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CROSS SECTION GENERATION STRATEGY FOR HIGHCONVERSION LIGHT WATER REACTORS
by
BRYAN R. HERMAN
B.S. Nuclear and Mechanical Engineering, 2009Rensselaer Polytechnic Institute
SUBMITTED TO THE DEPARTMENT OF NUCLEAR SCIENCEAND ENGINEERING
IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF
MASTER OF SCIENCE IN NUCLEAR SCIENCE AND ENGINEERINGAT THE
MASSACHUSETTS INSTITUTE OF TECHNOLOGY
SEPTEMBER 2011
02011 Massachusetts Institute of TechnologyAll rights reserved
Signature of Author.. ...............Department of Nuclear Science
. . .g * : . .f .... .- .............
Visiting Associate Professor of
...............and Engineering
July 29, 2011
Eugene Shwageraus, Ph.D.Nuclear Science and Engineering
Thesis Supervisor
..............................Benoit Forget, Ph.D.
Nuclear Science and EngineeringThesis Supervisor
Certified by ... .. .. .. ..
Certified by...........
y/Assistant Professor of
Certified by ... .. .. .. ... . .. . .. .f. . .. . . . . . . . . . . . .. . .. . . . . . . . . . . . . . . . . . .Mujid S. Kazimi, Ph.D.
TEPCO ofejssor f Nuclear Science and EngineeringProfessor of Mechanical Engineering
Thesis Reader
Mujid S. Kazimi, Ph.D.Chair, Department Committee on Graduate Students
2
CROSS SECTION GENERATION STRATEGY FOR HIGH CONVERSION LIGHT WATERREACTORS
by
BRYAN R. HERMAN
Submitted to the Department of Nuclear Science and Engineering onJuly 29, 2011 in Partial Fulfillment of the Requirements for the
Degree of Master of Science in Nuclear Science and Engineering
Abstract
High conversion water reactors (HCWR), such as the Resource-renewable Boiling Water Re-actor (RBWR), are being designed with axial heterogeneity of alternating fissile and blanket zonesto achieve a conversion ratio of greater than one and assure negative void coefficient of reactivity.This study assesses the generation of few-group macroscopic cross sections for neutron diffusiontheory analyses of this type of reactor, in order to enable three-dimensional transient simulations.The goal is to minimize the number of energy groups in these simulations to reduce computationaleffort.
A two-dimensional cross section generation methodology using the Monte Carlo code Serpent,similar to the traditional deterministic homogenization methodology, was used to analyze a singleRBWR assembly. Results from two energy group and twelve energy group diffusion analysesshowed an error in multiplication factor over 1000 pcm with errors in reaction rates between 10 and60%. Therefore, the traditional approach is not sufficiently accurate. Instead, a three-dimensionalhomogenization methodology using Serpent was developed to account for neighboring zones in thehomogenization process. A Python wrapper, SerpentXS, was developed to perform branch casecalculations with Serpent to parametrize few-group parameters as a function of reactor operatingconditions and to create a database for interpolation with the nodal diffusion theory code, PARCS.Diffusion analyses using this methodology also showed an error in multiplication factor over 1000pcm.
The three-dimensional homogenization capability in Serpent allowed for the introduction of ax-ial discontinuity factors in the diffusion theory analysis, needed to preserve Monte Carlo reactionrates and global multiplication factor. A one-dimensional finite-difference multigroup diffusiontheory code, developed in MATLAB, was written to investigate the use of axial discontinuity fac-tors for a single RBWR assembly. The application of discontinuity factors on either side of eachaxial interface preserved multiplication factor and reaction rate estimates between transport the-ory and diffusion theory analyses to within statistical uncertainty. Use of this three-dimensionalassembly homogenization approach in generating few-group macroscopic cross sections and axialdiscontinuity factors as a function of operating conditions will help further research in transientdiffusion theory simulations of axially heterogeneous reactors.
Thesis Supervisor: Eugene ShwagerausTitle: Visiting Associate Professor of Nuclear Science and Engineering
Thesis Supervisor: Benoit ForgetTitle: Assistant Professor of Nuclear Science and Engineering
3
Acknowledgments
I would like to express my sincere appreciation to my advisors, Professor Eugene Shwageraus,Professor Benoit Forget and Professor Mujid Kazimi. The enriching conversations with ProfessorShwageraus over the past two years have given me a great understanding of reactor physics andthe cross section generation process. I am very thankful that I was able to work and learn fromhim during his two year visitation at MIT. Professor Forget has provided invaluable insight duringthe course of this work. He has helped me through many tough problems and gave me new ideasto pursue that have made this work successful. Without Professor Kazimi, this project would nothave been possible. I would like to thank him for his guidance.
I would also like to thank Dr. Kord Smith. Without him, the formulation of axial discontinuityfactors for this work would not have been possible. I look forward to learning from him as hebegins his tenure at MIT.
I would also like to express my sincere gratitude to Dr. Brian Aviles, my fellowship mentor fromKnolls Atomic Power Laboratory. He has sparked my interest in multiphysics analyses of nuclearreactors and brought me into the naval lab family. He has been a great source of knowledge duringthe past few years and I look forward to working with him in the future.
Special thanks to Dr. Jaakko Leppanen for his assistance using the Serpent code. I would alsolike to thank Professor Downar and his research group at the University of Michigan for their helpwith PARCS.
Thanks to all of my friends who I have learned so much from over the past few years at MITand during my undergraduate study at RPI. Our study groups and daily interactions have helpedme get through challenging times.
I would also like to thank my closest friends, Robert Gibson and Matthew Mascelli. Their friend-ship and support throughout the years have helped me become the person I am today.
Without my loving family, especially my mother and father, I would not be where I am. Theyhave given me emotional support, encouragement, and financial means as I attain my personalgoals. My brother, Christopher, is one of my best friends who is also aspiring to become a nuclearengineer. I dedicate this thesis to him.
This research was performed under appointment to the Rickover Fellowship Program in NuclearEngineering sponsored by Naval Reactors Division of the U.S. Department of Energy.
4
Table of Contents
1 Introduction
1.1 Breeding in Light Water Reactors . . . . . . . . . . . .
1.2 M otivation . . . . . . . . . . . . . . . . . . . . . . . . .
1.3 Homogenization of Cross Sections . . . . . . . . . . . .
1.3.1 Deterministic Methods - Self Shielding Treatment
1.3.2 Deterministic Methods - Spatial Homogenization
1.3.3 Monte Carlo Methods . . . . . . . . . . . . . .
1.4 Full Core Calculations . . . . . . . . . . . . . . . . . .
1.5 O bjectives . . . . . . . . . . . . . . . . . . . . . . . . .
2 Serpent Reactor Physics Burnup Code
2.1 Using Serpent for Cross Section Generation . . . . . . .
2.1.1 Geometry Creation.....
2.1.2 Material Specification . . .
2.1.3 Burnup Calculations .
2.1.4 Detector Tallies . . . . . . .
2.1.5 Other Features . . . . . . .
2.2 Description of Lattice Codes .
2.2.1 Deterministic . . . . . . . .
2.2.2 Monte Carlo . . . . . . . .
2.3 Two-Dimensional Pin-cell Depletion
2.4 RBWR Serpent Assembly Model .
2.4.1 Geometry Specifications .
2.4.2 Material Specifications . . .
2.4.3 Operating Conditions . . .
2.4.4 Other Control Information .
Comparison
. . . . . . . . . . . . . . . . . 3 6
. . . . . . . . . . . . . . . . . 3 8
. . . . . . . . . . . . . . . . . 3 8
..... ............ 3 9
..... ............ 4 0
. . . . . . . . . . . . . . . . . 4 1
. . . . . . . . . . . . . . . . . 4 1
. . . . . . . . . . . . . . .. . 4 1
. . . . . . . . . . . . . . .. . 4 2
. . . . . . . . . . . . . . . .. 4 7
......... ........ 4 8
...... ........... 4 8
. . . . . . . . . . . . . . . . . 5 2
... .............. 5 5
5
14
14
15
16
16
22
23
31
35
36
36
2.4.5 Comparison with MCNP5 . . . . . . .
2.5 Neutron Balance in Monte Carlo Codes . . . .
2.6 Three-Dimensional Cross Sections . . . . . . .
3 Preparation of Homogenized Parameters
3.1 Branch Cases . . . . . . . . . . . . . . . . . .
3.1.1 Instantaneous Branch Cases . . . . . .
3.1.2 History Branch Cases . . . . . . . . . .
3.2 SerpentXS Wrapper . . . . . . . . . . . . . . .
3.2.1 Input to SerpentXS . . . . . . . . . . .
3.2.2 Framework of SerpentXS . . . . . . .
3.2.3 Generation of Homogenized Parameters
3.2.4 Creation of PMAXS Database . . . . .
3.3 Spatial Multigroup Diffusion Solver . . . . . .
3.4 PWR Lattice Test . . . . . . . . . . . . . . . .
55
57
65
73
. . . . . . . . . . . . . . . . . 7 3
. . . . . . . . . . . . . . . . . 74
. . . . . . . . . . . . . . . . . 77
... ....... ....... 7 9
.... ........ ..... 80
..... ....... ..... 8 5
...... ....... .... 8 8
....... ........ .. 9 2
........ ........ . 9 5
... ...... ........ 100
4 Diffusion Theory Analysis of RBWR
4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.2 Two-Zone Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.2.1 Fissile-Fissile System . . . . . . . . . . . . . . . . . . . . . . . . .
4.2.2 Fissile-Blanket System . . . . . . . . . . . . . . . . . . . . . . . . .
4.3 Axial Discontinuity Factors . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.3.1 Incorporation of Discontinuity Factors in Finite Difference Equations
4.3.2 Implementation of Discontinuity Factors into Analysis . . . . . . . .
4.4 Two-Zone Diffusion Analysis with Discontinuity Factors . . . . . . . . . . .
4.5 RBWR Single Assembly Analysis . . . . . . . . . . . . . . . . . . . . . . .
4.5.1 Reference Discontinuity Factors . . . . . . . . . . . . . . . . . . . .
4.5.2 Application of Discontinuity Factors to PARCS . . . . . . . . . . . .
6
107
. . . 107
. . . 107
. . . 108
. . . 108
. . . 115
. . . 116
. . . 118
. . . 121
125
126
. . . 134
4.5.3 Approximation of Discontinuity Factors . . . . . . . . . . . . . . . . . . . 136
4.5.4 Effect of Void Distribution on Discontinuity Factors . . . . . . . . . . . . 138
5 Conclusions and Future Work 141
5.1 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141
5.2 Future Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142
5.2.1 Serpent Improvement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143
5.2.2 SerpentXS Improvement . . . . . . . . . . . . . . . . . . . . . . . . . . . 143
5.2.3 Methodology Improvement . . . . . . . . . . . . . . . . . . . . . . . . . . 144
References 146
A Code Comparison Input Files 150
A. 1 Pin-cell Code Comparisons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 150
A.1.1 CASMO4E . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 150
A.1.2 Dragon . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 150
A.1.3 BGCORE-MCNP5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152
A .1.4 Serpent . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153
A.2 Serpent - RBWR Single Assembly . . . . . . . . . . . . . . . . . . . . . . . . . . 155
A.3 MCNP5 - RBWR Single Assembly . . . . . . . . . . . . . . . . . . . . . . . . . . 168
A.4 RBWR Two-Dimensional Example Input Files . . . . . . . . . . . . . . . . . . . 177
A.4.1 Lower Reflector . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 177
A.4.2 Lower Fissile Zone Sub-region 3 . . . . . . . . . . . . . . . . . . . . . . . 180
A.4.3 Upper Reflector . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183
B SerpentXS - PARCS Input Examples 186
B. 1 SerpentXS Branch Input File . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 186
B.2 SerpentXS PWR Geometry File . . . . . . . . . . . . . . . . . . . . . . . . . . . 187
B.3 SerpentXS to PMAXS Input File . . . . . . . . . . . . . . . . . . . . . . . . . . . 191
7
C MATLAB Multigroup Spatial Diffusion Solver 193
C.1 Example Input File .................................. 193
C.2 Source Code . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 194
C.2.1 Power Iteration Routine . . . . . . . . . . . . . . . . . . . . . . . . . . . 194
C.2.2 Build Loss Matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 195
C.2.3 Build Production Matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . 198
C.2.4 Fixed External Source . . . . . . . . . . . . . . . . . . . . . . . . . . . . 199
C.2.5 Extract Heterogeneous k-effective . . . . . . . . . . . . . . . . . . . . . . 199
C.2.6 Compute Interface Currents . . . . . . . . . . . . . . . . . . . . . . . . . 200
C.2.7 Coarse Mesh Homogeneous Flux Distribution . . . . . . . . . . . . . . . . 201
C.2.8 Compute Homogeneous Interface Flux . . . . . . . . . . . . . . . . . . . 202
C.2.9 Extract Heterogeneous Interface Flux . . . . . . . . . . . . . . . . . . . . 203
C.2.10 Compute Discontinuity Factors . . . . . . . . . . . . . . . . . . . . . . . 203
D RBWR Single Assembly Code Inputs 205
D .1 Serpent . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 205
D. 1.1 Branch Case Input File . . . . . . . . . . . . . . . . . . . . . . . . . . . . 205
D. 1.2 Geometry Input File . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 207
D.2 PARCS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 219
D.2.1 UF1 PMAXS File . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 219
D.2.2 Input File . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 219
8
List of Figures
1.1
1.2
1.3
1.4
1.5
1.6
2.1
2.2
2.3
2.4
2.5
2.6
2.7
2.8
2.9
2.10 Side-View of RBWR Assembly
Assembly Lattice Configuration
Cross-section of a Fuel Rod Unit
RBWR Axial Void Fraction Distri
Coolant Density Axial Distributio
Comparison of Axial Flux Distrib
Comparison of Power Density .
Comparison of Fission Neutron Pr
Comparison of Absorption Rate D
Computational Node n Nomencla
Hitachi Fuel Assembly and Core Layout . . . . . . . . . . . . . . .
Overall Reactor Analysis Calculation Scheme . . . . . . . . . . . .
Deterministic Cross Section Generation Procedure . . . . . . . . .
Free-flight Distance in Delta Tracking . . . . . . . . . . . . . . . .
PARCS Solution Scheme . . . . . . . . . . . . . . . . . . . . . . .
Quarter Core BWR Geometry used in Full Core Analyses . . . . .
Three-Dimensional Homogenization Diagram of a Single Assembly
Pin-cell Geom etry . . . . . . . . . . . . . . . . . . . . . . . . . .
Comparison of k-effective for Different Lattice Codes . . . . . . . .
Difference in k-effective between Several Codes and CASMO4E
Comparison of Uranium-235 Number Density versus CASMO4E
Comparison of Plutonium-239 Number Density versus CASMO4E
Comparison of Xenon- 135 Number Density versus CASMO4E . .
Dragon - Serpent Comparison of Total Macroscopic Cross Section
Dragon - Serpent Comparison of Fission Neutron Production Cross S
. . . . . . . . . . . . . . . . . . . . . . . . . . . 4 9
. . . . . . . . . . . . . . . . . . . . . . . . . . . 4 9
C ell . . . . . . . . . . . . . . . . . . . . . . . . 5 1
bution . . . . . . . . . . . . . . . . . . . . . . . 53
n in RBWR Assembly . . . . . . . . . . . . . . 57
ution . . . . . . . . . . . . . . . . . . . . . . . . 58
. . . . . . . . . . . . . . . . . . . . . . . . . . . 5 8
oduction Density . . . . . . . . . . . . . . . . . 59
ensity . . . . . . . . . . . . . . . . . . . . . . . 59
ture . . . . . . . . . . . . . . . . . . . . . . . . 6 1
2.20 Power Distribution Diagram of RBWR Assembly (side-view) . . . . . . . . . . 70
9
. . . . . . . . 15
. . . . . . . . 17
. . . . . .. . 17
. . . . . .. . 25
. . . . . .. . 32
. . . . . . . . 32
. . . . . . . . 37
. . . . . . . . 43
. . . . . . . . 43
........ 44
. . . . . . . . 45
. . . . . . . . 45
. . . . . .. . 46
. . . . . .. . 46
ection . .. . 47
2.11
2.12
2.13
2.14
2.15
2.16
2.17
2.18
2.19
2.21
2.22
2.23
3.1
3.2
Differences between 3-D and 2-D Transport Cross
Differences between 3-D and 2-D Fission Producti
Differences between 3-D and 2-D Absorption Cros
PMAXS "Tree-Leave" Structure . . . . . . . . .
Fuel Temperature Instantaneous Branch Case PMA
Sectio
on Cro
s Secti
XS Ex
3.3 Interpolation Example for History Cases Structure . . . .
3.4 Serpent to PARCS Flow Diagram . . . . . . . . . . . . .
3.5 Overall Flow of SerpentXS Branch Case Generator . . .
3.6 Data Structure Organization in SerpentXS . . . . . . . .
3.7 Discretization of Spatial Domain . . . . . . . . . . . . .
3.8 Orientation of Partial Currents at Reactor Boundaries . .
3.9 Homogenization Process of PWR 2-D Lattice . . . . . .
3.10 Geometry and Power Distribution Pictures from Serpent
3.11 Reference Case Results . . . . . . . . . . . . . . . . . .
3.12 Control Rod Results . . . . . . . . . . . . . . . . . . . .
3.13 Coolant Density Results . . . . . . . . . . . . . . . . .
3.14 Poison Concentration Results . . . . . . . . . . . . . . .
3.15 Fuel Temperature Results . . . . . . . . . . . . . . . . .
4.1 Two-zone Homogenization Process . . . . . . . . . . . .
4.2
4.3
Spatial Distribution of Reaction Densities for the Fissile-F
Spatial Distribution of Reaction Densities for Fissile-Blan
4.4 Spatial Flux Distribution for Fissile-Blanket System .
4.5
4.6
4.7
4.8
4.9
ns . . . . . . . . . . . . . 71
ss Sections . . . . . . . . 71
ons . . . . . . . . . . . . 72
. . . . . . . . . . . . . . 75
am ple . . . . . . . . . . 76
. . . . . . . . . . . . . . 79
. . . . . . . . . . . . . . 82
. ..... ........ 86
. ...... ....... 88
.. ..... ....... 95
.. ...... ...... 98
. . . . . . . . . . . . . . 101
. . . . . . . . . . . . . . 102
. . . . . . . . . . . . . . 104
. . . . . . . . . . . . . .104
..... ..... ....105
..... ....... ..105
...... ....... .106
...... ........ 109
issile System . . . . . . 109
ket System . . . . . . . . 113
. . . . . . . . . . . . . . 113
Spatial Distribution of Reaction Densities for Fissile-Blanket System - 12G
Coarse Region Homogeneous Flux Distribution . . . . . . . . . . . . . . .
Comparison of Reaction Densities for Fissile-Blanket System with ADFs .
Comparison of Group 1 Collapsed Flux . . . . . . . . . . . . . . . . . . .
RBWR Assembly Group I Flux from Two Group Calculation . . . . . . . .
. . . . 115
. . . . 122
. . . . 122
. . . . 126
. . . . 129
10
4.10
4.11
4.12
4.13
4.14
4.15
4.16
4.17
4.18
4.19
4.20
4.21 Approximated Discontinuity Factors for Two and Twelve Groups . 139
11
RBWR Assembly Group 2 Flux from Two Group Calculation . . . . . . . . . . . . 129
Assembly Fission Rate Density from Two Group Calculation . . . . . . . . . . . . 131
Fission Production Cross Section from Two Group Calculation . . . . . . . . . . . 131
Assembly Absorption Rate Density from Two Group Calculation . . . . . . . . . . 132
Absorption Cross Section from Two Group Calculation . . . . . . . . . . . . . . . 132
RBWR Assembly Group 3 Flux from Twelve Group Calculation . . . . . . . . . . 133
RBWR Assembly Group 12 Flux from Twelve Group Calculation . . . . . . . . . 133
Comparison of Group 3 Absorption Cross Section . . . . . . . . . . . . . . . . . . 134
Assembly Fission Rate Density from Twelve Group Calculation . . . . . . . . . . 135
Assembly Absorption Rate Density from Twelve Group Calculation . . . . . . . . 135
Assembly Absorption Rate Density with Vacuum Boundary Conditions . . . . . . 137
List of Tables
2.1 Geometric and Operating Conditions of Pin-cell . . . . . . . . . . . . . . . . . . . 42
2.2 Description of Sub-Regions in RBWR Assembly . . . . . . . . . . . . . . . . . . 50
2.3 Material Composition of Boron Carbide Rods in LR2 . . . . . . . . . . . . . . . . 52
2.4 Material Composition of Depleted UOX in Blanket Regions . . . . . . . . . . . . 52
2.5 Isotopic Composition of TRU Nuclides in Fuel Mixture . . . . . . . . . . . . . . . 53
2.6 Material Composition of Fuel in Fissile Regions . . . . . . . . . . . . . . . . . . . 54
2.7 Operating Conditions of RBWR Assembly . . . . . . . . . . . . . . . . . . . . . . 56
2.8 Comparison of Eigenvalues between Serpent and MCNP5 . . . . . . . . . . . . . 56
2.9 Comparison of Eigenvalues with Modification of Serpent . . . . . . . . . . . . . . 65
2.10 2-D and 3-D Transport Cross Section Comparison . . . . . . . . . . . . . . . . . . 67
2.11 2-D and 3-D Absorption Cross Section Comparison . . . . . . . . . . . . . . . . . 68
2.12 2-D and 3-D Neutron Fission Production Cross Section Comparison . . . . . . . . 69
3.1 Homogenized Parameters for PARCS Calculation . . . . . . . . . . . . . . . . . . 81
3.2 Logicals for PMAXS Database . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83
3.3 Parameters for PMAXS File . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93
3.4 PWR Geometric Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100
3.5 Instantaneous Branch Case Description . . . . . . . . . . . . . . . . . . . . . . . 102
4.1 Eigenvalue and Integral Reaction Rates for Fissile-Fissile System . . . . . . . . . . 110
4.2 Eigenvalue and Integral Reaction Rates for Fissile-Blanket System . . . . . . . . . 112
4.3 Twelve Group Energy Structure . . . . . . . . . . . . . . . . . . . . . . . . . . . 114
4.4 Eigenvalue and Reaction Rates for Fissile-Blanket System - Twelve Group . . . . . 114
4.5 Fissile-Blanket Interface Currents for Two Energy Groups . . . . . . . . . . . . . 121
4.6 Discontinuity Factors for Different Currents . . . . . . . . . . . . . . . . . . . . . 122
4.7 Current Comparison of Eigenvalue and Reaction Rates . . . . . . . . . . . . . . . 123
4.8 Current Comparison for Twelve Energy Groups . . . . . . . . . . . . . . . . . . . 124
4.9 Comparison of Twelve Group Discontinuity Factors . . . . . . . . . . . . . . . . . 124
12
4.10 Current Comparison of 12 Group ADF Eigenvalue and Integral Reaction Rates . . 125
4.11 Comparison of Multiplication Factors for RBWR Single Assembly . . . . . . . . . 127
4.12 Reference Two Group Axial Discontinuity Factors . . . . . . . . . . . . . . . . . . 128
4.13 Two Group Discontinuity Factors for Perturbed Void Distribution . . . . . . . . . . 140
13
1 Introduction
1.1 Breeding in Light Water Reactors
A High Conversion Water Reactor (HCWR) or Light Water Breeder Reactor (LWBR) is a nuclearreactor which is cooled by light water and can produce more fissile material than it consumes. Thefirst LWBR program was started by the U.S. Department of Energy in the mid-1960s to developa light water reactor to expand nuclear fuel resources. This core was designed and built at theShippingport Atomic Power Station in Shippingport, Pennsylvania and operated for five years. Atthe end of operation, an examination was completed and concluded that the fissile inventory ofthe expended core was 1.39 percent greater than the fissile inventory of the initial core (Atherton,1987).
A breeder reactor produces more fissile fuel than it consumes while it generates energy. Afissile isotope is an isotope that can undergo fission when interacting with thermal neutrons, whilefertile isotopes refer to materials that transform into fissile isotopes when interacting with a neu-tron. Each fissile isotope that undergoes fission produces on average two or three more neutrons.One of these neutrons is required for another fission to maintain the nuclear chain reaction, whilethe others are free to interact or leak from the reactor. To have a high breeding ratio, the remain-ing neutrons need to interact with fertile material to produce more fissile fuel. The conversion orbreeding ratio, if it is greater than unity, is defined as the ratio of the average rate of fissile isotopeproduction to the average rate of fissile isotope consumption (Duderstadt and Hamilton, 1976).Some of these neutrons will interact with other non-fuel core materials or eventually leak out ofthe system. In the Shippingport LWBR, the fissile material was uranium-233 while the fertile ma-terial was thorium-232. After the LWBR program was finished, this style of reactor core was notutilized in the commercial nuclear industry. All of the commercial nuclear reactors in the U.S. areof the Boiling Water Reactor (BWR) or Pressurized Water Reactor (PWR) type.
Two examples of programs in Japan have been established recently to design HCWRs to in-crease utilization of LWR fuel. Hitachi is designing the Resource-renewable Boiling Water Reactor(RBWR) model AC, while the Japanese Atomic Energy Agency is designing the Reduced Mod-eration Water Reactor (RMWR) as part of their innovative water reactor for flexible fuel cycle(FLWR) program (Takeda et al., 2007; Iwarnura et al., 2006; Uchikawa et al., 2007). These reac-tors operate with mixed oxide fuel that has a breeding ratio of 1.01. The core is characterized bytwo fissile zones sandwiched between axial internal blankets of depleted uranium dioxide (UOX).Unlike conventional BWRs, these advanced reactor designs are axially heterogeneous with fissilezones producing neutrons and blanket zones consuming them. To increase the breeding ratio, thesecores must operate at a higher heavy metal to water ratio. This is partially obtained by increasingthe void fraction, in some assemblies up to 80%, compared to conventional BWRs. In conventionalBWRs, the core average void fraction is about 40%, whereas, in the RBWR it is about 60%. Thishigher void fraction is obtained by reducing the flow to power ratio. The heavy metal to water ratiois also raised by reducing the pin pitch and arranging it in a hexagonal lattice. An illustration of afuel assembly and a full core arrangement is shown in Figs. 1. la and 1.1b, respectively. Similarto a conventional BWR assembly, the RBWR fuel assembly has a shroud surrounding it with acontrol rod inserted between assemblies. The fuel assembly contains five different enrichment fuelpins radially. An assembly of the RBWR contains five distinct active axial core zones: lower blan-ket, lower fissile, internal blanket, upper fissile and upper blanket. In these type of designs, a high
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( 10.7wt% 5
(13.5wt% 22e
( 16.8wt% 51
18.2wt% 70
199.2mm 19.5wt% 123
Average Fissile Pu Enrichment18.Owt%
Number of Fuel Rods 271-194.7mm Fuel Rod Diameter 10.1mm
Fuel Rod Gap 1.3mmThickness of Control Rod 6.5mm
(a) Hexagonal Fuel Assembly Configuration (b) Axial Zone Configuration in Full Core Design
Figure 1.1. Hitachi Fuel Assembly and Core Layout (Takeda et al., 2007)
axial leakage is needed to keep the void coefficient of reactivity negative. Instead of losing theseneutrons, they interact in spatially separate blanket zones to produce more fissile isotopes. Thisaxial configuration is shown in Fig. 1. 1b where zones are on top of one another in a parfait form.
A hexagonal assembly that is axially heterogeneous with a high void fraction in the upper partof the assembly, makes it a difficult problem to solve from a reactor physics standpoint. As willbe described in Section 1.3, macroscopic few-group cross sections and other parameters requiredfor solution of the neutron diffusion equation are conventionally generated from two-dimensionalradial lattice calculations. These cross sections may not be accurate because the RBWR doesnot have a distinct flux energy spectrum along the third dimension. Rather, this spectrum is con-tinuously changing along the axial direction because of the changing void fraction and materialzones. Thus, decoupling zones from each other may not be valid. As the void fraction increases,the moderator in the assembly becomes less dense and the mean free path of neutrons increases.Therefore, neutrons will travel further in the upper portion of the assembly and impact the behaviorof neighboring zones. Two-dimensional codes are therefore limited in capturing the effect of suchheterogeneity. In this work a Monte Carlo approach is proposed for homogenizing cross sectionsand other important parameters in this type of core using its three-dimensional neutron transportsimulation capability.
1.2 Motivation
The motivation for this work came out of ongoing efforts to perform void coefficient and transientanalyses for the RBWR. Since these reactors have a unique axial design, full three-dimensionalpower/void feedback is required. Before any transient calculations are performed, a steady statesolution of the core with a converged power and thermal-hydraulic distribution is needed. The fo-cus of this work is to lay a foundation on how to determine a three-dimensional power distributionof the RBWR. To determine this distribution, a three-dimensional neutronic calculation must beperformed which uses few-group homogenized parameters including macroscopic nuclear cross
15
sections as input. Current conventional two-dimensional homogenization methods do not work forhighly heterogeneous cores and therefore a new method is required. The large material discon-tinuities and spectral gradients in the axial direction in these reactors requires a new approach togenerating few-group homogenized parameters. This work investigates a three-dimensional ho-mogenization method involving axial discontinuity factors which may be required in generatingthese few-group homogenized parameters to obtain an accurate full core power distribution.
1.3 Homogenization of Cross Sections
To perform a full analysis of a reactor core, many steps need to be performed to accurately calcu-late the core multiplication factor, reaction rates and lifetime. Although computational power iscontinuously increasing, the ability for three-dimensional full core transport calculations in rou-tine reactor design is still far in the future. Performing a full core transport calculation wouldrequire a solution of a discretized problem of approximately 1012 unknowns. This is unrealistic forroutine design, optimization of fuel management and transient calculations. To reduce the compu-tational burden, spatial and energy condensation is performed (Sanchez, 2009). Figure 1.2 showsthe general overall process in flow chart form. The process starts with neutron cross section dataset preparation from evaluated nuclear data files, then lattice calculations to perform energy andspatial condensation to a few energy groups and ends with a full core reactor calculation.
The neutron cross sections are used to characterize the probability of interactions at differentneutron energies. The energy distribution of cross sections is commonly broken up into regionsbased on important physics that occur in that range. Below about 1 eV is referred to as the thermalenergy region where the neutron energy is on the order of the chemical binding energy and thermalmotion of molecules of a material. Therefore, it is necessary to take into account the thermal mo-tion of isotopes that are bound in a molecule. Above 1 eV, the resonances of heavy isotopes becomeimportant and self-shielding must be taken into account. At this stage, self-shielding is character-ized in terms of an equivalent dilution cross section. Cross sections are parametrized by dilutioncross section and by nuclide temperature since the Doppler effect will impact the shape of the res-onances. To perform these steps, a cross section processing code such as NJOY (Macfarlane andBoicourt, 1975) is used. From this point, there are two main methods that will be discussed in orderto perform few-group cross section generation: deterministic and stochastic (Monte Carlo). Theterm few-group cross section generation is equivalent to generation of homogenized parametersor generation of multigroup constants for full core deterministic calculations. In today's industry,deterministic methods are used to generate homogenized few-group cross sections. However, withthe increasing computational power, Monte Carlo methods can also be used for this purpose.
1.3.1 Deterministic Methods - Self Shielding Treatment
In nuclear engineering, deterministic methods are classified as methods in which the neutron trans-port equation and/or diffusion equation is discretized and solved directly with numerical methods.An illustration of the steps needed for a deterministic analysis are shown in Figure 1.3. As statedbefore, the beginning of the process involves processing neutron cross section data. For determin-istic methods, the module GROUPR in NJOY is used to generate a fine structure of multigroupcross sections that are a function of dilution and temperature. Depending on the application, thenumber of energy groups could range from hundreds to thousands of groups.
16
Figure 1.2. Overall Reactor Analysis Calculation Scheme (Hebert, 2009)
Basic data base:- cross-sections- decay chains
energy per Int.
-ission yields
Get basic cross-sectionsGenerate multi-group libraryUnit-cell CalculationsFuel Assembly/ Lattice CalculationsWhole Core Calculations
Unit cell1D transport ofequivalent cel r a a a a aa Mato 0 a C310 aaC73 coo 0 aa ma Do a0 am 0 am 0a 00 Mm 00 0 moo 0 0a
C30-
Fuel assembty20 transport or
diffusion
Reactor core
Figure 1.3. Deterministic Cross Section Generation Procedure
17
Depletionlcalculationl
Spitial kinetic",calcul"Itioll
Multigroup cross sections are generated by conserving reaction rates within a specific energygroup. To perform this spectral homogenization, GROUPR weights the energy dependent crosssection by a flux spectrum as described by
fEg _1 a, (E') p (E')dE'(ax)g g (1.1)
-1 $(E')dE'g
In Eq. (1.1), g is the energy group number, (ax), is the group-averaged microscopic cross sectionfor reaction type x, Eg and Eg_1 are the lower and upper boundaries of the energy group, respec-tively, ax (E) is the energy dependent microscopic cross section for reaction type x and 4 (E) isthe energy dependent flux spectrum. The equation is solvable if both the energy dependent crosssection and flux spectrum are known. Unfortunately, the flux spectrum is not known at this step asthe full problem would need to be solved in order to know this quantity. In deterministic methods,a flux spectrum must be assumed. Usually, a Maxwellian spectrum is chosen for thermal energygroups, an asymptotic slowing down spectrum is used in the resonance region and a Watt fissionspectrum is used in the fast range. The choice of flux spectrum is one of many approximations thatmust be made in the deterministic methodology.
The fine structure multigroup cross sections are then generated for different dilution cross sec-tions and temperatures. The dilution cross section is also referred to as the background crosssection, sigma zero cross section or Bondarenko cross section. This cross section characterizesself-shielding of isotopes and therefore must be generated specifically for isotopes that containresonances, referred to as resonant isotopes. Cross sections must be generated for a range of di-lution cross sections and temperatures as the dilution and temperature of the actual geometry inthe full core is not yet known. There are many complicated models for self-shielding (mutual self-shielding effects) and this is the source of another approximation needed in deterministic methods.The simplest self-shielding method for understanding dilution is the Bondarenko method (Bon-darenko, 1964). For a homogeneous mixture of two materials and isotropic scattering in center-of-mass system, the integral slowing down equation that GROUPR solves is
/E/ai ES1 (E')Er (E) # (E) j ( , (E') dE' (1.2)
/E/a2 s 2 E' (E') dE'.E (1 - a2) E '
In Eq. (1.2), Et (E) is the energy dependent macroscopic total cross section, # (E) is the energydependent scalar neutron flux, Es, (E) is the energy dependent scattering cross section and a, isrelated to the nuclear mass number (A) for material n, (A-1 )2. In this example, the subscript nrefers to material numbers where material 1 is the moderator that is a pure scatterer with a constantcross section and material 2 is a purely absorbing resonant isotope such as uranium-238. Themacroscopic cross sections can be divided by the number density of the resonant absorber, wherethe constant cross section of material 1 can be replaced with a dilution cross section, ao. Equation(1.2) becomes
18
[jO + ar2 (E)]# (E) = f (E') dE' (1.3)E (I- 1l) El
/E/a2 Gs2 (E') (E) dEE (I- a2) El
where Ct2 (E) is the energy dependent microscopic total cross section of material 2, Cas2 (E) is theenergy dependent scattering cross section of material 2 and ao is the dilution cross section. In this
formulation for a homogeneous system, the dilution cross section is defined as the macroscopic
potential scattering (equivalent to total for a pure scatterer) cross section over the number density
of the resonant isotope (material 2),
O - NES (1.4)N2
Therefore, if the number density of the resonant absorber is small compared to the macroscopic
potential scattering cross section of the moderator, the value of the dilution cross section is very
large. In the limit of very small number densities of resonant isotopes, an infinite dilution conditionwill exist. This means that the resonances of the resonant absorber are not important in shaping
the neutron flux spectrum and therefore the spectrum will take on the asymptotic form of 1/E.
However, if the number density of the resonant absorber is large, the resonances are not dilutedand impact the neutron energy spectrum. In the Bondarenko method, the flux energy spectrum
is represented by the asymptotic form. In the more general case this function can be denoted asC(E). Therefore the slowing down equation takes on the following form if a narrow resonance
approximation is assumed in the moderator (Macfarlane and Boicourt, 1975):
[o + ar2 (E)] # (E) C (E) o + 2 s (E) (E') d. (1.5)E (I -- a2) E
For heterogeneous systems, a dilution cross section can also be constructed. According toequivalence relationships, a heterogeneous system can be represented as a homogeneous system
if the dilution cross sections are equivalent (Duderstadt and Hamilton, 1976). For heterogeneoussystems such as a lattice, the slowing down equation can be constructed in the fuel, where the fuel
contains only a pure resonant absorber and the moderator contains a pure scatterer. In addition, it is
assumed that rod shadowing is not important such as the case where a single fuel pin is surrounded
by an infinite moderator. Thus, once the neutron leaves the fuel pin, it will interact in the moderator.
The slowing down equation can be formulated for this situation in collision probability form,
/E/af Ef('
VfYf (E) Of (E) VfPfaf (E) j( (E') )dE(1.6)t ~E (I - af) ElO
/E/an Em (E')+ VmPmng. (E) j ( / ,m (E') dE'.
fE (I -aUm) E
In Eq. (1.6), Vf is the fuel volume, Vm is the moderator volume, Pf f (E) is the energy depen-dent fuel-to-fuel collision probability and Pm>sf (E) is the moderator-to-fuel collision probability.
19
The subscripts, f and m represent the fuel and moderator, respectively. Using the reciprocity re-lation of collision probabilities, Pf-m (E) Ef (E) Vf = Pmsf (E) Em (E) Vm and that neutrons mustbe conserved, Pf- + Pf f = 1, Eq. (1.6) can be rewritten as,
VfEt{ (E) #f (E) VfPff (E) f ( # (E') dE' (1.7)E ( -- af) E'
1 - Pf_ f(E)] Etf(E) Vf E /am Em (E'),+ Ef (E) IE (1 - am) E' OM(E')dE'.
The moderator can be assumed to be a pure scatterer and flux energy spectrum in the moderatortakes the asymptotic form. The slowing down equation can be written in its final form as
/E/af E (E') [1 - Pfyf (E)] Ef (E)Ef (E) Of (E) = Pf _f(E) ,f (E' ) dE'+. (1.8)
t fE 1I - af) Ef E
Using the Wigner rational approximation, the fuel-to-fuel collision probability is
Ef (E)Pf -+f (E) = (1.9)
Eft (E) + Eewhere Ee is a fictitious cross section denoted as the macroscopic escape cross section. The macro-scopic escape cross section is a function of geometry given as the inverse of the mean chord lengthof the geometry, 1,
1Ee . (1.10)
Combining Eqs. (1.8), (1.9) and (1.10), and rearranging it into the form of Eq. (1.5), the dilutioncross section (neglecting bell factor, which is a small correction factor for the Wigner rationalapproximation (H6bert, 2009)) can be identified as
Go = - ==> Ge.(1)1Nf
Note that C (E) is taken as 1/E in Eq. (1.5). Therefore, for the situation where there is only apurely absorbing resonant isotope in the fuel and lattice effects are neglected, the dilution crosssection is the microscopic escape cross section. It is interesting to note that the dilution of theresonant isotope in the fuel for this situation is independent of the moderator. This is becauselattice effects were neglected.
To include lattice effects, it is common to characterize the probability that a neutron leaving thefuel will interact with another fuel element. This effect is usually called the Dancoff correction andis commonly denoted as C. The opposite probability is that the neutron interacts in the moderatorbefore colliding in another fuel element and this is known as the Dancoff factor and is denotedas D (Sugimura and Yamamoto, 2006). For the same situation described above, but now takinginto account lattice effects, the fuel-to-fuel collision probability is modified to account for rodshadowing where
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Ef(E)Pf-a_ (E) = (1.12)
Ef (E) +DEeTherefore, to first order, the dilution cross section is modified yielding
DGO = Due = - (1.13)
1Nf
In this formulation of the dilution cross section, the moderator dependence is taken into accountby the Dancoff factor, where different moderators will yield different Dancoff factors.
A more general situation is when the fuel is made up of both resonant and non-resonant isotopessuch as uranium 238 and oxygen. This situation represents a more realistic case where fuel is madeup of uranium dioxide. Again, the fuel is arranged in a lattice where lattice effects are taken intoaccount with the Dancoff factor. The slowing down equation for this situation is
/E/af,R f,R (E')Vf Et(E) Of (E) VfPf f (E) Of (E')dE'
]E (~ - cf,R) E' ()dE/El/af,NR Ef,N R E'
" V5Pfp g(E) J (F') ,pf (E') dE'E - af,NR) El
/E/am Em(E')" VmPm-nf (E) j ( OM (E') dE', (1.14)
fE (1 -- am) El
where the fuel, denoted with superscript f, is split into two components: the resonant isotope,denoted with superscript R and non-resonant isotope, denoted with superscript NR. Applying thereciprocity relation, conservation of neutrons and the asymptotic flux energy spectrum for the non-resonant isotope and the moderator, Eq. (1.14) becomes
Ef'R (E) + Ef'NR] Of (E) Pfrf (E) s (F) Pf(E')dE' (1.15)E (1- af,R) E
Pfa (F) Ef,NR [1 - Pff (E)] Ef'R (E) + Ef'NR
E E
Eq. (1.15) can be rearranged into the form of Eq. (1.5) where C(E) = 1/E. For the definition ofthe collision probability in Eq. (1.12), the dilution cross section in this form becomes
Ef,NR De(0 = NR + NR' 1'16)
f fThis form of the dilution cross section can be compared to the heterogeneous form in Eq. (1.13)and the homogeneous dilution cross section in Eq. (1.4). By inspection, the first term in Eq. (1.16)is similar to the homogeneous form and the second term is the heterogeneous form (Yamamoto,2008). Therefore, the general heterogeneous background cross section is made up of a homoge-neous part and a spatial dependent part. The dilution cross section can be represented in generalas
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ahet _ hon D2e (1.17)
where the homogeneous background cross section is a superposition of all of the potential crosssections, ay, of non-resonant isotopes (k) in the material,
h Nka k
k f
If the resonant isotopes of the fuel material are not pure absorbers, then the potential scatteringcross section should also be included in Eq. (1. 18).
1.3.2 Deterministic Methods - Spatial Homogenization
After the energy dependence of cross sections has been reduced in the range of hundreds to thou-sands of energy groups, further refinement of the energy groups are made on a pin cell level.As shown in Fig. 1.3, pin cell transport is performed via collision probability methods wherea material-specific spectrum can be calculated to account for spatial and energy self-shielding.Since the geometry is starting to be resolved, the dilution of the resonant isotopes can be calcu-lated with Eq. (1.17). The parametrized cross section sets from NJOY can then be interpolatedfor temperature and dilution so that the appropriate self-shielded cross sections are used in thetransport calculation. Cross sections are then further collapsed into generally tens to hundreds ofenergy groups. These cross sections are then utilized in lattice calculations (Smith, 1986). Fordeterministic analyses the lattice calculation involves a two-dimensional slice of an assembly withzero net current boundary conditions. By assuming zero net current boundary conditions, i.e. noleakage of neutrons, a distortion in the flux spectrum is present due to deviation from critical con-dition (k-effective of 1). To correct for this, leakage models can be used to force the assemblyto yield a multiplication factor that is representative of what would be seen in the actual core.The geometry is discretized and the the group flux is calculated in every spatial mesh via col-lision probability methods or method of characteristics (Hebert, 2009). Cross sections are thenhomogenized spatially over the whole lattice in a few energy groups (Rahnema and McKinley,2002). The cross sections are homogenized to the point where a full core calculation is feasiblein a reasonable amount of time. Before a full core calculation can be performed, cross sectionsmust also be generated for a wide range of possible operating conditions. This process is calledgenerating branch cases and is described in Section 3.1. A description of full core calculationsis presented in Section 1.4. The major assumption in deterministic methods at the lattice stage isthat the lattice being analyzed can be decoupled both radially and axially from the rest of the core.This approximation has been dealt with by introducing color-sets or supercells, and axial bucklingto attempt to capture the spectral effects of neighboring zones. Although this may work for somesituations, this assumption may not be applicable in general. The use of Monte Carlo for homog-enized cross section generation may help circumvent many of the needed assumptions that havebeen discussed for deterministic methods, including the use of equivalent dilution cross section tocapture self-shielding effects in the resolved resonance region.
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1.3.3 Monte Carlo Methods
Although not a new idea, there have been several recent studies using Monte Carlo methods togenerate few-group homogenized parameters for full core deterministic calculations (Leppanen,2007a; Pounders, 2006). In a Monte Carlo analysis, neutrons are simulated one at a time and ran-dom sampling of probability distributions is used to simulate the fundamental physics of neutrontransport. The general advantages of using a Monte Carlo code are continuous energy representa-tion of cross section data and the ability to handle arbitrary geometry. This is different from thedeterministic approach where continuous energy cross sections must be homogenized into a fewthousand groups in the first step. A Monte Carlo analysis does not need an initial flux spectrumguess to reduce the number of energy groups. Nuclear cross sections can naturally be defined fromthe cross section data depending on the neutron's energy. The dilution methodology presentedin the deterministic analysis for self-shielding does not need to be performed because continuous
energy cross sections are being used in the simulation of neutrons and their interactions. In theresolved resonance region, energy self-shielding is automatically taken into account with a con-tinuous energy representation of resonances. In addition, spatial self-shielding is also capturedbecause neutrons are being tracked individually through the geometry. Dilution cross sections are
still needed, however, in the unresolved resonance region where probability tables must be gen-erated (Levitt, 1972). Often in deterministic lattice codes, geometry is restricted because either
collision probability or characteristic tracks must be superimposed on the geometry for the analy-sis. If the geometry is restricted to a certain number of configurations, these collision probabilitiesand characteristic tracks can be optimized to increase computational efficiency. Therefore, it isnecessary for these codes to have built-in shapes for geometry. Although these codes can be ex-tended further for odd geometric shapes, it is relatively easy to handle arbitrary geometry in aMonte Carlo analysis. This is due to the simulation of path length of neutrons as they travel fromregion to region. A drawback of a Monte Carlo method is that it is a statistical process whereexpected values and associated variances are calculated. In order to get a good estimate of pa-rameters of interest, many particles need to be simulated and this takes computational time. Thislarge computational time factor prevents Monte Carlo methods from being used in routine designcalculations for commercial reactors.
Currently, full core Monte Carlo calculations coupled with thermal-hydraulic feedback are verydifficult because it would take a prohibitively large amount of time and computing resources. In
addition, a Monte Carlo neutronic analysis would also introduce stochasticity to the coupled ther-mal hydraulics analyses when performing both steady state and transient calculations. However,Monte Carlo methods can be used at the lattice level to generate a database of homogenized crosssections for use in full core analyses. Recently, VTT Technical Research Centre of Finland hasdeveloped a continuous energy neutron transport code with burnup capability for group constantgeneration called Serpent (Leppanen, 2007b). Serpent has been benchmarked by comparing re-sults against other neutron transport codes such as MCNP4C and CASMO-4E (Leppanen, 2005).In addition, some experimental benchmarking has been performed against axially-measured fissionrates of a VVER-440 (Leppdnen, 2007b). All comparisons and benchmarks yield good results andprove that the physics simulated in Serpent agrees well with established codes and experimentalmeasurements.
Since Serpent was designed specifically for homogenized group constant generation, it requiresspecialized tallies that are not present in many other codes. One of the limiting factors is that Monte
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Carlo codes may take an excessively long time to run to get adequate statistics on these tallies. Onemethod in Serpent that gives it a considerable speed-up is the use of an unionized energy grid forall point-wise cross section data. The main advantage is that the energy grid search needs to beperformed only once for each neutron energy. Once the location of this energy point is known,it can be used as an index for each reaction type for all isotopes. Another advantage is that themacroscopic total cross sections can be calculated before the transport cycle, which eliminates theneed for this process during the transport cycle. A main drawback of this method is that it requiresmore memory to store all of these cross section data. This procedure however cannot be donefor the unresolved resonance region where the total macroscopic cross section will be determinedfrom the sampling of probability tables. For the other energy regions, this is advantageous becausethe total cross section is needed to calculate the next flight distance. The free-flight distance that aneutron travels in Monte Carlo is calculated by
In()s = -,(1.19)
Et (E, ')
where s is the free-flight distance, 4 is a random number sampled uniformly between 0 and 1, andEt (E, ') is the precomputed total macroscopic cross section that is energy and material dependent.Finally, another major advantage of a unionized energy grid is that the reactions, dictated by therelative ratio of nuclear cross sections, can be sorted in ascending order. Therefore, when deter-mining a reaction type, the most probable reaction is checked first before all others (Leppanen,2009).
The other major source of speed-up in Serpent is the use of Woodcock delta tracking (Wood-cock et al., 1965). Delta tracking is an alternative to the conventional surface tracking methodthat is used to determine the free-flight distance in Eq. (1.19). Since the total cross section inthe denominator is a function of material region, it is not statistically valid to sample the neutronacross boundaries. Rather, the neutron is moved to the boundary and a new free-flight distance issampled for the new material region. In the delta tracking method, the material total cross sec-tions are sampled in such a way that the free-flight distances sampled are valid over the entiregeometry. To accomplish this, a cross section representing a delta collision is introduced such thatthe outgoing angle and energy are equivalent to the incoming angle and energy. A virtual totalcross section, E*, (E), can then be calculated as the summation of the material actual total crosssection, Etorm (E) and the delta scattering cross section, ES5m (E),
tot,m (E) = Etot,m (E) + ESm (E). (1.20)
The goal is to determine the smallest value of E,*o,, (E) for each energy such that it is spatiallyconstant, and therefore the delta scattering cross section is allowed to change. Figure 1.4 shows anillustration of how the virtual cross section and total cross section may appear. The value of thevirtual cross section corresponds to the maximum total cross section at each energy. Therefore, thefree-flight distance can be sampled using the virtual cross section and only one random number isneeded. At the collision another random number is sampled to determine if a "true" collision ora delta collision exists. If the delta tracking method is not efficient for the problem being studied,Serpent can use a conventional surface tracking method (Leppdnen, 2010a).
Serpent generates homogenized parameters automatically and lists them in the output file as afunction of homogenization region and burnup step. The most important of the parameters gener-
24
I*
XT
S
Figure 1.4. Free-flight Distance in Delta Tracking
ated are the geometry- and group-averaged homogenized cross sections. By simulating neutrons,Serpent is effectively solving Eq. (1.1) in each homogenization region. In Monte Carlo calcula-tions, the neutrons are simulated one at a time and therefore the homogenized cross section is ofthe form
(EX), = ' .j-~o (1.21)
In Eq. (1.21), (Ex)g is the group and region homogenized macroscopic cross section of reactionx, Exj is the macroscopic cross section that corresponds to the homogenization region and energygroup g for the j-th event, and 4j is the flux. In Monte Carlo, there are different estimators of theflux and therefore the j-th event may correspond to a collision or a free-flight travel. Therefore thenumerator is just a reaction rate which can easily be determined from built-in tallies that Serpentprovides. For all of its tallies, Serpent uses a collision estimator when operating in Delta trackingmode. The collision estimator of the flux (volume integrated and unnormalized) is calculated by
-Wi
# - (1.22)
where wj is the weight of the neutron and Et, is the total macroscopic cross section that corre-sponds to the region and energy group for the j-th event. If Serpent is using surface trackinginstead of Delta tracking, it uses a path length estimator of the flux given as
#j = wjdj, (1.23)
where dj is the path length that the neutron travels between collisions. Serpent is an analog MonteCarlo code and therefore the weight of the neutrons is unity and does not change throughout thecalculation. Note, in order to get the final estimate of the flux, #j in Eqs. (1.22) and (1.23) needsto be divided by total initial source weight and volume of the homogenization region.
The homogenization method discussed above works for general macroscopic cross sectionsneeded such as total, absorption, fission and fission neutron production. In addition to these crosssections, a group transfer scattering matrix is also needed to characterize the percentage of neu-trons scattering from one energy group to another. Serpent calculates the probability of a neutrontransferring from one group to another by using an analog estimator. It counts how many timesa neutron transfers from one energy group to another and normalizes it by the total amount ofscattering collisions from each energy group. The group transfer cross section in a certain homog-enization region can then be represented as
25
D 171
Es,h->g - EshPh-+g, (1.24)
where Es,h-g is the macroscopic group transfer scattering cross section between group h and groupg, Es,h is the homogenized macroscopic scattering cross section for energy group h and Phg is theprobability of scattering from group h to group g.
An important parameter to calculate in diffusion theory is the diffusion coefficient. This isprobably one of the most difficult parameters to calculate, especially in Monte Carlo. The rigorousdefinition of the group diffusion coefficient is
Dg9 v fVd 3rf dED(-,E)V# (-r,E)D =rdE (TE) .(1.25)Sfy d3 r f dEV 0 (-r, E)
In Eq. (1.25), Dg is the spatial and energy group homogenized diffusion coefficient, D ('r, E) is thediffusion coefficient as a function of space and energy and V# (', E) is the gradient of the spaceand energy dependent flux (Duderstadt and Hamilton, 1976). This relation is very similar to thedefinition of a homogenized cross section from Eq. (1.21) except for the gradient of the flux. Toget an accurate homogenized diffusion coefficient, the space and energy dependent diffusion co-efficient must be weighted by the current spectrum. Unfortunately, this is very difficult in latticecodes because of the imposition of zero net current boundary conditions. In Serpent, there are twomain methods for estimating the diffusion coefficient from flux-weighted homogenized parame-ters. The first definition is derived from the Pi Equations (Bell and Glasstone, 1970), where thegroup diffusion coefficient can be defined as
1 1D9 = (1.26)
3Etr,g 3 (Etg - Es1,g)
In Eq. (1.26), Dg is the group diffusion coefficient, Etr g is the macroscopic transport cross section,Et,g is the total macroscopic cross section and Esig is the first moment of the scattering crosssection. The first moment of the scattering cross section is defined as the average cosine of thescattering collision angle, jo,g, multiplied by the zeroth moment of the scattering cross section,Eso'g,
Es1,g = Ao19Eso1g. (1.27)
Since neutrons and scattering events are being simulated explicitly, the scattering cosine can betallied and the average can be taken at the end of the calculation for a specific homogenization re-gion and energy group. Once this parameter is known, the Pi definition of the diffusion coefficientcan be calculated.
The next definition of the diffusion coefficient in Serpent is more of a physical interpretationand is derived from the diffusion length. The diffusion length characterizes the distance that theneutron travels before it is absorbed. If this distance is denoted for energy group g by rg, then themean square distance is related to the diffusion length, Lg, by (Lamarsh, 1966)
= 6L 2 (1.28)
The group diffusion coefficient can then be calculated by
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Dg = (1.29)r1g
where Erg is the removal cross section defined as
Er g = g- Es-,g-+g, (1.30)
The within group scattering cross section, Eslg-,g, represents the probability of a neutron in energygroup g undergoing a scattering collision and leaving the collision with an energy also in energygroup g. If the homogenization were taking place over one group, the removal cross section wouldthen naturally reduce to the absorption cross section. Eqs. (1.26) and (1.29) were the original def-initions of the diffusion coefficient in Serpent. It was decided that the best diffusion coefficient formultiregion homogenization would be the Pi definition. This is because the mean square distancedefinition of the diffusion coefficient would correspond to neutrons traveling over multiple homog-enization zones, and therefore would be very difficult to characterize the diffusion coefficient forone of the regions. The Pi definition has more of a local definition and can be applied specificallyto a single zone.
The models for diffusion coefficients defined above are different than what deterministic latticephysics codes use. Current deterministic codes calculate the diffusion coefficient from a B1 fun-damental mode calculation (Stamm'ler and Abbate, 1983). This fundamental mode calculation,also called critical spectrum calculation, is important to get a better approximation of the neutronenergy spectrum that the geometry will see in the core. Since lattice calculations are performedwith zero net current boundary conditions, the system will not be critical and the contributionof neutrons produced from fission will be either overestimated (k > 1) or underestimated (k < 1)(Fridman and Leppanen, 2011). This will inevitably affect the homogenized parameters as they areweighted by the neutron flux spectrum. The overall goal of the B1 fundamental mode calculationis to iterate on a buckling term until the multiplication factor is unity. To accomplish this, crosssections are homogenized over a few thousand groups and used as a starting guess in the B1 equa-tions (Hebert, 2009). The critical flux and current spectrum are then determined from solving theB1 equations where k-effective is iterated to unity. The cross sections can then be re-homogenizedwith the critical spectrum using Eq. (1.21) and the group diffusion coefficients can be calculatedfrom the current spectrum with
JgD = (1.31)
where Jg is the group current, B is the buckling and #g is the group flux from the critical spectrum.This procedure has been implemented into Serpent and verified for steady state calculations (Frid-man and Leppanen, 2011). Unfortunately, the critical spectrum calculation is not available in thedepletion module yet and therefore the Pi definition will be used in this work.
The last parameters that are necessary for steady-state multigroup diffusion theory are the groupfission spectrum, assembly surface discontinuity factors, assembly corner discontinuity factors andgroup form functions. The fission spectrum parameter defines the fraction of neutrons that areemitted from fission into each energy group and is often referred to as the X - spectrum. The groupfission spectrum can be calculated as
27
fv d rg d E f7d E'L j zj (E) v2jf (r, E') ) (r, E')Xg = , (1.32)
fv d3r f; dE' E; vE2 (-, E') p (-, E')where Xg is the group fission spectrum, X' (E) is the energy dependent fission spectrum for isotopej, E' represents the energy of neutrons causing fission and E represents the energy of neutronsborn from fission. This is calculated in Serpent by counting the number of fission neutrons thatare emitted into each energy group and is similar to the calculation of group to group transferprobability in scattering.
As will be described in Section 1.4, surface discontinuity factors are necessary to get the bound-ary conditions correct in nodal diffusion theory. For conventional two-dimensional lattice calcula-tions, surface discontinuity factors and corner discontinuity factors are calculated as a ratio of theheterogeneous flux in an energy group to the spatially homogenized flux in that energy group. Ifzero net current boundary conditions are imposed, discontinuity factors can be expressed mathe-matically as
- v dr fg_1 dEO (, E)fi,g = (1.33)
V d3r ~g_1 dE p (-r, E )where fi,g is the surface or corner discontinuity factor, Vi is a small volume surrounding the surfaceor corner, V is the volume of the homogenization region and p (F, E) is the space and energy de-pendent flux. These parameters are relatively easy to calculate in Monte Carlo. The homogeneousflux (denominator) is just the total flux homogenized over the total geometry and within the energygroup. The heterogeneous flux can be calculated with the same methodology as the homogeneousflux except the volume is not the entire volume of the homogenization region; rather, it is justa small volume surrounding the surface or corner. There exists a trade-off between thickness ofthis small volume. If too small a volume is used, the variance of the estimate may be high andif too large a volume is used, the accuracy of the surface flux decreases. If there is a net leakagein or out of the homogenization region, Eq. (1.33) is not correct because the surface or cornerflux at an interface is not equal to the average flux in the homogenization region. This distinctionwill be important for the application of axial discontinuity factors presented in Section 4. In thesecases where leakage is present, the actual homogeneous flux at the surface or corner needs to becalculated.
The corner discontinuity factors and group form functions are used for pin power reconstruc-tion in the core simulator. As will be discussed in Section 1.4, nodal methods for full core analysesproduce limited information about intranodal flux distributions. Many reactor safety analyses re-quire detailed information about pin power distributions and therefore, pin power reconstructionmethods are needed to recover this information. The pin-by-pin distributions of flux or power areapproximated as
0g (X, y) reactor - g (xIY) hom 4Og (XY)form (1.34)In Eq. (1.34) 0g (X,y)reactor is the reconstructed pin-by-pin group flux distribution, 0g (x,y)hom isthe global homogenized intranodal group flux distribution and 0g (x, y) form is the local group formfunction. In the core simulator PARCS (Downar et al., 2009), a group-wise flux form function iscalculated using
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GFFg (x,y) = - -C2fg (Xy) ,g (Xy) (1.35)Kc2fgoPg
where GFFg (x,y) is the group-wise form function, KEfg (x,y) is the group g cell-averaged fissionenergy deposition at position (x, y), $g (x, y) is the group g cell-averaged flux at position (x, y),lcEfg is the group g assembly-averaged fission yield energy and Og is the assembly-averaged fluxof group g. The position (x,y) is defined as the center of the fuel pin (Xu and Downar, 2009). InPARCS, the group-wise form function can then be used to recover the local fission yield energy ofthe single assembly calculation. The corner discontinuity factors are needed to calculate the corner-point fluxes from the single assembly calculation and defined by Eq. (1.33). These corner-pointfluxes are needed as additional constraints in the pin-power reconstruction methodology (Rempeand Smith, 1989).
Serpent also generates important parameters for point kinetics and spatial kinetics calculations.For example, the point kinetics equations with six effective delayed precursor groups are
d (p(t)- (136dt n (t) = P A n (t) + AiCi (t) (1.36)
and
d #-Ci ( n (t) - AiCi (t), i 1, 2,..., 6. (1.37)
dt AIn Eqs. (1.36) and (1.37) n (t) is the neutron population as a function of time, p (t) is the reactivityof the system as a function of time, # is the total effective yield of delayed neutrons, Pi is theeffective yield of delayed neutrons for precursor group i, A is the mean neutron generation time,Ai is the decay constant of precursor group i and Ci (t) is the precursor concentration as a functionof time for group i. The mean neutron generation time is the average time between a neutronbeing produced from fission and subsequent absorption leading to a new fission reaction. This iscalculated in Serpent using
1A =(1.38)
VV~ 2f
where v is the velocity of a neutron and vEf is the fission neutron production cross section. Theneutron velocity, or in this case reciprocal velocity, can in general be calculated for an arbitraryenergy given by
1 fd r kg1 v(E) (7, E)- = (1.39)
vg fvdBr fgg_1 d Eo$(7,E)
where ± is the spatial and group homogenized reciprocal velocity and is the reciprocalvgv(Y,E) r cp o a
velocity as a function of space and energy. The decay constants for delay neutron precursor groupsare calculated from
fvd 3rfjdE~jZi# vEl (-,E)# (7,E)Ai =E.f (1.40)fv d3r ffdE Ej Xi vE- (ir, E) # (ir, E)
29
In Eq. (1.40) Aj is the decay constant of delayed neutron group i, #/ is the delayed neutron groupi delayed neutron yield fraction for isotope j, Aj is the decay constant of delayed neutron group ifor isotope j, vEJ (', E) is the fission neutron production cross section for isotope j and I (', E)is the neutron flux. The decay constant in Serpent is calculated by taking the average of sampleddelayed neutron emission times.
The last factor to be calculated is the delayed neutron yield fraction, #. For a given isotope, thedelayed neutron yield fraction can be calculated with
#i (E) = _ (1.41)vp (E) + vd
where #i (E) is the delayed neutron yield of precursor group i, Vd is the number of delayed neutronsemitted from precursor group i and V, (E) is the number of prompt neutrons emitted (Duderstadtand Hamilton, 1976). The total delayed neutron yield fraction is the sum of the individual yieldfractions for each precursor group, # (E) = Ei3i (E). Although these yields do not strongly dependon energy, it is suggested to average their values over the entire energy spectrum and geometry with
fv dr f0 dE E ;j x1i#/vYE (i, E) # (7, E)
fVd 3r LdE EX }vE (7,E) # (-,E)
In Eq. (1.42) 0,i is the physical delayed neutron fraction, vEf1 (', E) is the macroscopic fissionneutron production cross section for isotope j, which is a function of space and energy, Xi is thedelayed neutron fission spectrum, Xi is the total fission spectrum of isotope j and # (', E) is thespatial neutron flux energy spectrum (Henry, 1975). The reason the delayed neutron fraction isdenoted as physical is because it does not distinguish between the importance of delayed neu-trons relative to prompt neutrons. For thermal reactors, fission occurs primarily in thermal energygroups. To reach these low energy groups, neutrons must pass through the resonance region. Sincedelayed neutrons are emitted at relatively lower energies than prompt neutrons on average, they aremore likely to escape resonance absorption and are worth more to core multiplication. To calcu-late the appropriate average delayed neutron yield over the energy spectrum, the delayed neutronfractions must be weighted by the importance of these delayed neutrons at their lowest emittedenergies, given by the adjoint flux energy spectrum. The appropriate spectrum averaged delayedneutron yield, referred to as effective delayed neutron yield, is additionally weighted by the adjointflux,
fv d3 r fdE 0t (-, E) E Xi#jvEj (-, E) # (-, E)effi = , d3f(1.43)
f -fyd3 r fdEot (,E) Ex'vE (,E) # (F,E)
where #effi is the effective delayed neutron yield fraction for precursor group i and $ (', E) isthe adjoint flux energy spectrum (Bell and Glasstone, 1970). This method of calculating effectivedelayed neutron yield fractions is very difficult in Monte Carlo analysis, especially if continuousenergy data are used. To circumvent this, Serpent interprets this parameter differently by takingthe effective delayed neutron yield fraction as the contribution of delayed neutrons to the fissionreaction rate (Leppanen, 2007a). This method in Serpent can be performed by taking the ratio ofthe average number of fissions generated by the delayed neutrons to the average number of fissions
30
generated by all neutrons. This method has shown good results against experimental data and doesnot need any extra Monte Carlo runs (Meulekamp and van der Marck, 2006).
1.4 Full Core Calculations
The multigroup homogenized parameters discussed in Section 1.3 are used as input into a coresimulator. The core simulator used in the analyses presented in this work is the U.S. NRC codePARCS (Downar et al., 2009). Therefore, the homogenized parameters calculated from Serpent inthis work are tailored to this code. The PARCS simulator is used to predict the behavior of nuclearreactors under steady-state and transient conditions. It has the capability of solving core eigenvaluecalculations or eigenvalue searches by varying soluble boron concentration or control rod insertionheight. PARCS has many solution kernels such as time-dependent neutron diffusion solvers and
SP3 transport solvers. More importantly, it can solve the nodal diffusion equation in Cartesian andhexagonal geometry. For a typical PARCS calculation with coupled thermal hydraulic feedbackand depletion, many modules are needed as shown in Fig. 1.5.
The solution scheme presented in Fig. 1.5 is complex, but each component is necessary forthe solution and will be discussed. Section 1.3 described the process of homogenization of crosssections and focused on the lattice code process. The lattice code will need to be executed toproduce branch cases to create a cross section database. This is described separately in more detailin Section 3.1. Currently, PARCS accepts cross section databases from the lattice codes HELIOS(Wemple et al., 2008), CASMO (Studsvik, 2009) and TRITION (DeHart et al., 2003). The outputfrom these codes is sent to the processing module GenPMAXS (Xu and Downar, 2009) where aPMAXS cross section database is generated. This list of lattice codes has now been extended toinclude Serpent as described in Section 3. The PMAXS cross section database file can then beprocessed by the cross section module in PARCS. The cross section module is used to producenode-wise cross sections for each computational node. Since in general, each node will havedifferent neutronic and thermal hydraulic conditions, these node-wise cross sections will differ.The neutronics calculation can then use these node-wise cross sections to solve the nodal diffusionequations.
On the full core level, the nodal diffusion equations are solved over homogenized volumes asshown in Fig. 1.6. On this level, the fuel pins and spatial detail inside the assembly are not mod-eled, just the homogenized assemblies. Each box is a fuel assembly and extends axially into thepage. The axial length of the fuel assemblies is sub-divided even further in axial nodes. The ho-mogenized cross sections generated from the lattice calculations are then linked to specific nodesin the geometry. The three-dimensional flux distribution of the full core is solved with the multi-group diffusion equations. The time-dependent Pi form of the multigroup diffusion equation foran arbitrary node n can be represented by
vtogt 't) =-V -Jg ( $g (1.44)
Vg 9tt ,,t
ig (r, t) D" (t)rVg(,t) (1.45)
and
31
Figure 1.5. PARCS Solution Scheme for Coupled Thermal Hydraulic Feedback with CoreDepletion (Downar et al., 2009)
MUEn
EOl L k1 I k1
H~~~ FEE]l
soL W1-1 _uLLF 7on~_§ T nmfEET~f-Vi [ llHI67l I 7_7 I I I I I 71IE- i
LiL-uIE -]WDauu~_]-no ! _ II I l EE iF] '_.E1-IIIIIIIIIII- I Tm n I LMa M ImJ
171 1 J -O I 1! 11 i1 im1 1- IluL ud IN mI ILL1- [lMm
SMEME
Figure 1.6. Quarter Core BWR Geometry used in Full Core Analyses
d rG8tci(7,t) = -Xici(,t)+#eff,i E VE" (t)g (,t). (1.46)
g =1
Equation (1.44) represents the change in the neutron flux where vg is the group neutron velocity
for energy group g, #g (7, t) is the spatial and time dependent group flux, .g (', t) is the spatial and
time dependent group net current, ,g (', t) is the macroscopic removal cross section for node n as
a function of time, and qg (7, t) is the space and time dependent group source. Eq. (1.45) is Fick's
Law where the neutron current and gradient of neutron flux are related by the time dependent groupdiffusion coefficient, Dg (t). The removal cross section can be defined as the difference between the
group macroscopic total cross section, E" (t) and the within group macroscopic scattering cross
section, Eng, ()
Erg (t) -2g (t) g_,+ (t). (1.47)
The group source is represented by the scattering transfer matrix, fission source from prompt neu-trons and source from delayed neutrons where
32
; Code
G
qg (7,t) = I -iE"hg (t) Oh (, t) + (1 -feff) X0g Vfh (t) Oh (Vt) (1.48)h7 g h=1
I
+ A2Li ig Ci(, t).i=1
In Eq. (1.48), qg (', t) is the group source, E"h-g (t) is the macroscopic group transfer scatteringcross section, feff is the total effective delayed neutron fraction, Xog represents the fraction ofprompt neutrons emitted from fission in group g, vEfh (t) is the group fission neutron productioncross section, At is the decay constant for delayed neutron precursor group i, Xig is the fraction ofneutrons from group i delayed neutron precursors that are emitted in group g and ci (7, t) is the timedependent spatial distribution of delayed neutron precursors. In Eq. (1.46), the #eff,i representsthe fraction of effective delayed neutrons that are from delayed neutron precursor group i (Suttonand Aviles, 1996).
To solve these diffusion equations, it is common to employ a transverse integration procedure.In Cartesian coordinates, the diffusion equations are decoupled by integrating over the transversedirections. The solution of these one dimensional equations yields node-average fluxes and cur-rents that are necessary to solve the nodal equations. As an example, Eq. (1.44) can be integratedover the y- and z- directions to give the transverse integrated equation for the x- direction, yielding
+'x (xt+Eg Ox"(xt)+ v g -g = L" (xt). (1.49)
In Eq. (1.49), jg (x, t) is the transverse-integrated current in the x-direction, 'Pg (x, t) is thetransverse-integrated flux and q", is the transverse-integrated source. For example, the transverse-integrated flux in node n is calculated by
1 py"+-h"|2 z+' 1h/2-g , ,) = dy dzg (x, y,z,t), (1.50)
Yh y" "2 zh|2
where h, and hz are the dimensions of the node in the y- and z-directions, respectively. Although thespatial dimensions have been decoupled, a transverse leakage term appears, L", (x, t), as shown inEq. (1.49), to account for the neutrons leaving the node in the transverse directions. The transverseleakage term is represented by
1 z" + h"2 n + h,"|2, z, t) -- Jg, (x1yn - h,"|2,z, tL"x, t) =dzjg (1.51)
1h" z"-h|2 ha(1 y"+hn|2dy jgz (,yZ" +hn"2,t) - Jgz (x,y,z"n - h"|2,t)
hn Jn-h"| hnY fy hy/2
To solve this equation, approximations need to made for the transverse leakage along each di-rection. Some of these approximations include a flat transverse leakage, approximating it with abuckling, or a quadratic approximation (Sutton and Aviles, 1996).
33
The nodal equations (Eqs. (1.44)-(1.46)) can then be solved with various methods such as
polynomial expansion methods or analytic methods (Lawrence, 1986). PARCS uses nodal meth-
ods to obtain higher order solutions to the neutron diffusion equation. It uses a coarse mesh finite
difference (CMFD) formulation and then solves a two node problem with either the analytic nodal
method (ANM) or nodal expansion method (NEM). For hexagonal geometry, the nodal equations
become more difficult because the dimensions of the geometry are not in primary Cartesian direc-
tions. Therefore, the hexagonal module in PARCS does not use a transverse-integrated procedure
and solves the two-dimensional problem directly. The CMFD formulation is consistent with the
Cartesian method, however, the nodal coupling kernel uses a triangular polynomial expansion
nodal (TPEN) method (Downar et al., 2009; Pounders et al., 2007).The neutronics calculation produces a three-dimensional spatial distribution of node-averaged
group fluxes. Also supplied in the cross section database is group-wise energy deposition cross
sections. The spatial distribution of group fluxes is just an eigenvector of the system and therefore
must be normalized. To accomplish this, node-wise power can be calculated from these energy
deposition cross sections and compared to the actual power of the system. The power calculated
from the unnormalized fluxes is
N G
Pcalc = 1E K1 Vi, (1.52)n g
where Pcalc is the calculated power, KE",n is the energy deposition cross section interpolated from
the PMAXS database for each group and node, og" is the group flux in node n and Vn is the volume
of node n. In Eq. (1.52), the double sum represents the summation over all energy groups, G, and
over all nodes, N. To scale the fluxes appropriately, the normalization constant can be found by
dividing the true power of the reactor by the calculated power. Once known, the node-wise power
can be calculated summing up the energy deposition reaction rate over all energy groups in a single
node.The node-wise power is used for two different purposes. First, the power can be used as input
for a thermal hydraulic systems code such as RELAP5 (RELAP5-3D, 2005). The solution of the
thermal hydraulic equations in these codes results in a new set of thermal hydraulic operating
conditions for each node. These new node-wise operating conditions are then used by the cross
section module in PARCS to interpolate on new node-wise cross sections as a function of thermal
hydraulic conditions. This thermal hydraulic coupling is very important for reactor safety analyses
where the thermal hydraulic feedback on core reactivity is important. The node-wise power is
also important to calculate the burnup distribution of the core as well as the core-average burnup.
These analyses are extremely important for fuel-management calculations. Once the node-wise
power distribution is known, the local change in burnup of a node can be calculated by
ABn = PAT (1.53)Gn
In Eq. (1.53) ABn is the change in local burnup of the n-th node, Pn is the power of node n, Gn is
the initial mass of heavy metal present in node n and AT is the time step change. The initial mass
of heavy metal can be calculated from the volume of the node and heavy metal density of the node
which is present in the cross section database file. The node-wise change in burnup is then used
for a history calculation of all of the node-wise operating conditions. The history of each operating
34
condition of a node is taken into account in the interpolation schemes present in the cross sectionmodule. In order to perform this interpolation of cross sections, a cross section database must begenerated from a set of branch cases. The methodology for performing branch case calculations tohomogenize parameters is described in Section 3.
1.5 Objectives
The objectives of this thesis are as follows:
- determine the adequacy of Serpent in the generation of few-group homogenized parameters(Section 2),
- investigate different homogenization processes with the three-dimensional capability in Ser-pent (Section 2),
- create a tool to automate the generation of branch cases in Serpent and couple to PARCS(Section 3),
- determine whether three-dimensional cross sections can reduce the number of energy groups(Section 4),
- study the feasibility of generating axial discontinuity factors to preserve reaction rates andmultiplication factor (Section 4).
35
2 Serpent Reactor Physics Burnup Code
2.1 Using Serpent for Cross Section Generation
The Serpent Monte Carlo code is new to the nuclear research community. There have been rela-tively few efforts for its application to the generation of cross sections for full core calculations,which was one of the major reasons why the code was developed. Due to very little experiencein the research community, many lessons-learned and a few bugs have been identified. Therefore,in this section, an overview of how to generate an input file and run the code is presented for theapplication of cross section homogenization.
2.1.1 Geometry Creation
Serpent has a universe-based geometry structure and therefore, the user must be aware of how alluniverses relate to each other. The universe-based approach to geometry creation is very powerfuland allows for a logical integration of sub-geometries. All space in each universe must be specifiedand universes can be placed inside of one another. In addition to geometry creation, universes arealso used to specify the spatial region for detector tallies. This also includes the specification ofgroup constant homogenization regions, which is the ultimate goal of the calculation. Using theset gcu card, a vector of universe regions may be listed where group constant generation will beperformed. This statement also implies that these homogenization universes must be on the samelevel and not overlap. Therefore, the user must have an idea of which regions to homogenize andhow to build the geometry appropriately.
For conventional homogenization, the common approach is to treat each unique two- dimen-sional slice of an assembly separately, surrounded by reflective or periodic boundary conditions.This approach is rather simple since there is only one group constant homogenization region, whichis the whole geometry. The group constant universe 0 is reserved for the global geometry and there-fore this number would be listed on the group constant homogenization card in the input file. Forthe homogenization of cross sections for the RBWR, a three-dimensional approach to homogeniz-ing cross sections is used. This means that cross sections are being generated for a homogenizationregion in the presence of other regions. The homogenization region in this case is not the full ge-ometry, rather, a sub-region as shown in Fig. 2.1. In this example, a single assembly calculationis performed in Serpent and group constants are generated for multiple zones. The approach thatwas taken in generating the RBWR full assembly geometry in Serpent is to characterize each ho-mogenization zone with a two-dimensional representation and then stack these two-dimensionalslices into a three-dimensional assembly. The homogenization can then take place over the fulltwo-dimensional slice in the radial direction and only over a portion of the full assembly in theaxial direction.
Each two-dimensional slice is created separately and begins with the declaration of all typesof pin-cells that are present in the lattice that is contained in a homogenization zone. Serpent hasbuilt-in geometry structures that can be taken advantage of by users. One of these structures is thepin. The pin structure is used to create a number of concentric annuli of cylinders surroundedby a material that stretches the rest of the space. When creating pin-cell geometries, a pin onlyneeds to be created if it is unique to that lattice or should be treated as a separate depletion region.For example, if only one type of pin-cell is contained in a lattice, only one of these needs to be
36
Figure 2.1. Three-Dimensional Homogenization Diagram of a Single Assembly
created for the whole lattice assuming that all pins will be depleted as one material. There are
additional features for declaring materials which are explained in Section 2.1.2. Each pin-cell type
is given an unique number identifier. This number is actually a universe and the pin-cell structure
automatically defines all space. Note that a separate pin-cell should be created for each lattice
present in the geometry as Serpent will distribute the material volume on a lattice-by-lattice basis
for burnup calculations.Serpent contains an array of built-in lattice types including square, two hexagonal orientations
and an axial lattice. Similar to other Monte Carlo codes, the pin-cell universes can be placed in
these array types, which for the current application is a square or hexagonal lattice. This lattice
also has an unique universe identification. At this point in the geometry generation, a number of
two-dimensional slices of lattices exist and need to be stacked on top of one another to build the
full single assembly. These universes can be stacked using the axial lattice structure in Serpent.
However, in Serpent one lattice may not be directly placed in another and therefore all of the lattices
must be placed into a cell with a separate universe identifier. At this point, the radial direction is
not important and therefore the cell type inf is used, and each lattice is placed inside using the
f ill command. The universes representing these cells can then be stacked in an axial lattice.
When Serpent performs the homogenization, the universe identifier that is attached to the cell card
is used and the code homogenizes that specific three-dimensional space in the axial lattice.
After the axial lattice is constructed, global boundaries may now be created to surround the
geometry. There are two options for axial surrounding surfaces; reflective or vacuum boundary
conditions. It is assumed that the radial direction will have either reflective or periodic boundary
conditions, and this is chosen with the set bc card. If reflective boundaries are to be used axially,the surface types cube, cuboid, hexxprism or hexyprism must be used. If vacuum boundary
conditions are used axially, surface types sqc, hexxc or hexyc must be used. The universe of this
global cell must be labeled as 0. To declare void outside of the assembly, the outside command
is used for the material specification. An example of this method for geometry construction for the
RBWR application is highlighted further in Section 2.4 and an input file is listed in Appendix A.2.
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2.1.2 Material Specification
The specification of materials in cells is very similar to the MCNP5 Monte Carlo code (X-5 Monte
Carlo Team, 2003). In Serpent, a material can be specified with the mat card. Each material is
given an unique name instead of a number as in MCNP5. Following this, the mass density or atom
density is specified as a negative or positive value, respectively. In other codes such as MCNP5,this density is listed in the cell card. In Serpent, if the same material composition has multipledensities, multiple material cards need to be created.
A nice feature of Serpent is that if the same material with the same density is to be used for
multiple depletion regions, separate material cards do not need to be created; only multiple pin
structures need to be generated. This is because as a default, Serpent automatically duplicates the
material for each unique pin-cell type and renames it to < matname > p < pinu > r < ring# >.In this nomenclature, < matname > is the original material name, < pinu > is the pin-cell universe
where the material is located and < ring# > is the ring location. Another feature of Serpent is that
the inner most cylinder in the pin-cell structure can automatically be divided into equal volume
rings. This is especially convenient for gadolinium burnable absorbers where the self-shieldingduring depletion must be tracked with more detail. Finally, if the temperature of the material is
not the same as the included cross section libraries, a different temperature may be specified in the
material card. When Serpent begins, it Doppler broadens the cross section sets to this specified
temperature before the transport calculation begins. If a burnup calculation is being performed,the volume of the material must also be specified. When running Serpent with a two-dimensionalgeometry in burnup mode, the volume or area in this case need not to be specified as the code will
be able to calculate this parameter accurately. However, if a depletion case is being performed
with a three-dimensional geometry, the user must supply the appropriate volumes of depletablematerials.
Finally, the isotopic composition of materials is specified in a list along with the isotope's
weight or atom percent. Similar to MCNP5, the isotopics are identified in ZAID format with an
appropriate extension that directs the code to import the appropriate cross section library. Note
that as more cross section libraries are used, the memory needed to perform the calculation will
increase. This is further explained in Section 2.1.3 on burnup calculations. From experience usingSerpent, these calculations become memory limited very quickly. In addition, thermal scattering
libraries can also be specified on the material cards.
2.1.3 Burnup Calculations
The basic steps of burnup calculations is to (a) track the effective multiplication factor and isotopics
and (b) generate a set of homogenized cross sections for a list of burnup values. To perform thesecalculations, the Monte Carlo transport solver is used to generate a set of one group reaction
rates that are needed for the Batemann equations used in the depletion analysis. As is commonin the nuclear field, Monte Carlo codes are coupled to a separate depletion solver externally, and
a wrapper code is placed around them. Some notable code packages used at MIT are MCODE(Xu et al., 2002), BGCORE (Fridman et al., 2008b) and MONTEBURNS (Trellue and Poston,1999). All of these code packages use MCNP5 as the transport solver and ORIGEN (SCALE,2005) as the depletion tool, except for BGCORE which has its own depletion solver. Serpent alsohas a built-in subroutine for depletion analysis. The depletion analysis can be performed with a
38
predictor-corrector algorithm to get a better estimate of isotopics at each time step. These newisotopics are then used in the next set of transport calculations to get another estimate of one-groupreactions for the subsequent depletion time step. In Serpent, these time steps can be either specifiedin units of burnup or days. Both parameters are available in the output file and provide a good wayof checking for consistency against hand calculations.
In Serpent, there are two ways of tallying one group reaction rates and one group cross sections.The most accurate way is for the code to tally each type of reaction rate for each isotope in eachburnable material. Although accurate, this method requires a significant number of tallies and tendsto slow the calculation down. Despite the larger computational time, this method is one of the mostprevalent methods for depletion analysis. Another method that has proven successful in BGCOREis to just tally the energy spectrum and compute reaction rates and one group cross sections usingEq. (1.1) (Fridman et al., 2008a). Therefore, only one tally of the flux energy spectrum needs tobe placed in each burnable region. This results in fewer tallies and will be computationally fasterthan the other method.
Serpent tracks all nuclides in the depletion analysis and uses the results for isotopics in thenext transport cycle. Although all nuclides are tracked, only a portion of them are used in the nexttransport cycle according to their concentration with a minimum cutoff concentration specifiedby the user. To solve the depletion matrix, Serpent has three main methods: (1) TransmutationTrajectory Analysis (TTA), (2) Chebyshev Rational Approximation Method (CRAM), and (3) anextended TTA method. In the burnup applications present in this work, the CRAM method is usedas it is a very efficient way of solving the matrix exponential, which represents a solution for thesystem of the Batemann equations (Pusa and Leppanen, 2010).
To get the correct normalization of the flux, Serpent has a built-in method to scale the reactionrates and flux tallies appropriately. The user may set power density, total power or fission rate. Itis important when running the calculation to check the results to ensure that all normalizations areconsistent. The best method to do so is to compare the vector containing burnup values to the timevector. These values should be consistent against hand calculations using a burnup formulationsimilar to Eq. (1.53). The key is to make sure that the initial heavy metal mass computed bySerpent matches the expected heavy metal mass from hand calculations. Another useful tool isto run the checkvolumes routine in Serpent which gives a listing of the volume and fissile masscontent of each burnable region.
2.1.4 Detector Tallies
Serpent has the capability of user-specified detectors or tallies to gain additional information aboutthe calculation. This is similar to tallies in other Monte Carlo codes, except in Serpent only the fluxin a volume cell may be computed. When Serpent is in delta tracking mode, a collision estimator isused to calculate these tallies, when surface tracking is invoked, a path length tally is used. In bothof these tallies, the quantity needs to be divided by the volume of the cell. Currently, Serpent doesnot do this automatically and the user must supply the dv card with the appropriate cell volumelisted. This is a common confusion among users of the code when their flux estimates may be offby a factor of 100 or even 1000 for large geometries.
In addition to straight flux calculations, reaction rates may also be calculated using the dr card.There are two types of rates that can be calculated; those over a volume containing a number ofmaterials, and those for a certain isotope in a material. The different types of reaction modes are
39
presented in the user manual where a negative number is used for the first type of rate and a positivenumber, corresponding to the ENDF identifier, is used for the second type of rate. For most of theapplications, the total reaction modes over a whole cell containing different materials are needed.Two specific reaction modes are labeled total fission neutron production and total fission energydeposition. These reaction modes may be used to calculate the fission neutron production rate orenergy deposition rate. In both of these cases, the macroscopic fission cross section is needed andis automatically assumed in this reaction mode. Note that in the calculation of energy depositionrate, approximate heating values are used for each reaction with an isotope and may be changed ifneeded.
The detectors can also be discretized in energy and space. When computing the flux energyspectrum, an energy grid may be specified. In Serpent, there is a listing of common energy gridstructures that may be used. In addition, one can specify an energy grid of equal lethargy binsor specify a custom grid of arbitrary energy group boundaries. Finally, the spatial domain maybe meshed into smaller volumes over which tallies can be computed. In this work, tallies will bepredominately used to calculate reaction rates, power distributions and flux profiles.
2.1.5 Other Features
There are many other features that have been included in Serpent to relieve some of the work of theuser. One nice feature about Serpent is that the input file is not structured and input cards may bespecified in any order. As part of the input, the user can generate pictures of the geometry using theplot command and power distributions using the mesh command. The geometry pictures can beplotted before the execution of Serpent using the -plot command line option. These help the uservisualize the geometry that is being created. Unfortunately at this time, an interactive geometrygenerator has not been developed, but is an item for the future.
Another feature of Serpent is that it is "parallelized". The reason why parallelized is in quota-tions is because the current algorithm has some limitations. In Monte Carlo analyses, the phraseparallelized usually means that no matter how many processors are being used to solve the prob-lem, the results are exactly reproducible. This takes some extra effort to ensure that the randomnumber sequence is preserved for an arbitrary number of processors. This also means that theinput file does not need to be changed from run to run. These are very important qualities for aMonte Carlo code, especially for quality assurance purposes where results need to be reproduced.Unfortunately, Serpent does not do exactly this task. When Serpent runs in parallel, it actually willrun completely independent calculations and statistically combine the results among all of the runsfor the user and present them in the output file. Therefore, this approach gives the user an idea ofthe real variance of the mean values presented in the output file instead of the apparent variancefrom the statistics of one calculation. Another important fact in these calculations is that sourceneutrons per batch are divided equally among all processors. Therefore, source neutrons per batchmust be scaled with the number of processors to ensure that enough neutrons are sampled for eachcalculation.
Finally, Serpent has probability tables for sampling neutron cross sections in the unresolvedenergy region. Invoking these probability tables is a must to get accurate results, but this fea-tures slows the calculation down significantly. This problem is exacerbated in burnup calculationswhen there are hundreds of fission product and actinide isotopes used in each calculation. Sincethe concentration of these nuclides is relatively small compared to initial fuel constituents, it can
40
be assumed that these isotopes are infinitely diluted. Therefore, the user can list a vector of iso-topes to which probability tables are assigned to. This will allow accurate results and minimizecomputational time to an appropriate level.
2.2 Description of Lattice Codes
A number of lattice codes were chosen to compare with Serpent. The selected codes are estab-lished in the industrial and academic communities. Because Serpent is a relatively new code andhas a small user community, it was important to ensure that the code was being used properly.For the comparisons, both deterministic and Monte Carlo codes were selected. The deterministiccodes include CASMO-4E, developed by Studsvik Scandpower, and Dragon, developed by EcolePolytechnique Montr6al. The Monte Carlo code, MCNP5, developed by Los Alamos NationalLaboratory, was selected to compare against the Serpent Monte Carlo code.
2.2.1 Deterministic
CASMO-4E is an extended version of the CASMO-4 lattice code. It is a multigroup two-dimensional lattice code that solves the neutron transport equation with the method of character-istics. It is primarily used to generate homogenized cross sections of BWR and PWR assembliesas well as pin cells, for nodal core simulators. This code is used primarily by the nuclear industry,but has been made available for limited university use (Studsvik, 2009).
Dragon v4.0.3 is another lattice code that is used by both the industrial and academic com-munities. The Dragon code was developed for homogenization of CANDU assemblies, but hasthe capability to model pin cells and other types of lattices. Although it has three-dimensionalgeometry capability and multiple transport solvers, it is primarily used for two-dimensional ho-mogenization in which the neutron transport equation is solved using collision probability method(Marleau et al., 2011).
2.2.2 Monte Carlo
MCNP5-1.51 is a general-purpose Monte Carlo N-Particle code. It can be used for neutron,photon, or electron transport or any coupled transport of these particles. The code solves fixedsource as well as eigenvalue problems. Although it may not be considered a lattice code, it hasthe capability of analyzing reactor fuel lattices. Additionally, it is one of the most well-developedMonte Carlo code and enjoys a large user base. The physics in MCNP5 and in Serpent are verysimilar and this should be the most insightful comparison (X-5 Monte Carlo Team, 2003).
BGCORE is a software package for simulation of reactor systems and the fuel cycle. It in-terfaces with MCNP5 for transport analysis and uses a built-in module for performing burnupcalculations. It has been validated against well-established computer codes and has shown verygood agreement in k-effective and nuclide density predictions (Fridman et al., 2008b).
Serpent v1.1.14 is a reactor physics burnup calculation code that is designed specifically for thegeneration of homogenized multigroup constants for deterministic full core reactor simulations.Although it specializes in two-dimensional lattice physics, it has a universe-based geometry that
41
allows for three-dimensional geometries. Although Serpent is a newer code, it may become widelyused since it has many unique capabilities that make it attractive for generation of multigroupconstants (Leppdnen, 2010b).
2.3 Two-Dimensional Pin-cell Depletion Comparison
The purpose of this analysis is to show how the transport physics built into Serpent compares toother lattice physics codes described in the previous section, and that it is capable of performingaccurate few-group homogenization of cross sections. The comparison also shows that the method-ology of creating input files and overall execution of Serpent is correct. The eigenvalue, importantnuclide number densities and macroscopic cross sections are compared with other lattice physicscodes. All of the lattice codes use different initial neutron data libraries, all data libraries wereexecuted in Serpent and are presented in this work.
A simple pin-cell containing fuel and moderator, with pressurized water reactor-like conditionsand geometry was used to compute a thermal energy spectrum. The geometric dimensions andoperating conditions are listed in Table 2.1. A diagram of this geometry is shown in Fig. 2.2. Forthe Monte Carlo analyses, 500 active batches and 50 inactive batches were simulated with 5000neutrons per batch. The Monte Carlo runs required approximately a day and a half to completeusing one processor for the Serpent calculations and four processors for the MCNP5 calculations.The eigenvalue for each burnup step from all of the lattice codes was extracted and is shown inFig. 2.3. The plot of multiplication factors shows that the overall trend over depletion agreeswell. The initial drop in the multiplication factor in each data results from the insertion of negativereactivity due to xenon- 135 buildup. To understand differences between each of these calculations,the difference in k-effective is shown in Fig. 2.4 using CASMO4E as the reference. The differencesbetween the codes and neutron libraries range up to a significant 800 pcm. A positive differencerepresents a k-effective larger than CASMO4E. The Dragon results yield the best comparison withthe CASMO4E results. The transport physics in both of these codes are probably similar sincethey are both well-developed deterministic lattice physics codes. There is however a large increasein the difference at the beginning of the calculation and can likely be attributed to an incorrecttreatment of fission product buildup. The Dragon simulation was not run with a predictor-correctorprocess, but from the results, the system corrects itself and k-effective comes into good agreementonce fission products are in equilibrium. The Monte Carlo simulations do not agree as well with thedeterministic simulations. This is most likely due to differences in transport physics. However, the
Table 2.1. Geometric and Operating Conditions of Pin-cell
Fuel radius [cm] 0.4096
Fuel pitch [cm] 1.26
U0 2 enrichment [%] 4.5
Fuel density [g/cm 3] 10.4
Coolant Density [g/cm3 ] 0.660
Power density [W/g] 38.6
42
Figure 2.2. Pin-cell Geometry
1.45-CASMO4E_ENDF/B-VI.8
1.4 -DragonENDF/B-VII.0
-BGCORE(MCNP5_JEFF3.1)1.35 -Serpent ENDF/B-VI.8
-SerpentENDF/B-VII.01.3 -Serpent JEFF-2.2
1.25 -Serpent JEFF-3.1-Serpent JEFF-3.1.1
$ 1.2
1.15
1.1
1.05
1
0 5 10 15 20 25 30 35 40
Burnup [MWD/kglHM]
Figure 2.3. Comparison of k-effective for Different Lattice Codes
43
Moderator
Figure 2.4. Difference in k-effective between Several Codes and CASMO4E
Serpent calculation with the same neutron library as CASMO4E yields the best results. One majordifference is that CASMO4E treats resonance upscattering from uranium-238 whereas MCNP5and Serpent do not (Becker et al., 2009). To fully understand these differences, a more in-depthinvestigation is needed and is outside the scope of this work. This comparison gives an ideaof the differences that may be seen when comparing the multiplication factor from Serpent toconventional deterministic lattice codes.
In addition to the eigenvalue comparison, additional important nuclide densities were com-pared. The percent differences in nuclide densities from CASMO4E were compared for uranium-235, plutonium-239 and xenon-135. These differences are shown in Figs. 2.5, 2.6 and 2.7, re-spectively. In each of these diagrams, the CASMO4E results are larger than the other codes. Theunder-prediction of the xenon- 135 in Serpent and BGCORE lead to a larger capture absorption rateof neutrons resulting in higher multiplication factors. In the Serpent calculations, the ENDF/B-VI.8 and JEFF-2.2 neutron data libraries agree well for the uranium-235 and plutonium-239, butthe JEFF-2.2 has a lower xenon-135 prediction leading to a higher k-effective. In this work, theENDF/B-VII.0 neutron data library is primarily used with Serpent.
Finally, a comparison was made of the homogenized macroscopic total and fission neutronproduction cross sections over all space and energy. Since the ultimate goal of using Serpent is togenerate homogenized data for full core simulations, it is important to see how it compares to theconventional two-dimensional deterministic lattice codes. In this comparison, Serpent and Dragonare compared using the ENDF/B-VII.0 neutron data library. The same geometry and operatingconditions are used in this comparison and are listed in Table 2.1. In Serpent, the homogenizeddata is reported automatically for the entire geometry and energy by default. In Dragon, a reactordatabase had to be created manually using burnup as the global indexing parameter. A comparisonof the total cross section is shown in Fig. 2.8, and the fission neutron production cross section is
44
Comparison of Uranium-235 Number Density versus CASMO4E
CASMO4E
45
0.OOE+00
Rt -5.OOE-010
-1.00E+00
E2 -1.50E+00
-2.OOE+00
-2.50E+00EZ -3.OOE+00
-3.50E+00
0 5 10 15 20 25 30 35 40
Burnup [MWD/kglHM]
-BGCORE(MCNP5-JEFF3.1) -SerpentENDF/B-VI.8-SerpentENDF/B-Vil.0 -Serpent JEFF-2.2-Serpent JEFF-3.1 -Serpent JEFF-3.1.1
Figure 2.5.
Figure 2.6.
E -10
- -2
3
Ez4U)4
-5 - -----
3 0 5 10 15 20 25 30 35 40
Burnup [MWD/kgHM]-BGCORE(MCNP5_JEFF3.1) -SerpentENDF/B-VI.8-SerpentENDF/B-Vll.0 -SerpentJEFF-2.2-Serpent JEFF-3.1 -Serpent JEFF-3.1.1
Comparison of Plutonium-239 Number Density versus
,0
LURt -10
-2
E-_3
-4C
-5.nE
-6
8 -7
-8
0 5 10 15 20 25 30 35 40
Burnup [MWD/kg|HM]-BGCORE(MCNP5_JEFF3.1)-SerpentENDF/B-VII.0-Serpent JEFF-3.1
-SerpentENDF/B-VI.8-Serpent JEFF-2.2-Serpent JEFF-3.1.1
I 2
Figure 2.7. Comparison of Xenon- 135 Number Density versus CASM04E
6.42E-01
-' 6.40E-01E
6.38E-01
u 6.36E-01
6.34E-01
6.32E-010
2 6.30E-01U
2 6.28E-01
, 6.26E-01
6.24E-01
- ------- ------- -------- - - ------- - -.. 0 .70...0.70-Dragon
-Serpent0.60
-Difference-- --- -- -- - --------
0.50
0.400
- - -- - - - 0.30
C
0.20
0.10
0.00
0 5 10 15 20 25 30 35 40
Burnup [MWD/kgIHM]
Figure 2.8. Dragon - Serpent Comparison of Total Macroscopic Cross Section
46
E -Dragon-. 3.50E-02 -Serpent 1.40
0 -Difference
0
3.0E021.20
2.00E-02 -40.80MC00
2 1.50E-02 0.600
4" 1.OOE-02 0.40z
0 5.00E-03 0.20
T
0 1.00E0 0. 00
0 5 10 15 20 25 30 35 40
Burnup [MWD/kgIHM]
Figure 2.9. Dragon - Serpent Comparison of Fission Neutron Production Cross Section
presented in Fig. 2.9. The total cross section agrees well at the beginning of life, but diverges asthe burnup increases. However, the difference between these values is about 0.5%. The differencesin fission neutron production cross section are much smaller. Occurring shortly after the beginningof life, there is a larger deviation between these estimates. This same trend is observed in theeigenvalue comparison and is most likely due to the prediction of fission product build up. Allof these comparisons show that the Serpent code is capable of predicting similar physics to thatof other Monte Carlo and deterministic lattice codes. Although these differences are small, theyare important to understand before progressing to further analyses. The code input files for thesecalculations are listed in Appendix A. 1.
2.4 RBWR Serpent Assembly Model
A single assembly RBWR model was created in Serpent. The basis for this model is the HitachiRBWR and the JAEA RMWR (Iwamura et al., 2006; Fukaya et al., 2008; Takeda et al., 2007;Uchikawa et al., 2007). The objective of this work is not to analyze the exact assembly design ofthese reactors, but to choose geometric, material and operating conditions that have representativecharacteristics. These characteristics are:
- two distinct fission zones that produce the majority of the power and are surrounded byblankets for breeding,
- fissile zones composed of MOX fuel and blanket zones containing depleted UOX fuel,
- core average void fraction up to 60% with an axial coolant density distribution similar to theRBWR,
47
4.00E-02 1.60
- lower and upper reflectors.
All of these characteristics are incorporated into the Serpent model described in this section. Fi-nally, to determine if the model created properly reflected the input, the same model was run withthe MCNP5 Monte Carlo code. The description of the model will be divided into specific parts thatreflect the style of the Serpent input file. The geometry, material, and operating specifications willbe described and followed by other necessary control information for the Monte Carlo execution.The Serpent input file that will be used for the comparison with MCNP5 may be referenced duringthe description. It is listed in Appendix A.2.
2.4.1 Geometry Specifications
As described in Section 2.1.1, a two-dimensional slice approach was employed in generating thegeometry. For each axial zone, a two-dimensional lattice was created and stacked up in an axiallattice in Serpent. A side view of a single assembly is depicted in Fig. 2.10. Each color in thefigure represents a different material region. Other than the lowest zone, Lower Reflector Zone 1,each zone contains a lattice and the fuel rods are evident. Each coarse zone has a label and containsthree sub-regions. Each sub-region contains a different coolant density and material specificationsso that it would be treated as a different depletion region if a depletion calculation was performed.Table 2.2 lists each sub-region with the abbreviations that will be used for further identification. Inaddition, the axial thickness of each sub-region is listed.
Each fuel lattice in the fissile and blanket regions contains an array of 271 fuel rods. Thesefuel rods are arranged in a hexagonal lattice and surrounded by coolant, as shown in Fig. 2.11.In this diagram the minimal diameter dimension of the hexagonal assembly is also specified. Thisassembly does not represent the exact configuration that would be analyzed when performing cal-culations on a true RBWR assembly. In the true design, there is a duct surrounding the latticefollowed by a small layer of bypass fluid. It was assumed that the duct was made of material thatwould not perturb the neutron energy spectrum significantly and that the bypass flow gap was verysmall. One fuel rod surrounded by coolant represents a unit cell, as shown in Fig 2.12. In thisspecific fuel rod, there exists a cylinder containing fuel, then a small gap followed by claddingmaterial. The cladding is then surrounded by coolant. The outer diameter of each of these mate-rial regions is shown in the figure. These dimensions are then listed as radii in the pin structurecard. Finally, the pin pitch, defined as the closest distance between two pin centers in the hexag-onal array, equals 1.14 cm. This is all of the required information to build the geometry of thethree-dimensional RBWR lattice.
2.4.2 Material Specifications
In Serpent and other Monte Carlo codes, the compositions of materials are specified as eitheratom fractions or mass fractions of individual nuclides. In the Serpent input file, mass fractionswere used for all materials except water. The material water, which represents the coolant andmoderator surrounding the fuel rods shown in Fig. 2.11, is just a combination of hydrogen andoxygen. Since it is known that in water there are two hydrogen atoms for every oxygen atom,atom fractions were specified where hydrogen was listed with a 2.0 while oxygen was listed witha 1.0. Even though these atom fractions don't add up to unity, Serpent automatically re-normalizesthem. In addition, since the mass of a hydrogen atom is comparable to that of a neutron and
48
- Upper Reflector Zone
Upper Blanket Zones 1-3
Upper Fisslie Zones 1-3
Internal Blanket Zones 1-3
Lower Fissile Zones 1-3
Lower Blanket Zones 1-3
Lower Reflector Zone 2
Lower Reflector Zone 1
Figure 2.10. Side-View of RBWR Assembly
18.99cm
Figure 2.11. Assembly Lattice Configuration (Top-View)
49
Table 2.2. Description of Sub-Regions in RBWR Assembly
Name of Sub-region
Lower Reflector Zone 1
Lower Reflector Zone 2
Lower Blanket Zone 1
Lower Blanket Zone 2
Lower Blanket Zone 3
Lower Fissile Zone I
Lower Fissile Zone 2
Lower Fissile Zone 3
Internal Blanket Zone 1
Internal Blanket Zone 2
Internal Blanket Zone 3
Upper Fissile Zone 1
Upper Fissile Zone 2
Upper Fissile Zone 3
Upper Blanket Zone 1
Upper Blanket Zone 2
Upper Blanket Zone 3
Upper Reflector Zone 1
Identification Tag
LR1
LR2
LB1
LB2
LB3
LF1
LF2
LF3
IB1
1B2
IB3
UFl
UF2
UF3
UB1
UB2
UB3
UR
Thickness of Sub-region [cm]
23.0
7.0
7.75
7.75
2.5
5.0
12.4
5.0
7.0
31.0
7.0
5.0
12.6
5.0
2.0
8.0
2.0
30.0
50
0.875 cm 0.91 cm 1.01 cm
Figure 2.12. Cross-section of a Fuel Rod Unit Cell
since thermal neutrons will be colliding with this isotope, bound scattering effects must be takeninto account. Therefore, along with the water material specification, a thermal scattering librarycontaining S (a, #) information is specified. An example of one of these material cards is givenbelow with the thermal scattering library for light water listed:
mat waterUB3 -0.150 moder lwtr 1001 tmp 5601001.03c 2.0
8016.03c 1.0
therm lwtr lwe7.12t
In the input file, the material water must be specified each time it is in a different sub-region,unlike the MCNP5 format. This is because the absolute number or mass density must be specifiedon the material card. This will change for each sub-region, and each water material identifier isgiven a suffix which corresponds to the sub-region's identification tag (waterUB3).
The lowest sub-region is Lower Reflector Zone 1. This zone contains only water as a materialand there are no internal structures present. Above this sub-region, the Lower Reflector Zone 2contains boron carbide rods. These boron carbide rods are enriched with 90 weight percent ofboron- 10. The overall mass density and constituent mass fractions are listed in Table 2.3. The nextaxial set of sub-regions in the assembly make up the Lower Blanket Zone. This zone, along withthe Internal Blanket Zone and the Upper Blanket Zone, all contain the same type of fuel material.These are each listed separately for each sub-region so that they may be treated as individualdepletion regions. These regions contain depleted uranium dioxide where the mass percent ofuranium-235 relative to uranium metal is 0.25%. The detailed mass fractions and overall densityof this material are listed in Table 2.4. In the Lower and Upper Fissile Zones, the fuel is made up of
51
Table 2.3. Material Composition of Boron Carbide Rods in LR2
Constituents Mass Fraction (p = 2.394g/cm 3)
B-10 0.693962
B-11 0.077107
C-natural 0.228931
Table 2.4. Material Composition of Depleted UOX in Blanket Regions
Constituents Mass Fraction (p = 10.5 g/cm3 )
0-16 0.118466
U-235 2.23834E-3
U-238 0.87933
a plutonium, uranium, and minor actinide mixture in oxide form. In this design, it was assumed inboth coarse regions that 18% fissile plutonium by weight was present in this mixture. The isotopicvector of transuranics (TRUs) in this mixture is listed in Table 2.5. It is also assumed that themass percent of uranium-235 relative to uranium metal is 0.25% in this mixture as well. The finalcomputed mass fractions of the isotopes in this fuel material are shown in Table 2.6. The last typeof material that makes up this model is the cladding. For the purposes of the code comparison withMCNP5, this material is just made up of zirconium-90. For the actual model that will be used infuture calculations, this material will be switched to natural zirconium. Note, both of these areapproximations as the actual cladding material may be an alloy of zirconium and other metals inthe RBWR.
2.4.3 Operating Conditions
The operating conditions for the assembly must also be specified. These include coolant densitydistribution (DC), fuel temperatures (TF), coolant temperatures (TC), temperature of cladding (TS)and power. The RBWR has an unique axial design as shown in Fig. 1. 1b. Therefore, one wouldexpect this design to have an unique void fraction distribution due to the separation of powergenerating regions. The void fraction distribution for this reactor is shown in Fig. 2.13. Unlikeconventional void distributions as shown by the ABWR (Advanced Boiling Water Reactor) curve,the RBWR has a steep slope of the void fraction in fissile regions compared to blanket regions.In addition, the void fraction is much larger than the ABWR and even reaches 80% in the upperblanket and reflector.
In order to determine the coolant density in each sub-region, the void fraction in each sub-region was obtained from Fig. 2.13. To determine these void fractions, lines were drawn in thefigure corresponding to the boundaries of each sub-region. The void fraction at the midpoint ofthese boundaries was taken as the average void fraction in that sub-region. Once the void fractions
52
Table 2.5. Isotopic Composition of TRU Nuclides in Fuel Mixture
Nuclide Fraction in Overall TRU
Np-237 0.005
Pu-238 0.03
Pu-239 0.44
Pu-240 0.361
Pu-241 0.05
Pu-242 0.049
Am-241 0.037
Am-242m 0.001
Am-243 0.013
Cm-244 0.01
Cm-245 0.003
Cm-246 0.001
100
80
C0
70
0L
0 LZ._0
Bottom Relative Distance from Bottom of Core
120mm
Blanket
Top
Figure 2.13. RBWR Axial Void Fraction Distribution (Takeda et al., 2007)
53
Composition of Fuel in Fissile Regions
Constituent Mass Fraction (p = 10.5 g/cm3 )0-16 0.118182
U-235 1.394712E-3
U-238 0.556490
Np-237 1.619666E-3
Pu-238 9.717994E-3
Pu-239 0.142531
Pu-240 0.116940
Pu-241 0.016197
Pu-242 0.015873
Am-241 0.011986
Am-242m 3.23933 1E-4
Am-243 4.211131E-3
Cm-244 3.23933 1E-3
Cm-245 9.717994E-4
Cm-246 3.239331E-4
54
Table 2.6. Material
were known, the average mixture density in each coarse region was calculated with
Pm,i = pf (1 - ai) + pgai. (2.1)
In Eq. (2.1) pm,i is the average density in sub-region i, at is the corresponding void fraction, andpj /pg are the saturated liquid and vapor densities, respectively, which were evaluated at a pressureof 6.9 MPa to be pf = 736 g/cm3 and pg = 37.7 g/cm3 (Todreas and Kazimi, 1990). The densitydistribution in the blanket and fissile regions is shown in Fig. 2.14. The rest of the operatingconditions were chosen somewhat arbitrarily since the information has not been published in openliterature. These conditions are listed in Table 2.7. Note that different conditions exist for theSerpent comparison with MCNP5, that are different than the conditions that are used in calculationspresented in subsequent sections. This is denoted in the table with the Bench./Real heading. Thismainly affects the temperature distributions as MCNP5 does not have a built-in Doppler broadeningpre-processor routine. The power is not listed since only steady state calculations are performed inthis work. For depletion calculations this must be specified for flux-to-power normalization.
2.4.4 Other Control Information
In this section, other parameters needed for execution of this model will be described. For theboundary conditions, reflective conditions are used in the radial direction and zero incoming cur-rent conditions are used in the axial direction. For neutron interaction data, the ENDF/B-VII.0neutron library was used. Finally, for Monte Carlo calculations neutron history information isneeded. For this model, 25000 histories/cycle were used with 200 inactive cycles to converge thesource and 1000 active cycles. Also present in the input file, which are not needed for execution,are plotting and tally information that will be used for comparison against the MCNP5 results.
2.4.5 Comparison with MCNP5
To determine if the model was constructed correctly in Serpent, a comparison was made withMCNP5. The exact same geometry, materials and operating conditions were used to construct theMCNP5 model. Also, the input file was generated with the same method of two dimensional slices.One difference between the input files is that in MCNP5, the importance of each cell must be listed.This is primarily used for variance reduction techniques such as geometry splitting. However, it isalso used to kill neutrons when the importance is zero. This capability does not exist in Serpentsince it currently is an analog Monte Carlo code. Another difference between the two input decksis that in MCNP5, the cell temperature must be listed for free-gas thermal scattering using the tmpcommand. In Serpent, this cell temperature is assumed to be the same as the cross section librarytemperature or Doppler broadened temperature, unless otherwise specified. Lastly, the MCNP5 fileis shorter because the density of materials is allowed to change from cell-to-cell when specifyingthe same material card. Therefore, only one water material is needed in the MCNP5 input file. InSerpent, since the material density is given on material cards instead of cell cards, this materialmust be repeated. The MCNP5 input file for this calculation is listed in Appendix A.3.
In each code, k-effective and axial distributions of power density, fission neutron productiondensity, absorption density and flux were computed. A comparison of eigenvalues from the col-lision estimator is shown in Table 2.8. Difference in reactivity between the two systems is alsocomputed along with the combined statistical standard deviation. From the results, a reactivity
55
Table 2.7. Operating Conditions of RBWR Assembly
TF [K] Bench./Real TC [K] Bench./Real TS [K] Bench./Real
N/A
Sub-Region
LR1
LR2
LB1
LB2
LB3
LF1
LF2
LF3
IB1
IB2
IB3
UFI
UF2
UF3
UBI
UB2
UB3
UR
Table 2.8. Comparison of Eigenvalues between Serpent and MCNP5
Eigenvalue Standard Deviation (1 a)
kMCNP 1.08591 UMCNP 0.00013
kserpent 1.08554 aserpent 0.00021
Ap = 30pcm aAp = 20pcm
56
DC [g/cm3 ]
0.736
0.736
0.721
0.668
0.609
0.527
0.360
0.290
0.266
0.253
0.226
0.219
0.190
0.169
0.156
0.152
0.150
0.149
600/560
600/560
600/560
600/560
600/560
600/560
600/560
600/560
600/560
600/560
600/560
600/560
600/560
600/560
600/560
600/560
600/560
600/560
600/560 (B4C)
900/750
900/750
900/750
1200/1150
1200/1150
1200/1150
900/750
900/750
900/750
1200/1150
1200/1150
1200/1150
900/750
900/750
900/750
N/A
N/A
600/650
600/650
600/650
600/650
600/850
600/850
600/850
600/650
600/650
600/650
600/850
600/850
600/850
600/650
600/650
600/650
600/650
800
700
600
100
00 20 40 60 80 100 120
Axial Height from Bottom of Assembly [cm]
Figure 2.14. Coolant Density Axial Distribution in RBWR Assembly
difference of zeros lies within ±2u of the difference in reactivity. Figures 2.15 through 2.18 showa Serpent and MCNP5 comparison of flux distribution, power density, fission neutron productiondensity and absorption density, respectively.
In these plots, it is clear that there is an agreement in the overall trend between Serpent andMCNP5. The relative standard deviation in MCNP5 results is on the order of 5E-4 while therelative standard deviation in Serpent results are on the order of 2E-4. Most of the results arewithin ±2u which is approximately 0.5%. In the comparison of the sub-region fluxes (Fig. 2.15),the differences between the fluxes were all less than 1%. In the power density comparison (Fig.2.16), the differences range between 0 and 3%. This may be due to differences and definitions ofheating values of isotopes. Similar to the flux, the fission neutron production results are within1% of each other (Fig. 2.17). Finally, all differences are less than 1% in the absorption ratedensity comparison (Fig. 2.18) except for the lower and upper reflector. These differences are 1.3and 2.7%, respectively. Since the differences in the fluxes are small, this could be attributed tostatistics since these media have low absorption probabilities. The objective of this comparisonwas met, as the Serpent results and MCNP5 results compare well and it can be concluded that theSerpent model was correctly built with the assumptions above. This also gives confidence that theSerpent transport kernel generally matches the physics in MCNP5.
2.5 Neutron Balance in Monte Carlo Codes
The most important task in any neutronics code is to assess neutron balance. In the context of thiswork, neutron balance as calculated from a detailed transport simulation needs to be representedby generated multigroup constants. In Monte Carlo transport codes, this may be difficult dueto the statistical nature of individual simulations of neutrons. No matter the physics built-in to
57
0.16
0.14
0.12
0.1
0.08
0.06
0.04
0.02
0
Figure 2.15. Comparison of Axial Flux Distribution
0.18
-MCNP5-Serpent
-- -1
-30 0 30 60 90 120 150Axial Position [cm]
Figure 2.16. Comparison of Power Density
58
0.12
-MCNPS
0.1 -- Serpent
0.08
U
.2 0.06a
0.04
0.02
0-
-30 0 30 60 90 120 150
Axial Position [cm] (Bottom at -30 cm)
E
a,0
0
- -7-
r---E
4'
G
0
C
0-G
0.18
0.16-MCNP5
-Serpent
0.14
0.12
0.1
0.08
0.06
0.04
0.02
0
-30 0 30 60 90 120 150Axial Position [cm]
Figure 2.17. Comparison of Fission Neutron Production Density
0.14
-MCNP5
0.12 -Serpent
0.1
-30.08
00.06
0.04
)m.
CL
0 0.02
.'
0
-30 0 30 60 90 120 150
Axial Position [cm]
Figure 2.18. Comparison of Absorption Rate Density
59
the Monte Carlo code, the exact same neutron balance should be able to be reconstructed with
homogenized multigroup constants. The multigroup constants, which include macroscopic crosssections, surface currents and eigenvalue should satisfy the overall neutron balance equation for anarbitrary energy group g,
V -Jg (7) + Etg (7) Og (7) =
G
h=1
1 V E5, (') ph (V), (2.2)keff h=I
where
Ea (r) = Ea,h (r hK (2.3)
In this representation of neutron balance, Jg (7) is the group net current, 'g (') is the group flux,
Etg (7) is the group total macroscopic cross section, Xg (7) is the fission spectrum of group g,keff is the system eigenvalue and VEf,h (7) is the group macroscopic fission neutron production
cross section. The other macroscopic cross sections listed incorporate the probability of a neutron
interacting in group h and exiting in group g, as shown for an arbitrary reaction type a in Eq.
(2.3). Each of these macroscopic cross sections has a multiplicity associated with them. For
example, E (7) represents any neutron interaction where one neutron goes in and one comes
out. Therefore, the multiplicity of neutrons in this interaction is one. This macroscopic cross
section represents all elastic and inelastic reactions where there is a one-to-one interaction. The
others in this group represent one neutron going into an interaction and some arbitrary number,
v, coming out. This also implies that the multiplicity of neutrons may come out into different
energy groups and this needs to be recorded in a probability matrix similar to what Serpent does
for one-to-one scattering. This equation can be extended further for higher multiplicity reactions.
The main point in this process is to represent complex heterogeneous information in a few
homogeneous parameters. Therefore, every microscopic reaction that is sampled in the transport
calculation must be recorded somewhere in one of the homogenized parameters listed in Eq. (2.2).
The most difficult reactions to consider are multiplicity reactions in which one neutron enters the
reaction and more than one neutron comes out. There are a variety of methods to account for this
multiplicity including a pure analog approach where each multiplicity is treated separately and
a probability matrix is generated for each multiplicity separately. In this approach, tallying and
normalization of each probability matrix could automatically take into account this multiplicity.
Another approach is to record all neutron multiplicity in one energy transfer matrix. To do this, the
above energy transfer cross sections are represented in a lumped form with an effective multiplicity
so that
I~y-4gr r h-4g)V h h- g) -.
_S~ (7) ( EP7 {Xn (7) + 2E (7) + 3X( 3 ) (r) ,4 n) 0_r) +..j 'Ph (7r) (2.4)
This form is more familiar, other than the effective multiplicity v, since all of the (nxn) reactions
have been classified as scattering reactions. Eq. (2.2) can be re-written and integrated over an
60
Figur 2.19.j Computationa Node n Nomenlature/2
- ~~ 1/2) h~h2 + ,Yj 1/21 -kg+11 i2) hh (Xi1/gY+12Zk+/2)J,~12 ~~hih
(Xi h/,j+/h2 k 12
by
Figure 2.19. Computational Node n Nomenclature
arbitrary computational node n as shown in Fig. 2.19 so that
g,xi+ 1/2 1 ,xi- 1/2 )hyhz +{Z ( Y- 2_ ,yj+ 1/2) hxhz +(Z"9,zk+ 1/2 _ Ezk- 1/2 )hxhy+
G g G
Carlo ~ ~ E coe an shul satisfyo the reato towtinsaisis
Ein t' hxhyhz = h h Esnghhhhz + ( Ennhxhyhz. (2.5)Ag Y h=1 b o s t ceff h=1 n
In this formulation, the integral for the surface net currents for the x direction are defined as
-n~~~ f 2 dy f d,,k 12aznx -Zg9 (xi 1/p, Y,Z)g,xi+12 Y-1/2 hyhz, (2.6)
where the convention is that these integrated surface currents are positive in the +x direction andthe unit vector x is the outward surface normal (Stacey, 2007). This definition can be extended to
the y and z directions. The parameters in Eq. (2.5) can be represented with tallies in most MonteCarlo codes and should satisfy the relation to within statistics.
A number of simple tests can be created to determine if a code can represent neutron balancefrom the transport equation with homogenized multigroup constants. The easiest test is to eliminatethe spatial domain altogether and perform a transport calculation on an infinite and homogeneous
geometry. Also as an initial test, the energy domain can be collapsed into one energy group. Inthis case, the neutron balance equation in Eq. (2.5) becomes
Et # =VES# + VEf5#. (2.7)keff
61
It is then common to lump the group terms together so that
(Et - 1Es) # = vE . (2.8)keff
Therefore, the left hand side represents the destruction of neutrons while the right hand side rep-resents the production of neutrons. A more familiar form of this equation is to add and subtractthe macroscopic scattering cross section on the left hand side of the equation. Therefore, the totalmacroscopic cross section can be represented as the summation of the macroscopic absorption andmacroscopic scattering cross sections,
Et = Ea + Es. (2.9)
In the context of this derivation, the macroscopic scattering cross section represents the loss ofneutrons that enter into a scattering interaction. Although this was stated as a loss, one or moreneutrons would be produced from these interactions. The macroscopic scattering cross section isdefined as
Es E(n,in) + E(n,2n) +-(n,3n) -(n,4n) + ... (2.10)
Therefore, Eq. (2.8) becomes
1[Ea - (UEs - Es)] 4- = VEf$. (2.11)
The presentation of Eq. (2.11) allows for a good interpretation of the balance of neutrons. Theabsorption cross section represents the ultimate destruction of neutrons since there is no leakagepresent. If there was no multiplicity of neutrons in the system other than fission, the second ex-pression on the left hand side would be zero since the effective multiplicity, v would be unity.However, if there is multiplicity due to (n,xn) reactions then of course vEs > Es such that a netproduction of neutrons results. From this representation, the effective multiplicity could then becalculated from these tallies as
'I = . (2.12)
The (-) just represents a numerical result from a tally in a Monte Carlo code. A modified approach,that is used for input into nodal codes, is to create a modified absorption cross section, E*, suchthat the total macroscopic cross section can be represented as
t E* + UEs. (2.13)
This formulation should be used if the nodal code only accepts an absorption cross section. There-fore, this modified absorption cross section can be calculated from the Monte Carlo code as thedifference between the total cross section and the production scattering cross section. This method-ology can easily be extended to the multigroup form of the diffusion equation by defining a removalcross section instead of an absorption cross section as
Erg Sg - UE = E + (2.14)hpg
62
where E represents within group scattering. More in-depth test cases could be performed
to include spatial dependence. A simple example would be to have a l-D two-zone problem and
calculate the interface current between the two zones with the individual homogenized parametersfrom each zone separately. This current should be continuous at the interface to within statistics.Although the transport kernel in many of the Monte Carlo codes have been verified, this does not
ensure that neutron balance is met with homogenized parameters. These simple tests are a way of
verifying that at least neutron balance is preserved from the transport analysis. Even with neutron
balance from the transport calculation, results may not be able to be reproduced with diffusiontheory. This is the subject of Section 4.
After performing some of these tests in Serpent, it was clear that neutron balance is not pre-
served. The main reason why this balance is not preserved is that when generating multigroup con-stants, the production rate of multiplicity reactions is not taken into account correctly. Currently,in Serpent v 1.1.14, (n, xn) reactions are recorded in the same way as inelastic reactions. Each time
this reaction occurs, Serpent remembers the incoming energy, samples the outgoing energy and
tallies a one in the corresponding probability energy transfer matrix. After this is completed, thecode normalizes each column of this matrix to sum to unity. A column in this matrix represents all
of the possibilities of leaving an energy group and entering another energy group, including itself.
By doing this, the code effectively ignores the multiplicity of the (n,xn) reactions. Rather, it is
hypothesized that the code should normalize each column by the number of neutrons that entered
scattering collisions in that group, not the number of neutrons that exited collisions. This could be
an effective analog method for treating this multiplicity. After the transport calculations are com-
pleted, Serpent multiplies each column in this probability matrix by the corresponding scatteringcross section of that pre-collision energy group.
Currently, in v1. 1.14, the scattering macroscopic cross section is defined as
EtI1.14 I-If - Ec - En,2n, (2.15)
where Ec is the neutron capture macroscopic cross section which contains information about all
neutron absorptions other than those that yield one or more neutrons. Since the (n,xn) reactions
are taken into account in the energy transfer matrix, the (n, 2n) reactions should not be subtracted
here, but included in the scattering cross section. Therefore, to enforce neutron balance, the source
code of v1. 1.14 was modified. The following changes were made:
- the definition of the scattering cross section on line 949 in the subroutine collectresults.cwas changed from
scattxs = totxs - fissxs - captxs - n2nxs;
to
scattxs = totxs - fissxs - captxs;
- to have no multiplicity of neutrons other than fission, lines 31, 33 and 35 in nxn.c, whichset the multiplicity for (n, 2n), (n, 3n) and (n,4n), were set to zero so that they are just a
one-to-one scattering with
n=O
63
- since only (n, 2n) reactions are recorded into their own macroscopic cross section, (n, 3n)and (n, 4n) reactions were turned off completely in the transport code by changing line 270and 271 in transport.c from
else if ((mt == 16) || (mt == 17) || (mt == 37) |1(mt == 24) || (mt == 25))
to
else if (mt == 16)
- finally, to get the loss rate of neutrons correct for the group constant estimator of k-effective,line 146 in collectresults.c was changed from
if ((loss = abs + leak - n2n) > 0.0)
to
if ((loss = abs + leak) > 0.0)
and line 154 was changed from
if ((loss = abs - n2n) > 0.0)
to
if ((loss = abs) > 0.0)
To verify that these modifications yield neutron balance, a simple test was performed on an infiniteand homogeneous geometry containing pure U-235 at 19 g/cm 3 density. For the test, the collisionestimator of k-effective was compared to that of the another estimate involving one group reactionrates. Since there is no multiplicity of neutrons, the keff of the system can be solved from Eq.(2.11) with the multiplicity zero,
v25#keff = a, (2.16)EaOP
where the absorption reaction rate is the sum of capture and fission reaction rates,
Ya# = (-c + Ef) $. (2.17)
Table 2.9 shows the results comparing the standard vI. 1.14 and the modification. The reactivitydifference in the estimators for the standard version of Serpent is about 200 pcm, while the differ-ence for the modified version is 6 pcm. Therefore, with the modifications of Serpent, the transportcollision estimator of k-effective can be reproduced with homogenized reaction rates. These reac-tion rates are then used in the code to get macroscopic cross sections by dividing by the flux. Tobe fair, in one energy group, the standard version of Serpent can re-create neutron balance by amodified absorption treatment where the (n, 2n) macroscopic cross section can be subtracted from
64
Table 2.9. Comparison of Eigenvalues with Modification of Serpent
Serpentvl.1.14 Serpent v1.1.14ml
Capture Rate 1.16547E-01 1.16432E-01
Fission Rate 8.83453E-01 8.83568E-01
Fission Neutron Production Rate 2.26747E+00 2.26800E+00
keff from Eq. (2.16) 2.26747 2.26800
Collision keff 2.27673 2.26829
the absorption cross section, as indicated in Eq. (2.13). This can be determined by expanding thescattering cross section in Eq. (2.11) to just (n, In) and (n, 2n). Doing so yields a k-effective of2.2771 which is a difference of 7 pcm. Although this method may work in one energy group, itdoes not treat (n, 3n) and (n, 4n) reactions and is not applicable to multigroup.
It is clear from this investigation that more work needs to be completed to ensure this multi-plicity balance is taken into account correctly. Even if the eigenvalues are close between transportestimators and homogenized tally estimators, the energy transfer matrix needs to treat the multi-plicity of neutrons correctly. To prove the methodology in Section 4, it was easiest to implementthe changes above. Since neutron balance from the transport code is so crucial, the calculationspresented in Section 4 use the modified version of Serpent, now with incorrect transport physics(e.g. neglect of (n,xn) reactions). This problem of neutron balance has been addressed with thedevelopers and will be fixed in Serpent v1. 1.15.
2.6 Three-Dimensional Cross Sections
In the conventional deterministic methodology of homogenizing cross sections, each unique two-dimensional material is decoupled from the core and analyzed separately. In these models, zeronet current boundary conditions are imposed both radially and axially. The collapsed few-groupmacroscopic cross section sets that result from these analyses are referred to as two-dimensionalcross sections in this work. Since the RBWR has strong axial material discontinuities and a highervoid fraction, the method of fully decoupling zones from the core may be too crude of an approx-imation. This means the assumption that no neutrons flow across the boundary axially may notbe valid. Therefore, the homogenization process and generation of multigroup parameters will beperformed on a single assembly where zero net current boundary conditions are imposed radiallyand zero incoming current boundary conditions are imposed axially. In this method, cross sectionsfor specific homogenization zones are influenced by surrounding axial neighbors. The cross sec-tion sets resulting from these analyses are referred to as three-dimensional cross sections. A studyof this methodology is described in this section.
The process of this study will involve choosing homogenization zones with specific operatingconditions and then comparing cross sections resulting from the two-dimensional process againstthe three dimensional process. The coarse homogenization zone nodalization and specific operat-ing conditions are listed in Table 2.7. Since a comparison against MCNP5 is not being performed,restrictions on operating conditions are no longer required and therefore the real operating condi-
65
tions are used. For the two-dimensional transport calculations, a separate input file was constructedfor each axial blanket and fissile sub-region represented in Fig. 2.10, with corresponding fuel com-position and coolant density. These two-dimensional lattice calculations are run separately withreflective boundary conditions in all three dimensions with no influence from neighbors. For eachcalculation, the few-group parameters are homogenized over the full geometry. The process forreflectors is slightly different and is similar to the deterministic approach. To homogenize overthe reflector(s), the blanket zone immediately adjacent to the reflector region is also modeled.Examples of the lower reflector input file, lower fissile zone sub-region 3 (similar approach forblanket zones) input file and the upper reflector input file are listed in Appendix A.4. For thethree-dimensional approach, the full assembly was modeled similar to the analysis performed inthe MCNP5 comparison. Serpent allows for homogenization to take place over any universe. Sincethe geometry was created with this in mind, each sub-region was labeled with a unique universeover which Serpent homogenizes. The multigroup parameters are then output in a MATLAB ma-trix format for extraction.
Each two-dimensional calculation was performed on a single Intel Xeon 5620 Quad CoreProcessor (2.4 GHz) with 20,000 neutron histories per batch, 400 active batches and 20 inac-tive batches. The time to finish one two-dimensional calculation was on the order of 20 minutes.The three-dimensional calculations were performed with 25,000 neutron histories per batch, 8,000active batches and 200 inactive batches. This calculations took about 1 day to run. This increasein total neutrons simulated was necessary to get acceptable statistics on the tallies to investigateneutron balance effects in addition to this comparison. Serpent has a nice power distribution meshplotter to visualize some of the physics of the transport simulation. A side-view of the RBWRas plotted by Serpent is presented in Fig. 2.20. In this power distribution mesh plot, a range oftwo quantities is shown via different color shades. For the regions in which fissile or fissionablematerial is defined, the power distribution is shown where a dark red color indicates a low powerregion and a bright yellow color represents a high power region. On the other hand, for the regionswhere non-fissile material exists, the thermal flux is shown where a dark blue color represents alow thermal flux and a light blue to white represents a high thermal flux. The diagram shows thatthe lower and upper fissile zones are producing most of the power, while the blanket regions areproducing a lot less. Fig. 2.20 is for a beginning of life situation and as the assembly is depleted,power will start to shift toward these blanket zones as plutonium is being bred into the system. Inreflector regions, there is a high thermal flux due to the slowing-down of neutrons in these regions.
Since multigroup parameters are of ultimate interest, the total cross section, absorption crosssection, fission neutron production cross section, group scattering matrix and transport cross sec-tion were all compared. A conventional two-group structure was chosen where the group boundarywas placed at 0.625 eV. Since most of the comparisons yield the same results, only the transportcross section, absorption cross section and fission neutron production cross section are presented.These are listed in Tables 2.10, 2.11 and 2.12, respectively. In each table, the mean value ofthe homogenized cross section is presented followed by its relative uncertainty. Most of the rel-ative uncertainties are less than 1%. After the two- and three-dimensional parameters are listed,the percent difference between them is computed with the two-dimensional parameter being thereference. A positive value indicates that the three-dimensional cross section is larger. From thesetables, a few observations and trends can be seen. First, for the neutron balance terms, absorptioncross section and fission neutron production cross section, larger differences occur at the interfaceespecially in group 2. Differences between the two homogenization methods are also shown in
66
Table 2.10. Two- and Three-Dimensional Transport Cross Section Comparison
£3D ~3D
E3Dtr,Id,
0.2639
0.4156
0.3367
0.3288
0.3064
0.2905
0.2658
0.2639
0.2741
0.2883
0.2695
0.2558
0.2462
0.2487
0.2526
0.2616
0.2699
RE a
6.E-04
4.E-04
1.E-04
9.E-05
1.E-04
8.E-05
6.E-05
9.E-05
8.E-05
5.E-05
7.E-05
8.E-05
5.E-05
8.E-05
1.E-04
8.E-05
2.E-04
tr,2
1.4360
4.6451
0.6647
0.6305
0.5885
1.8295
1.3719
1.6506
0.3937
0.3901
0.3728
1.6052
1.4705
1.6324
0.3380
0.3403
0.3448
RE a
2.E-04
3.E-03
2.E-04
2.E-04
4.E-04
1.E-03
2.E-03
2.E-03
3.E-04
1.E-04
3.E-04
2.E-03
7.E-03
3.E-03
1.E-03
3.E-04
5.E-04
tr, 1
0.2705
0.3560
0.3268
0.3210
0.3148
0.2914
0.2726
0.2647
0.2791
0.2778
0.2752
0.2570
0.2538
0.2516
0.2687
0.2684
0.2682
RE a
5.E-04
4.E-04
9.E-05
9.E-05
9.E-05
1.E-04
I.E-04
1.E-04
9.E-05
9.E-05
9.E-05
1.E-04
1.E-04
1.E-04
1.E-04
9.E-05
1.E-04
tr,2
1.4366
4.3674
0.6695
0.6375
0.6022
1.4701
1.3429
1.2965
0.3991
0.3917
0.3759
1.2502
1.2465
1.2563
0.3364
0.3342
0.3332
RE a % diff Etr % diff Etr,2
2.E-04
5.E-03
2.E-04
1.E-04
2.E-04
3.E-03
5.E-03
6.E-03
2.E-04
3.E-04
3.E-04
9.E-03
1.E-02
1.E-02
4.E-04
4.E-04
4.E-04
-2.4
16.7
3.0
2.4
-2.7
-0.3
-2.5
0.3
-1.8
3.8
-2.0
-0.5
-3.0
-1.1
-6.0
-2.5
0.6
-0.04
6.4
-0.7
-1.1
-2.3
24.5
2.2
27.3
-1.4
-0.4
-0.8
28.4
18.0
29.9
0.5
1.8
3.5
UR 0.0577 2.E-04 0.1164 3.E-04 0.0437 5.E-04 0.1155 3.E-03 32.1 0.8
67
Zone
LR1
LR2
LB1
LB2
LB3
LF I
LF2
LF3
IB1
IB2
IB3
UFI
UF2
UF3
UBI
UB2
UB3
Table 2.11. Two- and Three-Dimensional Absorption Cross Section Comparison
RE ay3D
E 1
0.0004
0.1301
0.0105
0.0101
0.0084
0.0213
0.0168
0.0178
0.0074
0.0087
0.0068
0.0157
0.0134
0.0143
0.0056
0.0063
0.0082
0.0002
X2D ~2D
E 1
0.0003
0.0827
0.0100
0.0099
0.0097
0.0204
0.0174
0.0160
0.0081
0.0080
0.0078
0.0145
0.0138
0.0134
0.0072
0.0072
0.0071
0.0001
RE a
2.E-03
6.E-04
3.E-04
3.E-04
3.E-04
2.E-04
2.E-04
2.E-04
2.E-04
2.E-04
2.E-04
2.E-04
2.E-04
2.E-04
2.E-04
2.E-04
2.E-04
2.E-03
EMa,2
0.0100
3.3853
0.0317
0.0313
0.0308
0.9595
0.9558
0.9595
0.0268
0.0266
0.0260
0.9620
0.9761
0.9947
0.0244
0.0243
0.0242
0.0011
RE o % diff Ea,I % diff Ea, 2
1.E-04
6.E-03
2.E-04
2.E-04
2.E-04
4.E-03
6.E-03
7.E-03
3.E-04
4.E-04
4.E-04
1.E-02
1.E-02
2.E-02
5.E-04
5.E-04
5.E-04
3.E-03
9.7
57.3
4.9
2.1
-13.6
4.3
-3.6
11.3
-9.1
8.7
-13.3
8.7
-3.3
7.1
-22.3
-12.7
15.1
75.1
-0.03
8.1
-1.5
-2.8
-5.8
34.1
3.0
34.7
-5.1
-1.2
-2.9
34.9
22.3
36.3
0.4
7.1
13.5
1.8
68
Zone
LR1
LR2
LB1
LB2
LB3
LF1
LF2
LF3
IB1
IB2
IB3
UFI
UF2
UF3
UBI
UB2
UB3
UR
1.E-04
3.E-03
3.E-04
2.E-04
5.E-04
1.E-03
3.E-03
2.E-03
5.E-04
2.E-04
5.E-04
3.E-03
8.E-03
4.E-03
2.E-03
5.E-04
7.E-04
4.E-04
RE a
2.E-03
6.E-04
5.E-04
3.E-04
4.E-04
2.E-04
1.E-04
2.E-04
3.E-04
1.E-04
2.E-04
2.E-04
1.E-04
2.E-04
4.E-04
2.E-04
6.E-04
8.E-04
a,2
0.0100
3.6585
0.0313
0.0304
0.0290
1.2867
0.9847
1.2929
0.0254
0.0262
0.0253
1.2977
1.1941
1.3557
0.0245
0.0260
0.0275
0.0011
Table 2.12. Two- and Three-Dimensional Neutron Fission Production Cross Section Comparison
Zone vE3 RE a vE RE a vE RE a vE RE a % diff vYf,I % diff VEaf,2
LR1
LR2
LB 1
LB2
LB3
LF1
LF2
LF3
IB 1
IB2
IB3
UFI
UF2
UF3
UB1
UB2
UB3
0.0000
0.0000
0.0031
0.0030
0.0033
0.0297
0.0256
0.0262
0.0026
0.0021
0.0024
0.0240
0.0220
0.0225
0.0026
0.0022
0.0021
0.E+00
0.E+00
6.E-04
4.E-04
5.E-04
2.E-04
9.E-05
2.E-04
4.E-04
3.E-04
3.E-04
1.E-04
8.E-05
1.E-04
5.E-04
4.E-04
8.E-04
0.0000
0.0000
0.0215
0.0210
0.0201
1.9639
1.3930
1.9437
0.0181
0.0189
0.0181
1.9541
1.6699
2.0520
0.0177
0.0189
0.0201
0.E+00
0.E+00
4.E-04
3.E-04
6.E-04
1.E-03
3.E-03
2.E-03
6.E-04
2.E-04
6.E-04
3.E-03
8.E-03
4.E-03
2.E-03
6.E-04
8.E-04
0.0000
0.0000
0.0034
0.0033
0.0032
0.0288
0.0256
0.0241
0.0025
0.0025
0.0024
0.0224
0.0217
0.0211
0.0021
0.0021
0.0021
0.E+00
0.E+00
4.E-04
4.E-04
4.E-04
2.E-04
2.E-04
2.E-04
4.E-04
4.E-04
4.E-04
2.E-04
2.E-04
2.E-04
4.E-04
4.E-04
4.E-04
0.0000
0.0000
0.0218
0.0216
0.0214
1.3702
1.3515
1.3504
0.0192
0.0191
0.0187
1.3476
1.3642
1.3911
0.0176
0.0175
0.0175
0.E+00
0.E+00
2.E-04
2.E-04
2.E-04
4.E-03
6.E-03
7.E-03
4.E-04
4.E-04
4.E-04
1.E-02
1.E-02
2.E-02
5.E-04
6.E-04
5.E-04
N/A
N/A
-9.2
-10.2
2.9
3.1
-0.2
8.5
4.0
-15.6
2.5
7.1
1.4
6.5
24.6
5.1
1.2
N/A
N/A
-1.6
-3.1
-6.4
43.3
3.1
43.9
-5.7
-1.3
-3.2
45.0
22.4
47.5
0.4
7.8
15.1
UR 0.0000 0.E+00 0.0000 0.E+00 0.0000 0.E+00 0.0000 0.E+00 N/A N/A
69
Figure 2.20. Power Distribution Diagram of RBWR Assembly (side-view)
Figs. 2.21, 2.22, and 2.23 for the transport, fission production and absorption cross sections, re-spectively. For these parameters, there is a clear trend that sub-regions of a coarse zone that arenext to another coarse zone have a larger difference than those that are internal to the coarse zone.The blanket zones are receiving a net current-in of fast neutrons and net current-out of thermal neu-trons. Since the fissile zones are made of MOX fuel which contains isotopes of plutonium, thereis a very low flux of thermal neutrons. This source of thermal neutrons from the blanket zones hasa large impact on the group parameters representing the thermal energy region. There are zones,however, that do not follow this trend. One such zone is the group 1 fission neutron productioncross section of internal blanket zone 2. It is clear that this coarse region has a larger differencefor the sub-region that is internal to it. One explanation for this is that the two sub-regions sur-rounding this internal blanket zone are much smaller so that the fast neutrons coming in from thesurrounding fissile zones are making an impact in the internal sub-region. The group 1 transportcross section follows the opposite trend as well for all of its sub-regions. The last large differencesthat can be noticed from the tables are the group 1 transport and absorption cross sections. Recallthat in the two-dimensional calculation, the upper reflector zone was coupled to the upper blanketzone 3. The upper blanket zone in this case acts as a source of neutrons for the upper reflectorregion. It is clear that having the upper blanket region as the source of neutrons is incorrect. Thisalso indicates that the source of fast neutrons is the upper fissile zones. Since the void fraction ofthese upper regions of the assembly is large, neutrons will travel further between interactions andincrease the axial communication between zones. Therefore, one may have to model the reflectorregion with a fissile-blanket-reflector system. It can be concluded from this study that there are
70
3 Group 1 Transport XS
30 U Group 2 Transport XS
,25
x
E20 -
UC 15
0
-5 I
rT
-P -- _
-10
LR1 LR2 LB1 LB2 LB3 LF1 LF2 LF3 IB1 IB2 IB3 UF1 UF2 UF3 UB1 UB2 UB3 UR1I
Figure 2.21. Differences between Three-Dimensional and Two-Dimensional Transport CrossSections
60U Group 1 Fission Productior
50 E Group 2 Fission Productior
,,40x
30
020
-010
0
-10 Fr I-20
LR1 LR2 LB1 LB2 LB3 LF1 LF2 LF3 IB1 IB2 IB3 UFI UF2 UF3 UB1 UB2 UB3 UR1
Figure 2.22. Differences between Three-Dimensional and Two-Dimensional Fission ProductionCross Sections
71
0
35
XS
XS
800 Group 1 Absorption XS
M Group 2 Absorption XS
60
S40 -
-o
C 20 - - -----
0Al
-40 _
LR1 LR2 LB1 LB2 LB3 LF1 LF2 LF3 IB1 1B2 IB3 UF1 UF2 UF3 UB1 UB2 UB3 UR1
Figure 2.23. Differences between Three-Dimensional and Two-Dimensional Absorption CrossSections
large differences between homogenizing each zone with zero net current boundary conditions andhomogenizing them in the presence of neighboring zones. This study motivates in better under-standing of investing how the homogenization of three-dimensional cross sections will impact fullcore calculations.
72
3 Preparation of Homogenized Parameters
3.1 Branch Cases
To obtain Serpent-generated homogenized cross sections and other parameters in PARCS, a method-ology must be created to calculate and organize these data. A script was developed using the Pythonprogramming language to create Serpent input files, execute Serpent, extract output and organizePMAXS cross section data files. In this section, the created code wrapper, SerpentXS, is describedalong with its initial tests to verify that it was programmed correctly. In Section 4, SerpentXSwas applied to evaluate the use of three-dimensional instead of two-dimensional generated crosssections in PARCS.
The construction of a cross section database is one of the most important steps in the reactoranalysis procedure. To obtain accurate results from a core simulator, an accurate representationof materials and geometry must be provided in the form of a database. Within this database,homogenized macroscopic cross sections and other important parameters are listed to reflect dif-ferent possible operating state conditions that a specific region will experience as it operates inthe core. In this section, the PARCS-PMAXS methodology will be described as an example.The current methodology for PMAXS cross section preparation is handled by the interface codeGenPMAXS (Xu and Downar, 2009). In PARCS, a PMAXS file must be generated for each
unique two-dimensional region to be used in the full core calculation. An unique region meansthat if more than one node contains the same material, but different operating conditions, only onePMAXS cross section database file is needed. However, this PMAXS file must adequately coverall possible operating states that all nodes with that material may experience.
To represent operating state conditions of each node adequately, PARCS accounts for both in-stantaneous and history effects. The macroscopic cross sections are parametrized as a functionof state variables such as control rod insertion fraction (CR), coolant density (DC), fuel temper-ature (TF), coolant temperature (TC), coolant impurity (IC), moderator density (DM), moderatortemperature (TM), soluble poison concentration in moderator (PM), and moderator impurity (IM).These variables account for the instantaneous change of the state of the system. In addition, PARCSaccounts for the strong dependence on history effects of state conditions such as burnup, controlrod, coolant density, coolant soluble poison, fuel temperature and coolant temperature. These his-tory variables are important to capture the impact that state variables will have on the system overtime. For example, the energy spectrum and isotopics will be different at a given burnup depend-ing if the control rod was inserted or withdrawn during the operating time. This time-dependencecan be captured in the parametrization of history variables where burnup is always required. Fi-nally, PARCS has the capability of accounting for differences in the neighboring assembly coolantdensity (DN) and burnup (BN). These variables are treated as instantaneous branch cases.
Since xenon and samarium microscopic absorption cross sections are very large and the numberdensity of these isotopes is dependent on flux level, these macroscopic cross sections are treatedseparately. Therefore, the PMAXS database also has an option for including fission yields forisotopes, microscopic absorption cross section of these isotopes, and macroscopic fission crosssection for reaction rates. The representation of macroscopic cross sections in PARCS is
El (, S = (C,S,N,H) i E acoSp)+cN cro (C Sn, H) + SNHm (C S, N H) . (3.1)
In Eq. (3. 1), E' (C, S, N,H) is the macroscopic cross section, EEJi (C, S, N, H) is the macroscopic
73
cross section with xenon and samarium cross sections removed, N, is the number density ofxenon, ake (C, S, N, H) is the microscopic absorption cross section of xenon, N, is the numberdensity of samarium and ay, (C, S, N, H) is the microscopic absorption cross section of samarium,all for node i. Each cross section is a function of a combination of state variables. The state vari-able C represents the insertion fraction of the control rod, S represents all other instantaneous statevariables other than control rod, N represents neighbor assembly information and H represents thehistory. Therefore, if the explicit treatment of xenon and samarium is being used, a procedure forseparating out these cross sections must be created. A study was performed to rank the importanceof cross section dependencies on different state variables of the cross sections on different instan-taneous variables (Xu and Downar, 2009). From this study it was concluded that the importanceof instantaneous branches should be in the following (descending) order:
1. CR control rod fraction
2. DC density of coolant
3. PC soluble poison concentration in coolant
4. TF temperature of fuel
5. TC temperature of coolant
6. etc.
Only the first five of these variables are considered in this research, but more can easily be addedin the future.
The organization of the homogenized macroscopic cross section data stored in PMAXS can beconveniently presented as a "tree-leave" structure (Stalek and Demaziere, 2008). An illustrationof this structure is shown in Fig. 3.1. At the first level to the left, the dependency of homogenizedparameters is on the history effects. One can picture multiple of these tree-leave structures stackedon top of one another, each representing a different operating history. On the next level, in thecenter of the diagram, a history is parametrized by different operating conditions. Each path thatcan be drawn along the tree represents a combination of operating conditions that is calculated forthe assembly in the lattice code. Finally, the lattice code calculates homogenized parameters as afunction of burnup where the lowest level of the structure represents this dependency. The processfor generating instantaneous and history branch cases are treated differently. For all history casesin a PMAXS file, the structure of the branch cases must be the same, however, there may be adifferent number of burnup points in the branches of different histories.
3.1.1 Instantaneous Branch Cases
This type of branch case is used to characterize instantaneous changes in state values such ascontrol rod fraction, coolant density and fuel temperature. The process of creating instantaneousbranch cases begins with the creation of a reference branch. The reference branch contains a set ofoperating conditions that will be used as a base for the branch case calculations. The homogenizedparameters are stored directly in the PMAXS database. A new branch state is created by perturbingone of the state variables from its reference value. For example, if the reference state has the control
74
Upper 1"ed(depedency on Mso-
ry e si
D.LO --------
T bi 6=1_., No
T0 ------- ------
chi. IT1 I bsa=J], .Np
T o b(ai i]. ..No
t----------------
b-(ai= .No
Tj* bi 6=. . N)
----------------
statmwO twin aspT b0=1 N
Figure 3.1. PMAXS "Tree-Leave" Structure (Stalek and Demaziere, 2008)
rod withdrawn, a control rod branch state is created by inserting the control rod. Note that the nameof the branch state is labeled by the state value that is being perturbed. Unlike the reference state,the homogenized parameters of the branch are stored as partial derivatives in the PMAXS file. Ingeneral, this partial derivative is calculated by
dxk xi(3.2)E (X)-E (Xi)
Xi -x
where A is the partial derivative of the cross sections for state variable k evaluated at thedOk Xm1
midpoint X'. The variable X is a vector that represents the state variables of a particular node. InEq. (3.2), E (Xi) is the macroscopic cross section of arbitrary reaction that is calculated from thelattice code when the operating conditions are at state XI, which is the branch state, and E (Xi) isthe macroscopic cross section of arbitrary reaction at the reference state X'. The difference in themacroscopic cross sections is then divided by the difference between the state variable that wasperturbed, xi, and the reference state x. For control rods, the denominator of Eq. (3.2) is modifiedwith the control rod fraction of the branch state instead of the difference.
To illustrate these concepts, a fuel temperature branch will be described similar to the examplein the GenPMAXS Manual (see Fig. 3.2). The goal of this example is to explain how PARCSinterpolates on a node's operating state, given a cross section dataset in PMAXS format. In thisdiagram, macroscopic cross section data is represented for a reference state, control rod branchand fuel temperature branches. Each horizontal rectangle (in blue) with a branch identifier (Ref,
75
Rodded
Figure 3.2. Fuel Temperature Instantaneous Branch Case PMAXS Example
CR, TF) represents a macroscopic cross section such as the absorption cross section. The verticalrectangles represent the partial derivatives between the reference and branch states, calculated byEq. (3.2). For this simplified example, there is no bumup dependence and the example representsa steady state calculation. The group homogenized absorption cross sections represented by thehorizontal rectangles are generated at the lattice calculation stage of reactor analysis, with Serpentfor example. The goal of the interpolation is to calculate a group absorption cross section in whichthe node has a fraction of the control rod inserted and has a fuel temperature at point 1. Thismacroscopic cross section is calculated at point I with
d Ea dY~aEa (c, TF) = E' + C + (TF - TFr) 9TF (3.3)dCr (Cr1/2) dTF (c,TFm)
where Ea (c, TF) is the macroscopic absorption cross section at point 1, Ea is the absorption crosssection calculated at the reference conditions, c is the fraction that the control rod is inserted,
C ; (Cr'/2) represents the contribution from the inserted control rod that is labeled as point 2 in
the diagram, and (TF - TFr) (cT represents the contribution from the difference in fuelT(c,TmF))
temperature between points 1 and 2. The partial derivative, d r(Cr/2 is stored in the PMAXS file
in the control rod branch. The partial derivative for fuel temperature, -a-F ( ,) represented by
point 3, is more difficult to calculate and is obtained by a weighted linear interpolation representedwith
d Ea dEa dEadTF (c,TFm) dTF (0,TF1m) dTF (0,TF2m)
dLa d La± W3 dTF (1,TF3m) dTF (0,TF4m)
76
The interpolation weights in Eq. (3.4) are calculated as follows:
~zlCTF-TF2 (TF-FWl = - c) TF-TF2 F-TF 2 (3.5)
TF-TF ( TF -TF 4
W3=CTF3-TF4 W4 C TF 3 -TF 4 J
In the actual calculations, the cross sections may be parametrized in more variables which containseven more branches. In these cases, the expressions in Eqs. 3.3 and 3.4 become more complex.This process is repeated for all homogenized parameters for each node in the full core model.
3.1.2 History Branch Cases
The purpose of history branch cases is to capture the effects of operating with certain conditionsover time. Since operating conditions of a node in a full core will change with time, the model must
account for a range of operating conditions. To create a history branch case, operating conditionsare changed from the reference state at the lattice calculation stage. Unlike the instantaneousbranches, where the operating conditions are perturbed at select burnup steps, the history cases are
created by using the new conditions at the beginning of the calculation and depleting the lattice
with these conditions. All of the history cases located in the PMAXS file must have an identicalbranch structure although there may be a different number of burnup steps associated with differenthistories (Xu and Downar, 2009).
The cross section data for the reference case and the partial derivatives for all of the instan-
taneous branch cases are treated with partial derivatives with respect to history variables. Eachhistory variable that is present in the PMAXS file is tracked as a function of burnup in PARCS.At a given burnup, a history variable is calculated by an average from the initial time to a given
burnup. For example, to calculate the history of a control rod, the fraction of the control rod isaveraged over burnup
fB Cr (B) dBHCR(Bi) - (3.6)
In Eq. (3.6), HCR (Bi) is the control rod history at burnup Bi, Cr (B) is this control dependenceon burnup in past time steps. This equation is used for each history to be calculated. Since it
will be rare to exactly represent all of the histories explicitly for each node, a range of histories is
calculated at the lattice stage and can be interpolated on the fly during the full core analysis. To
calculate the appropriate homogenized cross section dataset to be used for the node, this dataset is
represented by
nh gE (H,B)= E(Hr,B)+[ Ahi dh , (3.7)
i=1 i (HT,B)
where E (H, B) represents the homogenized macroscopic cross sections and parameters at the ref-
erence state and also the partial derivative representation for the instantaneous branch states at the
specific combination of history conditions, H, of a node. In addition, E (Hr, B) is the cross section
dataset at the reference history with its specific combination of history variables, Hr, Ahi is the
77
difference between the history and the reference history value, and is the partial deriva-dhi (HB)tive with respect to the history evaluated at the midpoint between it and the reference history. Thispartial derivative is obtained through multi-dimensional piece-wise linear interpolation at neighborstates. If a cross section dataset is between burnup points calculated in the lattice code, the datasetmust be also treated by interpolation. This can be represented by
B 1 -B B-Bi 1E (H', B) =B -B E (H', B'1 - kBi- -I E (H', B') ,(3.8)
k k-1 k k-1
where E (Hi, B) is the cross section data set for a history at the target burnup B, B' is the burnup stepthat is greater than B where cross section data, E (Hi, B') is a cross section dataset generated by thelattice code at the larger burnup step and B'_ is the burnup step less than B with its correspondingcross section dataset E (Hi, B_ 1).
To illustrate this interpolation for history cases, an example is shown in Fig. 3.3 from theGenPMAXS Manual. This three-dimensional diagram represents three types of history variableswhich include a control rod history, several coolant density histories and dependence on burnup.The control rod history is represented on the vertical axis, and its dependence is shown as planesin the diagram. The coolant density is in the third dimension (into the page), and is representedby horizontal lines parallel to the horizontal axis and exists on both control rod planes. Finally,burnup dependence is shown on the horizontal axis and is represented by individual points alongcoolant density lines on both control rod planes. There are twelve burnup points, points 3, 4 and5 are the reference history, 10, 11 and 12 are the control rod history and the remaining points arecontained in the other 3 coolant density histories. Note that some of these coolant density historiesalso have the control rod inserted during its depletion.
During the PARCS calculation, at a target burnup, the control history and coolant density his-tory are computed similar to Eq. (3.6). These three parameters are sufficient to identify the targetinterpolation point 13. The first step is to determine the cross section dataset at the same burnupas point 13 (labeled as point 14). Point 14 is computed from linear interpolation between data atpoints 4 and 5. Similarly, at the control rod history, a cross section data set is computed for thesame burnup via linear interpolation of points 11 (1, 6) and 12 (2, 7) and is labeled as point 17(24, 25). The partial derivative of point 16 is needed to calculate point 15 and is determined frompoints 14 and 17. In order to calculate point 13, a partial derivative between 13 and 15 needs tobe calculated and is labeled as 18. This partial derivative can be linearly interpolated between thepartial derivatives of control rod histories at points 19 and 20. Point 19 is determined from points17 and 21, and point 20 is determined by the partial derivatives of points 22 and 23. The partialderivative at point 22 is calculated by points 24 and 14, and the partial derivative at point 23 iscalculated by points 14 and 25. The reason why the calculation of point 20 is more difficult than19 is because there is more history branches on that control rod history plane. This interpolationscheme is very complex and PARCS completes this task by defining weights, similar to Eq. (3.5),to the minimum number of burnup points needed to perform this calculation. This includes points1, 2, 4-9, 11 and 12. The rest of the points needed in the diagram can be derived from these basepoints.
78
HDcl 1 _ _ _W__ _Bumup
Figure 3.3. Interpolation Example for History Cases Structure (Xu and Downar, 2009)
3.2 SerpentXS Wrapper
In the general lattice code-to-PARCS methodology, the output from two-dimensional deterministiclattice codes, such as HELIOS and CASMO, are processed and organized into PMAXS data fileswith the code GenPMAXS. It was considered in this work to develop a similar processing capa-bility in GenPMAXS for Serpent, however, Serpent does not currently have built-in capability ofrunning instantaneous or history branch cases. Therefore, the main task in developing SerpentXSis to create a methodology to use Serpent to produce few-group homogenized cross sections. Theobjectives of the SerpentXS code are as follows:
1. develop a user friendly simple interface between Serpent and PARCS,
2. calculate instantaneous and history branch cases,
3. develop as a wrapper around Serpent without changing the source code,
4. make the code general so that it could be used with other lattice codes,
5. make the post processing general so that it can output in different cross section formats otherthan PMAXS,
6. have the capability of two-dimensional and three-dimensional cross section generation andprocessing.
The most important of these objectives is to make it user friendly. All of the complex data manipu-lation for this process is performed by SerpentXS, where the user just has to specify the geometryand the types of branch cases to generate. The input to the code is specified in two input files:a branch case input file and a geometry-specific file. The branch case input file contains all ofthe different instantaneous branches and burnable materials, whereas the geometry-specific file issimilar to a Serpent input layout and contains all other information required for executing Serpent.In order to perform history case calculations, the user can run SerpentXS multiple times with dif-ferent reference state variables. The PMAXS output of all of these runs can be combined into one
79
Y M-
PMAXS datafile to be used in PARCS. SerpentXS also has an option to either skip or perform par-tial derivative calculations. In any case, the PMAXS files in PARCS must be provided with partialderivatives. The user has the option of taking the SerpentXS generated PMAXS file and running itthrough GenPMAXS to calculate partial derivatives.
The wrapper is broken up into three files: a main file, RUN_SERPENT.py, a Serpent specificmodule, SERPENT.py and a PMAXS module, PMAXS.py. To make it general enough to use withother Monte Carlo codes, a data structure was developed to organize the large amount of dataoutput from Serpent. All of the Serpent-specific functions are contained in the SERPENT module.This module can be replaced in the code with a different module from another type of Monte Carlocode. As long as the functions within the module are the same and the end data structure is thesame, one can create a PMAXS formatted database. Conversely, if a cross section database formatother than PMAXS was desired, this module can be replaced. This general organization will allowthe code to expand in the future, especially for the latter purpose.
The overall process of incorporating Serpent with PARCS is illustrated in Fig. 3.4. After theSerpentXS input is created and the code is initiated, it creates input files for each branch case andexecutes them in Serpent. After Serpent is completed, SerpentXS extracts data from the output fileand organizes it in a data structure. This process is repeated for all history cases. When the historycases are completed, a PMAXS file is created for PARCS. If the user has not selected the partialderivative option, GenPMAXS may be used to prepare and verify the cross section set. A PARCSinput file and PMAXS files are necessary for full core analyses in PARCS. The data that PARCSaccepts as input from PMAXS files is listed in Table 3.1. The table is separated into a thermalhydraulic invariant part and a variant part. Thermal hydraulic invariant refers to parameters thatdo not change when the reference case conditions are instantaneously perturbed. Therefore, theseparameters do not need to be parametrized as a function of instantaneous operating conditions. Forthe variant block of data, these need to be extracted and organized for every instantaneous branchstate. From the table, it can be seen that Serpent outputs the majority of the parameters. If a "Yes"is listed, this means it is included in the standard output and is relatively easier to collect. If a "No,Tally" is listed, this means that the parameter is not listed in the standard output, but can be directlycalculated with a tally. These parameters are more difficult to obtain, but can still be calculatedfrom tally information. Finally, "No" means that it takes a large amount of effort to collect thedata. Of these types of parameters that are labeled with "No", the effective yields of I, Xe, and Pmare of importance.
3.2.1 Input to SerpentXS
A branch case input file and a Serpent geometry file are needed to execute the SerpentXS package.The purpose of this input file is to specify the structure of instantaneous branch cases and burnablematerials to be executed for a given set of history conditions specified in the reference branch. Thisinput file is free form and keywords can be listed in any order. To specify the Serpent geometryfile to include in the calculation, the keyword geomf ile is used. The syntax of the geometry cardis:
geomfile <filename>
where (f ilename) is the name of the geometry file that is present in the current working directory.
80
Table 3.1. Homogenized Parameters for PARCS Calculation
(a) Thermal Hydraulic Invariant Data
Parameter Description Serpent?
Chi
Chid
inV
YLDI, YLDXe, YLDPm
BETA
LAMBDA
DBET, DLAM
Parameter
Etr, Ea, Vyf, Ef
IcEr
~Xe Sin
Es,g -g
ADF
CDF
GFF
DED
J1F
Fission spectrum
Delayed fisson spectrum
Inverse neutron velocity
Effective yields of I, Xe and Pm
Effective delayed neutron fraction
Decay constant of precursors
Decay heat for delayed neutrons
(b) Thermal Hydraulic Variant Data
Description
Principle cross sections
Energy deposition xs
Micro absorption xs of Xe and Sm
Group scattering transfer matrix
Assembly discontinuity factor
Corner discontinuity factor
Group-wise form function
Direct energy deposition fractions
JI factors for CPR
81
Yes
Yes
Yes
No
Yes
Yes
No
Serpent?
Yes
No, Tally
Yes
Yes
Yes
Yes
No, Tally
No
No
CrCreate Serpent input
0 No
Figure 3.4. Serpent to PARCS Flow Diagram
The code has been written in such a way that different cross section database formats may begenerated. The syntax for specifying the format of the cross section database is:
<xsformat>
<option 1><option 2>
<option N>
where (xsf ormat) is the cross section database format in which the only option is pmaxs currently.Different formats of cross section databases may require different control options. These optionsare listed after the database format specification. For PMAXS formatted data, a set of logicalsare used to determine which parameters are to be included in the dataset. These logicals weredetermined from the GenPMAXS manual and are listed in Table 3.2. These logicals tell SerpentXSwhich data to calculate and output into PMAXS files. It also allows Serpent to know how manyenergy groups will be present and whether to calculate partial derivatives or not.
The next block of input contains information about the branch case structure. The only requiredbranch to be present is the reference branch which contains the reference operating state. Theother branch cases are optional, and they represent perturbations of state values from the referencebranch. The syntax of each case structure is:
case <dirname> <type> <number>burnup <buvect>CR <cfrac> CRU <cU>
82
Table 3.2. Logicals for PMAXS Database (Xu and Downar, 2009)
Description
Assembly discontinuity factor
Microscopic cross section of Xe and Sm
Direct energy deposition fraction
J 1 factor for MCPR
Fission spectrum
Delayed neutron fission spectrum
Inverse velocity
Detector response xs
Yield values of I, Xe, Pm
Corner discontinuity factor
Group-wise power form function
Effective delayed neutron fraction
Precursor decay constant
Decay heat beta and lambda
Number of energy groups
Calculate partial derivatives
Value
T/F
T/F
T/F
T/F
T/F
T/F
T/F
T/F
T/F
T/F
T/F
T/F
T/F
T/F
minimum 2
T/F
83
Parameter
Ladf
Lxes
Lded
Lj 1f
Lchi
Lchd
Linv
Ldet
Lyld
Lcdf
Lgf f
Lbet
Lamb
Ldec
NGROUP
Lderiv
DC <DCvec>PC <PCvec>
TF <TFvec>
TC <TCvec>
where (dirname) is the name of the directory that will be created in the current working directoryto contain the Serpent execution data, (type) is the type of branch case which is one of the follow-ing identifiers depending on which parameter is perturbed from the reference branch: CR, DC, PC,TF, TC. If the case is the reference branch, then the identifier REF must be used for (type). Theparameter (number) represents the current instance of that (type) of branch. Together, (number)and (type) uniquely identify a case structure. The parameter (buvect) is a list of cumulative bur-nup values to be calculated in ascending order, delimited by a space starting with 0.0, (cf rac) isthe fraction of the control rod inserted in the geometry, (cU) is the Serpent geometry universe thatcontains the control rod and (...vec) represents a vector of values corresponding to an operatingstate in the calculation. A vector is specified rather than just a single value when three-dimensionalcross sections are being generated. In this case, the operating states for homogenization regionsmay differ. These cases are listed in sequential order where the order does not matter. The onlyrequired case is (type) = REF with (number) = 1.
The last block of input corresponds to a listing of burnable materials. These burnable materialsmust be listed in this file because the concentrations and types of isotopes that these material cardswill contain will vary with burnup. By listing the materials in this file, SerpentXS will rememberwhich materials to recover at every burnup step for the instantaneous branch cases. To create aburnable material the syntax is:
mat <name> -<mdens> burn <nrings> vol <volume> gcu <u>
<ZAID> -<w%>
where (name) is the name of the burnable material, (mdens) is the mass density denoted with anegative sign in front of it for Serpent, (nrings) is the number of equal volume rings to subdividethe pin where this material is attributed, volume is the material's volume and (u) is the universenumber that corresponds to the homogenization region for cross sections. This universe identifiermust be contained in the gcu card in the Serpent geometry file. The volume specification is onlyneeded for three-dimensional files since Serpent cannot calculate these volumes automatically. Fortwo-dimensional calculations, the volume card may be left out and SerpentXS will omit it. Afterthe heading, a vertical list of ZAID-formatted (e.g. 92235 for uranium-235) isotope identifiersfollowed by weight percents must be provided. In this material card construct, cross section li-brary extensions do not need to be specified since the temperature of this material may changewith different instantaneous branch cases. SerpentXS automatically chooses the appropriate crosssection extension and places the Doppler broadening information in the input file. These burnablematerials are also listed in sequential order. This is the extent of the extra information that the usermust provide in order to use the branch case generator for Serpent. Although Serpent does Dopplerbroaden neutron-nuclide cross sections, it does not broaden thermal scattering data.
With any Serpent run, an input file must be created to include geometry, material and other datacontrol information. A variant of this file must also be included as part of the SerpentXS input and
84
fortunately does not vary too much from an actual Serpent input file. For specifics and syntax onhow to create a Serpent input file, the user's manual can be referenced (Leppinen, 2010b). Thesurfaces, cells, lattices and universes to create the geometry layout in the file is exactly the same asa Serpent input file. For the materials that are attributed to these cells, burnable materials are listedin the branch case input file and non-burnable materials are listed in this geometry file. In order topropagate the operating conditions specified in the branch case input file, modifications need to bemade to the geometry file. For the operating conditions DC, PC, and TC, a variable is listed in theSerpent input file in the form ( [type] [gcu]). For example, if the density of the coolant will changefrom the reference case to a branch case, the density listed in the material card should be listed as:- (DCO). The negative is in place to denote a mass density which must be specified, DC correspondsto the value listed in the DCvec which is indexed by the gcu card in the Serpent geometry file and0 corresponds to the group constant universe where this universe is being varied. Therefore, forthe gcu card in this geometry file, a 0 should be listed. For a calculation where multiple regionsare homogenized, a vector of universes is listed here instead.
To run branch cases with SerpentXS, the following cards are needed for control informationin Serpent: pop, acelib, declib, nfylib, fpcut, stabcut, bunorm, bumode, pcc, xscalc,printm 1, gcu, nf g, egrid, and powdens. The input card specifying the unit of burnup for deple-tion, and the burnup vector, are specified with the keyword but ot which is handled by Serpent XSautomatically. Most of the cards listed above can be referenced in the Serpent manual. The printmcard must be turned on so that material concentrations at each burnup step can be recovered forthe instantaneous branch case calculations. The gcu card is the most important card and lists thehomogenization regions. The operating condition vectors listed in the branch case input file forspecific geometry universes should correspond to this order. The nf g and egrid commands statethe energy group structure and are needed for the homogenized cross sections and other manualtallies that SerpentXS places in the input files. Examples of a branch case and geometry input fileare listed in Appendices B. 1 and B.2, respectively.
3.2.2 Framework of SerpentXS
The high level operations in SerpentXS are listed in a Python script which calls the different mod-ules that contain important functions to generate branch cases. An outline of the flow of the codeis shown in Fig. 3.5. The flow of the code is split into two main parts. The first part handles thereference case and the second part handles the branches cases. The code begins by reading theinput files and creating a data structure corresponding to all of the branch cases. The directorystructure of the reference case is then created, followed by the Serpent input file for this case. Thereference case must be executed first, separately from the branch cases, since the branch cases mustbe restarted from the reference case isotopics at each burnup step. The Serpent run is submittedvia the TORQUE job scheduling software (ClusterResources, 2011). A job scheduling software isrequired for the execution of SerpentXS because of the hundreds of potential files that can be runat one time and amount of memory that Serpent requires for each run. SerpentXS then waits untilthe Serpent job is completed to process the output into the data structure.
In order to run branch cases, SerpentXS processes the burnable materials that were present ateach burnup step in the reference case. These materials are output in a special file,
(name) .bumat < bustep >
85
au rac
Be in
L BeginGLetinn put ffii les]l
Build Reference CaseExecute Reference Case
Wait until finished
Process referenceoutput
into datastructure
Process Materials forinstantaneous cases
Create branch casestructure
build input files
W
Figure 3.5. Overall Flow of SerpentXS Branch Case Generator
86
W
WOMI
Run Branch N
Burnup step I
Run Branch 1
Burnup step 1
Run Branch N
Burnup step M
by Serpent, for every burnup step when the printm mode 1 option is activated. A directory is thencreated for each branch case with separate sub-directories corresponding to each burnup step. Aseparate Serpent input file is then created for each burnup step, in each branch case and placed inthese directories. To be consistent, the random number seed that was used for the reference case isalso placed in the branch case input files with the set seed input card. Although SerpentXS canrun on a table top machine, it makes use of cluster performance by submitting all of the branchcase files at once to a job queue. As cluster resources become available, these files can be executedin Serpent. This step is shown in Fig. 3.5, where all of the branch cases are run in parallel.When all of the branch cases are completed, the results are processed in the data structure. If thepartial derivative option is activated, SerpentXS will calculate partial derivatives for each branchcase with respect to the perturbation variable. Finally, a separate PMAXS file is generated foreach homogenization region. There may be multiple homogenization regions if cross sections arebeing generated from a subset of the total geometry, which is important for three-dimensional crosssections.
The data structure that SerpentXS uses is saved after the run is completed in a binary file namedresout.bin. A Python dictionary is used to organize all of the output data for the reference andbranch cases. The dictionary construct is a standard data type and can be found in most of thePython literature (Lutz, 2008). A simplified diagram of the data structure is illustrated in Fig.3.6. At the highest level, the data structure contains keys which refer to the unique identifier ofthe reference or branch case. This information is determined from the branch case input file inthe form of (type) (number). For example, the reference case which has a type REF with thenumber 1 would be stored as REF1. Under the branch case identifier key, there are a number ofsub-dictionaries and other keys. Under the key name, the directory name that was specified inthe SerpentXS branch case input file is listed. This is used by the code to create the directorystructure for that case. The oper block refers to a list of operating conditions for the case whichincludes the control rod position, coolant density, coolant temperature, fuel temperature and poisonconcentration. The values are substituted into the Serpent geometry file when the case input filefor each branch is created. The type key contains the type of case being analyzed such as REF,CR, etc. The burnup key contains a list of cumulative burnup values for this case, taken from theSerpentXS input file. Finally, the output sub-dictionary contains all of the output from the Serpentrun after the problem is analyzed. In this sub-dictionary, each key is a Serpent output variable andthat variable contains the values that Serpent calculated. For example in the reference case, theoutput sub-dictionary contains information about the group homogenized macroscopic absorptioncross section (ABSXS). Note the values in ABSXS are parametrized by burnup, homogenizationregion, and energy group. To retrieve the few-group macroscopic absorption cross section forhomogenization region 101 at burnup 2.5 MW-day/kglHM, the following command can be used:
caselist['REF1'1['output']['ABSXS'][buidxl[gcuidx]
where caselist is the name of the Python dictionary, buidx is the index corresponding to thatburnup and gcuidx is the index for the homogenization region. The result of this command wouldbe the line from the Serpent output file for this specific burnup and homogenization region inthe energy group order specified by the Serpent manual. For branch cases, the format is slightlymodified since these cases are run at different burnups separately. Under the output sub-dictionarythere is a listing of all of the burnup values for the case. The data can be retrieved with
87
Figure 3.6. Data Structure Organization in SerpentXS
caselist['PC1'] ['output'] ['0.1'] ['ABSXS'] [gcuidx]
where the burnup value is placed before the variable of interest. This data structure is saved in thelocal directory after the SerpentXS run and can be loaded into the interactive python shell with thefollowing commands:
import cPickleinfile = open('resout.bin')caselist = cPickle.load(infile)infile.close 0
3.2.3 Generation of Homogenized Parameters
Almost all of the homogenized few-group parameters that PARCS uses are calculated automati-cally by Serpent and are available for extraction from the output file. In this section, each of the
parameters will be addressed in the order they appear in the PMAXS file. The PMAXS file isseparated into thermal hydraulic invariant and variant sections. Each of these sections contain anumber of sub-blocks where the actual data is listed. This information is available in AppendixA of the GenPMAXS manual (Xu and Downar, 2009). The contents of each sub-block will beexplained along with how it is obtained from Serpent.
Invariant Sub-block 1: Chi, ChiD, inV The variables in sub-block 1 of the invariant data rep-resent the fission spectrum, delayed fission neutron spectrum and inverse of neutron velocity in(s/cm). In Serpent, these variables are labeled as CHI, CHD and RECIPVEL, respectively, and areavailable in the output file. This data is optional, but the fission spectrum should be included ifmore than two energy groups are used in the generation of homogenized parameters.
88
Data Structure
- I
,is mi1
7AB!S ,in 60s
7im
F1
7 N is
amel oper 7i petr
ADF
Invariant Sub-block 2: Yields of I, Xe and Pm The variables in sub-block 2 of the invariantdata represent the effective fission product yields of iodine-135, xenon-135 and promethium-149.These yields are not calculated directly by Serpent and must be determined manually. An effectiveyield of one of these isotopes for a homogenization region can be calculated by
(y = ( (3.9)
Y Nia piVJ
In Eq. (3.9), (yi) is the effective yield of isotope i, y/ is the yield of isotope i from fission of isotope
j (such as U-235, U-238, etc.), Ni is the atom density of isotope j, J is the effective microscopic
fission cross section of isotope j, pJ is the neutron flux in the material that isotope j is a part of andVi is the volume of that material. Therefore, the individual fission yields of isotopes in a materialmixture are weighted by their respective fission rates and then divided by the combined fission rateof the homogenization region.
The individual fission yields for each fissionable isotope are stored in ENDF data files (MT454 for independent yields and MT 459 for cumulative yields) and come with the Serpent code.Therefore, SerpentXS extracts the isotope specific yields from the fission yield file that is specifiedin the Serpent geometry file. A Python function was written to read in this fission yield file and
organize the yields by isotope, independent or cumulative yield, ground or metastable state andby energy. For the calculation of the above yields, the fast neutron incident energy data (500keV) or thermal energy data can be (0.0253 eV) used. For iodine and promethium, the cumulativeyields were used, whereas for xenon the independent or direct yield was used. The direct yield
was chosen for xenon since iodine decays into xenon. The number density, microscopic fission
cross section, neutron flux and volume of the isotopes are included in a Serpent depletion outputfile. A separate Python function was written to extract and store these values for the fission yield
calculations. This data is optional and should be included when the Xe/Sm option is activated inPARCS. This function is currently still under development in SerpentXS.
Invariant Sub-block 3 and 4: Beta and Lambda of Delayed Neutron The effective delayedneutron fraction for each precursor group, P, and its associated decay constant, A, is automaticallygenerated by Serpent. The delayed neutron fractions and decay constants are the same for eachhomogenization region since the method is applied for the entire geometry. The number of precur-
sor groups varies between six and eight and depends on the isotopes present in the materials. Thisdata is optional and is only required for transient cases in PARCS.
Invariant Sub-block 5: Decay Heat Data The decay heat data which includes the fraction of
total fission energy appearing as decay heat in each precursor group and their associated decayconstants are not calculated by Serpent. Currently, this feature has not been added to SerpentXSas there is no current application for it yet.
Variant Sub-block 1: Principle Cross Sections This sub-block consists of two parts. The firstincludes the macroscopic few group cross sections that PARCS requires for its calculations. These
include the transport, absorption, fission neutron production and fission energy deposition cross
sections. The first three cross sections in this list are automatically calculated by Serpent and
89
included in the output file. The energy deposition cross section is calculated by a detector tally thatSerpentXS automatically places in the Serpent input files. Recall from Eq. (1.52) that the energydeposition cross section is used to calculate node-average power, and it is defined as the energydeposition rate divided by the integrated flux in the homogenization region. To calculate this usingSerpent, the following tallies are set up by SerpentXS:
det fluxO du 0 de egriddet kfissO dr -8 void du 0 dt 3 fluxO de egrid
In this tally, det initiates a detector tally, f luxO and kf issO are the names of the tallies where duOspecifies the homogenization region, de is the command to list the energy bins which are given bythe user with the ene egrid command, dr specifies the reaction type where -8 multiplies the fluxby fission cross section and energy deposition per fission, void applies this to all materials in thehomogenization region and dt mode 3 divides a tally by another tally. Therefore, the tally namedkf issO contains the homogenized energy deposition cross section since the reaction rate is dividedby the flux tally. This is repeated for all homogenization universes that are specified by the gcucard in the Serpent geometry input file.
The second part this sub-block contains data to rebuild the number densities of xenon andsamarium during the PARCS calculation. This data includes microscopic capture cross sections ofxenon and samarium and the macroscopic homogenized fission cross section. These parametersare optional and must be included if the Xe/Sm option is activated in PARCS. The macroscopicfission cross section is directly available in the Serpent output file. The microscopic capture crosssections are available in the Serpent depletion output file.
Variant Sub-block 2: Scattering Cross Sections This sub-block contains the group-to-groupscattering matrix that characterizes the probability of a neutron's final energy after undergoinga scattering collision. Serpent automatically includes this group transfer matrix in the standardoutput file in vector format. The Serpent manual provides a formula to determine the index of thevector corresponding to the transfer cross section E,ji:
n = 2(i- 1)G+2--- 1, (3.10)where n is the index of the vector, i is the final energy group, j is the initial energy group and G isthe total number of energy groups. Eq. (3.10) is used by SerpentXS to appropriately organize theoutput from Serpent into the PMAXS format.
Variant Sub-block 3: ADF This sub-block contains energy group parametrized assembly dis-continuity factors. Depending on the geometry, these discontinuity factors can be in a Cartesianor hexagonal configuration. SerpentXS searches through the Serpent geometry file provided bythe user and determines the appropriate configuration. For a Cartesian configuration, Serpent au-tomatically outputs the discontinuity factor for each of the four sides of the geometry. ThereforeNADF = 4 is used in the PMAXS file. Serpent outputs the ADFs in vector format in the follow-ing orientation: W/S/E/N. These refer to the standard West, South, East and North directions. Toextract these from the output file the following formula is used:
n = 2(i - 1) G+2g - 1, (3.11)
90
where n is the index in the vector, G is the maximum number of energy groups, g is the energygroup number and i depends on the direction: 1-West, 2-South, 3-East, 4-North. The order in thePMAXS is also W/S/E/N. For hexagonal geometry, the order is E(1)/SE(2)/SW(3)/W(4)/NW(5)/NE(6). To extract these from the output, Eq. (3.11) can also be used with the values for i shownin parentheses in the previous statement. The hexagonal methodology described applies for hexxcsurface types only. In PMAXS, all six discontinuity factors can be reported and therefore NADF = 6.The order in the PMAXS database is W/NW/NE/E/SE/SW. SerpentXS performs the reorderingfrom Serpent output to PMAXS format.
Variant Sub-block 4: Direct Energy Deposition and J1 factors The direct energy deposition
parameters and J1 factors are not calculated by Serpent. The direct energy deposition parametersreflect the amount of energy deposited in the coolant, water rod, bypass and control rod regions.The J1 factors are used in critical power ratio (CPR) calculations with, for example, the Hench-Gillis correlation (Todreas and Kazimi, 1990). These factors characterize the local peaking in the
bundle and depend on the peaking factors of the fuel rods and mass flux. Although not currentlyavailable, methodology for determining J1 factors can be added in the future.
Variant Sub-block 5: CDF This sub-block contains information about energy group dependent
corner discontinuity factors. These parameters are automatically calculated by Serpent and varydepending on whether the geometry is in a Cartesian or hexagonal configuration. For rectangulargeometries, Serpent writes the CDFs in NW(1)/NE(2)/SE(3)/SW(4). These can be extracted from
the Serpent output with Eq. (3.11). In the PMAXS database, NCDF = 4 and the CDFs are orientedin NW/SW/SE/NE. For hexagonal geometries of surface type hexxyc, Serpent orders the output
in NW(1)/SW(2)/S(3)/SE(4)/NE(5)/N(6). Using Eq. (3.11), the CDFs can be extracted from the
vector in the Serpent output file and re-ordered in the PMAXS database as SW/NW/N/NE/SE/S.SerpentXS reorders data from Serpent output to PMAXS format.
Variant Sub-block 6: Group-wise Form Function Methodology for calculating the group-wise
form function (GFF) does not exist in Serpent. The GFF is calculated on a pin-by-pin basis as a
function of energy group by taking the ratio of the local energy production cross section to the
average energy production cross section of the homogenization region:
GFF' = _. (3.12)g KXfjg Og
In Eq. (3.12), GFFg is the group-wise form function of energy group g in pin i, ICE, is the energy
deposition cross section for energy group g in pin i, 4g is the flux in energy group g in pin i, KEf,g
is average energy deposition cross section in the homogenization region and $g is the average flux
in the homogenization region, all in group g. Therefore, this group-wise form function represents
local power peaking of the fuel rods compared to the average fuel rod power in the assembly.Although currently not developed in SerpentXS, it can be added using the lattice detector tally inSerpent.
91
3.2.4 Creation of PMAXS Database
After the SerpentXS run is completed, a PMAXS-like file is generated for that history, for eachseparate homogenization region. To generate the full PMAXS file that can be used with PARCS, aseparate input file and Python function must be executed. This stage is handled separately, becausethe user may want to generate many history cases for the same geometry. These history cases areappended together at this stage to generate the full PMAXS file. This input file is set up in thesame way the final PMAXS file will appear. The input file is separated into different blocks, whereeach block of input is invoked by an unique keyword. The file begins with the global variableswith the keyword GLOBALV. Table 3.3 shows the list of the keywords that must be listed in theGLOBALV command, their typical values and a description. The parameters shown in the table areused in PARCS to allocate memory and to determine which information is given in the PMAXSfile. Following this, a set of five comments can be listed under the keyword COMMENTS. Thesecomments do not impact the calculation and are there for the user only.
The next block in the input file is for the branch cases. This section is invoked with the keywordBRANCHES. Each branch is listed in order and the layout of the input is similar to the form in theSerpentXS branch case input file. An example of this is shown below:
BRANCHESbranch <dir> <type> <number>CR <CRval>DC <DCval>PC <PCval>TF <TFval>TC <TCval>branch <dir> <type> <number>
Each branch structure begins with the keyword branch followed by the directory name, the branchtype and number. Following this, each operating state is listed with its value. The next branchstructure continues right under the final line of the previous branch structure. Since a PMAXS fileis generated for each homogenization region separately, only scalar values are accepted for eachoperating condition. This differs from the SerpentXS input file where a vector was placed in theorder of the homogenization regions.
The burnup structure information is contained in the next block of the input file and is invokedwith the keyword BURNUPS. The syntax of this block of code is:
BURNUPSNBset <num>I <idx>NBP <bupts>Burns <buvec>
In this construct, (num) is the number of burnup set, (idx) is the index for the burnup set, (bupts)is the number of burnup points contained in the burnup vector and (buvec) is a list of cumulativeburnup steps.
92
Table 3.3. GLOBALV Parameters for PMAXS File (Xu and Downar, 2009)
Parameter
NSET
NGROUP
MDLAY
MDCAY
MADF
MCDF
MRODS
MCOLA
Ladf
Lxes
Lded
Ljlf
Lchi
Lchd
Linv
Ldet
Lyld
Lcdf
Lgf f
Lbet
Lamb
Ldec
Derivative
Description
93
Example Value(s)
1
2
6
6
6
6
289
17
T/F
T/F
T/F
T/F
T/F
T/F
T/F
T/F
T/F
T/F
T/F
T/F
T/F
T/F
T/F
number of history cases in PMAXS
number of energy groups
maximum number of delayed neutron groups
maximum number of decay heat groups
maximum number of ADFs in each group
maximum number of CDFs in each group
maximum number of rods in computed part of assembly
maximum number of rod columns in whole assembly
pmaxs file contains ADFs
pmaxs file contains Xe/Sm cross section
pmaxs file contains direct energy deposition info
pmaxs file contains JI factors
pmaxs file contains fission spectrum
pmaxs file contains delayed neutron fission spectrum
pmaxs file contains inverse velocities
pmaxs file contains detector xs
pmaxs file contains fission product yields of I, Xe, Pm
pmaxs file contains CDFs
pmaxs file contains group-wise form functions
pmaxs file contains delayed neutron fractions
pmaxs file contains precursor decay constants
pmaxs file contains decay heat data
pmaxs file contains partial derivatives
The next block of input contains cross section specific parameters and is invoked with thekeyword XSSET. The syntax of this block of code is:
XSSETSeries <num>IST <bridx>NADF <iadf>NCDF <icdf>NCOLA <icol>NROWA <irow>NPART <sym>PITCH <pitch>XBE <xi>YBE <yi>iHMD <rhoHM>Dsat <rhosat>ARWatR <arwatr>ARByPa <arbypa>ARConR <arconr>
In this block of input, (num) is the cross section series number, (bridx) is the index of the branchstructure, (iadf) is the number of assembly discontinuity factors per group and (icdf) is thenumber of corner discontinuity factors per group. In addition, information about the fuel rods isprovided where (icol) is the number of rod columns, (irow) is the number of rod rows, (sym) isthe symmetry of the lattice, (pit ch) is the rod pitch, (xi) is the start position for the first column ofrods in the form function structure and (yi) is the start position for the first row of rods in the formfunction structure. One of the most important parameters, especially for depletion calculations,is the initial heavy metal density specified in (rhoHM). The value listed here is different than theinitial heavy metal density of the fuel. This can be converted with
PIHMn PIHMf (3.13)
where PIHMn is the initial heavy metal of the node which is specified in the PMAXS file, PIHMfis the initial heavy metal density of the fuel, Af is the cross sectional area of fuel and An is thecross sectional area of the node (homogenization region). The remaining parameters listed are thesaturated moderator density, (Dsat), area ratio of water rods to coolant, (arwatr), area ratio ofbypass region to coolant, (arbypa), and the area ratio of control rods to coolant, (arconr).
The last block of input lists the absolute path to each history file and is invoked by the keywordHISTORYC. The first history file must be the reference history, and the rest can be in any order. Thesyntax for this input is:
HISTORYC<abs path to reference history><abs path to history case 2><abs path to history case 3>
94
An example of this input file is located in Appendix B.3.
3.3 Spatial Multigroup Diffusion Solver
A standalone spatial multigroup diffusion solver has been developed to verify results from thePARCS code as well as for the axial discontinuity factor methodology presented in Section 4. Thesteady-state neutron diffusion equation in multigroup form can be represented as
G Xg (7 G
V -Dg (7)Vg (7)+Eg (7)#g (7) = E (7) Oh / + Vfh (V Oh (7), (3.14)h=1 keff h=I
where Fick's Law has been applied to relate the neutron current to the neutron flux (see Eq. (1.45)).In this formulation, g is an arbitrary energy group, Dg is a matrix containing direction diffusion
coefficients, 4 g (7) is spatial neutron flux, Etg (') is the macroscopic total cross section, Eg (7) isthe group h to group g scattering transfer cross section, xg (') is the fission spectrum, keff is theeffective multiplication factor and VEJh (V) is the fission neutron production cross section (H6bert,2009). It is common, to move the within group scattering term on the left hand side of the equationto the right hand side to form the removal cross section, Erg (7). In this case, it will also be assumedthat there is only one effective diffusion coefficient so that Eq. (3.14) becomes
- V -Dg () Vog (-) + Erg (i) Og (r) = , () #h V) + L fh V) Oh )gkh eff h=1
In this work, only a one-dimensional Cartesian domain will be explored since the RBWR is moreheterogeneous in the axial direction. The neutron diffusion equation in one-dimension is reducedto
d Dg (X) dog +rg(XPg (X) X Oh (X) + kf (x) h (x) (3.16)dx gWdx g Si ik - 1fdX dXgg h kefh=1
The spatial domain will be taken such that the macroscopic cross sections and diffusion coefficientsare defined as constant in a mesh cell. An illustration of the discretized domain is shown in Fig.3.7. From this diagram, the neutron diffusion equation is solved for each region where Eq. (3.16)can be written for an arbitrary region i as
regioni i- I region i regioni i+ I
Fui-3/2 .71 Dci-1/2 zi of ia2 Di+n bi+ 3/2
Figure 3.7. Discretization of Spatial Domain (Hebert, 2009)
95
G-- D (x) = E h rhg± (x) - h f,i E Vfh~i'h (x),dx g"dx rgig g7h keff h-=I
xi-1/2 < X < Xi+1/2. (3.17)
To solve these equations, a mesh-centered finite difference approach was used. In the mesh-centered finite difference approach, it is assumed that the average neutron flux in a region is equalto the neutron flux at the center of that region. This is represented for region i as
I 1x 1/2
0g, i = og (Xi) = -f g (x) dx,xi 1/2
(3.18)
where Axi is the length of the region. Equation (3.17) can be integrated over a mesh cell so that
d2 og Xi+1/2
dx2 dx+ Ergi #g (X) dxX - 1/2
h-g fXi+1/2 dx + Xgi Gxi+12SJ Oh (X)dx+ ( vEfhli
g7h 'Xi 1/2 keff h=1 X- 1/2
Substituting Eq. (3.18) into Eq. (3.19) the neutron diffusion equation becomes
-- D g,i d x -
_ i+1/2
dg + + Erg,ig,iAXi = Es,ih-g,ih,iAXi+
Xki-12 ghh
XgiG
ke55h=
#h (x)dx. (3.19)
VXLfh,i'fh,iAXi. (3.20)
To finish the formulation, the differential terms on the left hand side of the equation are approxi-mated with finite difference relations. The first differential represents the derivative of the flux onthe right boundary of the region approaching from the left and the second differential represents thederivative of the flux on the left boundary of the region approaching from the right. The derivativeof the flux will need to be approximated for both of these differentials as well as the differentialsjust on the opposite side of each interface. On the right interface, the differentials are
dog
dx X-i+ 1/2
g,i+1/2 - 0gi
Axi/2dog
dx + /i+1/2
~#g,i+1 - #g,i+1/2Axi+ 1/2
(3.21)
and on the left interface of region i they are given as
dog
dx X-i-1/2
g,i-1/2 - Og,i- IAx i1/2
dog
dx X+i- 1/2
~ - #g,i-1/2
Axi/2(3.22)
Focusing on the left interface of the region, the neutron current is continuous and this continuitycondition can be represented by
96
- Dg,i f/ i+1/2
Xi 1/2
Dgi1 di_ g = DO d.g (3.23)dx - dx X+
i- 1/2 i-1/2
Substituting the relations of Eq. (3.22) and solving for the surface flux at the left interface, Eq.(3.23) becomes
00i-112 -AxiDg,i_ iPg,i_i -- Ax_ iDg,ig,i (3.24)AxiDg,i-i + Axi- iDg,i
This expression for the surface flux on the left boundary can then be substituted back into the Eq.(3.22) to get an expression for the derivative of the flux approaching from the right,
dog = 2Dgj I i g - O- . (3.25)dx X+ 1 ' AxiDg,i I + Axi iDg,i
--1/2
Using the same approach, the derivative of the flux on the right interface, approaching from theleft can be expressed as
dog 2 Dgji+1 Ogi+ -- Og,i (3.26)dx X- ' Axi+1Dg,i +AxiDg,i+ 1
i+1/2
The two expressions for the derivatives of the flux at the left and right boundary are only validwhen the interface is not a global boundary of the system. In this work, only zero incomingboundary conditions and/or reflective boundary conditions are needed. Therefore, an arbitraryalbedo boundary condition was placed into the solver. The albedo parameter is a ratio of theincoming partial current to the outgoing partial current at a global boundary,
J--p = g (3.27)
gwhere PT is the albedo parameter, assumed to be independent of energy group and is either ex-pressed at the left-most global boundary of the system (-) or the right-most global boundary of thesystem (+) (Hebert, 2009). The incoming partial current to a surface is Ji and the outgoing partialcurrent is J+. From the Pi expansion of angular flux in the neutron transport equation and applica-tion of Marshak boundary conditions, the incoming and outgoing partial neutron scalar currents ata boundary can be written respectively as
J- 1 Jg -n and J+ #g+ g-n. (3.28)
The net current at a boundary is taken as the difference of these partial currents
g -n = J+ - J- (3.29)
The net current is always positive to the right indicated by the dot product with the surface normal
vector because the outgoing partial current is always oriented in the same direction as the surface
normal vector, as shown in Fig. 3.8. For a one-dimensional slab reactor, Fick's Law (represented
by Eq. (1.45)) becomes
97
-lb 1j+
ii
gh
Figure 3.8. Orientation of Partial Currents at Reactor Boundaries
Jgx = -Dg dgdx
Equations (3.27), (3.28) and (3.30) can be combined to give
T- Dg og+ 1-Pg=0,dx 2 1 +#--
(3.30)
(3.31)
where the "-" sign is for the left boundary and the "+" sign is for the right global boundary. Thesign of the expression varies due to the dot product in Eq. (3.29). For the left global boundary, thederivative of the flux can be expressed as
dog
dx x+1/2
0g,1 - Og,1/2
Axi/2(3.32)
Equation (3.31) can be expressed for the left boundary as
dg- Dg,i dx
1/2
1 1 - #_~2 1 +/2 0 (3.33)
Combining Eqs. (3.34) and (3.33), the derivative of the flux at the surface can be expressed as
dg gdx X+
1/2
2(1- -)4 Dg,1 (1 +#+±) +±Ax1 (1 - #-)1'
(3.34)
A similar approach can be performed for the right boundary, where
doPgdx
1+1/2
_ g,1+1/2 - 0g,i
Axj/2 (3.35)
can be combined with
d XgDg,i x-1 1 - p+±+ 1 + 'I+1/2 = 0,
to yield the derivative of the flux at the surface,
2(1-p+)Axi (1 - P+)+ 4 Dg,i(1 +#+)
98
J-gIh
j9-
dog
dx X 1
(3.36)
(3.37)
The final expressions for the mesh-centered finite difference equations can be derived for anarbitrary internal region, the region containing the left global boundary and the region containingthe right global boundary. For the internal region, the derivatives of the fluxes at the left and rightinterface can be substituted into Eq. (3.20) yielding
-2 - Dg,iDg,i+1 Pg'i+1Axi+1Dg,i +AxiDgi+1
+2 [ Dg,iDg,i+ + Dg, iDg,i + Ax Ogj
Ai+1Dg,i+AxiDg,i+1 AxiDg,i_1+Axi__1DgJiDg- + Di~~gi~i_11-2 DggDg, 1i-I
AxiDg,i 1+Axi1IiDg,i
= E Eh, #h,JAXi + I: Vfhih,iAxi. (3.38)gph eff h=1
For the region that contains the left global boundary (i= 1), Eq. (3.34) is used instead of Eq. (3.25)when substituting into Eq. (3.20). The expression becomes
Dg,IDg,2 Dg,I (1 - +EAx Ax2Dg,1+Ax1Dg,2 4Dg,1(l+#)+Ax1(l -r_) g' 'g
-2 Dg,lDg,2 Og,2AX2Dg,1 + Ax1Dg,2
G
- E±h,1A X1 fh,1Gh,1AX - (3.39)gph eff h=1
Finally, for the region that contains the right global boundary (i= I), Eq. (3.37) is used in place of(3.26) when combining with Eq. (3.20) to yield
-2 Dg,iDg, (_ gAXiDg,I_1 + Ax_ iDg,i
+-2 [ Dg, i(1 - #+) + Dg iDgi 1 + _rg,1Axi (g,14Dg,i(1+#+)+Ax(1-+) AxiDg,j 1 +Ax iDg,i
-I A± V-fh,IPh,IAXI- (3.40)g7h keff h=1
The multigroup finite difference equations represented by Eqs. (3.38)-(3.40) can be cast in
matrix form as
A - H <D = 0. (3.41)ke55
To expand the matrix form of these equations, the arguments can be arranged in a block format asshown below for a 3 energy group example with N spatial regions:
99
A 1 1 -A 12 -A 13 i D1 B1 1 B 12 B 13 0-A 2 1 A 2 2 -A 2 3 42 B 2 1 B 22 B 23 - 01.-A 3 1 -A 3 2 A 3 3 _ 3. keff [ B 3 1 B 3 2 B 3 3 J L 0 _
The terms in the matrix A represent the loss of neutrons. In this matrix, diagonal terms, Agg, in thiseigenvalue formulation represent tridiagonal sub-matrices characterizing within group scattering,while the off diagonal terms, Ahg, represent energy group transfer between group h and groupg. The parameter CD9 represents a vector of N spatial group fluxes for group g. The matrix Brepresents the production of neutrons from fission. This eigenvalue matrix formulation can besolved in MATLAB using standard power-iteration techniques (Hebert, 2009). The convergencecriteria for both the eigenvalue and eigenvector are le-8. The source code that represents the abovederived equations is listed in Appendix C.2.
3.4 PWR Lattice Test
The purpose of this test case is to show the overall process of running the transport code Serpentto generate homogenized parameters and then reproduce the exact same results with a diffusioncode, PARCS. This process is illustrated in Fig. 3.9. This diagram shows the distinction betweenthe level of detail in the transport theory calculation to that of the diffusion theory calculation. Inthe transport code, the detailed lattice is modeled and a database of macroscopic cross sections andother parameters is generated. To verify this process, a homogenized version of this geometry willbe executed in PARCS with the exact same operating conditions. Unlike a full core calculation,where thermal hydraulic conditions are determined from the governing equations of fluid mechan-ics and heat transfer, in this test, these conditions are forced so that a one-to-one comparison canbe performed.
In this example, the PWR lattice depletion case from the Serpent manual is used (Leppanen,2010b). This PWR lattice calculation is based on a two-dimensional slice of an assembly whichcontains fuel, gadolinium rods and guide tubes for control rods placed in a 17x 17 lattice array. Forthe depletion, 1/8th symmetry is assumed when creating the lattice. This symmetry condition canbe observed in Fig. 3.9 where each unique fuel rod color represents a separate burnable region.The dark fuel rods represent the fuel rods loaded with gadolinium, modeled with ten spatial rings tocapture the self-shielding effects during burnup. The fuel is enriched to slightly more than 4% U-235. Information regarding the geometry of the lattice is listed in Table 3.4. Additional geometric
Table 3.4. PWR Geometric Conditions
fuel pellet radius [cm] 0.4025
fuel rod radius [cm] 0.4750
fuel rod pitch [cm] 1.265
assembly dimensions [cm] 21.612
bypass flow gap [cm] 0.054
100
Homogenize
MacroXS
Transport Theory Diffusion Theory
Figure 3.9. Homogenization Process of PWR 2-D Lattice
and material composition information is provided in the branch case and geometry input files listedin Appendices B. 1 and B.2, respectively.
An array of branch cases was made to test each type of instantaneous branch case. A descriptionof these branch cases is shown in Table 3.5. Each of these branches was executed for 42 burnupsteps until a burnup of 40 MWD/kgIHM was reached. As explained in Section 3.2.2, the referencecase is first executed in Serpent for all burnup steps (with predictor/corrector mode ON), whereisotopics of each material are recovered at each step. All of the instantaneous branch cases are thenrestarted at each bumup step by placing the recovered isotopics in the input file and perturbing theoperating conditions as listed in Table 3.5. Each calculation was executed with 5000 neutrons perbatch, with 20 inactive and 500 active batches on one processor with an initial random numberof 1306092941 for all branches. A modified version of Serpent v1. 1.14 was used due to neutronbalance issues which were explained in Section 2.5. The reference case calculation took aboutone day to complete while each branch case, at each restart point took about 40 minutes. Thewhole process was executed with the SerpentXS code and took about two days to complete. Thecalculation sequence produced a database of homogenized parameters in PMAXS format that wasused as input for PARCS. Pictures of the geometry and power distributions were generated to verifythat SerpentXS was correctly placing control rods into the geometry. An example of these picturesis shown in Figure 3.10 for beginning of life conditions. In the power distribution figures, regionscontaining fissile material show a bright yellow color (high power) and dark red to black (lowpower). In the regions not containing fissile material, a light blue to white color indicates a largethermal flux, while a dark blue to black indicates a small thermal flux. Looking at these powerdistributions, the lower power regions in dark red correspond to fuel rods loaded with gadolinium.In the rodded power distribution, it can clearly be seen that the thermal flux is depressed in thecontrol rod locations.
The PARCS code was then executed for each case by forcing the thermal hydraulic conditions,control rod fraction with the control rod cards and the poison concentration. Instead of running afull core with PARCS, only a 1 node volume was created with reflected boundary conditions every-where. Therefore, this is equivalent mathematically to the two-dimensional geometry calculationin Serpent. Figures 3.11-3.15 display the results for all of the branch cases comparing Serpent and
101
Table 3.5. Instantaneous Branch Case Description
Branch Case
reference
control rod
coolant density
coolant density
poison 1
poison 2
1
2
fuel temperature 1
fuel temperature 2
(a) Reference Geometry
(b) Rodded Geometry
Figure 3.10. Geometry and Power Distribution Pictures from Serpent
102
CR
0.0
1.0
0.0
0.0
0.0
0.0
0.0
0.0
DC [g/cc]
0.707
0.707
0.636
0.777
0.707
0.707
0.707
0.707
PC [pcm]
1000
1000
1000
1000
0.0
2000
1000
1000
TF [K]
900
900
900
900
900
900
582
1500
TC [K]
582
582
582
582
582
582
582
582
PARCS. The reference case is shown in Fig. 3.11, the control rod perturbation in Fig. 3.12, thetwo coolant density perturbations in Fig. 3.13, the two poison concentration perturbations in Fig.3.14 and finally the two fuel temperature cases in Fig. 3.15.
For the reference case, three calculations are shown: Serpent, PARCS and the results from thestandalone multigroup code described in Section 3.3. On the secondary y-axis the difference fromthe PARCS results is displayed. When the Serpent results are compared to PARCS, an oscillatorytrend is observed about zero and the magnitude is within +/- 20 pcm. This oscillatory trend isexpected and can be attributed to the statistical nature of the Monte Carlo transport calculation.These differences are within the statistics of the k-effective from the transport code in which themean value of collision estimator of k-effective has an uncertainty of +/- 70 pcm. To gain moreconfidence that the differences are not due to PARCS, the standalone multigroup code, describedin Section 3.3, was also used for a comparison. When using the same input database as PARCS,the differences between the multigroup code and PARCS cannot be resolved on the diagram andare less than one pcm.
The control rod results (Fig. 3.12) are very similar to the reference case results. The effect ofthe insertion of the control can be seen by a large reduction in k-effective. The uncertainties forall of the instantaneous branch cases are well within the statistics of the transport calculation. Thecoolant density results (Fig. 3.13) are an indication of the moderator temperature coefficient. Fora typical PWR it is known that this coefficient is negative and this is reflected in the results. Asthe temperature of the coolant/moderator increases and its density decreases, the k-effective of thesystem is lowered. Conversely, if the moderator becomes denser, more neutrons are thermalizedand the k-effective of the system increases. The soluble poison results (Fig. 3.14) also agree verywell and it is observed that the increase in this concentration will lower the k-effective. The fueltemperature results (Fig. 3.15) reflect the Doppler broadening phenomenon in the fuel isotoperesonances. As the temperature of the fuel increases, these resonances broaden, and since the fuelis predominately made up U-238, the resonance absorption due to neutron capture increases andlowers k-effective. Again, the results show good agreement between Serpent and PARCS.
This exercise shows that the homogenization of data in Serpent and sending them to PARCSis consistent using the methodology in the SerpentXS code. All of the results match well and arewithin the statistics of the Monte Carlo calculation. To get better agreement, a longer transportcalculation simulating more neutron histories and batches would be needed. However, the currentnumber of neutron histories is sufficient to show the intended comparison.
103
1.15 -
1.10
1.05
1.00
0.95
0.90
0.85
0.80 -
0.00
30.00= +-70 pcm
20.00
10.00
0.00
-10.00
-20.00
-30.00
-40.00
30.00 35.00 40.00
- -SERPENT-Serpent Diff
o PARCS-Multigroup Diff
-Multigroup
Figure 3.11. Reference Case Results
1.20
1.10
1.00
S0.90
0.80
0.70-
0.60
0.00 5.00 10.00 15.00 20.00 25.00
Burnup [MWd/kg]
30.00 35.00 40.00
- Serpent CR In- - Serpent Ref Out
o PARCS CR Ino PARCS Ref Out
Figure 3.12. Control Rod Results
104
5.00 10.00 15.00 20.00 25.00
Burnup [MWd/kg]
4.1
U
a.
a.
E
I
0.88
0.00 5.00 10.00 15.00 20.00 25.00 30.00 35.00 40.00
Burnup [MWd/kg]
-Serpent DC 0.636 g/cc- - Serpent Ref 0.707 g/cc- Serpent DC 0.777 g/cc
o PARCS DC 0.636 g/cco PARCS Ref 0.707 g/cco PARCS DC 0.777 g/cc
Figure 3.13. Coolant Density Results
0.00 5.00 10.00 15.00 20.00 25.00
Burnup [MWd/kg]
30.00 35.00 40.00
105
1.13
1.08
1.03
0.98
4'
U
0)
1.22
1.17
1.12
1.07
1.02
0.97
a,
4.'Ua,a,
0.92
0.87
0.82.4
-Serpent PC Oppm 0 PARCS PC Oppm - - Serpent Ref 1000ppm
o PARCS Ref 1000ppm - -Serpent PC 2000 ppm o PARCS PC 2000 ppm
Figure 3.14. Poison Concentration Results
I
4"
Figure 3.15. Fuel Temperature Results
106
1.13
1.08
1.03
0.98 -
0.93
0.88
0.00 5.00 10.00 15.00 20.00 25.00 30.00 35.00 40.00
Burnup [MWd/kg]
- Serpent TF 582 K 0 PARCS TF 582 K - - Serpent Ref 900K
* PARCS Ref 900K - - Serpent TF 1500 K o PARCS TF 1500 K
4 Diffusion Theory Analysis of RBWR
4.1 Introduction
In Section 2, the motivation for investigating three-dimensional homogenized parameters wasshown. The necessary tools for passing information from Serpent to PARCS was developed andpresented in Section 3. In this section, the performance of these multigroup constants will be ac-cessed for a single RBWR assembly. The goal of this process is to achieve an agreement betweenreference transport results from the Serpent Monte Carlo code (heterogeneous calculation) withan approximate result from the diffusion code PARCS (homogeneous calculation). First, the fullthree-dimensional geometry is run in Serpent to compute homogenized multigroup constants foreach sub-region similar to Section 2.6. To perform this analysis, the SerpentXS code is used to
generate PMAXS files for each homogenization region. These homogenized parameters are then
used in calculations based on diffusion theory for the single assembly model in PARCS. If ho-mogenization is performed correctly, integral reaction rates in each zone as well as multiplicationfactor will agree with the transport calculation.
The cross section set developed from the three-dimensional calculation will also be comparedto the cross section set from the traditional two-dimensional calculation for each axial sub-regionseparately. Before performing this analysis on a full assembly, simpler two-zone examples are cre-
ated to determine if this process will be successful. In all of the analyses presented in this section,a stand-alone diffusion solver, written in MATLAB, was used. The source code and example inputfile is listed in Appendix C.
4.2 Two-Zone Examples
The first two-zone example is comprised of two fissile zones with slightly different coolant densi-
ties. The second example is comprised of a fissile zone an a blanket zone to investigate the effect
of a strong material discontinuity since these two zones are comprised of two different fuel types.Geometry, material specifications and operating conditions are given in the RBWR model descrip-tion in Section 2.4. For these two-zone examples, each side of the geometry was reflected in both
the transport and diffusion theory calculations. In each example, the heterogeneous eigenvalue khet
(collision estimator of keff in Serpent), homogeneous eigenvalue from three-dimensional cross
sections with PARCS k omPARCS, homogeneous eigenvalue from three-dimensional cross sections
with the standalone finite difference multigroup code khmnlti and the homogeneous eigenvalue
from two-dimensional cross sections with PARCS k2ID were compared. The standalonefinite difference multigroup diffusion solver presented in Section 3.3 is also compared with the
HYBRID nodal solver in PARCS. In addition, the stand-alone diffusion solver was used to computeand compare integral absorption and fission neutron production reaction rates for the homogeneous
geometry. These integral reaction rates are of crucial importance for getting power distributioncorrect and will be compared to the results from the Serpent heterogeneous transport calculation.Since a finite difference code is being compared to a nodal method code, the mesh for the finitedifference calculation must be very fine so as to approach the nodal method result.
107
4.2.1 Fissile-Fissile System
A diagram of the geometry along with the process of homogenization is shown in Fig. 4.1. In thePARCS calculation, each sub-region is represented by a PMAXS cross section database file thatcontains multigroup homogenized parameters. The multigroup parameters were generated in twoenergy groups with the boundary at 0.625 eV. For this specific calculation, 25,000 neutron historieswere run per batch, 45,000 active cycles were performed along with 200 inactive batches. Such alarge number of neutron histories were tracked to check neutron balance and ensure good statisticson all of the homogenized cross sections for this base case. In the finite difference calculation,400 sub-meshes were used in each coarse region. All statistical uncertainties in the Monte Carloresults were less than 1%.
Table 4.1 shows a comparison of the eigenvalue and integral reaction rates between Serpent,PARCS, and the standalone finite difference code for the fissile-fissile system. In addition, esti-mates with two-dimensional and three-dimensional calculations are listed. Overall, the eigenvalueand reaction rate comparisons all agree very well between heterogeneous and homogeneous calcu-lations. The fact that the two-dimensional cross sections reproduce results from Serpent indicatesthat decoupling these zones from each other is, in this case, an acceptable approximation. The het-erogeneous and homogeneous spatial distributions of absorption reaction density and the fissionneutron production density along the slab is compared in Fig. 4.2. In the figure, "2-D Homo-geneous" represents the diffusion theory calculation using cross sections from the conventionaltwo-dimensional approach, whereas "3-D Homogeneous" represents three-dimensional cross sec-tions. The figure indicates that the shapes of these three curves do not compare well, but they arewithin approximately 1% which is as good as the results shown in Table 4.1. For all purposes, theflux is nearly flat since the range of the ordinate on the plot is very small.
In this case, the only difference between the two- and three-dimensional calculation is that theinterface reflects or transmits neutrons, respectively. Since the zones are almost identical (waterdensity varies slightly) the assumption of reflective or zero-net current boundary condition can beapproximately valid. This is one of the assumptions in the conventional deterministic method-ology of generating homogenized parameters for full core calculations. This system, therefore,represents a conventional case for cross section homogenization that can be referenced for thefurther calculations in this section.
4.2.2 Fissile-Blanket System
In addition to interfaces between two fissile zones in the RBWR, there are also interfaces thatsplit fissile and blanket sub-regions. Similar to the fissile-fissile system, a two-zone analysis withthe upper fissile zone and upper blanket zone was performed. The geometry is equivalent to thegeometry shown in Fig. 4.1 except with the properties of upper fissile zone 2 and upper blanketzone 2. In addition, lengths of each region were taken as the full upper fissile region and upperblanket region in the RBWR, 22.0 cm and 12.6 cm respectively. Again, the multigroup parameterswere generated in two energy groups in Serpent with the boundary at 0.625 eV. For this specificcalculation, 25,000 neutron histories were run per batch, 37,000 active cycles were performedalong with 200 inactive batches. In the finite difference calculation, 500 sub-meshes were used ineach coarse zone. All statistical uncertainties in the Serpent results were less than 1%.
Eigenvalues and integral reaction rates from the heterogeneous and homogeneous calculations
108
Serpent PARCS
PMAXS
UF2
PMAXS
U F1
Figure 4.1. Two-zone Homogenization Process
0.06 0.0915--- 3-D Homogeneous 3-D Homogeneo
0.0595 2-D Homogeneous 0.091 -.... 2-D HomogeneousHeterogeneous - Heterogeneous
0.059 U 0.0905
CC3 0.0585 0.09
0 _40.058 --- - 0.0895
a>(D CZ
M 0.0575 C 0.089C c
C 0.057 1 0.08850 LL ....---------(U-
-0 0.0565 0.088<...............................................
0.056 :. 0.08750 5 10 15 20 0 5 10 15 20Axial Position [cm] Axial Position [cm]
(a) Absorption Reaction Density (b) Fission Rate Density
Figure 4.2. Comparison of Heterogeneous and Homogeneous Spatial Distribution of ReactionDensities for the Fissile-Fissile System
109
Table 4.1. Comparison of Eigenvalue and Integral Reaction Rates for Fissile-Fissile System
Parameter Value* Difference from Serpent
khet 1.55908
kD 1.55913 2.1 pcmhorn PARCS
kDmmulti 1.55918 4.1 pcm
k2D 1.55900 3.3 pcmhorn,PARCS
UF1 Absorption Rate, Serpent 0.290437 -
3D UF1 Absorption Rate, Finite Difference 0.290276 0.056%
2D UFI Absorption Rate, Finite Difference 0.293148 0.933%
UF1 Fission Neutron Production Rate, Serpent 0.448699 -
3D UF1 Fission Rate, Finite Difference 0.448445 0.056%
2D UFI Fission Rate, Finite Difference 0.453195 1.002%
UF2 Absorption Rate, Serpent 0.709563 -
3D UF2 Absorption Rate, Finite Difference 0.709724 0.023%
2D UF2 Absorption Rate, Finite Difference 0.706852 0.382%
UF2 Fission Rate, Serpent 1.110440 -
3D UF2 Fission Rate, Finite Difference 1.110679 0.022%
2D UF2 Fission Rate, Finite Difference 1.105809 0.417%*Note: Eigenvalues are dimensionless and reaction rates are normalized such that absorption rate sums to 1.
Ito
are compared in Table 4.2. The results indicate that the heterogeneous eigenvalue and integralreaction rates have large discrepancies in the homogeneous calculation. In this specific case, theconventional method of homogenizing cross sections in two-dimensions performs better than thenew methodology. However, all of the eigenvalue comparisons are hundreds of pcm off, whichis unacceptable for such a simple test problem. The spatial distributions of absorption densityand fission neutron production density for the homogeneous finite difference and heterogeneouscalculations are shown in Fig. 4.3. In this plot, the integrals of the distributions are not equal.The sharp points between the curves do not line up due to the difference in mesh spacing betweenthe heterogeneous and homogeneous calculations. To understand these discrepancies further, aflux comparison for each energy group was performed since these spatial flux distributions willultimately be used in the reaction rate calculation. This comparison is shown in Fig. 4.4. The fluxplots show that there are significant flux gradients near the material interface. The large differencesin the homogeneous calculations are a result of Fick's Law. Fick's Law is a poor approximationwithin a few scattering mean free paths of a strongly absorbing medium (Henry, 1975). From Fig.
4.3, there is large change in the reaction rates between these regions. Therefore, it will be difficultto get the homogeneous calculation to agree with the heterogeneous transport results.
One method to get better agreement is to add more energy groups. This will reduce some of the
errors in the spectral condensation of multigroup parameters. An analysis was performed in whichthe number of homogenization groups was increased to twelve. The selection of energy groupsis shown in Table 4.3. The eigenvalue and integral reaction rate results for the twelve energygroups are shown in Table 4.4. Comparing Tables 4.2 and 4.4, the error between the eigenvalueand integral reaction rates diminishes for the twelve energy group case. Similar to Fig. 4.3, theabsorption density and neutron fission production density for the twelve group case is shown inFig. 4.5. Since more energy groups were used in the calculation, the spatial distribution of the
homogeneous reaction densities within each coarse zone matches the heterogeneous distributionsmore closely. Even though these distributions seem closer, they still do not reproduce the integral
correctly. This trend indicates that if more and more energy groups are added, better results will beobtained. However, as more groups are added, the number of histories filling these energy bins will
decrease and thus the variance in the estimates will increase. It is not desired to keep increasing thenumber of energy groups since transient simulations will eventually be performed for the RBWR.More energy groups will slow down these analyses.
This issue of large discrepancies in reaction rates between adjacent MOX and UOX assemblies
is not new and has been investigated for nodal methods applications (Palmtag, 1997). A largethermal flux gradient exists at the interface of these two zones due to spectrum differences. This
steep gradient can be observed in Fig. 4.4b. In addition, this steep thermal flux gradient mayalso introduce errors in the generation of multigroup cross sections. These errors will exist if
the two-dimensional methodology is used since the actual flux spectrum shape is not used in the
cross section homogenization. However, in the three-dimensional methodology, the actual flux of
that assembly is used to homogenize cross sections in specific sub-regions. To circumvent theseerrors, Palmtag developed a methodology to derive discontinuity factors, homogenized multigroup
cross sections and diffusion coefficients such that the heterogeneous reaction rates and currents arepreserved in the nodal calculation. In the RBWR case, the same basic problem exists. The strongspectrum differences between the MOX and UOX regions prevent the preservation of reaction rates
and multiplication factor. To preserve these parameters along with surface currents, discontinuityfactors are applied in the axial direction. By applying these discontinuity factors at the surface, it
111
Table 4.2. Comparison of Eigenvalue and Integral Reaction Rates for Fissile-Blanket System
Parameter Value* Difference from Serpent
kiet 1.36620
khmPARCS 1.38312 895 pcm
km imulti 1.38313 896 pcm
k2m,PARcs 1.37733 591 pcm
UF Absorption Rate, Serpent 0.846065 -
3D UF Absorption Rate, Finite Difference 0.859788 1.6%
2D UF Absorption Rate, Finite Difference 0.850490 0.52%
UF Fission Rate, Serpent 1.319184 -
3D UF Fission Rate, Finite Difference 1.340550 1.6%
2D UF Fission Rate, Finite Difference 1.329362 0.77%
UB Absorption Rate, Serpent 0.153935 -
3D UB Absorption Rate, Finite Difference 0.140212 8.9%
2D UB Absorption Rate, Finite Difference 0.149510 2.9%
UB Fission Rate, Serpent 0.047011 -
3D UB Fission Rate, Finite Difference 0.042574 9.4%
2D UB Fission Rate, Finite Difference 0.047966 2.0%*Note: Eigenvalues are dimensionless and reaction rates are normalized such that absorption rate sums t
112
o 1.
0.045
T 0.04:.
0.035!
0.03-.2
0.0250
4 0.02CL
0.
0.015
n ni
--- 3-D Homogeneous2-D Homogeneous
- Heterogeneous
0.07
0.06
0.05
0.04
(D
-C 0.03C
m 0.02
0.01 --- 32
--0
-D Homogeneous-D HomogeneousHeterogeneous
.00 5 10 15 20 25 30 35 0 5 10 15 20 25 30Axial Position [cm] Axial Position [cm]
(a) Absorption Reaction Density (b) Fission Rate Density
Figure 4.3. Comparison of Heterogeneous and Homogeneous Spatial Distribution of ReactionDensities for Fissile-Blanket System
3.2--- 3-D Homogeneous
- Heterogeneous-----. 2-D Homogeneous
2.8X
L- 2.6
0.~2.4
(D-o2.2N
ca 2Ez1.8
1.6,
1.40
Figure 4
0.035--- 3-D Homogeneous- Heterogeneous
0.03. - 2-D Homogeneous
x0.025
U-C\l
0.= 0.0200
.N
0.010z
0.005
5 10 15 20 25 30 35 ~0 5 10 15 20 25 30 3Axial Position[cm] Axial Position [cm]
(a) Group 1 Flux (b) Group 2 Flux
.4. Comparison of Heterogeneous and Homogeneous Spatial Flux Distribution forFissile-Blanket System
5
113
35
Table 4.3. Twelve Group Energy Structure
Group number
1
2
3
4
5
6
7
8
9
10
11
12
Energy range [MeV]
15.0 - 3.6788
3.6788 - 2.2313
2.2313 - 1.3534
1.3534 - 4.9787e-1
4.9787e-1 - 1.8316e-1
1.8316e-1 - 4.0868e-2
4.0868e-2 - 5.53084 e-3
5.53084e-3 - 1.3007e-4
1.3007e-4 - 3.9279e-6
3.9279e-6 - 1.445e-6
1.445e-6 - 6.25e-7
6.25e-7 - le-10
Table 4.4. Comparison of Eigenvalue and Integral Reaction Rates for Fissile-Blanket System - 12Group
Parameter Value* Difference from Serpent
ket 1.36620
kom,PARCS 1.35980 345 pcm
k 3 D 1.35963 354 pcmhorn,rnulti
k 2 D 1.35727 482 pcmhorn,PARCS
UF Absorption Rate, Serpent 0.846065 -
3D UF Absorption Rate, Finite Difference 0.842339 0.4%
UF Fission Rate, Serpent 1.319184 -
3D UF Fission Rate, Finite Difference 1.309415 0.7%
UB Absorption Rate, Serpent 0.153935 -
3D UB Absorption Rate, Finite Difference 0.157661 2.4%
UB Fission Rate, Serpent 0.047011 -
3D UB Fission Rate, Finite Difference 0.050214 6.8%*Note: Eigenvalues are dimensionless and reaction rates are normalized such that absorption rate sums to 1.
114
0.046c0.055 0.04
0.053 0.06
0.045 c
0.05C 0.04a)
.o -- -Ioognos001 -- 3DHooeeu
c 0.035 c 0.04o a)00
0.03 0 0) Ra0.03
C 0.025.2 .002,
0.02
Figure 3-0 Homogeneous 0.01 3-D Homogeneous< 0.015f Fsi-l 2-a Homogeneous 2-0 Homogeneous
m Heterogeneous -nHeterogeneous
'10 5 10 15 20 25 30 35 0 5 10 15 20 25 30 35Axial Position [cm] Axial Position [cm]
(a) Absorption Reaction Density (b) Fission Rate Density
Figurc 4.5. Comparison of Heterogeneous and Homogeneous Spatial Distribution of ReactionDensities for Fissile-Blanket System - Twelve Group
may not be necessary to increase the number of energy groups.
4.3 Axial Discontinuity Factors
The use of discontinuity factors to preserve neutron balance between high-order transport theorycalculations and low-order diffusion theory calculations has been well developed (Smith, 1980).Currently, in the commercial reactor industry, discontinuity factors are used only in the radial di-rection since only two-dimensional lattice calculations are performed. Since three-dimensionallattices calculations can be performed with the use of a Monte Carlo method, axial surface cur-rents can be calculated to compute axial discontinuity factors (ADFs). According to Smith, if onechooses an arbitrary diffusion coefficient and calculates two discontinuity factors for each side ofa computational node, in each direction, it is possible to preserve system multiplication factor andnode-integrated reaction rates. This statement is very important for the Monte Carlo homogeniza-tion technique since one can compute one flux-weighted diffusion coefficient for each node andthen calculate the appropriate discontinuity factors needed to preserve neutron balance. Therefore,even if there is an incorrect definition of the diffusion coefficient in Serpent that will inherentlynot preserve neutron balance, discontinuity factors can be calculated in each direction to preservethis balance. In this work, directional diffusion coefficients are not used and only one diffusioncoefficient is defined for both radial and axial directions. In general, the discontinuity factors aredefined on each side of a node as the ratio of the heterogeneous-to-homogeneous surface flux,
-Pgjiet (4.1)
-,hom
This definition of discontinuity factors is more general than Eq. (1.33) which is only valid fortwo-dimensional homogenization for a geometry surrounded by zero net current boundary con-
115
0.055, 0.07
ditions. This is because the homogeneous surface flux in these problems is equal to the averagehomogeneous flux in a homogenization region. In general, if leakage is present in or out of thehomogenization region, the homogeneous surface flux will not necessarily be equivalent to theaverage homogeneous flux in the region. Therefore, with the current definition of discontinuityfactors in Serpent, a separate code will have to be created to compute general surface disconti-nuity factors. Equation (4.1) leads to a flux continuity condition where the homogeneous surfacefluxes computed from each region around a interface do not need to be continuous, rather, the newcontinuity relation is
+ + - + (4.2)Si+1/2 g~o~i+1/2 = fgi+1/2 go~i+ 1/2
The +O is the group g surface flux on the right side of node i at xi+ 1/2 computed from the
homogeneous diffusion theory calculation for the node that connects to the interface from the left,
Som,x+1/ 2 is the surface flux on the left side of node i + 1 at xi+1/2 for the node that connects
to the interface from the right, and are the surface discontinuity factors approaching thefg- i+1_/2
interface from their respective sides. This relation indicates that heterogeneous fluxes will still becontinuous at an interface although homogeneous fluxes are allowed to be discontinuous.
Another important implication of Smith's work is that exact neutron balance can be preservedeven if the discontinuity factors are computed from an approximate method. Therefore, the dis-continuity factors account for all of the approximations in the homogenization process. This in-cludes group condensation errors, spatial approximation errors and the definition of flux- ratherthan current-weighted diffusion coefficient. An additional feature, important for Monte Carlo cal-culations, is that discontinuity factors will also account for some of the statistical uncertainty inthe multigroup constants. Note neutron balance will only be preserved to within statistics of theMonte Carlo calculation.
4.3.1 Incorporation of Discontinuity Factors in Finite Difference Equations
General one-dimensional mesh-centered finite difference equations were developed in Section 3.3.A separate equation was developed for an internal sub-mesh, left boundary sub-mesh and a right-boundary sub-mesh. Two more situations are constructed which include a sub-mesh to the leftof a coarse mesh interface and a sub-mesh to the right of a coarse mesh interface. In order todevelop these equations, new forms of the flux derivatives need to be generated. Equations (3.21)and (3.22) show the finite difference approximation to these derivatives for a right interface and aleft interface. Focusing on a left interface, discontinuity factors on either side are introduced as
dfg + / Ogi-1/2 - g,i--1 dfg O 1 2 4i-1/2dlg fgi 12 .lP ggi 1(43)
dx 1/ Axi_ 1 /2 dx x+ 1 Axi/2i- 1/2 i- 1/2
In Eq. (4.3), f+ is the surface discontinuity factor for the right-hand side node i - 1 andgi- 1/2
f -is the surface discontinuity factor for the left-hand side of node i. Applying continuity offg c1/2surface currents, shown in Eq. (3.23), the surface flux can be solved for
116
00-i 1/2 /XiDg,i- 1 g,i-1 + Axi-1 Dg,i Og,i (44)+1 AxiDgi--1 + 1 Ai-IDg,ig,i- 1/2 gi- 1/2
The surface flux can then be used to calculate the gradient of the flux at the left interface for nodeias
o 2 Dg,i_ 1 ''4. (4.5)dx X+ AxiDgji-i + _ x --i iDgO-1/2 f fg,i-1/2 Agi-1g2
Thus, the finite difference equation for this situation becomes
-2 Dg,iDg,i+1 Og'i±1Axi+IDg,i + AxiDg,i+ i
+2 Dg,iDg,i+ Igi 1/2 Dg,iDg,i 1 &Axi+ IDg,i+ AxiDg,i+ I + iDg,i- 1 + _ i-1Dg'i r j ,
fgi- 1/2 g9Ji-l/2
- Dg,iDg,i 1
-2 1 gi-1/2 1g'i-If AxiDg,i i + Axi-Dg,ig,i- 1/2 gi- 1/2
G
s hixi + __ L fhh,i i- (4.6)g7h eff h=I
The same derivation can be performed for a right interface between two coarse regions. The finitedifference relation in this situation becomes
+-Dg,iDg,i+1
-2 Agg, i + i 1 A gi i+I+ AAi+1Dg,i + 1Axi Dgi+1
fg, i+1I/ 2 fg- +1/ 2
-2 D /2DD,i+1 Dg,iDg,i_+1 Axi.+1Dg,i + AiDg,i+1 AxiDg,i- + Axi-1Dg,i g
. g,i+ 1/2 g i++1/2
-2 Dg iDg,i_j _i glAxiDg,i_1 +Axi_1D
G
E h i + Xgi Vfh,ih,iAxi- (4.7)g7h elf h=I
In this relation, f+ is the surface discontinuity factor for the right side of node i and fireato ,i g-1/2 g9J+ 1 /2is the surface discontinuity factor for the left side of node i + 1. These two situations are includedin the code and checked when computing coefficients for the finite difference matrices. The incor-poration of discontinuity factors into the source code is listed in Appendix C.2.2.
117
4.3.2 Implementation of Discontinuity Factors into Analysis
In this section, the overall analysis procedure to calculate and incorporate discontinuity factorsis described. A MATLAB code was developed to automate the whole analysis procedure oncethe Serpent run is completed. The high-level method of generating these discontinuity factors inone-dimension can be outlined as follows:
1. Obtain the heterogeneous solution with the Serpent Monte Carlo transport code with globalreflective boundary conditions,
2. Extract the flux-weighted homogenized parameters that Serpent calculates in each energygroup over coarse mesh regions (the sub-regions in RBWR geometry),
3. Compute interface currents using neutron balance and flux-weighted homogenized parame-ters
4. Perform a fixed source one-dimensional finite difference diffusion theory calculation with afine mesh for each coarse region with appropriate current boundary conditions and homoge-neous parameters obtained from heterogeneous calculation,
5. Compute discontinuity factors for each side of each coarse region as the ratio of the hetero-geneous surface flux from the Monte Carlo calculation to the homogeneous flux from thefinite difference calculation performed in step 3,
6. Run the entire homogeneous geometry with the computed discontinuity factors in a one-dimensional finite difference calculation with same spatial discretization of each coarse re-gion performed in step 2.
Only one-dimensional problems are considered in this work since the level of heterogeneity in theaxial direction is larger than the radial direction. Current methods for radial discontinuity factorscan still be applied since zero net current boundary conditions are imposed. In the axial direction,however, the effects of neighboring zones will be included. In step 1, the true heterogeneousgeometry is modeled in the Serpent code and cross sections are homogenized in user-specifiedcoarse homogenization regions. In each homogenization region, Serpent automatically generatesflux-weighted spatially-homogenized cross sections and diffusion coefficients. Therefore, steps 1and 2 are automatically performed by Serpent. Note that since the PARCS code will also be usedin the comparison, SerpentXS is also executed in the first two steps.
In steps 3 and 4, each coarse mesh region is analyzed separately to calculate the homogeneoussurface fluxes. To do so is not trivial mainly because Serpent does not have current tallies inversion 1.1.14. This means boundary and interface currents cannot be directly calculated from theSerpent code and must be computed manually. To do so, a current must be known at a surface in thegeometry. The only location where the current is absolutely known is at a zero net current boundarycondition (it is by definition zero). From knowing this boundary condition, the neutron balanceequations can be solved sequentially in one-dimension in order to calculate surface currents at allother interfaces. For example, if a zero net current boundary condition exists on the global leftboundary of the problem, the current at the other side of the first homogenization region with thisboundary is given by
118
jgnl ±n Xn G~ --+gz+ ~-- - Eg-->g) Oghx (4.8)Z" = Z + (_ vEh nh + E Eh n*pghx -- (Egan + Egsn - En**@~ 48gXi+ 1/2 gXi 1/2 keff h= I gh
This equation was derived from Eq. (2.5) where all multiplicity from scattering reactions is taken to
one, and the total cross section has been separated into an absorption and scattering cross section.
For the first region, Jf" will be zero by definition. For every subsequent coarse mesh, the left
surface current will be equivalent to the right surface current from the previous calculation. The
calculation can also be performed for a right global boundary condition by sweeping left through
the geometry to generate the surface currents. Since these parameters are all estimated from a
statistical process, it is not necessarily true that numerical results computed from a right sweep
and a left sweep be equivalent. This sensitivity is studied further in Section 4.4. After all surface
currents have been computed from a left and right sweep, the accepted surface current is taken as
an average of these two values at any given surface. This will introduce some error in reproducing
integral reaction rates and must be quantified. The source code that performs this step is listed in
Appendix C.2.6.Next, run fixed-source diffusion theory calculations are run with surface net currents as bound-
ary conditions for each homogenized region with their respective homogenized cross sections. To
perform this calculation, the diffusion equation must be solved with an external source vector.
For the fixed source calculation, the eigenvalue will be taken from the Serpent solution of the full
heterogeneous problem. The diffusion equation can be cast in a fixed-source form as
A - I BjD =Q, (4.9)ke55
where the flux vector <D, contains the spatial distribution of each group flux and is the only un-
known. The vector Q is the external source vector which contains current boundary conditions.
Therefore, the flux vector can be directly solved for
CD= A- E B Q. (4.10)ke55
The external source vector contains a value for each spatial mesh point of each energy group. It
can be shown that surface currents on each coarse mesh boundary are placed in their respective
positions in the external source vector. For example, if there are 2 energy groups and 5 spatial
mesh points in a coarse homogenization region, the external source vector would be
119
1/2
000
Q- 'X9/2 (4.11)12,x/2
000
J2,x9 /2
Recall that J 1,X1/2 represents group 1 x-direction surface averaged current at the left boundary
(xI/ 2). This process is performed for each coarse mesh region. At the end, mesh-centered fluxesnext to each boundary for each energy group will result. The source code that performs the ho-mogeneous flux calculations is listed in Appendix C.2.7. These mesh-centered fluxes must beextrapolated to the surface to define the discontinuity factors correctly. This can be determinedfrom Fick's Law (see Eq. (3.30)),
0gi±1/2 =g,i ± i- (4.12)2 DgOThis step is performed by the MATLAB code and is listed in Appendix C.2.8.
Discontinuity factors can now be computed from the homogeneous and heterogeneous surfacefluxes. To obtain the heterogeneous surface flux, a tally would need to be placed in the MonteCarlo calculation. Since Serpent does not currently support this capability, the flux is taken as1. The value of the heterogeneous surface flux is not truly needed since it will cancel out at theinterface. Surface discontinuity factors on either side of a right interface are then calculated as
1 i+1/2 + and f +1/2 (4.13)i+,hom,xi / ghom,x+
Note that in this equation, the homogeneous flux at the interface $±hT is equivalent to theg9_,hom,xi+1/
surface flux in the finite difference equations, 0g,i+1/2, except that the results will be different fromeach coarse mesh calculation that surround a given interface. Discontinuity factors are normalizedsuch that the ratio of them is held constant and their average is unity. Once discontinuity factorsare defined for each interface from each coarse region, they can be applied to the diffusion analysisof the global problem to preserve neutron balance. Step 6 is performed for verification that thetrue discontinuity factors were calculated. Since discontinuity factors will also correct for spatialdiscretization errors, the same spatial discretization should be used in the eigenvalue analysis. Thecalculation of discontinuity factors is listed in the source code presented in Appendix C.2. 10.
In PARCS, discontinuity factors may only be applied in the radial direction. In this application,axial discontinuity factors are being applied. To prove the concept, the vertical RBWR assemblywas put on its side for a one-dimensional calculation in the x-direction in PARCS. Now, the axialdiscontinuity factors that are calculated can be applied to PARCS in the radial direction. They arejust appended to PMAXS cross section database files for each coarse region.
120
4.4 Two-Zone Diffusion Analysis with Discontinuity Factors
In this section, the process of generating and applying discontinuity factors will be demonstratedin the two group fissile-blanket system presented in Section 4.2.2. This analysis will follow thesteps listed in the previous the section. Since the Serpent calculation was already performed forthis problem, the next step in the analysis is to determine the net current at the interface for eachenergy group. Each global boundary of this system is defined as reflective and Eq. (4.8) is appliedto compute group interface currents. The results starting from the left and right global boundaryare listed in Table 4.5. From the calculation of currents, it can be observed that perfect neutronbalance is not met. Since using both global boundaries does not yield the exact same results, thevalue of the group interface current is computed from the average. This assumption is analyzedfurther in this section.
The next step in the process is to calculate the homogeneous flux profile that would be neededin each coarse mesh region for the calculation of discontinuity factors such that reaction rates andneutron balance are preserved. In this example, there are two coarse mesh regions, the fissile andthe blanket region. Each of these regions was sub-divided into 500 sub-meshes to get convergeddiscontinuity factors for a nodal method calculation. A separate fixed-source finite differencecalculation was performed for each coarse zone and a plot of the homogeneous group 1 and group2 flux is shown in Fig. 4.6. These plots indicate that even though the heterogeneous flux from thetransport calculation is continuous, a discontinuity must be present in the homogeneous flux at theinterface between zones in order to preserve integral reaction rates in each coarse zone separately.At each interface, a discontinuity in the homogeneous flux is characterized by a discontinuity factorfor each energy group. To calculate these discontinuity factors, the mesh-centered homogeneousflux before and after the interface must be used to get the corresponding surface flux. These surfacefluxes can be computed from Eq. (4.12). After Eq. (4.13) is applied to calculate discontinuityfactors, they are normalized so that their ratio is preserved and their average is unity. For this case,the group 1 and group 2 discontinuity factors are listed in Table 4.6 under "Average Current". Thedata in the table indicates that for the energy group structure chosen, large discontinuities must existat the interface, especially in group 2. These discontinuity factors are then used in an eigenvalueanalysis for the entire two-zone problem. Comparisons of the absorption and fission rate densitiesare shown in Fig. 4.7. This figure is very similar to Fig. 4.3, except the homogeneous flux withADFs applied is added to the plot. This is the target distribution to enforce neutron balance for thisgiven energy group structure and spatial discretization. Even though this curve does not lie on topof the heterogeneous true distribution, the integral of the heterogeneous and homogeneous withADF curves should be exactly the same. However, due to statistics in the Monte Carlo analysis,a small error may exist. In Table 4.2, the Serpent calculated reaction rates are listed, and now the
Table 4.5. Fissile-Blanket Interface Currents for Two Energy Groups
Group 1 [cm-21 Group 2 [cm-2]
Left Global Boundary Reference 0.1236 -0.0041
Right Global Boundary Reference 0.1234 -0.0039
Average 0.1235 -0.0040
121
0.03
- Fissile Zone--- Blanket Zone
-Fissile Zone--Blanket Zone
0.025
0.02
U):30.015
0 .E 0.01,0"r
~'
2.8
2.6
2.4
2.2
2
1.8!
1.60
0.0
25 30 35
05
0 5 10 15 20Axial Position [cm]
25
(b) Group 2 Flux
Figure 4.6. Coarse Region Homogeneous Flux Distribution
Table 4.6. Discontinuity Factors for Different Currents
Average Current Left Current Right Current
Group 1 Fissile Zone 1.0713 1.0711 1.0716
Group IBlanket Zone 0.9287 0.9289 0.9284
Group 2 Fissile Zone 1.2470 1.2874 1.2045
Group 2 Blanket Zone 0.7530 0.7126 0.7955
0.07|
--- Homogeneous No ADF--- Homogeneous ADF.- 2-D Homogeneous
Heterogeneous
5 10 15 20 25Axial Position [cm]
0.06
0.05
0.04
3 0.031
S0.02,o 0.02 Homogeneous No ADF
-A2| -- Homogeneous ADF2-D Homogeneous
-Heterogeneous
30 35 00
(a) Absorption Reaction Density
5 10 15 20 25Axial Position [cm]
(b) Fission Rate Density
Figure 4.7. Comparison of Reaction Densities for Fissile-Blanket System with ADFs
122
UI)=30
E0r
5 10 15 20Axial Position [cm]
(a) Group 1 Flux
/
30 35
0.045E
0.04 -
0.0351
c 0.03.00O0.025
C 0.02i0.
0 0.0151
0.016 30 35
corresponding differences for the ADF case are shown in Table 4.7 under the column "AverageCurrent". Comparing the difference between the two tables, it is clear that the error is dramaticallyreduced when ADFs are applied.
In calculations performed thus far, the average predicted surface currents were used. In Table4.5, the surface currents calculated from the left boundary and from the right boundary are shown.There are differences between the surface currents, but they seem to be small. In Table 4.6, thediscontinuity factors are listed from each current calculation. Overall, the group 1 discontinuityfactors agree well between each of the estimates. The group 2 discontinuity factors have more of aspread and this is due to the statistics since the spectrum is very epithermal in this system. Table 4.7lists the multiplication factor and differences from Serpent reaction rates for each current estimate.The estimates of the eigenvalue are exactly the same, but there are noticeable differences in thereaction rate estimates. The results indicate that using the left global boundary as the referenceyields the best agreement and that using the right boundary would result in larger differences. Thereason for this behavior maybe attributed to the fact that most of the fast neutrons are producedin this zone and may have better estimates of the group parameters. This behavior is problemdependent and should be studied for each problem separately.
A twelve energy group analysis was also performed to study the sensitivity of currents and dis-continuity factors. Table 4.8 shows the left reference current compared with the average current.The differences range from almost no difference to about 6%. A positive difference indicates thatthe left reference current over-predicts the average. For the right reference current, the same differ-ence would be observed, except with the opposite sign. Some of these differences are larger thanin the two group comparison where the group 2 currents were different by about 3%. A compari-son of the discontinuity factors generated from these surface currents are shown in Table 4.9. Theresults indicate that there is still a large discontinuity in fast energy groups, small discontinuityin epithermal groups and depending on the current assumption there may or may not be a largediscontinuity in the thermal group. Since there are more energy groups present, spectral conden-sation approximations are reduced and thus discontinuity factors are closer to unity. Comparingwith the two group discontinuity factors in Table 4.6, it would seem that this is true except for invery fast energy groups. This is due to the fact that neutron flux in these groups is much smallerthan in epithermal groups and therefore, the estimates of the homogenized parameters are not as
Table 4.7. Current Comparison of Eigenvalues and Integral Reaction Rates for Fissile-BlanketSystem Calculated with the use of ADFs
Parameter Average current Left current Right current
khet 1.36620 1.36620 1.36620
khom,ADF 1.36620 1.36620 1.36620
UF Absorption Rate 0.0056% 0.0003% 0.0109%
UF Fission Rate 0.0043% 0.00002% 0.0087%
UB Absorption Rate 0.0309% 0.0019% 0.0599%
UB Fission Rate 0.1223% 0.0013% 0.2458%
123
Table 4.8. Current Comparison for Twelve Energy Groups
Group Average current [cm- 2] Left Current [cm- 2] % diff
1 0.0182 0.0182 -0.007
2 0.0277 0.0276 0.15
3 0.0280 0.0280 0.18
4 0.0447 0.0446 0.22
5
6
7
8
9
10
11
12
0.0195
0.0100
0.0023
-0.0121
-0.0111
-0.0019
-0.0018
-0.0040
0.0195
0.0100
0.0023
-0.0121
-0.0111
-0.0019
-0.0017
-0.0039
0.20
0.05
0.61
-0.04
0.14
1.53
6.12
2.38
Table 4.9. Comparison of Twelve Group Discontinuity Factors
Group Average current Left current Right current
0.9030/1.0970
0.9155/1.0845
0.9406/1.0594
0.9697/1.0303
1.0021/0.9979
1.0052/0.9948
1.0082/0.9918
0.9898/1.0102
1.0106/0.9894
1.0407/0.9593
1.0116/0.9884
0.9820/1.0180
0.9041/1.0959
0.9155/1.0845
0.9405/1.0595
0.9693/1.0307
1.0018/0.9982
1.0051/0.9949
1.0079/0.9921
0.9894/1.0106
1.0108/0.9892
1.0535/09465
1.0956/0.9044
1.0618/0.9382
0.9019/1.0981
0.9154/1.0846
0.9408/1.0592
0.9701/1.0299
1.0024/0.9976
1.0052/0.9948
1.0084/0.9916
0.9902/1.0098
1.0104/0.9896
1.0277/0.9723
0.9221/1.0779
0.8922/1.1078
124
1
2
3
4
5
6
7
8
9
10
11
12
good in reproducing neutron balance. Comparing the current approximations, the spread is notlarge except for groups 10, 11, and 12. In groups 11 and 12, the discontinuity factors actually flipfrom greater than 1.0 to less than 1.0 and could lead to poor approximations. This is attributed tostatistical uncertainty in these groups. The resulting reaction rate estimates for each of the currentapproximations as compared to the Serpent solution are listed in Table 4.10. Similar to the twogroup analysis, the current reference from the left boundary of the systems yields the best agree-ment. The concern here is that the results in reaction rate for each of these current approximationsmay not be as good as the two group analysis. If this was a fully deterministic case the twelvegroup calculation would perform better since there is a reduction of spectral approximations in theenergy group condensation. Since the transport calculation here is with Monte Carlo, the resultsare statistical. A possible reason why the twelve group case did not perform as well as the twogroup case is that the estimates have a higher variance due to finer energy bins. If one calculatesthe multiplication factor for the twelve energy groups straight from the group macroscopic neutronproduction and absorption cross sections, the result is 1.36592, which is 15 pcm from the collisionestimator and two group estimate of k-effective. More work needs to be done to understand thesediscrepancies.
To show the effect of more energy groups in the axial flux distribution, groups 1-11 werecollapsed into one energy group. This was compared to the group 1 flux from the two groupcalculation and is shown in Fig. 4.8. The shape of the 12 group estimate of the group 1 flux agreesbetter with the heterogeneous shape. All of the curves still have the same integral in each coarseregion. Similar behaviors would be observed in the reaction rate plots where the 12 group shapeis closer to the distribution from Serpent. These two-zone analyses allowed for an understandingof some of the intricacies involved in generating discontinuity factors from Monte Carlo transportcalculations where surface currents were not provided and parameters had a statistical uncertaintyassociated with them. These ideas need to be examined in the full RBWR assembly as the problemmay be amplified due to a greater number of zones.
4.5 RBWR Single Assembly Analysis
In this section, axial discontinuity factors will be generated for a RBWR single assembly with thegoal of matching Monte Carlo results with diffusion theory. First, reference discontinuity factors
Table 4.10. Current Comparison of 12 Group ADF Eigenvalue and Integral Reaction Rates
Parameter Average current Left current Right current
khet 1.366200 1.366200 1.366200
khom,ADF 1.366200 1.366200 1.366200
UF Absorption Rate 0.0286% 0.0032% 0.0541%
UF Fission Rate 0.0130% 0.0005% 0.02656%
UB Absorption Rate 0.1574% 0.0175% 0.2973%
UB Fission Rate 0.3615% 0.0186% 0.7417%
125
2G Homogeneous ADF- Heterogeneous
2.81 12G Homogeneous ADF
- 2.6
0
.8
7o 2.2%a) IN
o 2z
1.8
1.6.0 5 10 15 20 25 30 35
Axial Position [cm]
Figure 4.8. Comparison of Group 1 Collapsed Flux
will be calculated in order to exactly preserve reaction rates and global multiplication factor. Theseanalyses will be performed with both a two and twelve energy group structure. Comparisonswill be made between two-dimensional homogenization and three-dimensional homogenizationtechniques, with and without discontinuity factors. A branch case input file and Serpent geometryfile are listed for reference in Appendix D. 1 for a two energy group structure.
4.5.1 Reference Discontinuity Factors
The process of calculating reference discontinuity factors first involves a full detailed calculationof the RBWR assembly in Serpent. The characteristics of this assembly are listed in Section 2.4.In order to calculate discontinuity factors with Serpent, the global boundary conditions of thegeometry must be reflective. However, in a one-dimensional analysis of a RBWR single assembly,zero incoming current boundary conditions are imposed on top and bottom. In order to simulatethese boundary conditions, a purely absorbing medium was placed below and above the geometrywith reflective boundaries. To create this purely absorbing medium, the hydrogen ACE-formatteddata file was modified such that only absorption is possible. This isotope will be referred to asblack hydrogen. This medium was then made long enough such that the probability of a neutronreflecting off the boundary and returning back to the actual assembly was extremely small. Thedensity of the material was also increased to reduce the mean free path of neutrons in the materialto 10 g/cm3. The density, however, cannot be increased arbitrarily because of the delta trackingmethod efficiency (Leppanen, 2010a). A total of 4000 active cycles, 200 inactive cycles and 25,000neutrons per cycle were run in Serpent. All statistical uncertainties in the Monte Carlo results were
126
less than 1%.The results from Serpent were processed through the MATLAB code to compare two-dimensional
homogenization and three-dimensional homogenization with and without discontinuity factors.Each coarse mesh region contained 500 mesh cells. The two group and twelve group estimates ofthe multiplication factors are presented in Table 4.11. Results indicate that none of the estimateswithout discontinuity factors reproduce the heterogeneous multiplication factor. Cross sectiondatabase files were generated with SerpentXS to verify the estimates of multiplication factor withPARCS. In the two group results, without ADFs, the two-dimensional homogenization predictsthe multiplication factor better than the three-dimensional homogenization. Note that the distri-butions of reaction rates may still be inaccurate although the multiplication may agree well. Intwelve energy groups, the opposite is true and the two-dimensional results are very close to thethree-dimensional results without discontinuity factors due to the reduction of spectral approxima-tions in the energy condensation of few-group parameters. Calculating and applying discontinuityfactors for the exact detailed geometry results in an exact agreement in multiplication factor. Evenin twelve groups, results indicate that discontinuity factors must be applied. The reference dis-continuity factors are shown in Table 4.12. The first and last rows represent the interface betweenpurely absorbing black hydrogen slabs and reflector.
To understand the results, axial distributions of group fluxes are compared for the two groupcase. The group 1 flux is shown in Fig. 4.9 and the group 2 flux in Fig. 4.10. Each regioncontains the corresponding region identification tag from Table 2.2 and is further sub-divided inthree sub-regions in which cross sections are generated from Serpent. These axial flux comparisonsshow that neither two-dimensional nor three-dimensional homogenization without discontinuityfactors agree with the flux distribution from Serpent. In the group 1 flux, the fissile zones canclearly be noticed as neutrons are born from fission. The blanket zones do not show any significantproduction of fast neutrons from fission. At the top of the assembly in the upper reflector, the three-dimensional cross sections without ADFs show good agreement with the gradient of the flux. Theslope in the upper reflector cannot be captured in the two-dimensional results and therefore in orderto treat this upper reflector correctly, a three-dimensional homogenization must be used. Afterapplying ADFs, the homogeneous flux agrees very well with the heterogeneous flux from Serpent.Another interesting trend is the opposite nature of the predictions from the two-dimensional caseand three-dimensional without discontinuity factors. The two-dimensional cross section set overpredicts in the lower fissile zone and the three-dimensional set over predicts in the upper fissilezone. This is due to the way cross sections were homogenized and is explained further in thissection. In the group 2 flux, the peaks are reversed where thermal neutrons exist in the blankets,
Table 4.11. Comparison of Multiplication Factors for RBWR Single Assembly
Calculation 2-G k-effective Difference [pcm] 12-G k-effective Difference [pcm]
Serpent 1.081140 - 1.081140 -
2-D 1.102256 1770 1.064058 1490
3-D no ADFs 1.136686 4520 1.069002 1050
3-D ADFs 1.081140 0 1.081140 0
127
Table 4.12. Reference Two Group Axial Discontinuity Factors
Interface Group 1
BL/LR1 21.2183
LR1/LR2 -0.0062
LR2/LB1 0.8413
LB1/LB2 0.8811
LB2/LB3 0.9414
LB3/LF1 0.9610
LF1/LF2 0.9740
LF2/LF3 1.0221
LF3/IB1 1.0462
IB1/IB2 1.0687
IB2/IB3 0.9262
IB3/UF1 0.9558
UF1/UF2 0.9796
UF2/UF3 1.0214
UF3/UB1 1.0242
UB1/UB2 1.0424
UB2/UB3 1.0682
UB3/UR 1.0416
UR/BU 1.4350
Left/Right
-19.2183
2.0062
1.1587
1.11899
1.0586
1.0390
1.0260
0.9779
0.9538
0.9313
1.0738
1.0442
1.0204
0.9786
0.9758
0.9576
0.9318
0.9584
0.5650
Group 2
4.6986
0.7759
0.9521
0.9216
0.9349
0.9327
1.0229
0.7507
1.1505
1.0609
0.8340
0.7975
-2.8560
-6.8311
0.9536
0.9776
0.9988
1.0605
1.7396
Left/Right
-2.6986
1.2241
1.0479
1.0784
1.0651
1.0673
0.9771
1.2493
0.8495
0.9391
1.1660
1.2025
4.8560
8.8311
1.0464
1.0224
1.0012
0.9395
0.2604
128
1.6
--- Homogeneous No ADF1.4 -- Homogeneous ADF
- Heterogeneous---- 2-D Homogeneous
, 1.2
X=3
a. 1
3 0.80
o0.6
0.4 %i
0.21
0LR1 LR2 LB LF
-0.2-20 0 20 40 60 80
Axial Position [cm]
UF UB UR
100 120 140
Figure 4.9. Axial Distribution of RBWR Assembly Group I Flux from Two Group Calculation
0.08
--- Homogeneous No ADF0.07 -- Homogeneous ADF
-Heterogeneous.--- 2-D Homogeneous
0.06
/4.N
:3 0.040.02
0.01£0.0 .0i
N.............
z0 .01 I
LR1 LR2 LB LF IB UF UB UR
-20 0 20 40 60 80Axial Position [cm]
100 120 140
Figure 4.10. Axial Distribution of RBWR Assembly Group 2 Flux from Two Group Calculation
129
-0.01
but not in the fissile regions. This is due to the strong thermal absorption in the plutonium isotopes.It is observed that ADFs are needed in order to preserve reaction rates and multiplication factor.Although the flux distributions for the homogeneous case with application of discontinuity factorsdo not have the same shape as heterogeneous flux distributions, the integrals within each sub-regionare the same.
A comparison of the homogenization cases was made for the fission rate density and absorptionrate density. The fission rate density comparison is presented in Fig. 4.11. In Fig. 4.12 theone group fission neutron production cross section is shown from two-dimensional and three-dimensional homogenization in each sub-region. These one group cross sections are also providedin Serpent output. Similarly, absorption rate density and one group absorption cross sections arepresented in Figs. 4.13 and 4.14, respectively. In Fig. 4.11, discontinuous curves can be observedin both transport and diffusion theory results. In the transport theory fission rate density, suddenincreases are observed near interfaces of fissile and blanket regions. This is due to thermal neutronsentering fissile regions and causing fission in the plutonium in a very short distance from theinterface. These large increases are also observed in the diffusion theory results as well as in theabsorption rate results shown in Fig. 4.13. Additional discontinuities are observed within fissileregions in two-dimensional and three-dimensional diffusion results without discontinuity factors.These discontinuities are present due to macroscopic cross section differences in fission neutronproduction cross section presented in Fig. 4.12. In this figure, three colors are shown where blueindicates the three-dimensional homogenization, faded red represents two-dimensional and purplefor the overlapping of these data. In any fissile region, the only differences between sub-regions isthe void fraction. The void fraction in these fissile regions changes significantly as shown in Fig.2.13. This strong discontinuity in cross sections leads to the discontinuous behavior of reactionrate distributions even though the flux distributions are smooth. Larger differences are seen in thelower fissile region since the void fraction changes more rapidly. In the lower fissile region, thedifference between three-dimensional and two-dimensional homogenization is greater than in theupper fissile region. This is also true for absorption cross sections shown in Fig. 4.14. Therefore,since two-dimensional cross sections underestimate three-dimensional cross sections by a largeramount in the lower fissile zone, a higher flux from two-dimensional cross sections exists in thisregion. This is observed in group 1 flux in Fig. 4.9 and also for the corresponding reaction ratedistributions. Errors in the reactions rates between the analysis with discontinuity factors andwithout range from 10 to 60%.
The same calculation was performed with twelve energy groups. Axial distributions of group3 and 12 flux are shown in Figs. 4.15 and 4.16, respectively. These flux shapes, especially in thethermal group 12, show that the shape of the diffusion theory result with ADFs follows the shapeof the heterogeneous flux closer than in the two group calculation. In addition, the discontinuitybetween zones in these shapes has decreased. Near the lower reflector region in the group 12 figure(Fig. 4.16), a negative flux must be present in order to preserve reaction rates. This can happen neara reflector region especially if the flux is very low in a neighboring region, such as the second lowerreflector zone which contains boron. Two-dimensional and three-dimensional flux shapes followeach other more closely in the twelve group analysis compared to the two group calculation. Thiscan be observed by comparing group 3 in Fig. 4.15 and group 1 in Fig. 4.9 from the two groupcalculation. This agreement between homogenization methods is due to the finer treatment ofenergy groups. As further proof, group 3 absorption cross section estimates are shown in Fig. 4.17.In this graph, two-dimensional cross sections and three-dimensional cross sections overlap entirely
130
0.05;
E0.04
(I)
a),0 0.030)
o 0.02U,
LL
0.01
0'-20 0 20 40 60 80
Axial Position [cm]
--- Homogeneous No ADFHomogeneous ADF
- 2-D Homogeneous- Heterogeneous
UF UB UR
100
Figure 4.11. Comparison of Assembly Fission Rate Density from Two Group Calculation
0.035 '
D3- Homogenization
0.03 -*2-D Homogenization -
0
CO)0.025
2
0
a.2 0.01z
o 0.005
.0 LR1 LR2 U
0 -20 0 20 40 60 80 100 120 140
Axial Position [cm]
Figure 4.12. Comparison of Fission Production Cross Section from Two Group Calculation
131
0.06
LR1 LR2
--- Homogeneous No ADF--- Homogeneous ADF---- 2-D Homogeneous-Heterogeneous
0.04L
E0.035i
0.03Cc .o 0.025C0
-0.02
aljr 0.015!C0r-~ 0.01,0~
0
-0.005-20 0 20 40 60 80
Axial Position [cm]100 120 140
Figure 4.13. Comparison of Assembly Absorption Rate Density from Two Group Calculation
3-D Homogenization
2-D Homogenization
-20 0 20 40 60 80Axial Position [cm]
UR
100 120 140
Figure 4.14. Comparison of Absorption Cross Section from Two Group Calculation
132
LR1 LR2 LB LF iB UF UB UR
K
0.04
0.035
E0.030
00.0250
CI)
2 0.022
.20.0150.
L
-~0.01
0.0(
0
0.09--- Homogeneous No ADF
0.08 -- Homogeneous ADF--- Heterogeneous---- 2-D Homogeneous
0.07
1± 0.06cY)
0.0501
a0.041
-l 0.03
0Z 0.02
0.01LR1 LR2: LB LF
-20 0 20 40 60 80Axial Position [cm]
UF UB UR
100 120 140
Figure 4.15. Axial Distribution of RBWR Assembly Group 3 Flux from Twelve GroupCalculation
X
0
*0
N
0z
-0.005-20 0 20 40 60 80 100 120 140
Axial Position [cm]
Figure 4.16. Axial Distribution of RBWR Assembly Group 12 Flux from Twelve GroupCalculation
133
0.025 -3-D Homogenization2D Homogenization.
0.02-
Ul)CO)
2 0.015-
0CL
0.01
Cr)
5 0.005-
LR1 yR
-20 0 20 40 60 80 100 120 140Axial Position [cm]
Figure 4.17. Comparison of Group 3 Absorption Cross Section
such that only a purple color can be observed, signifying a combination of both red and blue plots.Finally, the fission rate density and absorption rate density are presented in Figs. 4.18 and 4.19,respectively. Comparing these reaction rate distributions with the two group results in Figs. 4.11and 4.13, the discontinuities in all curves are less. By dividing the group 1 energy range into11 energy groups the discontinuities in macroscopic cross sections between sub-regions becameless as shown in Fig. 4.17. This helped reduce discontinuities in the two-dimensional case andthree-dimensional case without ADFs. For diffusion theory results with ADFs, the discontinuityfactors are also closer to 1.0 to a reduction in energy condensation averaging, which also reducedthe discontinuity in macroscopic cross sections. As hinted by the comparison of multiplicationfactors for the twelve group analysis, the shapes of the two-dimensional and three-dimensionalreactions rates are very similar. This is why they yielded such similar multiplication factors. Alsothe magnitude of the error from the true reaction rate shape has also decreased. The errors nowrange from 2 to 25%. From these results, reference discontinuity factors can be generated foreither two energy groups or twelve energy groups as long as spatial discretization is constant whengenerating discontinuity factors. The effect of the reference boundary for the current calculationwas also checked and did not have any significant impact on the results and therefore the averagecurrent at each interface was used.
4.5.2 Application of Discontinuity Factors to PARCS
Ultimately, PARCS will be used to model a full RBWR core. In PARCS, the height of the assemblyis generated along the x direction, and East and West radial discontinuity factors are applied. Un-fortunately, PARCS cannot converge with the black hydrogen purely absorbing slabs. PARCS wasable to converge for an assembly without these slabs with zero incoming current boundary condi-
134
0.045
0.04
0 .0 35
0.03
U)
C0.025
0*- 0.02
C
o 0.015-
LL 0.01
0.005LR1 LR2 LB
-20 0
LF L._lB
20 40 60 80Axial Position [cm]
--- Homogeneous No ADF--- Homogeneous ADF.--- 2-D Homogeneous
- Heterogeneous
UF UR
100 120 140
Figure 4.18. Comparison of Assembly Fission Rate Density from Twelve Group Calculation
--- Homogeneous No ADF--- Homogeneous ADF- 2-D Homogeneous- Heterogeneous
IB UF UB UR
-20 0 20 40 60 80Axial Position [cm]
100 120 140
Figure 4.19. Comparison of Assembly Absorption Rate Density from Twelve Group Calculation
135
0.045
E 0.040.0
0.035
C 0.03
C 0.0250C.)Cz 0.02a)
C 0.0150
2- 0.010 .)
0.005L
-0.0051
LR1 LR2 LB LF
tions imposed. Before performing a PARCS calculation, axial discontinuity factors were generatedfrom the full assembly with black hydrogen. These axial discontinuity factors are first analyzedin the finite difference calculation without the black hydrogen slabs and zero incoming boundaryconditions imposed. The resulting k-effective for the two-group calculation from the finite differ-ence calculation is 1.08186 which is about 60 pcm different than the k-effective from Serpent. Toinvestigate these errors, the absorption rate density is shown in Fig. 4.20. Looking at the fissilezones, the integral is not the same as the heterogeneous distribution. Errors in the reaction ratesrange from 2-5% for both absorption and fission rate density. The same analysis was performedwith twelve energy groups. The k-effective was 1.08149 which is about 30 pcm different than thetarget multiplication factor from Serpent. As expected, errors in reaction rates are less and rangefrom 1-2%. The reason why the results are not exact is due to the approximation of the boundarycondition. Theoretically, the purely absorbing black hydrogen region should reproduce vacuumboundary conditions. At this interface, there is a very steep slope in the neutron flux and diffusiontheory cannot accurately calculate this. A small error in this leakage has a noticeable effect oneigenvalue and reaction rates.
These axial discontinuity factors were then added to the PMAXS cross section file for eachsub-region. For the North and South directions, a discontinuity factor of one was used. The fullassembly with zero incoming current boundary conditions was solved in two energy groups withthe HYBRID solver. The resulting multiplication factor was 1.08186 which agrees exactly withthe finite difference calculation. It has therefore been shown that the methodology of generatingdiscontinuity factors for a full RBWR assembly with a finite difference code can yield the sameanswer as a nodal calculation. The exact k-effective and reaction rates cannot be preserved whentaking out the black hydrogen slabs and replacing them with zero incoming current boundaryconditions. This occurs because Fick's Law is not valid for large flux gradients that exist at theinterfaces between the reflector and black hydrogen regions. This error then propagates to the restof the assembly and a small error exists. However, this error is less than the error resulting fromignoring axial discontinuity factors. Once Serpent has surface current tallies, the actual currentat the global boundaries of the system can be calculated and a reflective and simulated vacuumboundary would not be needed. An example of a PMAXS data file is listed in Appendix D.2 alongwith a corresponding PARCS input file for a full assembly.
4.5.3 Approximation of Discontinuity Factors
In the above analyses, discontinuity factors were generated for the same Serpent geometry. Inthe design of the RBWR, it is desirable to run full three-dimensional core calculations in PARCS.In order to preserve reaction rates and multiplication factor exactly, the same core that would berun in PARCS would need to be calculated in Serpent. Since a full core Monte Carlo analysisis not feasible for routine design calculations, axial discontinuity factors will need to be approxi-mated. Approximations of radial discontinuity factors are performed in routine design calculationsfor two-dimensional calculations with zero net current boundary conditions. The radial discon-tinuity factors are approximated by the ratio of heterogeneous flux at a boundary surface to theaverage flux in the geometry. These calculations do not involve neighbor cells and therefore truediscontinuity factors are not calculated. To apply this methodology to core analysis, single assem-bly calculations will be performed for ranges of operating conditions with SerpentXS. This willproduce a PMAXS cross section file for each sub-region. A PMAXS file will include all of the
136
0.035Homogeneous No ADFHomogeneous ADF
0.03: - Heterogeneous
0.025:1A=
C 0.02
00 0.015
0W 0.01
on
0LR1 LR2 LB LF 1B UF UB UR
-20 0 20 40 60 80 100 120 140Axial Position [cm]
Figure 4.20. Comparison of Assembly Absorption Rate Density with Vacuum BoundaryConditions
homogenized parameters along with radial discontinuity factors from Serpent output. The finitedifference MATLAB code can then process the homogenized parameters to generate appropriateaxial discontinuity factors. During the PARCS analysis, appropriate axial discontinuity factors canbe determined from interpolation of operating conditions. This interpolation process can be thesame as any homogenized few-group parameter.
In the current work, it is not feasible to show this process since PARCS currently does not havethe capability for handling axial discontinuity factors. Rather, a study is performed of approximat-ing single assembly discontinuity factors based on analyses of two-zone sub-geometries. This willhelp determine how sensitive discontinuity factors are to homogenized parameters. Only two-zonecalculations are performed for regions in which a strong material discontinuity exists. This situ-ation applies to BL-LR1-LR2-LB1, LB3-LF1, LF3-1B1, IB3-UF1, UF3-UB1, and UB3-UR-BU.The first and last items involve the reflector region and black hydrogen slabs and must be treatedwith additional zones for a neutron source. Each of these configurations were run in Serpent andhomogeneous parameters including axial discontinuity factors were calculated. This covers all ofthe sub-regions except for the interior ones. Axial discontinuity factors are not as essential forsimilar zones according to the analysis presented in Section 4.2.1. To get suitable cross sectionsfor these interior zones, they were coupled to the sub-region just below it. Therefore, LB 1-LB2,LF1-LF2, IB1-1B2, UFl-UF2, UB1-UB2 were also run in Serpent, but axial discontinuity factorswere not calculated. The resulting macroscopic cross sections and axial discontinuity factors werepieced together into a form that can be executed in the MATLAB finite difference solver.
Two group calculations yielded a multiplication factor of 1.10131 which has an error of about1700 pcm. Using twelve groups, a multiplication factor of 1.08124 which is only about 10 pcm
137
different than the target k-effective. Both of these values are closer to the true k-effective than with-out discontinuity factors as shown in Table 4.11. The twelve group results yield a multiplicationfactor that is very close to the target. To verify the preservation of reaction rates, axial distributionsof absorption rate density for these two cases along with the distribution from Serpent are shownin Fig. 4.21. This plot shows that neither the two group nor the twelve group calculations canreproduce true distributions of reaction rates. Even though the twelve group case yields an accu-rate k-effective, there is a cancellation of error in the reaction rates. This was verified integratingboth the fission production rate density and the absorption rate density and dividing the two. Thisyielded the same twelve group prediction of k-effective. The two group prediction tends to over es-timate absorption in the lower fissile zone and under estimate in the upper fissile zone. The twelvegroup calculation has the opposite trend. These errors are significant and are not acceptable andboth of these approaches are not recommended to approximate axial discontinuity factors. Thisshows that axial discontinuity factors are very sensitive to spatial homogenization. The eigenvalueand reaction rates are sensitive to these axial discontinuity factors even for twelve energy groups.It is recommended that full assemblies be used to homogenize cross sections and calculate axialdiscontinuity factors.
4.5.4 Effect of Void Distribution on Discontinuity Factors
In core simulations, homogenized parameters and radial discontinuity factors are interpolatedbased on local operating conditions. For a RBWR analysis, axial discontinuity factors can betreated with the same methodology and approximated over a suitable range of operating condi-tions. An analysis was performed to observe how ADFs change when a different void distributionis used in the analysis. The purpose of this analysis is to study how much the discontinuity factorschange when operating conditions are perturbed. This is important to study as ADFs will be ap-proximated by interpolation in a core simulation. The coolant density distribution listed in Table2.7 was increased by 10%. This is an example and is not an accurate perturbation as there is novoid fraction in lower reflector regions of the core. The same two group and twelve group analy-ses were performed (Section 4.5.1) with this perturbed void distribution. Resulting discontinuityfactors were compared to the reference discontinuity factors presented in Table 4.12. The percentdifferences of these values are shown in Table 4.13. A negative value indicates that the perturbeddiscontinuity factor is less than the corresponding reference discontinuity factor. The data showsthat large differences exist for the lower reflector regions. Since the void fraction in the lowerreflector region is not realistic, this sensitivity should be lower. In the active region of the core,the sensitivity is not large except for group 2 discontinuity factors in the upper fissile zones. Arelatively high coolant density in the surrounding neighbors of fissile zones will cause more ther-malization of neutrons and a larger current of these neutrons. This would explain the significantdifferences observed in the group 2 values. It is interesting that the interface between fissile andblanket zones do not show a significant difference, only the internal zones of the fissile regions. Inaddition, some statistical noise could be present since the thermal flux is very low in these fissileregions. The twelve group results show a similar trend and are therefore not presented. Largedifferences exist in the reflector regions as well the group 10, 11 and 12 values of the discontinu-ity factors. In the reflector regions, this difference was about 6-10% while in fissile regions wasbetween 10-20%. This is also an indication there may be a significant difference in the current ofthermal neutrons entering these fissile zones. This type of study should be performed for a range
138
of operating conditions to further study this sensitivity.
Homogeneous ADF 2G
0.04 -- Heterogeneous--- Homogeneous ADF 12G
EU 0.035
0.03a,)
o 0.025
0- 0.02a.)
( 0.015
'- 0.010 .
c')0.005 I. -< : ~ L
LR1 LR2 LB LF IB UF UB UR
-20 0 20 40 60 80Axial Position [cm]
100 120 140
Figure 4.21. Comparison of Absorption Rates Obtained with Approximated Discontinuity Factorsfor Two and Twelve Group Cases
139
-0.005
Table 4.13. Comparison of Two Group Discontinuity Factors for Perturbed Void Distribution
Interface Group 1 Left [%] Group 1 Right [%] Group 2 Left [%] Group 2 Right [%]
BL/LR1
LR1/LR2
LR2/LB 1
LB1I/LB2
LB2/LB3
LB3/LF1
LF1/LF2
LF2/LF3
LF3/IB1
IB 1/IB2
IB2/IB3
IB3/UF1
UF1/UF2
UF2/UF3
UF3/UB 1
UB1/UB2
UB2/UB3
UB3/UR
UR/BU
-60
1250
-2.5
-0.70
-0.24
-0.12
-0.11
-0.068
0.081
-0.048
-0.24
-0.11
-0.055
0.066
0.073
0.069
0.11
0.17
0.38
-66
3.9
1.8
0.55
0.21
0.11
0.10
0.071
-0.089
0.055
0.21
0.10
0.053
-0.069
-0.076
-0.075
-0.13
-0.18
-0.97
-4950
-11
-4.3
-0.85
0.096
-0.067
-18
-28
-0.26
-0.32
0.39
-0.30
-29
-27
1.9
1.4
0.19
0.35
0.79
-8625
7.1
3.9
0.73
-0.084
0.059
19
17
0.35
0.36
-0.28
0.20
-17
-21
-1.7
-1.3
-0.19
-0.39
-5.3
140
5 Conclusions and Future Work
5.1 Conclusions
The goal of this work was to provide a methodology for generating few-group homogenized crosssections for RBWR transient analyses. These time-dependent calculations cannot currently be per-formed with full spatial, energy and angle detail. Therefore, a homogenization methodology waspresented using the Monte Carlo code, Serpent. In recent years, there have been only a few ap-plications of cross section generation with Serpent. Since the RBWR is not a conventional typeof reactor, current methods of cross section homogenization are not adequate to treat strong axialmaterial discontinuities and large spectral gradients between axial regions. The Serpent MonteCarlo code was designed to handle arbitrary geometries and uses point-wise neutron data whichallows this reactor type to be analyzed. In this Monte Carlo process, no assumptions are made inspatial flux spectra and self-shielding effects as is done in conventional two-dimensional determin-istic methods. Lattice calculations can be performed directly from the point-wise neutron data andhomogenized data can be subsequently used in nodal core simulators.
An investigation was performed to determine how Serpent compares to other deterministic andstochastic lattice codes. The results indicated that the Serpent Monte Carlo code agrees well withBGCORE-MCNP5 depletion code, but there are significant differences in eigenvalue comparisonswith CASMO4E and Dragon. This is attributed to resonance upscattering treatment of uranium-238 in CASMO4E. However, a comparison of homogenized cross sections was also performedto show that two-dimensional homogenization in Serpent yielded similar results to Dragon. Thetwo-dimensional conventional cross section methodology was compared to a three-dimensionalmethod that involved homogenization over sub-regions of a RBWR single assembly. This analysisindicated that there are significant differences between these two approaches for this reactor de-sign. Most of the differences were seen in sub-regions that are located on either side of a strongmaterial discontinuity. This is due to a net current of fast neutrons leaving fissile zones and a netcurrent of thermal neutrons entering fissile zones. The small thermal neutron flux in the fissileregions is due to the presence of plutonium isotopes. This effect can be seen in reaction rates bylarge increases in fission and absorption rate densities near interfaces between fissile and blanketzones. Since Serpent agreed with Dragon for two-dimensional homogenization, three-dimensionalhomogenization results can also be accepted as an extension to Serpent's universe-based homoge-nization capability. These results provided motivation for investigating the use of few-group crosssections that were homogenized in the presence of axial neighbors.
The eventual goal is to perform full core transient simulations. In order to accomplish this, few-group homogenized parameters must be generated for a range of possible operating conditions.The process of history and instantaneous branch case calculations are essential to correctly predictchanges in operating conditions both instantaneously and throughout life of the simulation. Anefficient method of performing these analyses is not currently in the Serpent code. Recognizingthis need to perform these full core simulations, a branch case generation wrapper, SerpentXS,was developed to automate the creation of input files and organize output into a PMAXS crosssection database file for use in PARCS. This tool was tested on a typical PWR lattice where resultsfrom Serpent were compared to PARCS for a wide-range of branch cases. Results indicate thatdifferences between Serpent and PARCS are due primarily to statistical noise in the Monte Carloresults.
141
The conventional two-dimensional homogenization and Serpent three-dimensional homoge-nization were compared in a diffusion theory analysis of sub-regions of a RBWR assembly. Forsub-regions that are similar (i.e. different coolant void fraction but same fuel composition), bothhomogenization methods agree reasonably well with transport results. This was proved by ana-lyzing two upper fissile sub-regions together in a two-zone geometry with zero net current bound-ary conditions. However, for dissimilar sub-regions where large spectral gradients exist due tofuel composition, both homogenization procedures yielded unacceptable result using both two andtwelve energy groups. Errors in the range of hundreds of pcm were observed between diffusiontheory multiplication factors and transport multiplication factor from Serpent. To preserve reac-tion rates and multiplication factor, axial discontinuity factors were introduced on the interfacesbetween sub-regions. In order to calculate the true discontinuity factors, a net current at each inter-face is needed. Only the three-dimensional homogenization approach allows for the calculation ofsurface currents. A MATLAB code was written to generate discontinuity factors and incorporatethem into the finite difference equations. Results for the two-zone fissile-blanket system showedthat application of axial discontinuity factors preserved reaction rates and multiplication factor.This approach was extended to an analysis of a single RBWR assembly. Without discontinuityfactors, errors in multiplication factor were in the thousands of pcm. The analysis showed thataxial discontinuity factors could be applied to a full assembly. Even when using twelve energygroups, discontinuity factors were needed to preserve reaction rates and global multiplication fac-tor. The axial discontinuity factors were applied as radial discontinuity factors in PARCS to showtheir applicability to nodal methods. The results from the HYBRID nodal solver in PARCS agreedwith the finite difference calculation. This shows that PARCS could be used for core simulationswhen it is modified to use axial discontinuity factors.
In the process of analyzing a core, true axial discontinuity factors will not be calculated fora full core analysis in Serpent for every operating condition. Both radial and axial discontinuityfactors will be approximated and parametrized by a wide range of possible operating conditions.As an example, two group and twelve group axial discontinuity factors were approximated fromtwo-zone sub-geometries of a RBWR assembly. These approximate discontinuity factors and ho-mogenized cross sections were pieced together for a full assembly analysis. The results indicatethat this methodology does not work in two energy groups. For twelve energy groups, the eigen-value agrees well with the estimate from Serpent, however the axial distributions of reaction ratesdo not agree. It is concluded that these axial discontinuity factors are sensitive to spatial ho-mogenization and yield inaccurate eigenvalue and reaction rate estimates especially in two energygroups. It is recommended that this process be further investigated. A sensitivity study of dis-continuity factors was also performed for a +10% coolant density perturbation. Results indicatethat significant differences exist in reflector regions and in thermal groups of fissile regions in theRBWR. Although going to large number of energy groups would yield better results, this may beinefficient for full three-dimensional transient simulations with power/void feedback. It has beenshown that introduction of axial discontinuity factors in the RBWR could allow for a reduction ofthe number of energy groups needed to preserve multiplication factor and reaction rates.
5.2 Future Work
This work provides insight on the generation of few-group macroscopic homogenized parametersusing a three-dimensional Monte Carlo approach. A new methodology was developed and there is
142
a lot of potential for future work. This section includes suggestions for improvement of Serpentand the process to generate axial discontinuity factors.
5.2.1 Serpent Improvement
From experience using Serpent, there are some improvements that should be performed to help inthis new methodology. The first improvement should be enhanced capability for parallel calcula-tions. The parallel algorithm can be implemented in such a way so that source particles can bedivided among slave nodes and communication with the master node occurring after each sourcecycle. This will allow for multiple processor calculations so that statistics and random numbersequence can be preserved for an arbitrary number of processors. This is different than the currentimplementation of parallel capability which runs independent calculations and combines resultsstatistically when all calculations have been completed.
Another major improvement is the option of tallying surface currents. To calculate axial dis-continuity factors, these surface currents are necessary. In the current methodology, these surfacecurrents are computed from a neutron balance calculation. In order to perform this step, a surfacenet current somewhere in the geometry must be known. The only way to accomplish this currentlyis to impose a zero net current boundary condition on one or both of the global axial boundaries ofthe geometry. Due to statistics and possible neutron balance issues, this process may not yield theexact net current on each sub-region interface. This was seen in the current sensitivity studies andthe results of reaction rates since they weren't perfectly preserved. It is therefore recommendedthat a thorough investigation be performed to ensure that neutron balance can be re-created withthe homogenized output of Serpent once surface currents are calculated. It is expected that, forany sub-region of a RBWR assembly, surface net currents should balance the absorption and pro-duction of neutrons within statistics. Once this is proven, more studies of the calculation of axialdiscontinuity factors can be performed. By ensuring these steps and adding current tallies, thepurely absorbing black hydrogen slabs would not be needed and vacuum boundary conditions canbe imposed. This will be important when using PARCS since the nodal method did not convergewith the purely absorbing slabs.
Finally, it may be important to include an option of generating discontinuity factors from a finitedifference calculation within Serpent. Not only would this be important for axial discontinuityfactors, it is also a requirement for radial reflector calculations. Any time there is a coupled-zonecalculation at the lattice stage, discontinuity factors must be generated with the methodology shownin this work. The current methodology in Serpent is only valid for single-zone, two-dimensionalcalculations.
5.2.2 SerpentXS Improvement
The SerpentXS tool was developed to automate branch case calculations in Serpent for use withthe RBWR. Therefore, upgrades and improvements are necessary to make SerpentXS general. Thefirst improvement should be to handle any type of operating condition. Currently, the code onlyprocesses fuel temperature, coolant density, coolant temperature, poison concentration and controlrod fraction. There are more types of branches that can be used in PARCS and these options shouldeventually be included in the tool. Another upgrade should include the capability of processing
143
history branch case partial derivatives. Currently, in order to use history cases, GenPMAXS mustbe executed for the interpolation process. If this is built into SerpentXS, this step could be removed.
Features that have not been tested and implemented include fission yields of I, Xe, and Pm andgroup form functions. Currently in SerpentXS, there is a method for extracting this informationfrom tallies. However, the equations to process the data have not been implemented and the out-put routine to a PMAXS file has not been developed. These parameters are important for burnupapplications. In this work, these parameters were not used since the focus was to calculate a begin-ning of life power distribution. Another method of avoiding this upgrade in SerpentXS is to haveSerpent output these parameters automatically. This would limit the number of post-processingcalculations that SerpentXS must perform. Finally, there is no methodology present at all for di-rect energy deposition and J 1 factors. These parameters, used for thermal hydraulic analyses, arevery difficult to calculate in Serpent and would not be needed unless an application is identified.The last upgrade could include a graphical user interface that could aid the user in developingSerpent geometry and choosing branch cases accordingly. This is a long term improvement thatwould also benefit users of the Serpent code.
Another large improvement is the capability to handle "three-dimensional" branch cases. Sincehomogenization is now being performed in a three-dimensional calculation, this opens up newdefinitions for branch cases. One immediate application would be the insertion fraction of a controlrod. Now, geometries do not need to be either rodded or unrodded. Rather, partially roddedcases could be calculated. Using Serpent, branches could now be easily parametrized by operatingconditions of neighboring zones as well. PARCS has some capability for these types of branchesand this could be a useful extension for the RBWR. Finally, with any code, it could use a re-writingto take advantage of more object-oriented capabilities that Python programming language has tooffer.
5.2.3 Methodology Improvement
The methodology of generating axial discontinuity factors must be expanded and tested for a fullcore. There are many steps in this process that need to be further studied. The first is the methodof homogenizing cross sections in a three-dimensional calculation with neighboring zones. Thereare many parameters that need to be taken into account when subdividing the coarse regions of aRBWR assembly. The coolant void fraction discretization is one example. In an assembly, this voiddistribution is continuously changing and changes more rapidly in the fissile zones. The depletionzoning also needs to be taken into account when deciding how to subdivide regions. Finally, amesh sensitivity should be performed to capture the axial effects near interfaces. If the strongmaterial discontinuities can be captured better in the three-dimensional homogenization process,the discontinuity factors may not need to be as large to compensate for errors. However, in MonteCarlo analyses there are always tradeoffs between accuracy and confidence in an answer. A finemesh may be able to predict the interface effects, but the computational time may be larger in orderto simulate enough histories to get an acceptable variance of parameters.
Another study that should be performed is the energy group structure. By using axial discon-tinuity factors, one will be able to reduce the number of energy groups. However, with a coarsergroup structure, the discontinuity factors may become quite large and be very sensitive. This sen-sitivity is important when these discontinuity factors are approximated and interpolated in a nodalcode application. The method of how to approximate discontinuity factors and size of spatial
144
homogenization regions still needs to be investigated. In this work, ADFs were generated fromtwo-zone sub-geometries of an assembly. It is recommended to homogenize and generate discon-tinuity factors on a full single assembly level and parametrize this for full core analyses. Therefore,at least on an assembly basis, the true discontinuity factors are calculated. This idea would alsobe supported by investigating how these ADFs may change radially as operating conditions andburnup distribution differ. The sensitivity of discontinuity factors should be performed for all typesof operating conditions that will be used in the interpolation process in PARCS.
Finally, all of the homogenized parameters that are calculated by Serpent have statistical un-certainties associated with them. The effect of these uncertainties needs to be studied along with amethod for their propagation through calculations.
145
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149
A Code Comparison Input Files
A.1 Pin-cell Code Comparisons
A.1.1 CASMO4E
*************** *************** *****
* U02 Pin-cell Burnup Benchmark* Bryan Herman* CASMO4E
TTL * U02 Pin Cell
******** Operating Conditions****PDE 38.6, 'WGU * Specific Power of PinTFU=900.0 * Avg. Fuel Temperature (K)TMO=600.0 * Avg. Moderator Temperature (K)BOR=0.0 * Avg. Soluable Boron Conc. (ppm)VOI=0.0 * Void Fraction
******* Material Specs ******FUE 1,10.4,0.0,900.0/3.966646 92234=0.0 92238=84.181039 8000=11.852315
******* Geometry Specs *******^M
PIN 1 0.4096 /'1' * fuel RodPWR 2 1.26 2.52 0 0 0 0 1 1
THE 0 * all thermal expansion off
LPI1 11 1
DEP0.10.51.01.52.02.53.03.54.04.55.05.56.06.57.07.58.08.59.09.510.010.511.011.512.012.513.013.514.014.515.017.520.022.525.027.530.032.535.037.540.0
STAEND
A.1.2 Dragon
* PIN-CELL Benchmark - Dragon* U02 Fuel and Mod* 361 -GROUP ENDF7 DRAGLIB
150
* Bryan Herman
* Define STRUCTURES and MODULES used
LINKEDLISTPINCEL TRACK LIBRARY LIB2 CP FLUX BURNUP OUT DATABASE
SEQASCIIPSGEO XSSET
MODULELIB: GEO: NXT: USS: ASM: FLU: EVO: COMIO: EDI: PSP: DELETE: END:
Microscopic cross sections from ENDF7 Draglib 361 groups
LIBRARY := LIB:NMIX 2CIRA APOLANIS 2PTSLADED 4 NELAS N4N N2N N3NDEPL LIB: DRAGON FIL: DLIB E7MIXS LIB: DRAGON FIL: DLIBE7MIX 1 900.0 10.4
U235 = U235 3.966646 1 IRSET PTSL IU238 = U238 84.181039 1 IRSET PTSL I016 = 016 11.852315
MIX 2 600.0 0.660HIH20 = H1H20 11.189016 = 016 88.811
*----ASCII* Geometry PINCEL : PWR pin cell
PINCEL := GEO: :: CARCEL IX- REFL X+ REFL MESHX 0.0 1.26Y- REFL Y+ REFL MESHY 0.0 1.26RADIUS 0.0 0.4096 MIX 1 2
* Self-Shielding calculation USS" Tracking calculation NXT* Flux calculation for K no leakage
TRACK : NXT: PINCELTITLE 'PWR PINCELL'TISO 16 20. ;
LIB2 := USS: LIBRARY TRACK :: EDIT I TRAN PASS 2CP := ASM: LIB2 TRACK ;FLUX := FLU: CP LIB2 TRACK
TYPE K
* EDIT out macro xs
OUT := EDI: FLUX LIB2 TRACK PINCELEDIT 4 UPS MERGE COMP COND
SAVE
Create Reactor Database
DATABASE := COMPO:EDIT 5MAXCAL 50CDMM 'Multi-parameter reactor database ENDCPARA 'BURN' IRRAINIT
a Define Burnup Parameters
* Power
38.6 kW/kg
Burunup time interval2.590674 for 0 to 2.590674 days
= 10.362694 for 2.590674 to 12.953368 days= 12.953368 for 12.953368 to 388.601036 days= 64.766839 for 388.601036 to 1036.269430 days
Burnup control time variables Timei, Timef, TotalTime
Timei = initial timeTimef = final time
a TotalTime = Final time reached
REALPower Delt Timei Timef TotalTime38.6 2.590674 0.0 0.0 1036.269430
Burnup loop: for first step BURNUP is created* while for other steps it is modified
WHILE Timei TotalTime < DO
151
EVALUATE Timef := Timei Delt +IF Timei 0.0 = THEN
BURNUP LIB2 := EVO: LIB2 FLUX TRACKEDIT 3DEPL <<Timei>> <<Timef>> DAY
W/CC <<Power>>SET <<Timef>> DAYGLOB ;
* Record time 0 information and save it to databaseBURNUP LIB2 := EVO: BURNUP LIB2 FLUX TRACK
SAVE <<Timei>> DAY IOWR <<Power>> ;
DATABASE := (MPO: DATABASE OUT BURNUP LIB2EDIT 3SET <<Timei>> DAY
ELSEBURNUP LIB2 := EVO: BURNUP LIB2 FLUX TRACK
EDIT IDEPL <<Timei>> <<Timef>> DAYIOWR <<Power>>SET <<Timef>> DAYGLOB
ENDIF ;
* LIB2 := DELETE: LIB2LIB2 := USS: LIBRARY LIB2 TRACK :: EDIT 2 PASS 3CP := DELETE: CP ;CP : ASM: LIB2 TRACKFLUX := FLU: FLUX CP LIB2 TRACK
TYPE K ;
* Record information at end of timestepBURNUP LIB2 := EVO: BURNUP LIB2 FLUX TRACK
SAVE <<Timef>> DAY FOWR <<Power>>
* Edit out info at end of timesetpOUT := EDI: OUT FLUX LIB2 TRACK
EDIT 4 UPS MERGE COMP CONDSAVE ;
* Save end of timestep info in databaseDATABASE := CMPO: DATABASE OUT BURNUP LIB2EDIT 3SET <<Timef>> DAY
* change delta t for burnup
EVALUATE Timei := Timef Delt +
IF Timei 2.590674 > THENEVALUATE Delt := 10.362694
ENDIF
IF Timei 12.953368 > THENEVALUATE Delt := 12.953368
ENDIF
IF Timei 388.601036 > THENEVALUATE Delt := 64.766839
ENDIF
EVALUATE Timei := Timef
ENDWHILE
* Draw picturePSGEO := PSP: PINCEL
FILL RGBTYPE REGION
* Export databaseXSSET := DATABASE
ECHO " pincell completed"END: ;QUIT "LIST"
A.1.3 BGCORE-MCNP5
c U02 Burnup Benchmarkc MCNP5-BGCOREc Bryan Hermanc
c
152
c Cell CardsC
I 1 -10.4 -7 5 -6 tmp=7.75561E-8 inip:n=l $ fuel pellet2 4 -0.660 7 1 -2 3 -4 5 -6 tmp=5.17041E-8 imp:n=l $ moderator3 0 -1:2: -3:4: -5:6 imp:n=I $ void/boundary
c Surface Cardsc eI px -0.63 $ west boundary
*2 px 0.63 $ east boundary*3 py -0.63 $ south boundary*4 py 0.63 $ north boundary*5 pz -0.5 $ bottom boundary*6 pz 0.5 $ top boundary7 cz 0.4096 $ fuel pellet surface
c Data Cardse
mode nkcode 5000 1.0 20 500prdmp 520 520 520print -128
ec Materialscml 92235.37c -0.03966646
92238.37c -0.841810398016.37c -0.11852315
cm4 1001.34c -0.1119
8016.34c -0.8881cmt4 lwtr07 .31tc
LIB ./bgc data-polyval-ver- I Inew. matPOW 186.50965776SRC pin-srcMCR mpiexec -machinefile /hone/bherman/MASTERS/BENCH/BENCHI/MCNP_2/nodes -np 7 /opt /mcnp5/LANL/MCNP5/ Source/ src /mcnp5. mpi
TRS 0.999 1FYS thermal
MAT 1 0.52707179 900 -1
TMS 2.5906736 1.0 1TMS 10.3626943 1.0 1TMS 12.9533679 1.0 29TMS 64.7668394 1.0 10
A.1.4 Serpent
% Serpent% Bryan Herman
set title "U02 Pin Cell Burnup Benchmark No DBRC"
% -- Fuel pin:
Pitt 10fuel 0.4096water
% -- Lattice
lat I 1 0 0 1 1 1.26
10
% -- Cell:
cell pincell 0 fill 1 -1cell outer 0 outside I
% --- Surfaces:
surf I sqc 0 0 0.63
% --- Materials:
mat fuel -10.4 burn I92235.09c -0.0396664692238.09c -0.841810398016.09c -0.11852315
mat water 0.660 moder lwtr 1001
1001.06c -0.11198016.06c -0.8881
therm lwtr lwe7.12t
% --- Run parameters
153
% neutron population and criticality cycles
set pop 5000 500 50
% cross section library file path:
set acelib "/opt/serpent/xs/endfb7/sss-endfb7u.xsdata"
% unresolved resonance sampling
set ures I
% reflective boundary conditions (radial only)
set be 2
% group constants
set gcu 0
% plot optionsplot 3 1000 1000mesh 3 1000 1000
%- Decay and fission yield libraries:
set declib "/opt/serpent/xs/endfb7/sssendfb7.dec"set nfylib "/opt/serpent/xs/endfb7/sss endfb7.nfy"
%-- Cut-offs:
set fpcut IE-6set stabeut 1E-12
%-- Options for burnup calculation:
set bumode 2 % CRAM methodset pcc I % Predictor -corrector calculation onset xscalc 2 % Cross sections from spectrumset printm I % Right out burnup materials
%--- Set Normalization Condition
set powdens 38.6E-3
% -- Burnup Steps:
dep butot
0.10.51.01.52.02.53.03.54.04.55.05.56.06.57.07.58.08.59.09.510.010.511.011.512.012.513.013.514.014.515.017.520.022.525.027.530.032.535.037.540.0
set inventory
154
922350922380942380942390942400942410942420541350
A.2 Serpent - RBWR Single Assembly
######################################################################## MCNP5 and Serpent vl.l.14 Benchmark ## created by Bryan Herman #
# Steady Hot Unrodded ## Modified Thermal Conditions ##################8###################################################
set title "MIT RBWR 3D Assy for Benchmark"
# -- Begin Geometry Definitions with 2D slices
############################################################################################## Lower Reflector Layer 1 ################4t#####
############################ Universe # 101 ###########################
#################################################################### #
# Surfaces:
surf I inf
# Cells:
cell LRthom 101 Irlmat -1
# Materials
oat Irlmat -0.736 moder lwtr 10011001.06c 2.08016.06c 1.0
############################################################################################## Lower Reflector Layer 2 ###########t##########
############################ Universe # 102 ###########################
####################################################################8#8
# Pin Definitions:
pin Ilr2b4ccladbwaterlr2
# 1st Zone0.45500.5050
# Water Positions:
pin 2waterlr2
# Zone I
lat 3 3 0 0 21 21 1.14 # Lower Reflector Zone 2
2 2 2 2 2 2 2 2 211 I 1 1 2I I I I I 1 1 1 2
1 1 1 1 21 I I 1 1 1 2
1 1 1 1 1 211 1111 1 2111 1 1 1 2
1 1 1 1 1 1 2S 1 1 12
1 1 111 1 2I 1 I I 1 1 2 2
111 1 1 2 2 2111 1 2 2 2 2
I 1 1 2 2 2 2.21 1 1 2 2 2 2 2 21 1 2 2 2 2 2 2 21 2 2 2 2 2 2 2 22 2 2 2 2 2 2 2 22 2 2 2 2 2 2 2 22 2 2 2 2 2 2 2 2
155
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# Pin Definitions (allocate universes 20-39):
# 1st Zone0.43750.45500.5050
# 2nd Zone0.43750.45500.5050
enrichment 18.0%
enrichment 18.0%
pin 20fuelLFZ IvoidcladfwaterLFI
pin 21fuelLFZ2voidcladfwaterLF2
pin 22fuelLFZ3voidcladfwaterLF3
# Water Positions:
# Water in zone I
# Water in zone 2
# Water in zone 3
# Lattice Definitions
lat 300 3 0 0 21 21 1.14 # Lower Fissle Zone 1
373737373737373737373737373737373737373737
373737373737373737372020202020202020202037
373737373737373737202020202020202020202037
373737373737372020202020202020202020202037
373737373737202020202020202020202020202037
373737373720202020202020202020202020202037
373737372020202020202020202020202020202037
373737202020202020202020202020202020202037
373720202020202020202020202020202020202037
372020202020202020202020202020202020202037
372020202020202020202020202020202020203737
372020202020202020202020202020202020373737
372020202020202020202020202020202037373737
372020202020202020202020202020203737373737
372020202020202020202020202020373737373737
tat 301 3 0 0 21 21 1.14 # Lower Fissle Zone 2
383838383838383838383838383838383838383838
383838383838383838382121212121212121212138
3838383838383838382121212121
21212121212138
383838383838383821212121212121212121212138
383838383838382121212121212121212121212138
383838383838212121212121212121212121212138
383838383821212121212121212121212121212138
383838382121212121212121212121212121212138
383838212121212121212121212121212121212138
383821212121212121212121212121212121212138
382121212121212121212121212121212121212138
382121212121212121212121212121212121213838
382121212121212121212121212121212121383838
382121212121212121212121212121212138383838
382121212121212121212121212121213838383838
382121212121212121212121212121383838383838
tat 302 3 0 0 21 21 1.14 # Lower Fissle Zone 3
39393939393939
39393939393939
39393939393939
39393939393939
39393939393939
39393939393922
39393939392222
39393939222222
39393922222222
39392222222222
39222222222222
39222222222222
39222222222222
39222222222222
39222222222222
39222222222222
# 3rd Zone enrichment 18.0%0.43750.45500.5050
pin 37waterLFl
pin 38waterLF2
pin 39waterLF3
372020202020202020202020202037373737373737
382121212121212121212121212138383838383838
39222222222222
372020202020202020202020203737373737373737
382121212121212121212121213838383838383838
39222222222222
372020202020202020202020373737373737373737
382121212121212121212121383838383838383838
39222222222222
372020202020202020202037373737373737373737
382121212121212121212138383838383838383838
39222222222222
373737373737373737373737373737373737373737
383838383838383838383838383838383838383838
39393939393939
158
39 39 39 39 22 22 22 22 22 22 22 22 22 22 22 22 22 22 22 22 3939 39 39 22 22 22 22 22 22 22 22 22 22 22 22 22 22 22 22 22 3939 39 22 22 22 22 22 22 22 22 22 22 22 22 22 22 22 22 22 22 3939 22 22 22 22 22 22 22 22 22 22 22 22 22 22 22 22 22 22 22 3939 22 22 22 22 22 22 22 22 22 22 22 22 22 22 22 22 22 22 39 3939 22 22 22 22 22 22 22 22 22 22 22 22 22 22 22 22 22 39 39 3939 22 22 22 22 22 22 22 22 22 22 22 22 22 22 22 22 39 39 39 3939 22 22 22 22 22 22 22 22 22 22 22 22 22 22 22 39 39 39 39 3939 22 22 22 22 22 22 22 22 22 22 22 22 22 22 39 39 39 39 39 3939 22 22 22 22 22 22 22 22 22 22 22 22 22 39 39 39 39 39 39 3939 22 22 22 22 22 22 22 22 22 22 22 22 39 39 39 39 39 39 39 3939 22 22 22 22 22 22 22 22 22 22 22 39 39 39 39 39 39 39 39 3939 22 22 22 22 22 22 22 22 22 22 39 39 39 39 39 39 39 39 39 3939 39 39 39 39 39 39 39 39 39 39 39 39 39 39 39 39 39 39 39 39
# Surfaces:
surf 4 inf
# Cells for Lattice
cell LFZlpin 106 fill 300 -4 # Lower Fissile Zone I Pinscell LFZ2pin 107 fill 301 -4 # Lower Fissile Zone 2 Pinscell LFZ3pin 108 fill 302 -4 # Lower Fissile Zone 3 Pins
# Fuel and Water materials
mat fuelLFZI -10.5 burn I vol 814.789248016.12c -0.11818292235.12c -- 1.394712e-392238.12c -0.55649093237.12c -l.619666e-394238.12c -9.717994e-394239.12c -0.14253194240.12c -0.11694094241.12c -0.01619794242.12c -0.01587395241.12c -0.01198695342.12c -3.239331e-495243.12c -4.211131e -396244.12c -3.23933le-396245.12c -9.717994e -496246.12c -3.239331e-4
mat fuelLFZ2 -10.5 burn I vol 2020.67738016.12c -0.11818292235.12c -1.394712e-392238.12c -0.55649093237.12c -1.619666e-394238.12c -9.717994e-394239.12c -0.14253194240.12c -0.11694094241.12c -0.01619794242.12c -0.01587395241.12c -0.01198695342.12c -3.239331e-495243.12c -4.211131e -396244.12c -3.239331e-396245.12c -9.717994e-496246.12c -3.239331e-4
mat fuelLFZ3 -10.5 burn I vol 814.789248016.12c -0.11818292235.12c -l.394712e-392238.12c -0.55649093237.12c -l.619666e -394238.12c -9.717994e-394239.12c -0.14253194240.12c -0.11694094241.12c -0.01619794242.12c -0.01587395241.12c -0.01198695342.12c -3.239331e -495243.12c -4.211131e-396244.12c -3.239331e-396245.12c -9.7 17994e -496246.12c -3.239331e-4
mat waterLFl -0.527 moder lwtr 10011001.06c 2.08016.06c 1.0
mat waterLF2 -0.360 moder lwtr 10011001.06c 2.08016.06c 1.0
mat waterLF3 -0.290 moder lwtr 10011001.06c 2.08016.06c 1.0
159
#######################################################################
######################### Internal Blanket ################################################## Universe # 109-111 #############################################################################################
# Pin Definitions (allocate universes 40-49):
pin 40fuelIBZIvoidcladbwaterIB I
pin 41fuelIBZ2voidcladbwaterIB2
pin 42fuelIBZ3voidcladbwaterIB3
# 1st Zone0.43750.45500.5050
# 2nd Zone0.43750.45500.5050
# 3rd Zone0.43750.45500.5050
# Water Positions:
pin 47 # Zone IwaterIBI
pin 48waterIB2
pin 49waterIB3
# Zone 2
# Zone 3
# Lattice Definitions
tat 400 3 0 0 21 21 1.14
474747474747474747474747474747474747474747
474747474747474747474040404040404040404047
474747474747474740404040404040404040404047
474747474747474040404040404040404040404047
474747474747404040404040404040404040404047
474747474740404040404040404040404040404047
474747474040404040404040404040404040404047
# Internal Blanket Zone I
474747404040404040404040404040404040404047
474740404040404040404040404040404040404047
474040404040404040404040404040404040404047
474040404040404040404040404040404040404747
474040404040404040404040404040404040474747
474040404040404040404040404040404047474747
474040404040404040404040404040404747474747
474040404040404040404040404040474747474747
474040404040404040404040404047474747474747
474040404040404040404040404747474747474747
tat 401 3 0 0 21 21 1.14 # Internal Blanket Zone 2
484848484848484848484848484848484848484848
484848484848484848414141414141414141414148
484848484848484841414141414141414141414148
484848484848484141414141414141414141414148
484848484848414141414141414141414141414148
484848484841414141414141414141414141414148
484848484141414141414141414141414141414148
484848414141414141414141414141414141414148
484841414141414141414141414141414141414148
484141414141414141414141414141414141414148
484141414141414141414141414141414141414848
484141414141414141414141414141414141484848
484141414141414141414141414141414148484848
484141414141414141414141414141414848484848
484141414141414141414141414141484848484848
484141414141414141414141414148484848484848
484141414141414141414141414848484848484848
474040404040404040404040474747474747474747
484141414141414141414141484848484848484848
474040404040404040404047474747474747474747
484141414141414141414148484848484848484848
474747474747474747474747474747474747474747
484848484848484848484848484848484848484848
lat 402 3 0 0 21 21 1.14 # Internal Blanket Zone 3
160
494949494949494949494949494949494949494949
494949494949494949494242424242424242424249
49
4949494949494949424242424242424242424249
494949494949494942424242424242424242424249
494949494949494242424242424242424242424249
494949494949424242424242424242424242424249
494949494942424242424242424242424242424249
# Surfaces:
surf 5 inf
# Cells for Lattice
cell IBZlpin 109cell IBZ2pin 110cell IBZ3pin I ll
fill 400 -5fill 401 -5fill 402 -5
# Internal Blanket Zone I Pins
# Internal Blanket Zone 2 Pins# Internal Blanket Zone 3 Pins
# Blanket and Water materials
mat fuelIBZ I8016.09c92235.09c92238.09c
mat fuellBZ28016.09c92235.09c92238.09c
mat fuellBZ38016.09 c92235.09c92238.09c
-10.5 burn I-0.118466-2.203834e-30.87933
-10.5 burn I0.1184662.203834e-3
-0.87933
-10.5 burn I-0.118466-2.203834e-3-0.87933
vol 1140.70493
vol 5051.69326
vol 1140.70493
mat wateriB1 -0.266 moder lwtr 10011001.06c 2.08016.06c 1.0
mat waterIB2 -0.2531001.06c 2.08016.06c 1.0
moder lwtr 1001
mat waterIB3 -0.226 moder lwtr 10011001.06c 2.08016.06c 1.0
############################## #9 ########998######8######8##
########################## Upper Fissile #########################
########################## Universe # 112-114 #######################
##############################
# Pin Definitions ( allocate universes 50-69):
pin 50 # I st Zone enrichment 18%fuelUFZI 0.4375void 0.4550cladf 0.5050waterUFI
pin 51 # 2nd Zone enrichment 18%
fuelUFZ2 0.4375void 0.4550cladf 0.5050waterUF2
pin 52 # 3rd Zone enrichment 18%fuelUFZ3 0.4375void 0.4550cladf 0.5050waterUF3
161
494949494242424242424242424242424242424249
494949424242424242424242424242424242424249
494942424242424242424242424242424242424249
494242424242424242424242424242424242424249
494242424242424242424242424242424242424949
494242424242424242424242424242424242494949
49
4242424242424242424242424242424249494949
494242424242424242424242424242424949494949
494242424242424242424242424242494949494949
494242424242424242424242424249494949494949
494242424242424242424242424949494949494949
494242424242424242424242494949494949494949
494242424242424242424249494949494949494949
494949494949494949494949494949494949494949
# Water Positions:
# Water in zone 1
# Water in zone 2
# Water in zone 3
# Lattice Definitions
lat 500 3 0 0 21 21 1.14 # Upper Fissle Zone 1
676767676767675050505050505050505050505067
676767676767505050505050505050505050505067
676767676750505050505050505050505050505067
676767675050505050505050505050505050505067
676767505050505050505050505050505050505067
676750505050505050505050505050505050505067
675050505050505050505050505050505050505067
675050505050505050505050505050505050506767
675050505050505050505050505050505050676767
675050505050505050505050505050505067676767
675050505050505050505050505050506767676767
675050505050505050505050505050676767676767
675050505050505050505050505067676767676767
lat 501 3 0 0 21 21 1.14 # Upper Fissle Zone 2
686868686868685151515151515151515151515168
686868686868515151515151515151515151515168
686868686851515151515151515151515151515168
686868685151515151515151515151515151515168
686868515151515151515151515151515151515168
686851515151515151515151515151515151515168
685151515151515151515151515151515151515168
685151515151515151515151515151515151516868
685151515151515151515151515151515151686868
685151515151515151515151515151515168686868
685151515151515151515151515151516868686868
68515151
5151515151515151515151686868686868
lat 502 3 0 0 21 21 1.14 # Upper Fissle Zone 3
696969696969695252525252525252525252525269
696969696969525252525252525252525252525269
696969696952525252525252525252525252525269
696969695252525252525252525252525252525269
696969525252525252525252525252525252525269
696952525252525252525252525252525252525269
695252525252525252525252525252525252525269
6952525252525252525252525252525252525269
6952525252525252525252525252525252526969
6952525252525252525252525252525252696969
6952525252525252525252525252525269696969
6952525252525252525252525252526969696969
685151515151515151515151515168686868686868
6952525252525252525252525252696969696969
675050505050505050505050506767676767676767
685151515151515151515151516868686868686868
6952525252525252525252525269696969696969
675050505050505050505050676767676767676767
685151515151515151515151686868686868686868
6952525252525252525252526969696969696969
675050505050505050505067676767676767676767
685151515151515151515168686868686868686868
6952525252525252525252696969696969696969
69 69 69 69 69 69 69 69 69 69
# Surfaces:
surf 6 inf
162
pin 67waterUFt
pin 68waterUF2
pin 69waterUF3
676767676767676767676767676767676767676767
676767676767676767675050505050505050505067
676767676767676767505050505050505050505067
676767676767676750505050505050505050505067
686868686868686868686868686868686868686868
686868686868686868685151515151515151515168
686868686868686868515151515151515151515168
686868686868686851515151515151515151515168
676767676767676767676767676767676767676767
686868686868686868686868686868686868686868
6969696969696969696969696969696969696969
696969696969696969696969696969696969696969
696969696969696969695252525252525252525269
696969696969696969525252525252525252525269
696969696969696952525252525252525252525269
# Cells for Lattice:
cell UFZlpin 112 fill 500 -6 # Upper Fissile Zone I Pinscell UFZ2pin 113 fill 501 -6 # Upper Fissile Zone 2 Pinscell UFZ3pin 114 fill 502 -6 # Upper Fissile Zone 3 Pins
# Fuel and Water materials
mat fuelUFZI -10.5 burn I vol 814.789248016.12c -0.11818292235.12c -l.394712e-392238.12c -0.55649093237.12c -1.619666e-394238.12c -9.717994e-394239.12c -0.14253194240.12c -0.11694094241.12c -0.01619794242.12c -0.01587395241.12c -0.01198695342.12c -3.239331e-495243.12c -4.211131e-396244.12c -3.239331e-396245.12c -9.717994e-496246.12c -3.23933 1e -4
mat fuelUFZ2 -10.5 burn I vol 2053.268878016.12c -0.11818292235.12c -l.394712e -392238.12c -0.55649093237.12c -1 619666e-394238.12c -9.717994e-394239.12c -0.14253194240.12c -0.11694094241.12c -0.01619794242.12c -0.01587395241.12c -0.01198695342.12c -3.239331e -495243.12c -4.211131e -396244.12c -3.239331e -396245.12c -9.717994e-496246.12c -3.239331e-4
mat fuelUFZ3 -10.5 burn I vol 814.789248016.12c -0.11818292235.12c -1.394712e-392238.12c -0.55649093237.12c -l.619666e-394238.12c -9.7 17994e -394239.12c -0.14253194240.12c -0.11694094241.12c -0.01619794242.12c -0.01587395241.12c -0.01198695342.12c -3.239331e -495243.12c -4.211131e-396244.12c -3.239331e-396245. 12 c -9.7 17994e -496246.12c -3.239331e-4
mat waterUFl -0.219 moder lwtr 10011001.06c 2.08016.06c 1.0
mat waterUF2 -0.190 moder lwtr 10011001.06c 2.08016.06c 1.0
mat waterUF3 -0.169 moder lwtr 10011001.06c 2.08016.06c 1.0
##########################################8#8######8#8########################## Upper Blanket #######################
########################## Universe # 115-117 88###################
#####################################################
# Pin Definitions (allocate universes 70-89):
pin 70 # 1st ZonefuelUBZl 0.4375void 0.4550cladb 0.5050waterUB I
pin 71 # 2nd ZonefuelUBZ2 0.4375void 0.4550cladb 0.5050waterUB2
163
t791
6L 6L 6L 6L 6L 6L 6L 6L 6L 6L 6L 6L 6L 6L 6L 6L 6L 6L 6L 6L 6L6L 6L 6L 6L 6L 6L 6L 6L 6L 6L UL ZL ZL ZL ZL ZL ZL L ZL ZL 6L6L 6L 6L 6L 6L 6L 6L 6L 6L L ZL ZL ZL ZL ZL ZL ZL L ZL ZL 6L6L 6L 6L 6L 6L 6L 6L 6L L ZL EL ZL L ZL 1L ZL L L UL ZL 6L6L 6L 6L 6L 6L 6L 6L ZL 1L EL ZL ZL ZL ZL ZL ZL EL ZL ZL ZL 6L6L 6L 6L 6L 6L 6L ZL ZL ZL EL 1L ZL ZL ZL ZL ZL EZL1L ZL ZL 6L6L 6L 6L 6L 6L ZL ZL ZL ZL 1L 1L 2 L L ZL ZL ZL ZL ZL ZL ZL 6L6L 6L 6L 6L ZL ZL ZL ZL ZL ZL 1L L ZL ZL 1L ZL 1L ZL ZL ZL 6L6L 6L 6L 1L L ZL ZL ZL ZL ZL ZL L ZL ZL ZL ZL ZL 1L 1L 1L 6L6L 6L 1L 1L ZL ZL ZL ZL ZL ZL L EL ZL ZL ZL ZL 1L 1L ZL ZL 6L6L ZL 1L ZL L ZL ZL ZL l L L L ZL ZL ZL ZL EL L ZL L 6L6L 1L L ZL ZL ZL ZL ZL L L L L ZL ZL ZL L L ZL ZL 6L 6L6L ZL L ZL L ZL ZL ZL L L L L ZL L L L L L 6L 6L 6L6L ZL L L L ZL ZL ZL ZL L ZL L ZL L ZL ZL L 6L 6L 6L 6L6L ZL L ZL ZL ZL ZL ZL L ZL L L ZL ZL ZL ZL 6L 6L 6L 6L 6L6L ZL 1L ZL L ZL ZL L L L L ZL ZL ZL ZL 6L 6L 6L 6L 6L 6L6L ZL 1L L L ZL ZL ZL ZL L L L ZL ZL 6L 6L 6L 6L 6L 6L 6L6L ZL L L ZL ZL ZL ZL L L L ZL ZL 6L 6L 6L 6L 6L 6L 6L 6L6L 1l 2 L L ZL ZL ZL L ZL L ZL L 6L 6L 6L 6L 6L 6L 6L 6L 6L6L 1l 2 L ZL ZL L ZL L L L L 6L 6L 6L 6L 6L 6L 6L 6L 6L 6L6L 6L 6L 6L 6L 6L 6L 6L 6L 6L 6L 6L 6L 6L 6L 6L 6L 6L 6L 6L 6L
12 uoZ bvlunIu .i~ddf # t,[1 [1 112I 0 0 E2 1209 [l[
8L 8L 8L 8L L 8L1 8L 8L 81L L 8L 8L 8L 8L 8L 8L 8L SL 8L L 8L8L 8L 81 8L 8L SL 11 SL 8L 8L 1L 11 11 I IL L I. Ll IL L IL IL 8L8L 8L 8L 8 SL 8L SL 8 L IL 11.L L IL IL IL IL IL IL IL 8L8L 8L 8L 8L 8L SL 8L 8L 1L 1L IL L ILIL IL l IL IL IL IL 8L8L 8L 8L 8L 8L 8L 1L IL lI IL LIL IL IL IL IL IL IL IL 8 L 8LSL 8L 8L 8L 8L SL ILl IL ILl IL IL IL IL IL IL IL IL IL IL IL SLSL 8L 8L 1L 8L IL IL 1L IL IL 1L IL IL IL IL L L IL IL IL 8LSL 8L 9L 8L IL IL IL tL IL IL IL IL lL IL ILl L IL IL IL IL 8L8L 9L 8L IL IL IL IL IL IL IL IL 1L IL IL IL IL IL IL 8L 8L 8L8L 8L IL IL 1L IL IL IL IL IL IL IL IL IL IL L IL I L IL 8L8L IL IL IL IL IL IL IL IL IL IL IL IL ILl IL IL IL IL IL IL 8L8L IL IL IL IL IL IL ILl IL lL IL IL IL IL ILl IL IL IL IL 8L SL8L IL IL IL IL IL ILl IL IL IL IL IL ILl ILl IL IL IL IL 8L 8L 8L8L IL IL IL IL IL Ll IL IL IL IL IL IL ILl L Ll IL 8L 8L 8L 8LSL IL IL IL IL \L ILl IL IL IL IL IL IL iL IL IL SL 8L 8L 8L 8L8L IL IL IL IL IL ILl IL IL IL IL L ILl IL IL 8L RL 8L 8L 8L 8L8L IL IL IL IL IL IL IL IL IL IL IL IL IL 8L 8L 8L 8L 8L 8L 8L8L IL IL IL IL ILl [L IL IL IL IL IL IL 8L SL 8L 8L 8L 8L 8L 8L8L IL IL IL IL IL IL IL IL IL IL IL 8L 8L 8L 8L 8L 8L SL SL 8L8L L L IL IL IL IL IL IL IL IL 8L 8L 8L 8L 8L 8L 8L 8L 8L 8L8L 8L 8L 8L 8L 8L 8L 8L 8L SL 8L 8L 8L SL 8L 8L 8L L SL 8L 8L
12 DuoZ i~IuU[ ioddfl # 'l7 1 12 I 0 0 12 109 II
LL LL LL LL LL LL LL LL LL LL LL LL LL LL LL LL LL LL LL LL LLLL LL LL LL LL LL LL LL LL LL OL OL OL OL OL OL OL OL OL OL LLLL LL LL LL LL LL LL LL LL OL OL OL OL OL OL OL OL OL OL OL LLLL LL LL LL LL LL LL LL OL OL OL OL OL OL OL DL OL OL OL OL LLLL LL LL LL LL LL LL OL OL 0L OL OL OL OL OL OL OL OL OL OL LLLL LL LL LL LL LL OL OL OL OL OL OL OL OL OL OL OL OL OL OL LLLL LL LL LL LL OL OL OL OL OL OL OL OL OL OL OL OL OL OL OL LLLL LL LL LL OL OL OL OL OL OL OL OL OL OL OL 0L OL OL OL OL LLLL LL LL OL OL OL OL OL OL 0L OL OL OL OL OL OL 0L OL OL OL LLLL LL OL OL L OL OL OL OL OL OL OL OL OL OL L OL OL OL L LLLL DL OL OL OL OL OL OL OL OL OL OL OL OL OL OL OL OL OL OL LLLL OL OL OL OL OL OL OL OL OL OL OL OL OL OL OL OL OL OL LL LLLL OL OL OL OL OL OL OL OL OL OL OL OL OL OL OL OL OL LL LL LLLL OL OL OL OL OL OL OL OL OL OL OL OL OL OL 0L OL LL LL LL LLLL 0L OL OL OL OL OL OL OL OL OL OL OL OL OL 0L LL LL LL LL LLLL OL CL OL OL OL OL OL 0L OL OL OL OL OL OL LL LL LL LL LL LLLL OL OL OL OL OL OL OL OL OL 0L OL OL OL LL LL LL LL LL LL LLLL 0L OL OL OL OL OL OL OL OL OL OL OL LL LL LL LL LL LL LL LLLL L OL OL OL OL 0L L OL OL OL L LL LL LL LL LL LL LL LL LLLL OL OL OL OL OL OL OL OL OL OL LL LL LL LL LL LL LL LL LL LLLL LL LL LL LL LL LL LL LL LL LL LL LL LL LL LL LL LL LL LL LL
1.. .1 0. 1. 01 0. 1 .01.o 01.- 01. f 01 0 1.1 01 01. 01. 0 01 01 01. 1.1.
I~~~~~~~ 3u7I)U~I ~d I[2[20 0 12 009Id
I Eaflio8um
12 OUO7 # 61. uid
SUO!Ils6 1.1.M #
0!;6 t70 p!OA
12L 17'0 12ZgflIonJ;)Uoz PI1 6 121. uid
# Surfaces:
surf 7 inf
# Cells for Lattice:
cell UBZlpin 115cell UBZ2pin 116cell UBZ3pin 117
fill 600 -7fill 601 -7fill 602 -7
# Upper Blanket Zone I Pins# Upper Blanket Zone 2 Pins# Upper Blanket Zone 3 Pins
# Blanket and Water materials
mat fuelUBZ I8016.09c92235.09c92238.09c
mat fuelUBZ28016.09 c92235.09c922 38.09 c
mat fuelUBZ38016.09c92235.09c92238.09c
mat waterUB I1001.06c801 6.06c
-10.5 burn I-0.118466-2.203834e-3-0.87933
-10.5 burn 1-0.118466-2.203834e-3-0.87933
-10.5 burn 1-0.118466-2.203834e-3-0.87933
vol 325.91569
vol 1303.66278
vol 325.91569
-0.156 moder lwtr 10012.01.0
mat waterUB2 -0.152 moder lwtr 10011001.06c 2.08016.06c 1.0
mat waterUB3 -0.150 moder lwtr 10011001.06c 2.08016.06c 1.0
######################################################ftffff tttttfff ftt
######################## Upper Reflector Layer I ######################
############################ Universe # 118 ###########################
####################################################ftt f t f t f t f t f t f t f tt t
# Pin Definitions:
pin 4voidcladbwaterurl
# 1st Zone0.45500.5050
# Water Positions:
pin 5 # Zone Iwaterurl
lat 6 3 0 0 21 21 1.14 # Upper Reflector Zone I
# Surfaces
surf 8 inf
165
# Cell for Lattice
cell URIpin 118 fill 6 -8 # Upper Reflector Zone I Pins
# Materials
mat waterurl -0.149 moder lwtr 10011001.06c , 2.08016.06c 1.0
############################################################################################### Assembly Geometry ############################
######################################################################
# -- Stack Assembly Universes
lat 900 9 0 0 18 # Vertical Lattice 10 layers
-30.0 101 # Bottom of Lower Reflector 17.0 102 # Bottom of Lower Reflector 20.0 103 # Start of Core, Bottom of LBZI
7.75 104 # Bottom of Lower Blanket Zone 215.5 105 # Bottom of Lower Blanket Zone 318.0 106 # Bottom of Lower Fissile Zone 123.0 107 # Bottom of Lower Fissile Zone 235.4 108 # Bottom of Lower Fissile Zone 340.4 109 # Bottom of Internal Blanket Zone 147.4 110 # Bottom of Internal Blanket Zone 278.4 111 # Bottom of Internal Blanket Zone 385.4 112 # Bottom of Upper Fissile Zone 190.4 113 # Bottom of Upper Fissile Zone 2
103.0 114 # Bottom of Upper Fissile Zone 3108.0 115 # Bottom of Upper Blanket Zone I110.0 116 # Bottom of Upper Blanket Zone 2118.0 117 # Bottom of Upper Blanket Zone 3120.0 118 # Bottom of Upper Reflector 1
# -- Assembly Outer Surfaces
surf 31 hexxc 0.0 0.0 9.495 # outer unitcellsurf 32 pz -30.0 # bottom of assemblysurf 33 pz 150.0 # top of assembly
# -- Assembly cells
cell assyi 0 fill 900 -31 32 -33 # assembly latticecell outrf 0 outside 31 32 -33 # radially reflect becell outbb 0 outside -32 # bottom black bccell outbt 0 outside 33 # top black be
# -- Common Material Data Between Zones
mat cladf -6.5 # Fuel Clad - Zircaloy40090.06c 1.0
mat cladb -6.5 # Blanket Clad - Zircaloy40090.06c 1.0
# -- Thermal Scattering Data (take all at 600K)
therm lwtr lwe7.12t
#################################################################################### Control Information ##################################################################################################
# Neutron population and criticality cycles
set pop 25000 1000 200
# Cross section library file path
set acelib "/opt/serpent/xs/endfb7/sss-endfb7u.xsdata"
# reflective boundary conditions (radial only)
set bc 2
# turn on full unresolved resonance sampling
set ures I
166
# Group Constant Homogenization
set gcu 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118set nfg 2 0.625e-6ene egrid I I.E-10 0.625E-6 15.0
# random number seed
set seed 1300652262
# Flux Distribution
det LRI do 101det LR2 dn 102det LB1 du 103del LB2 du 104del LB3 du 105det LFI du 106det LF2 do 107del LF3 do 108del IBI du 109del IB2 du 110det IB3 du 111del UFl du 112del UF2 du 113del UF3 do 114det UBI do 115del UB2 du 116del UB3 du 117del UR du 118
# Power
del LRIpdel LR2pdel LBlpdel LB2pdel LB3pdel LFlpdel LF2pdel LF3pdel IBIpdel IB2pdel IB3pdel UFIpdel UF2pdel UF3pdel UBIpdel UB2pdel UB3pdel URp
Dis
dududododududodudutdudododududododudo
tribution
101 dr102 dr103 dr104 dr105 dr106 dr107 dr108 dr109 dr -110 dr -111 dr -112 dr -113 dr -114 dr -115 dr -116 dr -117 dr -118 dr -
# Absorption Di
del
del
deldeldel
deldel
del
deldel
del
del
deldeldeldel
deldel
LRIaLR2aLBlaLB2aLB3aLFlaLF2aLF3aIBIaLB2aIB3aUFlaUF2aUF3aUBIaUB2aUB3aURa
du
dododu
du
dudododudododudududodu
dudu
1011021031041051061071081091101111121131141151 16117118
voidvoidvoidvoidvoidvoidvoidvoidvoidvoidvoidvoidvoidvoidvoidvoidvoidvoid
s t ribut ion
dr -2 voiddr -2 voiddr -2 voiddr -2 voiddr -2 voiddr -2 voiddr -2 voiddr -2 voiddr -2 voiddr -2 voiddr -2 voiddr -2 voiddr -2 voiddr -2 voiddr -2 voiddr -2 voiddr -2 voiddr -2 void
# Fission Neutron Production Distribution
deldeldeldeldeldeldeldeldeldeldeldel
LRIfLR2fLB IfLB2fLB3fLFIfLF2fLF3fIBIfIB2fIB3fUFIf
dododu
dududu
dudududodudu
101102103104105106107108109110i 11
112
drdrdrdrdrdrdrdrdrdrdrdr
-7-7-7-7-7-7-7-7-7-7-7-7
voidvoidvoidvoidvoidvoidvoidvoidvoidvoidvoidvoid
167
det UF2f du 113 dr -7 voiddet UF3f du 114 dr -7 voiddet UBlf du 115 dr -7 voiddet UB2f du 116 dr -7 voiddet UB3f du 117 dr -7 voiddet URf do 118 dr -7 void
A.3 MCNP5 - RBWR Single Assembly
c MCNP Model of RBWR for Serpent Comparisonc Bryan Hermancc CELL CARDSc ccccccccccccccccccccccccccccccccccccccccccccccc
c cccccccc Lower Reflector Universe #101 cccccccec ccccccccccccccccccccccccccccccccccccccccccccce
I 1 -0.736 -3 -1 6 4 2 -5 -8 7 u=101 imp:n=I tmp=5.1704e-8 $ Lower Reflector2 0 3 :1 :-6 :-4 :-2 :5 :8 :-7 u=101 imp:n=0 $ Void
cc cccccccccccccccceccccccccccccccccccccccccccccc cccccccc Lower Reflector Universe #102 ccccccccc cccccccccccccccccccccccccccccccccccccccccccccc
3 0 -3 -1 6 4 2 -5 -9 8 u=102 fill=3 imp:n=14 0 -26 27 -28 29 -30 31 u=3 lat=2 imp:n=1
fill=-10:10 -10:10 0:02 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 $ROW 12 2 2 2 2 2 2 2 2 2 1 1 1 I 1 1 1 1 1 1 2 $ROW 22 2 2 2 2 2 2 2 2 1 1 1 1 1 1 1 1 1 1 1 2 $ROW 32 2 2 2 2 2 2 2 1 1 1 1 1 1 1 1 1 1 1 1 2 $ROW 42 2 2 2 2 2 2 1 1 1 1 1 1 1 1 1 1 1 1 1 2 $ROW 52 2 2 2 2 2 1 1 1 1 1 1 1 1 1 1 1 1 1 1 2 $ROW 62 2 2 2 2 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 2 $ROW 72 2 2 2 1 1 1 1 I 1 1 1 1 1 1 1 1 1 1 1 2 $ROW 82 2 2 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 2 $ROW 92 2 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 I 1 1 2 $ROW 102 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 12 $ROW 112 1 1 1 1 1 1 1 1 1 1 1 I 1 1 1 1 1 1 2 2 $ROW 122 1 1 1 1 1 1 1 1 I 1 1 1 1 1 1 1 1 2 2 2 $ROW 132 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 $ROW 142 1 1 1 1 I 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 $ROW 152 1 1 1 1 1 1 1 I 1 1 1 1 1 1 2 2 2 2 2 2 $ROW 162 1 1 1 1 1 I 1 1 1 1 1 1 1 2 2 2 2 2 2 2 $ROW 172 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 $ROW 182 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 $ROW 192 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 $ROW 202 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 $ROW 21
5 2 -2.394 -33 u=1 imp:n=I tmp=5.1704e-86 3 -6.5 33 -34 u= imp:n=1 tmp=5.1704e-87 1 -0.736 34 u=I imp:n=I tmp=5.1704e-88 1 -0.736 -26 27 -28 29 -30 31 u=2 imp:n= tmp=5.1704e-89 1 -0.736 26 :- 27 :28 : -29 :30 : -31 u=2 imp:n=1 tmp=5.1704e-810 0 3 :1 :-6 :-4 :-2 :5 :9 :-8 u=102 imp:n=0 $ Void
c cccccccccccccccccccccccccccccccccccccccccccccccc cccccccc Lower Blanket I Universe #103 ccccccccc ccccccccccccccccccccccccccccccccccccccccccccccc
11 0 -3 -1 6 4 2 -5 -10 9 u=103 fill=200 imp:n=I12 0 -26 27 -28 29 -30 31 u=200 lat=2 imp:n=I
fill=-10:10 -10:10 0:017 17 17 17 17 17 17 17 17 17 17 17 17 17 17 17 17 17 17 17 17 $ROW 117 17 17 17 17 17 17 17 17 17 10 10 10 10 10 10 10 10 10 10 17 $ROW 217 17 17 17 17 17 17 17 17 10 10 10 10 10 10 10 10 10 10 10 17 $ROW 317 17 17 17 17 17 17 17 10 10 10 10 10 10 10 10 10 10 10 10 17 $ROW 417 17 17 17 17 17 17 10 10 10 10 10 10 10 10 10 10 10 10 10 17 $ROW 517 17 17 17 17 17 10 10 10 10 10 10 10 10 10 10 10 10 10 10 17 $ROW 617 17 17 17 17 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 17 $ROW 717 17 17 17 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 17 $ROW 817 17 17 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 17 $ROW 917 17 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 17 $ROW 1017 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 17 $ROW 1117 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 17 17 $ROW 1217 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 17 17 17 $ROW 1317 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 17 17 17 17 $ROW 1417 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 17 17 17 17 17 $ROW 1517 10 10 10 10 10 10 10 10 10 10 10 10 10 10 17 17 17 17 17 17 $ROW 1617 10 10 10 10 10 10 10 10 10 10 10 10 10 17 17 17 17 17 17 17 $ROW 1717 10 10 10 10 10 10 10 10 10 10 10 10 17 17 17 17 17 17 17 17 $ROW 1817 10 10 10 10 10 10 10 10 10 10 10 17 17 17 17 17 17 17 17 17 $ROW 1917 10 10 10 10 10 10 10 10 10 10 17 17 17 17 17 17 17 17 17 17 $ROW 2017 17 17 17 17 17 17 17 17 17 17 17 17 17 17 17 17 17 17 17 17 $ROW 21
13 4 -10.5 -32 u=0 imp:n=1 tmp=7.7556e-814 0 32 -33 u=10 imp:n=115 3 -6.5 33 -34 u=10 imp:n=l tmp=5.1704e-816 1 -0.721 34 u=10 imp:n=l tmp=5.1704e-817 1 -0.721 -26 27 -28 29 -30 31 u=17 imp:n=l tmp=5.1704e-818 1 -0.721 26 :- 27 :28 :-29 :30 :- 31 u=17 imp:n=I tmp=5.1704e-819 0 3 :1 :-6 :-4 :-2 :5 :10 :-9 u=103 imp:n=0 $ Void
c cccccccccccccccccccccccccccccccccccccccccccccc
c ecccccc Lower Blanket 2 Universe #104 ccccccccc cecccccccccccccccccccccccceccccccccccccccccccc
168
20 0 -3 -1 6 4 2 -5 -11 1021 0 -26 27 -28 29 -30 31
fill=- 10:10 -10:10 0:018181818181818181111
1111
1111
111111
18
18181818181818111111
111111
11
111118
181818181818
11
11
11
11Il11
1111
1810.5 -32
32-6.5 33
0.668 34
1818181818
118
1111Il
1111
1111
111118
181818181111ll1111I1111llll11ll11ll11I111118
181818111111ll111111ll1111111I11111111111118
1818111111111111111111ll1ll11IIl11ll11111118
181111111l1ll1111llll1111111111111111111
1118
18111111ll111111
1111ll11111111111111111818
u=104 fill=201 imp:n=1u=201 lat=2 inip:n=I
18ll111111ll1111111111ll111111111111181818
18ll1111111111I11111111
11111111
11
ll18181818
1811
Is
11
1111
1111
1111
111818181818
181118
111111
111111
1111181818181818
u=ll inp:n= timp=7.7556e-8-33 u=ll imp:n=l
-34 u=ll imp:n=1 tmp=5.1704e-8
u=ll imp:n=I tmp=5.1704e-8
181118111111
11
11
18181818181818
1818
1818
11iil
1818
1111
181818181818181818
1811181111111111
11
1818181818181818181818
$ROW$ROW$ROW$ROW$ROW$ROW$ROW$ROW$ROW$ROW$ROW$ROW$ROW$ROW$ROW$ROW$ROW$ROW$ROW$ROW$ROW
123456789101112131415161718192021
0.668 -26 27 -28 29 -30 31 u=18 imp:n=l tnip=.l'/04e-8-0.668 26 :-27 :28 :-29 :30 :-31 u=l8 imp:n=I tmp=5.1704e-8
3 :1 :-6 :-4 :-2 :5 :11 :-10 u=104 imp:n=0 $ VoidccccccCcccccccCcccccccccccccccccccceccccccC
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29 0 -3 -1 6 4 2 -5 -12 11 u=105 fill=202 imp:n=l30 0 -26 27 -28 29 -30 31 u=202 lat=2 imp:n=I
fill I=-119 19 1919 19 1919 19 1919 19 1919 19 1919 19 1919 19 1919 19 1219 12 1212 12 1212 12 1212 12 1212 12 1212 12 1212 12 1212 12 1212 12 1212 12 1212 12 1212 12 1219 19 1
-10.5
-6.5-0.609-0.609-0.609
0:1019191919191912
121212121212
2 122 122 122 122 122 122 129 19-32323334-26263 :1
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191212121212121212121212121212121212121219
191212121212121212121212121212121212121919
191212121212121212121212121212121212191919
191212121212121212121212121212121219191919
191212121212121212121212121212191919191919
191212121212121212121212121919191919191919
191212121212121212121212191919191919191919
191212121212121212121219191919191919191919
191919191919191919191919191919191919191919
$ROW$ROW$ROW$ROW$ROW$ROWSROW$ROW$ROW$ROW$ROW$ROW$ROW$ROW$ROW$ROW$ROW$ROW$ROW$ROW$ROW
123456789101112131415161718192021
992222222222222222229
u=12 imp:n=I tmp=7.7556e-8-33 u=12 inip:n=l34 u=12 inp:n=l tnip=5.1704e-8u=12 imp:n=l tmp=5.1704e-827 -28 29 -30 31 u=19 imp:n= timp=5.1704e-8-27 :28 : -29 :30 : -31 u=19 imp:n=1 tmp=5.1704e-8
:-6 :-4 :-2 :5 :12 :-11 u=105 inp:n=0 $ Void
1 6 4 2 -5 -13 1227 -28 29 -30 31-10:10 0:037 37 37 37 37 3737 37 37 37 20 2037 37 37 20 20 2037 37 20 20 20 2037 20 20 20 20 2020 20 20 20 20 2020 20 20 20 20 2020 20 20 20 20 2020 20 20 20 20 2020 20 20 20 20 2020 20 20 20 20 2020 20 20 20 20 2020 20 20 20 20 2020 20 20 20 20 2020 20 20 20 20 2020 20 20 20 20 2020 20 20 20 20 2020 20 20 20 20 2020 20 20 20 20 2020 20 20 20 20 3737 37 37 37 37 37
u=106 fill=300 imp:n=lu=300 lat=2 imp:n= l
372020202020202020202020202020202020373737
372020202020202020202020202020202037373737
372020202020202020202020202020373737373737
372020202020202020202020202037373737373737
372020202020202020202020373737373737373737
181818181818181818181818181818181818181818
18181818181818181818111111
111111
1118
18181818181818181811
11111111
11
111118
22232425262728
40'3
0
191919191919191919191919191919191919191919
191919191919191919191212121212121212121219403
110
31323334353637
c ccccCeccccccccccccccccccccccccccccccccccccccccCc ccccccCC Lower Fissile I Universe #106 cceccccee CeCCCCccccccccccccccccccccccccccccccccccccccC
38 039 0
37 3737 3737 3737 3737 3737 3737 3737 3737 3737 3737 2037 2037 2037 2037 2037 2037 2037 2037 2037 2037 37
fill =37 3737 3737 3737 3737 3737 3737 3737 3737 2020 2020 2020 2020 2020 2020 2020 2020 2020 2020 2020 2037 37
-3 --26
-10:1037 3737 3737 3737 3737 3737 3737 2020 2020 2020 2020 2020 2020 2020 2020 2020 2020 2020 2020 2020 2037 37
373737373737373737373737373737373737373737
$ROW I$ROW 2$ROW 3$ROW 4$ROW 5$ROW 6$ROW 7$ROW 8$ROW 9$ROW 10$ROW 11$ROW 12$ROW 13$ROW 14$ROW I5$ROW 16$ROW 17$ROW 18$ROW 19$ROW 20$ROW 21
169
e
cc
40 5 -10.5 -32 u=20 imp:n=1 tmp=1.03408e-741 0 32 -33 u=20 imp:n=142 3 -6.5 33 -34 u=20 imp:n=I tmp=5.1704e-843 1 -0.527 34 u=20 imp:n=1 tmp=5.1704e-844 1 -0.527 -26 27 -28 29 -30 31 u=37 imp:n=1 tmp=5.1704e-845 1 -0.527 26 :-27 :28 :-29 :30 -31 u=37 imp:n=1 tmp=5.1704e-846 0 3 :1 :-6 :-4 :-2 :5 :13 :-12 u=106 imp:n=0 $ Void
c cccccccccccccccccccccccccccccccccccccccccccccccc cccccccc Lower Fissile 2 Universe #107 ccccccccc ccccccccccccccccccccccccccccccccccccccccccccccc
47 0 -3 -1 6 4 2 -5 -14 13 u=107 fill=301 imp:n=148 0 -26 27 -28 29 -30 31 u=301 lat=2 imp:n=1
fill =-10:10 -10:10 0:0383838383838383838383838383838383838383838
383838383838383838382121212121212121212138
383838383838383838212121212121212121212138
383838383838383821212121212121212121212138
383838383838382121212121212121212121212138
383838383838212121212121212121212121212138
383838383821212121212121212121212121212138
383838382121212121212121212121212121212138
383838212121212121212121212121212121212138
383821212121212121212121212121212121212138
382121212121212121212121212121212121212138
382121212121212121212121212121212121213838
382121212121212121212121212121212121383838
382121212121212121212121212121212138383838
382121212121212121212121212121213838383838
382121212121212121212121212121383838383838
382121212121212121212121212138383838383838
382121212121212121212121213838383838383838
382121212121212121212121383838383838383838
382121212121212121212138383838383838383838
383838383838383838383838383838383838383838
$ROW$ROW$ROW$ROW$ROW$ROW$ROW$ROW$ROW$ROW$ROW$ROW$ROW$ROW$ROW$ROW$ROW$ROW$ROW$ROW$ROW
49 5 -10.5 -32 u=21 imp:n=1 tmp=1.03408e-750 0 32 -33 u=21 imp:n=I51 3 -6.5 33 -34 u=21 imp:n=l tmp=5.1704e-852 1 -0.360 34 u=21 imp:n=1 tmp=5.1704e-853 1 -0.360 -26 27 -28 29 -30 31 u=38 imp:n=1 tmp=5.1704e-854 1 -0.360 26 :- 27 :28 : -29 :30 : -31 u=38 imp:n=l tmp=5.1704e-855 0 3 :1 :-6 :-4 :-2 :5 :14 :-13 u=107 imp:n=0 $ Voidccecccccccrcccccccccccccccccccccccccccccccccccccccccccc Lower Fissile 3 Universe #108 cccccccccccccccccccccccccccccccccccccccccccccccccccc cc56 0 -3 -1 6 4 2 -5 -15 14 u=108 fill=302 imp:n=l57 0 -26 27 -28 29 -30 31 u=302 Iat=2 imp: n=1
fill=-10:10 -10:10 0:039 39 39 39 39 39 39 39 39 39 39 39 39 39 39 39 39 39 39 3939 39 39 39 39 39 39 39 39 39 22 22 22 22 22 22 22 22 22 2239 39 39 39 39 39 39 39 39 22 22 22 22 22 22 22 22 22 22 2239 39 39 39 39 39 39 39 22 22 22 22 22 22 22 22 22 22 22 2239 39 39 39 39 39 39 22 22 22 22 22 22 22 22 22 22 22 22 2239 39 39 39 39 39 22 22 22 22 22 22 22 22 22 22 22 22 22 2239 39 39 39 39 22 22 22 22 22 22 22 22 22 22 22 22 22 22 2239 39 39 39 22 22 22 22 22 22 22 22 22 22 22 22 22 22 22 2239 39 39 22 22 22 22 22 22 22 22 22 22 22 22 22 22 22 22 2239 39 22 22 22 22 22 22 22 22 22 22 22 22 22 22 22 22 22 2239 22 22 22 22 22 22 22 22 22 22 22 22 22 22 22 22 22 22 2239 22 22 22 22 22 22 22 22 22 22 22 22 22 22 22 22 22 22 3939 22 22 22 22 22 22 22 22 22 22 22 22 22 22 22 22 22 39 3939 22 22 22 22 22 22 22 22 22 22 22 22 22 22 22 22 39 39 3939 22 22 22 22 22 22 22 22 22 22 22 22 22 22 22 39 39 39 3939 22 22 22 22 22 22 22 22 22 22 22 22 22 22 39 39 39 39 3939 22 22 22 22 22 22 22 22 22 22 22 22 22 39 39 39 39 39 3939 22 22 22 22 22 22 22 22 22 22 22 22 39 39 39 39 39 39 3939 22 22 22 22 22 22 22 22 22 22 22 39 39 39 39 39 39 39 3939 22 22 22 22 22 22 22 22 22 22 39 39 39 39 39 39 39 39 3939 39 39 39 39 39 39 39 39 39 39 39 39 39 39 39 39 39 39 39
58 5 -10.5 -32 u=22 imp:n=1 tmp=1.03408e-759 0 32 -33 u=22 imp:n=160 3 -6.5 33 -34 u=22 imp:n=1 tmp=5.1704e-861 1 -0.290 34 u=22 imp:n=l tmp=5.1704e-862 1 -0.290 -26 27 -28 29 -30 31 u=39 imp:n=I tmp=5.170463 1 -0.290 26 :- 27 :28 : -29 :30 :-31 u=39 imp:n=1 tmp=564 0 3 :1 :-6 :-4 :-2 :5 :15 :-14 u=108 imp:n=0 $
c ccccccccccccccccccccccccccccccccccceccccccCCc cccccccc Internal Blanket I Universe #109 cccccc cccccccccccccccccccccccccccccccccccccccccc
65 066 0
47 4747 4747 4747 4747 4747 4747 4747 4747 4747 4747 40
-3 -1 6 4 2 -5 -16 15-26 27 -28 29 -30 31
fill=-10:10 -10:10 0:04747474747474747404040
4747474747474740404040
4747474747474040404040
4747474747404040404040
4747474740404040404040
4747474040404040404040
4747404040404040404040
4740404040404040404040
4747474747474747474040
4740404040404040404040
393939393939393939393939393939393939393939
$ROW$ROW$ROW$ROW$ROW$ROW$ROW$ROW$ROW$ROW$ROW$ROW$ROW$ROW$ROW$ROW$ROW$ROW$ROW$ROW$ROW
e-8.1704e-8Void
u=109 fill=400 imp:n=1u=400 lat=2 imp:n=l
4740404040404040404040
4740404040404040404040
4740404040404040404040
4740404040404040404040
4740404040404040404040
4740404040404040404040
4740404040404040404040
4740404040404040404040
4747474747474747474747
$ROW$ROW$ROW$ROW$ROW$ROW$ROW$ROW$ROW$ROW$ROW
1234567891011
170
ccc
47 40 40 40 40 40 40 40 40 40 40 40 40 40 40 40 40 40 40 47 47 $ROW 1247 40 40 40 40 40 40 40 40 40 40 40 40 40 40 40 40 40 47 47 47 $ROW 1347 40 40 40 40 40 40 40 40 40 40 40 40 40 40 40 40 47 47 47 47 $ROW 1447 40 40 40 40 40 40 40 40 40 40 40 40 40 40 40 47 47 47 47 47 $ROW 1547 40 40 40 40 40 40 40 40 40 40 40 40 40 40 47 47 47 47 47 47 $ROW 1647 40 40 40 40 40 40 40 40 40 40 40 40 40 47 47 47 47 47 47 47 $ROW 1747 40 40 40 40 40 40 40 40 40 40 40 40 47 47 47 47 47 47 47 47 $ROW 1847 40 40 40 40 40 40 40 40 40 40 40 47 47 47 47 47 47 47 47 47 $ROW 1947 40 40 40 40 40 40 40 40 40 40 47 47 47 47 47 47 47 47 47 47 $ROW 2047 47 47 47 47 47 47 47 47 47 47 47 47 47 47 47 47 47 47 47 47 $ROW 21
67 4 -10.5 -32 u=40 imp:n=1 timp=7.7556e-868 0 32 -33 u=40 imp:n=69 3 -6.5 33 -34 u=40 imp:n=l tnip=5.1704e-870 I -0.266 34 u=40 imp:n=l tmp=5.1704e-871 1 -0.266 -26 27 -28 29 -30 31 u=47 imp:n=1 tmp=5.1704e-872 1 -0.266 26 : -27 :28 : -29 :30 : -31 u=47 imp:n= tmp=5.1704e-873 0 3 :1 :-6 :-4 :-2 :5 :16 :-15 u=109 imp:n=0 $ Void
c ccccccccccccccccccccccccccccccccccCCccccccCCCCC
c ccccccc Internal Blanket 2 Universe #110 cccccc cccccccccccccccCccCcccccccCCCccccccccccccccceC
74 0 -3 -1 6 4 2 -5 -17 16 u=110 fill=401 imp:n=l75 0 -26 27 -28 29 -30 31 u=401 lat=2 imp:n=1
fill=-10:10 -10:10 0:048 48 48 48 48 48 48 48 48 48 48 48 48 48 48 48 48 48 48 48 48 $ROW 1
48 48 48 48 48 48 48 48 48 48 41 41 41 41 41 41 41 41 41 41 48 $ROW 248 48 48 48 48 48 48 48 48 41 41 41 41 41 41 41 41 41 41 41 48 $ROW 348 48 48 48 48 48 48 48 41 41 41 41 41 41 41 41 41 41 41 41 48 $ROW 448 48 48 48 48 48 48 41 41 41 41 41 41 41 41 41 41 41 41 41 48 SROW 548 48 48 48 48 48 41 41 41 41 41 41 41 41 41 41 41 41 41 41 48 $ROW 648 48 48 48 48 41 41 41 41 41 41 41 41 41 41 41 41 41 41 41 48 $ROW 748 48 48 48 41 41 41 41 41 41 41 41 41 41 41 41 41 41 41 41 48 $ROW 848 48 48 41 41 41 41 41 41 41 41 41 41 41 41 41 41 41 41 41 48 $ROW 948 48 41 41 41 41 41 41 41 41 41 41 41 41 41 41 41 41 41 41 48 $ROW 1048 41 41 41 41 41 41 41 41 41 41 41 41 41 41 41 41 41 41 41 48 $ROW 1148 41 41 41 41 41 41 41 41 41 41 41 41 41 41 41 41 41 41 48 48 $ROW 1248 41 41 41 41 41 41 41 41 41 41 41 41 41 41 41 41 41 48 48 48 $ROW 1348 41 41 41 41 41 41 41 41 41 41 41 41 41 41 41 41 48 48 48 48 $ROW 14
48 41 41 41 41 41 41 41 41 41 41 41 41 41 41 41 48 48 48 48 48 SROW 1548 41 41 41 41 41 41 41 41 41 41 41 41 41 41 48 48 48 48 48 48 $ROW 1648 41 41 41 41 41 41 41 41 41 41 41 41 41 48 48 48 48 48 48 48 $ROW 1748 41 41 41 41 41 41 41 41 41 41 41 41 48 48 48 48 48 48 48 48 $ROW 1848 41 41 41 41 41 41 41 41 41 41 41 48 48 48 48 48 48 48 48 48 $ROW 1948 41 41 41 41 41 41 41 41 41 41 48 48 48 48 48 48 48 48 48 48 $ROW 2048 48 48 48 48 48 48 48 48 48 48 48 48 48 48 48 48 48 48 48 48 $ROW 21
76 4 -10.5 -32 u=41 imp:n=l tmp=7.7556e-877 0 32 -33 u=41 imp:n=I78 3 -6.5 33 -34 u=41 inip:n=I tmp=5.1704e-879 1 -0.253 34 u=41 imp:n= timp=5.1704e-880 1 -0.253 -26 27 -28 29 -30 31 u=48 imp:n=l tmp=5.1704e-881 1 -0.253 26 :- 27 :28 : -29 :30 :- 31 u=48 imp:n=I tmp=5.1704e-882 0 3 :1 :-6 :-4 :-2 :5 :17 :-16 u=110 inip:n=0 $ Void
c cccccccccccccccccCCCCcccccccccccccccccccccceCC
ce cccccc Internal Blanket 3 Universe #111 cccccc ccccccccccccccccccccccccccccccccccccccccccccC
83 0 -3 -1 6 4 2 -5 -18 17 u=111 fill=402 imp:n=I84 0 -26 27 -28 29 -30 3 1 u=402 lat=2 imp:n=l
fill=-10:10 -10:10 0:049 49 49 49 49 49 49 49 49 49 49 49 49 49 49 49 49 49 49 49 49 $ROW 1
49 49 49 49 49 49 49 49 49 49 42 42 42 42 42 42 42 42 42 42 49 $ROW 2
49 49 49 49 49 49 49 49 49 42 42 42 42 42 42 42 42 42 42 42 49 $ROW 3
49 49 49 49 49 49 49 49 42 42 42 42 42 42 42 42 42 42 42 42 49 $ROW 4
49 49 49 49 49 49 49 42 42 42 42 42 42 42 42 42 42 42 42 42 49 $ROW 549 49 49 49 49 49 42 42 42 42 42 42 42 42 42 42 42 42 42 42 49 $ROW 649 49 49 49 49 42 42 42 42 42 42 42 42 42 42 42 42 42 42 42 49 $ROW 7
49 49 49 49 42 42 42 42 42 42 42 42 42 42 42 42 42 42 42 42 49 $ROW 849 49 49 42 42 42 42 42 42 42 42 42 42 42 42 42 42 42 42 42 49 $ROW 949 49 42 42 42 42 42 42 42 42 42 42 42 42 42 42 42 42 42 42 49 $ROW 1049 42 42 42 42 42 42 42 42 42 42 42 42 42 42 42 42 42 42 42 49 $ROW 11
49 42 42 42 42 42 42 42 42 42 42 42 42 42 42 42 42 42 42 49 49 $ROW 12
49 42 42 42 42 42 42 42 42 42 42 42 42 42 42 42 42 42 49 49 49 SROW 1349 42 42 42 42 42 42 42 42 42 42 42 42 42 42 42 42 49 49 49 49 $ROW 14
49 42 42 42 42 42 42 42 42 42 42 42 42 42 42 42 49 49 49 49 49 $ROW 1549 42 42 42 42 42 42 42 42 42 42 42 42 42 42 49 49 49 49 49 49 $ROW 1649 42 42 42 42 42 42 42 42 42 42 42 42 42 49 49 49 49 49 49 49 $ROW 1749 42 42 42 42 42 42 42 42 42 42 42 42 49 49 49 49 49 49 49 49 SROW 1849 42 42 42 42 42 42 42 42 42 42 42 49 49 49 49 49 49 49 49 49 $ROW 1949 42 42 42 42 42 42 42 42 42 42 49 49 49 49 49 49 49 49 49 49 SROW 20
49 49 49 49 49 49 49 49 49 49 49 49 49 49 49 49 49 49 49 49 49 $ROW 21
85 4 -10.5 -32 u=42 imp:n=l tmp=7.7556e-886 0 32 -33 u=42 imp:n=I87 3 -6.5 33 -34 u=42 imp:n=l tmp=5.1704e-888 I -0.226 34 u=42 imp:n=l imp=5.1704e-889 1 -0.226 -26 27 -28 29 -30 31 u=49 imp:n= tmp=5.1704e-890 1 -0.226 26 :-27 :28 :-29 :30 :-31 u=49 imp:n=1 tmp=5.1704e-891 0 3 :1 :-6 :-4 :-2 :5 :18 :-17 u=ll imp:n=0 $ Void
c ccccccccccccccccccccccccccccccccccccccccccCCCC
c cccccccc Upper Fissile I Universe #112 cccccccc
c ccccccccccccccccccccccccccecccccccccCCccccccc92 0 -3 -1 6 4 2 -5 -19 18 u=112 fill=500 imp:n=93 0 -26 27 -28 29 -30 31 u=500 lat=2 imp:n=1
fill=-10:10 -10:10 0:067 67 67 67 67 67 67 67 67 67 67 67 67 67 67 67 67 67 67 67 67 $ROW 1
171
67 67 67 67 67 67 67 67 67 67 50 50 50 50 50 50 50 50 50 50 67 $ROW 267 67 67 67 67 67 67 67 67 50 50 50 50 50 50 50 50 50 50 50 67 $ROW 367 67 67 67 67 67 67 67 50 550 50 50 50 50 50 50 50 50 50 67 $ROW 467 67 67 67 67 67 67 50 50 50 50 50 50 50 50 50 50 50 50 50 67 $ROW 567 67 67 67 67 67 50 50 50 50 50 50 50 50 50 50 50 50 50 50 67 $ROW 667 67 67 67 67 50 50 50 50 50 50 50 50 50 50 50 50 50 50 50 67 $ROW 767 67 67 67 50 50 50 50 50 50 50 50 50 50 50 50 50 50 50 50 67 $ROW 867 67 67 50 50 50 50 50 50 50 50 50 50 50 50 50 50 50 50 50 67 $ROW 967 67 50 50 50 50 50 50 50 50 50 50 50 50 50 50 50 50 50 50 67 $ROW 1067 50 50 50 50 50 50 50 50 50 50 50 50 50 50 50 50 50 50 50 67 $ROW 167 50 50 50 50 50 50 50 50 50 50 50 50 50 50 50 50 50 50 67 67 $ROW 1267 50 50 50 50 50 50 50 50 50 50 50 50 50 50 50 50 50 67 67 67 $ROW 1367 50 50 50 50 50 50 50 50 50 50 50 50 50 50 50 50 67 67 67 67 $ROW 1467 50 50 50 50 50 50 50 50 50 50 50 50 50 50 50 67 67 67 67 67 $ROW 1567 50 50 50 50 50 50 50 50 50 50 50 50 50 50 67 67 67 67 67 67 $ROW 1667 50 50 50 50 50 50 50 50 50 50 50 50 50 67 67 67 67 67 67 67 $ROW 1767 50 50 50 50 50 50 50 50 50 50 50 50 67 67 67 67 67 67 67 67 $ROW 1867 50 50 50 50 50 50 50 50 50 50 50 67 67 67 67 67 67 67 67 67 $ROW 1967 50 50 50 50 50 50 50 50 50 50 67 67 67 67 67 67 67 67 67 67 $ROW 2067 67 67 67 67 67 67 67 67 67 67 67 67 67 67 67 67 67 67 67 67 $ROW 21
94 6 -10.5 -32 u=50 imp:n=1 tmp=1.03408e-795 0 32 -33 u=50 imp:n=196 3 -6.5 33 -34 u=50 imp:n=1 tmp=5.1704e-897 1 -0.219 34 u=50 imp:n=1 tmp=5.1704e-898 1 -0.219 -26 27 -28 29 -30 31 u=67 imp:n=1 tmp=5.1704e-899 1 -0.219 26 -27 :28 :-29 :30 -31 u=67 imp:n=1 tmp=5.1704e-8
100 0 3 :1 :-6 :-4 :-2 :5 :19 :-18 u=112 imp:n=0 $ Voidc ccccccccccccccccccccccccccccccccccccccccccccccc
c eccccccc Upper Fissile 2 Universe #113 ccccccccc cccccccccccccccccccccccccccccccccccccccccccccC
101 0 -3 1 6 4 2 -5 -20 19 u=113 fill=501 imp:n=1102 0 -26 27 -28 29 -30 31 u=501 lat=2 imp:n=1
fill=-10:10 -10:10 0:068 68 68 68 68 68 68 68 68 68 68 68 68 68 68 68 68 68 68 68 68 $ROW 168 68 68 68 68 68 68 68 68 68 51 51 51 51 51 51 51 51 51 51 68 $ROW 268 68 68 68 68 68 68 68 68 51 51 51 51 51 51 51 51 51 51 51 68 $ROW 368 68 68 68 68 68 68 68 51 51 51 51 51 51 51 51 51 51 51 51 68 $ROW 468 68 68 68 68 68 68 51 51 51 51 51 51 51 51 51 51 51 51 51 68 $ROW 568 68 68 68 68 68 51 51 51 51 51 51 51 51 51 51 51 51 51 51 68 $ROW 668 68 68 68 68 51 51 51 51 51 51 51 51 51 51 51 51 51 51 51 68 $ROW 768 68 68 68 51 51 51 51 51 51 51 51 51 51 51 51 51 51 51 51 68 $ROW 868 68 68 51 51 51 51 51 51 51 51 51 51 51 51 51 51 51 51 51 68 $ROW 968 68 51 51 51 51 51 51 51 51 51 51 51 51 51 51 51 51 51 51 68 $ROW 1068 51 51 51 1151151 51 151 51 51 51 151 51 51 51 68 $ROW 1168 51 51 51 51 51 51 51 51 51 51 51 51 51 51 51 51 51 51 68 68 $ROW 1268 51 51 51 51 51 51 51 51 51 51 51 51 51 51 51 51 51 68 68 68 $ROW 1368 51 51 51 51 51 51 51 51 51 51 51 51 51 51 51 51 68 68 68 68 $ROW 1468 51 51 51 51 51 51 51 51 51 51 51 51 51 51 51 68 68 68 68 68 $ROW 1568 51 51 51 51 51 51 51 51 51 51 51 51 51 51 68 68 68 68 68 68 $ROW 1668 51 51 51 51 51 51 51 51 51 51 51 51 51 68 68 68 68 68 68 68 $ROW 1768 51 51 51 51 51 51 51 51 51 51 51 51 68 68 68 68 68 68 68 68 $ROW 1868 51 51 51 51 51 51 51 51 51 51 51 68 68 68 68 68 68 68 68 68 $ROW 1968 51 51 51 51 51 51 51 51 51 51 68 68 68 68 68 68 68 68 68 68 $ROW 2068 68 68 68 68 68 68 68 68 68 68 68 68 68 68 68 68 68 68 68 68 $ROW 21
103 6 -10.5 -32 u=51 imp:n=1 tmp=1.03408e-7104 0 32 -33 u=51 imp:n=105 3 -6.5 33 -34 u=51 imp:n=1 tmp=5.1704e-8106 1 -0.190 34 u=51 imp:n=1 tmp=5.1704e-8107 1 -0.190 -26 27 -28 29 -30 31 u=68 imp:n=1 tmp=5.1704e-8108 1 -0.190 26 :- 27 :28 : -29 :30 :- 31 u=68 imp:n=1 tmp=5.1704e-8109 0 3 :1 :-6 :-4 :-2 :5 :20 :-19 u=113 imp:n=0 $ Void
c ccccccccccccccccccccccccccccccccccccccccccccccc
c cccccccc Upper Fissile 3 Universe #114 ccccccccc cccccccccccccccccccccccccccccccccccccccccccccc
110 0 -3 -1 6 4 2 -5 -21 20 u=114 fill=502 imp:n=I111 0 -26 27 -28 29 -30 31 u=502 lat=2 imp:n=1
fill=-10:10 -10:10 0:069 69 69 69 69 69 69 69 69 69 69 69 69 69 69 69 69 69 69 69 69 $ROW 169 69 69 69 69 69 69 69 69 69 52 52 52 52 52 52 52 52 52 52 69 $ROW 269 69 69 69 69 69 69 69 69 52 52 52 52 52 52 52 52 52 52 52 69 $ROW 369 69 69 69 69 69 69 69 52 52 52 52 52 52 52 52 52 52 52 52 69 $ROW 469 69 69 69 69 69 69 52 52 52 52 52 52 52 52 52 52 52 52 52 69 $ROW 569 69 69 69 69 69 52 52 52 52 52 52 52 52 52 52 52 52 52 52 69 $ROW 669 69 69 69 69 52 52 52 52 52 52 52 52 52 52 52 52 52 52 52 69 $ROW 769 69 69 69 52 52 52 52 52 52 52 52 52 52 52 52 52 52 52 52 69 $ROW 869 69 69 52 52 52 52 52 52 52 52 52 52 52 52 52 52 52 52 52 69 $ROW 969 69 52 52 52 52 52 52 52 52 52 52 52 52 52 52 52 52 52 52 69 $ROW 1069 52 52 52 52 52 252 52 52 52 52 52 52 52 52 52 52 52 52 69 $ROW 1169 52 52 52 52 52 52 52 52 52 52 52 52 52 52 52 52 52 52 69 69 $ROW 1269 52 52 52 52 52 52 52 52 52 52 52 52 52 52 52 52 52 69 69 69 $ROW 1369 52 52 52 52 52 52 52 52 52 52 52 52 52 52 52 52 69 69 69 69 $ROW 1469 52 52 52 52 52 52 52 52 52 52 52 52 52 52 52 69 69 69 69 69 $ROW 1569 52 52 52 52 52 52 52 52 52 52 52 52 52 52 69 69 69 69 69 69 $ROW 1669 52 52 52 52 52 52 52 52 52 52 52 52 52 69 69 69 69 69 69 69 $ROW 1769 52 52252 52 252 52 52 52 52 69 69 69 69 69 69 69 69 $ROW 1869 52 52 52 52 52 52 52 52 52 52 52 69 69 69 69 69 69 69 69 69 $ROW 1969 52 52 52 52 52 52 52 52 52 52 69 69 69 69 69 69 69 69 69 69 $ROW 2069 69 69 69 69 69 69 69 69 69 69 69 69 69 69 69 69 69 69 69 69 $ROW 21
112 6 -10.5 -32 u=52 imp:n= tmp=1.03408e-7113 0 32 -33 u=52 imp:n=1114 3 -6.5 33 -34 u=52 imp:n=1 tmp=5.1704e-8115 1 -0.169 34 u=52 imp:n=1 tmp=5.1704e-8
172
116 1 -0.169 -26 27 -28 29 -30 31 u=69 imp:n=1 tmp=5.1704e-8117 1 -0.169 26 :-27 :28 :-29 :30 :-31 u=69 imp:n=1 tmp=5.1704e-8118 0 3 :1 :-6 :-4 :-2 :5 :21 :-20 u=114 imp:n=0 $ Void
c cccccccccccccccccccccccccccccCccccccccccccCCc cccccccc Upper Blanket I Universe #115 ccccccccC ccccccccccccccccccccccccccccccccccccccccccc
119 0 -3 -1 6 4 2 -5 -22 21 a=115 fill=600 imp:n=1120 0 -26 27 -28 29 -30 31 u=600 lat=2 imp:n= I
fill=-10:10 -10:10 0:077 77 77 77 77 77 77 77 77 77 77 77 77 77 77 77 77 77 77 77 77 $ROW I77 77 77 77 77 77 77 77 77 77 70 70 70 70 70 70 70 70 70 70 77 $ROW 277 77 77 77 77 77 77 77 77 70 70 70 70 70 70 70 70 70 70 70 77 $ROW 377 77 77 77 77 77 77 77 70 70 70 70 70 70 70 70 70 70 70 70 77 $ROW 477 77 77 77 77 77 77 70 70 70 70 70 70 70 70 70 70 70 70 70 77 $ROW 577 77 77 77 77 77 70 70 70 70 70 70 70 70 70 70 70 70 70 70 77 $ROW 677 77 77 77 77 70 70 70 70 70 70 70 70 70 70 70 70 70 70 70 77 $ROW 777 77 77 77 70 70 70 70 70 70 70 70 70 70 70 70 70 70 70 70 77 $ROW 877 77 77 70 70 70 70 70 70 70 70 70 70 70 70 70 70 70 70 70 77 $ROW 977 77 70 70 70 70 70 70 70 70 70 70 70 70 70 70 70 70 70 70 77 $ROW 1077 70 70 70 70 70 70 70 70 70 70 70 70 70 70 70 70 70 70 70 77 $ROW 1177 70 70 70 70 70 70 70 70 70 70 70 70 70 70 70 70 70 70 77 77 $ROW 1277 70 70 70 70 70 70 70 70 70 70 70 70 70 70 70 70 70 77 77 77 $ROW 1377 70 70 70 70 70 70 70 70 70 70 70 70 70 70 70 70 77 77 77 77 $ROW 1477 70 70 70 70 70 70 70 70 70 70 70 70 70 70 70 77 77 77 77 77 $ROW 1577 70 70 70 70 70 70 70 70 70 70 70 70 70 70 77 77 77 77 77 77 $ROW 1677 70 70 70 70 70 70 70 70 70 70 70 70 70 77 77 77 77 77 77 77 $ROW 1777 70 70 70 70 70 70 70 70 70 70 70 70 77 77 77 77 77 77 77 77 $ROW 1877 70 70 70 70 70 70 70 70 70 70 70 77 77 77 77 77 77 77 77 77 $ROW 1977 70 70 70 70 70 70 70 70 70 70 77 77 77 77 77 77 77 77 77 77 $ROW 2077 77 77 77 77 77 77 77 77 77 77 77 77 77 77 77 77 77 77 77 77 $ROW 21
121 4 -10.5 -32 u=70 imp:n=l tmp=7.7556e-8122 0 32 -33 u=70 imp:n=I123 3 -6.5 33 -34 u=70 imp:n=1 tnip=5.1704e-8124 1 -0.156 34 u=70 imp:n=l Itnip=5.1704e-8125 1 -0.156 -26 27 -28 29 -30 31 u=77 imp:n= tmp=5.1704e-8126 1 -0.156 26 :- 27 :28 : -29 :30 :- 31 u=77 imp:n=1 tmp=5.1704e-8127 0 3 :1 :-6 :-4 :-2 :5 :22 :-21 u=115 imp:n=0 $ Void
c ccccccccccccccccCccccccccccccccccccccccCcccCCCC
c cccccCCC Upper Blanket 2 Universe #116 cccccccc
c ccccccccccccccccccCcccccccccccccccccccccccccCC128 0 -3 -1 6 4 2 -5 -23 22 u=116 fill=601 imp:n=1129 0 -26 27 -28 29 -30 31 u=601 lat=2 imp: n=1
fill=-10:10 -10:10 0:078 78 78 78 78 78 78 78 78 78 78 78 78 78 78 78 78 78 78 78 78 $ROW 178 78 78 78 78 78 78 78 78 78 71 71 71 71 71 71 71 71 71 71 78 $ROW 278 78 78 78 78 78 78 78 78 71 71 71 71 71 71 71 71 71 71 71 78 $ROW 378 78 78 78 78 78 78 78 71 71 71 71 71 71 71 71 71 71 71 71 78 $ROW 478 78 78 78 78 78 78 71 71 71 71 71 71 71 71 71 71 71 71 71 78 $ROW 578 78 78 78 78 78 71 71 71 71 71 71 71 71 71 71 71 71 71 71 78 $ROW 678 78 78 78 78 71 71 71 71 71 71 71 71 71 71 71 71 71 71 71 78 $ROW 778 78 78 78 71 71 71 71 71 71 71 71 71 71 71 71 71 71 71 71 78 $ROW 878 78 78 71 71 71 71 71 71 71 71 71 71 71 71 71 71 71 71 71 78 $ROW 978 78 71 71 71 71 71 71 71 71 71 71 71 71 71 71 71 71 71 71 78 $ROW 1078 71 71 71 71 71 71 71 71 71 71 71 71 71 71 71 71 71 71 71 78 $ROW 1178 71 71 71 71 71 71 71 71 71 71 71 71 71 71 71 71 71 71 78 78 $ROW 12
78 71 71 71 71 71 71 71 71 71 71 71 71 71 71 71 71 71 78 78 78 $ROW 1378 71 71 71 71 71 71 71 71 71 71 71 71 71 71 71 71 78 78 78 78 $ROW 1478 71 71 71 71 71 71 71 71 71 71 71 71 71 71 71 78 78 78 78 78 $ROW 1578 71 71 71 71 71 71 71 71 71 71 71 71 71 71 78 78 78 78 78 78 $ROW 1678 71 71 71 71 71 71 71 71 71 71 71 71 71 78 78 78 78 78 78 78 $ROW 1778 71 71 71 71 71 71 71 71 71 71 71 71 78 78 78 78 78 78 78 78 $ROW 1878 71 71 71 71 71 71 71 71 71 71 71 78 78 78 78 78 78 78 78 78 $ROW 1978 71 71 71 71 71 71 71 71 71 71 78 78 78 78 78 78 78 78 78 78 $ROW 2078 78 78 78 78 78 78 78 78 78 78 78 78 78 78 78 78 78 78 78 78 $ROW 21
130 4 -10.5 -32 u=71 imp:n=l tnip=7.7556e-8131 0 32 -33 u=71 imp:n=l132 3 -6.5 33 -34 u=71 imp:n=I tmp=5.1704e-8133 1 -0.152 34 u=71 imp:n=I tmp=5.1704e-8134 1 -0.152 -26 27 -28 29 -30 31 u=78 inip:n=l tmtp=5.1704e-8135 1 -0.152 26 :- 27 :28 : -29 :30 :- 31 u=78 imp:n=l tmp=5.1704e-8136 0 3 :1 :-6 :-4 :-2 :5 :23 :-22 u=116 imp:n=0 $ Void
c cccccccccccccccccccccccccccccccccccccCccccCCC
c ccccccC Upper Blanket 3 Universe #117 cccccccC
c cccccccCccccccccccccccccccccccccccccccccccccCC137 0 -3 -1 6 4 2 -5 -24 23 u=117 fill=602 imp:n=I138 0 -26 27 -28 29 -30 31 u=602 lat=2 imp:n=I
fill=-10:10 -10:10 0:079 79 79 79 79 79 79 79 79 79 79 79 79 79 79 79 79 79 79 79 79 $ROW 179 79 79 79 79 79 79 79 79 79 72 72 72 72 72 72 72 72 72 72 79 $ROW 279 79 79 79 79 79 79 79 79 72 72 72 72 72 72 72 72 72 72 72 79 $ROW 3
79 79 79 79 79 79 79 79 72 72 72 72 72 72 72 72 72 72 72 72 79 $ROW 479 79 79 79 79 79 79 72 72 72 72 72 72 72 72 72 72 72 72 72 79 $ROW 579 79 79 79 79 79 72 72 72 72 72 72 72 72 72 72 72 72 72 72 79 $ROW 679 79 79 79 79 72 72 72 72 72 72 72 72 72 72 72 72 72 72 72 79 $ROW 779 79 79 79 72 72 72 72 72 72 72 72 72 72 72 72 72 72 72 72 79 SROW 879 79 79 72 72 72 72 72 72 72 72 72 72 72 72 72 72 72 72 72 79 $ROW 979 79 72 72 72 72 72 72 72 72 72 72 72 72 72 72 72 72 72 72 79 $ROW 1079 72 72 72 72 72 72 72 72 72 72 72 72 72 72 72 272 72 72 79 $ROW 1179 72 72 72 72 72 72 72 72 72 72 72 72 72 72 72 72 72 72 79 79 $ROW 1279 72 72 72 72 72 72 72 72 72 72 72 72 72 72 72 72 72 79 79 79 $ROW 1379 72 72 72 72 72 72 72 72 72 72 72 72 72 72 72 72 79 79 79 79 $ROW 1479 72 72 72 72 72 72 72 72 72 72 72 72 72 72 72 79 79 79 79 79 $ROW 15
173
79 72 72 72 72 72 72 72 72 72 72 72 72 72 72 79 79 79 79 79 79 $ROW79 72 72 72 72 72 72 72 72 72 72 72 72 72 79 79 79 79 79 79 79 $ROW79 72 72 72 72 72 72 72 72 72 72 72 72 79 79 79 79 79 79 79 79 $ROW79 72 72 72 72 72 72 72 72 72 72 72 79 79 79 79 79 79 79 79 79 $ROW79 72 72 72 72 72 72 72 72 72 72 79 79 79 79 79 79 79 79 79 79 $ROW79 79 79 79 79 79 79 79 79 79 79 79 79 79 79 79 79 79 79 79 79 $ROW
139 4 -10.5 -32 u=72 imp:n=l tmp=7.7556e-8140 0 32 -33 u=72 imp:n=l141 3 -6.5 33 -34 u=72 imp:n=1 tmp=5.1704e-8142 1 -0.150 34 u=72 imp:n=I tmp=5.1704e-8143 1 -0.150 -26 27 -28 29 -30 31 u=79 imp:n=1 tmp=5.1704e-8144 1 -0.150 26 : -27 :28 : -29 :30 : -31 u=79 imp:n=I tmp=5.1704e-8145 0 3 :1 :-6 :-4 :-2 :5 :24 :-23 u=117 imp:n=0 $ Voidccccccccccccccccccccccccccccccccccccccccccccccc
cccccccC Upper Reflector Universe #118 cccccccccccCccccccccccccccccccccccccccccccCccCCCCceccce
146 0147 0
5 5 55 5 55 5 55 5 55 5 55 5 55 5 55 5 55 5 55 5 45 4 45 4 45 4 45 4 45 4 45 4 45 4 45 4 45 4 45 4 45 5 5
148 0149 3
-3 -1 6 4-26 27 -28
fill=- 10:10 -10:105 5 5 5 5 5 5 5 55 5 5 5 5 5 5 4 45 5 5 5 5 5 4 4 45 5 5 5 5 4 4 4 45 5 5 5 4 4 4 4 45 5 5 4 4 4 4 4 45 5 4 4 4 4 4 4 45 4 4 4 4 4 4 4 44 4 4 4 4 4 4 4 44 4 4 4 4 4 4 4 44 4 4 4 4 4 4 4 44 4 4 4 4 4 4 4 44 4 4 4 4 4 4 4 44 4 4 4 4 4 4 4 44 4 4 4 4 4 4 4 44 4 4 4 4 4 4 4 44 4 4 4 4 4 4 4 44 4 4 4 4 4 4 4 44 4 4 4 4 4 4 4 44 4 4 4 4 4 4 4 55 5 5 5 5 5 5 5 5
-33 u=4 imp:n=I-6.5 33 -34 u=
2 -5 -25 2429 -30 310:0
5 5 5 5 5 54 4 4 4 4 44 4 4 4 4 44 4 4 4 4 44 4 4 4 4 44 4 4 4 4 44 4 4 4 4 44 4 4 4 4 44 4 4 4 4 44 4 4 4 4 44 4 4 4 4 4
4444455555
4444555555
4445555555
4444444555
4444445555
4 im
4455555555
161718192021
o=118 fill=6 imp:n=lu=6 lat=2 imp:n=l
$ROW I$ROW 2$ROW 3$ROW 4$ROW 5$ROW 6$ROW 7$ROW 8$ROW 9$ROW 10$ROW II$ROW 12SROW 13$ROW 14$ROW 15$ROW 16$ROW 17$ROW 18$ROW 19$ROW 20SROW 21
4555555555
p:n= tmp=5.1704e-8150 1 -0.149 34 u=4 imp:n=1 tmp=5.1704e-8151 1 -0.149 -26 27 -28 29 -30 31 u=5 imp:n=1152 1 -0.149 26 :-27 :28 :-29 :30 :-31 u=5 imp153 0 3 :1 :-6 :-4 :-2 :5 :25 :-24 u=11
c cccccccccccccccccccccccccccccccccccccccccccccC
c cccccccc Attach 2D Together Universe 0 ccccccccc ccccccccccccccccccccccccccccccccccccccccccccccc
0000000000000000000
-3-3-3-3-3-3-3-3-3-3-3-3-3-3-3-3-3-33:
-1 6-1 6-1 6-1 6-1 6-1 6-1 6-1 6-1 6-1 6-1 61 6
-1 61 6
-1 61 61 61 61:-
4 24 24 24 24 24 24 24 24 24 24 24 24 24 24 24 24 24 2-6:
-5-5-5-5-5-5-5-5-5--5-5-5-5-5-5-5-5--5
-4:
-8 7-9 8-10 9-11 10-12 11-13 12-14 13-15 14-16 15-17 16-18 17-19 18-20 19-21 20-22 21-23 22-24 23-25 24-2: 5:
u=0u=0u=o0u=o0u=Ou=0u=0u=0u=Ou=o0u=o0u=0u=0u=0u=Ou=o0u=o0u=25:
imp: n=1imp:n=imp:n=imp:n=Iimp: n=1imp: n=1imp:n=imp:n=Iimp: n=Iimp: n=Iimp: n=1imp: n=imp:n=1Iimp: n=1imp: n=Iimp:n=1imp:n=Iimp: n=
fill =101fill=102fill=103fill=104fill=105fill=106fill=107fill=108fill=109fill 110fill-Illfill 2 I 12fill=113fill =114fill =115fill =116fill =117fill =118
-7 u=0 imp:n=0 $
tmp=5.1704e-8I:n= tmp=5.1704e-8
8 imp:n=0 $ Void
$ Lower Reflector 1$ Lower Reflector 2$ Lower Blanket 1$ Lower Blanket 2$ Lower Blanket 3$ Lower Fissile I$ Lower Fissile 2$ Lower Fissile 3$ Internal Blanket I$ Internal Blanket 2$ Interanl Blanket 3$ Lower Fissile I$ Lower Fissile 2$ Lower Fissile 3$ Upper Blanket I$ Upper Blanket 2$ Upper Blanket 3$ Lower ReflectorVoid Outside Radially Reflective
c SURFACE CARDSc
*1 px 9.495*2 px -9.495
p 0.p 0.p -p -
pz --3pz -7pz 0pz 7.pz 15pz 18pz 23pz 35pz 40pz 47pz 78pz 85pz 90pz 10pz 10
5 0.86602540378444 0 9.4955 0.86602540378444 0 -9.4950.51.50
75.5
.4
.4
.4
.4
.4
.438
0.86602540378444 0 9.4950.86602540378444 0 -9.495
154155156157158159160161162163164165166167168169170171172
*3*4*5*6789101112131415161718192021
174
ccc
22 pz 11023 pz 11824 pz 12025 pz 15026 p 0.86602540378444 -0.5 0 0.5727 p 0.86602540378444 -0.5 0 -0.5728 p 0.86602540378444 0.5 0 0.5729 p 0.86602540378444 0.5 0 -0.5730 py 0.5731 py -0.5732 cz 0.437533 cz 0.45534 cz 0.505
c DATA and MATERIAL CARDScmode nc Waterml 1001.71c 2
8016.71cintl lwtr.16tc B4C in Lower Reflector Layer 2m2 5010.71c -0.693962
50 lt.71 c -0.0771076000.71 c -0.228931
c Cladding on Blanket Rodsn3 40090.71 c Ic Blanket Depleted U02m4 8016.72c -0.118466
92235.72c -2.203834e-392238.72c -0.87933
c Lower Fissile Zone Materialsm5 8016.73c -0.118182
92235.73c -l.394712e-392238.73c -0.55649093237.73c -1.619666e-394238.73c -9.717994e-394239.73c -0.14253194240.73c -0.11694094241.73c -0.01619794242.73c -0.01587395241.73c -0.01198695242.73c -3.239331e-495243.73c -4.211131e -396244.73c -3.239331e -396245.73c -9.717994e-496246.73c -3.239331 e-4
c Upper Fissile Zone Materialsm6 8016.73c -0.118182
92235.73c -1.394712e-392238.73c -0.55649093237.73c -1.619666e-394238.73c -9.717994e-394239.73c -0.14253194240.73c -0.11694094241.73c -0.01619794242.73c -0.01587395241.73c -0.01198695242.73c -3.239331e-495243.73c -4.211131e -396244.73c -3.239331e-396245.73c -9.717994e-496246.73 c - 3.239331e -4
c Criticality Calculation Informationkcode 25000 1.0 200 1200ksrc 0.1 0.1 5.0
0.1 0.1 10.00.1 0.1 17.00.1 0.1 20.00.1 0. 1 25.00.1 0.1 38.00.1 0.1 45.00.1 0.1 65.00.1 0.1 82.00.1 0.1 87.00.1 0.1 95.00.1 0.1 105.00.1 0.1 109.00.1 0.1 114.00.1 0.1 119.0
ecc Tally Informationec Flux Talliesf114:n u=101fl24:n u=102f134:n u=103fl144:n u=104fl54:n u=105fl64:n u=106
175
fl74:n u=107fl84:n u=108fl94:n u=109f204:n u=110f214:n u=Illf224:n u=112f234:n u=113f244:n u=114f254:n u=115f264:n u=116f274:n u=117f284:n u=118Cc Rate Talliesfmesh364:n
IMESH = 20JMESH = 20ORIGIN -20.0 -20.0 -30.0KMESH = -7.0
0.07.7515.518.023.035.440.447.478.485.490.4
103.0108.01 10.01 18.0120.0150.0
fm364 -1 0 -6 -8fmesh374:n
IMESH = 20JMESH = 20ORIGIN -20.0 -20.0 -30.0KMESH = -7.0
0.07.7515.518.023.035.440.447.478.485.490.4
103.0108.0110.0118.0120.0150.0
fm374 -1 0 -6 -7fmesh384:n
IMESH = 20JMESH = 20ORIGIN -20.0 -20.0 -30.0KMESH = -7.0
0.07.7515.518.023.035.440.447.478.485.490.4
103.0108.0110.0118.0120.0150.0
fm384 -1 0 -2fmesh394:n
IMESH = 20JMESH = 20ORIGIN = -20.0 -20.0 -30.0KMESH = - 7.0
0.07.7515.518.0
176
fm394 -1 0 -6
23.035.440.447.478.485.490.4
103.0108.01 10.01 18.0120.0150.0
A.4 RBWR Two-Dimensional Example Input Files
A.4.1 Lower Reflector
############################################8###################### Lower Reflector Analysis ## created by Bryan Herman #######################################################W#################
set title "LRcase"
# --- Begin Geometry Definitions with 2D slices
########################################################################################### Lower Reflector Layer I ######################
############################ Universe # 101 ###########################
######################################8# #######8
# Surfaces:
surf I inf
# Cells:
cell LRlhont 101 Irlmat -1
# Materials
mat Irlmat -0.736 moder lwtr 1001 tmp 5601001.03c 2.08016.03c 1.0
################## #################################
######################## Lower Reflector Layer 2 ######################
############################ Universe # 102 #########################
###################### ################# ####################
# Pin Definitions:
pin Ilr2b4ccladbwaterlr2
# 1st Zone0.45500.5050
# Water Positions:
pin 2waterlr2
# Zone I
lat 3 3 0 0 21 21 1.14 # Lower Reflector Zone 2
177
1 1 2 2 2 2 2 2 2 21 1 1 2 2 2 2 2 2 2 2 21 1 2 2 2 2 2 2 2 2 2 22 2 2 2 2 2 2 2 2 2 2 2
# Surfaces
surf 2 inf
# Cell for Lattice
cell LR2pin 102 fill 3 -2 # Lower Reflector Zone 2 Pins
# Materials
mat waterlr2 -0.736 moder lwtr 1001 tmp 5601001.03c 2.08016.03c 1.0
mat Ir2b4c5010.03c5011.03 c6012.03c
-2.394 tmp 560 # B4C, 90w'Y% of B-10 in B-0.693962-0.077107-0.228931
#####################################################################
############################# Lower Blanket ##################################################### Universe # 103-105 #######################################################################
# Pin Definitions (allocate universes 10-19):
pin 10fuelLBZ IvoidcladbwaterLB I
# 1 st Zone
0.43750.45500.5050
# Water Positions:
pin 17waterLB I
# Zone I
# Lattice Definitions
lat 200 3 0 0 21 21 1.14
171717171717171717171717171717171717171717
171717171717171717171010101010101010101017
171717171717171717101010101010101010101017
171717171717171710101010101010101010101017
171717171717171010101010101010101010101017
171717171717101010101010101010101010101017
171717171710101010101010101010101010101017
171717171010101010101010101010101010101017
# Lower Blanket Zone I
171717101010101010101010101010101010101017
171710101010101010101010101010101010101017
171010101010101010101010101010101010101017
171010101010101010101010101010101010101717
# Surfaces:
surf 3 inf
# Cells for Lattice
cell LBZlpin 103 fill 200 -3
# Blanket and Water materials
171010101010101010101010101010101010171717
171010101010101010101010101010101017171717
171010101010101010101010101010101717171717
171010101010101010101010101010171717171717
171010101010101010101010101017171717171717
171010101010101010101010101717171717171717
171010101010101010101010171717171717171717
171010101010101010to1017171717171717171717
171717171717171717171717171717171717171717
# Lower Blanket Zone I Pins
vol 1262.92331 tmp 750
2 1 1 1 1 12 1 1 1 1 12 1 1 1 1 12 2 2 2 2 2 2
# mat fuelLBZ I# 8016.06c# 92235.06c# 92238.06c
-10.5 burn 1-0.118466-2.203834e-3-0.87933
178
2
mat waterLBl -0.721 moder lwtr 1001 tmp 5601001.03c 2.08016.03c 1.0
#######################################################################
######################## Assembly Geometry ############################
#######################################################################
# --- Stack Assembly Universes
lat 900 9 0 0 3 # Vertical Lattice 3 layers
30.0-7.0
0.0
101102103
# Bottom of Lower Reflector I# Bottom of Lower Reflector 2
# Start of Core, Bottom of LBZl
# --- Assembly Outer Surfaces
surf 31 hexxc 0.0 0.0 9.495surf 32 pz -30.0surf 33 pz 7.75
# outer unitcell# bottom of assembly# top of assembly
# - Assembly cells
cell assyicell outrfcell outbbcell outbt
0000
fill 900outsideoutsideoutside
-31 3231 32
-3233
-33 # assembly lattice-33 # radially reflect be
# bottom black be# top black bc
# --- Common Material Data Between Zones
ttat cladf -6.5 tnp 850Zr-nat.03c 1.0
mat cladb -6.5 tmp 650Zr-nat.03c 1.0
# Fuel Clad - Zircaloy
# Blanket Clad - Zircaloy
# -- Thermal Scattering Data (take all at 600K)
therm lwtr lwe7.12t
###############################################################888#8888####################### Control Information ###########################
############################################################8888#888888
# Neutron population and criticality cycles
set pop 20000 400 100
# Cross section library file path
set acelib "/opt/ serpent/xs/endfb7/sss endfb7u.xsdata"
# Decay and fission yield libraries:
set declib "/opt/ serpent /xs/endfb7/ sssendfb7 dec"set nfylib "/opt/serpent/xs/endfb7/sss endfb7.nfy"
# Cut-offs :
set fpcut IE-6set stabcut ILE-12
# Options for burnup calculation:
set bumode 2set pcc Iset xscalc 2
set printmt I
# CRAM method# Predictor -corrector calculation on# Cross sections from spectrum
# Right out burnup materials
# periodic boundary conditions (radial only)
set be 3
# turn on full unresolved resonance sampling
set ures I
# Group Constant Homogenization
179
set gcu 101 102 103set nfg 2 0.625e-6ene egrid I I.E-10 0.625E-6 15.0
# Power Density (kW/gm)
set powdens 7.291346e-3set seed 1300652262
# geometry and mesh
plotplotplot
plotmeshmesh
100010001000100010001000
100010001000100010001000
333232
plots
-10-5
5
# Record Nuclides
set inventory531350541350611490922350922380932370942380942390942400942410942420952410952421952430962440962450962460
% -- Burnable materials:
-10.5 vol 1262.92331-2.203834e-3-0.118466-0.87933
'7--Tallies:det fluxt0l du 101 de egriddet kfissl0l dr -8 void du 101det flux102 du 102 de egriddet kfiss102 dr -8 void du 102det flux103 du 103 de egriddet kfiss103 dr -8 void du 103
dt 3 fluxt0l de egrid
dt 3 flux102 de egrid
dt 3 flux103 de egrid
A.4.2 Lower Fissile Zone Sub-region 3
##################################### ### #########8######### Lower Reflector Sub-region 3 Analysis ## created by Bryan Herman ##############################################8888#################
set title "MIT RBWR 2D LFZ3"
#############################################88 8 8 8 8 # #8#########g############################# Lower Fissile #####################8#8############################# Universe # 106-108 #######################################################################8 8 8 8 8 8 ############
# Pin Definitions (allocate universes 20-39):
pin 22fuelLFZ3voidcladfwaterLF3
# 3rd Zone enrichment 18.0%0.43750.45500.5050
# Water Positions:
pin 39waterLF3
# Water in zone 3
# Lattice Definitions
lat 302 3 0 0 21 21 1.14 # Lower Fissle Zone 3
39 39 39 39 39 39 39 39 39 39 39 39 39 39 39 39 39 39 39 39 39
180
mat fuelLBZl92235.06c8016.06c92238.06 c
tmp 750
39393939393939393939393939393939393939
39393939393939393922222222222222222222
39393939393939392222222222222222222222
39393939393939222222222222222222222222
39393939393922222222222222222222222222
39393939392222222222222222222222222222
39393939222222222222222222222222222222
39393922222222222222222222222222222222
39392222222222222222222222222222222222
3922222222222222222222222222
2222222222
2222
2222222222
222222222222222222222222
222222222222222222
222222222222
22222239
22222222222222222222222222222222223939
22222222222222222222222222222222393939
222222
222222222222
22222222222239393939
22222222222222222222222222223939393939
22222222222222222222222222393939393939
39 39 39 39 39 39 39 39 39 39 39 39 39 39 39 39 39
# Surfaces:
surf 4 inf
# Cells for Lattice:
cell LFZ3pin 108 fill 302 -4
# Fuel and Water materials
#9#9#9#9#9#9#9#9#9#$#9#9#9#$#9#9
mat fuelLFZ38016.09c922 35.09 c92238.09c93237.09c942 38.09 c94239.09c94240.09c94241.09c94242.09c95241.09c95342.09 c95243.09c96244.09 c96245.09 c96246.09 c
-10.5 burn 10.1181821.394712e-30.5564901.619666e-39.717994e-30.1425310.1169400.016197
-0.0158730.01 19863.239331e-44.211131e -3
-3.23933 1 e-3-9.717994e-4-3.239331e-4
# Lower Fissile Zone 3 Pins
vol 814.78924 tmp 1150
mat waterLF3
1001.03c 2.08016.03c 1.0
0.290 moder lwtr 1001
###########################################################999999999999######################## Assembly Geometry ############################
#########################################################99999999999999
# -- Assembly Outer Surfaces
surf 31 hexxc 0.0 0.0 9.495 # outer unitcell
# --- Assembly cells
cell assyicell outrf
0 fill 302 -310 outside 31
# assembly lattice# radially reflect bc
# -- Common Material Data Between Zones
mat cladf -6.5 tmp 850Zr-nat.03c 1.0
mat cladb -6.5 tmp 650Zr-nat.03c 1.0
# Fuel Clad - Zircaloy
# Blanket Clad - Zircaloy
# --- Thermal Scattering Data (take all at 600K)
therin lwtr lwe7.12t
##############################9##################################99
####################### Control Information ###########################
###########################################999999999998898988999998989
# Neutron population and criticality cycles
set pop 20000 400 100
181
2222222222222222222222223939393939393939
2222222222222222222222393939393939393939
2222222222222222222239393939393939393939
3939393939393939393939393939393939393939
tmp 560
# Cross section library file path:
set acelih "/opt/ serpent /xs/endfb7 / sss endfb7u.xsdata"
# Decay and fission yield libraries:
set declib "/opt/serpent/xs/endfb7/sss endfb7.dec"set nfylib "/opt/serpent/xs/endfb7/sss endfb7.nfy"
# Cut-offs:
set fpcut IE-6set stabcut IE-12
# Options for burnup calculation
set bumode 2 # CRAM methodset pec I # Predictor -corrector calculation onset xscalc 2 # Cross sections from spectrumset printm I # Right out burnup materials
# periodic boundary conditions (radial only)
set be 3
# turn on full unresolved resonance sampling
set ures I
# Group Constant Homogenization
set gcu 0set nfg 2 0.625e-6ene egrid I I.E-10 0.625E-6 15.0
# Power Density (kW/gm)
set powdens 6.238571e-2set seed 1300652262
# Record Nuclides
set inventory922350922380932370942380942390942400942410942420952410952421952430962440962450962460
% --- Burnable materials:
mat fuelLFZ3 -10.5 vol 814.78924 tmp 115096244.09c -3.239331e-396245.09c -9.717994e -496246.09c -3.239331e-493237.09c -l.619666e-394239.09c -0.14253194238.09c -9.717994e-395241.09c -0.01198695243.09c -4.211131e -392238.09c -0.55649092235.09c -1.394712e-38016.09c -0.11818295342.09c -3.23
9331e -4
94242.09c -0.01587394240.09c -0.11694094241.09c -0.016197
%-T allies:det fluxO du 0 de egriddet kfiss0 dr -8 void du 0 dt 3 fluxO de egrid
182
A.4.3 Upper Reflector
############## ### ############################################## Upper Reflector Analysis ## created by Bryan Herman #
############################################## ############################
set title "MIT RBWR 3D UR"
# --- Begin Geometry Definitions with 2D slices
######t###### ############################################################################ Upper Blanket ##############################
########################## Universe # 115-117 #########################
######t###################################################
# Pin Definitions (allocate universes 70-89):
pin 72fuelUBZ3voidcladbwaterUB3
# 3rd Zone0.43750.45500.5050
# Water Positions:
pin 79waterUB3
# Zone 3
# Lattice Definitions
lat 602 3 0 0 21 21 1.14 # Upper Blanket Zone 3
797979797979797979797979797979797979797979
797979797979797979797272727272727272727279
797979797979797979727272727272727272727279
797979797979797972727272727272727272727279
797979797979797272727272727272727272727279
797979797979727272727272727272727272727279
797979797972727272727272727272727272727279
797979797272727272727272727272727272727279
797979727272727272727272727272727272727279
797972727272727272727272727272727272727279
797272727272727272727272727272727272727279
797272727272727272727272727272727272727979
797272727272727272727272727272727272797979
797272727272727272727272727272727279797979
797272727272727272727272727272727979797979
797272727272727272727272727272797979797979
# Surfaces:
surf 7 intf
# Cells for Lattice:
cell UBZ3pin 117 fill 602 -7
# Blanket and Water materials
# mat fuelUBZ3# 8016.06c# 92235.06c# 92238.06c
-10.5 burn 10.1184662.203834e-30.87933
# Upper Blanket Zone 3 Pins
vol 325.91569 tmp 750
mat waterUB3 -0.150 moder lwtr 1001 tmp 5601001.03c 2.08016.03c 1.0
########################################################
######################## Upper Reflector Layer I ####################t
############## ########## Universe # 118 #############################
##############################################
# Pin Definitions :
pin 4voidcladbwaterurl
# 1st Zone0.45500.5050
183
797272727272727272727272727279797979797979
797272727272727272727272727979797979797979
797272727272727272727279797979797979797979
797979797979797979797979797979797979797979
# Water Positions:
pin 5 # Zone Iwaterur1
lat 6 3 0 0 21 21 1.14 # Upper Reflector Zone I
44444445
44444445
44444445
44444445
44444445
44445555
# Surfaces
surf 8 inf
# Cell for Lattice
cell URlpin 118 fill 6 -8 # Upper Reflector Zone I Pins
# Materials
mat waterurl -0.149 moder lwtr 1001 tmp 5601001.03c 2.08016.03c 1.0
##########################################################
######################## Assembly Geometry ####################################################################################
# -- Stack Assembly Universes
tat 900 9 0 0 2 # Vertical Lattice 2 layers
118.0 117 # Bottom of Upper Blanket Zone 3120.0 118 # Bottom of Upper Reflector I
# --- Assembly Outer Surfaces
surf 31 hexxc 0.0 0.0 9.495surf 32 pz 118.0surf 33 pz 150.0
# outer unitcell# bottom of assembly# top of assembly
# -- Assembly cells
cell assyicell outrfcell outbbcell outbt
0000
fill 900outsideoutsideoutside
-31 3231 32
-3233
33 # assembly lattice33 # radially reflect bc
# bottom black bc# top black bc
# - Common Material Data Between Zones
mat cladf -6.5 tmp 850Zr-nat .03c 1.0
mat cladb -6.5 tmp 650Zr-nat.03c 1.0
# Fuel Clad - Zircaloy
# Blanket Clad - Zircaloy
# --- Thermal Scattering Data (take all at 600K)
therm lwtr lwe7.12t
############################################888888888888888888888888888####################### Control Information ###################################################################88 8 88 8 88 888 8 8# 888#######
184
# Neutron population and criticality cycles
set pop 20000 400 100
# Cross section library file path:
set acelib "/opt/ serpent/xs/endfb7/sss endfb7u.xsdata"
# Decay and fission yield libraries :
set declib "/opt/ serpent/xs/endfb7/ sss endfb7 . dec"set nfylib "/opt/serpent/xs/endfb7/sss endfb7.nfy"
# Cut-offs:
set fpeut IE-6set stabcut IE- 12
# Options for burnup calculation:
set bumode
set pecset xscalc
set printm
2 # CRAM methodI # Predictor -corrector calculation on
2 # Cross sections from spectrum1 # Right out burnup materials
# periodic boundary conditions (radial only)
set bc 3
# turn on full unresolved resonance sampling
set ures 1
# Group Constant Homogenization
set gcu 117 118set nfg 2 0.625e-6ene egrid I I.E-10 0.625E-6 15.0
# Power Density (kW/gm)
set powdens 1.076539e-2set seed 1300652262
# Record Nuclides
set inventory
531350541350611490922350922380932370942380942390942400942410942420952410952421952430962440962450962460
% --- Burnable materials:
mat fuelUBZ392235.06c8016.06c92238.06c
10.5 vol 325.91569-2.203834e-30.118466-0.87933
tmp 750
% -Tallies:det fluxl 17 du 117 de egriddet kfiss l 17 dr -8 void du 117 didet fluxI18 du 118 de egriddet kfiss 1 18 dr -8 void du 118 dt
3 flux 117 de egrid
3 fluxl 18 de egrid
185
981
11 01. 1.1 09.1 I 1.1I 'I1 01 1 1.90 1 0'01 1.*6 0'6 9'S 08S 9'L 01.
0' 0 t1.1. 01. I9 OZ1 01. 11 0 1 1.0'O1 0'01 1.6 0*6 1.8 0*8 9'. O'L
I Al I[JI aSej
006 I
LOL0 lxi06 f-1N 0'0 HD
1.9 0'9 9.9 01. *t 1.1' 01' 1.1. 01. 1. 01. [. 01 1.0~ 1 0 0,0 dnuinqZ. 3d !LIuos!od asoo
006 -l0 3d
LOL'0 DGI06 CR 006 HD 00't 9.L1. 01.1. 1.1. 001. 1.'L1 01.1. 9.Z1. 001. 9'L I 011 .' 1 O'il 9.E 1 01.1 'ZI ' 1 1.1 .1 0'1 1 9.0 1 0'01 9'6 0'6 1.8 0 8 9'L 0'L 9.9 0 9 1.1 01. 1't 1' WE 01. 1.. 01. 11 0'1 1.0 1 .0 0-0 dnuinq
I 3d oluosiod osuo
Z80 3D006 IL
0001 3d
9E9 0 30
06 3)13 00 HD3
1.. 1 ' ' 1 1 1 0 t 1.90 1 0'01 16 0'6 1.8 08S 9.1. 0L 1.9 0'9 1.1 01. 1.'t 0't 9.E 01 E.1 01. 1.9'1 01 1.90 1 .0 0-0 dnuinqZ. DG 113p st13
1.81 31006 Al
0001 3d990 3G
06 QU)1 00 HD3
9.E.1 01. 1 1.1I. 9'1 0 1 1~ 9.0 1 0 01 1.6 0 6 1.8 0'8 9.1. 0L 1.9 0'9 9'9. 01.9A 1.1' 01' 1.1 01 *.1 01 1Zg'1 0'1 g.0 1 .0 0-0 dnuinq
006 -410001 3d
LOL0 DIG16 fl?10'01 HD)
1.1.1 01.1 1.1.1 011 1.11 0*11 1.01 0'01 1.6 0'6 1.8 0'8 9'1. 01. 1.9 0'9 9.19 01. 1.'t7t 01' 1.1 01. 1.1. 01'11 0l 1.0 10 0-0 dnuiLnqiU 13 ppO 3Stu3
I.. 01. ' Z1 ' 01. 11 0'1 1 9'01 0'01 1.6 0'6 1.8 08 1'L 0'L 1.9 0"9 9.9 01. 9.'t 0't 1.. 01. 9.1. 01.
006 -l0001 3d
LOL'O ZG06 fLH3 0*0 )
g.1 01o 1. 10o 'o , dnuinqI 10)1 3OU3Ijol J u
##8 UOI~lIOJUJ aSe3 ###
I A!1J3I-
1. dHlONON
I jpo'i
I AU!']
I' p933I
,I jpu-3
syeiwd
##8 1t'WJOJ UO!)I3Us SSOIJ SXVIAd ##8
#88 UU3UH uieAi #88888 ZId- K(iUWO30 )1wM iuodjS ##8
al~ indul q;puejf SXjuadaaS leg
saldtx3l Indul SDHVd -SXjuadiaS qi
14.0 14.5 15.0 17.5 20.0 22.5 25.0 27.5 30.0 32.5 35.0 37.5 40.0CR 0.0 CRU 90DC 0.707PC 1000TF 582TC 582
case tfhi TF 2burnup 0.0 0.1 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0 5.5 6.0 6.5 7.0 7.5 8.0 8.5 9.0 9.5 10.0 10.5 11.0 11.5 12.0 12.5 13.0 13.5
14.0 14.5 15.0 17.5 20.0 22.5 25.0 27.5 30.0 32.5 35.0 37.5 40.0CR 0.0 CRU 90DC 0.707PC 1000TF 1500TC 582
### Burnable Materials ###
mat U02 -1.00699E+01 burn I gcu 092234 -3.52600E-0492235 -3.62295E-0292238 -8.44896E-018016 -1.18522E-01
mat UO2Gd -1.00700E+01 burn 10 gcu 092234 -1.65721E-0492235 -2.17927E-0292238 -8.06651E-0164154 -. 17196E-0364155 -7.59029E-0364156 1.06331E-0264157 8. 15593F-0364158 l.29653E-0264160 1. 15415E-028016 1. 19332E-01
B.2 SerpentXS PWR Geometry File
% Serpent% Bryan Herman
set title "PWR Burnup Calculation Based on NEA Benchmark"
% --- Fuel pins:
pin 10U02 0.4025clad 0.4750water
pin 11U02 0.4025clad 0.4750water
pin 12U02 0.4025clad 0.4750water
pin 13U02 0.4025clad 0.4750water
pin 14U02 0.4025clad 0.4750water
pin 15U02 0.4025clad 0.4750water
pin 16U02 0.4025clad 0.4750water
pin 17U02 0.4025clad 0.4750water
pin 18U02 0.4025clad 0.4750water
187
pin 19U02cladwater
pin 20U02cladwater
pin 21U02cladwater
pin 22U02cladwater
pin 23U02cladwater
pin 24U02cladwater
pin 25U02cladwater
pin 26U02cladwater
pin 27U02cladwater
pin 28U02cladwater
pin 29U02cladwater
pin 30U02cladwater
pin 31U02cladwater
pin 32U02cladwater
pin 33U02cladwater
pin 34U02cladwater
pin 35U02cladwater
pin 36U02cladwater
pin 37
0.40250.4750
0.40250.4750
0.40250.4750
0.40250.4750
0.40250.4750
0.40250.4750
0.40250.4750
0.40250.4750
0.40250.4750
0.40250.4750
0.40250.4750
0.40250.4750
0.40250.4750
0.40250.4750
0.40250.4750
0.40250.4750
0.40250.4750
0.40250.4750
188
UO2 0.4025clad 0.4750water
pin 38U02 0.4025clad 0.4750water
pin 39U02 0.4025clad 0.4750water
pin 40U02 0.4025clad 0.4750water
pin 41U02 0.4025clad 0.4750water
pin 42U02 0.4025clad 0.475(water
pin 43U02 0.402clad 0.475water
pin 44U02 0.402clad 0.475water
pin 45U02 0.402clad 0.475water
% --- Gd-pins
pin 50UO2Gd 0.402clad 0.475water
pin 51UO2Gd 0.402clad 0.475water
pin 52UO2Gd 0.402clad 0.475water
% --- Guide t
pin 90watertubewater
50
50
50
50
50
ube No CR:
0.57300.6130
% --- Guide tube w/CR:
pin 91aicss304water
cladwater
0.38250.48640.56340.6130
% -- Pin lattice:
lat 110 1 0.0 0.0 17 17 1.265
43 42 41 40 39 38 37 38 39 40 41 42 43 44 45
35 34 33 32 31 30 29 30 31 32 33 34 35 36 4428 27 52 <CRUO> 26 25 <CRUO> 25 26 <CRUO> 52 27 28 35 4327 <CRUO> 24 23 22 21 51 21 22 23 24 <CRUO> 27 34 4252 24 20 19 18 17 16 17 18 19 20 24 52 33 41<CRU0> 23 19 <CRU0> 15 14 <CRU0> 14 15 <CRUO> 19 23 <CRUO> 32 4026 22 18 15 50 13 12 13 50 15 18 22 26 31 39
25 21 17 14 13 11 10 11 13 14 17 21 25 30 38<CRUO> 51 16 <CRUO> 12 10 90 10 12 <CRU0> 16 51 <CRU0> 29 37
25 21 17 14 13 11 10 11 13 14 17 21 25 30 38
26 22 18 15 50 13 12 13 50 15 18 22 26 31 39
189
4544434241403938373839
4436353433323130293031
50
40 32 <CRUO> 23 19 <CRU0> 15 14 <CRU0> 14 15 <CRUO> 19 23 <CRU0> 32 4041 33 52 24 20 19 18 17 16 17 18 19 20 24 52 33 4142 34 27 <CRUO> 24 23 22 21 51 21 22 23 24 <CRUO> 27 34 4243 35 28 27 52 <CRUO> 26 25 <CRUO> 25 26 <CRU1O> 52 27 28 35 4344 36 35 34 33 32 31 30 29 30 31 32 33 34 35 36 4445 44 43 42 41 40 39 38 37 38 39 40 41 42 43 44 45
surf 1000 sqc 0.0 0.0 10.752surf 1001 sqc 0.0 0.0 10.806
cell 110 0 fill 110 -1000cell 111 0 water 1000 -1001cell 112 0 outside 1001
%--- Non-burnable Materials
mat clad 3.8510E-0226000.06c 1.3225E-0424000.06c 6.7643E-0540000.06c 3.8310E-02
mat tube 4.3206E-0226000.06c 1.4838E-0424000.06c 7.5891E-0540000.06c 4.2982E-02
mat water -<DCO> moder lwtr 10011001.06c 2.08016.06c 1.0
mat aic -10.1747107.06c -0.410947109.06c -0.389148110.06c -0.006148111.06c -0.006348112.06c -0.012048113.06c -0.006148114.06c -0.014548116.06c -0.003949113.06c -0.006349115.06c -0.1437
mat ss304 -7.766012.06c -0.000814000.06c -0.020024000.06c -0.195025055.06c -0.015026000.06c -0.673428000.06c -0.0950
mat boron 1.05010.06c 0.25011.06c 0.8
therm lwtr lwe7.12t
% -- Set soluble absorber in water:
set abs boron -<PCO>E-6 water
%--- Cross section library file path
set acelib '/opt/serpent/xs/endfb7/sss-endfb7u.xsdata"
% -- Periodic boundary condition
set bc 3
% -- Neutron population and criticality cycles:
set pop 5000 500 20
% --- Set group constants
set gcu 0set nfg 2 0.625e-6ene egrid I I.E-10 0.625E-6 15.0
% -- Set Unresolved Sampling
set ures I I 92235.06c 92235.09c 92238.09c 9
2238
.06
c 94239.06c 94239.09c 94240.06c 94240.09c
% -- Geometry plots:
plot 3 1000 1000mesh 3 1000 1000
%-- Decay and fission yield libraries:
set declib "/opt/ serpent /xs/endfb7/sss endfb7 . dec"set nfylib "/opt/serpent/xs/endfb7/sss endfb7.nfy
190
% --- Cut-offs :
set fpcut IE-6set stabcut 1E- 12
% --- Options for burnup calculation:
setset
setset
bumodepeexscalcprintm
% CRAM method% Predictor -corrector calculation on% Cross sections from spectrum% Right out burnup materials
% --- Set Normalization Condition
set powdens 38.6E-3
B.3 SerpentXS to PMAXS Input File
### PMAXS FILE CREATION### Bryan Herman
### XS Control Information ###GLOBAL VNSET INGROUP 2MDLAY 8MDCAY 8MADF 4MCDF 4MRODS 289MC!LA 17Ladf TLxes FLded FLjlf FLehi TLchd TLinv TLdet FLyld FLcdf TLgff FLbet TLamb TLdec FDerivative F
COMMENI'SPWR BenchmarkSerpent generated cross section comparisonBryan Herman3/27/201 1MIT
### Branch Information ###
BRANCHESbranch reference REF ICR 0.0DC 0.707065PC 1000TF 900TC 582branch rodded CR ICR 1.0DC 0.707065PC 1000TF 900TC 582branch dclow DC ICR 0.0DC 0.594193PC 1000TF 900TC 582branch dchi DC 2CR 0.0DC 0.740503PC 1000TF 900TC 582branch poisonlo PC ICR 0.0DC 0.707065PC 0TF 900TC 582branch poisonhi PC 2CR 0.0
191
DC 0.707065PC 2000TF 900TC 582branch tflo TF ICR 0.0DC 0.707065PC 1000TF 582TC 582branch tfhi TF 2CR 0.0DC 0.707065PC 1000TF 1500TC 582
### BurnupBURNUPSNBset II 1NBP 42
Information ###
Burns 0.0 0.1 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0 5.5 6.0 6.5 7.0 7.5 8.0 8.5 9.0 9.5 10.0 10.5 11.0 11.5 12.0 12.5 13.0 13.514.0 14.5 15.0 17.5 20.0 22.5 25.0 27.5 30.0 32.5 35.0 37.5 40.0
### XS Set identification ###XSSETSeries 1IST 1NADF 4NCDF 4NCOLA 17NROWA 17NPART 0PITCH 0.0XBE 0.0YBE 0.0iHMD 4.946188Dsat 1.000ARWatR 0.0ARByPa 0.0ARConR 0.0
### History Cases to append ###HISTORYCreference0.pmaxs
192
C MATLAB Multigroup Spatial Diffusion Solver
C.1 Example Input File
% Axial Discontinuity Factor Calculation% Full RBWR Assembly w/ Purely Absorbing Slabs% 2 Energy Group Calculation% Bryan Herman
% Variables Meaning
% BC cell array of boundary conditions% captid cell array of Serpent capture tally IDs% deppath name of Serpent detector file% f discontinuity factors% filepath name of Serpent output file% f issid cell array of Serpent fission tally IDs% fluxhet heterogeneous interface fluxes% fluxhom homogeneous interface fluxes
% G number of energy groups% grid thickness of regs% id cell array of Serpent flux tally IDs% J average interface current% JL interface current from left boundary% JR interface current from right boundary% keff-adf keff with discontinuity factors% keff noadf keff without discontinuity factors% keff-target target Serpent k-effective% mesh fine mesh in coarse mesh regs% offset coordinate shift amount% opt solve two-dimensional cross sections% phi flux vector w/adfs% phi h homogeneous flux distributions% phi-noadf flux vector w/o adfs% prefix file prefix - Serpent Notation% prodid cell array of Serpent production tally IDs% regs vector of region numbers corresponding to GCU% type type of calculation
% Inputsprefix = 'RBWR_2G';filepath strecat(prefix ,' res.m');deppath = strcat(prefix ,'_detO.m');
regs = [100,101 102,103,104,105,106,107,108,109,110,111,112,113 ,114,115,...I16,117,118,119];
grid = [30.0 ,23.0 ,7.0 7.75 ,7.75 2.5 ,5.0 , 12.4 ,5.0 ,7.0 31.0 ,7.0 ,5.0 , 12.6 ,...5.0 ,2.0 ,8.0 2.0 ,30.0 ,30.0];
mesh = 500*ones(1,20);opt 'yes';G = 2;
id = 'DETBLflux' ,'DETLRIflux' , 'DETLR2flux' , 'DETLBIflux' ,'DETLB2flux''DETLB3flux' ,' DETLF I flux' , 'DETLF2flux' , 'DETLF3flux' , 'DETIB1flux''DETIB2flux' ,'DETIB3flux ' , 'DETUFlflux' ,'DETUF2flux' , 'DETUF3flux''DETUBIflux' ,'DETUB2flux' 'DETUB3flux' 'DETURflux', 'DETBUflux' };
captid = { 'DETBLcapt' , 'DETLRIcapt' , 'DETLR2capt' , 'DETLBicapt', 'DETLB2capt'DETLB3capt ' , 'DETLFIcapt 'DETLF2capt' , ' DETLF3capt' , ' DETIBleapt ' , ..'DETIB2capt ' ,'DETIB3capt 'DETUFlcapt' ,'DETUF2capt' ,'DETUF3capt''DETUB I capt' ,'DETUB2capt' ,'DETUB3capt' 'DETURcapt', 'DETBUcapt' };
fissid = { 'DETBLfiss ' , ' DETLRIfiss ' , 'DETLR2fiss ' , 'DETLBIfiss ' , 'DETLB2fiss'DETLB3fiss' ,'DETLFIfiss' ,'DETLF2fiss' ,'DETLF3fiss' ,' DETIBlfiss'DETlB2fiss' ,'DETIB3fiss' ,'DETUFlfiss' ,'DETUF2fiss' ,'DETUF3fiss'DETUB I fiss' ,'DETUB2fiss' , 'DETUB3fiss 'DETURfiss ', 'DETBUfiss ' };
prodid = {'DETBLprod' ,'DETLRlprod' ,'DETLR2prod' , 'DETLBlprod' , 'DETLB2prod'DETLB3prod' 'DETLFI prod' ,'DETLF2prod' ,'DETLF3prod' 'DETIB I prod ' ..'DETIB2prod' ,'DETIB3prod' 'DETUFIprod' ,'DETUF2prod' ,'DETUF3prod''DETUBlprod' ,'DETUB2prod' ,'DETUB3prod' , 'DETURprod', 'DETBUprod' };
offset = -60;
% Calculate k-effective before= ones( length (regs) -- 1,2*G)
fb = ones(2,2*G);type = ' forward ;BC = cell(l ,G+2); for i=l:2*G+2 BC{i [=1.0; end; BC[l}='ALB'; BC{G+2}='ALB';[keff-noadf ,phi-noadf] = spatial-group ( filepath , grid ,mesh, regs ,G,BC, type , f)
% Calculate Target Eigenvaluekeff target = get-keff(filepath);
% Calculate Interface Currents from Neutron BalanceI J,JL,JR ] = calc-currents( filepath ,regs ,G,keff-target )
% Calculate Homogenized Fluxes for Each Region
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phih = homogeneous-flux( filepath ,grid mesh ,regs ,G, keff-target ,J, fb
% Get Homogeneous Surface Fluxesfluxhom = surface-flux ( filepath regs ,mesh, grid phih ,J,G
% Get Heterogeneous Surface Fluxesfluxhet = het flux( deppath regsG );
% Calculate Discontinuity Factorsf = calc adfs ( regs ,G, fluxhet , fluxhom )
% Calculate Eigenvalue Problem[ keff-adf , phi ] = spatial-group ( filepath ,grid , mesh , regs ,G,BC, type f , fb
C.2 Source Code
C.2.1 Power Iteration Routine
function [ keig , phi ] spatial-group ( filepath , grid , mesh ,regs ,G,BC, type , f , keig
% Author: Bryan Herman% Description: Power iteration routine
% Variable Meaning
% A loss matrix% ABSXS absorption xs from Serpent% avez mesh cell centers% B production matrix% BC cell array of boundary conditions% CHI group fission spectrum from Serpent% dx vector of all fine meshes% dz thickness of fine mesh only coarse meshes% ferr error in flux% filepath name of Serpent output file% G number of energy groups% GCUNI vector of group homogenization universes% GTRANSFXS scattering matrix from Serpent% grid thickness of regs% f discontinuity factors% iter iteration number% keig keffective eigenvalue% keignew updated keffective eigenvalue% kerr error in eigenvalue% mesh fine mesh in coarse mesh regs% meshtot cumulative sum vector of mesh% N total number of fine mesh cells% NSF nusigmaf xs from Serpent% P1_TRANSPXS pl transport xs from Serpent% phi flux vector% phi-new updated flux vector% Q external source vector% reg current coarse mesh region% regs vector of region numbers corresponding to GCU% SCATTXS scattering xs from Serpent% type type of calculation% z mesh boundaries
% Read in Serpent Data[stat , attrib] = fileattrib (filepath); % Get absolute pathif stat == 0 % check if file exists
disp('File does not exist')stop
endrun( attrib Name); % Run serpent file
% ParametersN = sum(mesh); % number of total mesh cells
% Set up mesh and geometrymeshtot = zeros(length(mesh),1);for i = 1: length(mesh)
meshtot(i) = sum(mesh(1:i));end
% z locations for node boundaries and mesh centersz = zeros(sum(mesh)+ll);avez = zeros(sum(mesh),l);dz = grid ./(mesh);for i=2:length(z)
reg = find(i <= meshtot+1l);z(i) =z(i-l)+dz(reg);avez(i -1) = (z(i) + z(i-1))/2;
end
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% Flux and Eig Guess
if -strcmp(type , ' fixed')
phi = ones(G*N,l);keig 1.0;
end
% Calculate Loss Matrixdisp ( 'Building system matrices...[A,dxJ = LOSSTERM( N,G,PITRANSPXS,ABSXS,GTRANSFXS,SCATTXS, grid ,meshtot ...
meshBC, regs ,GCUNI,f );
% Calculate Production MatrixB = PRODTERM( N, meshtot ,NSF, regs ,GCUNI,G, CHI, dx );disp ( '. . . system matrices completed.
if strcmp(type , adjoint ')A = A.';B B.';
end
% Calculate Fixed Source Vector
if strcip (type , ' fixed ')Q = FIXEDSOURCE( N,BC,Gphi = (A - /keig*B)\Q;iter = 1
elsefor iter 1:10000
% Update Fluxphinew = A\( I / keig*Bphi)
% Update Keff
keignew = keig*suim((B*phinew) .*(B*phinew))/sui((B*phi) .*(B*phinew));
% Calculate Differenceferr = norm(phinew-phi)/length(phi);kerr = abs(keignew-keig)/keignew;
fprintf('Iteration: %1od k-error: %d flux-error: /d\n', iter kerr, ferr);
% Check Convergenceif ferr < l.Oe-8 && kerr < t.Oe-8
disp( 'Converged ')break
elsephi phinewkeig keignew;
end
end
end
end
C.2.2 Build Loss Matrix
function I Mdx ] LOSSTERM( N,G,PI TRANSPXS,ABSXS,GTRANSFXS,SCA'ITXS, grid ...ineshtot ,mesh BC, regs ,GCU, f
% Author: Bryan Herman% Description: Calculates the Loss Matrix M in Mx=S
% Variables Meaning
% ABSXS absorption xs from Serpent% BC cell array of boundary conditions% colidx sparse matrix column id% count counter for sparse matrix vectors% DO diffusion coefficient for mesh cell%/o DL diffusion coefficient for left mesh cell% DR diffusion coefficient for right mesh cell% dx vector of all fine meshes% dx0 thickness of mesh% dxL thickness of left mesh% dxR thickness of right mesh% f discontinuity factors% fOl region index for discontinuity factor-current mesh
% 102 group index for discontinuity factor-current mesh% fLl region index for discontinuity factor-left mesh
% fL2 group index for discontinuity factor-left mesh
% fRI region index for discontinuity factor-right mesh% fR2 group index for discontinuity factor-right mesh% G number of energy groups% GCU vector of group homogenization universes% gidx group index% grid thickness of regs% GTRANSFXS scattering matrix from Serpent% i iteration counter for group g% idx0 index for current mesh cell in diffusion mesh
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% idxc index for column in sparse matrix% idxL index for mesh cell to the left% idxR index for mesh cell to the right% idxr row index for sparse matrix% idxrc starting row and column index in matrix% j iteration counter for group h% k spatial mesh iteration counter% M loss matrix% mesh fine mesh in coarse mesh regs% meshtot cumulative sum vector of mesh% N total number of fine mesh cells% PlTRANSPXS pl transport xs from Serpent% regO index of region in Serpent output% regL index of left region in Serpent output% regR index of right region in Serpent output% reggcu current homogenization region% regs vector of region numbers corresponding to GCU% rowidx sparse matrix row vector id% SCATTXS scattering xs from Serpent% sidx h-->g scattering xs index% SigA local var for absorption cross section% Sigs local var for scattering cross section% SigR removal cross section% SigSig local var for within group scattering% Sz total size of vector% validx sparse matrix value id vector
% Calculate SizeSz = N*G;
% Generate Vectorsrowidx = zeros (G*(N+2*(N- 1))+(G^2-G)*N, I);colidx = zeros (G*(N+2*(N-l))+(GA2-G)*N, I);validx = zeros(G*(N+2*(N-l))+(G'2-G)*N,I);
% Initialize Global Countercount = 1;
for i = 1:Gfor j = t:G
disp(horzcat('Building Group ',num2str(i), to Group ',num2str(j)));
% If a diagonal block -- > within group scatteringif i == j
% Initialize index counteridxrc = (i -1)*N+1; % starting idxdx = zeros(N,I);
for k = 1:N
% Indexing Scheme for Region Pointers to XS dataidxO = find (k <= meshtot ,1)reggcu = regs(idxO);regO = find (GCU == reggcu , I
idxL = find(k-I <= meshtot .1)reggcu = regs(idxL);regL = find (GCU == reggcu , 1;
if k NidxR = find (k+t <= meshtot ,1);reggcu = regs(idxR);regR = find (GCU == reggcu ,1);
end
% Cross Sections for CalculationDO = 1/(3*PlTRANSPXS(regO,2*i+l)); % Diffusion Coef. of center nodeDL = 1/(3*PlTRANSPXS(regL,2*i+l)); % Diffusion Coef. of left nodeDR = 1/(3*PTRANSPXS(regR,2*i+])); % Diffusion Coef. of right nodeSigA = ABSXS(regO 2* i +1); % Absorption Cross sectionSigSgg = GTRANSFXS(regO ,2*(i -1)*G+2*i -1); % Within group scatteringSigS = SCATTXS(regO ,2* i+1); % Total Scattering Cross Sectionif G ==I
SigR = SigA; % no out of group scatteringelse
SigR = SigA + SigS - SigSgg; % Removal Cross Sectionend
% X-spacingdxO = grid(idxO)/mesh(idxO);dxL = grid(idxL)/mesh(idxL);dxR = grid(idxR)/mesh(idxR);
% Handle Left Boundaryif k == I
% Get Group Indxgidx = i+t;
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% Index the row column vectors
% Handle Group-wise Albedo Boundary Conditionsif strcmp(BC(I),'ALB')
rowidx(count) = idxrc;colidx(count) = idxrc;validx(count) = 2*(((DO*DR)/(dxR*DO+dxODR))+...
((D0O*(I-BC(gidx}))/(4*DO*(I+BC(gidx})+...dx0*(l -BCgidx})))) + dxO*SigR;
count=count+l;
rowidx(count) = idxrccolidx(count) = idxrc+l;validx(count) = -2*((DO*DR)/(dxR*DO+dx0*DR));count = count+l;
% Handle Group-wise Fixed Net Currentelseif strcmp(BC(l},'CUR')
rowidx(count) = idxrc
colidx (count) = idxrcvalidx (count) = 2*((DO*DR)/(dxR*DO+dx0*DR)) + dx0*SigR;count = count + 1;rowidx(count) = idxrccolidx (count) = idxrc+l;validx (count) = -2*((DO*DR) /(dxR*DO+dxO*DR));count = count + 1;
end
% Handle Right Boundaryelseif k == N
% Get Group Indxgidx = (G+I) + i+1;
% Handle Group-wise Albedo Boundary Conditionsif strcmp(BC{G+2},'ALB')
rowidx(count) = idxrc
colidx (count) = idxrcvalidx ( count) = 2*(((D0*(l-BC( gidx )/(4*D0O*(l+BC{gidx))+...
dx0*(l -BC(gidx })))+...((DO*DL)/(dxO*DL+dxL*DO))) + dxO*SigR;
count = count+l;
rowidx(count) = idxrc;colidx (count) = idxrc Ivalidx (count) = -2*((DO*DL) /(dxO*DL+dxL*DO));count = count+I;
% Handle Group-wise Fixed Net Currentelseif strcmp(BC(G+2),'CUR')
rowidx(count) = idxrccolidx (count) = idxrcvalidx(count) = 2*((DO*DL)/(dxO*DL+dxL*D0)) + dx0*SigR;count = count+];rowidx(count) = idxrccolidx(count) = idxrc 1;validx(count) = -2*((D0*DL)/(dx0*DL+dxL*D0));count = count + 1;
end
% Interior mesh cell left of boundaryelseif regR == regO+t
f0l = regO;f02 = i;fRI = regO;fR2 = G+i;rowidx(count) = idxrc
colidx(count = idxrc
validx(count) = 2*(((DO*DR*f(fRI ,fR2) 1)/(dxR*D0f(ffO ,f02)^ ...+dx0O*DR*f(fRl,fR2)^ -))+((DO*DL)/(dxO*DL+dxL*DO))) + dx0O*SigR;
count = count + 1;rowidx(count) = idxrc;
colidx(count) = idxrc+l;validx(count) = -2*(DO*DR*f(f01,f02)^-l)/(dxR*D0O*f(f0 ,1f02)^-1 ...
+dxO*DR*f(fRl ,fR2) -1) ;
count = count+l;rowidx (count) idxrc
colidx (count) = idxrc 1;validx(count) = 2*(DO*DL) /(dxO*DL+dxL*DO);count = count+l;
% Interior mesh cell right of boundaryelseif regL == regO-I
fO= regO - 1;102 = G+i;
fl I regO - 1;fL2 i;rowidx(count) = idxrc
colidx ( count) = idxrcvalidx (count ) = 2*(((DO*DR) /(dxR*D0+dx0+DR))+((DO*DL*f(fL, fL2)^ 1) /
(dxO*DL*f(fLi ,fL2)^-l+dxL*DO*f(f0t ,f02)^-I))) + dx0O*SigR;
count = count+t;
rowidx (count) idxrc
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colidx (count) = idxrc+1;validx (count) = -2*(DO*DR)/(dxR*DO+dxO*DR);count = count + 1;rowidx(count) idxrc;colidx (count) = idxrc 1;validx(count) -2*(DO*DL*f(f01, f02)^- )/(dxO*DL*f(fLt ,fL2)^-]
+dxL*DO*f(fO , f02)^-I);count = count+t;
elserowidx(count) = idxrccolidx (count) = idxrcvalidx(count) = 2*(((DO*DR)/(dxR*D+dxO*DR))+((DO*DL)/ ...
(dxO*DL+dxL*DO))) + dxO*SigR:count = count+t;rowidx(count) = idxrc;colidx (count) = idxrc+l;validx (count) = -2*(DO*DR)/(dxR*DO+dxO*DR);count = count+1;rowidx(count) = idxrc;colidx (count) = idxrc - Ivalidx(count) = -2*(DO*DL) (dxODL+dxL*DO)count = count+t;
end
% Bank discretizationdx(k) = dxO;
% Update Counteridxrc = idxrc + 1;
end
% Off Diagonal Block For Group to Group Couplingelse
sidx = 2*(i -t)*G + 2*j - 1; % idx for scattering xsidxr = (i -)*N+l; % starting idxidxc = (j -l)*N+l;for k = t:N
% Get regionidxO = find(k <= meshtotl);reggcu = regs(idxO);reg = find (GCU == reggcu ,1);dxO = grid(idxO)/mesh(idxO);
% Get scattering xsrowidx(count) = idxr;colidx(count) = idxc:validx(count) = -GTRANSFXS(reg ,sidx)*dxO;count = count+t;
% Update indiciesidxr = idxr + 1;idxc = idxc + 1;
endend
endend
% Create Sparse MatrixM = sparse (rowidx colidx validx);
end
C.2.3 Build Production Matrix
function [ B I = PRODTERM( Nmesh,NSF, regs ,GCU,GCHI,dx
% Author: Bryan Herman% Description: Create fission production matrix
% Variable Meaning
% B production matrix% CHI group fission spectrum from Serpent% colidx sparse matrix column id% count counter for sparse matrix vectors% dx vector of all fine meshes% G number of energy groups% GCU vector of group homogenization universes% idxc index for column in sparse matrix% idxr row index for sparse matrix% mesh fine mesh in coarse mesh regs% N total number of fine mesh cells% NSF nusigmaf xs from Serpent% reggcu current homogenization region% reg index in Serpent output for region% regs vector of region numbers corresponding to GCU% rowidx sparse matrix row vector id% validx sparse matrix value id vector
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% Initialize vectors
rowidx = zeros(G*N,I);
colidx = zeros(G*N,l);
validx = zeros(G*N,1);count = 1;
% Loop around Groupsfor i = I:G
for j = l:G
% Initialize indicesidxr = (i -)*N+l;idxc = (j -1)*N+t;
% Loop around spatial pointsfor k = l:N
reggcu = regs (find (k <= mesh , I));reg = find (GCU == reggcu ,);rowidx(count) = idxr;colidx (count) = idxc;validx (count) = CHI(reg ,2*(i -1)+1)*NSF(reg ,2*j +)*dx(k)count count+t;idxr = idxr + 1;idxc = idxc + 1;
end
end
end
% Create Sparse MatrixB = sparse ( rowidx , colidx , vatidx
end
C.2.4 Fixed External Source
function [ Q ] = FDEDSOURCE( N,BC,G )
% Author: Bryan Herman% Desciption: Calculation External Fixed Source
% Variable Meaning
% BC% G% gidx% i
% N
% Q
cell array of boundary conditionsnumber of energy groupsgroup indexiteration countertotal number of fine mesh cellsexternal source vector
% Allocate Vector for fixed source
Q = zeros(N*G,t);
for i = I:G% Put Fixed Source at boundaries if need% Check Left Boundaryif strcmp(BC{ t } 'CUR')
c% Get group indexgidx = i+1;
% Put in fixed source
Q((i-t)*N+t) = BC{gidx};
end
%, Check Right Boundaryif strcmp(BC{G+2},'CUR')
% Get group index
gidx G+t + i+l;
% Put in fixed sourceQ(N* = BC{gidx };
endend
end
C.2.5 Extract Heterogeneous k-effective
function [keff] = get-keff(filepath)
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% Author: Bryan Herman% Description : Extract Collision estimator of keff from Serpent
% Variable Meaning
% COLKEFF collision estimator of keff in Serpent% filepath name of Serpent output file% keff keffective
% Read in Serpent File[stat attrib] = fileattrib(filepath); % Get absolute pathif stat == 0 % check if file exists
disp('File does not exist ')stop
endrun( attrib Name); % Run serpent file
% Get collision estimatorkeff = COLKEFF( I1);
end
C.2.6 Compute Interface Currents
function [ J.JLJR ] = calc-currents( filepath regs ,Gkeff
% Author: Bryan Herman% Description : Calculate interface currents
% Variable Meaning
% ABSXS absorption xs from Serpent% BC cell array of boundary conditions% CHI group fission spectrum from Serpent% filepath name of Serpent output file% FLUX integrated region flux in Serpent output% G number of energy groups% GTRANSFXS scattering matrix from Serpent% i group g counter% idxin h-->g scattering index in GTRANSFXS% J average interface current% j group h counter% JL interface current from left boundary% JR interface current from right boundar% k iteration coarse mesh counter% keff keffective% NSF nusigmaf xs from Serpent% reg index of region in Serpent output% reggcu current homogenization region% regs vector of region numbers corresponding to GCU% S total fission source into group g% scattin overall group scattering from h to g
% Read in Serpent File[stat , attrib] = fileattrib(filepath); % Get absolute pathif stat == 0 % check if file exists
disp('File does not exist ')stop
endrun(attrib Name); % Run serpent file
% Start loop through regions from rightJR = zeros(length(regs)+l,G);BC(t:G) = 0.0;JR(l ,l:G) = BC(1:G):for k = 1:length(regs)
% Get material pointerreggcu = regs(k);reg = find(GCUNI == reggcuI);
% Start loop through groupsfor i = l:G
% Get Scattering Source into current groupscattin = 0;S = 0;
if G > Ifor j = t:G
% Get index in Serpent vectorif i-=j
idxin = 2*(i -)*G + 2*j - 1;scattin = scattin + GTRANSFXS(reg ,idxin)*FLUX(reg,2*j+l);
end
% Get total source
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S = S + NSF(reg ,2*j+1)*FLUX(reg ,2*j+1)
endelse
scattin = SCATTXS(reg ,3)*FLUX(reg 3)end
% Calculate Current on Right Boundarygidx = 2*(i-l)*G + 2*i - 1;JR(k+l, i) = CHI(reg ,2*(i -l)+l),S/keff - (ABSXS(reg ,2* i+l) +
SCATTXS(reg,2*i+l) - GTRANSFXS(reg,gidx))*FLUX(reg,2*i+l) ...+ scattin + BC(i);
end
% Get next left boundaryfor i 1:G
BC(i) = JR(k+l,i);end
end
% Start loop through regions from leftJL = zeros(length(regs)+lG);BC(l:G) = 0.0;JL(length(regs),1:G) = BC(I:G);for k = length(regs):-l:1
% Get material pointerreggcu = regs(k);reg = find (GCUNI == reggcu ,1);
% Start loop through groupsfor i = 1:G
% Get Scattering Source into current groupscattin = 0;S = 0;if G > I
for j = 1:G
% Get index in Serpent vectorif i-=j
idxin = 2*(i-l)*G + 2*j - 1;scattin = scattin + GTRANSFXS(reg idxin)*FLUX(reg 2*j+1);
end
% Get total source
S = S + NSF(reg,2*j+l)*FLUX(reg,2*j+l);
end
else
sc attin = SCATFXS(reg ,3)*HLJX(reg ,3);end
% Calculate Current on Right Boundarygidx = 2*(i-l)*G + 2*i - 1;JL(k, i) = -(CHI(reg ,2*(i -l)+l)*S/keff - (ABSXS(reg ,2* i+) +
SCATrXS(reg,2*i+l) - GTRANSFXS(ireg,gidx))*FLUX(reg,2*i+) ...+ scattin + BC(i));
end
% Get next left boundary
for i = 1:GBC(i) = -JL(k,i);
end
end
J = (JR + JL)./2;
end
C.2.7 Coarse Mesh Homogeneous Flux Distribution
function [ phih I = homogeneous flux( filepath grid ,mesh,regs G,keigJ
% Author: Bryan Herman
% Description : Solve Fixed Source Diffusion for homogeneous flux in each% coarse mesh homogenization region
% Variable Meaning%% BC cell array of boundary conditions% filepath name of Serpent output file% G number of energy groups
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% grid% i
% J
% j% keig% mesh% phih% regs% type
thickness of regsiteration counter , coarse meshinterface currentiteration counter, group
keffective eigenvaluefine mesh in coarse mesh regshomogeneous flux arrayvector of region numbers corresponding to GCUtype of calculation
if ~exist('fb ','var ')error('No Boundary Factors')
end
for i = 1:tength(regs)BC = {};
% Setup boundary identifierif i == I
BC{l} = 'ALB';BC{G+2} = 'CUR';
elseif i == length(regs)BC(l} = 'CUR';BC{G+2} = 'ALB';
elseBC( I} = 'CUR';BC{G+2} = 'CUR';
end
% Loop through groups andfor j = I:G
if i == 1BC{j+t} = 1;BC{G+2+j} = -J(i+t
elseif i == length(regBC{j+t} = J(i,j);BC{G+2+j} = 1;
set up boundary conditions
elseBC{j+1} = J(i j);BC(G+2+j} = -J(i+t,j);
endend
% Set up remaining varsf = ones(length ( regs) G)type = 'fixed ';
% Run Fixed source and get homogeneous fluxeskeig phi ] = spatial-group( filepath grid(i),mesh(i) ...
regs (i) ,GBC, type f ,keig
for j = 1:Gphih (( i - )*mesh( i )+1: i *mesh ( i ) = phi ((j - )*mesh( i ) +l:j *mesh( i)
endend
end
C.2.8 Compute Homogeneous Interface Flux
function [ fluxhom 1 = surface-flux ( filepath , regs ,mesh, grid , phi. J ,G
% Author: Bryan Herman% Desciption: Compute interface flux from mesh-centered flux
% Variable Meaning
% aa% D% dx% filepath% fluxhom% fluxmesh%G% grid% i% J% j% k% mesh% PTRANSPXS% phi% regs
overall row counter
diffusion coefficient for mesh cellcoarse mesh thicknessname of Serpent output filehomogeneous interface fluxesmesh-centered homogeneous flux next to interfacenumber of energy groupsthickness of regscoarse mesh iteration counterinterface currentgroup counterleft /right iteration counterfine mesh in coarse mesh regspl transport xs from Serpentflux vectorvector of region numbers corresponding to GCU
% Get Homogeneous Flux near Interfacefor i 1:length(regs)
aa 0;
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% Loop around groupsfor k = 1:2
% Left and Right of Regionfor j = 1:G
aa = aa+l;if k == I
fluxnesh(i,aa) = phi((i-l)*mesh(i)+l,j);else
fluxmesh(iaa) = phi(i*mesh(i),j);end
endend
end
% Read in Serpent Filet stat , attrib ] = fileattrib(filepath ); % Get absolute pathif stat == 0 % check if file exists
disp(' File does not existstop
endrun( attrib Name); % Run serpent file
% Extrapolate Mesh Centered Fluxes to Interfacefor i = 1:length(regs)
% Get region indexreg =find (GCU NI == regs( i) ,);aa = 0;for k = 1:2
for j = l:Gaa aa+l;
% Calculate Diffusion CoefficientD = 1/(3nPlTRANSPXS(reg,2*j+l));
% Calculate Spacingdx = grid(i)/mesh(i);
% Calculate Surface Fluxif k == I
fluxhom(i,aa) = fluxmesh(i,aa) + (J(i,j)*dx)/(2*D);else
fluxhom(i ,aa) fluxnesh(i ,aa) (J( i+I,j )*dx)/(2*D)
end
endend
end
end
C.2.9 Extract Heterogeneous Interface Flux
function [ flux-het ] = het-flux ( deppath , regs ,G
% Author: Bryan Herman% Description : set heterogeneous flux to unity , does not matter
% Variable Meaning
% deppath name of Serpent detector file% fluxhet heterogeneous interface fluxes% G number of energy groups% i coarse mesh counter% j group counter% regs vector of region numbers corresponding to GCU
/v Can extract tallies instead of set to unity itn this nested loopfor i = 1:length(regs)-l
for j = l:Gflux het(i ,j )=1.0;
endend
end
C.2.10 Compute Discontinuity Factors
function [ f ] = calcadfs ( regs ,G, fluxhet ,fluxhot )
% Author : Bryan Herman% Description : Calculation axial discontinuity factors
203
% Variable Meaning
% C ratio of interface ADFs% f discontinuity factors% fluxhet heterogeneous interface fluxes% fluxhom homogeneous interface fluxes% G number of energy groups% i loop around coarse mesh regions% j loop around groups% regs vector of region numbers corresponding to GCU
for i = I:length(regs)-Ifor j = l:G
f(i,j) = fluxhet(i,j)/fluxhom(i,G+j);f(iG+j) = fluxhet (i ,j)/fluxhom(i+1,j);
endend
% Renormalize Discontinuity Factorsfor i = I:length(regs)-I
for j=l:GC = f(i,j)/f(i,G+j);f(ij ) = 2*C/( I +C)f(i ,G+j) = 2/(1 +C);
endend
end
204
D RBWR Single Assembly Code Inputs
D.1 Serpent
D.1.1 Branch Case Input File
### Serpent Geometry File ###
geomtfile RBWRAssy-geom
### PMAXS Cross section format ###
pmaxs
Ladf FLxes FLded FLjlf FLehi TLchd FLinv FLdet FLyld FLcdf FLgff FLbet FLamb FLdec FNGROUP 2Lderiv F
### Case Information ###
case RBWR_2G REF Iburnup 0.0CR 0.0 CRU 90DC 0.736 0.736 0.736PC 0.0 0.0 0.0 0.0 0IF 560 560 560 750 7TC 560 560 560 560 5
### Burnable Materials ###
mat fuelLBZI -10.5 burn I8016 -0.11846692235 -2.203834e -392238 -0.87933
mat fuelLBZ2 -10.5 burn 18016 -0.11846692235 -2.203834e -392238 -0.87933
mat fuelLBZ3 -10.5 burn I8016 -0.11846692235 -2.203834e -392238 -0.87933
mat fuelLFZ1 -10.58016 -0.11818292235 -1.394712e92238 -0.55649093237 -1.619666e94238 -9.717994e94239 -0.14253194240 -0.11694094241 -0.01619794242 -0.01587395241 -0.01198695342 -3.239331e-95243 -4.211131e-96244 -3.239331e-96245 -9.7 17994e96246 -3.239331e-
burn I
-3
-3-3
0.721 0.668 0.609 0.527 0.360 0.290 0.266 0.253 0.226 0.219 0.190 0.169 0.156 0.152 0.150 0.149 0.1490 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.00 750 1150 1150 1150 750 750 750 1150 1150 1150 750 750 750 560 5600 560 560 560 560 560 560 560 560 560 560 560 560 560 560 560
vol 1262.92331 gcu 103
vol 1262.92331 gcu 104
vol 407.39462 gcu 105
vol 814.78924 gcu 106
-4-3-3-4-4
mat fuelLFZ2 -10.5 burn 18016 -0.11818292235 -1.394712e-392238 -0.55649093237 -1.619666e-394238 -9.7 17994e -394239 -0.14253194240 -0.11694094241 -0.01619794242 -0.01587395241 -0.01198695342 -3.239331e-4
vol 2020.6773 gcu 107
205
56
95243 -4.211131e96244 -3.239331e96245 -9.717994e96246 -3.239331 e
mat fuelLFZ3 -10.58016 -0.11818292235 -l.394712e92238 -0.55649093237 -1.619666e94238 -9.717994e94239 -0.14253194240 -0.11694094241 -0.01619794242 -0.01587395241 -0.01198695342 -3.239331e95243 -4.211131e96244 -3.239331e96245 -9.717994e96246 -3.239331e
-3-3-4-4
burn 1
-3
-3-3
vol 814.78924 gcu 108
-4-3-3-4-4
mat fuelIBZl -10.58016 -0.11846692235 -2.203834e -392238 -0.87933
burn 1 vol 1140.70493
mat fuelIBZ2 -10.5 burn 18016 -0.11846692235 -2.203834e -392238 -0.87933
mat fuelIBZ3 -10.5 burn 18016 -0.11846692235 -2.203834e-392238 -0.87933
mat fuelUFZI -10.58016 -0.11818292235 -1.394712e-92238 -0.55649093237 -1.619666e-94238 -9.717994e-94239 -0.14253194240 -0.11694094241 -0.01619794242 -0.01587395241 -0.01198695342 -3.239331e-95243 -4.211131e-96244 -3.239331 e-96245 -9.717994e-96246 -3.239331e-
mat fuelUFZ2 -10.58016 -0.11818292235 -1.394712e-92238 -0.55649093237 -1.619666e-94238 -9.717994e-94239 -0.14253194240 -0.11694094241 -0.01619794242 -0.01587395241 -0.01198695342 -3.23933Ie-95243 -4.211131e-96244 -3.239331e-96245 -9.717994e-96246 -3.23933 1e-
mat fue]UFZ3 -10.58016 -0.11818292235 -1.394712e-92238 -0.55649093237 -1.619666e-94238 -9.717994e94239 -0.14253194240 -0.11694094241 -0.01619794242 -0.01587395241 -0.01198695342 -3.239331e-95243 -4.211131e-96244 -3.239331e-96245 -9.717994e-96246 -3.239331e-
burn 1
-3
-3-3
-4-3-3-4-4
burn 1
-3
-3-3
-4-3-3-4-4
burn 1
-3
-3-3
vol 5051.69326 gcu 110
vol 1140.70493 gcu 111
vol 814.78924 gcu 112
vol 2053.26887 gcu 113
vol 814.78924 gcu 114
-4-3-3-4-4
mat fuelUBZl -10.5 burn 18016 -0.11846692235 -2.203834e-392238 -0.87933
vol 325.91569 gcu 115
gcu 109
206
mat fuelUBZ2 -10.5 burn I vol 1303.66278 gcu 1168016 -0.11846692235 -2.203834e-392238 -0.87933
mat fuelUBZ3 -10.5 burn I vol 325.91569 gcu 1178016 -0.11846692235 -2.203834e-392238 -0.87933
D.1.2 Geometry Input File
######################################################################## created by Bryan Herman ########################################################################
set title "MIT RBWR 3D Assy"
# -- Begin Geometry Definitions with 2D slices
############################################################################################### Black Hydrogen Bottom Layer ############################################## Universe # 100 ##################################################################################################
# Surfaces:
surf 60 inf
# Cells:
cell Blhom 100 blackl -60
# Materials:
mat blackl -10.09999.03c 1.0
############################################################################################### Lower Reflector Layer I ################################################## Universe # 101 ##################################################################################################
# Surfaces:
surf I inf
# Cells:
cell LRlhon 101 Irlmat -1
# Materials
mat Irlmat -<DCIOI> moder lwtr 1001 tmp <TCIOI>1001.03c 2.08016.03c 1.0
############################################################################################### Lower Reflector Layer 2 ################################################## Universe # 102 ##################################################################################################
# Pin Definitions:
pin Ilr2b4ccladbwaterlr2
# I st Zone0.45500.5050
# Water Positions:
pin 2 # Zone Iwaterlr2
]at 3 3 0 0 21 21 1.14 # Lower Reflector Zone 2
2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 22 2 2 2 2 2 2 2 2 2 1 1 1 1 1 1 1 1 1 1 22 2 2 2 2 2 2 2 2 1 1 1 1 1 1 1 1 1 1 1 2
207
# Surfaces
surf 2 inf
# Cell for Lattice
cell LR2pin 102 fill 3 -2 # Lower Reflector Zone 2 Pins
# Materials
mat waterlr2 -<DC102>1001.03c 2.08016.03c 1.0
mat Ir2b4c5010.03c501 1.03c6012.03c
-2.394 tmp-0.693962-0.077107-0.228931
moder lwtr 1001 tmp <TC102>
<TF102> # B4C, 90w'% of B-10 in B
#######################################################################
######################### Lower Blanket ##################################################### Universe # 103-105 ############################88###########88##############################################
# Pin Definitions (allocate universes 10-19):
pin 10fuelLBZ IvoidcladbwaterLB I
pin 11fuelLBZ2voidcladbwaterLB2
pin 12fueILBZ3voidcladbwaterLB3
# I st Zone0.43750.45500.5050
# 2nd Zone0.43750.45500.5050
# 3rd Zone0.43750.45500.5050
# Water Positions:
pin 17waterLB I
pin 18waterLB2
pin 19waterLB3
# Zone I
# Zone 2
# Zone 3
# Lattice Definitions
lat 200 3 0 0 21 21 1.14
1717171717171717
1717171717171717
1717171717171717
1717171717171717
1717171717171710
1717171717171010
1717171717101010
1717171710101010
# Lower Blanket Zone I
1717171010101010
1717101010101010
1710101010101010
1710101010101010
1710101010101010
1710101010101010
1710101010101010
1710101010101010
1710101010101010
1710101010101010
1710101010101010
1710101010101010
1717171717171717
208
00
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--
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c
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--
--
--
--
--
# 92238.06c -0.87933
mat waterLBl -<DC103>1001.03c 2.08016.03c 1.0
mat waterLB2 -<DCI04>1001.03c 2.08016.03c 1.0
mat waterLB3 -<DC105>1001.03c 2.08016.03c 1.0
moder lwtr 1001 tmp <TC103>
moder lwtr 1001 tmp <TCI04>
moder lwtr 1001 tmp <TC105>
################################################################################################# Lower Fissile #################################################### Universe # 106-108 ################################################################################################
# Pin Definitions (allocate universes 20-39):
# 1st Zone
0.43750.45500.5050
# 2nd Zone0.43750.45500.5050
# 3rd Zone0.43750.45500.5050
enrichment 18.0%
enrichment 18.0%
enrichment 18.0%
# Water Positions:
# Water in zone I
# Water in zone 2
# Water in zone 3
# Lattice Definitions
tat 300 3 0 0 21 21 1.14 # Lower Fissle Zone 1
373737373737202020202020202020202020202037
373737373720202020202020202020202020202037
373737372020202020202020202020202020202037
373737202020202020202020202020202020202037
373720202020202020202020202020202020202037
372020202020202020202020202020202020202037
372020202020202020202020202020202020203737
372020202020202020202020202020202020373737
372020202020202020202020202020202037373737
372020202020202020202020202020203737373737
372020202020202020202020202020373737373737
372020202020202020202020202037373737373737
lat 301 3 0 0 21 21 1.14 # Lower Fissle Zone 2
3838383838382121212121
3838383838212121212121
38 38 38 38 38 38 38 38 38 3838 38 38 21 21 21 21 21 21 2138 38 21 21 21 21 21 21 21 2138 21 21 21 21 21 21 21 21 2121 21 21 21 21 21 21 21 21 2121 21 21 21 21 21 21 21 21 2121 21 21 21 21 21 21 21 21 2121 21 21 21 21 21 21 21 21 2121 21 21 21 21 21 21 21 21 2121 21 21 21 21 21 21 21 21 2121 21 21 21 21 21 21 21 21 21
372020202020202020202020203737373737373737
3821212121212121212121
3821212121212121212121
372020202020202020202037373737373737373737
3821212121212121212121
3838383838383838383838
210
pin 20fuelLFZ IvoidcladfwaterLFl
pin 21fuelLFZ2voidcladfwaterLF2
pin 22fuelLFZ3voidcladfwaterLF3
pin 37waterLFI
pin 38waterLF2
pin 39waterLF3
373737373737373737373737373737373737373737
373737373737373737372020202020202020202037
373737373737373737202020202020202020202037
373737373737373720202020202020202020202037
373737373737372020202020202020202020202037
3838383838383838383838
3838383838383838383821
3838383838383838382121
3838383838383838212121
3838383838383821212121
21212121212121212138
21212121212121212138
21212121212121212138
21212121212121212138
21212121212121212138
21212121212121212138
21212121212121212138 38 38 38 38 38 38 38 38 38
lat 302 3 0 0 21 21 1.14 # Lower Fissle Zone 3
393939393939393939392222222222222222222239
393939393939393939222222222222222222222239
3939393939393922222222222222222222222222
39
393939393939222222222222222222222222222239
3939393922222222222222222222222222
22222239
# Surfaces :
surf 4 inf
# Cells for Lattice:
cell LFZlpin 106cell LFZ2pin 107cell LFZ3pin 108
fill 300 -fill 301 -fill 302 -
Fuel and Water materials
mat fuelLFZl -10.5 burn I8016.09c -0.11818292235.09c -1.394712e -392238.09c -0.55649093237.09c -l.619666e-394238.09c -9.717994e-394239.09c -0.14253194240.09c -0.11694094241.09c -0.01619794242.09c -0.01587395241.09c -0.01198695342.09c -3.239331e -495243.09c -4.211131e-396244.09c -3.239331e -396245.09c -9.717994e -496246.09c -3.239331e -4
mat fuelLFZ2 -10.5 burn 18016.09c -0.11818292235.09c -l.394712e -392238.09c -0.55649093237.09c -1.619666e -394238.09c -9.717994e-394239.09c -0.14253194240.09c -0.11694094241.09c -0.01619794242.09c -0.01587395241.09c -0.01198695342.09c -3.239331e-495243.09c -4.211131e -396244.09c -3.239331e-396245.09c -9.717994e-496246.09c -3.239331e-4
mat fuelLFZ3 -10.5 burn 18016.09c -0.11818292235.09c -1.394712e-392238.09c -0.55649093237.09c -1.619666e -394238.09c -9.717994e -394239.09c -0.14253194240.09c -0.116940
4 # Lower Fissile Zone I Pins4 # Lower Fissile Zone 2 Pins4 # Lower Fissile Zone 3 Pins
vol 814.78924 tnip 1150
vol 2020.6773 tmp 1150
vol 814.78924 tip 1150
38383838383838383838
212121212121212121
21212121212121212138
212121212121212121
212121212121212138
212121212121213838
212121212121383838
212121212138383838
212121213838383838
21
2121383838383838
212138383838383838
21383838383838383838
38383838383838383838
393939393939393939393939393939393939393939
38383838383838383838
392222222222222222222239393939393939393939
393939222222222222222222222222222222222239
39392222222222
2222222222222222222222222239
392222222222222222222222222222222222222239
3922
22222222222222222222222222222222393939
392222222222222222222222222222223939393939
392222222222222222222222222222393939393939
392222222222222222222222222239393939393939
392222222222222222222222223939393939393939
211
94241.09c94242.09c95241.09c95342.09c95 243.09 c96244.09c96245.09c96246.09c
-0.016197-0.015873-0.011986-3.239331-4.211131-3.239331-9.717994-3.239331
mat waterLFl -<DC106>1001.03c 2.08016.03c 1.0
mat waterLF2 -<DC107>1001.03c 2.08016.03c 1.0
mat waterLF3 -<DCI08>1001.03c 2.08016.03c 1.0
e-4
e-3e-3e-4e-4
moder lwtr 1001
moder lwtr 1001
moder lwtr 1001
#######################################################################
############################# Internal Blanket ################################################## Universe # 109-111 ######################
#######################################################################
# Pin Definitions (allocate universes 40-49):
pin 40fuelIBZlvoidcladbwaterIBI
pin 41fuelIBZ2voidcladbwaterIB2
pin 42fuelIBZ3voidcladbwaterIB3
# 1st Zone0.43750.45500.5050
# 2nd Zone0.43750.45500.5050
# 3rd Zone0.43750.45500.5050
# Water Positions:
pin 47waterIBI
# Zone I
pin 48 # Zone 2waterIB2
pin 49 # Zone 3waterIB3
# Lattice Definitions
lat 400 3 0 0 21 21 1.14 # Internal Blanket Zone I
474747474747474747474747474747474747474747
474747474747474747474040404040404040404047
474747474747474747404040404040404040404047
474747474747474740404040404040404040404047
474747474747474040404040404040404040404047
474747474747404040404040404040404040404047
474747474740404040404040404040404040404047
474747474040404040404040404040404040404047
474747404040404040404040404040404040404047
474740404040404040404040404040404040404047
474040404040404040404040404040404040404047
474040404040404040404040404040404040404747
474040404040404040404040404040404040474747
474040404040404040404040404040404047474747
474040404040404040404040404040474747474747
474040404040404040404040404747474747474747
474040404040404040404047474747474747474747
474747474747474747474747474747474747474747
lat 401 3 0 0 21 21 1.14 # Internal Blanket Zone 2
48 48 48 48 48 48 48 48 48 48 48 48 48 48 48 48 48 48 48 48 4848 48 48 48 48 48 48 48 48 48 41 41 41 41 41 41 41 41 41 41 4848 48 48 48 48 48 48 48 48 41 41 41 41 41 41 41 41 41 41 41 48
212
tmp <TC106>
tmp <TC107>
tmp <TC108>
484848484848484848484848484848484848
484848484848484141414141414141414148
484848484848414141414141414141414148
484848484841414141414141414141414148
484848484141414141414141414141414148
484848414141414141414141414141414148
4848414141414141414141414141414'4148
484141414141414141414141414141414148
414141414141414141414141414141414148
414141414141414141414141414141414148
4141
4141
4141414141414141414141414148
414141414141414141414141414141414848
414141414141414141414141414141484848
414141414141414141414141414148484848
414141414141414141414141414848484848
414141414141414141414141484848484848
414141414141414141414148484848484848
lat 402 3 0 0 21 21 1.14 # Internal Blanket Zone 3
494949494949494949494949494949494949494949
494949494949494949424242424242424242424249
494949494949494942424242424242424242424249
494949494949494242424242424242424242424249
494949494949424242424242424242424242424249
494949494242424242424242424242424242424249
494949424242424242424242424242424242424249
494242424242424242424242424242424242424249
494242424242424242424242424242424242494949
494242424242424242424242424242424949494949
4942
42424242424242424242424249494949494949
# Surfaces
surf 5 inf
# Cells for Lattice
cell IBZIpin 109 fill 400 -5cell IBZ2pin 110 fill 401 -5cell IBZ3pin 111 fill 402 -5
# Blanket and Water materials
#8#8#8#8
#8#8#8#8
mat fuelIBZ I8016.06c92235.06c92238.06c
mat fuellBZ28016.06c92235.06c92 23 8.06 c
mat fuellBZ38016.06c92235.06c92238.06c
-10.5 burn 10.118466
-2.203834e -30.87933
-10.5 burn I0.1184662.203834e-30.87933
-10.5 burn 10.1184662.203834e-30.87933
# Internal Blanket Zone 1 Pins# Internal Blanket Zone 2 Pins# Internal Blanket Zone 3 Pins
vol 1140.70493 tnp 750
vol 5051.69326 tmp 750
vol 1140.70493 tnp 750
mat waterIBI -<DC109> moder lwtr 1001 tutp <TC109>1001.03c 2.08016.03c 1.0
mat waterIB2 -<DCIIO> moder lwtr 1001 tnp <TCIIO>1001.03c 2.08016.03c 1.0
mat waterIB3 -<DCIII> moder lwtr 1001 tmp <TCIIl>1001.03c 2.08016.03c 1.0
#################################################################################################### Upper Fissile ##################################################### Universe # 112-114 ################################################################################################
213
414141414141414141414848484848484848
414141414141414141484848484848484848
414141414141
414148484848484848484848
494242424242424242424249494949494949494949
484848484848484848484848484848484848
494949494949494949494949494949494949494949
# Pin Definitions (allocate universes 50-69):
# Ist Zone enrichment 18%0.43750.45500.5050
# 2nd Zone enrichment 18%0.43750.45500.5050
# 3rd Zone enrichment 18%0.43750.45500.5050
# Water Positions:
# Water in zone I
# Water in zone 2
# Water in zone 3
# Lattice Definitions
lat 500 3 0 0 21 21 1.14 # Upper Fissle Zone I
676767676767676767676767676767676767676767
676767676767676767675050505050505050505067
676767676767676767505050505050505050505067
676767676767676750505050505050505050505067
676767676767675050505050505050505050505067
676767676750505050505050505050505050505067
676767505050505050505050505050505050505067
676750505050505050505050505050505050505067
675050505050505050505050505050505050506767
675050505050505050505050505050505067676767
675050505050505050505050505050676767676767
lat 501 3 0 0 21 21 1.14 # Upper Fissle Zone 2
686868686868686868686868686868686868686868
686868686868686868685151515151515151515168
686868686868686868515151515151515151515168
686868686868686851515151515151515151515168
686868686868685151515151515151515151515168
686868686868515151515151515151515151515168
686868686851515151515151515151515151515168
686868685151515151515151515151515151515168
686868515151515151515151515151515151515168
686851515151515151515151515151515151515168
685151515151515151515151515151515151515168
685151515151515151515151515151515151516868
685151515151515151515151515151515151686868
685151515151515151515151515151515168686868
685151515151515151515151515151516868686868
685151515151515151515151515151686868686868
Lat 502 3 0 0 21 21 1.14 # Upper Fissle Zone 3
69696969696969
69696969696969
69696969696969
69696969696969
69696969696969
69696969696952
69696969695252
69696969525252
69696952525252
69695252525252
69525252525252
69525252525252
69525252525252
69525252525252
69525252525252
69525252525252
pin 50fuclUFZ1voidcladfwaterUF1
pin 51fuelUFZ2voidcladfwaterUF2
pin 52fuelUFZ3voidcladfwaterUF3
pin 67waterUFl
pin 68waterUF2
pin 69waterUF3
675050505050505050505050505067676767676767
685151515151515151515151515168686868686868
69525252525252
675050505050505050505050506767676767676767
685151515151515151515151516868686868686868
69525252525252
675050505050505050505050676767676767676767
685151515151515151515151686868686868686868
69525252525252
676767676767676767676767676767676767676767
686868686868686868686868686868686868686868
69696969696969
685151515151515151515168686868686868686868
69525252525252
214
6969696969696969696969696969
6969695252525252525252525269
6969525252525252525252525269
6952525252525252525252525269
5252525252525252525252525269
5252525252525252525252525269
5252525252525252525252525269
# Surfaces:
surf 6 inf
# Cells for Lattice
cell UFZlpin 112cell UFZ2pin 113cell UFZ3pin 114
fill 500 -fill 501 -fill 502 -6
6 #6 #
# 9
# Fuel and Water materials
# mat fuelUFZI -10.5 burn I vol 814.# 8016.09c -0.118182# 92235.09c -l.394712e -3# 92238.09c -0.556490# 93237.09c -1.619666e-3# 94238.09c -9.717994e-3# 94239.09c -0.142531# 94240.09c -0.116940# 94241.09c -0.016197# 94242.09c -0.015873# 95241.09c -0.011986# 95342.09c -3.239331e-4# 95243.09c -4.211131e-3# 96244.09c -3.239331e-3# 96245.09c -9.717994e-4# 96246.09c -3.239331e-4
# mat fuelUFZ2 -10.5 burn 1 vol 2053# 8016.09c -0.118182# 92235.09c -l.394712e-3# 92238.09c -0.556490# 93237.09c -l.619666e-3# 94238.09c -9.717994e-3# 94239.09c -0.142531# 94240.09c -0.116940# 94241.09c -0.016197# 94242.09c -0.015873# 95241.09c -0.011986# 95342.09c -3.239331e--4# 95243.09c -4.211131e-3# 96244.09c -3.239331e-3# 96245.09c -9.717994e-4# 96246.09c -3.239331e-4
# mat fuelUFZ3 -10.5 burn I vol 814.# 8016.09c -0.1 18182# 92235.09c -l.394712e-3# 92238.09c -0.556490# 93237.09c - 1.619666e -3# 94238.09c -9.717994e-3# 94239.09c -0.142531# 94240.09c -0.116940# 94241.09c -0.016197# 94242.09c -0.015873# 95241.09c -0.011986# 95342.09c -3.239331e -4# 95243.09c -4.211131e -3# 96244.09c -3.239331 e-3# 96245.09c -9.717994e-4# 96246.09c -3.239331e-4
mat waterUFI -<DC112> moder lwtr 10011001.03c 2.08016.03c 1.0
mat waterUF2 -<DCll3> moder lwtr 10011001.03c 2.08016.03c 1.0
mat waterUF3 -<DCII4> moder lwtr 10011001.03c 2.08016.03c 1.0
Upper Fissile Zone I PinsUpper Fissile Zone 2 PinsUpper Fissile Zone 3 Pins
78924 tnp 1150
.26887 tmp 1150
78924 tnp 1150
tmp <TCll2>
tmp <TCIl3>
tmp <TCI 14>
5252525252525252525252525269
5252525252525252525252525269
5252525252525252525252525269
5252525252525252525252525269
5252525252525252525252526969
5252525252525252525252696969
5252525252525252525269696969
5252525252525252526969696969
5252525252525252696969696969
5252525252525269696969696969
5252525252526969696969696969
5252525252696969696969696969
5252525269696969696969696969
6969696969696969696969696969
215
91Z
6L 6L 6L 6L 6L 6L 6L 6L 6L 6L 6L 6L 6L 6L 6L 6L 6L 6L 6L 6L 6L
£ ~ouZ 1oluelE0 Loddfl # El1 1 1. El o o 6 1.09 lei
8L8LSL8L8L8L8L8L81L8L81L81L81L81L8L8L8L81L81LSL81L
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8L8L8LSLSL8L8L8LSL8L8L8L8L8L8L8L8L8L8L8L8L
1. auoZ Ej)EurIg jzddfl # 17'V1 EL EL 0 0 £ Eog lei
LLLLLLLLLLLLLLOL01LOLOLOLOLOLOLOLOLOLOL0LLL
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LLLLLLOLOLOLOLOLOLOL0LOLOLOLOLOLOLOLOLOLLL
LLLLOLOLOLOLOLOLOLOLOLOL0LOLOLOLOLOLOLOLLL
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LLOLOLOLOLOLOLOLOLOL0LOLOLOLOLOLLLLLLLLLLL
LLOLOLOLOLOL01L01LOLOLOLOLOL01LLLLLLLLLLLLLLL
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LLOLOLOLOLOLOLOLOLOLOLOLLLLLLLLLLLLLLLLLLL
LL01LOLOLOLOLOLOLOLOL01LLLLLLLLLLLLLLLLLLLLL
LLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLL
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£ HLIOZ 4 61. urd
ZE~flHeHmZ. 3Oz # 81. uid
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#######################################################################
###################### LEE#LEE # ZIOA!Ua ########################################8###W######### ti~uuI[E Ijddfl ###################I#####~####g
7979797979797979797979797979797979797979
7979797979797979797272727272727272727279
7979797979797979727272727272727272727279
7979797979797972727272727272727272727279
7979797979797272727272727272727272727279
7979797979727272727272727272727272727279
7979797972727272727272727272727272727279
# Surfaces:
surf 7 inf
# Cells for Lattice:
cell UBZlpin 115cell UBZ2pin 116cell UBZ3pin 117
fill 600 -7fill 601 -7fill 602 -7
# Upper Blanket Zone I Pins# Upper Blanket Zone 2 Pins# Upper Blanket Zone 3 Pins
# Blanket and Water materials
mat fuelUBZ I8016.06c92235.06c92238.06c
mat fuelUBZ280 16.06c92235.06c92238.06c
mat fuelUBZ38016.06c92235.06c92238.06c
-10.5 burn I-0.118466-2.203834e-3-0.87933
-10.5 burn 1-0.118466-2.203834e-3-0.87933
-10.5 burn I0.1 18466
-2.203834e-30.87933
vol 325.91569 tntp 750
vol 1303.66278 tmp 750
vol 325.91569 tmp 750
mat waterUBl -<DCI 15> moder lwtr 1001 tmp <TCll5>1001.03c 2.08016.03c 1.0
mat waterUB2 -<DCI 16> moder1001.03c 2.08016.03c 1.0
mat waterUB3 -<DC 117> moder1001.03c 2.08016.03c 1.0
lwtr 1001 tmp <TCI16>
lwtr 1001 tiip <TCII7>
############################################################################################## Upper Reflector Layer I ################# ############################### Universe # 118 #########################################################W###########################
# Pin Definitions:
pin 4voidcladbwaterurl
# 1st Zone0.45500.5050
# Water Positions:
pin 5 # Zone Iwaterurl
lat 6 3 0 0 21 21 1.14 # Upper Reflector Zone I
55555555
55555555
55555555
55555554
55555544
55444444
54444444
54444444
54444444
54444444
54444444
54444444
55555444
55554444
5554444
4
54444444
54444444
54444444
54444444
55555555
217
7979797272727272727272727272727272727279
7979727272727272727272727272727272727279
7972727272727272727272727272727272727279
7272727272727272727272727272727272727279
7272727272727272727272727272727272727979
7272727272727272727272727272727272797979
7272727272727272727272727272727279797979
7272727272727272727272727272797979797979
7272727272727272727272727979797979797979
7272727272727272727272797979797979797979
7272727272727272727279797979797979797979
7979797979797979797979797979797979797979
55555555
5 5 5 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 5
5 5 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 5
5 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 5
5 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 5 5
5 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 5 5 5
5 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 5 5 5 5
5 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 5 5 5 5 5
5 4 4 4 4 4 4 4 4 4 4 4 4 4 4 5 5 5 5 5 5
5 4 4 4 4 4 4 4 4 4 4 4 4 4 5 5 5 5 5 5 5
5 4 4 4 4 4 4 4 4 4 4 4 4 5 5 5 5 5 5 5 5
5 4 4 4 4 4 4 4 4 4 4 4 5 5 5 5 5 5 5 5 5
5 4 4 4 4 4 4 4 4 4 4 5 5 5 5 5 5 5 5 5 5
5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5
# Surfaces
surf 8 inf
# Cell for Lattice
cell URlpin 118 fill 6 -8 # Upper Reflector Zone I Pins
# Materials
mat waterurl -<DC11 I8> moder lwtr 1001 tmp <TC118>
1001.03c 2.0
8016.03c 1.0
#######################################################W################
######################## Black Hydrogen Top Layer ####################
############################ Universe # 119 ###########################
#######################################################################
# Surfaces :
surf 61 inf
# Cells :
cell B32hom 119 black2 -61
# Materials :
mat black2 -10.09999.03c 1.0
#######################################################################
######################## Assembly Geometry ############################
##########################################################W#############
# -- Stack Assembly Universes
lat 900 9 0 0 20 # Vertical Lattice 10 layers
60.0 100 # Bottom of BH bottom layer-30.0 101 # Bottom of Lower Reflector 1-7.0 102 # Bottom of Lower Reflector 2
0.0 103 # Start of Core , Bottom of LBZI
7.75 104 # Bottom of Lower Blanket Zone 2
15.5 105 # Bottom of Lower Blanket Zone 3
18.0 106 # Bottom of Lower Fissile Zone 1
23.0 107 # Bottom of Lower Fissile Zone 2
35.4 108 # Bottom of Lower Fissile Zone 3
40.4 109 # Bottom of Internal Blanket Zone 1
47.4 1 10 # Bottom of Internal Blanket Zone 2
78.4 111 # Bottom of Internal Blanket Zone 3
85.4 112 # Bottom of Upper Fissile Zone 1
90.4 113 # Bottom of Upper F iss ilIe Zone 2
103.0 114 # Bottom of Upper Fissile Zone 3
108.0 115 # Bottom of Upper Blanket Zone I110.0 116 # Bottom of Upper Blanket Zone 2
118.0 117 # Bottom of Upper Blanket Zone 3
120.0 118 # Bottom of Upper Reflector 1
150.0 119 # Bottom of BH top layer
# -- Assembly Outer Surfaces
surf 31 hexxc 0.0 0.0 9.4 95 # outer unitcell
surf 32 pz -60.0 # bottom of assemblysurf 33 pz 180.0 # top of assembly
# -- Assembly cells
218
cell assyi 0 fill 900cell outrf 0 outsidecell outbb 0 outsidecell outbt 0 outside
-31 32 -33 # assembly lattice31 32 -33 # radially reflect be
-32 # bottom black be33 # top black bc
# --- Common Material Data Between Zones
mat cladf -6.5 tmp 850Zr-nat.03c 1.0
mat cladb -6.5 tnp 650Zr-natt.03c 1.0
# Fuel Clad - Zircaloy
# Blanket Clad - Zircaloy
# --- Thermal Scattering Data (take all at 600K)
therm lwtr lwe7.12t
##################################### ################################################ Control Info rmation ######################################################88#######################################
# Neutron population and criticality cycles
set pop 25000 4000 200
# Cross section library file path:
set acelib "/opt/ serpent /xs/endfb7/sss-endfb7u . xsdata"
# periodic boundary conditions (radial only)
set be 2
# turn on full unresolved resonance sampling
set ures I
# Group Constant Homogenization
set gcu 100 101 102 103 104 105 106set nfg 2 6.25e-7ene egrid I I.E-10 6.25e-7 15.0
# random number seed
set seed 1300652262
D.2 PARCS
D.2.1 UF1 PMAXS File
107 108 109 110 111 112 113 114 115 116 117 118 119
GLOBALV I 2 8 8 4 4 289 17 T F F F T F F F FRBWR 3D Assy - UF2SerpentBryan Herman5/12/201 1MITBRANCHES 1 0 0 0 0 0REFE I 0.00000 0.19000 0.00000BURNUPS I0 1 0.000XSSET I 1 4 4 17 17 0 0.00000 0.00000HSTCASE 0.00000 0.19000 0.000001.00000E+00 0.OOOOOE+00REFERENC 1 12.46198E-01 1.47051E+00 1.33889E-02 1.19414E+2.91847E-01 6.23719E-03 3.83517E-05 3.49757E-1.02041E+00 4.85995E+00 1.00000E+00 1.00000E+
F F F F F
1150.00000 560.00000
0.00000 4.83129 1.000001 150.00000 560.00000
0.00000 0.00000 0.00000
00 2.19520E-02 1.66992E+00 1.93295E-11 2.44975E-130100 1.02139E+00 -6.83111E+00 1.00000E+00 1.00000E+00
D.2.2 Input File
CASEID RBWR 3D Assy Calc
CNTLcore-type BWRcore-power 100.0thfdbkxe sm
F0 0
219
depletion T 1Oe-3 FTREE, nset , adf , xes , ene , jI f , chi , chd , vel , det , yld , cdf , gff , bet , iam , dht
TREEXS T t T F F F T F F F F F F F F F
input iterationedit table
print opt T Tfdbk flux
rho precursprint opt F F
Idconst
print-opt F
pkdata
F
planarpower
Fplanar
fluxT
rad pwrshape
F
pinF
xe/smF
rad fluxshape
F
adjreac
F
T/HF
assyconst
PARAMnodal-kernel HYBRIDn iters 1 1000 default is 1 500CONVSS 1.0e-8 1.0e-8 5.0e-8 0.001 def 1.0e-6 1.0e-5 5.0e-5 0.001Isolver 2 1 20
WIELANDT 1.0 1.0 1.0nlupd-ss 10 10 1 default is 3 3 1init-guess 1.13 default is I
XSECgroup-spec 2 1
GEOMgeo-dim 18 1 1 0 0 !nrow, nz fuel onRADCONF radial configuratic
1 1 15*2 1GRIDX 23.0 7.0 7.75 7.75 2.5 5.0 12.4 5.0 7.0NEUTMESHX 18*1 ! neutronic meshGRIDY 15.5 mesh in y. cmNEUTMESHY I neutronic mesh pgrid-z 1.0bouncond 2 2 0 0 0 0PLANARREG 1
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18PR ASSIGN I
lyin of core
31.0 7.0 5.0 12.6 5.0 2.0 8.0per assy in x
er assy in y
2.0 30.0 ! mesh in x, cm
THn pingt 271 0 npin ,ngtfa-powpit 5.723993 15.5 !assembly power(Mw) and pitch (cm)
T T T F F ! PPOW, PHST, PTHS, PXESM. PXSS3.1625553 12.6502214 29*15.8127767 10*79.0638836
PMAXS File Name Br-struct.. \xsec\RBWRLRIADF.PMAXS' I.. \xsec\RBWRLR2_ADF.PMAXS' IS..\xse c \RBWRLBI ADF.PMAXS' I..\x s e c \ RBWRLB2_ADF. PMAXS' I..\x s c \RBWRLB3_ADF. PMAXS' 1
'..\xsec\RBWR LFlADF.PMAXS' I.. \x s e c \ RBWR LF2 ADF. PMAXS' I
'..\x s e c \ RBWRLF3ADF. PMAXS' I.. \x s e c \ RBWR EIt ADF. PMAXS' I
\ xsec\RBWR-IB2_ADF.PMAXS' I\x s e c \ RBWR-IB3_ADF. PMAXS' I
xsec \RBWRUFlADF.PMAXS' I\x s ec \ RBWRUF2_ADF. PMAXS'
S\ xse c \RBWR UF3 ADF. PMAXS'\ xsec \RBWR UBI ADF.PMAXS' I\x s e c \ RBWR-UB2_ADF. PMAXS'
x s e c \RBWRUB3_ADF. PMAXS' I\xsec \RBWR UR ADF.PMAXS' I
220
DEPLOUTOPT
TIMESTPIndex
PMAXSF 1PMAXSF 2PMAXSF 3PMAXSF 4PMAXSF 5PMAXSF 6PMAXSF 7PMAXSF 8PMAXSF 9PMAXSF 10PMAXSF 11PMAXSF 12PMAXSF 13PMAXSF 14PMAXSF 15PMAXSF 16PMAXSF 17PMAXS F 18
