperature of metal top free surface ε the emissivity of

metal ε119904 the emissivity of internal structures 119860119898119905119906119901 the area of metal top surface

and 119860119904 the total area of internal structures Sometimes the internal water cooling may be available above metal layer In this case Eq (47) is unnecessary and 119879119898119905119905119900119901 can be set to a constant value

For the oxide layer top surface Eq (48) is used if not crust is formed in the interface between two layers ie 119879119894119899119905 gt 119879119898119890119897119905119894119899119892119900119909 Otherwise Eq (49) is used

119879119900119909119906119901 = 119879119894119899119905 (48)

119879119900119909119906119901 = 119879119898119890119897119905119894119899119892119900119909 (49)

For the metal layer top surface Eq (50) is used if not crust is formed on top of the metal ie 119879119906119901119904119906119903119891 gt 119879119898119890119897119905119894119899119892119898119905 Otherwise Eq (51) is used

119879119898119905119906119901 = 119879119898119905119905119900119901 (50)

39

119879119898119905119906119901 = 119879119898119890119897119905119894119899119892119898119905 (51)

Overall this part involves 6 equations regarding 6 unknows 119879119900119909 119879119900119909119906119901 119879119894119899119905 119879119898119905

119879119898119905119906119901 and 119879119898119905119905119900119901

513 Energy balance in the RPV wall

The heat transfer in the PRV wall can be modelled as a simple 1D heat conduction problem that heat only transfers in the wall thickness direction or 2D axisymmetric heat conduction problem

For 1D model the treatment in [72] was adopted As illustrated in Figure 21 the heat from the oxide pool transfers to inner vessel surface (119879119894) then to the interior vessel (119879119907119890119904) then to outer vessel (119879119900) and finally to environment

Figure 21 1D heat transfer model in RPV [74]

Unknown variables are the heat flux through inner vessel surface 119902119894119899 the crust thickness 119909119888119903 inner vessel temperature 119879119894 vessel center temperature 119879119907119890119904 and outer vessel temperature 119879119900 Eq (52)~(56) give a closed equation set for these variables considering a radiation boundary condition on the external vessel surface Eq (52)~(53) describe the heat transfer through the crust where an internal heat source 119876119888119903 is also considered Eq (52) can be easily derived from the one-dimensional heat transfer equation with heat source Eq (54)~(55) describe the energy balance on the inner vessel surface and vessel interior respectively Eq (56) describes the energy balance on the outer vessel surface regarding radiation boundary condition For Dirichlet boundary condition Eq (56) is replaced with Eq (57) For other boundary conditions we only need to correspondingly modify the terms on the right-hand side of Eq (56) For situations that no crust is formed on the surface of the vessel Eq (52)~(53) are eliminated and 119902119894119899 in Eq (54) can directly be determined from calculations in Section 512 and Eq (54)~(56) are solved for the unknows 119879119894 119879119907119890119904 and 119879119900 Once 119879119894 exceeds the vessel melting temperature (with or without crust formation) wall ablation is considered to occur Then 119879119894 is fixed to the vessel melting temperature and vessel thickness 119909119907119890119904 becomes the new unknown variable

119879119900119909119889119899 119879119894 119879119907119890119904 119879119900 119879119890119899119907

119909119888119903 119909119907119890119904

crust RPV

Qcr

119902119894119899 119902119897119900119888119886119897

40

119902119894119899 = 119896119888119903(119879119900119909119889119899 minus 119879119894)

119909119888119903+ 1198761198881199031199091198881199032

(52)

119902119894119899 = 119902119897119900119888119886119897 + 119876119888119903119909119888119903

(53)

119902119894119899 = 119896119907119890119904(119879119894 minus 119879119907119890119904)

05119909119907119890119904

(54)

120588119907119890119904119862119901119907119890119904119889119879119907119890119904119889119905

= 119896119907119890119904(minus2 lowast 119879119907119890119904 + 119879119894 + 119879119900)

(05119909119907119890119904)2

(55)

119896119907119890119904(119879119907119890119904 minus 119879119900)

05119909119907119890119904= 휀120590(119879119900

4 minus 1198791198901198991199074 )

(56)

119879119900 = 119879119900119888119900119899119904119905

(57)

For the 2D representation the heat conduction in the RPV is solved using FDM considering the heat transfer in both the thickness and the angular directions in the RPV wall The RPV lower head is treated as a spherical shell The corresponding heat conduction equation in the 2D spherical coordinate is given as Eq (58) The cylindrical part of the vessel is treated as a cylindrical shell The corresponding heat conduction equation in the 2D cylindrical coordinate is given as Eq (59) Eq (58) and (59) are coupled and solved together In default the mesh size is 5deg in angular direction and 0025119909119888119903 in radius direction This mesh discretization can be changed up to user demands

ρ119862119901part119879

part119905=

1

1199032part

part119903(1198961199032

part119879

part119903) +

1

r2sinθ

part

partθ(119896119904119894119899θ

part119879

partθ)

(58)

ρ119862119901part119879

part119905=1

119903

part

part119903(119896119903

part119879

part119903) +

part

part119911(119896part119879

part119911) (59)

52 Hybrid Coupling Approach

The hybrid coupling approach to assess the RPV integrity is illustrated in Figure 22 transIVR and ANSYS Mechanical are coupled in a one-way manner transIVR conducts the conjugate heat transfer of melt pool and RPV and provides transient thermal loads to the ANSYS Mechanical where structural analysis of RPV deformation is performed The 2D heat transfer of RPV is solved in transIVR to better capture the temperature distribution In RPV structural analysis same considerations are taken as in Section 422 for 16MND5 steel the modified theta projection model for creep and MISO for plasticity and temperature-dependent Young modulus etc The VLM is adopted for the transfer of transient thermal loads

41

Figure 22 Framework of the hybrid coupling approach [74]

53 Verification and Validation

The transIVR code was first benchmarked against the UCSB FIBS case of two-layer pool configuration and then validated against the LIVE-7V experiment showing the codersquos capability in predicting two-layer and transient melt pool heat transfers respectively Finally the hybrid coupling approach encompassing transIVR and ANSYS Mechanical was validated against the RPV failure experiment FOREVER-EC2 and the results showed acceptable agreements with those of both the experiment and the previous approach using CFD in Section 43 Hence the transIVR code is an efficient tool for melt pool heat transfer and the hybrid approach an efficient way for RPV failure analysis

531 UCSB FIBS benchmark case

The UCSB FIBS benchmark case [78] was calculated to demonstrate the capability of the transIVR code in simulating two-layer pool heat transfer problems The considered pool configuration was similar to that shown in Figure 20 with the following specific dimensions a two-layer melt pool with a 16 m oxide layer and a 08 m metallic pool on top and an RPV lower head with a thickness of 015 m The internal volumetric source in the oxide layer was 13 MWm3 The calculation results were compared with those of the DOE study [78] and the INEEL study [79]

The predicted results were compared with the digitalized results of the DOE and INEEL studies in Figure 23 All three studies show similar profiles of the heat flux distribution along the polar angle (Figure 23(a)) it increased with the polar angle in the oxide layer to the maximum value at around 77deg and then dropped to a significantly lower value in the metal layer The transIVR result agreed well with the other results with small discrepancy in the metal layer the DOE study gave a

heat flux of 500 MWm2 the INEEL study of 400 MWm2 and the transIVR of somewhere in between Figure 23(b) shows the predicted corium crust thickness distribution along the RPV polar angle also agreed well the DOE and INEEL

transIVR ANSYS Mechanical

Transient

thermal loads

42

studies The crust thickness decreased with polar angle in the oxide layer and no crust was formed in the metal layer As the crust formation is closely related to the heat flux distribution the crust thickness profile was reasonable regarding the heat flux distribution shown in Figure 23(a) Vessel wall ablation would occur due to the high thermal loads Figure 23(c) gives the profile of the remaining vessel wall thickness along the RPV polar angle considering the wall ablation It shows that no ablation occurred in low angle region (0deg~36deg) where heat flux was low and the wall thickness changed from 15 cm to around 6 cm as angle increased in high angle region of the oxide layer Because of the discrepancy in the heat flux of metal layer the predicted wall thicknesses also differ from those of the DOE and INEEL studies the thinnest wall was 73 cm predicted by the DOE study corresponding to the maximum heat flux followed by 88 cm of the present transIVR calculation and 94 cm of the INEEL study Generally speaking the prediction of transIVR is in an acceptable level compared with the other studies for the same benchmark case

Figure 23 Simulation results of the UCSB FIBS benchmark case [74]

(a) heat flux distribution (b) crust thickness distribution

(c) wall thickness distribution

43

532 LIVE-7V experiment

LIVE-7V is an experiment under the LIVE program conducted in KIT to study the core melt phenomena As Figure 24(a) shows it employed the LIVE-3D hemispherical facility with an inner diameter of 1 m and a wall thickness of ~25 mm The non-eutectic binary salt mixture of 20 mol NaNO3- 80 mol KNO3 was used as the simulant material The decay heat in the melt was simulated by cable-type heating with 8 planes of electrical resistance heating wires Water cooling was applied to the top and vessel surfaces Top surface cooling was realized by a cooling lid mounted at height 413 mm from the vessel bottom and vessel cooling by the flow channel between the test vessel and the outer cooling vessel The inlet and outlet water temperature for both top and vessel cooling surfaces were kept ~293 K and ~303 K throughout the whole experiment respectively A 413 mm height pool corresponds to a total volume of ~0194 m3 simulant material The experiment procedure can be divided into four stages with the heating power sequentially shifted from 29 kW to 24 kW 18 kW and 9 kW Figure 24(b) indicates the thermocouple positions in experiment

The simulation results of LIVE-7V using transIVR were compared with experimental data [80] and numerical study using REALPSCDAPSIM code [81] For simplicity the RELAPSCDAPSIM results in the following are labelled as lsquoCOUPLErsquo which is the name of the module in RELAPSCDAPSIM for melt pool heat transfer

Figure 24 LIVE test facility [80] and thermocouple positions [82]

Figure 25 compares the predicted heat removal upward to the top lid and downward to the bottom vessel surface with those of the experiment and COUPLE The black curve indicates the total input power at the four stages The experimental results (red and blue dots) were calculated from the Table 3 of [80] assuming no extra heat losses For the first three stages the transIVR predictions agreed well with the COUPLE results while the predicted upward heat flow was slightly lower and the downward heat flow higher than the experimental data For the last stage the transIVR agreed well with the experimental result while the COUPLE predicted lower upward and higher downward heat flows The short

(a) (b)

44

transition periods between two stages with different total inputs taken in transIVR also agreed with those in the COUPLE predictions

Figure 25 Comparison of heat removal upward to the top melt surface cooling lid and downward to bottom vessel wall [74]

Figure 26 shows the comparison of heat flux distribution along the vessel wall All curves show similar trend that a higher total input power would result in a higher heat flux profile As the COUPLE predictions had relatively flatter profiles transIVR predictions generally agreed better with the experimental results especially when total input was high A discrepancy is seen that the minimum value of a curve occurred at angle 0degfor both transIVR and COUPLE and at angle 30deg for the experimental data It is noted that the models in default assume heat flux monotonically increases with angle (cf Eq (37) and (38)) resulting it impossible to predict higher heat flux at angle 0deg than angle 30deg as in experimental results

45

Figure 26 Heat flux profiles along the vessel polar angle [74]

533 FOREVER-EC2 experiment

In this section we performed calculations against the FOREVER-EC2 experiment to validate the performance of the coupled transIVR and ANSYS Mechanical in assessing the RPV integrity The applied boundary conditions were identical to the settings of previous coupling approach in Section 4 (ANSYS Fluent+ Mechanical) as indicated in Figure 11 The initial bulk temperature of the melt pool was set to be 1261 K in transIVR which corresponds to the maximum value of the initial temperature field in CFD calculation of previous coupling approach Both the experimental data and the simulation results using previous coupling approach are included for comparison In the following results from current hybrid approach are labelled lsquotransIVRrsquo (and referred as transIVR results) results from previous approach labelled lsquoCFDrsquo (and referred as CFD results) and results from experiment labelled lsquoexperimentrsquo

Figure 27 compares the bulk temperature predicted by transIVR with the averaged and maximum temperatures of the CFD results All curves show the same trend that the temperature gradually increased with time till reaching a steady state and then kept unchanged till the end The times needed for reaching the steady state were ~5000 s for the CFD and ~4000 s for the transIVR which are close to each other It indicates again a good performance of the transIVR in predicting the transient behavior of melt pool Compared with the steady-state averaged temperature 1478 K and maximum temperature 1609 K of the CFD results the steady-state temperature of transIVR 1599 K is a reasonable prediction

0

3000

6000

9000

12000

15000

18000

0 20 40 60 80 100

Hea

t fl

ux

(W

m2)

Polar angle (deg)

29 kW-EXP24 kW-EXP18 kW-EXP9 kW-EXP29 kW-COUPLE24 kW-COUPLE18 kW-COUPLE9 kW-COUPLE29 kW-transIVR24 kW-transIVR18 kW-transIVR9 kW-transIVR

46

Figure 27 Temperature evolution with time [74]

Figure 28 Temperature fields at time (a) 01 s (b) 3038 s (c) 6038 s and (d) 13289 s [74]

Figure 28 shows the temperature fields of the vessel at time 01 s 3038 s 6038 s and 13289 s The temperature fields were visualized in the ANSYS Mechanical It also indicates that the transient temperature loads were properly transferred from transIVR to ANSYS Mechanical Figure 28(a-d) show both the minimum and maximum temperatures increased with time In the lower head region the temperature gradually increased with the height The maximum temperature occurs at the location slightly lower than the melt pool top surface This is because thermal loadheat flux imposed on the vessel generally increases with polar angle in the pool natural convection so the code transIVR inherently also predicted higher heat flux at higher angle positions The higher flux at higher position then correspondingly resulted in higher temperature In the cylindrical region the

(a) (b) (c) (d)

[K] [K] [K] [K]

47

temperature decreased as the height since the bottom part would be more significantly heated by the high-temperature lower head region

Figure 29 shows the steady-state temperature profiles of the external vessel surface along the polar angle The transIVR result agreed well with other two results in trend that the temperature increased with the angle and reach the peak values at angle ~80deg then gradually decreased with angle But slightly lower temperature was predicted in the low angle region Possible reason may be that the local heat flux distribution determined by Eq (37) and (38) is steeper than that in the experiment and CFD study which then resulted in a steeper increase of temperature along the angle and a lower temperature at low angle region In the cylindrical part (gt90deg) the transIVR result generally agreed with the experimental and CFD results The discrepancy with the experimental data may be caused by the simplified boundary conditions the ambient temperature inside the vessel was set to uniform constant value while it could decrease as height Therefore predicted heat loss in the higher position of cylindrical part could be smaller than that with height-dependent ambient temperature resulting in higher vessel temperature Overall the prediction was reasonable for a lumped-parameter model

Figure 29 Temperature profiles of the external vessel surface along the vessel polar angle [74]

The transIVR predicted a failure time of 24959 s which is close to 24180 s in experiment and 23959 s in the CFD results Figure 30 shows the total deformation of vessel bottom point with time All three curves increased gradually with time and experienced drastic accelerations at the end The

400

600

800

1000

1200

1400

0 40 80 120 160

Tem

per

ature

(K

)

Polar angle (deg)

transIVRExperimentCFDWeilding Line

48

transIVR curve agreed well with the CFD curve The final total deformation at failure time is also close to that of the CFD result but lower than the experiment data The discrepancy might be related to the simplifications in the boundary conditions accuracy of the thermal load prediction and the modelling errors in plasticity and creep

Figure 30 Total deformation in vertical direction of the vessel bottom point [74]

Figure 31 shows the profile of wall thickness change The transIVR curve was compared with the CFD curve and four experimental curves measured at vessel left right front and back sides The simulation result fitted the experimental and CFD results well in the region 0deg~50deg In the region 50deg~90deg wall thickness gradually decreased with angle till around 75deg and then increased till 90deg The maximum thickness change was -3 mm for the simulation which is close to the experiment left side curve and slightly larger than the -6~ -45 mm of the other curves from experiment and CFD results Concerning possible errors and uncertainties in the modelling and numerical settings the new approach generally gives reasonable predictions

0

5

10

15

20

25

11670 15170 18670 22170 25670

Tota

l def

orm

atio

n (

mm

)

Time (s)

Experiment

CFD

transIVR

49

Figure 31 Wall thickness changes along the vessel polar angle [74]

-6

-5

-4

-3

-2

-1

0

1

0 20 40 60 80 100

Wal

l th

icknes

s ch

ange

(mm

)

Polar angle (deg)

CFD

transIVR

Experiment (left)

Experiment (right)

Experiment (front)

Experiment (back)

50

51

6 Summary and Outlook

Reactor pressure vessel (RPV) is one of the key physical barriers in the defence-in-depth approach of nuclear power safety and its failure is also the transition point of core melt (corium) progression from in-vessel to ex-vessel phase Quantification of RPV failure during a hypothetical severe accident is therefore paramount to the specific qualification of severe accident mitigation strategies as well as the general assessment of corium risk in light water reactors The RPV failure is a problem of structural performance of RPV under the influence of thermal loads imposed by corium melt pool ie it is a transient thermo-fluid-structure coupled problem per se whose solution requires simultaneous modelling of both highly turbulent melt pool heat transfer and large-scale nonlinear RPV creep Due to the complexity of the problem only limited capabilities have been developed in the previous studies under different simplifications and assumptions in models and coupling methods The present study was motivated to advance modelling and coupling approaches for RPV performance analysis considering new understanding of physical phenomena and latest coupling platform of simulation tools

For the new development of coupling approaches and models for physical phenomena related to RPV failure analysis during severe accidents this thesis work was conducted in the following order (i) a new creep model for the RPV steel 16MND5 was developed which features the capability to cover all three creep stages (ii) a new thermo-fluid-structure coupling approach was developed for RPV failure analysis and validated against FOREVER-EC2 experiment (iii) the performance of the new coupling approach using volume loads mapping (VLM) for data transfer was further compared with another approach using surface loads mapping (SLM) (iv) a hybrid coupling approach was proposed for efficient analysis of RPV failure which employs a lumped-parameter code for quick estimate of thermal load and a mechanistic FE code for detailed structural analysis and (v) for the hybrid coupling approach the code transIVR was developed for calculation of transient melt pool heat transfer

The developed creep model for 16MND5 is called lsquomodified theta projection modelrsquo and describes all three stages of a creep process Creep curves are expressed as a function of time with five model parameters 120579119894(119894 = 14 119886119899119889 119898) A dataset for the model parameters was constructed by fitting experimental creep curves into this function and an iterative parameter adjustment process to satisfy the monotonicity assumption of creep strain vs temperature and stress Model parameters for new curves under different temperature and stress loads are directly interpolated from this dataset in contrast to the conventional ldquotheta projection modelrdquo that employs an extra fitting process The new treatment was proven to better accommodate all experimental curves over the wide ranges of temperature and stress loads The model was implemented into the ANSYS Mechanical code The simulation results indicate that the model successfully captured all three creep stages and a good agreement was achieved between each experimental creep curve and corresponding modelrsquos prediction Regarding the model response to transient changes of temperature and stress loads the widely used time hardening and strain hardening models were implemented and

52

performing reasonably well in the new method These properties fulfil the requirements of a creep model for structural analysis

The transient thermo-fluid-structure coupling approach was developed using the ANSYS Fluent code for thermofluid dynamics of the melt pool and ANSYS Structural code for structural mechanics of RPV An extension tool was introduced to realize transient load transfer from ANSYS Fluent to Structural and minimize the user effort Both turbulence models and the effective model PECM can be used for melt pool heat transfer The developed creep model was used for RPV failure analysis of the RPV Since it is insensitive to geometry the coupling approach allows more complex geometries such as a BWR lower head with a forest of penetrations (CRGTs and IGTs) modelled in detail Moreover the coupling approach does not only reflect the transient response of thermo-fluid-structure interaction but also facilitates the implementation of advanced (high-fidelity) models in both fluid mechanics and structural mechanics The validation of the coupling approach against the FOREVER-EC2 experiment showed that the temperature profile vessel failure time total displacement of vessel bottom point and wall thickness profile in the experiment were well predicted

Following the variation of the coupling approach from VLM to SLM for data transfer between ANSYS Fluent and ANSYS Structural the respective coupling approaches were investigated by comparative calculations against the FOREVER-EC2 experiment and a postulated severe accident of a reference BWR For the FOREVER-EC2 experiment both VLM and SLM approaches predicted generally well the total deformation in vertical direction of bottom point and the wall thickness change though the former predicted a slightly later failure time For the reference BWR case good agreements were achieved in predicting the maximum creep strain and the total deformation in vertical direction of bottom point For both experiment and reactor cases the VLM approach predicted slightly later failure times than SLM but the differences were negligible compared to the long time-scale of vessel failure (~104 s) The deformation rate of the vessel wall was slow at the early stage and accelerated with time till a drastic deformation and final vessel failure Generally speaking the VLM and SLM coupling approaches had the similar performance in terms of their predictability of experiment and applicability to reactor case but the VLM was computationally more efficient

In the development of the hybrid coupling approach the transIVR code was developed for quick estimate of transient melt pool heat transfer and determination of thermal loads on RPV transIVR does not only support transient analysis of one- and two- layer melt pool but also the heat transfer in RPV being resolved with 2D finite difference method which captures more spatial thermal details for RPV structural analysis in addition to 1D heat transfer model The capabilities of transIVR in modelling two-layer pool heat transfer and transient pool heat transfer were demonstrated by calculations against the UCSB FIBS benchmark case and the LIVE-7V experiment respectively It was then coupled to the mechanical solver ANSYS Mechanical which together form the hybrid coupling framework for detailed RPV failure analysis The calculation against the FOREVER-EC2 experiment did not only reproduce the major results of the experiment but also had a good agreement with the thermo-fluid-structure

53

coupling approach using CFD This indicated the new coupling approach is capable of capturing the vessel creep failure characteristics and can be employed as an efficient tool addressing the complex vessel failure problem

As a prospective more research topics can be addressed in future either as a continuation of the current work toward an effective tool for vessel failure analysis or from the general view of scientific interest in melt pool heat transfer and RPV failure The following topics are recommended

bull Extended applications to the reactor cases For instance the thermo-fluid-structure coupling approach can be used to study penetration failures in the RPV of an BWR with a detailed representation of penetrations (IGTs and CRGTs)

bull Synthesis or update of heat transfer correlations for melt pool Heat transfer correlations directly benefit the parametric codes such as transIVR in accurate quantification of heat flux distribution around a melt pool So far many empirical correlations have been proposed based on experiments which had a variety of geometries and melt simulants with a wide range of Rayleigh number (Ra) Sensitivity studies have also been performed on the influences of Ra Pa and other material properties Therefore a comprehensive synthesis or update of the existing heat transfer correlations will be a meaningful step for making reasonable extrapolations to the reactor applications with prototypic materials

bull Modelling of pool stratification The motivation starts with the improvement of the PECMrsquos modelling capability such that the dynamic stratification and resulting pool heat transfer can be modelled altogether Simulation of melt stratification has been supported in some system codes such as ASTEC using a predefined parametric approach The improvement of PECM can be considered in the similar approach or more mechanistic modelling

bull Development of a structural module for RPV The idea is to develop a simplified module for structural analysis so it can be coupled with transIVR ultimately forming a standalone code for quick estimate of vessel failure of LWRs during postulated severe accidents

bull Extension of vessel analysis capability The current version of tranIVR is limited to melt pool Although the precursory debris bed and melt ejection after vessel failure were not the focus of the present study they are equally important to assessment of corium risk especially for Swedish BWRs where the melt release conditions and history are crucial to qualification of their severe accident mitigation strategies

54

55

Bibliography

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[2] IAEA Safety of Nuclear Power Plants Design Vienna INTERNATIONAL ATOMIC ENERGY AGENCY 2012

[3] M Fischer ldquoGen II and III In-Vessel and Ex-Vessel Melt Retentionrdquo The Short Course on Severe Accident Phenomenology Oct 2017

[4] B R Sehgal Nuclear Safety in Light Water Reactors Severe Accident Phenomenology 1st ed Academic Press 2011

[5] IAEA Approaches and Tools for Severe Accident Analysis for Nuclear Power Plants Vienna INTERNATIONAL ATOMIC ENERGY AGENCY 2008

[6] WENRA ldquoPractical Elimination Applied to New NPP Designs - Key Elements and Expectationsrdquo Sep 2019 [Online] Available httpwwwwenraorgmediafiler_public20191111practical_elimination_applied_to_new_npp_designs_-_key_elements_and_expectations_-_for_issuepdf

[7] P Matejovic M Barnak M Bachraty and L Vranka ldquoASTEC applications to VVER-440V213 reactorsrdquo Nuclear Engineering and Design vol 272 pp 245ndash260 2014 doi 101016jnucengdes201308077

[8] AREVA ldquoUS EPR Severe Accident Evaluationrdquo Topical Report ANP-10268NP Oct 2006 [Online] Available httpswwwnrcgovdocsML0631ML063100157pdf

[9] C T Martin ldquoCoupled 3D Thermo-mechanical Analysis of Nordic BWR Lower Head Failure in case of Core Melt Severe Accidentrdquo master thesis KTH Stockholm 2013

[10] IAEA ldquoAdvanced Large Water Cooled Reactors A supplement to the IAEArsquos Advanced Reactor Information System (ARIS)rdquo Sep 2015 [Online] Available httpsarisiaeaorgPublicationsIAEA_WRC_Bookletpdf

[11] B Pershagen Light Water Reactor Safety Pergamon Press 1989 [12] Ninos Garis M Agrell H Glanneskog and L Agrenius ldquoAPRI 7 ndash Accident

Phenomena of Risk Importancerdquo SSM SSM 201212 2012 [Online] Available httpswwwstralsakerhetsmyndighetensecontentassets2a9f299f46b949b3b209e780cf71436a201212-apri-7-accident-phenomena-of-risk-importance-en-lagesrapport-om-forskningen-inom-omradet-svara-haverier-under-aren-2009-2011-

[13] F Fichot et al ldquoSome considerations to improve the methodology to assess In-Vessel Retention strategy for high-power reactorsrdquo Annals of Nuclear Energy vol 119 pp 36ndash45 2018 doi 101016janucene201803040

[14] T G Theofanous C Liu S Additon S Angelini O Kymaumllaumlinen and T Salmassi ldquoIn-vessel coolability and retention of a core meltrdquo Nuclear Engineering and Design vol 169 no 1 pp 1ndash48 1997 doi 101016S0029-5493(97)00009-5

[15] C Sainte Catherine ldquoTensile and creep tests material characterization of pressure vessel steel(16MND5) at high temperatures (20 up to 1300 degC)rdquo CEA France SEMTLISNRT98-009A 1998

56

[16] E Altstadt H-G Willschuetz B R Sehgal and F-P Weiss ldquoModelling of in-vessel retention after relocation of corium into the lower plenumrdquo Forschungszentrum Rossendorf FZR-437 2005

[17] H-G Willschuumltz and E Altstadt ldquoGeneration of a High Temperature Material DataBase and its Application to Creep Tests withFrench or German RPV-steelrdquo FORSCHUNGSZENTRUM ROSSENDORF FZR-353 ISSN 1437-322X Aug 2002

[18] J L Rempe S A Chavez and G L Thinnes ldquoLight water reactor lower head failure analysisrdquo Nuclear Regulatory Commission NUREGCR--5642 1993 Accessed Aug 23 2019 [Online] Available httpinisiaeaorgSearchsearchaspxorig_q=RN25025905

[19] S Brosi et al ldquoCORVIS Investigation of light water reactor lower head failure modesrdquo Nuclear Engineering and Design vol 168 no 1 pp 77ndash104 1997 doi 101016S0029-5493(96)00008-8

[20] T Y Chu M M Pilch J H Bentz J S Ludwigsen W Y Lu and L L Humphries ldquoLower Head Failure Experiments and Analysesrdquo Sandia National Laboratories Albuquerque NM USA NUREGCR-5582 SAND98-2047 1999

[21] L L Humphries et al ldquoOECD Lower Head Failure Project Final Reportrdquo Sandia National Laboratories Albuquerque NM 87185-1139 2002

[22] B R Sehgal R R Nourgaliev T N Dihn A Karbojian J A Green and V A Bui ldquoFOREVER Experiments on Thermal and Mechanical Behavior of a Reactor Pressure Vessel during a Severe Accidentrdquo in Proceedings of the Workshop on in-vessel core debris retention and coolability Garching near Munich Germany Mar 3-6 1998

[23] S Bhandari M Fischer B R Sehgal and A Theerthan ldquoPost Test Analysis of the FOREVER C2 Experimentrdquo The 9th International conference on Nuclear Engineering (ICONE-9) Nice France Apr 8-12 2001

[24] J F Mao J W Zhu S Y Bao L J Luo and Z L Gao ldquoCreep deformation and damage behavior of reactor pressure vessel under core meltdown scenariordquo International Journal of Pressure Vessels and Piping vol 139ndash140 pp 107ndash116 2016 doi 101016jijpvp201603009

[25] J Mao L Hu S Bao L Luo and Z Gao ldquoInvestigation on the RPV structural behaviors caused by various cooling water levels under severe accidentrdquo Engineering Failure Analysis vol 79 pp 274ndash284 2017 doi 101016jengfailanal201704029

[26] J F Mao J W Zhu S Y Bao L J Luo and Z L Gao ldquoStudy on structural failure of RPV with geometric discontinuity under severe accidentrdquo Nuclear Engineering and Design vol 307 pp 354ndash363 2016 doi 101016jnucengdes201607027

[27] B R Sehgal V A Bui T N Dinh and R R Nourgaliev ldquoHeat Transfer Processes in Reactor Vessel Lower Plenum During Late Phase of In-Vessel Core Melt Progressionrdquo in Advances in Nuclear Science and Technology J Lewins M Becker R W Albrecht E J Henley J D McKean K Oshima A Sesonske H B Smets and C P L Zaleski Eds Boston MA Springer US 1999 pp 103ndash135

[28] C-T Tran and T-N Dinh ldquoThe effective convectivity model for simulation of melt pool heat transfer in a light water reactor pressure vessel lower head Part I Physical processes modeling and model implementationrdquo Progress

57

in Nuclear Energy vol 51 no 8 pp 849ndash859 2009 doi 101016jpnucene200906007

[29] C-T Tran and T-N Dinh ldquoThe effective convectivity model for simulation of melt pool heat transfer in a light water reactor pressure vessel lower head Part II Model assessment and applicationrdquo Progress in Nuclear Energy vol 51 no 8 pp 860ndash871 2009 doi 101016jpnucene200906001

[30] H-G Willschuumltz E Altstadt B R Sehgal and F-P Weiss ldquoCoupled thermal structural analysis of LWR vessel creep failure experimentsrdquo Nuclear Engineering and Design vol 208 no 3 pp 265ndash282 2001 doi 101016S0029-5493(01)00364-8

[31] E Altstadt and T Moessner ldquoExtension of the ANSYS creep and damage simulation capabilitiesrdquo Forschungszentrum Rossendorf Technical report FZR-296 2000

[32] H-G Willschuumltz E Altstadt B R Sehgal and F-P Weiss ldquoRecursively coupled thermal and mechanical FEM-analysis of lower plenum creep failure experimentsrdquo Annals of Nuclear Energy vol 33 no 2 pp 126ndash148 2006 doi 101016janucene200508006

[33] H-G Willschuumltz E Altstadt B R Sehgal and F-P Weiss ldquoSimulation of creep tests with French or German RPV-steel and investigation of a RPV-support against failurerdquo Annals of Nuclear Energy vol 30 no 10 pp 1033ndash1063 2003 doi 101016S0306-4549(03)00036-7

[34] H Madokoro A Miassoedov and T Schulenberg ldquoA thermal structural analysis tool for RPV lower head behavior during severe accidents with core meltrdquo Mechanical Engineering Letters vol 4 pp 18-00038-18ndash00038 2018 doi 101299mel18-00038

[35] W Villanueva C-T Tran and P Kudinov ldquoCoupled thermo-mechanical creep analysis for boiling water reactor pressure vessel lower headrdquo Nuclear Engineering and Design vol 249 pp 146ndash153 2012 doi 101016jnucengdes201107048

[36] W Villanueva C T Tran and P Kudinov ldquoA Computational Study on Instrumentation Guide Tube Failure During a Severe Accident in Boiling Water Reactorsrdquo in The 14th International Topical Meeting on Nuclear Reactor Thermal hydraulics NURETH-14 Toronto Ontario Canada Sep25-30 2011

[37] W Villanueva C T Tran and P Kudinov ldquoAssessment with Coupled Thermo-Mechanical Creep Analysis of Combined CRGT and External Vessel Cooling Efficiency for A BWRrdquo The 14th International Topical Meeting on Nuclear Reactor Thermal hydraulics NURETH-14 Toronto Ontario Canada Sep 25-30 2011

[38] F R Larson and J Miller ldquoA Time-Temperature Relationship for Rupture and Creep Stressesrdquo Trans ASME vol 74 p 765minus775 1952

[39] C T Tran ldquoDevelopment validation and application of an effective convectivity model for simulation of melt pool heat transfer in a light water reactor lower headrdquo Licentiate thesis KTH Stockholm 2007

[40] ANSYS ldquoANSYS Fluent Theory Guide Release 162rdquo ANSYS Inc Jul 2015 [Online] Available httpwwwansyscom

[41] P Zeng Finite element analysis and application (in Chinese) Tsinghua University Press amp Springer 2004

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[42] ANSYS ldquoANSYS Mechanical | Finite Element Analysis Softwarerdquo httpswwwansyscomproductsstructuresansys-mechanical-enterprise (accessed Sep 26 2019)

[43] ANSYS ldquoANSYS Mechanical APDL Theory Reference Release 162rdquo ANSYS Inc Jul 2015 [Online] Available httpwwwansyscom

[44] U Steinberner and H H Reineke ldquoTurbulent Buoyancy Convection Heat Transfer with Internal Heat Sourcesrdquo in Proceedings of the 6th Int Heat Transfer Conference Toronto Canada 1978 vol Vol2 pp 305ndash310 doi 101615IHTC63350

[45] R W Evans and B Wilshire Introduction to Creep London Institute of Materials 1993

[46] DoITPoMS ldquoCreep Deformation of Metalsrdquo Creep Deformation of Metals University of Cambridge 2006 httpswwwdoitpomsacuktlplibcreepindexphp (accessed Jul 09 2019)

[47] ANSYS ANSYS Mechanical APDL Material Reference Release 162 2015 [48] SIMULIA Abaqus 612 Analysis Userrsquos Manualvol 3 Materials Chapter

2324 2012 [49] S Imaoka ldquoUser Creep Subroutine STI0704A ANSYS Release110rdquo 2007 [50] R W Evans J D Parker and B Wilshire ldquoThe θ projection conceptmdashA

model-based approach to design and life extension of engineering plantrdquo International Journal of Pressure Vessels and Piping vol 50 no 1 pp 147ndash160 1992 doi 1010160308-0161(92)90035-E

[51] ECCC ldquoRecommendations and Guidance for the Assessment of Creep Strain and Creep Strength Data ECCC Recommendations Volume 5 Part 1 b[Issue 2]rdquo 2013

[52] W D Day and A P Gordon ldquoLife Fraction Hardening Applied to a Modified Theta Projection Creep Model for a Nickel-Based Super-Alloyrdquo The ASME Turbo Expo 2014 Turbine Technical Conference and Exposition Duumlsseldorf Germany Jun16-20 2014 vol 7A-Structures and Dynamics p V07AT29A010 doi 101115GT2014-25881

[53] M Kumar I V Singh B K Mishra S Ahmad A Venugopal Rao and V Kumar ldquoA Modified Theta Projection Model for Creep Behavior of Metals and Alloysrdquo J of Materi Eng and Perform vol 25 no 9 pp 3985ndash3992 2016 doi 101007s11665-016-2197-y

[54] P Yu and W Ma ldquoA modified theta projection model for creep behavior of RPV steel 16MND5rdquo Journal of Materials Science amp Technology vol 47 pp 231ndash242 2020 doi 101016jjmst202002016

[55] ANSYS ANSYS Mechanical Advanced Nonlinear Materials v160 Lecture 2 Rate Dependent Creep ANSYS training materials 2015

[56] Q Y Li N C Wang and D Y Yi Numerical Analysis (in Chinese) 5th ed Tsinghua University Press 2008

[57] T N Dinh R R Nourgaliev and B R Sehgal ldquoOn heat transfer characteristics of real and simulant melt pool experimentsrdquo Nuclear Engineering and Design vol 169 no 1 pp 151ndash164 1997 doi 101016S0029-5493(96)01283-6

[58] O Kymaumllaumlinen H Tuomisto O Hongisto and T G Theofanous ldquoHeat flux distribution from a volumetrically heated pool with high Rayleigh numberrdquo

59

Nuclear Engineering and Design vol 149 no 1ndash3 pp 401ndash408 1994 doi 1010160029-5493(94)90305-0

[59] F J Asfia and V K Dhir ldquoNatural convection heat transfer in volumetrically heated spherical poolsrdquo 1994 Accessed Aug 23 2019 [Online] Available httpinisiaeaorgSearchsearchaspxorig_q=RN26037727

[60] M Jahn and H H Reineke ldquoFree Convection Heat Transfer with Internal Heat Sources Calculations and Measurementsrdquo in Proceedings of the 5th Int Heat Transfer Conference Tokyo Japan 1974 vol Vol3 paper NC-28

[61] M Gustafsson ldquoFSI Transient extension V1704rdquo EDR MEDESO 2016 [62] EDR MEDESO httpsedrmedesocom [63] H Kim and I Namgung ldquoThermo-mechanical Analysis of RVLH for

APR1400 during a Severe Accidentrdquo in Transactions SMiRT-23 Manchester United Kingdom Aug10-14 2015 [Online] Available httpwwwlibncsueduresolver18402034197

[64] P Yu and W Ma ldquoComparative Analysis of Reactor Pressure Vessel Failure using Two Thermo-Fluid-Structure Coupling Approachesrdquo submitted to journal

[65] F R Menter ldquoTwo-equation eddy-viscosity turbulence models for engineering applicationsrdquo AIAA Journal vol 32 no 8 pp 1598ndash1605 1994 doi 102514312149

[66] T N Dinh and R R Nourgaliev ldquoTurbulence modelling for large volumetrically heated liquid poolsrdquo Nuclear Engineering and Design vol 169 no 1 pp 131ndash150 1997 doi 101016S0029-5493(96)01281-2

[67] ANSYS Inc ldquoIntroduction to ANSYS Fluent Lecture 7 Turbulence Modelling Fluent Tutorial Materialrdquo 2016

[68] A Horvat and M Borut ldquoNumerical investigation of natural convection heat transfer in volumetrically heated spherical segmentsrdquo in Proceedings of the International Conference Nuclear Energy for New Europe 2004

[69] Y Li CFD Pre-test Analysis of SIMECO-2 Experiment 2016 [70] A Theerthan A Karbojian and B R Sehgal ldquoEC-FOREVER experiments

on thermal and mechanical behavior of a reactor pressure vessel during a severe accident Technical report-1 EC-FOREVER-2 testrdquo SAM-ARVI-D008 Mar 2001

[71] P Yu W Ma W Villanueva A Karbojian and S Bechta ldquoValidation of a thermo-fluid-structure coupling approach for RPV creep failure analysis against FOREVER-EC2 experimentrdquo Annals of Nuclear Energy vol 133 pp 637ndash648 2019 doi 101016janucene201906067

[72] T G Theofanous C Liu S Additon S Angelini O Kymaumllaumlinen and T Salmassi ldquoIn-vessel coolability and retention of a core meltrdquo Nuclear Engineering and Design vol 169 no 1ndash3 pp 1ndash48 1997 doi 101016S0029-5493(97)00009-5

[73] H P Langtangen ldquoSolving nonlinear ODE and PDE problemsrdquo Simula Research Laboratory 2016 [Online] Available httpshplgitgithubionum-methods-for-PDEsdocpubnonlinpdfnonlin-4printpdf

[74] P Yu and W Ma ldquoDevelopment of a Lumped-Parameter Code for Efficient Assessment of In-Vessel Melt Retention Strategy of LWRsrdquo submitted to journal

60

[75] T G Theofanous M Maguire S Angelini and T Salmassi ldquoThe first results from the ACOPO experimentrdquo Nuclear Engineering and Design vol 169 no 1 pp 49ndash57 1997 doi 101016S0029-5493(97)00023-X

[76] S W Churchill and H H S Chu ldquoCorrelating equations for laminar and turbulent free convection from a vertical platerdquo International Journal of Heat and Mass Transfer vol 18 no 11 pp 1323ndash1329 1975 doi 1010160017-9310(75)90243-4

[77] S Globe and D Dropkin ldquoNatural-Convection Heat Transfer in Liquids Confined by Two Horizontal Plates and Heated from Belowrdquo J Heat Transfer vol 81 no 1 pp 24ndash28 1959 doi 10111514008124

[78] T G Theofanous C Liu S Additon S Angelini O Kymaelaeinen and T Salmassi ldquoIn-vessel coolability and retention of a core melt Volume 2rdquo Argonne National Lab DOEID--10460-VOL2 1996 Accessed Apr 18 2019 [Online] Available httpinisiaeaorgSearchsearchaspxorig_q=RN30000459

[79] J L Rempe D L Knudson C M Allison G L Thinnes and C L Atwood ldquoPotential for AP600 in-vessel retention through ex-vessel floodingrdquo United States INEELEXT--97-00779 1997 Accessed Apr 18 2019 [Online] Available httpinisiaeaorgSearchsearchaspxorig_q=RN30025872

[80] A Miassoedov X Gaus-Liu T Cron and B Fluhrer ldquoLive experiments on melt pool heat transfer in the reactor pressure vessel lower headrdquo The 10th International Conference on Heat Transfer Fluid Mechanics and Thermodynamics (HEFAT 2014) Orlando FL Jul 14-16 2014

[81] H Madokoro A Miassoedov and T Schulenberg ldquoAssessment of a lower head molten pool analysis module using LIVE experimentrdquo Nuclear Engineering and Design vol 330 pp 51ndash58 2018 doi 101016jnucengdes201801036

[82] B Fluhrer et al ldquoThe LIVE-L1 and LIVE-L3 experiments on melt behaviour in RPV lower headrdquo Forschungszentrum Karlsruhe GmbH Technik und Umwelt (Germany) Inst fuer Kern- und Energietechnik FZKA--7419 2008 Accessed Nov 12 2019 [Online] Available httpinisiaeaorgSearchsearchaspxorig_q=RN40036867

IX

Table of Contents

AbstracthelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphellipI

SammanfattninghelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphellipIII

List of PublicationshelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphellipV

AcknowledgementshelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphellipVII

NomenclaturehelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphellipXI

1 Introductionhelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphellip1

11 Nuclear Power Safety 1

12 RPV Failure 2

13 Literature Review Creep Behaviour and RPV Failure Analysis 4

131 Experiments on creep behaviour and RPV failure 4

132 Numerical studies on RPV failure 5

14 Motivation Objectives and Main Achievements 7

2 Numerical Solvers and Modelshelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphellip9

21 ANSYS Fluent 9

22 ANSYS Mechanical 9

23 PECM 11

3 Creep Modelling of 16MND5 Steelhelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphellip13

31 Creep and Creep Modelling 13

32 A Modified Theta Projection Model 14

321 Original theta projection model and its variations 14

322 A modified theta projection model 16

4 Coupled Thermo-Fluid-Structure Analysis of RPV Failurehelliphelliphelliphellip23

41 Thermo-Fluid-Structure Coupling Approach 23

42 Numerical Modelling 25

421 Conjugate heat transfer 25

422 Mechanical analysis 25

43 Validation against the FOREVER-EC2 Experiment 26

431 FOREVER-EC2 experiment 26

X

432 Conjugate heat transfer 27

433 Mechanical analysis 29

44 Application to BWR Case 32

45 Computational Efficiency 34

5 Hybrid Coupling Approach for Efficient RPV Failure Analysishelliphellip35

51 A Lumped-parameter Code for Melt Pool Heat Transfer 35

511 Heat transfer correlations 36

512 Energy balance in the pools 37

513 Energy balance in the RPV wall 39

52 Hybrid Coupling Approach 40

53 Verification and Validation 41

531 UCSB FIBS benchmark case 41

532 LIVE-7V experiment 43

533 FOREVER-EC2 experiment 45

6 Summary and Outlookhelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphellip51

Bibliographyhelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphellip55

XI

Nomenclature

Latin

119860 area of a surface

119862119901 heat capacity

119863 damage factor

119892 gravity

ℎ heat transfer coefficient

H Melt pool height

119896 thermal conductivity

119871 characteristic length

119873119906 Nusselt number

119875119903 Prandtl number

119902 heat flux

119902119894119899 heat flux through inner vessel surface

119876119881 internal volumetric heat source

119903 radial distance in a cylindrical or spherical coordinate

119877119886 external Rayleigh number

119877119886119894119899 internal Rayleigh number

119877119886119910 local Rayleigh number

119905 time

119879 temperature

119881119900119897 total volume

119909 thickness (of crust or RPV wall)

119911 height in a cylindrical coordinate

Greek

120572 thermal diffusivity

120573 thermal expansion

Δ119879 temperature difference

ε emissivity (in thermal analysis) strain (in structural analysis)

휀119888119903 creep strain

휀119903 creep strain rate

120579 polar angle

120579119894 parameters of a creep model 119894 = 1234119898

120584 kinematic viscosity

σ stress

XII

Super and subscripts

119886119898119887 ambient

119888119903 creep (in Section 3) crust (in Section 5)

119889119899 downward

119890119899119907 environment

119894 inner surface of RPV

119894119899119905 interface

119897119900119888119886119897 local

119898119886119909 maximum value

119898119890119897119905119894119899119892 melting point

119898119905 metal layer (bulk)

119898119905119889119899 downward surface of metal pool

119898119905119904119889 sideward surface of metal pool

119898119905119905119900119901 top surface of metal layer

119898119905119906119901 upward surface of metal pool

119900 outer surface of RPV

119900119909 oxide layer (bulk)

119900119909119889119899 downward surface of oxide layer

119900119909119906119901 upward surface of oxide layer

119903119890119891 reference

119904 internal structures inside the RPV

119904119889 sideward

119906119901 upward

119907119890119904 vessel

Acronyms

BWR boiling water reactor

CBD creep database

CFD computational fluid dynamics

CRGT control rod guide tube

ECCM effective convectivity conductivity model

FDM finite difference method

FE finite element

FEA finite element analysis

FEM finite element method

FVM finite volume method

IGT instrument guide tube

IVR in-vessel retention

XIII

IVMR in-vessel melt retention EU project

LES large eddy simulation

MPS moving particle semi-implicit method

NPP nuclear power plant

PECM phase-change effective convectivity model

RANS Reynolds-averaged Navier-Stokes

RPV reactor pressure vessel

SBO station blackout

SLM surface loads mapping

SPH smoothed-particle hydrodynamics

UDF user-defined function

UPF user programmable features

VLM volume loads mapping

XIV

1

1 Introduction

11 Nuclear Power Safety

Greenhouse gas emissions contribute to the global warming problem Phasing out fossil fuels by switching to low-carbon energy sources is considered as a mitigation strategy With this regard clean energy should be increased in the world energy consumption structure Being clean and stable nuclear energy is an attractive alternative to fossil fuels After the development over decades it also proves to be a mature and safe technology According to IAEA 2018 annual report [1] there were 450 nuclear power reactors under operation worldwide at the end of 2018 which had a record generating capacity of 3964 GW(e) 9 reactors were connected to the grid and 55 reactors under construction around the world

Due to the potential risk of the release of radioactivity nuclear power safety is emphasized throughout the stages of design construction operation and decommission of a nuclear power plant (NPP) The principal technical requirements [2] specify that the three fundamental safety functions should be fulfilled for all plant states of an NPP (i) control of reactivity (ii) removal of heat from the reactor and from the fuel store and (iii) confinement of radioactive material shielding against radiation and control of planned radioactive releases as well as limitation of accidental radioactive releases The defence-in-depth concept is also applied to provide several levels of defence to prevent consequences of accidents The fuel claddings reactor pressure vessel (RPV) and primary coolant pressure boundaries and the containment are the three physical barriers to prevent the release of radioactive materials to the environment

Severe accidents are events that cause significant damage to reactor fuel and result in core meltdown It may be caused by insufficient cooling of the reactor core due to various reasons or by the rapid introduction of huge reactivity which causes sharp increases in the power and fuel fragmentation eg Chernobyl accident During a severe accident the failure of the fuel claddings will result in the radioactive materials previously enclosed inside the cladding released to the primary loop If cooling is not sufficient to remove the decay heat the core materials will melt migrate down and relocate into the RPV lower head after which the melt will react with water and partially form fragments quench solidify and form a debris bed The debris bed formed by the melt may continue to melt to form a melt pool due to the decay heat As the corium contacts the RPV it would continuously heat the RPV and threaten its integrity Once the RPV fails containment becomes the last barrier to prevent the release of radioactivity The released melt would result in an increase of the pressure and temperature due to the high temperature fractions oxidation and hydrogen combustion which is called direct containment heating The melt would fall onto the reactor cavity Without sufficient cooling the molten core would interact with basemat concrete (MCCI) This process can last for few days It can result in containment failure due to either containment overpressure or the basemat melt-through of containment In addition containment is also subjected to isolation failure and early containment failure due to missiles and dynamic pressure [3] More

2

detailed severe accident progression or phenomena could also be found in eg [4] [5]

Though severe accidents are unlikely to occur the consequences can be severe if no proper measure is taken Related researches have been continuously going on aiming at mitigating the accident consequences and improving safety After the Fukushima Accident safety inspections and re-evaluation actions have been carried out worldwide especially on safety measurements regarding severe accidents The concept lsquoPractical Eliminationrsquo applied to new NPP designs demands that new nuclear installations be designed with the objective of mitigating the consequences of accidents and avoiding early radioactive releases and large radioactive releases [6] Severe accident mitigation strategies are applied to the existing and new NPPs (both Generation II and III) The strategies can generally be classified into two categories [3] the in-vessel retention (IVR) and ex-vessel retention (EVR) In simple words in-vessel retention tries to retain the core melt inside the RPV (second barrier) by applying external water cooling on the vessel as shown in Figure 1(a) Such method minimizes the dispersal of the melt within the containment and hence simplifies the later recovery work Examples of the application are AP1000 HPR1000 APR 1400 The idea of the EVR is to let the RPV fail but to retain the core melt inside the containment Two principle approaches of EVR are the retention within a dedicated lsquocore catcherrsquo and melt fragmentation Examples of core catcher would be EPR that directs the melt to a spreading compartment (Figure 1(b)) and VVER1000 that uses metallic crucible below the RPV An example of the melt fragmentation is the Nordic BWRs that the corium released from RPV would relocate in the cavity with a deep-water pool (7-12 m) and is expected to fragment quench and form a coolable debris bed (Figure 1(c))

Figure 1 Examples of IVR [7] EVR with core catcher [8] and melt fragmentation [9]

12 RPV Failure

RPV which contains the reactor core core shroud and coolant is a key component to the reactor system RPV and the primary loop constitute the primary coolant system which conveys the fission heat from the reactor core to the secondary loop through steam generator (PWR) or to the turbine (BWR) The pressure under

(a) IVR (b) Core catcher (c) Melt fragmentation

3

operation can be ~155 MPa for a 1000 MW PWR [10] and ~70 MPa for a 1000 MW BWR [11] It also has to withstand the neutron radiation damage The thickness of RPV [11] can be ~20 cm for PWR and ~16 cm for BWR

As the transition of in-vessel progression to ex-vessel progression RPV failure is important for analysis of accident progression and assessment of the effectiveness of severe accident mitigation strategies In the melt fragmentation strategy of Nordic BWR the melt-coolant interaction and debris bed coolability after the corium poured into the deep-water pool are essentially dependent on the vessel failure mode including the characteristic time of melt release rupture size and location and amount and superheat of melt available for release [12] New concern also arose [13] on the necessity of performing structural analysis for the RPV in the IVR strategy In the initial methodology and further development of the IVR the success of the strategy ie the integrity of the RPV was demonstrated (eg in [14]) with the following two conclusions (i) even for a local heat flux on the vessel as high as critical heat flux (CHF) the resulted very thin external ldquocold layerrdquo of the vessel would be sufficient to withstand the internal stress load and hold the vessel integrity (ii) as CHF is the critical value for the thermal load the main specific goal for assessing the effectiveness of IVR becomes to demonstrate that the local heat flux along RPV be lower than the CHF The new concern doubts that local lsquohot partrsquo of the vessel would have negative effects on the overall resistance due to creep and dilatation which could result in a vessel failure and hence violate the first conclusion in particular for ldquolong termrdquo retention of corium in RPV If so a structural integrity analysis of RPV would be needed in addition to assessing local heat flux especially for high-power reactors Therefore accurate prediction of vessel failure could be meaningful for both strategies

Figure 2 Illustration of RPV under thermal loads from corium

As the structural performance is highly affected by the thermal loads from the corium RPV failure is essentially a transient thermo-fluid-structure coupled problem Figure 2 illustrates the situation that a corium debris bedmelt pool formed in the RPV lower head The decay heat of nuclear fission products could continuously be generated inside the corium leading to the formation of melt pools as the debris melts The heated debris bedmelt pool imposes thermal and

4

mechanical loads on the RPV Such loads can be transient (change with time) and spatially non-uniform Complex phenomena are also involved in this process Strong turbulent natural convection induced by the decay heat may be expected in the melt pool under IVR The corresponding Rayleigh number 119877119886 can be 1016minus17 for prototypic reactor cases Material structural properties can become non-linear under high temperature and stress loads The creep deformation also becomes significant at high temperature Detailed modelling of turbulent natural convection and non-linear properties could be computationally expensive In addition large-scale geometry (hemisphere with a radius of 20-30 m) long-term transient deformation process (~10000 s) and required small time steps (eg 001 s) for numerical convergence consideration further increase the computational costs

13 Literature Review Creep Behaviour and RPV Failure Analysis

In this section the most relevant studies concerning the experiments modelling and analysis of creep behaviour and RPV failure are presented

131 Experiments on creep behaviour and RPV failure

Various experiments have been performed on the RPV failure problem some of which were to generate data of mechanical properties (especially the creep properties) and some to study the overall vessel failure with scaled or simplified geometries

The REVISA experiment [15] generated creep curves and stress strain curves of French steel 16MND5 ranging from 600 to 1300 degC This data is accessible in [16] as well as [17] which also documents other material properties eg density thermal expansion coefficient conductivity heat capacity Youngrsquos Modulus and plasticity Similar experiments were carried out to generate high-temperature creep and tensile test data for American RPV steels and Inconel [18]

The CORVIS experiments [19] investigated the lower head failure modes with and without penetrations Each experiment was carried out on a steel plate 100 mm thick representing a pressure vessel lower head The penetration case represented a boiling water reactor lower head carrying a tube penetration in the form of a drain line

The LHF (lower head failure) experiments [20] and the succeeding OLHF tests [21] investigated the vessel failure with a 15 scaled vessel heated by an electrical radiator allowing to locate the hot focus at different position of the vessel with a maximum wall temperature of around 900 degC Internal pressure ranging from 53 to 123 bar was also imposed to initiate the vessel failure

The FOREVER experiments [22] investigated the phenomena of melt pool convection vessel creep possible failure processes and failure modes occurred during the late phase of the in-vessel progression by heating up a melt pool in a vessel and observing possible vessel failures The study covers cases with and without penetration different melt level existence of gap cooling and in-vessel melt coolability

5

132 Numerical studies on RPV failure

Various numerical approaches have been employed on RPV failure Since a two-way thermo-fluid-structure coupling approach is challenging these approaches take simplifications and can generally be classified into two categories (i) RPV failure decoupled from melt pool heat transfer and (ii) RPV failure coupled with melt pool heat transfer in simplified manners

1) Decoupled approaches In the decoupled approaches only RPV is modelled Simplified constant thermal loads are imposed on RPV considering the melt pool heat transfer

In a post-test analysis of FOREVE-C2 experiment [23] it was treated as a thermo-structural coupled analysis of the vessel with the thermal load estimated by the outer thermocouples and the normal heat flux on the external vessel wall The analysis was done using finite element (FE) code SYSTUS A coupled creepdamage law based on a viscosity-hardening creep law describing primary and secondary creep was employed for creep analysis

Mao et al conducted various studies on reactor cases assessing the IVR strategy using FE code ABAQUS Thermal boundary conditions were imposed on the RPV which was then solved as a thermal-structural coupled problem The strain hardening creep model (Eq (1)) was mainly used Their studies cover the general RPV failure [24] as well as the effects of cooling water levels under IVR [25] and geometric discontinuity (ablated RPV) [26]

ε119888119903 = 1198891σ1198892ε1198893119890119909119901 [minus

1198894119879] (1)

In the EU IVMR project [13] the RPV integrity was also studied through a benchmark calculation which deals with the RPV of a PWR with a prescribed ablated profile with 2D axisymmetric geometry Prescribed temperature gradient was set as thermal load The objective was to calculate deformations for various pressure loads up to vessel rupture A smoother profile was planned in further research

The advantage of such approaches is once thermal loads are reasonably assumed the focus only needs to be put on the RPV The disadvantage is a reasonable thermal load should be consistent with the melt pool heat transfer while sometimes it is not well-noted eg constant uniform temperature boundary conditions were assumed for inner and outer RPV (with uniform thickness in lower head) surfaces [25] which ignored the non-uniform heat flux distribution caused by the turbulent convective corium pool As thermal loads strongly affect the structural performance they should be quantified carefully

2) Coupled approaches In coupled approaches both melt pool heat transfer and RPV failure are modelled and coupled in one-way or simplified manner With melt pool heat transfer modelled the coupled approaches are more flexible to provide reasonable thermal loads on the RPV In existing approaches CFD and the correlation-based distributed-parameter models eg the Effective Convectivity Conductivity Model

6

(ECCM) [27] and the Phase-change Effective Convectivity Model (PECM) [28] [29] are general methods for melt pool heat transfer

Sehgal et al [22] presented an approach that employed the MVITA code to calculate the thermal processes and ANSYS for structural analysis MVITA code employed the ECCM Bailey-Norton creep law was used as shown in Eq (2)

ε119888119903 = 6055 times 10minus26 sdot σ472119905085119890minus

37200

119879 (2)

This approach was used for the pre-test simulation of FOREVER experiment to determine the thermal and mechanical loadings Parametric investigations also helped in the identification of important factors that could affect the test procedure and test conditions

Willschutz et al [30] employed the CFD-module Flotran for the melt pool heat transfer and ANSYS for structural analysis of RPV Standard k-ε turbulence model was selected for the turbulent natural convection in the melt pool A user-defined three-stage creep model [31] was employed In this model experimental creep curves were discretized into a creep database (CDB) under given temperature 119879 and stress 120590 the creep curve was discretized into finite point pairs

(휀 ε) indicating under such thermal and mechanical loads the creep strain rate ε is subjected to creep strain 휀 The creep strain rate for any other case can be interpolated over known temperature stress and strain from the database This method was applied to the post-test calculation of the FOREVE-C2 experiment It was concluded that the unexpected deformation behaviour during the experiment was caused by the combination of the transient temperature field in the vessel wall and the three creep stages

They later replaced the Flotran solver with the ECCM for melt pool heat transfer in a following study [32] By implementing it into FEM-solver a recursively coupled analysis was constructed Moreover a modified creep model [33] was used with damage measures the CDB itself did not contain tertiary creep instead a factor 1(1 minus 119863) was multiplied to the original creep strain rate to get the modified creep strain rate where 119863 is a damage factor (0 lt 119863 lt 1) defined with creep strain information and plastic strain information As 119863 grows close to 1 the modified creep strain rate would be enlarged significantly by this factor 1(1 minus 119863) which then behaves like the accelerating tertiary stage This method was applied to pre- and post- test calculations for the FOREVER experiments proving to be closer to reality than the previous method

Madokoro et al [34] developed a thermal structural analysis tool based on open source solver OpenFOAM and numerically investigated the coupled RPV failure and melt pool convection The PECM was used in the melt pool heat transfer In structural part the plasticity and damage parameters were similar to Willschutz et al [32] except for that strain hardening creep model (Eq (3)) was used instead of a CDB Simulations of FOREVER-EC2 and -EC4 were investigated It was concluded from the simulations that although the emissivity and mesh type (updated or fixed) affect the results significantly the approach generally has the capability of predicting the vessel deformation and failure

7

ε119888119903 = 1198891 sdot σ1198892 sdot ε1198893 (3)

Villanueva et al [35] performed a coupled thermo-mechanical creep analysis of the reactor scale RPV (ABB-Atom design BWR) to evaluate efficacy of the cooling measures for the in-vessel melt coolability and retention The PECM was used for melt pool heat transfer and ANSYS for RPV integrity analysis The modified time hardening model (Eq (4)) of the American RPV steel SA533B was selected as creep model The transient thermal load from the melt pool was applied as a thermal boundary condition to the vessel wall Further studies were done with this approach on investigations of instrumentation guide tube (IGT) failure [36] and control rod guide tube (CRGT) cooling [37] of a Nordic BWR

ε119888119903 =1198881σ

11988821199051198883+1

1198883 + 1 1198881 gt 0 (4)

It is noted that only RPV analysis with finite element analysis (FEA) are reviewed here Engineering methods like the Larson-Miller method [38] are not covered in this thesis One can refer to theory manuals of severe accident codes like MELCOR ASTEC for more details Comprehensive description on modelling of melt pool heat transfer is not covered either One can refer to either [39] for a general view or to any severe accident code manuals for engineering methods

14 Motivation Objectives and Main Achievements

The existing experimental and numerical studies provide us a good understanding of RPV failure involved phenomena and valuable simulation experiences These numerical methods may vary in solvers (which may be related to code accessibility) and models (which may be related to model accessibility understanding of phenomena computational power) However improvements of modelling approaches are still possible as the improvements in any of the related aspects eg

bull Simulation tools are continuously being upgraded and providing new possibilities just like that PECM appeared and ECCM was no longer used

bull Better understanding of the phenomena and more simulation experiences from in-vessel retention and vessel failures researches benefit the further modelling and simulations

bull Increasing computational power allows more detailed modelling eg turbulence modelling of melt pool convection

bull There is still room for the improvement of creep modelling

In addition implementation difficulty of a coupling approach is another important issue to consider Since it is a TRANSIENT thermo-fluid-structure coupled problem it is almost impossible to replicate an approach simply by referring to a paper with limited details For BWRs it would be even better for the coupling approaches to support complex geometries considering forest of penetrations (CRGTs and IGTs) on the RPV lower head without significant increase of user effort in the coupling process

8

The objectives of this thesis are to investigate new approaches and develop new models for the thermo-fluid-structure coupled problem of RPV failure and involved phenomena These objectives are accomplished with the following achievements

1) A new creep model for the 16MND5 steel was developed which covers all three creep stages Good agreement was achieved between model predictions and experimental curves Both time and strain hardening models were implemented (Paper I)

2) A new thermo-fluid-structure coupling approach was developed for RPV failure analysis It supports the advanced models in both melt pool convection and structural mechanics to improve fidelity and facilitates implementation Validation of the approach with the new creep model was performed against FOREVER-EC2 experiment This approach is flexible for complex geometries and can be used in further analysis eg lower head with detailed modelling of penetrations in BWRs (Paper II)

3) The performance of this new approach which uses volume loads mapping for data transfer was compared with the previous approach using surface loads mapping showing good agreement with small discrepancy (Paper III)

4) A lumped-parameter code transIVR for melt pool heat transfer was developed which supports one- and two-layer pool configuration transient analysis and well-resolved RPV wall (Paper IV)

5) A hybrid coupling approach was developed for efficient analysis of RPV failure where transIVR is coupled with ANSYS Mechanical (Paper IV)

In the following general descriptions on the codes ANSYS Fluent Mechanical and the PECM are given in Section 2 the new creep model in Section 3 the new coupling approach as well as its comparison with the previous approach in Section 4 the efficient hybrid coupling approach in Section 5 and summary amp outlook in Section 6

9

2 Numerical Solvers and Models

In this section we briefly introduce the ANSYS Fluent as the CFD solver for melt pool heat transfer the ANSYS Mechanical as the mechanical solver for RPV failure and PECM as an efficient alternative for melt pool heat transfer that are heavily used in the following sections

21 ANSYS Fluent

In the fluid mechanics the fluid is treated as a continuum which is generally described by the conservative mass momentum and energy equations [40]

120597120588

120597119905+ 120571 sdot (120588) = 119878119898 (5)

120597(120588)

120597119905+ 120571 sdot (120588) = minus120571119901 + 120571 sdot (120591) + 120588 + (6)

120597120588119864

120597119905+ 120571 sdot ((120588119864 + 119901)) = 120571 sdot (119896120571119879 minussumℎ119895119869119895

119895

+ (120591 sdot )) + 119878ℎ

(7)

Computational Fluid Dynamics (CFD) studies topics on the numerical solutions of above equations Typical methods are the Finite Volume Method (FVM) Finite Difference Method (FDM) and Finite Element Method (FEM) as well as particle methods that are mesh-free eg SPH and MPS

ANSYS Fluent is a general purpose CFD code which solves above equations with FVM Turbulence modelling and multiphase flows are supported Turbulence can be modelled with RANS LES or hybrid LESRANS Customizations eg adding sourcesink time or spatial dependent boundary conditions are realized through the user-defined functions (UDF) which is programmed in C language

22 ANSYS Mechanical

In the continuum mechanics of solids three sets of variables are considered to describe the structure performance namely element displacement field strain field and stress field (119906 ε σ) A general 3-dimensional problem is described by 3 sets of equations the equilibrium equations (Eq (8)) strain-displacement relationship (Eq (9)) and constitute equations (Eq (10)) [41] For Eq (8) the reciprocal theorem of shear stress also gives τ119894119895 = τ119895119894 where 119894 119895 = 119909 119910 119911

indicating the stress tensor being symmetric For very simple applications eg 1D beam it can be solved analytically For complex problems where analytical solutions are not available approximation methods eg Galerkin weighted residual method principle of virtual work and principle of minimum potential energy would be used

10

partσ119909119909part119909

+partτ119909119910

part119910+partτ119909119911part119911

+ 119909 = 0

partτ119910119909

part119909+partσ119910119910

part119910+partτ119910119911

part119911+ 119910 = 0

partτ119911119909part119909

+partτ119911119910

part119910+partσ119911119911part119911

+ 119911 = 0

(8)

ε119909119909 =part119906

part119909 휀119910119910 =

120597119907

120597119910 휀119911119911 =

120597119908

120597119911

120574119909119910 =120597119906

120597119910+120597119907

120597119909 120574119910119911 =

120597119908

120597119910+120597119907

120597119911 120574119911119909 =

120597119908

120597119909+120597119906

120597119911

(9)

ε119909119909 =1

119864[σ119909119909 minus μ(σ119910119910 + σ119911119911)]

ε119910119910 =1

119864[σ119910119910 minus μ(σ119909119909 + σ119911119911)]

ε119911119911 =1

119864[σ119911119911 minus μ(σ119909119909 + σ119910119910)]

γ119909119910 =1

119866τ119909119910 γ119910119911 =

1

119866τ119910119911 γ119911119909 =

1

119866τ119911119909

(10)

ANSYS Mechanical [42] is a mechanical engineering software that uses finite element analysis (FEA) for structural analysis The principle of virtual work is adopted in the solver [43] The idea of this principle is that a virtual (very small) change of the internal strain energy must be offset by an identical change in external work due to the applied loads which reads

δ119880 = δ119881 (11) where 119880 is the strain energy or the internal work 119881 the external work and δ the virtual operator The virtual internal work can be expressed as

δ119880 = int δεσd(vol)T

vol

+int 119886119903119890119886119891

δ119908119899119879σ119889(119886119903119890119886119891) (12)

where 휀 is the strain vector 120590 stress vector vol volume of element 119908119899 motion normal to the surface 119886119903119890119886119891 area of the distributed resistance The virtual

external work is contributed from three parts the inertial effects the pressure force on the surface and nodal forces applied to the element which then can be expressed as

δ119881 = minusint 119907119900119897

δ119908119879119865119886

119907119900119897119889(119907119900119897) + int

119886119903119890119886119901

δ119908119899119879119875119889(119886119903119890119886119901)

+ δ119906119879119865119890119899119889

(13)

11

where 119908 is the vector of displacements 119865119886 acceleration force vector 119875 applied pressure vector 119886119903119890119886119901 area over which pressure acts and 119865119890

119899119889 nodal

forces applied to the element

Plugging Eq (12) and (13) into (11) would result in the equilibrium equation Since the virtual displacement can be arbitrary the final equations used to describe the equilibrium of an element is reduced to Eq (14) with multiple operations

([119870119890] + [119870119890119891])119906 minus 119865119890

119905ℎ = [119872119890] + 119865119890119901119903 + 119865119890

119899119889

(14)

where [119870119890] [119870119890119891] 119865119890

119905ℎ [119872119890] and 119865119890119901119903 are the element stiffness matrix

foundation stiffness matrix thermal load vector mass matrix acceleration vector and pressure vector respectively It represents the equilibrium equation on a one element basis Alternatively using a minimum total potential energy principle could result in a consistent expression to the principle of virtual work [41]

Customizations eg on material properties are realized through the user programmable features (UPF) which is programmed in FORTRAN language

23 PECM

The PECM [28] [29] developed in KTH is a correlation-based method to model the melt pool heat transfer It is implemented in ANSYS Fluent through UDF and has been validated against many experiments such as ACOPO COPO LIVE etc In the PECM the flow velocities 119906119909 119906119910 and 119906119911 are replaced by characteristic

velocities 119880119909 119880119910 and 119880119911 respectively in order to eliminate the necessity of solving

the Navier-Stokes equation

The characteristic velocities are calculated using heat transfer correlations

119880119906119901 =α

ℎ119901119900119900119897times (119873119906119906119901 minus

ℎ119901119900119900119897

ℎ119906119901) (15)

119880119889119899 =α

ℎ119901119900119900119897times (119873119906119889119899 minus

ℎ119901119900119900119897

ℎ119889119899) (16)

119880119904119889 =α

ℎ119901119900119900119897times (119873119906119904119889 minus

ℎ119901119900119900119897

119882119901119900119900119897) (17)

where ℎ119901119900119900119897 is the melt pool depth ℎ119906119901 the thickness of well mixed layer of the

pool ℎ119889119899 the thickness of lower stratified region of melt pool 119882119901119900119900119897 the width of

pool 120572 the thermal diffusivity The PECM employs reduced characteristic velocities to describe mushy zone convection heat transfer For internally heated pool natural convection the Steinberner-Reineke correlations (Eq (18)~(19)) [44] are employed The profile of sideward 119873119906 is described using Eckertrsquos type correlation (Eq (20)) for a vertical boundary layer where the local Rayleigh

number at pool height 119910 is defined as 119877119886119910 =119892βΔ1198791199103

αν

12

119873119906119906119901 = 03451198771198861198941198990233

(18)

119873119906119889119899 = 13891198771198861198941198990095

(19)

119873119906119904119889 = 050811987511990314 (

20

21+ 119875119903)

minus14

11987711988611991014

(20)

As the characteristic velocities are calculated the energy equation is the only transport equation that needs to be solved and can be expressed as follows

partρ119862119901119879

part119905+ nabla(ρ119862119901119932119879) = 119896nabla2119879 + 119876119907 (21)

where from left to right are the transient term convection term diffusion term and volumetric heat source respectively The convection term then can be calculated explicitly with use of values in the last time step and then can be treated together with the volumetric source term Therefore the PECM finally simplifies the energy equation to a conduction equation which can be easily solved and as a result will significantly increase the computational efficiency in simulations

In addition solidificationmelting process can also be modelled and the temperature-dependent physical properties are supported

13

3 Creep Modelling of 16MND5 Steel

Creep deformation becomes significant when a component is subjected to high temperature and stress Modelling of creep behaviour of the RPV is paramount since it is considered as a main mechanism leading to the RPV failure In this section we present the development of a three-stage creep model (called lsquomodified theta projection modelrsquo ) for the French RPV steel 16MND5 This model was successfully implemented into the ANSYS Mechanical code

31 Creep and Creep Modelling

Creep is an irreversible time-dependent inelastic deformation process Diffusion and dislocation are the two molecular mechanisms of creep The former normally occurs due to the presence of vacancies within the crystal lattice (or the grain) while the latter occurs if there is a slip by the passage of crystal dislocations [45] There are two types of diffusion creep [46] Coble creep (favoured at lower temperatures) if the diffusion paths are predominantly through the grain boundaries and Nabarro-Herring creep (favoured at higher temperatures) if through the grains themselves Dislocation creep tends to dominate at high stresses and relatively low temperatures Dislocations can move by gliding in a slip plane which is a process that requires little thermal activation But the rate-determining step for their motion is often a climb process which also requires diffusion and is thus a time-dependent process and favoured by higher temperatures Creep behaviour is highly affected by temperature and stress generally creep is more significant under higher temperature and stress

Creep curves describe the creep strain development with time Figure 3 shows a creep curve under constant temperature and stress which typically experiences three creep stages primary stage where the creep strain rate ε119888119903 (the change of creep strain ε119888119903 in unit time) decreases with time secondary stage where the creep strain rate almost keeps constant and tertiary stage where the creep strain rate drastically increases with time

Figure 3 Creep curve under constant temperature and stress

14

The built-in creep models in commercial software eg ANSYS [47] and ABAQUS [48] in default only cover the primary andor the secondary stage(s) Table 1 [47] gives the default creep models supported in ANSYS for example Possible reason why the tertiary stage is not modelled might be that for most industrial applications the failure analysis is beyond interest and therefore the built-in creep laws in the commercial software was supposed to be sufficient [49] Though the two codes offer the interfaces through either ANSYS UPF or ABAQUS user subroutines to support more general user-defined creep laws eg complex models that also include the tertiary creep stage

Table 1 Default creep models in ANSYS Mechanical

Creep model Equations Type

Strain hardening

ε119888119903 = 1198621σ1198622ε119888119903

11986231198901198624119879 1198621 gt 0 Primary

Time hardening ε119888119903 = 1198621σ11986221199051198623119890minus1198624119879 1198621 gt 0 Primary

Generalized exponential

ε119888119903 = 1198621σ1198622119903119890minus119903119905 r = 1198625σ

11986231199051198623119890minus1198624119879 1198621 gt 0

1198625 gt 0

Primary

Generalized Graham

ε119888119903 = 1198621σ1198622(1199051198623 + 1198624119905

1198625 + 11986261199051198627)119890minus1198628119879 1198621 gt 0 Primary

Generalized Blackburn

ε119888119903 = 119891(1 minus 119890minus119903119905) + 119892119905 119891 = 1198621119890

1198622σ 119903 = 1198623(σ1198624)1198625 119892 = 1198626119890

1198627σ

1198621 gt 0

1198623 gt 0

1198626 gt 0

Primary

Modified time hardening

ε119888119903 = 1198881σ11988821199051198883+1(1198883 + 1) 1198621 gt 0 Primary

Modified strain hardening

ε119888119903 = 1198621σ1198622[(1198623 + 1)ε119888119903]

1198623119890minus1198624119879 1198621 gt 0 Primary

Generalized Garofalo

ε119888119903 = 1198621[119904119894119899ℎ(1198622σ)]1198623119890minus1198624119879 1198621 gt 0 Secondary

Exponential form

ε119888119903 = 1198621119890σ1198622119890minus1198623119879 1198621 gt 0 Secondary

Norton ε119888119903 = 1198621σ1198622119890minus1198623119879 1198621 gt 0 Secondary

Combined time hardening

ε119888119903 = 1198621σ11986221199051198623+1119890minus1198624119879(1198883 + 1)

+ 1198625σ1198626119905119890minus1198627119879

1198621 gt 0

1198625 gt 0

Primary+ Secondary

Rational polynomial ε119888119903 = 1198621

partε119888part119905 ε119888 =

119888119901119905

1 + 119901119905+ ε119898119905

ε119898 = 1198622101198623σσ1198624 119888 = 1198627ε

1198628σ1198629

119901 = 11986210ε11986211σ11986212

1198622 gt 0 Primary+ Secondary

Generalized time hardening

ε119888119903 = 119891119905119903119890minus1198626119879 119891 = 1198621σ + 1198622σ2 + 1198623σ

3 119903 = 1198624 + 1198625

Primary

32 A Modified Theta Projection Model

321 Original theta projection model and its variations

The original theta projection model [50] was first proposed to accurate predict long-term creep and creep-fracture properties It assumes that a creep curve under constant temperature and stress can be described as

15

휀119888119903 = 1205791(1 minus 119890minus1205792119905) + 1205793(119890

1205794119905 minus 1) (22)

where 휀119888119903 is the creep strain 119905 the time and 1205791~1205794 the model parameters Figure 4 illustrates how the first term and second term on right hand side develop with time and constitute a creep curve which has obvious primary secondary and tertiary stages Model parameters 120579119894(119894 = 1234) are temperature and stress dependent and are described with the following expression

119897119899thinsp120579119894 = 119886119894 + 119887119894120590 + 119888119894119879 + 119889119894120590119879 (23)

where 119886119894 119887119894 119888119894 and 119889119894 ( 119894 = 1234 ) are constants It is noted [45] that this equation is suitable when the temperature range of experiment data sets is very narrow such that minus1119879 is almost proportional to 119879 These constants (119886119894 119887119894 119888119894 and 119889119894 (119894 = 1234)) are the final model parameters which are generated with two fitting procedures

(i) Fitting the experimental creep curves to Eq (22) generates several groups of 1205791~1205794 for each creep curve under certain temperature and stress We denote (119879119895 120590119895 1205791119895 1205792119895 1205793119895 1205794119895 ) as parameters for the 119895119905ℎ

experimental curve under temperature 119879119895 and stress 120590119895

(ii) Fitting 1205791 to Eq (23) as a function of temperature and stress with the obtained data ( 119879119895 120590119895 1205791119895 ) from ( 119879119895 120590119895 1205791119895 1205792119895 1205793119895 1205794119895 ) generates

parameters 1198861 1198871 1198881 and 1198891 Repeat this fitting procedure for 1205792 1205793 and 1205794 to get parameters 119886119894 119887119894 119888119894 and 119889119894 (119894 = 234)

Variations of the model were developed and applied to other different materials to better accommodate the experimental data which are all then called a lsquomodified theta projection methodrsquo independently Examples are models in [51] for lowhigh alloy ferritic austenitic steels Al-alloys Al-matrix composites [52] for Nickel-based Super-alloy [53] for pure metals and class M alloys with intermediate range of stress (40-70 of the yield stress) and temperature (40-60 of the melting temperature)

Figure 4 Illustration of theta projection method [54]

16

322 A modified theta projection model

The new modified theta projection model we proposed [54] was mainly adapted from the original theta projection method [50] and the modified version [51] The new model describes the creep curve with the following equation

휀119888119903 = 1205791(1 minus 119890minus1205792119905) + 120579119898119905 + 1205793(119890

1205794119905 minus 1) (24)

An additional term 120579119898119905 is seen compared with Eq (22) in the original method The reason for adding this term is that the 16MND5 steel shows fairly long secondary stages in many experimental creep curves and the creep curves can be better fitted by Eq (24) than Eq (22) As the structure undergoes different creep process under different temperature and stress the model parameters θ119894 (119894 =1~4 and 119898) are also temperature and stress dependent

The main difference compared with the original and all other modified projection models is that numerical interpolation is used to replace the second numerical fitting in previous models In other words instead of fitting the model parameters as functions of stress and temperature with Eq (23) the distributions of the model parameters are assumed to be piecewise in the temperature-stress space and they will be interpolated directly from known parameters obtained from experiments The consideration is that the experimental data ([16] [17]) for 16MND5 covers a wide temperature range varying from 600 degC to 1300 degC and a wide stress load range varying from 08 MPa to 248 MPa while Eq (23) in the existing methods is based on a narrow temperature span Poor parameter fitting results would be expected if we directly fit the parameters 1205791~1205794 with Eq (23) as used in the original methods

The creep model was developed using the following steps

(i) Fitting Each creep curve can be represented by a group of model parameters [1205791 1205792 120579119898 1205793 1205794] with respect to Eq (24) by fitting the creep curves to Eq (24) All parameters of the 32 experimental curves form an

initial dataset of the model parameters [119879119895 120590119895 1205791119895 1205792119895 120579119898119895 1205793119895 1205794119895] where j (0ltjle32) is the jth creep curve

(ii) Interpolation Parameters of a creep curve under any [119879 σ] are directly interpolated from the dataset

(iii) Parameter adjustment The dataset was further updated to satisfy the monotonicity assumption we have taken that creep strain monotonically increases with temperature and stress

(iv) Different hardening models for dynamic loads

Step (i) was done using MATLAB In the following steps (ii)-(iv) are described in detail

Model parameters for new creep curves are interpolated from the dataset over the 2D 119879 minus 120590 space The interpolation is divided into two sub-interpolation steps interpolate over stress with fixed temperature and then interpolate over temperature with fixed stress We take the interpolation of model parameters under (119879prime σprime) for example In the first interpolation we first find out the two

17

temperatures 119879119896119899119900119908119899119904119898119886119897119897 and 119879119896119899119900119908119899119897119886119903119892119890 most close to 119879prime from the experimental

temperature set 600 700 800900 1000 1100 1200 1300 Under each temperature there are four sets of model parameters in the dataset corresponding to four different stress loads Then the first interpolation is done over stress load from the known dataset of model parameters under each

temperature to output the parameters for (119879119896119899119900119908119899119904119898119886119897119897 120590rsquo) and (119879119896119899119900119908119899119897119886119903119892119890 120590rsquo)

In the second interpolation the model outputs parameters for (119879rsquo 120590rsquo) from

(119879119896119899119900119908119899119904119898119886119897119897 120590rsquo) and (119879119896119899119900119908119899119897119886119903119892119890 120590rsquo) For 120579i(119894 = 12 3 4) we assume a logarithm

relation between 120579119894 temperature 119879 and stress 120590 which is similar to Eq (23)

119897119899120579119894 = (119886 + 119887120590)(119888 + 119889119879) (25)

It reduces to 120579119894 = 11986011198901198871σ with fixed temperature in the first interpolation and to

θ119894 = 11986021198901198872T with fixed stress in the second interpolation The interpolation

expression for θ119894 = 11986011198901198871σwith two known points (1205901 1205791198941) and (1205902 1205791198942) is given

by Eq (26) The interpolation expression for θ119894 = 11986021198901198872T with two known points

(1198791 1205791198941) and (1198792 1205791198942) is given by Eq (27)

120579119894 = 120579i1

120590minus12059021205901minus1205902120579i2

1205901minus1205901205901minus1205902 (26)

120579119894 = 120579i1

119879minus11987921198791minus1198792120579i2

1198791minus1198791198791minus1198792 (27)

Similarly we assume the following relation between 120579119898 temperature 119879 and stress 120590

120579119898 = 119860120590119899119890minus119887 119879frasl (28)

The interpolation is done over the stress with 120579119898 = 1198603120590n by fixing temperature

and then over temperature with 120579119898 = 1198604119890minus1198874119879 by fixing stress The interpolation

expression over stress for 120579119898 = 1198603120590119899 with two known points (1205901 1205791198981) and

(1205902 1205791198982) would be given by Eq (29) The interpolation expression over

temperature for 120579119898 = 1198604119890minus1198874119879 with two known points (1198791 1205791198981) and (1198792 1205791198982)

would be given by Eq (30)

120579119898 = 119890119909119901(119897119899120590 minus 11989711989912059011198971198991205902 minus 1198971198991205901

1198971198991205791198982 +1198971198991205902 minus 119897119899120590

1198971198991205902 minus 11989711989912059011198971198991205791198981) (29)

120579119898 = 120579m2(11198791minus1119879) (

11198791minus11198792)fraslsdot 120579m1

(1119879minus11198792) (

11198791minus11198792)frasl (30)

In this model we also take the assumption that the creep strain monotonically increases with temperature and stress loads1 Or specifically with the same creep time the structural component under higher temperature or stress experiences

1 Intuitively materials deform more under higher stress and temperature This is also mostly observed in tensile experiments Though some exceptions exist one instance of which is the effect of phase change some materials under certain temperature around the phase change temperature may show much weaker structural performance Here we just simply assumed that the creep strain monotonically increases with temperature and stress

18

higher creep strain than that with lower temperature or stress (the creep curve under higher temperature or stress should be above the curve with lower temperature or stress) When generating the parameter dataset (Step (i)) we already kept the trend that under the same temperature cases under higher stress loads would have larger creep model parameters and give higher creep strains This trend would also hold for the interpolated temperatures which thus guarantees that creep strain monotonically increases with stress load under same temperature Then the question becomes if creep strain also monotonically increases with temperature given fixed stress However the answer could not always be a yes especially in the extrapolation range of the stress load Figure 5(a) illustrates an example of the interpolation of model parameters under (119879119872 σ119872) The red dots and blue dots are known parameters from the parameter dataset for corresponding high temperature 119879119867 and low temperature 119879119871 respectively The green dots (119879119867 σ119872) and (119879119871 σ119872) are interpolated from the red and blue dots in the first sub-interpolation respectively The black dot (119879119872 σ119872) is interpolated from the green dots in the second sub-interpolation It is seen that when σ119872 lt σ119862 the value of (119879119871 σ119872) is higher than that of (119879119867 σ119872) It means the higher 119879119872 is the smaller the model parameter will be ie the model would predict larger creep strain for lower temperatures which violates our monotonicity assumption One simple solution used here as shown in Figure 5(b) is to minorly adjust the values of the four red and blue dots such that the intersection point C can be sufficiently small and the situation σ119872 lt σ119862 would barely occur in the extrapolation range

Figure 5 Illustration of model parameter interpolation (a) before and (b) after parameter adjustment [54]

A corresponding parameter adjustment process was introduced to verify that the model predicts higher creep strain under higher temperature under fixed stress Figure 6 illustrates the general verification process for any (119879 120590) in the interested load range the creep strain of (119879 + 120575119879 120590) should be larger than that of (119879 120590) Otherwise parameters would be minorly adjusted and the verification process be repeated This process would be iterated several times until the requirement is reached The achieved verification shows

Stress

σC

High temperature TH

MPa

a

0

Model

par

amet

er θ

Low temperature TL

MPa

a

0

Model

par

amet

er θ

High temperature TH

Low temperature TL

σC

Stress

C C

(a) (b)

σM

σM

19

(i) For any (119879 120590) in the temperature range 273~ 1073 K with an interval of 1 K and the stress range 1 ~260 MPa with an interval of 1 MPa its creep strain is higher than that of (119879 minus 1119870 120590) for the first 12000000 s

(ii) For any (119879 120590) in the temperature range 1073~1673 K with an interval of 1 K and the stress range 1 ~80 MPa with an interval of 1MPa its creep strain is higher than that of (119879 minus 1119870 120590) for the first 12000000 s

(iii) For any (119879 120590) in the stress range 01~26 MPa with a smaller interval of 01 MPa for the full temperature range 273~1673 K with an interval of 1 K its creep strain is higher than that of (119879 minus 1 120590) for the first 12000000 s

Above verification covers a wide range of temperature and stress especially where the stress and temperature are low and extrapolation is used Further verification to cover even higher stress or temperature ranges is not considered as creep increases drastically under such high stress and temperature and the results would become meaningless With these regards we generally think the monotonicity requirement is reached The model parameter dataset was updated along with the adjustment The final dataset was given as Table 2 in Paper I

Figure 6 Parameter adjustment process [54]

Figure 7(a-d) present the model predictions of selected creep curves in ANSYS Mechanical (labelled as lsquoANSYSrsquo) compared with experimental curves (labelled as lsquoREVISArsquo) at temperature 600 800 1000 and 1200 degC respectively The predicted curves proved to clearly show the three-stage creep process and agreed well with the experimental creep curves

Initial parametersdataset

θ1 θ2 θm θ3 θ4 from creep

curve-fitting

Parameters interpolation

Is the monotonicity

assumption satisfied

IsMon= =1

Final model parameters

Adjust the

parametersdataset

θ1 θ2 θm θ3 θ4

No

Yes

IsMon=1 a flag is monotonic achieved

For σ in the stress range in MPa

For T in range Tstart Tend in K

For t in range 112000000 in s

If ε(σ T+1t)lt ε(σTt) IsMon=0

Iteration loop of

parameter adjustment

20

Figure 7 Selected model predictions compared with experimental creep curves [54]

Instead of being constant temperature and stress loads also dynamically change with time in real cases Hardening models determine how creep strain rate reacts to the change of loads during a creep process The time hardening model and strain hardening model2 are supported in our model implementation in ANSYS Mechanical which are two mostly used hardening models

Time hardening assumes that the following creep strain rate after a load change depends only on the current creep time from the beginning of the creep process (and also temperature and stress if used in the creep model) and ignores the current creep strain Strain hardening assumes that the creep strain rate depends only upon the current creep strain (and also temperature and stress if used in the creep model) [55]

2 The time hardening and strain hardening here are more general terms than the specific time hardening creep model and the strain hardening creep model mentioned in Table 1

(a) 600 degC (b) 800 degC

(c) 1000 degC (d) 1200 degC

21

Eq (24) in default is a time hardening model as creep strain is directly described by the time 119905 Eq (31) gives the creep strain increment 119889휀119888119903 at time 119905 during time interval 119889119905

119889휀119888119903 = (12057911205792119890minus1205792119905 + 120579119898 + 12057931205794119890

1205794119905)119889119905 (31)

The strain hardening model is derived from Eq (24) in an implicit form Specifically when temperature and stress load change to (119879 120590) at the time 119905 and current creep strain 휀1198881199030 the model will get the creep curve of constant (119879 120590) and

output how this curve will react under the current strain 휀1198881199030 Since the creep curve shows creep strain monotonically increases with time given the creep strain 휀1198881199030 under creep curve of (119879 120590) we can uniquely find this reference time

119905119903119890119891 satisfying Eq (32) One should notice that the reference time 119905119903119890119891 can be

different from the current physical time 119905

휀1198881199030 = 1205791(1 minus 119890minus1205792119905119903119890119891) + 120579119898119905119903119890119891 + 1205793(119890

1205794119905119903119890119891 minus 1) (32)

where all 120579119894(119894 = 1~4119898) are model parameters of the new creep curve (119879 120590) It can be solved for 119905119903119890119891 with methods eg Newton-Ramphson [56] Then the creep

strain increment at 119905119903119890119891 during time interval 119889119905 is given as

119889휀119888119903 = (12057911205792119890minus1205792119905119903119890119891 + 120579119898 + 12057931205794119890

1205794119905119903119890119891)119889119905 (33)

Therefore combining Eq (32) and (33) gives the creep strain rate at creep strain 휀1198881199030 In summary when loads are changed to (119879 120590) with current creep strain 휀1198881199030

the model traces the creep curve of (119879 120590) and calculates the creep strain rate of this curve at strain 휀1198881199030

One calculation was done to demonstrate the performance of the hardening models The stress load was set to 151 MPa and kept unchanged A step temperature load was applied as the orange dash curve shows in Figure 8 it was 900 degC during 0-36000 s and suddenly jumped to 1000 degC at 36000 s which was then kept till the end of the simulation The blue solid curve (the part before the temperature change was identical to and therefore covered by the green solid curve in the figure) and blue dash curve are the creep curves under constant temperature T=900 degC and T=1000 degC respectively The green solid curve and green dash curve represent the model predictions considering this temperature change with time hardening model and strain hardening model respectively Before this change the creep curves with both hardening models were identical to the creep curve of the T=900 degC till point A After that the creep strain rate then shifted from Point A to Point B (which had the same time with A) in the time hardening model and behaved the same as the curve T=1000 degC as shown in the rest part of the green solid curve In the strain hardening model the creep strain rate shifted from Point A to Point C (which had the same creep strain with A) and behaved the same as the curve T=1000 degC as shown in the rest part of the green dash curve The large difference between two models was partially caused by the drastic temperature jump In real cases the difference would be less significant as the temperature changes more mildly Overall this prediction is exactly the expected behaviors of the time hardening model and strain hardening model

22

Figure 8 Time hardening and strain hardening in ANSYS Mechanical [54]

23

4 Coupled Thermo-Fluid-Structure Analysis of RPV Failure

As described in Section 13 RPV failure is coupled with melt pool heat transfer leading it into a transient thermo-fluid-structure coupled problem The goal of this section is to develop and validate a reliable coupling approach which supports transient coupled thermo-fluid-structure analysis as well as advanced models and computational platforms so as to improve fidelity and facilitate implementation The approach was validated against the FOREVER-EC2 experiment and then applied to an RPV failure analysis of a Nordic BWR As the approach adopts the volume loads mapping for load transfer comparative analysis was also performed against the coupling approach using surface loads mapping Future work is considered in analysis of penetration failures (IGTs and CRGTs) of a BWR RPV since the approach is flexible to complex geometries Results of Paper II ampIII are jointly presented in this section

41 Thermo-Fluid-Structure Coupling Approach

Considering the involved phenomena a coupling approach should at least address three issues (i) melt pool heat transfer (ii) RPV deformation and (iii) the interaction between melt pool heat transfer and vessel deformation A two-way strong coupling considering both the transient thermal load from melt pool on vessel and the influence of vessel deformation on melt pool heat transfer involves complex data exchange and is still full of challenges Therefore the new coupling approach is realized in a one-way manner with the assumptions that the influence of vessel deformation on melt pool heat transfer can be ignored and thus only the thermal loads from melt pool to vessel are considered It is based on the considerations that on one hand the deformation is a slow process regarding the relatively long failure time and on the other hand it was pointed out [57] that the physics of natural convection heat transfer in internally heated liquid pools is not sensitive to the geometrical factors based on their calculation results and experimental data from [58] [59] as well as previous experiments in semicircular and square cavities eg [44] [60] In addition one-way coupling is also computationally more efficient especially when heat transfer turbulence structural non-linearity and large-scale geometries are involved The developed approach is based on the ANSYS platform which generally consists of three parts with respect to the three issues respectively

(i) conjugate heat transfer of the melt pool and vessel using ANSYS Fluent (ii) transient thermal loads transfer from ANSYS Fluent to ANSYS

Structural (the ANSYS Mechanical in ANSYS Workbench) (iii) structural analysis of the vessel in ANSYS Structural

ANSYS Structural in default only supports direct transfer of constantsteady-state spatial loads from ANSYS Fluent Transfer of transient loads is realized with the extension tool FSI Transient [61] It is a pre-processing tool (with good documentation) developed by a third-party CAE vendor EDRTM MEDESO [62] to map transient CFD results (heat transfer coefficient temperature and surface pressure) to thermal or structural analysis including the definition of load steps

24

This new procedure proves to be straightforward and reduces efforts in the load transfer

Since structural performance of RPV is highly dependent on temperature the transfer of transient temperature loads from conjugate heat transfer to mechanical solver is a core step of the coupling framework There are two general approaches to realize the transfer as shown in Figure 9 Both approaches contain three parts a conjugate heat transfer analysis for fluid and structure where flow and heat transfer are coupled a mechanical analysis for the structure and a load mapping process that transfers the transient loads The conjugate heat transfer part can be solved with any CFD solver and is treated in the same manner in both approaches In the first approach (Figure 9(a)) transient volume thermal loads are transferred and mapped into the mechanical simulation domain The structure domain reads the essential thermal loads from the mapping after which a structure analysis of this domain is performed In the second approach (Figure 9(b)) transient surface thermal loads are mapped onto the boundary surfaces of structure domain as thermal boundary conditions Then a thermo-structure coupled analysis of the structure domain in the mechanical part is performed To distinguish them we call the former approach volume loads mapping (VLM) and the latter surface loads mapping (SLM) In the RPV failure analysis the fluid and structure domains correspond to the melt pool and RPV respectively Examples of these methods are the studies of [63] and [30] using VLM and [35] using SLM The VLM is adopted in current approach

Figure 9 Frameworks of the thermo-fluid-structure coupling approaches (a) volume loads mapping (b) surface loads mapping [64]

fluid structure structure

fluid structure structure

Conjugate heat transfer problem Mechanical problem

(structure)

Conjugate heat transfer problem Mechanical problem

(thermo-structure coupled)

Transient volume

loads mapping

(VLM)

Transient surface

loads mapping

(SLM)

(a)

(b)

25

42 Numerical Modelling

421 Conjugate heat transfer

In the conjugate heat transfer analysis main focus is put on the modelling of melt pool heat transfer Within the ANSYS Fluent solver two options are supported CFD with turbulence modelling and PECM CFD is the choice when the model goes to detail and more mechanistic analysis are needed while PECM provides an alternative that is faster and possibly more suitable for large-scale problems

Within the scope of RANS models the SST (Shear Stress Transport) model [65] is selected as the turbulence model in present study This model is a combination of k-ε model and k-ω model The k-ω formulation is used in the inner parts of the boundary layer which makes the model directly usable all the way down to the wall through the viscous sub-layer It also switches to k-ε model in the free-stream and thereby avoids the common k-ω problem that the model is too sensitive to the inlet free-stream turbulence properties The considerations on choosing this model are

bull Several two-equation low-Re k-ε models have been proved in [66] not good enough for melt pool heat transfer Therefore these mentioned two-equation models as well as other one-equation models are not considered

bull SST is highly recommended [67] for simulating most industrial heat transfer problems

bull Convincing results ([68] [69]) using SST have been obtained in capturing the thermal behaviour of the natural convection in an internally heated fluid Especially good agreements were achieved between the predicted heat flux distribution along the vessel and experimental data which would be one of the most important factors to estimate model performance in this problem

The description of PECM as an effective model for melt pool heat transfer is given in Section 23

422 Mechanical analysis

The material for the RPV failure analysis considered in this study is the French RPV steel 16MND5 The three-stage modified theta projection model in Section 3 is used as the creep model Since the definition of an RPV failure criterion is not universal (it can be the time when damage variable reaching 1 or either specified value of creep strain creep strain rate or total deformation is reached) we do not explicitly define a failure criterion Instead the simulation is kept running until the creep converge criterion cannot be satisfied even with the defined minimum time step (indicating a fairly large creep strain rate) and the simulation will be automatically terminated The time the simulation stops is taken as the failure time This treatment implicitly implies a criterion of a large creep strain rate We refer to [17] for other material properties Youngrsquos modulus and thermal conductivity are temperature dependent Multilinear Isotropic Hardening (MISO) is used as the plasticity model

26

43 Validation against the FOREVER-EC2 Experiment

In this section the validation against the FOREVER-EC2 experiment is presented (indicated as lsquoVLMrsquo in the following) In addition the simulation results using the SLM approach (indicated as lsquoSLMrsquo in the following) is also included for comparison purpose

431 FOREVER-EC2 experiment

The FOREVER-EC2 experiment [70] is one of the FOREVER experiments that achieved a vessel failure Figure 10 shows the experimental setup A 110 scale pressure vessel with a wall thickness of 15 mm and an outer diameter of ~400 mm was employed The lower head was made of 16MND5 steel Since it was difficult to use prototypical corium materials in the experiment the binary salt 30 CaO-70 B2O3 was employed as simulant To generate the internal heat which corresponds to the nuclear decay heat a specially designed heater was used The power input was maintained ~38 kW during the experiment such that maximum melt pool temperature would be ~1300 degC The pressure inside the vessel was maintained at ~25 MPa Thermocouples were used to measure the melt pool and vessel wall temperatures and linear displacement transducers used to measure the vessel wall displacements The designed experimental conditions were reached at experimental time 11670 s Vessel failure was observed about 12510 s later (at experiment time 24180 s)

Figure 10 Scheme of the experimental setup [22]

27

Figure 11 Computational domains and boundary conditions (a) conjugate heat transfer (b) mechanical analysis [71]

Computational domains and boundary conditions are indicated in Figure 11 SST turbulence modelling was used for pool heat transfer The simulations started at the moment when experimental conditions were reached (experiment time 11670 s) So simulation time 0 s corresponds to experiment time 11670 s

432 Conjugate heat transfer

Figure 12 shows the temperature fields at simulation time 0 s 1225 s 2475 s 4975 s and 9975 s All the time points in this part refer to simulation time The field at time 0 s (Figure 12(a)) was the initial temperature that was reconstructed based on the thermocouple readings The temperature increased with time and reached a steady state when the maximum temperature reached ~1607 K Figure 12(b-f) show clear thermal stratification which is an expected phenomenon of natural convection

Internal pressure

25 bar

Internal pressure 25e6-2500times981timesy Pa

External pressure

1 bar

Zero displacement

y=0 m

Gravity 400

mm

R=203 mm

15 m

m

Tamb

=323 K radiation with ε = 08

+ convection with h=10Wm2K

Tamb

=800 K radiation with ε=02

(a) (b)

Tamb

=800 K radiation with ε=08

+ convection with h=10 Wm2K

28

Figure 12 Temperature fields at simulation time (a) 0 s (b)1225 s (c) 2475s (d) 4975 s (e) 7475 s and (f) 9975 s [64]

Figure 13 shows the temperature profiles of the external vessel along the polar angle in comparison with experimental data Regions with a polar angle larger than 90deg are those in the cylindrical part of the vessel It also shows the temperature profile increased with time until reaching a steady state The profile almost did not change after 4975 s which indicates that a steady state of the temperature was reached As time increased the temperature in the lower head region 0~90deg (where the vessel was contacted with the melt) evolved from a relatively flat distribution to a distribution that temperature at high angle was higher than that at low angle by more than 200 K Generally the steady state results agreed well with the experimental data except for the low angle part of vessel lower head and the high angle part of the cylindrical vessel In addition to modelling errors the discrepancy in the low angle part of the lower head may be caused by the crust and gap possibly formed at vessel bottom in the experiment which can reduce the downward heat transfer and result in lower temperature while they cannot be modelled in simulations As to the high angle cylindrical vessel part simplified boundary conditions were applied to the inner and outer vessel wall since it was far from the melt and not of interest which may result in some discrepancy

(a) (b)

(c) (d) (e) (f)

29

Figure 13 Temperature profiles of external vessel surface along the vessel polar angle [64]

433 Mechanical analysis

In the mechanical analysis the VLM predicted a failure time of 23959 s and the SLM predicted a failure time of 22445 s both are close to the experimental failure time 24180 s All the time points in this part refer to the experiment time Figure 14 shows the creep strain fields at failure time Both approaches predicted maximum creep strains at same location where the temperature was high (cf Figure 12) This is related to the fact that creep behavior is highly influenced by temperature Maximum values for both cases were also close to each other 0393 in VLM and 0366 in SLM

Figure 15 presents the total deformation of the vessel bottom point (also called ldquosouth polerdquo in some literature) in the vertical direction Since there was an initial displacement reading of ~355 mm at time 0 s which almost did not change during 0~11670 s (the whole period before the experimental conditions were reached) an initial value of 355 mm was added to the simulation result such that experimental and simulation results had the same total displacement at t=11670 s Similar trend was seen in the VLM SLM and experimental data that the deformation increased with time and experienced a drastic acceleration at the end Generally both simulation approaches agreed well with the experimental data though they both slightly under predicted the value of total deformation The discrepancy may possibly be related to the accuracy of the thermal predictions modelling errors of plasticity and creep and simplified boundary conditions

400

600

800

1000

1200

1400

1600

0 20 40 60 80 100 120 140 160

Tem

per

ature

(K

)

Polar angle (deg)

Experiment

25 s

1225 s

2475 s

4975 s

7475 s

9975 s

Weilding line

30

Figure 14 Creep strain fields at failure time (a) VLM (b) SLM [64]

Figure 15 Total deformation in vertical direction of the vessel bottom point [64] [71]

The wall thickness changes are plotted together with experimental results in Figure 16 The wall thickness change δ119909 is defined as

δ119909 = 119909119890119899119889 minus 119909119900119903119894119892119894119899119886119897 (34)

000

500

1000

1500

2000

2500

11670 13670 15670 17670 19670 21670 23670

Tota

l def

orm

atio

n (

mm

)

Time (s)

ExperimentVLMSLM

(a) (b)

31

where 119909119890119899119889 and 119909119900119903119894119892119894119899119886119897 are the wall thickness at failure time and the original wall

thickness respectively Since 119909119890119899119889 cannot directly be post-processed it was estimated in the following way (cf the subfigure of Figure 16) for the same polar angle θ find the corresponding points Pin (1199091 1199101) and Pout (1199092 1199102) on the inner and outer surfaces of the deformed vessel respectively the distance of Pin and

Pout radic(1199091 minus 1199092)2 + (1199101 minus 1199102)2 is calculated as 119909119890119899119889 at polar angle 120579 Such estimation of the wall thickness is more accurate when the line PinPout is closer to be orthogonal to the deformed local surface Considering the final deformed vessel as shown in Figure 14 the estimation should be fine at regions close to 0deg and 90deg and can be slightly overestimated at medium angle region The VLM curve was compared with the SLM curve and 4 experimental curves measured at vessel left right front and back sides Generally VLM curve agreed well with the experimental and SLM curves wall thickness gradually decreases with angle till around 75deg and then increases till 90deg the maximum thickness change occurred at around 75deg would be around -5 mm The positive values at low angle may be caused by the thermal expansion It also shows that the thickness transitions in experimental curves were smoother than those in both simulation curves while simulation curves showed narrower regions with large thickness changes (at angle 70deg~80deg) Possible reasons might be (i) sharper temperature profile predicted in simulation (ii) input uncertainty and (iii) modelling errors The SLM predicted slightly thinner vessel than the VLM except for the necking region The discrepancy might be caused by various reasons eg the interior vessel temperature can be slightly different between two approaches which would then result in different thermal stress and different deformation the final deformed vessel shapes cannot be exactly the same which also results in different errors in estimating PinPout

Figure 16 Wall thickness changes along the vessel polar angle [64] [71]

-6

-5

-4

-3

-2

-1

0

1

0 10 20 30 40 50 60 70 80 90

Wal

l th

icknes

s ch

ange

(mm

)

Polar angle (deg)

VLMSLMExperiment (left)Experiment (right)Experiment (front)Experiment (back)

32

44 Application to BWR Case

In this section the coupling approach (indicated as lsquoVLMrsquo in the following) was applied to an RPV failure analysis of a BWR A simulation using the SLM coupling approach (indicated as lsquoSLMrsquo in the following) was also performed for comparison purpose

A hypothetical severe accident of a Nordic BWR initiated by an SBO was considered The core materials would melt and migrate downward to the RPV lower head The total amount of corium relocated in the lower head was considered to form a 19 m corium melt pooldebris bed assuming the corium distribution was homogeneous

Figure 17 Computational domains and boundary conditions (a) conjugate heat transfer (b) mechanical analysis [64]

Figure 17(a) shows the computational domain of the conjugate heat transfer PECM was used in this calculation A general mesh with a pool height of 23 m was used to accommodate most of the relocation situations with the same mesh Since the corium only had a height of 19 m the top 04 m region was treated as injected water covering the corium The top water was expected to be boiling and we assumed a uniform temperature of 383 K Under the implementation of PECM this region was damped with artificially high heat capacity and high conductivity such that it would efficiently remove heat from the corium and suppress its own temperature increase For the corium part below water a volumetric heat source 11 MWm3 was implemented as the nuclear decay heat For the in-vessel surfaces a constant temperature of 383K was applied as they are covered by water which can be boiling Without water cooling on the external vessel wall a very small heat loss of 20 Wm2 was applied Melting and solidification of the corium were also considered with a solidus temperature of 2750 K and a liquidus temperature of 2770 K The initial temperature was set as

q=20 Wm2

T=383 K

Qv=11

MWm3

315 bar

315 bar+ hydrostatic

pressure of corium

Gravity

248 bar

Zero

displacement

Water

Corium

(a) (b)

33

1100 K for the corium 383 K for the water and 450 K for the vessel For the mechanical analysis (Figure 17(b)) an inner pressure of 315 bar and an outer pressure of 248 bar were considered For the region covered by corium an extra hydrostatic pressure was added on the vessel wall A 2 m long vessel extension was considered as well as the standard earth gravity

Figure 18 presents the wall temperature profile along the inner vessel surface at 50 s 2500 s 7500 s and 10000 s The temperature increased with time for the vessel surface contacted with corium and the distribution was flat which was due to that large part of the corium was still in solidus status It also shows that for the surface region that contacted the water the temperature was basically unchanged which indicates that the damping treatment indeed suppressed the temperature increment and guaranteed the heat removal upward by water

Figure 18 Temperature profiles of inner vessel wall along the vessel polar angle [64]

The predicted RPV failure time was 9869 s by VLM and 9650 s by SLM Compared with the FOREVER-EC2 case in Section 43 the VLM predicted a slightly later failure time than the SLM in this BWR case too Figure 19(a) compares the maximum creep strain developments with time of both approaches Before 9000 s the predicted values by both approaches are close to each other Afterward the creep acceleration occurred and the strain increased drastically The acceleration of the VLM occurred 200 s later than that of the SLM which agreed with the predicted later failure time Similar trend was observed for the total deformation in vertical direction of vessel bottom point as shown in Figure 19(b) The deformations of both approaches are close except that the acceleration of VLM occurred later than that of SLM Overall results of VLM agreed well with those of SLM

0

500

1000

1500

2000

2500

0 10 20 30 40 50 60 70 80 90 100

Tem

per

ature

(K

)

Polar angle (deg)

50 s

2500 s

5000 s

7000 s

10000 s

34

Figure 19 Results of structural analysis (a) creep strain and (b) total deformation in vertical direction of vessel bottom point [64]

45 Computational Efficiency

Computational efficiency is also an important consideration regarding the performance of numerical approaches The computational efficiencies of both VLM and SLM are analysed and compared in this section As parallel computing technique was used to speed up the simulations the number of parallel cores and corresponding costed computational time for each simulation are given together in Table 2 as a generally reflection of the computational efficiency All the simulations were done in a Lenovo P700 workstation equipped with 2 Intel Xeon E5-2630 v3 processors (8 cores per processor) and 32 G RAM For the simulations of FOREVER-EC2 SLM employed twice the parallel cores and costed 286 more time compared with VLM For the simulations of prototypic BWR case both VLM and SLM employed the same number of parallel cores but VLM only took 13 of the computational time compared with SLM Overall it can be concluded that VLM has higher computational efficiency than the SLM The reason is a structural analysis is performed in the VLM approach while a coupled thermo-structural analysis is performed in the SLM approach (cf Figure 9) a standalone structural analysis is computed faster than a coupled thermo-structural analysis

Table 2 Comparison of computational efficiency [64]

Case FOREVER-EC2 BWR VLM SLM VLM SLM

Number of cores 4 8 4 4 Computational time (hours) 21 27 18 72

35

5 Hybrid Coupling Approach for Efficient RPV Failure Analysis

In this section we propose a hybrid coupling approach for the efficient RPV failure analysis The idea is to replace the ANSYS Fluent in previous coupling approach with a lumped-parameter code for quick estimate of the melt pool heat transfer With the least modelling details (compared with CFD methods and distributed-parameter methods eg PECM) lumped-parameter codes prove to be the most efficient for melt pool heat transfer The new hybrid approach characterizes 1) utilization of efficient models for melt pool heat transfer and 2) the support of detailed structural modelling of RPV deformation allowing studies with different RPV geometric shapes under effects like local thinning due to ablation For studies of which the focus is mainly on structural behaviour of RPV eg [13] [26] while mechanistic simulation of melt pool convection is computationally expensive and unnecessary the lumped-parameter code could help providing reasonable thermal loads

The development of the approach consists of two tasks (i) development of a lumped-parameter code and (ii) its coupling with ANSYS Mechanical to form this hybrid coupling framework In the following we describe both tasks and the corresponding verification and validation processes respectively This section is mainly based on Paper IV

51 A Lumped-parameter Code for Melt Pool Heat Transfer

A lumped-parameter code transIVR was developed It features (i) quick estimate of transient melt pool heat transfer with reasonable accuracies and (ii) two-dimensional representation of heat conduction in the RPV wall facilitating the coupled analysis of the RPV failure under thermal loads The following assumptions are taken in the transIVR modelling

bull The crust between the oxide pool and metal pool is thin

bull The heat flux has uniform distributions on the bottom top and side surfaces of metal pool

bull Heat source only exists in oxide layer

bull Latent heat of fusion is ignored

Figure 20 illustrates the modelled three domains oxide layer metal layer and the RPV wall In addition the internal structure inside the vessel can also be considered as a radiative body to the top metal surface as in [72] An internal volumetric heat source generated by nuclear decay heat is considered in the oxide layer Heat in oxide layer transfers downward to the RPV wall and upward to the top metal layer Heat in metal layer can be transferred sideward to the RPV wall and upward to the surrounding internal structure through radiation (or to water if inside water flooding is applied) The heat transferred to RPV will be removed from outer RPV surface Possible formations of oxide crust in the top and bottom surface are illustrated as the black regions Possible formation of metal crust on the top surface is shown as the grey region The code automatically detects the

36

existence of the crust on oxide top and metal top surfaces during the simulation and solves the problem correspondingly withwithout crust Boundary conditions are applied to the external and internal surfaces of RPV and (if needed) top surface of metal layer Arbitrary boundary conditions are supported If a boundary condition is nonlinear eg mixed heat convection (linear dependent on 119879 ) and heat radiation (proportional to 1198794) the Picard method [73] would be used for solving this problem

This model consists of three main parts

(i) empirical heat transfer correlations (mostly from experiments) to determine global and local heat transfer coefficients

(ii) energy balances in oxide layer and metal layer where the heat transfer coefficients would be used and

(iii) energy balance in RPV

Figure 20 Two-layer melt pool configuration and RPV wall [74]

511 Heat transfer correlations

For internally heated natural convection (oxide layer) in transIVR the mini-ACOPO correlations [75] are used for the averaged heat transfer as expressed in Eq (35) and (36) and for the angular variation of the heat flux as expressed in Eq (37) and (38)

119873119906119906119901 = 03451198771198861198941198990233 (35)

119873119906119889119899 = 00038119877119886119894119899035(119867119877)025 (36)

119873119906119889119899(θ)

119873119906119889119899= 01 + 108(

θ

θ119898119886119909) minus 45 (

θ

θ119898119886119909)2

+ 86 (θ

θ119898119886119909)3

01 leθ

θ119898119886119909le 06 (37)

RPV

Oxide pool

Metal pool

37

119873119906119889119899(θ)

119873119906119889119899= 041 + 035 (

θ

θ119898119886119909) + (

θ

θ119898119886119909)2

06 ltθ

θ119898119886119909le 1 (38)

The 119877119886119894119899 that appears in Eq (35)~(36) is the internal Rayleigh number which is defined as

119877119886119894119899 =119892β119876119881119867

5

α119896ν (39)

For Rayleigh-Bernard convection in the metal layer heat transfer occurs in all the top bottom and side surfaces To side surface the reduced form [72] of the Churchill-Chu correlation [76] for 119875119903 = 013 is employed as shown in Eq (40)

119873119906119904119889 = 007611987711988613 (40)

To top and bottom surfaces the reduced form [72] of Globe-Dropkin correlation [77] for

119875119903 = 013 is used as expressed in Eq (41)

119873119906119906119901 = 005911987711988613 (41)

The 119877119886 that appears in Eq (40) and (41) is the external Rayleigh number which is defined

as

119877119886 =119892βΔ1198791198673

να (42)

Heat transfer coefficients can be calculated from Nusselt number by Eq (43)

ℎ =119873119906119896

119871 (43)

where 119896 and 119871 are the heat conductivity and characteristic length respectively 119873119906 can be the Nusselt numbers expressed in Eq (35)~(38) and (40)~(41)

512 Energy balance in the pools

The energy balance of oxide layer can be described as Eq (44) The total generated heat of oxide pool equals the sum of the increase of enthalpy (transient temperature change) upward heat flow to metal layer and downward heat flow to RPV

120588119862119901119881119900119897119889119879119900119909119889119905

+ ℎ119906119901119860119900119909119906119901(119879119900119909 minus 119879119900119909119906119901) + ℎ119889119899119860119900119909119889119899(119879119900119909 minus 119879119900119909119889119899) minus 119876119881119881119900119897 = 0 (44)

where ρ is the oxide density 119862119901 the heat capacity 119881119900119897 the total volume 119879119900119909 the bulk temperature 119905 the time ℎ119906119901 the upward heat transfer coefficient 119860119900119909119906119901

upward heat transfer area 119879119900119909119906119901 upward surface temperature ℎ119889119899 downward

heat transfer coefficient 119860119900119909119889119899 downward heat transfer area 119879119900119909119889119899 downward surface temperature and 119876119881 volumetric heat source

38

On the interface between oxide layer and metal layer energy balance requires that the heat transferred from oxide layer should equal the heat to the metal layer which can be described as Eq (45)

ℎ119906119901119860119900119909119906119901(119879119900119909 minus 119879119900119909119906119901) = ℎ119898119905119889119899119860119898119905119889119899(119879119894119899119905 minus 119879119898119905) (45)

where ℎ119898119905119889119899 is the downward heat transfer coefficient of metal layer 119860119898119905119889119899 the downward heat transfer area 119879119894119899119905 the interface temperature and 119879119898119905 the bulk temperature of metal layer

The energy balance of metal pool can be described as Eq (46) The increase of enthalpy (transient temperature change) equals the sum of upward downward and sideward heat flows to surrounding surfaces

ρ119898119905119862119901119898119905119881119900119897119898119905119889119879119898119905119889119905

+ ℎ119898119905119906119901119860119898119905119906119901(119879119898119905 minus 119879119898119905119906119901) + ℎ119898119905119904119889119860119898119905119904119889(119879119898119905 minus 119879119898119905119904119889)

+ ℎ119898119905119889119899119860119898119905119889119899(119879119898119905 minus 119879119894119899119905) = 0

(46)

where ρmt is the metal density 119862119901mt the heat capacity 119881119900119897mt the total volume 119879119898119905 the bulk metal temperature 119905 the time ℎ119898119905119906119901 the upward heat transfer

coefficient 119860119898119905119906119901 upward heat transfer area 119879119898119905119906119901 upward surface

temperature ℎ119898119905119904119889 sideward heat transfer coefficient 119860119898119905119904119889 sideward heat transfer area 119879119898119905119904119889 sideward surface temperature ℎ119898119905119889119899 downward heat transfer coefficient 119860119898119905119889119899 downward heat transfer area and 119879119894119899119905 oxide-metal interface temperature

The heat transferred from metal layer to the top surface should equal the heat radiated from this surface to the environment Considering the radiation and reflections of internal structures to this surface following energy balance equation [72] is achieved

ℎ119898119905119906119901(119879119898119905 minus 119879119898119905119906119901) = σ(1198791198981199051199051199001199014 minus 119879119904

4) (1

ε+1 minus ε119904ε119904

119860119898119905119906119901

119860119904)minus1

(47)

where 119879119898119905119905119900119901 is the temperature of metal top free surface ε the emissivity of

metal ε119904 the emissivity of internal structures 119860119898119905119906119901 the area of metal top surface

and 119860119904 the total area of internal structures Sometimes the internal water cooling may be available above metal layer In this case Eq (47) is unnecessary and 119879119898119905119905119900119901 can be set to a constant value

For the oxide layer top surface Eq (48) is used if not crust is formed in the interface between two layers ie 119879119894119899119905 gt 119879119898119890119897119905119894119899119892119900119909 Otherwise Eq (49) is used

119879119900119909119906119901 = 119879119894119899119905 (48)

119879119900119909119906119901 = 119879119898119890119897119905119894119899119892119900119909 (49)

For the metal layer top surface Eq (50) is used if not crust is formed on top of the metal ie 119879119906119901119904119906119903119891 gt 119879119898119890119897119905119894119899119892119898119905 Otherwise Eq (51) is used

119879119898119905119906119901 = 119879119898119905119905119900119901 (50)

39

119879119898119905119906119901 = 119879119898119890119897119905119894119899119892119898119905 (51)

Overall this part involves 6 equations regarding 6 unknows 119879119900119909 119879119900119909119906119901 119879119894119899119905 119879119898119905

119879119898119905119906119901 and 119879119898119905119905119900119901

513 Energy balance in the RPV wall

The heat transfer in the PRV wall can be modelled as a simple 1D heat conduction problem that heat only transfers in the wall thickness direction or 2D axisymmetric heat conduction problem

For 1D model the treatment in [72] was adopted As illustrated in Figure 21 the heat from the oxide pool transfers to inner vessel surface (119879119894) then to the interior vessel (119879119907119890119904) then to outer vessel (119879119900) and finally to environment

Figure 21 1D heat transfer model in RPV [74]

Unknown variables are the heat flux through inner vessel surface 119902119894119899 the crust thickness 119909119888119903 inner vessel temperature 119879119894 vessel center temperature 119879119907119890119904 and outer vessel temperature 119879119900 Eq (52)~(56) give a closed equation set for these variables considering a radiation boundary condition on the external vessel surface Eq (52)~(53) describe the heat transfer through the crust where an internal heat source 119876119888119903 is also considered Eq (52) can be easily derived from the one-dimensional heat transfer equation with heat source Eq (54)~(55) describe the energy balance on the inner vessel surface and vessel interior respectively Eq (56) describes the energy balance on the outer vessel surface regarding radiation boundary condition For Dirichlet boundary condition Eq (56) is replaced with Eq (57) For other boundary conditions we only need to correspondingly modify the terms on the right-hand side of Eq (56) For situations that no crust is formed on the surface of the vessel Eq (52)~(53) are eliminated and 119902119894119899 in Eq (54) can directly be determined from calculations in Section 512 and Eq (54)~(56) are solved for the unknows 119879119894 119879119907119890119904 and 119879119900 Once 119879119894 exceeds the vessel melting temperature (with or without crust formation) wall ablation is considered to occur Then 119879119894 is fixed to the vessel melting temperature and vessel thickness 119909119907119890119904 becomes the new unknown variable

119879119900119909119889119899 119879119894 119879119907119890119904 119879119900 119879119890119899119907

119909119888119903 119909119907119890119904

crust RPV

Qcr

119902119894119899 119902119897119900119888119886119897

40

119902119894119899 = 119896119888119903(119879119900119909119889119899 minus 119879119894)

119909119888119903+ 1198761198881199031199091198881199032

(52)

119902119894119899 = 119902119897119900119888119886119897 + 119876119888119903119909119888119903

(53)

119902119894119899 = 119896119907119890119904(119879119894 minus 119879119907119890119904)

05119909119907119890119904

(54)

120588119907119890119904119862119901119907119890119904119889119879119907119890119904119889119905

= 119896119907119890119904(minus2 lowast 119879119907119890119904 + 119879119894 + 119879119900)

(05119909119907119890119904)2

(55)

119896119907119890119904(119879119907119890119904 minus 119879119900)

05119909119907119890119904= 휀120590(119879119900

4 minus 1198791198901198991199074 )

(56)

119879119900 = 119879119900119888119900119899119904119905

(57)

For the 2D representation the heat conduction in the RPV is solved using FDM considering the heat transfer in both the thickness and the angular directions in the RPV wall The RPV lower head is treated as a spherical shell The corresponding heat conduction equation in the 2D spherical coordinate is given as Eq (58) The cylindrical part of the vessel is treated as a cylindrical shell The corresponding heat conduction equation in the 2D cylindrical coordinate is given as Eq (59) Eq (58) and (59) are coupled and solved together In default the mesh size is 5deg in angular direction and 0025119909119888119903 in radius direction This mesh discretization can be changed up to user demands

ρ119862119901part119879

part119905=

1

1199032part

part119903(1198961199032

part119879

part119903) +

1

r2sinθ

part

partθ(119896119904119894119899θ

part119879

partθ)

(58)

ρ119862119901part119879

part119905=1

119903

part

part119903(119896119903

part119879

part119903) +

part

part119911(119896part119879

part119911) (59)

52 Hybrid Coupling Approach

The hybrid coupling approach to assess the RPV integrity is illustrated in Figure 22 transIVR and ANSYS Mechanical are coupled in a one-way manner transIVR conducts the conjugate heat transfer of melt pool and RPV and provides transient thermal loads to the ANSYS Mechanical where structural analysis of RPV deformation is performed The 2D heat transfer of RPV is solved in transIVR to better capture the temperature distribution In RPV structural analysis same considerations are taken as in Section 422 for 16MND5 steel the modified theta projection model for creep and MISO for plasticity and temperature-dependent Young modulus etc The VLM is adopted for the transfer of transient thermal loads

41

Figure 22 Framework of the hybrid coupling approach [74]

53 Verification and Validation

The transIVR code was first benchmarked against the UCSB FIBS case of two-layer pool configuration and then validated against the LIVE-7V experiment showing the codersquos capability in predicting two-layer and transient melt pool heat transfers respectively Finally the hybrid coupling approach encompassing transIVR and ANSYS Mechanical was validated against the RPV failure experiment FOREVER-EC2 and the results showed acceptable agreements with those of both the experiment and the previous approach using CFD in Section 43 Hence the transIVR code is an efficient tool for melt pool heat transfer and the hybrid approach an efficient way for RPV failure analysis

531 UCSB FIBS benchmark case

The UCSB FIBS benchmark case [78] was calculated to demonstrate the capability of the transIVR code in simulating two-layer pool heat transfer problems The considered pool configuration was similar to that shown in Figure 20 with the following specific dimensions a two-layer melt pool with a 16 m oxide layer and a 08 m metallic pool on top and an RPV lower head with a thickness of 015 m The internal volumetric source in the oxide layer was 13 MWm3 The calculation results were compared with those of the DOE study [78] and the INEEL study [79]

The predicted results were compared with the digitalized results of the DOE and INEEL studies in Figure 23 All three studies show similar profiles of the heat flux distribution along the polar angle (Figure 23(a)) it increased with the polar angle in the oxide layer to the maximum value at around 77deg and then dropped to a significantly lower value in the metal layer The transIVR result agreed well with the other results with small discrepancy in the metal layer the DOE study gave a

heat flux of 500 MWm2 the INEEL study of 400 MWm2 and the transIVR of somewhere in between Figure 23(b) shows the predicted corium crust thickness distribution along the RPV polar angle also agreed well the DOE and INEEL

transIVR ANSYS Mechanical

Transient

thermal loads

42

studies The crust thickness decreased with polar angle in the oxide layer and no crust was formed in the metal layer As the crust formation is closely related to the heat flux distribution the crust thickness profile was reasonable regarding the heat flux distribution shown in Figure 23(a) Vessel wall ablation would occur due to the high thermal loads Figure 23(c) gives the profile of the remaining vessel wall thickness along the RPV polar angle considering the wall ablation It shows that no ablation occurred in low angle region (0deg~36deg) where heat flux was low and the wall thickness changed from 15 cm to around 6 cm as angle increased in high angle region of the oxide layer Because of the discrepancy in the heat flux of metal layer the predicted wall thicknesses also differ from those of the DOE and INEEL studies the thinnest wall was 73 cm predicted by the DOE study corresponding to the maximum heat flux followed by 88 cm of the present transIVR calculation and 94 cm of the INEEL study Generally speaking the prediction of transIVR is in an acceptable level compared with the other studies for the same benchmark case

Figure 23 Simulation results of the UCSB FIBS benchmark case [74]

(a) heat flux distribution (b) crust thickness distribution

(c) wall thickness distribution

43

532 LIVE-7V experiment

LIVE-7V is an experiment under the LIVE program conducted in KIT to study the core melt phenomena As Figure 24(a) shows it employed the LIVE-3D hemispherical facility with an inner diameter of 1 m and a wall thickness of ~25 mm The non-eutectic binary salt mixture of 20 mol NaNO3- 80 mol KNO3 was used as the simulant material The decay heat in the melt was simulated by cable-type heating with 8 planes of electrical resistance heating wires Water cooling was applied to the top and vessel surfaces Top surface cooling was realized by a cooling lid mounted at height 413 mm from the vessel bottom and vessel cooling by the flow channel between the test vessel and the outer cooling vessel The inlet and outlet water temperature for both top and vessel cooling surfaces were kept ~293 K and ~303 K throughout the whole experiment respectively A 413 mm height pool corresponds to a total volume of ~0194 m3 simulant material The experiment procedure can be divided into four stages with the heating power sequentially shifted from 29 kW to 24 kW 18 kW and 9 kW Figure 24(b) indicates the thermocouple positions in experiment

The simulation results of LIVE-7V using transIVR were compared with experimental data [80] and numerical study using REALPSCDAPSIM code [81] For simplicity the RELAPSCDAPSIM results in the following are labelled as lsquoCOUPLErsquo which is the name of the module in RELAPSCDAPSIM for melt pool heat transfer

Figure 24 LIVE test facility [80] and thermocouple positions [82]

Figure 25 compares the predicted heat removal upward to the top lid and downward to the bottom vessel surface with those of the experiment and COUPLE The black curve indicates the total input power at the four stages The experimental results (red and blue dots) were calculated from the Table 3 of [80] assuming no extra heat losses For the first three stages the transIVR predictions agreed well with the COUPLE results while the predicted upward heat flow was slightly lower and the downward heat flow higher than the experimental data For the last stage the transIVR agreed well with the experimental result while the COUPLE predicted lower upward and higher downward heat flows The short

(a) (b)

44

transition periods between two stages with different total inputs taken in transIVR also agreed with those in the COUPLE predictions

Figure 25 Comparison of heat removal upward to the top melt surface cooling lid and downward to bottom vessel wall [74]

Figure 26 shows the comparison of heat flux distribution along the vessel wall All curves show similar trend that a higher total input power would result in a higher heat flux profile As the COUPLE predictions had relatively flatter profiles transIVR predictions generally agreed better with the experimental results especially when total input was high A discrepancy is seen that the minimum value of a curve occurred at angle 0degfor both transIVR and COUPLE and at angle 30deg for the experimental data It is noted that the models in default assume heat flux monotonically increases with angle (cf Eq (37) and (38)) resulting it impossible to predict higher heat flux at angle 0deg than angle 30deg as in experimental results

45

Figure 26 Heat flux profiles along the vessel polar angle [74]

533 FOREVER-EC2 experiment

In this section we performed calculations against the FOREVER-EC2 experiment to validate the performance of the coupled transIVR and ANSYS Mechanical in assessing the RPV integrity The applied boundary conditions were identical to the settings of previous coupling approach in Section 4 (ANSYS Fluent+ Mechanical) as indicated in Figure 11 The initial bulk temperature of the melt pool was set to be 1261 K in transIVR which corresponds to the maximum value of the initial temperature field in CFD calculation of previous coupling approach Both the experimental data and the simulation results using previous coupling approach are included for comparison In the following results from current hybrid approach are labelled lsquotransIVRrsquo (and referred as transIVR results) results from previous approach labelled lsquoCFDrsquo (and referred as CFD results) and results from experiment labelled lsquoexperimentrsquo

Figure 27 compares the bulk temperature predicted by transIVR with the averaged and maximum temperatures of the CFD results All curves show the same trend that the temperature gradually increased with time till reaching a steady state and then kept unchanged till the end The times needed for reaching the steady state were ~5000 s for the CFD and ~4000 s for the transIVR which are close to each other It indicates again a good performance of the transIVR in predicting the transient behavior of melt pool Compared with the steady-state averaged temperature 1478 K and maximum temperature 1609 K of the CFD results the steady-state temperature of transIVR 1599 K is a reasonable prediction

0

3000

6000

9000

12000

15000

18000

0 20 40 60 80 100

Hea

t fl

ux

(W

m2)

Polar angle (deg)

29 kW-EXP24 kW-EXP18 kW-EXP9 kW-EXP29 kW-COUPLE24 kW-COUPLE18 kW-COUPLE9 kW-COUPLE29 kW-transIVR24 kW-transIVR18 kW-transIVR9 kW-transIVR

46

Figure 27 Temperature evolution with time [74]

Figure 28 Temperature fields at time (a) 01 s (b) 3038 s (c) 6038 s and (d) 13289 s [74]

Figure 28 shows the temperature fields of the vessel at time 01 s 3038 s 6038 s and 13289 s The temperature fields were visualized in the ANSYS Mechanical It also indicates that the transient temperature loads were properly transferred from transIVR to ANSYS Mechanical Figure 28(a-d) show both the minimum and maximum temperatures increased with time In the lower head region the temperature gradually increased with the height The maximum temperature occurs at the location slightly lower than the melt pool top surface This is because thermal loadheat flux imposed on the vessel generally increases with polar angle in the pool natural convection so the code transIVR inherently also predicted higher heat flux at higher angle positions The higher flux at higher position then correspondingly resulted in higher temperature In the cylindrical region the

(a) (b) (c) (d)

[K] [K] [K] [K]

47

temperature decreased as the height since the bottom part would be more significantly heated by the high-temperature lower head region

Figure 29 shows the steady-state temperature profiles of the external vessel surface along the polar angle The transIVR result agreed well with other two results in trend that the temperature increased with the angle and reach the peak values at angle ~80deg then gradually decreased with angle But slightly lower temperature was predicted in the low angle region Possible reason may be that the local heat flux distribution determined by Eq (37) and (38) is steeper than that in the experiment and CFD study which then resulted in a steeper increase of temperature along the angle and a lower temperature at low angle region In the cylindrical part (gt90deg) the transIVR result generally agreed with the experimental and CFD results The discrepancy with the experimental data may be caused by the simplified boundary conditions the ambient temperature inside the vessel was set to uniform constant value while it could decrease as height Therefore predicted heat loss in the higher position of cylindrical part could be smaller than that with height-dependent ambient temperature resulting in higher vessel temperature Overall the prediction was reasonable for a lumped-parameter model

Figure 29 Temperature profiles of the external vessel surface along the vessel polar angle [74]

The transIVR predicted a failure time of 24959 s which is close to 24180 s in experiment and 23959 s in the CFD results Figure 30 shows the total deformation of vessel bottom point with time All three curves increased gradually with time and experienced drastic accelerations at the end The

400

600

800

1000

1200

1400

0 40 80 120 160

Tem

per

ature

(K

)

Polar angle (deg)

transIVRExperimentCFDWeilding Line

48

transIVR curve agreed well with the CFD curve The final total deformation at failure time is also close to that of the CFD result but lower than the experiment data The discrepancy might be related to the simplifications in the boundary conditions accuracy of the thermal load prediction and the modelling errors in plasticity and creep

Figure 30 Total deformation in vertical direction of the vessel bottom point [74]

Figure 31 shows the profile of wall thickness change The transIVR curve was compared with the CFD curve and four experimental curves measured at vessel left right front and back sides The simulation result fitted the experimental and CFD results well in the region 0deg~50deg In the region 50deg~90deg wall thickness gradually decreased with angle till around 75deg and then increased till 90deg The maximum thickness change was -3 mm for the simulation which is close to the experiment left side curve and slightly larger than the -6~ -45 mm of the other curves from experiment and CFD results Concerning possible errors and uncertainties in the modelling and numerical settings the new approach generally gives reasonable predictions

0

5

10

15

20

25

11670 15170 18670 22170 25670

Tota

l def

orm

atio

n (

mm

)

Time (s)

Experiment

CFD

transIVR

49

Figure 31 Wall thickness changes along the vessel polar angle [74]

-6

-5

-4

-3

-2

-1

0

1

0 20 40 60 80 100

Wal

l th

icknes

s ch

ange

(mm

)

Polar angle (deg)

CFD

transIVR

Experiment (left)

Experiment (right)

Experiment (front)

Experiment (back)

50

51

6 Summary and Outlook

Reactor pressure vessel (RPV) is one of the key physical barriers in the defence-in-depth approach of nuclear power safety and its failure is also the transition point of core melt (corium) progression from in-vessel to ex-vessel phase Quantification of RPV failure during a hypothetical severe accident is therefore paramount to the specific qualification of severe accident mitigation strategies as well as the general assessment of corium risk in light water reactors The RPV failure is a problem of structural performance of RPV under the influence of thermal loads imposed by corium melt pool ie it is a transient thermo-fluid-structure coupled problem per se whose solution requires simultaneous modelling of both highly turbulent melt pool heat transfer and large-scale nonlinear RPV creep Due to the complexity of the problem only limited capabilities have been developed in the previous studies under different simplifications and assumptions in models and coupling methods The present study was motivated to advance modelling and coupling approaches for RPV performance analysis considering new understanding of physical phenomena and latest coupling platform of simulation tools

For the new development of coupling approaches and models for physical phenomena related to RPV failure analysis during severe accidents this thesis work was conducted in the following order (i) a new creep model for the RPV steel 16MND5 was developed which features the capability to cover all three creep stages (ii) a new thermo-fluid-structure coupling approach was developed for RPV failure analysis and validated against FOREVER-EC2 experiment (iii) the performance of the new coupling approach using volume loads mapping (VLM) for data transfer was further compared with another approach using surface loads mapping (SLM) (iv) a hybrid coupling approach was proposed for efficient analysis of RPV failure which employs a lumped-parameter code for quick estimate of thermal load and a mechanistic FE code for detailed structural analysis and (v) for the hybrid coupling approach the code transIVR was developed for calculation of transient melt pool heat transfer

The developed creep model for 16MND5 is called lsquomodified theta projection modelrsquo and describes all three stages of a creep process Creep curves are expressed as a function of time with five model parameters 120579119894(119894 = 14 119886119899119889 119898) A dataset for the model parameters was constructed by fitting experimental creep curves into this function and an iterative parameter adjustment process to satisfy the monotonicity assumption of creep strain vs temperature and stress Model parameters for new curves under different temperature and stress loads are directly interpolated from this dataset in contrast to the conventional ldquotheta projection modelrdquo that employs an extra fitting process The new treatment was proven to better accommodate all experimental curves over the wide ranges of temperature and stress loads The model was implemented into the ANSYS Mechanical code The simulation results indicate that the model successfully captured all three creep stages and a good agreement was achieved between each experimental creep curve and corresponding modelrsquos prediction Regarding the model response to transient changes of temperature and stress loads the widely used time hardening and strain hardening models were implemented and

52

performing reasonably well in the new method These properties fulfil the requirements of a creep model for structural analysis

The transient thermo-fluid-structure coupling approach was developed using the ANSYS Fluent code for thermofluid dynamics of the melt pool and ANSYS Structural code for structural mechanics of RPV An extension tool was introduced to realize transient load transfer from ANSYS Fluent to Structural and minimize the user effort Both turbulence models and the effective model PECM can be used for melt pool heat transfer The developed creep model was used for RPV failure analysis of the RPV Since it is insensitive to geometry the coupling approach allows more complex geometries such as a BWR lower head with a forest of penetrations (CRGTs and IGTs) modelled in detail Moreover the coupling approach does not only reflect the transient response of thermo-fluid-structure interaction but also facilitates the implementation of advanced (high-fidelity) models in both fluid mechanics and structural mechanics The validation of the coupling approach against the FOREVER-EC2 experiment showed that the temperature profile vessel failure time total displacement of vessel bottom point and wall thickness profile in the experiment were well predicted

Following the variation of the coupling approach from VLM to SLM for data transfer between ANSYS Fluent and ANSYS Structural the respective coupling approaches were investigated by comparative calculations against the FOREVER-EC2 experiment and a postulated severe accident of a reference BWR For the FOREVER-EC2 experiment both VLM and SLM approaches predicted generally well the total deformation in vertical direction of bottom point and the wall thickness change though the former predicted a slightly later failure time For the reference BWR case good agreements were achieved in predicting the maximum creep strain and the total deformation in vertical direction of bottom point For both experiment and reactor cases the VLM approach predicted slightly later failure times than SLM but the differences were negligible compared to the long time-scale of vessel failure (~104 s) The deformation rate of the vessel wall was slow at the early stage and accelerated with time till a drastic deformation and final vessel failure Generally speaking the VLM and SLM coupling approaches had the similar performance in terms of their predictability of experiment and applicability to reactor case but the VLM was computationally more efficient

In the development of the hybrid coupling approach the transIVR code was developed for quick estimate of transient melt pool heat transfer and determination of thermal loads on RPV transIVR does not only support transient analysis of one- and two- layer melt pool but also the heat transfer in RPV being resolved with 2D finite difference method which captures more spatial thermal details for RPV structural analysis in addition to 1D heat transfer model The capabilities of transIVR in modelling two-layer pool heat transfer and transient pool heat transfer were demonstrated by calculations against the UCSB FIBS benchmark case and the LIVE-7V experiment respectively It was then coupled to the mechanical solver ANSYS Mechanical which together form the hybrid coupling framework for detailed RPV failure analysis The calculation against the FOREVER-EC2 experiment did not only reproduce the major results of the experiment but also had a good agreement with the thermo-fluid-structure

53

coupling approach using CFD This indicated the new coupling approach is capable of capturing the vessel creep failure characteristics and can be employed as an efficient tool addressing the complex vessel failure problem

As a prospective more research topics can be addressed in future either as a continuation of the current work toward an effective tool for vessel failure analysis or from the general view of scientific interest in melt pool heat transfer and RPV failure The following topics are recommended

bull Extended applications to the reactor cases For instance the thermo-fluid-structure coupling approach can be used to study penetration failures in the RPV of an BWR with a detailed representation of penetrations (IGTs and CRGTs)

bull Synthesis or update of heat transfer correlations for melt pool Heat transfer correlations directly benefit the parametric codes such as transIVR in accurate quantification of heat flux distribution around a melt pool So far many empirical correlations have been proposed based on experiments which had a variety of geometries and melt simulants with a wide range of Rayleigh number (Ra) Sensitivity studies have also been performed on the influences of Ra Pa and other material properties Therefore a comprehensive synthesis or update of the existing heat transfer correlations will be a meaningful step for making reasonable extrapolations to the reactor applications with prototypic materials

bull Modelling of pool stratification The motivation starts with the improvement of the PECMrsquos modelling capability such that the dynamic stratification and resulting pool heat transfer can be modelled altogether Simulation of melt stratification has been supported in some system codes such as ASTEC using a predefined parametric approach The improvement of PECM can be considered in the similar approach or more mechanistic modelling

bull Development of a structural module for RPV The idea is to develop a simplified module for structural analysis so it can be coupled with transIVR ultimately forming a standalone code for quick estimate of vessel failure of LWRs during postulated severe accidents

bull Extension of vessel analysis capability The current version of tranIVR is limited to melt pool Although the precursory debris bed and melt ejection after vessel failure were not the focus of the present study they are equally important to assessment of corium risk especially for Swedish BWRs where the melt release conditions and history are crucial to qualification of their severe accident mitigation strategies

54

55

Bibliography

[1] IAEA ldquoIAEA Annual Report 2018rdquo IAEA 2019 [Online] Available httpswwwiaeaorgsitesdefaultfilesgcgc63-5pdf

[2] IAEA Safety of Nuclear Power Plants Design Vienna INTERNATIONAL ATOMIC ENERGY AGENCY 2012

[3] M Fischer ldquoGen II and III In-Vessel and Ex-Vessel Melt Retentionrdquo The Short Course on Severe Accident Phenomenology Oct 2017

[4] B R Sehgal Nuclear Safety in Light Water Reactors Severe Accident Phenomenology 1st ed Academic Press 2011

[5] IAEA Approaches and Tools for Severe Accident Analysis for Nuclear Power Plants Vienna INTERNATIONAL ATOMIC ENERGY AGENCY 2008

[6] WENRA ldquoPractical Elimination Applied to New NPP Designs - Key Elements and Expectationsrdquo Sep 2019 [Online] Available httpwwwwenraorgmediafiler_public20191111practical_elimination_applied_to_new_npp_designs_-_key_elements_and_expectations_-_for_issuepdf

[7] P Matejovic M Barnak M Bachraty and L Vranka ldquoASTEC applications to VVER-440V213 reactorsrdquo Nuclear Engineering and Design vol 272 pp 245ndash260 2014 doi 101016jnucengdes201308077

[8] AREVA ldquoUS EPR Severe Accident Evaluationrdquo Topical Report ANP-10268NP Oct 2006 [Online] Available httpswwwnrcgovdocsML0631ML063100157pdf

[9] C T Martin ldquoCoupled 3D Thermo-mechanical Analysis of Nordic BWR Lower Head Failure in case of Core Melt Severe Accidentrdquo master thesis KTH Stockholm 2013

[10] IAEA ldquoAdvanced Large Water Cooled Reactors A supplement to the IAEArsquos Advanced Reactor Information System (ARIS)rdquo Sep 2015 [Online] Available httpsarisiaeaorgPublicationsIAEA_WRC_Bookletpdf

[11] B Pershagen Light Water Reactor Safety Pergamon Press 1989 [12] Ninos Garis M Agrell H Glanneskog and L Agrenius ldquoAPRI 7 ndash Accident

Phenomena of Risk Importancerdquo SSM SSM 201212 2012 [Online] Available httpswwwstralsakerhetsmyndighetensecontentassets2a9f299f46b949b3b209e780cf71436a201212-apri-7-accident-phenomena-of-risk-importance-en-lagesrapport-om-forskningen-inom-omradet-svara-haverier-under-aren-2009-2011-

[13] F Fichot et al ldquoSome considerations to improve the methodology to assess In-Vessel Retention strategy for high-power reactorsrdquo Annals of Nuclear Energy vol 119 pp 36ndash45 2018 doi 101016janucene201803040

[14] T G Theofanous C Liu S Additon S Angelini O Kymaumllaumlinen and T Salmassi ldquoIn-vessel coolability and retention of a core meltrdquo Nuclear Engineering and Design vol 169 no 1 pp 1ndash48 1997 doi 101016S0029-5493(97)00009-5

[15] C Sainte Catherine ldquoTensile and creep tests material characterization of pressure vessel steel(16MND5) at high temperatures (20 up to 1300 degC)rdquo CEA France SEMTLISNRT98-009A 1998

56

[16] E Altstadt H-G Willschuetz B R Sehgal and F-P Weiss ldquoModelling of in-vessel retention after relocation of corium into the lower plenumrdquo Forschungszentrum Rossendorf FZR-437 2005

[17] H-G Willschuumltz and E Altstadt ldquoGeneration of a High Temperature Material DataBase and its Application to Creep Tests withFrench or German RPV-steelrdquo FORSCHUNGSZENTRUM ROSSENDORF FZR-353 ISSN 1437-322X Aug 2002

[18] J L Rempe S A Chavez and G L Thinnes ldquoLight water reactor lower head failure analysisrdquo Nuclear Regulatory Commission NUREGCR--5642 1993 Accessed Aug 23 2019 [Online] Available httpinisiaeaorgSearchsearchaspxorig_q=RN25025905

[19] S Brosi et al ldquoCORVIS Investigation of light water reactor lower head failure modesrdquo Nuclear Engineering and Design vol 168 no 1 pp 77ndash104 1997 doi 101016S0029-5493(96)00008-8

[20] T Y Chu M M Pilch J H Bentz J S Ludwigsen W Y Lu and L L Humphries ldquoLower Head Failure Experiments and Analysesrdquo Sandia National Laboratories Albuquerque NM USA NUREGCR-5582 SAND98-2047 1999

[21] L L Humphries et al ldquoOECD Lower Head Failure Project Final Reportrdquo Sandia National Laboratories Albuquerque NM 87185-1139 2002

[22] B R Sehgal R R Nourgaliev T N Dihn A Karbojian J A Green and V A Bui ldquoFOREVER Experiments on Thermal and Mechanical Behavior of a Reactor Pressure Vessel during a Severe Accidentrdquo in Proceedings of the Workshop on in-vessel core debris retention and coolability Garching near Munich Germany Mar 3-6 1998

[23] S Bhandari M Fischer B R Sehgal and A Theerthan ldquoPost Test Analysis of the FOREVER C2 Experimentrdquo The 9th International conference on Nuclear Engineering (ICONE-9) Nice France Apr 8-12 2001

[24] J F Mao J W Zhu S Y Bao L J Luo and Z L Gao ldquoCreep deformation and damage behavior of reactor pressure vessel under core meltdown scenariordquo International Journal of Pressure Vessels and Piping vol 139ndash140 pp 107ndash116 2016 doi 101016jijpvp201603009

[25] J Mao L Hu S Bao L Luo and Z Gao ldquoInvestigation on the RPV structural behaviors caused by various cooling water levels under severe accidentrdquo Engineering Failure Analysis vol 79 pp 274ndash284 2017 doi 101016jengfailanal201704029

[26] J F Mao J W Zhu S Y Bao L J Luo and Z L Gao ldquoStudy on structural failure of RPV with geometric discontinuity under severe accidentrdquo Nuclear Engineering and Design vol 307 pp 354ndash363 2016 doi 101016jnucengdes201607027

[27] B R Sehgal V A Bui T N Dinh and R R Nourgaliev ldquoHeat Transfer Processes in Reactor Vessel Lower Plenum During Late Phase of In-Vessel Core Melt Progressionrdquo in Advances in Nuclear Science and Technology J Lewins M Becker R W Albrecht E J Henley J D McKean K Oshima A Sesonske H B Smets and C P L Zaleski Eds Boston MA Springer US 1999 pp 103ndash135

[28] C-T Tran and T-N Dinh ldquoThe effective convectivity model for simulation of melt pool heat transfer in a light water reactor pressure vessel lower head Part I Physical processes modeling and model implementationrdquo Progress

57

in Nuclear Energy vol 51 no 8 pp 849ndash859 2009 doi 101016jpnucene200906007

[29] C-T Tran and T-N Dinh ldquoThe effective convectivity model for simulation of melt pool heat transfer in a light water reactor pressure vessel lower head Part II Model assessment and applicationrdquo Progress in Nuclear Energy vol 51 no 8 pp 860ndash871 2009 doi 101016jpnucene200906001

[30] H-G Willschuumltz E Altstadt B R Sehgal and F-P Weiss ldquoCoupled thermal structural analysis of LWR vessel creep failure experimentsrdquo Nuclear Engineering and Design vol 208 no 3 pp 265ndash282 2001 doi 101016S0029-5493(01)00364-8

[31] E Altstadt and T Moessner ldquoExtension of the ANSYS creep and damage simulation capabilitiesrdquo Forschungszentrum Rossendorf Technical report FZR-296 2000

[32] H-G Willschuumltz E Altstadt B R Sehgal and F-P Weiss ldquoRecursively coupled thermal and mechanical FEM-analysis of lower plenum creep failure experimentsrdquo Annals of Nuclear Energy vol 33 no 2 pp 126ndash148 2006 doi 101016janucene200508006

[33] H-G Willschuumltz E Altstadt B R Sehgal and F-P Weiss ldquoSimulation of creep tests with French or German RPV-steel and investigation of a RPV-support against failurerdquo Annals of Nuclear Energy vol 30 no 10 pp 1033ndash1063 2003 doi 101016S0306-4549(03)00036-7

[34] H Madokoro A Miassoedov and T Schulenberg ldquoA thermal structural analysis tool for RPV lower head behavior during severe accidents with core meltrdquo Mechanical Engineering Letters vol 4 pp 18-00038-18ndash00038 2018 doi 101299mel18-00038

[35] W Villanueva C-T Tran and P Kudinov ldquoCoupled thermo-mechanical creep analysis for boiling water reactor pressure vessel lower headrdquo Nuclear Engineering and Design vol 249 pp 146ndash153 2012 doi 101016jnucengdes201107048

[36] W Villanueva C T Tran and P Kudinov ldquoA Computational Study on Instrumentation Guide Tube Failure During a Severe Accident in Boiling Water Reactorsrdquo in The 14th International Topical Meeting on Nuclear Reactor Thermal hydraulics NURETH-14 Toronto Ontario Canada Sep25-30 2011

[37] W Villanueva C T Tran and P Kudinov ldquoAssessment with Coupled Thermo-Mechanical Creep Analysis of Combined CRGT and External Vessel Cooling Efficiency for A BWRrdquo The 14th International Topical Meeting on Nuclear Reactor Thermal hydraulics NURETH-14 Toronto Ontario Canada Sep 25-30 2011

[38] F R Larson and J Miller ldquoA Time-Temperature Relationship for Rupture and Creep Stressesrdquo Trans ASME vol 74 p 765minus775 1952

[39] C T Tran ldquoDevelopment validation and application of an effective convectivity model for simulation of melt pool heat transfer in a light water reactor lower headrdquo Licentiate thesis KTH Stockholm 2007

[40] ANSYS ldquoANSYS Fluent Theory Guide Release 162rdquo ANSYS Inc Jul 2015 [Online] Available httpwwwansyscom

[41] P Zeng Finite element analysis and application (in Chinese) Tsinghua University Press amp Springer 2004

58

[42] ANSYS ldquoANSYS Mechanical | Finite Element Analysis Softwarerdquo httpswwwansyscomproductsstructuresansys-mechanical-enterprise (accessed Sep 26 2019)

[43] ANSYS ldquoANSYS Mechanical APDL Theory Reference Release 162rdquo ANSYS Inc Jul 2015 [Online] Available httpwwwansyscom

[44] U Steinberner and H H Reineke ldquoTurbulent Buoyancy Convection Heat Transfer with Internal Heat Sourcesrdquo in Proceedings of the 6th Int Heat Transfer Conference Toronto Canada 1978 vol Vol2 pp 305ndash310 doi 101615IHTC63350

[45] R W Evans and B Wilshire Introduction to Creep London Institute of Materials 1993

[46] DoITPoMS ldquoCreep Deformation of Metalsrdquo Creep Deformation of Metals University of Cambridge 2006 httpswwwdoitpomsacuktlplibcreepindexphp (accessed Jul 09 2019)

[47] ANSYS ANSYS Mechanical APDL Material Reference Release 162 2015 [48] SIMULIA Abaqus 612 Analysis Userrsquos Manualvol 3 Materials Chapter

2324 2012 [49] S Imaoka ldquoUser Creep Subroutine STI0704A ANSYS Release110rdquo 2007 [50] R W Evans J D Parker and B Wilshire ldquoThe θ projection conceptmdashA

model-based approach to design and life extension of engineering plantrdquo International Journal of Pressure Vessels and Piping vol 50 no 1 pp 147ndash160 1992 doi 1010160308-0161(92)90035-E

[51] ECCC ldquoRecommendations and Guidance for the Assessment of Creep Strain and Creep Strength Data ECCC Recommendations Volume 5 Part 1 b[Issue 2]rdquo 2013

[52] W D Day and A P Gordon ldquoLife Fraction Hardening Applied to a Modified Theta Projection Creep Model for a Nickel-Based Super-Alloyrdquo The ASME Turbo Expo 2014 Turbine Technical Conference and Exposition Duumlsseldorf Germany Jun16-20 2014 vol 7A-Structures and Dynamics p V07AT29A010 doi 101115GT2014-25881

[53] M Kumar I V Singh B K Mishra S Ahmad A Venugopal Rao and V Kumar ldquoA Modified Theta Projection Model for Creep Behavior of Metals and Alloysrdquo J of Materi Eng and Perform vol 25 no 9 pp 3985ndash3992 2016 doi 101007s11665-016-2197-y

[54] P Yu and W Ma ldquoA modified theta projection model for creep behavior of RPV steel 16MND5rdquo Journal of Materials Science amp Technology vol 47 pp 231ndash242 2020 doi 101016jjmst202002016

[55] ANSYS ANSYS Mechanical Advanced Nonlinear Materials v160 Lecture 2 Rate Dependent Creep ANSYS training materials 2015

[56] Q Y Li N C Wang and D Y Yi Numerical Analysis (in Chinese) 5th ed Tsinghua University Press 2008

[57] T N Dinh R R Nourgaliev and B R Sehgal ldquoOn heat transfer characteristics of real and simulant melt pool experimentsrdquo Nuclear Engineering and Design vol 169 no 1 pp 151ndash164 1997 doi 101016S0029-5493(96)01283-6

[58] O Kymaumllaumlinen H Tuomisto O Hongisto and T G Theofanous ldquoHeat flux distribution from a volumetrically heated pool with high Rayleigh numberrdquo

59

Nuclear Engineering and Design vol 149 no 1ndash3 pp 401ndash408 1994 doi 1010160029-5493(94)90305-0

[59] F J Asfia and V K Dhir ldquoNatural convection heat transfer in volumetrically heated spherical poolsrdquo 1994 Accessed Aug 23 2019 [Online] Available httpinisiaeaorgSearchsearchaspxorig_q=RN26037727

[60] M Jahn and H H Reineke ldquoFree Convection Heat Transfer with Internal Heat Sources Calculations and Measurementsrdquo in Proceedings of the 5th Int Heat Transfer Conference Tokyo Japan 1974 vol Vol3 paper NC-28

[61] M Gustafsson ldquoFSI Transient extension V1704rdquo EDR MEDESO 2016 [62] EDR MEDESO httpsedrmedesocom [63] H Kim and I Namgung ldquoThermo-mechanical Analysis of RVLH for

APR1400 during a Severe Accidentrdquo in Transactions SMiRT-23 Manchester United Kingdom Aug10-14 2015 [Online] Available httpwwwlibncsueduresolver18402034197

[64] P Yu and W Ma ldquoComparative Analysis of Reactor Pressure Vessel Failure using Two Thermo-Fluid-Structure Coupling Approachesrdquo submitted to journal

[65] F R Menter ldquoTwo-equation eddy-viscosity turbulence models for engineering applicationsrdquo AIAA Journal vol 32 no 8 pp 1598ndash1605 1994 doi 102514312149

[66] T N Dinh and R R Nourgaliev ldquoTurbulence modelling for large volumetrically heated liquid poolsrdquo Nuclear Engineering and Design vol 169 no 1 pp 131ndash150 1997 doi 101016S0029-5493(96)01281-2

[67] ANSYS Inc ldquoIntroduction to ANSYS Fluent Lecture 7 Turbulence Modelling Fluent Tutorial Materialrdquo 2016

[68] A Horvat and M Borut ldquoNumerical investigation of natural convection heat transfer in volumetrically heated spherical segmentsrdquo in Proceedings of the International Conference Nuclear Energy for New Europe 2004

[69] Y Li CFD Pre-test Analysis of SIMECO-2 Experiment 2016 [70] A Theerthan A Karbojian and B R Sehgal ldquoEC-FOREVER experiments

on thermal and mechanical behavior of a reactor pressure vessel during a severe accident Technical report-1 EC-FOREVER-2 testrdquo SAM-ARVI-D008 Mar 2001

[71] P Yu W Ma W Villanueva A Karbojian and S Bechta ldquoValidation of a thermo-fluid-structure coupling approach for RPV creep failure analysis against FOREVER-EC2 experimentrdquo Annals of Nuclear Energy vol 133 pp 637ndash648 2019 doi 101016janucene201906067

[72] T G Theofanous C Liu S Additon S Angelini O Kymaumllaumlinen and T Salmassi ldquoIn-vessel coolability and retention of a core meltrdquo Nuclear Engineering and Design vol 169 no 1ndash3 pp 1ndash48 1997 doi 101016S0029-5493(97)00009-5

[73] H P Langtangen ldquoSolving nonlinear ODE and PDE problemsrdquo Simula Research Laboratory 2016 [Online] Available httpshplgitgithubionum-methods-for-PDEsdocpubnonlinpdfnonlin-4printpdf

[74] P Yu and W Ma ldquoDevelopment of a Lumped-Parameter Code for Efficient Assessment of In-Vessel Melt Retention Strategy of LWRsrdquo submitted to journal

60

[75] T G Theofanous M Maguire S Angelini and T Salmassi ldquoThe first results from the ACOPO experimentrdquo Nuclear Engineering and Design vol 169 no 1 pp 49ndash57 1997 doi 101016S0029-5493(97)00023-X

[76] S W Churchill and H H S Chu ldquoCorrelating equations for laminar and turbulent free convection from a vertical platerdquo International Journal of Heat and Mass Transfer vol 18 no 11 pp 1323ndash1329 1975 doi 1010160017-9310(75)90243-4

[77] S Globe and D Dropkin ldquoNatural-Convection Heat Transfer in Liquids Confined by Two Horizontal Plates and Heated from Belowrdquo J Heat Transfer vol 81 no 1 pp 24ndash28 1959 doi 10111514008124

[78] T G Theofanous C Liu S Additon S Angelini O Kymaelaeinen and T Salmassi ldquoIn-vessel coolability and retention of a core melt Volume 2rdquo Argonne National Lab DOEID--10460-VOL2 1996 Accessed Apr 18 2019 [Online] Available httpinisiaeaorgSearchsearchaspxorig_q=RN30000459

[79] J L Rempe D L Knudson C M Allison G L Thinnes and C L Atwood ldquoPotential for AP600 in-vessel retention through ex-vessel floodingrdquo United States INEELEXT--97-00779 1997 Accessed Apr 18 2019 [Online] Available httpinisiaeaorgSearchsearchaspxorig_q=RN30025872

[80] A Miassoedov X Gaus-Liu T Cron and B Fluhrer ldquoLive experiments on melt pool heat transfer in the reactor pressure vessel lower headrdquo The 10th International Conference on Heat Transfer Fluid Mechanics and Thermodynamics (HEFAT 2014) Orlando FL Jul 14-16 2014

[81] H Madokoro A Miassoedov and T Schulenberg ldquoAssessment of a lower head molten pool analysis module using LIVE experimentrdquo Nuclear Engineering and Design vol 330 pp 51ndash58 2018 doi 101016jnucengdes201801036

[82] B Fluhrer et al ldquoThe LIVE-L1 and LIVE-L3 experiments on melt behaviour in RPV lower headrdquo Forschungszentrum Karlsruhe GmbH Technik und Umwelt (Germany) Inst fuer Kern- und Energietechnik FZKA--7419 2008 Accessed Nov 12 2019 [Online] Available httpinisiaeaorgSearchsearchaspxorig_q=RN40036867

X

432 Conjugate heat transfer 27

433 Mechanical analysis 29

44 Application to BWR Case 32

45 Computational Efficiency 34

5 Hybrid Coupling Approach for Efficient RPV Failure Analysishelliphellip35

51 A Lumped-parameter Code for Melt Pool Heat Transfer 35

511 Heat transfer correlations 36

512 Energy balance in the pools 37

513 Energy balance in the RPV wall 39

52 Hybrid Coupling Approach 40

53 Verification and Validation 41

531 UCSB FIBS benchmark case 41

532 LIVE-7V experiment 43

533 FOREVER-EC2 experiment 45

6 Summary and Outlookhelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphellip51

Bibliographyhelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphellip55

XI

Nomenclature

Latin

119860 area of a surface

119862119901 heat capacity

119863 damage factor

119892 gravity

ℎ heat transfer coefficient

H Melt pool height

119896 thermal conductivity

119871 characteristic length

119873119906 Nusselt number

119875119903 Prandtl number

119902 heat flux

119902119894119899 heat flux through inner vessel surface

119876119881 internal volumetric heat source

119903 radial distance in a cylindrical or spherical coordinate

119877119886 external Rayleigh number

119877119886119894119899 internal Rayleigh number

119877119886119910 local Rayleigh number

119905 time

119879 temperature

119881119900119897 total volume

119909 thickness (of crust or RPV wall)

119911 height in a cylindrical coordinate

Greek

120572 thermal diffusivity

120573 thermal expansion

Δ119879 temperature difference

ε emissivity (in thermal analysis) strain (in structural analysis)

휀119888119903 creep strain

휀119903 creep strain rate

120579 polar angle

120579119894 parameters of a creep model 119894 = 1234119898

120584 kinematic viscosity

σ stress

XII

Super and subscripts

119886119898119887 ambient

119888119903 creep (in Section 3) crust (in Section 5)

119889119899 downward

119890119899119907 environment

119894 inner surface of RPV

119894119899119905 interface

119897119900119888119886119897 local

119898119886119909 maximum value

119898119890119897119905119894119899119892 melting point

119898119905 metal layer (bulk)

119898119905119889119899 downward surface of metal pool

119898119905119904119889 sideward surface of metal pool

119898119905119905119900119901 top surface of metal layer

119898119905119906119901 upward surface of metal pool

119900 outer surface of RPV

119900119909 oxide layer (bulk)

119900119909119889119899 downward surface of oxide layer

119900119909119906119901 upward surface of oxide layer

119903119890119891 reference

119904 internal structures inside the RPV

119904119889 sideward

119906119901 upward

119907119890119904 vessel

Acronyms

BWR boiling water reactor

CBD creep database

CFD computational fluid dynamics

CRGT control rod guide tube

ECCM effective convectivity conductivity model

FDM finite difference method

FE finite element

FEA finite element analysis

FEM finite element method

FVM finite volume method

IGT instrument guide tube

IVR in-vessel retention

XIII

IVMR in-vessel melt retention EU project

LES large eddy simulation

MPS moving particle semi-implicit method

NPP nuclear power plant

PECM phase-change effective convectivity model

RANS Reynolds-averaged Navier-Stokes

RPV reactor pressure vessel

SBO station blackout

SLM surface loads mapping

SPH smoothed-particle hydrodynamics

UDF user-defined function

UPF user programmable features

VLM volume loads mapping

XIV

1

1 Introduction

11 Nuclear Power Safety

Greenhouse gas emissions contribute to the global warming problem Phasing out fossil fuels by switching to low-carbon energy sources is considered as a mitigation strategy With this regard clean energy should be increased in the world energy consumption structure Being clean and stable nuclear energy is an attractive alternative to fossil fuels After the development over decades it also proves to be a mature and safe technology According to IAEA 2018 annual report [1] there were 450 nuclear power reactors under operation worldwide at the end of 2018 which had a record generating capacity of 3964 GW(e) 9 reactors were connected to the grid and 55 reactors under construction around the world

Due to the potential risk of the release of radioactivity nuclear power safety is emphasized throughout the stages of design construction operation and decommission of a nuclear power plant (NPP) The principal technical requirements [2] specify that the three fundamental safety functions should be fulfilled for all plant states of an NPP (i) control of reactivity (ii) removal of heat from the reactor and from the fuel store and (iii) confinement of radioactive material shielding against radiation and control of planned radioactive releases as well as limitation of accidental radioactive releases The defence-in-depth concept is also applied to provide several levels of defence to prevent consequences of accidents The fuel claddings reactor pressure vessel (RPV) and primary coolant pressure boundaries and the containment are the three physical barriers to prevent the release of radioactive materials to the environment

Severe accidents are events that cause significant damage to reactor fuel and result in core meltdown It may be caused by insufficient cooling of the reactor core due to various reasons or by the rapid introduction of huge reactivity which causes sharp increases in the power and fuel fragmentation eg Chernobyl accident During a severe accident the failure of the fuel claddings will result in the radioactive materials previously enclosed inside the cladding released to the primary loop If cooling is not sufficient to remove the decay heat the core materials will melt migrate down and relocate into the RPV lower head after which the melt will react with water and partially form fragments quench solidify and form a debris bed The debris bed formed by the melt may continue to melt to form a melt pool due to the decay heat As the corium contacts the RPV it would continuously heat the RPV and threaten its integrity Once the RPV fails containment becomes the last barrier to prevent the release of radioactivity The released melt would result in an increase of the pressure and temperature due to the high temperature fractions oxidation and hydrogen combustion which is called direct containment heating The melt would fall onto the reactor cavity Without sufficient cooling the molten core would interact with basemat concrete (MCCI) This process can last for few days It can result in containment failure due to either containment overpressure or the basemat melt-through of containment In addition containment is also subjected to isolation failure and early containment failure due to missiles and dynamic pressure [3] More

2

detailed severe accident progression or phenomena could also be found in eg [4] [5]

Though severe accidents are unlikely to occur the consequences can be severe if no proper measure is taken Related researches have been continuously going on aiming at mitigating the accident consequences and improving safety After the Fukushima Accident safety inspections and re-evaluation actions have been carried out worldwide especially on safety measurements regarding severe accidents The concept lsquoPractical Eliminationrsquo applied to new NPP designs demands that new nuclear installations be designed with the objective of mitigating the consequences of accidents and avoiding early radioactive releases and large radioactive releases [6] Severe accident mitigation strategies are applied to the existing and new NPPs (both Generation II and III) The strategies can generally be classified into two categories [3] the in-vessel retention (IVR) and ex-vessel retention (EVR) In simple words in-vessel retention tries to retain the core melt inside the RPV (second barrier) by applying external water cooling on the vessel as shown in Figure 1(a) Such method minimizes the dispersal of the melt within the containment and hence simplifies the later recovery work Examples of the application are AP1000 HPR1000 APR 1400 The idea of the EVR is to let the RPV fail but to retain the core melt inside the containment Two principle approaches of EVR are the retention within a dedicated lsquocore catcherrsquo and melt fragmentation Examples of core catcher would be EPR that directs the melt to a spreading compartment (Figure 1(b)) and VVER1000 that uses metallic crucible below the RPV An example of the melt fragmentation is the Nordic BWRs that the corium released from RPV would relocate in the cavity with a deep-water pool (7-12 m) and is expected to fragment quench and form a coolable debris bed (Figure 1(c))

Figure 1 Examples of IVR [7] EVR with core catcher [8] and melt fragmentation [9]

12 RPV Failure

RPV which contains the reactor core core shroud and coolant is a key component to the reactor system RPV and the primary loop constitute the primary coolant system which conveys the fission heat from the reactor core to the secondary loop through steam generator (PWR) or to the turbine (BWR) The pressure under

(a) IVR (b) Core catcher (c) Melt fragmentation

3

operation can be ~155 MPa for a 1000 MW PWR [10] and ~70 MPa for a 1000 MW BWR [11] It also has to withstand the neutron radiation damage The thickness of RPV [11] can be ~20 cm for PWR and ~16 cm for BWR

As the transition of in-vessel progression to ex-vessel progression RPV failure is important for analysis of accident progression and assessment of the effectiveness of severe accident mitigation strategies In the melt fragmentation strategy of Nordic BWR the melt-coolant interaction and debris bed coolability after the corium poured into the deep-water pool are essentially dependent on the vessel failure mode including the characteristic time of melt release rupture size and location and amount and superheat of melt available for release [12] New concern also arose [13] on the necessity of performing structural analysis for the RPV in the IVR strategy In the initial methodology and further development of the IVR the success of the strategy ie the integrity of the RPV was demonstrated (eg in [14]) with the following two conclusions (i) even for a local heat flux on the vessel as high as critical heat flux (CHF) the resulted very thin external ldquocold layerrdquo of the vessel would be sufficient to withstand the internal stress load and hold the vessel integrity (ii) as CHF is the critical value for the thermal load the main specific goal for assessing the effectiveness of IVR becomes to demonstrate that the local heat flux along RPV be lower than the CHF The new concern doubts that local lsquohot partrsquo of the vessel would have negative effects on the overall resistance due to creep and dilatation which could result in a vessel failure and hence violate the first conclusion in particular for ldquolong termrdquo retention of corium in RPV If so a structural integrity analysis of RPV would be needed in addition to assessing local heat flux especially for high-power reactors Therefore accurate prediction of vessel failure could be meaningful for both strategies

Figure 2 Illustration of RPV under thermal loads from corium

As the structural performance is highly affected by the thermal loads from the corium RPV failure is essentially a transient thermo-fluid-structure coupled problem Figure 2 illustrates the situation that a corium debris bedmelt pool formed in the RPV lower head The decay heat of nuclear fission products could continuously be generated inside the corium leading to the formation of melt pools as the debris melts The heated debris bedmelt pool imposes thermal and

4

mechanical loads on the RPV Such loads can be transient (change with time) and spatially non-uniform Complex phenomena are also involved in this process Strong turbulent natural convection induced by the decay heat may be expected in the melt pool under IVR The corresponding Rayleigh number 119877119886 can be 1016minus17 for prototypic reactor cases Material structural properties can become non-linear under high temperature and stress loads The creep deformation also becomes significant at high temperature Detailed modelling of turbulent natural convection and non-linear properties could be computationally expensive In addition large-scale geometry (hemisphere with a radius of 20-30 m) long-term transient deformation process (~10000 s) and required small time steps (eg 001 s) for numerical convergence consideration further increase the computational costs

13 Literature Review Creep Behaviour and RPV Failure Analysis

In this section the most relevant studies concerning the experiments modelling and analysis of creep behaviour and RPV failure are presented

131 Experiments on creep behaviour and RPV failure

Various experiments have been performed on the RPV failure problem some of which were to generate data of mechanical properties (especially the creep properties) and some to study the overall vessel failure with scaled or simplified geometries

The REVISA experiment [15] generated creep curves and stress strain curves of French steel 16MND5 ranging from 600 to 1300 degC This data is accessible in [16] as well as [17] which also documents other material properties eg density thermal expansion coefficient conductivity heat capacity Youngrsquos Modulus and plasticity Similar experiments were carried out to generate high-temperature creep and tensile test data for American RPV steels and Inconel [18]

The CORVIS experiments [19] investigated the lower head failure modes with and without penetrations Each experiment was carried out on a steel plate 100 mm thick representing a pressure vessel lower head The penetration case represented a boiling water reactor lower head carrying a tube penetration in the form of a drain line

The LHF (lower head failure) experiments [20] and the succeeding OLHF tests [21] investigated the vessel failure with a 15 scaled vessel heated by an electrical radiator allowing to locate the hot focus at different position of the vessel with a maximum wall temperature of around 900 degC Internal pressure ranging from 53 to 123 bar was also imposed to initiate the vessel failure

The FOREVER experiments [22] investigated the phenomena of melt pool convection vessel creep possible failure processes and failure modes occurred during the late phase of the in-vessel progression by heating up a melt pool in a vessel and observing possible vessel failures The study covers cases with and without penetration different melt level existence of gap cooling and in-vessel melt coolability

5

132 Numerical studies on RPV failure

Various numerical approaches have been employed on RPV failure Since a two-way thermo-fluid-structure coupling approach is challenging these approaches take simplifications and can generally be classified into two categories (i) RPV failure decoupled from melt pool heat transfer and (ii) RPV failure coupled with melt pool heat transfer in simplified manners

1) Decoupled approaches In the decoupled approaches only RPV is modelled Simplified constant thermal loads are imposed on RPV considering the melt pool heat transfer

In a post-test analysis of FOREVE-C2 experiment [23] it was treated as a thermo-structural coupled analysis of the vessel with the thermal load estimated by the outer thermocouples and the normal heat flux on the external vessel wall The analysis was done using finite element (FE) code SYSTUS A coupled creepdamage law based on a viscosity-hardening creep law describing primary and secondary creep was employed for creep analysis

Mao et al conducted various studies on reactor cases assessing the IVR strategy using FE code ABAQUS Thermal boundary conditions were imposed on the RPV which was then solved as a thermal-structural coupled problem The strain hardening creep model (Eq (1)) was mainly used Their studies cover the general RPV failure [24] as well as the effects of cooling water levels under IVR [25] and geometric discontinuity (ablated RPV) [26]

ε119888119903 = 1198891σ1198892ε1198893119890119909119901 [minus

1198894119879] (1)

In the EU IVMR project [13] the RPV integrity was also studied through a benchmark calculation which deals with the RPV of a PWR with a prescribed ablated profile with 2D axisymmetric geometry Prescribed temperature gradient was set as thermal load The objective was to calculate deformations for various pressure loads up to vessel rupture A smoother profile was planned in further research

The advantage of such approaches is once thermal loads are reasonably assumed the focus only needs to be put on the RPV The disadvantage is a reasonable thermal load should be consistent with the melt pool heat transfer while sometimes it is not well-noted eg constant uniform temperature boundary conditions were assumed for inner and outer RPV (with uniform thickness in lower head) surfaces [25] which ignored the non-uniform heat flux distribution caused by the turbulent convective corium pool As thermal loads strongly affect the structural performance they should be quantified carefully

2) Coupled approaches In coupled approaches both melt pool heat transfer and RPV failure are modelled and coupled in one-way or simplified manner With melt pool heat transfer modelled the coupled approaches are more flexible to provide reasonable thermal loads on the RPV In existing approaches CFD and the correlation-based distributed-parameter models eg the Effective Convectivity Conductivity Model

6

(ECCM) [27] and the Phase-change Effective Convectivity Model (PECM) [28] [29] are general methods for melt pool heat transfer

Sehgal et al [22] presented an approach that employed the MVITA code to calculate the thermal processes and ANSYS for structural analysis MVITA code employed the ECCM Bailey-Norton creep law was used as shown in Eq (2)

ε119888119903 = 6055 times 10minus26 sdot σ472119905085119890minus

37200

119879 (2)

This approach was used for the pre-test simulation of FOREVER experiment to determine the thermal and mechanical loadings Parametric investigations also helped in the identification of important factors that could affect the test procedure and test conditions

Willschutz et al [30] employed the CFD-module Flotran for the melt pool heat transfer and ANSYS for structural analysis of RPV Standard k-ε turbulence model was selected for the turbulent natural convection in the melt pool A user-defined three-stage creep model [31] was employed In this model experimental creep curves were discretized into a creep database (CDB) under given temperature 119879 and stress 120590 the creep curve was discretized into finite point pairs

(휀 ε) indicating under such thermal and mechanical loads the creep strain rate ε is subjected to creep strain 휀 The creep strain rate for any other case can be interpolated over known temperature stress and strain from the database This method was applied to the post-test calculation of the FOREVE-C2 experiment It was concluded that the unexpected deformation behaviour during the experiment was caused by the combination of the transient temperature field in the vessel wall and the three creep stages

They later replaced the Flotran solver with the ECCM for melt pool heat transfer in a following study [32] By implementing it into FEM-solver a recursively coupled analysis was constructed Moreover a modified creep model [33] was used with damage measures the CDB itself did not contain tertiary creep instead a factor 1(1 minus 119863) was multiplied to the original creep strain rate to get the modified creep strain rate where 119863 is a damage factor (0 lt 119863 lt 1) defined with creep strain information and plastic strain information As 119863 grows close to 1 the modified creep strain rate would be enlarged significantly by this factor 1(1 minus 119863) which then behaves like the accelerating tertiary stage This method was applied to pre- and post- test calculations for the FOREVER experiments proving to be closer to reality than the previous method

Madokoro et al [34] developed a thermal structural analysis tool based on open source solver OpenFOAM and numerically investigated the coupled RPV failure and melt pool convection The PECM was used in the melt pool heat transfer In structural part the plasticity and damage parameters were similar to Willschutz et al [32] except for that strain hardening creep model (Eq (3)) was used instead of a CDB Simulations of FOREVER-EC2 and -EC4 were investigated It was concluded from the simulations that although the emissivity and mesh type (updated or fixed) affect the results significantly the approach generally has the capability of predicting the vessel deformation and failure

7

ε119888119903 = 1198891 sdot σ1198892 sdot ε1198893 (3)

Villanueva et al [35] performed a coupled thermo-mechanical creep analysis of the reactor scale RPV (ABB-Atom design BWR) to evaluate efficacy of the cooling measures for the in-vessel melt coolability and retention The PECM was used for melt pool heat transfer and ANSYS for RPV integrity analysis The modified time hardening model (Eq (4)) of the American RPV steel SA533B was selected as creep model The transient thermal load from the melt pool was applied as a thermal boundary condition to the vessel wall Further studies were done with this approach on investigations of instrumentation guide tube (IGT) failure [36] and control rod guide tube (CRGT) cooling [37] of a Nordic BWR

ε119888119903 =1198881σ

11988821199051198883+1

1198883 + 1 1198881 gt 0 (4)

It is noted that only RPV analysis with finite element analysis (FEA) are reviewed here Engineering methods like the Larson-Miller method [38] are not covered in this thesis One can refer to theory manuals of severe accident codes like MELCOR ASTEC for more details Comprehensive description on modelling of melt pool heat transfer is not covered either One can refer to either [39] for a general view or to any severe accident code manuals for engineering methods

14 Motivation Objectives and Main Achievements

The existing experimental and numerical studies provide us a good understanding of RPV failure involved phenomena and valuable simulation experiences These numerical methods may vary in solvers (which may be related to code accessibility) and models (which may be related to model accessibility understanding of phenomena computational power) However improvements of modelling approaches are still possible as the improvements in any of the related aspects eg

bull Simulation tools are continuously being upgraded and providing new possibilities just like that PECM appeared and ECCM was no longer used

bull Better understanding of the phenomena and more simulation experiences from in-vessel retention and vessel failures researches benefit the further modelling and simulations

bull Increasing computational power allows more detailed modelling eg turbulence modelling of melt pool convection

bull There is still room for the improvement of creep modelling

In addition implementation difficulty of a coupling approach is another important issue to consider Since it is a TRANSIENT thermo-fluid-structure coupled problem it is almost impossible to replicate an approach simply by referring to a paper with limited details For BWRs it would be even better for the coupling approaches to support complex geometries considering forest of penetrations (CRGTs and IGTs) on the RPV lower head without significant increase of user effort in the coupling process

8

The objectives of this thesis are to investigate new approaches and develop new models for the thermo-fluid-structure coupled problem of RPV failure and involved phenomena These objectives are accomplished with the following achievements

1) A new creep model for the 16MND5 steel was developed which covers all three creep stages Good agreement was achieved between model predictions and experimental curves Both time and strain hardening models were implemented (Paper I)

2) A new thermo-fluid-structure coupling approach was developed for RPV failure analysis It supports the advanced models in both melt pool convection and structural mechanics to improve fidelity and facilitates implementation Validation of the approach with the new creep model was performed against FOREVER-EC2 experiment This approach is flexible for complex geometries and can be used in further analysis eg lower head with detailed modelling of penetrations in BWRs (Paper II)

3) The performance of this new approach which uses volume loads mapping for data transfer was compared with the previous approach using surface loads mapping showing good agreement with small discrepancy (Paper III)

4) A lumped-parameter code transIVR for melt pool heat transfer was developed which supports one- and two-layer pool configuration transient analysis and well-resolved RPV wall (Paper IV)

5) A hybrid coupling approach was developed for efficient analysis of RPV failure where transIVR is coupled with ANSYS Mechanical (Paper IV)

In the following general descriptions on the codes ANSYS Fluent Mechanical and the PECM are given in Section 2 the new creep model in Section 3 the new coupling approach as well as its comparison with the previous approach in Section 4 the efficient hybrid coupling approach in Section 5 and summary amp outlook in Section 6

9

2 Numerical Solvers and Models

In this section we briefly introduce the ANSYS Fluent as the CFD solver for melt pool heat transfer the ANSYS Mechanical as the mechanical solver for RPV failure and PECM as an efficient alternative for melt pool heat transfer that are heavily used in the following sections

21 ANSYS Fluent

In the fluid mechanics the fluid is treated as a continuum which is generally described by the conservative mass momentum and energy equations [40]

120597120588

120597119905+ 120571 sdot (120588) = 119878119898 (5)

120597(120588)

120597119905+ 120571 sdot (120588) = minus120571119901 + 120571 sdot (120591) + 120588 + (6)

120597120588119864

120597119905+ 120571 sdot ((120588119864 + 119901)) = 120571 sdot (119896120571119879 minussumℎ119895119869119895

119895

+ (120591 sdot )) + 119878ℎ

(7)

Computational Fluid Dynamics (CFD) studies topics on the numerical solutions of above equations Typical methods are the Finite Volume Method (FVM) Finite Difference Method (FDM) and Finite Element Method (FEM) as well as particle methods that are mesh-free eg SPH and MPS

ANSYS Fluent is a general purpose CFD code which solves above equations with FVM Turbulence modelling and multiphase flows are supported Turbulence can be modelled with RANS LES or hybrid LESRANS Customizations eg adding sourcesink time or spatial dependent boundary conditions are realized through the user-defined functions (UDF) which is programmed in C language

22 ANSYS Mechanical

In the continuum mechanics of solids three sets of variables are considered to describe the structure performance namely element displacement field strain field and stress field (119906 ε σ) A general 3-dimensional problem is described by 3 sets of equations the equilibrium equations (Eq (8)) strain-displacement relationship (Eq (9)) and constitute equations (Eq (10)) [41] For Eq (8) the reciprocal theorem of shear stress also gives τ119894119895 = τ119895119894 where 119894 119895 = 119909 119910 119911

indicating the stress tensor being symmetric For very simple applications eg 1D beam it can be solved analytically For complex problems where analytical solutions are not available approximation methods eg Galerkin weighted residual method principle of virtual work and principle of minimum potential energy would be used

10

partσ119909119909part119909

+partτ119909119910

part119910+partτ119909119911part119911

+ 119909 = 0

partτ119910119909

part119909+partσ119910119910

part119910+partτ119910119911

part119911+ 119910 = 0

partτ119911119909part119909

+partτ119911119910

part119910+partσ119911119911part119911

+ 119911 = 0

(8)

ε119909119909 =part119906

part119909 휀119910119910 =

120597119907

120597119910 휀119911119911 =

120597119908

120597119911

120574119909119910 =120597119906

120597119910+120597119907

120597119909 120574119910119911 =

120597119908

120597119910+120597119907

120597119911 120574119911119909 =

120597119908

120597119909+120597119906

120597119911

(9)

ε119909119909 =1

119864[σ119909119909 minus μ(σ119910119910 + σ119911119911)]

ε119910119910 =1

119864[σ119910119910 minus μ(σ119909119909 + σ119911119911)]

ε119911119911 =1

119864[σ119911119911 minus μ(σ119909119909 + σ119910119910)]

γ119909119910 =1

119866τ119909119910 γ119910119911 =

1

119866τ119910119911 γ119911119909 =

1

119866τ119911119909

(10)

ANSYS Mechanical [42] is a mechanical engineering software that uses finite element analysis (FEA) for structural analysis The principle of virtual work is adopted in the solver [43] The idea of this principle is that a virtual (very small) change of the internal strain energy must be offset by an identical change in external work due to the applied loads which reads

δ119880 = δ119881 (11) where 119880 is the strain energy or the internal work 119881 the external work and δ the virtual operator The virtual internal work can be expressed as

δ119880 = int δεσd(vol)T

vol

+int 119886119903119890119886119891

δ119908119899119879σ119889(119886119903119890119886119891) (12)

where 휀 is the strain vector 120590 stress vector vol volume of element 119908119899 motion normal to the surface 119886119903119890119886119891 area of the distributed resistance The virtual

external work is contributed from three parts the inertial effects the pressure force on the surface and nodal forces applied to the element which then can be expressed as

δ119881 = minusint 119907119900119897

δ119908119879119865119886

119907119900119897119889(119907119900119897) + int

119886119903119890119886119901

δ119908119899119879119875119889(119886119903119890119886119901)

+ δ119906119879119865119890119899119889

(13)

11

where 119908 is the vector of displacements 119865119886 acceleration force vector 119875 applied pressure vector 119886119903119890119886119901 area over which pressure acts and 119865119890

119899119889 nodal

forces applied to the element

Plugging Eq (12) and (13) into (11) would result in the equilibrium equation Since the virtual displacement can be arbitrary the final equations used to describe the equilibrium of an element is reduced to Eq (14) with multiple operations

([119870119890] + [119870119890119891])119906 minus 119865119890

119905ℎ = [119872119890] + 119865119890119901119903 + 119865119890

119899119889

(14)

where [119870119890] [119870119890119891] 119865119890

119905ℎ [119872119890] and 119865119890119901119903 are the element stiffness matrix

foundation stiffness matrix thermal load vector mass matrix acceleration vector and pressure vector respectively It represents the equilibrium equation on a one element basis Alternatively using a minimum total potential energy principle could result in a consistent expression to the principle of virtual work [41]

Customizations eg on material properties are realized through the user programmable features (UPF) which is programmed in FORTRAN language

23 PECM

The PECM [28] [29] developed in KTH is a correlation-based method to model the melt pool heat transfer It is implemented in ANSYS Fluent through UDF and has been validated against many experiments such as ACOPO COPO LIVE etc In the PECM the flow velocities 119906119909 119906119910 and 119906119911 are replaced by characteristic

velocities 119880119909 119880119910 and 119880119911 respectively in order to eliminate the necessity of solving

the Navier-Stokes equation

The characteristic velocities are calculated using heat transfer correlations

119880119906119901 =α

ℎ119901119900119900119897times (119873119906119906119901 minus

ℎ119901119900119900119897

ℎ119906119901) (15)

119880119889119899 =α

ℎ119901119900119900119897times (119873119906119889119899 minus

ℎ119901119900119900119897

ℎ119889119899) (16)

119880119904119889 =α

ℎ119901119900119900119897times (119873119906119904119889 minus

ℎ119901119900119900119897

119882119901119900119900119897) (17)

where ℎ119901119900119900119897 is the melt pool depth ℎ119906119901 the thickness of well mixed layer of the

pool ℎ119889119899 the thickness of lower stratified region of melt pool 119882119901119900119900119897 the width of

pool 120572 the thermal diffusivity The PECM employs reduced characteristic velocities to describe mushy zone convection heat transfer For internally heated pool natural convection the Steinberner-Reineke correlations (Eq (18)~(19)) [44] are employed The profile of sideward 119873119906 is described using Eckertrsquos type correlation (Eq (20)) for a vertical boundary layer where the local Rayleigh

number at pool height 119910 is defined as 119877119886119910 =119892βΔ1198791199103

αν

12

119873119906119906119901 = 03451198771198861198941198990233

(18)

119873119906119889119899 = 13891198771198861198941198990095

(19)

119873119906119904119889 = 050811987511990314 (

20

21+ 119875119903)

minus14

11987711988611991014

(20)

As the characteristic velocities are calculated the energy equation is the only transport equation that needs to be solved and can be expressed as follows

partρ119862119901119879

part119905+ nabla(ρ119862119901119932119879) = 119896nabla2119879 + 119876119907 (21)

where from left to right are the transient term convection term diffusion term and volumetric heat source respectively The convection term then can be calculated explicitly with use of values in the last time step and then can be treated together with the volumetric source term Therefore the PECM finally simplifies the energy equation to a conduction equation which can be easily solved and as a result will significantly increase the computational efficiency in simulations

In addition solidificationmelting process can also be modelled and the temperature-dependent physical properties are supported

13

3 Creep Modelling of 16MND5 Steel

Creep deformation becomes significant when a component is subjected to high temperature and stress Modelling of creep behaviour of the RPV is paramount since it is considered as a main mechanism leading to the RPV failure In this section we present the development of a three-stage creep model (called lsquomodified theta projection modelrsquo ) for the French RPV steel 16MND5 This model was successfully implemented into the ANSYS Mechanical code

31 Creep and Creep Modelling

Creep is an irreversible time-dependent inelastic deformation process Diffusion and dislocation are the two molecular mechanisms of creep The former normally occurs due to the presence of vacancies within the crystal lattice (or the grain) while the latter occurs if there is a slip by the passage of crystal dislocations [45] There are two types of diffusion creep [46] Coble creep (favoured at lower temperatures) if the diffusion paths are predominantly through the grain boundaries and Nabarro-Herring creep (favoured at higher temperatures) if through the grains themselves Dislocation creep tends to dominate at high stresses and relatively low temperatures Dislocations can move by gliding in a slip plane which is a process that requires little thermal activation But the rate-determining step for their motion is often a climb process which also requires diffusion and is thus a time-dependent process and favoured by higher temperatures Creep behaviour is highly affected by temperature and stress generally creep is more significant under higher temperature and stress

Creep curves describe the creep strain development with time Figure 3 shows a creep curve under constant temperature and stress which typically experiences three creep stages primary stage where the creep strain rate ε119888119903 (the change of creep strain ε119888119903 in unit time) decreases with time secondary stage where the creep strain rate almost keeps constant and tertiary stage where the creep strain rate drastically increases with time

Figure 3 Creep curve under constant temperature and stress

14

The built-in creep models in commercial software eg ANSYS [47] and ABAQUS [48] in default only cover the primary andor the secondary stage(s) Table 1 [47] gives the default creep models supported in ANSYS for example Possible reason why the tertiary stage is not modelled might be that for most industrial applications the failure analysis is beyond interest and therefore the built-in creep laws in the commercial software was supposed to be sufficient [49] Though the two codes offer the interfaces through either ANSYS UPF or ABAQUS user subroutines to support more general user-defined creep laws eg complex models that also include the tertiary creep stage

Table 1 Default creep models in ANSYS Mechanical

Creep model Equations Type

Strain hardening

ε119888119903 = 1198621σ1198622ε119888119903

11986231198901198624119879 1198621 gt 0 Primary

Time hardening ε119888119903 = 1198621σ11986221199051198623119890minus1198624119879 1198621 gt 0 Primary

Generalized exponential

ε119888119903 = 1198621σ1198622119903119890minus119903119905 r = 1198625σ

11986231199051198623119890minus1198624119879 1198621 gt 0

1198625 gt 0

Primary

Generalized Graham

ε119888119903 = 1198621σ1198622(1199051198623 + 1198624119905

1198625 + 11986261199051198627)119890minus1198628119879 1198621 gt 0 Primary

Generalized Blackburn

ε119888119903 = 119891(1 minus 119890minus119903119905) + 119892119905 119891 = 1198621119890

1198622σ 119903 = 1198623(σ1198624)1198625 119892 = 1198626119890

1198627σ

1198621 gt 0

1198623 gt 0

1198626 gt 0

Primary

Modified time hardening

ε119888119903 = 1198881σ11988821199051198883+1(1198883 + 1) 1198621 gt 0 Primary

Modified strain hardening

ε119888119903 = 1198621σ1198622[(1198623 + 1)ε119888119903]

1198623119890minus1198624119879 1198621 gt 0 Primary

Generalized Garofalo

ε119888119903 = 1198621[119904119894119899ℎ(1198622σ)]1198623119890minus1198624119879 1198621 gt 0 Secondary

Exponential form

ε119888119903 = 1198621119890σ1198622119890minus1198623119879 1198621 gt 0 Secondary

Norton ε119888119903 = 1198621σ1198622119890minus1198623119879 1198621 gt 0 Secondary

Combined time hardening

ε119888119903 = 1198621σ11986221199051198623+1119890minus1198624119879(1198883 + 1)

+ 1198625σ1198626119905119890minus1198627119879

1198621 gt 0

1198625 gt 0

Primary+ Secondary

Rational polynomial ε119888119903 = 1198621

partε119888part119905 ε119888 =

119888119901119905

1 + 119901119905+ ε119898119905

ε119898 = 1198622101198623σσ1198624 119888 = 1198627ε

1198628σ1198629

119901 = 11986210ε11986211σ11986212

1198622 gt 0 Primary+ Secondary

Generalized time hardening

ε119888119903 = 119891119905119903119890minus1198626119879 119891 = 1198621σ + 1198622σ2 + 1198623σ

3 119903 = 1198624 + 1198625

Primary

32 A Modified Theta Projection Model

321 Original theta projection model and its variations

The original theta projection model [50] was first proposed to accurate predict long-term creep and creep-fracture properties It assumes that a creep curve under constant temperature and stress can be described as

15

휀119888119903 = 1205791(1 minus 119890minus1205792119905) + 1205793(119890

1205794119905 minus 1) (22)

where 휀119888119903 is the creep strain 119905 the time and 1205791~1205794 the model parameters Figure 4 illustrates how the first term and second term on right hand side develop with time and constitute a creep curve which has obvious primary secondary and tertiary stages Model parameters 120579119894(119894 = 1234) are temperature and stress dependent and are described with the following expression

119897119899thinsp120579119894 = 119886119894 + 119887119894120590 + 119888119894119879 + 119889119894120590119879 (23)

where 119886119894 119887119894 119888119894 and 119889119894 ( 119894 = 1234 ) are constants It is noted [45] that this equation is suitable when the temperature range of experiment data sets is very narrow such that minus1119879 is almost proportional to 119879 These constants (119886119894 119887119894 119888119894 and 119889119894 (119894 = 1234)) are the final model parameters which are generated with two fitting procedures

(i) Fitting the experimental creep curves to Eq (22) generates several groups of 1205791~1205794 for each creep curve under certain temperature and stress We denote (119879119895 120590119895 1205791119895 1205792119895 1205793119895 1205794119895 ) as parameters for the 119895119905ℎ

experimental curve under temperature 119879119895 and stress 120590119895

(ii) Fitting 1205791 to Eq (23) as a function of temperature and stress with the obtained data ( 119879119895 120590119895 1205791119895 ) from ( 119879119895 120590119895 1205791119895 1205792119895 1205793119895 1205794119895 ) generates

parameters 1198861 1198871 1198881 and 1198891 Repeat this fitting procedure for 1205792 1205793 and 1205794 to get parameters 119886119894 119887119894 119888119894 and 119889119894 (119894 = 234)

Variations of the model were developed and applied to other different materials to better accommodate the experimental data which are all then called a lsquomodified theta projection methodrsquo independently Examples are models in [51] for lowhigh alloy ferritic austenitic steels Al-alloys Al-matrix composites [52] for Nickel-based Super-alloy [53] for pure metals and class M alloys with intermediate range of stress (40-70 of the yield stress) and temperature (40-60 of the melting temperature)

Figure 4 Illustration of theta projection method [54]

16

322 A modified theta projection model

The new modified theta projection model we proposed [54] was mainly adapted from the original theta projection method [50] and the modified version [51] The new model describes the creep curve with the following equation

휀119888119903 = 1205791(1 minus 119890minus1205792119905) + 120579119898119905 + 1205793(119890

1205794119905 minus 1) (24)

An additional term 120579119898119905 is seen compared with Eq (22) in the original method The reason for adding this term is that the 16MND5 steel shows fairly long secondary stages in many experimental creep curves and the creep curves can be better fitted by Eq (24) than Eq (22) As the structure undergoes different creep process under different temperature and stress the model parameters θ119894 (119894 =1~4 and 119898) are also temperature and stress dependent

The main difference compared with the original and all other modified projection models is that numerical interpolation is used to replace the second numerical fitting in previous models In other words instead of fitting the model parameters as functions of stress and temperature with Eq (23) the distributions of the model parameters are assumed to be piecewise in the temperature-stress space and they will be interpolated directly from known parameters obtained from experiments The consideration is that the experimental data ([16] [17]) for 16MND5 covers a wide temperature range varying from 600 degC to 1300 degC and a wide stress load range varying from 08 MPa to 248 MPa while Eq (23) in the existing methods is based on a narrow temperature span Poor parameter fitting results would be expected if we directly fit the parameters 1205791~1205794 with Eq (23) as used in the original methods

The creep model was developed using the following steps

(i) Fitting Each creep curve can be represented by a group of model parameters [1205791 1205792 120579119898 1205793 1205794] with respect to Eq (24) by fitting the creep curves to Eq (24) All parameters of the 32 experimental curves form an

initial dataset of the model parameters [119879119895 120590119895 1205791119895 1205792119895 120579119898119895 1205793119895 1205794119895] where j (0ltjle32) is the jth creep curve

(ii) Interpolation Parameters of a creep curve under any [119879 σ] are directly interpolated from the dataset

(iii) Parameter adjustment The dataset was further updated to satisfy the monotonicity assumption we have taken that creep strain monotonically increases with temperature and stress

(iv) Different hardening models for dynamic loads

Step (i) was done using MATLAB In the following steps (ii)-(iv) are described in detail

Model parameters for new creep curves are interpolated from the dataset over the 2D 119879 minus 120590 space The interpolation is divided into two sub-interpolation steps interpolate over stress with fixed temperature and then interpolate over temperature with fixed stress We take the interpolation of model parameters under (119879prime σprime) for example In the first interpolation we first find out the two

17

temperatures 119879119896119899119900119908119899119904119898119886119897119897 and 119879119896119899119900119908119899119897119886119903119892119890 most close to 119879prime from the experimental

temperature set 600 700 800900 1000 1100 1200 1300 Under each temperature there are four sets of model parameters in the dataset corresponding to four different stress loads Then the first interpolation is done over stress load from the known dataset of model parameters under each

temperature to output the parameters for (119879119896119899119900119908119899119904119898119886119897119897 120590rsquo) and (119879119896119899119900119908119899119897119886119903119892119890 120590rsquo)

In the second interpolation the model outputs parameters for (119879rsquo 120590rsquo) from

(119879119896119899119900119908119899119904119898119886119897119897 120590rsquo) and (119879119896119899119900119908119899119897119886119903119892119890 120590rsquo) For 120579i(119894 = 12 3 4) we assume a logarithm

relation between 120579119894 temperature 119879 and stress 120590 which is similar to Eq (23)

119897119899120579119894 = (119886 + 119887120590)(119888 + 119889119879) (25)

It reduces to 120579119894 = 11986011198901198871σ with fixed temperature in the first interpolation and to

θ119894 = 11986021198901198872T with fixed stress in the second interpolation The interpolation

expression for θ119894 = 11986011198901198871σwith two known points (1205901 1205791198941) and (1205902 1205791198942) is given

by Eq (26) The interpolation expression for θ119894 = 11986021198901198872T with two known points

(1198791 1205791198941) and (1198792 1205791198942) is given by Eq (27)

120579119894 = 120579i1

120590minus12059021205901minus1205902120579i2

1205901minus1205901205901minus1205902 (26)

120579119894 = 120579i1

119879minus11987921198791minus1198792120579i2

1198791minus1198791198791minus1198792 (27)

Similarly we assume the following relation between 120579119898 temperature 119879 and stress 120590

120579119898 = 119860120590119899119890minus119887 119879frasl (28)

The interpolation is done over the stress with 120579119898 = 1198603120590n by fixing temperature

and then over temperature with 120579119898 = 1198604119890minus1198874119879 by fixing stress The interpolation

expression over stress for 120579119898 = 1198603120590119899 with two known points (1205901 1205791198981) and

(1205902 1205791198982) would be given by Eq (29) The interpolation expression over

temperature for 120579119898 = 1198604119890minus1198874119879 with two known points (1198791 1205791198981) and (1198792 1205791198982)

would be given by Eq (30)

120579119898 = 119890119909119901(119897119899120590 minus 11989711989912059011198971198991205902 minus 1198971198991205901

1198971198991205791198982 +1198971198991205902 minus 119897119899120590

1198971198991205902 minus 11989711989912059011198971198991205791198981) (29)

120579119898 = 120579m2(11198791minus1119879) (

11198791minus11198792)fraslsdot 120579m1

(1119879minus11198792) (

11198791minus11198792)frasl (30)

In this model we also take the assumption that the creep strain monotonically increases with temperature and stress loads1 Or specifically with the same creep time the structural component under higher temperature or stress experiences

1 Intuitively materials deform more under higher stress and temperature This is also mostly observed in tensile experiments Though some exceptions exist one instance of which is the effect of phase change some materials under certain temperature around the phase change temperature may show much weaker structural performance Here we just simply assumed that the creep strain monotonically increases with temperature and stress

18

higher creep strain than that with lower temperature or stress (the creep curve under higher temperature or stress should be above the curve with lower temperature or stress) When generating the parameter dataset (Step (i)) we already kept the trend that under the same temperature cases under higher stress loads would have larger creep model parameters and give higher creep strains This trend would also hold for the interpolated temperatures which thus guarantees that creep strain monotonically increases with stress load under same temperature Then the question becomes if creep strain also monotonically increases with temperature given fixed stress However the answer could not always be a yes especially in the extrapolation range of the stress load Figure 5(a) illustrates an example of the interpolation of model parameters under (119879119872 σ119872) The red dots and blue dots are known parameters from the parameter dataset for corresponding high temperature 119879119867 and low temperature 119879119871 respectively The green dots (119879119867 σ119872) and (119879119871 σ119872) are interpolated from the red and blue dots in the first sub-interpolation respectively The black dot (119879119872 σ119872) is interpolated from the green dots in the second sub-interpolation It is seen that when σ119872 lt σ119862 the value of (119879119871 σ119872) is higher than that of (119879119867 σ119872) It means the higher 119879119872 is the smaller the model parameter will be ie the model would predict larger creep strain for lower temperatures which violates our monotonicity assumption One simple solution used here as shown in Figure 5(b) is to minorly adjust the values of the four red and blue dots such that the intersection point C can be sufficiently small and the situation σ119872 lt σ119862 would barely occur in the extrapolation range

Figure 5 Illustration of model parameter interpolation (a) before and (b) after parameter adjustment [54]

A corresponding parameter adjustment process was introduced to verify that the model predicts higher creep strain under higher temperature under fixed stress Figure 6 illustrates the general verification process for any (119879 120590) in the interested load range the creep strain of (119879 + 120575119879 120590) should be larger than that of (119879 120590) Otherwise parameters would be minorly adjusted and the verification process be repeated This process would be iterated several times until the requirement is reached The achieved verification shows

Stress

σC

High temperature TH

MPa

a

0

Model

par

amet

er θ

Low temperature TL

MPa

a

0

Model

par

amet

er θ

High temperature TH

Low temperature TL

σC

Stress

C C

(a) (b)

σM

σM

19

(i) For any (119879 120590) in the temperature range 273~ 1073 K with an interval of 1 K and the stress range 1 ~260 MPa with an interval of 1 MPa its creep strain is higher than that of (119879 minus 1119870 120590) for the first 12000000 s

(ii) For any (119879 120590) in the temperature range 1073~1673 K with an interval of 1 K and the stress range 1 ~80 MPa with an interval of 1MPa its creep strain is higher than that of (119879 minus 1119870 120590) for the first 12000000 s

(iii) For any (119879 120590) in the stress range 01~26 MPa with a smaller interval of 01 MPa for the full temperature range 273~1673 K with an interval of 1 K its creep strain is higher than that of (119879 minus 1 120590) for the first 12000000 s

Above verification covers a wide range of temperature and stress especially where the stress and temperature are low and extrapolation is used Further verification to cover even higher stress or temperature ranges is not considered as creep increases drastically under such high stress and temperature and the results would become meaningless With these regards we generally think the monotonicity requirement is reached The model parameter dataset was updated along with the adjustment The final dataset was given as Table 2 in Paper I

Figure 6 Parameter adjustment process [54]

Figure 7(a-d) present the model predictions of selected creep curves in ANSYS Mechanical (labelled as lsquoANSYSrsquo) compared with experimental curves (labelled as lsquoREVISArsquo) at temperature 600 800 1000 and 1200 degC respectively The predicted curves proved to clearly show the three-stage creep process and agreed well with the experimental creep curves

Initial parametersdataset

θ1 θ2 θm θ3 θ4 from creep

curve-fitting

Parameters interpolation

Is the monotonicity

assumption satisfied

IsMon= =1

Final model parameters

Adjust the

parametersdataset

θ1 θ2 θm θ3 θ4

No

Yes

IsMon=1 a flag is monotonic achieved

For σ in the stress range in MPa

For T in range Tstart Tend in K

For t in range 112000000 in s

If ε(σ T+1t)lt ε(σTt) IsMon=0

Iteration loop of

parameter adjustment

20

Figure 7 Selected model predictions compared with experimental creep curves [54]

Instead of being constant temperature and stress loads also dynamically change with time in real cases Hardening models determine how creep strain rate reacts to the change of loads during a creep process The time hardening model and strain hardening model2 are supported in our model implementation in ANSYS Mechanical which are two mostly used hardening models

Time hardening assumes that the following creep strain rate after a load change depends only on the current creep time from the beginning of the creep process (and also temperature and stress if used in the creep model) and ignores the current creep strain Strain hardening assumes that the creep strain rate depends only upon the current creep strain (and also temperature and stress if used in the creep model) [55]

2 The time hardening and strain hardening here are more general terms than the specific time hardening creep model and the strain hardening creep model mentioned in Table 1

(a) 600 degC (b) 800 degC

(c) 1000 degC (d) 1200 degC

21

Eq (24) in default is a time hardening model as creep strain is directly described by the time 119905 Eq (31) gives the creep strain increment 119889휀119888119903 at time 119905 during time interval 119889119905

119889휀119888119903 = (12057911205792119890minus1205792119905 + 120579119898 + 12057931205794119890

1205794119905)119889119905 (31)

The strain hardening model is derived from Eq (24) in an implicit form Specifically when temperature and stress load change to (119879 120590) at the time 119905 and current creep strain 휀1198881199030 the model will get the creep curve of constant (119879 120590) and

output how this curve will react under the current strain 휀1198881199030 Since the creep curve shows creep strain monotonically increases with time given the creep strain 휀1198881199030 under creep curve of (119879 120590) we can uniquely find this reference time

119905119903119890119891 satisfying Eq (32) One should notice that the reference time 119905119903119890119891 can be

different from the current physical time 119905

휀1198881199030 = 1205791(1 minus 119890minus1205792119905119903119890119891) + 120579119898119905119903119890119891 + 1205793(119890

1205794119905119903119890119891 minus 1) (32)

where all 120579119894(119894 = 1~4119898) are model parameters of the new creep curve (119879 120590) It can be solved for 119905119903119890119891 with methods eg Newton-Ramphson [56] Then the creep

strain increment at 119905119903119890119891 during time interval 119889119905 is given as

119889휀119888119903 = (12057911205792119890minus1205792119905119903119890119891 + 120579119898 + 12057931205794119890

1205794119905119903119890119891)119889119905 (33)

Therefore combining Eq (32) and (33) gives the creep strain rate at creep strain 휀1198881199030 In summary when loads are changed to (119879 120590) with current creep strain 휀1198881199030

the model traces the creep curve of (119879 120590) and calculates the creep strain rate of this curve at strain 휀1198881199030

One calculation was done to demonstrate the performance of the hardening models The stress load was set to 151 MPa and kept unchanged A step temperature load was applied as the orange dash curve shows in Figure 8 it was 900 degC during 0-36000 s and suddenly jumped to 1000 degC at 36000 s which was then kept till the end of the simulation The blue solid curve (the part before the temperature change was identical to and therefore covered by the green solid curve in the figure) and blue dash curve are the creep curves under constant temperature T=900 degC and T=1000 degC respectively The green solid curve and green dash curve represent the model predictions considering this temperature change with time hardening model and strain hardening model respectively Before this change the creep curves with both hardening models were identical to the creep curve of the T=900 degC till point A After that the creep strain rate then shifted from Point A to Point B (which had the same time with A) in the time hardening model and behaved the same as the curve T=1000 degC as shown in the rest part of the green solid curve In the strain hardening model the creep strain rate shifted from Point A to Point C (which had the same creep strain with A) and behaved the same as the curve T=1000 degC as shown in the rest part of the green dash curve The large difference between two models was partially caused by the drastic temperature jump In real cases the difference would be less significant as the temperature changes more mildly Overall this prediction is exactly the expected behaviors of the time hardening model and strain hardening model

22

Figure 8 Time hardening and strain hardening in ANSYS Mechanical [54]

23

4 Coupled Thermo-Fluid-Structure Analysis of RPV Failure

As described in Section 13 RPV failure is coupled with melt pool heat transfer leading it into a transient thermo-fluid-structure coupled problem The goal of this section is to develop and validate a reliable coupling approach which supports transient coupled thermo-fluid-structure analysis as well as advanced models and computational platforms so as to improve fidelity and facilitate implementation The approach was validated against the FOREVER-EC2 experiment and then applied to an RPV failure analysis of a Nordic BWR As the approach adopts the volume loads mapping for load transfer comparative analysis was also performed against the coupling approach using surface loads mapping Future work is considered in analysis of penetration failures (IGTs and CRGTs) of a BWR RPV since the approach is flexible to complex geometries Results of Paper II ampIII are jointly presented in this section

41 Thermo-Fluid-Structure Coupling Approach

Considering the involved phenomena a coupling approach should at least address three issues (i) melt pool heat transfer (ii) RPV deformation and (iii) the interaction between melt pool heat transfer and vessel deformation A two-way strong coupling considering both the transient thermal load from melt pool on vessel and the influence of vessel deformation on melt pool heat transfer involves complex data exchange and is still full of challenges Therefore the new coupling approach is realized in a one-way manner with the assumptions that the influence of vessel deformation on melt pool heat transfer can be ignored and thus only the thermal loads from melt pool to vessel are considered It is based on the considerations that on one hand the deformation is a slow process regarding the relatively long failure time and on the other hand it was pointed out [57] that the physics of natural convection heat transfer in internally heated liquid pools is not sensitive to the geometrical factors based on their calculation results and experimental data from [58] [59] as well as previous experiments in semicircular and square cavities eg [44] [60] In addition one-way coupling is also computationally more efficient especially when heat transfer turbulence structural non-linearity and large-scale geometries are involved The developed approach is based on the ANSYS platform which generally consists of three parts with respect to the three issues respectively

(i) conjugate heat transfer of the melt pool and vessel using ANSYS Fluent (ii) transient thermal loads transfer from ANSYS Fluent to ANSYS

Structural (the ANSYS Mechanical in ANSYS Workbench) (iii) structural analysis of the vessel in ANSYS Structural

ANSYS Structural in default only supports direct transfer of constantsteady-state spatial loads from ANSYS Fluent Transfer of transient loads is realized with the extension tool FSI Transient [61] It is a pre-processing tool (with good documentation) developed by a third-party CAE vendor EDRTM MEDESO [62] to map transient CFD results (heat transfer coefficient temperature and surface pressure) to thermal or structural analysis including the definition of load steps

24

This new procedure proves to be straightforward and reduces efforts in the load transfer

Since structural performance of RPV is highly dependent on temperature the transfer of transient temperature loads from conjugate heat transfer to mechanical solver is a core step of the coupling framework There are two general approaches to realize the transfer as shown in Figure 9 Both approaches contain three parts a conjugate heat transfer analysis for fluid and structure where flow and heat transfer are coupled a mechanical analysis for the structure and a load mapping process that transfers the transient loads The conjugate heat transfer part can be solved with any CFD solver and is treated in the same manner in both approaches In the first approach (Figure 9(a)) transient volume thermal loads are transferred and mapped into the mechanical simulation domain The structure domain reads the essential thermal loads from the mapping after which a structure analysis of this domain is performed In the second approach (Figure 9(b)) transient surface thermal loads are mapped onto the boundary surfaces of structure domain as thermal boundary conditions Then a thermo-structure coupled analysis of the structure domain in the mechanical part is performed To distinguish them we call the former approach volume loads mapping (VLM) and the latter surface loads mapping (SLM) In the RPV failure analysis the fluid and structure domains correspond to the melt pool and RPV respectively Examples of these methods are the studies of [63] and [30] using VLM and [35] using SLM The VLM is adopted in current approach

Figure 9 Frameworks of the thermo-fluid-structure coupling approaches (a) volume loads mapping (b) surface loads mapping [64]

fluid structure structure

fluid structure structure

Conjugate heat transfer problem Mechanical problem

(structure)

Conjugate heat transfer problem Mechanical problem

(thermo-structure coupled)

Transient volume

loads mapping

(VLM)

Transient surface

loads mapping

(SLM)

(a)

(b)

25

42 Numerical Modelling

421 Conjugate heat transfer

In the conjugate heat transfer analysis main focus is put on the modelling of melt pool heat transfer Within the ANSYS Fluent solver two options are supported CFD with turbulence modelling and PECM CFD is the choice when the model goes to detail and more mechanistic analysis are needed while PECM provides an alternative that is faster and possibly more suitable for large-scale problems

Within the scope of RANS models the SST (Shear Stress Transport) model [65] is selected as the turbulence model in present study This model is a combination of k-ε model and k-ω model The k-ω formulation is used in the inner parts of the boundary layer which makes the model directly usable all the way down to the wall through the viscous sub-layer It also switches to k-ε model in the free-stream and thereby avoids the common k-ω problem that the model is too sensitive to the inlet free-stream turbulence properties The considerations on choosing this model are

bull Several two-equation low-Re k-ε models have been proved in [66] not good enough for melt pool heat transfer Therefore these mentioned two-equation models as well as other one-equation models are not considered

bull SST is highly recommended [67] for simulating most industrial heat transfer problems

bull Convincing results ([68] [69]) using SST have been obtained in capturing the thermal behaviour of the natural convection in an internally heated fluid Especially good agreements were achieved between the predicted heat flux distribution along the vessel and experimental data which would be one of the most important factors to estimate model performance in this problem

The description of PECM as an effective model for melt pool heat transfer is given in Section 23

422 Mechanical analysis

The material for the RPV failure analysis considered in this study is the French RPV steel 16MND5 The three-stage modified theta projection model in Section 3 is used as the creep model Since the definition of an RPV failure criterion is not universal (it can be the time when damage variable reaching 1 or either specified value of creep strain creep strain rate or total deformation is reached) we do not explicitly define a failure criterion Instead the simulation is kept running until the creep converge criterion cannot be satisfied even with the defined minimum time step (indicating a fairly large creep strain rate) and the simulation will be automatically terminated The time the simulation stops is taken as the failure time This treatment implicitly implies a criterion of a large creep strain rate We refer to [17] for other material properties Youngrsquos modulus and thermal conductivity are temperature dependent Multilinear Isotropic Hardening (MISO) is used as the plasticity model

26

43 Validation against the FOREVER-EC2 Experiment

In this section the validation against the FOREVER-EC2 experiment is presented (indicated as lsquoVLMrsquo in the following) In addition the simulation results using the SLM approach (indicated as lsquoSLMrsquo in the following) is also included for comparison purpose

431 FOREVER-EC2 experiment

The FOREVER-EC2 experiment [70] is one of the FOREVER experiments that achieved a vessel failure Figure 10 shows the experimental setup A 110 scale pressure vessel with a wall thickness of 15 mm and an outer diameter of ~400 mm was employed The lower head was made of 16MND5 steel Since it was difficult to use prototypical corium materials in the experiment the binary salt 30 CaO-70 B2O3 was employed as simulant To generate the internal heat which corresponds to the nuclear decay heat a specially designed heater was used The power input was maintained ~38 kW during the experiment such that maximum melt pool temperature would be ~1300 degC The pressure inside the vessel was maintained at ~25 MPa Thermocouples were used to measure the melt pool and vessel wall temperatures and linear displacement transducers used to measure the vessel wall displacements The designed experimental conditions were reached at experimental time 11670 s Vessel failure was observed about 12510 s later (at experiment time 24180 s)

Figure 10 Scheme of the experimental setup [22]

27

Figure 11 Computational domains and boundary conditions (a) conjugate heat transfer (b) mechanical analysis [71]

Computational domains and boundary conditions are indicated in Figure 11 SST turbulence modelling was used for pool heat transfer The simulations started at the moment when experimental conditions were reached (experiment time 11670 s) So simulation time 0 s corresponds to experiment time 11670 s

432 Conjugate heat transfer

Figure 12 shows the temperature fields at simulation time 0 s 1225 s 2475 s 4975 s and 9975 s All the time points in this part refer to simulation time The field at time 0 s (Figure 12(a)) was the initial temperature that was reconstructed based on the thermocouple readings The temperature increased with time and reached a steady state when the maximum temperature reached ~1607 K Figure 12(b-f) show clear thermal stratification which is an expected phenomenon of natural convection

Internal pressure

25 bar

Internal pressure 25e6-2500times981timesy Pa

External pressure

1 bar

Zero displacement

y=0 m

Gravity 400

mm

R=203 mm

15 m

m

Tamb

=323 K radiation with ε = 08

+ convection with h=10Wm2K

Tamb

=800 K radiation with ε=02

(a) (b)

Tamb

=800 K radiation with ε=08

+ convection with h=10 Wm2K

28

Figure 12 Temperature fields at simulation time (a) 0 s (b)1225 s (c) 2475s (d) 4975 s (e) 7475 s and (f) 9975 s [64]

Figure 13 shows the temperature profiles of the external vessel along the polar angle in comparison with experimental data Regions with a polar angle larger than 90deg are those in the cylindrical part of the vessel It also shows the temperature profile increased with time until reaching a steady state The profile almost did not change after 4975 s which indicates that a steady state of the temperature was reached As time increased the temperature in the lower head region 0~90deg (where the vessel was contacted with the melt) evolved from a relatively flat distribution to a distribution that temperature at high angle was higher than that at low angle by more than 200 K Generally the steady state results agreed well with the experimental data except for the low angle part of vessel lower head and the high angle part of the cylindrical vessel In addition to modelling errors the discrepancy in the low angle part of the lower head may be caused by the crust and gap possibly formed at vessel bottom in the experiment which can reduce the downward heat transfer and result in lower temperature while they cannot be modelled in simulations As to the high angle cylindrical vessel part simplified boundary conditions were applied to the inner and outer vessel wall since it was far from the melt and not of interest which may result in some discrepancy

(a) (b)

(c) (d) (e) (f)

29

Figure 13 Temperature profiles of external vessel surface along the vessel polar angle [64]

433 Mechanical analysis

In the mechanical analysis the VLM predicted a failure time of 23959 s and the SLM predicted a failure time of 22445 s both are close to the experimental failure time 24180 s All the time points in this part refer to the experiment time Figure 14 shows the creep strain fields at failure time Both approaches predicted maximum creep strains at same location where the temperature was high (cf Figure 12) This is related to the fact that creep behavior is highly influenced by temperature Maximum values for both cases were also close to each other 0393 in VLM and 0366 in SLM

Figure 15 presents the total deformation of the vessel bottom point (also called ldquosouth polerdquo in some literature) in the vertical direction Since there was an initial displacement reading of ~355 mm at time 0 s which almost did not change during 0~11670 s (the whole period before the experimental conditions were reached) an initial value of 355 mm was added to the simulation result such that experimental and simulation results had the same total displacement at t=11670 s Similar trend was seen in the VLM SLM and experimental data that the deformation increased with time and experienced a drastic acceleration at the end Generally both simulation approaches agreed well with the experimental data though they both slightly under predicted the value of total deformation The discrepancy may possibly be related to the accuracy of the thermal predictions modelling errors of plasticity and creep and simplified boundary conditions

400

600

800

1000

1200

1400

1600

0 20 40 60 80 100 120 140 160

Tem

per

ature

(K

)

Polar angle (deg)

Experiment

25 s

1225 s

2475 s

4975 s

7475 s

9975 s

Weilding line

30

Figure 14 Creep strain fields at failure time (a) VLM (b) SLM [64]

Figure 15 Total deformation in vertical direction of the vessel bottom point [64] [71]

The wall thickness changes are plotted together with experimental results in Figure 16 The wall thickness change δ119909 is defined as

δ119909 = 119909119890119899119889 minus 119909119900119903119894119892119894119899119886119897 (34)

000

500

1000

1500

2000

2500

11670 13670 15670 17670 19670 21670 23670

Tota

l def

orm

atio

n (

mm

)

Time (s)

ExperimentVLMSLM

(a) (b)

31

where 119909119890119899119889 and 119909119900119903119894119892119894119899119886119897 are the wall thickness at failure time and the original wall

thickness respectively Since 119909119890119899119889 cannot directly be post-processed it was estimated in the following way (cf the subfigure of Figure 16) for the same polar angle θ find the corresponding points Pin (1199091 1199101) and Pout (1199092 1199102) on the inner and outer surfaces of the deformed vessel respectively the distance of Pin and

Pout radic(1199091 minus 1199092)2 + (1199101 minus 1199102)2 is calculated as 119909119890119899119889 at polar angle 120579 Such estimation of the wall thickness is more accurate when the line PinPout is closer to be orthogonal to the deformed local surface Considering the final deformed vessel as shown in Figure 14 the estimation should be fine at regions close to 0deg and 90deg and can be slightly overestimated at medium angle region The VLM curve was compared with the SLM curve and 4 experimental curves measured at vessel left right front and back sides Generally VLM curve agreed well with the experimental and SLM curves wall thickness gradually decreases with angle till around 75deg and then increases till 90deg the maximum thickness change occurred at around 75deg would be around -5 mm The positive values at low angle may be caused by the thermal expansion It also shows that the thickness transitions in experimental curves were smoother than those in both simulation curves while simulation curves showed narrower regions with large thickness changes (at angle 70deg~80deg) Possible reasons might be (i) sharper temperature profile predicted in simulation (ii) input uncertainty and (iii) modelling errors The SLM predicted slightly thinner vessel than the VLM except for the necking region The discrepancy might be caused by various reasons eg the interior vessel temperature can be slightly different between two approaches which would then result in different thermal stress and different deformation the final deformed vessel shapes cannot be exactly the same which also results in different errors in estimating PinPout

Figure 16 Wall thickness changes along the vessel polar angle [64] [71]

-6

-5

-4

-3

-2

-1

0

1

0 10 20 30 40 50 60 70 80 90

Wal

l th

icknes

s ch

ange

(mm

)

Polar angle (deg)

VLMSLMExperiment (left)Experiment (right)Experiment (front)Experiment (back)

32

44 Application to BWR Case

In this section the coupling approach (indicated as lsquoVLMrsquo in the following) was applied to an RPV failure analysis of a BWR A simulation using the SLM coupling approach (indicated as lsquoSLMrsquo in the following) was also performed for comparison purpose

A hypothetical severe accident of a Nordic BWR initiated by an SBO was considered The core materials would melt and migrate downward to the RPV lower head The total amount of corium relocated in the lower head was considered to form a 19 m corium melt pooldebris bed assuming the corium distribution was homogeneous

Figure 17 Computational domains and boundary conditions (a) conjugate heat transfer (b) mechanical analysis [64]

Figure 17(a) shows the computational domain of the conjugate heat transfer PECM was used in this calculation A general mesh with a pool height of 23 m was used to accommodate most of the relocation situations with the same mesh Since the corium only had a height of 19 m the top 04 m region was treated as injected water covering the corium The top water was expected to be boiling and we assumed a uniform temperature of 383 K Under the implementation of PECM this region was damped with artificially high heat capacity and high conductivity such that it would efficiently remove heat from the corium and suppress its own temperature increase For the corium part below water a volumetric heat source 11 MWm3 was implemented as the nuclear decay heat For the in-vessel surfaces a constant temperature of 383K was applied as they are covered by water which can be boiling Without water cooling on the external vessel wall a very small heat loss of 20 Wm2 was applied Melting and solidification of the corium were also considered with a solidus temperature of 2750 K and a liquidus temperature of 2770 K The initial temperature was set as

q=20 Wm2

T=383 K

Qv=11

MWm3

315 bar

315 bar+ hydrostatic

pressure of corium

Gravity

248 bar

Zero

displacement

Water

Corium

(a) (b)

33

1100 K for the corium 383 K for the water and 450 K for the vessel For the mechanical analysis (Figure 17(b)) an inner pressure of 315 bar and an outer pressure of 248 bar were considered For the region covered by corium an extra hydrostatic pressure was added on the vessel wall A 2 m long vessel extension was considered as well as the standard earth gravity

Figure 18 presents the wall temperature profile along the inner vessel surface at 50 s 2500 s 7500 s and 10000 s The temperature increased with time for the vessel surface contacted with corium and the distribution was flat which was due to that large part of the corium was still in solidus status It also shows that for the surface region that contacted the water the temperature was basically unchanged which indicates that the damping treatment indeed suppressed the temperature increment and guaranteed the heat removal upward by water

Figure 18 Temperature profiles of inner vessel wall along the vessel polar angle [64]

The predicted RPV failure time was 9869 s by VLM and 9650 s by SLM Compared with the FOREVER-EC2 case in Section 43 the VLM predicted a slightly later failure time than the SLM in this BWR case too Figure 19(a) compares the maximum creep strain developments with time of both approaches Before 9000 s the predicted values by both approaches are close to each other Afterward the creep acceleration occurred and the strain increased drastically The acceleration of the VLM occurred 200 s later than that of the SLM which agreed with the predicted later failure time Similar trend was observed for the total deformation in vertical direction of vessel bottom point as shown in Figure 19(b) The deformations of both approaches are close except that the acceleration of VLM occurred later than that of SLM Overall results of VLM agreed well with those of SLM

0

500

1000

1500

2000

2500

0 10 20 30 40 50 60 70 80 90 100

Tem

per

ature

(K

)

Polar angle (deg)

50 s

2500 s

5000 s

7000 s

10000 s

34

Figure 19 Results of structural analysis (a) creep strain and (b) total deformation in vertical direction of vessel bottom point [64]

45 Computational Efficiency

Computational efficiency is also an important consideration regarding the performance of numerical approaches The computational efficiencies of both VLM and SLM are analysed and compared in this section As parallel computing technique was used to speed up the simulations the number of parallel cores and corresponding costed computational time for each simulation are given together in Table 2 as a generally reflection of the computational efficiency All the simulations were done in a Lenovo P700 workstation equipped with 2 Intel Xeon E5-2630 v3 processors (8 cores per processor) and 32 G RAM For the simulations of FOREVER-EC2 SLM employed twice the parallel cores and costed 286 more time compared with VLM For the simulations of prototypic BWR case both VLM and SLM employed the same number of parallel cores but VLM only took 13 of the computational time compared with SLM Overall it can be concluded that VLM has higher computational efficiency than the SLM The reason is a structural analysis is performed in the VLM approach while a coupled thermo-structural analysis is performed in the SLM approach (cf Figure 9) a standalone structural analysis is computed faster than a coupled thermo-structural analysis

Table 2 Comparison of computational efficiency [64]

Case FOREVER-EC2 BWR VLM SLM VLM SLM

Number of cores 4 8 4 4 Computational time (hours) 21 27 18 72

35

5 Hybrid Coupling Approach for Efficient RPV Failure Analysis

In this section we propose a hybrid coupling approach for the efficient RPV failure analysis The idea is to replace the ANSYS Fluent in previous coupling approach with a lumped-parameter code for quick estimate of the melt pool heat transfer With the least modelling details (compared with CFD methods and distributed-parameter methods eg PECM) lumped-parameter codes prove to be the most efficient for melt pool heat transfer The new hybrid approach characterizes 1) utilization of efficient models for melt pool heat transfer and 2) the support of detailed structural modelling of RPV deformation allowing studies with different RPV geometric shapes under effects like local thinning due to ablation For studies of which the focus is mainly on structural behaviour of RPV eg [13] [26] while mechanistic simulation of melt pool convection is computationally expensive and unnecessary the lumped-parameter code could help providing reasonable thermal loads

The development of the approach consists of two tasks (i) development of a lumped-parameter code and (ii) its coupling with ANSYS Mechanical to form this hybrid coupling framework In the following we describe both tasks and the corresponding verification and validation processes respectively This section is mainly based on Paper IV

51 A Lumped-parameter Code for Melt Pool Heat Transfer

A lumped-parameter code transIVR was developed It features (i) quick estimate of transient melt pool heat transfer with reasonable accuracies and (ii) two-dimensional representation of heat conduction in the RPV wall facilitating the coupled analysis of the RPV failure under thermal loads The following assumptions are taken in the transIVR modelling

bull The crust between the oxide pool and metal pool is thin

bull The heat flux has uniform distributions on the bottom top and side surfaces of metal pool

bull Heat source only exists in oxide layer

bull Latent heat of fusion is ignored

Figure 20 illustrates the modelled three domains oxide layer metal layer and the RPV wall In addition the internal structure inside the vessel can also be considered as a radiative body to the top metal surface as in [72] An internal volumetric heat source generated by nuclear decay heat is considered in the oxide layer Heat in oxide layer transfers downward to the RPV wall and upward to the top metal layer Heat in metal layer can be transferred sideward to the RPV wall and upward to the surrounding internal structure through radiation (or to water if inside water flooding is applied) The heat transferred to RPV will be removed from outer RPV surface Possible formations of oxide crust in the top and bottom surface are illustrated as the black regions Possible formation of metal crust on the top surface is shown as the grey region The code automatically detects the

36

existence of the crust on oxide top and metal top surfaces during the simulation and solves the problem correspondingly withwithout crust Boundary conditions are applied to the external and internal surfaces of RPV and (if needed) top surface of metal layer Arbitrary boundary conditions are supported If a boundary condition is nonlinear eg mixed heat convection (linear dependent on 119879 ) and heat radiation (proportional to 1198794) the Picard method [73] would be used for solving this problem

This model consists of three main parts

(i) empirical heat transfer correlations (mostly from experiments) to determine global and local heat transfer coefficients

(ii) energy balances in oxide layer and metal layer where the heat transfer coefficients would be used and

(iii) energy balance in RPV

Figure 20 Two-layer melt pool configuration and RPV wall [74]

511 Heat transfer correlations

For internally heated natural convection (oxide layer) in transIVR the mini-ACOPO correlations [75] are used for the averaged heat transfer as expressed in Eq (35) and (36) and for the angular variation of the heat flux as expressed in Eq (37) and (38)

119873119906119906119901 = 03451198771198861198941198990233 (35)

119873119906119889119899 = 00038119877119886119894119899035(119867119877)025 (36)

119873119906119889119899(θ)

119873119906119889119899= 01 + 108(

θ

θ119898119886119909) minus 45 (

θ

θ119898119886119909)2

+ 86 (θ

θ119898119886119909)3

01 leθ

θ119898119886119909le 06 (37)

RPV

Oxide pool

Metal pool

37

119873119906119889119899(θ)

119873119906119889119899= 041 + 035 (

θ

θ119898119886119909) + (

θ

θ119898119886119909)2

06 ltθ

θ119898119886119909le 1 (38)

The 119877119886119894119899 that appears in Eq (35)~(36) is the internal Rayleigh number which is defined as

119877119886119894119899 =119892β119876119881119867

5

α119896ν (39)

For Rayleigh-Bernard convection in the metal layer heat transfer occurs in all the top bottom and side surfaces To side surface the reduced form [72] of the Churchill-Chu correlation [76] for 119875119903 = 013 is employed as shown in Eq (40)

119873119906119904119889 = 007611987711988613 (40)

To top and bottom surfaces the reduced form [72] of Globe-Dropkin correlation [77] for

119875119903 = 013 is used as expressed in Eq (41)

119873119906119906119901 = 005911987711988613 (41)

The 119877119886 that appears in Eq (40) and (41) is the external Rayleigh number which is defined

as

119877119886 =119892βΔ1198791198673

να (42)

Heat transfer coefficients can be calculated from Nusselt number by Eq (43)

ℎ =119873119906119896

119871 (43)

where 119896 and 119871 are the heat conductivity and characteristic length respectively 119873119906 can be the Nusselt numbers expressed in Eq (35)~(38) and (40)~(41)

512 Energy balance in the pools

The energy balance of oxide layer can be described as Eq (44) The total generated heat of oxide pool equals the sum of the increase of enthalpy (transient temperature change) upward heat flow to metal layer and downward heat flow to RPV

120588119862119901119881119900119897119889119879119900119909119889119905

+ ℎ119906119901119860119900119909119906119901(119879119900119909 minus 119879119900119909119906119901) + ℎ119889119899119860119900119909119889119899(119879119900119909 minus 119879119900119909119889119899) minus 119876119881119881119900119897 = 0 (44)

where ρ is the oxide density 119862119901 the heat capacity 119881119900119897 the total volume 119879119900119909 the bulk temperature 119905 the time ℎ119906119901 the upward heat transfer coefficient 119860119900119909119906119901

upward heat transfer area 119879119900119909119906119901 upward surface temperature ℎ119889119899 downward

heat transfer coefficient 119860119900119909119889119899 downward heat transfer area 119879119900119909119889119899 downward surface temperature and 119876119881 volumetric heat source

38

On the interface between oxide layer and metal layer energy balance requires that the heat transferred from oxide layer should equal the heat to the metal layer which can be described as Eq (45)

ℎ119906119901119860119900119909119906119901(119879119900119909 minus 119879119900119909119906119901) = ℎ119898119905119889119899119860119898119905119889119899(119879119894119899119905 minus 119879119898119905) (45)

where ℎ119898119905119889119899 is the downward heat transfer coefficient of metal layer 119860119898119905119889119899 the downward heat transfer area 119879119894119899119905 the interface temperature and 119879119898119905 the bulk temperature of metal layer

The energy balance of metal pool can be described as Eq (46) The increase of enthalpy (transient temperature change) equals the sum of upward downward and sideward heat flows to surrounding surfaces

ρ119898119905119862119901119898119905119881119900119897119898119905119889119879119898119905119889119905

+ ℎ119898119905119906119901119860119898119905119906119901(119879119898119905 minus 119879119898119905119906119901) + ℎ119898119905119904119889119860119898119905119904119889(119879119898119905 minus 119879119898119905119904119889)

+ ℎ119898119905119889119899119860119898119905119889119899(119879119898119905 minus 119879119894119899119905) = 0

(46)

where ρmt is the metal density 119862119901mt the heat capacity 119881119900119897mt the total volume 119879119898119905 the bulk metal temperature 119905 the time ℎ119898119905119906119901 the upward heat transfer

coefficient 119860119898119905119906119901 upward heat transfer area 119879119898119905119906119901 upward surface

temperature ℎ119898119905119904119889 sideward heat transfer coefficient 119860119898119905119904119889 sideward heat transfer area 119879119898119905119904119889 sideward surface temperature ℎ119898119905119889119899 downward heat transfer coefficient 119860119898119905119889119899 downward heat transfer area and 119879119894119899119905 oxide-metal interface temperature

The heat transferred from metal layer to the top surface should equal the heat radiated from this surface to the environment Considering the radiation and reflections of internal structures to this surface following energy balance equation [72] is achieved

ℎ119898119905119906119901(119879119898119905 minus 119879119898119905119906119901) = σ(1198791198981199051199051199001199014 minus 119879119904

4) (1

ε+1 minus ε119904ε119904

119860119898119905119906119901

119860119904)minus1

(47)

where 119879119898119905119905119900119901 is the temperature of metal top free surface ε the emissivity of

metal ε119904 the emissivity of internal structures 119860119898119905119906119901 the area of metal top surface

and 119860119904 the total area of internal structures Sometimes the internal water cooling may be available above metal layer In this case Eq (47) is unnecessary and 119879119898119905119905119900119901 can be set to a constant value

For the oxide layer top surface Eq (48) is used if not crust is formed in the interface between two layers ie 119879119894119899119905 gt 119879119898119890119897119905119894119899119892119900119909 Otherwise Eq (49) is used

119879119900119909119906119901 = 119879119894119899119905 (48)

119879119900119909119906119901 = 119879119898119890119897119905119894119899119892119900119909 (49)

For the metal layer top surface Eq (50) is used if not crust is formed on top of the metal ie 119879119906119901119904119906119903119891 gt 119879119898119890119897119905119894119899119892119898119905 Otherwise Eq (51) is used

119879119898119905119906119901 = 119879119898119905119905119900119901 (50)

39

119879119898119905119906119901 = 119879119898119890119897119905119894119899119892119898119905 (51)

Overall this part involves 6 equations regarding 6 unknows 119879119900119909 119879119900119909119906119901 119879119894119899119905 119879119898119905

119879119898119905119906119901 and 119879119898119905119905119900119901

513 Energy balance in the RPV wall

The heat transfer in the PRV wall can be modelled as a simple 1D heat conduction problem that heat only transfers in the wall thickness direction or 2D axisymmetric heat conduction problem

For 1D model the treatment in [72] was adopted As illustrated in Figure 21 the heat from the oxide pool transfers to inner vessel surface (119879119894) then to the interior vessel (119879119907119890119904) then to outer vessel (119879119900) and finally to environment

Figure 21 1D heat transfer model in RPV [74]

Unknown variables are the heat flux through inner vessel surface 119902119894119899 the crust thickness 119909119888119903 inner vessel temperature 119879119894 vessel center temperature 119879119907119890119904 and outer vessel temperature 119879119900 Eq (52)~(56) give a closed equation set for these variables considering a radiation boundary condition on the external vessel surface Eq (52)~(53) describe the heat transfer through the crust where an internal heat source 119876119888119903 is also considered Eq (52) can be easily derived from the one-dimensional heat transfer equation with heat source Eq (54)~(55) describe the energy balance on the inner vessel surface and vessel interior respectively Eq (56) describes the energy balance on the outer vessel surface regarding radiation boundary condition For Dirichlet boundary condition Eq (56) is replaced with Eq (57) For other boundary conditions we only need to correspondingly modify the terms on the right-hand side of Eq (56) For situations that no crust is formed on the surface of the vessel Eq (52)~(53) are eliminated and 119902119894119899 in Eq (54) can directly be determined from calculations in Section 512 and Eq (54)~(56) are solved for the unknows 119879119894 119879119907119890119904 and 119879119900 Once 119879119894 exceeds the vessel melting temperature (with or without crust formation) wall ablation is considered to occur Then 119879119894 is fixed to the vessel melting temperature and vessel thickness 119909119907119890119904 becomes the new unknown variable

119879119900119909119889119899 119879119894 119879119907119890119904 119879119900 119879119890119899119907

119909119888119903 119909119907119890119904

crust RPV

Qcr

119902119894119899 119902119897119900119888119886119897

40

119902119894119899 = 119896119888119903(119879119900119909119889119899 minus 119879119894)

119909119888119903+ 1198761198881199031199091198881199032

(52)

119902119894119899 = 119902119897119900119888119886119897 + 119876119888119903119909119888119903

(53)

119902119894119899 = 119896119907119890119904(119879119894 minus 119879119907119890119904)

05119909119907119890119904

(54)

120588119907119890119904119862119901119907119890119904119889119879119907119890119904119889119905

= 119896119907119890119904(minus2 lowast 119879119907119890119904 + 119879119894 + 119879119900)

(05119909119907119890119904)2

(55)

119896119907119890119904(119879119907119890119904 minus 119879119900)

05119909119907119890119904= 휀120590(119879119900

4 minus 1198791198901198991199074 )

(56)

119879119900 = 119879119900119888119900119899119904119905

(57)

For the 2D representation the heat conduction in the RPV is solved using FDM considering the heat transfer in both the thickness and the angular directions in the RPV wall The RPV lower head is treated as a spherical shell The corresponding heat conduction equation in the 2D spherical coordinate is given as Eq (58) The cylindrical part of the vessel is treated as a cylindrical shell The corresponding heat conduction equation in the 2D cylindrical coordinate is given as Eq (59) Eq (58) and (59) are coupled and solved together In default the mesh size is 5deg in angular direction and 0025119909119888119903 in radius direction This mesh discretization can be changed up to user demands

ρ119862119901part119879

part119905=

1

1199032part

part119903(1198961199032

part119879

part119903) +

1

r2sinθ

part

partθ(119896119904119894119899θ

part119879

partθ)

(58)

ρ119862119901part119879

part119905=1

119903

part

part119903(119896119903

part119879

part119903) +

part

part119911(119896part119879

part119911) (59)

52 Hybrid Coupling Approach

The hybrid coupling approach to assess the RPV integrity is illustrated in Figure 22 transIVR and ANSYS Mechanical are coupled in a one-way manner transIVR conducts the conjugate heat transfer of melt pool and RPV and provides transient thermal loads to the ANSYS Mechanical where structural analysis of RPV deformation is performed The 2D heat transfer of RPV is solved in transIVR to better capture the temperature distribution In RPV structural analysis same considerations are taken as in Section 422 for 16MND5 steel the modified theta projection model for creep and MISO for plasticity and temperature-dependent Young modulus etc The VLM is adopted for the transfer of transient thermal loads

41

Figure 22 Framework of the hybrid coupling approach [74]

53 Verification and Validation

The transIVR code was first benchmarked against the UCSB FIBS case of two-layer pool configuration and then validated against the LIVE-7V experiment showing the codersquos capability in predicting two-layer and transient melt pool heat transfers respectively Finally the hybrid coupling approach encompassing transIVR and ANSYS Mechanical was validated against the RPV failure experiment FOREVER-EC2 and the results showed acceptable agreements with those of both the experiment and the previous approach using CFD in Section 43 Hence the transIVR code is an efficient tool for melt pool heat transfer and the hybrid approach an efficient way for RPV failure analysis

531 UCSB FIBS benchmark case

The UCSB FIBS benchmark case [78] was calculated to demonstrate the capability of the transIVR code in simulating two-layer pool heat transfer problems The considered pool configuration was similar to that shown in Figure 20 with the following specific dimensions a two-layer melt pool with a 16 m oxide layer and a 08 m metallic pool on top and an RPV lower head with a thickness of 015 m The internal volumetric source in the oxide layer was 13 MWm3 The calculation results were compared with those of the DOE study [78] and the INEEL study [79]

The predicted results were compared with the digitalized results of the DOE and INEEL studies in Figure 23 All three studies show similar profiles of the heat flux distribution along the polar angle (Figure 23(a)) it increased with the polar angle in the oxide layer to the maximum value at around 77deg and then dropped to a significantly lower value in the metal layer The transIVR result agreed well with the other results with small discrepancy in the metal layer the DOE study gave a

heat flux of 500 MWm2 the INEEL study of 400 MWm2 and the transIVR of somewhere in between Figure 23(b) shows the predicted corium crust thickness distribution along the RPV polar angle also agreed well the DOE and INEEL

transIVR ANSYS Mechanical

Transient

thermal loads

42

studies The crust thickness decreased with polar angle in the oxide layer and no crust was formed in the metal layer As the crust formation is closely related to the heat flux distribution the crust thickness profile was reasonable regarding the heat flux distribution shown in Figure 23(a) Vessel wall ablation would occur due to the high thermal loads Figure 23(c) gives the profile of the remaining vessel wall thickness along the RPV polar angle considering the wall ablation It shows that no ablation occurred in low angle region (0deg~36deg) where heat flux was low and the wall thickness changed from 15 cm to around 6 cm as angle increased in high angle region of the oxide layer Because of the discrepancy in the heat flux of metal layer the predicted wall thicknesses also differ from those of the DOE and INEEL studies the thinnest wall was 73 cm predicted by the DOE study corresponding to the maximum heat flux followed by 88 cm of the present transIVR calculation and 94 cm of the INEEL study Generally speaking the prediction of transIVR is in an acceptable level compared with the other studies for the same benchmark case

Figure 23 Simulation results of the UCSB FIBS benchmark case [74]

(a) heat flux distribution (b) crust thickness distribution

(c) wall thickness distribution

43

532 LIVE-7V experiment

LIVE-7V is an experiment under the LIVE program conducted in KIT to study the core melt phenomena As Figure 24(a) shows it employed the LIVE-3D hemispherical facility with an inner diameter of 1 m and a wall thickness of ~25 mm The non-eutectic binary salt mixture of 20 mol NaNO3- 80 mol KNO3 was used as the simulant material The decay heat in the melt was simulated by cable-type heating with 8 planes of electrical resistance heating wires Water cooling was applied to the top and vessel surfaces Top surface cooling was realized by a cooling lid mounted at height 413 mm from the vessel bottom and vessel cooling by the flow channel between the test vessel and the outer cooling vessel The inlet and outlet water temperature for both top and vessel cooling surfaces were kept ~293 K and ~303 K throughout the whole experiment respectively A 413 mm height pool corresponds to a total volume of ~0194 m3 simulant material The experiment procedure can be divided into four stages with the heating power sequentially shifted from 29 kW to 24 kW 18 kW and 9 kW Figure 24(b) indicates the thermocouple positions in experiment

The simulation results of LIVE-7V using transIVR were compared with experimental data [80] and numerical study using REALPSCDAPSIM code [81] For simplicity the RELAPSCDAPSIM results in the following are labelled as lsquoCOUPLErsquo which is the name of the module in RELAPSCDAPSIM for melt pool heat transfer

Figure 24 LIVE test facility [80] and thermocouple positions [82]

Figure 25 compares the predicted heat removal upward to the top lid and downward to the bottom vessel surface with those of the experiment and COUPLE The black curve indicates the total input power at the four stages The experimental results (red and blue dots) were calculated from the Table 3 of [80] assuming no extra heat losses For the first three stages the transIVR predictions agreed well with the COUPLE results while the predicted upward heat flow was slightly lower and the downward heat flow higher than the experimental data For the last stage the transIVR agreed well with the experimental result while the COUPLE predicted lower upward and higher downward heat flows The short

(a) (b)

44

transition periods between two stages with different total inputs taken in transIVR also agreed with those in the COUPLE predictions

Figure 25 Comparison of heat removal upward to the top melt surface cooling lid and downward to bottom vessel wall [74]

Figure 26 shows the comparison of heat flux distribution along the vessel wall All curves show similar trend that a higher total input power would result in a higher heat flux profile As the COUPLE predictions had relatively flatter profiles transIVR predictions generally agreed better with the experimental results especially when total input was high A discrepancy is seen that the minimum value of a curve occurred at angle 0degfor both transIVR and COUPLE and at angle 30deg for the experimental data It is noted that the models in default assume heat flux monotonically increases with angle (cf Eq (37) and (38)) resulting it impossible to predict higher heat flux at angle 0deg than angle 30deg as in experimental results

45

Figure 26 Heat flux profiles along the vessel polar angle [74]

533 FOREVER-EC2 experiment

In this section we performed calculations against the FOREVER-EC2 experiment to validate the performance of the coupled transIVR and ANSYS Mechanical in assessing the RPV integrity The applied boundary conditions were identical to the settings of previous coupling approach in Section 4 (ANSYS Fluent+ Mechanical) as indicated in Figure 11 The initial bulk temperature of the melt pool was set to be 1261 K in transIVR which corresponds to the maximum value of the initial temperature field in CFD calculation of previous coupling approach Both the experimental data and the simulation results using previous coupling approach are included for comparison In the following results from current hybrid approach are labelled lsquotransIVRrsquo (and referred as transIVR results) results from previous approach labelled lsquoCFDrsquo (and referred as CFD results) and results from experiment labelled lsquoexperimentrsquo

Figure 27 compares the bulk temperature predicted by transIVR with the averaged and maximum temperatures of the CFD results All curves show the same trend that the temperature gradually increased with time till reaching a steady state and then kept unchanged till the end The times needed for reaching the steady state were ~5000 s for the CFD and ~4000 s for the transIVR which are close to each other It indicates again a good performance of the transIVR in predicting the transient behavior of melt pool Compared with the steady-state averaged temperature 1478 K and maximum temperature 1609 K of the CFD results the steady-state temperature of transIVR 1599 K is a reasonable prediction

0

3000

6000

9000

12000

15000

18000

0 20 40 60 80 100

Hea

t fl

ux

(W

m2)

Polar angle (deg)

29 kW-EXP24 kW-EXP18 kW-EXP9 kW-EXP29 kW-COUPLE24 kW-COUPLE18 kW-COUPLE9 kW-COUPLE29 kW-transIVR24 kW-transIVR18 kW-transIVR9 kW-transIVR

46

Figure 27 Temperature evolution with time [74]

Figure 28 Temperature fields at time (a) 01 s (b) 3038 s (c) 6038 s and (d) 13289 s [74]

Figure 28 shows the temperature fields of the vessel at time 01 s 3038 s 6038 s and 13289 s The temperature fields were visualized in the ANSYS Mechanical It also indicates that the transient temperature loads were properly transferred from transIVR to ANSYS Mechanical Figure 28(a-d) show both the minimum and maximum temperatures increased with time In the lower head region the temperature gradually increased with the height The maximum temperature occurs at the location slightly lower than the melt pool top surface This is because thermal loadheat flux imposed on the vessel generally increases with polar angle in the pool natural convection so the code transIVR inherently also predicted higher heat flux at higher angle positions The higher flux at higher position then correspondingly resulted in higher temperature In the cylindrical region the

(a) (b) (c) (d)

[K] [K] [K] [K]

47

temperature decreased as the height since the bottom part would be more significantly heated by the high-temperature lower head region

Figure 29 shows the steady-state temperature profiles of the external vessel surface along the polar angle The transIVR result agreed well with other two results in trend that the temperature increased with the angle and reach the peak values at angle ~80deg then gradually decreased with angle But slightly lower temperature was predicted in the low angle region Possible reason may be that the local heat flux distribution determined by Eq (37) and (38) is steeper than that in the experiment and CFD study which then resulted in a steeper increase of temperature along the angle and a lower temperature at low angle region In the cylindrical part (gt90deg) the transIVR result generally agreed with the experimental and CFD results The discrepancy with the experimental data may be caused by the simplified boundary conditions the ambient temperature inside the vessel was set to uniform constant value while it could decrease as height Therefore predicted heat loss in the higher position of cylindrical part could be smaller than that with height-dependent ambient temperature resulting in higher vessel temperature Overall the prediction was reasonable for a lumped-parameter model

Figure 29 Temperature profiles of the external vessel surface along the vessel polar angle [74]

The transIVR predicted a failure time of 24959 s which is close to 24180 s in experiment and 23959 s in the CFD results Figure 30 shows the total deformation of vessel bottom point with time All three curves increased gradually with time and experienced drastic accelerations at the end The

400

600

800

1000

1200

1400

0 40 80 120 160

Tem

per

ature

(K

)

Polar angle (deg)

transIVRExperimentCFDWeilding Line

48

transIVR curve agreed well with the CFD curve The final total deformation at failure time is also close to that of the CFD result but lower than the experiment data The discrepancy might be related to the simplifications in the boundary conditions accuracy of the thermal load prediction and the modelling errors in plasticity and creep

Figure 30 Total deformation in vertical direction of the vessel bottom point [74]

Figure 31 shows the profile of wall thickness change The transIVR curve was compared with the CFD curve and four experimental curves measured at vessel left right front and back sides The simulation result fitted the experimental and CFD results well in the region 0deg~50deg In the region 50deg~90deg wall thickness gradually decreased with angle till around 75deg and then increased till 90deg The maximum thickness change was -3 mm for the simulation which is close to the experiment left side curve and slightly larger than the -6~ -45 mm of the other curves from experiment and CFD results Concerning possible errors and uncertainties in the modelling and numerical settings the new approach generally gives reasonable predictions

0

5

10

15

20

25

11670 15170 18670 22170 25670

Tota

l def

orm

atio

n (

mm

)

Time (s)

Experiment

CFD

transIVR

49

Figure 31 Wall thickness changes along the vessel polar angle [74]

-6

-5

-4

-3

-2

-1

0

1

0 20 40 60 80 100

Wal

l th

icknes

s ch

ange

(mm

)

Polar angle (deg)

CFD

transIVR

Experiment (left)

Experiment (right)

Experiment (front)

Experiment (back)

50

51

6 Summary and Outlook

Reactor pressure vessel (RPV) is one of the key physical barriers in the defence-in-depth approach of nuclear power safety and its failure is also the transition point of core melt (corium) progression from in-vessel to ex-vessel phase Quantification of RPV failure during a hypothetical severe accident is therefore paramount to the specific qualification of severe accident mitigation strategies as well as the general assessment of corium risk in light water reactors The RPV failure is a problem of structural performance of RPV under the influence of thermal loads imposed by corium melt pool ie it is a transient thermo-fluid-structure coupled problem per se whose solution requires simultaneous modelling of both highly turbulent melt pool heat transfer and large-scale nonlinear RPV creep Due to the complexity of the problem only limited capabilities have been developed in the previous studies under different simplifications and assumptions in models and coupling methods The present study was motivated to advance modelling and coupling approaches for RPV performance analysis considering new understanding of physical phenomena and latest coupling platform of simulation tools

For the new development of coupling approaches and models for physical phenomena related to RPV failure analysis during severe accidents this thesis work was conducted in the following order (i) a new creep model for the RPV steel 16MND5 was developed which features the capability to cover all three creep stages (ii) a new thermo-fluid-structure coupling approach was developed for RPV failure analysis and validated against FOREVER-EC2 experiment (iii) the performance of the new coupling approach using volume loads mapping (VLM) for data transfer was further compared with another approach using surface loads mapping (SLM) (iv) a hybrid coupling approach was proposed for efficient analysis of RPV failure which employs a lumped-parameter code for quick estimate of thermal load and a mechanistic FE code for detailed structural analysis and (v) for the hybrid coupling approach the code transIVR was developed for calculation of transient melt pool heat transfer

The developed creep model for 16MND5 is called lsquomodified theta projection modelrsquo and describes all three stages of a creep process Creep curves are expressed as a function of time with five model parameters 120579119894(119894 = 14 119886119899119889 119898) A dataset for the model parameters was constructed by fitting experimental creep curves into this function and an iterative parameter adjustment process to satisfy the monotonicity assumption of creep strain vs temperature and stress Model parameters for new curves under different temperature and stress loads are directly interpolated from this dataset in contrast to the conventional ldquotheta projection modelrdquo that employs an extra fitting process The new treatment was proven to better accommodate all experimental curves over the wide ranges of temperature and stress loads The model was implemented into the ANSYS Mechanical code The simulation results indicate that the model successfully captured all three creep stages and a good agreement was achieved between each experimental creep curve and corresponding modelrsquos prediction Regarding the model response to transient changes of temperature and stress loads the widely used time hardening and strain hardening models were implemented and

52

performing reasonably well in the new method These properties fulfil the requirements of a creep model for structural analysis

The transient thermo-fluid-structure coupling approach was developed using the ANSYS Fluent code for thermofluid dynamics of the melt pool and ANSYS Structural code for structural mechanics of RPV An extension tool was introduced to realize transient load transfer from ANSYS Fluent to Structural and minimize the user effort Both turbulence models and the effective model PECM can be used for melt pool heat transfer The developed creep model was used for RPV failure analysis of the RPV Since it is insensitive to geometry the coupling approach allows more complex geometries such as a BWR lower head with a forest of penetrations (CRGTs and IGTs) modelled in detail Moreover the coupling approach does not only reflect the transient response of thermo-fluid-structure interaction but also facilitates the implementation of advanced (high-fidelity) models in both fluid mechanics and structural mechanics The validation of the coupling approach against the FOREVER-EC2 experiment showed that the temperature profile vessel failure time total displacement of vessel bottom point and wall thickness profile in the experiment were well predicted

Following the variation of the coupling approach from VLM to SLM for data transfer between ANSYS Fluent and ANSYS Structural the respective coupling approaches were investigated by comparative calculations against the FOREVER-EC2 experiment and a postulated severe accident of a reference BWR For the FOREVER-EC2 experiment both VLM and SLM approaches predicted generally well the total deformation in vertical direction of bottom point and the wall thickness change though the former predicted a slightly later failure time For the reference BWR case good agreements were achieved in predicting the maximum creep strain and the total deformation in vertical direction of bottom point For both experiment and reactor cases the VLM approach predicted slightly later failure times than SLM but the differences were negligible compared to the long time-scale of vessel failure (~104 s) The deformation rate of the vessel wall was slow at the early stage and accelerated with time till a drastic deformation and final vessel failure Generally speaking the VLM and SLM coupling approaches had the similar performance in terms of their predictability of experiment and applicability to reactor case but the VLM was computationally more efficient

In the development of the hybrid coupling approach the transIVR code was developed for quick estimate of transient melt pool heat transfer and determination of thermal loads on RPV transIVR does not only support transient analysis of one- and two- layer melt pool but also the heat transfer in RPV being resolved with 2D finite difference method which captures more spatial thermal details for RPV structural analysis in addition to 1D heat transfer model The capabilities of transIVR in modelling two-layer pool heat transfer and transient pool heat transfer were demonstrated by calculations against the UCSB FIBS benchmark case and the LIVE-7V experiment respectively It was then coupled to the mechanical solver ANSYS Mechanical which together form the hybrid coupling framework for detailed RPV failure analysis The calculation against the FOREVER-EC2 experiment did not only reproduce the major results of the experiment but also had a good agreement with the thermo-fluid-structure

53

coupling approach using CFD This indicated the new coupling approach is capable of capturing the vessel creep failure characteristics and can be employed as an efficient tool addressing the complex vessel failure problem

As a prospective more research topics can be addressed in future either as a continuation of the current work toward an effective tool for vessel failure analysis or from the general view of scientific interest in melt pool heat transfer and RPV failure The following topics are recommended

bull Extended applications to the reactor cases For instance the thermo-fluid-structure coupling approach can be used to study penetration failures in the RPV of an BWR with a detailed representation of penetrations (IGTs and CRGTs)

bull Synthesis or update of heat transfer correlations for melt pool Heat transfer correlations directly benefit the parametric codes such as transIVR in accurate quantification of heat flux distribution around a melt pool So far many empirical correlations have been proposed based on experiments which had a variety of geometries and melt simulants with a wide range of Rayleigh number (Ra) Sensitivity studies have also been performed on the influences of Ra Pa and other material properties Therefore a comprehensive synthesis or update of the existing heat transfer correlations will be a meaningful step for making reasonable extrapolations to the reactor applications with prototypic materials

bull Modelling of pool stratification The motivation starts with the improvement of the PECMrsquos modelling capability such that the dynamic stratification and resulting pool heat transfer can be modelled altogether Simulation of melt stratification has been supported in some system codes such as ASTEC using a predefined parametric approach The improvement of PECM can be considered in the similar approach or more mechanistic modelling

bull Development of a structural module for RPV The idea is to develop a simplified module for structural analysis so it can be coupled with transIVR ultimately forming a standalone code for quick estimate of vessel failure of LWRs during postulated severe accidents

bull Extension of vessel analysis capability The current version of tranIVR is limited to melt pool Although the precursory debris bed and melt ejection after vessel failure were not the focus of the present study they are equally important to assessment of corium risk especially for Swedish BWRs where the melt release conditions and history are crucial to qualification of their severe accident mitigation strategies

54

55

Bibliography

[1] IAEA ldquoIAEA Annual Report 2018rdquo IAEA 2019 [Online] Available httpswwwiaeaorgsitesdefaultfilesgcgc63-5pdf

[2] IAEA Safety of Nuclear Power Plants Design Vienna INTERNATIONAL ATOMIC ENERGY AGENCY 2012

[3] M Fischer ldquoGen II and III In-Vessel and Ex-Vessel Melt Retentionrdquo The Short Course on Severe Accident Phenomenology Oct 2017

[4] B R Sehgal Nuclear Safety in Light Water Reactors Severe Accident Phenomenology 1st ed Academic Press 2011

[5] IAEA Approaches and Tools for Severe Accident Analysis for Nuclear Power Plants Vienna INTERNATIONAL ATOMIC ENERGY AGENCY 2008

[6] WENRA ldquoPractical Elimination Applied to New NPP Designs - Key Elements and Expectationsrdquo Sep 2019 [Online] Available httpwwwwenraorgmediafiler_public20191111practical_elimination_applied_to_new_npp_designs_-_key_elements_and_expectations_-_for_issuepdf

[7] P Matejovic M Barnak M Bachraty and L Vranka ldquoASTEC applications to VVER-440V213 reactorsrdquo Nuclear Engineering and Design vol 272 pp 245ndash260 2014 doi 101016jnucengdes201308077

[8] AREVA ldquoUS EPR Severe Accident Evaluationrdquo Topical Report ANP-10268NP Oct 2006 [Online] Available httpswwwnrcgovdocsML0631ML063100157pdf

[9] C T Martin ldquoCoupled 3D Thermo-mechanical Analysis of Nordic BWR Lower Head Failure in case of Core Melt Severe Accidentrdquo master thesis KTH Stockholm 2013

[10] IAEA ldquoAdvanced Large Water Cooled Reactors A supplement to the IAEArsquos Advanced Reactor Information System (ARIS)rdquo Sep 2015 [Online] Available httpsarisiaeaorgPublicationsIAEA_WRC_Bookletpdf

[11] B Pershagen Light Water Reactor Safety Pergamon Press 1989 [12] Ninos Garis M Agrell H Glanneskog and L Agrenius ldquoAPRI 7 ndash Accident

Phenomena of Risk Importancerdquo SSM SSM 201212 2012 [Online] Available httpswwwstralsakerhetsmyndighetensecontentassets2a9f299f46b949b3b209e780cf71436a201212-apri-7-accident-phenomena-of-risk-importance-en-lagesrapport-om-forskningen-inom-omradet-svara-haverier-under-aren-2009-2011-

[13] F Fichot et al ldquoSome considerations to improve the methodology to assess In-Vessel Retention strategy for high-power reactorsrdquo Annals of Nuclear Energy vol 119 pp 36ndash45 2018 doi 101016janucene201803040

[14] T G Theofanous C Liu S Additon S Angelini O Kymaumllaumlinen and T Salmassi ldquoIn-vessel coolability and retention of a core meltrdquo Nuclear Engineering and Design vol 169 no 1 pp 1ndash48 1997 doi 101016S0029-5493(97)00009-5

[15] C Sainte Catherine ldquoTensile and creep tests material characterization of pressure vessel steel(16MND5) at high temperatures (20 up to 1300 degC)rdquo CEA France SEMTLISNRT98-009A 1998

56

[16] E Altstadt H-G Willschuetz B R Sehgal and F-P Weiss ldquoModelling of in-vessel retention after relocation of corium into the lower plenumrdquo Forschungszentrum Rossendorf FZR-437 2005

[17] H-G Willschuumltz and E Altstadt ldquoGeneration of a High Temperature Material DataBase and its Application to Creep Tests withFrench or German RPV-steelrdquo FORSCHUNGSZENTRUM ROSSENDORF FZR-353 ISSN 1437-322X Aug 2002

[18] J L Rempe S A Chavez and G L Thinnes ldquoLight water reactor lower head failure analysisrdquo Nuclear Regulatory Commission NUREGCR--5642 1993 Accessed Aug 23 2019 [Online] Available httpinisiaeaorgSearchsearchaspxorig_q=RN25025905

[19] S Brosi et al ldquoCORVIS Investigation of light water reactor lower head failure modesrdquo Nuclear Engineering and Design vol 168 no 1 pp 77ndash104 1997 doi 101016S0029-5493(96)00008-8

[20] T Y Chu M M Pilch J H Bentz J S Ludwigsen W Y Lu and L L Humphries ldquoLower Head Failure Experiments and Analysesrdquo Sandia National Laboratories Albuquerque NM USA NUREGCR-5582 SAND98-2047 1999

[21] L L Humphries et al ldquoOECD Lower Head Failure Project Final Reportrdquo Sandia National Laboratories Albuquerque NM 87185-1139 2002

[22] B R Sehgal R R Nourgaliev T N Dihn A Karbojian J A Green and V A Bui ldquoFOREVER Experiments on Thermal and Mechanical Behavior of a Reactor Pressure Vessel during a Severe Accidentrdquo in Proceedings of the Workshop on in-vessel core debris retention and coolability Garching near Munich Germany Mar 3-6 1998

[23] S Bhandari M Fischer B R Sehgal and A Theerthan ldquoPost Test Analysis of the FOREVER C2 Experimentrdquo The 9th International conference on Nuclear Engineering (ICONE-9) Nice France Apr 8-12 2001

[24] J F Mao J W Zhu S Y Bao L J Luo and Z L Gao ldquoCreep deformation and damage behavior of reactor pressure vessel under core meltdown scenariordquo International Journal of Pressure Vessels and Piping vol 139ndash140 pp 107ndash116 2016 doi 101016jijpvp201603009

[25] J Mao L Hu S Bao L Luo and Z Gao ldquoInvestigation on the RPV structural behaviors caused by various cooling water levels under severe accidentrdquo Engineering Failure Analysis vol 79 pp 274ndash284 2017 doi 101016jengfailanal201704029

[26] J F Mao J W Zhu S Y Bao L J Luo and Z L Gao ldquoStudy on structural failure of RPV with geometric discontinuity under severe accidentrdquo Nuclear Engineering and Design vol 307 pp 354ndash363 2016 doi 101016jnucengdes201607027

[27] B R Sehgal V A Bui T N Dinh and R R Nourgaliev ldquoHeat Transfer Processes in Reactor Vessel Lower Plenum During Late Phase of In-Vessel Core Melt Progressionrdquo in Advances in Nuclear Science and Technology J Lewins M Becker R W Albrecht E J Henley J D McKean K Oshima A Sesonske H B Smets and C P L Zaleski Eds Boston MA Springer US 1999 pp 103ndash135

[28] C-T Tran and T-N Dinh ldquoThe effective convectivity model for simulation of melt pool heat transfer in a light water reactor pressure vessel lower head Part I Physical processes modeling and model implementationrdquo Progress

57

in Nuclear Energy vol 51 no 8 pp 849ndash859 2009 doi 101016jpnucene200906007

[29] C-T Tran and T-N Dinh ldquoThe effective convectivity model for simulation of melt pool heat transfer in a light water reactor pressure vessel lower head Part II Model assessment and applicationrdquo Progress in Nuclear Energy vol 51 no 8 pp 860ndash871 2009 doi 101016jpnucene200906001

[30] H-G Willschuumltz E Altstadt B R Sehgal and F-P Weiss ldquoCoupled thermal structural analysis of LWR vessel creep failure experimentsrdquo Nuclear Engineering and Design vol 208 no 3 pp 265ndash282 2001 doi 101016S0029-5493(01)00364-8

[31] E Altstadt and T Moessner ldquoExtension of the ANSYS creep and damage simulation capabilitiesrdquo Forschungszentrum Rossendorf Technical report FZR-296 2000

[32] H-G Willschuumltz E Altstadt B R Sehgal and F-P Weiss ldquoRecursively coupled thermal and mechanical FEM-analysis of lower plenum creep failure experimentsrdquo Annals of Nuclear Energy vol 33 no 2 pp 126ndash148 2006 doi 101016janucene200508006

[33] H-G Willschuumltz E Altstadt B R Sehgal and F-P Weiss ldquoSimulation of creep tests with French or German RPV-steel and investigation of a RPV-support against failurerdquo Annals of Nuclear Energy vol 30 no 10 pp 1033ndash1063 2003 doi 101016S0306-4549(03)00036-7

[34] H Madokoro A Miassoedov and T Schulenberg ldquoA thermal structural analysis tool for RPV lower head behavior during severe accidents with core meltrdquo Mechanical Engineering Letters vol 4 pp 18-00038-18ndash00038 2018 doi 101299mel18-00038

[35] W Villanueva C-T Tran and P Kudinov ldquoCoupled thermo-mechanical creep analysis for boiling water reactor pressure vessel lower headrdquo Nuclear Engineering and Design vol 249 pp 146ndash153 2012 doi 101016jnucengdes201107048

[36] W Villanueva C T Tran and P Kudinov ldquoA Computational Study on Instrumentation Guide Tube Failure During a Severe Accident in Boiling Water Reactorsrdquo in The 14th International Topical Meeting on Nuclear Reactor Thermal hydraulics NURETH-14 Toronto Ontario Canada Sep25-30 2011

[37] W Villanueva C T Tran and P Kudinov ldquoAssessment with Coupled Thermo-Mechanical Creep Analysis of Combined CRGT and External Vessel Cooling Efficiency for A BWRrdquo The 14th International Topical Meeting on Nuclear Reactor Thermal hydraulics NURETH-14 Toronto Ontario Canada Sep 25-30 2011

[38] F R Larson and J Miller ldquoA Time-Temperature Relationship for Rupture and Creep Stressesrdquo Trans ASME vol 74 p 765minus775 1952

[39] C T Tran ldquoDevelopment validation and application of an effective convectivity model for simulation of melt pool heat transfer in a light water reactor lower headrdquo Licentiate thesis KTH Stockholm 2007

[40] ANSYS ldquoANSYS Fluent Theory Guide Release 162rdquo ANSYS Inc Jul 2015 [Online] Available httpwwwansyscom

[41] P Zeng Finite element analysis and application (in Chinese) Tsinghua University Press amp Springer 2004

58

[42] ANSYS ldquoANSYS Mechanical | Finite Element Analysis Softwarerdquo httpswwwansyscomproductsstructuresansys-mechanical-enterprise (accessed Sep 26 2019)

[43] ANSYS ldquoANSYS Mechanical APDL Theory Reference Release 162rdquo ANSYS Inc Jul 2015 [Online] Available httpwwwansyscom

[44] U Steinberner and H H Reineke ldquoTurbulent Buoyancy Convection Heat Transfer with Internal Heat Sourcesrdquo in Proceedings of the 6th Int Heat Transfer Conference Toronto Canada 1978 vol Vol2 pp 305ndash310 doi 101615IHTC63350

[45] R W Evans and B Wilshire Introduction to Creep London Institute of Materials 1993

[46] DoITPoMS ldquoCreep Deformation of Metalsrdquo Creep Deformation of Metals University of Cambridge 2006 httpswwwdoitpomsacuktlplibcreepindexphp (accessed Jul 09 2019)

[47] ANSYS ANSYS Mechanical APDL Material Reference Release 162 2015 [48] SIMULIA Abaqus 612 Analysis Userrsquos Manualvol 3 Materials Chapter

2324 2012 [49] S Imaoka ldquoUser Creep Subroutine STI0704A ANSYS Release110rdquo 2007 [50] R W Evans J D Parker and B Wilshire ldquoThe θ projection conceptmdashA

model-based approach to design and life extension of engineering plantrdquo International Journal of Pressure Vessels and Piping vol 50 no 1 pp 147ndash160 1992 doi 1010160308-0161(92)90035-E

[51] ECCC ldquoRecommendations and Guidance for the Assessment of Creep Strain and Creep Strength Data ECCC Recommendations Volume 5 Part 1 b[Issue 2]rdquo 2013

[52] W D Day and A P Gordon ldquoLife Fraction Hardening Applied to a Modified Theta Projection Creep Model for a Nickel-Based Super-Alloyrdquo The ASME Turbo Expo 2014 Turbine Technical Conference and Exposition Duumlsseldorf Germany Jun16-20 2014 vol 7A-Structures and Dynamics p V07AT29A010 doi 101115GT2014-25881

[53] M Kumar I V Singh B K Mishra S Ahmad A Venugopal Rao and V Kumar ldquoA Modified Theta Projection Model for Creep Behavior of Metals and Alloysrdquo J of Materi Eng and Perform vol 25 no 9 pp 3985ndash3992 2016 doi 101007s11665-016-2197-y

[54] P Yu and W Ma ldquoA modified theta projection model for creep behavior of RPV steel 16MND5rdquo Journal of Materials Science amp Technology vol 47 pp 231ndash242 2020 doi 101016jjmst202002016

[55] ANSYS ANSYS Mechanical Advanced Nonlinear Materials v160 Lecture 2 Rate Dependent Creep ANSYS training materials 2015

[56] Q Y Li N C Wang and D Y Yi Numerical Analysis (in Chinese) 5th ed Tsinghua University Press 2008

[57] T N Dinh R R Nourgaliev and B R Sehgal ldquoOn heat transfer characteristics of real and simulant melt pool experimentsrdquo Nuclear Engineering and Design vol 169 no 1 pp 151ndash164 1997 doi 101016S0029-5493(96)01283-6

[58] O Kymaumllaumlinen H Tuomisto O Hongisto and T G Theofanous ldquoHeat flux distribution from a volumetrically heated pool with high Rayleigh numberrdquo

59

Nuclear Engineering and Design vol 149 no 1ndash3 pp 401ndash408 1994 doi 1010160029-5493(94)90305-0

[59] F J Asfia and V K Dhir ldquoNatural convection heat transfer in volumetrically heated spherical poolsrdquo 1994 Accessed Aug 23 2019 [Online] Available httpinisiaeaorgSearchsearchaspxorig_q=RN26037727

[60] M Jahn and H H Reineke ldquoFree Convection Heat Transfer with Internal Heat Sources Calculations and Measurementsrdquo in Proceedings of the 5th Int Heat Transfer Conference Tokyo Japan 1974 vol Vol3 paper NC-28

[61] M Gustafsson ldquoFSI Transient extension V1704rdquo EDR MEDESO 2016 [62] EDR MEDESO httpsedrmedesocom [63] H Kim and I Namgung ldquoThermo-mechanical Analysis of RVLH for

APR1400 during a Severe Accidentrdquo in Transactions SMiRT-23 Manchester United Kingdom Aug10-14 2015 [Online] Available httpwwwlibncsueduresolver18402034197

[64] P Yu and W Ma ldquoComparative Analysis of Reactor Pressure Vessel Failure using Two Thermo-Fluid-Structure Coupling Approachesrdquo submitted to journal

[65] F R Menter ldquoTwo-equation eddy-viscosity turbulence models for engineering applicationsrdquo AIAA Journal vol 32 no 8 pp 1598ndash1605 1994 doi 102514312149

[66] T N Dinh and R R Nourgaliev ldquoTurbulence modelling for large volumetrically heated liquid poolsrdquo Nuclear Engineering and Design vol 169 no 1 pp 131ndash150 1997 doi 101016S0029-5493(96)01281-2

[67] ANSYS Inc ldquoIntroduction to ANSYS Fluent Lecture 7 Turbulence Modelling Fluent Tutorial Materialrdquo 2016

[68] A Horvat and M Borut ldquoNumerical investigation of natural convection heat transfer in volumetrically heated spherical segmentsrdquo in Proceedings of the International Conference Nuclear Energy for New Europe 2004

[69] Y Li CFD Pre-test Analysis of SIMECO-2 Experiment 2016 [70] A Theerthan A Karbojian and B R Sehgal ldquoEC-FOREVER experiments

on thermal and mechanical behavior of a reactor pressure vessel during a severe accident Technical report-1 EC-FOREVER-2 testrdquo SAM-ARVI-D008 Mar 2001

[71] P Yu W Ma W Villanueva A Karbojian and S Bechta ldquoValidation of a thermo-fluid-structure coupling approach for RPV creep failure analysis against FOREVER-EC2 experimentrdquo Annals of Nuclear Energy vol 133 pp 637ndash648 2019 doi 101016janucene201906067

[72] T G Theofanous C Liu S Additon S Angelini O Kymaumllaumlinen and T Salmassi ldquoIn-vessel coolability and retention of a core meltrdquo Nuclear Engineering and Design vol 169 no 1ndash3 pp 1ndash48 1997 doi 101016S0029-5493(97)00009-5

[73] H P Langtangen ldquoSolving nonlinear ODE and PDE problemsrdquo Simula Research Laboratory 2016 [Online] Available httpshplgitgithubionum-methods-for-PDEsdocpubnonlinpdfnonlin-4printpdf

[74] P Yu and W Ma ldquoDevelopment of a Lumped-Parameter Code for Efficient Assessment of In-Vessel Melt Retention Strategy of LWRsrdquo submitted to journal

60

[75] T G Theofanous M Maguire S Angelini and T Salmassi ldquoThe first results from the ACOPO experimentrdquo Nuclear Engineering and Design vol 169 no 1 pp 49ndash57 1997 doi 101016S0029-5493(97)00023-X

[76] S W Churchill and H H S Chu ldquoCorrelating equations for laminar and turbulent free convection from a vertical platerdquo International Journal of Heat and Mass Transfer vol 18 no 11 pp 1323ndash1329 1975 doi 1010160017-9310(75)90243-4

[77] S Globe and D Dropkin ldquoNatural-Convection Heat Transfer in Liquids Confined by Two Horizontal Plates and Heated from Belowrdquo J Heat Transfer vol 81 no 1 pp 24ndash28 1959 doi 10111514008124

[78] T G Theofanous C Liu S Additon S Angelini O Kymaelaeinen and T Salmassi ldquoIn-vessel coolability and retention of a core melt Volume 2rdquo Argonne National Lab DOEID--10460-VOL2 1996 Accessed Apr 18 2019 [Online] Available httpinisiaeaorgSearchsearchaspxorig_q=RN30000459

[79] J L Rempe D L Knudson C M Allison G L Thinnes and C L Atwood ldquoPotential for AP600 in-vessel retention through ex-vessel floodingrdquo United States INEELEXT--97-00779 1997 Accessed Apr 18 2019 [Online] Available httpinisiaeaorgSearchsearchaspxorig_q=RN30025872

[80] A Miassoedov X Gaus-Liu T Cron and B Fluhrer ldquoLive experiments on melt pool heat transfer in the reactor pressure vessel lower headrdquo The 10th International Conference on Heat Transfer Fluid Mechanics and Thermodynamics (HEFAT 2014) Orlando FL Jul 14-16 2014

[81] H Madokoro A Miassoedov and T Schulenberg ldquoAssessment of a lower head molten pool analysis module using LIVE experimentrdquo Nuclear Engineering and Design vol 330 pp 51ndash58 2018 doi 101016jnucengdes201801036

[82] B Fluhrer et al ldquoThe LIVE-L1 and LIVE-L3 experiments on melt behaviour in RPV lower headrdquo Forschungszentrum Karlsruhe GmbH Technik und Umwelt (Germany) Inst fuer Kern- und Energietechnik FZKA--7419 2008 Accessed Nov 12 2019 [Online] Available httpinisiaeaorgSearchsearchaspxorig_q=RN40036867