investigation and validation of void and pressure drop...

MASTER OF SCIENCE THESIS

Investigation and validation of void and

pressure drop correlations in BWR fuel

assemblies

MANUEL AULIANO

Supervisors

Prof. Henryk Anglart

Dr. Jean-Marie Le Corre

Prof. Bruno Panella

Division of Nuclear Reactor Technology

Royal Institute of Technology

Stockholm, Sweden, June 2014

TRITA-FYS 2014:47 ISSN 0280-316X ISRN KTH/FYS/--14:47—SE

ii

To Damiano, Antonella and Michele

iii

Preface

This thesis was submitted to Kungliga Tekniska Högskolan in fulfilment of the

requirements for obtaining the double master degree from Kungliga Tekniska högskolan

and Politecnico di Torino, under the supervision of Prof. Henryk Anglart, Dr. Jean-

Marie Le Corre and Prof. Bruno Panella.

The present work was performed within the BWR Methods and Technology department

(BTE) in Westinghouse Electric Sweden AB in Västerås in the period January - May

2014.

iv

Abstract

This thesis presents the review and the assessment of void and pressure drop

correlations against experimental data (internal and external to Westinghouse) that

cover a wide range of operating conditions typical of those in a nuclear reactor.

It confirms the drift-flux models as the most recommended choice for predicting void

and it provides the opportunity to extend the applicability of void correlations to the

high void region. The recommended void correlations are finally selected. A survey of

correlations regarding one-phase and two-phase pressure drops has been conducted: an

optimized correlation for the friction factor has been proposed and the grid pressure loss

coefficients have thus been adjusted; a review of two-phase frictional pressure drop

correlations has been performed and an optimized correlation for the friction two-phase

multiplier has been proposed. Finally the two-phase pressure drops over the entire

assembly have been evaluated comparing the grid two-phase multipliers derived from

the homogeneous and the separated flow model.

v

Acknowledgements

Professionally, I would like to thank my supervisors Prof. Henryk Anglart , Prof. Bruno

Panella, and especially Dr. Jean-Marie Le Corre: all the work could not be done without

his help and his patience. In addition, I should also thank the whole personal of the BTE

department, particularly Mr. Juan Casal for having accepted me within Westinghouse

Electric Sweden AB and for his warm welcome: it was a pleasure for me to work with

all of you.

Personally, I would like to thank my Italian best friend Amir Al Ghatta for the last two

years in Stockholm: it has been an amazing and unforgettable experience. Then I thank

my colleagues in Västerås, Didier Bourgin, Raphal Barawnoski, Håkan Carlsson, Ante

Hultgren and Karolina Olofsson for the enjoyable Swedish lunch and fika we took

during this period in Westinghouse. I would like to express my gratitude to my uncle

Daniele Auliano for his valuable advices during my academic path.

Of course it would not be possible to make this without the complete moral and

economic support of my family under all my studies: ―Grazie di tutto, vi amo: senza di

voi non avrei conseguito questi risultati che dedico a voi‖.

Manuel Auliano

Västerås, 6th

June 2014

vi

Contents

1. OVERVIEW .................................................................................................................................... 1

1.1 INTRODUCTION ....................................................................................................................................... 1

1.2 OBJECTIVES AND METHODOLOGY ............................................................................................................... 1

1.3 OUTLINE ............................................................................................................................................... 2

2. THEORY AND MODELS .................................................................................................................. 4

2.1 TREATMENT OF TWO-PHASE FLOW ............................................................................................................. 4

2.1.1 The homogeneous equilibrium model ........................................................................................ 4

2.1.2 Separate flow model .................................................................................................................. 5

2.2 VOID FRACTION MODELS .......................................................................................................................... 6

2.3 SUBCOOLED BOILING MODEL ..................................................................................................................... 7

2.3.1 Levy's model ............................................................................................................................... 8

2.3.2 EPRI model ............................................................................................................................... 12

2.4. PRESSURE DROPS ................................................................................................................................. 13

2.4.1 Gravity pressure drop ............................................................................................................... 14

2.4.2 Friction pressure drop .............................................................................................................. 14

2.4.3 Local pressure drop .................................................................................................................. 14

2.4.4 Acceleration pressure drop ...................................................................................................... 15

3. MATLAB STEADY-STATE TH CODE ............................................................................................... 18

4. EXPERIMENTAL MEASUREMENTS ............................................................................................... 21

4.1 INTERNAL TEST FACILITY .......................................................................................................................... 21

4.2 VOID MEASUREMENTS ........................................................................................................................... 22

4.3 PRESSURE DROP MEASUREMENTS ............................................................................................................. 23

5. RESULTS AND DISCUSSION ......................................................................................................... 24

5.1 VOID .................................................................................................................................................. 24

5.1.1 Experimental void data ............................................................................................................ 24

5.1.2 Whole range ............................................................................................................................. 28

5.1.3 Subcooled boiling region .......................................................................................................... 34

5.1.4 High void region ....................................................................................................................... 36

5.1.5 Recommended void correlations .............................................................................................. 41

5.2 PRESSURE DROP .................................................................................................................................... 48

5.2.1 Experimental pressure data ..................................................................................................... 48

5.2.2 Friction factor ........................................................................................................................... 50

vii

5.2.3 Grid pressure loss coefficients .................................................................................................. 54

5.2.4 Friction two-phase multipliers ................................................................................................. 56

5.2.5 Grid two-phase multipliers ....................................................................................................... 60

6. CONCLUSIONS AND FURTHER WORK .......................................................................................... 62

BIBLIOGRAPHY ..................................................................................................................................... 64

APPENDIXES......................................................................................................................................... 66

A. DATABASE INFORMATION ................................................................................................................... 66

FRIGG loop ........................................................................................................................................ 66

Sub-bundle section ....................................................................................................................................... 66

Location of pressure taps along the channel................................................................................................ 67

bfbt .................................................................................................................................................... 69

psbt ................................................................................................................................................... 70

rdipe .................................................................................................................................................. 71

B. VOID CORRELATIONS .......................................................................................................................... 72

aa69 .................................................................................................................................................. 72

aa78 .................................................................................................................................................. 72

Bestion .............................................................................................................................................. 72

Chexal ................................................................................................................................................ 72

EPRI ................................................................................................................................................... 73

Inoue ................................................................................................................................................. 73

Maier and Coddington ...................................................................................................................... 73

scp ..................................................................................................................................................... 74

Smith ................................................................................................................................................. 74

Toshiba .............................................................................................................................................. 74

vann96 .............................................................................................................................................. 74

vann97 .............................................................................................................................................. 74

Zuber-Findlay .................................................................................................................................... 74

C. ONE-PHASE FRICTION FACTOR CORRELATIONS ......................................................................................... 75

Blasius ............................................................................................................................................... 75

Churchill ............................................................................................................................................ 75

Coolebrook ........................................................................................................................................ 75

Fang .................................................................................................................................................. 76

Filonenko ........................................................................................................................................... 76

Haaland ............................................................................................................................................. 76

Moody ............................................................................................................................................... 76

viii

Nikuradse .......................................................................................................................................... 76

Westinghouse ................................................................................................................................... 76

D. TWO-PHASE FRICTION MULTIPLIERS CORRELATIONS .................................................................................. 77

aa69 .................................................................................................................................................. 77

aa74 .................................................................................................................................................. 77

Cavallini ............................................................................................................................................. 77

Chisholm............................................................................................................................................ 77

Friedel ............................................................................................................................................... 78

Gronnerud ......................................................................................................................................... 78

Muller-Steinhagen and Heck ............................................................................................................. 78

scp ..................................................................................................................................................... 79

Souza and Pimenta ........................................................................................................................... 79

Tran et al. .......................................................................................................................................... 79

Wilson ............................................................................................................................................... 79

Zhang and Webb ............................................................................................................................... 79

E. ADDITIVE PLOTS AND TABLES ............................................................................................................... 80

Void fraction...................................................................................................................................... 80

Friction factor .................................................................................................................................... 85

Grid pressure loss coefficients ........................................................................................................... 87

Friction two-phase multiplier ............................................................................................................ 88

Grid two-phase multiplier ................................................................................................................. 90

ix

Nomenclature

Latin notations

Parameter Description Definition Selected

unit

A Area m2

Ablock Blocked area m2

Afuel Fuel pin area m2

a Coefficient in eq (5.3) and (5.4) -

b Coefficient in eq (5.3) and (5.4) -

C Coefficient introduced in eq. (2.17) -

CB Buoyancy force coefficient in eq. (2.15) -

Cc Coefficient in eq (2.52) -

CD Drag force coefficient in eq. (2.15) -

CDB Coefficient in eq (2.26) Eq. (2.27) -

CHN Coefficient in eq. (2.34) Eq (2.35) -

CS Surface force coefficient in eq. (2.15) -

C0 Drift-flux distribution parameter -

Cv Grid drag coefficient in eq. (2.41) -

C' Coefficient introduced in eq. (2.17)

cp Heat capacity at constant pressure J / kg / K

DH Equivalent heated diameter

m

DW Equivalent wetted diameter

m

Dr Rod diameter m

x

err error -

f Darcy-Weisbach friction factor -

flo Darcy-Weisbach friction factor based on total

flow assumed liquid

-

fgo Darcy-Weisbach friction factor based on total

flow assumed vapor

-

G Mass flux kg / m2 / s

g Gravitational acceleration constant 9.81 m / s2

h Specific enthalpy J / kg

hDB Dittus-Bolter heat transfer coefficient Eq. (2.26) W / m2 / K

hHN Hancol-Nicox heat transfer coefficient Eq. (2.34) W / m2 / K

hevap Evaporation heat transfer coefficient Eq. (2.33) W / m2 / K

hlg Vaporization latent heat J / kg

hThom Thom heat transfer coefficient Eq. (2.29) W / m2 / K

j Superficial velocity m / s

k Thermal conductivity W / m / K

LC Characteristic length m

P Perimeter m

Pr Prandtl number cp μ / k -

p Pressure Pa

pc Critical pressure Pa

q'' Heat flux W / m2

Re Reynolds number G DW / μ -

rB Bubble radius m

S Slip ratio ug / ul -

SB Spacing between bubbles m

xi

s Ratio between the upstream flow area and

downstream flow area

Aup / Adown -

T Temperature °C

TB Liquid temperature at bubble tip °C

TB+

Dimensionless liquid temperature at bubble

tip

-

Tl (z) Bulk liquid temperature at the axial position °C

ΔTsub Subcooling temperature °C

u Phase velocity m / s

ugj Drift-flux velocity m / s

u* Friction velocity Eq. (2.18) m /s

W Mass flow kg / s

xa Actual flow quality Gg / G -

xe Thermodynamic equilibrium quality ( h – hls) /

hfg

-

YB Distance between the wall and the tip bubble m

YB+

Dimensionless distance between the wall and

the tip bubble

-

z Axial elevation m

zd Bubble detachment elevation m

xii

Greek notations

Parameter Description Definition Selected

unit

α Void fraction Ag / A -

αB Void fraction on the point of the detachment Eq. (2.11) -

Γcond Condensation mass rate Eq. (2.31) kg / s

Γevap Evaporation mass rate Eq. (2.30) kg / s

ε Local pressure loss coefficient -

ε block Fraction of the unblocked flow area available

for flow

Ablock/A -

εsurf Surface roughness -

ρ Density kg / m3

ρm Static mixture density ρls (1-α) +

ρgs α

kg / m3

σ Surface tension N / m

τ Shear stress N / m2

μ Dynamic viscosity Pa s

Φ2

g Two-phase multiplier based on pressure

gradient for gas alone flow

-

Φ2

go Two-phase multiplier based on pressure

gradient for total flow assumed gas

-

Φ2

l Two-phase multiplier based on pressure

gradient for liquid alone flow

-

Φ2

lo Two-phase multiplier based on pressure

gradient for total flow assumed liquid

-

xiii

Subscripts

Parameter Description

A Acceleration

A0 Acceleration

(single-phase only)

ArCh Area change

Cont Contraction

down Downstream

Exp Expansion

e Thermodynamic

equilibrium

F Friction

F0 Friction (single-

phase only)

G Gravity

g Local gas state

gs Saturated steam

H Heated

homo Homogeneous

model

Irr Irreversible

K Local obstruction

K0 Local loss (single

phase only)

l Local liquid state

ls Saturated liquid

meas measured

xiv

PhCh Phase change

pred predicted

Rev Reversible

sat Saturation

condition

sp Single-phase

tp Two-phase

up Upstream

W Wetted

w wall

xv

Abbreviations and acronyms

Parameter Description

BHWR Boiling Heavy Water Reactor

BWR Boiling Water Reactor

DPRESS Pressure drop

EPRI European Power Research Institute

ext external

GTP Grid two-phase

homo Homogeneous flow model

LWHCR Light Water High Conversion Reactor

M-C Maier - Coddington

M-N Martinelli-Nelson

MS-N Muller-Stenhagen and Heck

meas measured

PSI Paul Sherrer Institute

PWR Pressurized Water Reactor

pred predicted

RBMK Reaktor Bolshoy Moshchnosti Kanalnyy

("High Power Channel-type Reactor")

RDIPE Research and Development Institute of Power Engineering

SCB Subcooled Boiling

SQP Sequential Quadratic Programming

separ Separated flow model

spdp Single-phase pressure drop

TH Thermal-hydraulic

tpdp Two-phase pressure drop

VF Void fraction

Z-F Zuber - Findlay

xvi

List of figures

Figure 2.1: Void in subcooled boiling region [5] ............................................................. 8

Figure 2.2: Forces acting on a bubble [5] ....................................................................... 10

Figure 3.1: Matlab TH code structure ............................................................................ 18

Figure 3.2: Input ............................................................................................................. 19

Figure 3.3: TH calculations ............................................................................................ 19

Figure 4.1: FRIGG loop [6] ............................................................................................ 21

Figure 4.2: The void measuring table [6] ....................................................................... 22

Figure 4.3: Location of pressure taps [13] ...................................................................... 23

Figure 5.1: Void experimental data ............................................................................... 27

Figure 5.2: Statistical analysis - sf24va and sf24vb ....................................................... 29

Figure 5.3: Pred. Vs. Meas. - sf24a and sf24vb ............................................................. 29

Figure 5.4: Pred. vs. Meas. - sf24va and sf24vb ........................................................... 30

Figure 5.5: Pred. vs. Meas. - sf24va and sf24vb ............................................................ 30

Figure 5.6: Statistical analysis - ft36 of36 of64a of64b ................................................. 31

Figure 5.7: Statistical analysis - psbt .............................................................................. 32

Figure 5.8: Statistical analysis - rdipe ............................................................................ 32

Figure 5.9: Subcooled boiling models - sf24va and sf24vb ........................................... 34

Figure 5.10: Characteristic length in the subcooled boiling region ................................ 35

Figure 5.11: High void region ........................................................................................ 36



Figure 5.14: Different slip ratio ...................................................................................... 38

Figure 5.15: Different slip ratio ...................................................................................... 39

Figure 5.16: Bestion correlation - C0 .............................................................................. 40

Figure 5.17: Pred. vs Meas - FRIGG data ...................................................................... 42

Figure 5.18: Pred. vs Meas. - FRIGG data ..................................................................... 42

Figure 5.19: Pred. vs Meas. - psbt ................................................................................. 43

Figure 5.20: Pred. vs. Meas. - psbt ................................................................................. 43

Figure 5.21: Pred. vs. Meas. - Steady-state experiments (PSI) ...................................... 44

Figure 5.22: Pred. vs. Meas. - Steady-state experiments (PSI) ...................................... 44

Figure 5.23: Pred. vs. Meas. - Transient experiments (PSI) ........................................... 45

xvii

Figure 5.24: Pred. vs. Meas. - Transient experiments (PSI) ........................................... 45

Figure 5.25: aa69 void correlation - Error vs Mass flux ................................................ 46

Figure 5.26: Recommended void correlations ................................................................ 47

Figure 5.27: Recommended void correlations ................................................................ 47

Figure 5.28: Statistical analysis - bfbt_spdp ................................................................... 51

Figure 5.29: Statistical analysis - sf24h .......................................................................... 51

Figure 5.30: Pred. vs. Meas. - sf24h ............................................................................... 52


Figure 5.32: Grid pressure loss coefficients - sf24x ....................................................... 55

Figure 5.33: Grid pressure loss coefficients - sf24s ....................................................... 55

Figure 5.34: Statistical analysis - sf24ec ........................................................................ 57

Figure 5.35: Statistical analysis - sf24h .......................................................................... 58




Figure 5.39: Grid two-phase multipliers - Pred. vs. Meas. ............................................. 60

Figure A.1: Sub-bundle section [6] ................................................................................ 66

Figure A.2: Sub-bundle section [6] ................................................................................ 67

Figure A.3: Location of pressure taps at single-phase pressure drop measurements [6] 67

Figure A.4: Location of pressure taps at two-phase pressure drop measurements [6] ... 68

Figure A.5: Location of pressure taps at single-phase pressure drop measurements [6] 68

Figure A.6: Location of pressure taps at two-phase pressure drop measurements [6] ... 69

Figure A.7: Location of pressure taps - bfbt [13] ........................................................... 70

Figure A.8: Cross section of the experimental channel - rdipe1 [6]............................... 71

Figure A.9: Cross section of the experimental channel - rdipe2 rdipe3 [6] ................... 71

Figure E.1: Statistical analysis - bfbt .............................................................................. 80

Figure E.2: Statistical analysis - bwr8x8 neptun pwr5x5 tptf ........................................ 80

Figure E.3: Recommended void correlations. - bfbt ...................................................... 81

Figure E.4: Recommended void correlations - bfbt ....................................................... 81


xviii


Figure E.7: Recommended void correlations - rdipe ...................................................... 83

Figure E.8: Recommended void correlations - rdipe ...................................................... 83

Figure E.9: Statistical analysis - sf24i ............................................................................ 85

Figure E.10: Statistical analysis - sf24ec ........................................................................ 85

Figure E.11: Statistical analysis - sf24et ........................................................................ 86

Figure E.12: Grid pressure loss coefficients - sf24vc ..................................................... 87

Figure E.13: Grid pressure loss coefficients - bfbt ......................................................... 87

Figure E.14: Statistical analysis - sf24et ........................................................................ 88

Figure E.15: Statistical analysis - sf24i .......................................................................... 88

Figure E.16: Statistical analysis - bfbt_tpdp ................................................................... 89

xix

List of tables

Table 2.1: Void correlations ............................................................................................. 6

Table 5.1: Experimental void databases ( FRIGG loop ) ............................................... 25

Table 5.2: Experimental void databases ......................................................................... 26

Table 5.3: Experimental void databases [2] ................................................................... 26

Table 5.4: Experimental pressure drop database ............................................................ 49

Table A.1: Geometry and Power distribution - bfbt [13] ............................................... 69

Table A.2: Geometry and Power distribution - psbt [14] ............................................... 70

Table E.1: Recommended void correlations - Mean Error [-] ........................................ 84

Table E.2: Recommended void correlations - Standard deviation [-] ............................ 84

Table E.3: Homo vs Separ GTP multiplier - Mean error [bar]....................................... 90

Table E.4: Homo vs Separ GTP multiplier - Standard deviation [bar] .......................... 90

1

1. Overview

1.1 Introduction

Presence of the steam in the nuclear reactors affects significantly the value of the

coolant density, thus its moderation power decreases and in turn influences the local

neutron flux and thus the local power. Due to the feedback between the local power and

the local void fraction it is important to predict accurately its local value in order to

predict the correct response of nuclear reactors [1] by using models predicting the

energy transfer and the transport of the vapor phase along the system [2]. The void

prediction is a required input for computing many key flow parameters, it is important

in the modeling of the two-phase flow pattern transitions, heat transfer and pressure

drops and it plays a crucial role in many thermal-hydraulic simulations.

1.2 Objectives and methodology

The main objective of this work is to review and optimize Westinghouse methods to

compute void fraction and pressure drop in BWR fuel assemblies. In preparation of this

project a large number of void fraction and pressure drop databases have been collected.

In addition to the available internal FRIGG databases, other databases in rod bundle

from the open literature have been compiled. Primarily the void benchmark analysis has

been conducted in order to select the recommended void predictive correlations overall

the whole range: it is important to predict accurately the void in order to predict

accurately the pressure drop. Then the attention has been moved to the single-phase

pressure drop: an optimized friction factor is proposed. Once the best friction factor is

known, the grid pressure loss coefficients have been adjusted by minimizing the

statistical objective functions such as mean error and standard deviation. A comparison

between several two-phase friction multipliers has been conducted and an optimized

correlation has been proposed. Finally the total pressure drops overall the fuel assembly

have been computed including the grid two-phase pressure drop: the homogeneous and

separated multipliers have been compared.

2 1. Overview

1.3 Outline

The thesis is divided in three parts.

This chapter represents the introductory part (Part 0).

Part I presents the theory (predictive models and correlations) and the structure of the

Matlab code used in the work to perform the different thermal-hydraulics simulations.

Part II contains the description of experimental measurements and the numerical

investigations made as a part of this work.

Part III presents final remarks, conclusions and further works.

In the following, an overview of the topics discussed in each of the chapters is

presented.

Part I: Theory and numerical framework

Chapter 2: The main predictive correlations regarding void fraction and

pressure drop are introduced and a general description of the subcooled

boiling models is presented.

Chapter 3: The structure of the Matlab code used to perform steady-state

thermal-hydraulics calculations is schematically described: a user manual has

been written during the project.

Part II: Experimental and numerical investigations

Chapter 4: The test facility owned by Westinghouse Electric Sweden AB and

the measurement techniques for void fraction and pressure drops are

described.

Chapter 5: The benchmark analysis regarding void and pressure drop is

presented: the experimental data are compared with the results of the main

correlations available from Westinghouse and open literature. The

recommended void correlations are selected. The optimization is carried out

for the friction factor, the grid pressure loss coefficients and the two-phase

friction multiplier. Homogeneous and separated grid two-phase multipliers are

compared.

Part III: Conclusions and further work

Chapter 6: Final remarks and observations are presented, the main

conclusions are drawn and further research directions are suggested.

Part I

Theory and numerical framework

4 2. Theory and models

2. Theory and models

2.1 Treatment of two-phase flow

Currently the two-phase flows are widely modeled by using the homogeneous and

separated flow approaches. Furthermore empirical and semi-empirical approaches that

model the hydrodynamics features of the flow have been developed. In the present work

the simplest approach has been adopted for the TH steady-state calculations: the

conservation equations are based on the homogeneous equilibrium model (that is three-

equation model) and that the non-homogeneity and non-equilibrium is accounted for

using additional constitutive relations.. The experience has taught that this simple

approach supported by constitutive models approaches reasonably the separated flow

model.

2.1.1 The homogeneous equilibrium model

The homogeneous equilibrium model is classified as one-fluid model of two-phase

flows: the mixture is considered as a single phase with averaged properties of the liquid

and vapor phase. The term homogeneous allows considering the flow as a homogeneous

mixture with no relative motion between vapor and liquid (slip ratio equal to 1); the

term equilibrium refers to the thermodynamic equilibrium between the two phases.

Under the assumption of thermodynamic equilibrium the actual non equilibrium quality

is equal to the thermodynamic equilibrium quality. The single-phase basic equations

(mass, momentum and energy) are used for the mixture.[3]

The homogeneous approach computes the two-phase pressure drop by using the same

formula for the single-phase flow but with the averaged properties defined by

homogeneous models: the two-phase density is defined as

(

)

(2.1)

for the mixture dynamic viscosity different models can be used as

McAdams (

)

(2.2)

5

Cicchitti ( ) (2.3)

Dukler (

( )

) (2.4) . [4]

2.1.2 Separate flow model

The two phases vapor and liquid are considered separated into two streams each with a

mean velocity. They have a constant but not necessarily equal velocity (slip ratio not

equal to one) and are in thermodynamic equilibrium quality.[5]

The separated flow model introduces the two-phase multipliers to compute the two-

phase pressure drops.

The most common approach (used in the present work) is to first compute the single-

phase liquid pressure drop assuming that the two-phase mixture is entirely in the liquid

phase and then to multiply it by the two-phase pressure drop multiplier ϕlo2 as

.

/

.

/

(2.5)

A second approach less used is the ϕl, ϕg based method that computes the two-phase

pressure gradient as

.

/

.

/

(2.6)

where the single-phase pressure gradient is computed assuming the liquid phase to flow

alone.

For both the approaches ϕ2

lo and ϕ2

l can be replaced respectively by ϕ2

go and ϕ2

g by

considering the gas phase instead of the liquid phase. [4]


2.2 Void fraction models

The table 2.1 shows the used void prediction correlations that have can be classified in

three groups having in common the reference homogeneous model that often over-

predicts the void fraction. The first group is represented by the slip ratio models based

on empirical relationships to compute the slip between the two phases. The second

group is given by the drift.-flux models (the most used and recommended) that compute

the distribution parameter and the drift-flux velocity by using empirical relations. The

third group is represented by the so-called miscellaneous correlations that are empirical

relations not included into any of the other groups. A fourth group not used in the

benchmark analysis is given by the so-called Kαhomo models that correct empirically the

void fraction predicted by the homogeneous model by a factor K. [7]

Table 2.1: Void correlations

Homogeneous: 𝛼

( )

(2.7) [1]

Slip:

( )

(2.8) [1]

Smith [5]

EPRI, SCP [6]

Drift-Flux: 𝛼

(2.9) [1]

Zuber-Findlay (Z-F), Bestion, Chexal et al., Toshiba, Inoue,

Maier-Coddington (M-C) [2]

aa69, aa78 [6]

Miscellaneous: experimental

vann96, vann97 [6]

All the void correlations are defined in the appendix B.

The drift-flux model

The drift-flux formulation developed by Zuber and Findlay for the void fraction is given

by

7

𝛼

(2.10)

where j and jg are respectively the mixture and vapor superficial velocity, C0 is the drift-

flux distribution parameter that is a covariance coefficient for cross-section distributions

of void fraction and total superficial velocity and ugj is the drift-flux velocity defined as

cross-section averaged difference between gas velocity and total superficial velocity [1]:

this model is able to take into account both the vapor production and the effect of the

relative velocity between the two phases included respectively in jg and ugj. [2]

2.3 Subcooled boiling model

Although the bulk boiling is prevalent in the thermal-hydraulics performance of BWR

reactors, accurate models are required to predict the void in the subcooled region.

Considering a tube heated with axial uniform flux and introducing a subcooled liquid at

the inlet, the void fraction will vary with the axial position as the curve ABCDE in

figure 2.1. [5]

The subcooled boiling process can be divided into two regions, namely the wall voidage

(region AB) and the detached voidage (region BCD): in the former with high degrees of

subcooling (partial subcooled boiling) the vapor generated travels in a narrow bubble

layer attached to the wall whilst growing and collapsing until the void departure point zd

is achieved; in the latter with lower degree of subcooling (fully developed subcooled

boiling) bubbles detach from the heated surface and an appreciable void fraction begins

to appear into the subcooled core; the region DE represents the saturated nucleate

boiling.

The mechanistic and profile-fit approaches have been analyzed in order to predict the

forced convection subcooled void fraction: the former postulates a phenomenological

description of the boiling heat transfer process and so computes the subcooled flow

quality and void fraction, the latter postulates a convenient mathematical fit to the data

for the flow quality or the enthalpy profile between the void departure point zd and the


point at which thermodynamic equilibrium is reached ze. For steady-state calculations a

profile-fit method is recommended since it is accurate though it is easier to use than a

Figure 2.1: Void in subcooled boiling region [5]

mechanistic method. The main drawback is that it is based on a fit to uniform axial heat

flux data, so the predictions of subcooled void fraction in case of non-uniform axial heat

flux have to be confirmed. For transient calculations the mechanistic model is

recommended. Some of the more used references for each method are

1. Mechanistic models: Griffith et al., Bowring, Rouhani and Axelsson, Larsen and

Tong, Hancox and Nicoll, EPRI.

2. Profile-Fit Models: Zuber et. Al., Staub, Levy, Saha and Zuber. [8]

2.3.1 Levy's model

The highly subcooled region

The Levy´s model is based on the assumption that at point B (transition between the

first and second subcooled regions) the bubbles are spherical with radius r and the

9

distance between them is equal to SB. Around the heated perimeter there is a number of

bubbles equal to PH/SB and the volume of vapor in a section of channel SB in length is

(PH/SB)(4/3 π rB3), so the void fraction at the point B is given by

𝛼 .

/ .

/ .

/

.

/

(2.11)

Assuming the bubbles to be wrapped in a squared array and to interfere with each other,

if rB/SB ≈ 0.25, then the void fraction on the point of detachment is given by

𝛼

(2.12)

where YB is the distance from the wall to the tip of the vapour bubble (shown in figure

2.2) given by the equation

0

1

(2.13)

where τw is the wall shear stress computed as

(2.14) . [5]

Departure of vapor bubbles from the heated surface

In order to compute the subcooling at the incipient bubble departure, primarily the size

of the bubble has to be computed. Figure 2.2 depicts the forces acting on the bubble at

the moment of the departure: the surface tension and the inertia forces (negligible) hold

the bubble to the surface, while the buoyancy and frictional drag forces attempt to

remove it. From the force balance

( )

(2.15)


the bubble radius in the incipient departure rB is computed as

.

/

.

/

(2.16)

and then the distance to the tip of the bubble YB, assumed to be proportional to rB, is

computed as

0

1

0 . ( )

/1

(2.17)

where the constants C and C' were computed from the experimental data.

Figure 2.2: Forces acting on a bubble [5]

The dimensionless distance YB+ is expressed in terms of the parameter

√

(2.18)

as

11

0

1

, -

0 . ( )

/1

(2.19)

Necessary condition for the growing or equilibrium of the bubble is that the liquid

temperature TB at the distance YB exceeds the saturation temperature: the Levy´s model

simplifies the analysis and assumes that TB equals the saturation temperature.

By using the Martinelli's universal temperature profile the dimensionless temperature

TB+ at the position YB

+ is computed as:

, ( 2

3)-

(2.20)

, ( ) 2

3-

Under the assumption of the single-phase temperature profile in the liquid

( )

(2.21)

and expressing the dimensionless temperature as

( ) (2.22)

where Tw is the wall temperature, q'' is the wall heat flux and hDB is the heat transfer

coefficient computed by the Dittus-Bolter correlation, the subcooling at the bubble

departure point can be expressed as:

( ) ( ) 0

1 (2.23) [5]

If the fluid bulk temperature distribution is computed by applying the heat balance, the

nonlinear equation can be solved to compute the bubble detachment elevation:


( ) ( ) (2.24)

2.3.2 EPRI model

The mechanistic subcooled boiling model developed by the EPRI is based on the

following physics: the evaporation process leads to the formation of bubbles at the

surface of the cladding, then the condensation occurs due to the transport of the bubble

in the subcooled water. The transition to the subcooled boiling occurs when the

evaporation rate exceeds the condensation rate: heating comes only from fission,

contributions from direct gamma and neutron heating are neglected. The saturated

boiling occurs when the bulk temperature of the fluid reaches its saturation value. As

long as the cladding wall superheat is negative, the single-phase subcooled heat transfer

is described by the Newton's cooling law

( ) (2.25)

where the Dittus-Bolter's heat transfer coefficient is computed as

(2.26)

{

(2.27)

with εblock defined as fraction of the unblocked flow area available for flow.

When the cladding wall superheat is not negative anymore, the heat transfer is ruled by

( ) ( ) (2.28)

where the Thom heat transfer coefficient is given by

( ) (2.29)

The evaporation and condensation rates are computed respectively as

13

( )

(2.30)

( )

(2.31)

( )

( ) (2.32)

where the evaporation and the Hancox-Nicol coefficient are computed respectively as

( ) (2.33)

(2.34)

with

{

(2.35)

When , the subcooled boiling begins. Net bulk boiling begins when

: all the heat removed by the coolant is used to produce steam. [6] [9] [10]

2.4. Pressure drops

The total axial pressure loss for the two-phase mixture in a channel along the flow

direction can be split into four contributions (gravity, friction, local flow obstructions

and acceleration) as:

.

/

.

/

.

/

.

/

.

/

(2.36)


The ϕ2

lo based approach is used to compute the two-phase friction and grid pressure

drop.

2.4.1 Gravity pressure drop

Assuming the gravity as the only external volume force, the gravitational contribution is

expressed as:

.

/

(2.37)

2.4.2 Friction pressure drop

The frictional two-phase contribution is computed according to the separated flow

model as

.

/

.

/

(2.38)

where the pressure loss for the single-phase liquid is given by

.

/

(2.39)

All the friction factor and two-phase friction multiplier correlations are defined

respectively in the appendix A3 and A4.

2.4.3 Local pressure drop

The local pressure loss is due to a local geometric obstruction within the fluid flow

region around a grid or an orifice. In single-phase it is computed as

.

/

(2.40)

15

where ε is the pressure loss coefficient for the local perturbation.

For rod bundles with part-length rods, to convert the spacer loss coefficients to different

local area in each zone, the following "rule of thumb" is used:

.

/

(2.41)

where Cv is the grid drag coefficient and εblock is the relative blockage defined as the

ratio between the area blocked in the axial direction Ablock that is the same for all the

axial zones and the local flow area A is in each zone. [11]

The two-phase local pressure loss is calculated by using a two-phase spacer multiplier

.

/

.

/

(2.42)

The two-phase local multipliers used are derived from the homogeneous and separated

flow models and are respectively defined as:

[ (

)] (2.43) [1]

( )

( )

(2.44) [12]

2.4.4 Acceleration pressure drop

Considering the flow incompressible, the flow changes velocity in a channel due to

phase change and/or area change. The acceleration pressure contribution is due to the

flow acceleration that affects the amount of net momentum in and out of the considered

fluid volume.

The contribution due to the phase change (evaporation/condensation) is computed as:


.

/

(

) (2.45)

where ϕ2

A0,PhCh is the phase change acceleration two-phase multiplier defined as

( )

( )

(2.46)

Under the conditions of constant mass flow rate and flow cross-section area, it becomes

.

/

.

/

(

) (2.47) [1]

The acceleration contribution due to the area change (sharp expansion or sudden

contraction) is computed as:

( )

(2.48)

where ϕ2

A0,ArCh is the area change acceleration two-phase multiplier.

The pressure loss due to the area change has the reversible and the irreversible

contributions.

For the reversible contribution, the term ε,ArCh is computed as

(2.49)

where the parameter s is the ration between the upstream flow area and the downstream

flow area and the two-phase multiplier is computed as

[ (

)] (2.50)

17

For a sharp expansion the reversible contribution is negative since s is lower than 1,

therefore it represents a pressure gain.

The irreversible pressure drop due to sudden expansion is computed as

( ) {[( )

( )

]

( )[

( )

( ) ]

[

]} (2.51) [12]

For the irreversible pressure drop due to sudden contraction the local pressure loss

coefficient is computed as

.

/

(2.52)

where the parameter Cc is function of the parameter s, and the homogeneous two-phase

local multiplier is used. [12]

18 3. Matlab steady-state TH code

3. Matlab steady-state TH code

The steady-state thermal-hydraulic code developed in Matlab presents a simple structure

shown in figure 3.1.

Figure 3.1: Matlab TH code structure

The inputs shown in figure 3.2 are defined by the database, the channel and the

geometry. The database contains information about the operating conditions (outlet

pressure, mass flux, subcooling inlet temperature and total power) and the

measurements (void fraction or pressure drops). The channel provides information

about the geometry and the local axial power distribution. The model defines the axial

grid along the channel, the models and correlations used in the calculations (void

correlation, subcooled boiling model, friction factor, friction two-phase multiplier, grid

two-phase multiplier) and the pressure option for the TH fluid property calculations,

that is pressure can be kept constant along the channel and equal to the outlet pressure

or the pressure distribution can be computed iteratively.

19

Figure 3.2: Input

Once the input have been defined, for each experimental run the code reads the

boundary conditions, performs the steady-state thermal-hydraulics calculations (shown

in figure 3.3) and save the results (post-processing). As output it provides a comparison

between the predictions and the measurements needed for the benchmark analysis.

Figure 3.3: TH calculations

Part II

Experimental and numerical investigations

21

4. Experimental measurements

4.1 Internal test facility

The internal databases have been collected from experiments performed at the FRIGG

loop (only for internal databases) at the previously ABB Atom laboratories, now

Westinghouse, in Västerås (Sweden). The loop (shown schematically in figure 4.1) is

designed for a pressure of 10 MPa and a temperature of 311 °C and it covers all

requirements for BWR fuel heat transfer and pressure drop testing at two-phase

conditions including thermal-hydraulics stability: it consists of a main circulation loop

including the test section, a cooling circuit and a purification system. The test section

consists of a pressure vessel, a Zircaloy flow channel and a sub-bundle with heater rods

representing the fuel design with full and part-length fuel rods. Pressure sensors are

connected to the flow channel at different elevation taps and thermocouples are

accommodated in the heater rods. The steam drum is used to separate the steam

produced in the test section from the saturated water: the steam is transported to the

condenser and the saturated water back to the main circulating pump. The cooling

circuit is composed of the condenser, heat exchanger and a circulating pump.[6]

Figure 4.1: FRIGG loop [6]

22 4. Experimental measurements

4.2 Void measurements

Void measurements have been carried out at two-phase flow and different operating

conditions defined by mass flux, the system pressure, the inlet subcooling temperature,

the bundle power and the local power distribution. Databases from void measurement

have been used to validate the void correlations needed in the core design methods.

The technique used to detect the void distribution is the transmission tomography

equipment shown in the figure 4.2: the intensity of the radiation beam emitted from a

Cesium-137 source is attenuated as it passes through some material, particularly it

decreases exponentially according to the Beer´s law and the coefficient of attenuation of

a gamma ray through a bubbly flow may be determined by measuring the intensity

before and after its passage through the channel flow. The measured coefficient of

attenuation is directly proportional to the mean density of the mixture that is function of

the void fraction. [6]

Figure 4.2: The void measuring table [6]

23

4.3 Pressure drop measurements

In order to license a new fuel type pressure drop measurements are carried out at both

single- and two-phase flow and different operating conditions defined by mass flux, the

system pressure, the inlet subcooling temperature, the bundle power and differential

pressures [6]. The main goal is to obtain pressure loss coefficients representative of the

fuel assembly main components.

In the experiments the bundle pressure drop has been monitored at several locations as

depicted in figure 4.3.

Figure 4.3: Location of pressure taps [13]

24 5. Results and discussion

5. Results and discussion

5.1 Void

Due to its relevance in characterizing two-phase flows, several void predictive

correlations have been proposed and assessed by comparing the predictions against

experimental data. Despite their use limited to co-current flow, the drift-flux models are

considered the most recommended considering their simplicity and predictive accuracy.

A review of a wide range of void correlations based on the Zuber-Findlay drift-flux

model has been conducted by Paul Coddington and Rafael Macian, evaluating them

against experimental PWR and BWR steady-state and transient (boil-off experiments)

data obtained from facilities in France, Japan, Switzerland, the UK and the USA: the

large size of the experimental database allowed a detailed statistical analysis that

compared the different correlations and has pointed out that the iterative correlations do

not increase so significantly the accuracy of the prediction. The present work assesses

the predictive capability of the available void correlations (internal to Westinghouse and

from open literature) against a much larger experimental database (including FRIGG

data, RBMK data and channel geometry), extends their applicability to the high void

region and investigates on the subcooled boiling region comparing two different models

as Levy and EPRI.

5.1.1 Experimental void data

A wide range of experimental void fraction data internal (table 5.1) and external (table

5.2 and table 3.3) to Westinghouse at various pressure and mass flux has been collected

and provides the opportunity to assess the predictive capability and the overall

applicability of the void correlations (internal and external to Westinghouse). The data

covers pressure from 0.1 MPa to 16.9 MPa and mass fluxes from 2.8 kg/m2/s to 4138.9

kg/m2/s and provides information on void fractions in sub-channels and rod bundles

including BWR, PWR and RBMK normal operating conditions and small and large

break transient conditions for both PWRs and BWRs. The experimental data can be

split in 3 different groups according to the type of the experiment performed and are

labeled as steady-state, boil-up and boil-down experiments. The majority of the

25

experiments were performed under steady-state conditions with the inlet subcooling,

mass flux and power at constant values: this first group includes all the databases from

table 5.1, table 5.2 and some from table 5.3 (bwr8x8, neptun, pwr5x5, tptf )[15]. The

boil-up experiments, where the inlet flow has been varied to keep constant the collapsed

liquid level, include the pericles, thetis and lstf databases[15]. The last group of the boil-

down experiments where the liquid inventory in the test facility is gradually boiled

away includes the achilles and thetis data[15]. Figure 5.1 provides an indication of the

wide range of pressure and mass fluxes covered by the experimental data.

Table 5.1: Experimental void databases ( FRIGG loop )

sf24va

sf24vb

of36

ft36

of64a

of64b

Reference

[6]

[6]

[6]

[6]

[6]

[6]

Type

BWR

BWR

BWR

BHWR

BWR

BWR

Length [m]

3.74

2.37 –3.74

3.65

4.365

3.65

3.65

Rods (heated)

24(24)

24 (24)

36(36)

36(36)

64(64)

64(64)

D

r [mm]

9.62 9.62 - 10.32

12.27 13.8 11.78 – 12. 25

11.78 – 12. 25

D

w [mm]

10.22 9.88 13.5 26.9 14.07 14.07

Axial Power distribution

Uniform

Uniform

Uniform

Uniform

Top Peak

Top Peak

ΔT

sub [K]

5.4 – 21.8

8.3 – 33.8

8.2 – 62.3

3.0 – 16.5

9.1 – 39.0

7.7 – 38.7

p [MPa]

5.5 – 7.1

5.4 - 8.0

3.0 – 9.0

7.0

4.8 – 6.8

6.7 – 6.9

G [kg/ m2

/s]

374 - 1653

390 - 1730

548 - 2919

495 - 1967

494 - 2479

513 - 2006

q'' [kW/m2

]

129 - 752

105 - 972

187 - 958

220 - 664

222 - 570

383 - 554


Table 5.2: Experimental void databases

bfbt [11 - 21

- 31]

bfbt [1071 - 2081 - 3091]

bfbt

[4101]

psbt

rdipe [1]

rdipe [2,3]

Reference

[13]

[13]

[13]

[14]

[6]

[6]

Type

BWR

BWR

BWR

PWR

RBMK

RBMK

Length [m]

3.71

3.71 /1.75/

3.71

3.71

1.56

2.5

7.00

Rods (heated) 64 (62/60/55)

64 (62) 64 (60) Subchannel [1-0.75-0.5-

0.25]

1 7

Dr [mm] 12.3 12.3 12.3 4.8 14 13.5

Dw [mm] 13.0 13.0 12.4 5.1 – 7.8 8.8 7.7

Axial power distribution

Uniform

Cosine / Cosine /

Inlet peak

Uniform

Uniform

Uniform

Uniform

ΔTsub

[K] 4.24 – 25.79

4.46 – 28.01 4.69 – 26.51

6.8 – 102.2 10.07 – 277.37

8.26-192.96

p [MPa] 0.98 – 8.69 0.96 – 8.68 0.97 – 8.71

5.0 – 16.91 2.99 – 10.03

3.01-14.38

G [kg/m2

/s] 288 - 1987 284 – 1978 296 - 2046

500 - 4139 989 - 2038

491 - 2349

q'' [kW/m2

]

24 - 824 25 - 827 26 - 853 429 – 4301 486 - 1043

125 - 402

Table 5.3: Experimental void databases [2]

ACHILLES

THETIS

PERICLES

NEPTUN

PWR 5x5

BWR 8X8

LSTF

TPTF

Reference

[2]

[2]

[2]

[2]

-

[2]

[2]

[2]

Type

PWR

BWR

PWR

LWHCR

PWR

BWR

PWR

PWR

27

Length

[m]

3.7

3.6

3.7

1.7

3.66

3.7

3.7

3.7

Rods

(heated)

69

(69)

49

(49)

357

(357)

37

(37)

25(25)

64

(62)

1104

(1008)

32

(24)

D

r [mm]

9.5

12.2

9.5

10.7

9.5 12.3

9.5

9.5

D

w [mm]

13

13

11

4

15.6 13

13

10

Axial Power

distribu-tion

Chopped cosine

Chopped cosine

Chopped cosine

Chopped cosine

Uniform

Uniform / Chopped

cosine

Chopped cosine

Uniform

ΔT

sub [K]

18 / 24

25-157

20/60

0.5-3

20.36 – 90.48

9-12

0

5-35

p [MPa]

0.1/0.2

0.2 – 4.0

0.3/ 0.6

0.4

7.4 – 16.6

1.0 – 8.6

1.0-7.3-

15.0

3.0/6.9/

11.8

G

[kg/m2

/s]

0.08

2.5-3.1

21-48

42/91

2222 - 3056

284-1988

2.2-84

11-189

q''

[kW/m2

]

11

11/12

11-40

5/10

1465 - 2014

225-3377

5-45

9-170

Figure 5.1: Void experimental data


5.1.2 Whole range

The experimental data have been used to assess the predictive capability of the void

correlations used widely in thermal-hydraulic analysis codes. In order to determine the

quality of the predictions for each experimental run the absolute error has been

computed as the difference between the measured and predicted value

𝛼 𝛼 (5.1)

The comparison of the void prediction correlations is based on the mean absolute error

and the standard deviation. The simulations have first been run by using the "reference"

model (Levy subcooled boiling model with the equivalent wetted diameter and 50 axial

nodes along the channel grid).

The statistical analysis has been applied firstly to the databases sf24va and sf24vb: the

results for sf24va are slightly better than the ones for sf24vb. Figure 5.2 shows the void

mean error and the void standard deviation over the whole void range: regarding the

mean error the homogeneous model over-predicts the measurements as expected [1]; the

original Zuber-Findlay model does not give good results when it is applied overall the

void range; unlike the vann97, vann96 gives good results as expected since it is fitted to

these experimental data: there are some deviations for some data around void fraction

equal to 0.68. The iterative void correlations, such as EPRI and Chexal, increase the

complexity of the solution without giving a dramatic increase in the quality of the

prediction. Bestion (one of the simplest tested) and aa78 have a low mean error.

Regarding the void standard deviation, the unreliable behavior of the homogeneous

model and vann97 is confirmed; the aa69, Bestion, aa78, scp, Toshiba, Maier-

Coddington and Inoue give good results and the iterative correlations do not provide a

significant improvement in the quality of the prediction.

Figures 5.3, 5.4 and 5.5 show the values predicted by the void correlations versus the

measured values for the databases sf24va and sf24vb. It shows the under-prediction of

the void in the subcooled region and the deviation for some correlations towards higher

void fractions.

29

Figure 5.2: Statistical analysis - sf24va and sf24vb

Figure 5.3: Pred. Vs. Meas. - sf24a and sf24vb


Figure 5.4: Pred. vs. Meas. - sf24va and sf24vb

Figure 5.5: Pred. vs. Meas. - sf24va and sf24vb

31

Other FRIGG (see section 4.1) data regarding BWR rod bundles have been analyzed

statistically in figure 5.6 and confirm the previous analysis.

Then the statistical analysis has been applied to the databases bfbt described in the

appendix A (see table A.1): figure E.1 confirms the good statistic behavior from

Bestion, aa78, aa69, scp and the iterative correlations.

Figure 5.7 and figure 5.8 show the statistical analysis performed respectively for the

databases psbt and rdipe (see table A.2 and figure A.8, A.9): it has been confirmed the

unreliable behavior from correlations purely empirical like the vann96 that even

presents not physical results by over-predicting the measurements more than the

homogeneous model in both the cases. For PWR operating conditions the aa78

correlation has given not physical results by exceeding the predictions of the

homogeneous model.

Figure 5.6: Statistical analysis - ft36 of36 of64a of64b


Figure 5.7: Statistical analysis - psbt

Figure 5.8: Statistical analysis - rdipe

33

Figure E.2 (see appendix E) depicts the statistical analysis applied to the experimental

data from PSI (only steady-state experiments are considered as bwr8x8, neptun, pwr5x5,

tptf): the good behavior from Bestion, Maier-Coddington, Inoue, scp, EPRI and Chexal

is confirmed, vann96 and aa78 exceed the homogeneous predictions in the case of high

pressure operating conditions (pwr5x5).


5.1.3 Subcooled boiling region

After assessing the predictive capability of the void correlations over the whole range of

steady-state data, the attention has been focused on the subcooled boiling region in

order to investigate the under-prediction pointed out in the section 5.1.2.

An open issue regarding the subcooled boiling models is the correct characteristic

length to be used in the Nusselt number.

Figure 5.9 shows the deviation between predicted and measured values in the low void

region by using equivalent wetted and heated diameter for both the Levy and EPRI

subcooled boiling models. For the databases sf24va and sf24vb it seems that the

characteristic length to be used in the Levy model is the heated equivalent diameter, for

the EPRI model it is not clear which is the correct characteristic length to be used.

A parametric study regarding the characteristic length has been performed: the database

psbt (see table A.2) has been considered owing to the significant variation of the

geometry.

Figure 5.9: Subcooled boiling models - sf24va and sf24vb

35

Figure 5.10: Characteristic length in the subcooled boiling region

The TH simulations have been run by using the EPRI subcooled boiling model and the

aa69 void correlation and varying the characteristic length in the Nusselt number.

Figure 5.10 depicts on the left the variation of the standard deviation and the mean error

with the characteristic length: there is an optimum value for which the objective

functions have a minimum value. On the right the optimum value found for each

database has been plotted versus the heated equivalent diameter, both referred to the

wetted equivalent diameter. This parametric study is intended to prove that the heated

equivalent diameter works well in some geometry as shown for sf24va and sf24vb, but it

is not always the correct characteristic length to be used.


5.1.4 High void region

One of the most important goals is to extend the predictive applicability of the void

correlations to the high void region.

Due to the lack of experimental data simulated experiments have been performed: the

maximum total power in the void database sf24va and sf24vb has been increased in

order to reach the saturated conditions, the power of the remainder experimental run has

been proportionally increased and the void fraction has been "measured" at the outlet of

the rod bundle so that the virtual void databank covers the high void region.

Figure 5.11, 5.12 and 5.13 show the void predicted by the correlations versus the void

predicted by the homogeneous model taken as reference. It is interesting to look at how

they approach the saturated vapor conditions. Most of them reach the unity, Bestion

under-predicts slightly the void, vann96, Maier-Coddington, Toshiba and Inoue do not

reach the unity and are quite far from the ideal behavior. Scatter is evident for Bestion,

aa78, scp, Inoue, Toshiba, Maier-Coddington correlations.

Figure 5.11: High void region

37




Figure 5.14 and figure 5.15 show the void fraction versus the non-equilibrium quality:

the predicted curve and the curves with different slip ratios are plotted. By increasing

the slip ratio the gas phase velocity becomes dominant over the liquid phase: the gas

phase flows in the central part of the channel creating the gas core and the liquid phase

flows as a thin film on the wall forming an annular ring of the liquid: as the slip ratio

increases, the liquid film flows more slowly and it is more unlikely that liquid may be

entrained in the gas core as small droplets. It is interesting to look at where the predicted

void is located compared with the references: for most of the correlations the prediction

is located in the expected slip ratio range (2 ; 3) in agreement with [16], the aa69 is

located beyond S = 4 and the Bestion seems to have an irregular behavior for high void

fractions.

Figure 5.14: Different slip ratio

39

Figure 5.15: Different slip ratio

It is worth to investigate more on the Bestion correlation developed for use in the

thermal hydraulic code CATHARE. Due to the absence of a value for the distribution

parameter in the reference available to the authors it was set to 1. [2] A parametric study

is shown in figure 5.16: other values of C0 lead to a decrease of the overall prediction

quality.

The original drift flux correlation proposed and assessed by many CATHARE

calculations is

√

(5.2) [2]

In agreement with [17], the Wallis annular flow correlation is used for very high void

fractions: a new simple void correlation could be developed and extended to the whole

range, thus avoiding specifying the transition between different correlations (one for the

low void region, one for the high void region).


Figure 5.16: Bestion correlation - C0

41

5.1.5 Recommended void correlations

The statistical analysis performed overall the wide range of experimental data,

supported by the extension to the high void region and the assessment against some

boil-off experiments, allows selecting the recommended void correlations to be used in

the TH codes.

The empirical correlations like vann96 and vann97 are not considered due to their

unreliable behavior: particularly vann96 gives not reasonable predictions when it is

applied to the rdipe and psbt as shown in figure 5.7 and 5.8 since it exceeds the

homogeneous model. The aa78 correlation shows one of the best performances overall

the benchmark analysis: it works not well for PWR since it gives a not physical

behavior when it is applied to psbt as shown in figure 5.7.

On the one hand the void correlations aa69, Smith, Toshiba, scp, Maier-Coddington and

Inoue show a good behavior overall the statistical analysis, sometimes close to the

performance of the iterative correlations and even better (see figure 5.8); on the other

hand figure 5.3, 5.4, 5.5, 5.17 and 5.18 point out deviation towards higher void fraction

[0.8 ÷ 0.9].

According to the analysis performed in the section 5.1.4, Maier-Coddington, Inoue and

Toshiba cannot be used in the high void region.

The present section depicts the measured versus the predicted void for the void

correlations: the simulations have been run by using the Levy subcooled boiling model

with the equivalent heated diameter. Statistical results regarding the internal databases

are included in the table E.1 and E.2.

Figure 5.19 and 5.20 confirm the under-prediction in the subcooled region and the

investigation carried out in the section 5.1.3.

Figure 5.21 and 5.22 point out that the aa69, aa78, scp, Toshiba and Smith does not

give reasonable predictions for several data from neptun and tptf database. For the

pwr5x5 experiments performed at high pressures, high mass fluxes and significant

subcooling (see table 5.3), all the correlations over-predict significantly the void in the

subcooled region, that confirms the investigation regarding the characteristic length (see

section 5.1.3).


Figure 5.17: Pred. vs Meas - FRIGG data

Figure 5.18: Pred. vs Meas. - FRIGG data

43

Figure 5.19: Pred. vs Meas. - psbt

Figure 5.20: Pred. vs. Meas. - psbt


Figure 5.21: Pred. vs. Meas. - Steady-state experiments (PSI)

Figure 5.22: Pred. vs. Meas. - Steady-state experiments (PSI)

45

Figure 5.23: Pred. vs. Meas. - Transient experiments (PSI)

Figure 5.24: Pred. vs. Meas. - Transient experiments (PSI)


Transient data from boil-off experiments have finally been used to assess the predicitve

capability of the correlations: figure 5.23 and 5.24 point out that aa69, aa78, scp and

Smith over-predict excessively the measurements, this can be explained due the fact that

the TH calculations performed assume a steady-state configuration. It has been verified

that the bias increases as the inlet mass flux decreases as shown in figure 5.25 for the

aa69 void correlation (as well for scp and Smith).

Despite the transient conditions, the Bestion, epri, Chexal, Maier-Coddington, Inoue

and Toshiba give good results.

The correlations that have shown their robustness overall the benchmark analysis are

those of Bestion, EPRI and Chexal: figure 5.26 and figure 5.27 show the statistical

analysis for these correlations. It is quite clear that the iterative correlations do not

improve too much the quality of the prediction increasing furthermore the complexity of

the solution. Although not applicable with low mass fluxes and transient conditions,

aa69, aa78, scp and Smith are confirmed as reliable void correlations for BWRs fuel

assemblies; Maier-Coddington, Toshiba and Inoue have shown a good behavior overall

the benchmark analysis even in the case of the boil-off experiments, but are not

applicable to the high void region that plays a crucial role in BWRs operating

conditions.

Figure 5.25: aa69 void correlation - Error vs Mass flux

47

Figure 5.26: Recommended void correlations

Figure 5.27: Recommended void correlations


5.2 Pressure drop

Due to the higher energy efficiency compared with the single-phase flow, the two-phase

flow is applied to several fields, but the penalty to pay is represented by the higher

pressure drops. A review of two-phase frictional pressure drop correlations has been

conducted by the Institute of Air Conditioning and Refrigeration that evaluated them

against experimental data covering operating conditions of industrial refrigerants [4].

The scope of the present work is to assess the predictive capability of the correlations

(internal and external to Westinghouse) against experimental data covering BWR

operating conditions and to adjust the grid pressure loss coefficients once the best

friction factor has been found.

5.2.1 Experimental pressure data

A wide range of experimental pressure drop single- and two-phase data (internal and

external) at various pressure and mass flux has been collected and provides the

opportunity to assess the predictive capability and the overall applicability of the

pressure drop correlations (internal and external to Westinghouse). The data covers

pressure from 0.2 MPa to 8.6 MPa and mass fluxes from 291 kg/m2/s to 2560 kg/m

2/s

and provides information on pressure drops in BWR fuel bundles.

The single-phase databases sf24ec, sf24et, sf24h, sf24i and bfbt are those that allow

removing the grid pressure loss so that only friction can be considered.

The single-phase databases sf24s, sf24x, sf24vc and bfbt have been used for evaluating

the grid pressure loss coefficients.

Table 5.4 provides information about the operating conditions and geometry of the

experimental databases.

49

Table 5.4: Experimental pressure drop database

sf24ec

sf24et

sf24h

sf24i

sf24s

sf24vc

sf24x

bfbt

Reference

[6]

[13]

Type

BWR

Length [m]

3.77

3.77

3.77

3.77

3.74

3.75

3.76

3.71

Rods

(heated)

24

(24)

24

(24/23/24)

64 (60)

D

r [mm]

9.66

9.66

9.66

9.66

9.84

9.85

9.85

12.30

D

W [mm]

9.05

9.50

9.53

9.53

9.93

9.29 - 11.25

9.26 - 11-25

12.80

Axial Power distribution

Cosine

Top peak

Cosine

Cosine

Bottom / Cosine /

Top

Cosine

Bottom / Cosine /

Top

Bottom peak

ΔT

sub [K]

7.5 - 195

9.3 - 127

8.1-

109.6

8.0 -120.6

50.7 - 198.2

6.6 - 142.6

80.3 - 157.6

1.7 - 85.8

p [MPa]

2.2 - 8.4

2.5 - 5.0

2.0 - 6.9

2.3 - 7.1

2.1 - 6.0

3.9 - 4.0

3.9 - 6.9

0.2 - 8.6

G [kg/m2

/s]

530-2560

549-2516

533-2510

543-2511

924 - 2522

955 - 2483

915 - 2488

291 - 2061

q'' [kW/m2

]

53 - 739

57 - 750

49 - 748

53 - 754

-

97 - 790


5.2.2 Friction factor

The attention has been focused on the single-phase databases sf24h, sf24ec, sf24et, sf24i

and bfbt that allow removing the local grid contribution from the measured pressure

drops so that it is possible to compare the predicted frictional pressure drops against the

experimental ones.

A multi-objective non-linear constrained optimization has been performed in order to

minimize both the mean error and the standard deviation by using the bfbt single-phase

database: the location of the pressure channel is depicted in figure 4.3, particularly the

pressure taps T3-T1 and T4-T2 have been used to remove the grid pressure loss.

It is proposed to optimize a friction factor correlation under the explicit form

(5.3)

by using the optimization toolbox available in Matlab: the goal attainment method SQP

(implemented function fgoalattain) has been used and it has given results very close to

the ones provided by the least square method despite the different optimization

criterion.

The coefficients a and b provided by the optimization are respectively equal to 0.3029

and 0.2521.

Figure 5.28 depicts the statistical analysis performed for the databases bfbt by

computing the mean error and the standard deviation when different friction factor

correlations are used, included the proposed correlation called optimum. The

optimization has tried to minimize both the objective functions, achieving a trade-off:

the Moody correlation remains the best considering the mean error, but the optimum

correlation has the lowest standard deviation equal to 4.46e-4 bar.

Figure 5.29 depicts the statistical analysis performed for the databases sf24h, figure 5.30

and figure 5.31 show the measured against the predicted values. Additive plots for the

other considered databases are attached to the appendix E and highlight the best

behavior of the optimum correlation.

51

Figure 5.28: Statistical analysis - bfbt_spdp

Figure 5.29: Statistical analysis - sf24h


Figure 5.30: Pred. vs. Meas. - sf24h


53

The statical analysis highlights that it is not convenient to use the Colebrook and

Nikuradse correlations that involve iterative calculations increasing the complexity of

the solution since there are many explicit correlations that can give better results as the

Haaland formula.


5.2.3 Grid pressure loss coefficients

Once the best friction factor has been found, the grid pressure loss coefficients for the

databases bfbt, sf24s, sf24vc and sf24x have been adjusted by performing the

optimization as done for the friction factor: it is important to underline that only single-

phase data from measurements at 200 ˚C have been used to develop the single-phase

spacer loss correlations since they are the most representative compared to reactor

conditions.

It is proposed to optimize the grid loss coefficient correlation under the form

(5.4)

that includes the dependence on the Reynolds number.

For the rod bundles with part-length rods, the "rule of thumb" introduced in the section

2.4.3 has been applied. Considering for instance the database sf24x, the spacer loss

coefficients in the zone 2 and 3 (respectively with 23 and 21 rods compared with the

zone 1 with 24 rods) have been computed respectively as:

.

/

(5.5)

.

/

(5.6)

Figure 5.32 and figure 5.33 depict the results given by the optimization applied to the

databases sf24x and sf24s.

Additive plots concerning the other databases are attached to the appendix E.

55

Figure 5.32: Grid pressure loss coefficients - sf24x

Figure 5.33: Grid pressure loss coefficients - sf24s


5.2.4 Friction two-phase multipliers

The attention has been focused on the two-phase databases sf24h, sf24ec, sf24et, sf24i

and bfbt that allow removing the grid pressure drop from the measured pressure drop so

that it is possible to compare the predicted two-phase pressure drops, including friction

and acceleration due to the phase change (see section 2.4.4), against the experimental

ones.

The simulations have been run with the aa69 void correlation, the Levy subcooled

boiling model (the equivalent heated diameter has been used), the optimum friction

factor and the separated flow model approach based on the ϕ2

lo method has been used to

compute the two-phase friction pressure drop (see section 2.4.2).

A multi-objective non-linear constrained optimization has been performed in order to

minimize both the mean error and the standard deviation by using the sf24ec two-phase

database: the location of the pressure channel is depicted in figure A.4, particularly the

pressure taps T3-T4 and T5-T6 have been used to remove the grid pressure loss.

It is proposed to optimize a friction two-phase multiplier under the form of the aa69

two-phase multiplier, by using the optimization toolbox available in Matlab: the goal

attainment method SQP (implemented function fgoalattain) has been used.

Figure 5.34 depicts the statistical analysis performed for the databases sf24ec by

computing the mean error and the standard deviation when different friction two-phase

multiplier correlations are used, included the proposed correlation called aa69opt that

present the best performance. Figure 5.35 depicts the statistical analysis performed for

the databases sf24h: in this case the aa69opt has a mean error slightly larger than the

one from aa69, but a lower standard deviation.

Figure 5.36 5.37 and 5.38 show the measured against the predicted values. Additive

plots for the other considered databases are attached to the appendix E and confirm the

good behavior from this correlation, except for bfbt which the standard deviation is a bit

larger for.

Most of the two-phase friction multiplier correlations, except Cavallini and Chen that

present the biggest deviations by under- and over-predicting respectively the data,

provide acceptable results under-predicting the two-phase pressure drop BWR

experimental data under steady-state operating conditions.

57

The internal correlations aa69, aa74 and scp show a low mean error and standard

deviation for the entire experimental data. The correlations from open literature like

Martinelli-Nelson, Muller-Steinagen and Heck, Becker, Chisholm, Friedel, Gronnerud

reasonably predict the entire database (particularly the first three listed): the Muller-

Steinagen and Heck is one of the top correlations suggested by the Institute of Air

Conditioning and Refrigeration for predicting the two-phase frictional pressure drop of

refrigerants as R134a, CO2, R410A, R22 and ammonia [4].

It has been verified that unlike for void fraction, the homogeneous equilibrium model

for pressure drop can either under-predict or over-predict the experimental results,

mainly depending on the flow regime. There is no simple explanation of this (like for

void prediction) and for each flow regime this should be investigated separately. For

two-phase flows with known flow regime it is thus better to use an empirical correlation

instead of homogeneous equilibrium model to get better accuracy.

Figure 5.34: Statistical analysis - sf24ec


Figure 5.35: Statistical analysis - sf24h


59




5.2.5 Grid two-phase multipliers

Once the friction pressure drop has been optimized (friction factor and two-phase

multiplier respectively in section 5.2.2 and 5.2.4), the two-phase databases bfbt, sf24s,

sf24vc and sf24x, which the grid pressure loss coefficients have been adjusted for in

section 5.2.3 for, have been considered for comparing the grid two-phase multipliers

derived from the homogeneous and separated flow model (see section 2.4.3).

Figure 5.39 depicts the total predicted pressure drop against the measured and points out

that the homogeneous model predicts better the grid two-phase pressure drop compared

with the separated flow model, even if a slight under-prediction is evident at high

pressure drop for the databases sf24vc and sf24x. The statistical results comparing the

two models are included in the table E.3 and E.4.

Figure 5.39: Grid two-phase multipliers - Pred. vs. Meas.

Part III

Final Remarks & Conclusions

62 6. Conclusions and further work

6. Conclusions and further work

A survey of void fraction and pressure drop correlations in rod bundles and sub-channel

has carried out.

From the results given by the numerical evaluations of the various void correlations it

has been observed that

1. Surprisingly a simple void drift-flux correlation, namely Bestion, gives very

good results for most of the experimental data: it produces standard deviations

and mean error that are very close to the ones of the iterative correlations,

sometimes even better. The drawback of this correlation is the slight under-

prediction in the high-void region as in agreement with [2], therefore a new

correlation valid over the whole range could be developed without applying the

Wallis correlation for high void region.

2. Void correlations that require iterative calculations (EPRI and Chexal-

Lellouche.) increase the complexity of the solutions without increasing

dramatically the quality of the prediction if compared with simple correlations

such as Bestion.

3. Bestion, Chexal and EPRI are the void correlations that have shown their

robustness overall the void bankdata.

4. The unreliable behavior from purely empirical correlations like vann96 and

vann97 has been confirmed.

5. The good performance from aa69, aa78, scp and Smith has been confirmed for

BWRs fuel assemblies.

6. Maier-Coddington, Inoue and Toshiba have shown a good statistical

performance even when tested with the boil-off experimental data, but are not

applicable to the high void region.

7. The assessment of the drift-flux correlations against transient data confirms the

applicability to the transient analysis of the drift-flux models developed from

steady-state considerations.

8. The heated equivalent diameter seems to not be always the right characteristic

length to be used in the Nusselt number in the subcooled boiling model,

especially when the geometry has significantly been varied, therefore more

investigation is needed and a new mechanistic model could be developed.

63

Concerning the pressure drop the following conclusions have been drawn:

1. An optimized explicit friction factor correlation depending only on the Reynolds

number and not on the surface roughness has been proposed and it gives better

results for all the available databases that only the friction pressure drop can be

separately evaluated for.

2. With the proposed friction factor the grid pressure loss coefficients have been

adjusted thus getting a better accuracy for the grid pressure loss.

3. Most of the present two-phase friction multiplier correlations present stable

predictions and an optimized correlation is proposed.

4. New friction two-phase multiplier correlations could be developed for get a

better accuracy of the two-phase frictional pressure drop: since the homogeneous

equilibrium model can either under-predict or over-predict the experimental

results, with known flow regime it is better to develop appropriate empirical

correlations.

5. The grid two-phase multiplier derived from the homogeneous flow model is

recommended for predicting the grid two-phase pressure drop: some

improvements could be done at high pressure drops.

64 Bibliography

Bibliography

[1] Henryk Anglart. "Thermal-Hydraulics in Nuclear Systems". KTH, 2010

[2] Paul Coddington, Rafael Macian. "A study of the performance of void fraction

correlations used in the context of drift-flux two-phase flow models".

Laboratory for Reactor Physics and System Behavior, PSI, Switzerland.

[3] Henryk Anglart, ―Fundamentals of Multi-phase flows‖, Lectures in Thermal-

Hydraulics in Nuclear Engineering

[4] Yu Xu, Xiande Fang, Xianghui Su, Zhanru Zhou, Weiwei Chen, "Evaluation of

frictional pressure drop correlations for two-phase flow in pipes", Nuclear

Engineering and Design 253(2012) 86-97

[5] John G. Collier and John R. Thome. "Convective boiling and condensation",

Oxford Science Publications (1996)

[6] Westinghouse electric Sweden AB, proprietary information

[7] A.Cioncolini, J.R.Thome. "Void fraction prediction in annular two-phase flow".

International Journal of Multiphase Flow 43(2012) 72-84

[8] Lahey, R.T. and F.J. Moody. ―The Thermal Hydraulics of Boiling Water

Nuclear Reactor‖. ANS (1977)

[9] G.S. Lellouche and B.A. Zolotar. "Mechanistic Modeling for Predicting Two-

Phase Void Fraction for Water in Vertical Tubes, Channels & Rod bundles",

Electric Power research Institute (California 1982)

[10] B. Chexal, G. Lellouche, J. Horowitz and J. Healzer. "A void fraction

correlation for generalized applications". Progress in nuclear energy (1992)

[11] K. Rehme and G. Trippe, "Pressure drop and velocity distribution in rod

bundles with spacer grids", Nuclear Engineering Design 62(1980) 349-359

[12] G.F. Hewitt and N.S. Hall-Taylor. "Annular Two-Phase Flow". Chemical

Engineering Division, A.E.R.E., Harwell, England

[13] B. Neykov, F. Aydogan, L. Hochreiter, K. Ivanov, H. utsuno, F. Kasahara, E.

Sartori, M. Martin. "NUPEC BWR Full-size Fine-mesh Bundle test (BFBT)

Benchmmark". Nuclear Science NEA/NSC/DOC(2005)5

[14] A. Rubin, A. Schoedel, M. Avramova. "OECD/NRC Benchmark beased on

NUPEC PWR sunchannel and bundle tests (PSBT). Nuclear Science

NEA/NSC/DOC(2010)1

[15] RETRAN maintenance group, "Review of RETRAN-3D", NRC

65

[16] Joseph S.Miller, P.E. and Dr. John Bickel, "Two Phase heat Transfer and Fluid

flow", NRC

[17] D. Bestion. "The physical closure laws in the CATHARE code". Nuclear

Engineering and Design 124 (1990) 229 - 245

66 APPENDIX A

APPENDIXES

A. Database information

FRIGG loop

Sub-bundle section

sf24ec - sf24et - sf24h - sf24i -sf24va - sf24vb

Figure A.1: Sub-bundle section [6]

sf24sb - sf24sc - sf24st - sf24vc - sf24xb - sf24xc - sf24xt

Three fuel rods are part length rods: the rods 20 and 24 are two-thirds of the length of a

full-length rod and are located close to the central channel of the water cross; the rod 1

is one-third of the length of a full-length rod and is located in the outer corner of the

sub-bundle.

67

Figure A.2: Sub-bundle section [6]

Location of pressure taps along the channel

sf24ec - sf24et

Figure A.3: Location of pressure taps at single-phase pressure drop measurements [6]

68 APPENDIX A

Figure A.4: Location of pressure taps at two-phase pressure drop measurements [6]

sf24h - sf24i

Figure A.5: Location of pressure taps at single-phase pressure drop measurements [6]

69

Figure A.6: Location of pressure taps at two-phase pressure drop measurements [6]

bfbt

Table A.1: Geometry and Power distribution - bfbt [13]

Database

No.

11 21 31 1071 2081 3091 4101 spdp

tpdp

Fuel

Type

Axial

Power

Profile

Uniform

Uniform

Uniform

Cosine

Cosine

Inlet

peak

Uniform

Cosine

Heated

length

Full Full Full Full Half Full Full Full

Heated

rods

62 60 55 62 60 60

70 APPENDIX A

Figure A.7: Location of pressure taps - bfbt [13]

psbt

Table A.2: Geometry and Power distribution - psbt [14]

Database No. s1 s2 s3 s4

Assembly

(subjected

subchannel)

Subchannel type Center

(typical)

Center

(thimble)

Side Corner

No. heaters 4 x 1/4 3 x 1/4 2 x 1/4 1 x 1/4

Axial power shape Uniform

71

rdipe

Figure A.8: Cross section of the experimental channel - rdipe1 [6]

Figure A.8 depicts the geometry of the elementary scale model: the numbers 1, 2 and 3

are respectively the vessel, the displacer and the steel heating elements.

Figure A.9: Cross section of the experimental channel - rdipe2 rdipe3 [6]

72 APPENDIX B

B. Void correlations

aa69

The correlation is classified and available in [6].

aa78


Bestion

√

[2]

Chexal

( )𝛼

4

5

( 𝛼)

( )

( )

( ) (

)

73

(

(

)

( )

{

( (

))

{

. .

//

[2]

EPRI


Inoue

( )( ) [2]

Maier and Coddington

( ) (

)

74 APPENDIX B

[2]

scp


Smith

[(

) .

/

.

/

]

[5]

Toshiba

[2]

vann96


vann97


Zuber-Findlay

(

)

[2]

75

C. One-phase friction factor correlations

Blasius

The one-phase friction factor depends on Reynold´s number and two constants A and B

as:

[6]

Churchill

6.

/

( )

7

[6]

A and B are constant depending on Reynold's number.

Coolebrook

The one-phase friction factor depends on Reynold´s number, the surface roughness and

the friction factor itself as:

( [

√ ]) [4]

The friction factor has to be computed iteratively. As start guess the friction factor

according to Blasius is used.

76 APPENDIX C

Fang

0 .

/1

[4]

Filonenko

( ) [4]

Haaland

√ [ .

/

] [1]

Moody

The one-phase friction factor depends on Reynold´s number and the surface roughness

as:

6 √

7 [4]

Nikuradse

√ ( √ ) [4]

Westinghouse


77

D. Two-phase friction multipliers correlations

aa69


aa74


Cavallini

( )

4

5

(

)

(

)

[4]

Chisholm

( )* , ( )-

+

.

/

.

/

If B = {

If B = {

[4]

78 APPENDIX D

Friedel

( )

( )

4

5

(

)

(

)

[

]

[4]

Gronnerud

.

/

[

.

/

]

(

)

, (

)-

{

0 .

/1

[4]

Muller-Steinhagen and Heck

( )

, ( )- [4]

79

scp


Souza and Pimenta

(Γ )

( )

(

)

.

/

.

/

[4]

Tran et al.

( )*, ( )-

+

√

( ) [4]

Wilson

( )

.

/

.

/

(

)

[4]

Zhang and Webb

( )

.

/

( )

.

/

[4]

80 APPENDIX E

E. Additive plots and tables

Void fraction

Figure E.1: Statistical analysis - bfbt

Figure E.2: Statistical analysis - bwr8x8 neptun pwr5x5 tptf

81

Figure E.3: Recommended void correlations. - bfbt

Figure E.4: Recommended void correlations - bfbt

82 APPENDIX E



83

Figure E.7: Recommended void correlations - rdipe

Figure E.8: Recommended void correlations - rdipe

84 APPENDIX E

Table E.1: Recommended void correlations - Mean Error [-]

Database\Void aa69 Bestion aa78 Chexal EPRI scp Smith

sf24va 0.0025 -0.0248 -0.0141 -0.0042 -0.0053 0.0052 0.0309

sf24vb 0.0057 -0.0205 -0.0046 0.0021 -0.0012 0.0057 0.0326

of36 -0.0065 -0.0310 -0.0200 -0.0119 -0.0118 0.0033 0.0277

ft36 0.0300 0.0414 0.0630 0.0265 0.0266 0.0507 0.0625

of64a -0.0234 -0.0438 -0.0292 -0.0298 -0.0304 -0.0108 0.0089

of64b 0.0062 -0.0176 -0.0037 -0.0014 -0.0017 0.0089 0.0400

Table E.2: Recommended void correlations - Standard deviation [-]

Database\Void aa69 Bestion aa78 Chexal EPRI scp Smith

sf24va 0.0218 0.0207 0.0230 0.0227 0.0228 0.0419 0.0385

sf24vb 0.0265 0.0330 0.0324 0.0260 0.0267 0.0394 0.0367

of36 0.0541 0.0588 0.0593 0.0538 0.0537 0.0599 0.0549

ft36 0.0461 0.0445 0.0398 0.0452 0.0450 0.0430 0.0408

of64a 0.0369 0.0367 0.0387 0.0378 0.0382 0.0516 0.0476

of64b 0.0330 0.0311 0.0365 0.0319 0.0321 0.0476 0.0446

85

Friction factor

Figure E.9: Statistical analysis - sf24i

Figure E.10: Statistical analysis - sf24ec

86 APPENDIX E

Figure E.11: Statistical analysis - sf24et

87

Grid pressure loss coefficients

Figure E.12: Grid pressure loss coefficients - sf24vc

Figure E.13: Grid pressure loss coefficients - bfbt

88 APPENDIX E

Friction two-phase multiplier

Figure E.14: Statistical analysis - sf24et

Figure E.15: Statistical analysis - sf24i

89

Figure E.16: Statistical analysis - bfbt_tpdp

90 APPENDIX E

Grid two-phase multiplier

Table E.3: Homo vs Separ GTP multiplier - Mean error [bar]

Database \ GTP

multiplier

Homo Separ

bfbt 0.0053 0.0142

sf24sb 0.0096 0.0584

sf24sc 0.0205 0.0572

sf24st 0.0078 0.0363

sf24vc 0.0295 0.0426

sf24xb 0.0032 0.0289

sf24xc 0.0261 0.0457

sf24xt 0.0212 0.0363

Table E.4: Homo vs Separ GTP multiplier - Standard deviation [bar]

Database \ GTP

multiplier

Homo Separ

bfbt 0.0152 0.0233

sf24sb 0.0424 0.0817

sf24sc 0.0356 0.0722

sf24st 0.0309 0.0609

sf24vc 0.0190 0.0349

sf24xb 0.0255 0.0561

sf24xc 0.0249 0.0528

sf24xt 0.0327 0.0555

investigation and validation of void and pressure drop...

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