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A Hybrid Fine-Coarse Mesh Computational Fluid Dynamics and Heat Transfer Model for Advanced Nuclear Energy Systems

Permanent Link: http://ufdc.ufl.edu/UFE0021523/00001

Material Information

Title: A Hybrid Fine-Coarse Mesh Computational Fluid Dynamics and Heat Transfer Model for Advanced Nuclear Energy Systems
Physical Description: 1 online resource (228 p.)
Language: english
Creator: Charmeau, Anne Dominique
Publisher: University of Florida
Place of Publication: Gainesville, Fla.
Publication Date: 2007

Subjects

Subjects / Keywords: Nuclear and Radiological Engineering -- Dissertations, Academic -- UF
Genre: Nuclear Engineering Sciences thesis, Ph.D.
bibliography   ( marcgt )
theses   ( marcgt )
government publication (state, provincial, terriorial, dependent)   ( marcgt )
born-digital   ( sobekcm )
Electronic Thesis or Dissertation

Notes

Abstract: The next generation of nuclear reactor systems is designed with strong emphasis on safety, efficiency, economy, and reducing proliferation threats. New designs include new operational conditions and new types of working fluids. This doctoral research aims at developing a robust simulation tool for thermal fluid simulation of advanced high temperature gas-cooled reactors. It results in a hybrid fine and coarse mesh numerical approach to solve Navier-Stokes equations with real gas properties. The model uses coarse mesh points to simulate the thermal fluid behavior of the entire nuclear system. Complex geometries and flow are modeled with a fine mesh Computational Fluid Dynamics (CFD) solver for higher accuracy. The CFD code is based on two-dimensional axisymmetric RANS approach combined to a TVD MacCormack implicit-explicit discretization scheme. Spalart-Allmaras and k-epsilon models are selected to account for turbulence. The solver models the real properties of gases of interest (helium and hydrogen) in gas-cooled nuclear systems. The real-property CFD model development is motivated for the study of low pressure thrust chambers for the Nuclear Thermal Propulsion (NTP): at low pressure and high temperature, hydrogen molecules dissociate, molecular weight, heat capacity and viscosity show strong dependence with temperature and pressure. Such drastic changes cannot be modeled with traditional CFD codes. In the real property solver the properties are discretized in parallel with the Navier-Stokes equations. The time-dependent CFD solver iteratively sets the steady-state condition in the fluid systems of interest. It is applied to the study of high temperature reactor systems portions. Results of the analysis of the helium-cooled channel for High Temperature Gas-Cooled Reactors and hydrogen-cooled channel for NTP show significant differences compared to frozen chemistry approaches, validating the need for a real property CFD solver development. Once validated on the coolant channels, the real property fine-mesh solver is implemented into a coarse mesh system code. The coupled fine and coarse-mesh solver is applied to model the entire NTP system: high resolution solution is obtained by modeling the core with the fine-mesh solver. Simulation of the system with the new coupled code shows great improvement in accuracy, speed and stability.
General Note: In the series University of Florida Digital Collections.
General Note: Includes vita.
Bibliography: Includes bibliographical references.
Source of Description: Description based on online resource; title from PDF title page.
Source of Description: This bibliographic record is available under the Creative Commons CC0 public domain dedication. The University of Florida Libraries, as creator of this bibliographic record, has waived all rights to it worldwide under copyright law, including all related and neighboring rights, to the extent allowed by law.
Statement of Responsibility: by Anne Dominique Charmeau.
Thesis: Thesis (Ph.D.)--University of Florida, 2007.
Local: Adviser: Anghaie, Samim.

Record Information

Source Institution: UFRGP
Rights Management: Applicable rights reserved.
Classification: lcc - LD1780 2007
System ID: UFE0021523:00001

Permanent Link: http://ufdc.ufl.edu/UFE0021523/00001

Material Information

Title: A Hybrid Fine-Coarse Mesh Computational Fluid Dynamics and Heat Transfer Model for Advanced Nuclear Energy Systems
Physical Description: 1 online resource (228 p.)
Language: english
Creator: Charmeau, Anne Dominique
Publisher: University of Florida
Place of Publication: Gainesville, Fla.
Publication Date: 2007

Subjects

Subjects / Keywords: Nuclear and Radiological Engineering -- Dissertations, Academic -- UF
Genre: Nuclear Engineering Sciences thesis, Ph.D.
bibliography   ( marcgt )
theses   ( marcgt )
government publication (state, provincial, terriorial, dependent)   ( marcgt )
born-digital   ( sobekcm )
Electronic Thesis or Dissertation

Notes

Abstract: The next generation of nuclear reactor systems is designed with strong emphasis on safety, efficiency, economy, and reducing proliferation threats. New designs include new operational conditions and new types of working fluids. This doctoral research aims at developing a robust simulation tool for thermal fluid simulation of advanced high temperature gas-cooled reactors. It results in a hybrid fine and coarse mesh numerical approach to solve Navier-Stokes equations with real gas properties. The model uses coarse mesh points to simulate the thermal fluid behavior of the entire nuclear system. Complex geometries and flow are modeled with a fine mesh Computational Fluid Dynamics (CFD) solver for higher accuracy. The CFD code is based on two-dimensional axisymmetric RANS approach combined to a TVD MacCormack implicit-explicit discretization scheme. Spalart-Allmaras and k-epsilon models are selected to account for turbulence. The solver models the real properties of gases of interest (helium and hydrogen) in gas-cooled nuclear systems. The real-property CFD model development is motivated for the study of low pressure thrust chambers for the Nuclear Thermal Propulsion (NTP): at low pressure and high temperature, hydrogen molecules dissociate, molecular weight, heat capacity and viscosity show strong dependence with temperature and pressure. Such drastic changes cannot be modeled with traditional CFD codes. In the real property solver the properties are discretized in parallel with the Navier-Stokes equations. The time-dependent CFD solver iteratively sets the steady-state condition in the fluid systems of interest. It is applied to the study of high temperature reactor systems portions. Results of the analysis of the helium-cooled channel for High Temperature Gas-Cooled Reactors and hydrogen-cooled channel for NTP show significant differences compared to frozen chemistry approaches, validating the need for a real property CFD solver development. Once validated on the coolant channels, the real property fine-mesh solver is implemented into a coarse mesh system code. The coupled fine and coarse-mesh solver is applied to model the entire NTP system: high resolution solution is obtained by modeling the core with the fine-mesh solver. Simulation of the system with the new coupled code shows great improvement in accuracy, speed and stability.
General Note: In the series University of Florida Digital Collections.
General Note: Includes vita.
Bibliography: Includes bibliographical references.
Source of Description: Description based on online resource; title from PDF title page.
Source of Description: This bibliographic record is available under the Creative Commons CC0 public domain dedication. The University of Florida Libraries, as creator of this bibliographic record, has waived all rights to it worldwide under copyright law, including all related and neighboring rights, to the extent allowed by law.
Statement of Responsibility: by Anne Dominique Charmeau.
Thesis: Thesis (Ph.D.)--University of Florida, 2007.
Local: Adviser: Anghaie, Samim.

Record Information

Source Institution: UFRGP
Rights Management: Applicable rights reserved.
Classification: lcc - LD1780 2007
System ID: UFE0021523:00001


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A HYBRID FINE-COARSE MESH COMPUTATIONAL FLUID DYNAMICS AND
HEAT TRANSFER MODEL FOR ADVANCED NUCLEAR ENERGY SYSTEMS


















By
ANNE CHARMEAU


A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL
OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT
OF THE REQUIREMENTS FOR THE DEGREE OF
DOCTOR OF PHILOSOPHY

UNIVERSITY OF FLORIDA

2007







































S2007 Anne C'I 1) is. .0
































To my parents, Pierre and Martine

To my brother, Nicolas










ACKENOWLED GMENTS

I particular thank my advisor and mentor during my graduate studies, Dr Santin

Anghaie. He constantly showed support, appreciation of my work and ah-- .1-< gave

me great advice. I am very grateful for all the research opportunities he offered me. I

acknowledge the nienters of my coninittee, Dr Ed Dugan, Dr Glen Sjoden and Dr Renwei

Mei.

Thank you to Dr Frederique Drullion for her great advices and tips for my CFD

solvers developments. Alany heartfelt thanks belong to Thierry Dubroca who ah-- 1-~

helieved in me and helped me go through the ups and downs of doctoral research. Thank

you to Melanie Heller for her invaluable gift of friendship. I also thank Fabien Gerard

and Rint Rouached for their meticulous reading of the manuscript and great el__o----- -H~e

Thank you to Lynne Schreiber for all her logistic help and kindness.

Above all, thank you to nly parents, Pierre and Martine, my brother Nicolas, and my

family for being so supportive, loving and encouraging to me. I am very grateful for their

sacrifice of seeing me leave my home country and understanding my choices.











TABLE OF CONTENTS

page

ACK(NOWLEDGMENTS .......... . .. .. 4

LIST OF TABLES ......... ... . 9

LIST OF FIGURES ......... .. . 10

LIST OF ACRONYMS ......... . 1:3

NOMENCLATURE ......... .. . 15

ABSTRACT ......... ...... 17

CHAPTER

1 INTRODUCTION: ANSWERING THE NEEDS OF THE FITTIRE IN THE
NUCLEAR INDITSTRY AND SYSTEM MODELING .. .. .. 19

1.1 The Nuclear Power Scene of the 21st Century .... .. .. 19
1.2 C'!s 11! !. ng. in Thernial-Hydraulics ... .. .. .. .. .. 20
1.3 Issues Faced with the Current Computational Tools for Thernial-Hydraulics 21
1.4 Objectives ............. ........... 2:3
1.5 The Different Approaches ......... ... 24
1.6 Overview of the Dissertation ........ .. 25

2 APPLICATION FUNDAMENTALS . ...... .. 28

2.1 Space Propulsion and Nuclear Thermal Propulsion ... .. .. 28
2.2 The Low-Pressure Thrust CHI I.M.I~er ...... .... :30
2.3 Computational Fluid Dynanxics: An Overview ... .. :31

:3 A SITRVEY ON STATE-OF-THE-ART COMPUTATIONAL FLITID DYNAMICS
DEVELOPMENTS AND APPLICATIONS ..... .. :34

:3.1 System Coupling ......... ... :39
:3.2 Numerical Techniques for CFD Solvers ..... .. . 41
:3.2.1 Considerations on Conipressible Flows ... ... .. .. 41
:3.2.2 Discussion on the Macdorniack Methods .. .. .. .. 4:3
:3.2.3 Considerations on Turbulence ..... .... . 46
:3.2.4 Discussion on the Spalart-Allmaras Model ... .. .. 49

4 MATERIAL PROPERTIES ......... .. 51

4.1 Hydrogen ......... .. . 51
4.1.1 Note on Dissociation ......... .. 52
4.1.2 Thernio-Physical Properties ...... .. 5:3
4.2 Helium ......... . .. . 56











4.3 Carbon Dioxide.

METHODS FOR THE CFD SOLVER DEVELOPMENT

5.1 CFD Solver Development .....
5.1.1 Requirements ..........
5.1.2 Governing Equations for Compressible Flow .....
5.1.3 Need for the Equation of State ...
5.1.4 Reynolds-Averaged Navier-Stokes Equations .....
5.1.5 Closure Problem: the Need for Turbulence Model .....
5.1.5.1 Turbulence models: classification and overview .
5.1.5.2 k-e model ..... ..
5.1.5.3 The Spalart-Allmaras model .......
5.1.6 Space Discretization ..... .. .. .. .
5.1.7 Time and Space Discretization: TVD MacCormack Scheme .
5.1.8 Time Stepping .......
5.1.9 Boundary Conditions ....... ....
5.1.10 Limiters and Total Variation Diminishing Condition .....
5.1.10.1 Full MacCormack scheme ......
5.1.10.2 Modified Causon limiter ........ ..


... 72
... 72
... 73
... 75
... 76
... 77
... 78
... 79
... 80
... 82
... 85
... 87
... 87
... 90
... 92
... 93


rithm 94
96i
96i
97
98
98


5.2 Remark on the Practical Implementation of the Real Gas Properties Algfo
5.3 Validation Cases: The Riemann Shock Tube ..........
5.3.1 Presentation of the Test Case .....
5.3.2 Numerical Method ..........
5.3.3 Results ..............
5.3.3.1 On the influence of the gas ..........

6 CFD STUDY OF THE COOLANT CHANNELS OF ADVANCED GASEOUS
REACTORS ..........

6.1 The Very High Temperature Gas-Cooled Reactor and the Helium ('ll ,Ill., I
6.1.1 Grid Generation ...... .... .....
6.1.2 Application of the TVD MacCormack Scheme Derivation of the
Viscous Term ..........
6.1.3 Simulation Results ............
6.1.3.1 Validation of the model ..... ......
6.1.3.2 Modeling of the VHTR Coolant ('I! .Ill., I ..........
6.1.4 Conclusion on the Helium Channel Study with Real Property Fine
CFD Solver .. ... ... ..
6.2 The Nuclear Thermal Propulsion and the hydrogen channel .........
6.2.1 NERVA Derivative Hot ('I! .Ill., I Analysis ...........
6.2.2 Hydrogen dissociation effects ...... ..... .....
6.2.2.1 Conclusion on the Hydrogen ('I! .Ill., I Modeling ......


108

108
110


110
112
112
114


114
115
115
116
117












7 COUPLING OF COARSE AND FINE MESH SOLVERS .. .. .. .. 1:32


7.1 NuRok: a 1-D System Code for Simulation of NTP Thernial-Hydraulics .1:32
7.1.1 Description of NuRok Core Subroutine and One-Dintensional Correlations1:34
7.1.2 NuRok Upgrading . ..... .... .. 1:37
7.1.2.1 Update to FORTR AN 90 .... .. .. 1:38
7.1.2.2 Levels of thrust ...... ... . 1:38
7.1.2.3 Foant fuel ........ .. .. 1:39

7.1.2.4 Hydrogen dissociation .... ... .. 1:39
7.1.2.5 Carbon dioxide . ..... .. .. 1:39
7.1.3 Reconinendations for NuRok Users .. .. .. 140
7.1.3.1 Mass Flow Rate . ..... .. .. 140

7.1.3.2 Pump Outlet Pressure ..... ... .. 140
7.1.3.3 Core Power . . .. .. .. 140

7.2 Thentis: A Coupled Coarse-Fine Mesh Simulation Tool for Simulation of
NTP Thernial-Hydraulics ......... ... .. 141
7.2.1 Treatment of the Interfaces ... .. .. ... .. 141


7.2.1.1 One-dintension to two-dintension axisyninetric ..
7.2.1.2 Two-dintension axisyninetric to one-dintension ..
Thentis Results in the Case of NERVA Derivative Systems .
Conclusion on the Coupled fine-coarse niesh solver ......


.... 142
. 14:3
.... 144
.... 146


7.2.2
7.2.3


8 CFD STITDY OF THE LOW-PRESSITRE THRITST CHAMBER WITH FROZEN
CHEMISTRY ............ ............ 158


Presentation of the Low-Pressure Thrust C'I! I11.1>er ..... ...

Quasi-one-dintension Isentropic All lli-;- of the Low Thrust C'I! In!l~er .
Two-Dintensional Contour Profile of the Thrust Chamber Walls ....
Grid Generation .. .. .. .

Low-Pressure Thrust C'I Inl~er Analysis using FLITENT and Frozen C!.


. .
. .
. .
. .
. .
. .
. .
. .
. .
. .


.. 158
.. 159
.. 161
..16:3


.. 16:3
.. 16:3
..165s
..165s
.. 166
.. 167
.. 167
.. 168
.. 170
.. 170


8.5.1 FLITENT ......
8.5.2 Numerical Procedure ......

8.5.3 Analysis results .........
8.5.3.1 Investigfation of the nozzle representation ......
8.5.3.2 Which governing equation to use? ......
8.5.:3.3 Investigfation of the turbulence model .......
8.5.3.4 Sensitivity study ......
8.5.4 Design Parameters Sensitivity Study .....
8.5.4.1 Influence of the thermal formulation ..... .

8.6 Conclusion on the Fr-ozen C!. ~!!I s-1y Analysis and Perspectives ...


9 CONCLUSIONS AND RECOMMENDATIONS ... .. .. .. 179


APPENDIX


A PROGRAM FOR COMPILATION OF THE EXACT RIEMANN PROBLEM
SOLUTION ............ ............ 18:3











B FLUENT ANALYZES ......... . .. 190

B.1 Low-Pressure Thrust C'I Inl~er: Design Parameter Analysis .. .. .. .. 190
B.2 Flow Area Contraction ... . . .. .. .. .. 191
B.3 Comparison of the Hydrogen Channel Simulation with a FLUENT Fr-ozen
ClaIns !!-iy Analysis ............ ......... 191
B.4 Definitions ......... . .. .. 192

C THEMIS SUBROUTINE LISTING: NERVA DERIVATIVE .. .. .. .. 197

C.1 Core subroutine from the system code ..... .. . 197
C.2 Interface subroutine ......... .. .. 202

D THE PROPERTY LOOP: INFLUENCE OF THE HYDROGEN DISSOCIATION203

D.1 No Dissociation ......... . .. 203
D.2 Small Dissociation ......... . .. 204
D.3 High Levels of Dissociation ........ ... .. 204
D.4 Program ......... . ... .. 204

REFERENCES ............. ........... 220

BIOGRAPHICAL SK(ETCH ....._._. .. .. 228










LIST OF TABLES


Table page

1-1 C'I I) Il-teristics and operating conditions of different nuclear systems [1] .. .. 27

4-1 Error on the density formulation for P=0.1 bar .... .. .. 61

4-2 Error on the density formulation for P=1 bar .. .. .. 62

4-3 Error on the density formulation for P=10 bar .... .. .. 6:3

4-4 Error on the density formulation for P=100 bar .... .. .. 64

4-5 Error on the density formulation for P=150 bar .. .. .. 65

4-6 VHTR operating conditions and features ..... .. . 66

4-7 Comparison between experimental data and ideal-gas law for helium density 71

7-1 Fanning friction factor correlations and their range of validity [2] .. .. .. .. 147

7-2 Nusselt number correlation for turbulent gas flow in a pipe [:3] .. .. .. .. 147

7-3 Pressure drop over a flow area contraction. Comparison between FLITENT simulation
and 1-D correlation results. ......... .. .. 147

7-4 Properties at diverse locations of the square lattice honeycomb system for thrust
chamber inlet temperature and pressure of 2,800 K( and 500 psi (=:3.45 1\Pa),
respectively. ......... ... .. 149

7-5 Pressure drop across the SLHC core for different densities of foam fuel. .. .. 149

7-6 Comparison of the results obtained with NuRok and Thentis .. .. .. .. .. 150

8-1 Hydrogen thermal properties (heat capacity,specific heat ratio and thermal conductivity)
for diverse sets of operating conditions ...... .. . 159

8-2 CPIT time (s) and number of iteration to reach convergence of the steady-state
for the four different grids . .. ... .. .. 171

8-:3 Test cases for different design parameters .... .. .. 17:3










LIST OF FIGURES


Figure page

2-1 The serial cooling system (NERVA type) ...... .. 3:3

4-1 Level of hydrogen molecule dissociation as a function of temperature .. .. 60

4-2 Hydrogen heat capacity vs. temperature at low and high pressure (0.1 and 150
bars) .... ........ ............... 61

4-3 Hydrogen density at 0.1 bar and high temperature ... .. .. .. 62

4-4 Hydrogen heat capacity vs. temperature ...... .. . 6:3

4-5 Hydrogen viscosity vs. temperature . ..... .. 64

4-6 Helium viscosity vs. pressure for different temperatures .. .. .. 65

4-7 Helium thermal conductivity vs. pressure for different temperatures .. .. .. 66

4-8 Carbon dioxide density vs. pressure for different temperatures .. .. .. .. 67

4-9 Carbon dioxide heat capacity vs. temperature for different pressures .. .. .. 68

4-10 Carbon dioxide viscosity vs. temperature for different pressures .. .. .. .. 69

4-11 Carbon dioxide thermal conductivity vs. pressure for different temperatures 70

5-1 Sketch of the axisyninetric representation of the :3-D channel .. .. .. .. .. 100

5-2 Location of the numerical nodes for a 5x5 domain .. . .. 101

5-3 Elementary volume (blue area) for integration in the axisyninetric configuration 102

5-4 Details of the notations of the axisvninetric cell (i, j) ... .. .. .. 10:3

5-5 Nomenclature for the surface vectors . ..... .. 104

5-6 Algorithm which shows how the real properties are intpleniented and computed 105

5-7 The shock tube experiment ......... .. .. 106

5-8 Shock tube profile at t= 6.1 nis in case 5-62. The blue curves are the theoretical
profiles and the green curves the profiles computed with our Euler solver .. 106

5-9 Properties profile at t= 6.1 nis in case 5-62 for air, helium, carbon dioxide and
uranium tetrafluoride. ......... . .. 107

6-1 VHTR coolant channel .. ... .. .. 118

6-2 Dual cell for the approximation of the derivatives at the cell interface i + i 119










6-3 Helium channel mesh: 100x1000 quadrilateral grid zoomed at the inlet of the
channel ......... .... . 119

6-4 Static pressure (Pa) profile on the axis of the test-section. Comparison between
FLUENT simulations and our solver. . ..... .. 120

6-5 Velocity (m/s) profile on the axis of the test-section. Comparison between FLUENT
simulations and our solver. ......... .. .. 121

6-6 Temperature (K() profile on the axis of the test-section. Comparison between
FLUENT simulations and our solver. . ..... .. 122

6-7 Static pressure (Pa) profile on the axis of the test-section. Comparison between
constant and real property formulations. ...... .. . 123

6-8 Velocity (m/s) profile on the axis of the test-section. Comparison between constant
and real property formulations. ......... ... .. 124

6-9 Temperature (K() profile on the axis of the test-section. Comparison between constant
and real property formulations. ......... ... .. 125

6-10 Property profiles across the helium cooling channel. ... .. .. .. 126

6-11 Heat profile on the channel walls ........ ... .. 127

6-12 Absolute pressure (j! Pa) profile on the axis of the NTP coolant channel .. 128

6-13 Temperature (K() profiles on the axis and the wall of the NTP coolant channel .129

6-14 Density (kg/m3) profile on the axis of the NTP coolant channel .. .. .. .. 130

6-15 Molecular weight (g/mol) profile on the NTP axis channel for Ti = 1, 500 K( (no
dissociation) and Ti = 1, 700 K( (very few dissociation) ... .. .. 131

7-1 The different NTP systems modeled with NuRok: NVTR, NERVA derivative
and P&W XNR 2000 I & II ......... .. .. 148

7-2 3-D grid of the contraction test section for study with FLUENT. .. .. .. .. 149

7-3 Static pressure profile along the axis of the contraction test section .. .. .. 150

7-4 Identification and location of portions of the NTP system which would benefit
from a more detailed solver ......... .. .. 151

7-5 Structure of the code coupling when the CFD is used to simulate the hot channel 152

7-6 Extrapolation of the temperature nodes from the axisymmetric geometry to a
single value. ......... ... .. 153

7-7 Pressure evolution in MPa at the inlet and exit of the hot channel vs. the iteration
number .... ........ .............. 154










7-8 Thermal characteristics of the hot channel vs. the iteration number .. .. .. 155

7-9 Coolant channel pressure profiles calculated with NuRok and Thentis .. .. 156

7-10 Coolant channel temperature profiles calculated with NuRok and Thentis (on
the axis and the wall) ......... . .. 157

8-1 Thrust chamber and its principal characteristics ... .. .. .. 171

8-2 Coarser grid ......... . .. .. 172

8-3 Comparison of the Mach number profile for the axisyninetric and 2-D models .172

8-4 Alach number profiles for an inviscid and viscous flow of hydrogen .. .. .. 174

8-5 Alach number profiles obtained for different turbulence models .. .. .. .. 175

8-6 Relative difference in the nmaxiniunt Aach number between a given grid and the
finest grid ......... ... . 176

8-7 Influence of the design parameters on the nmaxiniun velocity on the exhaust plane 177

8-8 Velocity magnitude profile on the exhaust plane of the thrust chamber .. .. 178

B-1 Meshes for the two contraction experiments ..... .. . 19:3

B-2 Velocity profile on the contraction plane in the single channel configuration. Comparison
between computed and theoretical profiles ..... .. . 194

B-:3 Velocity profile on the contraction plane in the four-channel contraction experiment.
Comparison between computed and theoretical profiles ... .. .. 195

B-4 Pressure profile (jl!pa) on the axis of the coolant channel. Comparison between
the real property solver and FLUENT with frozen chemistry .. .. .. .. .. 196

D-1 The property loop ......... . .. 216

D-2 Evolution of the 1 its jur properties with the number of iteration in the hydrogen
pipe with no dissociation ......... .. .. 217

D-:3 Evolution of the 1 its jur properties with the number of iteration in the hydrogen
pipe with low levels of dissociation . ...... .. 218

D-4 Evolution of the 1 its jur properties with the number of iteration in the hydrogen
pipe with higher levels of dissociation . ..... .. .. 219









LIST OF ACRONYMS


AMG Algebraic niultigrid

BWR Boiling water reactor

CFD Computational Fluid Dynamics

DNS Direct numerical simulation

FDM Finite difference method

FEM Finite element method

FVAI Finite volume method

FOIT First order upwind

GFR Gas-cooled fast reactor

HTGCR Higfh temperature gas-core reactor

INL Idaho National Laboratory

INSPI Innovative Nuclear Space Power and Propulsion Institute

ISP Specific impulse

LES Large eddy simulation

LWR Light water reactor

MITSCL Monotone upstrean1-centered schemes for conservation laws

NASA National Aeronautics Space Agency

NEP Nuclear electrical propulsion

NTP Nuclear thermal propulsion

PDE Partial differential equation

P&W Pratt and Whitney

PWR Pressurized water reactor

QUICK( Quadratic upwind differencing

R ANS Reynolds averaged Navier-Stokes

RNG Renornialized group theory










R S1\ Reynolds stress model

SLHC Square lattice honeycomb

SOU Second order upwind

VHTR Very high temperature gas reactor









LIST OF NOMENCLATURE


English Symbols

a speed of sound (nt/s)
A flow cross-sectional area ni2

y gravitational constant
U vector of conservative variables

V vector of primitive variables

R universal gas constant

T temperature (K()

p pressure (Pa)

(u, v) velocity components in artesian coordinates

(ur, u,) velocity components in polar coordinates

k thermal conductivity (W/K(.s)

At Alach number

At molecular weight

k turbulent kinetic energy

V velocity vector

cp heat capacity

Qc Heat source

E total energy per unit volume

t tinte(s)

f body forces

r radius (cm)

re throat radius (cm)

n number of moles (nlol)

Re Reynolds number

Pe Peclet number










D diameter (nt)

dH hydraulic diameter (cm)

(.r, y, x) position in artesian coordinates

(r, 8, x) position in polar coordinates

R,. throat radius of curvature

Yk nozzle exit slope

L contoured nozzle length

T, wall temperature (K()




Greek Symbols
Specific heat ratio

p density (kg/n?)

Turbulent dissipation

I-1 viscosity (Pa.s)

I-t, turbulent viscosity (Pa.s)

-r shear stress tensor

II total shear stress tensor

6.r characteristic length (nt)

Diffusion coefficient

8, subsonic approach angle

8 nozzle inflection angle










Abstract of Dissertation Presented to the Graduate School
of the University of Florida in Partial Fulfillment of the
Requirements for the Degree of Doctor of Philosophy

A HYBRID FINE-COARSE MESH COMPUTATIONAL FLITID DYNAMICS AND
HEAT TR ANSFER MODEL FOR ADVANCED NUCLEAR ENERGY SYSTES HS

By



December 2007

C'I I!1-: Santin Angfhaie
Major: Nuclear Engineering Sciences


The next generation of nuclear reactor systems is designed with strong emphasis

on safety, efficiency, economy, and reducing proliferation threats. New designs include

new operational conditions and new types of working fluids. This doctoral research aints

at developing a robust simulation tool for thermal fluid simulation of advanced high

temperature gas-cooled reactors. It results in a hybrid fine and coarse niesh numerical

approach to solve Navier-Stokes equations with real gas properties.

The model uses coarse niesh points to simulate the thermal fluid behavior of the

entire nuclear system. Complex geometries and flow are modeled with a fine niesh

Computational Fluid Dynamics (CFD) solver for higher accuracy. The CFD code is based

on two-dintensional axisyninetric R ANS approach combined to a TVD Macdorniack

intplicit-explicit discretization scheme. Spalart-Allmaras and k-e models are selected to

account for turbulence.

The solver models the real properties of gases of interest (helium and hydrogen) in

gas-cooled nuclear systems. The real-property CFD model development is motivated for

the study of low pressure thrust chambers for the Nuclear Thermal Propulsion (NTP): at

low pressure and high temperature, hydrogen molecules dissociate, molecular weight, heat

capacity and viscosity show strong dependence with temperature and pressure. In the real










property solver the properties are discretized in parallel with the T li-; -Stokes equations.

Such drastic changes cannot he modeled with traditional CFD codes.

The tinte-dependent CFD solver iteratively sets the steady-state condition in the

fluid systems of interest. It is applied to the study of high temperature reactor systems

portions. Results of the analysis of the heliuni-cooled channel for High Temperature

Gas-Cooled Reactors and hydrogen-cooled channel for NTP show significant differences

compared to frozen chemistry approaches, validating the need for a real property CFD

solver development.

Once validated on the coolant channels, the real property fine-niesh solver is

intpleniented into a coarse niesh system code. The coupled fine and coarse-niesh solver is

applied to model the entire NTP system: high resolution solution is obtained by modeling

the core with the fine-niesh solver. Simulation of the system with the new coupled code

shows great intprovenient in accuracy, speed and stability.









CHAPTER 1
INTRODUCTION: ANSWERING THE NEEDS OF THE FUTURE IN THE NUCLEAR
INDUSTRY AND SYSTEM MODELING

1.1 The Nuclear Power Scene of the 21st Century

Research and development in the field of nuclear power production is a very dynamic

field. The Generation IV International Forum was established in January 2000 to

investigate innovative nuclear energy system concepts to meet future energy challenges.

The members include Argentina, Brazil, Canada, Euratom, France, Japan, South Africa,

South K~orea, Switzerland, United K~ingfdom, and United States, with the OECD-Nuclear

Energy Agency and the International Atomic Energy Agency as permanent observers.

The forum serves to coordinate international research and development on promising new

nuclear energy systems for meeting future energy challenges: The new power plants should

meet the increased energy demands, while being resistant to diversion into weapons of

mass destruction and safe from terrorist attacks. In 2002, the final Generation IV concepts

have been selected to shape the future of nuclear energy. The international task force

agreed on six nuclear technologies, which represent advances in sustainability, economics,

safety, reliability and proliferation-resistance Out of the six concepts to be developed in

the scope of Generation IV nuclear reactors, two reactors are gas-cooled: the very high

temperature gas reactor (VHTR) and the gas cooled fast reactors (GFR). They are aimed

at energy and hydrogen production. Both systems are helium-cooled reactors with outlet

temperatures of 1,000 and 850 degrees Celsius, respectively. On June 28, 2005, the US

Senate voted a wide-ranging energy policy bill which includes the authorization for reactor

development work at Idaho National Lab in support of the Generation IV initiative

and the nuclear hydrogen initiative. While the US Department of Energy is supporting

research on several reactor concepts, priority is being given to the VHTR.

In December 2006, the National Aerospace and Space Agency (NASA) announced

its will to build a base on the Moon by 2020. This implies not only manned-missions

to the Moon, but also to Mars and of further planets, for which the Moon would serve










as a launch hase.The project has given a new start for research in the area of space

nuclear propulsion. Started in the late 50's in the USA, the project has been progressively

abandoned after the end of the cold war and the end of the space race with the Soviet

U~nion. Nuclear space propulsion allows for faster and longer space travels. Therefore,

there is a renewed and growing interest for the concept.

1.2 Challenges in Thermal-Hydraulics

1\ost of operating nuclear reactor are Light Water Reactors (LWR): As of December

2005 85' of worldwide nuclear plants are cooled with light water, which represents 91

of the total electrical power generated from nuclear technology [4]. Within the scope of

Gen IV development, the United States is developing the VHTR. They also are leading

research on Nuclear Thermal Propulsion. Both concepts use inert gases as working fluid,

separating themselves from the more traditional systems operating with water. Operating

characteristics of VHTR and NTP are compiled in Table 1-1 as well as the Pressure Water

Reactor (PWR) and Boiling Water Reactors (BWR) for comparison. The type of coolant

and operating conditions are greatly different between designs of the future (VHTR

and NTP) and last generation of reactors (PWR and LWR). Besides, the difference in

coolant nature (helium or hydrogen vs. water), new reactors are operated at much higher

temperatures .

Over the years, the nuclear industry has largely relied on simulation tools to design

systems, analyze fuel cycle, as well as studying the behavior of reactors in case of severe

accidents. Besides neutronics and transport codes, thermal-hydraulics codes represent

a large part of the simulation tools used by engineers and researchers. Some of the

established thermal-hydraulics computer codes currently used in the nuclear industry










are RELAP-3D1 TRACE2 and CATHARE3 They are power plant simulation codes,

generally applied to extensive analysis of severe accidents.

Simulation of the flow behavior is obtained by solving the equations of conservation

of mass, momentum and energy. The software use heat transfer, generic component

and control systems models to describe the behavior of nuclear power plants with water

(liquid and vapor) and other non compressible gases. The entire power plant is defined

using volumes and junctions. Pump, turbine, pipes, pressurizers, branches and so on, are

examples of volumes. Each volume is decomposed into axial nodes on which the algebraic

governing equations are solved. We refer to this type of codes by the name of system

codes and to the characteristics of the meshing by the appellation coarse mesh. The

momentum equations solved by these codes are the Euler inviscid equations instead of the

basic Navier-Stoke equations because they are much more simple. Wall friction and heat

transfer are evaluated from empirical correlations and then applied to model turbulence

effects. In their most updated versions, they comprise 3-D fluid dynamics modules. Due

to coarse nodalization, these modules require volume-averaged coefficients for turbulent

exchange of mass, energy and momentum. Such coefficients are not well known. This

type of system code is not adequate in the case of complex geometries. These system

codes are intended for a limited range of fluids (principally liquid water and vapor) and

temperatures. They are not developed for high temperature gases.

1.3 Issues Faced with the Current Computational Tools for
Thermal-Hydraulics

The industry needs powerful tools for the design of the future generation of nuclear

reactors. Because experiments are very costly, developers of the new concepts need to rely




1 LWR accident analysis code Developed at Idaho National Laboratory

2 LWR accident analysis code Developed at Nuclear Regulatory Commission

3 LWR accident analysis code Developed at CEA-Grenoble, France










mostly on accurate numerical tools. The current thernial-hydraulics codes cannot he used

for the fluids and operation ranges suitable to the next generation of reactors. They need

to be modified in order to rely partially on empirical correlations and models because we

do not have the experimental resources to do so.

For a decade, nuclear engineers investigating original designs or particular portions

of the nuclear system have been relying on Computational Fluid Dynamics (CFD).

Compared to the system codes previously discussed, volumes of interest are decomposed

into a fine grid in all directions to be analyzed with CFD: CFD models any flow nmap by

solving the full set of Navier-Stokes equations on the discretized domains. We then refer

to this type of solver by the name of fine niesh solver. Historically, CFD software used

to be primarily developed for aerospace applications. Because it remains very versatile,

applications of CFD have expanded to a large variety of engineering fields. Nuclear

industry has been increasingly relying on CFD modeling since it makes possible a reduced

reliance on empirical correlations and experiments.

Advances in the numerical techniques have made feasible the model of realistic

reactor geometries for any type of fluid and any range of operational pressures with a

few limitations still: the flow domains needs to be discretized into very small nieshes.

Therefore, Computational Fluid Dynamics can only model a part of the entire nuclear

plant. The main reason is the extreme computational cost. It still remains a partial

answer to the need in computational tools for the simulation of new generation of nuclear

power generation systems involving gaseous flows at high temperature. Helium is heated

and cooled over a 700 K( range in the VHTR (cf. Table 1-1) and hydrogen undergoes

a 3,000K( increase in temperature in the NTP. This condition on temperature is very

different front the conditions which can occur in aerospace applications: initially, the CFD

techniques would consider constant properties for the gases of interest. In gaseous nuclear

systems, some properties such as heat capacity for hydrogen see variations of up to 5011' .










The CFD codes which are available are not built to handle such variations and can only

be used with caution for the analysis of high temperature gaseous reactors.

1.4 Ob jectives

In the previous section, we highlighted the shortconlings of both thernial-hydraulics

(coarse-niesh) codes and CFD (fine-niesh) codes:

Safety codes: only allow one- and two-dintensional description of the geometry of the
flow. They are only accurate for specific ranges of pressure and temperature.

CFD solvers: coniputationally expensive. They have limited definition of the
temperature and pressure dependence of the fluid properties.

Since system codes cannot he accurate in certain areas of the new nuclear systems

and CFD codes cannot he used to study the entire system, we -II---- -1 to couple a

system code with a CFD solver to combine both advantages of the two numerical

tools: the coupling technique answers the industry needs in accuracy while limiting

the computational expense. On one hand, the system code and its coarse niesh provide

a fast and accurate simulation of most parts of the nuclear system. Simultaneously,

Computational Fluid Dynamics nunterics is used to analyze portions of the system. These

portions exhibit either particular geometries or specific operational conditions, for which

the traditional numerical tools lack accuracy in flow prediction.

The first objective of the doctoral work presented here is to demonstrate that it is

possible to obtain a powerful computational tool which couples fine and coarse niesh

simulations for thernial-hydraulics predictions. To prove the feasibility of this project,

the 1-D coarse niesh code NuRok [5] which is developed at the Innovative Nuclear Space

Power and Propulsion Institute at the University of Florida is expanded. NuRok aints at

modeling the thernial-hydraulics of different types of Nuclear Thermal Propulsion systems.

The entire path of the fluid flow is modeled. NuRok aints at designing NTP systems

as well as dintensioning each various component. A more detailed overview of the code

functionality and composition is given in C'!s Ilter 7. NuRok is particularly adapted to the

coupling study because:










The entire source code is accessible, making it easy to add components to it.

It is a perfect example of 1-D coarse niesh simulator which gives excellent results in
the overall system but lacks accuracy in some particular areas.

To meet the requirements of the coupling, CFD solvers particularly adapted to the

gaseous nuclear reactors need to be developed. Therefore, the second objective of this

dissertation is to develop a CFD solver which accounts for the real properties of fluids.

1.5 The Different Approaches

The ultimate goal is to build a coupled simulation tool for increased accuracy in gas

reactors flow predictions. Several approaches have been investigated in the domain of

coupling coarse one-dintensional codes to fine-niesh CFD solvers.

To illustrate the coupling techniques, the reference case is the study of the Nuclear

Thermal Propulsion. The NuRok simulation code is written in order to study NTP: it is

written in a modular fashion with FORTR AN subroutines. Each portion of the system

being described with a different subroutine: if one of the portions needs refined and

detailed CFD analysis, its corresponding subroutine can he easily modified. Here is a list

of approaches for intplenientation of the CFD models in the coarse niesh solver and a

discussion of each of the techniques:

Sensitivity study: The section of interest is modeled with FLITENT. An extensive
analysis of the influence of the operational parameters is led and a trend is
extrapolated front the results. This type of work had been started for the study
of the Square Lattice HoneyComb design of the propulsion core [6]. This technique
require extensive preliminary work to cover as many operating conditions as possible:
the final product is restricted by the operational conditions which have been covered
in the sensitivity study. Also, the behavioral correlations obtained could introduce
further errors. This solution does not require a lot of takedown but it proves to be
a limited answer to the modularity of the numerical tool developed. Therefore, this
technique will not he covered in the case of the doctoral work

Full coupling with FLITENT: in this approach, the models of the portion under
consideration are still developed with FLITENT. This time, an interface would
be created in between the system code and FLITENT. This approach exhibits
several limitations: first the interface is challenging to create because FLITENT is
a coninercial code for which the source is not accessible. Second, the time steps
inerrant to the two codes are not controllable which could cause instability of the










overall coupled code. Third, the CFD commercial codes are limited in the area
of modeling of the real properties of any gas. Therefore, this technique which
is considered in several institutions will nor be used is our attempt to build a
fully-coupled thermal-hydraulics simulation tool. Other CFD codes were considered
for this technique such as STAR-CD or FEMLAB but the issues are the same will all
the CFD codes.

*Full coupling with in-house CFD code: For a better control of the coupling between
the two codes and to increase accuracy of the fine mesh simulations, a CFD code
could be written which would be specific to the systems of consideration in Nuclear
Thermal Propulsion and which would account for the real properties of the gases.

Because the last approach enables the most flexibility and accuracy, it is the approach

which is considered for the work presented in this dissertation.

1.6 Overview of the Dissertation

Before the coupling techniques and results can be presented, the dissertation

describes the numerical techniques chosen to develop the real-property CFD solver.

After demonstrating the relevance of the implementation of the gas property exact

description, the CFD solver developed is applied to simulations of coolant channels in

High Temperature Gas-Cooled Reactors and Nuclear Thermal Propulsion. First, a coolant

channel with helium flow and linear temperature profile on the wall is studied with the

in-house CFD solver just developed. Then, the effects of hydrogen dissociation on the

numerical technique are investigated on the Nuclear Thermal Propulsion coolant channel:

1. at high pressure and high temperature: no dissociation.

2. at low pressure and high temperature: low dissociation.

3. at low pressure and very high temperature: high dissociation

Then, the document describes the techniques to couple the CFD solver to the one-dimensional

coarse mesh solver. The coupled numerical tool, referred as Themis, is applied to the

study of the NERVA derivative systems. The hot channel of the system is modeled with

the CFD solver and coupled to the one-dimensional code.










The final chapter presents the frozen chemistry CFD analysis of the low-pressure

thrust chamber for application in the NTP system and an estimation of the performance

of such a design.































Table 1-1:
type of
reactor
pressurized
water reactor
boiling
water reactor
gas-cooled
reactor
NTP


C'I I) Il:teristics and
coolant phase


conditions of different
inlet exit
temp(K() temp(K()


nuclear systems [1]
pressure rjh
(\!P'a) (1:/h


operating
working
fluid


liquid water
liquid
vapor water

gas helium
SGL hydrogen


603

559

828
3200


15.3

7


water

water

helium
hydrogen


5 5
0.01-5 5.10-4










CHAPTER 2
APPLICATION FUNDAMENTALS

To better understand the challenges of the doctoral work, the chapter gives a short

description of the domains of application of interests as well as a quick presentation of the

1!! li.r~ tool used through the dissertation: computational fluid dynamics. Applications of

the research work include nuclear space propulsion and more particularly Nuclear Thermal

Propulsion. Basic principles of a low-pressure thrust chamber for combination to space

propulsion are also explained.

2.1 Space Propulsion and Nuclear Thermal Propulsion

Historically, space rockets have been powered by chemical propulsion. Specific impulse

(Isp), also called specific thrust, is a measure of the rocket performance. It is defined as

the level of thrust divided by the propellant mass flow rate (Equation 2-1).


thrust vezhaust
Isp (2-1)
propellant mass flow rate g

where vezhause is the velocity at the exit of the thrust chamber and g is the gravitational

constant.

C'I. inu! d1 propulsion is the only technology used to launch rockets as of tod .v. Its

performances are limited compared to other systems under investigations such as space

nuclear propulsion: chemical propulsion Isps are no greater than 475 s-l. In contrast to

traditional chemical propulsion, nuclear propulsion could provide Isps at least twice as

big. First estimations evaluated that nuclear propulsion would reduce a trip to Mars from

600 d .rs to about 200 d 7i~ Space nuclear propulsion is hoped to create a breakthrough

in space exploration, allowing manned missions to Mars or rocket exploration of outer

planets.

One of the current approaches to space nuclear propulsion is the nuclear thermal

propulsion (NTP), which stands as the main focus of this study. In the NTP design, cold

hydrogen is heated by a compact nuclear core. Temperature of the fluid at the exit of










the reactor is about :3,000K(. Hydrogen then enters the thrust chamber where it expands.

Finally, it accelerates to velocities as large as 10,000 nt/s.

The investigation of the performances and design of the NTP started in 1955 with a

joint effort between NASA and the Atomic Energy Coninission [7]. It was terminated in

197:3 due to the change in national priorities. Over the twenty years of active work on the

project, nineteen cores were investigated combining various types of solid fuel and core

design.

The general layout of a NTP system is shown Figure 2-1. Liquid hydrogen is stored in

a tank at a temperature of 20-22 K(. Over the years, it as been established by the nuclear

engineering coninunity that redundancy of the turbo-machineries is the first step for plant

safety. In cases when one of the turbo-punip system would fail, the redundant systems

would help safe operation of the system. The same principle is applied to NTP and the

main hydrogen flow line is separated into at least two parallel and identical systems. First,

hydrogen flows through a pump which extracts the fluid front the tank. Since the fluid is

heated, compressed and is in a single gaseous phase at the exit of the pump. Hydrogen

cools down the walls of the nozzle and of the core reflector. Then, it enters turbines which

are coupled to the first pumps. The different flow lines are recombined before they enter

the core of the propulsion. At the exit of the nuclear core, temperature of the fluid reaches

:3,000 K(. The very hot hydrogen then enters the thrust chamber, a diverging-converging

nozzle. The gas expands and accelerates, thrusting the rocket.

The Innovative Nuclear Space Power and Propulsion Institute at the University of

Florida developed a system code to model the flow of the gas in NTP systems [8]. The 1-D

code aints at sizing the different parts of the system for given operational thermal power

and core exit pressure. It has been developed such that a large variety of NTP designs can

he studied, as well as a panel of system organizations. This coarse niesh simulation tools

will be referred in this dissertation as NuRok (Nuclear Rocket). The skeleton of the code

relies on the resolution of the montentunt equation. However, it required experintent-hased










relations for characteristics such as friction factor or Nusselt number. NuRok users can

monitor pressure and temperature at any point of the NTP system of consideration.

This code is very competitive to get a good qualitative idea of the performances

of a given system within a set of operating conditions. Since it is a one-dimensional

representation of the NTP, it gives results on the Isp, mass flow rate and other characteristics

within seconds. Several factors limit the accuracy of the results obtained. First, the

one-dimensional representation of highly non-symmetric geometries biases the simulation.

Besides, relations on friction factor or Nusselt number for example would require

experimental data on hydrogen in temperature and pressure conditions comparable to

the ones of NTP systems. However, experiments with hydrogen-gas for temperatures

as high as 3,000 K( is costly and hazardous: high temperature hydrogen is extremely

flammable and corrosive. Hence, very few information on the behavior of hydrogen in

conditions comparable to the ones of NTP is available. It is then difficult to estimate if

the relations available in the literature for different gases or operating conditions apply to

our system.

2.2 The Low-Pressure Thrust Chamber

The previous section highlighted the main interest of space nuclear propulsion.

We believe that using nuclear propulsion will increase space rockets Isps. Typically,

the nuclear core reactor exit temperature and pressure are equal to 3,000 K( and 1850

psi (about 12 MPa), respectively [9]. In the case of a lower pressure with constant

temperature, the molecules of hydrogen (H2) Start dissociating. As they recombine, there

is extra energy which is liberated in the gas. The dissociation energy of a molecule of

hydrogen H2 is equal to 4.52 eV [10]. This can be observed when considering the heat

capacity cp which is discontinuous at the pressure/temperature of dissociation [11].

The properties of hydrogen and dissociated hydrogen will be described more in detail in

C'!s Ilter 4. However, the behavior of the hydrogen molecule as it dissciates/recombines

leads to the following statement: We believe the dissociation/recombination effects of










the hydrogen molecule at high temperature and low pressure is transmitted to the gas

under kinetic energy. Hence, if we operate a Nuclear Thermal Propulsion system at low

pressure, hydrogen molecules will dissociate in the core. As the flow passes through the

thrust chamber, hydrogen temperature decreases, molecules recombine, transmitting

kinetic energy to the flow. Hence, If we operate the core at low pressure, we think that the

exhaust velocity of the hydrogen flow through the system will be greatly increased, leading

to high values of the Isp, and thus increase the performances of the system.

2.3 Computational Fluid Dynamics: An Overview

Fluid (gas and liquid) flows are governed by partial differential equations (PDEs)

which represent conservation laws for the mass, momentum, and energy. The system of

equations is referred to as the Navier-Stokes equations. It can only be solved numerically.

To get the flow map, the solution also requires an equation of state, for example ideal-gas

law if it applies. Computational Fluid Dynamics (CFD) is the technique of replacing such

PDE systems by a set of algebraic equations which can be solved using computers. A CFD

solution is obtained after performing the following steps:

Discretization of the domain

Discretization in time and space of the partial differential equations

Iteration to obtain a converged numerical solution

Post-processing

Once a discrete solution in space and time is obtained a flow map is provided: it

consists of the values of density, velocity components, energy and pressure or temperature

at every mesh point. In most flow patterns, the geometry is complex and parts of the

Navier-Stokes equation components require modeling (e.g. turbulence effects). The

accuracy of the numerical solution depends on the quality of the discretization technique

and relevance in the model of choice. Several numerical methods and algorithms have

been proposed and developed over the last three decades. Despite several decades of

investigation no single method or approach is robust enough for simulation of any










type of flow. A compromise between simplicity, ease of implementation, accuracy and

computational efficiency ought to be made. Appropriate approximations and models are

developed to reach the highest quality possible of the information generated by CFD.

We now detail the differences between CFD simulation and experiment. Traditionally,

CFD gives an insight into flow patterns that are difficult or expensive to study using

traditional experimental techniques. Experiments give a quantitative description of the

flow for one quantity at a time (e.g. temperature, pressure or, velocity), at a given number

of points, for a laboratory-scale model and for a limited range of problems and operating

conditions. Errors can arise from measurement errors or flow disturbances by the probes.

Similarly, numerical simulations result in a quantitative prescription of the flow for all

desired quantities with a high resolution in space and time, for the actual flow domain and

virtually any realistic problem and set of operating conditions. However, CFD does not

replace the measurements completely since numerical techniques require benchmarking in

order to assess their accuracy.

For the type of systems we want to analyze here, simulation is a much more adapted

method. CFD is much faster, broader and cheaper technique than experimentation.














L


Iii


Figure 2-1: The serial cooling system (NERVA type)


r'


'Thrust
chamber









CHAPTER 3
A SURVEY ON STATE-OF-THE-ART COMPUTATIONAL FLITID DYNAMICS
DEVELOPMENTS AND APPLICATIONS

Since the early 1970s, commercial software packages have become available,

making CFD an important component of engineering design and analysis [12]. Ten

years later, the use of commercial CFD software started to become accepted by 1 in r

companies around the world rather than their continuing to develop in-house CFD codes.

Historically, the nuclear industry has relied on CFD capabilities in its first stages of

development: CFD was applied to safety analysis and core reload design of nuclear power

plants. Over the last twenty years dramatic improvements in both numerical techniques

and computing power have offered the possibility of performing large-scale, high-fidelity

core thermal-hydraulics analysis.

The most commonly used commercial CFD codes are the finite volume solvers

FLUENT [13], CFX [14] and Star-CD [15]. These codes offer a large variety of solvers to

discretize and solve the N l.-;1 r-Stokes partial differential equations which describe most

flow behaviors. They also offer a large variety of modeling capabilities, including but

not restricted to turbulence, radiative heat transfer and two-phase modeling. To a large

degree, these codes offer very similar performances and modeling options.

Fr-eitas [16] has compiled a comprehensive benchmarking of the different commercial

CFD codes. These benchmark problems included: the backward-facing step problem,

the uniform flow past a circular cylinder, the three-dimensional shear-driven cavity flow

(all three laminar flows), the turbulent flow around a square cross-section tube and the

developing turbulent flow in a 180 degree elbow. The main drawbacks of the Freitas [16]

study were the lack of independence and completeness. The study was performed by

CFD code vendors and to a large extent did not include turbulence modeling. The

k-e model [17] is the most commonly used turbulence model in practical industrial

applications. A detailed formulation of k-e model is described in Chapter 5. It is this

particular model which accounted for turbulence modeling in the benchmark study of










Freitas. The benchmark test cases in Freitas study are extreme in terms of application of

this particular model. The study concluded that in general, the different commercial CFD

codes produce similar results. The Fr-eitas study, at the time, recommended an increase

in the spacial discretization order and a wider variety of turbulent models to choose from.

These features have been added to most commercial CFD codes over the last decade,

but due to highly competitive nature of the business,CFD vendors have not agreed to

participate in another benchmark exercise.

MI lIly of the nuclear reactor research programs around the world are focused on the

development of numerical tools for safety analysis of Light Water Reactors (LWR) with

emphasis on increasing the simulation accuracy. In 2003, Yadigaroglu [18] presented

the trends in LWR safety analysis. The article underlined the shortcomings of the

traditional system codes and proposed to move toward multi-dimensional analysis by

taking advantage of single and two-phase detailed CFD capabilities. The main issue

addressed by the paper was the limitation in computing power and skills for use of CFD

in practical problems. The main areas of interest for use of CFD capabilities are the

primary loop and containment of nuclear power plants. To date, CFD 2.1, lli-~ performed

in nuclear power plant systems included boron dilution in the primary loop of Pressurize

Water Reactors (PWR), mixing of various gases, stratification of containment pools, etc.

Boyd [19], [20] analyzed a 1/7th and full-scale steam generator inlet plenum mixing

during a PWR severe accident. One-sixth of the inlet plenum was modeled with FLUENT

6.0 using the Reynolds Stress Model (RSM) associated with non-equilibrium wall function.

Experiments on the 1/7th scale model validated the accuracy of the turbulence model for

the inlet plenum geometry and flow. However, numerical results of the study exhibited

an non-physical oscillatory behavior of the plume for the geometry associated with a

symmetry plane. This problem had already been identified with the use of FLUENT's on

several test cases. The non-physical behavior can be attributed to the shortcomings of










the earlier versions of FLUENT. Besides a few discrepancies in the area of the symmetry

plane, computed results were in good agreement with experimental data.

K~eheley [21] modeled a horon dilution transient during the shut down phase of a

PWR plant. The analysis was based on STAR-CD's Algebraic Multigrid Solver with the

Re-Normalization Group (RNG) k-e turbulence model. Simulation results were compared

to test data and it was shown that temperature dependent properties and huoi- ill i- effects

modeling could increase the accuracy of the simulation. The K~eheley's work demonstrated

that STAR-CD, could adequately model a typical horon dilution transient in PWR cores.

Tinoco [22] showed the feasibility of analysis of the steam line break transient in

Boiling Water Reactors (BWR) using STAR-CD. Since turbulences do not have time

to develop in such fast transients, the laminar solver was considered. Such a transient

actually occurred in Unit 1 and 2 at the Forsmark nuclear power plant. The incident was

caused by defective welding and led to the replacement of the core grids. Tinoco's CFD

simulation helped to develop a better understanding of incidents which could occur while

the plant is in operation.

Behavior of the water flow and heat transfer around rod bundles in LWR cores is

also a 1!! linr~ subject of interest in reactor safety study. The rod bundle is a standard

geometry in LWRs because of the fuel pin arrangement. Modeling of rod bundles in

LWR cores is a great test case for assessment of CFD codes. References [23] and [24]

present a wide v .1'i. r ii of experimental results involving flow and heat transfer in LWR fuel

bundles. Using the rod bundle test data, Tzanos [25] demonstrated the shortcomings of

the STAR-CD and it's associated linear k-e models for the simulation of such problems in

PWR cores. Tzanos simulation results showed a significant discrepancy with experimental

data. This was particularly true in areas of flow deflection such as spacer grids. The

results highlighted the need for other R ANS turbulence models or LES.

In et al. [26] led a more detailed analysis of the same problem by investigating

various R ANS turbulence models with the CFX code. These turbulence models included










the standard k-e model, the non-linear quadratic Speziale k-e model, the SSG (Speziale,

Sarkar, and Gatski) model and the RSM-w model. It is important to note that not all of

these models are available in the original source code, and the solver had to be modified to

allow for incorporation of such models. The RSM model showed best agreement with the

test results by accurately accounting for the anisotropy of turbulence and by predicting,

with great accuracy, the turbulence-driven secondary flows. The R SM model also enabled

computation of Reynolds stresses, wall shear stresses and wall temperature values very

close to experimental results. However, the RSM model lacked accuracy in prediction

of flow and heat transfer in tight lattice rod bundles. Other turbulence models had to

be investigated for cases where there is a large difference in geometrical scales. In's

simulation results for flow around a bundle with split-vane spacer [27] showed that the

standard and Speziale k-e models gave the most accurate predictions. Results of In et. al.

studies [26]-[27] proved that the choice of turbulence model must he carefully considered

depending on the flow situation.

The k-e model usually remains the safest choice in turbulence modeling. Hence, in

2001, a consortium of twelve European partners started the project [28] under sponsoring

of the European Coninission. ECOR A stands for Evaluation of Computational Fluid

Dynamics Methods for Reactor Safety All lli--- The project aimed at assessing the

performances of numerous CFD codes (CFX, FLITENT, Saturne, STAR-CD and TRIO-IT)

for simulating flows in the primary system and containment of nuclear reactors. The

project also highlighted and defined CFD code intprovenients that were necessary for

nuclear engineering applications. The interest in the three-dintensional flow effects in some

reactor components arose front the inability of one-dintensional system codes to predict

accurately flow behavior in specific areas of the reactor systems. Examples of systems

under investigation were:

fuid-fluid mixing in the cold leg and downcomer taking place during a small break
LOCA,










coolant mixing in the rod-to-rod and rod-to-wall gaps,

effects of spacers on coolant mixing,

mixing of both forced and huois ,ut jets/plumes, jet impingement on curved surfaces,
plunging jets, and interaction of parallel jets/plume,

horon-dilution events,

contact condensation in the two-phase stratified steam-water flow.

All the work we just cited testifies of the dynamic activity surrounding CFD modeling

in the area of LWR safety analysis. It is of great importance since we saw that 85' of

the nuclear power plants worldwide are Light Water Reactors. However, the system of

the future which is the most investigated tod .< in the world and in the United States

in particular is the VHTR. In 2004, Schultz [29] simulated the mixing of Helium in the

lower plenum of a prismatic Gas-Cooled Reactor. The standard k-e model of FLUENT 6.1

was considered along with buoi- .vwy modeling. The predicted hot and cold temperatures

were within 1.5 to 2 K( of the measured values. This result shows the good accuracy of the

simulation. Comparison was made possible by McEligot report [:30] which compiled a few

experiments that have been developed at INL to assess the performances of the CFD codes

in the case of the VHTR. More recently, Ahmad et al. [:31] studied the :3-D distribution

and pressure drop at the inlet plenum and rising channel of a pehhle-bed modular high

temperature reactor with CFX-5.7.1. The fluid was modeled by an incompressible and

isothermal flow and once again the k-e model accounted for turbulent effects. The model

was not benchmarked but Ahmad formulated recommendations for similar CFD studies.

However, considering an incompressible flow in the VHTR seems like a very limiting

assumption.

CFD studies in the nuclear industry are not limited to applications to the primary

loop of power plants. CFD is of great help for other fields of study related to nuclear

applications. For example, Lee [:32] used CFX capabilities to model the heat generated

by radioactive decay of dry spent fuel in a canister. The analysis required to account










for the buoi- .wv--induced natural convection and for the laminar flow of both Helium

and Nitrogen. The boundary condition of consideration at the wall was a thermal flux

derived from the power density generated by radioactivity. The computed results were

consistent with natural convection phenomena found in literature. Great resolution of the

computational grid close to the walls helped improve accuracy of the solver.

Therefore, CFD is widely used in the nuclear industry for simulations of very detailed

portions of nuclear systems. The review shows that CFD codes can he used to model

flow conditions of almost any nuclear system designs from traditional PWRs to designs

of the near future such as VHTR. CFD simulations are limited by the size of the domain

of consideration: because a power plant geometry is composed of so many different

length scales, a potential meshing of the entire system would require several million of

computational cells as we will see in the next section. Also, the review highlights the

shortcomings of some of the classical turbulence models available in commercial CFD

codes. When leading our own CFD analysis, much attention will be given to the turbulent

effects and their modeling in the systems we consider.

3.1 System Coupling

For all the previous applications we reviewed, only targeted portions of the systems

for analysis with CFD codes. On the opposite, Weber et al. [33] developed a high fidelity

representation of a current generation PWR with STAR-CD. The study modeled the

entire core for which the grid was composed of 240 million CFD computational zones.

Calculations required 50 hours of computation on 200 IBlM processors (in 2000) to get the

steady-state solution. The analysis proved that CFD can he used to assess fluid-dynamics

and thermo-mechanical issues that were not possible with more simplified approaches.

Even if such an extensive study is possible, it does not remain realistic as we want to

study transients of the core. This demonstrates again that CFD an~ ll-k- in nuclear reactor

systems should be restricted to small, targeted portions. In order for us to simulate the










entire Nuclear Thermal Propulsion system on a thermal-hydraulics point of view, we will

have to consider coupling codes.

Lots of recent international efforts have been focused toward the development of

coupled simulation codes which combine advantages of different computational software:

rapidity of 1-D system codes and accuracy of CFD codes. Research on coupling between

system-codes and CFD started in the late 1990's with the publication of Yoon and Suh's

work [:34]. It consisted of an indirect coupling applied to the K~orean next generation

reactor. A CFX simulation analyzed in detail the flow for the direct vessel injection,

while the bulk of the system was simulated with the thermal-hydraulics code TR AC. The

system code returned a boundary condition which became the inlet boundary for the CFD

simulation. This coupling technique is limited to steady-state cases.

Accident analyzes generally require full-coupling, capable of handling transients. With

that in mind, the Idaho National Laboratory (INL) started a co-operative project: couple

FLUENT with RELAP5-:3D [:35]. The 1 in r~ part of the nuclear system is simulated with

RELAP. One or more particular portion of the system is modeled with FLUENT. In the

particular example of Weaver the system is composed of a simple linear association of

volumes with a by-pass. The by-pass is composed of a heated channel modeled with CFD

and a simple pipe modeled with RELAP. RELAP starts calculating the map flow in the

first cells until it reaches the by-pass. The properties computed by RELAP at the inlet of

the by-pass are input as the inlet boundary condition for the FLUENT model. FLUENT

is run and returns a set of properties at the exit of the pipe which are implemented back

into RELAP as inlet conditions for the rest of the system. This type of coupling is referred

to as dynamic coupling. It is the optimum coupling for obtaining accurate solutions: this

coupling combines the advantages of the two types of codes.

The nuclear industry also needs similar coupling between CFD and neutronics

codes. This type of work is currently under investigation. The NURESIM project [:36]

is a very promising collaboration which aims at combining a pin-based Monte-Carlo










or deterministic transport calculation with a full CFD Thernial-hydraulics core-wide

sub-channel simulation code. This work should be noted however not relevant to this

research.

Most of the coupling difficulties when using coninercial CFD codes arise front the

data interpretation, front the module interface. Such coupling should be made easier by a

standardization of the coninercial CFD code architecture and module interfaces [:37].

The first part of the review shows that coninercial CFD codes when used properly

can give satisfying results to compute portions of flows in nuclear systems. However,

Weber's study [:33] showed that it is not realistically possible to simulate the transient

of an entire system because of the computational cost. Several attempts are made at

coupling system codes with coninercial CFD codes but the module interface makes the

task really difficult. Also, the coninercial codes are not built to consider large variations

of properties with temperatures and pressure. For all the previous reasons, we decided

to develop out own solvers as we explained in the introduction. Our solver should be

applicable to compressible high Reynolds number flows over all ranges of Alach numbers

(subsonic, transonic and supersonic flows) as well as turbulent flows. We will now review

the numerical methods developed which apply to the type of flow we want to model.

3.2 Numerical Techniques for CFD Solvers

3.2.1 Considerations on Compressible Flows

At high Reynolds number, the convective effects are predominant compared to

diffusive effects. Therefore, the fluid behaves like an inviscid flow [:38]. The governing

equations for a viscous fluid are the system of Navier-Stokes equations: they reduce

to the Euler equations in the case of inviscid flow. The numerical techniques for the

two types of equations are very similar except for the discretization of the shear stress

which arise in viscous flows. For compressible and supersonic flows, there is formation of

shocks, expansion fans and contact discontinuities. Specific numerical schemes should be

considered to capture properly pressure and properties discontinuities.










Numerical techniques developments on Euler equations started in the early 1950s

with the work of Courant et al. [39], Lax and Friedrichs [40]. Lax-Friedrichs methods

are easy to evaluate and smooth, however they are excessively dissipative [41]. The

milestone for the modern development is to be found in the work of Lax and Wendroff [42]

which resulted in the type of solver named after them. Their main characteristic is to be

second-order accurate, space-centered and explicit in time.

A variant of this method, expanded to the system of Navier-Stokes equations, was

developed later: it is the widely-used explicit MacCormack scheme [43]. Most of the

work in the early 1960s happened at the Courant Institute on hyperbolic systems. The

numerical schemes tended to exhibit non-physical oscillations and it was difficult to obtain

converged steady-state solutions. At the same time, in Russia, Godunov introduced the

concept of representing the flow as piecewise constant in each computational cells and

solving a Riemann solver at each interface [44]. The scheme was first order accurate and

avoided non-physical features such as expansion shocks: expansion shocks cannot happen

because they violate the Third Law of Thermodynamics, meaningly entropy can only

increase (AS > 0). Because Godunov's pioneering work was not translated as it was

published, it took time for the West to acknowledge the numerical technique [45]. However

very accurate, the scheme was very costly in terms of calculations of the complex fluxes at

the interfaces.

In 1981, Roe [46] developed a technique which enabled a practical and efficient

implementation of Godunov-type schemes: he introduced the concept of locally linearizing

the governing equations through a mean value Jacobian: his work still has aml ri ~ impact

on modern computational techniques. Roe's technique belongs to the Approximate

Riemann Solvers class, along with Osher's method [47]. Roe's concept is based on a

flux-difference splitting scheme and yields a single-point numerical shock structure for

stationary normal shocks.










ENO (Essentially Non-Oscillatory) scheme of Harten et al. [48] is another example of

widely used flux-splitting solvers. Godunov's work also recognized that numerical schemes

might benefit from distinguishing various wave speeds leading to characteristic-based

schemes. Early high-order characteristic-based solvers used flux vector-splitting, the most

famous being the scheme of Steger and Warming [49]. It still led to oscillations near

discontinuities.

The Monotone Upstream-centered Schemes for Conservation Laws (j!USCL) of Van

Leer [50] extended Godunov's scheme to higher orders: the scheme was second order

accurate in space thanks to the introduction of flux-limiters. Limiters are functions which

smoothes the numerical solutions in areas where high order scheme would normally create

oscillations. Their first use dates back to 1973 with the work of Boris and Book [51] who

demonstrated perfect propagation of discontinuities. Harten [52] demonstrated that Total

Variation Diminishing (TVD) schemes (see Section 5.1.10) are monotonicity-preserving

and enable oscillation control in the solution of non-linear problems. An other very

significant work is the Runge-K~utta algorithm by Jameson [53] which is based on an

alternative-direction implicit approximate factorization.

During the eighties, development of compressible Euler solvers was an essential step

toward simulation of the more complex flows.It was made possible thanks to techniques

such as combination of upwinding, artificial methods, finite element or finite volume

approximations and flux limiting concepts, for example. We now present the MacCormack

methods which are of great importance for our research. It belongs to the family of block

methods which consist in solving all the equations simultaneously. It usually requires more

computations at each grid point but improves dramatically the convergence rate [54].

3.2.2 Discussion on the MacCormack Methods

In 1969, MacCormack published pioneering work for solving the system of Navier-Stokes

equations in compressible, high Reynolds number flows [43]. The solver was explicit,

second-order accurate and used the predictor-corrector method. Its performances were










well-proven. However, explicit schemes are convergent and stable as long as the time step

is small enough to satisfy the stability condition. In 1975, AlacCormack published an

improved version of his own technique [55].

Continuity of his work led to a combined implicit-explicit MacCormack scheme in

1981 [56]. Explicit schemes are ahr-li- numerically stable and convergent but usually more

numerically intensive than the explicit method. The explicit predictor-corrector method

which is faster is used over most of computational domain. However, it is not adequate

for points in the flow at which the local CFL number exceeds the stability limit [57]: the

method adds an implicit procedure to the predictor-corrector sequence.Thne method is

implicit in nature and thus allows a much larger time marchingf step size.

The method posses three advantages over fully-implicit methods. First, the method

uses two-point, one-sided differences which lead to block hidiagonal systems, easier to

invert that block tridiagonal systems found in traditional methods. Secondly, the method

uses inviscid jacohians and corrects them using representative viscous terms added to

the eigenvalues. This maintains stability while avoiding the expensive calculation of

the viscous jacohian. Finally, the algorithm allows the implicit step to be skipped in

regions where the stability restriction condition is satisfied. The expressions given for the

predictor and corrector steps are adequate for applications with the finite volume method.

Snyder [58] compared the original 2-2 MacCormack scheme from 1969 [43] to a 2-4

modified version of the method and other similar explicit finite difference methods. The

methods where benchmarked against a Gaussian pulse diffusion and the shock tube: the

two test cases study non-linear wave propagations of an inviscid flow, i.e. they require

to solve for the system of Euler equations only. Both experiments are well documented

and allow detailed validation of numerical techniques. The methods included artificial

damping or filtering models to reduce spurious oscillations. The 2-2 MacCormack scheme

is second order accurate in space and time. The 2-4 scheme is fourth order accurate in

space: the spacial accuracy is improved by increasing the stencil size from three to five.










A Macdormack-Baldwin [59] damping function was added to both numerical methods to

suppress spurious pulses. The study shows that the 2-2 MacCormack is the fastest method

of all investigated. However, it gives spurious oscillations and is highly diffusive. The 2-4

method captures better the sharpness of the shock but largely overshoots the value of

the discontinuity. The analysis concludes that each method has its own unique feature

and that the order of accuracy of a method does not necessarily is a good indicator of the

method performances: To reduce oscillations in areas of discontinuities one might prefer a

2-2 MacCormack over a 2-4 MacCormack even if the first is more dissipative.

After the release of AlacCormack implicit-explicit predictor-corrector scheme [56],

somle analyzes showed that the imnplicit vac~orr mack mnethod is unr~eliable such that

the steady-state solutions depend on time increments [60]. The severity of such

dependence was assessed by Ong [61] in a comprehensive study in 1987. The scheme

was also compared to the fully-implicit scheme of Beam and Warming [62]. The study

was validated over a supersonic turbulent flow past a two-dimensional compression corner.

Turbulence was simulated by the algebraic turbulent eddy-viscosity of Baldwin and

Lomax [63]. The study concluded that the steady-state solution of the explicit-implicit

method is relatively insensitive to the Courant number, therefore the steady-state

appeared to be independent of the size of the time step. Moreover, the accuracy of

the steady-state solution using Macdormacks hybrid algorithm was comparable to that

of the Beam-Warming method for all cases, and was observed to reduce the computing

time by a factor of up to three. Lawrence [57] also investigated the performances of the

implicit-explicit form of MacCormack technique: results of the simulation of a laminar

flat-plate boundary-1 w-;r and hypersonic laminar flow over a 15-degree compression corner

were compared to experimental results and the Beam-Warming scheme [62]. The results

compared pretty well with published results and experimental data. However, the authors

identified a source of error in the boundary treatment, particularly visible in the flat-plate

simulation: the reflective condition at the boundary led to instabilities at the beginning of










the iteration. Therefore, the authors recommended the use of explicit boundary condition

treatment in the case of the Macdormack simulation.

Although not very sophisticated, AlacCormack techniques are perfectly satisfactory

for ]rn lny fluid flow applications especially in the case of 2D flows [64]. The Macdormack

schemes are particularly adapted for the applications of interest in this dissertation: they

are fast and easy to implement, and the accuracy is satisfactory for the type of flow we

investigate.

3.2.3 Considerations on Turbulence

All of the techniques we just described can he applied to the system of Navier-Stokes

equations. However the N l.-;1 r-Stokes equations are only valid for laminar flows. Most

practical engineering flows are turbulent. In turbulent flows, large and small scales of

continuous energy spectrum proportional to the size of the eddy motions are mixed.

Eddies are overlapping in space, larger ones carrying smaller ones. In this process,

turbulent kinetic energy transfers from larger to smaller eddies and the smallest eddies

eventually dissipate into heat through molecular viscosity. A typical 0.1 by 0.1 m flow

domain with relatively high Reynolds number might contain eddies down to 10-100 pm.

To compute all length scale a grid of 109 up to 1012 mesh points would be required to

describe all scale lengths [65]. Engineers need access to viable tools capable of representing

the effects of turbulence. Here are the three methods which are commonly used:

Direct Numerical Simulation (DNS)

Reynolds Averaged Navier-Stokes (R ANS)

Large Eddy Simulations (LES)

In Direct Numerical Simulation (DNS), resolution of the grid is chosen such that

all turbulent scales, large and small, are resolved. This is a deterministic method,

which requires prohibitive computer code and cannot undertake large-scale industrial

problems. Since turbulence is characterized by random fluctuations, statistical methods

are being extensively studied. In the context of the Reynolds-Averaged Navier-Stokes










(R ANS) method, the system of conservation equations is time-averaged. The various

flow properties are decomposed into mean and fluctuating components. It results in

new variables appearing in the governing equations: the average value of the product of

velocity components fluctuations. These terms are models. It corresponds to modeling of

all turbulence scales. Therefore, R ANS method does not require as much mesh refinement

as for DNS. The Large Eddy Simulation (LES) technique is a compromise between the

DNS and the R ANS method. For this technique, large eddies are computed and small

eddies are modeled. Small scale eddies are associated with the dissipation range of

turbulence and are simpler to model than in R ANS. This method requires more refinement

than R ANS since the large eddies are computed but not as much as the DNS because the

small-scale turbulence is modeled. In DNS, the K~olmogorov microscale [66] defines the size

of the meshes and in 3-D the number of grid points ends up being equal to NV = Re9/4

Therefore, DNS is not adapted to the type of solvers we aim at building. The development

of the LES methods remains particularly marginal because the computation of the large

scale eddies requires an amount of computational resources which were not available

until a few years ago. Moreover the nuclear industry is just beginning to introduce this

technique for analysis of turbulent flows.

Most of the time, engineers are interested in information about the time-averaged

properties of the flow. This is particularly true for the studies presented in this dissertation.

Hence, the R ANS technique is often privileged and historically, has been the most

developed technique, thus is very well documented. The 1 in 4 Gry of the CFD simulations

for nuclear reactor applications we previously quoted rely on the k-e model for turbulence

considerations. Even if some research work show the limitations of this model in certain

cases [25], it is not clear if this model is suitable for all researches. Besides, the previous

review showed the importance of accounting for the real properties of the fluids [29].

However, the commercial CFD codes which are traditionally used have limited capabilities

in real properties modeling. Moreover Direct Numerical Methods, Large Eddy simulations










and other advanced CFD methods have not yet found many applications in nuclear

engineering. However, they will become more and more use as precise two-phase

simulation with interface tracking is one of the main research focus. Also since all the

flows in nuclear engineering applications are internal, modeling of the fluid close to the

walls is of great importance. The industry would gain from the use of a combination of

advantages of both RANS and LES turbulence modeling in high Reynolds number flows.

This technique known as hybrid RANS-LES solves for different scales of turbulence while

being still computationally affordable. In this method unsteady RANS is used near walls

and away from walls LES is used. The matching between RANS and LES takes place in

the inner log-regionl Unfortunately, this model is still in its development phase and not

available in commercial CFD codes: there are still some issues on the area of transition

between RANS and LES treatment of the turbulent viscosity [67]. Georgiadis [68] solved

this problem by using a RANS and a LES mixing-length model. However, when applied to

compressible turbulent mixing lI evers it only enables predictions of the main characteristics

of the boundary 1.>. ris and the authors themselves concluded that there is large room for

calculations improvement.

Therefore, the hybrid RANS-LES methods for turbulence modeling are in a too early

stage for us to consider in our solvers. Hence, we will build our CFD solvers using the

RANS method that is widely documented and primarily used for flow analyzes related to

the nuclear engineering field. In the review of CFD analysis in nuclear systems, the models

of predilection are primarily the k-e and RSM models. The k-e seems very well-adapted

to the VHTR coolant channel study. It is a two-equation model and therefore is one of

the RANS models which requires relatively low computational capabilities. It has shown

good performances in the study of subsonic pipe flows, which is our interest for the coolant



1 Stefan Wallin: Turbulence Modeling and Flow Control Seminar University of
Florida April, 26 2007










channel. In the case of the NTP thrust chamber, we study a transonic flow which is wall

hounded. As we detail in the next section, the Spalart-Allmaras model [69] seems to be

more adapted in the particular case of the thrust chamber.

3.2.4 Discussion on the Spalart-Allmaras Model

In 1994, Spalart and Allmaras developed a turbulence model for applications to

wall-bounded transonic flows: the model was based primarily on empiricism and on

dimensional analysis arguments [70]. It is a very recent model, but we review the work

which has been done on the validity of the model to determine if it is applicable to our

particular case of thrust chamber.

Paciorri [71] investigated the validity of the Spalart-Allmaras turbulence model for

hypersonic flows. The model was validated on a variety of test cases: flat plate turbulent

boundary 111-c- v fow, hypersonic wind-tunnel flow, a Mach number equals 5 flow over a

hollow cylinder and a Mach number equals 6.8 flow over a hyperbolic flare. The study

concluded that the model, implemented in a finite volume solver with a finite difference

technique, overall gave good predictions of the flow behavior. Exception was made in

cases which involved turbulent separation The author recommended the use of the model

for attached flows, which is the case of the thrust chamber flow we investigate. Fink and

Pirzadeh [72] also investigated the validity of their solver on a flat-plate boundary-1 e--
using the Spalart-Allmaras technique to model turbulence. The Euler equations were

discretized using Roe's scheme and flux limiters Superbee or Minmod. Bother Euler

equation and the Spalart-Allmaras transport equations were discretized using an implicit

backward Euler time-stepping method. Therefore, it allowed a loose coupling between

the different equations. The Spalart-Allmaras model was coupled with a wall function

formulation to eliminate the need for resolving the flow in the sublayer portion of a

turbulent boundary 111-c vr. The paper showed that this technique gave good results both in

the flow field and the boundary 11s-< r, and had the advantage of reducing the computing

memory requirement and improving overall convergence. Bonfiglioli et al. [73] studied










the two-dimensional steady transonic turbo-machinery flow, where the eddy viscosity

was modeled with the Spalart-Allmaras model. The analysis proved the good behavior of

the turbulent model to predict the transonic flow. Finally, Cavicchi [74] led an extensive

study of nine turbulence models applied to the secondary flow in an accelerating 90" elbow

and the performances of each models were assessed with the exit passage vortex. The

Spalart-Allmaras model gave the best predictions along with the Baldwin-Lomax and the

Shear Stress Transport models.

In conclusion, we observe that CFD has become one of the preferred simulation tools

in thermal-hydraulics studies for portions of nuclear systems. The nuclear industry relies

mostly on commercial CFD-codes, but these codes exhibit a few drawbacks: They were

not aimed at modeling large pressure and temperature variations which lead inevitably

to 1!! ri ~ property variations. Also, a very critical issue in CFD simulations is turbulence

modeling: the turbulence models often offered in commercial codes were developed for

aeroacoustic flows. Some studies ([16], [26], [27]...) aimed at evaluating the validity of

such models for nuclear applications. It resulted that, besides its relative simplicity, the

k-e turbulence model showed good performances. We then will be able to use this model

for most of our simulations. However, in the case of the NTP thrust chamber, our review

shows that a simpler model, the Spalart-Allmaras model, would give better results while

being computationally less expensive. Also, we reviewed the main numerical techniques

for compressible flows which are competitive in shocks and discontinuities capture.

Usually, commercial codes only allows one type of solver for this type of flow: for example,

FLUENT 6.1 is limited to a third order 1\USCL scheme. Although very accurate in most

flow types, the user might want more flexibility. In our case, we want to get results as

accurate as possible but with an emphasis on calculation rapidity. In the previous review,

we showed that the 1\acdormack methods might he the one most relevant for our study.

In the scope of the development of CFD solvers for VHTR and NTP study, this review

helps us make informed choices of each of the different techniques.










CHAPTER 4
MATERIAL PROPERTIES

The present analyses focus on the hydrogen flow in NTP, helium and carbon dioxide

flow in gas-cooled reactors. The fluids are gases in the operating conditions we consider.

They undergo a large range of pressure and temperature. To get accurate modeling

of the flow the CFD solvers need to consider a variety of gases over a large range of

operating conditions. Therefore, meticulous modeling of the properties behavior with

temperature and pressure is critical. We model the density, heat capacity, viscosity,

thermal conductivity and molecular weight. All the gas data which are available give

discrete values for various sets of operating conditions. We would like to have a continuous

description of all properties to avoid computationally expensive table reading. Besides,

continuous definition better describes the property gradients involved in the governing

equations. This section aims at describing the properties models as polynomial functions

of T and P. The models should match the experimental data within 5' for our simulations

to give a realistic model of the actual fluid flows. Models for hydrogen, helium and carbon

dioxide properties are given. The properties of considerations are:

Density

Heat Capacity

Viscosity

and Thermal Conductivity

4.1 Hydrogen

Hydrogen is considered for applications in the space nuclear propulsion only. For

example, for the square lattice 1,. .1,. v.inh~ ~! design of the NTP, when the core thermal

power is equal to 600 MW hydrogen operating conditions in the system range from

22 K( and 2 bars in the storage tank to 3,000 K( and 110 bars [9]. This is the starting

point for the range of operating conditions. We saw in section 2.2 that we expect great

performances of the rocket thrust chamber if the system is operated at low pressure.










Therefore, we want to derive all hydrogen property correlations over the range:


22 < T < 3, 500K(

0.1 < p<150bars(41

This range covers all the operating conditions for the possible NTP designs, plus a safety

niargmn.

4.1.1 Note on Dissociation

When the core is operated at low pressure, there is the possibility that the molecules

of hydrogen start dissociating as the fluid temperature reaches a threshold in temperature [75].

The dissociation follows the scheme:


H2 H + H


The level of dissociation is plotted Figure 4-1 as a function of temperature for different

pressures. Dissociation can only occur at temperatures greater than 1,500 K(. At a given

temperature, the lower the gas pressure the more molecules dissociate. Dissociation

primarily impacts the molecular weight of the gas and its heat capacity. The molecular

weight of hydrogen at room temperature and atmospheric pressure is M~ = 2.016
mol
As the gas start dissociating the statistical average of the gas molecular weight drops.

The values of the molecular weight is obtained front Vergaftig [11]. Figure4-2 shows the

evolution of the heat capacity with temperature when there is a high level of dissociation

(p=0.1 bar) and when the impact of dissociation is negligible (p=150 bars). At the lowest

pressure the heat capacity of the dissociated gas abruptly increases whereas at 150 bars,

the heat capacity remains roughly constant. At 0.1 bars, the heat capacity reaches a

nmaxiniun at T = 3, 000 K(.










4.1.2 Thermo-Physical Properties

The hydrogen property packages provides discrete values of all hydrogen properties:

enthalpy, entropy, speed of sound, density, thermal conductivity, viscosity, specific

heat and specific heat ratio. For temperatures below 3,000 K(, data are based on the

National Bureau of Standards. Above 3,000 K( and up to 10,000 K( data are obtained

from NASA computer programs for para- and dissociated hydrogen [76] and the National

Institute of Standard and Technology. The property data provided by the hydrogen

package are the best available as of tod w-. A computer routine developed by C'I. 1. [77]

conveniently interpolates thermodynamic and transport properties of hydrogen within the

temperature range above. Different techniques are used to develop the hydrogen properties

package: Linear interpolation (LI), natural cubic spline interpolation (NCSI), and multiple

polynomial curve-fitting with polynomial bridging between the curve fits. The package

allows the user to get hydrogen properties for a discrete set of temperature and pressure.

For use in the CFD solver, we would like to have a continuous description of the different

properties as a function of temperature and pressure to avoid cpu-expensive table reading

and be able to describe as well as possible property gradients which appear in the fluid

governing equations. Each set of properties is now curve-fitted to obtain a correlation in

good agreement with experimental data (about 5' .).

*Density and molecular weight:
By definition, an ideal gas is a hypothetical gas with molecules of negligible size that
exert no intermolecular forces. A molecule of hydrogen, composed of only two atoms,
and moreover the two lightest ones, should be very close to an ideal gas. For an ideal
gas, the density is defined by

P=RT
where p is the pressure, R the perfect gas constant and T the temperature.
A comparison between real thermo-physical data and ideal gas approximation is
presented Tables 4-1 to 4-5. The level of dissociation of the hydrogen molecules is



I http: //www. inspi .ufl. edu/data/h_prop_package .html










taken from Vargaftik [11] and shown Figure 4-1. We compare density values obtained
from Given [78] to the theoretical density calculated with the ideal gas law corrected
for molecular weight variations due to the dissociations of hydrogen molecules.
The relative errors are compiled in the last column of the tables. Over most of the
temperature and pressure range the ideal gas law applies well since the relative
errors remain smaller than ;:' At the extreme range of pressure, when the pressure
is small (P = 0.1 bars and T > 3, 000 K() the relative errors increases up to t:I' The
error is too large to consider the ideal gas law is accurate under these conditions.
Figure 4-3 shows the difference between the data from Given [78], the data from
Vargaftik [11] and those computed from the ideal gas law: density is plotted as of
function of the inverse of temperature. It seems that the values from Given at high
temperature are not accurate: the error has been identified as coming from a digit
inversion when the density values were input in the code. The graph shows the good
agreement between the ideal-gas law model and the density values from Vergaftik.
Therefore, the ideal-gas law correctly describes the hydrogen density behavior for all
ranges of pressure and temperature.
All hydrogen thermo-physical properties (besides density that we just discussed) are
implemented in the solver through correlations which are a function of both pressure
and temperature. Those correlations are polynomial functions of the two extensive
properties.

* Heat Capacity:
The heat capacity is very sensitive to both pressure and temperature and is defined
as follow:

cp(kJ/kg.K() = al(p)T4 2 aa 3) 3 + as )2 + 4 ~)T 85(p) (4-2)

where T is in K~elvins, p in bars and

al(p) = 0.560 10-llp-0.4
a2(p) = -2.630 10-8p-0.35
a3(p) = 4.349 + 10-5p-0.30
a4(p) = -2.730 10-2 -0.25
as(p) = 20.57p~-oos

Figure 4-4 shows the evolution of the heat capacity with temperature for diverse
pressures. As the gas dissociates there is a sharp increase in the heat capacity value.
The previous correlation is continuous with pressure and temperature. However,
this correlation only gives satisfying results (within 15' .) for temperatures up to
3,000K( at low pressures. In fact, at 0.1 bar, heat capacity reaches a maximum at
T=3,250 K( as seen Figure 4-2 and a simple, continuous correlation for all pressures
is not obtainable. Therefore, the trend of the heat capacity behavior is favored in
the approach of hydrogen dissociation. Note that the formulation above is for heat
capacity expressed in kJ/kg.K(. The SI units are J/kg.K(.

* Viscosity:










For pressures greater or equal to 1 bar, viscosity of hydrogen is pres sure-indep endent
and can be modeled by:

p-(kg/m.s) = -1.44 10-12 2 + 1.69 10-sT + 4.64 10-6 (3

where T is the temperature expressed in K~elvins. We can use this approximation
in all ranges of pressure and temperature: viscosity calculated with Equation 4-3
remains within 5' of the measured viscosity. Figure 4-5 shows the good agreement
between the model and the experimental data.

* Thermal Conductivity:
For temperatures smaller than 2,000K( thermal conductivity of hydrogen is pressure
independent and linear with temperature. The following linear approximation

k(W/m.K) = 3.54 10-4T + 9.13 + 10-2 T < 2, 000K (4-4)

leads to an error on the calculated values no greater than "'
When temperature is greater than 2,000 K( thermal conductivity is pressure-dependent.
An interpolation is given below, where the pressure is expressed in bars. This
formulation gives an error no greater than Io' compared to the measured values

k(W/m.K) = al(p) T2 2 a~p) T + 3(p) T '> 2, 000K (4-5)

(al(p) = 2.43 + 10-11* p2 9.28 + 10-9 + p + 1.46 10-6
a2(p) = -1.04 10-7 s p2 +3.98 + 10-s + p 5.87 10-3 _46)
a3(p) = 1.11 10-4 ~ 2 4.24 10-2 p +6.72
In Equations 4-4 and 4-5 temperature T is expressed in K~elvin and the pressure p in
bars.

* Implementation of the dissociation effects the case of hydrogen molecular weight:
Since the ideal gas law is an accurate model for hydrogen density modeling, we use

p = (4-7)
M~(p, T)

to derive the density at any point of the domain. The universal gas constant (sR) is
equal to 8314.3 J/kg.mole.K. M~(p, T) is the molecular weight of hydrogen which
varies with temperature and pressure.
No simple correlation was found which could describe accurately the variation of
molecular weight with variable operational conditions (Figure 4-1).
Therefore, we develop a linear extrapolation between each discrete point. If we look
for the molecular weight at pressure p and temperature T such that

TiIp 21 < T < T2










we compile the molecular weight at p and T such that:

T T1
T2 T1
Af (T, pl) = Lc/ Af(T2, pl) +(1 ~t)~ A(T1, pl)
< Af(T, p2) 0, Lc/ Af(T2, p2) + (1 0,)~ + A(T1, p2)
p pl
Sp2 pl
Af (T, p) ap L+ Af n(T) p2) + (1 Lcp) + Af (T, pl)

With the correlations above on temperature and pressure dependance of hydrogen

properties, we are able to intplenient real gas properties in the CFD solver with an

accuracy within 5' of the literature data.

4.2 Helium

Helium gas is the coolant of predilection for the VHTR. Table 4-6 compiles VHTR

operating conditions and features. Helium circulates in a closed loop. Typically, the

coldest point of the loop, at the inlet of the core, is at a temperature of 490 "C. In order

to get the best efficiency out of the system, helium is heated up to 950-1,000"C. The usual

operating pressure is estimated at 7.12 1\Pa. The range of operating conditions over which

we derive the helium properties correlations is therefore:


1 < p(bars) < 140

200 < T("C) < 1, 500 (8

Helium atomic weight is constant through the range of pressure and temperature we

consider and equal to Af = 4.003 g/nlol. The specific heat ratio is then R 2077.

We refer to Vargaftig [11] for helium thernial-properties.

Density:
Since helium is a niono-atomic gas, it should behave closely to an ideal gas. We
compare the experimental data to the ideal-gas law within range 4-8. The results
are presented in Table 4-7 and show that the ideal-gas law properly describes the
behavior of helium density: the theoretical data fall within lI' of the real values.

Heat capacity:
Vergaftig [11] only lists heat capacity of helium at low temperature. We refer to
Leninon (et. al.) [79] for temperatures ranging front 200 to 1,500 K( and pressures









between 1 and 100 bars. The value of helium heat capacity can be modeled by a
constant, independent of temperature, with great accuracy (within 0."' of the
experimental data). Helium heat capacity in not dependent on T and p because
helium is a mono-atomic gas. Compared to hydrogen there is no dissociation
occurring, which has been identified as the source of heat capacity variations. We
model helium heat capacity by:

cp(J/g.K() = 5.191 (4-9)


Viscosity:
The correlation to model viscosity as a function of pressure and temperature is

p-(p- Pa.s) = al(T) p + a2(T) (4-10)

where T is expressed in K~elvins and p in bars and


Ial(T) = 2.25 10-2 -1.129
a2 (T) =3.69 + 10-4+ +.04 *10-2

The values obtained with Equation 4-10 fall within 0. !' of the experimental data
for temperatures within range 4-8. The good agreement between the model and the
data can be seen Figure 4-6.

Thermal Conductivity:
Values of helium thermal conductivity are pressure and temperature dependent. We
define :
k(W/(m.K) = al(T) p + a2()(41
where ,

Ial(T) = 4.92 + 10-12 2 1.20 10-7T + 8.41 10-s
a2(T) = 2.75 10-4T +7.43 10-2
where T is expressed in K~elvins an p in bars. Through the range of interest for
the VHTR the calculated values are within 5' of the experimental data and the
comparison between the two can be seen Figure 4-7.
4.3 Carbon Dioxide

A few VHTR designs in the early stage of development consider using carbon dioxide

(CO2) aS a COolant and working fluid. For example, carbon dioxide is investigated in some

pebble-bed reactors configurations. One of the criteria of the code development is its

modularity. Therefore, it is important for our solvers to model carbon dioxide and thus

get an estimation of its performances as a coolant compared to helium. We derive the









behavior of the gas for the same range of pressure and temperature as helium (range 4-8).

We use the data provided by Lemmon (et.al.) [79].

Density:
The ideal gas law gives good results to model carbon dioxide density. Isothermal
experimental and computed densities are plotted as a function of pressure (Figure 4-8).
The model does not give good results for temperatures of 400 K(. For temperatures
equal or greater than 800 K( the computed results fall within "' of the experimental
data. To derive the ideal gas law model, we use M~ = 44.011g/mol and R=
188.9.

Heat Capacity:
Between 400 and 1,100 K( the heat capacity increases with temperature at lower
pressure. For pressures greater than 30 bars, heat capacity first decreases as
temperature increases and then increases. To account for this phenomenon we
use this higher order modeling:

cp(J/g.K() = al(p)T4 2 aa 3) 3 + as )2 + 4 p)T 85(p) (4-12)

where T is expressed in K~elvin p in bars and

at (p) = 10-12 [6.61 10-4 2 + 3.07 10-2p 0.179]
a2(p) = 10-s [-2.18 + 10-4 2 1.07* 10-2p +8.06 10-2]
a3(p) = 10-s [2.62 + 10-4 2 +1.41 + 10-2p 0.159]
a4(p) = 10-2 [-1.38 + 10-4 2 8.35 10-3p +0.175]
as(p) = 2.66 10-4 2 +1.94 10-2p +0.447

The model leads to calculated values of the heat capacity within 5' of the NIST
data. Both calculated and experimental data are plotted Figure 4-9

Viscosity:
Over the range of operating conditions 4-8, CO2 V1SCOSity is pressure independent
and varies linearly with temperature as seen Figure 4-10. We model carbon dioxide
viscosity by:
p-(Pa.s) = 3.43 + 10-sT + 7.01 10-6 43
where T is expressed in K~elvin. The modeled values fall within ;:' of the NIST data.
They are also computed on Figure 4-10.

Thermal Conductivity:
Carbon dioxide thermal conductivity can be modeled by:

k(W/(m.K) = al(p) T + a2(p (44









where, T is in K~elvins ad p in bars.


Ial(p) = -6.10 10-sp + 7.48 + 10-s
a2(p) = 7.64 10-sp 4.51 10-3

The model gives results within ;:' of experimental data. Plot of modeled and
experimental data are given Figure 4-11

Development of real property CFD solver requires continuous descriptions of all

thermal-physical properties of the gas of considerations. Temperature- and pressure-dependent

correlations are derived for hydrogen, helium and carbon dioxide for density, heat capacity,

viscosity and thermal conductivity. All the modeled values fall within 5' of the data from

our literature sources. The 1!! ri ~ remark after this study is that all three gases behave

like ideal gases.
























o

-mF 0.6-
O
vl

O

e 0.4-




0.2-





01 0 20 20 30 30 40 40 50 50 60
Temperature (K)



Figure 4-1: Level of hydrogen molecule dissociation as a function of temperature
























.d150-




S100






50


0 500 1000 1500 2000 2500 3000 3500
Temperature (K)


Figure 4-2: Hydrogen heat capacity vs. temperature at low and high pressure (0.1 and 150
bars)


Table 4-1: Error on the density formulation for P=0.1 bar


P (bar)
0.1
0.1
0.1
0.1
0.1
0.1
0.1
0.1
0.1
0.1


T(K()
1500
1700
1900
2000
2200
2400
2800
3000
3200
36;00


M(g/mol)
2.016
2.016
2.013
2.011
1.998
1.966
1.786
1.622
1.433
1.146


p -IGL
1.62 + 10-3
1.43 + 10-3
1.27 10-3
1.21 +10-3
1.092 + 10-3
9.85 10-4
7.67 10-4
6.50 10-4
5.39 + 10-4
3.83 + 10-4


p compiled
1.60 10-3
1.39 + 10-3
1.25 10-3
1.20 10-3
1.10 *10-3
1.00 10-3
8.00 10-4
7.00 10-4
3.91 10-4
3.30 10-4


error

2.3
2.2
0.8
0.7
1.5

7.6
27.4
















x x data from Given
d f V fik


-


0.3 0.4 0.5 0.6(
1/T*1e-3


0.0018


0.0016


**estimation


+ x
+4

x
t


x

x

x* x

xx


0.0014-


0.0012


0.0010 -


0.0008-


0.0006 -


0.0004 -


0.0002 -


0.0000


Figure 4-3: Hydrogen density at 0.1 bar and high temperature. Differences between
the literature data, Given [78] and Vergaftig [11] and the data computed with the ideal
gas-law


Table 4-2:
P(bar) T(K()
1 1700
1 1900
1 2000
1 2200
1 2400
1 3000
1 3200
1 3600


Error on the density formulation for P=


:1 bar
error ( )>
0.4
0.0
0.1
0.1
0.2
0.1
2.9


M(g/mol)
2.016
2.015
2.014
2.010
2.000
1.869
1.769
1.498


p -IGL
1.43 + 10
1.28 + 10
1.21 +10
1.10 *10
1.00 10
7.49 + 10
6.65 10
5.00 10


p compiled
1.43 + 10-2
1.28 + 10-2
1.21 +10-2
1.10 *10-2
1.00 10-2
7.50 10-3
6.84 10-3
5.07 10-3





















































error ( )>
0.1
0.1
0.1
0.1
0.1
0.0
0.0
1.6


S150 **p=1 bar
p10 bars
p=10 bars
p100 bars
id a p=10 bars
o 100-








5 0


0 500 1000 1500 2000
Temperature (K)


Figure 4-4: Hydrogen heat capacity vs. temperature: Comparison
data (solid lines) and the model (dots).


I


2500 3000



between experimental


Table 4-3:
P(bar) T(K()
10 1900
10 2000
10 2200
10 2400
10 2800
10 3000
10 3200
10 3600


Error on the
M(g/mol)
2.016
2.015
2.014
2.011
1.990
1.966
1.930
1.775


density formulation for P=10 bar


p -IGL
1.28 + 10
1.21 +10
1.10 *10
1.01 *10
8.55 10
7.88 + 10
7.25 10
5.93 + 10


p compiled
1.28 + 10-1
1.21 +10-1
1.10 *10-1
1.01 *10-1
8.54 10-2
7.88 + 10-2
7.25 10-2
6.03 + 10-2
















* p=0.1 bars
* p=1 bar
* p=10 bars I *. I.~
e p=100 bars
- correlation


4.0E


2.5t


0.50


1000 1500 2000
Temperature (K)


2500 3000 3500


Figure 4-5: Hydrogen viscosity vs.
(dots) and the model (solid line).


temperature: Comparison between experimental data


Table 4-4: Error on the density formulation for P=100 bar


P (bar)
100
100
100
100
100
100
100


T(K()
2000
2200
2400
2800
3000
3200
3600


M(g/mol)
2.016
2.015
2.014
2.008
2.000
1.988
1.944


p -IGL
1.21
1.10
1.01
8.63 + 1(
8.02 + 1(
7.47 1(
6i.49 + 1(


p compiled
1.20
1.09
1.00
8.56 10-1
7.96 10-1
7.46 10-1
6.49 + 10-1


error ( )>
1.0
1.0
0.8
0.8
0.7
0.1
0.0

















**T=200 K
5.0t T= 200 K
**T= 500 K
T= 500 K
4.5-l T=750 K
T=750 K
ST=1000 K
S4.0t ---------- T=1000 K
*I T=1250 K *
T=1250 K



2~ 3.0-





2.0


50 50 100 150 200
Pressure (bars)


Figure 4-6: Helium viscosity vs. pressure for different temperatures: Comparison between
experimental data (dots) and the model (solid line).






Table 4-5: Error on the density formulation for P=150 bar
P(bar) T(K() M(g/mol) p -IGL p -compiled error ( .)
150 2200 2.016 1.65 1.63 1.5
150 2400 2.014 1.51 1.50 1.3
150 2800 2.010 1.30 1.28 1.2
150 3000 2.003 1.20 1.19 1.0
150 3200 1.993 1.12 1.12 0.1
150 3600 1.957 0.98 0.98 0.1
















450


400






300-


**T=300K
- T=300 K
**T=500 K
- T=500 K
**T=1,000
- T=1,000
e T=1,200
----------- T= 1,200
**T=1,500
- T=1,500


250E


200





150


150
Pressure (bars)


250


300


Figure 4-7: Helium thermal conductivity vs. pressure for different temperatures:
Comparison between experimental data (dots) and the model (solid line).


Table 4-6: VHTR operating conditions and features
Condition or feature VHTR
power output (\!Wt) 600 900
average power density (W/cm3) 4 6.5
coolant and pressure helium e 7.12 Mpa
moderator graphite
core geometry annular
safety design philosophy passive
plant design life 60 years
core outlet temp. 1,000 oC
core inlet temp. 490 oC
fuel coated particles a) LEU-PyC/SiC
b) LEU-PyC/ZrC
fuel max temp. a) ~ 1,250 oC
normal operation b) ~1,400 oC

























- T=400K
- T= 800K
- T=1000K
T=1100K
T=400K
T= 800K
T=1000K
o T=1100K


100E


10 20 30 40
Pressure (bar)


50 60 70 80


Figure 4-8: Carbon dioxide density vs. pressure for different
represent the NIST data and the dots the ideal-gas law.


temperatures. The solid lines




































s*
1.1 -*



o 1.10



1.05



1.00-



0.9 00 00 60 70 800900 000
Temperature (K)



Figure 4-9: Carbon dioxide heat capacity vs. temperature for different pressures:
Comparison between experimental data (dots) and the model (solid line).

























* p=1 bar
* p=25 bars
* p= 50 bars
a p=70 bars
* p=74 bars
- model


4.0 E


3.0E


600


700 800
Temperature (K)


900


1000


Figure 4-10: Carbon dioxide viscosity vs. temperature for different
between experimental data (dots) and the model (solid line).


pressures: Comparison
























0.08



0.07


0.06
e **


0.05t


**T=400 K
- T=400 K
**T=800 K
- T=800 K
**T=1,000 K
- T=1,000 K
o T= 1, 100 K
T= 1,100 K


0.04t


0.03E


~C


0.02L
O


10 20 30 40 50 60 70
Pressure (bars)


Figure 4-11: Carbon dioxide thermal conductivity vs. pressure for different temperatures:
Comparison between experimental data (dots) and the model (solid line).


























Table 4-7: Comparison between experimental data and ideal-gas law for helium density


P (bar)
1
1
10
10
40
40
80
80
80
80
100
100
100
140
140
140


T (oC)
200
800
200
800
200
800
400
800
1000
1400
200
800
1000
200
800
1000


p(exp)
0.10172
0.044855
1.01463
0.44814
4.025
1.7871
5.6387
3.5597
3.0053
2.2914
9.8996
4.4406
3.7506
13.713
6.1919
5.2339


p(IGL)
0.10176
0.04486
1.01757

4.07028
1.79458
5.72192
3.58916
3.02534
2.30207
10.17571
4.486345
3.78167
14.24599
6.28103
5.29434


error( .)
0.04
0.02
0.29
0.11
1.13
0.42
1.48
0.83
0.67
0.47
2.79
1.03
0.83
3.89
1.44
1.15










CHAPTER 5
METHODS FOR THE CFD SOLVER DEVELOPMENT

This chapter presents the methods we chose and we used to develop the real-property

CFD solvers we use to analyze the diverse systems in C'!s Ilter 6 and 7. We justify each

choice of method based on our own experience and an extensive literature search. The

final product is a combination of proven CFD techniques which are particularly well

adapted to analyze gas flow in nuclear systems. The solver is validated on benchmark

cases. Since we well-document our sources but do not develop our own numerical methods,

we will not analyze consistency, stability, convergence, conservation and boundedness of

our final solver. If the reader is interested in such an analysis, he is invited to consult the

cited references.

5.1 CFD Solver Development

5.1.1 Requirements

The two systems we study with our CFD solver are the VHTR coolant channels

and the NTP thrust chamber. Both systems exhibit an axial symmetry and allow a

two-dimensional axisymmetric representation, as sketched Figure 5-1. The geometry

of the system are relatively simple: the grids are composed of quadrilateral cells. Both

flow system require a real-property analysis and are compressible. We are interested in

the steady-state solution of such compressible and turbulent flows. First, we present the

assumptions and derive the Reynolds Averaged N l.-;1 r-Stokes equations. Then we present

the time and space discretization techniques based on a Finite Volume method and a

MacCormack implicit technique. We also give a description of the models which account

for turbulence: k-e and Spalart-Allmaras. The two systems require similar boundary

condition treatment: inlet velocity and temperature, exit pressure, heated walls and

symmetry axis.









5.1.2 Governing Equations for Compressible Flow

The governing equations for a compressible, viscous and turbulent flow are the set of

Navier-Stokes equations. The local form of the equations is derived from the equations

of conservation: conservation of mass, momentum and energy. The compact form of the

system of equations is written [80]:


+div(p V) = 0



+ div(p V E + pl) = div(-r)(51


iBE
+ div(H V) = div(-r V) div(s).

where, t is the time, p the density, V the velocity vector, p the pressure, I the unity

matrix, -r the shear stress tensor, E the volumetric energy, H the volumetric enthalpy and

s the heat flux. The total volumetric energy is linked to the enthalpy by:


H =E +p (5-2)


The heat source flux is defined by:


s = (T)V V(5-3)

cpp~
and -A(T) = where, cp is the heat capacity at constant pressure, p is the molecular
Pr
viscosity and Pr is the Prandlt number.

The gases of considerations are hydrogen, helium and carbon dioxide: all three are

Newtonian and the shear stress tensor in compact tensor notation becomes [38]:

ii0u =iu p + b~v3iUei, (5-4)


We derive the equations for an axisymmetric geometry: the polar coordinates are

(r, 8, z) and the velocity components are Vi = (u,, no, uz)T. For an axisymmetric model,










differentiation along 8 is omitted. Because the flow is identical in any axial plane of the

systems, the swirl velocity can be considered negligible, and hence I,. = 0. The system of

Navier-Stokes equations becomes [81]:


8p 8, -<; 18rpur
++
dt 8z dr


8ipur ,asa
dt 8z


r dr


8p 8,z 1Brrr Too pui
+ + + ~e+
dr 8z r dr r r


(5-5)


8, d pu 1 Brpur,
iit 8z~ r iir


8z~ 8zi ~ r iir


dE 8 (E + p)n,
dt 8z


1 Br (E + p) or
r dr


+ Uzrz + Ur,, + norzo]

-+ Uzrrz + 7,~, + uearro)


r dr


and the tensor of shear stress -r is expressed by:

Buz 2 duz 84
Tz zrTe2 V-V+
a8 3 Br 8z
84u 84, 84 2
7 To = p- + 2 V-V
rz rr Br 8z Br 3
due due
~~ez8 Brr ~

divergence of velocity vector becomes:


due


Br
or 2
-2 V -V
r 3


(5-6)


B uz
V -V =+
8z


1 Bru, Bu u, $ u,
r dr 8z Br


(5-7)


The momentum equation in the 0-direction is redundant and shall be ignored. We can

write the previous set of equations in a compact form:


+V-F=V VF"+S (5-8)
iit










where


puruz



ur (E + p)


pu, +

uz (E + p)








CPl~tot iif
-Pr 8z~


CP~tot 3T
-Pr iir


5.1.3 Need for the Equation of State

The system of Equations 5-1 is composed of four equations. There are five unknowns:

p, ur, uz, E and p. Therefore an equation of state is required to relate the density to the

thermodynamic variables. In the previous chapter we determined that the ideal gas law

describes properly density variations with temperature and pressure for all three gases of

consideration. The equation of state for our solver will be:


p = pRT


(5-9)


where R is the universal gas constant. We then have the extra equation linking energy,

pressure and velocity:
pl1
E =+ || Vi ||2 (5-10)









5.1.4 Reynolds-Averaged Navier-Stokes Equations

The system of governing equations 5-1 only holds for laminar flows. In Reynolds-averaged

approaches to turbulence, all the unsteadiness is averaged out. In a statistically steady

flow, every variable #(xi, t) can be written as the sum of a time averaged value (mean flux)

and fluctuation about that value (fluctuating fulux:


~(Xi, t) = ~(Xi) + 'I(Xi, t) (5-11)

where
1 r
~(xi) =lim I (xi, t)dt (5-12)
T->+oo T Jo

If T is large enough, 4 does not depend on the time at which the averaging started.

From equation 5-11, it follows that #' = 0. Reynolds average is better suited for

incompressible flows (p=constant). Compressible flows introduce extra fluctuations, in

particular fluctuations in density p, which add extra correlation terms. In the situation of

compressible flow, Favre-averaging is used for ui, nyj, T, h, qj... and Reynolds-averaging is

used for p and p. Favre-averaging (represented by (.)) is obtained by mass-averaging:


q = (q) + q"l (q) = pq


Favre-averaging obeys the following rules:


pg"1 = 0 pq = p 4 pqr = p ( r + pg"r"


We now apply these calculations rules to derive the RANS equation in polar

coordinates, for axisymmetric geometry. We use the assumption that the mean density can

be written as p and that the swirl velocity is equal to zero. However, its fluctu~~uation L'

need to be accounted for. We neglect the mass-average of the shear stress fluctuations over

the mass-average of the shear stress: -r}
between viscous shear stress and turbulent mass flux are neglected. Thus, uiy =









(us) (ny) + U('ny + (Ui)rff After mass-averaging, the mean-flow conservation equations
in 2-D axisymmetric becomes [81]:


a p 8p(ur) 8p(uz)
++t
iit iir 8z~


p(ur)


dp(u,) dp(U,) 2 dpGr) Ez)
+ +
dt dr 8z


ii8 a Trr)> aTr~z
+ +
Br dr 8z

r r iir
1P~r: U?~


2 ro))
(p((ur I) (ne


8p(uz) 8p(ur) (uz) 8p(uz) (uz)
+ +
dt dr 8z


aST a(Trz) S(z)1
+ + -p(u,) (uz )
8z Br 8zr
1 8~? ipu~u" 8ipu~u'f 1
r Br 8zr

8 ((UZ)((E) + p))1







1813


8(E) 8 ((ur) ((E) + p))
+
iit iir


Equations 5-13 are the system of Reynolds Averaged Navier-Stokes (RANS)

equations .
5.1.5 Closure Problem: the Need for Turbulence Model

Derivation of the time-averaged equations introduce four unknowns in the time-averaged

momentum equations: namely, the turbulent stresses -, as,'ug. In order to obtain closure of

the system of equations, the turbulent stresses need to be modeled. The double correlation

between of and itself is the kinetic energy per unit volume (k) of the turbulent velocity









fluctuations. Thus, we define:

pk = 2, -II)IU' (5-14)

The terms as''ug are called the Reynolds-stress tensor and are denoted by:


po-, -, -a~ug(5-15)


The approach to model the turbulent stresses defines the type of turbulence model chosen.

There are a lot of turbulence models which have been developed since the early 70s. Each

one has its particular domain of application, computational requirements. We classify and

present the main models of Reynolds Averaged T li-; v Stokes turbulence.
5.1.5.1 Turbulence models: classification and overview

RANS models can be divided into two broad approaches: the first category is based

on the Boussinesq hypothesis, the second one on the Reynolds Stress Models.

1. The Boussinesq hypothesis relates the Reynolds stresses to the mean velocity
gradients:
7.,,~ ~~ ('C/ =i dui + k+p (5-16)

This method involves using an algebraic equation for the Reynolds stresses
which include determining the turbulent viscosity, and depending on the level
of sophistication of the model, solving transport equations for determining the
turbulent kinetic energy, k, and dissipation, e. The advantage of this approach is the
relatively low computational cost associated with the computation of the turbulent
viscosity Pt. The models available in this approach are often referred to by the
number of transport equations:

(a) Zero-equation: In a zero-equation models or algebraic models no transport
equations are solved. The models are calculated directly from flow properties.
The two most common algebraic models are the Baldwin-Lomax [63] and
Cebeci-Smith models [82] which are based on:


Pt = inne if z< zb(5-17)
oster x Xbiz > zb

where z is the normal distance from the wall, and zb is the smallest value of z
at which values from inner and outer formulae are equal. The formulation of
I-inner and Ipower depends of the model of choice. The algebraic models are often
too simple for use in general situations, but can be quite useful for simpler flow










geometries, in start-up situations and they are quite accurate for high-speed
flows with thin attached boundary 1.vrlis.

(b) One-equation: One-equation models solve one turbulent transport equation,
usually the turbulent kinetic energy k. Historically, the first model was the
Prandtl's one-equation model [83]. In 1945, Prandtl introduced the notion
that differential equations must be introduced to overcome the limitations of
algebraic models [84]. The equations should have both convective (first-order)
and diffusive (second-order) terms to explain how turbulence created at one
point could have an effect somewhere else. However, Prandtl introduced only
one such equation: his model therefore had the disadvantage that the length
scale still had to be guessed. The differential transport equation to solve for is:

8ik 8k 8iUi k3/2 k
+ U j CD 1 i)j[" t i)j


where CD and ark are COnStantS and Vt = kl/21
Another one-equation, which use is growing particularly in the area of
supersonic wall-bounded flows, is the Spalart-Allmaras model [69]. Detailed
description of this model is given in the following section.

(c) Two-equation: Two-equation models are the RANS models which are most
commonly used in turbulence modeling for CFD techniques. Models like the
k-e model [85] and the k-co [86] model have become industry standard models
and are commonly used for most types of engineering problems. Two equation
turbulence models are also very much still an active area of research and new
refined two-equation models are still being developed.

2. The Reynolds Stress Models (RSM) attempts to actually solve transport equations
for the Reynolds stresses. This means introduction of several transport equations for
all the Reynolds stresses and hence this approach is much more costly in CPU effort.
For the cases we study in the scope of this doctoral research RSM models do not
bring a lot more accuracy, but lengthen significantly the computational time.

5.1.5.2 k-e model

K(-e model is a two-equation level of closure which attempt to develop transport

equations for both the turbulent velocity and length scales of the flow [70]. The k-e is one

attempt to solve for the transport equations, and uses the turbulent kinetic energy for the

turbulent velocity scale equation and the turbulent energy dissipation rate for the length










scale equation. The transport equation in their differential form are [65]:


+ div(pk V) =div .Tradl: +2/LF ; F ; pe (5-18)




+div(peV) = div -grIiade +,, C1 F F ;-Ce (5-19)

where ,

pt = pC,_ (5-20)

and E is the mean rate of deformation of a fluid element [65].





Equations 5-18, 5-19 and 5-20 contain five adjustable constants. We will consider the

standard k-e model for which:
C', = 0.09

ak = 1.00


C1, = 1.44

Ce = 1.92

5.1.5.3 The Spalart-Allmaras model

In the thrust chamber study hydrogen flow is wall-bounded: the flow is subsonic

at the inlet and as the gas expands in the converging-diverging nozzle the flow becomes

supersonic. The Spalart-Allmaras turbulence model was created and developed for

analysis of this particular type of flow [69]: it is a one-equation model based primarily

on empiricism and on dimensional analysis arguments [70]. First publication of the

model dates back to 1992. Although it is a quite recent model, it has been the object

of numerous validations and shows great efficiency in prediction of wall-bounded flows

with a reduced computational time expense. For example, Paciorri [87] validated the









model for boundary 1 e. ris in the flow on a plate with Mach numbers of 0.3, 2.0 and

6.0. It shows that for flows with no abrupt geometrical transitions the Spalart-Allmaras

turbulence model gives satisfactory results. Lorin et al. [88] also prove the quality of the

flow predictions on the flat plate benchmark exercise when they use the Spalart-Allmaras

model, as long as the positivity of the turbulent viscosity is preserved. This model is local,

i.e. the equation a one point does not depend on the solution at other points: the model is

easily usable on all types of grids. Therefore this model is easy to implement and is fast.

In the Spalart-Allmaras model, the eddy or apparent turbulent viscosity pt is

computed through a partial differential equation. The intermediate value v follows the

relation:

Pt=T v~>

where X is the ratio



and fAt is a damping function:
X3
f3 C 1

The variable v is obtained through the transport equation:

Dv= Cb1l ft2]Sv + l -[ ((v y)Vv) + Cb2 VV2] (5-21)
D t o-

car~ f -ai~ Iv] ft2 12 (5-22)

The four terms of the right hand of the equation are the production term, the diffusion

term, the destruction term and the source term, respectively. For the production term, the

vorticity magnitude S is replaced by S defined as:


S -S + fu2
K~2d2

where the function fu2 is

fu2=1
1 + X fvl










and d is the distance to the closest wall. The function ft2 1S Simply used to ensure that Vv

is a stable solution:

ft2 = Ct36-ct4X2

No particular function appears in the diffusion term, for the destruction term of the
function

f,(r) ~1 = g 1 + c6>3 4 C27
g6 C 3 Sm~2 2

Finally, the trip term or source term aU represents the velocity difference from the field

to the trip point. The function ftl is computed as follows:


-ct2 t
ftl = ctlSte AU/2


where we is the wall vorticity at the trip point and dt the distance from the field to the

trip point. The function gt writes:


gt -- min(0.1, Axt AU


As a starting point the default numbers given in [69] are used:


Cb1 = 0.1355 o- = 2/3 cb2 = 0.633

m = 0.41 car = 2.8 cw2 = 0.3
(5-23)
cw3 = 2 cry = 7.1 ctl = 1

ct2 = 2 ct3 = 1.2 ct4 = 0.5


5.1.6 Space Discretization

Using the fact that we only consider time-averaged quantities, we can rewrite the

Reynolds-averaged equations in the compact form, omitting the <> notation with

Equation 5-8: where we now have

(84 Buj 2 duk,
Tij = Iprot + d bs (5-24)
8x xi a 8Zk~










and prtot = Ip+lt. All the variables are time-averaged following Favre or Reynolds-averaging

as described above. The method to derive pt depends on the RANS turbulence model

chosen:
I-t = P~ful (X) Spalart-Allmaras
k2 (5 25)
I-t = pC, k-e

First, we use aN ?- li-;-Stokes two-dimensional conservative solver. Then we add

corrections to account for axisymmetric behavior and finally we present the algorithm to

solve the turbulent model [89].

Discretization methods are used to convert the set of governing partial differential

equations 5-8 and turbulent transport equations into algebraic equations at each

grid node. It is the heart of numerical methods and particular attention is focuses on

discretization methods research: The equivalent algebraic relations should faithfully

represent the original PDEs. Each differential term is transformed into an approximate

algebraic relation particularly adapted to the nature of the term. F'inite Difference Method

(FDM), Finite Volume Method (FVM) and Fint lmnsMto FM r h

most used and developed methods in the solution of partial differential equations for

numerical methods in general and for CFD in particular. In a FDM, the derivatives are

approximated at the selected grid points using Taylor series expansion. More details

are given in [64]. FVM is based on the control volume formulation of analytical fluid

dynamics. The domain is divided into small control volumes and the variable is located at

the centroid of the control volumes. Approximations are made on the surface and volume

integrals. The FEM belongs to the class of weighted residual methods: The transport

variable is approximated by a polynomial function = ao + a1x + a2 2 + ... + am m. By

choosing a succession of weight functions, one can generate as many equations as there are

unknowns (ai) thus yielding a algebraic system of equations.

The most compelling feature of the FVM is that the resulting solution satisfies the

conservation of quantities such as mass, momentum, energy, and species. This is exactly









satisfied for any control volume as well as for the whole computational domain and

for any number of control volumes. Even a coarse grid solution exhibits exact integral

balances and FVAI is the ideal method for computing discontinuous solutions arising

in compressible flows. Therefore it is the most commonly used method in engineering

numerical method studies. Hence, techniques, applications, limitations and advancements

are well documented. Naturally, this is the method we choose to develop our CFD solvers.

The Finite Volume Method

The key feature of the FVAI is that the governing equations in their integral form are

integrated over a control volume, in general the cell volume. Control points are located

at the centroid of the volume, midway between successive node points. Location of the

numerical nodes in a quadrilateral 2-D domain composed of 5x5 cells are given Figure 5-2.

The derivation of the algebraic governing equations for each computational cell requires

the use of Gauss' theorem or divergence theorem:


Vii~ FdV=~ F -dS (5-26)

Figure 5-3 shows the infinitesimal volume expressed in cylindrical coordinates. We get
dV = I ./; 1,.1: and

dS = ;1,31 e + drd:4 + I;. i,.ie

In the case where we apply the FVAI to the divergence of F on the control volume/cell

(ig)). Using the notations from Figure 5-4 we get:



fff V -F1.11.10.1 =: F -dS

= T k[ Frdl:II / + S[ drd;J' + T[ Ff+rdrde F~ rll:/C -/ -


=257 [ft (Fr(i+ f .j)r~dz + Fz(+ f 1 rdr), + S2 (Fr ij )Fd 1\, z I c )Fd ?,,5
{ (r~i ..j~d; +Fz(i 1 .j~rdr) r Fri" -o )rd + Fzis, )dr

(5-27)










Now, we apply the FVM technique to the axisymmetric Navier-Stokes equations 5-8.

The different terms are 8-independent and thus f de = 2xr and we divide all the terms

of the governing equations by 2xr. We derive the system of algebraic equations from the

model by Lyra [90] and Guardone [91]. We write Voli,j = ff Rdrdz the volume of the

control volume or cell of consideration, where R is the distance of the cell centroid to the

axis. Here, we wil consider a normalized grid, i.e. Ar = 0 on edges 1 and 3 and az = 0 on

edges 2 and 4. For each control volume (i, j) of the grid we get the algebraic equation:







(5-28)

5.1.7 Time and Space Discretization: TVD MacCormack Scheme

We now useth TVD Mac """ormack method presented in Fiirst [92] that we adapt to

the axisymmetric geometry. We solve the equations using a body-fitted structured mesh

with quadrilateral cells. We denote Si+1/2,j the normal vector to the interface between the

cells (i, j) and (i + 1, j) with the length of the vector equal to the area of the interface

times the radius as defined above. It accounts for axisymmetric effects. In 2-D, the length

of the vector is actually equal to the length of the interface. We suppose that the vector is

oriented from cell (i, j) to (i + 1, j) (Figure 5-5). We operate similarly for Si-1/2,j, Si,j+1/2

and Si-12

We write:

Si+1/2,j = S,z+1/2,j z" + S,T+1/2,~j 7

Si,j+1/2 = St j+1/2 z + ST j+1/2 7

S i-1/2,j = S"z1/, z +S,r-1/2,j e

Si~-1/ = t -1/2 z + t -1/2










Note that writing the numerical method using this particular formulation of the

elementary volume and surface vectors enables the use of the exact same solver in

two-dimensional geometry by simply dividing the latest quantities by Ri,j.

To make the derivation easier to read, we will now write that F, = F, F," = F",

Fz = G and Ff = G"

The finite volume IVacCormack scheme is then:

Predictor


U" at SE Fr SI .G"(

+ SE. 1F".n+ SZ cn S 1G -SE F


+ ,-1 At 3 [Vs(,", + S]~3 2- '



Corrector

1~ At
U," 2n U,,+ ,,1/ ol, E
+ S sF"+ SzxG -S
i2j+, 2,+ i2'+ 23 li,j+ z_ sy 5-0




1


-SZ 1F .G+ -3 Sz IG' z n S' i F" -S xG
ST. (5-31)
For ~ ~ 2-' the evauaio ofte icusfuesF adG a h intrae w edt
evaluate ~ ~ ~ ~ z th eoiyvcora h nefc a ela h rssrvlct n

temperature dervatve.Weue enrl el cem o cieeboh

TVDsosTr









After the predictor and corrector, we add the TVD-type terms:


Un+1 T U'+1 DW1. +DWF. (5-32)


computed as for a one dimensional TVD-type viscosity terms in the direction of

change of i and j. Several definitions are available to derive the DW terms. The

various model are presented further in this chapter.

5.1.8 Time Stepping

The TVD MacCormack scheme is implicit in time. However, to make sure that the

scheme is ahr-l-w stable, we implement the so-called CFL condition:


at = CFLmin (-3



where cs~y is the local speed of sound and CFL is chosen between 0 and 1.

5.1.9 Boundary Conditions

The types of boundary conditions are the same for all the systems under study. The

design imposes a condition on mass-flow-rate (or temperature+ velocity) at the inlet, a

condition on pressure at the exit and a condition on temperature (or heat flux) on the

wall. Also the gases of considerations are viscous, so we can assume a no-slip condition

along the wall. Since the configuration chosen is axisymmetric, an condition should be

expressed on the axis of rotation. In the approach to CFD solver we chose for our research

project we use 'ghost' cells to impose the values at the geometrical boundaries.

*Inlet Boundary Condition At the inlet is given the velocity or mass-flow rate which

depends on the reactors design. The inlet temperature is also given. An initial guess

for the inlet pressure can be interpreted from typical operation pressure in systems.

We create a ghost well downstream of the inlet plane where we impose the axial










velocity equal to the inlet velocity and the radial velocity equal to zero


Ur = 0 UZ = Uinlte (5-34)


In the case we would want to consider an already developed flow, the

uz () =niner + .44 + 215 ogl 1 (5-35)


where ff is the friction factor defined by


fy = 1. e301 Re < 2000 (Blasius Law)
Re (5-36)
ff = 0.0014 +Re > 2000 (K~oo's relation)

* Exit Boundary Condition

At the exit we suppose given a static pressure


p = pezit


and the normal derivative of the temperature

dT dT
8n 8z=

* Wall Boundary Condition

At an impermeable boundary, the flow of fluid relative to the boundary must be

tangential to it. Since the flow is viscous, we consider the slip condition for which

the tangential velocity equals the velocity of the wall.



Ur = 0 Uz = U,,II (5-37)

For the systems we consider, the walls are immobile and therefore the tangential

velocity equals zero.

To implement the numerical models we create fictive cells behind the wall.

At the wall we also prescribe two types of thermal boundary conditions:









1. Temperature

2. Heat Flux

Based on Fourier's law

q" =( Ta -,, km (5-3)

where k, is the conductivity of the gas estimated at the wall.

Then the gradient of Equation 5-38 is rewritten

T~ T
q" = kw (5-39)


where subscripts 1 and 2 refer to the outside and inside wall boundary mesh

points, respectively. Because non-constrained integration of the first derivative of

temperature provides an infinite number of solutions it is very difficult to implement

the boundary condition. To overcome the shortcut, we use the enthalpy-rebalancing

technique developed by Anghaie and Chen [93]. It is based on the observation that

under steady-state the gas enthalpy increase along the wall surface is equal to the

heat removed from the wall:

P (Az); qi'
(Tb)i+1 = (Tb)i + (5-40)
S (cp); (pu)i

where Tb is the bulk temperature, P = 2xrR is the perimeter of the channel and

S = ;,R2 the flow cross-sectional area. The previous bulk ((Tb)i+1)0 iS calculated

from temperature fields for each iteration. We then define a convergence parameter


S=(Tb)i+1 5-1
((Tb)i+1)0
During the transitional state, the product pu is smaller that in the steady-state.

Therefore, 6 > 1. When the mass flux approximates its steady state value, 6 starts

approaching its .I-i-inpind ilcal value of 1. At this point can we calculate the wall

boundary temperature from Equation 5-39. Then, as soon as he condition 6 = 1 is

verified, the solution is converged.










*Axis Boundary Condition

we suppose that there is no fluid flow and scalar flux across the boundary: the

normal velocity is set to be zero at the boundary. Practically, we implement this

boundary condition by using a fictive axis cell, as for the wall. The fluxes in this cell

are equal to the fluxes in the .Il11 Il:ent cell.



Frictive = Fad and, Grictive = G,, (5-42)

where Fadj and Gadj are the fluxes in the cell .Il11 Il:ent to the boundary.

5.1.10 Limiters and Total Variation Diminishing Condition

The concept of limiters has been introduced to prevent the generation of potential

oscillations rather than damping them. If we consider the simple conservative form of the
dU 8F
E~uler eqluation + = 0~ the total variation quantity of a variable such ats Ui is defined

by:

TV(U)= | dx

where in a discrete form, the total variation of numerical solution may be expressed by:


OO

A numerical scheme is said to be Total Variation Diminishing (TVD) in time if


TV u"+l < TV U")


If a scheme is monotone at time 'n' it has to remain monotone at time 'n+1'. When

numerical oscillations are generated the global rate of variation of the function over

the domain increases, and so is the total variation quantity. TVD states that the total

variation quantity cannot grow in time. For a MUSCL second-order scheme to satisfy the

TVD condition the fluxes terms in the differential equations are multiplied by a non-linear

function. The function is to be chosen such that the PDE satisfies the TVD condition.










The flux limiter function takes values between 0 and 1. When in a smooth region, the

gradients are small and the flux limiter should be closed to 1. Near a discontinuity, the

gradients can take values close to infinity and the value of the limiter should be close to

zero. For a discretization at point i given by:


lI = 1- ,. (1 )( > 2


the flux limiter W(r) is introduced such that





with rtz = 6 1 and rf$= fi _
i 6 i_i i-2
In the stenciling formulas, the limited gradients are in all cases a function of the

appropriate gradient ratios. The gradient limiters can be selected from the list given below

(5-43 5-46). Note that all the limiters return 0 if the gradient ratio is negative. The case

where the gradient switches direction indicates that there is a shock within the cell range

and the local reconstruction is reduced for first order in this instance. The most common

limiters are [38]:

Min-mod limiter

m~r) = min(r, 1) if r > 0(-3
0 if r <

Superbee limiter

W,(r) = max [0, min(2r, 1), min(r, 2)] (5-44)

Van Albada Limiter

1+r

Van Leer Limiter
2r
Wa (r) = (5-46)
1+r









In the MacCormack method presented by Fiirst there are three types of methods to

derive the TVD-type terms DW. We are interested in two of them: the full MacCormack

scheme gives very accurate results but is computationally expensive. A modified Causon

scheme [94] is also proposed which accelerates consequently convergence of the solution at

the expense of accuracy.

5.1.10.1 Full MacCormack scheme

The full MacCormack scheme is close to a Roe scheme [46]. We derive the TVD-term

in direction 1, in the case of a two-dimensional problem. The governing equation in its

compact form was derived Equation 5-8. We build a matrix A such that


Ai+i), i+) F\,,i

A has m real eigenvalues and m linearly independent right vectors. The eigenvalues of











i-A 3 RAR (5 48)



whee, cA isth pe diaona matrix copoe inofteignausoA.r revcrswh




components and the lth components are written




R- (Us U,)














The vectors of conservative variables U are taken at the old time step n. To evaluate the

values of A and R at the interfaces, we can use Roe's averaging technique

4Ui + ~piUi,l
Ui- 1 = (5-49)

lThe viscousity couefficilents G are the mxm diagonal matrices which elements are given by

1 |a (l) |at |1 la () |a )!"
G*r*)(,t = 1 1 (r*)1

and a(l) is the lth eigenvalue of matrix A given by system 5-47. 9 is the limiter function.

We typically choose the min-mod limiter defined in Equation 5-43. Finally,


i+,,3(5-50)
R ( r + ( R- z1Us_1


5.1.10.2 Modified Causon limiter

The TVD-type limiter in the direction 1 is given by:


DW = ( I(r )
Tu 2~,3(5-51)
~ ~ [G (r )+G- ]U Up_)


ls i U U Un Un



ri i -U_ U (5-53)

zJ Tin U Un_ 1

G *"(r )= 0(< )1 ( (5 541)









where # is the flux limiter function. Fr-om the 1-D validation case, we chose the Van

Albada technique (see equation).


C~us4) = 0,; 1 ,4)for vi,j < 0.5 (-5
C(Y~j> 0.25 for vi,4 > 0.5 5 )





where ul is the velocity in the direction of the change of the index 1, c is the local speed of

sound, az is the approximation of the cell length in the direction of the change and W is

the entropy correction function:


~(X = |z|X forlx |z|o ~> ee
W z)= z2 62 (5 57)


with ee = 10-3

5.2 Remark on the Practical Implementation of the Real Gas Properties -
Algorithm

The real challenge when implementing the real gas properties is to get convergence

of the solver, while the diverse properties such as heat capacity, thermal conductivity and

viscosity are very sensitive to small temperature and pressure variations.

Figure 5-6 presents a schematics of the algorithm that we used to implement the real

property values. At time n, we have the values of U which were derived at the previous

time step, as well as the values of p and T and all the properties (k, c, and p).

Wet thlen derive: ~lthe dIifusionI fuxes F andU G~ and ~lthe viscous~ fuxes F" and Gatm

a which is the initial time for the iteration step.

Using the TVD-MacCormack scheme we get the value of the vector of conservative

variables at the intermediate time step, classically noted n + ~: U"+I

We then derive the value of the predicted pressure using the ideal-gas relation 5-10.

However, we note that in the relation p is a function of total energy, velocity vector and










specific heat ratio. Both energy and velocity at the prediction step are derived with

exactitude. However, the specific heat is a function of pressure and temperature. We do

not know the exact value of it at n + and therefore, the value of pressure obtained is also

an estimation. The same way we derive the temperature at the predictor step using the

ideal gas law 5-9.

Fr-om these predicted values of pressure and temperature, we run the property

subroutine where the correlations of all properties are expressed as a function of

temperature and pressure and are continuous over the range of operations.

With the new value of the specific heat ratio (y) we obtain, we recalculate an

estimation of the pressure and temperature.

If the update value of both pressure and temperature is the same as the previous

prediction, then we can consider that the loop on the properties is over. Otherwise, the

properties are re-calculated and the loop on properties iterated until the values of pressure

and temperature are converged in all the grid calculation nodes.

Once the loop on properties is converged, the fluxes are updated with the all the

predicted values in the predictor step.

We now are ready to perform the corrector step, and derive the values of U at time

n + 1, using the values of the vector of conservative variables and fluxes at time n + .

Then, the flux limiter is determined. We calculate one flux limiter per modeling

dimension, therefore, for our two-dimensional axisymmetric model, we will have to derive

two flux limiters one in the axial and one in the radial dimension.

Fr-om the values of the corrector and the flux limiters, we derive the value of the

vector of conservative variables at time n + 1 with equation 5-32.

To get the values of pressure and temperature as well as fluid properties at time n + 1

we need to iterate again on the properties, in the same fashion as for the predictor step.

Once the convergence of the property loop is obtained, the complete solution and flow

map at time n+ 1 is calculated.










We iterate on the time-loop until steady-state is reached: Once steady-state is

reached, the flow maps remains the same even when more iterations are implemented.

5.3 Validation Cases: The Riemann Shock Tube

5.3.1 Presentation of the Test Case

The shock tube or Riemann problem constitutes a particularly interesting test case

since it presents an exact solution to the one-dimensional Euler equation and involves

simultaneously a shock wave, a contact discontinuity and an expansion fan. The Riemann

problem can be realized experimentally by the sudden breakdown of a diaphragm in a long

one-dimensional tube. Before time of breakdown (t=0) the diaphragm separates two gas

states at two different pressures and densities. We consider air at room temperature in the

tube. The governing equations for this test case are the one-dimensional time-dependent

Euler equations with an ideal Equation of State. Their conservative form is:

BU 8F(U)
+ =0
dt 8x

where U and F(U) are the vectors of conserved variables and fluxes, given respectively by:


p pu

U = pu F = pU2 +

E u(E + p)

The ideal-gas law applies and the specific heat ratio of air remain constant: y = 1.4. If the

diaphragfm is located a position xo then the initial condition are the following:


u = U p =p p =p LX < XO

u = U p =p p =p RX > Xo

Schematic of the experiment is presented Figure 5-7. The analytical solution is interfered

form the theory of the compressible fluid flows. Hirsch [38] gives the theoretical profiles for

different initial pressure ratios and at different times. We also derive a FORTRAN code

which computes the theoretical answer. It can be found Appendix A.










5.3.2 Numerical Method

A one-dimensional code using the theory presented in the previous section is written

and results are compared to the analytical solution. We use the TVD MacCormack

implicit scheme to derive the flow behavior. For the one-dimensional solver the predictor

and corrector steps are now written




UPr+ = Up +xU" -x F (5-58)

8F
In the case of the full TVD-MacCormack we need to derive the matrix A = ii its

eigenvalues and right eigenvectors:

Matrix A


0 1 0

Ai' = af (Y 3) (3 y)ui y 1 (5-59)



Eigfenvalues
8 1)

a 2) = ui Ci (5-60)
(3)
ti us C i

Right Eigfenvectors

1 1 1

Re = ui ui + ci Ui ci (5-61)
22~ 2 2

.22 -12 7 1










5.3.3 Results

The CFD solver results are compared to analytical results of the following Riemann

case:
left side : p = 1.0 a = 0 p =1.0
(5-62)
right side : p = 0.125 u = 0 p = 0.1

The analytical and numerical profiles of density, velocity and pressure are compared. They

are given Figure 5-8. We see that the gas behavior trendline is perfectly followed when

using with our TVD MacCormack solver with modified Causon scheme. However, the

limiter lacks a little bit of accuracy since a few numerical oscillations are generated in

the viscidity of the property discontinuities. It would be a problem in applications where

shocks are generated. However, in the system we study, no shocks should be generated.

We need a solver which can simulate accurately flow as they become close to M~ = 1 but

we should never have formation and propagation of discontinuities in the gas.

This case on the very traditional Riemann Shock tube benchmark validates the

numerical technique we chose for applications to advanced nuclear systems such as the two

systems presented (I Ilpter 6 and 7.

5.3.3.1 On the influence of the gas

We run the Riemann solver for diverse gases of interest at 300 K(. Their heat ratios

are the following



YHe = 1.67 (-3

Too, = 1.19

YUF4 = 1.08

The profiles of density, velocity and pressure at t=6.1 ms with the initial condition 5-62

are shown figure 5-9. The graph shows the importance of the value of the heat ratio on the

evolution of the properties in the shock tube. Shock fronts and expansion waves propagate

faster for gases with higher heat ratios. They lead to smaller amplitude of the shock fields,










as seen on the velocity graph: For Helium, which has the higher heat ratio, the resulting

velocity burst is smaller than for uranium tetrafluoride, which has the smallest value of

ganina. Also, for higher values of ganina, there are almost no non-physical oscillations

generated in areas of discontinuity. For y = 1.08 the solver is not able to capture the

shocks properly. Therefore the linliter does not work as well. However, since the system

we study should not exhibit propagating shocks and discontinuities in normal operation,

we will consider that the results obtained with the solvers show the adequacy of out solver

for the systems we will consider in our study.





d


Figure 5-1: Sketch of the axisymmetric representation of the 3-D channel



































(5,1) (5,2) (5,3) (5,4) (5,5)




(4,1) (4,2) (4,3) (4,4) (4,5)




(3, 1) (3, 2) (3, 3) (3, 4) (3, 5)




(2,1) (2,2) (2,3) (2,4) (2,5)




(1,1) (1,2) (1,3) (1,4) (1,5)


Figure 5-2: Location of the numerical nodes for a 5x5 domain





~


dr-`T


_ _~I


I

t--

i....
''""'.

1


I
'-

~


Figure 5-3: Elementary volume (blue area)
configuration


for integration in the axisvninetric
















Fz(idj+1/2)


Fr ~i-1/2,j)











z L 4 Fzci-j-1/2)





Figure 5-4: Details of the notations of the axisymmetric cell (i, j)




















































Figure 5-5: Nomenclature for the surface vectors













104









Time t-n
U, p, T, c, p and k are known
Calculate F, G, FY and G'


Have the properties
converged?


Have the properties
converged?


Figure 5-6: Algforithm which shows how the real properties are implemented and
compulted 105




















State L State R



Diaphragm

Initial position of the diaphragm



L 1'2 *2 3R


Figure 5-7: The shock tube experiment: Above, at time t=0 a diaphragm separates the
gases in two different states. Below, when the diaphragm is removed, a shock wave travels
down to the low pressure section of the tube. Simultaneously, an expansion fan travels
in the opposite direction. Between the gases which propagate in opposite directions, a
contact surface moves rapidly along the tube behind the shock front.


























Figure 5-8: Shock tube profile at t=6.1 ms in case 5-62. The blue curves are the
theoretical profiles and the green curves the profiles computed with our Euler solver


initial state t= o







flow state t>O


Expansion
wave


Contact Shock
discontinuity wave






































100 200 300 400 500
x (m)


100000
90000
80000
70000
60000
50000
40000
30000
20000
10000
-


100 200 300
x (m)


400 500




=6.1 ms in case 5-62 for air, helium, carbon dioxide and


Figure 5-9: Properties profile at t
uranium tetrafluoride.









CHAPTER 6
CFD STUDY OF THE COOLANT CHANNELS OF ADVANCED GASEOUS
REACTORS

6.1 The Very High Temperature Gas-Cooled Reactor and the Helium
Channel

In the scope of the Generation IV of reactors, the United States have decided to

develop the Very High Temperature Gas-Cooled Reactor. On a thermal-hydraulics

standpoint, some of the main characteristics are:

Coolant: helium under single phase (gas)

1\oderator: Graphite block matrix

Helium heat transfer and transport coefficients (such as specific heat, thermal

conductivity) are one of the highest of all gases. Helium is also chemically neutral and

does not exhibit phase changes. It is nearly transparent to neutrons, which limits the

reactivity effects associated with variations in coolant density. Therefore, helium is the

preferred candidate for VHTR coolant and working fluid.

The operating pressure of the core is determined by considerations of core energy

extraction (low density of helium) and thermodynamic efficiency. Preliminary studies

determined the optimal value to be approximately 7 1\Pa.

The core of prismatic VHTRs is composed of a fuel and moderator matrix. The

matrix mostly consists in graphite. Helium flows through cylindrical channels pierced

inside the matrix. Typically, the gas enters the core at 400"C and is heated to temperatures

as high as 1,000"C.

C'I I) Il-teristics of the flow of helium in the cooling channel is critical: the pressure

drop over the core influences the rest of the system, the heat extracted from the fuel

should be high enough to limit the maximum fuel temperature to 1,600"C to insure its

integrity.

Figure 6-1 shows a schematic representation of the system. The geometrical

parameters were determined at from Argonne National Laboratory [95]. The diameter










and active length of the coolant channel are respectively:


D = 0.0159 m

L = 7.93 m

The mass flow rate value is design specific but can be estimated such that the Mach

number of the flow is no greater than 0.1. If we used an aerospace modeling approach,

we could typically consider the flow as incompressible since the Mach number is smaller

than 0.3. However, in the case of the flow in a coolant channel, this rule-of-thumb does not

apply: if we consider that the walls of the channel are made of aluminum, therefore the

roughness of the walls is approximately equal to 0.0013 mm. The pressure drop resulting

in a simple Bernoulli analysis is

AP = 4.43 bar

Therefore, the pressure variation across the length of the core is large and will greatly

influence the density of the gas. Even if the Mach number is small, we have to consider

a compressible flow because of the large variation in pressure and temperature across the

system.

For a set of inlet temperature and pressure of 4000C and 7 MPa respectively, the

helium properties are:


p- = 3.86 10-s Pa.s

p = 4.31kg/m3 (6-1)

a = 1544.4 m/s

which leads to a Reynolds number of:


Re = 3.46 10s


Even in the most conservative version of the channel (11 0.1) the flow is undoubtedly

turbulent. Thus, we model the turbulent effects in the coolant channel with the k-e model

(Section 5.1.5.2).










The k-e axisymmetric solver presented in (I Ilpter 5 and the helium property

correlations from Chapter 3 are applied to the analysis of the prismatic VHTR coolant

channel.

6.1.1 Grid Generation

The axisymmetric representation of the VHTR coolant channel, given Figure 5-1 is

a rectangle. Therefore, we use a quadrilateral orthogonal uniform grid to solve for the

Navier-Stokes equations. The grid (100x1000) is given Figure 6-3. The grid (100x1000)

corresponds to 100 intervals in the radial direction and 1,000 intervals in the axial

direction. The red boundary corresponds to the inlet plane, the blue one to the exit

plane, the yellow edge to the solid wall and the green one to the symmetry axis.

6.1.2 Application of the TVD MacCormack Scheme Derivation of the
Viscous Term

On the uniform orthogonal grid described in the previous section, the terms

ST ,ST 1,S" and SZ are equal to zero which simplifies greatly the predictor,

corrector and viscous terms (Equations 5-29 to 5-31).

Computation of the viscous terms require the derivation of velocity and velocity

gradients at the cells interfaces i + ~,j (or i, j + )/. We\ rewrite, Euat*CionII 5-3 for the

uniform, orthogonal mesh as:

Visc~i, j) = E F) + SF. 1G" STL 1 .F' 1 SZ 1 GV 1 (62


The velocity is evaluated using:

1 (,)+ (+ 1,)
u,(i + )=(6-3)
2 2

1 uz(,j z(,j+1)
z(,j+) = (6-4)
2 2

From the definition of F" and G" we need to derive the values of -r,, and -rz, on

the interfaces in the i-direction and the values of -rzz and -rz, on the interfaces in the









j-direction.


(6-5)


1
T i+ j)
2


1
2


(6-6)



(6-7)



(6-8)


1 iii48iiuz 2 84~u ur


Therefore, for the source term, we need an evaluation of the shear stress tensor

component 'roo at the center of the cell.


To8s u) = poor2iiu,1 (6-9)
3 r 3 B~r 3 8z

For the evaluation of the derivatives we use dual cells as seen Figure 6-2, with the

vertices located at the centers of gravity of the cells and at the end point of the interfaces.
We use


ur~ii+ ,;+4) = 4 [Ilr(i'i)f + r(i

which reduces at the solid. walls to up ~.+;

we get:


+1,y) + ur(iy+1) + ur(i+1,y+1)]


(6-10)


0. Usingf the nomenclature of Figfure 6-2


VO AC1 A n i, ,) + n, C u, 4 ,
VoACD\2 2

+ C'D + DA
2 2
(6-11)


We use a similar formulae for the derivatives of n, and T.


+,(i+, j) = prot + iu 2d U
2 r38


ipros + i
Bir 8z~


B~~ [iuzi~ 84u~i


Bur
8 z










When using a uniform normal grid, the surface vectors S at the interfaces reduce to:


Si+1/2,j = S,T+1/2,~j T

S i, j+1/2 = St ~j+1/2 z

Si-1/2,j = S,F-1/2,~j T



where






St =-R 1Azc

SEZj_/ = -RiAr

6.1.3 Simulation Results

6.1.3.1 Validation of the model

To validate our model on the coolant channel specific geometry, we perform a

code-to-code benchmark: The system is modeled both with FLUENT and our code. Since

we do not model temperature and pressure property dependence with FLUENT, the

comparison will be done with constant properties, except for density which is modeled

with the ideal-gas-law.

To limit the computation time, the benchmark is done on a 10 cm long section.

Specification of the numerical method is given in Appendix B. It is the same for all

the FLUENT studies which are presented through the document.

The boundaries for this model are:

Operating pressure: Poperating 7M~

Properties: They are considered constant and given by Equation 6-1 and k = 0.304
W/K(.m and c, = 5.1888 J/g.K(

Inlet boundary: Typically, the velocity of the fluid in the coolant channel is of
the order of a couple of meters per second. Here, we will consider the value of the










Advanced High Temperature Reactor developed at Oak Ridge National Lab [96]. We
consider a purely axial velocity:

uz = 2.32 m/s, and u, = 0.0 m/s

.For FLUENT compressible solver, we need to input the inlet mass flow rate, in
other words rjh = 0.00231 kg/s. The inlet temperature for this particular analysis is
equal to 4000C (i.e. 773 K().

Wall: We consider a constant temperature wall for ease of implementation in both
models. The average value is equal to 9000C.

Outlet boundary: The outlet pressure is equal to 7 MPa.

The profiles of pressure, temperature and velocity on the axis of the channel section

are plotted Figure 6-4 to 6-6. Each figure shows the profiles obtained with FLUENT and

our in-house solver.

The pressure drop across a 10-cm section of the coolant channel is equal to 10.24 Pa.

Our solver, when implemented with constant properties, gives results of the same order of

magnitude as the test-case run with FLUENT and therefore shows the good performance

of our numerical method.

The same test-section in now run with our in-house solver, accounting for the helium

property dependence with pressure and temperature. Figures 6-7 to 6-8 show the profiles

of pressure, temperature and velocity respectively, on the axis of the test section for both

frozen chemistry and real properties. The differences are not drastic because helium is the

gas which behaves closest to the ideal gas law. However, the variable properties should not

be neglected during a high temperature gas-cooled reactor study: On a section as small as

10-cm long, the pressure drop calculated with the two solvers differs of !I' Since the real

channel is 7.93-meter long, we can imagine that the difference will grow proportionally.

Therefore, it is important to include real property formulation of helium when analyzing

the flow in a high temperature gas cooled reactor.










6.1.3.2 Modeling of the VHTR Coolant Channel

The test case is shown Figure 6-1. At the inlet, we impose a constant axial velocity

with corresponds to a Mach number of 0.1: uz = 2.32 m/s, an inlet temperature of 4000C.

The wall temperature is linearly dependent with the axial position and varies between

4000C at the inlet and 1,000oC at the exit. We impose the exit pressure, PEzit a

7.106 Pa. The time step is automatically implemented following relation 5-33. Practically,

the time step is typically of the order of at = 4.10-8 s. The steady-state is obtained after

12,500 time iterations which corresponds about to 25 min CPU. Figure 6-10 shows the

profiles of absolute pressure, temperature of the wall and at the axis, velocity magnitude

and density.

The pressure drop across the high temperature gas-cooled reactor core is equal to

5,500 Pa. The flow, driven by the hot wall, is accelerated and the exit velocity is 1.4 times

the inlet velocity. The exit temperature on the axis is about 1,000oC which shows that the

integrality of the heat generated at the wall is transmitted to the fluid.

6.1.4 Conclusion on the Helium Channel Study with Real Property Fine
CFD Solver

The study of the coolant channel in an typical helium-cooled reactor shows the

validity of the fine-mesh CFD solver that was developed for real property models. The

simulation of a small section with constant property with both FLUENT and our solver

demonstrates the accuracy of the CFD solver developed.

The comparison between the frozen chemistry model and the real property simulation

shows that even if the profiles calculated are very similar, the numerical differences are

significant enough to see benefits of using the variable property solver.

The model of the helium coolant channels gives great confidence in the overall

performances of the fine-mesh CFD solver with variable or real fluid properties.









6.2 The Nuclear Thermal Propulsion and the hydrogen channel

6.2.1 NERVA Derivative Hot Channel Analysis

Out of all the NTP designs, we select the NERVA derivative design because it will be

then studied with the coupled solver (ChI Ilpter 7). Hydrogen flows in the channel to cool

down the walls which model the fuel. Heat released from fission is modeled by applying

a heat flux to the walls of the model. The hot channel of the system is analyzed and the

heat flux profile is plotted Figure 6-11. The weight function is then multiplied by the

average heat flux in the core.

The dimensions of the coolant channels in the NERVA derivative system are:

D=2.54 mm

L = 1.27 m

The system is studied in the typical conditions of operation. NuRok, the one-dimensional

simulator for NTP hydrogen flow is run for the following input:

100, 000 lbf of thrust

thrust chamber temperature: 3, 000 K( (6-12)

thrust chamber pressure: 1, 000 psi = 6.89 MPa

The mass flow rate, inlet temperature, inlet pressure and average heat flux are extracted

from the run output for steady-state. The characteristics are:

mj = 46.20 kg/s

pi = 74.3 MPa (-3

Ti = 302. K(


The results of the simulation are presented Figures 6-12 to 6-14. Figure 6-12 shows

the pressure profile on the axis of the channel. The pressure drop across the channel is

equal to 1.4 MPa. The profiles of temperature on the channel axis and along the wall are










plotted Figure 6-1:3. The exit temperature at the center of the exhaust plane 2,799 K(.

Therefore, hydrogen through the core of the NTP under the conditions of average flux


the density evolution on the axis of the hydrogen channel is shown Figure 6-14. As the

temperature increases and pressure drops, the density decreases rapidly. At the exit of

the channel, the density is more than one order of magnitude smaller than at the inlet.

Steady-state is reached after 17,000 time steps.

6.2.2 Hydrogen dissociation effects

If the NTP systems were to be operated at low pressure, hydrogen would start

dissociating in the coolant channels of the nuclear core. We study here the influence

of hydrogen dissociation on a small channel test-section. The system has the following

characteristics:

radius: equivalent to the NERVA derivative coolant channel r = 1.27 mm

length: to minimize the number of mesh length, we study a :$-cm long channel:

L = :3 cm.

To study the influence of the hydrogen dissociation, we will start the analysis with a case

where there is no dissociation. Then we will increase the inlet temperature such that the

dissociation becomes more and more important. For this study, we will consider the walls

adiahatic: the pressure drop across the channel is important enough to reach pressures

for which the dissociation starts. The inlet pressure is equal to 0.1 bars and the starting

temperature is equal to 1,500 K(. Then the case is run for an inlet temperature of 1,700

K(. The profile of molecular weight on the axis of the channel is given Figure 6-15 for the

diverse inlet temperatures investigated.

As dissociation effects start prevailing, convergence of the property loop cannot

he reached anymore. This is demonstrated in Appendix D. Because the heat capacity

gradient vs temperature is high, the smallest change in temperature leads to a large

variation in the heat capacity, the specific heat ratio and therefore the pressure. The










solution at each computational node oscillates between two extrenia of pressure and

temperature. The loop on properties does not converge. Even if an average value between

the two extrenia were to be chosen, the error at each node becomes so great that the

solution is not stable and then diverges.

6.2.2.1 Conclusion on the Hydrogen Channel Modeling

The CFD solver proved to be robust in the modeling of non-dissociated hydrogen. It

is also used to model systems with small levels of dissociation. However, the subroutine

with the loop on properties do not converge when the level of dissociation is greater than

a few percent and therefore, a new technique should be developed to assess hydrogen

dissociation in the NTP systems.








Exit Condition
- Pressure


400


Inlet Condition
- Temperature
- Velocity

Figure 6-1: VHTR coolant channel


Telad (K)




















(i-1J)


zT
r


Figure 6-2: Dual cell for the approximation of the derivatives at the cell interface i +






















Figure 6-3: Helium channel mesh: 100x1000 quadrilateral grid zoomed at the inlet of the
channel. The yellow edge represents the axis, the blue one the inlet and the black one the
heated wall.







































a 4







2-




s.00 0.02 0.04 0.06 0.08 0.10 0.12
Position(m)



Figure 6-4: Static pressure (Pa) profile on the axis of the test-section. Comparison
between FLUENT simulations and our solver.




































S2.8-


S2.7-






2.5


2.4-



2#00 0.02 0.04




Figure 6-5: Velocity (m/s) profile on
FLUENT simulations and our solver.


0.06
Position(m)


0.12


the axis of the test-section. Comparison between





























690-




685-




S680




675-




678.30 0.02 0.04




Figure 6-6: Temperature (K() profile (
FLUENT simulations and our solver.


0.06
Position(m)


0.12


on the axis of the test-section. Comparison between




































10-




8-





a 4



2-


9.000 .400600 .001
Poiinm


Fi ur6-:SaiprsuePaprflonteaioftets-eto.C m ain
bewe osatadralpoet omltos



































S2.9-


S2.8-


a 2.7-


S2.6


2.5


2.4-


21#00 0.02 0.04 0.06 0.08 0.10 0.12
Position(m)



Figure 6-8: Velocity (m/s) profile on the axis of the test-section. Comparison between
constant and real property formulations.




























690-




685-




S680-




675-




678.30 0.02 0.04 0.06 0.08 0.10 0.12
Position(m)


Figure 6-9: Temperature (K() profile on the axis of the test-section. Comparison between
constant and real property formulations.


























6000 I I I I I 1300
axis
? 5000~ 1200 1- al
1100
S4000-
I \ ~1000-
S3000-

a 2000-
a 800-
3; 1000~ 700-

001234567860012345678
Position(m) Position(m)
5.5 .


4.5 -1 3.0


S4.0 2.8-

El 3.5 -2.6-

3.0~ --30 2.4

2. 123456782. 12345678
Position(m) Position(m)



Figure 6-10: Property profiles across the helium cooling channel.

































X XXXXX X

x xxx
X
x
X
x
X
x
X
X
X
x
X
X
x
x
x
x
x
x
x
x


XX
Xx
x
Xx
x


0.6E


0.4E


0.2L-


0.6 0.8
Position(m)


1.0 1.2


Figure 6-11: Heat profile on the channel walls































7.4


7.2


S7.0


~i6.8


S6.6


6.4


6.2



08.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4
Axial Position (m)



Figure 6-12: Absolute pressure (\ Pa) profile on the axis of the NTP coolant channel
























3000


2500



2000



E 1500



1000



500




8.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4
Axial Position (m)


Figure 6-13: Temperature (K) profiles on the axis and the wall of the NTP coolant
channel




























G




5-




4-























13





















































3.0




= 1, 500 K(


1.5
Axial Position (cm)


Figure 6-15: Molecular weight (g/mol) profile on the NTP axis channel for T
(no dissociation) and Ti = 1, 700 K( (very few dissociation)


+r


2.01598


-


0


0.000020


0.000018


0.000016


0.000014


0.000012


0.000010


0.000008


0.000006


0.000004


- ,00

- 170









CHAPTER 7
COUPLING OF COARSE AND FINE MESH SOLVERS

7. 1 NuRok: a 1-D System Code for Simulation of NTP Thermal-Hydraulics

NuRok goal is to simulate complete nuclear thermal engine systems in order to

compare performances of several types of engines. The code was developed in 1993 [5]-[78]

and Plancher [6] brought additions in 2002. Since the nineties, there has been a renewed

interest in thermal nuclear propulsion. It came after a long research hiatus after the

NERVA program cancelation in 1972. The research in this area therefore has been

focusing on regaining the knowledge gained in the NERVA program. It resulted in a large

ne1'~ir iv of core and engine system designs. Most of them are a variation of the NERVA

design: prismatic fuel elements and full topping expander cycle. Therefore, he NuRok

system code aims at simulating the different designs under development and to assess how

the different components of the system are affected by each other and by the operating

conditions .

The NuRok program is composed of several FORTRAN subroutines, each simulates a

component of the NTP system. The program accounts for the real properties of hydrogen.

Typically, hydrogen is the working fluid, coolant and propellant at the same time. The

code models the designs described below and sketched Figure 7-1. The schematics does

not show the actual piping network but shows the connections between the propulsion

components and how the flow is regulated.

Rocketdynes' nuclear vapor thermal rocket: This expander engine cycle is called a
vapor rocket because the uranium fuel is in the form of uranium tetrafluoride at high
temperature. The gas is embedded in fuel channels.

NERVA derivative: It consists of the NERVA design adapted to 'Md w~i's technology.
Typically it is a dual turbo-pump topping cycle with direct cooling. The main
particularity of this design is the use of structural tie-tubes inside the nuclear core.

Pratt and Whitney XNR2000: This system is meant to work at much lower thrust
levels (25,000 lbf instead of 75,000 lbf for NERVA derivative and the nuclear vapor
thermal rocket). The core uses a neutron fast spectrum and a two-pass coolant
configuration is emploi-e I The gas goes through a pre-heater core on the outer










diameter of the nuclear core and then though the central core without being mixed.
This design enables flexibility and lower fuel peak temperatures. Two types of
XNR2000 systems are considered. They differ by the system connections as seen
Figure 7-1.

*Square lattice honeycomb (SLHC): In the three previous designs, the fuel is an
hexagonal matrix of fissile materials pierced with thousands of holes to enable
the hydrogen flow, which removes the heat from fissions. manufacturing of such
fuel is very challenging. A new type of fuel, called the Square Lattice Honeycomb,
facilitates the manufacturing steps: the fuel originally under the form of a powder
is compacted into 1 to 2 nin thick grooved wafers which then are interconnected to
form a "honey puck" type fuel subassenthly. The subassemblies are inserted into a
graphite matrix to form the core of the propulsion. The flow nmap is the same as for
the Nuclear Vapor Thermal Rocket, except in the core where the cross-sectional area
of a coolant channel is a square.

All of these designs are full-topping expander cycles. The term 'full-toppingf' is used

because all of the propellant is exhausted through the nozzle at the core exit temperature.

It differs front the 'hot-bleed' cycle where the coolant front the thrust chamber and

pump outlet are mixed and used to drive the turbine and which sends the cooler turbine

exhaust through a separate nozzle. Originally, the NERVA program considered the hot

bleed configuration for its simplicity. The newer designs favor the full topping expander

configuration because performs better.

The NuRok core returns information about the entire engine system such as mass

flow rate, pressure at any location, nmaxiniun fuel temperature, for example, once a set

of operating conditions is converged upon. The operating conditions are defined by the

rocket thrust and the thrust chamber inlet pressure and temperature.

Each of the components of the engine are described by an independent subroutine: for

example, turbine and pump are each described by a subroutine. The core is represented

by its hot channel. The core subroutine calls for other subroutines such as the ones for the

pressure drop in a tube, the heat flux profile, the clad temperature, the friction factor...

The subroutines work as modules and the different design configurations are described by

simply linking the different subroutine/modules accordingly.










7.1.1 Description of NuRok Core Subroutine and One-Dimensional Correla-
tions

The pressure drop across a nuclear rocket core is of 1! in r~ importance since it

significantly influences the other engine components. Its analysis should be as accurate as

possible. However, it is not an easy task because of the complex nature of the core flow.

The ultimate goal of the core subroutines in the system architecture of NuRok is to adjust

the core power until a desired exit temperature is reached. Then, the code derives the

resulting pressure drop across the system. The thermodynamic properties of the propellant

are known at the inlet of the core. The core subroutines must analyze the flow transition

from the inlet chamber to the coolant channels, flow through these coolant channels, and

the transition from the channels to the thrust chamber. The sudden contraction from the

inlet chamber into the coolant channels, and the expansion that occurs as the hydrogen

coolant exits these channels are the plenum effects. The contraction and expansion

of the flow is considered instantaneous and thus adiahatic. A sharp pressure drop is

considered at the inlet and exit plenums. In the 1D model developed, the pressure drop is

a function of the area difference between the core inlet or outlet chambers and the coolant

sub-channels. The pressure drop is modeled in the upper and lower plenum by:



ap = 7. ,I f (7-1)

for an area contraction and

ap3 = kep (<' <'2 2 (7-2)

for an area expansion, where k,. is the contraction loss coefficient defined by:


ke. =1 + l-AA +A (7-3)

Since all of the coolant channels are geometrically identical and mean radial conditions

are used in the analysis, only one channel must he analyzed to obtain an axial flow

profile though the core. This axial flow profile includes information about the propellant










temperature, pressure, velocity, Alach number, and thermodynamic properties. The profile

is generated by discretizing a one-dimensional axial representation of the square channel

into 50 elements. The conditions derived at the previous element's exit node are set equal

to the current element's inlet node and used to calculate the exit node conditions of this

current element. The desired inlet thrust chamber temperature is input by the user and

equals the core exit temperature. It dictates the core power and the mass flow rate at a

given thrust level. To calculate the propellant temperature rise, the system is considered

under steady-state conditions. A linear heat generation rate (LHGR) is derived from the

heat flux profile at each node element. The enthalpy rise across a coolant channel element

can he calculated from:

ah = LG (7-4)

The inlet node pressure is used in conjunction with the exit enthalpy to find the exit

temperature and hence the bulk temperature profile is derived. The temperature rise

in the core is of the order of 2,500 K( so hydrogen properties across the core cannot he

considered constant. Also because each coolant channel is very small, Alach number of the

flow can he greater than 0.3 and the compressible nature of the flow has to be accounted

for. Finally, heat exchanges in the boundary lIwa-r of the channel must he considered. The

one-dimensional correlation accounts for boundary 1,:ce- heat transfer and compressibility

of the fluid. The equation:

(1 1 ax
Ap = -772 + 2f f7 (7-5)
P1 P2 D p>

agrees best with experiment and computer simulations and is usually recommended (K~ays

and Perkins, 1985). It is the equation used for pressure drop calculations in NuRok. The

value 1 is the integrated mean value across the duct section ad can he approximated by:

1 1 1 1
Pro 2 p1 P2









where ff is the fanning friction factor. Different correlations can be found in the

literature. The most commons are summarized in Table 7-1. The definition implemented

in NuRok is the correlation by K~oo.

Finally the fuel surface temperature profile must be computed: once the surface

temperature is known it is possible to derive the maximum fuel temperature by considering

conductive heat transfer in the fuel. It is important to compute the maximum fuel

temperature since it limits the core operating power: for a given operating condition one

has to make sure that the maximum fuel temperature is low enough compared to the fuel

melting point. To compute the surface temperature at any axial location of the core we

need the information on bulk temperature and heat transfer coefficient:





where & is the heat transfer coefficient defined by


& (7-8)


Colburn, Dittus-Boelter, Seider-Tate and many others developed correlations to find the

Nusselt number. The most common correlations are given in Table 7-2. The relation used
in NuRok is:

Nu =Nass (7-9)


wheren = -log 0.3 (K~ays and Perkins, 1985) and the isotropic Nusselt
number is derived with the Petukov (1970) relation and an axial distance correction. This

correction is needed since there are entry effects due to the thin thermal boundary le rli

at the inlet of the channel. A correction function by Pierce is implemented, and little

further research will be done on the axial correction factor since for the channel of concern

x/D ~ 750. The Pierce correction is:
( z-0.7
Nuz)=Nu1+ (7-10)
Dj










To illustrate the shortcomings of the one-dimensional correlations, we compare

relation 7-1 to a three-dimensional CFD simulation of a channel with sudden area loss.

The model represents a circular tank which area reduces to a square channel as seen

Figure 7-2. The radius if the tank is 5 mm and the edge of the square channel is equal to

1 mm. The simulation is done using the k-e segregated solver of a tetrahedral mesh with

118,042 computational cells. The test section is 5 cm long and the walls are considered

adiahatic. The operating conditions are typical of the hydrogen conditions at the inlet of

the SLHC core: 1 atm and 186 K(. Figure 7-3 shows the static pressure evolution along

the axis of the test-section. The inlet of the test section is located at x = -2 cm and

the area contraction at x = 0. The left-hand portion corresponds to the pressure drop

over the tank. It is almost negfligfible. The middle portion corresponds to the abrupt

pressure drop due to the flow contraction and the right-hand portion to the pressure drop

over the channel. For comparison with the analytical formula 7-1, we only consider the

abrupt pressure drop. Results for diverse inlet velocities is given Table 7-:3. Overall, the

correlation underestimates the importance of the pressure drop at the area contraction.

As the inlet velocity increases, the error reduces, however, the difference is still very

consequent (";-' .).

This simple example illustrates how the 1-D correlations may fail to predict the flow

behavior accurately. This is why this ChI Ilpter will present further down the coupling

technique used to increase NuRok accuracy. Beforehand, let us review the upgrades we

brought to the system code itself.

7.1.2 NuRok Upgrading

The first version of NuRok was developed in 199:3 and was becoming obsolete. We

made several adjustments to update the old version.










7.1.2.1 Update to FORTRAN 90

First, we updated the program to FORTRAN 90 to bring more stability to the solver

(obsolescence of the COMMON procedure for example) and help compatibility with the

fine mesh subroutines.

7.1.2.2 Levels of thrust

We also modified the code such that it could handle larger thrust values and thus

larger core powers. The initial code would consider the diameter of the core to be constant

and equal to the design diameter. The NuRok versions by Given and Plancher would

enable runs for thrusts ranging between 10,000 and 70,000 lbs. If the thrust levels were

greater than 70,000 lbs, the solver would diverge and the Mach number in the core coolant

channels would increase to critical values. Critical Mach number values in nuclear systems

corresponds to flows which are close to being transonic. If the Mach number in the core

is too high, we can imagine that with fuel expansion and swallowing the flow area is

reduced, leading to a local increase in Mach number. The flow reduction might even

accelerate the flow such that it becomes supersonic and generates shocks in the core of

the propulsion. Such a flow behavior is very dangerous for the core stability and to be

prohibited. Therefore, greater levels of thrusts for the Nuclear Thermal Propulsion require

greater flow area in the core. This means larger core diameters with the same density of

coolant channel. Increasing the core diameter increases the number of coolant channels

and then increases the flow area available for the coolant, reducing the local velocity. This

technique avoids critical Mach numbers of the flow in the core. The critical Mach number

is set to 0.4. This value provides sufficient safety margin in the core.

Practically, the flow Mach number is computed in the core with its initial design

diameter. Then the flow is iterated such that, as long as the Mach number is greater than

0.4 (critical Mach number) the core diameter is increased by 5' Table 7-4 presents the

different characteristics of the SLHC systems for diverse thrust levels. The thrust chamber

inlet temperature and pressure are 2,800 K( and 500 psi, respectively.










7.1.2.3 Foam fuel

Researchers and Engineers at NASA's Georgfe C Marshall Space Flight Center and

INSPI are interested in tri-carbide, open-cell foam fuel matrix for nuclear propulsion.

Usually, density of the foam fuel is of the order of 211' The core subroutine is modified

to allow modeling of such fuel. The user can model a NTP system with foam fuel by

selecting the flow area ratio, corresponding to the foam density. The code works for

densities between 17 and 501' Table 7-5 presents the system characteristics for diverse

foam fuel densities. This is only an approximation since very few research has been done

on pressure drop and heat transfer. The foam fuel still need to be qualified on a materials

standpoint first, before leading further research. However, the simple study presented

here shows that the only property the variation in density influences is the pressure drop

across the core. The cases of comparison exhibit a level of thrust equal to 50,000 lbs and

nozzle inlet temperature and pressure equal 2,800 K( and 500 psi, respectively. The power

transferred to the fluid is equal to 1024 MW for all densities and the maximum cladding

temperature in the core is 2, 807 + 1 K(. The pressure drop across the core increases as the

foam density (and fluid flow area) decreases.

7.1.2.4 Hydrogen dissociation

Older versions of NuRok account for the real properties, except for dissociation. Since

we are interested in performances of the NTP associated to low-pressure thrust chamber

(CI! Ilpter 7), it becomes important to model properly hydrogen molecular weight. The

implementation is the same as for the fine mesh solvers (Section 4.1.2).

7.1.2.5 Carbon dioxide

Some nuclear propulsion designs in the early stages of development considered

substituting Hydrogen with Carbon Dioxide. To investigate the qualities of carbon dioxide

as a working fluid in nuclear thermal propulsion NuRok is modified: CO2 properties are

given in Chapter 4 and are implemented in the programs. The user is now asked to choose










a coolant to perform his calculations. The user can choose between helium, hydrogen and

carbon dioxide.

7.1.3 Recommendations for NuRok Users

When running NuRok for a set of operating conditions, namely exhaust thrust

(Th (lbf)), thrust chamber pressure (PrC (psi)) and thrust chamber temperature

(Tec (K()), the solver asks for user guesses on the mass flow rate in the system (ri (kg/s)),

the pump outlet pressure (Post (psi)) and the core power (Q (\!W)). If the guess is not

appropriate, the solver will not converge. Therefore, we recommend that the user follows

the hints below. It guarantees NuRok convergence. Quantitative values are given as an

example for the SLHC system under the following conditions (which corresponds to an ISP

of 950 s-l)


6 = 1 mm; Tec = 2, 800 K(; Pro = 500 psi; Th = 50, 000 lbf


where, 6 is the wafer thickness.

7.1.3.1 Mass Flow Rate

A low value of the mass flow rate helps the convergence. In this case ri = 24.02 (kg/s).

The convergence is reached for guesses up to 30 kg/s. By default, we would recommend an

initial guess of 15 kg/s.

7.1.3.2 Pump Outlet Pressure

The solver is very sensitive to this value. It is recommended to input a pressure close

but no greater than twice the thrust chamber inlet pressure. In our example, the pump

outlet pressure we derive is equal to 1048 psi. However, the code only converges if the

initial guess is within 700 and 1000 psi. Note that in the newer versions of NuRok the

pressure is automatically guessed by the code.

7.1.3.3 Core Power

An overestimated guess of the core power helps the convergence. In our example, Q=

1041.6 MW. The solver converges if the initial guess is within 1,000 and 2,000 MW.










7.2 Themis: A Coupled Coarse-Fine Mesh Simulation Tool for Simulation of
NTP Thermal-Hydraulics

NuRok is composed of several subroutines. We can identify areas of the code where

the one dimensional simulation lacks accuracy. On the schematics of a full-toping,

by-passed NTP system Figure 7-4 we identified several portions of the system which

would benefit from a mesh and solver refinement. They are the inlet and exit plenum,

mixing junctions, coolant channels and thrust chamber. Chapter 7 detailed the reasons for

a 2-D analysis of the thrust chamber. In OsI Ilpter 6 we saw the case of the coolant channel

in the VHTR. Since the NTP is also a nuclear system with gas flow, the remarks we made

also apply to the NTP coolant channels.

At the inlet and exit of the core there is a sudden restriction and expansion of the

flow area, respectively. The particular geometry induces turbulent effects as well as

anisotropy of the flow. The nodal code of NuRok cannot account correctly for these

multi-dimensional phenomena. A detailed CFD analysis would be required to get an

estimation of the importance of the flow anisotropy in the exit plenum, for example.

Besides, hydrogen is not heated uniformly in all the core channels. Because the thermal

flux generated by neutron fission is not isentropic over the core, certain channels will be

hotter: the gas enters the exit plenum with different temperatures. The temperature gap

could be over dozens of degrees. Typically, a compact nuclear core is composed of 24,000

coolant channels. Representation of even 1/6th of the core would require several million of

computational cells. This is two computationally expensive for our feasibility study here.

However, future work could be led in this direction.

The coolant channel which is solved here with the fine mesh solver was presented in



7.2.1 Treatment of the Interfaces

There are two types of interfaces, the interface from the system code to the fine mesh

solver and the interface from the fine mesh solver to the system code. Therefore, we need










to treat the interface from 1-D to 2-D axisymmetric and vice versa. Figure 7-5 shows the

coupling schematics of a fine mesh solver for the hot channel of the core to the rest of the

NTP system code.

7.2.1.1 One-dimension to two-dimension axisymmetric



(system code) 4 (CFD solver)

This type of interface is used to input the boundary conditions extracted from

the system code calculations into the CFD solver. It is the case of the inlet velocity

components, inlet density and exit pressure. Since the system code returns only one values

of each parameter at the boundaries of the system of consideration, we will implement

uniform boundary condition values in the CFD solver. This is a trivial but accurate way

to treat the interface 1-D to 2-D axisymmetric.

In reality, it is very unlikely that the profile at the inlet of the cooling channel

is uniform. However, the flow entering the coolant channel goes first through an area

contraction and redistribution as it enter the core inlet plenum. Figure B-2 shows the

profile of velocity at the very inlet of the smaller pipe in the system presented in Section

B.2 (Appendix B). The flow before the contraction is fully developed. The second curve

shows the theoretical velocity profile at the inlet of the coolant channel if the flow were

fully developed. This Figure shows that the actual profile of velocity does not correspond

exactly to a fully-developed velocity profile. To obtain a very rigorous simulation of the

entire system, the actual flow in the inlet and exit plenum should also be modeled with

a fine-mesh CFD code. Since this ChI Ilpter aims at showing the feasibility of a coupled

technique, we do not consider the level of turbulence development as an issue in the

treatment of the interface. Only a uniform velocity profile is implemented at the inlet of

the system. For the cases of the hydrogen coolant channel presented in Section 6.2. the

developing length will be equal to Le ~ 4.4 tRer D ~ 7.6 cm. This distance only represents

5 '.of the total length of a typical NERVA derivative nuclear core.









However, in cases where the inlet flow were to be demonstrated fully-developed the

'develop.f' subroutine (Appendix C) would be used to assess the velocity profile at the

inlet of the pipe. If anode were to be the velocity value calculated at the last coarse node

before the CFD module, the velocity profile on the inlet plane would be given by


uz () =unoe 1+ 144 2.5 lg(1- r/R)(7i-11)

where, ff is the friction factor and R the radius of the pipe. This factor is already

calculated by NuRok through the K~oo formulation.

7.2.1.2 Two-dimension axisymmetric to one-dimension



(CFD solver) 4 (system code)

This type of interface is used to extrapolate the values computed by the fine mesh

solver and input them back into the system code. This type of interface is needed to

implement the pressure at the inlet of the system, the velocity and temperature at the exit

of the system.

Typically, the pressure profile in a heated tube is constant on the radial profile.

Therefore, the pressure that we compute on the inlet plane of the hot channel can

be averages and the averaged pressure input in the coarse mesh solver for a realistic

representation of the pressure profiles. At the exit of the hot channel, we need to

interpolate the velocity components and temperature. Usually, for a turbulent flow,

the radial velocity profile is almost flat. Thus, we obtain a single value of the velocity

magnitude at the exit of the hot channel by extracting the maximum value derived on the

exit plane and inputting it in the coarse mesh.

Input of the temperature is a little bit more challenging: As seen on Figure 7-6, one

computational node in the axisymmetric model corresponds to an annulus area on the

exit plane of the channel. Therefore, to get an idea of the average temperature at the










exit of the channel, we weight the temperature derived at each computational node by its

associated area. The temperature which is input in the coarse mesh therefore is:




Tezit__i=1 el (7-12)

where Aemit is the cross-sectional area of the channel, NV the number of computational

nodes, Ti the temperature derived at node i and ri the interface between node i 1 and

node i radius.

7.2.2 Themis Results in the Case of NERVA Derivative Systems

To study the feasibility of coupling coarse and fine mesh solvers, we study the NERVA

derivative system with the following characteristics:


100, 000 lbf of thrust

thrust chamber temperature: 3, 000 K( (7-13)

thrust chamber pressure: 1, 000 psi = 6.89 MPa

The overall system is modeled with the coarse mesh solver and the coolant channel

with the fine mesh solver. The channel model is the one which was developed in OsI Ilpter 6

in the Section with no dissociation.

The average core heat flux applied on the channel walls is updated at every core

iteration by the coarse mesh solver, such that the channel exit temperature and pressure

match the design requirements specified by the user for each run. NuRok obtains the

convergence on the pressure in the system by deriving the pressure drop across the coolant

channel. At each iteration, the inlet pressure is considered known. On the first iteration it

is interfered with the guess of the pressure drop across the pumps. The pressure is derived

through the coolant channel and the exit channel is compared to the user input for the

thrust chamber pressure. Depending on the difference between the two values, the pressure

drop across the pumps is re-estimated and a new iteration starts. For the CFD model

which is coupled to the system code, the boundary condition on pressure is located at the










inlet. Traditionally, the exit pressure is known. This modification does not influence the

performances of the fine mesh solver.

Figures 7-7 to 7-8 show the evolution of the different property characteristics at each

iteration step. The vertical lines corresponds to a new system-loop iterations. During a

system loop the inlet pressure and inlet temperature are calculated by the system code

and kept constant. At each core iteration, the heat flux is updated such that the exit

temperature matches the temperature input by the user. When the heat flux is derived,

the pressure drop is calculated and the exit pressure is compared to the input pressure.

The core loop is then over and the channel inlet pressure is modified such that the inlet

pressure is equal to the input pressure plus the pressure drop value. The system loop is

updated with the new value of the channel inlet pressure. A new core loop then begins.

Steady-state is obtained when the value of inlet pressure, inlet temperature and heat flux

do not evolve and the exit pressure and temperature match the user input.

Table 7-6 compiles the results of the simulation. The results are compared to

those obtained with NuRok only. Figure 7-9 and 7-10 show the profiles of pressure

and temperature across the hot channel calculated with NuRok and Themis. The core

thermal power derived with Themis is larger by 15' from the NuRok simulation. This is

mainly due to the large difference in the pressure drop across the channel that is computed

with the two methods. The pressure profiles are significantly different: not only the

pressure prod prediction obtained with the coupled solver is three times larger than the

prediction obtained with NuRok, but also the trends of the two curves seen Figure 7-9

are quite different. It is difficult to compare the temperature profiles (Figure 7-10) since

the coarse model only gives one profile and the fine model gives a couple of profiles: wall

temperature and temperature at the axis. The temperature profiles are very similar for the

two approaches. The pressure profiles (Figure 7-9) are very different. The pressure drop

calculated with the CFD solver is three time large than the pressure drop obtained with

simpler one-dimensional correlations.










Since the heat fluxes and temperature profiles are very similar in both simulations,

we can interfere that he pressure drop calculated in the one-dimensional core simulation

lacked accuracy. To make sure that the problem does not arise from the CFD solver,

a simulation of the coolant channel in its steady-state configuration is done with

frozen chemistry using FLUENT. Result of this simulation is presented in Appendix

B. Figure B-4 shows that the pressure profile obtained with our CFD solver is most likely

more accurate than the one derived by the 1-D solver.

The difference between the two simulations is very sensible. It shows the importance

of using the fine mesh solver. With NoRok, the overall code converges within 8 iteration,

whereas with Themis, the code converges after 10 iterations. The computational time

is multiplied by three. However, the overall CPU still remains under 1 min to obtain

steady-state.

7.2.3 Conclusion on the Coupled fine-coarse mesh solver

Themis highlights the shortcomings of using a one-dimensional representation of the

cooling channel. The one-dimensional representation definitively gives a very good order

of magnitude of the phenomena happening in the coolant channel and the system overall.

However, the calculations are more qualitative. The coupled fine-coarse mesh code with

real properties descriptions proved to be efficient and greatly increase the accuracy of the

calculation while the steady-state computation remains very fast.














N.A. 16Laminar flow Re < 2300
Re
Schlichting [97] 1.33 + 10-3R60.52 TTrI1Sitional flow 2300 < Re < 4000
KaronNikrase 1 4o Re -0.4 Turbulent flow 2000 < Re < 3 +106

Filonenko (1.581n(Re) 3.28)-2 Turbulent flow 104 < Re < 10'
Blasius 0.079Re-2.5 Turbulent flow 3000 < Re < 105
McAdams 0.046Re-02 Turbulent flow 6000 < Re < 2 +10s
K~oo 0.0014 + IRe-0.32 Turbulent flow 2000 < Re < 10s


Table 7-1: Fanning friction factor correlations and their range of validity [2]


Investigator


Correlation


Range of Validity


Reynolds Range


Table 7-2: Nusselt number correlation for turbulent gas flow in a pipe [3]
Investigator Correlation Range of Validity

SRe > 104
0.7 < Pr < 160

Dittus-Boelter,1930 0.023Reos~r Re 0.40. r


0.027Reos P 1/3 0


Re > 104
0.7 < Pr < 16, 700

0.1 < Pr < 10414< R 0


0.5 < Pr < 200R 0


Seider-Tate, 1936


Notter-Sleicher, 1975


Petukov, 1970


0.24
0.88 ~
4.0 + PT
0.33 + 0.5e-0.6Pr


a=
5.0 + 0.015Re Prb Where

RePr ( f/2)


1.07 + 12.7 (Pr2/3 1) g /


Table 7-3: Pressure drop over a flow area contraction. Comparison between FLUENT


simulation and 1-D correlation results.
Inlet Velocity(m/s) AP (Pa) CFD
0.2 23.2
0.37 39.5
0.5 122
0.75 261


AP (Pa) calc
4.3
8.5
27
30.7
108
243
571
10794


Relative difference ( ~)
440
365
352
333
314
301
195
272


1687
40,142




















i 80.5W V


D) 35W'


sWsr






i1ar I
-s r


y tr


< f


1C


c taral



I


/ i


n. ~


e.Isw


oa.SW


Figure 7-1: The different NTP systems modeled with NuRok: NVTR, NERVA derivative
and P&W XNR 2000 I & II







148


": '~' 0425W.

.075W



0 i 50
C.ln


--r































Fiue72:3Dgido h cnrtion tetscin o td wt LET

Table~~~~~~~~~~~~~~. 7-4 Prprisa ies oain fte qaeltiehnyobsse o
thrust~IE chme ne eprtr adpesr f280Kan 0 s =34 i)


respectively.
Thrust (lbf)
Core transferred power(\!W)
Mass flow rate (kg/s)
Turbo-pump outlet temperature(K()
Turbo-pump outlet pressure (psi)
core inlet temperature (K()
core inlet pressure (psi)
turbine inlet temperature (K()
turbine inlet pressure (psi)
Maximum Mach number


25,000 50,000 75,000
512 1024 1535
12.01 24.02 36.02
27.3 29.7 32.8
801 1151 1627
186.4 185.7 184.5
6322 8635 1148
192.8 194.8 197.3
728 1074 1547
0.13 0.25 0.36


100,000
2045
48.03
37.0
2248
182.9
1447
200.6
21633
0.46


Table 7-5: Pressure drop across the SLHC
Core Area Flow Fraction ( .) 17 20
AP(Pa) 1181.5 872.2


core for
25
552.0


different densities of foam fuel.
30 40 50
365.0 171.9 87.06


























S50-


S40-


S30-
en,

20-


10-


-8.02 -0.01 0.00 0.01 0.02 0.03 0.04
Axial Position (m)


Figure 7-3: Static pressure profile along the axis of the contraction test section


Table 7-6: Comparison of the results obtained with NuRok and Thentis
NuRok Thentis relative difference( .)
Pressure drop (Pa) 3.56.105 1.19.106 234
Element power density ( \!W) 2269.74 2626.34 15.7
Core inlet temperature (K() 302.03 372.1






































Figure 7-4: Identification and location of portions of the NTP system which would benefit
from a more detailed solver


Thrust Chamber
















,Sytembefre hecore
I I

I cI I

I I

I I

I iI

I I



I I

I I
I ThutcabrI
I,,,,,,I


4

O



/3


Direction of the flow

Interface 1D/2D


Hot channel






10r
I


Figure 7-5: Structure of the code coupling when the CFD is used to simulate the hot
channel



















TI T2 T3 T4 T5
I~- 17-I III* Computational node


Figure 7-6: Extrapolation of the temperature nodes from the axisymmetric geometry to a
single value.






























(d8.4-



S8.0-

S7.8-

a 7.6

7.4
0 5 10 15 20 25 30 35 40 45
Iteration number
7.4






~6.6-

6.4-

w 6.2-

6.0
0 5 10 15 20 25 30 35 40 45
Iteration number



Figure 7-7: Pressure evolution in MPa at the inlet and exit of the hot channel vs. the
iteration number


























-400


M 340-
t 320
ai 300-
280-
260
S0 5 10 15 20 25 30 35 40 45

2800
27300 -
~2600
~2500-

2400
2300-
~2200
0 5 10 15 20 25 30 35 40 45

S1.5-







Iteration number



Figure 7-8: Thermal characteristics of the hot channel vs. the iteration number



































S7600000-


S7400000-


7200000-


7000000-



6 0 .080 0.2 0.4 0.6 0.8 1.0 1.2 1.4
Position(m)


Figure 7-9: Coolant channel pressure profiles calculated with NuRok and Themis
























3500
Themis axis
Themis wall
3000 NuRok


2500-



a 2000-


S1500-


1000-


500-



B.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4
Position(m)



Figure 7-10: Coolant channel temperature profiles calculated with NuRok and Themis (on
the axis and the wall)










CHAPTER 8
CFD STUDY OF THE LOW-PRESSURE THRUST CHAMBER WITH FROZEN
CHEMISTRY

8.1 Presentation of the Low-Pressure Thrust Chamber

The core of the nuclear propulsion heats up hydrogen gas to temperatures ranging

from 2,800 K( to 3,250 K( depending on the design. The hot hydrogen then enters a

converging-diverging nozzle or thrust chamber where the gas is accelerated and provides

thrust to the system. In typical NTP designs, the thrust chamber inlet pressure is of

the order of 1850 psi, or 12 MPa. Table 8-1 shows the main thermal characteristics of

hydrogen at the inlet of the thrust chamber.

If the pressure of the gas is decreased by 3 orders of magnitude (0.1 bar for example),

hydrogen at the exit of the core is dissociated. Its properties are also compiled in

Table 8-1 and a schematics of the system is shown Figure 8-1 For the dissociated gas

the heat capacity is eight times greater than the one of H2. Also, the specific ratio is

much smaller. We saw Figure 5-9 that lower specific heat ratios lead to greater velocities.

Therefore, we expect the velocities on the thrust chamber exhaust plane to be greater

for hydrogen at low pressure. Moreover, as the gas travels through the chamber, its

temperature decreases. Hydrogen at the inlet of the thrust chamber is highly dissociated

(up to t:I' .) and recombines as it flows through the nozzle. The energy balance is

transferred under the form of kinetic energy, which helps increase the exhaust velocity

and the system ISP.

To validate our assumption, we first use a quasi one-dimensional approach to derive

the basics geometrical information on the nozzle (throat dimension) and the nozzle

characteristics (exhaust velocity). Then, we present the method for the nozzle contour

design. Once all the geometrical parameters are derived, we present two studies: the first

one is a FLUENT analysis of the nozzle with frozen chemistry, the second analysis will use

our solver.









Table 8-1: Hydrogen thermal properties (heat capacity,specific heat ratio and thermal
conductivity) for diverse sets of operating conditions
Pressure (bars) Temperature (K() cp (kJ/kg.K() y, k (W/m.K()
120 :3,000 2:3.48 1.250 1.696
0.1 :3,000 179.5 1.1:31 15.31


8.2 Quasi-one-dimension Isentropic Analysis of the Low Thrust Chamber

First we derive the throat radius and the exit Mach number with the isentropic

relation [98]:
y+1
AA 1~ 2-~ -* y- ~i1 2 (rl (*- 1


where, ,4 is the area of the nozzle at a given point and At the Mach number of the flow at

this point. ,4* is the area at the throat, where At = 1.

The relation is accurate for one-dimensional steady flow where body forces and

gravitational effects are negligible, for a frictionless ideal gas with constant specific

heat and in the case of an adiahatic flow with no shear and shaft work. Some of the

assumptions such as constant specific heat are not verified. However, in this first study

we look for an order of magnitude for the nozzle performances. Therefore, we will assume

that all the conditions of the isentropic study are verified.

The inlet Mach number in the system is equal to 0.3 and the inlet radius of the thrust

chamber is equal to :30 cm. Hence, the throat radius is:


r* = 20.77 cm


We now calculate the Mach number on the exit plane for an exit radius of 1 m. We

solve Equation 8-1 for the ratio =2:3.19. We get:


S. I. = :3.569










To derive the exhaust velocity knowing 3. I. we need the exhaust speed of sound

tezit:
Dexha~ust
tezit

The speed of sound for an ideal gas is given by:


temiit = Txit~


Therefore, we need to derive the exit temperature. To do so, we use the Rayleigh relations.

The Rayleigh relations are accurate here, since they are aimed at modeling flows with

heat exchanges where the viscosity remains negligible. The ratio of temperature T at the

location where the Mach number is equal to M~ over the temperature at the throat T* is

given by:
T (1 + )2M\2
(8-2)
T* (1 + 73 F)2

We get:

T\I = 0.336676 and T = 0.243820
M n=0.3 M=3.568

Then ,
(T\
Texit M=3.568Tinlet = 2, 172.65K

M n=0.3


We approximate the molecular weight of the hydrogen of the flow to the hydrogen

molecular weight at 3,000 K( and 0.1 bar: M~ = 1.622 g/mol.

Finally, the speed of sound at the exit is aemit = z/7RTexit = 3549 m/s.

Using the quasi-one-dimensional relation for isentropic flow, and the Rayleigh

relations to account for heat transfers, we expect the gas velocity at the exhaust of the

thrust chamber to be equal to 12,663 m/s for an inviscid flow.










As the Mach number of a flow increases, the effects of viscosity cannot be neglected

anymore. We can account for viscous effects when considering a Fanno flow. The ratio of

temperatures becomes:
T y+1
(8-3)
T* 2 + (y 1) O1

Now,

T\I = 1.05926 and T = 0.581016
M /=0.3 M=3.568

Then ,

Texit = 1645.53K

When considering an adiabatic viscous flow the exhaust velocity becomes:


vezit = 11, 020 m/s


Depending on the approach when leading the quasi-one-dimensional we get different

exhaust velocity values:

Inviscid adiabatic flow vezit = 12, 663 m/s

Viscous adiabatic flow vezit = 11, 020 m/s

In both cases, we derive an ISP value greater than the ISP for thrust chamber at high

pressure.

For the real low-pressure thrust chamber, the assumptions to use Reigfhleigfh

flow, Fanno flow and isentropic flow relations are not all verified. To get a more

accurate description of the nozzle flow we need to model the thrust chamber with a

two-dimensional solver.

8.3 Two-Dimensional Contour Profile of the Thrust Chamber Walls

To design the contours of the low pressure nozzle the method of Rao [99] extended

to real-gas is chosen, coupled to a CFD-based optimization procedure elaborated by

K~orte [100]. The converging diverging nozzle is divided into a subsonic-throat region, a

transonic region and a supersonic region. The specific heat ratio at the inlet of the nozzle










is equal to 1.131. The throat radius is estimated for a Mach number of 0.3 by using the

isentropic relationships for converging nozzles. The throat radius is equal to 15.9 cm. The

minimum area of the nozzle is located at z = 0.

Converging region:
In that portion the flow remains subsonic. The nozzle radius at a position z is given
by:

y *C 1 l-e 2C1Rc


where ,
Ci = Retan2 8a)~ e



Transonic region:
The transonic region is described by a circle of radius Rer*

Supersonic region:
The supersonic contour is described by a contoured nozzle geometry-cubic
polynomial [101]. The nozzle radius y at an axial position z is given by:

Y = Yo + o~z zo) + c(z zo)2 ~ 0 Xo3

with
3(yL yo ygL) (yLI yoI)
c= -o
L2 L3
To design the optimum converging-diverging nozzle, five parameters have to be determined:

subsonic approach angle 8,

nozzle inflection angle 8

throat radius of curvature Re

slope at the exit of the chamber yE

length of the diverging portion L

In the following analysis, we make the assumption that the slope at the exit of the

nozzle is equal to zero: y~ = 0. The best nozzle shape is determined by looking at the










performances associated to each design. Uniform velocity profile and maximum velocity on

the exit plane are a proof of optimum performances.

8.4 Grid Generation

We generate a structured grid: the interfaces in the axial direction are parallel to the

inlet and exhaust plane. The radial interfaces are aligned with the flow path. In the area

of the throat, the density of cells is greatly increased.

For the coarse mesh shown Figure 8-2, the length of the meshes is equal to 10 of

the nozzle radius at that location. We also create a boundary 1... -r in the area of the wall

to resolve properly the boundary 1 ... r. The grid we show Figure 8-2 is very coarse and

does not resolve the boundary 1... -r. Therefore the nodes are equidistant in the radial

direction.

8.5 Low-Pressure Thrust Chamber Analysis using FLUENT and Frozen
Chemistry

This section aims at having a preliminary look at the numerics choices possible for the

computational case. Once the suitable model is chosen, we do a sensitivity study on the

design parameters of the nozzle contours.

8.5.1 FLUENT

FLUENT is a commercial CFD code which uses a finite volume method. A

pre-processor, GAMBIT allows mesh generation. All modeling capabilities are accessible

through a GUI. The versions we use are GAMBIT 2.3.16 and FLUENT 6.3.26. Modeling

details and definitions are given in Appendix B.

8.5.2 Numerical Procedure

Description of hydrogen properties with temperature and pressure are complex to

model in a simple fashion. Therefore, we present here an analysis of the low-pressure

thrust chamber with frozen chemistry. Frozen chemistry consists in keeping all hydrogen

properties constant and equal to the inlet properties, except for density which is modeled

with the ideal-gas law.










The design base for this study is the nozzle with:


8, = 200


Rci = 2m

< remi = 1m(8-5)
8 = 40

L = 3m

y~ ',=0

Our interest in this section is not the nozzle performance but the determination of

the most accurate numerical methods and models. Therefore, we will use the geometrical

parameters 8-5, regardless their influence on the flow map.

The nozzle of interest is the low pressure thrust chamber at the exit of the SLHC

nuclear core, i.e.

ri = 30cm

ifl 0.3
<~(8-6)
Ti = 3, 000K

Pi = 0.1bars

In reality, the gas exits the nozzle and is released in space, where the surrounding

pressure is close to 0. Hence, the exit pressure in the simulation is set to 10 Pa. In the

following study, we will limit our model to an adiabatic system with isothermal walls.

The fluid used in the system is hydrogen as described in OsI Ilpter 4. Since FLUENT

does not enable modeling of the fluid properties as a function of both pressure and

temperature, we will consider frozen chemistry: we use the ideal-gas law formulation of

density and all the other properties are constant and equal to the properties of hydrogen










at the inlet of the system:
m = 1.622 g/mol

c, = 179, 500 J/(kg.K()

p- = 4.47 *10- Pa.s

k= 15.31 W/(m.K()

To capture properly the pressure in the supersonic fluid, we use the third order

MITSCL scheme on density, momentum components and energy. We also use a second

order upwind scheme on pressure and turbulence components.

In the case of the system of N l.-;1 r-Stokes equations we use the Spalart-Allmaras

turbulence model.

8.5.3 Analysis results

8.5.3.1 Investigation of the nozzle representation

FLUENT allows for three type of representations of any fluid system: 2D, 2D

axisymmetric and :3D. A three-dimensional representation ahr-l- .- gives the best simulations,

however it is computationally expansive: :3D models require a lot of computational cells.

In systems which are symmetric with respect to an axis, such as a tube, or here the thrust

chamber, a 2D axisymmetric representation is often sufficient. This is true provided there

are no three-dimensional effects occurring in the flow. The two-dimensional representation

of a system is the simplest one, but one need to be careful when using this representation:

in axisymmetrical geometries, a two-dimensional representation neglects some effects due

to symmetry of rotation.

We investigate the accuracy of each of the representations offered by FLUENT.

For a given set of nozzle design parameter we create one two-dimensional model, one

axisymmetric model and on three-dimensional model with the same numerical cell density.

The goal is to see if a two-dimensional model of the thrust chamber would be accurate

enough. Figure 8-3 shows the profile of Alach number on the axis of each of the models. It

shows clearly that the two-dimensional model under-estimates the exhaust Mach number










and is not adapted for the thrust chamber study. Three-dimensional and axisymmetric

models show very similar profiles. Therefore, the axisymmetric solver is chosen for this

study. This preliminary an sh ll--i confirms our choice to develop a single axisymmetric

solver for the thrust chamber analysis.

8.5.3.2 Which governing equation to use?

Flows are considered inviscid when the viscous or friction forces are small in

comparison to the inertial force. The Reynolds number quantifies the ratio of the inertial

forces over the viscous forces. Therefore, when the Reynolds number is large, the flow can

he considered inviscid. In the case of the low-pressure nozzle, the inlet Reynolds number is

equal to 1.7 104 and the Navier-Stokes equations can he simplified to the Euler equations.

FLUENT enables modeling of inviscid flow by selecting 'inviscid' in the turbulence

panel. Besides the governing equation, the principal difference between inviscid and

viscous flow 1I .v in the wall boundary condition treatment. For an inviscid flow, the slip

condition applies: Since there is no viscosity the fluid tangential velocity at the wall is not

equal to zero. On the opposite, in the case of a viscous fluid, the velocity at the wall is

equal to the velocity of the wall itself: the relative velocity between the wall and the fluid

.Il1i Il-ent is equal to zero.

Figure 8-4 shows the 1\ach number profile on the axis of the thrust chamber for both

Euler and N .vi.-; 1-Stokes solvers. The profiles are similar, except at the exit of the nozzle.

This is consistent with the results we obtained in the quasi-one-dimensional analysis: the

exhaust velocity is slightly higher(11'.) for an inviscid flow than for a viscous flow.

Therefore, in a first approach, the Euler solver could be used to model the flow in

the low-pressure thrust chamber. However, as the hydrogen travels trough the chamber,

the viscous effects cannot he neglected anymore. Thus, the Navier-Stokes solvers are more

adapted to model the flow in the low-pressure thrust chamber.










8.5.3.3 Investigation of the turbulence model

The literature review presented in OsI Ilpter 3 showed the adequacy of the Spalart-Allmaras

turbulence in the particular case of the thrust chamber modeling. FLUENT allows us to

compare easily the performances of several RANS turbulence models and assess their

performances. The following models are investigated: Spalart-Allmaras, standard k-e,

realizable k-e with enhanced wall-treatment, k-co and RSM. The profiles of pressure on the

walls of the nozzle (not shown) are very similar for all the models. The profiles of Mach

number on the nozzle axis is given for each turbulence model Figure 8-5. We notice that

the k-e model with enhanced wall treatment gives the worst results. Because RSM is the

most advanced turbulence model within the RANS technique, we can consider the profiles

for this model as being the reference: RSM models all the fluctuatingf products within the

RANS approach by soling a transport equation, and therefore, modeling of the turbulent

effects is much more restricted than in the other models. The Spalart-Allmaras model

shows very good agreement with the RSM predictions. This analysis confirms the good use

of the Spalart-Allmaras model in the case of the nozzle modeling.

8.5.3.4 Sensitivity study

As the number of computational cells increase towards infinity the numerical

solution becomes more and more similar to the real flow. There is a threshold where

the number of cells is large enough that if the number of cells is increased, the solution

remains unchanged. The goal of the sensitivity study is to derive the minimum number

of computational cells required to obtain a solution as close as possible from the real flow

field. To allow for faster computations, the grid is built such as the edges of the meshes

are parallel to the flow direction.

Six meshes are generated with GAMBIT, the grid processor associated to FLUENT.

The following grids are investigated:

Gl: 6 x 51 (306)computational nodes.

G2: 10 x 51 (510) computational nodes.










G3: 19 x 101 (1 ,919)computational nodes.

G4: 28 x 151 (4,228) computational nodes.

G5: 37 x 201 (7,437) computational nodes.

G6: 46 x 251 (11,546) computational nodes.

Results of the analysis are presented Figure 8-6: the relative differences between the

maximum Mach number compiled with a given grid and the finest grid are plotted as a

function of the number of computational nodes for the given grid. As the number of mesh

points increase, the difference becomes smaller and smaller, and tends towards zero. The

variation as the number of nodes increases follows an exponential decay curve. Table 8-2

compiles the CPU time and number of iteration required to reach convergence for grids G1

to G5.

We need to make a compromise between accuracy and CPU time. Fr-om Figure 8-6

and Table8-2 we can interfere that grids G2 is well adapted to get fast results. We might

want to use grid G3 in cases where we want to increase accuracy of the simulation.

8.5.4 Design Parameters Sensitivity Study

All the previous analyzes considered a single nozzle design, characterized by the

design parameters 8-5. Before we start our own CFD study of the nozzle, we want

to investigate which combination of the design parameters lead to the best nozzle

performance. We use the frozen chemistry model above to investigate the best design.

The performance criteria are:

maximum velocity in the nozzle should be localized on the exhaust plane

the best design exhibits the highest exhaust velocity

the profile of velocity on the exhaust plane should be as uniform as possible

Bullets 1 and 3 ensure that the gas is properly expanded in the nozzle and bullet 2 that

the thrust chamber exhibit great ISP values.

We realize an axiomatic design theory, i.e. we investigate the different design

parameters independently. These parameters are Remit, 8,, 8, Rc and L. Through this










analysis, the slope of the nozzle at the exit is considered equal to zero: yi = 0. This is

particularly true in all the designs we reviewed. This is a parameter which value should

be addressed in a more detailed analysis. Table 8-3 lists the different cases investigated

and the design parameters associated. A 'x' in the last column indicates that the case is

geometrically impossible to create, and a 'o' means that the gas is not properly expanded

in the chamber. As a rule of thumb, the gas usually does not expand in the chamber when

the diverging nozzle is either too short or too long. The table also lists the maximum

velocity in each case.

Figure 8-7 shows the influence of an isolated parameter on the nozzle performances

while all other parameters remain constant.

Logically, high throat to exit ratios (A4/,*) lead to higher velocities and 1\ach

numbers on the exhaust plane. An exit radius of 1.5 m corresponds to a ratio of about

100. This is the type of ratio we want to investigate. The thorough investigation exhibits a

set of parameters which lead to highest performances of the nozzle with A4/,* = 100:


Re = 1



0 = 40"

L = 3.3 m

The optimum design corresponds to 'Grid45' and leads to velocities of 14,:358 m/s, which

corresponds to Isp of 1,464 s l. The Isp is also a measure of the available working fluid

over the molecular weight of that fluid. Since hydrogen properties are accounted for with

frozen chemistry, the molecular weight of hydrogen is equal to 1.662 in the exhaust plane.

When accounting for the real properties, there is hydrogen recombination which occurs

in the chamber. Therefore, the actual molecular weight of the exhausting gas might he

slightly greater and the study with FLUENT might overpredict the real value of the Isp.










8.5.4.1 Influence of the thermal formulation

In all the cases investigated so far, the walls were considered adiahatic. In the real

system, the cold hydrogen which exits the tank flows on the walls of the thrust chamber.

A simple run with NuRok shows that the wall temperature can he modeled with a

constant equal to 1,500 K(. The changes in the exit velocity value is negligible (0.;:' .) when

we change the adiahatic into constant temperature walls. Figure 8-8 shows the exhaust

velocity profiles for the two formulations on the wall condition.

8.6 Conclusion on the Frozen Chemistry Analysis and Perspectives

The frozen chemistry analysis is a limited analysis of the thrust chamber because it

does not account for the property variations with temperature and pressure. However,

it gives a good qualitative idea of the behavior of the flow in the low pressure thrust

chamber. The study reveals that exhaust velocities could be of the order or 14,000

m/s, leading roughly to 1,400 s- in specific impulse. With the frozen chemistry, the

recombination and decrease in the heat capacity cannot he modeled. However, lower heat

capacities with equivalent inlet energy and pressure fields should lead to even greater

velocities as the term -- increases with recombination.
y-1
It was seen in ChI Ilpter 6 that the high dissociation hydrogen provides the property

loop to converge. However, as a new technique to account for dissociated hydrogen is

developed, finer analyses of the low pressure thrust chamber should be available and

quantitative results should support the theory that low pressure chamber leads to higher

levels of performances for the NTP systems.








Table 8-2: CPU time (s) and number
for the four different grids
Grid
CPU (s)
Iteration Nb
Time per iteration


of iteration to reach convergence of the steady-state


G2
2:51:04
72
0:02:23


G3
5:24:09
107
0:03:02


G4
7:08:16
124
0:03:27


G5
10:03:35
146
0:04:08


~cf~F~c~/~


E
r ~r)
?D ~~
c, L
-( a~ O
>r
r
C)


~ ~Jt'

L~"

r-~R-c~
IrJ"


Figure 8-1: Thrust chamber and its principal characteristics

























Figure 8-2: Coarser grid


Figure 8-3: Comparison of the Mach number profile for the axisymmetric and 2-D models




172















Table 8-3: Test cases for different design parameters


Case name
Grid1
Grid2
Grid3
Grid4
Grid5
Grid6
Grid7
Grid8
Grid9
Gridl6
Gridl7
Gridl8
Gridl9
Grid20
Grid21
Grid22
Grid23
Grid24
Grid25
Grid26
Grid27
Grid28
Grid29
Grid30
Grid37
Grid38
Grid39
Grid40
Grid41
Grid42
Grid43
Grid44
Grid45
Grid46


Rc
2
2
3
2
3
3
2
3
4
1
1.5
0.5
0.8
1.25
0.99
1.1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1


8, Aexu
20 1
40 1
20 1
40 1
20 1
40 1
30 1.5
30 1.5
30 1.5
30 1.5
30 1.5
30 1.5
30 1.5
30 1.5
30 1.5
30 1.5
30 1.5
30 1.5
30 1.5
30 1.5
30 1.5
30 1.5
30 1.5
30 1.5
30 1.5
30 1.5
30 1.5
30 1.5
30 1.5
30 1.5
30 1.5
25 1.5
35 1.5
45 1.5


0 L
20 0.7
20 1
20 1
20 2
20 2
20 2
20 3
20 3
20 3
20 3
20 3
20 3
20 3
20 3
20 3
20 3
10 3
15 3
25 3
30 3
40 3
50 3
60 3
55 3
40 4
40 2
40 3.5
40 2.5
40 3.2
40 3.3
40 3.4
40 3.3
40 3.3
40 3.3


vmax note

12724o
12713o
12621
12598
12594
13513
13509
13510
13550
13517


13522
13535
13526
13544
13567
13540
13619
14353
146348o
14701o
14710
14333
13732
14351
14172
14355
14367
14342


14358
14340































- Euler
1 Navier-Stokes


I I I I I


4.0 -


3.5-


3.0


0.0L


-0.5 0.0 0.5 1.0 1.5
Axial Position (m)


2.0 2.5 3.0 3.5


Figure 8-4: Mach number profiles for an inviscid and viscous flow of hydrogen



















































* Standard k-epsilon
* k-epsilon Enhanced Walls
* k-omega
RSM
* Spalart-Allmaras


-----
...(..-...I
..e-* ))
..r-
..-r-
...*



Jl


0.5E


0 (


-0.5 0.0 0.5 1.0 1.5
Axial Position (m)


2.0 2.5 3.0 3.5


Figure 8-5: Mach number profiles obtained for different turbulence models






























5-














1-

0





0 1000 2000 3000 4000 5000 6000 7000
Number of nodes



Figure 8-6: Relative difference in the maximum Mach number between a given grid and
the finest grid
























*





-,
- 0 2 0 40 5 0 7
Inletin nge der*s


13600


-4800

14600

14400

14200

14000

13800

13600
4.5

114400


, I I I I


13580 _

13560-

13540 -

13520-





*
'


13500 L
0.5

14500
14400-
14300
14200 -
14100-
14000 -
13900-
13800-

13700 S


1.0 1.5 2.0 2.5 3.0 3.5 4.0
Throat Radius of Curvature













2.0 2.5 3.0 3.5 4.0
Contoured Length (m)


14 3 8 0

14360-

14340-

14320-


11.430nu~


30 35 40
Approach Angle (degrees)


Figure 8-7: Influence of the design parameters on the maximum velocity on the exhaust
plane a) Influence of the throat ratio of curvature (Rc) 8, = 300: 0 = 200 and L = 3
m b) Influence of the inflection angle (0): the black dots corresponds to designs where
the exhaust velocity is not uniform Rc = 1, O = 300 and L = 3 m -c) Influence of the
contoured length (L): Rc = 1, On = 300 and 8 = 400 d) Influence of the approach angle
(8,): Rc = 1, O = 400 and L = 3.3 m

























16000


14000-


12000-


S10000-


~ 800- adiabatic
rd Tw=1500 K

O


4000-


2000



B.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6
Radial Position (m)



Figure 8-8: Velocity magnitude profile on the exhaust plane of the thrust chamber










CHAPTER 9
CONCLUSIONS AND RECOMMENDATIONS

The work presented in this dissertation is a contribution to improved accuracy and

speed of thernial-fluid simulation tools for high temperature gas cooled reactors. The

numerical tools are applied to advanced high temperature nuclear reactors where the

coolant and/or the working fluid is a gas such as helium, hydrogen, or carbon dioxide.

High temperature gas-cooled reactors are considered for the next generation of nuclear

electric power plants as well as for therntochentical production of hydrogen. Another

important application of high temperature gas cooled reactors is the nuclear thermal

propulsion that is the primary system for powering a future manned mission to Mars.

Nuclear thermal rockets due to their high specific impulse and high thrust to weight ration

are also considered for many other robotic and manned exploration of the Solar System.

To increase the accuracy and efficiency of the current generation thernial-fluid

simulation tools, the technique developed for this work combines the advantages of

two coninonly used techniques. A system simulation code known for its speed and

ability to handle large flow systems is coupled to a fine-niesh CFD solver for which the

accuracy and resolution of the calculation are rather high. The fine-niesh code is based

on computational fluid dynamics techniques and uses real gas properties. Inmplenientation

of real gas property formulation in the fine-niesh CFD solver was motivated by the need

to simulate hydrogen flow in a low pressure nuclear thermal propulsion system. At low

pressure and high temperatures the hydrogen propellant partially dissociates. Hydrogen

dissociation results in reduction of molecular weight that increases the specific impulse. To

assess the effect of hydrogen dissociation on specific impulse, a temperature and pressure

description of the dissociated hydrogen is needed in the CFD solver. The model should

account for the variation of thermal and transport properties as a function of dissociation.

Coninercial CFD codes lack capability to model temperature and pressure dependent










hydrogen properties including the molecular weight that pll we- a key role in the calculation

of specific impulse. In this work, the following issues are demonstrated:

1. The feasibility of developing a real property computational fluid dynamics solver
to model fluid dynamics and heat transfer of hydrogen and helium cooled high
temperature gas cooled systems.

2. The feasibility of coupling a coarse-mesh system simulation code to a fine mesh CFD
solver.

The fine-mesh Computational Fluid Dynamics solver developed in this work is

directly adaptable to all types of high temperature gas cooled reactors. It consists in a

two-dimensional axisymmetric solver that is compatible with the geometry of most high

temperature gas cooled reactor systems. The TVD MacCormack solver combined with

RANS turbulence models such as k-e and Spalart-Allmaras are used to develop a robust

CFD solver that incorporates real gas properties. The CFD solver has been applied to the

analysis of the coolant channel of a helium cooled High Temperature Gas-Cooled Reactor

In the case of the core-length channel the temperature applied on the walls linearly

increase from 400 C to 1,000 C. The predicted pressure drop across the 7.93 m channel is

28 kPa Using a simple one-dimensional model the calculated pressure drop in the channel

is 443 kPA, which is a gross overestimation. The flow in the tube is accelerated such that

the exit velocity is twice larger than the inlet velocity. The change in the helium gas

pressure and temperature at the core exit result in 47

Then, the coupling of the NuRok coarse-mesh simulation code to the fine mesh CFD

solver is done through a simple interfacing procedure which consists of a mathematical

interpolation of the extensive properties from 1D code to 2D axisymmetric code and vice

a versa. The coupling is applied to the study of the NERVA derivative nuclear thermal

propulsion system. The hot channel in the NERVA core is modeled with the fine mesh

CFD solver. The use of the fine mesh CFD solver significantly increases the computational

time. However, the total computation time still is less than one minute In the case of a

100,000 lbf thrust NERVA system the pressure drop across the channel is equal to 1.19










1\Pa. The coarse mesh simulation alone predicted a pressure drop three times smaller

than the values derived by the fine-mesh CFD solver. Therefore, the accuracy of the

simulation has been greatly increased by adding the fine mesh simulation of the coolant

channel. The computational time remains considerably small since the technique used at

the interfaces between the fine- and coarse-mesh solvers leads to fast convergence.

The modeling approach presented in this work is the first attempt in coupling coarse-

and fine-mesh CFD solvers using real properties of gases. Now, that the basis has been

set for such numerical tools, many improvements could be incorporated in the coupled

code. Since NuRok is modular, it would be easy to implement fine mesh solvers for other

portions of the system which require a detailed thermal-hydraulics analysis. Examples

of such areas in the system include mixing junctions or the core inlet and exit plenums.

A robust discretization scheme is used to allow for handling of all types of flows such as

subsonic, transonic, or supersonic flow. Therefore, the code could be used for simulation

of steady state operation as well as for accidental analysis. NuRok could be modified to

simulate the entire domain of closed loop High Temperature Gas Cooled systems that

would require closing the flow loop and adding a heat exchanger module. A full property

package for carbon dioxide is developed. Therefore, the same code could be modified to

study supercritical carbon dioxide cooled High Temperature Gas Cooled Reactors.

The development of the real property CFD solver development was motivated hv

the acknowledgment that in the case of the low-pressure thrust chamber of the NTP, the

dissociation process could not he modeled with traditional CFD solvers. The FLUENT

study of this particular thrust chamber with frozen chemistry showed great promise as to

the performances of the overall system operating at low pressure. The analysis enabled

the determination of the converging-diverging nozzle optimum design and estimated the

specific impulse of the system to be in the order of 1,400 s at core exit temperature of

3,000 K(. This would represent a great increase in performance compared to original NTP

designs which traditionally operate at higher pressures with specific impulse of about 925










s at 3,000 K( exit temperature. 1\odeling of the low-pressure thrust chamber with the real

property fine niesh CFD solver remains very challenging because of the sharp variation of

the hydrogen heat capacity due to dissociation. Further work should be done to determine

a more accurate estimation of the Isp associated with the low pressure nuclear thermal

rocket design. In a future work, two approaches could be tested to obtain convergence of

the CFD solver with dissociated hydrogen: first, a set of continuous functions could be

used to express the hydrogen properties as a function of pressure and temperature. During

the first time steps, properties could be coarsely approxiniated with linear functions and as

the solution converges to a steady-state solution, the property formula could be modified

to match the real values. The other technique would be to use the look-up tables which

were developed for NuRok. This approach is computationally very expensive, however,

it may prove to be the only viable option for achieving convergence in cases involving

dissociated hydrogen.










APPENDIX A
PROGRAM FOR COMPILATION OF THE EXACT RIEMANN PROBLEM
SOLUTION


program riemann
implicit none

.solves the exact riemann problem
.original code from bruce fryxell
.can vary length of the tube xl,time at which solution is desired t
.location of discontinuity at initial time
.initial pressure, density and velocity


c..declare
integer
parameter
real


n,npts,itmax,iter,i
(n = 1000)
x(n),rho(n),u(n),p(n),
rhol,pl,ul,rhor,pr,ur,gamma,xi,t,xl,x~r,
rhol,pi,ul,rho5,p5,u5,p40,p41,f0,f,eps,
fi,p4,error,z,c5,gmi,gpi,gmfaci,gmfac2,fct
u4,rho4,w,p3,u3,rho3,c1,c3,xsh,xcd,xft,xhdd


c..define initial conditions
c..state at left of discontinuity
rhol = 1.0
pl = 1.e5
ul = 0.


c..state at right of
rhor = 0.01
pr = 1.e3
ur = 0.


if (ul .ne.
write (6,*
stop
endif


discontinuity


0. .0r. ur .ne. 0.) then
)'must have ul = ur = 0.'


c.. equation of state
gamma = 1.4


c..location of discontinuity at t = 0
xi = 5.

c..time at which solution is desired











t = 0.0039


c ..number of points in solution
npts = 500
if (npts .gt. n) then
write (6,*) 'number of points exceeds array size'
stop
endif


c..spatial interval over which to compute solution
xl = 0.0
xr = 10.0
if (xr .1t. xl) then
write (6,*) 'xr must be greater than xl'
stop
endif


c..begin solution
if (pl gt pr) then
rhol = rhol
pl = pl
ui = ul
rho5 = rhor
p5 = pr
u5 = ur
else
rhol = rhor
pl = pr
ui = ur
rho5 = rhol
p5 = pl
u5 = ul
endif


c..solve for post-shock pressure by secant method
c .initial guesses


p40 = pl
p41 = p5
f 0 = f (p40, pi, p5, rhol, rho5, gamma)

c..maximum number of iterations and maximum allowable relative error
itmax = 20
eps = 1.e-5










do iter = 1, itmax
fl = ftp41, pi, p5, rhol, rho5, gamma)
if (fl .eq. f0) go to 10

p4 = p41 (p41 p40) fi / (fi f0)

error = abs (p4 p41) / p41
if (error .It. eps) go to 10

p40 = p41
p41 = p4
f0 = fi
enddo
write (6,*) 'iteration failed to converge'
stop 'abnormal termination'

10 continue

c..compute post-shock density and velocity
z = (p4 / p5 1.)
c5 = sqrt (gamma p5 / rho5)

gmi = gamma 1.
gpi = gamma + 1.
gmfaci = 0.5 gmi / gamma
gmfac2 = 0.5 gpi / gamma

fact = sqrt (1. + gmfac2 z)

u4 = c5 z / (gamma fact)
rho4 = rho5 (1. + gmfac2 z)
1 / (1. + gmfaci z)

c..shock speed
w = c5 fact

c..compute values at foot of rarefaction
p3 = p4
u3 = u4
rho3 = rhol (p3 / pi)**(1. /gamma)

c..compute positions of waves
if (pl .gt. pr) then
cl = sqrt (gamma pl / rhol)
c3 = sqrt (gamma p3 / rho3)












xsh =
xcd =
xft =
xhd =


+ w *t
+ u3 t
+ (u3 c3) t
-cl t


c..and do say
write
write
write
write
write
write

write
write
write
write


what we found
(6, 500)
(6, 501) rhol, pi, ui
(6, 502)
(6, 503) rho3, p3, u3
(6, 504) rho4, p4, u4
(6, 505) rho5, p5, u5


506)
507)
508)
509)


xhd
xft
xcd
xsh


500 format (// 2x,
1


'Region', 4x, 'Density', 8x, 'Pressure',
8x, 'Velocity')
,3(2x,1pel4.7))
,20x, RAREFACTIONN')
,3(2x,1pel4.7))
,3(2x,1pel4.7))
,3(2x,1pel4.7)//)


501
502
503
504
505

506
507
508
509


(5x,
(5x,
(5x,
(5x,
(5x,

(2x,
(2x,
(2x,
(2x,


format
format
format
format
format

format
format
format
format


x pel4.7)
x pel4.7)
x pel4.7)
x = ', 1pel4.7//)


'Head Of Rarefaction
'Foot Of Rarefaction
'Contact Discontinuity
'Shock


c..compute solution as a function of position
dx = (xr xl) / (npts 1)
do i = 1, npts
x(i) = xl + dx (i 1)
enddo


do i = 1, npts
if (x(i) .It. xhd)
rho(i) = rhol
p(i) = pl
u(i) = ui
else if (x(i) .It.
u(i) = 2. / gpi


xft) then
* (ci + (x(i) xi) / t)










= 1. 0.5 gmi u(i) / ci
= rhol fact ** (2. / gmi)
= pl fact ** (2. gamma / gmi)
(x(i) .1t. xcd) then
=rho3
=p3
=u3
(x(i) .1t. xsh) then
=rho4
= p4
= u4

= rho5
= p5
= u5


fact
rho(i)
p(i)
else if
rho(i)
p(i)
u(i)
else if
rho(i)
p(i)
u(i)
else
rho(i)
p(i)
u(i)
endif
enddo
endif


c..if pr > pl, reverse solution
if (pr .gt. pl) then
cl = sqrt (gamma pl / rhol)
c3 = sqrt (gamma p3 / rho3)


xsh = xi w t
xcd = xi u3 t
xft = xi (u3 c3)
xhd = xi + cl t


c..and do say what we found
write (6, 500)
write (6, 501) rho5,
write (6, 602) rho4,
write (6, 503) rho3,
write (6, 604)
write (6, 505) rhol,


p5, u5
p4, u4
p3, u3

pi, ui


xsh
xcd
xft
xhd


609)
508)
507)
606)


write
write
write
write


602
604
606


format (5x, '2', 3(2x,1pel4.7))
format (5x, '4' ,20x, 'RAREFACTION')
format (2x, 'Head Of Rarefaction x = ', 1pel4.7//)










609 format (2x, 'Shockx= ,1e47

dx = (xr xl) / (npts 1)
do i = 1, npts
x(i) = xl + dx (i 1)
enddo

do i = 1, npts
if (x(i) .It. xsh) then
rho(i) = rho5
p(i) = p5
u(i) = -u5
else if (x(i) .1t. xcd) then
rho(i) = rho4
p(i) = p4
u(i) = -u4
else if (x(i) .It. xft) then
rho(i) = rho3
p(i) = p3
u(i) = -u3
else if (x(i) .1t. xhd) then
u(i) = -2. / gpi (ci + (xi x(i)) / t)
fact = 1. + 0.5 gmi u(i) / ci
rho(i) = rhol fact ** (2. / gmi)
p(i) = pl fact ** (2. gamma / gmi)
else
rho(i) = rhol
p(i) = pl
u(i) = -ul
endif
enddo
endif



open (unit=1, file = 'output',status='unknown')
write (1, 1000)
1000 format (2x, 'i', 10x, 'x', 12x, 'density', 8x, 'pressure',
1 8x, 'velocity'/)
1001 format (i4, 4(2x, 1pel4.7))
do i = 1, npts
write (1,1001) i, x(i), rho(i), p(i), u(i)
enddo
close (1)
stop 'normal termination'
end












real function f(p4, pi, p5, rhol, rho5, gamma)
implicit none

c..shock tube equation

c..declare the pass
real p4,pi,p5,rhol,rho5,gamma

c..local variables
real z,c1,c5,gmi,gpi,g2,fact

z = (p4 / p5 1.)
cl = sqrt (gamma pl / rhol)
c5 = sqrt (gamma p5 / rho5)


gmi = gamma 1.
gpi = gamma + 1.
g2 = 2. gamma

fact = gmi / g2 (c5 / ci) z
1 / sqrt (1. + gpi / g2 z)
fact = (1. fact) ** (g2 / gmi)

f = pl fact p4

return
end










APPENDIX B
FLUENT ANALYZES

All preliminary analyzes are performed with FLUENT 6.3.26. The cases we describe

here are compatible with any version of FLUENT 6. Definitions of terms particular to the

FLUENT solver are given at the end of the appendix.

B.1 Low-Pressure Thrust Chamber: Design Parameter Analysis

The axisymmetric study is done with a simple 2-D pressure-based (segregated) solver

and Spalart-Allmaras model accounts for turbulence. Frozen properties of hydrogen are

considered as well as the ideal-gas formulation for density. The operating pressure is set at

0.1 bar (104 Pa). The inlet mass flow rate is equal to 0.228 kg/s with the velocity normal

to the boundary and the inlet temperature equal to :3,000 K(. The turbulent method is

initially specified with the turbulent intensity (1(' .) and the hydraulic diameter (0.6 m).

The exit pressure is set to -9990 Pa, which corresponds to an absolute pressure of 10 Pa.

The backflow total temperature is equal to :3,000K(. The discretization scheme is set to a

third order MITSCL method for all variables except for pressure for which it is set to a

second-order upwind. To improve convergence performances, the pressure is limited to 0.4

hars and the temperature is bounded between 1,500 and :3,500 K(. The convergence criteria

is set to 10".

pressure-based solver and the Spalart-Allmaras model to account for turbulence.

Hydrogen properties are given by Equation 8-7and the ideal-gas law describes the density

behavior. The inlet mass flow rate is set to 0.228 kg/s and the inlet temperature and

pressure to :3,000 K( and 0.1 bars respectively. This corresponds to an inlet Mach number

of 0.:3. The adiahatic behavior is studied, therefore the thermal flux applied to the walls

is equal to zero. By default, the walls are considered to be in Aluminum. Finally, the exit

pressure is set at 10 Pa. We use a second order solver on pressure and a :$rd order MITSCL

on all the other conservative variables.









B.2 Flow Area Contraction

In order to estimate the profile of velocity at the inlet of a typical nuclear reactor

coolant channel two steady-state FLUENT cases are run with an area contraction:

circular tank contracts into a single NERVA coolant channel

exact same tank contracts into four identical NERVA coolant channel

The two mesh grids are plotted Figure B-1. The profiles obtained at the inlet of

the coolant pipes are plotted Figures B-2 and B-:3 and compared to the theoretical

fully-developed turbulent flow profiles for both geometries. Generally, the fully-developed

flow profile is a good estimation of the overall hot channel inlet profile but this approximation

only is qualitative. On a quantitative point of view, the flow profile does not match the

computed results.

B.3 Comparison of the Hydrogen Channel Simulation with a FLUENT
Frozen Chemistry Analysis

Modeling of the coolant channel of the NERVA-derivative system with Themis gave

result significantly different from the results of the NuRok analysis. To get a qualitative

validation of the results obtained with our new fine mesh CFD code, we quickly simulated

the channel with FLUENT using a frozen chemistry formulation and an ideal gas law

definition of the hydrogen density. In FLUENT, we cannot set the inlet pressure of the

system but its outlet pressure. Therefore, the characteristics of the model are given by

the system 6-13 with an exit pressure of 6.9:3 MPa. Since the inlet pressure is estimated

to be equal to 8.12 MPa, hydrogen properties which are considered constant in the frozen

chemisrty approach are equal to:


c, =15, 020 J/kg.K(

k = 0.197 W/m.K (B-1)

p- = 9.078 + 10-6 kg/m.s










The profile of pressure obtained with the frozen chemistry in plotted figure B-4 and

compared to the prediction with the real property solver. The difference between the two

simulations is only of 6.'7' compared with 2 :' between Themis and NuRok.

Thanks to the frozen chemistry study, we can be very confident that the predictions

with Themis are accurate and that the NuRok predictions were lacking accuracy in

handling the pressure drop across the coolant channel.

B.4 Definitions

Turbulent intensity: The turbulent intensity is defined by the ratio of the fluctuating

velocity over the mean velocity.



In high turbulence flow, the turbulent intensity usually varies between 5 and 211' For

fully-developed pipe flows the turbulent intensity can be estimated by:


I" ~ e~ (B-2)


where Reoh is the Reynolds number based on the hydraulic diameter Dh.

Backflow total temperature[13]:The backflow temperature should be input when

we use the pressure-outlet boundary condition in FLUENT. The specified values will be

used only if flow is pulled in through the outlet. The value might be used in the first few

iterations for which FLUENT often computes velocities coming from the exit. Hence,

the absolute value of the backflow total temperature itself is not of great importance but

should be set with a realistic value to help computation of the first iterations.



























,'


cp~
,s'
1~7~11
;i~
'
r-
II


.1.1-

:+ilcr


+C


.r
lV~'


I I?~.
r
E" `i~e


r'
r
... .~i~
F'
r


I'


II
I


L*
r' l


1,


Figure B-1: 1\eshes for the two contraction experiments










193


Muttiple Ch=~annel


Single Channl ~

























400


350-


300-


250-


200-


150-


100-


50



-00E015


-0.0010 -0.0005 0.0000 0.0005 0.0010 0.0015


Figure B-2: Velocity profile on the contraction plane in the single channel configuration.
Comparison between computed and theoretical profiles

























80


70-


60-


50-


40-


30-


20~ channel 1
channel 2
channel 3
10
--- channel 4
fully-developped

0.8020 0.0025 0.0030 0.0035 0.0040 0.0045 0.0050 0.0055 0.0360




Figure B-3: Velocity profile on the contraction plane in the four-channel contraction
experiment. Comparison between computed and theoretical profiles





































S7.6-



v,7.4-



7.2-



7.0-



6.0.0 0.2 0.4 0.6 0.8 1.0
Position (m)



Figure B-4: Pressure profile (jl!pa) on the axis of the coolant channel.
between the real property solver and FLUENT with frozen chemistry


1.2 1.4




Comparison










APPENDIX C:
THEMIS SUBROUTINE LISTING: NERVA DERIVATIVE

C.1 Core subroutine from the system code

SUBROUTINE NCORE(ITERCHC,TIN,PIN,W,QT,THCT,N,TOUT,POT

USE CONSTANTS
IMPLICIT NONE
REAL, DIMENSION(100) :: Z,P,T,H,Mn,VEL,APSF,TFS,TFC,TFC1
INTEGER :: I,K,N,ITERCHC
REAL :: TIN,PIN,W,QT,THCT,TOUT,POUT,TARL,BARL,ADICDFN
REAL :: NH,NHCC,KCOND,RHO,LHGR,FLUX,Q,APF,ZC,ZMNVLCUP
REAL :: P1,DELP,Cc,TC,THCA,ACIC,CHTA,CHTSAE,FUELVO
REAL :: FUELA,FLOWA,FLOWAPE,APE,ATOT,ECCW,EW,TFCMXTSAVEDP
REAL :: TD,HD,DELH,ZD,TCL,TWALL,VHGR,DPCORE,CPD,FPPEMCPH
REAL :: VELCH,RHOCH,LRPD,TCH,SLOPE,QTa,QTb,TCa,T bTFmaDEZV
REAL :: CHP,TERR,FPD
real :: bla

!VARIABLES THAT DESCRIBE THE AXIAL REFLECTOR ARE SET BELOW


TARL = 0.0
BARL = 0.0254


! AXIAL REFL. TOP LENGTH
! AXIAL REFL. BOTTOM LENGTH


!VARIABLES THAT SPECIFY THE CORE GEOMETRY ARE SET BELOW


RPF
TTPF


!REFLECTOR POWER FRACTION
!TIE-TUBE POWER FRACTION
!LENGTH OF FLAT SIDE
!COOLANT HOLE DIAMETER

!TOTAL NUMBER OF FUEL ELEMENTS

!NUMBER OF HOLES PER FUEL ELEMENT
!NUMBER OF HOLES BETWEEN ELEMENT CORNERS
!CORE OUTER DIAMETER
!COMPOSITE FUEL CONDUCTIVITY (W/M.K)
!TO GET N DATA POINTS
!ELEMENT WIDTH
IM**2) ELEMENT CORNER-TO-CORNER WIDTH
!COOLANT HOLE PITCH
!X-SECT ELEM AREA
!TOTAL ELEM AREA
D**2 COOLANT F.A. PER ELEM


= 0.02
= 0.04


ADIM = .01905
CD = .00254

FANT = 1265

NH = 19
NHCC = 5
DOUT = 0.7595
KCOND = 15.0
N =N + 1
EW = 1.7321*ADIM
ECCW = SQRT(EW**2 + AD
CHP = ECCW/NHCC
APE = 2.5981*ADIM**2
Atot = FANT*APE
FLOWAPE = 0.25*NH*PI*C










VF = FLOWAPE/APE
FLOWA = FANT*FLOWAPE
FUELA = Atot FLOWA
FUELVOL = FUELA*ALF
CHTSAE = NH*PI*CD*ALF
CHTA = FANT*CHTSAE
ACIC = 0.25*PI*DOUT**2
THCA = ACIC


CORE VOID FRACTION
TOTAL COOLANT X-SEC FLOW AREA
TOTAL X-SECT FUEL AREA
TOTAL FUEL VOLUME
COOLANT HEAT TRANSFER S.A. PER ELEM
TOTAL COOLANT HEAT TRANSFER AREA
FLOW AREA OF CORE INLET CMBR (APPRX)
FLOW AREA OF THRUST CMBR (APPRX)


!CALCULATE THE CORE POWER DISTRIBUTION

CALL AXIAL_PWR_DIST(N,APSF)


!START CORE CALCULATIONS

IF (OUTPUT.NE.0) THEN
PRINT*,'
PRINT*,'BEGINNING FINAL CORE DISTRIBUTION CALCULATIONS.
ENDIF


DELZ = ALF/REAL(N-1)


! ITERATION STEPSIZE


!CALCULATE FUEL ASSEMBLY INLET CONDITIONS

T(1) = TIN
TC = TIN
CALL DENSITY(1,TIN,PIN,RHO)
VEL(1) = W/(RHO*FLOWA)
Cc = 1./(1. + ABS((ACIC+FLOWA)/(ACIC-FLOWA))) CONTRACTION-LOSS COEFF
DELP = 0.5*RHO*Cc*VEL(1)**2 PRESSURE LOSS AT SUDDEN CONTRACTION
P1 = PIN DELP


CALL DP(TIN,TIN,P1,CD,W,FLOWA,TARL,URPD)
P(1) = P1 URPD
PC = P(1)
VELC = VEL(1)
CALL FINDMN(TIN,PC,VELC,ZMN)
Mn(1) = ZMN
Z(1) = 0.0
ZC = Z(1)
CALL ENTHAL(1,T(1),P(1),H(1))
APF = APSF(1)


! PRESSURE LOSS IN TOP REFL












!BEGIN ITERATIVE LOOP TO SOLVE FOR DESIRED THCT


DO 999 K=1,100 100 = MAX ITERATIONS
Q = QT QT*(RPF+TTPF) CORE POWER TRANSFERRED TO PROPELLANT

FLUX = Q/CHTA AVE CORE HEAT FLUX
LHGR = Q/ALF AVE CORE LINEAR HEAT GEN RATE
VHGR = QT/FUELVOL AVE CORE VOLUMETRIC HEAT GEN RATE

call density(1,t(1),pc,bla)

call channel_solver(flux,pc,t(1),velc*bla*PI*(13e3*2pN,()


IF (ITERCHC .EQ. 1) THEN
GOTO 1000 ONE PASS WITH GIVEN CORE POWER
ELSE
TERR = ABS((T(N)-THCT)/THCT)
IF (TERR .LT. 0.0002) GOTO 1000
ENDIF

TCb = TCa SET THESE UP FOR SECANT METHOD
TCa = T(N)
QTb = QTa
QTa = QT

IF (I.EQ.1) THEN
QT = QT (THCT-TIN)/(T(N)-TIN) NEW TOTAL CORE POWER GUESS
GOTO 999
ENDIF NOW USE SECANT METHOD
SLOPE = (TCb-TCa)/(QTb-QTa)
QT = QTa + (THCT-TCa)/SLOPE
999 CONTINUE
1000 CONTINUE CORE CALCS ARE COMPLETE
IF (K .GE. 100) PRINT*,' ERROR: CORE CALC LOOP DID NOT CONVERGE'

IF (OUTPUT.NE.0) WRITE(35,47) VF,CD,CHP,TFCmax,TFC~max
47 FORMAT(1X,F8.5,1X,F8.5,1X,F8.5,1:K,F8.2,1K,82
IF (OUTPUT.NE.0) WRITE(36,*) 'FLUX = ',FLUX,' VF = ',VF

!CALC THRUST CHAMBER INLET CONDITIONS

TCH = T(N)










PC = P(N)
CALL DP(TCH,TCH,PC,CD,W,FLOWA,BARL,LRPD) LOWER REFL PRES DROP
PC = PC LRPD
CALL DENSITY(1,TCH,PC,RHOCH)
VELCH = W/(RHOCH*THCA)
DELP = 0.5*RHOCH*( VEL(N)**2 VELCH**2 ) SUDDEN EXPANSION
PCH = PC DELP


CALL FINDMN(TCH,PCH,VELCH,ZMN)
MnCH = ZMN
TOUT = TCH
POUT = PCH

!CALCULATE POWER DENSITIES

QPE = QT/FANT
FPD = QT/FUELVOL
CPD = QT/(APE*ANT*ALF)
DPcore = (PIN-POUT)/6894.76


POWER PER ELEMENT
FUEL POWER DENSITY
CORE POWER DENSITY


!THE CORE CYCLE IS NOW COMPLETE. NOW OUTPUT THE DATA.

PRINT*, CORE ITERATIONS = ',K
IF (OUTPUT.NE.0) PRINT*,'CORE ITERATIONS = ',K,&
& '---> CORE CALCS ARE COMPLETE'
IF (OUTPUT.NE.0) THEN
WRITE (37,*) '
WRITE (37,*) '
WRITE(37,*) '******************************************
WRITE(37,*) '*************** CORE MODULE ***************'
WRITE(37,*) '******************************************
WRITE (37,*) '
WRITE(37,*) 'CORE INLET CONDITIONS:'
W RITE (37,*) - - - - - --
WRITE (37,*) '
WRITE(37,601) TIN
WRITE(37,602) PIN, PIN/6894.76 Pa --> psi
WRITE(37,603) W
601 FORMAT(6X,'TEMPERATURE = ',F8.2,' (K)')
602 FORMAT(6X,'PRESSURE = ',E10.5,' (Pa) = ',F9.2,' (
603 FORMAT(6X,'FLOW RATE = ',F7.2,' (kg/S)')
WRITE (37,*) '
WRITE(37,*) 'CORE AXIAL FLOW PROFILE:'
WRITE (37,*) '
WRITE(37,*) Z V Mn P T


psi)')


Tc1 APSF'


Tfs










WRITE(37,*) (m) (m/S) (Pa) (K) (K) (K)'
W R ITE (3 7,* ) - - - - - - - - -
DO 10 I=1,N
WRITE(37,604) Z(I),VEL(I),Mn(I),P(I),T(I),TFS(I),TFC(I),PFI
604 FORMAT(F8.5,' ',F8.3,' ',F8.4,' ',E12.4,' ',3F10.2,1X,F7.3)
10 CONTINUE
WRITE (37,*) '
WRITE(37,*) 'CORE OUTLET (THRUST CHAMBER INLET) CONDITIONS:'
WRITE (37,*) '
WRITE(37,*) Temp Pres Vel Mn'
WRITE(37,*) (K) (Pa) (M/S)'
W RITE (37,*) - - - - -
WRITE(37,605) TCH,PCH,VELCH,MnCH
605 FORMAT(1X,F9.2,1X,E10.5,1X,F9.2,1X,F8.4)
WRITE (37,*) '
WRITE(37,*) 'FINAL CORE CHARACTERISTICS:'
W RITE (37,*) - - - - - - --
WRITE(37,606) QT/1.0E6 CONVERT W --> MW
WRITE(37,610) Q/1.0E6
WRITE(37,613) QPE/1.0E6
WRITE(37,615) FPD/1.0E6
WRITE(37,614) CPD/1.0E6
WRITE(37,616) TFSmax
WRITE(37,617) TFCmax
WRITE(37,618) DPcore
WRITE (37,*) '
WRITE(37,607) TCH
WRITE(37,608) PCH, PCH/6894.76 CONVERT Pa --> PSI
WRITE(37,609) W
606 FORMAT(6X,'CORE THERMAL POWER = ',F9.2,' (MW)')
610 FORMAT(6X,'CORE TRANSFERRED PWR = ',F9.2,' (MW)')
613 FORMAT(6X,'ELEMENT PWR DENSITY = ',F9.2,' (MW/elem)')
615 FORMAT(6X,'FUEL POWER DENSITY = ',F9.2,' (MW/m^3)')
614 FORMAT(6X,'CORE POWER DENSITY = ',F9.2,' (MW/m^3)')
616 FORMAT(6X,'MAX FUEL SURF TEMP = ',F10.2,' (K)')
617 FORMAT(6X,'MAX FUEL TEMP = ',F10.2,' (K)')
618 FORMAT(6X,'CORE PRESSURE DROP = ',F10.2,' (psi)')
607 FORMAT(6X,'CHAMBER TEMPERATURE = ',F9.2,' (K)')
608 FORMAT(6X,'CHAMBER PRESSURE = ',E10.5,' (Pa) =' F9.2,' (psi)')
609 FORMAT(6X,'MASS FLOW RATE = ',F9.2,' (kg/S)')
ENDIF

!INCLUDE THE LINE BELOW FOR A SYSTEM PARAMETRIC STUDY

IF (0UTPUT.EQ.2) THEN










WRITE(39,701) Mn(N),TFSmax,TFCmax,DPcore
701 FORMAT(1X,F8.5,1X,F8.2,1X,F8.2,1X,F8.2)



ENDIF



END MAIN PROGRAM

C.2 Interface subroutine
subroutine interfacer(p,t,pe,te)
use channel
implicit none

integer :: i
real :: A
real : : weight

A=3. 14159*rpos (Nnode+1)**2



pe=p (1)/A*rpos (1)**2
te= t (1)/A*rpos (1)**2

do i=2,Nnode
weight=1. /At (rpos (i)**2-rpos (i-1)**2)
pe=pe+p(i)*weight
te=te+weight
end do
return

end subroutine










APPENDIX D
THE PROPERTY LOOP: INFLUENCE OF THE HYDROGEN DISSOCIATION

The property loop is at the heart of the methodology which is developed for this

research. It allows in a few iterations to update the real value of all the gases properties

at each point of the niesh. A schematics of the pressure loop is given Figure D-1. To

illustrate the challenges of obtaining a converged solution of the property loop when there

is a high level of dissociation, the property loop is applied to a single point of the niesh, at

the center of the channel. The three hydrogen cases presented in ChI Ilpter 6 are presented:

no-dissociation

small dissociation on the order of "'

high levels of dissociations

Figures D-2 to D-4 show the evolution of the pressure, temperature and specific heat ratio

for the node selected.

D.1 No Dissociation

The initial characteristics are:


p = 78 hars
(D-1)
T = 1, 840 K(

After the predictor step, the vector of conservative variables have the following numerical

values:
1.03

4634.5
U/ = (D-2)
1.13 +10-2

3.65 10'

Convergence is obtained within only 5 iterations on the loop. Figure D-2 shows the

evolution of the properties for 40 iterations: it proves that we obtained a stable solution.










D.2 Small Dissociation

The temperature of the previous case is increased such that the hydrogen is

dissociated. The initial characteristics are then:


p = 78 hars
(D-3)
T = 2, 400 K(

which corresponds to a level of dissociation of "' Figure D-3 shows that convergence of

the property loop is obtained but it requires more iterations. Properties are converged

within 10 iterations to levels of residuals of 10-4. The solution obtained is stable.

D.3 High Levels of Dissociation

The pressure is now decreased and temperature increased to obtain higher levels of

dissociation of ~' .

p = 0.1 bars
(D-4)
T = 2, 800 K(

Figure D-4 shows the evolution of pressure, temperature and specific heat ratio with the

iteration number. The solution does not converges but oscillates between two extrenmun

values with 100I' difference for pressure. As it is the case for all points of the 100x1,000

grid, the solution cannot reach a convergence when levels of dissociation are high. The

property loop technique introduces numerical discontinuities in the pipe where pressure

should be continuous given the low 1\ach number and the geometry. Therefore, the

property loop cannot he used when hydrogen dissociation is high and other techniques

have to be investigated to reach convergence in such gas configuration.

D.4 Program

module channel

integer :: nature
real, parameter :: epsilon=1.e-3
real, dimension(14,10) :: disso
REAL, dimension(10) :: press
real, dimension(14) :: tempe












end module


program looptemp

!loop on property to derive pressure and temperature from the vector of conservative va
use channel
implicit none
real :: oldgamma, oldpre,01dtemp
real :: rho,rhour,rhouz,nrj,newp,p,rgas,t,gamma
integer :: i



open(unit=10,file='property_100p.dat',stats'nnw'

nature=2


call dis initialization of the arrays


oldpre=7.8e6
oldtemp=1840.

call heat_ratio(oldpre,01dtemp,oldgamma)



!vector of conservative variables

rho=1.03
rhour=1.1e-2*1.03
rhouz=451*1.03
nrj=3.652e7

do i=1,40

!calculation of p with the old value of gamma


newp=(oldgamma-1)*(nrj-0.5/rho*(rhour**2+roz*)

!update on the gas constant with the new pressure

call Rideal(newp,01dtemp,rgas)










!calculation of the corresponding temperature

t=newp/(rhoargas)

if(abs(t-oldtemp)/oldtemp.gt.0.001) then limiter
print*, '*'
t=(1+(t-oldtemp)/abs(t-oldtemp)*0.001)*oltm
endif

!calculation of the new gamma with the new values of p and t

call heat_ratio(newp,t,gamma)

!calculation of the corresponding p


p=(gamma-1)*(nrj-0.5/rho**2*(rhour**2+rhoz2)

if(abs(newp-p)/p.1t~epsilon) then check on convergence
goto 100 if convergence exit loop

else
oldgamma=gamma
oldtemp=t
endif
write(10,*) i,newp,p,t,gamma



enddo
100 continue



end program






subroutine heat_cap(temperature,pressure,heatcap)

subroutinee to calculate the heat capacity of helium or hydrogen knowing p and t

use channel
implicit none
real :: pressure, temperature,heatcap,tp,ps
real :: al,a2,a3,a4,a5












tp=temperature
ps=pressure

if (nature==1) then helium
heatcap=5.191e3
elseif (nature==2) then
ps=pressure/1. e5
al=0.560*ps**(-0.4)
a2=-2.630*ps**(-0.35)
a3=4.349*ps**(-0.3)
a4=-2.730*ps**(-0.25)
a5=20.57*ps**(-0.05)
heatcap=aidle-11*tp**4+a2*1e-8*tp**3+a3*1-*p*+41-*pa
heat cap=heat cap*1e3
endif

return
end subroutine






subroutine density(temperature,pressure,dens)

!density calculation

use channel
implicit none
real :: temperature, pressure, dens,R

if (nature==1) then
R=2077.
elseif (nature==2) then
call Rideal (pressure, temperature, R)
endif


dens=pressure/ (R*temperature)

return
end subroutine




subroutine viscous (temperature, visc)












! viscosity calculation


use channel
implicit none
real :: temperature, visc

if(nature==1) then
visc=3.30e-8*temperature+1.125e-5
visc=2.e-5
elseif(nature==2) then
visc=3.43e-8*temperature+7.01e-6
endif
return
end subroutine



subroutine therm_cond(temperature,pressure,k)


!thermal conductivity calculation

use channel
implicit none
real :: pressure, temperature,k
real :: ps,tp,al,a2


ps=pressure*1e-5
tp=temperature

if(nature==1) then
al=4.92e-12*tp**2+1.20e-7*tp + 8.41e-5
a2=2.75e-4*tp + 7.43e-2
k=al*ps+a2
k=.152
elseif(nature==2) then
al=-6.1e-8*ps+7.48e-5
a2= 7.64e-5*ps-4.52e-3
k=al*tp+a2
endif
return
end subroutine



subroutine heat_ratio(pressure, temperature,g)











!specific heat ratio (gamma) calculation for hydrogen
use channel
implicit none
real :: pressure,temperature,g,cp,R

call heat_cap(temperature,pressure,cp)

call Rideal(pressure, temperature,R)


g=cp/(cp-R)
return
end subroutine



subroutine Rideal(pressure,temperature,Rgas)


!gas constant calculation for hydrogen

use channel
implicit none
real :: pressure,temperature, Rgas,mw





call mol_weight(nature,temperature,pressure,m)


Rgas=8314./mw

return
end subroutine






SUBROUTINE MOL_WEIGHT(NAT,TEMP,PRES,MW)

!molecular weight calculation for hydrogen

use channel
implicit none
INTEGER :: I,J,K,K1,K2,IT,IP,NAT

REAL :: TEMP,PRES,MW
real :: wt,wp












real :: M_LT_LP,M_LT_HP,M_HT_LP,M_HT_HP


IF(TEMP.LE.1500.OR.TEMP.GE.3000) THEN
!print*, 'temperature out of range'
GOTO 100

elselF(PRES.LE.0.1E5.OR.PRES.GE.100E5) THEN
MW=2.016
GOTO 100

else



!DETERMINE THE ARRAY OF PRESSURE AND TEMPERATURE
IF(TEMP.LE.TEMPE(2)) THEN
IT=2
ELSEIF(TEMP. GT.TEMPE(2).AND.TEMP.LE.TEMPE() THEN
IT=3
ELSEIF(TEMP. GT.TEMPE(3).AND.TEMP.LE.TEMPE() THEN
IT=4
ELSEIF(TEMP. GT.TEMPE(4).AND.TEMP.LE.TEMPE() THEN
IT=5
ELSEIF(TEMP. GT.TEMPE(5).AND.TEMP.LE.TEMPE() THEN
IT=6
ELSEIF(TEMP. GT.TEMPE(6).AND.TEMP.LE.TEMPE() THEN
IT=7
ELSEIF(TEMP. GT.TEMPE(7).AND.TEMP.LE.TEMPE() THEN
IT=8
ELSEIF(TEMP. GT.TEMPE(8).AND.TEMP.LE.TEMPE() THEN
IT=9
ELSEIF(TEMP. GT.TEMPE(9).AND.TEMP.LE.TEMPE1) THEN
IT=10
ELSEIF(TEMP. GT.TEMPE(10).AND.TEMP.LE.TEMPE1) THEN
IT=11
ELSEIF(TEMP. GT.TEMPE(11).AND.TEMP.LE.TEMPE1) THEN
IT=12
ELSEIF(TEMP. GT.TEMPE(12).AND.TEMP.LE.TEMPE1) THEN
IT=13
ELSEIF(TEMP.GT.TEMPE(13)) THEN
IT=14
ENDIF










IF(PRES.LE.PRESS(2)) THEN
IP=2
ELSEIF(P RES.GT.P RESS(2).AND.PRES.LE.PRESS() THEN
IP=3
ELSEIF(P RES.GT.P RESS(3).AND.PRES.LE.PRESS() THEN
IP=4
ELSEIF(P RES.GT.P RESS(4).AND.PRES.LE.PRESS() THEN
IP=5
ELSEIF(P RES.GT.P RESS(5).AND.PRES.LE.PRESS() THEN
IP=6
ELSEIF(PRES.GT.PRESS(6)AN .PRES PRSLE.PRES(7) THEN
IP=7
ELSEIF(P RES.GT.P RESS(7).AND.PRES.LE.PRESS() THEN
IP=8
ELSEIF(P RES.GT.P RESS(8).AND.PRES.LE.PRESS() THEN
IP=9
ELSEIF(PRES.GT.PRESS(9)) THEN
IP=10
ENDIF



!WEIGHT OF TEMPERATURE
WT=(TEMP-TEMPE(IT-1))/(TEMPE(IT)-TEMPE(I T-)
!WEIGHT OF PRESSURE
WP=(PRES-PRESS(IP- 1))/(PRESS(IP)-PRESS(I P-)
M_LT_LP=DISSO(It-1,Ip-1)
M_LT_HP=DISSO(It-1 I)
M_HT_LP=DISSO(It,Ip-1)
M_HT_HP=DISSO(It I)



MW=(M_LT_LP*WT+M_HT_LP*(1-WT))*WP
MW=MW+(M_LT_HP*WT+M_HT_HP*(1-WT))*(1-WP)

endif
100 CONTINUE
RETURN
END



subroutine dis

!arrays for molecular weight calculations

use channel











disso(1,1) = 2.016
disso(2,1) = 2.016
disso(3,1) = 2.016
disso(4,1) = 2.016
disso(5,1) = 2.015
disso(6,1) = 2.013
disso(7,1) = 2.011
disso(8,1) = 1.998
disso(9,1) = 1.966
disso(10,1) = 1.901
disso(11,1) = 1.786
disso(12,1) =1.622
disso(13,1) = 1.433
disso(14,1) = 1.264
disso(1,2) = 2.016
disso(2,2) = 2.016
disso(3,2) = 2.016
disso(4,2) = 2.016
disso(5,2) = 2.016
disso(6,2) = 2.015
disso(7,2) = 2.014
disso(8,2) = 2.010
disso(9,2) = 2.0
disso(10,2) =1.978
disso(11,2) =1.937
disso(12,2) =1.869
disso(13,2) =1.769
disso(14,2) =1.642
disso(1,3) =2.016
disso(2,3) =2.016
disso(3,3) =2.016
disso(4,3) =2.016
disso(5,3) =2.016
disso(6,3) =2.016
disso(7,3) =2.015
disso(8,3) =2.014
disso(9,3) = 2.011
disso(10,3) = 2.004
disso(11,3) = 1.990
disso(12,3) = 1.966
disso(13,3) = 1.930
disso(14,3) = 1.877
disso(1,4) = 2.016
disso(2,4) = 2.016










disso(3,4) = 2.016
disso(4,4) = 2.016
disso(5,4) = 2.016
disso(6,4) = 2.016
disso(7,4) = 2.016
disso(8,4) = 2.015
disso(9,4) = 2.012
disso(10,4) = 2.007
disso(11,4) = 1.998
disso(12,4) = 1.981
disso(13,4) = 1.954
disso(14,4) = 1.916
disso(1,5) = 2.016
disso(2,5) = 2.016
disso(3,5) = 2.016
disso(4,5) = 2.016
disso(5,5) = 2.016
disso(6,5) = 2.016
disso(7,5) = 2.016
disso(8,5) = 2.015
disso(9,5) = 2.013
disso(10,5) = 2.009
disso(11,5) = 2.001
disso(12,5) = 1.987
disso(13,5) = 1.965
disso(14,5) = 1.933
disso(1,6) = 2.016
disso(2,6) = 2.016
disso(3,6) = 2.016
disso(4,6) = 2.016
disso(5,6) = 2.016
disso(6,6) =2.016
disso(7,6)= 2.016
disso(8,6)= 2.015
disso(9,6)= 2.013
disso(10,6) = 2.010
disso(11,6) = 2.003
disso(12,6) = 1.991
disso(13,6) = 1.972
disso(14,6) = 1.944
disso(1,7) = 2.016
disso(2,7) = 2.016
disso(3,7) = 2.016
disso(4,7) = 2.016
disso(5,7) = 2.016










disso(6,7) = 2.016
disso(7,7) = 2.016
disso(8,7) = 2.016
disso(9,7) = 2.014
disso(10,7) = 2.011
disso(11,7) = 2.004
disso(12,7) = 1.994
disso(13,7) = 1.977
disso(14,7) = 1.951
disso(1,8) = 2.016
disso(2,8) = 2.016
disso(3,8) = 2.016
disso(4,8) = 2.016
disso(5,8) = 2.016
disso(6,8) = 2.016
disso(7,8) = 2.016
disso(8,8) = 2.015
disso(9,8) = 2.014
disso(10,8) = 2.011
disso(11,8) = 2.005
disso(12,8) = 1.996
disso(13,8) =1.980
disso(14,8)= 1.957
disso(1,9) = 2.016
disso(2,9) = 2.016
disso(3,9) = 2.016
disso(4,9) = 2.016
disso(5,9) = 2.016
disso(6,9) = 2.016
disso(7,9) = 2.016
disso(8,9) = 2.015
disso(9,9) = 2.014
disso(10,9) = 2.012
disso(11,9) = 2.007
disso(12,9) = 1.998
disso(13,9) = 1.985
disso(14,9) = 1.964
disso(1,10) = 2.016
disso(2,10) = 2.016
disso(3,10) = 2.016
disso(4,10) = 2.016
disso(5,10) = 2.016
disso(6,10) = 2.016
disso(7,10) = 2.016
disso(8,10) = 2.015










disso(9,10) = 2.014
disso(10,10) = 2.012
disso(11,10)= 2.008
disso(12,10)= 2.000
disso(13,10) = 1.988
disso(14,10) = 1.970

press(1)=10000.
press(2)=100000.
press(3)=10.E5
press(4)=20.E5
press(5)=30.E5
press(6)=40.E5
press(7)=50.E5
press(8)=60.E5
press(9)=80.E5
press(10)=100.E5

t emp e(1) =300 .
tempe(2)=1500.
tempe(3)=1600.
tempe(4)=1700.
tempe(5)=1800.
tempe(6)=1900.
tempe(7)=2000.
tempe(8)=2200.
tempe(9)=2400.
t empe (10) =2600 .
tempe (11)=2800 .
tempe (12)=3000 .
tempe(13)=3200.
t empe (14) =3400 .


end subroutine













Predictor Step
Derive Unw2


Calculate prediction for p and T

Update y(p,T), cp(pT), k(pT),

ave the properties
converged?

Figure D-1: The property loop














































0.0007 ~'~
I~0.0006
S0.0005
a~0.0004
u0.0003
E 0.0002
am 0.0001
0 5 10 15 20 25 30 35 40
Iteration Number



Figure D-2: Evolution of the 1!! I i r properties with the number of iteration in the
hydrogen pipe with no dissociation
























7220000
S7200000
a- 7180000
a, 7160000
8 7140000
8 7120000
2 7100000
a 7080000
7060000
0 5 10 15 20 25 30 35 4r
2400
S2380
8 2360
e 2340
0. 2 3 2 0
E "


15 20 25
Iteration Number


Figure D-3: Evolution of the 1!! r ~ properties with the number of iteration in the
hydrogen pipe with low levels of dissociation





IU
a
a,
~ 104
vl
vl
a,
L
a


1.04

1.03

1.02

1.01

1.00
0


10 20 30
Iteration Number


40 50





number of iteration in the


Figure D-4: Evolution of the 1!! r ~ properties with the

hydrogen pipe with higher levels of dissociation









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BIOGRAPHICAL SKETCH

Anne Charmeau was born May 13, 1980, in Versailles, France. She graduated from

Edouart Herriot High School in Voiron, France. After two years of pre-engineering

classes, she enrolled at the National School for Physics in Grenoble and obtained her

engineer diploma in September 2003. As part of an academic exchange she joined the

University of Florida in August 2002. She graduated with a Masters of Science in 2004.

Since then, she has been pursuing her doctoral research in the Nuclear and Radiologfical

Engineering Department and has been working at the Innovative Nuclear Space Power

and Propulsion Institute as a research and teaching assistant. Her fields of specialty are

thermal-hydraulics, Computational Fluid Dynamics and reactor safety.





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Iparticularthankmyadvisorandmentorduringmygraduatestudies,DrSamimAnghaie.Heconstantlyshowedsupport,appreciationofmyworkandalwaysgavemegreatadvice.Iamverygratefulforalltheresearchopportunitiesheoeredme.Iacknowledgethemembersofmycommittee,DrEdDugan,DrGlenSjodenandDrRenweiMei.ThankyoutoDrFrederiqueDrullionforhergreatadvicesandtipsformyCFDsolversdevelopments.ManyheartfeltthanksbelongtoThierryDubrocawhoalwaysbelievedinmeandhelpedmegothroughtheupsanddownsofdoctoralresearch.ThankyoutoMelanieHellerforherinvaluablegiftoffriendship.IalsothankFabienGerardandRimRouachedfortheirmeticulousreadingofthemanuscriptandgreatsuggestions.ThankyoutoLynneSchreiberforallherlogistichelpandkindness.Aboveall,thankyoutomyparents,PierreandMartine,mybrotherNicolas,andmyfamilyforbeingsosupportive,lovingandencouragingtome.Iamverygratefulfortheirsacriceofseeingmeleavemyhomecountryandunderstandingmychoices. 4

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page ACKNOWLEDGMENTS ................................. 4 LISTOFTABLES ..................................... 9 LISTOFFIGURES .................................... 10 LISTOFACRONYMS .................................. 13 NOMENCLATURE .................................... 15 ABSTRACT ........................................ 17 CHAPTER 1INTRODUCTION:ANSWERINGTHENEEDSOFTHEFUTUREINTHENUCLEARINDUSTRYANDSYSTEMMODELING .............. 19 1.1TheNuclearPowerSceneofthe21stCentury ................ 19 1.2ChallengesinThermal-Hydraulics ....................... 20 1.3IssuesFacedwiththeCurrentComputationalToolsforThermal-Hydraulics 21 1.4Objectives .................................... 23 1.5TheDierentApproaches ........................... 24 1.6OverviewoftheDissertation .......................... 25 2APPLICATIONFUNDAMENTALS ........................ 28 2.1SpacePropulsionandNuclearThermalPropulsion ............. 28 2.2TheLow-PressureThrustChamber ...................... 30 2.3ComputationalFluidDynamics:AnOverview ................ 31 3ASURVEYONSTATE-OF-THE-ARTCOMPUTATIONALFLUIDDYNAMICSDEVELOPMENTSANDAPPLICATIONS .................... 34 3.1SystemCoupling ................................ 39 3.2NumericalTechniquesforCFDSolvers .................... 41 3.2.1ConsiderationsonCompressibleFlows ................. 41 3.2.2DiscussionontheMacCormackMethods ............... 43 3.2.3ConsiderationsonTurbulence ..................... 46 3.2.4DiscussionontheSpalart-AllmarasModel .............. 49 4MATERIALPROPERTIES ............................. 51 4.1Hydrogen .................................... 51 4.1.1NoteonDissociation .......................... 52 4.1.2Thermo-PhysicalProperties ...................... 53 4.2Helium ...................................... 56 5

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................................. 57 5METHODSFORTHECFDSOLVERDEVELOPMENT ............ 72 5.1CFDSolverDevelopment ............................ 72 5.1.1Requirements .............................. 72 5.1.2GoverningEquationsforCompressibleFlow ............. 73 5.1.3NeedfortheEquationofState ..................... 75 5.1.4Reynolds-AveragedNavier-StokesEquations ............. 76 5.1.5ClosureProblem:theNeedforTurbulenceModel .......... 77 5.1.5.1Turbulencemodels:classicationandoverview ....... 78 5.1.5.2k-model ........................... 79 5.1.5.3TheSpalart-Allmarasmodel ................. 80 5.1.6SpaceDiscretization ........................... 82 5.1.7TimeandSpaceDiscretization:TVDMacCormackScheme ..... 85 5.1.8TimeStepping .............................. 87 5.1.9BoundaryConditions .......................... 87 5.1.10LimitersandTotalVariationDiminishingCondition ......... 90 5.1.10.1FullMacCormackscheme .................. 92 5.1.10.2ModiedCausonlimiter ................... 93 5.2RemarkonthePracticalImplementationoftheRealGasProperties-Algorithm 94 5.3ValidationCases:TheRiemannShockTube ................. 96 5.3.1PresentationoftheTestCase ..................... 96 5.3.2NumericalMethod ........................... 97 5.3.3Results .................................. 98 5.3.3.1Ontheinuenceofthegas .................. 98 6CFDSTUDYOFTHECOOLANTCHANNELSOFADVANCEDGASEOUSREACTORS ..................................... 108 6.1TheVeryHighTemperatureGas-CooledReactorandtheHeliumChannel 108 6.1.1GridGeneration ............................. 110 6.1.2ApplicationoftheTVDMacCormackScheme-DerivationoftheViscousTerm .............................. 110 6.1.3SimulationResults ........................... 112 6.1.3.1Validationofthemodel ................... 112 6.1.3.2ModelingoftheVHTRCoolantChannel .......... 114 6.1.4ConclusionontheHeliumChannelStudywithRealPropertyFineCFDSolver ............................... 114 6.2TheNuclearThermalPropulsionandthehydrogenchannel ......... 115 6.2.1NERVADerivativeHotChannelAnalysis ............... 115 6.2.2Hydrogendissociationeects ...................... 116 6.2.2.1ConclusionontheHydrogenChannelModeling ...... 117 6

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............. 132 7.1NuRok:a1-DSystemCodeforSimulationofNTPThermal-Hydraulics .. 132 7.1.1DescriptionofNuRokCoreSubroutineandOne-DimensionalCorrelations 134 7.1.2NuRokUpgrading ............................ 137 7.1.2.1UpdatetoFORTRAN90 .................. 138 7.1.2.2Levelsofthrust ....................... 138 7.1.2.3Foamfuel ........................... 139 7.1.2.4Hydrogendissociation .................... 139 7.1.2.5Carbondioxide ........................ 139 7.1.3RecommendationsforNuRokUsers .................. 140 7.1.3.1MassFlowRate ........................ 140 7.1.3.2PumpOutletPressure .................... 140 7.1.3.3CorePower .......................... 140 7.2Themis:ACoupledCoarse-FineMeshSimulationToolforSimulationofNTPThermal-Hydraulics ........................... 141 7.2.1TreatmentoftheInterfaces ....................... 141 7.2.1.1One-dimensiontotwo-dimensionaxisymmetric ....... 142 7.2.1.2Two-dimensionaxisymmetrictoone-dimension ...... 143 7.2.2ThemisResultsintheCaseofNERVADerivativeSystems ..... 144 7.2.3ConclusionontheCoupledne-coarsemeshsolver .......... 146 8CFDSTUDYOFTHELOW-PRESSURETHRUSTCHAMBERWITHFROZENCHEMISTRY ..................................... 158 8.1PresentationoftheLow-PressureThrustChamber .............. 158 8.2Quasi-one-dimensionIsentropicAnalysisoftheLowThrustChamber ... 159 8.3Two-DimensionalContourProleoftheThrustChamberWalls ...... 161 8.4GridGeneration ................................. 163 8.5Low-PressureThrustChamberAnalysisusingFLUENTandFrozenChemistry 163 8.5.1FLUENT ................................. 163 8.5.2NumericalProcedure .......................... 163 8.5.3Analysisresults ............................. 165 8.5.3.1Investigationofthenozzlerepresentation .......... 165 8.5.3.2Whichgoverningequationtouse? .............. 166 8.5.3.3Investigationoftheturbulencemodel ............ 167 8.5.3.4Sensitivitystudy ....................... 167 8.5.4DesignParametersSensitivityStudy .................. 168 8.5.4.1Inuenceofthethermalformulation ............ 170 8.6ConclusionontheFrozenChemistryAnalysisandPerspectives ....... 170 9CONCLUSIONSANDRECOMMENDATIONS .................. 179 APPENDIX APROGRAMFORCOMPILATIONOFTHEEXACTRIEMANNPROBLEMSOLUTION ...................................... 183 7

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................................ 190 B.1Low-PressureThrustChamber:DesignParameterAnalysis ......... 190 B.2FlowAreaContraction ............................. 191 B.3ComparisonoftheHydrogenChannelSimulationwithaFLUENTFrozenChemistryAnalysis ............................... 191 B.4Denitions .................................... 192 CTHEMISSUBROUTINELISTING:NERVADERIVATIVE ........... 197 C.1Coresubroutinefromthesystemcode .................... 197 C.2Interfacesubroutine ............................... 202 DTHEPROPERTYLOOP:INFLUENCEOFTHEHYDROGENDISSOCIATION 203 D.1NoDissociation ................................. 203 D.2SmallDissociation ............................... 204 D.3HighLevelsofDissociation ........................... 204 D.4Program ..................................... 204 REFERENCES ....................................... 220 BIOGRAPHICALSKETCH ................................ 228 8

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Table page 1-1Characteristicsandoperatingconditionsofdierentnuclearsystems[ 1 ] ..... 27 4-1ErroronthedensityformulationforP=0.1bar .................. 61 4-2ErroronthedensityformulationforP=1bar ................... 62 4-3ErroronthedensityformulationforP=10bar ................... 63 4-4ErroronthedensityformulationforP=100bar .................. 64 4-5ErroronthedensityformulationforP=150bar .................. 65 4-6VHTRoperatingconditionsandfeatures ...................... 66 4-7Comparisonbetweenexperimentaldataandideal-gaslawforheliumdensity .. 71 7-1Fanningfrictionfactorcorrelationsandtheirrangeofvalidity[ 2 ] ......... 147 7-2Nusseltnumbercorrelationforturbulentgasowinapipe[ 3 ] .......... 147 7-3Pressuredropoveraowareacontraction.ComparisonbetweenFLUENTsimulationand1-Dcorrelationresults. ............................. 147 7-4Propertiesatdiverselocationsofthesquarelatticehoneycombsystemforthrustchamberinlettemperatureandpressureof2,800Kand500psi(=3.45MPa),respectively. ...................................... 149 7-5PressuredropacrosstheSLHCcorefordierentdensitiesoffoamfuel. ..... 149 7-6ComparisonoftheresultsobtainedwithNuRokandThemis ........... 150 8-1Hydrogenthermalproperties(heatcapacity,specicheatratioandthermalconductivity)fordiversesetsofoperatingconditions ....................... 159 8-2CPUtime(s)andnumberofiterationtoreachconvergenceofthesteady-stateforthefourdierentgrids .............................. 171 8-3Testcasesfordierentdesignparameters ...................... 173 9

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Figure page 2-1Theserialcoolingsystem(NERVAtype) ...................... 33 4-1Levelofhydrogenmoleculedissociationasafunctionoftemperature ...... 60 4-2Hydrogenheatcapacityvs.temperatureatlowandhighpressure(0.1and150bars) .......................................... 61 4-3Hydrogendensityat0.1barandhightemperature ................ 62 4-4Hydrogenheatcapacityvs.temperature ...................... 63 4-5Hydrogenviscosityvs.temperature ......................... 64 4-6Heliumviscosityvs.pressurefordierenttemperatures .............. 65 4-7Heliumthermalconductivityvs.pressurefordierenttemperatures ....... 66 4-8Carbondioxidedensityvs.pressurefordierenttemperatures .......... 67 4-9Carbondioxideheatcapacityvs.temperaturefordierentpressures ....... 68 4-10Carbondioxideviscosityvs.temperaturefordierentpressures ......... 69 4-11Carbondioxidethermalconductivityvs.pressurefordierenttemperatures .. 70 5-1Sketchoftheaxisymmetricrepresentationofthe3-Dchannel ........... 100 5-2Locationofthenumericalnodesfora5x5domain ................. 101 5-3Elementaryvolume(bluearea)forintegrationintheaxisymmetricconguration 102 5-4Detailsofthenotationsoftheaxisymmetriccell(i;j) ............... 103 5-5Nomenclatureforthesurfacevectors ........................ 104 5-6Algorithmwhichshowshowtherealpropertiesareimplementedandcomputed 105 5-7Theshocktubeexperiment ............................. 106 5-8Shocktubeproleatt=6.1msincase 5{62 .ThebluecurvesarethetheoreticalprolesandthegreencurvestheprolescomputedwithourEulersolver .... 106 5-9Propertiesproleatt=6.1msincase 5{62 forair,helium,carbondioxideanduraniumtetrauoride. ................................ 107 6-1VHTRcoolantchannel ................................ 118 6-2Dualcellfortheapproximationofthederivativesatthecellinterfacei+1 2;j 119 10

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........................................ 119 6-4Staticpressure(Pa)proleontheaxisofthetest-section.ComparisonbetweenFLUENTsimulationsandoursolver. ........................ 120 6-5Velocity(m/s)proleontheaxisofthetest-section.ComparisonbetweenFLUENTsimulationsandoursolver. .............................. 121 6-6Temperature(K)proleontheaxisofthetest-section.ComparisonbetweenFLUENTsimulationsandoursolver. ........................ 122 6-7Staticpressure(Pa)proleontheaxisofthetest-section.Comparisonbetweenconstantandrealpropertyformulations. ...................... 123 6-8Velocity(m/s)proleontheaxisofthetest-section.Comparisonbetweenconstantandrealpropertyformulations. ........................... 124 6-9Temperature(K)proleontheaxisofthetest-section.Comparisonbetweenconstantandrealpropertyformulations. ........................... 125 6-10Propertyprolesacrosstheheliumcoolingchannel. ................ 126 6-11Heatproleonthechannelwalls .......................... 127 6-12Absolutepressure(MPa)proleontheaxisoftheNTPcoolantchannel .... 128 6-13Temperature(K)prolesontheaxisandthewalloftheNTPcoolantchannel 129 6-14Density(kg/m3)proleontheaxisoftheNTPcoolantchannel ......... 130 6-15Molecularweight(g/mol)proleontheNTPaxischannelforTi=1;500K(nodissociation)andTi=1;700K(veryfewdissociation) .............. 131 7-1ThedierentNTPsystemsmodeledwithNuRok:NVTR,NERVAderivativeandP&WXNR2000I&II ............................. 148 7-23-DgridofthecontractiontestsectionforstudywithFLUENT. ......... 149 7-3Staticpressureprolealongtheaxisofthecontractiontestsection ....... 150 7-4IdenticationandlocationofportionsoftheNTPsystemwhichwouldbenetfromamoredetailedsolver ............................. 151 7-5StructureofthecodecouplingwhentheCFDisusedtosimulatethehotchannel 152 7-6Extrapolationofthetemperaturenodesfromtheaxisymmetricgeometrytoasinglevalue. ...................................... 153 7-7PressureevolutioninMPaattheinletandexitofthehotchannelvs.theiterationnumber ........................................ 154 11

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....... 155 7-9CoolantchannelpressureprolescalculatedwithNuRokandThemis ...... 156 7-10CoolantchanneltemperatureprolescalculatedwithNuRokandThemis(ontheaxisandthewall) ................................ 157 8-1Thrustchamberanditsprincipalcharacteristics .................. 171 8-2Coarsergrid ...................................... 172 8-3ComparisonoftheMachnumberprolefortheaxisymmetricand2-Dmodels 172 8-4Machnumberprolesforaninviscidandviscousowofhydrogen ........ 174 8-5Machnumberprolesobtainedfordierentturbulencemodels .......... 175 8-6RelativedierenceinthemaximumMachnumberbetweenagivengridandthenestgrid ....................................... 176 8-7Inuenceofthedesignparametersonthemaximumvelocityontheexhaustplane 177 8-8Velocitymagnitudeproleontheexhaustplaneofthethrustchamber ..... 178 B-1Meshesforthetwocontractionexperiments .................... 193 B-2Velocityproleonthecontractionplaneinthesinglechannelconguration.Comparisonbetweencomputedandtheoreticalproles ..................... 194 B-3Velocityproleonthecontractionplaneinthefour-channelcontractionexperiment.Comparisonbetweencomputedandtheoreticalproles .............. 195 B-4Pressureprole(Mpa)ontheaxisofthecoolantchannel.ComparisonbetweentherealpropertysolverandFLUENTwithfrozenchemistry ........... 196 D-1Thepropertyloop .................................. 216 D-2Evolutionofthemajorpropertieswiththenumberofiterationinthehydrogenpipewithnodissociation ............................... 217 D-3Evolutionofthemajorpropertieswiththenumberofiterationinthehydrogenpipewithlowlevelsofdissociation ......................... 218 D-4Evolutionofthemajorpropertieswiththenumberofiterationinthehydrogenpipewithhigherlevelsofdissociation ........................ 219 12

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AMGAlgebraicmultigridBWRBoilingwaterreactorCFDComputationalFluidDynamicsDNSDirectnumericalsimulationFDMFinitedierencemethodFEMFiniteelementmethodFVMFinitevolumemethodFOUFirstorderupwindGFRGas-cooledfastreactorHTGCRHightemperaturegas-corereactorINLIdahoNationalLaboratoryINSPIInnovativeNuclearSpacePowerandPropulsionInstituteISPSpecicimpulseLESLargeeddysimulationLWRLightwaterreactorMUSCLMonotoneupstream-centeredschemesforconservationlawsNASANationalAeronauticsSpaceAgencyNEPNuclearelectricalpropulsionNTPNuclearthermalpropulsionPDEPartialdierentialequationP&WPrattandWhitneyPWRPressurizedwaterreactorQUICKQuadraticupwinddierencingRANSReynoldsaveragedNavier-StokesRNGRenormalizedgrouptheory 13

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4 ].WithinthescopeofGenIVdevelopment,theUnitedStatesisdevelopingtheVHTR.TheyalsoareleadingresearchonNuclearThermalPropulsion.Bothconceptsuseinertgasesasworkinguid,separatingthemselvesfromthemoretraditionalsystemsoperatingwithwater.OperatingcharacteristicsofVHTRandNTParecompiledinTable 1-1 aswellasthePressureWaterReactor(PWR)andBoilingWaterReactors(BWR)forcomparison.Thetypeofcoolantandoperatingconditionsaregreatlydierentbetweendesignsofthefuture(VHTRandNTP)andlastgenerationofreactors(PWRandLWR).Besides,thedierenceincoolantnature(heliumorhydrogenvs.water),newreactorsareoperatedatmuchhighertemperatures.Overtheyears,thenuclearindustryhaslargelyreliedonsimulationtoolstodesignsystems,analyzefuelcycle,aswellasstudyingthebehaviorofreactorsincaseofsevereaccidents.Besidesneutronicsandtransportcodes,thermal-hydraulicscodesrepresentalargepartofthesimulationtoolsusedbyengineersandresearchers.Someoftheestablishedthermal-hydraulicscomputercodescurrentlyusedinthenuclearindustry 20

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1-1 )andhydrogenundergoesa3,000KincreaseintemperatureintheNTP.Thisconditionontemperatureisverydierentfromtheconditionswhichcanoccurinaerospaceapplications:initially,theCFDtechniqueswouldconsiderconstantpropertiesforthegasesofinterest.Ingaseousnuclearsystems,somepropertiessuchasheatcapacityforhydrogenseevariationsofupto500%. 22

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5 ]whichisdevelopedattheInnovativeNuclearSpacePowerandPropulsionInstituteattheUniversityofFloridaisexpanded.NuRokaimsatmodelingthethermal-hydraulicsofdierenttypesofNuclearThermalPropulsionsystems.Theentirepathoftheuidowismodeled.NuRokaimsatdesigningNTPsystemsaswellasdimensioningeachvariouscomponent.AmoredetailedoverviewofthecodefunctionalityandcompositionisgiveninChapter7.NuRokisparticularlyadaptedtothecouplingstudybecause: 23

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6 ].Thistechniquerequireextensivepreliminaryworktocoverasmanyoperatingconditionsaspossible:thenalproductisrestrictedbytheoperationalconditionswhichhavebeencoveredinthesensitivitystudy.Also,thebehavioralcorrelationsobtainedcouldintroducefurthererrors.Thissolutiondoesnotrequirealotoftakedownbutitprovestobealimitedanswertothemodularityofthenumericaltooldeveloped.Therefore,thistechniquewillnotbecoveredinthecaseofthedoctoralwork 24

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1. athighpressureandhightemperature:nodissociation. 2. atlowpressureandhightemperature:lowdissociation. 3. atlowpressureandveryhightemperature:highdissociationThen,thedocumentdescribesthetechniquestocoupletheCFDsolvertotheone-dimensionalcoarsemeshsolver.Thecouplednumericaltool,referredasThemis,isappliedtothestudyoftheNERVAderivativesystems.ThehotchannelofthesystemismodeledwiththeCFDsolverandcoupledtotheone-dimensionalcode. 25

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Characteristicsandoperatingconditionsofdierentnuclearsystems[ 1 ] typeofcoolantphaseworkinginletexitpressure_mreactoruidtemp(K)temp(K)(MPa)(Mkg/h) pressurizedwaterreactorwaterliquidwater30060315.365boilingliquidwaterreactorwatervaporwater269559747gas-cooledreactorheliumgashelium33782855NTPhydrogenSGLhydrogen2232000.01-55:104

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2{1 ). propellantmassowrate'vexhaust 28

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7 ].Itwasterminatedin1973duetothechangeinnationalpriorities.Overthetwentyyearsofactiveworkontheproject,nineteencoreswereinvestigatedcombiningvarioustypesofsolidfuelandcoredesign.ThegenerallayoutofaNTPsystemisshownFigure 2-1 .Liquidhydrogenisstoredinatankatatemperatureof20-22K.Overtheyears,itasbeenestablishedbythenuclearengineeringcommunitythatredundancyoftheturbo-machineriesistherststepforplantsafety.Incaseswhenoneoftheturbo-pumpsystemwouldfail,theredundantsystemswouldhelpsafeoperationofthesystem.ThesameprincipleisappliedtoNTPandthemainhydrogenowlineisseparatedintoatleasttwoparallelandidenticalsystems.First,hydrogenowsthroughapumpwhichextractstheuidfromthetank.Sincetheuidisheated,compressedandisinasinglegaseousphaseattheexitofthepump.Hydrogencoolsdownthewallsofthenozzleandofthecorereector.Then,itentersturbineswhicharecoupledtotherstpumps.Thedierentowlinesarerecombinedbeforetheyenterthecoreofthepropulsion.Attheexitofthenuclearcore,temperatureoftheuidreaches3,000K.Theveryhothydrogenthenentersthethrustchamber,adiverging-convergingnozzle.Thegasexpandsandaccelerates,thrustingtherocket.TheInnovativeNuclearSpacePowerandPropulsionInstituteattheUniversityofFloridadevelopedasystemcodetomodeltheowofthegasinNTPsystems[ 8 ].The1-Dcodeaimsatsizingthedierentpartsofthesystemforgivenoperationalthermalpowerandcoreexitpressure.IthasbeendevelopedsuchthatalargevarietyofNTPdesignscanbestudied,aswellasapanelofsystemorganizations.ThiscoarsemeshsimulationtoolswillbereferredinthisdissertationasNuRok(NuclearRocket).Theskeletonofthecodereliesontheresolutionofthemomentumequation.However,itrequiredexperiment-based 29

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9 ].Inthecaseofalowerpressurewithconstanttemperature,themoleculesofhydrogen(H2)startdissociating.Astheyrecombine,thereisextraenergywhichisliberatedinthegas.ThedissociationenergyofamoleculeofhydrogenH2isequalto4.52eV[ 10 ].ThiscanbeobservedwhenconsideringtheheatcapacitycPwhichisdiscontinuousatthepressure/temperatureofdissociation[ 11 ].ThepropertiesofhydrogenanddissociatedhydrogenwillbedescribedmoreindetailinChapter4.However,thebehaviorofthehydrogenmoleculeasitdissciates/recombinesleadstothefollowingstatement:Webelievethedissociation/recombinationeectsof 30

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12 ].Tenyearslater,theuseofcommercialCFDsoftwarestartedtobecomeacceptedbymajorcompaniesaroundtheworldratherthantheircontinuingtodevelopin-houseCFDcodes.Historically,thenuclearindustryhasreliedonCFDcapabilitiesinitsrststagesofdevelopment:CFDwasappliedtosafetyanalysisandcorereloaddesignofnuclearpowerplants.Overthelasttwentyyearsdramaticimprovementsinbothnumericaltechniquesandcomputingpowerhaveoeredthepossibilityofperforminglarge-scale,high-delitycorethermal-hydraulicsanalysis.ThemostcommonlyusedcommercialCFDcodesarethenitevolumesolversFLUENT[ 13 ],CFX[ 14 ]andStar-CD[ 15 ].ThesecodesoeralargevarietyofsolverstodiscretizeandsolvetheNavier-Stokespartialdierentialequationswhichdescribemostowbehaviors.Theyalsooeralargevarietyofmodelingcapabilities,includingbutnotrestrictedtoturbulence,radiativeheattransferandtwo-phasemodeling.Toalargedegree,thesecodesoerverysimilarperformancesandmodelingoptions.Freitas[ 16 ]hascompiledacomprehensivebenchmarkingofthedierentcommercialCFDcodes.Thesebenchmarkproblemsincluded:thebackward-facingstepproblem,theuniformowpastacircularcylinder,thethree-dimensionalshear-drivencavityow(allthreelaminarows),theturbulentowaroundasquarecross-sectiontubeandthedevelopingturbulentowina180degreeelbow.ThemaindrawbacksoftheFreitas[ 16 ]studywerethelackofindependenceandcompleteness.ThestudywasperformedbyCFDcodevendorsandtoalargeextentdidnotincludeturbulencemodeling.Thek-model[ 17 ]isthemostcommonlyusedturbulencemodelinpracticalindustrialapplications.Adetailedformulationofk-modelisdescribedinChapter5.Itisthisparticularmodelwhichaccountedforturbulencemodelinginthebenchmarkstudyof 34

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18 ]presentedthetrendsinLWRsafetyanalysis.Thearticleunderlinedtheshortcomingsofthetraditionalsystemcodesandproposedtomovetowardmulti-dimensionalanalysisbytakingadvantageofsingleandtwo-phasedetailedCFDcapabilities.ThemainissueaddressedbythepaperwasthelimitationincomputingpowerandskillsforuseofCFDinpracticalproblems.ThemainareasofinterestforuseofCFDcapabilitiesaretheprimaryloopandcontainmentofnuclearpowerplants.Todate,CFDanalysesperformedinnuclearpowerplantsystemsincludedborondilutionintheprimaryloopofPressurizeWaterReactors(PWR),mixingofvariousgases,straticationofcontainmentpools,etc.Boyd[ 19 ],[ 20 ]analyzeda1/7thandfull-scalesteamgeneratorinletplenummixingduringaPWRsevereaccident.One-sixthoftheinletplenumwasmodeledwithFLUENT6.0usingtheReynoldsStressModel(RSM)associatedwithnon-equilibriumwallfunction.Experimentsonthe1/7thscalemodelvalidatedtheaccuracyoftheturbulencemodelfortheinletplenumgeometryandow.However,numericalresultsofthestudyexhibitedannon-physicaloscillatorybehavioroftheplumeforthegeometryassociatedwithasymmetryplane.ThisproblemhadalreadybeenidentiedwiththeuseofFLUENT'sonseveraltestcases.Thenon-physicalbehaviorcanbeattributedtotheshortcomingsof 35

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21 ]modeledaborondilutiontransientduringtheshutdownphaseofaPWRplant.TheanalysiswasbasedonSTAR-CD'sAlgebraicMultigridSolverwiththeRe-NormalizationGroup(RNG)k-turbulencemodel.Simulationresultswerecomparedtotestdataanditwasshownthattemperaturedependentpropertiesandbuoyancyeectsmodelingcouldincreasetheaccuracyofthesimulation.TheKeheley'sworkdemonstratedthatSTAR-CD,couldadequatelymodelatypicalborondilutiontransientinPWRcores.Tinoco[ 22 ]showedthefeasibilityofanalysisofthesteamlinebreaktransientinBoilingWaterReactors(BWR)usingSTAR-CD.Sinceturbulencesdonothavetimetodevelopinsuchfasttransients,thelaminarsolverwasconsidered.SuchatransientactuallyoccurredinUnit1and2attheForsmarknuclearpowerplant.Theincidentwascausedbydefectiveweldingandledtothereplacementofthecoregrids.Tinoco'sCFDsimulationhelpedtodevelopabetterunderstandingofincidentswhichcouldoccurwhiletheplantisinoperation.BehaviorofthewaterowandheattransferaroundrodbundlesinLWRcoresisalsoamajorsubjectofinterestinreactorsafetystudy.TherodbundleisastandardgeometryinLWRsbecauseofthefuelpinarrangement.ModelingofrodbundlesinLWRcoresisagreattestcaseforassessmentofCFDcodes.References[ 23 ]and[ 24 ]presentawidevarietyofexperimentalresultsinvolvingowandheattransferinLWRfuelbundles.Usingtherodbundletestdata,Tzanos[ 25 ]demonstratedtheshortcomingsoftheSTAR-CDandit'sassociatedlineark-modelsforthesimulationofsuchproblemsinPWRcores.Tzanossimulationresultsshowedasignicantdiscrepancywithexperimentaldata.Thiswasparticularlytrueinareasofowdeectionsuchasspacergrids.TheresultshighlightedtheneedforotherRANSturbulencemodelsorLES.Inetal.[ 26 ]ledamoredetailedanalysisofthesameproblembyinvestigatingvariousRANSturbulencemodelswiththeCFXcode.Theseturbulencemodelsincluded 36

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27 ]showedthatthestandardandSpezialek-modelsgavethemostaccuratepredictions.ResultsofInet.al.studies[ 26 ]-[ 27 ]provedthatthechoiceofturbulencemodelmustbecarefullyconsidereddependingontheowsituation.Thek-modelusuallyremainsthesafestchoiceinturbulencemodeling.Hence,in2001,aconsortiumoftwelveEuropeanpartnersstartedtheproject[ 28 ]undersponsoringoftheEuropeanCommission.ECORAstandsforEvaluationofComputationalFluidDynamicsMethodsforReactorSafetyAnalysis.TheprojectaimedatassessingtheperformancesofnumerousCFDcodes(CFX,FLUENT,Saturne,STAR-CDandTRIO-U)forsimulatingowsintheprimarysystemandcontainmentofnuclearreactors.TheprojectalsohighlightedanddenedCFDcodeimprovementsthatwerenecessaryfornuclearengineeringapplications.Theinterestinthethree-dimensionaloweectsinsomereactorcomponentsarosefromtheinabilityofone-dimensionalsystemcodestopredictaccuratelyowbehaviorinspecicareasofthereactorsystems.Examplesofsystemsunderinvestigationwere: 37

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29 ]simulatedthemixingofHeliuminthelowerplenumofaprismaticGas-CooledReactor.Thestandardk-modelofFLUENT6.1wasconsideredalongwithbuoyancymodeling.Thepredictedhotandcoldtemperatureswerewithin1.5to2Kofthemeasuredvalues.Thisresultshowsthegoodaccuracyofthesimulation.ComparisonwasmadepossiblebyMcEligotreport[ 30 ]whichcompiledafewexperimentsthathavebeendevelopedatINLtoassesstheperformancesoftheCFDcodesinthecaseoftheVHTR.Morerecently,Ahmadetal.[ 31 ]studiedthe3-Ddistributionandpressuredropattheinletplenumandrisingchannelofapebble-bedmodularhightemperaturereactorwithCFX-5.7.1.Theuidwasmodeledbyanincompressibleandisothermalowandonceagainthek-modelaccountedforturbulenteects.ThemodelwasnotbenchmarkedbutAhmadformulatedrecommendationsforsimilarCFDstudies.However,consideringanincompressibleowintheVHTRseemslikeaverylimitingassumption.CFDstudiesinthenuclearindustryarenotlimitedtoapplicationstotheprimaryloopofpowerplants.CFDisofgreathelpforothereldsofstudyrelatedtonuclearapplications.Forexample,Lee[ 32 ]usedCFXcapabilitiestomodeltheheatgeneratedbyradioactivedecayofdryspentfuelinacanister.Theanalysisrequiredtoaccount 38

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33 ]developedahighdelityrepresentationofacurrentgenerationPWRwithSTAR-CD.Thestudymodeledtheentirecoreforwhichthegridwascomposedof240millionCFDcomputationalzones.Calculationsrequired50hoursofcomputationon200IBMprocessors(in2000)togetthesteady-statesolution.TheanalysisprovedthatCFDcanbeusedtoassessuid-dynamicsandthermo-mechanicalissuesthatwerenotpossiblewithmoresimpliedapproaches.Evenifsuchanextensivestudyispossible,itdoesnotremainrealisticaswewanttostudytransientsofthecore.ThisdemonstratesagainthatCFDanalysisinnuclearreactorsystemsshouldberestrictedtosmall,targetedportions.Inorderforustosimulatethe 39

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34 ].ItconsistedofanindirectcouplingappliedtotheKoreannextgenerationreactor.ACFXsimulationanalyzedindetailtheowforthedirectvesselinjection,whilethebulkofthesystemwassimulatedwiththethermal-hydraulicscodeTRAC.ThesystemcodereturnedaboundaryconditionwhichbecametheinletboundaryfortheCFDsimulation.Thiscouplingtechniqueislimitedtosteady-statecases.Accidentanalyzesgenerallyrequirefull-coupling,capableofhandlingtransients.Withthatinmind,theIdahoNationalLaboratory(INL)startedaco-operativeproject:coupleFLUENTwithRELAP5-3D[ 35 ].ThemajorpartofthenuclearsystemissimulatedwithRELAP.OneormoreparticularportionofthesystemismodeledwithFLUENT.IntheparticularexampleofWeaverthesystemiscomposedofasimplelinearassociationofvolumeswithaby-pass.Theby-passiscomposedofaheatedchannelmodeledwithCFDandasimplepipemodeledwithRELAP.RELAPstartscalculatingthemapowintherstcellsuntilitreachestheby-pass.ThepropertiescomputedbyRELAPattheinletoftheby-passareinputastheinletboundaryconditionfortheFLUENTmodel.FLUENTisrunandreturnsasetofpropertiesattheexitofthepipewhichareimplementedbackintoRELAPasinletconditionsfortherestofthesystem.Thistypeofcouplingisreferredtoasdynamiccoupling.Itistheoptimumcouplingforobtainingaccuratesolutions:thiscouplingcombinestheadvantagesofthetwotypesofcodes.ThenuclearindustryalsoneedssimilarcouplingbetweenCFDandneutronicscodes.Thistypeofworkiscurrentlyunderinvestigation.TheNURESIMproject[ 36 ]isaverypromisingcollaborationwhichaimsatcombiningapin-basedMonte-Carlo 40

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37 ].TherstpartofthereviewshowsthatcommercialCFDcodeswhenusedproperlycangivesatisfyingresultstocomputeportionsofowsinnuclearsystems.However,Weber'sstudy[ 33 ]showedthatitisnotrealisticallypossibletosimulatethetransientofanentiresystembecauseofthecomputationalcost.SeveralattemptsaremadeatcouplingsystemcodeswithcommercialCFDcodesbutthemoduleinterfacemakesthetaskreallydicult.Also,thecommercialcodesarenotbuilttoconsiderlargevariationsofpropertieswithtemperaturesandpressure.Forallthepreviousreasons,wedecidedtodevelopoutownsolversasweexplainedintheintroduction.OursolvershouldbeapplicabletocompressiblehighReynoldsnumberowsoverallrangesofMachnumbers(subsonic,transonicandsupersonicows)aswellasturbulentows.Wewillnowreviewthenumericalmethodsdevelopedwhichapplytothetypeofowwewanttomodel. 3.2.1ConsiderationsonCompressibleFlowsAthighReynoldsnumber,theconvectiveeectsarepredominantcomparedtodiusiveeects.Therefore,theuidbehaveslikeaninviscidow[ 38 ].ThegoverningequationsforaviscousuidarethesystemofNavier-Stokesequations:theyreducetotheEulerequationsinthecaseofinviscidow.Thenumericaltechniquesforthetwotypesofequationsareverysimilarexceptforthediscretizationoftheshearstresswhichariseinviscousows.Forcompressibleandsupersonicows,thereisformationofshocks,expansionfansandcontactdiscontinuities.Specicnumericalschemesshouldbeconsideredtocaptureproperlypressureandpropertiesdiscontinuities. 41

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39 ],LaxandFriedrichs[ 40 ].Lax-Friedrichsmethodsareeasytoevaluateandsmooth,howevertheyareexcessivelydissipative[ 41 ].ThemilestoneforthemoderndevelopmentistobefoundintheworkofLaxandWendro[ 42 ]whichresultedinthetypeofsolvernamedafterthem.Theirmaincharacteristicistobesecond-orderaccurate,space-centeredandexplicitintime.Avariantofthismethod,expandedtothesystemofNavier-Stokesequations,wasdevelopedlater:itisthewidely-usedexplicitMacCormackscheme[ 43 ].Mostoftheworkintheearly1960shappenedattheCourantInstituteonhyperbolicsystems.Thenumericalschemestendedtoexhibitnon-physicaloscillationsanditwasdiculttoobtainconvergedsteady-statesolutions.Atthesametime,inRussia,GodunovintroducedtheconceptofrepresentingtheowaspiecewiseconstantineachcomputationalcellsandsolvingaRiemannsolverateachinterface[ 44 ].Theschemewasrstorderaccurateandavoidednon-physicalfeaturessuchasexpansionshocks:expansionshockscannothappenbecausetheyviolatetheThirdLawofThermodynamics,meaninglyentropycanonlyincrease(S0).BecauseGodunov'spioneeringworkwasnottranslatedasitwaspublished,ittooktimefortheWesttoacknowledgethenumericaltechnique[ 45 ].Howeververyaccurate,theschemewasverycostlyintermsofcalculationsofthecomplexuxesattheinterfaces.In1981,Roe[ 46 ]developedatechniquewhichenabledapracticalandecientimplementationofGodunov-typeschemes:heintroducedtheconceptoflocallylinearizingthegoverningequationsthroughameanvalueJacobian:hisworkstillhasamajorimpactonmoderncomputationaltechniques.Roe'stechniquebelongstotheApproximateRiemannSolversclass,alongwithOsher'smethod[ 47 ].Roe'sconceptisbasedonaux-dierencesplittingschemeandyieldsasingle-pointnumericalshockstructureforstationarynormalshocks. 42

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48 ]isanotherexampleofwidelyusedux-splittingsolvers.Godunov'sworkalsorecognizedthatnumericalschemesmightbenetfromdistinguishingvariouswavespeedsleadingtocharacteristic-basedschemes.Earlyhigh-ordercharacteristic-basedsolversuseduxvector-splitting,themostfamousbeingtheschemeofStegerandWarming[ 49 ].Itstillledtooscillationsneardiscontinuities.TheMonotoneUpstream-centeredSchemesforConservationLaws(MUSCL)ofVanLeer[ 50 ]extendedGodunov'sschemetohigherorders:theschemewassecondorderaccurateinspacethankstotheintroductionofux-limiters.Limitersarefunctionswhichsmoothesthenumericalsolutionsinareaswherehighorderschemewouldnormallycreateoscillations.Theirrstusedatesbackto1973withtheworkofBorisandBook[ 51 ]whodemonstratedperfectpropagationofdiscontinuities.Harten[ 52 ]demonstratedthatTotalVariationDiminishing(TVD)schemes(seeSection5.1.10)aremonotonicity-preservingandenableoscillationcontrolinthesolutionofnon-linearproblems.AnotherverysignicantworkistheRunge-KuttaalgorithmbyJameson[ 53 ]whichisbasedonanalternative-directionimplicitapproximatefactorization.Duringtheeighties,developmentofcompressibleEulersolverswasanessentialsteptowardsimulationofthemorecomplexows.Itwasmadepossiblethankstotechniquessuchascombinationofupwinding,articialmethods,niteelementornitevolumeapproximationsanduxlimitingconcepts,forexample.WenowpresenttheMacCormackmethodswhichareofgreatimportanceforourresearch.Itbelongstothefamilyofblockmethodswhichconsistinsolvingalltheequationssimultaneously.Itusuallyrequiresmorecomputationsateachgridpointbutimprovesdramaticallytheconvergencerate[ 54 ]. 43 ].Thesolverwasexplicit,second-orderaccurateandusedthepredictor-correctormethod.Itsperformanceswere 43

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55 ].Continuityofhisworkledtoacombinedimplicit-explicitMacCormackschemein1981[ 56 ].Explicitschemesarealwaysnumericallystableandconvergentbutusuallymorenumericallyintensivethantheexplicitmethod.Theexplicitpredictor-correctormethodwhichisfasterisusedovermostofcomputationaldomain.However,itisnotadequateforpointsintheowatwhichthelocalCFLnumberexceedsthestabilitylimit[ 57 ]:themethodaddsanimplicitproceduretothepredictor-correctorsequence.Themethodisimplicitinnatureandthusallowsamuchlargertimemarchingstepsize.Themethodpossesthreeadvantagesoverfully-implicitmethods.First,themethodusestwo-point,one-sideddierenceswhichleadtoblockbidiagonalsystems,easiertoinvertthatblocktridiagonalsystemsfoundintraditionalmethods.Secondly,themethodusesinviscidjacobiansandcorrectsthemusingrepresentativeviscoustermsaddedtotheeigenvalues.Thismaintainsstabilitywhileavoidingtheexpensivecalculationoftheviscousjacobian.Finally,thealgorithmallowstheimplicitsteptobeskippedinregionswherethestabilityrestrictionconditionissatised.Theexpressionsgivenforthepredictorandcorrectorstepsareadequateforapplicationswiththenitevolumemethod.Snyder[ 58 ]comparedtheoriginal2-2MacCormackschemefrom1969[ 43 ]toa2-4modiedversionofthemethodandothersimilarexplicitnitedierencemethods.ThemethodswherebenchmarkedagainstaGaussianpulsediusionandtheshocktube:thetwotestcasesstudynon-linearwavepropagationsofaninviscidow,i.e.theyrequiretosolveforthesystemofEulerequationsonly.Bothexperimentsarewelldocumentedandallowdetailedvalidationofnumericaltechniques.Themethodsincludedarticialdampingorlteringmodelstoreducespuriousoscillations.The2-2MacCormackschemeissecondorderaccurateinspaceandtime.The2-4schemeisfourthorderaccurateinspace:thespacialaccuracyisimprovedbyincreasingthestencilsizefromthreetove. 44

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59 ]dampingfunctionwasaddedtobothnumericalmethodstosuppressspuriouspulses.Thestudyshowsthatthe2-2MacCormackisthefastestmethodofallinvestigated.However,itgivesspuriousoscillationsandishighlydiusive.The2-4methodcapturesbetterthesharpnessoftheshockbutlargelyovershootsthevalueofthediscontinuity.Theanalysisconcludesthateachmethodhasitsownuniquefeatureandthattheorderofaccuracyofamethoddoesnotnecessarilyisagoodindicatorofthemethodperformances:Toreduceoscillationsinareasofdiscontinuitiesonemightprefera2-2MacCormackovera2-4MacCormackeveniftherstismoredissipative.AfterthereleaseofMacCormackimplicit-explicitpredictor-correctorscheme[ 56 ],someanalyzesshowedthattheimplicitMacCormackmethodisunreliablesuchthatthesteady-statesolutionsdependontimeincrements[ 60 ].TheseverityofsuchdependencewasassessedbyOng[ 61 ]inacomprehensivestudyin1987.Theschemewasalsocomparedtothefully-implicitschemeofBeamandWarming[ 62 ].Thestudywasvalidatedoverasupersonicturbulentowpastatwo-dimensionalcompressioncorner.Turbulencewassimulatedbythealgebraicturbulenteddy-viscosityofBaldwinandLomax[ 63 ].Thestudyconcludedthatthesteady-statesolutionoftheexplicit-implicitmethodisrelativelyinsensitivetotheCourantnumber,thereforethesteady-stateappearedtobeindependentofthesizeofthetimestep.Moreover,theaccuracyofthesteady-statesolutionusingMacCormackshybridalgorithmwascomparabletothatoftheBeam-Warmingmethodforallcases,andwasobservedtoreducethecomputingtimebyafactorofuptothree.Lawrence[ 57 ]alsoinvestigatedtheperformancesoftheimplicit-explicitformofMacCormacktechnique:resultsofthesimulationofalaminarat-plateboundary-layerandhypersoniclaminarowovera15-degreecompressioncornerwerecomparedtoexperimentalresultsandtheBeam-Warmingscheme[ 62 ].Theresultscomparedprettywellwithpublishedresultsandexperimentaldata.However,theauthorsidentiedasourceoferrorintheboundarytreatment,particularlyvisibleintheat-platesimulation:thereectiveconditionattheboundaryledtoinstabilitiesatthebeginningof 45

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64 ].TheMacCormackschemesareparticularlyadaptedfortheapplicationsofinterestinthisdissertation:theyarefastandeasytoimplement,andtheaccuracyissatisfactoryforthetypeofowweinvestigate. 65 ].Engineersneedaccesstoviabletoolscapableofrepresentingtheeectsofturbulence.Herearethethreemethodswhicharecommonlyused: 46

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66 ]denesthesizeofthemeshesandin3-DthenumberofgridpointsendsupbeingequaltoN=Re9=4.Therefore,DNSisnotadaptedtothetypeofsolversweaimatbuilding.ThedevelopmentoftheLESmethodsremainsparticularlymarginalbecausethecomputationofthelargescaleeddiesrequiresanamountofcomputationalresourceswhichwerenotavailableuntilafewyearsago.Moreoverthenuclearindustryisjustbeginningtointroducethistechniqueforanalysisofturbulentows.Mostofthetime,engineersareinterestedininformationaboutthetime-averagedpropertiesoftheow.Thisisparticularlytrueforthestudiespresentedinthisdissertation.Hence,theRANStechniqueisoftenprivilegedandhistorically,hasbeenthemostdevelopedtechnique,thusisverywelldocumented.ThemajorityoftheCFDsimulationsfornuclearreactorapplicationswepreviouslyquotedrelyonthek-modelforturbulenceconsiderations.Evenifsomeresearchworkshowthelimitationsofthismodelincertaincases[ 25 ],itisnotclearifthismodelissuitableforallresearches.Besides,thepreviousreviewshowedtheimportanceofaccountingfortherealpropertiesoftheuids[ 29 ].However,thecommercialCFDcodeswhicharetraditionallyusedhavelimitedcapabilitiesinrealpropertiesmodeling.MoreoverDirectNumericalMethods,LargeEddysimulations 47

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67 ].Georgiadis[ 68 ]solvedthisproblembyusingaRANSandaLESmixing-lengthmodel.However,whenappliedtocompressibleturbulentmixinglayersitonlyenablespredictionsofthemaincharacteristicsoftheboundarylayersandtheauthorsthemselvesconcludedthatthereislargeroomforcalculationsimprovement.Therefore,thehybridRANS-LESmethodsforturbulencemodelingareinatooearlystageforustoconsiderinoursolvers.Hence,wewillbuildourCFDsolversusingtheRANSmethodthatiswidelydocumentedandprimarilyusedforowanalyzesrelatedtothenuclearengineeringeld.InthereviewofCFDanalysisinnuclearsystems,themodelsofpredilectionareprimarilythek-andRSMmodels.Thek-seemsverywell-adaptedtotheVHTRcoolantchannelstudy.Itisatwo-equationmodelandthereforeisoneoftheRANSmodelswhichrequiresrelativelylowcomputationalcapabilities.Ithasshowngoodperformancesinthestudyofsubsonicpipeows,whichisourinterestforthecoolant 48

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69 ]seemstobemoreadaptedintheparticularcaseofthethrustchamber. 70 ].Itisaveryrecentmodel,butwereviewtheworkwhichhasbeendoneonthevalidityofthemodeltodetermineifitisapplicabletoourparticularcaseofthrustchamber.Paciorri[ 71 ]investigatedthevalidityoftheSpalart-Allmarasturbulencemodelforhypersonicows.Themodelwasvalidatedonavarietyoftestcases:atplateturbulentboundarylayerow,hypersonicwind-tunnelow,aMachnumberequals5owoverahollowcylinderandaMachnumberequals6.8owoverahyperbolicare.Thestudyconcludedthatthemodel,implementedinanitevolumesolverwithanitedierencetechnique,overallgavegoodpredictionsoftheowbehavior.ExceptionwasmadeincaseswhichinvolvedturbulentseparationTheauthorrecommendedtheuseofthemodelforattachedows,whichisthecaseofthethrustchamberowweinvestigate.FinkandPirzadeh[ 72 ]alsoinvestigatedthevalidityoftheirsolveronaat-plateboundary-layer,usingtheSpalart-Allmarastechniquetomodelturbulence.TheEulerequationswerediscretizedusingRoe'sschemeanduxlimitersSuperbeeorMinmod.BotherEulerequationandtheSpalart-AllmarastransportequationswerediscretizedusinganimplicitbackwardEulertime-steppingmethod.Therefore,itallowedaloosecouplingbetweenthedierentequations.TheSpalart-Allmarasmodelwascoupledwithawallfunctionformulationtoeliminatetheneedforresolvingtheowinthesublayerportionofaturbulentboundarylayer.Thepapershowedthatthistechniquegavegoodresultsbothintheoweldandtheboundarylayer,andhadtheadvantageofreducingthecomputingmemoryrequirementandimprovingoverallconvergence.Bongliolietal.[ 73 ]studied 49

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74 ]ledanextensivestudyofnineturbulencemodelsappliedtothesecondaryowinanaccelerating90oelbowandtheperformancesofeachmodelswereassessedwiththeexitpassagevortex.TheSpalart-AllmarasmodelgavethebestpredictionsalongwiththeBaldwin-LomaxandtheShearStressTransportmodels.Inconclusion,weobservethatCFDhasbecomeoneofthepreferredsimulationtoolsinthermal-hydraulicsstudiesforportionsofnuclearsystems.ThenuclearindustryreliesmostlyoncommercialCFD-codes,butthesecodesexhibitafewdrawbacks:Theywerenotaimedatmodelinglargepressureandtemperaturevariationswhichleadinevitablytomajorpropertyvariations.Also,averycriticalissueinCFDsimulationsisturbulencemodeling:theturbulencemodelsoftenoeredincommercialcodesweredevelopedforaeroacousticows.Somestudies([ 16 ],[ 26 ],[ 27 ]...)aimedatevaluatingthevalidityofsuchmodelsfornuclearapplications.Itresultedthat,besidesitsrelativesimplicity,thek-turbulencemodelshowedgoodperformances.Wethenwillbeabletousethismodelformostofoursimulations.However,inthecaseoftheNTPthrustchamber,ourreviewshowsthatasimplermodel,theSpalart-Allmarasmodel,wouldgivebetterresultswhilebeingcomputationallylessexpensive.Also,wereviewedthemainnumericaltechniquesforcompressibleowswhicharecompetitiveinshocksanddiscontinuitiescapture.Usually,commercialcodesonlyallowsonetypeofsolverforthistypeofow:forexample,FLUENT6.1islimitedtoathirdorderMUSCLscheme.Althoughveryaccurateinmostowtypes,theusermightwantmoreexibility.Inourcase,wewanttogetresultsasaccurateaspossiblebutwithanemphasisoncalculationrapidity.Inthepreviousreview,weshowedthattheMacCormackmethodsmightbetheonemostrelevantforourstudy.InthescopeofthedevelopmentofCFDsolversforVHTRandNTPstudy,thisreviewhelpsusmakeinformedchoicesofeachofthedierenttechniques. 50

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9 ].Thisisthestartingpointfortherangeofoperatingconditions.Wesawinsection2.2thatweexpectgreatperformancesoftherocketthrustchamberifthesystemisoperatedatlowpressure. 51

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75 ].Thedissociationfollowsthescheme:H2!H+HThelevelofdissociationisplottedFigure 4-1 asafunctionoftemperaturefordierentpressures.Dissociationcanonlyoccurattemperaturesgreaterthan1,500K.Atagiventemperature,thelowerthegaspressurethemoremoleculesdissociate.Dissociationprimarilyimpactsthemolecularweightofthegasanditsheatcapacity.ThemolecularweightofhydrogenatroomtemperatureandatmosphericpressureisM=2:016g molAsthegasstartdissociatingthestatisticalaverageofthegasmolecularweightdrops.ThevaluesofthemolecularweightisobtainedfromVergaftig[ 11 ].Figure 4-2 showstheevolutionoftheheatcapacitywithtemperaturewhenthereisahighlevelofdissociation(p=0.1bar)andwhentheimpactofdissociationisnegligible(p=150bars).Atthelowestpressuretheheatcapacityofthedissociatedgasabruptlyincreaseswhereasat150bars,theheatcapacityremainsroughlyconstant.At0.1bars,theheatcapacityreachesamaximumatT=3;000K. 52

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76 ]andtheNationalInstituteofStandardandTechnology.Thepropertydataprovidedbythehydrogenpackagearethebestavailableasoftoday.AcomputerroutinedevelopedbyChen[ 77 ]convenientlyinterpolatesthermodynamicandtransportpropertiesofhydrogenwithinthetemperaturerangeabove.Dierenttechniquesareusedtodevelopthehydrogenpropertiespackage:Linearinterpolation(LI),naturalcubicsplineinterpolation(NCSI),andmultiplepolynomialcurve-ttingwithpolynomialbridgingbetweenthecurvets.Thepackageallowstheusertogethydrogenpropertiesforadiscretesetoftemperatureandpressure.ForuseintheCFDsolver,wewouldliketohaveacontinuousdescriptionofthedierentpropertiesasafunctionoftemperatureandpressuretoavoidcpu-expensivetablereadingandbeabletodescribeaswellaspossiblepropertygradientswhichappearintheuidgoverningequations.Eachsetofpropertiesisnowcurve-ttedtoobtainacorrelationingoodagreementwithexperimentaldata(about5%). RTwherepisthepressure,RtheperfectgasconstantandTthetemperature.Acomparisonbetweenrealthermo-physicaldataandidealgasapproximationispresentedTables 4-1 to 4-5 .Thelevelofdissociationofthehydrogenmoleculesis

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11 ]andshownFigure 4-1 .WecomparedensityvaluesobtainedfromGiven[ 78 ]tothetheoreticaldensitycalculatedwiththeidealgaslawcorrectedformolecularweightvariationsduetothedissociationsofhydrogenmolecules.Therelativeerrorsarecompiledinthelastcolumnofthetables.Overmostofthetemperatureandpressurerangetheidealgaslawapplieswellsincetherelativeerrorsremainsmallerthan3%.Attheextremerangeofpressure,whenthepressureissmall(P=0:1barsandT>3;000K)therelativeerrorsincreasesupto30%.Theerroristoolargetoconsidertheidealgaslawisaccurateundertheseconditions.Figure 4-3 showsthedierencebetweenthedatafromGiven[ 78 ],thedatafromVargaftik[ 11 ]andthosecomputedfromtheidealgaslaw:densityisplottedasoffunctionoftheinverseoftemperature.ItseemsthatthevaluesfromGivenathightemperaturearenotaccurate:theerrorhasbeenidentiedascomingfromadigitinversionwhenthedensityvalueswereinputinthecode.Thegraphshowsthegoodagreementbetweentheideal-gaslawmodelandthedensityvaluesfromVergaftik.Therefore,theideal-gaslawcorrectlydescribesthehydrogendensitybehaviorforallrangesofpressureandtemperature.Allhydrogenthermo-physicalproperties(besidesdensitythatwejustdiscussed)areimplementedinthesolverthroughcorrelationswhichareafunctionofbothpressureandtemperature.Thosecorrelationsarepolynomialfunctionsofthetwoextensiveproperties. 4-4 showstheevolutionoftheheatcapacitywithtemperaturefordiversepressures.Asthegasdissociatesthereisasharpincreaseintheheatcapacityvalue.Thepreviouscorrelationiscontinuouswithpressureandtemperature.However,thiscorrelationonlygivessatisfyingresults(within15%)fortemperaturesupto3,000Katlowpressures.Infact,at0.1bar,heatcapacityreachesamaximumatT=3,250KasseenFigure 4-2 andasimple,continuouscorrelationforallpressuresisnotobtainable.Therefore,thetrendoftheheatcapacitybehaviorisfavoredintheapproachofhydrogendissociation.NotethattheformulationaboveisforheatcapacityexpressedinkJ/kg.K.TheSIunitsareJ/kg.K. 54

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4{3 remainswithin5%ofthemeasuredviscosity.Figure 4-5 showsthegoodagreementbetweenthemodelandtheexperimentaldata. 4{4 and 4{5 temperatureTisexpressedinKelvinandthepressurepinbars. 4-1 ).Therefore,wedevelopalinearextrapolationbetweeneachdiscretepoint.IfwelookforthemolecularweightatpressurepandtemperatureTsuchthatp1p
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4-6 compilesVHTRoperatingconditionsandfeatures.Heliumcirculatesinaclosedloop.Typically,thecoldestpointoftheloop,attheinletofthecore,isatatemperatureof490oC.Inordertogetthebesteciencyoutofthesystem,heliumisheatedupto950-1,000oC.Theusualoperatingpressureisestimatedat7.12MPa.Therangeofoperatingconditionsoverwhichwederivetheheliumpropertiescorrelationsistherefore: 11 ]forheliumthermal-properties. 4{8 .TheresultsarepresentedinTable 4-7 andshowthattheideal-gaslawproperlydescribesthebehaviorofheliumdensity:thetheoreticaldatafallwithin4%oftherealvalues. 11 ]onlylistsheatcapacityofheliumatlowtemperature.WerefertoLemmon(et.al.)[ 79 ]fortemperaturesrangingfrom200to1,500Kandpressures 56

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4{10 fallwithin0.4%oftheexperimentaldatafortemperatureswithinrange 4{8 .ThegoodagreementbetweenthemodelandthedatacanbeseenFigure 4-6 4-7 57

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4{8 ).WeusethedataprovidedbyLemmon(et.al.)[ 79 ]. 4-8 ).Themodeldoesnotgivegoodresultsfortemperaturesof400K.Fortemperaturesequalorgreaterthan800Kthecomputedresultsfallwithin2%oftheexperimentaldata.Toderivetheidealgaslawmodel,weuseM=44:011g/molandR=< 4-9 4{8 ,CO2viscosityispressureindependentandvarieslinearlywithtemperatureasseenFigure 4-10 .Wemodelcarbondioxideviscosityby: 4-10 58

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4-11 DevelopmentofrealpropertyCFDsolverrequirescontinuousdescriptionsofallthermal-physicalpropertiesofthegasofconsiderations.Temperature-andpressure-dependentcorrelationsarederivedforhydrogen,heliumandcarbondioxidefordensity,heatcapacity,viscosityandthermalconductivity.Allthemodeledvaluesfallwithin5%ofthedatafromourliteraturesources.Themajorremarkafterthisstudyisthatallthreegasesbehavelikeidealgases. 59

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Levelofhydrogenmoleculedissociationasafunctionoftemperature 60

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Hydrogenheatcapacityvs.temperatureatlowandhighpressure(0.1and150bars) Table4-1: ErroronthedensityformulationforP=0.1bar P(bar)T(K)M(g/mol)-IGL-compilederror(%) 0.115002.0161:621031:601031.30.117002.0161:431031:391032.30.119002.0131:271031:251032.20.120002.0111:211031:201030.80.122001.9981:0921031:101030.70.124001.9669:851041:001031.50.128001.7867:671048:001044.30.130001.6226:501047:001047.60.132001.4335:391043:9110427.40.136001.1463:831043:3010413.7 61

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Hydrogendensityat0.1barandhightemperature.Dierencesbetweentheliteraturedata,Given[ 78 ]andVergaftig[ 11 ]andthedatacomputedwiththeidealgas-law Table4-2: ErroronthedensityformulationforP=1bar P(bar)T(K)M(g/mol)-IGL-compilederror(%) 117002.0161:431021:431020.4119002.0151:281021:281020.0120002.0141:211021:211020.1122002.0101:101021:101020.1124002.0001:001021:001020.2130001.8697:491037:501030.1132001.7696:651036:841032.9136001.4985:001035:071031.4 62

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Hydrogenheatcapacityvs.temperature:Comparisonbetweenexperimentaldata(solidlines)andthemodel(dots). Table4-3: ErroronthedensityformulationforP=10bar P(bar)T(K)M(g/mol)-IGL-compilederror(%) 1019002.0161:281011:281010.11020002.0151:211011:211010.11022002.0141:101011:101010.11024002.0111:011011:011010.11028001.9908:551028:541020.11030001.9667:881027:881020.01032001.9307:251027:251020.01036001.7755:931026:031021.6 63

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Hydrogenviscosityvs.temperature:Comparisonbetweenexperimentaldata(dots)andthemodel(solidline). Table4-4: ErroronthedensityformulationforP=100bar P(bar)T(K)M(g/mol)-IGL-compilederror(%) 10020002.0161.211.201.010022002.0151.101.091.010024002.0141.011.000.810028002.0088:631018:561010.810030002.0008:021017:961010.710032001.9887:471017:461010.110036001.9446:491016:491010.0 64

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Heliumviscosityvs.pressurefordierenttemperatures:Comparisonbetweenexperimentaldata(dots)andthemodel(solidline). Table4-5: ErroronthedensityformulationforP=150bar P(bar)T(K)M(g/mol)-IGL-compilederror(%) 15022002.0161.651.631.515024002.0141.511.501.315028002.0101.301.281.215030002.0031.201.191.015032001.9931.121.120.115036001.9570.980.980.1 65

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Heliumthermalconductivityvs.pressurefordierenttemperatures:Comparisonbetweenexperimentaldata(dots)andthemodel(solidline). Table4-6: VHTRoperatingconditionsandfeatures ConditionorfeatureVHTR poweroutput(MWt)600-900averagepowerdensity(W/cm3)4-6.5coolantandpressurehelium@7.12Mpamoderatorgraphitecoregeometryannularsafetydesignphilosophypassiveplantdesignlife60yearscoreoutlettemp.1,000oCcoreinlettemp.490oCfuel-coatedparticlesa)LEU-PyC/SiCb)LEU-PyC/ZrCfuelmaxtemp.a)1,250oCnormaloperationb)1,400oC 66

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Carbondioxidedensityvs.pressurefordierenttemperatures.ThesolidlinesrepresenttheNISTdataandthedotstheideal-gaslaw. 67

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Carbondioxideheatcapacityvs.temperaturefordierentpressures:Comparisonbetweenexperimentaldata(dots)andthemodel(solidline). 68

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Carbondioxideviscosityvs.temperaturefordierentpressures:Comparisonbetweenexperimentaldata(dots)andthemodel(solidline). 69

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Carbondioxidethermalconductivityvs.pressurefordierenttemperatures:Comparisonbetweenexperimentaldata(dots)andthemodel(solidline). 70

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Comparisonbetweenexperimentaldataandideal-gaslawforheliumdensity P(bar)T(oC)(exp)(IGL)error(%) 12000.101720.101760.0418000.0448550.044860.02102001.014631.017570.29108000.448140.448650.11402004.0254.070281.13408001.78711.794580.42804005.63875.721921.48808003.55973.589160.838010003.00533.025340.678014002.29142.302070.471002009.899610.175712.791008004.44064.486451.0310010003.75063.781670.8314020013.71314.245993.891408006.19196.281031.4414010005.23395.294341.15 71

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5.1.1RequirementsThetwosystemswestudywithourCFDsolveraretheVHTRcoolantchannelsandtheNTPthrustchamber.Bothsystemsexhibitanaxialsymmetryandallowatwo-dimensionalaxisymmetricrepresentation,assketchedFigure 5-1 .Thegeometryofthesystemarerelativelysimple:thegridsarecomposedofquadrilateralcells.Bothowsystemrequireareal-propertyanalysisandarecompressible.Weareinterestedinthesteady-statesolutionofsuchcompressibleandturbulentows.First,wepresenttheassumptionsandderivetheReynoldsAveragedNavier-Stokesequations.ThenwepresentthetimeandspacediscretizationtechniquesbasedonaFiniteVolumemethodandaMacCormackimplicittechnique.Wealsogiveadescriptionofthemodelswhichaccountforturbulence:k-andSpalart-Allmaras.Thetwosystemsrequiresimilarboundaryconditiontreatment:inletvelocityandtemperature,exitpressure,heatedwallsandsymmetryaxis. 72

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80 ]: @t+div(!V)=0@!V @t+div(!V!V+pI)=div()@E @t+div(H!V)=div(!V)div(s):(5{1)where,tisthetime,thedensity,!Vthevelocityvector,pthepressure,Itheunitymatrix,theshearstresstensor,Ethevolumetricenergy,Hthevolumetricenthalpyandstheheatux.Thetotalvolumetricenergyislinkedtotheenthalpyby: Prwhere,cPistheheatcapacityatconstantpressure,isthemolecularviscosityandPristhePrandltnumber.Thegasesofconsiderationsarehydrogen,heliumandcarbondioxide:allthreeareNewtonianandtheshearstresstensorincompacttensornotationbecomes[ 38 ]: 3ij@uk 73

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81 ]: @t+@uz @r+@rz @z+@zz @t+@(E+p)uz @z[cP Pr@T @z+uzzz+urzr+uz]+1 @r[r(cP Pr@T @r+uzrz+urrr+ur)] (5{5) andthetensorofshearstressisexpressedby: 3r!V@uz 3r!V@u 3r!V377775(5{6)divergenceofvelocityvectorbecomes: @t+rF=rFv+S(5{8) 74

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@r+urrr+uzrz+ur377777775;Fvz=2666666640zrzzcPtot @z+urzr+uzzz+uz377777775;S=26666666400 5{1 iscomposedoffourequations.Thereareveunknowns:,ur,uz,Eandp.Thereforeanequationofstateisrequiredtorelatethedensitytothethermodynamicvariables.Inthepreviouschapterwedeterminedthattheidealgaslawdescribesproperlydensityvariationswithtemperatureandpressureforallthreegasesofconsideration.Theequationofstateforoursolverwillbe: 1+1 2k!Vk2(5{10) 75

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5{1 onlyholdsforlaminarows.InReynolds-averagedapproachestoturbulence,alltheunsteadinessisaveragedout.Inastatisticallysteadyow,everyvariable(xi;t)canbewrittenasthesumofatimeaveragedvalue(meanux)anductuationaboutthatvalue(uctuatingux): 5{11 ,itfollowsthat 76

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81 ]: @t+@huri p @r+@hrri u00ru00r u00ru00z p @z+@hrzi u00ru00z u00zu00z u00ru00z@hEi @zhuzi(hzzi @z Pr+t @rrhuzi(hrzi @rr Pr+t 5{13 arethesystemofReynoldsAveragedNavier-Stokes(RANS)equations. 77

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2 1. TheBoussinesqhypothesisrelatestheReynoldsstressestothemeanvelocitygradients: 3k+t@ui (a) Zero-equation:Inazero-equationmodelsoralgebraicmodelsnotransportequationsaresolved.Themodelsarecalculateddirectlyfromowproperties.ThetwomostcommonalgebraicmodelsaretheBaldwin-Lomax[ 63 ]andCebeci-Smithmodels[ 82 ]whicharebasedon: 78

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(b) One-equation:One-equationmodelssolveoneturbulenttransportequation,usuallytheturbulentkineticenergyk.Historically,therstmodelwasthePrandtl'sone-equationmodel[ 83 ].In1945,Prandtlintroducedthenotionthatdierentialequationsmustbeintroducedtoovercomethelimitationsofalgebraicmodels[ 84 ].Theequationsshouldhavebothconvective(rst-order)anddiusive(second-order)termstoexplainhowturbulencecreatedatonepointcouldhaveaneectsomewhereelse.However,Prandtlintroducedonlyonesuchequation:hismodelthereforehadthedisadvantagethatthelengthscalestillhadtobeguessed.Thedierentialtransportequationtosolveforis:@k @t+Uj@k @xj=ij@Ui @xj+t @xjwhereCDandkareconstantsandt=k1=2l.Anotherone-equation,whichuseisgrowingparticularlyintheareaofsupersonicwall-boundedows,istheSpalart-Allmarasmodel[ 69 ].Detaileddescriptionofthismodelisgiveninthefollowingsection. (c) Two-equation:Two-equationmodelsaretheRANSmodelswhicharemostcommonlyusedinturbulencemodelingforCFDtechniques.Modelslikethek-model[ 85 ]andthek-![ 86 ]modelhavebecomeindustrystandardmodelsandarecommonlyusedformosttypesofengineeringproblems.Twoequationturbulencemodelsarealsoverymuchstillanactiveareaofresearchandnewrenedtwo-equationmodelsarestillbeingdeveloped. 2. TheReynoldsStressModels(RSM)attemptstoactuallysolvetransportequationsfortheReynoldsstresses.ThismeansintroductionofseveraltransportequationsforalltheReynoldsstressesandhencethisapproachismuchmorecostlyinCPUeort.ForthecaseswestudyinthescopeofthisdoctoralresearchRSMmodelsdonotbringalotmoreaccuracy,butlengthensignicantlythecomputationaltime. 70 ].Thek-isoneattempttosolveforthetransportequations,andusestheturbulentkineticenergyfortheturbulentvelocityscaleequationandtheturbulentenergydissipationrateforthelength 79

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65 ]: @t+div(!V)=divt k2t~Ei;j~Ei;jC22 65 ].~Ei;j~Ei;j1 2@huii 5{18 5{19 and 5{20 containveadjustableconstants.Wewillconsiderthestandardk-modelforwhich:8>>>>>>>>>><>>>>>>>>>>:C=0:09k=1:00=1:30C1=1:44C2=1:92 69 ]:itisaone-equationmodelbasedprimarilyonempiricismandondimensionalanalysisarguments[ 70 ].Firstpublicationofthemodeldatesbackto1992.Althoughitisaquiterecentmodel,ithasbeentheobjectofnumerousvalidationsandshowsgreateciencyinpredictionofwall-boundedowswithareducedcomputationaltimeexpense.Forexample,Paciorri[ 87 ]validatedthe 80

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88 ]alsoprovethequalityoftheowpredictionsontheatplatebenchmarkexercisewhentheyusetheSpalart-Allmarasmodel,aslongasthepositivityoftheturbulentviscosityispreserved.Thismodelislocal,i.e.theequationaonepointdoesnotdependonthesolutionatotherpoints:themodeliseasilyusableonalltypesofgrids.Thereforethismodeliseasytoimplementandisfast.IntheSpalart-Allmarasmodel,theeddyorapparentturbulentviscositytiscomputedthroughapartialdierentialequation.Theintermediatevalue~followstherelation:t=t andfv1isadampingfunction:fv1=3 Dt=cb1[1ft2]~S~+1 d2+ft1U2 Thefourtermsoftherighthandoftheequationaretheproductionterm,thediusionterm,thedestructiontermandthesourceterm,respectively.Fortheproductionterm,thevorticitymagnitudeSisreplacedby~Sdenedas:~SS+ 2d2f2wherethefunctionf2isf2=1

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wt)Asastartingpointthedefaultnumbersgivenin[ 69 ]areused: (5{23) 5{8 :wherewenowhave 3ij@uk 82

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89 ].Discretizationmethodsareusedtoconvertthesetofgoverningpartialdierentialequations 5{8 andturbulenttransportequationsintoalgebraicequationsateachgridnode.Itistheheartofnumericalmethodsandparticularattentionisfocusesondiscretizationmethodsresearch:TheequivalentalgebraicrelationsshouldfaithfullyrepresenttheoriginalPDEs.Eachdierentialtermistransformedintoanapproximatealgebraicrelationparticularlyadaptedtothenatureoftheterm.FiniteDierenceMethod(FDM),FiniteVolumeMethod(FVM)andFiniteElementsMethod(FEM)arethemostusedanddevelopedmethodsinthesolutionofpartialdierentialequationsfornumericalmethodsingeneralandforCFDinparticular.InaFDM,thederivativesareapproximatedattheselectedgridpointsusingTaylorseriesexpansion.Moredetailsaregivenin[ 64 ].FVMisbasedonthecontrolvolumeformulationofanalyticaluiddynamics.Thedomainisdividedintosmallcontrolvolumesandthevariableislocatedatthecentroidofthecontrolvolumes.Approximationsaremadeonthesurfaceandvolumeintegrals.TheFEMbelongstotheclassofweightedresidualmethods:Thetransportvariableisapproximatedbyapolynomialfunction 83

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5-2 .ThederivationofthealgebraicgoverningequationsforeachcomputationalcellrequirestheuseofGauss'theoremordivergencetheorem: 5-3 showstheinnitesimalvolumeexpressedincylindricalcoordinates.WegetdV=rdrddzandd!S=rddz!er+drdz!e+rdrd!ezInthecasewhereweapplytheFVMtothedivergenceofFonthecontrolvolume/cell(i;j).UsingthenotationsfromFigure 5-4 weget: 2;j)rdz+Fz(i+1 2;j)rdr+R2Fr(i;j+1 2)rdz+Fz(i;j+1 2)rdrR3Fr(i1 2;j)rdz+Fz(i1 2;j)rdrR4Fr(i;j1 2)rdz+Fz(i;j1 2)rdr(5{27) 84

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5{8 .Thedierenttermsare-independentandthusRd=2andwedivideallthetermsofthegoverningequationsby2.WederivethesystemofalgebraicequationsfromthemodelbyLyra[ 90 ]andGuardone[ 91 ].WewriteVoli;j=RRRdrdzthevolumeofthecontrolvolumeorcellofconsideration,whereRisthedistanceofthecellcentroidtotheaxis.Here,wewilconsideranormalizedgrid,i.e.r=0onedges1and3andz=0onedges2and4.Foreachcontrolvolume(i;j)ofthegridwegetthealgebraicequation: @ti;jVoli;j+R1Fr(+)ri+1 2;jdzR3Fr()ri1 2;jdz+R2Fz(+)rdrR4Fz()rdr=R1Fvr(+)ri+1 2;jdzR3Fvr()ri1 2;jdz+R2Fvz(+)rdrR4Fvz()rdr+Si;jVoli;j(5{28) 92 ]thatweadapttotheaxisymmetricgeometry.Wesolvetheequationsusingabody-ttedstructuredmeshwithquadrilateralcells.Wedenote~Si+1=2;jthenormalvectortotheinterfacebetweenthecells(i;j)and(i+1;j)withthelengthofthevectorequaltotheareaoftheinterfacetimestheradiusasdenedabove.Itaccountsforaxisymmetriceects.In2-D,thelengthofthevectorisactuallyequaltothelengthoftheinterface.Wesupposethatthevectorisorientedfromcell(i;j)to(i+1;j)(Figure 5-5 ).Weoperatesimilarlyfor~Si1=2;j,~Si;j+1=2and~Si;j1=2.Wewrite:8>>>>>>><>>>>>>>:!Si+1=2;j=Szi+1=2;j!ez+Sri+1=2;j!er!Si;j+1=2=Szi;j+1=2!ez+Sri;j+1=2!er!Si1=2;j=Szi1=2;j!ez+Sri1=2;j!er!Si;j1=2=Szi;j1=2!ez+Sri;j1=2!er

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2i;j=Uni;jt Voli;jhSri+1 2;jFni;j+Szi+1 2;jGni;j+Sri;j+1 2Fni;j+Szi;j+1 2Gni;jSri1 2;jFni1;jSzi1 2;jGni1;jSri;j1 2Fni;j1Szi;j1 2Gni;j1i+tViscUni;j+S(5{29) 2Uni;j+Un+1=2i;jt Voli;jhSri+1 2;jFn+1 2i+1;j+Szi+1 2;jGn+1 2i+1;j+Sri;j+1 2Fn+1 2i;j+1+Szi;j+1 2Gn+1 2i;j+1Sri1 2;jFn+1 2i;jSzi1 2;jGn+1 2i;jSri;j1 2Fn+1 2i;jSzi;j1 2Gn+1 2i;jii+t 2i;j)+Si(5{30) 2;jFvi+1 2;j+Szi+1 2;jGvi+1 2;j+Sri;j+1 2Fvi;j+1 2+Szi;j+1 2Gvi;j+1 2Sri1 2;jFvi1 2;jSzi1 2;jGvi1 2;jSri;j1 2Fvi;j1 2Szi;j1 2Gvi;j1 2i(5{31)FortheevaluationoftheviscousuxesFvandGvattheinterfacesweneedtoevaluatethevelocityvectorattheinterfaceaswellasthepressure,velocityandtemperaturederivatives.Weuseacentralcellschemetoachieveboth. 86

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t=CFLmin0BB@1 87

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R(5{35)whereffisthefrictionfactordenedby @n=@T @z=0 88

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Temperature 2. HeatFluxBasedonFourier'slaw @rgas(5{38)wherekwistheconductivityofthegasestimatedatthewall.ThenthegradientofEquation 5{38 isrewritten 93 ].Itisbasedontheobservationthatundersteady-statethegasenthalpyincreasealongthewallsurfaceisequaltotheheatremovedfromthewall: (Tb)i+1=(Tb)i+P(z)iq00i 5{39 .Then,assoonashecondition=1isveried,thesolutionisconverged. 89

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@t+@F @x=0thetotalvariationquantityofavariablesuchasUisdenedby:TV(U)=Zj@U @xjdxwhereinadiscreteform,thetotalvariationofnumericalsolutionmaybeexpressedby:TV(Un)=+1Xjuni+1unijAnumericalschemeissaidtobeTotalVariationDiminishing(TVD)intimeifTV(un+1)TV(Un)Ifaschemeismonotoneattime'n'ithastoremainmonotoneattime'n+1'.Whennumericaloscillationsaregeneratedtheglobalrateofvariationofthefunctionoverthedomainincreases,andsoisthetotalvariationquantity.TVDstatesthatthetotalvariationquantitycannotgrowintime.ForaMUSCLsecond-orderschemetosatisfytheTVDconditiontheuxestermsinthedierentialequationsaremultipliedbyanon-linearfunction.ThefunctionistobechosensuchthatthePDEsatisestheTVDcondition. 90

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5{43 5{46 ).Notethatallthelimitersreturn0ifthegradientratioisnegative.Thecasewherethegradientswitchesdirectionindicatesthatthereisashockwithinthecellrangeandthelocalreconstructionisreducedforrstorderinthisinstance.Themostcommonlimitersare[ 38 ]: m(r)=8><>:min(r;1)ifr>00ifr0(5{43) s(r)=max[0;min(2r;1);min(r;2)](5{44) va(r)=r+r2 vl(r)=2r 91

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94 ]isalsoproposedwhichacceleratesconsequentlyconvergenceofthesolutionattheexpenseofaccuracy. 46 ].WederivetheTVD-termindirection1,inthecaseofatwo-dimensionalproblem.ThegiverningequationinitscompactformwasderivedEquation 5{8 .WebuildamatrixAsuchthatAi+1 2;j=Sri+1 2;j@F @Ui+1 2;j+Szi+1 2;j@G @Ui+1 2;jAhasmrealeigenvaluesandmlinearlyindependentrightvectors.TheeigenvaluesofAi+1 2;jare: 2;j=Sri+1 2;j(ur)i+1 2;j+Szi+1 2;j(uz)i+1 2;jai+1 2;jj!Si+1 2;jja(2)i+1 2;j=Sri+1 2;j(ur)i+1 2;j+Szi+1 2;j(uz)i+1 2;ja(3)i+1 2;j=Sri+1 2;j(ur)i+1 2;j+Szi+1 2;j(uz)i+1 2;ja(4)i+1 2;j=Sri+1 2;j(ur)i+1 2;j+Szi+1 2;j(uz)i+1 2;j+ai+1 2;jj!Si+1 2;jj(5{47)WecandecomposethematrixAinto 2;j(Ui;jUi1;j)(l) 2;j(Ui+1;jUi;j)(l)

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2;j(Ui+1;jUi;j)(l) 2;j(Ui;jUi1;j)(l)ThevectorsofconservativevariablesUaretakenattheoldtimestepn.ToevaluatethevaluesofAandRattheinterfaces,wecanuseRoe'saveragingtechnique 2=p 2ja(l)jt 5{47 .isthelimiterfunction.Wetypicallychoosethemin-modlimiterdenedinEquation 5{43 .Finally, 2;jh~G+(r+i;j)+~G(r+i+1;j)iR1i+1 2;j(Ui+1;jUi;j)Ri1 2;jh~G+(r+i1;j+~G(r+i;j)iR1i1 2;j(Ui;jUi1;j)(5{50) Uni+1;jUni;j;Uni+1;jUni;j(5{52) Uni;jUni1;j;Uni;jUni1;j(5{53) G1(r1i;j)=1 2C(i;j)1(r1i;j)(5{54) 93

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(z)=8><>:jzjforjzj>ez2+2e 5-6 presentsaschematicsofthealgorithmthatweusedtoimplementtherealpropertyvalues.Attimen,wehavethevaluesofUwhichwerederivedattheprevioustimestep,aswellasthevaluesofpandTandalltheproperties(k,cpand).WethenderivethediusionuxesFandGandtheviscousuxesFvandGvattimenwhichistheinitialtimefortheiterationstep.UsingtheTVD-MacCormackschemewegetthevalueofthevectorofconservativevariablesattheintermediatetimestep,classicallynotedn+1 2:Un+1 2.Wethenderivethevalueofthepredictedpressureusingtheideal-gasrelation 5{10 .However,wenotethatintherelationpisafunctionoftotalenergy,velocityvectorand 94

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2andtherefore,thevalueofpressureobtainedisalsoanestimation.Thesamewaywederivethetemperatureatthepredictorstepusingtheidealgaslaw 5{9 .Fromthesepredictedvaluesofpressureandtemperature,werunthepropertysubroutinewherethecorrelationsofallpropertiesareexpressedasafunctionoftemperatureandpressureandarecontinuousovertherangeofoperations.Withthenewvalueofthespecicheatratio()weobtain,werecalculateanestimationofthepressureandtemperature.Iftheupdatevalueofbothpressureandtemperatureisthesameasthepreviousprediction,thenwecanconsiderthatthelooponthepropertiesisover.Otherwise,thepropertiesarere-calculatedandthelooponpropertiesiterateduntilthevaluesofpressureandtemperatureareconvergedinallthegridcalculationnodes.Oncethelooponpropertiesisconverged,theuxesareupdatedwiththeallthepredictedvaluesinthepredictorstep.Wenowarereadytoperformthecorrectorstep,andderivethevaluesofUattimen+1,usingthevaluesofthevectorofconservativevariablesanduxesattimen+1 2.Then,theuxlimiterisdetermined.Wecalculateoneuxlimiterpermodelingdimension,therefore,forourtwo-dimensionalaxisymmetricmodel,wewillhavetoderivetwouxlimitersoneintheaxialandoneintheradialdimension.Fromthevaluesofthecorrectorandtheuxlimiters,wederivethevalueofthevectorofconservativevariablesattimen+1withequation 5{32 .Togetthevaluesofpressureandtemperatureaswellasuidpropertiesattimen+1weneedtoiterateagainontheproperties,inthesamefashionasforthepredictorstep.Oncetheconvergenceofthepropertyloopisobtained,thecompletesolutionandowmapattimen+1iscalculated. 95

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5.3.1PresentationoftheTestCaseTheshocktubeorRiemannproblemconstitutesaparticularlyinterestingtestcasesinceitpresentsanexactsolutiontotheone-dimensionalEulerequationandinvolvessimultaneouslyashockwave,acontactdiscontinuityandanexpansionfan.TheRiemannproblemcanberealizedexperimentallybythesuddenbreakdownofadiaphragminalongone-dimensionaltube.Beforetimeofbreakdown(t=0)thediaphragmseparatestwogasstatesattwodierentpressuresanddensities.Weconsiderairatroomtemperatureinthetube.Thegoverningequationsforthistestcasearetheone-dimensionaltime-dependentEulerequationswithanidealEquationofState.Theirconservativeformis:@U 5-7 .Theanalyticalsolutionisinterferedformthetheoryofthecompressibleuidows.Hirsch[ 38 ]givesthetheoreticalprolesfordierentinitialpressureratiosandatdierenttimes.WealsoderiveaFORTRANcodewhichcomputesthetheoreticalanswer.ItcanbefoundAppendixA. 96

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2i=Unit 2Uni+Un+1 2it 2i+1Fn+1 2i(5{58)InthecaseofthefullTVD-MacCormackweneedtoderivethematrixA=@F @U,itseigenvaluesandrighteigenvectors: Ai=2666640101 2u2i(3)(3)ui1u3i(1)EiuiEi3 2(1)u2iui377775(5{59) 97

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5-8 .WeseethatthegasbehaviortrendlineisperfectlyfollowedwhenusingwithourTVDMacCormacksolverwithmodiedCausonscheme.However,thelimiterlacksalittlebitofaccuracysinceafewnumericaloscillationsaregeneratedintheviscidityofthepropertydiscontinuities.Itwouldbeaprobleminapplicationswhereshocksaregenerated.However,inthesystemwestudy,noshocksshouldbegenerated.WeneedasolverwhichcansimulateaccuratelyowastheybecomeclosetoM=1butweshouldneverhaveformationandpropagationofdiscontinuitiesinthegas.ThiscaseontheverytraditionalRiemannShocktubebenchmarkvalidatesthenumericaltechniquewechoseforapplicationstoadvancednuclearsystemssuchasthetwosystemspresentedChapter6and7. 5{62 areshowngure 5-9 .Thegraphshowstheimportanceofthevalueoftheheatratioontheevolutionofthepropertiesintheshocktube.Shockfrontsandexpansionwavespropagatefasterforgaseswithhigherheatratios.Theyleadtosmalleramplitudeoftheshockelds, 98

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99

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Sketchoftheaxisymmetricrepresentationofthe3-Dchannel 100

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Locationofthenumericalnodesfora5x5domain 101

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Elementaryvolume(bluearea)forintegrationintheaxisymmetricconguration 102

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Detailsofthenotationsoftheaxisymmetriccell(i;j) 103

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Nomenclatureforthesurfacevectors 104

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Algorithmwhichshowshowtherealpropertiesareimplementedandcomputed 105

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Theshocktubeexperiment:Above,attimet=0adiaphragmseparatesthegasesintwodierentstates.Below,whenthediaphragmisremoved,ashockwavetravelsdowntothelowpressuresectionofthetube.Simultaneously,anexpansionfantravelsintheoppositedirection.Betweenthegaseswhichpropagateinoppositedirections,acontactsurfacemovesrapidlyalongthetubebehindtheshockfront. Figure5-8: Shocktubeproleatt=6.1msincase 5{62 .ThebluecurvesarethetheoreticalprolesandthegreencurvestheprolescomputedwithourEulersolver 106

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Propertiesproleatt=6.1msincase 5{62 forair,helium,carbondioxideanduraniumtetrauoride. 107

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6-1 showsaschematicrepresentationofthesystem.ThegeometricalparametersweredeterminedatfromArgonneNationalLaboratory[ 95 ].Thediameter 108

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109

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5-1 isarectangle.Therefore,weuseaquadrilateralorthogonaluniformgridtosolvefortheNavier-Stokesequations.Thegrid(100x1000)isgivenFigure 6-3 .Thegrid(100x1000)correspondsto100intervalsintheradialdirectionand1,000intervalsintheaxialdirection.Theredboundarycorrespondstotheinletplane,theblueonetotheexitplane,theyellowedgetothesolidwallandthegreenonetothesymmetryaxis. 2,Sri;j1 2,Szi+1 2;jandSzi1 2;jareequaltozerowhichsimpliesgreatlythepredictor,correctorandviscousterms(Equations 5{29 to 5{31 ).Computationoftheviscoustermsrequirethederivationofvelocityandvelocitygradientsatthecellsinterfacesi+1 2;j(ori;j+1 2).WerewriteEquation 5{31 fortheuniform,orthogonalmeshas: 2;jFvi+1 2;j+Szi;j+1 2Gvi;j+1 2Sri1 2;jFvi1 2;jSzi;j1 2Gvi;j1 2i(6{2)Thevelocityisevaluatedusing: 2;j)=ur(i;j)+ur(i+1;j) 2(6{3) 2)=uz(i;j)+uz(i;j+1) 2(6{4)FromthedenitionofFvandGvweneedtoderivethevaluesofrrandrzontheinterfacesinthei-directionandthevaluesofzzandzrontheinterfacesinthe 110

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2;j)=tot4 3@ur 3@uz 2;j(6{5) 2;j)=tot@uz 2;j(6{6) 2)=tot@uz 2(6{7) 2)=tot4 3@uz 3@ur 2(6{8)Therefore,forthesourceterm,weneedanevaluationoftheshearstresstensorcomponentatthecenterofthecell. 3ur 3@ur 3@uz 6-2 ,withtheverticeslocatedatthecentersofgravityofthecellsandattheendpointoftheinterfaces.Weuse 2;j+1 2)=1 4ur(i;j)+ur(i+1;j)+ur(i;j+1)+ur(i+1;j+1)(6{10)whichreducesatthesolidwallstour(i+1 2;j+1 2)=0.UsingthenomenclatureofFigure 6-2 weget: 2;j=1 2;j1 2) 2;j1 2)+ur(i+1;j 2;j+1 2) 2;j+1 2)+ur(i;j) 111

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2zSzi;j+1=2=RirSri1=2;j=Ri1 2zSzi;j1=2=Rir 6.1.3.1ValidationofthemodelTovalidateourmodelonthecoolantchannelspecicgeometry,weperformacode-to-codebenchmark:ThesystemismodeledbothwithFLUENTandourcode.SincewedonotmodeltemperatureandpressurepropertydependencewithFLUENT,thecomparisonwillbedonewithconstantproperties,exceptfordensitywhichismodeledwiththeideal-gas-law.Tolimitthecomputationtime,thebenchmarkisdoneona10cmlongsection.SpecicationofthenumericalmethodisgiveninAppendixB.ItisthesameforalltheFLUENTstudieswhicharepresentedthroughthedocument.Theboundariesforthismodelare: 6{1 andk=0:304W/K.mandcp=5:1888J/g.K 112

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96 ].Weconsiderapurelyaxialvelocity:uz=2:32m/s,andur=0:0m/s.ForFLUENTcompressiblesolver,weneedtoinputtheinletmassowrate,inotherwords_m=0:00231kg/s.Theinlettemperatureforthisparticularanalysisisequalto400oC(i.e.773K). 6-4 to 6-6 .EachgureshowstheprolesobtainedwithFLUENTandourin-housesolver.Thepressuredropacrossa10-cmsectionofthecoolantchannelisequalto10.24Pa.Oursolver,whenimplementedwithconstantproperties,givesresultsofthesameorderofmagnitudeasthetest-caserunwithFLUENTandthereforeshowsthegoodperformanceofournumericalmethod.Thesametest-sectioninnowrunwithourin-housesolver,accountingfortheheliumpropertydependencewithpressureandtemperature.Figures 6-7 to 6-8 showtheprolesofpressure,temperatureandvelocityrespectively,ontheaxisofthetestsectionforbothfrozenchemistryandrealproperties.Thedierencesarenotdrasticbecauseheliumisthegaswhichbehavesclosesttotheidealgaslaw.However,thevariablepropertiesshouldnotbeneglectedduringahightemperaturegas-cooledreactorstudy:Onasectionassmallas10-cmlong,thepressuredropcalculatedwiththetwosolversdiersof4%.Sincetherealchannelis7.93-meterlong,wecanimaginethatthedierencewillgrowproportionally.Therefore,itisimportanttoincluderealpropertyformulationofheliumwhenanalyzingtheowinahightemperaturegascooledreactor. 113

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6-1 .Attheinlet,weimposeaconstantaxialvelocitywithcorrespondstoaMachnumberof0.1:uz=2:32m/s,aninlettemperatureof400oC.Thewalltemperatureislinearlydependentwiththeaxialpositionandvariesbetween400oCattheinletand1,000oCattheexit.Weimposetheexitpressure,PExit=7MPa=7:106Pa.Thetimestepisautomaticallyimplementedfollowingrelation 5{33 .Practically,thetimestepistypicallyoftheorderoft=4:108s.Thesteady-stateisobtainedafter12,500timeiterationswhichcorrespondsaboutto25minCPU.Figure 6-10 showstheprolesofabsolutepressure,temperatureofthewallandattheaxis,velocitymagnitudeanddensity.Thepressuredropacrossthehightemperaturegas-cooledreactorcoreisequalto5,500Pa.Theow,drivenbythehotwall,isacceleratedandtheexitvelocityis1.4timestheinletvelocity.Theexittemperatureontheaxisisabout1,000oCwhichshowsthattheintegralityoftheheatgeneratedatthewallistransmittedtotheuid. 114

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6.2.1NERVADerivativeHotChannelAnalysisOutofalltheNTPdesigns,weselecttheNERVAderivativedesignbecauseitwillbethenstudiedwiththecoupledsolver(Chapter7).Hydrogenowsinthechanneltocooldownthewallswhichmodelthefuel.Heatreleasedfromssionismodeledbyapplyingaheatuxtothewallsofthemodel.ThehotchannelofthesystemisanalyzedandtheheatuxproleisplottedFigure 6-11 .Theweightfunctionisthenmultipliedbytheaverageheatuxinthecore.ThedimensionsofthecoolantchannelsintheNERVAderivativesystemare:8><>:D=2:54mmL=1:27mThesystemisstudiedinthetypicalconditionsofoperation.NuRok,theone-dimensionalsimulatorforNTPhydrogenowisrunforthefollowinginput: 6-12 to 6-14 .Figure 6-12 showsthepressureproleontheaxisofthechannel.Thepressuredropacrossthechannelisequalto1.4MPa.Theprolesoftemperatureonthechannelaxisandalongthewallare 115

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6-13 .Theexittemperatureatthecenteroftheexhaustplane2,799K.Therefore,hydrogenthroughthecoreoftheNTPundertheconditionsofaverageux_q=10:4MW/m2seesatemperatureincreaseof2,497Koveralengthof1.32m.Finally,thedensityevolutionontheaxisofthehydrogenchannelisshownFigure 6-14 .Asthetemperatureincreasesandpressuredrops,thedensitydecreasesrapidly.Attheexitofthechannel,thedensityismorethanoneorderofmagnitudesmallerthanattheinlet.Steady-stateisreachedafter17,000timesteps. 6-15 forthediverseinlettemperaturesinvestigated.Asdissociationeectsstartprevailing,convergenceofthepropertyloopcannotbereachedanymore.ThisisdemonstratedinAppendixD.Becausetheheatcapacitygradientvstemperatureishigh,thesmallestchangeintemperatureleadstoalargevariationintheheatcapacity,thespecicheatratioandthereforethepressure.The 116

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117

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VHTRcoolantchannel 118

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Dualcellfortheapproximationofthederivativesatthecellinterfacei+1 2;j Heliumchannelmesh:100x1000quadrilateralgridzoomedattheinletofthechannel.Theyellowedgerepresentstheaxis,theblueonetheinletandtheblackonetheheatedwall. 119

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Staticpressure(Pa)proleontheaxisofthetest-section.ComparisonbetweenFLUENTsimulationsandoursolver. 120

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Velocity(m/s)proleontheaxisofthetest-section.ComparisonbetweenFLUENTsimulationsandoursolver. 121

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Temperature(K)proleontheaxisofthetest-section.ComparisonbetweenFLUENTsimulationsandoursolver. 122

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Staticpressure(Pa)proleontheaxisofthetest-section.Comparisonbetweenconstantandrealpropertyformulations. 123

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Velocity(m/s)proleontheaxisofthetest-section.Comparisonbetweenconstantandrealpropertyformulations. 124

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Temperature(K)proleontheaxisofthetest-section.Comparisonbetweenconstantandrealpropertyformulations. 125

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Propertyprolesacrosstheheliumcoolingchannel. 126

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Heatproleonthechannelwalls 127

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Absolutepressure(MPa)proleontheaxisoftheNTPcoolantchannel 128

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Temperature(K)prolesontheaxisandthewalloftheNTPcoolantchannel 129

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Density(kg/m3)proleontheaxisoftheNTPcoolantchannel 130

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Molecularweight(g/mol)proleontheNTPaxischannelforTi=1;500K(nodissociation)andTi=1;700K(veryfewdissociation) 131

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5 ]-[ 78 ]andPlancher[ 6 ]broughtadditionsin2002.Sincethenineties,therehasbeenarenewedinterestinthermalnuclearpropulsion.ItcameafteralongresearchhiatusaftertheNERVAprogramcancelationin1972.TheresearchinthisareathereforehasbeenfocusingonregainingtheknowledgegainedintheNERVAprogram.Itresultedinalargevarietyofcoreandenginesystemdesigns.MostofthemareavariationoftheNERVAdesign:prismaticfuelelementsandfulltoppingexpandercycle.Therefore,heNuRoksystemcodeaimsatsimulatingthedierentdesignsunderdevelopmentandtoassesshowthedierentcomponentsofthesystemareaectedbyeachotherandbytheoperatingconditions.TheNuRokprogramiscomposedofseveralFORTRANsubroutines,eachsimulatesacomponentoftheNTPsystem.Theprogramaccountsfortherealpropertiesofhydrogen.Typically,hydrogenistheworkinguid,coolantandpropellantatthesametime.ThecodemodelsthedesignsdescribedbelowandsketchedFigure 7-1 .Theschematicsdoesnotshowtheactualpipingnetworkbutshowstheconnectionsbetweenthepropulsioncomponentsandhowtheowisregulated. 132

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7-1 133

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p=1 2kcv22(7{1)foranareacontractionand p=1 2kc(v1v2)2(7{2)foranareaexpansion,wherekcisthecontractionlosscoecientdenedby: 134

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h=LHGRz p=_m21 D1 1 21 135

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7-1 .ThedenitionimplementedinNuRokisthecorrelationbyKoo.Finallythefuelsurfacetemperatureprolemustbecomputed:oncethesurfacetemperatureisknownitispossibletoderivethemaximumfueltemperaturebyconsideringconductiveheattransferinthefuel.Itisimportanttocomputethemaximumfueltemperaturesinceitlimitsthecoreoperatingpower:foragivenoperatingconditiononehastomakesurethatthemaximumfueltemperatureislowenoughcomparedtothefuelmeltingpoint.Tocomputethesurfacetemperatureatanyaxiallocationofthecoreweneedtheinformationonbulktemperatureandheattransfercoecient: D(7{8)Colburn,Dittus-Boelter,Seider-TateandmanyothersdevelopedcorrelationstondtheNusseltnumber.ThemostcommoncorrelationsaregiveninTable 7-2 .TherelationusedinNuRokis: 4+0:3(KaysandPerkins,1985)andtheisotropicNusseltnumberisderivedwiththePetukov(1970)relationandanaxialdistancecorrection.Thiscorrectionisneededsincethereareentryeectsduetothethinthermalboundarylayerattheinletofthechannel.AcorrectionfunctionbyPierceisimplemented,andlittlefurtherresearchwillbedoneontheaxialcorrectionfactorsinceforthechannelofconcernx=D750.ThePiercecorrectionis: D0:7(7{10) 136

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7{1 toathree-dimensionalCFDsimulationofachannelwithsuddenarealoss.ThemodelrepresentsacirculartankwhichareareducestoasquarechannelasseenFigure 7-2 .Theradiusifthetankis5mmandtheedgeofthesquarechannelisequalto1mm.Thesimulationisdoneusingthek-segregatedsolverofatetrahedralmeshwith118,042computationalcells.Thetestsectionis5cmlongandthewallsareconsideredadiabatic.TheoperatingconditionsaretypicalofthehydrogenconditionsattheinletoftheSLHCcore:1atmand186K.Figure 7-3 showsthestaticpressureevolutionalongtheaxisofthetest-section.Theinletofthetestsectionislocatedatz=2cmandtheareacontractionatz=0.Theleft-handportioncorrespondstothepressuredropoverthetank.Itisalmostnegligible.Themiddleportioncorrespondstotheabruptpressuredropduetotheowcontractionandtheright-handportiontothepressuredropoverthechannel.Forcomparisonwiththeanalyticalformula 7{1 ,weonlyconsidertheabruptpressuredrop.ResultsfordiverseinletvelocitiesisgivenTable 7-3 .Overall,thecorrelationunderestimatestheimportanceofthepressuredropattheareacontraction.Astheinletvelocityincreases,theerrorreduces,however,thedierenceisstillveryconsequent(272%).Thissimpleexampleillustrateshowthe1-Dcorrelationsmayfailtopredicttheowbehavioraccurately.ThisiswhythisChapterwillpresentfurtherdownthecouplingtechniqueusedtoincreaseNuRokaccuracy.Beforehand,letusreviewtheupgradeswebroughttothesystemcodeitself. 137

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7-4 presentsthedierentcharacteristicsoftheSLHCsystemsfordiversethrustlevels.Thethrustchamberinlettemperatureandpressureare2,800Kand500psi,respectively. 138

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7-5 presentsthesystemcharacteristicsfordiversefoamfueldensities.Thisisonlyanapproximationsinceveryfewresearchhasbeendoneonpressuredropandheattransfer.Thefoamfuelstillneedtobequaliedonamaterialsstandpointrst,beforeleadingfurtherresearch.However,thesimplestudypresentedhereshowsthattheonlypropertythevariationindensityinuencesisthepressuredropacrossthecore.Thecasesofcomparisonexhibitalevelofthrustequalto50,000lbsandnozzleinlettemperatureandpressureequal2,800Kand500psi,respectively.Thepowertransferredtotheuidisequalto1024MWforalldensitiesandthemaximumcladdingtemperatureinthecoreis2;8071K.Thepressuredropacrossthecoreincreasesasthefoamdensity(anduidowarea)decreases. 139

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7-4 weidentiedseveralportionsofthesystemwhichwouldbenetfromameshandsolverrenement.Theyaretheinletandexitplenum,mixingjunctions,coolantchannelsandthrustchamber.Chapter7detailedthereasonsfora2-Danalysisofthethrustchamber.InChapter6wesawthecaseofthecoolantchannelintheVHTR.SincetheNTPisalsoanuclearsystemwithgasow,theremarkswemadealsoapplytotheNTPcoolantchannels.Attheinletandexitofthecorethereisasuddenrestrictionandexpansionoftheowarea,respectively.Theparticulargeometryinducesturbulenteectsaswellasanisotropyoftheow.ThenodalcodeofNuRokcannotaccountcorrectlyforthesemulti-dimensionalphenomena.AdetailedCFDanalysiswouldberequiredtogetanestimationoftheimportanceoftheowanisotropyintheexitplenum,forexample.Besides,hydrogenisnotheateduniformlyinallthecorechannels.Becausethethermaluxgeneratedbyneutronssionisnotisentropicoverthecore,certainchannelswillbehotter:thegasenterstheexitplenumwithdierenttemperatures.Thetemperaturegapcouldbeoverdozensofdegrees.Typically,acompactnuclearcoreiscomposedof24,000coolantchannels.Representationofeven1/6thofthecorewouldrequireseveralmillionofcomputationalcells.Thisistwocomputationallyexpensiveforourfeasibilitystudyhere.However,futureworkcouldbeledinthisdirection.ThecoolantchannelwhichissolvedherewiththenemeshsolverwaspresentedinChapter6. 141

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7-5 showsthecouplingschematicsofanemeshsolverforthehotchannelofthecoretotherestoftheNTPsystemcode. B-2 showstheproleofvelocityattheveryinletofthesmallerpipeinthesystempresentedinSectionB.2(AppendixB).Theowbeforethecontractionisfullydeveloped.Thesecondcurveshowsthetheoreticalvelocityproleattheinletofthecoolantchanneliftheowwerefullydeveloped.ThisFigureshowsthattheactualproleofvelocitydoesnotcorrespondexactlytoafully-developedvelocityprole.Toobtainaveryrigoroussimulationoftheentiresystem,theactualowintheinletandexitplenumshouldalsobemodeledwithane-meshCFDcode.SincethisChapteraimsatshowingthefeasibilityofacoupledtechnique,wedonotconsiderthelevelofturbulencedevelopmentasanissueinthetreatmentoftheinterface.Onlyauniformvelocityproleisimplementedattheinletofthesystem.ForthecasesofthehydrogencoolantchannelpresentedinSection6.2.thedevelopinglengthwillbeequaltoLe'4:4Re1 6D'7:6cm.Thisdistanceonlyrepresents5.8%ofthetotallengthofatypicalNERVAderivativenuclearcore. 142

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7-6 ,onecomputationalnodeintheaxisymmetricmodelcorrespondstoanannulusareaontheexitplaneofthechannel.Therefore,togetanideaoftheaveragetemperatureatthe 143

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7-7 to 7-8 showtheevolutionofthedierentpropertycharacteristicsateachiterationstep.Theverticallinescorrespondstoanewsystem-loopiterations.Duringasystemlooptheinletpressureandinlettemperaturearecalculatedbythesystemcodeandkeptconstant.Ateachcoreiteration,theheatuxisupdatedsuchthattheexittemperaturematchesthetemperatureinputbytheuser.Whentheheatuxisderived,thepressuredropiscalculatedandtheexitpressureiscomparedtotheinputpressure.Thecoreloopisthenoverandthechannelinletpressureismodiedsuchthattheinletpressureisequaltotheinputpressureplusthepressuredropvalue.Thesystemloopisupdatedwiththenewvalueofthechannelinletpressure.Anewcoreloopthenbegins.Steady-stateisobtainedwhenthevalueofinletpressure,inlettemperatureandheatuxdonotevolveandtheexitpressureandtemperaturematchtheuserinput.Table 7-6 compilestheresultsofthesimulation.TheresultsarecomparedtothoseobtainedwithNuRokonly.Figure 7-9 and 7-10 showtheprolesofpressureandtemperatureacrossthehotchannelcalculatedwithNuRokandThemis.ThecorethermalpowerderivedwithThemisislargerby15%fromtheNuRoksimulation.Thisismainlyduetothelargedierenceinthepressuredropacrossthechannelthatiscomputedwiththetwomethods.Thepressureprolesaresignicantlydierent:notonlythepressureprodpredictionobtainedwiththecoupledsolveristhreetimeslargerthanthepredictionobtainedwithNuRok,butalsothetrendsofthetwocurvesseenFigure 7-9 arequitedierent.Itisdiculttocomparethetemperatureproles(Figure 7-10 )sincethecoarsemodelonlygivesoneproleandthenemodelgivesacoupleofproles:walltemperatureandtemperatureattheaxis.Thetemperatureprolesareverysimilarforthetwoapproaches.Thepressureproles(Figure 7-9 )areverydierent.ThepressuredropcalculatedwiththeCFDsolveristhreetimelargethanthepressuredropobtainedwithsimplerone-dimensionalcorrelations. 145

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B-4 showsthatthepressureproleobtainedwithourCFDsolverismostlikelymoreaccuratethantheonederivedbythe1-Dsolver.Thedierencebetweenthetwosimulationsisverysensible.Itshowstheimportanceofusingthenemeshsolver.WithNoRok,theoverallcodeconvergeswithin8iteration,whereaswithThemis,thecodeconvergesafter10iterations.Thecomputationaltimeismultipliedbythree.However,theoverallCPUstillremainsunder1mintoobtainsteady-state. 146

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Fanningfrictionfactorcorrelationsandtheirrangeofvalidity[ 2 ] InvestigatorCorrelationRangeofValidityReynoldsRange N.A.16 97 ]1:33103Re0:52Transitionalow2300
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ThedierentNTPsystemsmodeledwithNuRok:NVTR,NERVAderivativeandP&WXNR2000I&II 148

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3-DgridofthecontractiontestsectionforstudywithFLUENT. Table7-4: Propertiesatdiverselocationsofthesquarelatticehoneycombsystemforthrustchamberinlettemperatureandpressureof2,800Kand500psi(=3.45MPa),respectively. Thrust(lbf)25,00050,00075,000100,000 Coretransferredpower(MW)512102415352045Massowrate(kg/s)12.0124.0236.0248.03Turbo-pumpoutlettemperature(K)27.329.732.837.0Turbo-pumpoutletpressure(psi)801115116272248coreinlettemperature(K)186.4185.7184.5182.9coreinletpressure(psi)62286511481447turbineinlettemperature(K)192.8194.8197.3200.6turbineinletpressure(psi)728107415472163MaximumMachnumber0.130.250.360.46 Table7-5: PressuredropacrosstheSLHCcorefordierentdensitiesoffoamfuel. CoreAreaFlowFraction(%)172025304050P(Pa)1181.5872.2552.0365.0171.987.06 149

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Staticpressureprolealongtheaxisofthecontractiontestsection Table7-6: ComparisonoftheresultsobtainedwithNuRokandThemis NuRokThemisrelativedierence(%) Pressuredrop(Pa)3.56.1051.19.106234Elementpowerdensity(MW)2269.742626.3415.7Coreinlettemperature(K)302.03372.1 150

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IdenticationandlocationofportionsoftheNTPsystemwhichwouldbenetfromamoredetailedsolver 151

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StructureofthecodecouplingwhentheCFDisusedtosimulatethehotchannel 152

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Extrapolationofthetemperaturenodesfromtheaxisymmetricgeometrytoasinglevalue. 153

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PressureevolutioninMPaattheinletandexitofthehotchannelvs.theiterationnumber 154

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Thermalcharacteristicsofthehotchannelvs.theiterationnumber 155

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CoolantchannelpressureprolescalculatedwithNuRokandThemis 156

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CoolantchanneltemperatureprolescalculatedwithNuRokandThemis(ontheaxisandthewall) 157

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8-1 showsthemainthermalcharacteristicsofhydrogenattheinletofthethrustchamber.Ifthepressureofthegasisdecreasedby3ordersofmagnitude(0.1barforexample),hydrogenattheexitofthecoreisdissociated.ItspropertiesarealsocompiledinTable 8-1 andaschematicsofthesystemisshownFigure 8-1 ForthedissociatedgastheheatcapacityiseighttimesgreaterthantheoneofH2.Also,thespecicratioismuchsmaller.WesawFigure 5-9 thatlowerspecicheatratiosleadtogreatervelocities.Therefore,weexpectthevelocitiesonthethrustchamberexhaustplanetobegreaterforhydrogenatlowpressure.Moreover,asthegastravelsthroughthechamber,itstemperaturedecreases.Hydrogenattheinletofthethrustchamberishighlydissociated(upto30%)andrecombinesasitowsthroughthenozzle.Theenergybalanceistransferredundertheformofkineticenergy,whichhelpsincreasetheexhaustvelocityandthesystemISP.Tovalidateourassumption,werstuseaquasione-dimensionalapproachtoderivethebasicsgeometricalinformationonthenozzle(throatdimension)andthenozzlecharacteristics(exhaustvelocity).Then,wepresentthemethodforthenozzlecontourdesign.Onceallthegeometricalparametersarederived,wepresenttwostudies:therstoneisaFLUENTanalysisofthenozzlewithfrozenchemistry,thesecondanalysiswilluseoursolver. 158

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Hydrogenthermalproperties(heatcapacity,specicheatratioandthermalconductivity)fordiversesetsofoperatingconditions Pressure(bars)Temperature(K)cP(kJ/kg.K)k(W/m.K) 1203,00023.481.2501.6960.13,000179.51.13115.31 98 ]: A=1 2M2+1 2(1)(8{1)where,AistheareaofthenozzleatagivenpointandMtheMachnumberoftheowatthispoint.Aistheareaatthethroat,whereM=1.Therelationisaccurateforone-dimensionalsteadyowwherebodyforcesandgravitationaleectsarenegligible,forafrictionlessidealgaswithconstantspecicheatandinthecaseofanadiabaticowwithnoshearandshaftwork.Someoftheassumptionssuchasconstantspecicheatarenotveried.However,inthisrststudywelookforanorderofmagnitudeforthenozzleperformances.Therefore,wewillassumethatalltheconditionsoftheisentropicstudyareveried.TheinletMachnumberinthesystemisequalto0.3andtheinletradiusofthethrustchamberisequalto30cm.Hence,thethroatradiusis:r=20:77cmWenowcalculatetheMachnumberontheexitplaneforanexitradiusof1m.WesolveEquation 8{1 fortheratioA A=23:19.Weget:Mexit=3:569 159

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T=(1+)2M2 TM=0:3=0:336676andT TM=3:568=0:243820Then,Texit=T TM=3:568 TM=0:3Tinlet=2;172:65K.Weapproximatethemolecularweightofthehydrogenoftheowtothehydrogenmolecularweightat3,000Kand0.1bar:M=1:622g/mol.Finally,thespeedofsoundattheexitisaexit=p 160

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T=+1 2+(1)M2(8{3)Now,T TM=0:3=1:05926andT TM=3:568=0:581016Then,Texit=1645:53KWhenconsideringanadiabaticviscousowtheexhaustvelocitybecomes:vexit=11;020m/sDependingontheapproachwhenleadingthequasi-one-dimensionalwegetdierentexhaustvelocityvalues: 99 ]extendedtoreal-gasischosen,coupledtoaCFD-basedoptimizationprocedureelaboratedbyKorte[ 100 ].Theconvergingdivergingnozzleisdividedintoasubsonic-throatregion,atransonicregionandasupersonicregion.Thespecicheatratioattheinletofthenozzle 161

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101 ].Thenozzleradiusyatanaxialpositionzisgivenby:y=y0+y00(zz0)+c(zz0)2+d(zz0)3with 162

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8-2 ,thelengthofthemeshesisequalto10%ofthenozzleradiusatthatlocation.Wealsocreateaboundarylayerintheareaofthewalltoresolveproperlytheboundarylayer.ThegridweshowFigure 8-2 isverycoarseanddoesnotresolvetheboundarylayer.Thereforethenodesareequidistantintheradialdirection. 163

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8{5 ,regardlesstheirinuenceontheowmap.ThenozzleofinterestisthelowpressurethrustchamberattheexitoftheSLHCnuclearcore,i.e. 164

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8.5.3.1InvestigationofthenozzlerepresentationFLUENTallowsforthreetypeofrepresentationsofanyuidsystem:2D,2Daxisymmetricand3D.Athree-dimensionalrepresentationalwaysgivesthebestsimulations,howeveritiscomputationallyexpansive:3Dmodelsrequirealotofcomputationalcells.Insystemswhicharesymmetricwithrespecttoanaxis,suchasatube,orherethethrustchamber,a2Daxisymmetricrepresentationisoftensucient.Thisistrueprovidedtherearenothree-dimensionaleectsoccurringintheow.Thetwo-dimensionalrepresentationofasystemisthesimplestone,butoneneedtobecarefulwhenusingthisrepresentation:inaxisymmetricalgeometries,atwo-dimensionalrepresentationneglectssomeeectsduetosymmetryofrotation.WeinvestigatetheaccuracyofeachoftherepresentationsoeredbyFLUENT.Foragivensetofnozzledesignparameterwecreateonetwo-dimensionalmodel,oneaxisymmetricmodelandonthree-dimensionalmodelwiththesamenumericalcelldensity.Thegoalistoseeifatwo-dimensionalmodelofthethrustchamberwouldbeaccurateenough.Figure 8-3 showstheproleofMachnumberontheaxisofeachofthemodels.Itshowsclearlythatthetwo-dimensionalmodelunder-estimatestheexhaustMachnumber 165

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8-4 showstheMachnumberproleontheaxisofthethrustchamberforbothEulerandNavier-Stokessolvers.Theprolesaresimilar,exceptattheexitofthenozzle.Thisisconsistentwiththeresultsweobtainedinthequasi-one-dimensionalanalysis:theexhaustvelocityisslightlyhigher(14%)foraninviscidowthanforaviscousow.Therefore,inarstapproach,theEulersolvercouldbeusedtomodeltheowinthelow-pressurethrustchamber.However,asthehydrogentravelstroughthechamber,theviscouseectscannotbeneglectedanymore.Thus,theNavier-Stokessolversaremoreadaptedtomodeltheowinthelow-pressurethrustchamber. 166

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8-5 .Wenoticethatthek-modelwithenhancedwalltreatmentgivestheworstresults.BecauseRSMisthemostadvancedturbulencemodelwithintheRANStechnique,wecanconsidertheprolesforthismodelasbeingthereference:RSMmodelsalltheuctuatingproductswithintheRANSapproachbysolingatransportequation,andtherefore,modelingoftheturbulenteectsismuchmorerestrictedthanintheothermodels.TheSpalart-AllmarasmodelshowsverygoodagreementwiththeRSMpredictions.ThisanalysisconrmsthegooduseoftheSpalart-Allmarasmodelinthecaseofthenozzlemodeling. 167

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8-6 :therelativedierencesbetweenthemaximumMachnumbercompiledwithagivengridandthenestgridareplottedasafunctionofthenumberofcomputationalnodesforthegivengrid.Asthenumberofmeshpointsincrease,thedierencebecomessmallerandsmaller,andtendstowardszero.Thevariationasthenumberofnodesincreasesfollowsanexponentialdecaycurve.Table 8-2 compilestheCPUtimeandnumberofiterationrequiredtoreachconvergenceforgridsG1toG5.WeneedtomakeacompromisebetweenaccuracyandCPUtime.FromFigure 8-6 andTable 8-2 wecaninterferethatgridsG2iswelladaptedtogetfastresults.WemightwanttousegridG3incaseswherewewanttoincreaseaccuracyofthesimulation. 8{5 .BeforewestartourownCFDstudyofthenozzle,wewanttoinvestigatewhichcombinationofthedesignparametersleadtothebestnozzleperformance.Weusethefrozenchemistrymodelabovetoinvestigatethebestdesign.Theperformancecriteriaare: 168

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8-3 liststhedierentcasesinvestigatedandthedesignparametersassociated.A'x'inthelastcolumnindicatesthatthecaseisgeometricallyimpossibletocreate,anda'o'meansthatthegasisnotproperlyexpandedinthechamber.Asaruleofthumb,thegasusuallydoesnotexpandinthechamberwhenthedivergingnozzleiseithertooshortortoolong.Thetablealsoliststhemaximumvelocityineachcase.Figure 8-7 showstheinuenceofanisolatedparameteronthenozzleperformanceswhileallotherparametersremainconstant.Logically,highthroattoexitratios(A=A)leadtohighervelocitiesandMachnumbersontheexhaustplane.Anexitradiusof1.5mcorrespondstoaratioofabout100.Thisisthetypeofratiowewanttoinvestigate.ThethoroughinvestigationexhibitsasetofparameterswhichleadtohighestperformancesofthenozzlewithA=A=100: 169

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8-8 showstheexhaustvelocityprolesforthetwoformulationsonthewallcondition. 170

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CPUtime(s)andnumberofiterationtoreachconvergenceofthesteady-stateforthefourdierentgrids GridG2G3G4G5 CPU(s)2:51:045:24:097:08:1610:03:35IterationNb72107124146Timeperiteration0:02:230:03:020:03:270:04:08 Figure8-1: Thrustchamberanditsprincipalcharacteristics 171

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Coarsergrid Figure8-3: ComparisonoftheMachnumberprolefortheaxisymmetricand2-Dmodels 172

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Testcasesfordierentdesignparameters CasenameRcaAexitLvmaxnote Grid12201200.7xGrid2240120112724oGrid3320120112713oGrid4240120212621Grid5320120212598Grid6340120212594Grid72301.520313513Grid83301.520313509Grid94301.520313510Grid161301.520313550Grid171.5301.520313517Grid180.5301.5203xGrid190.8301.5203xGrid201.25301.520313522Grid210.99301.520313535Grid221.1301.520313526Grid231301.510313544Grid241301.515313567Grid251301.525313540Grid261301.530313619Grid271301.540314353Grid281301.550314648oGrid291301.560314701oGrid301301.555314710oGrid371301.540414333Grid381301.540213732Grid391301.5403.514351Grid401301.5402.514172Grid411301.5403.214355Grid421301.5403.314367Grid431301.5403.414342Grid441251.5403.3xGrid451351.5403.314358Grid461451.5403.314340 173

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Machnumberprolesforaninviscidandviscousowofhydrogen 174

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Machnumberprolesobtainedfordierentturbulencemodels 175

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RelativedierenceinthemaximumMachnumberbetweenagivengridandthenestgrid 176

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Inuenceofthedesignparametersonthemaximumvelocityontheexhaustplane-a)Inuenceofthethroatratioofcurvature(Rc)a=30o:=20oandL=3m-b)Inuenceoftheinectionangle():theblackdotscorrespondstodesignswheretheexhaustvelocityisnotuniformRc=1,=30oandL=3m-c)Inuenceofthecontouredlength(L):Rc=1,a=30oand=40od)Inuenceoftheapproachangle(a):Rc=1,=40oandL=3:3m 177

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Velocitymagnitudeproleontheexhaustplaneofthethrustchamber 178

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179

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1. Thefeasibilityofdevelopingarealpropertycomputationaluiddynamicssolvertomodeluiddynamicsandheattransferofhydrogenandheliumcooledhightemperaturegascooledsystems. 2. Thefeasibilityofcouplingacoarse-meshsystemsimulationcodetoanemeshCFDsolver.Thene-meshComputationalFluidDynamicssolverdevelopedinthisworkisdirectlyadaptabletoalltypesofhightemperaturegascooledreactors.Itconsistsinatwo-dimensionalaxisymmetricsolverthatiscompatiblewiththegeometryofmosthightemperaturegascooledreactorsystems.TheTVDMacCormacksolvercombinedwithRANSturbulencemodelssuchask-andSpalart-AllmarasareusedtodeveloparobustCFDsolverthatincorporatesrealgasproperties.TheCFDsolverhasbeenappliedtotheanalysisofthecoolantchannelofaheliumcooledHighTemperatureGas-CooledReactorInthecaseofthecore-lengthchannelthetemperatureappliedonthewallslinearlyincreasefrom400Cto1,000C.Thepredictedpressuredropacrossthe7.93mchannelis28kPaUsingasimpleone-dimensionalmodelthecalculatedpressuredropinthechannelis443kPA,whichisagrossoverestimation.Theowinthetubeisacceleratedsuchthattheexitvelocityistwicelargerthantheinletvelocity.Thechangeintheheliumgaspressureandtemperatureatthecoreexitresultin47Then,thecouplingoftheNuRokcoarse-meshsimulationcodetothenemeshCFDsolverisdonethroughasimpleinterfacingprocedurewhichconsistsofamathematicalinterpolationoftheextensivepropertiesfrom1Dcodeto2Daxisymmetriccodeandviceaversa.ThecouplingisappliedtothestudyoftheNERVAderivativenuclearthermalpropulsionsystem.ThehotchannelintheNERVAcoreismodeledwiththenemeshCFDsolver.TheuseofthenemeshCFDsolversignicantlyincreasesthecomputationaltime.However,thetotalcomputationtimestillislessthanoneminuteInthecaseofa100,000lbfthrustNERVAsystemthepressuredropacrossthechannelisequalto1.19 180

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181

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182

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8{7 andtheideal-gaslawdescribesthedensitybehavior.Theinletmassowrateissetto0.228kg/sandtheinlettemperatureandpressureto3,000Kand0.1barsrespectively.ThiscorrespondstoaninletMachnumberof0.3.Theadiabaticbehaviorisstudied,thereforethethermaluxappliedtothewallsisequaltozero.Bydefault,thewallsareconsideredtobeinAluminum.Finally,theexitpressureissetat10Pa.Weuseasecondordersolveronpressureanda3rdorderMUSCLonalltheotherconservativevariables. 190

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B-1 .TheprolesobtainedattheinletofthecoolantpipesareplottedFigures B-2 and B-3 andcomparedtothetheoreticalfully-developedturbulentowprolesforbothgeometries.Generally,thefully-developedowproleisagoodestimationoftheoverallhotchannelinletprolebutthisapproximationonlyisqualitative.Onaquantitativepointofview,theowproledoesnotmatchthecomputedresults. 6{13 withanexitpressureof6.93MPa.Sincetheinletpressureisestimatedtobeequalto8.12MPa,hydrogenpropertieswhichareconsideredconstantinthefrozenchemisrtyapproachareequalto: 191

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B-4 andcomparedtothepredictionwiththerealpropertysolver.Thedierencebetweenthetwosimulationsisonlyof6.7%comparedwith243%betweenThemisandNuRok.Thankstothefrozenchemistrystudy,wecanbeverycondentthatthepredictionswithThemisareaccurateandthattheNuRokpredictionswerelackingaccuracyinhandlingthepressuredropacrossthecoolantchannel. 13 ]:Thebackowtemperatureshouldbeinputwhenweusethepressure-outletboundaryconditioninFLUENT.Thespeciedvalueswillbeusedonlyifowispulledinthroughtheoutlet.ThevaluemightbeusedintherstfewiterationsforwhichFLUENToftencomputesvelocitiescomingfromtheexit.Hence,theabsolutevalueofthebackowtotaltemperatureitselfisnotofgreatimportancebutshouldbesetwitharealisticvaluetohelpcomputationoftherstiterations. 192

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Meshesforthetwocontractionexperiments 193

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Velocityproleonthecontractionplaneinthesinglechannelconguration.Comparisonbetweencomputedandtheoreticalproles 194

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Velocityproleonthecontractionplaneinthefour-channelcontractionexperiment.Comparisonbetweencomputedandtheoreticalproles 195

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Pressureprole(Mpa)ontheaxisofthecoolantchannel.ComparisonbetweentherealpropertysolverandFLUENTwithfrozenchemistry 196

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D-1 .Toillustratethechallengesofobtainingaconvergedsolutionofthepropertyloopwhenthereisahighlevelofdissociation,thepropertyloopisappliedtoasinglepointofthemesh,atthecenterofthechannel.ThethreehydrogencasespresentedinChapter6arepresented: D-2 to D-4 showtheevolutionofthepressure,temperatureandspecicheatratioforthenodeselected. D-2 showstheevolutionofthepropertiesfor40iterations:itprovesthatweobtainedastablesolution. 203

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D-3 showsthatconvergenceofthepropertyloopisobtainedbutitrequiresmoreiterations.Propertiesareconvergedwithin10iterationstolevelsofresidualsof104.Thesolutionobtainedisstable. D-4 showstheevolutionofpressure,temperatureandspecicheatratiowiththeiterationnumber.Thesolutiondoesnotconvergesbutoscillatesbetweentwoextremumvalueswith100%dierenceforpressure.Asitisthecaseforallpointsofthe100x1,000grid,thesolutioncannotreachaconvergencewhenlevelsofdissociationarehigh.ThepropertylooptechniqueintroducesnumericaldiscontinuitiesinthepipewherepressureshouldbecontinuousgiventhelowMachnumberandthegeometry.Therefore,thepropertyloopcannotbeusedwhenhydrogendissociationishighandothertechniqueshavetobeinvestigatedtoreachconvergenceinsuchgasconguration.

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Thepropertyloop 216

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Evolutionofthemajorpropertieswiththenumberofiterationinthehydrogenpipewithnodissociation 217

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Evolutionofthemajorpropertieswiththenumberofiterationinthehydrogenpipewithlowlevelsofdissociation 218

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Evolutionofthemajorpropertieswiththenumberofiterationinthehydrogenpipewithhigherlevelsofdissociation 219

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AnneCharmeauwasbornMay13,1980,inVersailles,France.ShegraduatedfromEdouartHerriotHighSchoolinVoiron,France.Aftertwoyearsofpre-engineeringclasses,sheenrolledattheNationalSchoolforPhysicsinGrenobleandobtainedherengineerdiplomainSeptember2003.AspartofanacademicexchangeshejoinedtheUniversityofFloridainAugust2002.ShegraduatedwithaMastersofSciencein2004.Sincethen,shehasbeenpursuingherdoctoralresearchintheNuclearandRadiologicalEngineeringDepartmentandhasbeenworkingattheInnovativeNuclearSpacePowerandPropulsionInstituteasaresearchandteachingassistant.Hereldsofspecialtyarethermal-hydraulics,ComputationalFluidDynamicsandreactorsafety. 228


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