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TRANSCRIPT
Validation of a mathematical model for the simulationof loss of coolant accidents in nuclear power plants
Pedro Pupo Sa da Costa
Thesis to obtain the Master of Science Degree in
Mechanical Engineering
Supervisors: Prof. Luıs Rego da Cunha EcaProf. Carlo Benocci
Examination Committee
Chairperson: Prof. Viriato Sergio de Almeida SemiaoSupervisor: Prof. Luıs Rego da Cunha Eca
Member of the Committe: Prof. Pedro Jorge Martins Coelho
November 2016
Acknowledgments
I would like to thank my supervisors, Professor Luis Eca and Professor Carlo Benocci as well as
my advisor Professor Philippe Planquart. A huge thank you to Amanda, who had the patience to hear
me complain every step of the way.
Einmal muss jeder gehen und wenn dein Herz zerbricht,davon wird die Welt nicht untergehn Mensch arger dich nicht!
Andreas Frege
i
Abstract
The fallout of Loss of Coolant Accidents (LOCA) in nuclear power plants can be minimised by
predicting their evolution and consequences. Currently, lumped parameter codes, which under predict
the heating of nuclear fuel cells, are used for the simulation of LOCAs. A research project was
proposed to the Von Karman Institute to validate a mathematical model capable of simulating LOCAs.
This work considers the total LOCA scenario, which is a combined heat transfer problem where
radiative heat and a laminar natural convective flow developing inside a bundle of rods cool a solid
domain. A steady state solution was sought. A mathematical model describing this scenario was
defined and implemented with OpenFOAM.
The model was found to accurately predict the temperature profile of a single cylinder being cooled
in analogous conditions to a LOCA. The FVDOM radiation model was chosen as the most accurate.
A convergence criteria of 10−4 was deemed acceptable. The results were shown to be independent of
the grid refinement. Convergence of the results was not achieved for a seven cylinder bundle domain.
Despite an analysis of different parameters of the problem, it was not possible to determine the source
of the instability.
The mathematical model is considered as not validated. Further testing to resolve the residual
convergence instability, and an experimental measurement that better suits the needs of this project,
are proposed.
Keywords
LOCA, CHT, laminar natural convection, FVDOM, OpenFOAM
iii
Resumo
As consequencias de um acidente de perda de refrigeracao, em ingles LOCA, em centrais nu-
cleares podem ser minimizadas prevendo a sua evolucao. Actualmente, sao usados codigos de
parametros concentrados para a simulacao de LOCAs que tendem a sobrestimar o arrefecimento
de celulas de combustivel nuclear nestas condicoes. Neste contexto, foi proposto ao Von Karman
Institute que fosse validado um modelo matematico capaz de simular LOCAs.
O caso de uma perda completa de refrigeracao e considerado neste trabalho. Trata-se de um
problema CHT onde um escoamento por conveccao natural laminar se desenvolve no interior de
um conjunto de cilindros que, juntamente com o calor dissipado por radiacao, arrefecem um dominio
solido. Neste trabalho, foi admitido um escoamento em regime estacionario. Um modelo matematico
que descreve este cenario foi definido. Este modelo foi implementado no OpenFOAM.
O modelo matematico previu correctamente a temperatura ao longo de um cilindro arrefecido em
condicoes analogas as de um LOCA. O modelo Finite Volume Discrete Ordinates Method (FVDOM),
foi considerado o mais preciso. Um criterio de convergencia de 10−4 foi considerado aceitavel. Foi
demonstrado que os resultados sao independentes da malha usada.
Nao foi possıvel convergir os resultados para o caso de um conjunto de sete cilindros. Um estudo
para determinar a causa da instabilidade nas simulacoes foi inconclusivo.
Conclui-se que o modelo matematico nao se encontra validado. Sao propostos como trabalho
de continuacao a correcao possıveis fontes de erro numerico, juntamente com um novo trabalho
experimental mais adequado para o objetivo deste projeto.
Palavras Chave
LOCA, CHT, conveccao natural laminar, FVDOM, OpenFOAM
v
Table of Contents
Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . i
Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iii
Resumo . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . v
Table of Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vii
List of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . x
List of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xi
List of Symbols . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xiii
List of Acronyms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xv
1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.1 Loss of Coolant Accidents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
1.2 Validation of a Mathematical Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
1.3 State of the art . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
1.4 Objectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
1.5 Outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
2 Mathematical model for the loss of coolant accident scenario . . . . . . . . . . . . . . . 11
2.1 Partial differential equations for the fluid domain . . . . . . . . . . . . . . . . . . . . . . . 11
2.1.1 Continuity Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
2.1.2 Momentum balance equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
2.1.3 Energy balance equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
2.2 Partial differential equation for the solid domain . . . . . . . . . . . . . . . . . . . . . . . 16
2.3 Boundary Conditons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
2.4 Initialisation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
2.5 Radiation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
2.5.1 Radiative transfer equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
2.5.2 P1 model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
2.5.3 Discrete ordinates model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
2.5.4 View factor model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
2.6 Summary of the mathematical model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
3 Computational implementation of the mathematical model . . . . . . . . . . . . . . . . . 25
3.1 Selection of a CFD software . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
3.2 Overview of chtMultiRegionSimpleFoam . . . . . . . . . . . . . . . . . . . . . . . . . . 26
vii
3.3 Computational implementation of the radiation models . . . . . . . . . . . . . . . . . . . 30
3.3.1 Reference model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
3.3.2 P1 Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
3.3.3 Finite volume discrete ordinates method model . . . . . . . . . . . . . . . . . . . 31
3.3.4 View Factor model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
3.4 Boundary Conditons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
3.5 Convergence and solution control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
3.6 Numerical Schemes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
3.7 Linear solvers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
3.8 Equation and solution relaxation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
4 Numerical simulations setting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
4.1 Experimental data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
4.1.1 Single cylinder . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
4.1.2 Seven cylinder bundle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
4.2 Computational domain and boundary conditions . . . . . . . . . . . . . . . . . . . . . . 44
4.2.1 Single cylinder . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
4.2.2 Seven cylinder bundle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48
4.2.3 Domain for the code verification . . . . . . . . . . . . . . . . . . . . . . . . . . . 51
4.3 Selected test cases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51
5 Results and discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
5.1 Code verification . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
5.2 Radiation models analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
5.3 Results for the single cylinder case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
5.4 Grid refinement study . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61
5.5 Numerical limitations of the model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63
6 Conclusions and Future Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69
Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71
Appendix A Modified buoyantSimpleFoam . . . . . . . . . . . . . . . . . . . . . . . . . . . . A-1
Appendix B ExtendedBuoyancyBoundaries . . . . . . . . . . . . . . . . . . . . . . . . . . . B-1
Appendix C blockMesh.py . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . C-1
viii
List of Figures
1.1 Schematic of a Nuclear Power Plant. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
1.2 PWR 17x17 assembly. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
1.3 The processes of validation and verification. . . . . . . . . . . . . . . . . . . . . . . . . . 5
2.1 Pressure drops along BWR and PWR as function of the Reynolds number. . . . . . . . 12
2.2 Schematic of a LOCA domain. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
2.3 View Factor geometry schematic. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
3.1 chtMultiRegionSimpleFoam algorithm. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
3.2 Angular coordinate system. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
3.3 FVM generic descritization for a 1D domain. . . . . . . . . . . . . . . . . . . . . . . . . . 36
4.1 Experimental Installation for the single rod testing. . . . . . . . . . . . . . . . . . . . . . 42
4.2 Temperature profile along the rod surface. . . . . . . . . . . . . . . . . . . . . . . . . . . 43
4.3 Experimental installation of the seven cylinder bundle. . . . . . . . . . . . . . . . . . . . 43
4.4 Single cylinder computational domain. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
4.5 Overview of the top section of the Mesh for the single Cylinder case. . . . . . . . . . . . 47
4.6 Seven cylinder rod bundle computational domain. . . . . . . . . . . . . . . . . . . . . . . 48
4.7 Schematic meshing for an arbitrary cylinder surface section. . . . . . . . . . . . . . . . . 49
4.8 Final Mesh for the 7 cylinder scenario. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50
4.9 Schematic of the domain for the verification case. . . . . . . . . . . . . . . . . . . . . . . 51
5.1 Temperature profile along the rod surface, reference model. . . . . . . . . . . . . . . . . 54
5.2 Temperature profile along the rod surface, P1 model. . . . . . . . . . . . . . . . . . . . . 55
5.3 Temperature profile along the rod surface, FVDOM model. . . . . . . . . . . . . . . . . . 55
5.4 Temperature profile along the rod surface, view factor model. . . . . . . . . . . . . . . . 56
5.5 Temperature variation within the rod surface, view factor model. . . . . . . . . . . . . . . 56
5.6 Temperature profile along the rod surface, simulation with CHT. . . . . . . . . . . . . . . 57
5.7 Temperature and velocity profiles for different heights for qs = 264Wm−2. . . . . . . . . 58
5.8 Residual and control variables progression for qs = 264Wm−2. . . . . . . . . . . . . . . 59
5.9 Mass flow and energy balance in the domain for qs = 264Wm−2. . . . . . . . . . . . . . 60
5.10 Grid refinement study of the entrance section, qs = 264Wm−2. . . . . . . . . . . . . . . 61
5.11 Grid refinement study in the radial direction at z = 1.5m, qs = 264Wm−2. . . . . . . . . 62
5.12 Temperature profile in the solid domain at z = 1.0m. . . . . . . . . . . . . . . . . . . . . 63
ix
5.13 Residual convergence and control variables values in the seven cylinder domain. . . . . 64
5.14 Residual convergence for an outer wall radius R = 33mm. . . . . . . . . . . . . . . . . . 67
5.15 Variation of velocity profile at the outlet for different outer wall radius (R). . . . . . . . . . 68
x
List of Tables
2.1 Summary of the mathematical model for the fluid domain. . . . . . . . . . . . . . . . . . 22
2.2 Summary of the mathematical model for the fluid domain (cont.). . . . . . . . . . . . . . 23
2.3 Summary of the mathematical model for the solid domain. . . . . . . . . . . . . . . . . . 24
3.1 Thermodynamics packages choice for the fluid domain. . . . . . . . . . . . . . . . . . . 28
3.2 Thermodynamics packages choice for the solid domain. . . . . . . . . . . . . . . . . . . 28
3.3 Numerical boundary conditions for the fluid domain. . . . . . . . . . . . . . . . . . . . . 33
3.4 Numerical boundary conditions for the solid domain. . . . . . . . . . . . . . . . . . . . . 35
3.5 Numerical discretization schemes for the fluid domain. . . . . . . . . . . . . . . . . . . . 37
3.6 Numerical discretisation schemes for the solid domain. . . . . . . . . . . . . . . . . . . . 37
4.1 Mesh quality parameters. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
4.2 Mesh quality parameters. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50
5.1 Attempted solutions for the lack of convergence of the cylinder bundle case. . . . . . . . 65
5.2 Attempted solutions for the lack of convergence of the cylinder bundle case (cont.). . . . 66
5.3 Results for the varying outer wall radius test. . . . . . . . . . . . . . . . . . . . . . . . . 67
xi
List of Symbols
Roman symbols
A Area m2
a Absorptivity −an Discretised linear weight of cell n −ap Discretised linear weight of cell p −C1 Sutherland coefficient 1 kg/(msK1/2)C2 Sutherland coefficient 2 Kcp Specific heat capacity J kg−1 K−1
D Diameter mDh Hydraulic Diameter mFAi→Aj View factor −~fe External force Nm−1
~e Unit vector −G Incident Radiation Wm−2
~g Gravity acceleration vector ms−2
Gr Grashof number −H Total enthalpy J kg−1
H(Ψ) Discretised linear coefficients of neighbouring cells −h Specific enthalpy J kg−1
I Radiation Intensity Wsr−1
I Identity Matrix (only page 14) -k Thermal conductivity Wm−1 K−1
M Molecular mass kgmol−1
~m Mass flow kg s−1
n Refractive index −~n Surface normal vector −Nθ Number of solid angles in θ −Nφ Number of solid angles in φ −Nu Nusselt number −P Perimeter mp Pressure Pap0 Total pressure Papρgh Modified pressure PaPr Prandtl number −Q Power WQs Volumetric Heat source Wm−3
Qr Radiative heat source Wm−3
q Heat flux Wm−2
qr Radiative heat flux Wm−2
qs Surface heat flux Wm−2
R Ideal gas constant JK−1 mol−1
r Radius mRe Reynolds number −S Distance m
xiii
Roman symbols
~s Direction vector −T temperature KT0 Total temperature K~U Velocity vector ms−1
~x Position vector m
Greek symbols
α Thermal diffusivity m2 s−1
αe Explicit under relaxation factor −αi Implicit under relaxation factor −β Compressibility K−1
δ Boundary layer thickness mε Emissivity −θ Azimuthal angle −µ Dynamic viscosity Pa s−1
ν Kinematic Viscosity m2 s−1
ρ Density kgm−3
σ Stefan-Boltzmann constant Wm−2 K−4
σs Scattering coefficient −τ Viscous stress tensor kgm2 s−2
Φ Scattering phase function −Φl Linear-anisotropic phase function −φ Flux kgm−2 s−1
φ Polar angle (FVDOM model) −Ω Directional vector of radiative intensity −∇ Gradient −∇· Divergence −∇2· Laplacian −
xiv
List of Acronyms
API Application Programming Interface
BWR Boiling Water Reactor
CFD Computational Fluid Dynamics
CHT Conjugate Heat Transfer
CPU Central Processing Unit
DIC Diagonal Incomplete Cholesky
DILU Diagonal Incomplete LU
FANC (Belgian) Federal Agency for Nuclear Control
FVDOM Finite Volume Discrete Ordinates Method
FVM Finite Volume Method
GAMG Geometric-Algebraic Multi-Grid
GUI Guided User Interface
LOCA Loss Of Coolant Accident
LHS Left Hand side
NRC Nuclear Regulatory Commission
PBiCG Preconditioned bi-conjugate gradient
PCG Preconditioned conjugate gradient
PDE Partial Differential Equation
PWR Pressurised Water Reactor
RHS Right Hand Side
RTE Radiative Transfer Equation
SFP Spent Fuel Pool
xv
SIMPLE Semi-Implicit Method for Pressure Linked Equations
VKI Von Karman Institute
xvi
1 IntroductionAs of March 2016 there are in the world 440 operable nuclear reactors that, combined, produce
11.5% of the world’s energy supply [1]. The advantages of these facilities, such as their low emissions
and high efficiency, are frequently overshadowed by the security concerns raised by the fallout of a
catastrophic event. It is therefore necessary to study both prevention methods and the possible
outcomes of the failure of the systems that compose a nuclear power plant.
In a nuclear power plant, nuclear fuel cells are constantly producing heat through radioactive
decay. This heat is then converted to electrical energy by means of a steam cycle. The nuclear fuel
cells need to be kept at stable temperatures by an active cooling system to avoid a core meltdown.
Additionally, even after being depleted, the spent nuclear fuel cells have to be cooled for a number of
years before being moved to a permanent storage in order to protect both the environment and the
health and safety of the public [2].
The failure of any of the cooling systems for the fuel cells in nuclear power-plants is called a Loss
Of Coolant Accident (LOCA). The most well known incident of a LOCA occurred in the Fukushima
Daiichi power plant in 2011, when an earthquake followed by a tsunami caused significant damage to
the installation. A loss of coolant in both the reactors and the spent fuel cooling systems caused the
water present in these systems to overheat, resulting in the evaporation of the cooling water and the
emission of radioactive elements into the atmosphere.
Knowing how to predict the consequences and the progression of a LOCA is vital in order to reduce
the fallout of accidents in nuclear power plants. The state of the art simulations on this matter involve
the use of lumped parameter models that require experimental data to tune simulation parameters [3].
These experimental measurements are expensive and require complete fuel cell assemblies and their
instrumentation for measurements to be made. To reduce costs, empirical correlations for the flow
through a bundle of rods with low Reynolds numbers are often used to estimate flow parameters.
However, it has been shown [3] that this leads to an underestimation of the resistance to flow in the
assembly and, consequently, an overestimation of the cooling effects of naturally induced flows that
occur during LOCAs.
A research project has been proposed to the Von Karman Institute (VKI) to establish a Computational
Fluid Dynamics (CFD) method to accurately simulate the evolution of LOCA scenarios. This approach
would remove the necessity of routine experimental measurements and therefore greatly reduce the
cost of predicting LOCA progression and consequences. The contribution of CFD could possibly
range from a more accurate estimation of flow parameters, replacing the usage of problematic em-
piric correlations, to the complete replacement of lumped parameter codes.
The present project aims to establish a mathematical model that is validated for the simulation
of LOCA accidents. This venture is too large for a single thesis and as such this work continues
previous work done at the VKI and focuses on the validation of the mathematical model against
simple academic cases analogous to the total LOCA scenario.
1
Before the mathematical model and its validation process is introduced it is important to give a
brief introduction to the LOCA scenario as well as the validation procedure. These two topics will
be approached in Section 1.1 and 1.2 respectively, followed by a review of the state of the art in
Section 1.3. Finally, clear objectives for this work will be defined in Section 1.4 and an outline of this
document is given in Section 1.5.
1.1 Loss of Coolant Accidents
Nuclear power plants in western Europe use Pressurised Water Reactors (PWRs) to power a
steam cycle to produce electrical energy. Figure 1.1 shows a simplified schematic of a nuclear power
plant. The installation contains three different cycles that do not share fluid.
1
2
3
Figure 1.1: Schematic of a Nuclear Power Plant [4].
The main cycle (1) uses pressurised water circulating through a nuclear fuel cell. Inside this cell,
enriched nuclear elements produce heat to power the reactor. For the purpose of this work, it is not
necessary to fully understand the nuclear physics underlying this process, beyond understanding that
these cells continuously and predictably generate heat through radioactive decay. The heat generated
in this cycle is transferred to a steam cycle (2) where electricity is produced [5]. A third cycle (3),
commonly using a river or an ocean as its source, cools the vapour in the steam cycle.
A nuclear fuel cell consists of a bundle of nuclear fuel rods, grouped in assemblies like the one
shown in Figure 1.2. These rods are made of hundreds of small pellets of enriched uranium placed
end-to-end inside hollow tubes of Zirconium alloy [3].
As shown in Figure 1.2(a), the structure is supported by 24 guiding tubes, marked with G/T in
Figure 1.2(a), and is capable of housing 264 different nuclear fuel rods. The central tube is an instru-
mentation tube. Figure 1.2(b) shows a prototypical installation with the nuclear fuel cells not installed.
The spacers placed along the assembly give the structure rigidity. The dimensions of a PWR fuel as-
sembly vary with the dimensions of the spacers, resulting in different areas for fluid circulation inside
the rod bundle.
After its designated lifetime, a fuel cell is removed from the nuclear reactor vessel to be replaced.
At this time, due to the nature of these cells, they still release residual amounts of radioactive elements
and heat and still require cooling to be kept stable. For this purpose, the Nuclear Regulatory Com-
2
(a) Rod configuration. (b) Prototypical 17x17 Fuel Assembly.
Figure 1.2: PWR 17x17 Fuel assembly [3].
mission (NRC) requires all nuclear power plants to have a Spent Fuel Pool (SFP) [2]. Inside a SFP,
decommissioned nuclear fuel cells are continuously cooled. Pumps circulate water from the SFP to
heat exchangers, then back to the spent fuel pool. The water pool serves a secondary purpose of
containing residual radioactive emissions. It is important to note that a fuel rod’s heat and radioactive
element emissions are a function of its lifetime. The maximum temperature of the spent fuel bundles
decreases significantly between 2 and 4 years.
Once most of the residual radioactive elements have decayed, around 10 years after being placed
in a SFP, the nuclear fuel cells are moved to a permanent storage in a dry cask. At this stage,
water cooling is no longer necessary, and radioactive element emissions can safely be shielded by
a concrete shell. When moved to a dry cask storage, the life cycle of a nuclear fuel rod can be
considered finished.
A Loss Of Coolant Accident (LOCA) is defined as a partial or complete damage to the active
cooling systems for any of the systems where a nuclear fuel cell is located [6]. These accidents
can occur at any stage of the lifetime of a nuclear fuel cell where active cooling is required, with
different consequences. The worst possible outcome from a LOCA is the event of core meltdown.
This happens when the zirconium shelling ignites, resulting in the emission of radioactive elements
into the air, endangering people and the environment.
A total loss of coolant accident is the worst case scenario for these types of accidents. In this
scenario, a complete loss of coolant for a particular system (either a reactor or a SFP) is verified and
the nuclear fuel cells of said system are left exposed to air. This drastically reduces the cooling of
the nuclear cells. However, a natural convective flow develops axially along the bundle of nuclear fuel
cell rods, which provides some degree of cooling to the system. Additionally, heat dissipation through
radiation becomes significant and can possibly account for around 50% of the total heat dissipation
during a LOCA [7]. Note that this heat dissipation by radiation is due to the fact that any heated
surface emits radiation and should not be confused with the radioactive emissions due to the nuclear
decay inside the nuclear fuel rod.
In the present work several simplifications were made in the study of the LOCA phenomenon.
First, only the total LOCA scenario is considered. It is also admitted that the entire cooling of the
3
system is removed instantly when in reality, LOCAs are inherently a transient phenomenon where the
water in the cooling system must completely evaporate before the system is cooled only by natural
convection of air.
The LOCA scenario therefore involves both the heating of a solid domain, the bundle of rods, and
a fluid domain, the surrounding air. This is commonly referred to in the literature as a Conjugate Heat
Transfer (CHT) problem [8].
When simulating a case where a core meltdown will happen, the most important information to be
assessed is the time it takes to reach the meltdown temperature. However, the physical phenomenon
being studied, natural convective flow through a bundle of rods, is expected to reach a steady solution.
As mentioned, the heat dissipated through the lifetime of a nuclear fuel cell varies significantly, and as
such not all nuclear fuel cells will reach the meltdown temperature during a total LOCA. In this work
the steady state solution of the flow will be sought as if in an attempt to determine whether or not a
given fuel cell will ignite.
The flow developed inside the nuclear fuel assemblies due to natural convection is assumed to be
laminar. This assumption is based on available experimental data since no critical transition number
for the specific case of a natural convection flow along a bundle of rods was found in the reviewed
literature. This assumption will be justified in Chapter 2. To further simplify the flow, at this stage struc-
tural elements of the nuclear fuel cell, such as the grid spacers, are not considered. This simplification
reduces the complexity of the domain being studied.
As evident in Figure 1.2, the structure of a fuel cell is by nature symmetric. Due to the predicted
large size of the domain to be modelled, it is important to take advantage of the symmetry of the
domain.
The source of heat inside a nuclear fuel cell is a complex radioactive phenomenon. However, an
understanding of nuclear physics is not necessary to comprehend the LOCA scenario. It is preferable
to interpret the heat dissipation as a uniform heat source inside each of the nuclear fuel rods.
For the radiative heat dissipation the medium is considered as non participating. As a simpli-
fication, dry air is considered, which means that the medium, air, is constituted by 99% diatomic
molecules and can be approximated as a non absorbing, non emitting and non scattering medium.
This brief analysis allows the abstraction of the total LOCA scenario into a much broader class of
problems. A short summary of the conditions is given below:
• The problem is a CHT problem since it combines a solid domain with a fluid domain;
• The heat being generated inside the solid domain can be treated as a uniform heat source;
• The flow develops in the axial direction through a bundle of rods by natural convection;
• The developed flow is expected to be laminar throughout the domain;
• The domain is characterised by several degrees of symmetry that are easily exploited;
• The energy dissipation through radiation on the rod surface must be considered;
• The fluid is considered as a non-participating medium for radiation emissions;
• The flow has a steady state solution;
4
The above short summary will first be used to assess if a given experimental measurement can
be used as a physical model for the LOCA scenario. It is also needed to establish a mathematical
model that describes the LOCA scenario as summarised. In the following section, the importance of
these two parallel processes will be further clarified with an explanation of the validation process.
1.2 Validation of a Mathematical Model
The main goal with any computational simulation is to obtain a numerical solution from a mathe-
matical model that describes a specific problem. The mathematical model comprises a set of Partial
Differential Equations (PDEs) that govern the system being studied, the thermodynamic equations
to calculate the media properties, and the boundary conditions of the domain. Typically, in the case
of fluid dynamics and heat transfer problems, this set of PDEs is fairly complex and exact analytic
solutions are only available for the simplest cases. For this reason, to solve more complex problems,
a numerical method is used to iteratively obtain an approximate solution to the problem.
The numerical solution however is only relevant if it is possible to guarantee its correctness, within
a margin of error. As such, the mathematical model being used must be validated. Validation can be
defined as ”The process of determining the degree to which a (mathematical) model is an accurate
representation of the real world from the perspective of the intended uses of the model” [9]. The
process of validation is schematised in Figure 1.3.
Reality(No data Available)
Physical Model(Experiments)
MathematicalModel
Measurements Numerical Solutions
Data ± U Data ± UValidation
verification
Figure 1.3: The processes of validation and verification [9].
Ideally, the validation is done by comparing the results obtained from the numerical simulations
with experimental data from the actual phenomena it is intended to model. However, this scenario
5
seldom happens since the reason why a numerical simulation is sought is due to the impracticability
of simulating the actual phenomena being studied. In the case of LOCA simulation, the accurate
experimental measurements of these accidents involve the destructive testing of an entire nuclear
fuel cell which, although possible, would be prohibitively expensive if it were to be systematically
performed.
It is therefore preferable to validate the mathematical model against a physical model of the reality
it is intended to simulate. This physical model comes in the form of experiments that closely resemble
the conditions of a LOCA. A summary of these conditions was given at the end of Section 1.1.
Once data from experimental measurements is retrieved, it must be compared with the numerical
solution. If both sets of data match within their respective measurement and numerical errors (U ),
the mathematical model is considered validated. After validation, it is possible to perform numerical
simulations of the real domain of a LOCA and guarantee the results obtained are correct within a
margin of error.
In addition to the process of validation, the process by which it is guaranteed that the numerical
solution obtained is the correct solution to the established mathematical model is called verification.
Verification is formally defined as ”The process of determining that a model implementation accurately
represents the underlying mathematical model and its solutions.” [9]. The verification process can be
split into code and solution verification. Code verification aims at removing bugs in the code that cause
errors between the numerical solution and the ideal solution obtained from the mathematical model.
Solution verification is the process by which uncertainty of the numerical solution can be assessed.
Both of these processes are essential to obtaining a valid numerical simulation. Although the main
objective of this work is the validation of a mathematical model, a code verification exercise will be
performed.
1.3 State of the art
As mentioned, lumped parameter models are used for the simulation of accident scenarios in
nuclear power plants. A lumped parameter model describes the behaviour of physical systems with a
topology consisting of discrete entities that approximate the behaviour of a real system under certain
assumptions. The most prevalent software being used for these simulations is MELCOR [10], a
fully integrated, engineering-level computer code capable of simulating a broad spectrum of severe
accident phenomena. It has the ability to model core heat-up and degradation, fission product release
and transport within the primary system and SFPs. This program is able to model LOCA scenarios.
Work to replace lumped parameter codes has been on going for over 30 years. A paper from
1978 describes a numerical simulation limited to a 2D slice of the fully developed section for the
flow along a bundle of rods [11]. In order to achieve these results, several tricks, such as using a
modified set of PDEs that included only dimensionless numbers and a massive oversimplification of
the thermophysical properties of the flow, were necessary. These results do not have any practical
use today but laid the first stone for the simulation of flows inside nuclear fuel reactors.
6
A more recent work also uses a 2D slice of the domain to predict temperatures inside a horizontal
nuclear fuel cell cooled by air [12]. These simulations were not for the LOCA scenario specifically, but
rather a study of the viability of dry cask storage for used nuclear fuel cells. Both the solid domain
and the radiation energy dissipation were considered. The results showed a good correlation with
experimental data and the suggested work of this paper proposed the modulation of the complete
domain when it is vertical. However, a paper presenting the results for this proposed continuation
work was not found in the reviewed literature.
Another work was found where radiation and the solid domain were considered, but a 3D do-
main was used [13]. This work however considered the Boussinesq approximation which, as will be
discussed in Section 2.1, is not valid for the simulation of LOCAs. Additionally, this work used an
enclosed bundle instead of an open bundle.
The most advanced work reviewed included a simulation of an entire 5x5 fuel assembly [14]. This
simulation was validated successfully against experimental data. The mesh used contained 20 million
cells and the simulation time for a steady state solution was of 18 hours in a 14×2.0GHz CPU cluster.
However, this simulation did not take into consideration radiation, nor the solid domain. In addition,
this simulation considered the existence of forced convection. The key takeaway from this work is
that with the existing hardware it is now possible to simulate an entire fuel assembly in a reasonable
amount of time.
All of the above mentioned simulations of the flow inside nuclear fuel rod assemblies, did not
specifically study the LOCA scenario. For this reason, the boundary conditions and the assumptions
made about the thermophysical properties from the working fluid don’t always match the ones for the
LOCA scenario.
On the experimental side, two papers containing experimental measurements of cases analogous
to the LOCA scenario were chosen for the validation procedure. The first paper contains experimental
data for the natural convective flow around a single heated cylinder [15]. The second paper details
the experimental measurement of the laminar steady-state free convection in a open-ended vertical
seven cylinder bundle [16]. These papers will be reviewed in greater detail in Section 4.1.
Another relevant paper describes the validation of a mathematical model against the experimental
data for the seven cylinder bundle of the aforementioned paper [17]. This paper had to resort to using
several approximations of the exact geometry of the problem and the Boussinesq approximation, as
well as not considering radiation or the solid domain to obtain a numerical solution.
Finally, a paper describing the experimental measurements of a total LOCA scenario performed
in a nuclear fuel rod assembly was reviewed [2]. This paper has been cited before since it discusses
the critical problem that motivates this project: state of the art simulations of LOCAs using MELCOR
underestimate the temperatures reached inside a nuclear fuel cell. Other important data was collected
from this paper, such as the maximum allowable temperature before ignition occurs, the pressure
drop along the fuel cell, and the type of flow that develops. This data will be reviewed more thoroughly
further in this document.
At the VKI the work performed in this area started in 2014. In the first work on this project, the
7
author validates the solver against the simplest case of a single heated cylinder [7, 15]. Transient
simulations that did not consider either a radiation model or the solid part were performed. Although
the results obtained were considered by the author as having an acceptable fit with the experimental
data used for the validation, there is still a significant error that was not justified. Instead, the author
named as the main challenge faced the excessive simulation time due to the high number of cells in
the designed mesh. The simulations present in this work were performed using OpenFOAM.
Another work where OpenFOAM was used for the simulation of the flow in a heated cylinder
annulus was reviewed [18]. In this report, the author failed to achieve convergence with an inlet and
an outlet in the domain when compressible flow was considered and for this reason he resorted to
simulating a closed casket instead. This academical case is relevant since it hints at the possibility of
OpenFOAM not being able to numerically resolve simulations of natural convective flows in annulus
and subsequently bundles of rods.
The follow-up work performed in the VKI focused on the inclusion of radiation and CHT in the
numerical simulations. Due to the excessive simulation time of the previous work, a decision was
made to change from OpenFOAM to Numeca’s FineOPEN. This work concluded that this software
package was capable of integrating CHT and radiation only if a stationary solver was used. However,
the results obtained were not satisfactory [19]. Additionally, similar simulation time constraints to
OpenFOAM were faced by the author.
Both of the previous works done in this project used numerical software as a ”block box”, and a
study of the implemented mathematical model was not done. Understanding the implemented model
is vital for the validation procedure since it allow to determine if eventual errors are due to bugs in the
numerical implementation or shortcomings of the model itself. For this reason, the definition of the
mathematical model to be validated must be a main focus in this work.
As a summary to this section, a valid mathematical model for the simulation of a LOCA scenario
was not found in the reviewed literature. However, other projects with similar topics indicate that, if a
mathematical model is found to provide accurate solutions, the computational power limitations that
handicapped simulations of this magnitude have now been overcome.
1.4 Objectives
This thesis continues the work started in 2014 in the VKI [7, 19] with the objective of simulating
Loss Of Coolant Accidents (LOCAs) in nuclear power plants. The main goal established by the VKI
was the validation of a CHT solver that considered heat dissipation through radiation, convection and
conduction as well as the existence of heat sources in the solid domain. This is virtually the same goal
pursued by the previous work done since the results obtained were not satisfactory [19]. To achieve
this goal, three objectives were defined:
• Establish a complete mathematical model for the total LOCA scenario;
• Verify the numerical code that implements the established mathematical model;
8
• Validate the mathematical model against experimental data analogous to the LOCA scenario.
1.5 Outline
This thesis is divided in six chapters. This first chapter should have provided the reader with a
working knowledge of the LOCA scenario and of the validation procedure, as well as the current state
of the project developed by the VKI. From this review, clear objectives for this work were established.
In Chapter 2, the mathematical model that describes the total LOCA scenario will be explained.
This will cover the Navier-Stokes equations to describe the fluid domain, the energy equation for a
solid domain and three distinct equations for the inclusion of the radiation effects. The boundary
conditions and supporting thermophysical models for each of these equations will also be discussed.
Following that, Chapter 3 will present an overview of the computational implementation of the
mathematical model. The decision to revert to using OpenFOAM for the project will be justified. The
chosen solver will be explained together with the numerical boundary conditions of the problem and
the adopted thermophysical models. Key simulation topics, such as the numerical schemes, linear
solvers and convergence criteria will also be discussed.
Chapter 4 is dedicated to explaining the specific numerical simulations that will be performed. The
experimental data used for the validation process will be reviewed and the computational domains
for the simulations will be defined. Finally, the selected test cases will be listed. These include a
verification exercise to guarantee the code reproduces the simplest flow, an exhaustive test of the
three radiation models to determine the most appropriate one for the simulation of LOCA scenarios,
and the replication of the experimental measurements numerically.
Chapter 5 will present and discuss the results obtained for the numerical simulations performed.
The results obtained will allow the selection of the most appropriate radiation model. The model’s
numerical limitations will be discussed in light of the results obtained for the more complex case
studied.
Finally, in Chapter 6 summarises the work carried out in this project. Conclusions on the mathe-
matical model’s accuracy at representing a LOCA scenario will be made. A future continuation work
will be proposed regarding the validation of the solver as well as a possible experimental measure-
ment to improve feasibility of the validation step.
9
10
2 Mathematical model for the loss ofcoolant accident scenario
This chapter presents a review of the mathematical model that describes a total LOCA. As men-
tioned before, this scenario is a CHT problem where a fluid domain interacts with multiple independent
solid domains.
In Section 2.1 the Navier-Stokes equations that describe the fluid domain will be presented along
with the equations that allow to explicitly calculate relevant fluid properties. Section 2.2 will present a
similar analysis for the set of equations used for the solid domains.
In Section 2.3 the boundary conditions that complement the system of PDEs will be discussed,
while in Section 2.4 the initialisation fields for the system variables are explained.
Section 2.5 presents the complementary models that allow the inclusion of radiative heat transfer
in the mathematical model of the fluid domain. The P1 model and the discrete ordinates models
based on the Radiative Transfer Equation (RTE) and the view factor model will be briefly explained.
Finally, for convenience and clarity, a summary of all the equations and variables that constitute
the mathematical model for the LOCA scenario is presented in Section 2.6.
2.1 Partial differential equations for the fluid domain
Before a set of PDEs that describes the fluid domain can be chosen it is vital to review the type of
flow being studied. In general, any flow can be categorised as laminar, turbulent, or a combination of
both these flows when a transition region exists. The type of flow that develops is typically correlated
with the Reynolds number, Re, of the flow. The Reynolds number is a dimensionless quantity defined
as the ratio of inertial forces and viscous forces of a flow, calculated as shown in Equation (2.1), where
Dh is the hydraulic diameter, ν is the kinematic viscosity and ~U is the free flow velocity [20].
Re =|~U |Dh
ν(2.1)
Experimental data regarding the transition Reynolds number for a PWR assembly was not found
in the reviewed literature, however experimental measurements in a Boiling Water Reactor (BWR)
fuel assembly exhibited laminar flow with Reynolds numbers spanning from 70 to 900 [2]. In the more
generic case of a flow along a bundle of rods, interpolation of the collected data from experimental re-
sults shows a relation between the rod distance, P , and rod diameter, D, with the transition Reynolds
number [21]. For the values typical of a PWR, P/D ≈ 1.5, transition effects are not observed below
Retrans = 800.
To estimate the maximum Reynolds number inside a nuclear fuel cell during a LOCA, let’s consider
a single fuel assembly. In natural convection the flow is driven by a pressure difference caused by a
density gradient. The total pressure variation in the domain is generically given by Equation (2.2).
11
∆ptotal = −∫ h2
h1
g(ρh2 − ρh1)dz (2.2)
During a LOCA, the pressure inside the system is reduced to the pressure of the outside environ-
ment. Let’s consider for this calculation the STP condition of p = 101 325Pa, and the height from the
nuclear fuel cell shown in Figure 1.2b of 4.19m.
To account for the most extreme case, let’s consider the temperature at the top of the fuel cell
as being the temperature at which a core ignition would happen. No experimental data was found
regarding the ignition of PWR fuel cells, however extensive testing was performed in BWR assemblies,
which have similar rod design but different dimensions to PWR fuel cells [2]. It was concluded that the
ignition of these rods is a complex event requiring a temperature of 1273K and a heat rate of 0.1K s−1.
However, around 1173K oxidation of the zirconium alloy outer layer occurs. This makes ignition more
difficult, by reducing the conductivity of the material, but implies an undesired destruction of the fuel
cell rods. For this reason, for all intents and purposes, it is considered that the temperature inside of
a nuclear fuel rod must not reach 1173K.
For the present calculation, temperature at the inlet at the domain is admitted as 298K and the
maximum permissible temperature of 1173K is used as the outlet temperature. Given that R/M =
287.058 J kg−1 K−1 for air, and the perfect gas law, Equation (2.8), the maximum expected pressure
gradient can be calculated as shown in Equation (2.3).
∆ptotal = 9.81× 4.19×(
101325
287.058× 1173− 101325
287.058× 298
)= 36.319Pa (2.3)
Figure 2.1 shows the results for the pressure drop inside a BWR and a PWR fuel assemblies as a
function of Reynolds Number, Re, defined in Equation (2.1) obtained through experimental measure-
ments [3]. For the worst case scenario calculated in Equation (2.3), the Reynolds number of the PWR
fuel assembly is 380.
0 200 400 600 800 1,000 1,2000
20
40
60
80
100
120
140
Re
∆p(N
m−2)
BWR PWR
Figure 2.1: Pressure drops along BWR and PWR as function of the Reynolds number [3].
It is therefore assumed that the developed flow is laminar in the entire domain during a LOCA
since the predicted Reynolds number for the worst case scenario, Re = 380 is still considerably below
12
the estimated transitional Reynolds, Retrans = 800.
Combined with the fact that the flow is laminar, it has been established in Section 1.1 that a steady
state solution will be sought. Therefore, in order to describe the motion of the fluid domain, the
Navier-Stokes equations for a steady state laminar problem must be solved.
The Navier-Stokes Equations describe the motion of viscous fluid substances, and can be written
as three distinct conservative equations [9]. Equation (2.4) describes the principle of mass conserva-
tion, Equation (2.5) the conservation of momentum, and Equation (2.6) conservation of energy. The
equations presented are written for the steady state case where all time derivatives are equal to zero.
In the following subsections each of these equations will be briefly analysed and their terms adjusted
to the problem being solved.
∇ · (ρ~U) = 0 (2.4)
∇ · (ρ~U ~U) = ∇ · τ + ρ~fe −∇p (2.5)
∇ · (ρ~UH) = ∇ · (k∇T ) +∇ · (τ ~U) + ρ~fe~U +QS (2.6)
2.1.1 Continuity Equation
The continuity equation, Equation (2.4), presents the fundamental principle of mass conservation,
that the mass entering the system must be equal to the mass exiting the system. The only variable in
the continuity equation is the velocity, ~U . The density, ρ, is calculated through an equation of state.
An equation of state is a thermodynamic equation relating variables that describe the state of
matter under a given set of physical conditions. The flow generated during a LOCA has buoyancy
as its primary driver. Buoyancy force is generated by density differences in the working fluid. It is
therefore necessary to define what is the relation between the density, ρ, the fluid temperature, T ,
and pressure, p. Given the fact that the working fluid is air and the relevance of buoyancy forces, the
working fluid must be considered as compressible. However, it is pertinent to consider the use of the
Boussinesq approximation.
The Boussinesq approximation uses a constant density, therefore independent from temperature,
T , and pressure, p, for all terms except for the buoyancy force term in the momentum equation [22].
The purpose of this approximation is to reduce the computational load to obtain a numerical solution
by using an incompressible flow solver for natural convection problems.
The criteria to validate the Boussinesq approximation is presented in Equation (2.7). The temper-
ature range is admitted as the worst case scenario, where the temperature at the top is the maximum
allowed temperature 1173K, and the inlet temperature is 298K. The air expansion coefficient β is
considered as 3.33× 10−3 K−1. This results, for the problem studied in this thesis, in a relative density
variation β(T − T∞) ≈ 3, not verifying the necessary criteria of the Boussinesq approximation.
β(Tmax − Tmin) 1 (2.7)
Physically speaking, this means the density variation is of the same order of magnitude as the
13
density itself. For this reason, using the Boussinesq approximation would result in a severe modelling
error.
Since the Boussinesq approximation is not valid, the working fluid is considered as obeying the
perfect gas law. The density field can therefore be calculated implicitly from the pressure field, P ,
the fluid molar mass, M , the ideal gas constant, R, and the temperature field, T , as shown in Equa-
tion (2.8).
ρ =pM
RT(2.8)
2.1.2 Momentum balance equations
The momentum balance equation, Equation (2.5), is originally derived from Newton’s second law
of motion. It is important to notice that, despite its brief and compact aspect, Equation (2.5) is a vector
equation. This means that, for a 3D problem, each of the components of this equation contains nine
terms and a total of three distinct momentum equations must be solved [20].
The Left Hand side (LHS) of the equation represents the variation of momentum inside of the
domain. Contrary to the mass continuity equation, the momentum of the fluid may vary within the
domain; as such, the Right Hand Side (RHS) of Equation (2.5) is an aggregate of all the possible
sources of momentum variations inside the domain.
The first term on the RHS, ∇τ , represents the work done by internal friction forces of fluid layers
against each other, where τ is the viscous shear stress tensor [9]. In the case of a Newtonian fluid, a
fluid in which the viscous stress arising from its flow at every point is linearly proportional to the local
strain rate, the shear stress tensor can be calculated as shown in Equation (2.9) [9]. Note that in the
term, (∇~U)T , the symbol T refers to the transpose algebraic operation and not to the temperature
field, while the symbol I in the term, − 23∇ · ~UI, refers to the identity matrix.
τ = µ
[−2
3∇ · ~UI +∇U − (∇ · ~U)T
](2.9)
The viscous shear stress tensor is therefore a function of the velocity, ~U , and the dynamic viscosity,
µ. For the present case, the dynamic viscosity of the fluid was considered as obeying Sutherland’s
Law [22]. The dynamic viscosity can be computed as shown in Equation (2.10), where C1 and C2 are
the Sutherland’s constants.
µ =C1T
3/2
T + C2(2.10)
The second term on the RHS of Equation (2.5), ρ~fe, represents the contribution of external forces,~fe. In this problem the only external force that needs to be considered is gravity, ~g.
Finally, the last term on the RHS, −∇p, represents the work done by pressure forces. To facili-
tate the application of the boundary conditions later, a new quantity, pρgh, was defined as shown in
14
Equation (2.11), where ~x is the position vector1.
pρgh = p− ρ~g × ~x (2.11)
When Equation (2.11) is replaced into the momentum equation and the only external force con-
sidered is gravity, Equation (2.5) is rewritten as shown in Equation (2.12). When this alternative form
of the momentum balance equation is used, the system variable being calculated changed from the
pressure, p, to the modified pressure, pρgh.
∇ · (ρ~U ~U) = ∇ · τ −∇pρgh − (~g.~x)∇ρ (2.12)
2.1.3 Energy balance equation
The energy equation, Equation (2.6), is used to determine the temperature field. The equation is
however solved for the total enthalpy, H. The total enthalpy is a sum of the enthalpy and the fluid’s
kinetic energy as shown in Equation (2.13).
H = cpT +|~U |2
2(2.13)
The LHS term of the equation, ∇(ρ~UH), represents a balance of the energy inside the domain.
Much like the momentum equation, the energy of the flow may vary within the domain. The LHS terms
of the energy equation account for the possible sources of energy variation within the domain.
The first term on the LHS, ∇(k∇T ), represents the energy diffusion. This term can be rewritten
considering the fluid thermal diffusivity, α, calculated as shown in Equation (2.14), where the specific
heat capacity, cP , is a fluid property and the thermal conductivity, k, is calculated through the modified
Euken thermal conductivity model from Equation (2.15) [23].
α =k
ρcp(2.14)
k = µ
(cp −
R
M
)(1.32 +
1.77 RM
cp − RM
)(2.15)
The second term on the RHS, ∇(τ ~U), represents the viscous energy dissipation rate. Given the
small velocity gradients expected for this type of problem, this term can be considered as negligible
and completely removed from the final equation.
The third term on the RHS, ρ~fe~U , represents the contribution of external forces to the energy
balance. The only relevant external force to be considered is, once again, gravity, ~g.
The final term on the RHS, QS , accounts for the inclusion of energy sources in the domain. This
heat source can be used to include the effects of radiative heat dissipation.1To be mathematically consistent, this variable symbol should be pρ~g~x, however, pρgh was chosen for being more similar
with its OpenFOAM implementation name ”p rgh”.
15
The final form of the energy balance equation is written in Equation (2.16). Once the total enthalpy,
H, field has been calculated, the temperature field can be calculated using equation (2.13).
∇ · (ρ~UH) = ∇ · (αρ∇H) + ρ~g~U +QS (2.16)
2.2 Partial differential equation for the solid domain
The only variable in a solid domain is the temperature field, T . Therefore, the energy equation,
Equation (2.17), needs to be solved in order to determine the temperature field in the domain [24].
∇.(k∇T ) = Qs (2.17)
The density, ρ, the specific heat capacity, cp, and thermal conductivity, k, are considered as fixed
properties of the domain that do not vary with the temperature. Note that these variables may not
be constant throughout the domain. As discussed in Section 1.1, a nuclear fuel rod has an inner
layer with nuclear fuel and an outer layer made of zirconium alloy. To avoid an excessive amount
of separate domains, different layers can be defined in the same domain by modifying the material
properties matrices.
The term Qs allows for the inclusion of an energy generation source within the domain. Through
this term it is possible to include in the system the energy generated by the nuclear reaction occurring
inside the rods of a nuclear fuel cell.
To simplify the interactions between the domains, instead of solving directly the temperature field, it
is preferable to solve Equation (2.17) for total enthalpy, H, according to Equation (2.18). The thermal
diffusivity can be calculated as shown in Equation (2.14). The temperature field can be calculated
from the total enthalpy using Equation (2.19).
∇ · (αρ∇H) = Qs (2.18)
H = cpT (2.19)
2.3 Boundary Conditons
Complementary to the system of equations that describe the physical reality being studied, a set
of boundary conditions must be imposed so as to determine the constants of integration of the PDE.
The domain of a LOCA is schematised in Figure 2.2. The inlet, outlet and solid surfaces all require
boundary conditions for all the system variables.
Fundamentally, two types of boundary conditions can be imposed. A fixed value condition, called
Dirichlet condition and a fixed gradient condition, called von Neuman condition. These conditions,
written for an arbitrary quantity, Ψ, when applied at an arbitrary boundary face, bf , are given in Equa-
tions (2.20) and (2.21). It is relevant to note that the von Neuman condition only defines a gradient in
the direction normal to the surface.
16
Inlet
Outlet
CHT wall
outer wall
Flow
dire
ctio
n
Figure 2.2: Schematic of a LOCA domain [25].
Ψbf = Ψfixed (2.20)
(∇Ψ.~n)bf =
(δΨ
δn
)fixed
(2.21)
The system variables, the velocity, ~U , modified pressure, pρgh, and both the solid and fluid domains
temperatures, T , must have clearly defined boundary conditions at all the surfaces, inlets and outlets
of the system.
In the case of a solid surface a no slip condition is imposed for the velocity, ~U . As a fluid flows
past a wall, neighbouring fluid particles tend to stick to the surface slowing down the surrounding fluid.
This is because, immediately at the wall, the velocity of the flow must be the same as the velocity of
the wall. Mathematically, this is a Dirichlet condition where the fixed value imposed is zero as shown
in Equation (2.22). This boundary condition results in the formation of what is called a boundary layer.
Due to the viscosity of the flow, a slim layer of fluid is formed close to the surface, inside which the
velocity quickly varies [26]. The correct calculation of the flow development inside the boundary layer
is fundamental to obtain a correct solution for the entire domain.
(~U)bf = 0 (2.22)
The modified pressure, pρgh, boundary condition for solid walls is a zero gradient von Neuman
condition so long as the walls are vertical.
The temperature, T , boundary condition at an adiabatic solid surface is a simple von Neuman
condition of zero gradient. This dictates that no energy is exchanged through this surface.
For a heated solid surface, as in, a surface that serves as interface between the solid and fluid
domains, a fixed temperature condition must be applied as shown in Equation (2.23). This condition
comes from the continuity condition that, much like the velocity of the fluid at the wall surface must be
17
zero, there must be a continuity of the temperature at the wall. Additional to this condition, it is also
possible to relate the temperature gradients at the interface. This second condition, Equation (2.24)
comes from the energy balance at the surface saying the heat flux exiting the solid domain must be
the same as the heat flux entering the fluid domain. Note that this equation considers the radiative
heat dissipation at the wall in the term qr.
(Tsolid)bf = (Tfluid)bf (2.23)
(ksolid∇Tsolid)bf = (kfluid∇Tfluid)bf + qr (2.24)
An additional auxiliary boundary can be defined for cases where only the energy dissipation at
the wall is considered without including the solid domain. When this simpler boundary condition is
considered only the temperature gradient is imposed by means of a fixed heat flux, qs, at the wall.
This is shown in Equation (2.25).
(∇T )bf =(qs − qr)kfluid
(2.25)
The top and bottom surfaces of the solid domain are considered as adiabatic and as such a zero
gradient condition is applied for the temperature.
In the case of an inlet for the fluid domain, the boundary condition must be specified in such a way
that the total energy entering the system is fully defined. This can be achieved by imposing a total
temperature, T0 and total pressure, p0, boundary condition. When these two quantities are imposed,
as defined in Equations (2.26) and (2.27), the total energy of the system is fixed at the inlet, and the
actual temperature, pressure and velocities can be calculated. Given that at the inlet the expected
velocity field has a magnitude of |~U | ≈ 1m s−1, compressibility effects can be neglected and the
incompressible forms of these equations used.
(T0)bf = T + |~U |2/(2Cp) (2.26)
(p0)bf = p+ 0.5ρ|~U |2 (2.27)
It is important to notice that these equations only give information about the magnitude of the
velocity vector. The orientation of the velocity vector must come from the flow inside the domain by
means of the continuity equation. The pressure, p, can be converted to the modified pressure, pρgh,
using Equation (2.11).
At an outlet, a temperature and velocity profile are expected to be fully developed. For this reason,
a zero gradient von Neuman condition should be applied at the outlet for these two quantities. Since
a zero gradient velocity profile at the outlet is considered, the momentum equation, Equation (2.12),
reduces to the equation shown in Equation (2.28). This condition can be used as the outlet condition
for the modified pressure in the outlet.
(∇pρgh)bf = −(~g.~x)∇ρ (2.28)
18
Finally, a symmetry boundary condition can be applied to simplify the domain. When this condition
is applied, a zero gradient von Neuman condition is applied to all the variables of the domain in the
symmetry surface.
2.4 Initialisation
The strategy followed to initialise the fluid domain is setting the entire temperature field in the fluid
domain to the total temperature, T0, imposed at the inlet. Additionally, the velocity field, ~U , is initialised
as zero in the entire domain.
The pressure field resulting from these two assumptions is shown Equation (2.29).
p = p0 exp
(~g.~x
RT
)(2.29)
Once these three fields are initialised, all the explicitly calculated fields, such as the density, ρ, in
the domain can be calculated explicitly.
The initialisation of the solid domain is made by setting an arbitrary temperature in the entire
domain. For a faster convergence, the temperature should be as close to the expected final value as
possible.
2.5 Radiation
Complementary to the equations that describe the heat transfer by conduction and convection, it
is necessary to include a radiation model to account for the radiative heat transfer in the system. For
this purpose three independent models will be reviewed. The first two models, the P1 model and the
discrete ordinate model, are derived from the Radiative Transfer Equation (RTE), while the third is the
view factor model.
These three models will calculate the radiative energy fluxes in the domain from the temperature
field, T , calculated with the energy equation, Equation (2.6). Once calculated, the radiative energy
dissipation can be included in the energy equation through the source term, QS , and the radiative
heat flux at the surfaces, qr.
2.5.1 Radiative transfer equation
The RTE, Equation (2.30), is a PDE describing the energy balance of a radiation beam as it travels
through an absorbing, emitting, and scattering media [27]. The equation calculates the radiation
intensity, I, as function of both the position, ~x, and the direction, ~s. The absorption coefficient, a,
scattering coefficient, σs, refractive index, n, are properties of the media being modelled, while σ is
the Stefan-Boltzmann constant. The function, Φ, is called the scattering phase function and describes
the probability that a ray from one direction, ~s, will be scattered into a certain other direction, ~s′. Finally,
Ω is the directional vector of radiative intensity. The presented form of the RTE considers the radiation
intensity independent from the wave-number.
19
dI(~x,~s)
ds= −aI(~x,~s)− σsI(~x,~s) + an2σT
4
π+σs4π
∫ 4π
0
I(~x,~s)Φ(~s · ~s′)dΩ′ (2.30)
The LHS term of the equation represents the net variation of radiation intensity for a given ray.
The first term of the RHS of the equation represents the attenuation by absorption, the second term
the attenuation by scattering, the third term the augmentation by emission, and the fourth term the
augmentation by scattering. Deriving or further explaining the RHS terms of the RTE goes beyond
the scope of this work since the working fluid, air, was considered as a non absorbing, non scattering
and non emitting media. The RTE however is still valid for this scenario even if inside the domain it is
reduced to the LHS of the equation being equal to zero.
2.5.2 P1 model
The P1 radiation model is based on the expansion of the RTE, Equation (2.30), into an orthogonal
series of spherical harmonics [28]. This model is derived to be used in small chambers where the
heat transfer by radiation through a participating medium is the dominant mode of heat transfer.
The model solves a transport equation, Equation (2.31), for the incident radiation, G, where the
absorptivity, a, the scattering coefficient, σ, the refractive index, n, and the linear-anisotropic phase
function coefficient, Φl, are fixed properties of the medium where radiation is propagating. The equa-
tion includes a source term, Qr, for the inclusion of radiative heat sources in the domain.
∇ ·(
1
3(a+ σs)− Φlσs)∇G
)= aG+ 4aσT 4 +Qr (2.31)
The boundary condition for this model for diffuse grey surfaces is shown in Equation (2.32). This
condition requires that an emissivity, ε, is given for every surface. After solving the transport equation
for the incident radiation, the radiative heat flux can be included in the source term of the energy
equation as shown in Equation (2.33).
(qr)bf = − εbf2(2− εbf )
(4σT 4
bf −Gbf)
(2.32)
Qs = a(4σT 4 −G) (2.33)
2.5.3 Discrete ordinates model
The discrete ordinates model solves the RTE for a finite number of discrete solid angles. Each
of these angles has an associated vector direction, ~s. Equation (2.30) is rewritten as a transport
equation of the radiation intensity, I, in the spatial coordinate system, Equation (2.34). This model
solves as many equations as the number of considered discrete solid angles.
∇ · (I(~x,~s)~s) = −aI(~x,~s)− σsI(~x,~s) + an2σT4
π+σs4π
∫ 4π
0
I(~x,~s)Φ(~s · ~s′)dΩ′ (2.34)
The boundary condition for this model for diffuse grey surfaces is shown in Equation (2.35). This
condition requires that an emissivity, ε, is given for every surface. After solving the transport equation
20
for the incident radiation, the radiative heat flux can be included in the source term of the energy
equation as shown in Equation (2.33).
(qr)bf = (1− εbf )
∫~s~n>0
I(~x,~s)~s~ndΩ + n2εσsT4bf (2.35)
Qs = a
(4σT 4 −
∫4π
I(~x,~s)~sdΩ
)(2.36)
2.5.4 View factor model
An alternative to solving the RTE when a non participating media is considered is the usage of
the view factor model. As mentioned, the working media is non absorbing, non scattering and non
emitting. This makes the view factor treatment of the radiation heat transfer more coherent with the
type of problem being solved. The view factor model however is derived for enclosed domains.
The view factor model consists of a balance of the emitted and absorbed energy between all the
surfaces in the domain without considering the media [27]. This means that the photons emitted at
the heated surfaces will travel unimpeded until they reach another surface, where they will either be
absorbed or reflected. For a given surface, i, the energy balance, as shown in Equation (2.37), where
qr is the radiative energy flux emitted by the surface and qin the radiative energy flux received, must
be respected. This equation is written considering only diffuse grey surfaces. It is necessary to know,
as a material propriety, the emissivity, ε of all the surfaces in the domain.
qr,i = qemission,i − qabsorption,i = εiσT4i − (1− εi)qin,i (2.37)
To calculate the radiative energy flux received by a surface it is necessary to calculate the view
factor of that surface to all the other surfaces in the domain. The view factor is a ratio between the
diffuse energy leaving surface Ai that reaches surfaces Aj and the total diffuse energy leaving Ai. A
schematic of a view factor between two arbitrary surfaces is presented in Figure 2.3. The angles θi
and θj are the angles between the surface normals, ni and nj , and the line connecting the centre of
both surfaces, while S is the distance between the two surfaces.
Figure 2.3: View Factor geometry schematic [24].
The view factor between two arbitrary surfaces can be calculated as shown in Equation (2.38).
FAi→Aj =1
Ai
∫Ai
∫Aj
cos θi cos θjπS2
dAjdAi (2.38)
21
Once the view factors from all the surfaces are calculated, it is possible to rewrite Equation (2.37)
as shown in Equation (2.39) [24].
qr,i = εiσT4i + (1− εi)
N∑j=1
(qr,jFAi→Aj
)(2.39)
The contribution from the radiative heat flux, qr, is then included in the boundary condition for the
interface of solid and fluid domains as shown in Equation (2.24). In all the remaining surfaces, as
long as the value of emissivity is ε = 1, all the incident radiation is considered as absorbed by the
boundaries and removed from the system as a loss. This way, by defining the inlet and outlet of the
system as surfaces with an emissivity of ε = 1, the domain is simulated as an enclosure from the
radiative point of view without affecting the resolution of the fluid domain equations.
2.6 Summary of the mathematical model
The complete set of equations, variables and boundary conditions that compose the mathematical
model is given in Tables 2.1, 2.2 and 2.3 so as to serve as a reference for the reader in later chapters.
Table 2.1: Summary of the mathematical model for the fluid domain.
Variables
T Temperature~U Velocity
pρgh Modified Pressure
Constantproperties
M Molar Mass
CP Heat Capacity
C1, C2 Sutherland’s Coefficients
ε Emissivity (surfaces)
Implicitequations
∇ · (ρ~U) = 0 (2.4 ) Mass balance
∇ · (ρ~U ~U) = ∇ · τ −∇pρgh − (~g.~x)∇ρ (2.12) Momentum balance
∇ · (ρ~UH) = ∇ · (αρ∇H) + ρ~g~U +QS (2.16) Energy balance
Explicitequations
ρ = pMRT (2.8) Equation of state
τ = µ[−2−3∇ · ~UI +∇~U − (∇~U)T
](2.9) Viscous stress tensor
µ = C1T3/2
T+TS(2.10) Dynamic Viscosity
H = Tcp + |~U |22 (2.13) Total enthalpy
α = kρcp
(2.14) Thermal diffusivity
k = µ(cp − R
M
) (1.32 +
1.77 RMcp− R
M
)(2.15) Thermal conductivity
p = pρgh + ρ~g.~x (2.11) Pressure
22
Table 2.2: Summary of the mathematical model for the fluid domain (cont.).
P1 ∇ ·(
1(3(a+σs)−Φlσs)
∇G)
= aG+ 4aσT 4 +Qr (2.31) transport Equation
DiscreteOrdinates
∇ · (I(~x,~s)~s) = −aI(~x,~s)− σsI(~x,~s) + ...(2.34) RTE transport equation
...+ an2 σT 4
π + σs4π
∫ 4π
0I(~x,~s)Φ(~s · ~s′)dΩ′
View factorqr,i = εiσT
4i + (1− εi)
N∑j=1
(qr,jFAi→Aj
)(2.39) Radiative Energy Balance
FAi→Aj = 1Ai
∫Ai
∫Aj
cos θi cos θjπS2 dAjdAi (2.38) View factor
BoundaryConditions
∇U = 0 (2.21)
Outlet∇T = 0 (2.21)
∇pρgh = −(~g~x)∇ρ (2.28)
T0 = T + |~U |2/(2Cp) (2.26)Inlet
p0 = p+ 0.5ρ|~U |2 (2.27)
~U = 0 (2.22)
Unheated wall∇T = 0 (2.21)
∇pρgh = 0 (2.21)
~U = 0 (2.22)
CHT wallTfluid = Tsolid (2.23)
kfluid∇Tfluid = ksolid∇Tsolid − qr (2.24)
∇pρgh = 0 (2.21)
~U = 0 (2.22)
Heated wall∇T = (qs − qr)/kfluid (2.25)
∇pρgh = 0 (2.21)
∇U = 0 ∇T = 0 ∇pρgh = 0 (2.21) Symmetry
qr = − ε2(2−ε)
(4n2σT 4 −G
)(2.32) P1 model
qr = (1− ε)∫~s~n>0
I(~x,~s)~s~ndΩ + n2εσsT4 (2.35) DO model
23
Table 2.3: Summary of the mathematical model for the solid domain.
Variables T Temperature
Constantproperties
CP Heat Capacity
ρ Density
k Thermal conductivity
Implicitequation ∇ · (αρ∇H) = Qs (2.18) Energy balance
Explicitequations
H = Tcp (2.19) Total enthalpy
α = kρCp
(2.14) Thermal diffusivity
BoundaryConditions
Tsolid = Tfluid (2.20)CHT wall
ksolid∇Tsolid = kfluid∇Tfluid + qr (2.24)
∇T = 0 (2.21) Symmetry
∇T = 0 (2.21) Adiabatic surfaces
24
3 Computational implementation ofthe mathematical model
With the mathematical model that describes the studied problem now fully defined, the next step
is finding or developing a computational implementation of said model. Section 3.1 addresses this
task, justifying the choice of OpenFOAM as the software package to be used in this project.
OpenFOAM ’s application chtMultiRegionSimpleFoam was chosen for the validation process. This
solver will be overviewed in Section 3.2. In Section 3.3, the computational implementation of the
radiation models discussed in Section 2.5 will be analysed, while numeric implementation of the
boundary conditions for the system is presented in Section 3.4.
The chosen software provides a discretisation method that approximates the PDE from the mathe-
matical model by a system of algebraic equations for the variables at a finite number of points. For the
discretisation of the Navier-Stokes equations the Finite Volume Method (FVM) is commonly used [9].
The FVM will not be explained in this text, however it is recommended that the reader is familiarised
with this method to comprehend some of the topics discussed in this chapter.
The numerical solution is obtained iteratively and as such it is necessary to monitor the simulation
and define stoppage criteria. These topics are covered in Section 3.5.
The numerical schemes used for the discretisation of the variables is briefly presented in Sec-
tion 3.6 while the linear solvers used for the generated system of equations are discussed in Sec-
tion 3.7. Due to the complexity of these two topics, these two sections will be oriented to justify the
choices made more than to explain the intricacies of the schemes or solvers.
Section 3.8 will explain the types of relaxation used to ensure iterative solution convergence.
Several modifications had to be made to OpenFOAM ’s source code to correctly implement the
mathematical model for the LOCA scenario. Changes made to the solver used are discussed in
Annex A, while the boundary conditions that had to be implemented are discussed in Annex B.
3.1 Selection of a CFD software
There are two options when it comes to transitioning from a mathematical model to a numerical
code from which a solution can be obtained: either a software package is found that already imple-
ments the defined model or one is programmed from scratch. The second approach is not ideal given
the time constraints of this project and the fact that software to solve the Navier-Stokes equations is
rather common.
Inside the premade software packages one can still distinguish between proprietary and open
source software. The first functions as a black box. The user never actually sees the code imple-
mented by the numerical solver and has to blindly trust the developers of the code to have correctly
implemented the mathematical model. Additionally, with proprietary software, if a certain feature or a
small detail of the model is not implemented according to the needs of the project there is no possi-
25
bility for the user to implement it themselves. To overcome such a problem, it is necessary to request
for the necessary feature to be implemented by the developers of the software, which can take a
prohibitive amount of time.
The alternative is using open source software. With open source code, it is possible to make
adjustments in the code as necessary for the current project. This however requires some technical
expertise, knowledge of programming and an understanding of the code. The downside of open
source software is that there is no liability from the developers to fix their code if something is amiss.
The user is expected to fix their own problems and if possible share the solution to become part of
the software. Another big advantage of open source software is the fact that it is free to use. This
contrasts with proprietary software, which can have prohibitive price tags.
Numeca’s FINE/Open was chosen for the previous work done in this project [19, 29]. FINE/Open is
a proprietary CFD flow Integrated Environment dedicated to complex internal and external flows that
includes all the computational models necessary to model both the solid and fluid domains, including
the treatment of radiation. However, these models are not all compatible at the same time [19].
Initially this project was developed using FINE/Open because it is possible to include in the solver
small modifications to the source code using the Python Application Programming Interface (API)
OpenLabs.
However, during the initial stages of this work it was discovered that FINE/Open is not capable
of applying the correct outlet boundary conditions defined in Section 2.3. The resulting numerical
simulations would have some degree of forced convection in the domain, compromising the final
results. This problem was brought forward to Numeca, who communicated that the implementation of
the correct boundary condition on their software would only be concluded after the time frame set for
this project.
For this reason, a decision was made to change from FINE/Open to OpenFOAM [30]. Choosing
an open source software would eliminate any problems with bad implementations of the mathematical
model since these can be corrected.
OpenFOAM can be seen as a toolbox with many independent solvers from which the user has
the responsibility to choose the most appropriate one for the defined mathematical model. The cht-
MultiRegionSimpleFoam solver was designed for CHT problems and implements all the equations
exhibited in Tables 2.1 and 2.3. In the following section, this solver will be reviewed along with the
numerical implementation of these equations.
For this work, OpenFOAM ’s version 4.0, the most recent stable release of OpenFOAM at the date
of this report, October 2016, was used.
3.2 Overview of chtMultiRegionSimpleFoam
The selected application, chtMultiRegionSimpleFoam, is a steady state solver for CHT problems.
The generalised algorithm from chtMultiRegionSimpleFoam is shown in Figure 3.1.
chtMultiRegionSimpleFoam is a combination of two different independent solvers, heatConduc-
26
Start
Initialisation
i=1
1. Solve momentum equation
2. Solve energy equation
3. Update thermophysical models
4. Pressure corrector
5. Momentum corrector
i = Nfluid ?
j=1
1. Solve energy equation
j = Nsolid ?
Converged?
Stop
yes
yes
yes
no
no
no
Update variables
i = i +1
j = j + 1
Fluid Domains
Solid Domains
Figure 3.1: chtMultiRegionSimpleFoam algorithm.
tionFoam and buoyantSimpleFoam. This can be observed in the algorithm, where each domain is
solved completely independently from the others. In fact, the only interaction between the domains
happens through boundary conditions of the energy equations.
Before the iterative process starts, the solver performs an initialisation step for all the domains.
This step starts with importing the domain mesh. The process of meshing, through which a domain
is discretised into a mesh, is crucial in order to correctly capture the development of the flow in the
fluid domain, namely in the boundary layer section as discussed in Section 2.3, or the temperature
27
gradients inside a solid domain. Since the meshing process is very dependent of the domain be-
ing modelled, this topic is diverted to Section 4.2 where the meshing process for each of the three
domains studied in this work will be explained in detail.
After importing a mesh, the initialisation variables are mapped into the domain. The default solver
does not calculate the pressure in the domain as described in Equation (2.29); as such, this equation
was implemented for this work.
From the initialisation fields, all the thermophysical properties of the flow are calculated using the
explicit equations from Table 2.1. A listing of the models that implement these equations in Open-
FOAM is given in Table 3.1. For the solid domain, an equivalent model list must be provided, given in
Table 3.2.
Table 3.1: Thermodynamics packages choice for the fluid domain.
Option Model Description
type heRhoThermo Energy for a mixture based on density.
mixture pureMixture Defines a pure mixture (M = 28.96).
transport sutherland Transport package using Sutherland’s formula.
Thermo hConst Constant properties thermodynamics package
equationOfState perfectGas Perfect gas equation of state.
specie specie Base class of the thermophysical property types.
energy sensibleEnthalpy Defines sensible enthalpy as the standard enthalpy function.
Table 3.2: Thermodynamics packages choice for the solid domain.
Option Model Description
type heSolidThermo Energy for a solid mixture.
mixture pureMixture Defines a pure mixture.
transport constIso Constant thermal conductivity.
Thermo hConst Constant properties thermodynamics package.
equationOfState rhoConst Constant density.
specie specie Base class of the thermophysical property types.
energy sensibleEnthalpy Defines sensible enthalpy as the standard enthalpy function.
Additionally, a new variable is calculated for the fluid domains. The flux, φ, is defined in Equa-
tion (3.1) for an arbitrary surface, s. The flux is a surface scalar variable, which means that for a given
control volume, only the value of the flux normal to the control surfaces holds physical meaning.
φ =
∫s
ρ~Uds (3.1)
The momentum and energy equations can be rewritten considering the flux, φ as shown in Equa-
tions 3.2 and 3.3. When solving these equations, the flux is considered as a fixed variable, linearising
28
these equations for a single unknown variable, ~U and T respectively [31].
∇ · (φ~U) = ∇ · τ −∇pρgh − (~g~x)∇ρ (3.2)
∇ · (φH) = ∇ · (αρ∇H) + ρ~g~U +QS (3.3)
Finally, the PDE will also be discretised using the domain mesh. This step will convert the lin-
earised PDEs into a system of linear algebraic equations. In OpenFOAM this is done according to
the FVM [9, 30]. As mentioned, the FVM will not be explained in this text, however it is convenient to
explain the resulting system of equations produced by this method. Generically, a linearised PDE can
be written as a function of a generic variable, Φ as shown in Equation (3.4).
F (Φ) = b (3.4)
The FVM will convert this equation into an algebraic system of equations which can be written in
the matrix form of Equation (3.5). This process is done using numerical schemes that convert the
differential terms, such as the gradient operator, ∇·, into simple algebraic operations using only infor-
mation from the variable at an arbitrary cell, Φp and the values from the neighbour cells, Φn=1,2,...N
[M ]Ψ = b (3.5)
As such, for a given cell, p, an equation from the system can be written as shown in Equation (3.6).
The terms ap and an represent the relative weights for the cell p and the neighbour cells, n = 1, 2, ..N ,
respectively. These weights come from the discretisation procedure and depend on the numerical
schemes used. From an algebraic point of view, it is is convenient to notice that ap terms will figure
on the diagonal of the matrix, [M ], while the an terms will appear outside the diagonal.
apΨp −∑n
anΨn = bp ⇔ apΨp +H(Ψp) = bp (3.6)
Function H(Ψp) represents the matrix coefficients of the contributions of neighbour cells to the
equation.
This concludes the initialisation step after which the solver has a discretised grid of the domain,
an initial field for all the variables in the domain and the PDE sytem that will be solved written as a
system of linear algebraic equations. The iterative process can now begin.
The solver first iterates through all the fluid domains. For each fluid domain, there are three
equations that need to be solved implicitly, as discussed in Section 2.1. These equations can’t be
solved independently from each other since all three share common variables. The solution must
therefore be found iteratively, where the solution from one equation is used to solve the next and
so on until an acceptable solution is found. Fortunately, this iterative process has been optimised
through the usage of algorithms. In the case of the chosen solver, OpenFOAM ’s Semi-Implicit Method
for Pressure Linked Equations (SIMPLE) algorithm, named after the method used to calculate the
pressure in the domain, is used [32].
29
The first step of the iterative process for the fluid domain calculates the velocity field based solely
on solving the momentum equation. This means that this velocity field will not necessarily respect the
equation of mass conservation, Equation (2.4). For this reason this step is often called the momentum
predictor step, as the velocity field obtained is only temporary.
The second step solves the energy equation to obtain the new temperature field of the domain.
Subsequently, the third step updates the thermophysical properties of the fluid. In this step, in addition
to all the explicit variables from Table 2.1, the radiation model is also updated.
The fourth step is known as the pressure corrector step. The equation solved in this step can be
derived by combining the semi discretised form of the momentum equation with the mass balance
equation. The resulting pressure equation, Equation (3.7), can be considered as a Poisson equation
for the pressure for a given velocity field [9].
∇ ·(ρ
ap∇pρgh
)= ∇ ·
(ρ
apH(~U)
)−∇ ·
(ρ
ap(~g.~x)∇ρ
)(3.7)
Once Equation (3.7) is solved, the next step is to correct the velocity field in order to guarantee
consistency with the mass flow equation. The momentum corrector step implements equations 3.8
and 3.9.
~U =1
apH(~U)− 1
ap(∇pρgh + (~g~x)∇ρ) (3.8)
φ =
∫s
ρ
apH(~U)ds−
∫s
ρ
ap(∇pρgh + (~g~x)∇ρ)ds (3.9)
This concludes one step of the iterative process for one fluid domain. This sequence is repeated
for all the existing fluid domains.
After all the fluid domains have been iterated, the solver will cycle through all the solid domains.
For each domain, the Laplacian Equation (2.18) for the heat diffusion in a solid domain is solved.
Finally, after all the solid and fluid domains are computed, the convergence of the solution is anal-
ysed and, if the convergence criteria is reached, the iterative procedure is stopped. The convergence
criteria is discussed in Section 3.5.
3.3 Computational implementation of the radiation models
As discussed in Section 2.5, the radiation effects are included in the energy equation through
a source term. OpenFOAM natively has in its code three different models implemented to ac-
count for radiation effects within a domain: the P1 model, the Finite Volume Discrete Ordinates
Method (FVDOM) and the view factor model. In addition to these models, a reference model will
be established with the intent of providing a benchmark from which it is possible to compare the
different models objectively.
With the exception of the reference model, which is implemented as a boundary condition, all
the other models are updated in the ”Update themophysical models” step of OpenFOAM ’ SIMPLE
algorithm.
30
3.3.1 Reference model
For the case of a single diffuse surface at a fixed temperature, Ts, surrounded by a non partic-
ipating media and losing heat to the infinity, the radiative energy dissipation, qr, can be calculated
according to Stefan-Boltzmann Law expressed in Equation (3.10). The surface emissivity, ε, is a
material property, σ is the Stefan-Boltzmann constant, and T∞ the temperature at infinity.
qr = σε(T 4s − T 4
∞)As (3.10)
This radiative flux can then be included in the heat boundary condition for a heated wall, removing
the energy by radiation from the system. This simplified model is valid for very simple cases where
only one surface is heated and is losing heat by radiation to infinity exclusively. However, provided a
valid domain is used, this model should accurately predict the heat dissipation by radiation.
This model is not natively implemented in OpenFOAM and as such was implemented in the form
of a boundary condition. OpenFOAM ’s externalWallHeatFluxTemperature was edited to include an
option to accommodate it. The modified boundary condition was baptised radiatingWall.
3.3.2 P1 Model
The P1 model is a direct implementation of the mathematical model discussed in Section 2.5.2.
The transport equation, Equation (2.31), is discretised using the same mesh as the one used to
solve the Navier-Stokes equations. When a non-participating media is considered, this Equation
can be rewritten as Equation (3.11). The physical interpretation of this equation is that the incident
radiation is treated as a scalar being diffused with a infinitely large diffusivity. This however is not a
good approach from a computational point of view since, numerically, infinity is a concept that does
not exist. To avoid numeric problems it is preferable to define the domain as having a very small
absorptivity, a = 10−6, to obtain the desired effect without risking an overflow.
∇(∞∇G) = 0 (3.11)
The boundary condition for this model, defined in Equation (2.32), is implemented in OpenFOAM
under the name MarshakRadiation. This condition allows for an emissivity, ε, to be defined at every
surface.
The main advantage of this method is its simplicity: only one Laplace equation is added to the
system to calculate the radiative heat flux in the domain. Its main limitation is its tendency to overpre-
dict the radiative heat dissipation where sources are located. This translates to a high possibility of
an overprediction of the heat dissipation by radiation in a heated surface.
3.3.3 Finite volume discrete ordinates method model
The Finite Volume Discrete Ordinates Method (FVDOM) is a Finite Volume Method (FVM) imple-
mentation of the discrete ordinates method from Section 2.5.3. Equation (2.34) is spatially discretised
31
using the FVM for the same mesh used to solve the Navier-Stokes equations. Additionally, each oc-
tant of the angular space 4π is discretised in a finite number of solid angles, Nθ and Nφ. The angular
coordinate system is shown in Figure 3.2. Due to this discretisation, a total of 4×Nθ ×Nφ equations
must be solved in this model.
Figure 3.2: Angular coordinate system.
The boundary condition for this model, defined in Equation (??) , is implemented in OpenFOAM
under the name greyDiffusiveRadiation;. This condition allows for an emissivity, ε, to be defined at
every surface.
The implementation of this model was also done considering a participating medium, however,
contrary to the P1 model, this model is compatible with a non-absorbing, non-scattering media, pro-
vided that the angular discretisation is fine enough. It is important to note that at least one previous
work obtained good results using this model [12].
This model represents a significant increase in the computational load of the solver. The equation
added to the system is a complex PDE that needs to be solved for, at the very least, 4 different
directions. For this reason, selecting the minimum amount of angular discretisations for which an
acceptable solution is obtained is preferable .
To compensate for the high computational demand, it is possible to reduce the frequency at which
the radiation contribution is calculated. OpenFOAM ’s parameter solverFreq allows to define how
many iterations of the SIMPLE algorithm are performed before the radiation contribution is recalcu-
lated. A value of 10 was used to mitigate the computational load imposed by this model.
3.3.4 View Factor model
The view factor model solves the matrix system from Equation (3.12) derived from Equation (2.39).
1ε1
(1− 1
ε1
)FA1→A2
. . .(
1− 1ε1
)FA1→Aj
(1− 1
ε2
)FA2→A1
1ε2
. . .(
1− 1ε2
)FA2→Aj
......
. . ....
(1− 1
εi
)FAi→A1
(1− 1
εi
)FAi→A2 . . . 1
εi
q1
q2
...
qi
=
σT 41
σT 42
...
σT 4j
(3.12)
32
The domain solved in this model is completely independent from the mesh used for the Navier-
Stokes equations. The dimension of the matrix from Equation (3.12) is jxj, where j is the number
of radiating surfaces considered. This means that this matrix’s actual dimensions are usually smaller
than the dimensions of the matrices used to solve the Navier-Stokes equations. The result is a much
faster computation of the radiative heat transfer between surfaces compared to the methods derived
from the RTE. The downside of this approach is the necessity of computing all the view factors before
the simulation, which is a time-consuming procedure.
This model includes a boundary condition, greyDiffusiveRadiationViewFactor, allowing for the in-
clusion of an emissivity, ε, for each surface.
The main limitation of this model comes from its implementation within OpenFOAM. The model
is not currently compatible with symmetry boundary conditions. This results in the necessity of a
complete three-dimensional mesh of the domain in order to be able to use this model, significantly
increasing the computation time.
3.4 Boundary Conditons
A summary of the numerical implementation of the boundary conditions discussed in Section 2.3
for the fluid domain is given in Table 3.3. Most of these conditions are direct implementations of the
boundary conditions summarised in Section 2.2.
Table 3.3: Numerical boundary conditions for the fluid domain.
T ~U pρgh p
Inlet totalTemperature pressureInletVelocity prghTotalPressure Calculated
Outlet inletOutlet inletOutlet outletPrghPressure Calculated
Unheated wall zeroGradient noSlip fixedFluxPressure Calculated
Heated wall radiatingWall noSlip fixedFluxPressure Calculated
CHT wallturbulentTemperature
RadCoupledMixednoSlip fixedFluxPressure Calculated
Symmetry symmetryPlane
Wedge wedge
The boundary conditions prghTotalPressure, totalTemperature and pressureInletVelocity com-
bined implement the total temperature and pressure inlet boundary condition. After the momentum
corrector step, the inlet velocity is updated through Equation (3.13), while the temperature and modi-
fied pressure are calculated trough the total pressure and total temperature conditions assuming the
velocity as a constant.
~Ubf =
(φ
ρA
)bf
· ~nbf (3.13)
The boundary condition, outletPrghPressure, which implements Equation (2.28), was not natively
available in OpenFOAM. As such, it was implemented for this work. However, this boundary condition
33
is prone to instability. After convergence, the modified pressure gradient at the outlet is expected to be
a negative value since the density gradient is a negative value. However, if during the convergence
process of the simulation flow reversal occurs, the density gradient may become positive and the
solution will diverge. Equation (3.14) shows how to derive an alternate form of this condition based on
the ideal gas law, Equation (2.8), the fact that the temperature gradient at the outlet is zero, and the
buoyancy term from the unmodified momentum equation, Equation (2.12), when the velocity gradient
is zero.
∇Pρgh = −(~g~x)∇ρ = −(~g~x)∇(Mp
RT
)since ρ =
(Mp
RT
)= −(~g~x)
(M
RT
)∇p since ∇T = 0 (3.14)
= −(~g~x)
(M
RT
)(~gρ) since ∇p = (~gρ)
The resulting expression is independent of the density gradient and only dependent on system
variables or constants. This implies that the pressure gradient will always have the correct sign even
if the developed flow is non physical, reducing the instabilities of this boundary condition.
Additionally, the boundary condition described in Section 3.3.1 for the calculation of the radiative
heat dissipation in a diffuse surface was implemented under the name of radiatingWall.
The numerical condition inletOutlet switches between a Von Neuman and a Direchlet boundary
condition depending on the flux, as shown in Equation (3.15). This allows to better control the solution
development if punctually, the outlet of the domain becomes an inlet by providing physically plausible
inlet values. Note that the flux, φ, through a boundary face, bf , is positive when mass is leaving the
control volume through that face.
∇(Ψ)bf = 0 if φbf > 0Ψbf = Ψfixed if φbf < 0
(3.15)
The boundary condition turbulentTemperatureRadCoupledFixed is a special boundary condition
that applies Equation (2.23) and 2.24 between domains. It is also able to account for radiative emis-
sion and absorption at the wall.
The condition fixedFluxPressure sets the modified pressure gradient to the provided value such
that the flux on the boundary is that specified by the velocity boundary condition. This is a more
general implementation of the zero gradient boundary condition for cases where body forces such as
gravity and surface tension are present in the solution equations. The condition adjusts the gradient
accordingly.
Despite not being a system variable, all OpenFOAM solvers expect a pressure field to be given for
the initialisation step. The boundary condition calculated accounts for this by translating the modified
pressure boundary condition to its pressure equivalent.
For the solid domain, boundary conditions must be given for the temperature, T , and pressure, p.
The need for a pressure field is the same as the one in the fluid domain. The pressure field is not
34
used at all, so the values used are completely indifferent, but if not supplied the solver will crash. The
applied boundary conditions are given in Table 3.4
Table 3.4: Numerical boundary conditions for the solid domain.
T p
CHT wallturbulentTemperature
RadCoupledMixedFixedValue
Symmetry symmetryPlane symmetryPlane
Wedge wedge wedge
3.5 Convergence and solution control
Due to the nature of iterative processes, the solution is obtained through convergence of variables
obtained at every iteration. Solution convergence is achieved when for all variables defined in Sec-
tion 2.1 and 2.2 the value obtained for every point of the domain does not change from one iteration
to the other. Mathematically, this condition is written as shown in Equation (3.16) for an arbitrary
variable, Ψ [33].
Ψn+1 = Ψn = Ψ (3.16)
The convergence condition is however an ideal condition that is impossible to reach using numeri-
cal methods. The machine being used to solve the linear algebraic system of equations has a limited
numerical precision. In the case of OpenFOAM, double precision is used. This limit the capacity to
obtain a solution where the difference between two consecutive iterations to a theoritical minimum
of 10−16. To monitor convergence, a residual of the equation is evaluated by substituting the cur-
rent solution into the equation and taking the magnitude of the difference between the left and right
hand sides. The residual is also normalised to make it independent of the scale of the problem being
analysed [30].
In the case of natural convective flows, a residual of 10−3 is usually deemed acceptable to con-
sider the solution as converged. In a simulation similar to the one being performed in this work, a
convergence rate of the order of 10−4 for the energy equation was reached while the other quantities
only reached a convergence of the order of 10−2 [13].
In this work the initial target will be to converge the residuals of the system variables to an order
of 10−6.
Additionally, in order to quickly assess that the final results obtained are physically plausible, and
to better visualise the progress of the simulation, it is convenient to define control variables. These
control variables should be easily calculated values, so as to not induce a significant computational
load in the solver.
It was decided to monitor the maximum, average, and minimum velocity in the axial direction, ~Uz,
pressure, p, modified pressure, pρgh, and temperature, T . The maximum temperature in the domain
35
is an especially interesting control variable, since it allows to determine if the maximum permissible
temperature in a LOCA is reached without any further post-processing. Controlling the pressure
and modified pressure values for non-physical values allows for a quick determination of diverging
simulations.
Complementing the aforementioned variables, a calculation of mass and energy flow was per-
formed at every iteration. Controlling the mass flow at the inlet and the outlet is used to guarantee
that, in global terms, the mass continuity principle, shown in Equation (2.4), is respected. The mass
flow for a given boundary face, bf , is calculated as shown in Equation (3.17).
Controlling the energy balance, as shown in Equation (3.18), allows to guarantee that the energy
being added at the wall is being dissipated by the flow and that Equation (2.6) is respected.
mbf =
∫bf
φds (3.17)
∆Qtotal =
∫outlet
φHds−∫inlet
φHds (3.18)
Due to the assumption of a steady state solution, it is expected for all the aforementioned variables
to reach a stable, steady value when the convergence criterion is reached.
3.6 Numerical Schemes
In the discretised form of the PDEs from the mathematical model, the values of the variables
being studied are stored either on the control volume centre or corners. As such, when information is
necessary at a point where the variable is not implicitly stored it must be interpolated using the known
values. This is done through numerical schemes. Numerical schemes are also used to calculate
Laplacian, gradient, and divergence operators present in the PDE system using the variables stored
in discrete points [9].
The correct choice of numerical schemes is not a trivial problem and has direct consequences on
the convergence of the numerical solution. The main criteria in the choice of numerical schemes was
to use second order accurate schemes. The final list of numerical schemes used for the fluid domain
is presented in Table 3.5. The nomenclature used for the schemes and parameters is the same as
used in OpenFOAM ’s fvSchemes files [30]. The mathematical formulation of the schemes is written
for a generic unidimensional, equally spaced domain and for an arbitrary quantity, Ψ, as schematised
in Figure 3.3.
Ψi−1 Ψi Ψi+1Ψi− 1
2Ψi+1
2
φ
∆x
Figure 3.3: FVM generic descritization for a 1D domain.
In the fluid domain, the flow is expected to develop exclusively from the inlet to the outlet, without
any return flow. For these reasons, the linearupwind scheme was used for the divergence operator,
36
Table 3.5: Numerical discretization schemes for the fluid domain.
Operator Scheme Order Mathematical formulation
GradientGauss
upwind phi2nd Order
∇Ψ = Ψi−Ψi−1
∆x for φ > 0
∇Ψ = Ψi+1−Ψi∆x for φ < 0
Surface Normal
GradientOrthogonal 2nd Order
∇Ψ = Ψi−Ψi−1
∆x for φ > 0
∇Ψ = Ψi+1−Ψi∆x for φ < 0
InterpolationlinearUpwind phi
Gauss upwind phi2nd Order
Ψi+ 12
= Ψi + ∆x2 ∇Ψ for φ > 0
Ψi+ 12
= Ψi+1 + ∆x2 ∇Ψi+1 for φ < 0
LaplacianGauss linear
Orthogonal2nd Order ∇2 ·Ψi = Ψi+1−2∗Ψi+Ψi−1
∆x2
Divergence
default
Gauss
linearUpwind Phi2nd Order
∇ ·Ψi = 3Ψi−4Ψi−1+Ψi−2
2∆x for φ > 0
∇ ·Ψi = 3Ψi−4Ψi+1+Ψi+2
2∆x for φ < 0
Divergence
(∇τ )Gauss Linear 2nd Order ∇ ·Ψi = Ψi+1−2Ψi+Ψi−1
2∆x
with the exception of the viscous stress tensor term, where a linear scheme must be used. For the
same reason, a linear orthogonal scheme is also used for the Laplacian terms. The usage of orthog-
onal schemes is only possible if the generated mesh is completely orthogonal, otherwise corrected
schemes must be used.
The solid domain contains only one equation of an elliptic nature, as such linear second order
schemes were used for the discretisation of this domain. The scheme choice for the solid domain is
presented in Table 3.6.
Table 3.6: Numerical discretisation schemes for the solid domain.
Operator Scheme Order Mathematical formulation
Gradient Gauss Linear 2nd Order ∇Ψ = Ψi+1−2Ψi+Ψi−1
2
Interpolation linear 2nd Order Ψi+ 12
= Ψi+1−Ψi−1
2
LaplacianGauss linear
Orthogonal2nd Order ∇2 ·Ψi =, Ψi+1−2∗Ψi+Ψi−1
∆x2
Divergence Gauss linear 2nd Order ∇ ·Ψi = Ψi+1−2Ψi+Ψi−1
2∆x
3.7 Linear solvers
The discretised form of the equations of the mathematical model that are solved in the algorithm
of Figure 3.1 are written as a system of linear equations. These systems are generically written in
37
Equation (3.5) for an arbitrary variable, Ψ. Each of these systems could be solved directly, however
this is a prohibitively expensive computational procedure. As such, linear solvers are used to solve
these matrices iteratively.
In OpenFOAM the only criteria for choosing an appropriate linear solver is the type of matrix
being solved [34]. The iteration of these procedures has three stop criteria: an absolute tolerance, a
relative tolerance, and a maximum number of iterations. The absolute tolerance defines the maximum
precision sought in relation to the initial residual. It is convenient to have this value a few orders
below the residual convergence target so that if one variable converges faster than the others, the
iterative procedure can proceed without being artificially dampened. The relative tolerance dictates
the maximum convergence to be sought by the iterative process of the solver. A very tight relative
tolerance will increase the time per iteration without significantly reducing the total number of iterations
that will have to be performed. Finally, a maximum number iterations is given to avoid having a linear
solver iterating indefinitely. This last value was left to the default of 1000 for all the solvers.
For the fluid domain, the (Generalised) Geometric-Algebraic Multi-Grid (GAMG) linear solver was
used for the modified pressure equation, the radiative flux equation of the FVDOM model and the
incident radiation equation of the P1 model. GAMG generates a quick solution on a mesh with a
small number of cells and maps this solution onto a finer mesh using it as an initial guess to obtain an
accurate solution. GAMG is faster than alternative methods when the increase in speed by solving
first a coarser mesh outweighs the additional costs of mesh refinements.
For the modified pressure and the incident radiation equations, a DICGaussSeidel smoother was
used with an absolute tolerance of 10−6 and a relative tolerance (maximum convergence sought in a
single iteration) of 10−3.
For the radiative heat flux equation, a symGaussSeidel smoother was used with an absolute
tolerance of 10−4 and a relative tolerance of 0.1. Additionally, a limit of 5 iterations was imposed.
In the case of the energy and momentum predictor equations in the fluid domain, the Preconditioned
bi-conjugate gradient (PBiCG) linear solver was used with a Diagonal Incomplete LU (DILU) pre-
conditioner. PBiCG is a Krylov space solver for asymmetric matrices. An absolute tolerance of 10−8
and a relative tolerance of 10−3 were used.
For the solid domain, the Preconditioned conjugate gradient (PCG) linear solver is used combined
with a Diagonal Incomplete Cholesky (DIC) pre-conditioner. PCG is also a Krylov subspace solver
but for symmetric matrices. An absolute tolerance of 10−8 and a relative tolerance of 10−3 were used.
3.8 Equation and solution relaxation
The iterative process from Figure 3.1 is not inherently stable. This instability comes from the fact
that a linear system of equations using a discrete and finite number of points is being used to calculate
the solution to a non-linear, continuous system. To promote convergence of the results an under-
relaxation of the system is used to control the change in the variables done in every iteration [35].
The system of linear of equations from Equation (3.5) can be written generically in a discretised
38
form as shown in Equation (3.19).
apΨp −∑n
anΨn = bp (3.19)
The first term on the LHS of the equation represents the diagonal of the matrix while the second
term represents the non diagonal terms. A relaxation factor, αi ∈ [0, 1[, can then be applied giving
origin to Equation (3.20). In this equation, a new field for the the variable, Ψ∗, is calculated from the
value of the previous solution Ψk−1.
apαi
Ψ∗p −∑n
anΨ∗k = bp +1− αiαi
anΨk−1p (3.20)
This method, known as Patankar implicit under relaxation, changes the diagonal of matrix [M ]
and vector b without modifying the equation mathematically. A smaller relaxation factor increases the
diagonal dominance of the algebraic system which increases the stability of the linear solver used [36].
Additionally, an explicit relaxation can be made to the solution obtained from the algebraic system.
Equation (3.20) solution, Ψ∗, can be under-relaxed as shown in Equation (3.21). The relaxation factor,
αe ∈ [0, 1[, is used to weight in the solution of the previous iteration, Ψk−1, with the new solution
calculated by the linear solver.
Ψk = (1− αe)Ψ∗ + αeΨn (3.21)
Each individual system variable for each domain can have its own explicit and implicit relaxation
factors defined. The explicit relaxation for the velocity, ~U , and temperature, T , are not available
natively in the used solver but were added for this work. Generically, a higher value of the relaxation
factors leads to instability and a lower value to an excessive amount of iterations.
39
40
4 Numerical simulations settingIn this chapter the specific numerical simulations performed in this work will be explained.
In previous works the outlet boundary condition for the pressure was not the one from Equa-
tion (2.28) [7, 19]. Instead, a zero gradient von Neuman condition was used. This creates a spurious
flow due to a forced convection enforced at the outlet with a maximum velocity of |~U | ≈ 10−3 ms−1.
The problem with this spurious flow, which in magnitude is not significant, originates from the fact
that the flow develops from the outlet to the inlet. This causes an instability in the inlet and outlet
boundary conditions which results in a bad convergence of the numerical solution. This is a code
verification error since it is an error between the obtained numerical solution and what is predicted by
the mathematical model.
As such, complementary to the validation procedure, a verification exercise will be performed to
verify the correct implementation of the mathematical model.
For the validation procedure, two sets of experimental data were selected. These will be reviewed
in Section 4.1 with the objective of establishing clear parallels with the LOCA scenario.
Section 4.2 will describe the computational domains generated. Three computational domains will
be considered, one domain will be designed for the verification of the code, and two domains, one for
each of the experimental installations, will be used in the validation procedure.
Finally, Section 4.3 will specify the numerical simulations performed for this work and their objec-
tives.
4.1 Experimental data
For the validation process two distinct sets of experimental data were chosen. The first case, a
single heated cylinder through natural convection and radiative heat emission, was chosen as the sim-
plest possible case that resembles the LOCA scenario. The second case concerns a more complex
flow through a bundle of rods.
In the following two subsections each of these cases will be reviewed in detail.
4.1.1 Single cylinder
The experimental installation consists of a 1.5m cylinder with a 3.2mm radius, made of a resistance
consisting of a 80% Nickel-20% Chromium alloy tube, a layer of compressed Magnesium oxide and
an outer tube manufactured in inox steel (321). This cylinder is heated through the Joule effect to
simulate the heat generated inside a nuclear fuel rod. To shield the generated natural convective flow,
the cylinder is placed inside a parallelepipedic transparent box. An entrance section shaped like a
honeycomb is placed at the domain entrance to guarantee an uniform flow at the inlet. The described
domain is schematised in figure 4.1.
This experimental set-up matches the abstracted class of problems where the total LOCA is in-
cluded as established in Section 1.1. A short summary of these conditions is given below:
41
• The problem is a CHT problem since it combines both a solid and fluid domain;
• The heating of the solid domain by the Joule effect can be modelled as a uniform heat source;
• The flow develops along a single rod, this can be perceived as the simplest possible bundle of rods;
• The flow develops in the axial direction along the rod by natural convection;
• Only the laminar sections are considered as regions of interest for the validation.
• The effects of radiation are significant [15];
• The working fluid, air, can be modelled as a non-participating medium;
• The measurements were made after a steady state was reached;
200mm
1500mm
Inlet section withhoneycomb
Figure 4.1: Experimental Installation for the single rod testing [15].
The temperature in the cylinder is measured through infra-red imaging. The results include a plot
of the temperature along the rod for different values of heat dissipation at the rod surface. Since
the focus of this validation is in the laminar flow region, only the curves with a significant amount
of laminar flow through the rod were considered. The considered experimental data is presented in
Figure 4.2. The transition line shows the height of the rod where the flow becomes turbulent. The
present mathematical model does not account for turbulence, therefore the curves obtained through
numerical simulation are only expected to match the laminar part of the flow.
4.1.2 Seven cylinder bundle
The chosen data for this validation consists of the experimental measurement of the laminar
steady-state free convection flow in a open-ended vertical seven cylinder bundle [16]. Each of the
rods, made of stainless steel, is heated by the Joule effect. The outer wall is insulated in order to
42
0 0.25 0.5 0.75 1 1.25 1.5
300
325
350
375
400
425
450
475
500
525
Turbulent regime
z(m)
T(K
)264W/m2 807W/m2
2031W/m2 3636W/m2
transition
Figure 4.2: Temperature profile along the rod surface [15].
avoid, as much as possible, heat losses. To avoid radiative losses, the outer wall is covered with an
aluminium layer. The experimental installation used is schematised in Figure 4.3.
300mm
1200mm
300mm
A A
(a) Side view
54.5mm
32.5mm
12mm
(b) Section A-A
Figure 4.3: Experimental installation of the seven cylinder bundle [16].
Similarly to the single rod experimental installation, an entrance section exists to guarantee that
the flow at the inlet is uniform. An outlet section is used to reduce disturbances from the environment.
This experimental set-up matches the abstracted conceptual model for a LOCA described in Sec-
tion 1.1:
43
• The problem is a CHT problem since it combines both a solid and fluid domain;
• The heating of the solid domain by the Joule effect can be modelled as a uniform heat source;
• The flow develops along a bundle of rods by natural convection;
• The developed flow is laminar throughout the domain.
• Radiation effects were considered by isolating the outer wall for radiative losses;
• The working fluid, air, can be modelled as a non-participating medium;
• The measurements were made after a steady state was reached;
The temperature of a rod is measured using three thermocouples placed 20mm from the inlet,
in the middle of the tube and 20mm from the outlet. Given the symmetry of the problem, only three
tubes were instrumentalised. The velocity at the inlet and outlet of the domain were measured using
hot wire sensors.
The collected data was translated to non-dimensional numbers and an empirical correlation be-
tween the Grashof, Gr, and Nusselt, Nu, numbers was found. The Grashof number, Equation (4.1)
defines the ratio of the buoyancy to viscous force acting on a fluid, where the Nusselt number, Equa-
tion (4.2) is the ratio of convective to conductive heat transfer across a boundary [24].
GrDH =|~g|qsD4
h
kν2T0(4.1)
NuDH =hDh
k(4.2)
The hydraulic diameter, Dh, can be calculated as shown in Equation (4.3), where A is the cross-
sectional area, and P is the wetted perimeter of the cross-section.
Dh =4A
P(4.3)
The heat transfer coefficient, h, is calculated as shown in Equation (4.4) where the average tem-
perature at the cylinder walls, Tw, and the average temperature at the domain, Tb, are used.
h =qs
Tw − Tb(4.4)
The empirical correlation for a bundle of seven cylinders is shown in Equation (4.5). This correla-
tion has a standard deviation of 0.06.
NuDh = 0.115Gr0.287Dh
(4.5)
4.2 Computational domain and boundary conditions
The computational domains generated for this work based on the experimental installations were
designed with the objective of having the least amount of cells as possible to guarantee a faster
convergence process.
44
For both meshes, the main focus was in exploiting as much as possible the symmetry of the do-
mains and guaranteeing the boundary layer developing along significant solid surfaces was properly
captured by the numerical solver. As such, the objective is to have at least ten points in the direction
perpendicular to the wall inside the boundary layers along the entire domain [9].
Other key simplification is modelling neither the inlet nor the outlet sections of the experimental
installations. These sections were omitted from the domains since their effects can be achieved by
imposing the inlet and outlet boundary conditions described in Section 2.3.
The following three sub sections will review in more detail the three computational domains con-
sidered and the meshes used in the numerical simulations performed in this work.
4.2.1 Single cylinder
To achieve symmetry in the domain the outer wall was approximated as a cylindrical surface. Due
to its sufficiently large distance to the heated rod, this surface does not influence the flow being stud-
ied. In order to maintain the surface area of the inlet and outlet sections, the radius of the outer wall
was calculated as R = 113mm. The resulting domain is axisymmetric and is sketched in Figure 4.4.
3.2mm 109.8mm
Inlet
Outlet
Outer wallWedgeWedge
Wed
geW
edge
Bottom
Top
Solid domain Fluid domain
5
15000mm
x
z
y
Figure 4.4: Single cylinder computational domain.
The chosen strategy to mesh the fluid domain was to have a mesh expanding from the interface
with the solid domain. In the proximity of the heated cylinder, it is vital to correctly capture the devel-
opment of the boundary layer. Given that the working fluid, air, has a Prandtl number of Pr = 0.7,
the thermal boundary layer is expected to be thicker than the momentum boundary layer. This means
45
that a point density that correctly captures the momentum boundary layer will also correctly capture
the thermal boundary layer.
The solution of the momentum boundary layer equations for the flow along a slender cylinder was
not found in the literature reviewed. Therefore, the boundary layer development over a flat wall was
considered for the estimation of the boundary layer thickness.
In order to calculate the mesh density, the domain was arbitrarily defined to have 100 cells in the
axial direction, z. Only by doing a grid refinement study study is it possible to evaluate if this is enough
points to correctly calculate the boundary layer development near the inlet of the domain. This grid
refinement study is shown in Section 5.4, but for now it is convenient to know that this point density
will be sufficient.
Since the boundary layer is at its minimum thickness near the inlet, the following calculations were
made for the first point inside the domain, z = 0.015m. The numerical simulations performed in
previous works showed that the maximum velocity in the boundary layer varies from |~U | = 0.5m s−1
for qs = 264Wm−2 to |~U | = 1.5m s−1 for qs = 3636Wm−2 [7]. Since the boundary layer thickness is
inversely proportional to the flow velocity, a velocity of |~U | = 1.5m s−1 was considered.
The boundary layer for a flat plate relates to the Reynolds number, Re, defined in Equation (4.6),
according to Equation (4.7) [26]. Air at PTN conditions has a viscosity, ν = 17.55× 10−6 m2 s−1 which
results in Rez = 1282 for an outer flow of |~U | = 1.5m s−1 at z = ∆z. The resulting boundary layer
thickness is (δx)z=0.015m = 0.002m.
Rez =|~U |zν
(4.6)
δxz
= 4.92Re−1/2z (4.7)
The mesh cell size along the radial direction of the cylinder was defined to increase at a geometric
rate of 1.1. Considering this, Equation (4.8) shows how to calculate the height of the smallest cell in
the domain, ∆x0.
δx = ∆x0
9∑k=0
1.1k ⇔ ∆x0 = 0.0001m (4.8)
Therefore, the minimum cell size should be ∆x0 = 0.0001m combined with a geometric growth
ratio of 1.1. The tool used to generate the mesh, blockMesh, takes as an input for the grading algorithm
the number of cells and the total expansion ratio. 50 cells over the radial direction combined with a
total expansion rate of 107 were used to achieve the desired meshing definition.
The equation that describes the solid domain, Equation (2.18), is expected to have a parabolic
solution. For a section where the axial variations of temperature are negligible, the temperature
inside of a rod with constant conductivity, k, is expected to be described by Equation (4.9), where c1
and c2 are undetermined constants.
T (x) =Qs2kx2 + c1x+ c2 x ∈ [0, r] (4.9)
46
It is therefore interesting to have a higher mesh definition near the interface with the fluid domain,
where higher temperature variations are expected, and a lower mesh definition near the centre of the
rod. A first guess of 10 points in the radial direction was used to discretise the solid domain and the
same grading as the fluid mesh was used.
A detail of the final mesh used in this Chapter is shown in Figure 4.5. The most relevant mesh
quality parameters are shown in Table 4.1. The high values of aspect ratio come from the cells
in the interface between the domains having a large ∆z for a relatively small ∆x. This values are
still considered as OK by OpenFOAM ’s utility CheckMESH, but should be improved in case of bad
numerical results. The fact that the mesh is completely orthogonal means the orthogonal schemes
discussed in Section 3.6 can be used and orthogonal correctors are not needed for the SIMPLE
algorithm.
Figure 4.5: Overview of the top section of the Mesh for the single Cylinder case.
Table 4.1: Mesh quality parameters.
Cells Max aspect ratio Max non-orthogonality Max Skewness
Solid900 Hexahedra
100 Prism439.19898 0 0.33079647
Fluid 5000 Hexahedra 161.80449 0 0.10513444
The boundary conditions used in the domain are the ones discussed in Section 2.3 and 3.4 with
the outer wall of the domain being considered as an unheated wall and the interface between the solid
and fluid domain as a CHT wall. A total temperature of T0 = 293.7K and total pressure p0 = 101 325Pa
were imposed at the inlet.
For the properties of air , the Sutherland coefficients, C1 = 1.458× 10−5 kg/(msK1/2) and C2 =
110.4K, a molar mass of M = 28.96 kgmol−1 and a specific heat capacity of cp = 1006 J kg−1 K−1
were used.
The heated rod of the experimental installation was fabricated in order to be a diffuse surface with
an emission coefficient, ε = 0.96. The surface of the outer wall is transparent to radiation, including
infra-red radiation. For this reason, the outer wall, the inlet, and the outlet were modelled as diffuse
surfaces with an emission coefficient of ε = 1. This approximates the background radiation from the
room at the reference temperature.
Since no specific dimensions on the thickness of each layer in the heated cylinder, it was opted
47
to approximate the solid domain as being a uniform 80% Nickel-20% Chromium alloy. As such, a
density ofρ = 8300 kgm−3, a thermal conductivity of k = 15Wm−1 K−1 and a specific heat capacity of
cp = 480 J kg−1 K−1 were used.
The experimental data does not provide the values for the volumetric heat source in the rod, but
rather the heat dissipated per surface area. The conversion from a heat flux, qs to the source term
values, Qs, to be imposed in the solid domain is done using Equation (4.10), where r is the rod radius.
Qs =2qsr
(4.10)
4.2.2 Seven cylinder bundle
In this set of numerical simulations, the objective is to obtain a correlation similar to Equation (4.5).
It is technically possible to use any dimensions for the computational model of the seven cylinder
bundle, since the numerical results will be converted to dimensionless numbers. As such, the height
of the domain was changed to 1000mm but all other dimensions of the bundle were kept the same as
the ones from the experimental installation.
Due to symmetry, it is possible to simplify the domain to a 3D 30 wedge. Figure 4.6 shows the
computational domain, containing only two solid and one fluid sub-domains.
1000
mm
Cen
trecy
linde
r
Sym
met
ry
Rad
ialc
ylin
der
Sym
met
ry
Out
erw
all
Inlet
outlet
(a) Perspective.
Symmetry
6mm
16.25mm
27.25mm
30
c1 r1
r2 r3
r4 o1
o2
(b) Top view.
Figure 4.6: Seven cylinder rod bundle computational domain.
Meshing the fluid domain has once again as the main target achieving a sufficiently fine grid near
the solid surfaces in order to correctly capture the boundary layer development. The main difference
between the approach made in the single cylinder domain is that the boundary layer development in
the outer wall must also be considered. The outer wall is no longer at a sufficiently large distance
from the main flow that its boundary layer development no longer influences the flow being studied.
48
Typically, for this kind of problem, unstructured mesh tools are used. Although an unstructured
mesh usually produces lower quality meshes with an overall excessive amount of cells when com-
pared to a structured mesh, the use of such a tool can save many hours of development. However,
keeping the mesh size as low as possible was considered as a priority in this work. Since a solver
validation is being performed, many iterations of different parameters have to be done until a working
solution is found, and a mesh with fewer cells allows for fast prototyping of new solutions. For this
reason a structured mesh was designed.
To guarantee an equal treatment between all surfaces, mesh guidelines in the form of hexagons
were established around each cylinder. Figure 4.6 shows 7 distinct mesh zones (c1,p1,p2,...). Sec-
tions r2 and r3 are equal and each covers a 60 angle of the cylinder wall. Sections c1, r1 and r4
are a cut in half version of the former due to symmetry of the domain. As such, the meshing method
derived for one of these sections is valid for all.
A schematic of the meshing performed for one section is given in Figure 4.7. The angular direction
is discretised considering ∆θ = 5. This is the same angle used for the wedge in the single cylinder
case and a common angle for the discretisation of simple circular geometries where the flow occurs
axially in the domain.
60
∆θ
graded ∆x
Figure 4.7: Schematic meshing for an arbitrary cylinder surface section.
The discretisation in the direction perpendicular to the wall followed the same methodology as the
one for the single cylinder domain. For this domain however, the Reynolds number was calculated
using Equation (2.1). The used geometry has an hydraulic diameter ofDh = 14.17mm. The maximum
velocity measured experimentally was of |~U | ≈ 0.4m s−1, which results in a Reynolds number of
Re = 313.
It is important to note that the grading inside the domain is not uniform. Due to the geometry of the
section, the discretisation closer to the section edge is finer than the one in the middle of the section.
For this reason, the grading to be used must be designed for the side wall dimension, x = 2.125mm.
In the axial direction, z, a constant grading of ∆z = 10mm was used for both the solid and
fluid domain. Using the same calculations as in the previous subsection, a minimum cell size of
∆x0 = 0.2mm is obtained for the first cell in the domain. To achieve this minimum cell size, combined
49
with a geometric growth ratio of 1.1, a total expansion ratio of 1.95 and 8 cells were used.
In the case of the outer wall, the less refined section will occur in the top wedge of section o2.
Here, a distance of 8.77mm must be meshed, also with a minimum cell size of ∆x0 = 0.2mm and a
geometric growth ratio of 1.1. This results in a total expansion ratio of 5.05 with 18 cells.
For the solid domains the same angular discretisation of the fluid domain boundary layers was
applied. A mesh with 5 cells with a total expansion ration of 8.95 were used to discretise the solid
domain in the radial direction.
The final mesh, viewed from the top, is exhibited in figure 4.8
Figure 4.8: Final Mesh for the 7 cylinder scenario.
To improve the meshing procedure, a small script written in Python was created to generate the
mesh for this case. This script is available in Annex C.
A few limitations of the mesh can be observed straight away. The meshing of section o1 is clearly
excessive. This section, which is not the most critical section of the flow, has the highest point density
of the domain. It can also be observed that the mesh is not orthogonal, implying that orthogonal
schemes can not be used. Finally, some undesirable cell size jumps can be observed between the
interfaces of section o2 and p3.
The results obtained from the utility checkMesh are given in Table 4.2.
Table 4.2: Mesh quality parameters.
Cells Max aspect ratio Max non-orthogonality Max Skewness
Centre
Cylinder
2400 Hexahedra
600 Prism123.70941 0.0000157 0.330796
Radial
Cylinder
14400 Hexahedra
3600 Prism123.70941 0.0001706 0.330796
Fluid 66000 hexahedra 183.16804 28.686408 1.343365
The boundary conditions used in the domain are the ones discussed in Section 2.3 and 3.4 with
the outer wall of the domain being considered as an unheated wall and the interface between the solid
and fluid domain as a CHT wall. A total temperature of T0 = 300K and total pressure p0 = 101 325Pa
were imposed at the inlet. The solid surfaces of the cylinders were considered as having an emissivity
50
of ε = 1, and the outer wall an emissivity of ε = 0.1 since the outer wall is covered with an aluminium
sheet.
The solid and fluid properties are the same as used in the single cylinder domain.
4.2.3 Domain for the code verification
For the verification case the simplest possible conceivable domain was designed so that, if errors
are detected, it can be immediately excluded that the domain is the source of the error. As such, a
parallelepiped fluid domain as shown in Figure 4.9 was used.
0.10m
1.00m
0.01m
Solid wall Solid wall
Inlet
Outlet
SymmetrySymmetry
Figure 4.9: Domain for the verification case.
For this particular exercise a simple discretisation of 100 cells in the x and z directions without any
grading was used. Since for this problem it is intended to calculate a case where no flow develops,
and a case where there is only interest in the global energy balance, there is no necessity to correctly
mesh the boundary layer at the heated surfaces.
Two distinct boundary conditions for the temperature, T , are applied at the solid wall. In a first case
a zero gradient condition is applied. In a second case a heat flux of qs = 100Wm−2 was imposed at
both the solid walls. This amounts to a total energy being supplied to the system of Qtotal = 2W. At
the inlet a total temperature of T0 = 300K and total pressure p0 = 101 325Pa were imposed.
The fluid properties used are the same as the previous two domains.
4.3 Selected test cases
The first set of numerical simulations to be made will use the verification case domain.
The numerical code must be able to reproduce the simplest of flows, where no heat is provided
in the domains’ surfaces. In this case, it is expected for the fluid domain to remain stationary. In a
second case, a pre determined quantity of heat will be supplied at the system walls and the energy
balance from the inlet and outlet will be monitored. The total heat supplied at the heated surfaces
must be equal to the heat balance between the domain inlet and outlet. If the solver obtains the
51
correct numeric solution for these two cases, the problem observed in the previous works can be
considered as no longer existing.
The computational domain for the single cylinder case is the simplest of the two domains designed
for the validation procedure. For this reason, it was chosen to determine which of the radiation models
is better suited for this problem using this domain. A first set of simulations, where only the fluid
domain is considered, was performed with the intent of comparing the radiation models against the
experimental data, as well as against the reference radiation model defined in Section 3.3.1. A total of
4 simulations were performed for each model, one for each imposed heat flux at the cylinder surface
from Figure 4.2. From the results obtained, the most suitable radiation model must be chosen to be
used in the remaining simulations.
Once the most appropriate radiation model was chosen, a complete simulation of the single cylin-
der case domain was performed. These simulations should be compared to the experimental results
and the error of the numerical simulation will be assessed. Following, a study to guarantee the nu-
merical results are not dependent on the grid refinement will be done.
The first simulations on the seven cylinder domain should consider only the fluid domain without
the radiation model. After, the radiation model should be added and finally, the full CHT case should
be considered. Given that the empirical correlation from Equation (4.5) is exponential, at least four
simulations should be performed. A heat flux of qs = 50Wm−2, qs = 100Wm−2, qs = 200Wm−2,
qs = 300Wm−2 are to used. The values of Grashof and Nusselt will be obtained by calculating the
average values of the fluid properties after the numerical solution is obtained. The numerical results
obtained should have an error of no more than the standard deviation from the empirical correlation.
52
5 Results and discussionA summary of the results obtained in this work is presented in this chapter.
The results of the verification exercise are discussed in Section 5.1. Section 5.2 will show a
comparison of the different radiation models from which the most suitable model will be chosen.
The results for the complete domain of the single cylinder case using this model are presented in
Section 5.3, followed by a grid refinement study in section 5.4.
Convergence of the results was not obtained for the cylinder bundle case. Section 5.5 will discuss
possible sources of instability as well as the different approaches attempted to fix the problem.
5.1 Code verification
For the case where no heat is supplied at the walls, the solution for this domain, according to the
mathematical model, is the same as the initialisation defined in Section 2.4.
The result obtained under the present model has a maximum velocity of |~U | ≈ 10−8 ms−1. This
spurious flow can be admitted to originate from the numerical precision of the solver. The modified
pressure, pρgh, and temperature, T , fields remain at the same value as their initialisation fields with
a difference of less than 1%. The difference between the total energy at the inlet and outlet is of an
order of ∆Qtotal =≈ 10−10 W, which is consistent with the fact that no energy is being supplied to the
system at the walls.
The results are however not perfect. It takes around 500 iterations, which in turn require 1000
iterations of the pressure equation alone, to reach this no flow scenario. On the first iteration a velocity
field with maximum velocity of |~U | ≈ 10−5 ms−1 is created. This is not consistent with the initialisation
defined nor the results this simulation eventually reaches. This happens because the initialisation
field does not change the boundary face values for the first iteration. This is a minor problem that can
be fixed by rewriting the initialization for the boundary condition outletPrghPressure. After this change
was made, the no flow conditions were obtained from the first iteration and remained with a maximum
velocity of |~U | ≈ 10−8 ms−1 until the simulation was stopped.
The second numerical simulation performed, where a heat flux of qs = 100Wm−2 was imposed at
both solid walls, resulted in a net energy balance between the inlet and outlet of ∆Qtotal = 1.9971W.
This is a difference of 0.14% relative to the correct solution of ∆Qtotal = 2W.
These short numerical simulations allow to establish a degree of confidence in the numerical
solver that did not exist in previous works. The modified OpenFOAM solver being used in this work
can be considered as verified against the mathematical model for the two simplest scenarios.
5.2 Radiation models analysis
The results for the reference model are presented in Figure 5.1. The obtained numerical solution
for a heat flux qs = 264Wm−2 has a maximum relative error of 1.44% and a maximum absolute error
53
of 4.5K. When compared with the laminar sections only, the maximum absolute error of the four
numerical simulations is of 13.2K for the curve with qs = 2031Wm−2. Globally, the average error is
of 1.11%. The numerical simulations with this model took around 800 iterations to obtain and all the
monitored residuals converged to an order of 10−6.
0 0.25 0.5 0.75 1 1.25 1.5
300
325
350
375
400
425
450
475
500
525
z(m)
T(K
)
Experimental 264Wm−2 Experimental 807Wm−2 Experimental 2031Wm−2 Experimental 3636Wm−2
Numerical 264Wm−2 Numerical 807Wm−2 Numerical 2031Wm−2 Numerical 3636Wm−2
Figure 5.1: Temperature profile along the rod surface, reference model.
The results obtained using the P1 model are presented in Figure 5.2. Globally, this model under-
predicts the rod temperature for all the imposed heat fluxes at the heated wall. The results obtained
once again took an average of 800 iterations and all the monitored residuals converged to an or-
der of 10−6 except for the curve of heat flux qs = 264Wm−2 where it was only possible to obtain a
convergence of an order of 10−5.
As discussed in Subsection 3.3.2, the P1 model has a tendency to overpredict radiative fluxes
from sources. This shortcoming was clearly observed in the numerical results obtained. This error
compromises the fidelity of the model and for this reason, the P1 model should not be used in the
simulation of LOCAs.
The results obtained using the FVDOM model are presented in Figure 5.3. The presented results
were obtained with Nφ = 1 and Nθ = 1 resulting in 4 total discretised solid angles. Once again,
convergence of an order of 10−6 was reached in approximately 800 iterations.
When compared with the reference model a maximum absolute difference of 10.0K exists in the
numerical simulations. In average, the results have a error of 0.11%. When compared with the
experimental results, for a heat flux qs = 264Wm−2, a maximum absolute error of 5.2K exists. For
all the curves, considering only the laminar regions of the flow, the maximum error is of 14.8K for the
curve where a heat flux qs = 3636Wm−2 was applied. Globally, the average error is 1.2%.
The results obtained with the view factor model are presented in Figure 5.4. Although slightly
54
0 0.25 0.5 0.75 1 1.25 1.5
300
325
350
375
400
425
450
475
500
525
z(m)
T(K
)Experimental 264Wm−2 Experimental 807Wm−2 Experimental 2031Wm−2 Experimental 3636Wm−2
Numerical 264Wm−2 Numerical 807Wm−2 Numerical 2031Wm−2 Numerical 3636Wm−2
Figure 5.2: Temperature profile along the rod surface, P1 model.
0 0.25 0.5 0.75 1 1.25 1.5
300
325
350
375
400
425
450
475
500
525
z(m)
T(K
)
Experimental 264Wm−2 Experimental 807Wm−2 Experimental 2031Wm−2 Experimental 3636Wm−2
Numerical 264Wm−2 Numerical 807Wm−2 Numerical 2031Wm−2 Numerical 3636Wm−2
Figure 5.3: Temperature profile along the rod surface, FVDOM model.
more imprecise than the results obtained using the FVDOM, this model still shows an acceptable
match with the experimental data for the heat flux curves of qs = 264Wm−2 and qs = 807Wm−2.
The major limitation of this model is the necessity of using a three-dimensional mesh of the full
domain. To accomplish this, the designed wedge mesh was revolved, resulting in a domain with
100000 cells where the wedge angle had to be increased to 36. This makes obtaining a numerical
solution significantly slower when compared with the 5000 cell domain used in the other models.
55
0 0.25 0.5 0.75 1 1.25 1.5
300
325
350
375
400
425
450
475
500
525
z(m)
T(K
)
Experimental 264Wm−2 Experimental 807Wm−2 Experimental 2031Wm−2 Experimental 3636Wm−2
Numerical 264Wm−2 Numerical 807Wm−2 Numerical 2031Wm−2 Numerical 3636Wm−2
Figure 5.4: Temperature profile along the rod surface, view factor model.
However, this model is faster than either the P1 Model or the FVDOM model if the same domain is
used.
Although the problem is symmetric, the results obtained vary angularly within the rod as shown in
Figure 5.5. The profile presented in Figure 5.4 is for the angle where the maximum temperature is
observed.
0 π2
π 3π2
2π295
300
305
310
315
320
θ(rad)
T(K
)
x = 0.10m x = 1.00m
Figure 5.5: Temperature variation within the rod surface, view factor model.
The view factor model must be discarded mainly due to its faulty implementation in OpenFOAM. In
its current state, the model is limited to 10000 view factors, which forced the revolved wedge to have
a 36 angle, which is far superior than the ideal 5 angle. The results observed in Figure 5.5 clearly
show that this model was not properly verified and that bugs exist in the code. On top of that, the
constraint of not being compatible with symmetry boundary conditions would make this model unfit
for more complex cases.
Given the unsatisfactory performance of the remaining models, the FVDOM model was chosen to
be used for the simulation of LOCA scenarios. This model difference in relation to the reference model
56
is in average 0.1%. This means that, numerically, it is almost impossible to obtain better results, since
the reference model is the most accurate description of the radiative heat dissipation for this problem.
When compared with the experimental data, the error of the numerical results in the cases with
higher heat flux is probably due to influences of the turbulent region downstream. These errors are
however not significant since it is expected that no turbulent region will exist in the domain of LOCA
scenario.
There are however some setbacks to this model. The precision of the model comes at the expense
of a significant increase in computation time, with the average simulation time rising from 25 s in the
reference model to 60 s when the FVDOM model is used. Another disadvantage of this model is the
inclusion of two new discretisation parameters, namely the number of discretised angular rays, Nθ
and Nφ. It is imperative that an independence study is performed in regards to the number of solid
angles used to discretise the angular domain.
5.3 Results for the single cylinder case
In this section the results of the CHT numerical simulations using the FVDOM radiation model are
presented. The typical simulation time is of 108 s for each 2000 iterations in a 3.30GHz CPU with a
memory consumption of 70MB.
There results were obtained using an explicit relaxation of the modified pressure of 0.7 and an
implicit relaxation of the velocity and enthalpy of 0.6 and 0.9 respectively. No relaxation factors were
used for the solid domain.
A comparison between the experimental results and the simulation results is given in Figure 5.6.
0 0.25 0.5 0.75 1 1.25 1.5
300
325
350
375
400
425
450
475
500
525
z(m)
T(K
)
Experimental 264Wm−2 Experimental 807Wm−2 Experimental 2031Wm−2 Experimental 3636Wm−2
Numerical 264Wm−2 Numerical 807Wm−2 Numerical 2031Wm−2 Numerical 3636Wm−2
Figure 5.6: Temperature profile along the rod surface, simulation with CHT.
The results are similar to the ones observed in Figures 5.1 and 5.3. The relative errors for the
57
curve with heat flux qs = 264Wm−2 have an average value of 0.8% with a maximum value of 1.5%
while the absolute errors have an average of 2.6K and a maximum of 4.5K.
Globally, the maximum absolute error occurs in the curve of heat flux qs = 2031Wm−2 with a
value of 11.5K, while the maximum relative error occurs in the curve of heat flux qs = 807Wm−2
with a value of 2.8%. For all the curves the average error is of 1.1%. These errors only consider the
laminar regions of the flow.
These errors represent an improvement of 0.1% in comparison to the numerical results presented
in Figure 5.3. In absolute terms, the maximum error was reduced from 14.6K to 11.5K. This means
that the inclusion of the solid domain to the simulation improved the precision of the model.
Figure 5.7 shows the temperature and velocity profiles at different heights for qs = 264Wm−2. It
illustrates the fact that, as predicted in Section 4.2.1, the thermal boundary layer is thicker than the
velocity boundary layer. The variation of temperature in the radial direction inside the solid domain is
almost insignificant when compared to the variation of temperatures that occurs immediately after the
interface surface.
0 0.01 0.02 0.030
0.2
0.4
x(m)
Uz(m
s−1)
z = 0.25mz = 0.50mz = 0.75m
(a) Velocity profiles.
0 0.01 0.02 0.03
295
300
305
310
315
320
x(m)
T(K
)
z = 0.25mz = 0.5mz = 0.75m
(b) Temperature profiles.
Figure 5.7: Temperature and velocity profiles for different heights for qs = 264Wm−2.
All the residuals converged to an order of 10−6 within 2800 iterations, with the slowest convergence
ratio being observed for the case with heat flux qs = 264Wm−2 and the fastest for the case with a
heat flux qs = 3636Wm−2. Figure 5.8 shows the residual convergence and the progression of control
variables for the qs = 264Wm−2 numerical simulation.
Figure 5.8(a) shows that the slowest converging residue is the enthalpy, h, residue for the solid
domain. The residue of the modified pressure, pρgh, converges to an order of 10−6 and oscillates
between this value and 10−5, since the GAMG linear solver used was set up to a maximum precision
of 10−6. Although the linear solver precision could be increased, the gained precision would come
at the cost of a significant increase of computational time since around 1000 iterations would be
necessary for the computation of the modified pressure at each iteration.
Figure 5.8(b) shows that the temperature, T , in the domain is stabilised at a steady value from
around 400 iterations. At this stage the residual of the enthalpy equation for the fluid domain is
10−4. Since there are no significant radial variations of temperature inside the cylinder, as shown in
Figure 5.7(b), there is an overlap between the maximum temperature at in the solid and fluid domain.
58
0 200 400 600 800 1,000 1,200 1,400 1,600 1,800 2,000 2,200 2,400 2,600 2,800
0
−1
−2
−3
−4
−5
−6
−7
Iterations
Residual(
10n
)
pρgh Uz
hfluid hsolid
(a) Residual evolution
0 200 400 600 800 1,000 1,200 1,400 1,600 1,800 2,000 2,200 2,400 2,600 2,800290
300
310
320
330
340
Iterations
T(K
)
Maximum Centre cylinder Average fluid domain Minimum fluid domain
(b) Temperature evolution.
0 200 400 600 800 1,000 1,200 1,400 1,600 1,800 2,000 2,200 2,400 2,600 2,8000
0.2
0.4
0.6
0.8
Iterations
~ Uz(m
s−1)
Maximum Average
(c) Velocity evolution.
0 200 400 600 800 1,000 1,200 1,400 1,600 1,800 2,000 2,200 2,400 2,600 2,800101323
101324
101325
101326
Iterations
pρgh(Pa)
Maximum Average Minimum
(d) Modified pressure evolution.
Figure 5.8: Residual and control variables progression for qs = 264Wm−2.
59
0 200 400 600 800 1,000 1,200 1,400 1,600 1,800 2,000 2,200 2,400 2,600 2,8000
2
4
Iterations
m(g
s−1)
(a) Mass flow evolution.
0 200 400 600 800 1,000 1,200 1,400 1,600 1,800 2,000 2,200 2,400 2,600 2,8000
2
4
Iterations
∆Q
(W)
(b) Energy balance between the inlet and outlet evolution.
Figure 5.9: Mass flow and energy balance in the domain for qs = 264Wm−2.
For this reason, only the curve for the solid domain temperature is presented.
The velocity control variables take significantly more iterations to stabilise than the temperature
control variables, as show in Figure 5.8(c(. Only from 9000 iterations can these quantities be consid-
ered as converged. At this iteration the residual is of approximately 10−4.
Finally, Figure 5.8(d) shows that the modified pressure, pρgh, control variables have stabilised
around 600 iterations. This occurs when the modified pressure residuals abruptly reach a value of
10−4.
Figure 5.9 shows the mass flow and energy balance in the domain calculated according to Equa-
tion (3.17) and (3.18) respectively. Both these values are calculated considering the full, 360 domain.
The mass flow is the only control variable that is only stabilised from around 18600 iterations.
The value obtained for the energy balance can be compared with the total energy dissipated at
the wall. For a wall heat flux of qs = 264Wm−2, a total of Q = 7.96W is supplied to the domain. This
means that 51% of the heat flux is dissipated through radiation. This testifies for the importance of
correctly modelling the radiative heat transfer in the domain if accurate temperature predictions are
desired.
As a summary, the results obtained validate the mathematical model against the chosen experi-
mental data since the numerical results have an error no larger than 2.8% in the laminar sections of
the domain. The residual convergence analysis shows that a convergence of the system variables
residuals to an order of 10−4 is sufficient to correctly predict rod temperature.
60
5.4 Grid refinement study
To guarantee that the results presented in the previous section are not affected with the grid refine-
ment and that the assumptions made in Section 4.2.1 during the mesh creation process were correct,
a series of numerical simulations using meshes with different refinement levels were performed.
Regarding the refinement in the axial direction, the critical section is the entrance section. An
alternative mesh was designed where, instead of a constant mesh spacing of ∆Z = 0.125m, a graded
mesh refined with 50 points between 0 ≤ X ≤ 0.3m was used. The results obtained for a numerical
simulation where qs = 264Wm−2 are presented in Figure 5.10.
0.000 0.005 0.010 0.015 0.020 0.025 0.030 0.035 0.0400
0.1
0.2
z = 0.3mz = 0.15m
z = 0m
x(m)
|~ Uz|(m
s−1)
Original meshRefined entry section
(a) Velocity.
0.000 0.005 0.010 0.015 0.020 0.025 0.030 0.035 0.040
300
310
320
z = 0.15m
z = 0.3m
x(m)
T(K
)
Original meshRefined entry section
(b) Temperature.
Figure 5.10: Grid refinement study of the entrance section, qs = 264Wm−2.
As can be seen, only at the very inlet of the domain, z = 0m, is there a variation of the velocity
profile between the two meshes. This difference is no longer present at z = 0.15m and beyond.
Conversely, the temperature profiles along the domain do not change with a refinement of the inlet
section. This serves to justify the assumption made in Section 4.2.1 that 100 points along the axial
direction would be enough to discretise the domain without loss of solution quality.
Regarding the meshing in the radial direction, a comparison was made between four different
meshes. In addition to the designed mesh with 50 cells in the radial direction, a mesh with 25 and
a mesh with 100 cells were created. These two meshes have the same total expansion ratio as the
61
default mesh, resulting in a mesh with theoretically insufficient points to correctly capture boundary
layer development and a mesh with an excessive amount of points. Complementarily, a mesh with
100 cells where the boundary layer of the outer wall was also discretised using the same criteria as
the cylinder wall (marked as ”50 cells (OW)”).
The results presented in Figure 5.11 show that the results obtained are completely independent
from the grading in the radial direction. It shows that the assumptions made in the mesh design
process were correct regarding both the number of points discretising the boundary layer at the heated
cylinder wall and the fact that refining the mesh in the outer cylinder wall would not improve the results
obtained.
0 0.02 0.04 0.06 0.08 0.1 0.120
0.2
0.4
0.6
x(m)
Uz(m
s−1)
25 cells50 cells
100 cells50 cells (OW)
(a) Velocity.
0 0.02 0.04 0.06 0.08 0.1 0.12
300
310
320
x(m)
T(K
)
25 cells50 cells100 cells
50 cells (OW)
(b) Temperature.
Figure 5.11: Grid refinement study in the radial direction at z = 1.5m, qs = 264Wm−2.
A comparison of the results obtained between the designed mesh for the solid domain, and a
mesh with no discretisation in the radial direction is shown in Figure 5.12. It can be seen that the
predicted solution for a temperature parabola profile for a fixed height is verified. However, the overall
temperature variation along the rod radius is of less than 0.05K and the removal of the discretisation
completely does not significantly affect the maximum temperature observed within the solid domain.
Finally, it is necessary to study the independence of the radiation model in regards to the number
of discretised solid angles, Nθ and Nφ. In this test, instead of the default Nθ = 1 and Nφ = 1 for a
total of 4 discretised solid angles, Nθ = 5 and Nφ = 5 for a total of 100 discretised solid angles were
used. This extra discretisation resulted in a five fold increase of simulation time, from 65 s to 305 s.
The results obtained only improve the results presented in Figure 5.3 by a maximum of 0.05% when
compared with the experimental data. In absolute values, the biggest difference between the two
results set is of 0.3K. For this reason, the results are considered as independent of the discretisation
of the solid angles in the FVDOM model.
Globally, There are no significant variations of the results with the usage of slightly more or slightly
less refined meshes. As such, the results obtained in the previous section can be considered as not
depending on the grid definition.
62
0 1 2 3 3.2314.4
314.45
314.5
314.55
314.6
x(mm)
T(K
)
10 cells 1 cell
Figure 5.12: Temperature profile in the solid domain at X = 1.0m.
5.5 Numerical limitations of the model
As mentioned, it was not possible to achieve result convergence for the seven cylinder bundle
case. Similarly to the single cylinder case, the very first simulations were performed using only the
fluid domain and no radiation model. The residual convergence and control variable progression for
the first 500 iterations of the simulation with a heat flux at the heated surfaces of qs = 100Wm−2 is
shown in Figure 5.13.
After the first 500 iterations the solution repeats the pattern seen from 400 to 500 iterations until
the solution is stopped. It can be seen that the most probable cause of instabilities is the solution the
solver obtains for the velocity field. From the first iteration, negative velocities in the axial direction
exist in the domain. This is not physical since the flow developed inside the seven cylinder bundle is
expected to be unidirectional, from the inlet to the outlet.
After approximately 80 iterations, the maximum value of the modified pressure peaks at values
larger than the total pressure imposed at the inlet, p0 = 101325. This is also non physical.
A residual convergence of 10−2 is reached for the temperature field within the first 30 iterations.
Contrary to all the other control variables, the temperature maximum and average temperature are
stable until 300 iterations, after which the maximum and averages values will also begin to oscillate.
From this it can be concluded that the source of instability is more likely in the prediction of the
momentum or pressure fields.
The most obvious causes for a solution to not converge are using too tight relaxation factors, a low
quality of the domain mesh, or a poor initialisation of the domain fields. It was also considered that
the solution might not be completely laminar and an unstable transition section could exist. Tables 5.1
and 5.2 summarise the attempts made to determine if any of the aforementioned probable causes is
indeed the source of the instability in the numerical simulations.
The most obvious causes for instability are therefore excluded. It’s not a problem with mesh-
ing used, numerical schemes, or the relaxation factors. The tests done with the smaller and larger
meshes, also allow to exclude the possibility that the problem is not fully laminar or that the flow at the
outlet is not fully developed.
The main difference between the bundle of cylinders problem and the single cylinder case where
convergence was achieved is that new surfaces where boundary layers develop are added to the
domain. Hence, it is suspected that this may be the source of the numerical instability. As such,
63
0 50 100 150 200 250 300 350 400 450 500
1
0
−1
−2
−3
Iterations
Residual(
10n
)
pρgh Uz hfluid
(a) Residual evolution
0 50 100 150 200 250 300 350 400 450 500290
320
350
380
410
Iterations
T(K
)
Maximum Average Minimum
(b) Temperature evolution.
0 50 100 150 200 250 300 350 400 450 500−3
−2
−1
0
1
2
3
Iterations
~ Uz(m
s−1)
Maximum Average Minimum
(c) Velocity evolution.
0 50 100 150 200 250 300 350 400 450 500101000
101200
101400
101600
101800
Iterations
pρgh(Pa)
Maximum Average Minimum
(d) Modified pressure evolution.
Figure 5.13: Residual convergence and control variables values in the seven cylinder domain.
64
Table 5.1: Attempted solutions for the lack of convergence of the cylinder bundle case.
Objective: Guarantee that lack of mesh definition in the axial direction was not the root causeof the problem;
Change: Doubled the mesh refinement in the axial direction;
result: The convergence did not improve;
Objective: Guarantee that lack of mesh definition in the axial direction of the entrance sectionwas not the root cause of the problem without increasing the total number of cellsin the mesh;
Change: Defined a mesh grading with a total expansion ratio of 10 in the axial direction;
Result: The convergence did not improve;
Objective: Guarantee that lack of mesh definition in the cylinder boundary layers was not theroot cause of the problem;
Change: Doubled the points inside the cylinder boundary layers;
Result: The convergence did not improve, while increasing the simulation time signifi-cantly;
Objective: Guarantee that lack of mesh definition in the outer wall boundary layer was notthe root cause of the problem;
Change: Doubled the points inside the outer wall boundary layers;
Result: The convergence did not improve, while increasing the simulation time signifi-cantly;
Objective: Define if the root problem was in the outer wall boundary layer;
Change: removed the outer wall boundary layer meshing replacing the boundary conditionfor the velocity in this surface to a slip condition;
Result: The convergence did not improve;
Objective: Define if an unstable section exists within the domain;
Change: Reduced the domain height to 0.4m so that if a unstable section exists it is re-moved from the domain;
Result: The convergence did not improve;
Objective: Guarantee that the flow is fully developed at the outlet;
Change: Increased the domain height to 2m to guarantee the flow is fully developed at theoutlet;
Result: The convergence did not improve;
Objective: Guarantee that the flow is fully developed at the outlet;
Change: Added an outlet section of 0.2m where the cylinders had a zero gradient boundarycondition for the temperature;
Result: The convergence did not improve;
Objective: Increase stability;
Change: Numerical schemes for gradient and divergence were changed from upwind tolinear;
Result: Stability of the problem increased however residual convergence did not go below10−3. The results obtained where however non physical with a maximum speedin the order of |~U | ≈ 10−2 and temperatures increasing gradually beyond themaximum expected temperature in the domain;
65
Table 5.2: Attempted solutions for the lack of convergence of the cylinder bundle case (cont.).
Objective: Increase stability;
Change: Numerical schemes for gradient and divergence were changed from upwind tolinear and after solution is stabilized reverted to upwind schemes;
Result: After the upwind schemes were enabled solution quickly diverged;
Objective: Increase linear solver stability
Change: Reduction of implicit relaxation factor for the velocity and explicit relaxation factorfor the modified pressure;
Result: Stability was achieved in the first iterations but the solution eventually diverged;
Objective: Increase linear solver stability
Change: Add explicit under relaxation for the velocity;
Result: Stability was achieved in the first iterations but the solution eventually diverged;
Objective: Confirm the energy equation is not the source of instabilities;
Change: Reduced the heat imposed at the heated surfaces to a value of 2Wm−2;
Result: Solution did not converge;
Objective: Provide a better initialization to the problem
Change: Added a velocity profile to the starting solution;
Result: Stability was achieved in the first iterations but the solution eventually diverged;
Objective: Provide a better initialization to the problem
Change: Added a temperature profile to the starting solution;
Result: The convergence did not improve;
Objective: Provide a better initialization to the problem
Change: Iterated the energy equation 100 times before starting the actual iterative algo-rithm;
Result: Stability was achieved in the first iterations but the solution eventually diverged;
numerical simulations were made using the single cylinder case domain where the outer wall radius
was progressively diminished. The mesh used was the one created to test the independence of
the results when the outer wall boundary layer was refined, so that both walls’ boundary layers are
correctly captured.
Table 5.3 summarises the results obtained for this progressive test. Convergence was not achieved
once the outer wall was significantly close to the the heated cylinder. Some interesting results are
the fact that wall temperature and maximum velocity do not vary significantly with the diameter re-
duction. However, the energy balance error increases with a decrease of radius. A total energy of
Q = 7.96W is supplied at the heat wall. For the measurements where convergence was not achieved,
the Reynolds numbers were calculated assuming the maximum velocity remained constant. For the
calculation of the Reynolds number, Equation 2.1 was used.
Convergence of the case with an outer radius, R = 33mm, already showed signs of instability, but
convergence was eventually achieved. This can be observed in Figure 5.14. it’s important to notice
that, once again, the instability is not present for a heat flux of qs = 100Wm−2 in the residuals of the
energy equation.
66
Table 5.3: Results for the varying outer wall radius test.
R(m) Dh(m) ~Uz,max(ms−1) Tmax(K) m(kg s−1) Q(W) Iterations Re
0.113 0.109 0.70 339.7 2.61× 10−5 7.99 429 4239
0.103 0.099 0.70 339.1 2.71× 10−5 7.91 551 3850
0.063 0.059 0.70 338.3 2.64× 10−5 7.83 690 2294
0.043 0.039 0.72 336.7 2.19× 10−5 7.69 931 1560
0.033 0.029 0.74 335.7 1.69× 10−5 7.56 1187 1192
0.023 0.019 - - - - - 781*
0.013 0.009 - - - - - 370*
0 100 200 300 400 500 600 700 800 900 1,000 1,100
1
0
−1
−2
−3
−4
−5
−6
Iterations
Residual(
10n
)
pρgh Uz hfluid
Figure 5.14: Residual convergence for an outer wall radius R = 33mm.
The velocity and temperature profiles at the outlet for these measurements can be observed in
Figures 5.15. As expected, as the outer wall is closer to the heated wall, the boundary layer on this
surface starts being significant. This seems to be the source of instability since, as the shear wall
stress in this surface becomes significant, the iterative process becomes unstable and the solution no
longer converges.
A possible source of the instability is the way the inlet boundary condition is implemented. The
boundary condition, pressureInletVelocity, calculates the velocity in the inlet based on the flux, φ,
using Equation (3.13). This, combined with the fact that at the outlet a zero gradient condition is
applied for the velocity, implies that the velocity is never explicitly given in the inlet or the outlet. The
entire velocity field, including the inlet and outlet, are calculated only through the momentum balance
equation, where the mass conservation equation is not necessarily respected.
This can justify the appearance of instability when the velocity profile that needs to be calculated
at the inlet increases in complexity, as velocity gradients exist all over the domain. Furthermore, since
a thermal boundary layer does not exist in the outer wall, these instabilities are not reflected in the
energy equation.
67
0 0.02 0.04 0.06 0.08 0.10
0.2
0.4
0.6
0.8
x(m)
Uz(m
s−1)
(a) Velocity, R = 113mm.
0 0.02 0.04 0.06 0.08 0.1
300
320
340
x(m)
T(K
)
(b) Temperature, R = 113mm.
0 0.02 0.04 0.06 0.08 0.10
0.2
0.4
0.6
0.8
x(m)
Uz(m
s−1)
(c) Velocity, R = 83mm.
0 0.02 0.04 0.06 0.08 0.1
300
320
340
x(m)T
(K)
(d) Temperature, R = 83mm.
0 0.02 0.04 0.06 0.08 0.10
0.2
0.4
0.6
0.8
x(m)
Uz(m
s−1)
(e) Velocity, R = 63mm.
0 0.02 0.04 0.06 0.08 0.1
300
320
340
x(m)
T(K
)
(f) Temperature, R = 63mm.
0 0.02 0.04 0.06 0.08 0.10
0.2
0.4
0.6
0.8
x(m)
Uz(m
s−1)
(g) Velocity, R = 43mm.
0 0.02 0.04 0.06 0.08 0.1
300
320
340
x(m)
T(K
)
(h) Temperature, R = 43mm.
0 0.02 0.04 0.06 0.08 0.10
0.2
0.4
0.6
0.8
x(m)
Uz(m
s−1)
(i) Velocity, R = 33mm.
0 0.02 0.04 0.06 0.08 0.1
300
320
340
x(m)
T(K
)
(j) Temperature, R = 33mm.
Figure 5.15: Variation of velocity profile at the outlet for different outer wall radius (R).
68
6 Conclusions and Future WorkIn this report a mathematical model that describes the total Loss Of Coolant Accident (LOCA) sce-
nario was defined. This model was based on experimental measurements made using nuclear fuel
cells that determined that the developed natural convective flow is laminar and that compressibility
effects must be considered. Appropriate thermophysical models were chosen based on the expected
temperature range within the domain, as well as boundary conditions appropriate for natural convec-
tion problems.
The OpenFOAM application that implements this mathematical model was reviewed and the de-
tails from the model that were not natively present in the code were implemented for this work. These
include a new boundary condition for the modified pressure at the outlet, a boundary condition that
allows to remove radiation effects from simple domains, a new initialisation for the solver and a small
script that calculates the maximum, minimum, and average of all the variables in the system as well
as the mass flow and energy balance between the domains inlet and outlet.
A small verification exercise was performed where it was shown that the solver correctly predicts
the solution for a case where no heat is given to a domain.
Preliminary numerical simulations using experimental data of the temperature of a single cylinder
under natural convection showed the Finite Volume Discrete Ordinates Method (FVDOM) model to
be the most suitable for the simulation of radiation. The view factor model, which has the most
appropriate mathematical description of the reality it is intended to simulate, did not have positive
results due to its faulty implementation in OpenFOAM.
Complete testing performed for the single cylinder case showed that the results obtained for the
temperature profiles of laminar natural convective flows have an acceptable fit with the experimental
data with a maximum relative error of 2.8%. A residual convergence of an order of 10−4 was deemed
acceptable for future cases as long as the chosen control variables are stabilised. The obtained
results were also shown to be completely independent of the grid refinement.
The numerical simulations for the cylinder bundle case could not be converged. Extensive testing
was performed where neither the mesh quality, the relaxation factors used, nor the numerical schemes
chosen were found to be the root cause of the solver instability.
Due to these instabilities, the defined mathematical model for a LOCA scenario can not be con-
sidered as validated. As such, only the fist two of the three objectives established for this work can be
considered as accomplished.
The continuation to this work should focus in removing the existing instabilities in the solver. A
set of simulations using a simpler domain allows to correlate the appearance of instability with the
presence of significant velocity gradient in the boundary layer of a second surface within the domain.
It is suspected that this could lead to an instability in the inlet boundary condition.
In the inlet boundary condition, the velocity field is calculated using the flux, φ, calculated in the
momentum corrector step, while in the outlet a zero gradient condition is applied. This means that
69
the velocities at the inlet and outlet of the system are never given explicitly, but rather calculated
iteratively. This makes the solver prone to instabilities if the velocity field of the numerical solution has
a high gradient throughout the domain.
The implementation of a new boundary condition is proposed. Instead of calculating the velocity
through the flux, the velocity should be calculated using the total pressure, p0, as shown in Equa-
tion (6.1) while the modified pressure at inlet was to be obtained by interpolation from the value inside
the domain.
~Ubf =
√2(p0 − pbf )
ρbf· ~nbf (6.1)
The experimental data used in Chapter 4.1.2 was not collected within the scope of this project.
The main purpose of the measurements was to define empirical correlations for the flow inside a
bundle of rods and, for this reason, the results presented are based in dimensionless numbers [16].
By the authors’ estimates, the calculated Grashof numbers, Gr, have an uncertainty of ≈ 5%,
and Nusselt numbers, Nu, of ≈ 7%. Additionally, information about the fluid conditions such as
temperature, viscosity, and density, are not explicitly given in the article. Therefore, it is not known
what the conditions present during the experimental measurements were, what approximations were
used, and what assumptions were made during the post processing of the experimental data. This
makes replicating the exact measurements performed in the experimental installation impossible.
It is therefore interesting to consider doing an experimental measurement similar to the one used
in Chapter 4.1.2 but that more closely reflects the necessary data for the validation of a solver. The
experimental measurement should take the following topics in consideration:
• The presented results should be the cylinder temperature along the rod’s surface;
• Results should be in function of heat dissipated inside the rod [Wm−3];
• Radiative heat flux should be considered and properly accounted for;
• Rod materials and dimensions should be registered;
• If possible, temperatures should be measured at different radii inside the rod;
• Fluid temperature at the inlet should be measured;
This experimental data measured in this set-up closely resembles the inputs and outputs of the
numerical simulation and should allow for a more accurate validation of the mathematical model.
It was also noted before that the view factor model is mathematically the better suited model for
this problem. However, its OpenFOAM implementation is currently not verified, it has a limitation of
10000 view factors in the domain and is not able to account for symmetry boundary conditions. The
correct implementation of this model would be a major contribution to the OpenFOAM community and
to this project. At the very least, it is highly recommended re-evaluate the usage of the view factor
model as soon as an updated version of OpenFOAM that deals with these limitations is released.
70
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[31] T. Holzmann. (2016, May) Mathematics, numerics, derivations and OpenFOAM. Holzmann-CFD.
[Online]. Available: http://www.Tobias.HolzmannHolzmann-cfd.de
[32] C. J. Greenshields. (2015, December) OpenFOAM the open source cfd tool programmer’s guide.
OpenFOAM Foundation Ltd. [Online]. Available: http://foam.sourceforge.net/docs/Guides-a4/
ProgrammersGuide.pdf
[33] J. H. Ferziger and M. Peric, Computational methods for fluid dynamics. New York: Springer
Science & Business Media, 2012.
[34] T. Behrens. (2009, February) OpenFOAM’s basic solvers for linear systems of equations. Dept
of Thermo and Fluid Dynamics Chalmers. [Online]. Available: http://www.tfd.chalmers.se/∼hani/
kurser/OS CFD 2008/TimBehrens/tibeh-report-fin.pdf
[35] F. Moukalled, L. Mangani, and M. Darwish, The Finite Volume Method in Computational Fluid
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vol. 113.
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73
74
A Modified buoyantSimpleFoamSome changes were made to the solver used in this work. A new initialisation for the pressure field
was added, a script to evaluate control variables was implemented, and the option to explicitly relax
the velocity and enthalpy fields were added. The changes presented in this annex were also made
to chtMultiRegionSimpleFoam, the solver used for CHT cases. The complete code for these solver is
too long to be displayed, for this reason only snippets of the modifications made are presented. To im-
plement these conditions the reader should first familiarise themselves with the basics of OpenFOAM
programming [32].
Listing A.1 displays how the new pressure initialisation, discussed in Equation (2.29), was imple-
mented. Additionally, a field was created to store the compressibility of the fluid to be used by the
outlet boundary condition for the modified pressure.
Listing A.1: createFields.C (snippet)
specie theSpecie ( thermo . subDict ( ” mix ture ” ) ) ;dimensionedScalar RperfectGas (
” RperfectGas ” ,dimensionSet (0 ,2 ,−2 ,−1 ,0 ,0 ,0) ,sca la r ( theSpecie .R ( ) )
) ;
/ / I n i t i a l i z e s pressurethermo . p ( ) = thermo . p ( ) ∗ Foam : : exp ( gh / RperfectGas / thermo . T ( ) ) ;thermo . c o r r e c t ( ) ;
v o l S c a l a r F i e l d& p = thermo . p ( ) ;
/ / . . .
v o l S c a l a r F i e l d ps i \\ necessary for the out le tPrghPressure c o n d i t i o n(
IOob jec t ( ” ps i ” , runTime . timeName ( ) , mesh , IOob jec t : : READ IF PRESENT , IOob jec t : : AUTO WRITE) ,thermo . ps i ( )
) ;
/ / . . .
v o l S c a l a r F i e l d p rgh(
IOob jec t ( ” p rgh ” , runTime . timeName ( ) , mesh , IOob jec t : : MUST READ, IOob jec t : : AUTO WRITE) ,mesh
) ;
p rgh = p − rho∗gh ;
Listing A.2 shows how to enable explicit under relaxation of a system variable. The same code
was added in the energy equation file to under relax explicitly the enthalpy.
Listing A.2: Ueqn.C (snippet)
MRF. cor rec tBoundaryVe loc i t y (U ) ;
/ / . . .
i f ( s imple . momentumPredictor ( ) )
/ / . . .
U. re l ax ( ) ; / / e x p l i c i t l y re laxes U
A-1
Listing A.3 presents the code used to calculate the control variables on the domain, and print them
on the console so that the user can monitor the progress of the simulation.
Listing A.3: buoyantSimpleFoam.C (snippet)
l a b e l i n l e t = mesh . boundaryMesh ( ) . f indPatch ID ( ” I n l e t ” ) ;l a b e l o u t l e t = mesh . boundaryMesh ( ) . f indPatch ID ( ” Ou t l e t ” ) ;
/ / . . .
while ( s imple . loop ( ) )
/ / . . .
In fo<< ” ExecutionTime = ” << runTime . elapsedCpuTime ( ) << ” s ”<< ” ClockTime = ” << runTime . elapsedClockTime ( ) << ” s ” << endl ;
In fo<< ” I n f l o w : ” << −1.0∗ gSum( ph i . boundaryFie ld ( ) [ i n l e t ] )<<” [ kg / s ] ” << endl ;
In fo<< ” Outf low : ” << gSum( ph i . boundaryFie ld ( ) [ o u t l e t ] )<<” [ kg / s ] ” << endl ;
In fo<< ” EnergyInf low : ” << −1.0∗ gSum( ph i . boundaryFie ld ( ) [ i n l e t ] ∗( thermo . he ( ) . boundaryFie ld ( ) [ i n l e t ] + 0.5∗magSqr (U. boundaryFie ld ( ) [ i n l e t ] ) ) )<<” [W] ” << endl ;
In fo<< ” EnergyOutf low : ” << gSum( ph i . boundaryFie ld ( ) [ o u t l e t ] ∗( thermo . he ( ) . boundaryFie ld ( ) [ o u t l e t ] + 0.5∗magSqr (U. boundaryFie ld ( ) [ o u t l e t ] ) ) )<<” [W] ” << endl ;
In fo<< ” EnergyBalance : ” << gSum( ph i . boundaryFie ld ( ) [ o u t l e t ] ∗( thermo . he ( ) . boundaryFie ld ( ) [ o u t l e t ] + 0.5∗magSqr (U. boundaryFie ld ( ) [ o u t l e t ] ) ) )+1.0∗ gSum( ph i . boundaryFie ld ( ) [ i n l e t ] ∗ ( thermo . he ( ) . boundaryFie ld ( ) [ i n l e t ] +0.5∗magSqr (U. boundaryFie ld ( ) [ i n l e t ] ) ) ) <<” [W] ” << endl ;
In fo<< ” rho max / avg / min : ” << gMax( thermo . rho ( ) ) << ” ” << gAverage ( thermo . rho ( ) )<< ” ” << gMin ( thermo . rho ( ) ) << endl ;
In fo<< ”T max / avg / min : ” << gMax( thermo . T ( ) ) << ” ” << gAverage ( thermo . T ( ) )<< ” ” << gMin ( thermo . T ( ) ) << endl ;
In fo<< ”P max / avg / min : ” << gMax( thermo . p ( ) ) << ” ” << gAverage ( thermo . p ( ) )<< ” ” << gMin ( thermo . p ( ) ) << endl ;
In fo<< ” Prg max / avg / min : ” << gMax( p rgh ) << ” ” << gAverage ( p rgh )<< ” ” << gMin ( p rgh ) << endl ;
In fo<< ”U max / avg / min : ” << gMax(U) . component ( 2 ) << ” ” << gAverage (U) . component ( 2 )<< ” ” << gMin (U ) . component ( 2 ) << endl ;
In fo<< ” ” << endl ;
A-2
B ExtendedBuoyancyBoundariesIn order to perform the numerical simulations presented in this work, two distinct boundary con-
ditions had to be implemented into OpenFOAM. These were compiled into a single library baptised
extendedBuoyancyBoundaries. Similarly to the listings of Appendix A, only the most relevant parts of
the implementation are presented in this annex.
The outlet boundary condition for the modified pressure, is a subclass of the fixedGradientFv-
PatchScalarField class. The update operator, which updates the boundary values of the gradient
in the outlet, is given in listing B.1. Notice that the compressibility field is called directly, instead of
fetching the molar mass, temperature and the ideal gas constant.
Listing B.1: outletPrghPressure.C (snippet)
void Foam : : out le tPrghPressure : : updateCoeffs ( )
i f ( updated ( ) )
return ;
const f vPatchF ie ld<sca lar>& rhop =patch ( ) . lookupPatchFie ld<vo lSca la rF ie ld , sca lar >( ” rho ” ) ;
const f vPatchF ie ld<sca lar>& psip =patch ( ) . lookupPatchFie ld<vo lSca la rF ie ld , sca lar >( ” ps i ” ) ;
const s c a l a r F i e l d& ghfp =patch ( ) . lookupPatchFie ld<sur faceSca la rF ie ld , sca lar >( ” ghf ” ) ;
/ / g rad ien t ( ) = −1.0 ∗ rhop . snGrad ( ) ∗ ghfp ; / / Unstable due to sur face normal g rad ien t
grad ien t ( ) = −1.0 ∗ ghfp ∗ ps ip ∗ rhop ∗ −9.81 ; / / s tab le
For the boundary condition, radiatingWall, is a modification of the class externalWallHeatFluxTem-
peratureFvPatchScalarField. A new operation mode is added, implicitRadiation, which, when a wall
emissivity and a temperature at infinity are given at the patch, calculate the radiative heat dissipation.
The code where the radiative heat flux is calculated is shown in listing B.2.
Listing B.2: radiatingWall.C (snippet)
void Foam : : r a d i a t i n g W a l l : : updateCoeffs ( )
i f ( updated ( ) )
return ;
const s c a l a r F i e l d Tp(∗ th is ) ;s c a l a r F i e l d hp ( patch ( ) . s i ze ( ) , 0 . 0 ) ;
s c a l a r F i e l d Qr ( Tp . s ize ( ) , 0 . 0 ) ;i f ( QrName != ” none ” )
Qr = patch ( ) . lookupPatchFie ld<vo lSca la rF ie ld , sca lar >(QrName ) ;
Qr = QrRelaxat ion ∗Qr + (1 .0 − QrRelaxat ion )∗QrPrevious ;QrPrevious = Qr ;
switch ( mode )
B-1
/ / . . .
case i m p l i c i t R a d i a t i o n :
QrPrevious = E ∗ constant : : physicoChemical : : sigma . value ( ) ∗( pow4( Tp ) − pow4( T i n f ) ) ;
refGrad ( ) = ( q − QrPrevious ) / kappa ( Tp ) ;re fVa lue ( ) = 0 . 0 ;va lueFrac t ion ( ) = 0 . 0 ;
break ;
/ / . . .
mixedFvPatchScalarFie ld : : updateCoeffs ( ) ;
/ / . . .
B-2
C blockMesh.pyTo speed up the mesh creation process in this work, a small python script was designed. This
script will create a new mesh for the seven cylinder domain and split them into solid and fluid domains
automatically. Using this script it was possible to quickly update the boundary layer definition of all the
surfaces in the domain, add new refinement sections or change global domain dimensions.
Listing C.1: blockMesh.py
# ! / usr / b in / env pythonimport osimport math
############################################################################################## Funct ion d e f i n i t i o n s #################################################################################################################################################################
def addpoint ( p , z ) :i f [ p [ 0 ] , p [ 1 ] , z ] not in p o i n t l i s t : p o i n t l i s t . append ( [ p [ 0 ] , p [ 1 ] , z ] )return p o i n t l i s t . index ( [ p [ 0 ] , p [ 1 ] , z ] ) ;
def addarc ( arc , z ) :arc = [ addpoint ( arc [ 0 ] , z ) , addpoint ( arc [ 1 ] , z ) , [ arc [ 2 ] [ 0 ] , arc [ 2 ] [ 1 ] , z ] ]i f arc not in a r c l i s t : a r c l i s t . append ( arc )return a r c l i s t . index ( arc ) ;
def getmidpo in t ( c , p1 , p2 ) :r = math . s q r t ( ( p1[0]−c [ 0 ] )∗∗2 + ( p1[1]−c [ 1 ] )∗∗2 )m = [ ( p1 [ 0 ] + p2 [ 0 ] ) / 2 − c [ 0 ] , ( p1 [ 1 ] + p2 [ 1 ] ) / 2 − c [ 1 ] ]r oo t = math . s q r t ( m[0 ]∗∗2 + m[1 ]∗∗2 )m = [ m[ 0 ] / r oo t , m[ 1 ] / r oo t ]return [ m[ 0 ]∗ r + c [ 0 ] , m[ 1 ]∗ r + c [ 1 ] ] ;
def createHexBlock ( p1 , p2 , p3 , p4 , top , bot , po in ts , grade ) :hexblocks . append ( [
addpoint ( p1 , bot ) , addpoint ( p2 , bot ) , addpoint ( p3 , bot ) , addpoint ( p4 , bot ) ,addpoint ( p1 , top ) , addpoint ( p2 , top ) , addpoint ( p3 , top ) , addpoint ( p4 , top ) ,po in ts , grade ] )
return ;
def createboundary (name, faces , h , type ) :l i s t o f f a c e s = [ ]for f in faces :
i f len ( f ) == 2:for x in xrange (0 , len ( h )−1):
l i s t o f f a c e s . append ( [ addpoint ( f [ 0 ] , h [ x ] ) ,addpoint ( f [ 1 ] , h [ x ] ) ,addpoint ( f [ 1 ] , h [ x +1 ] ) ,addpoint ( f [ 0 ] , h [ x +1 ] )] )
e l i f len ( f ) == 4:l i s t o f f a c e s . append ( [ addpoint ( f [ 0 ] , h ) , addpoint ( f [ 1 ] , h ) ,
addpoint ( f [ 2 ] , h ) , addpoint ( f [ 3 ] , h ) ] )b o un d l i s t . append ( [ name, l i s t o f f a c e s , type ] )return [ name, faces , type ] ;
def wr i teb lockmeshd ic t ( p o i n t l i s t , hexblocks , a r c l i s t , bound l i s t , mergepatches ) :blockMeshDict = open ( ” system / blockMeshDict ” , ”w” )
# Wri te the headingblockMeshDict . w r i t e ( ” FoamFile\n\n vers ion 2 .0 ;\n format a s c i i ;\n ” )blockMeshDict . w r i t e ( ” c lass d i c t i o n a r y ;\n ob jec t blockMeshDict ;\n\n ” )blockMeshDict . w r i t e ( ” convertToMeters 0.001;\n ” )
# Wri te down the v e r t i c e sblockMeshDict . w r i t e ( ” \ n v e r t i c e s \n (\n ” )for po in t in p o i n t l i s t :
blockMeshDict . w r i t e ( ” ( ” +repr ( po i n t [ 0 ] ) + ” ” +repr ( po i n t [ 1 ] ) + ” ” +
C-1
repr ( po i n t [ 2 ] ) + ” ) / / ” +repr ( p o i n t l i s t . index ( po in t ) )+ ” \n ” )
# Wri te down the blocksblockMeshDict . w r i t e ( ” ) ; \ n\nblocks\n (\n ” )for hex in hexblocks :
blockMeshDict . w r i t e ( ” hex ( ” +repr ( hex [ 0 ] ) + ” ” +repr ( hex [ 1 ] ) + ” ” +repr ( hex [ 2 ] ) + ” ” + repr ( hex [ 3 ] ) + ” ” +repr ( hex [ 4 ] ) + ” ” +repr ( hex [ 5 ] ) + ” ” +repr ( hex [ 6 ] ) + ” ” + repr ( hex [ 7 ] ) + ” ) ( ” +repr ( hex [ 8 ] [ 0 ] ) + ” ” +repr ( hex [ 8 ] [ 1 ] ) + ” ” +repr ( hex [ 8 ] [ 2 ] ) + ” ) simpleGrading ( ” +repr ( hex [ 9 ] [ 0 ] ) + ” ” +repr ( hex [ 9 ] [ 1 ] ) + ” ” +repr ( hex [ 9 ] [ 2 ] ) + ” ) \n ” )
# Wri te down the edgesblockMeshDict . w r i t e ( ” ) ; \ n\nedges\n (\n ” )for arc in a r c l i s t :
blockMeshDict . w r i t e ( ” arc ” +repr ( arc [ 0 ] ) + ” ” +repr ( arc [ 1 ] ) + ” ( ” +repr ( arc [ 2 ] [ 0 ] ) +” ” +repr ( arc [ 2 ] [ 1 ] ) + ” ” +repr ( arc [ 2 ] [ 2 ] ) + ” )\n ” )
# Wri te down the edgesblockMeshDict . w r i t e ( ” ) ; \ n\nboundary\n (\n ” )for b in b o un d l i s t :
q = ” ”for f in b [ 1 ] :
q=q+ ” ( ” +repr ( f [ 0 ] ) + ” ” +repr ( f [ 1 ] ) + ” ” +repr ( f [ 2 ] ) + ” ” +repr ( f [ 3 ] ) + ” ) ”blockMeshDict . w r i t e ( ” ” +b [ 0 ] + ” type ” +b [ 2 ] + ” ; faces ( ” +q+ ” ) ;\ n ” )
# w r i t e down meged patchesblockMeshDict . w r i t e ( ” ) ; \ n\nmergePatchPairs\n (\n ” )for m in mergepatches :
blockMeshDict . w r i t e ( ” ( ” + m[ 0 ] + ” ” + m[ 1 ] + ” )\n ” )blockMeshDict . w r i t e ( ” ) ; \ n ” )blockMeshDict . c lose ( )return ;
############################################################################################## S c r i p t ##############################################################################################################################################################################
p o i n t l i s t , a r c l i s t , hexblocks , bound l i s t , mergepatches = ( [ ] for i in range ( 5 ) )
r = 6 # Radius o f c y l i n d e r sR = 54.5 /2 # Radius o f ou ts ide c y l i n d e rd = 1 # Boundary l aye r minimum th ickness f o r ou ts ide c y l i n d e rdg = math . p i ∗30/180 # Degree of the mesh (30o )pepdis = 32.5 /2 # Distance between 2 c y l i n d e r sl = [200 ,0 ] # Levels o f the Meshl f = [ [ 2 0 ] , [ 1 ] ] # Mesh po in t s and grading f o r the f l u i d mesh l e v e l s [ 5 0 ] , [ 1 ]l s = [ [ 2 0 ] , [ 1 ] ] # Mesh po in t s and grading f o r the s o l i d mesh l e v e l sl p i = 36 # number o f mesh po in t s per p i (180o )expf = [12 ,12 ] # Number o f po in t s i n the mesh and t h e i r re f inement [27 ,12 ]exps = [ 5 , 0 . 1 ] # Number o f po in t s i n the s o l i d mesh and t h e i r re f inement
# po in t s f o r the center c y l i n d e rc0 = [ 0 , 0 ]a1 = [ r , 0 ]a2 = [ r ∗math . cos ( dg ) , r ∗math . s in ( dg ) ]
# po in t s f o r the top c i l i n d e rp0 = [ pepdis , 0 ]p1 = [ pepdis+r , 0 ]p2 = [ pepdis+ r ∗math . cos ( dg ) , r ∗math . s in ( dg ) ]p3 = [ pepdis , r ]p4 = [ pepdis−r ∗math . cos ( dg ) , r ∗math . s in ( dg ) ]p5 = [ pepdis−r , 0 ]
# g r i d po in t ss1 = [ pepdis /2 , 0 ]s2 = [ pepdis /2 , pepdis /2∗math . tan ( dg ) ]s3 = [ pepdis , pepdis∗math . tan ( dg ) ]s4 = [ pepdis ∗3/2 , pepdis /2∗math . tan ( dg ) ]s5 = [ pepdis ∗3/2 , 0 ]
# po in t s f o r the o u t t e r wa l lo1 = [R, 0 ]
C-2
o2 = [ R ∗math . cos ( dg ) , R ∗math . s in ( dg ) ]o3 = [ math . s q r t (R ∗∗ 2 − ( pepdis /2∗math . tan ( dg ) ) ∗∗ 2) , pepdis /2∗math . tan ( dg ) ]
#Def ine Hexblocks i npu tfor x in xrange (0 , len ( l )−1):
# boundary l aye r cen c i l i n d e rcreateHexBlock ( a1 , s1 , s2 , a2 , l [ x ] , l [ x +1] , [ expf [ 0 ] , l p i / 6 , l f [ 0 ] [ x ] ] , [ expf [ 1 ] , 1 , l f [ 1 ] [ x ] ] )
# boundary l aye r top c i l i n d e rcreateHexBlock ( p1 , s5 , s4 , p2 , l [ x ] , l [ x +1] , [ expf [ 0 ] , l p i / 6 , l f [ 0 ] [ x ] ] , [ expf [ 1 ] , 1 , l f [ 1 ] [ x ] ] )createHexBlock ( p2 , s4 , s3 , p3 , l [ x ] , l [ x +1] , [ expf [ 0 ] , l p i / 3 , l f [ 0 ] [ x ] ] , [ expf [ 1 ] , 1 , l f [ 1 ] [ x ] ] )createHexBlock ( p3 , s3 , s2 , p4 , l [ x ] , l [ x +1] , [ expf [ 0 ] , l p i / 3 , l f [ 0 ] [ x ] ] , [ expf [ 1 ] , 1 , l f [ 1 ] [ x ] ] )createHexBlock ( p4 , s2 , s1 , p5 , l [ x ] , l [ x +1] , [ expf [ 0 ] , l p i / 6 , l f [ 0 ] [ x ] ] , [ expf [ 1 ] , 1 , l f [ 1 ] [ x ] ] )
# boundary l aye r o u t t e r wa l lcreateHexBlock ( o3 , s4 , s5 , o1 , l [ x ] , l [ x +1] , [ expf [ 0 ] , l p i / 6 , l f [ 0 ] [ x ] ] , [ expf [ 1 ] , 1 , l f [ 1 ] [ x ] ] )createHexBlock ( o2 , s3 , s4 , o3 , l [ x ] , l [ x +1] , [ expf [ 0 ] , l p i / 3 , l f [ 0 ] [ x ] ] , [ expf [ 1 ] , 1 , l f [ 1 ] [ x ] ] )
# top c i lcreateHexBlock ( p0 , p2 , p3 , p0 , l [ x ] , l [ x +1] , [ exps [ 0 ] , l p i / 3 , l s [ 0 ] [ x ] ] , [ exps [ 1 ] , 1 , l s [ 1 ] [ x ] ] )createHexBlock ( p0 , p3 , p4 , p0 , l [ x ] , l [ x +1] , [ exps [ 0 ] , l p i / 3 , l s [ 0 ] [ x ] ] , [ exps [ 1 ] , 1 , l s [ 1 ] [ x ] ] )createHexBlock ( p0 , p4 , p5 , p0 , l [ x ] , l [ x +1] , [ exps [ 0 ] , l p i / 6 , l s [ 0 ] [ x ] ] , [ exps [ 1 ] , 1 , l s [ 1 ] [ x ] ] )createHexBlock ( p0 , p1 , p2 , p0 , l [ x ] , l [ x +1] , [ exps [ 0 ] , l p i / 6 , l s [ 0 ] [ x ] ] , [ exps [ 1 ] , 1 , l s [ 1 ] [ x ] ] )createHexBlock ( c0 , a1 , a2 , c0 , l [ x ] , l [ x +1] , [ exps [ 0 ] , l p i / 6 , l s [ 0 ] [ x ] ] , [ exps [ 1 ] , 1 , l s [ 1 ] [ x ] ] )
#Def ine arcsfor z in l :
addarc ( [ a1 , a2 , getmidpo in t ( c0 , a1 , a2 ) ] , z )addarc ( [ o1 , o3 , getmidpo in t ( c0 , o1 , o3 ) ] , z )addarc ( [ o3 , o2 , getmidpo in t ( c0 , o3 , o2 ) ] , z )addarc ( [ p1 , p2 , getmidpo in t ( p0 , p1 , p2 ) ] , z )addarc ( [ p2 , p3 , getmidpo in t ( p0 , p2 , p3 ) ] , z )addarc ( [ p3 , p4 , getmidpo in t ( p0 , p3 , p4 ) ] , z )addarc ( [ p4 , p5 , getmidpo in t ( p0 , p4 , p5 ) ] , z )
createboundary ( ” Ou t t e rwa l l ” , [ [ o2 , o3 ] , [ o3 , o1 ] ] , l , ” patch ” )
createboundary ( ” c i l t o p t o p ” , [ [ p0 , p1 , p2 , p0 ] , [ p0 , p2 , p3 , p0 ] ,[ p0 , p3 , p4 , p0 ] , [ p0 , p4 , p5 , p0 ] ] , l [ 0 ] , ” patch ” )
createboundary ( ” c i l b o t t o p ” , [ [ p0 , p2 , p1 , p0 ] , [ p0 , p3 , p2 , p0 ] ,[ p0 , p4 , p3 , p0 ] , [ p0 , p5 , p4 , p0 ] ] , l [−1] , ” patch ” )
createboundary ( ” c i l t o p c e n ” , [ [ c0 , a1 , a2 , c0 ] ] , l [ 0 ] , ” patch ” )createboundary ( ” c i l b o t c e n ” , [ [ c0 , a2 , a1 , c0 ] ] , l [−1] , ” patch ” )
createboundary ( ” symmetry le f t ” , [ [ s1 , a1 ] , [ p5 , s1 ] , [ s5 , p1 ] , [ o1 , s5 ] ] , l , ” symmetry ” )createboundary ( ” symmetryr ig t ” , [ [ a2 , s2 ] , [ s2 , s3 ] , [ s3 , o2 ] ] , l , ” symmetry ” )
createboundary ( ” symmetrytop1 ” , [ [ p1 , p0 ] , [ p0 , p5 ] ] , l , ” symmetry ” )
createboundary ( ” symmetrycen1 ” , [ [ a1 , c0 ] ] , l , ” symmetry ” )createboundary ( ” symmetrycen2 ” , [ [ c0 , a2 ] ] , l , ” symmetry ” )
createboundary ( ” Ou t l e t ” , [ [ a1 , s1 , s2 , a2 ] , [ p5 , p4 , s2 , s1 ] , [ p4 , p3 , s3 , s2 ] ,[ p3 , p2 , s4 , s3 ] , [ p2 , p1 , s5 , s4 ] , [ s4 , o3 , o2 , s3 ] , [ o1 , o3 , s4 , s5 ] ] , l [ 0 ] , ” patch ” )
createboundary ( ” I n l e t ” , [ [ a2 , s2 , s1 , a1 ] , [ s1 , s2 , p4 , p5 ] , [ s2 , s3 , p3 , p4 ] ,[ s3 , s4 , p2 , p3 ] , [ s4 , s5 , p1 , p2 ] , [ s4 , s3 , o2 , o3 ] , [ s5 , s4 , o3 , o1 ] ] , l [−1] , ” patch ” )
wr i teb lockmeshd ic t ( p o i n t l i s t , hexblocks , a r c l i s t , bound l i s t , mergepatches )os . system ( ” blockMesh ” )os . system ( ” checkMesh ” )os . system ( ” topoSet ” )os . system ( ” spl i tMeshRegions −cel lZones −ove rwr i t e ” )
C-3
C-4