The Pennsylvania State University
The Graduate School
College of Engineering
TRANSPORT MODEL BASED ON 3-D CROSS-SECTION
GENERATION FOR TRIGA CORE ANALYSIS
A Thesis in
Nuclear Engineering
by
Nateekool Kriangchaiporn
© 2006 Nateekool Kriangchaiporn
Submitted in Partial Fulfillment
of the Requirements
for the Degree of
Doctor of philosophy
May 2006
The thesis of Nateekool Kriangchaiporn was reviewed and approved* by the following:
Kostadin Ivanov
Professor of Nuclear Engineering
Thesis Advisor
Chair of Committee
Alireza Haghighat
Professor of Nuclear Engineering
C. Frederick Sears
Senior Scientist
Affiliate Professor of Nuclear Engineering
Ludmil Zikatanov
Assistant Professor of Mathematics
Yoursry Azmy
Professor of Nuclear Engineering
Jack Brenizer
Professor of Nuclear Engineering
Chair of Nuclear Engineering
*Signatures are on file in the Graduate School.
ii
ABSTRACT
This dissertation addresses the development of a reactor core physics model based
on 3-D transport methodology utilizing 3-D multigroup fuel lattice cross-section
generation and core calculation for PSBR.
The proposed 3-D transport calculation scheme for reactor core simulations is
based on the TORT code. The methodology includes development of algorithms for 2-D
and 3-D cross-section generation. The fine- and broad- group structures for the TRIGA
cross-section generation problems were developed based on the CPXSD (Contributon
and Point-wise Cross-Section Driven) methodology that selects effective group structure.
Along with the study of cross section generation, the parametric studies for SN
calculations were performed to evaluate the impact of the spatial meshing, angular, and
scattering order variables and to obtain the suitable values for cross-section collapsing of
the TRIGA cell problem.
The TRIGA core loading 2 is used to verify and validate the selected effective
group structures. Finally, the 13 group structure was selected to use for core calculations.
The results agree with continuous energy for eigenvalues and normalized pin power
distribution. The Monte Carlo solutions are used as the references.
iii
TABLE OF CONTENTS
List of Tables
vii
List of Figures
xii
Acknowledgement xiv
CHAPTER 1
1.1
1.2
Introduction ............................................................................................... 1
Background .......................................................................................................... 1
Research Objectives ............................................................................................. 3
CHAPTER 2
Literature Review and Methodology......................................................... 5
2.1 TRIGA Review .................................................................................................... 5
2.2 The Forward Neutron Transport Equation ........................................................... 9
2.3 The Multigroup Discrete Ordinates Equations................................................... 11
2.4 Discrete Ordinate Quadrature Sets..................................................................... 13
2.4.1
Level (Fully) Symmetric (LQN) Quadrature ............................................... 13
2.4.2
Square Legendre-Chebyschev (SLC) Quadrature Set................................. 14
2.5 Resonance Treatment ......................................................................................... 16
2.5.1
Flux Calculator............................................................................................ 16
2.5.2
The Bondarenko Method ............................................................................ 19
2.5.3
CENTRM.................................................................................................... 20
2.6 Group Structure Selection Methodology............................................................ 20
2.7 Code Description................................................................................................ 22
2.7.1 DORT.......................................................................................................... 22
2.7.2 TORT .......................................................................................................... 23
2.8 Applications of Discrete Ordinates Method to Criticality Calculations ............ 24
2.8.1 A Sub-Critical ‘C28’ and a Critical Assembly ........................................... 24
2.8.2
The C5G7 MOX Benchmark ...................................................................... 24
CHAPTER 3
Cross-Section Generation Methodology ................................................. 27
3.1 Cross Section Generation Procedure and Studies .............................................. 27
3.1.1
The Weight Function Study ........................................................................ 30
3.1.2
The Corner-Material Study ......................................................................... 31
3.1.3
Resonance Treatment Study ....................................................................... 33
3.2 Fine Group Structure Selection .......................................................................... 47
3.2.1
Extension of the CPXSD Methodology to Criticality Problem .................. 48
3.3 Cross-Section Collapsing and Homogenization................................................. 49
3.3.1
Fine- to Broad-Group Collapsing ............................................................... 49
3.3.2
Cross-Section Homogenization .................................................................. 50
3.4 Summary ............................................................................................................ 51
CHAPTER 4
Two-Dimensional Cross Section Generation.......................................... 52
4.1 Two-Dimensional Model for Cross-Section Generation ................................... 52
4.2 Fine Group Structure for TRIGA ....................................................................... 54
4.2.1
Fast Range Group Refinement.................................................................... 55
4.2.2
Epithermal Range-Group Refinement ........................................................ 57
iv
4.2.3
Thermal Range-Group Refinement............................................................. 60
4.3 Parametric Studies.............................................................................................. 63
4.3.1
Spatial Mesh, Angular Quadrature, and Scattering Order Studies ............. 64
4.3.2
Qudrature Order Determination.................................................................. 74
4.4 Cross-Section Collapsing ................................................................................... 79
4.4.1
Fast Range-Group Collapsing..................................................................... 80
4.4.2
Epithermal Range-Group Collapsing.......................................................... 81
4.4.3
Thermal Range-Group Collapsing.............................................................. 82
4.5 Two-Dimensional Cross Section Generation for Other Materials ..................... 90
4.5.1
Graphite....................................................................................................... 90
4.5.2
Control Rod................................................................................................. 95
4.6 Cross-Section Homogenization........................................................................ 102
4.7 Summary .......................................................................................................... 105
CHAPTER 5
Three-Dimensional Cross Section Generation...................................... 107
5.1 Three-dimensional model for Cross-Section Generation for Fuel Element..... 107
5.2 Parametric Studies............................................................................................ 109
5.2.1
Spatial Mesh, Angular Quadrature, and Scattering Order Studies ........... 109
5.2.2
Qudrature Order Determination................................................................ 115
5.3 Fine- Group Structure for TRIGA.................................................................... 118
5.3.1
Fast Group Refinement ............................................................................. 118
5.3.2
Epithermal-Group Refinement.................................................................. 121
5.3.3
Thermal-Group Refinement...................................................................... 123
5.4 Cross-Section Collapsing ................................................................................. 128
5.4.1
Fast-Group Collapsing and Axial Nodalization Study ............................. 129
5.4.2
Epithermal Energy Range:........................................................................ 135
5.4.3
Thermal Energy Range: ............................................................................ 136
5.5 Three-Dimensional Cross Section Model for Materials with Non-Fissile
Element ............................................................................................................ 143
5.5.1
Control Rod............................................................................................... 143
5.6 Two-Dimensional vs. Three-Dimensional Cross Sections .............................. 146
5.6.1
Two-Dimensional vs. Three-Dimensional Flux Distribution Collapsing. 146
5.6.2
Two-Dimensional vs. Three-Dimensional Group Structure..................... 148
5.7 Summary .......................................................................................................... 149
CHAPTER 6
Core Simulation..................................................................................... 150
6.1 Mini-Core Simulation ...................................................................................... 150
6.1.1
Mesh Size Study ....................................................................................... 151
6.1.2
Mini-Core Results..................................................................................... 154
6.2 Coarse Group Study ......................................................................................... 157
6.3 Core loading 2 Simulations .............................................................................. 161
6.3.1
Mesh Size Study ....................................................................................... 162
6.3.2
Core Reflector Thickness Study ............................................................... 171
6.3.3 Core Loading 2 – ARI............................................................................... 173
6.3.4 Core Loading 2 – ARO ............................................................................. 177
6.4 Summary .......................................................................................................... 180
v
CHAPTER 7
7.1
7.2
Conclusions and Future Research ......................................................... 181
Conclusions ...................................................................................................... 181
Future Research................................................................................................ 183
References....................................................................................................................... 185
APPENDIX A. TORT INPUT SAMPLE FOR TRIGA................................................. 188
APPENDIX B. TRIGA FISSION SPECTRUM............................................................. 196
vi
LIST OF TABLES
Table 3-1: Results of eigenvalue calculation using MCNP ...............................................32
Table 3-2: Reaction rates with continuous energy cross-section library in MCNP...........36
Table 3-3: Reaction rates with 238-group cross-section library using the Bondarenko
method..............................................................................................................37
Table 3-4: Percent deviations from MCNP .......................................................................37
Table 3-5: Reaction rates with 238-group cross-section library using Flux- Calculator
in NJOY ...........................................................................................................38
Table 3-6: Percent deviations from MCNP .......................................................................38
Table 3-7: Reaction rates with 238-group cross-section library using CENTRM.............39
Table 3-8: Percent deviations from MCNP .......................................................................39
Table 3-9: Reaction rates with 238-group cross-section library using Flux Calculator
in NJOY for U238............................................................................................40
Table 3-10: Percent deviations from MCNP .....................................................................41
Table 3-11: Reaction rates with 238-group cross-section library using Centrm
treatment for Zr and U238 .................................................................................41
Table 3-12: Percent deviations from MCNP .....................................................................42
Table 3-13: Reaction rates with 238-group cross-section library using Centrm
treatment in Zr, U238, and Fe56 .........................................................................43
Table 3-14: Percent deviations from MCNP .....................................................................43
Table 3-15: Reaction rates with 238-group cross-section library, 24192 cells: ................44
Table 3-16: Percent deviations from MCNP .....................................................................45
Table 3-17: Reaction rates with 253-group cross-section library......................................46
Table 3-18: Percent deviations from MCNP .....................................................................46
Table 4-1: Material density of the fuel elements ...............................................................53
Table 4-2: Cladding composition.......................................................................................54
Table 4-3: Fine groups selected in the fast energy range...................................................55
Table 4-4: Eigenvalue results of fine group energy for 8.5% wt. case..............................57
Table 4-5: Eigenvalue results of fine group energy for 12% wt. case...............................57
Table 4-6: Fine groups generated in the epithermal energy range.....................................58
Table 4-7: Eigenvalue results of fine group energy...........................................................59
Table 4-8: Eigenvalue results of fine group energy...........................................................59
Table 4-9: Reaction rate comparison for 8.5% wt. case ....................................................59
Table 4-10: Reaction rate comparison for 12% wt. case ...................................................59
Table 4-11: Fine groups generated in the thermal energy range .......................................60
Table 4-12: Eigenvalue results for fine group energy .......................................................61
Table 4-13: Eigenvalue results for fine group energy .......................................................62
Table 4-14: Reaction rate comparison of 8.5% wt. case....................................................62
Table 4-15: Reaction rate comparison of 12% wt. case.....................................................62
Table 4-16: DORT results with 280-energy group XS and 1554 cells, LevelSymmetric ........................................................................................................65
Table 4-17: DORT results with 280-energy group XS and 6132 cells, LevelSymmetric ........................................................................................................65
vii
Table 4-18: DORT results with 280-energy group XS and 13625 cells, LevelSymmetric ........................................................................................................65
Table 4-19: DORT results with 280-energy group XS and 24192 cells, LevelSymmetric ........................................................................................................66
Table 4-20 DORT results with 280-energy group XS and 1554 cells, LevelSymmetric ........................................................................................................66
Table 4-21: DORT results with 280-energy group XS and 6132 cells, LevelSymmetric ........................................................................................................66
Table 4-22: DORT results with 280-energy group XS and 13625 cells, LevelSymmetric ........................................................................................................67
Table 4-23: DORT results with 280-energy group XS and 24192 cells, LevelSymmetric ........................................................................................................67
Table 4-24: DORT results with 280-energy group XS and 1554 cells, SLC ....................69
Table 4-25: DORT results with 280-energy group XS and 6132 cells, SLC ....................70
Table 4-26: DORT results with 280-energy group XS and 13625 cells, SLC ..................70
Table 4-27: DORT results with 280-energy group XS and 24192 cells, SLC ..................70
Table 4-28: DORT results with 280-energy group XS and 1554 cells, SLC ....................71
Table 4-29: DORT results with 280-energy group XS and 6132 cells, SLC ....................71
Table 4-30: DORT results with 280-energy group XS and 13625 cells, SLC ..................71
Table 4-31: DORT results with 280-energy group XS and 24192 cells, SLC ..................72
Table 4-32: DORT results with 280-energy group XS and 6132 cells, SLC ....................74
Table 4-33: Neutron-production reaction rates from MCNP and DORT ..........................78
Table 4-34: Percentage of relative deviation from MCNP ................................................78
Table 4-35: Comparison between 229G and 280G for 8.5% wt. case...............................80
Table 4-36: Comparison between 229G and 280G for 12% wt. case................................81
Table 4-37: kinf comparison between 229G and 127G for 8.5% wt. case..........................81
Table 4-38: kinf comparison between 229G and 127G for 12% wt. case...........................82
Table 4-39: Reaction rate comparison between 229G and 127G for 8.5% wt. case .........82
Table 4-40: Reaction rate comparison between 229G and 127G for 12% wt. case ..........82
Table 4-41: Result comparison in thermal energy range for 8.5% wt. case ......................83
Table 4-42: Result comparison in thermal energy range for 12% wt. case .......................84
Table 4-43: DORT results with 280-energy group XS and 6132 cells for 8.5% wt.
case...................................................................................................................85
Table 4-44: DORT results with 12-energy group XS, 6132 cells for 8.5% wt. case.........85
Table 4-45: DORT results with 280-energy group XS and 6132 cells for 12% wt. case..86
Table 4-46: DORT results with 12-energy group XS, 6132 cells for 12% wt. case..........86
Table 4-47: DORT calculation with 280-group cross section library for 8.5% wt. case...87
Table 4-48: DORT calculation with 12-group cross section library for 8.5% wt. case.....88
Table 4-49: Reaction rates deviation between 280G and 12G for 8.5% wt. case..............88
Table 4-50: DORT calculation with 280-group cross section library for 12% wt. case....89
Table 4-51: DORT calculation with 12-group cross section library for 12% wt. case......89
Table 4-52: Reaction rates deviation between 280G and 12G for 12% wt. case...............90
Table 4-53: Eigenvalue results for graphite cross section generation model ....................92
Table 4-54: MCNP reaction rates ......................................................................................92
Table 4-55: DORT, 280GP3 reaction rates........................................................................92
Table 4-56: DORT, 280GP1 reaction rates........................................................................93
viii
Table 4-57: DORT, 12GP1 reaction rates..........................................................................93
Table 4-58: Percentage deviation between DORT, 280GP3 and MCNP ..........................93
Table 4-59: Percentage deviation between DORT, 280GP1 and MCNP ..........................94
Table 4-60: Percentage deviation between DORT, 12GP1 and MCNP ............................94
Table 4-61: Percentage deviation between DORT, 12GP1 and 280GP1 ..........................94
Table 4-62: Eigenvalues calculated by DORT and MCNP ...............................................96
Table 4-63: Reaction rates calculated by MCNP...............................................................96
Table 4-64: The reaction rates calculated by DORT with 280 groups, S10 quadrature
order .................................................................................................................96
Table 4-65: Percent deviation of reaction rates between DORT 280G and MCNP ..........97
Table 4-66: Percent deviation of reaction rates between DORT 280GP1 and 280GP3 ....97
Table 4-67: Kinf results predicted by DORT with 280G....................................................99
Table 4-68: Reaction rates calculated by DORT with 280 groups, S10 quadrature
order for Model#2 ............................................................................................99
Table 4-69: Reaction rates calculated by DORT with 280 groups, S10 quadrature
order for Model#3 ..........................................................................................100
Table 4-70: Percent deviation of reaction rates between DORT 280G S10 Model #2
and MCNP .....................................................................................................100
Table 4-71: Percent deviation of reaction rates between DORT 280G S10 Model #3
and MCNP .....................................................................................................101
Table 4-72: Eigenvalues calculated by DORT and MCNP .............................................101
Table 4-73: Reaction rates calculated by DORT with 12 groups, S10 quadrature order
and P3 scattering order...................................................................................102
Table 4-74: Percent deviation of reaction rates between DORT 12GP3 and 280GP3 ....102
Table 4-75: Kinf calculated by DORT (3–region combination) .......................................104
Table 4-76: Kinf calculated by DORT (4–region combination) .......................................105
Table 5-1: Material density of the fuel elements .............................................................108
Table 5-2: TORT results with 238-energy group XS, Level-Symmetric ........................110
Table 5-3: TORT results with 238-energy group XS, SLC .............................................110
Table 5-4: TORT results with 238-energy group XS, S8 (SLC), P1...............................113
Table 5-5: TORT results with 238-energy group XS, S8 (SLC), P1...............................114
Table 5-6: TORT results with 238-energy group XS and 48x59x55 cells ......................115
Table 5-7: Neutron production reaction rate and percent deviations...............................116
Table 5-8: Fine groups generated in the fast energy range..............................................119
Table 5-9: Eigenvalue results of fine group energy for 8.5% wt. case............................120
Table 5-10: Fine groups generated in the epithermal energy range.................................121
Table 5-11: Eigenvalue results of fine group energy.......................................................123
Table 5-12: Reaction rate comparison for 8.5% wt. case ................................................123
Table 5-13: Fine groups generated in the thermal energy range .....................................124
Table 5-14: Eigenvalue results for fine group energy .....................................................125
Table 5-15: Reaction rate comparison of 8.5% case........................................................125
Table 5-16: Group structure of the 280 fine groups ........................................................126
Table 5-17: Number of groups for each energy range.....................................................129
Table 5-18: Eigenvalue results for 3D, 8.5% fuel cell.....................................................131
Table 5-19:The minimum and maximum of mesh-wise reaction rate deviations for
each layer between case 2 and case1 .............................................................131
ix
Table 5-20:The minimum and maximum of mesh-wise reaction rate deviations for
each layer between case 3 and case2 .............................................................132
Table 5-21: Number of groups for each energy range.....................................................134
Table 5-22: Eigenvalue results for 3D, 8.5% fuel cell.....................................................134
Table 5-23: Number of groups for each energy range.....................................................135
Table 5-24: Eigenvalue results for 3D, 8.5% fuel cell.....................................................135
Table 5-25: Reaction rate comparison for broad group in epithermal range...................136
Table 5-26: Number of groups for each energy range.....................................................137
Table 5-27: Eigenvalue results for 3D, 8.5% fuel cell.....................................................137
Table 5-28: Result comparison in thermal energy range .................................................138
Table 5-29: Energy boundaries of 26-group structures ...................................................138
Table 5-30: Eigenvalues calculated by TORT and MCNP..............................................140
Table 5-31: MCNP calculation with continuous cross section library ............................140
Table 5-32: TORT calculation with 280-group cross section library ..............................141
Table 5-33: TORT calculation with 26-group cross section library ................................141
Table 5-34: Reaction rates deviation between 280G and MCNP ....................................142
Table 5-35: Reaction rates deviation between 26G and MCNP ......................................142
Table 5-36: Reaction rates deviation between 26G and 280G ........................................143
Table 5-37: Eigenvalues calculated by TORT and MCNP..............................................144
Table 5-38: Reaction rates calculated by MCNP.............................................................145
Table 5-39: Reaction rates calculated by TORT with 26 groups, S8 quadrature order
and P1 scattering order...................................................................................145
Table 5-40: Reaction rates calculated by TORT with 26 groups, S8 quadrature order
and P3 scattering order...................................................................................145
Table 5-41: Percent deviation of reaction rates between TORT 26GP1 S8 and MCNP .146
Table 5-42: Percent deviation of reaction rates between TORT 26GP3 S8 and MCNP .146
Table 5-43: Eigenvalue results 2-D vs 3-D flux distribution collapsing cases................147
Table 5-44: Percentage deviation of reaction rates between 2-D and 3-D cross-section
collapsing cases..............................................................................................147
Table 5-45: Number of groups placed in each energy range ...........................................148
Table 5-46: Eigenvalue results 2-D vs 3-D group structure cases...................................148
Table 5-47: Percentage deviation of reaction rates between 2-D and 3-D group
structure cases ................................................................................................149
Table 6-1: Eigenvalues calculated from TORT ...............................................................151
Table 6-2: Percentage deviation of reaction rates between 2nd model and 1st model ......152
Table 6-3: Percentage deviation of reaction rates between 3rd model and 1st model.......152
Table 6-4: Percentage deviation of reaction rates between 4th model and 1st model.......152
Table 6-5: Percentage deviation of reaction rates between 5th model and 1st model.......153
Table 6-6: Eigenvalues calculated from TORT ...............................................................153
Table 6-7: Percentage deviation of reaction rates between 2nd model and 1st model ......154
Table 6-8: Percentage deviation of reaction rates between 3rd model and 1st model.......154
Table 6-9: Eigenvalues calculated from TORT and MCNP............................................155
Table 6-10: MCNP reaction rates ....................................................................................155
Table 6-11: TORT reaction rates for P1 case ..................................................................156
Table 6-12: TORT reaction rates for P3 case ..................................................................156
Table 6-13: Percentage deviation of reaction rates between Tort-P1 and MCNP...........156
x
Table 6-14: Percentage deviation of reaction rates between Tort-P3 and MCNP...........157
Table 6-15: Eigenvalues calculated from TORT .............................................................158
Table 6-16: Percentage deviation of reaction rates between 12G and 26G cases ...........159
Table 6-17: Energy boundaries of 12-group structures ...................................................159
Table 6-18: Eigenvalues calculated by TORT.................................................................160
Table 6-19: Percentage deviation of reaction rates between 13G and 26G cases ...........160
Table 6-20: Energy boundaries of 13-group structure.....................................................160
Table 6-21: Eigenvalues calculated by TORT.................................................................164
Table 6-22: Percentage deviation of reaction rates between 2nd model and 1st model ....164
Table 6-23: Percentage deviation of reaction rates between 3rd model and 1st model.....164
Table 6-24: Percentage deviation of reaction rates between 4th model and 1st model.....165
Table 6-25: Eigenvalues calculated from TORT .............................................................167
Table 6-26: Percentage deviation of reaction rates between 2nd model and 1st model ....167
Table 6-27: Percentage deviation of reaction rates between 3rd model and 1st model.....167
Table 6-28: Percentage deviation of reaction rates between 4th model and 1st model.....168
Table 6-29: Eigenvalues calculated by TORT.................................................................169
Table 6-30: Percentage deviation of reaction rates between 2nd model and 1st model ....169
Table 6-31: Percentage deviation of reaction rates between 3rd model and 1st model.....169
Table 6-32: Percentage deviation of reaction rates between 4th model and 1st model.....170
Table 6-33: Eigenvalues calculated from TORT .............................................................170
Table 6-34: Percentage deviation of reaction rates between 2nd model and 1st model ....171
Table 6-35: Percentage deviation of reaction rates between 3rd model and 1st model.....171
Table 6-36: Eigenvalues calculated by TORT.................................................................173
Table 6-37: Eigenvalues calculated from TORT and MCNP..........................................176
Table 6-38: Eigenvalues calculated from TORT and MCNP..........................................178
xi
LIST OF FIGURES
Figure 2-1 : Core cycle 52 ...................................................................................................6
Figure 3-1: Procedure for generating cross section library................................................28
Figure 3-2: Unit cell for TRIGA fuel element ...................................................................30
Figure 3-3: MCNP-predicted TRIGA spectrum ................................................................31
Figure 3-4: Cells Models for MCNP .................................................................................32
Figure 3-5: Flux distribution in fuel Region ......................................................................33
Figure 3-6: Pointwise absorption cross section of Zr ........................................................34
Figure 3-7: Pointwise absorption cross section of U238 ...................................................35
Figure 3-8: Mesh Model from 1554 cells to 24192 cells...................................................44
Figure 3-9 Fuel cell homogenization .................................................................................50
Figure 4-1: Cross section generation model ......................................................................53
Figure 4-2: Importance of groups of 238G and 246G libraries .........................................56
Figure 4-3: Importance in groups of 246G library.............................................................58
Figure 4-4: Importance in groups of 246G, 254G and 280G libraries...............................61
Figure 4-5: P1 scattering order with level symmetric quadrature order ............................68
Figure 4-6: P3 scattering order with level symmetric quadrature order ............................68
Figure 4-7: P1 scattering order with Square Legendre-Chebyshev quadrature order .......72
Figure 4-8: P3 scattering order with Square Legendre-Chebyshev quadrature order .......73
Figure 4-9: Flux distribution of group 23rd ........................................................................75
Figure 4-10: Flux distribution for group 242nd ..................................................................76
Figure 4-11: Detector locations .........................................................................................77
Figure 4-12: 2-D model for graphite XS generation..........................................................91
Figure 4-13: 2-D model for control rod XS generation .....................................................95
Figure 4-14: Absorption reaction rate as a function of B4C radius....................................98
Figure 4-15: Three-Region Homogenization...................................................................103
Figure 4-16: Four-Region Homogenization.....................................................................103
Figure 5-1: 3D cross section generation model ...............................................................108
Figure 5-2 Eigenvalue behavior under variation of scattering order and level
symmetric quadrature.....................................................................................111
Figure 5-3 Eigenvalue behavior under variation of scattering order and Square
Legendre-Chebyshev quadrature ...................................................................111
Figure 5-4: Eigenvalue behavior with different radial-mesh model................................113
Figure 5-5: Eigenvalue behavior with different axial-mesh model .................................115
Figure 5-6: Flux distribution for each quadrature order ..................................................117
Figure 5-7: Importance in groups of 238G and 246G libraries .......................................120
Figure 5-8: Importance in groups of 246G libraries ........................................................122
Figure 5-9: Importance in groups of 246G, 254G and 280G libraries.............................124
Figure 5-10 Axial mesh size used in nodal length collapsing study................................130
Figure 5-11: 3-D model for control rod XS generation ...................................................144
Figure 5-12: A pin cell model in axial direction..............................................................147
Figure 6-1: Configuration of Mini-core...........................................................................150
Figure 6-2: Importance distribution of 26-group structure ..............................................158
Figure 6-3: TRIGA core loading 2...................................................................................161
xii
Figure 6-4: Pin cell in axial direction ..............................................................................162
Figure 6-5: The studied models .......................................................................................163
Figure 6-6: The studied models .......................................................................................166
Figure 6-7: Core loading 2 with 15 cm reflector thickness .............................................172
Figure 6-8: Radial-cross-section view of ARI .................................................................174
Figure 6-9: Axial-cross-section view of ARI...................................................................175
Figure 6-10: Normalized pin-power distribution for ARI ...............................................176
Figure 6-11: Radial-cross-section view of ARO .............................................................177
Figure 6-12: Axial-cross-section view of ARO ...............................................................178
Figure 6-13: Normalized pin-power distribution for ARO..............................................179
Figure B-1: 8.5% wt. fuel fission spectrum of 280 groups.............................................. 196
Figure B-2: 8.5% wt. fuel fission spectrum of 26 groups……………………………… 196
Figure B-3: 8.5% wt. fuel fission spectrum of 13 groups……………………………… 197
Figure B-4: 12% wt. fuel fission spectrum of 280 groups……………………………... 197
Figure B-5: 12% wt. fuel fission spectrum of 26 groups………………………………. 198
Figure B-6: 12% wt. fuel fission spectrum of 13 groups………………………………. 198
xiii
ACKNOWLEDGEMENTS
I express deep thanks to my family, especially my father, Kriang Kriangchaiporn,
and my mother, Sangiam Kriangchaiporn, for their love, prayer, support and
encouragement throughout long journey of my study.
I also would like to express my gratitude to my academic advisor, Dr. Kostadin
Ivanov, for his motivation, enthusiasm and guidance, which played a major role in the
successful completion of the work. I would like to especially thank Dr. Frederick Sears
for all his questions during the meetings, which have been very useful for this research
project. I gratefully acknowledge for all the suggestions of the committee, Dr. Alireza
Haghighat, Dr. Yoursry Azmy, and Dr. Ludmil Zikatanov.
I would like to thank the Radiation Science and Engineering Center (RSEC) for
financial support of this research project.
I would like to extend my thanks to my best friend, Dr. Sathaporn Opasanon, my
wonderful roommates Hathairat Maneetes (Bell) and Tianboon Soh (Chris) for their
friendship, help, comfort and always being there whenever I needed.
Lastly, I thank God who has provided me with all opportunities and blessings.
xiv
CHAPTER 1
Introduction
The demand for accurate simulations of nuclear reactors is increasing to enable
improving the reactor design, safety and economy. The computer simulation of a reactor
core is an important aspect of both designing new reactors and analyzing the safety of
existing reactors. Innovative three-dimensional (3-D) core models are necessary to
achieve the desired accuracy. Since these types of numerical simulations tend to be
computationally expensive, further developments are needed to address both accuracy
and efficiency. Recent progress in computer technology combined with new methods and
code developments makes feasible new calculation schemes capable of providing
accurate solutions in an efficient manner.
1.1
Background
Generally, the reactor core physics calculation process contains two main steps.
The first step is to compute the group cross sections for the various regions of a nuclear
reactor. The second is to employ these cross sections by using varying methods to
analyze the reactor core. This modeling approach is applied to both steady state and
transient calculations. Most of the core analysis methodologies utilize approximate
methods to simplify the complex problems associated with reactor core modeling. The
cross sections are generated in two-dimensional (2-D) instead of three-dimensional (3-D)
geometries. In addition, the diffusion theory methodology, which is derived from a
transport equation, is used to analyze the full core using the 2-D cross-section library.
These approaches have three main weaknesses. The first weakness is in the cross-section
1
generation. Current lattice physics codes suffer from a combination of some of the
following shortcomings: 2-D geometry approximation, shape leakage approximation, and
approximated self-shielding trestment for the burnable-poison-containing fuel rods. The
reason for the aforementioned shortcomings is in the fact that lattice physics codes are
generally based on the collision probability method (CPM). This method is practical only
in 2-D geometries, since it becomes cumbersome and impractical in 3-D geometries for
arbitrary boundary conditions, combined geometric shapes, or where detailed information
of problems is required. The second weakness is in the current cross-section modeling
approach, which is based on cross-section parameterization and functionalization
techniques. These techniques cause uncertainties in the evaluated cross sections from the
cross-section libraries, which are used later in the core simulations. The third weakness is
the diffusion approximation in the full core calculations. This approximation is not
accurate at the interfaces between different dissimilar assemblies. Transport effects are of
particular importance in highly heterogeneous cores where the traditional procedure of
applying various transport corrections is unsatisfactory.
In order to achieve the desired accuracy, a three-dimensional (3-D) model based
on the exact transport theory method is necessary to simulate the real problems. In the
past, 3-D numerical simulations were computationally expensive and impractical.
However, recent advancements in computer technology, combined with new methods and
code developments makes it feasible to develop novel calculation schemes capable of
providing accurate solutions in an efficient manner.
The Pennsylvania State University Breazeale Reactor (PSBR) is a TRIGA Mark
III research reactor designed for 1 MWt power generation. It is a light water cooled, pool
2
type reactor, which utilizes U-ZrH 20% enriched fuel elements containing 8.5 wt% and
12 wt% uranium [Ref.14]. The uniform lattice in PSBR has a hexagonal shape and the
PSBR core has relatively small dimensions as compared to the commercial light water
reactors (LWRs). The other differences from LWRs especially those affecting the
neutronics characteristics are as follows:
•
TRIGA is an over-moderated reactor, where the majority of neutron moderation
occurs in the fuel meat (UZrH),
•
TRIGA core has more pronounced upscattering effects, due to the Zr-H mixture
crystalline structure, which cause hardening of the neutron spectrum as compared
to LWRs
The above-mentioned facts show the uniqueness of this reactor type and require
the development of its own core analysis methodology and cross-section library. Since
TRIGA has a relatively small reactor core, it’s modeling even using higher order
transport methods does not require prohibitly large computer resources. This fact along
with the availability of measured data makes TRIGA core an appropriate test
environment for testing new methodologies.
1.2
Research Objectives
This research addresses the development of a state-of-the-art reactor core physics
model based on 3-D transport methodology utilizing 3-D multigroup fuel lattice crosssection generation and core calculation. The focus of the proposed research is a new
methodology for enhanced core physics simulation of the PSBR.
The proposed 3-D transport calculation scheme for reactor core simulations is
based on the TORT code. The complete methodology includes development of
3
algorithms for 3-D cross-section generation and modeling. This will solve several major
weaknesses of the current reactor core analysis methodology (the diffusion
approximation of the whole core calculations), the shortcomings of generation of
multigroup cross section, and the approximations introduced with cross-section
parameterization and functionalization. In fact, in this research instead of proposing
incremental improvements to the TRIGA (research reactor) analysis methodology, the
performed research results in a new generation of reactor core analysis methods. The
objectives of the proposed research are formulated as development and implementation
of
(a) An efficient transport method for 3-D reactor core simulation for steady-state
calculations.
(b) An innovative algorithm for 3-D cross-section generation and modeling.
(c) An effective group structure for the TRIGA reactor.
(d) A systematic validation of the new calculation scheme against Monte Carlo
results for the PSU TRIGA reactor.
The expected outcome of the above-described objectives is the development of
new methodology for accurate 3-D reactor core analysis in an efficient manner. This
methodology will be validated for the TRIGA reactor and can be later expanded for
power reactor applications. Increasing the accuracy and efficiency of core analysis
methodologies can directly improve both safety and economy of nuclear power reactors.
4
CHAPTER 2
Literature Review and Methodology
This chapter describes the TRIGA reactor and several research studies that have
been performed for analysis of this reactor. The linear Boltzmann equation in multigroup
form is presented along with the discrete ordinates method used for its solution. Two
types of quadrature order techniques are discussed: i) level (full) symmetry, and ii)
Legendre-Chebyshev. Several techniques of resonance treatment are explained for
accounting of self-shielding effect in cross sections. Finally, the group-structure selection
to generate a multigroup cross section library is discussed.
2.1
TRIGA Review
The PSBR is a TRIGA Mark III research reactor manufactured by General
Atomic. It has been operated since 1965, when the core was upgraded from MTR type
fuel. The PSBR is a light water cooled, pool type reactor designed for 1 MW (t) steadystate power operation (up to 2000 MW when pulsing) with natural circulation cooling. It
is used for experimental, training, educational and service purposes.
The PSBR core system was first loaded in 1965 with only 8.5 % wt ZrHx-U fuel.
Since July 1972, the core has been reloaded with fresh 12 % wt ZrHx-U fuel elements, six
at each reload. Currently, the PSBR is operated using core cycle 52 as shown in Figure
2-1. The uniform lattice in PSBR is formed in hexagonal shape. The center of the core is
the location of the central thimble (the water rod), which is surrounded by hexagonal
rings. The rings running from the center outward are designated B, C, D, E, F and so on,
respectively. There are 102 fuel rods; 34 of them are 12 % wt. and 68 of them are 8.5 %
wt. both with a 20% uranium-235 enrichment. Three control rods (shim, regulating and
safety) are fuel follower control rods driven by motor. They are composed of graphite at
5
the top and bottom, fuel and absorber (borated graphite) are in the middle. The fourth
control rod is the transient rod (air rod), the only control rod without fuel material driven
by an electro-pneumatic during the steady state. The neutron source used in PSBR is a 3Curie americium-beryllium (Am-Be) neutron source doubly encapsulated in type 304L
stainless steel.
A
B
C
D
air
E
F
G
H
I
air
SA
SH
RR
TR
8.5 wt %
12 wt%
C.R.
Source
Figure 2-1 : Core cycle 52
6
In the past, the TRIGA core management model (TRICOM) [Ref.18] was
developed based on old codes like PSU-LEOPARD, EXTERMINATOR2 and MCRAC.
The core fuel management plan of PSBR has been developed and verified based on
TRICOM during the years by the researchers and staff of the reactor for fuel management
and safety analyses. However, these outdated tools have modeling limitations that
introduce large uncertainties in the calculated parameters. Subsequently, the calculated
results have to be normalized to the measured data in order to be used for analysis.
In 1994, analytical models of the TRIGA core configuration based on the Monte
Carlo Method were developed and applied in the framework of Y.S. Kim’s Master thesis
[Ref.20]. The reactor core power distribution was examined using MCNP code for
criticality simulations and ORIGEN2 for the depletion calculations. The results indicated
that a maximum of 21% of the U235 was depleted in 8.5% fuel rods, and a maximum of
15% of the U235 was depleted in 12% fuel rods. In the analyzed core configuration, the
power peaking factor was extremely high, but it can be reduced by using a proper core
configuration. Thus, the improvement of the core configuration was investigated with a
goal to gain lower peaking factor (lower maximum temperature) and minimal change of
reactivity relative to previous configuration. However, the Monte Carlo based calculation
method is not practical for routine use because it is very time-consuming, but it can be
used to generate reference results for verification of the more efficient deterministic
codes.
In 2000, a new Advanced Fuel Management System (AFMS) [Ref.14] was
developed based on the HELIOS lattice-physics code and the multi-dimensional nodal
diffusion code ADMARC-H. The modeling deficiencies of the old TRICOM code system
7
are corrected on both levels; the cross-section generation and the core simulation. The
HELIOS code was used to generate the cross-section library. HELIOS improved the
geometry modeling by explicitly modeling the hexagonal unit cell, and therefore allowing
for a better thermalization model. The transport theory and CCCP methods in HELIOS
are superior to slowing down theory approximations in LEOPARD, especially in the case
of TRIGA, which uses a hydride fuel. The ADMARC-H code uses a 3-D full core
hexagonal geometry and a 3-D macroscopic semi-implicit burnup model. It yields more
accurate results than those predicted by the 2-D finite-difference MCRAC code in onequarter rectangular core geometry.
The cross-section generation and modeling in the aforementioned research was
developed in 2-D geometry approximation and using off-line calculations. Furthermore,
the diffusion approximation in the full core calculations is the cause of degradation in
accuracy at the interfaces between different regions. In summary, the following
shortcomings of the current core analysis methodology have to be addressed: the use of
diffusion approximation of the whole core calculations, 2-D cross-section generation and
depletion, and cross-sections parameterization. Hence, we further develop new
algorithms and methods for fuel management based on 3-D transport theory. These
methodologies can be applied for accurate determination of flux/power distribution and
isotropic depletion of PSBR in an efficient manner, which is the goal for this research.
8
2.2
The Forward Neutron Transport Equation
The neutron transport equation is given by the linear form of the Boltzmann
equation. The linear form is derived by ignoring neutron-neutron interactions. The timeindependent neutron transport equation with no external source is given below [Ref.5].
∞
v
v
v
ˆ
ˆ
ˆ
ˆ ′σ (rv, E ′ → E , Ω
ˆ ′⋅Ω
ˆ )Ψ (rv , E ′, Ω
ˆ ′)
Ω ⋅ ∇ Ψ ( r , E , Ω ) + σ t ( r , E ) Ψ ( r , E , Ω ) = ∫ dE ′ ∫ dΩ
s
0
4π
χ (E) ∞
v
v
ˆ ′)
+
dE ′νσ f (r , E ′) ∫ dΩ′Ψ (r , E ′, Ω
∫
4π 0
4π
Equation 2-1
The terms on the left hand side of Equation 2-1 represents the loss, and the right
hand side represents the gain of the neutrons in a phase space. Each term is explained
[Ref.5] as follows.
ˆ ⋅ ∇Ψ (rv , E , Ω
ˆ )dEdΩdV
Streaming Term: Ω
This term gives the flow of neutrons. Ω̂ is the unit vector that gives the direction
v
ˆ ) is the angular flux. Angular flux is defined as the expected
of a particle and Ψ (r , E , Ω
v
rate of particles crossing a d 2 r at position r , with energies between E and E+dE,
traveling in directions dΩ̂ about Ω̂ .
v
v
ˆ )dEdΩdV
Collision Term: σ t (r , E )Ψ (r , E , Ω
This term gives the removal rate of neutrons due to all types of interactions in a
v
volume element d 3 r , about r , with energies between E and E+dE, traveling in
directions dΩ̂ about Ω̂ . Interactions include scattering (elastic and inelastic) and
9
v
absorption ((n,f),(n,2n),(n,p),(n,γ),etc.). σ t (r , E ) is the total interaction macroscopic
v
cross section at position r and energy E. It gives the probability per unit length that a
neutron will have an interaction of any type.
∞
ˆ ′σ (rv, E ′ → E , μ )Ψ (rv, E ′, Ω
ˆ ′)dEdΩdV
Scattering Term: ∫ dE ′ ∫ dΩ
s
0
0
4π
This term gives the rate of scattering of particles (in a volume element d 3 r , about
v
ˆ ′ ) into energies between E and E+dE,
r , with energy between E ′ and direction Ω
v
traveling in directions dΩ̂ about Ω̂ , in dV about r . Integration is performed over all
v
incoming energies and directions. σ s (r , E ′ → E , μ 0 ) is the macroscopic differential
scattering cross section and defines the probability per unit length that neutrons, at
v
ˆ ′ are scattered into dE about E , and dΩ̂ about Ω̂ .
position r , energy E ′ , direction Ω
Note that the scattering cross section does not depend on the initial and final directions
separately, but rather on the angle between the incident and emerging particle (i.e.,
ˆ ′⋅Ω
ˆ ).
μ0 = Ω
χ (E) ∞
v
v
dE ′νσ f (r , E ′)Φ (r , E ′)dEdΩdV
Fission Term:
∫
4π 0
This term gives the rate of fission neutrons generated in dE about E, dΩ̂ about
v
Ω̂ , in dV about r . χ ( E ) is the fraction of fission neutrons emitted per unit energy. ν is
v
the average number of neutrons emitted per fission event. σ f (r , E ′) is the macroscopic
v
fission cross section. The scalar flux is formulated as Φ(r , E ′) =
v
∫ dΩˆ ′Ψ (r , E ′, Ωˆ ′) .
4π
10
2.3
The Multigroup Discrete Ordinates Equations
In order to solve the transport equation with a deterministic computational
method, discretization of the energy, angular and spatial variables are applied to the
transport equation (Equation 2-1). First, we present the multigroup equations for timeindependent criticality or eigenvalue problems. It is derived by integrating the linear
Boltzmann equation over each energy interval g, as given below.
ˆ ⋅ ∇ dEΨ (rv , E , Ω
ˆ ) + dEσ (rv, E )Ψ (rv, E , Ω
ˆ)
Ω
∫
∫ t
g
=
g
G
v
v
∑ ∫ dE ∫ dE ′ ∫ dΩˆ ′σ s (r , E ′ → E , μ 0 )Ψ (r , E ′, Ωˆ ′)
g ′=1g
+
g′
4π
G
1
dE
χ
(
E
)
∑
4π ∫g
g ′=1
v
v
∫ dE ′νσ f (r , E ′)Φ(r , E ′)
Equation 2-2
g′
Equation 2-2 is re-written by preserving reaction rates in each term:
ˆ ⋅ ∇Ψ (rv, Ω
ˆ ) + σ (rv )Ψ (rv , Ω
ˆ)=
Ω
g
t
g
+
G
v
v
∑ ∫ dΩˆ ′σ s , g ′→ g (r , μ 0 )Ψg ′ (r , Ωˆ ′)
g ′=14π
χg
4π
G
v
v
∑νσ f , g ′ (r )Φ g′ (r )
Equation 2-3
g ′=1
v ˆ
The group flux Ψg (r , Ω
) is defined as:
v ˆ
v
ˆ)
Ψg (r , Ω
) = ∫ dEΨ (r , E , Ω
Equation 2-4
g
The total, scattering and fission group constants are given by Equations 2-5, 2-6
and 2-7, respectively.
11
v
σ t , g ( r ) ∫ dΩ =
∫ dEσ
g
t
v
v
ˆ)
(r , E )Ψ (r , E , Ω
v
∫ dΩ∫ dEΨ (r , E, Ωˆ )
Equation 2-5
g
v
σ s , g ′ → g ( r , μ 0 ) ∫ dΩ ′ =
∫ dE ∫ dE ′σ
g ′′
g
s
v
v
ˆ ′)
(r , E ′ → E , μ 0 )Ψ (r , E ′, Ω
v
∫ dΩ′ ∫ dE ′Ψ (r , E ′, Ωˆ ′)
Equation 2-6
g′
v
v
νσ f , g ′′ (r ) =
v
∫ dE ′νσ f (r , E ′)Φ(r , E ′)
g′
v
∫ dE ′Φ(r , E ′)
Equation 2-7
g′
Finally, the group fission spectrum is defined in Equation 2-8.
χ g = ∫ dEχ ( E )
Equation 2-8
g
The Discrete Ordinate Method (Sn) is one of the most widely used techniques to
solve the Linear Boltzmann equation in terms of discretization of the angular variable. In
this method, the Boltzmann equation is solved for a number of discrete directions Ω̂ m , to
each of which is associated a weight wm . Each weight represents a segment or area
ˆ on the unit directional sphere. Normally, these areas are expressed in units of 4π, so
ΔΩ
m
that wm =
ˆ
ΔΩ
m
and
4π
∑ wm = 1
Equation 2-9
m
12
The Boltzmann equation for arbitrary direction Ω̂ m is given by
ˆ ⋅ ∇Ψ (rv, Ω
ˆ ) + σ (rv, )Ψ (rv, Ω
ˆ ) = q (rv, Ω
ˆ ),
Ω
m
m
t
m
m
Equation 2-10,
v ˆ
where the group index g is suppressed and q (r , Ω
m ) includes scattering from other
energy groups, scattering at the given energy from other directions, fission and any other
particle sources.
2.4
Discrete Ordinate Quadrature Sets
The choice of ordinate sets for discrete ordinates approximation is one of the
major modeling methods used to apply the SN method for solving different problems.
There are several techniques for the generation of discrete ordinates and associated
weights. Here, we present two techniques of discrete quadrature orders, level (fully)
symmetric quadrature (LQN) and Square Legendre-Chebyshev quadrature (SLC).
2.4.1 Level (Fully) Symmetric (LQN) Quadrature
Level symmetric quadratures are used for general applications. Full symmetry
requires that Ω̂ be invariant under all 90° rotations about any axis. Hence, each set of
coordinates must be symmetric with respect to the origin and the set of points on each
axis must be the same. For N levels, the total of N(N+2) directions are on the unit sphere
(N(N+2)/8 per octant) with the same set of N/2 positive values of the direction cosines
with respect to each of the three axes. There is only one degree of freedom in determining
the direction cosines of the ordinates, the choice of μ1. Then, the other values of μn are
determined
based
on
Equation
2-11
by
considering
μ i2 + η 2j + ξ k2 = 1
and
13
i+ j+k =
N
+ 2 , where N refers to the number of levels and i,j,k are indices for
2
direction cosines.
μ i2 = μ12 + (i − 1)Δ
Equation 2-11
2(1 − 3μ12 )
N
1
where Δ =
and 2 ≤ i ≤ ;0 < μ12 ≤
( N − 2)
2
3
The weights associated with directions are obtained as follows:
M
∑ wi
= 1.0
Equation 2-12
i =1
M
M
M
i =1
i =1
i =1
M
M
M
i =1
i =1
i =1
∑ wi μ in =∑ wiη in =∑ wiξ in
=0.0
∑ wi μ in =∑ wiη in =∑ wiξ in =
1
n +1
for n odd
Equation 2-13
for n even
Equation 2-14
Equation 2-12 is a normalization condition for the weights, Equation 2-13 and
Equation 2-14 represent the odd-moment and even-moment conditions, respectively. The
odd-moment condition is automatically satisfied over the entire range of μ because of
symmetry. The even-moment condition in Equation 2-14 is required in order to properly
integrate the Legendre polynomials. This technique is limited to order 20, because
beyond this order some of weights become negative.
2.4.2 Square Legendre-Chebyschev (SLC) Quadrature Set
The SLC methodology has been derived in order to relax the constraints imposed
by the LQN method. The ξ levels are set on the z-axis equal to the roots of Legendre
14
polynomials and the azimuthal angles on each level are calculated by the roots of the
Chebyschev polynomials. Points lie on the unit sphere on ξ levels but not on μ or η
levels, and point weights are the product of Legendre and Chebyschev weights. The use
of the same Chebyschev quadrature on each ξ level gives μ and point weights as the
following formulation:
μ i 0 = − 1 − ξ i2
μ ij = 1 − ξ i2 cos(
pi = 0
2n − 2 j + 1
π)
2n
pi =
wi
n
Equation 2-15
i = 1,2,…,n/2 and j = 1,2,…,n
The μ points with zero weights are those incoming directions used as starting
directions in the current version of the SN discrete ordinates transport code. For the same
ξi and wi the use of a different order Chebyschev quadrature on each ξ level gives μ and
point weights in Equation 2-16.
μ i 0 = − 1 − ξ i2
μ ij = 1 − ξ i2 cos(
pi = 0
2n − 4i − 2 j + 5
π)
2n − 4i + 4
pi =
wi
Equation 2-16
n + 2 − 2i
i = 1,2,…,n/2 and j = 1,2,…,(n+2-2i)
This SLC technique gives a significant improvement over level symmetric
quadrature, since the SLC quadraure set completely satisfies the even-moment condition
for all axes. It is another option to study the effect of types of quadrature set on our
problem.
15
2.5
Resonance Treatment
In the “resonance energy” region, from roughly 1 eV to 100 keV, the main
absorption of neutrons by heavy nuclei takes place at pronounced peaks or resonances of
cross section. The shielding effects are presented in this region because of the flux dip at
resonances. The resonance structure can be separated into two regions, resolved and
unresolved.
In resolved resonance region, the resonances are wide when compared to the
scattering ranges for the mixtures in a particular configuration. It is in the range of eV up
to a few keV. This region is significant for thermal reactors. In the unresolved resonance
region, the resonances are not able to achieve adequate resolution of the individual
resonances. The neutron absorption in this region is important for fast reactors.
An appropriate treatment of the resonance absorption is needed in order to obtain
more accurate solutions. The three selected methods for resonance shielding treatment are
explained as follows.
2.5.1 Flux Calculator
The narrow resonance approach is quite useful for practical fast reactor problems.
However, for nuclear systems sensitive to energies from 1 to 500 eV, there are many
broad- and intermediate-width resonances, which cannot be self-shielded with sufficient
accuracy using the Bondarenko approach. The flux calculator option of GROUPR
module in NJOY is designed to solve such problems.
The infinite-medium neutron spectrum equation is expressed as
∞
Σ t ( E )Φ ( E ) = ∫ dE ' Σ s ( E ' → E )Φ ( E ' ) + S ( E )
Equation 2-17
0
16
where the term on the left hand side of Equation 2-17 represents the collision, the integral
on the right hand side is the scattering source, and S(E) the external source.
Next, Equation 2-17 is written considering a homogeneous medium consisting of
two materials: an absorber and a moderator, represented by A and M, respectively in
Equation 2-18. Elastic scattering cross sections that are isotropic in the center of mass are
used. Neutron slowing down in a single resonance of the absorber material is assumed.
Σ t ( E )Φ ( E ) =
E /αM
∫ dE '
E
Σ Ms ( E ' )
Φ( E ' ) +
(1 − α M ) E '
E /αA
∫ dE '
E
Σ sA ( E ' )
Φ ( E ' ) Equation 2-18
(1 − α A ) E '
where α M and α A are the moderator and absorber collision parameters, respectively,
defined as:
⎛ A −1⎞
⎟
⎝ A + 1⎠
2
α =⎜
Equation 2-19
where A is the atomic mass in Equation 2-19
The following approximations are introduced to Equation 2-18 :
•
The moderator scattering cross-section is assumed to be constant and
equal to the potential scattering cross-section: i.e. Σ Ms ( E ' ) = Σ Mp
•
The moderator absorption cross-section is assumed to be negligible; i.e.
Σ tM ( E ' ) = Σ Mp
•
The narrow resonance approximation is used for the moderator. This states
that the resonance width is very small compared to the energy loss from scattering with
the moderator nucleus. Therefore, the flux distribution is the moderator integral is
17
assumed to have an asymptotic form. In general, the moderator integral is assumed to be
a smooth function of energy represented as C(E).
•
The moderator is assumed to represent all nuclides other than the absorber.
This enables the inclusion of the dilution microscopic cross-section of the absorber, σo, in
Equation 2-18. The dilution (or background) cross section of an isotope i is defined to be
all cross sections representing isotopes other than the isotope i. The dilution cross-section
is a measure of energy self-shielding. It determines the significance of a resonance
compared to other cross sections. If the dilution cross-section (σo) is small, it indicates
that the resonance has a significant impact on the flux and a large self-shielding effect
exists. If σo is very large (infinite dilution), the cross sections of the absorber do not
affect the flux spectrum, and the flux may be represented as a smooth function of energy.
Including the above approximations, Equation 2-18 becomes:
[σ
o
]
+ σ tA ( E ) Φ ( E ) = C ( E )σ o +
E /αA
∫ dE '
E
σ sA ( E ' )
Φ ( E ' ) Equation 2-20
(1 − α A ) E '
The dilution cross-section for an isotope i is given as:
σo =
1
ρi
∑ρ σ
j ≠i
j
j
t
Equation 2-21
Where i and j represent isotope indexes and ρ is atomic density. Equation 2-20 is
the simplest form used in NJOY for computing the flux with the flux calculator option. In
NJOY, several dilution cross sections are provided as input. Depending on a system of
interest, the cross sections corresponding to the appropriate dilution cross-section are
used.
18
2.5.2 The Bondarenko Method
The Bondarenko method is obtained by using the narrow resonance
approximation in the absorber scattering integral of Equation 2-22, which is derived from
neutron slowing down equation in Equation 2-18.
E /αA
σ sA ( E ' )
[σ o + σ ( E )]Φ( E ) = C ( E )σ o + ∫ dE ' (1 − α ) E ' Φ( E ' ) Equation 2-22
A
E
A
t
The practical width of a resonance of the absorber is considered to be much
smaller than the energy loss due to a collision with the absorber. This enables the
absorber integral to be represented as a smooth function of energy. Therefore, the flux is
represented by:
Φ( E ) =
C(E)
σ (E) + σ o
(
A
t
)
Equation 2-23
If σ 0 is larger than the tallest peaks in σ t , the weighting flux φ is approximately
proportional to the smooth weighting function C(E). This is called infinite dilution; the
cross section in the material of interest has little or no effect on the flux. On the other
hand, if σ 0 is small with respect to σ t , the weighting flux will have large dips at the
locations of the peaks in σ t , and a large self-shielding effect will be expected. This
treatment is good for the unresolved region (high energy resonances). Since resonance
width in this region is very small.
19
2.5.3 CENTRM
CENTRM (Continuous Energy Transport Module) is the new method existing in
SCALE 5.0 (Ref.17). It computes continuous-energy neutron spectra in zero- or onedimensional systems, by solving the Boltzmann Transport Equation using a combination
of pointwise and multigroup nuclear data. Several calculational options are available,
including discrete ordinates in slab, spherical, or cylindrical geometry; collision
probabilities in slab or cylindrical coordinates; and zone-wise or homogenized infinite
media. In SCALE, CENTRM is used mainly to calculate problem-specific fluxes on a
fine energy mesh (>10000 points), which may be used to generate self-shielded
multigroup cross section for subsequent criticality or shielding analysis.
CENTRM avoids many of the inherent assumptions by calculating a problemdependent flux profile, thus making it a far more rigorous cross-section treatment. Effects
from overlapping resonances, fissile material in the fuel and surrounding moderator, and
inelastic level scattering are explicitly handled in CENTRM. Another advantage of
CENTRM is that it can explicitly model rings in a fuel pin to more precise model the
spatial effect on the flux and cross sections. CENTRM enables problem-dependent
multigroup cross sections to have the flexibility and accuracy of pointwise-continuousenergy cross sections for criticality analyses.
2.6
Group Structure Selection Methodology
In 2003, Alplan and Haghighat developed the Contributon and Point-wise
Cross Section Driven (CPXSD) methodology and its application focused on the
shielding problem [Ref.2]. The CPXSD methodology constructs fine- and broad-
20
group structures considering two criteria: i) importance of groups and ii) pointwise
cross sections of an isotope/material mixture of interest. The importance of the groups
is determined using the group-dependent response flux formulation (or contributon)
given by Equation 2-24.
Cg =
2l + 1 m
Ψl , g , s Ψl+, g,m,s
l =0 m =0 4π
L
l
∑ Vs ∑ ∑
s∈D
Equation 2-24
In Equation 2-24, Vs is the volume of the sub-domain, l and m are azimuthal and
polar indices for the spherical harmonic polynomial, g refers to energy group, Ψ is the
angular flux, and Ψ+ is the adjoint (“importance”) function.
The CPXSD methodology constructs group structures by refining an initial
arbitrary group structure, considering the two aforementioned criteria. First, the
objectives are calculated using the cross section library having the initial group structure.
The importance values of all groups are calculated and the most important group is
identified. Depending on the point-wise cross sections of the important isotope/mixture
and/or group of isotopes/mixture, sub-groups are placed in the most important group,
considering the resonance and non-resonance behavior of cross sections. After the
number of sub-divisions in the most important group is obtained, the number of subdivisions in other groups are determined based on the ratio of their Cg, to the maximum
Cg. Sub-divisions in other groups are performed and a new group structure is generated.
The refinement process continues until a convergence criterion on the objectives is
achieved.
The CPXSD methodology was applied to a reactor pressure vessel problem using
TMI-1 to generate new group structures for the fast neutron dosimetry applications. It
21
was demonstrated that the broad-group libraries containing the CPXSD generated group
structures are in close agreement with their fine-group libraries (within 1-2%). Also,
comparing with continuous energy Monte Carlo predictions, Alplan and Haghighat used
the CPXSD methodology to generate new broad-group libraries, which have significantly
fewer groups, but yield more accurate results than the standard BUGLE libraries. Their
analyses demonstrated that the group structures constructed by the CPXSD methodology
can significantly improve the efficiency and accuracy of shielding calculations.
2.7
Code Description
2.7.1 DORT
DORT [Ref.4] is a 2-D discrete ordinates code (it also has a 1-D slab option) that
is suitable for XZ, RZ, or R-Θ geometry. It can be used to solve either the forward or the
adjoint form of the Boltzmann transport equation. The Boltzmann transport equation is
solved, using either the method of discrete ordinates or diffusion theory approximation.
In the discrete ordinates method, the primary mode of operation, balance equations are
solved for the flow of particles moving in a set of discrete directions in each cell of a
space mesh and in each group of a multigroup energy structure. Iterations are performed
until all implicitness in the coupling of cells, directions, groups, and source regeneration
has been resolved. Several methods are available to accelerate convergence i.e., single
group-wise rebalance factor, diffusion acceleration, and partial current rebalance.
Anisotropic cross sections can be expressed in a Legendre expansion of arbitrary order.
Output data sets can be used to provide an accurate restart of a previous problem or to
deliver information to other codes. Several techniques are available to remove the effects
22
of negative fluxes caused by the finite difference approximation and of negative
scattering sources due to truncation of the cross-section expansion. The space mesh can
be described such that the number of first-dimensional (i) intervals varies with the second
dimension (j). The number of discrete directions can vary across the space mesh and with
energy. Direction sets can be biased, with discrete directions concentrated such as to give
fine detail to streaming phenomena.
2.7.2 TORT
TORT [Ref.4] is a 3-D discrete ordinates code that is suitable for cylindrical
(RΘZ) or Cartesian (XYZ) geometry, as well as several two-dimensional subsets. It
calculates the neutron flux and/or photons throughout three-dimensional systems due to
particles incident upon the system's external boundaries, due to fixed internal sources, or
due to sources generated by interaction with the system materials. The Boltzmann
transport equation is solved using the method of discrete ordinates to treat the directional
variable. The weighted difference, nodal, or characteristic methods are available to treat
spatial variables. Energy dependence is treated using a multigroup formulation.
Anisotropic scattering is treated using a Legendre expansion. Iterations are used to
resolve implicitness caused by scattering between directions within a single energy
group, by scattering from one-energy group to another group previously calculated, by
fission, and by certain boundary conditions. Methods are available to accelerate
convergence. Fixed sources can be specified at either external or internal mesh
boundaries, or distributed within mesh cells.
23
2.8
Applications of Discrete Ordinates Method to Criticality Calculations
2.8.1 A Sub-Critical ‘C28’ and a Critical Assembly
Sjoden and Haghighat selected two criticality safety problems from KENO
multigroup Monte Carlo code standard set [Ref.13]. The first problem is a sub-critical
‘2C8’ enriched uranium cylinder, and the second problem is a critical assembly
composed of an enriched uranium annular ring with an offset cylindrical inside the ring.
These two problems were solved using the PENTRAN 3-D Cartesian parallel discrete
ordinates code with 16-group Hansen-Roach multigroup cross section library, assuming a
zero potential dilute absorber treatment.
Besides considering the eigenvalue of the problems, the calculations were also
performed to examine the effect of quadrature set, spatial differencing scheme, and grid
refinement on the eigenvalue solutions. PENTRAN results were compared to KENO
Monte Carlo Code, MCNP code in Multigroup mode (using an independent 30-Group
Library), and MCNP using the standard Continuous Energy mode. The results are quite
consistent with all Monte Carlo code results for both problems. It is also found that
PENTRAN computed keff values depend on the order of the angular quadrature, the
spatial grid interval, and the spatial differencing scheme used.
2.8.2 The C5G7 MOX Benchmark
The seven-group form of the C5 MOX fuel assembly (C5G7MOX) is a transport
benchmark problem [Ref.11]. The model includes two MOX-fuel assemblies and two
UO2-fuel assemblies, which are partitioned in a square lattice and surrounded by water.
24
Each fuel assembly is 17x17 lattice of fuel cells. This benchmark was designed to test the
performance of deterministic transport methods and codes in solving reactor physics
problems. Three papers about application of SN methods to this benchmark problem have
been reviewed as follows.
In the first paper [Ref.1], Haghighat, Ce Yi, and Sjoden developed models for
three study cases used in PENTRAN; (i) a fuel cell model (ii) a fuel cell assembly and
(iii) a full C5G7MOX model. The problems were solved in 2-D geometry by using
reflective boundary conditions on the top and bottom boundaries. They examined
different angular qudrature orders between S8 and S20, and various mesh sizes between
0.1260 cm. and 0.0785 cm. for a fuel cell case. The PENTRAN results were compared
with the reference MCNP solutions for keff. The fuel cell results indicate that S16 with
0.09 cm mesh size model is adequate for this simulation. For a fuel cell assembly and a
full C5G7MOX model cases, PENTRAN yields accurate solutions with an error less than
0.1% on keff. The relative differences of power distribution in C5G7MOX model vary in
a range of ~-3% to ~+2%, the larger differences can be attributed to higher uncertainties
in the Monte Carlo predictions.
In the second paper [Ref.8], Klingensmith, Azmy, Gegin, and Orsi used TORTMPI to solve the 3-D C5G7MOX problem and compared with the KENO (Monte Carlo
reference solution). The problem was solved on a sequence of refined spatial grids using
increasing orders of angular quadratures (S6, S12, and S16), and it was observed that the
eigenvalue converges. The results show that the obtained eigenvalues on various meshes
and angular quadrature orders is accurate if compared to the expectations in reactor
applications with an error less than 0.2%.
25
In the third paper [Ref.10], Dahl and Alcouffe used PARTISN (PARallel Time
Dependent SN) to perform this problem in 2-D and 3-D Cartesian grid with various mesh
refinement. They used the angular quadrature orders of the square TChebyshev-Legendre
to solve each problem with diamond spatial differencing. The results were compared for
different mesh refinement and quadrature orders but not compared with any reference
code. It was found that the angular dependence was stronger than the spatial.
At present, there is no such a complete calculation scheme for core analysis that
performs both cross-section generation and core simulation based on 3-D SN transport
method in a consistent manner.
26
CHAPTER 3
Cross-Section Generation Methodology
The multigroup cross sections are very important data for the nuclear reactor
analyses. Standard cross-section generation techniques involve three major steps. The
first one is to generate a fine-group cross-section library from the ENDF/B-VI data using
a piecewise linear energy weighting function generated from theoretical spectrum
approximations. The cross sections are processed with the appropriate resonance
treatment method. Second, infinite array unit cell calculations using the fine-group library
are performed to get the spatial flux distribution. These weighting flux functions are used
to collapse the fine-group library to a broad-group library. The third step involves spatial
homogenization of the unit cell in the framework of the broad group structure. In this
chapter the developed cross-section generation methodology for the TRIGA core analysis
is described, including the selection of fine and broad energy group cross-section
structures for the TRIGA core analysis.
3.1
Cross Section Generation Procedure and Studies
The NJOY code version 99.81 [Ref.12] is used for cross section processing
followed with the AMPX module from the SCALE code package [Ref.17] for postprocessing of cross sections. The standard cross section generation procedure contains
several steps as follows:
Step 1: NJOY generates multigroup cross section in Group-wise Evaluated Nuclear Data
Format (GENDF) format.
Step 2: SMILER converts the NJOY (GENDF) files to the AMPX master library format
Step 3: AJAX combines each AMPX master library file of isotopes to a single file.
27
Step 4: BONAMI performs resonance self-shielding effect with Bondarenko factors.
Step 5: NITAWL converts AMPX master library to AMPX working library format.
Step 6: ALPO converts AMPX working library format to standard ANISN format.
Step 7: GIP generates mixture cross-section library.
Step 8: Utilize the multigroup cross section library with transport code e.g., DORT and
TORT
The cross section generation flow chart is presented in Figure 3-1.
NJOY
Fine group XS
SMILER
AMPX MASTER LIBRARY
AJAX
AMPX MASTER LIBRARY
BONAMI
AMPX MASTER LIBRARY (self-shielding XS
for selected region)
NITAWL
AMPX WORKING LIBRARY
ALPO
ANISN FORMAT
GIP
Mixture XS
TRANSPORT CODE
Figure 3-1: Procedure for generating cross section library
28
The isotopes that are used to generate a multigroup cross-section library for
TRIGA core analysis are listed below categorized by the elements.
1) Zirconium – Zr
2) Boron Carbide –B10, B11, C12
3) UZrH – U234, U235, U236, U238, Zr, H1
4) Graphite – C12
5) SS304 – Fe54, Fe56, Fe57, Fe58, Cr50, Cr52, Cr53, Cr54, Ni58, Ni60, Ni61,
Ni62, Ni64, Si, Mn
6) H2O – H1, O16
7) Al – Al27
In the NJOY process, all nuclides, except for hydrogen, zirconium in UZrH, and
graphite, are processed at 300°, 600°, 1000°, and 2100° K. Hydrogen, zirconium in UZrH
and graphite are processed for the temperatures available in the ENDF tape that contains
thermal neutron scattering data. These temperatures are 296°, 400°, 500°, 600°, 700°,
800°, 1000° and 1200° K. It should be noted that in this phase of study, calculations are
performed based on an 8.5% wt. single unit fuel element cell model (shown in Figure 32) with fresh fuel and at cold conditions (300° K). This model is used to establish the
cross-section generation methodology for deterministic transport SN-based TRIGA core
analysis, which later will be applied to other material compositions in the present TRIGA
core.
29
Clad
Fuel
Zr
Coolant
Figure 3-2: Unit cell for TRIGA fuel element
3.1.1 The Weight Function Study
The accuracy of a set of multigroup constants is determined by the selected
energy group structure and the utilized weight function. It is necessary to have a weight
function that represent as accurate as possible the flux distribution as a function of energy
in the nuclear reactor core of interest. GROUPR in NJOY provides the in-code built
weight functions that represent a few typical nuclear systems including the thermal
reactor spectrum. The later weight function combines a thermal Maxwellian at low
energies, a 1/E function at intermediate energies, and a fission spectrum at high energies.
In GROUPR, user has freedom to choose the temperatures of the Maxwellian and fission
parts and the energies where the spectra join.
A quarter of 8.5%wt. fuel TRIGA cell was modeled in MCNP to study for the
weighting function spectrum that will be applied in NJOY. The energy tally card was
used to tally neutron flux for each of 238-group energy bins. This 238-group structure is
the group structure of the library in the SCALE package. Figure 3-3 shows the predicted
30
with MCNP neutron flux distribution per unit lethargy as function of energy. It has the
shape of the thermal reactor spectrum that is available in GROUPR.
The cutoff energies between spectra were determined by using the MaxwellBoltzmann distribution function for low energies and a 1/E function for intermediate
energies. As a result, the function consists of
1. A Maxwellian spectrum (peak at 0.07eV) from 10-5 to 0.3 eV
2. An 1/E spectrum from 0.3 eV to 20.0 keV
3. A fission spectrum from 20.0 keV to 20 MeV.
1.E+01
Flux per unit lethargy
1.E+00
1.E-01
1.E-02
1.E-03
1.E-04
Maxwellian
Spectrum
1/E Spectrum
Fission
1.E-05
1.E-03 1.E-02 1.E-01 1.E+00 1.E+01 1.E+02 1.E+03 1.E+04 1.E+05 1.E+06 1.E+07 1.E+08
Energy (eV)
Figure 3-3: MCNP-predicted TRIGA spectrum
3.1.2 The Corner-Material Study
TRIGA core has a hexagonal unit cell lattice type, which cannot be modeled
explicitly in the SN transport codes such as DORT, TORT or PENTRAN. Therefore, the
model has to be generated in a rectangular geometry. This study is performed to
31
determine the suitable material that should be used in the corner of the TRIGA cell for
the cross-section generation process. Four study cases have been considered as shown in
Figure 3-4.
1) None (Real Model)
2) Void
3) Water
4) Graphite
Figure 3-4: Cells Models for MCNP
All cases were modeled in a quarter sector of symmetry. Case 1 represents the
real model in the TRIGA cell; the reflective boundary was applied on the hexagonal
surface. In cases 2, 3 and 4, the void, water, and graphite were filled in the corner,
respectively. The reflective boundaries were applied on the rectangular surface.
The calculations for all cases were performed by using MCNP4C2 with 10000
cycles with 100 inactive cycles and 5000 histories/cycles. Table 3-1 shows the kinf and the
percentage of deviations from the real model. Figure 3-5 depicts the neutron spectrum in
fuel region for each case.
Table 3-1: Results of eigenvalue calculation using MCNP
Case
kinf
Dev. in pcm
Real
1.40168±0.00018(3σ)
-
Void
1.40186±0.00018(3σ)
18
Water
1.30137±0.00018(3σ)
-10031
Graphite
1.40106±0.00018(3σ)
-62
32
Flux in fuel region
2.E+00
1.E+00
1.E+00
Flux/lethargy
1.E+00
Void
Water
Real
Graphite
8.E-01
6.E-01
4.E-01
2.E-01
0.E+00
1.E-10
1.E-09
1.E-08
1.E-07
1.E-06
1.E-05
1.E-04
1.E-03
1.E-02
1.E-01
1.E+00
1.E+01
1.E+02
Energy (MeV)
Figure 3-5: Flux distribution in fuel Region
The relative differences of eigenvalue results from the real cell model are 18 pcm
for void case, -10031 pcm for water case, and -62 pcm for graphite case. Besides the kinf,
the neutron spectrum of the void case is closer to the real case than other cases. Hence,
the model with void is selected for cross-section generation.
3.1.3 Resonance Treatment Study
One of the most important issues to be considerated in criticality calculations is
the energy self-shielding in the resonance region for multigroup cross sections. The
method utilized for treatment of energy self-shielding is one of the factors in a multigroup
cross-section generation that may have a significant impact on the multiplication factor
and also on the absorption reaction rate predictions, mostly in the epithermal region.
33
Here, we study the effect of different self-shielding methods for Zr and U238 that
are present in the TRIGA fuel cell. These two isotopes have significant resonances in the
energy range of eV to KeV as illustrated in Figure 3-6 and Figure 3-7. The calculations
involve the use of GROUPR module in NJOY code (version 99.81) to calculate Zr and
U238 self-shielded cross sections in an infinite homogeneous medium. The Bondarenko,
flux calculator and CENTRM methods were used for self-shielding calculations. The
238-group structure of the library in the SCALE package was utilized in NJOY.
Figure 3-6: Pointwise absorption cross section of Zr
34
Figure 3-7: Pointwise absorption cross section of U238
The study is performed with DORT using S10 quadrature order and P1 scattering
order. The kinf and reaction rates from DORT were compared with the reaction rates
obtained using the continuous energy MCNP calculation. The reaction rates calculated by
MCNP have less than 0.1% of statistical uncertainty. Table 3-2 shows the reaction rates
for each energy range from the MCNP calculation.
35
Table 3-2: Reaction rates with continuous energy cross-section library in MCNP
Reaction
Energy range
Zr rod
Absorption
Fast
1.37E-03
3.06E-03
2.99E-03
7.67E-04
Epithermal
4.10E-03
3.86E-02
1.23E-02
7.72E-04
Thermal
1.08E-02
2.94E-01
4.77E-01
4.63E-02
Total
1.62E-02
3.36E-01
4.92E-01
4.79E-02
Fast
0.00E+00
4.27E-03
0.00E+00
0.00E+00
Epithermal
0.00E+00
2.25E-02
0.00E+00
0.00E+00
Thermal
0.00E+00
5.19E-01
0.00E+00
0.00E+00
Total
0.00E+00
5.46E-01
0.00E+00
0.00E+00
Fast
8.28E-01
1.53E+00
8.28E-01
1.18E+00
Epithermal
6.83E-01
3.04E+00
1.89E+00
3.08E+00
Thermal
6.67E-01
6.21E+00
2.99E+00
8.64E+00
Total
2.18E+00
1.08E+01
5.70E+00
1.29E+01
nu-fission
Total
Fuel meat
Cladding
Water
The resonance treatments for Zr and U238 were studied separately. First, we
concentrated on the Zr in the Zr rod region. The Bondarenko, flux calculator, and
CENTRM methods were utilized to calculate self-shielded cross sections of Zr from
ENDF/B-VI. For other nuclides, the Bondarenko Method was used. Table 3-3 shows the
reaction rates for each energy range from 238-group cross section library with the
Bondarenko method for Zr. Table 3-4 presents the percent deviations from the MCNP
calculation.
36
Table 3-3: Reaction rates with 238-group cross-section library using the
Bondarenko method
Reaction
Energy range
Zr rod
Fuel meat Cladding
Water
Absorption
Fast
1.37E-03
3.04E-03
2.93E-03
7.59E-04
Epithermal
4.95E-03
3.95E-02
1.14E-02
7.67E-04
Thermal
1.10E-02
2.95E-01
4.69E-01
4.51E-02
Total
1.73E-02
3.37E-01
4.83E-01
4.66E-02
nu-fission
Fast
4.20E-03 0.00E+00 0.00E+00
0.00E+00
Epithermal
2.23E-02 0.00E+00 0.00E+00
0.00E+00
Thermal
5.20E-01 0.00E+00 0.00E+00
0.00E+00
Total
5.47E-01 0.00E+00 0.00E+00
0.00E+00
Total
Fast
8.30E-01
1.53E+00
8.20E-01 1.19E+00
Epithermal
7.08E-01
3.03E+00 1.84E+00 3.09E+00
Thermal
6.61E-01
6.18E+00 2.96E+00 8.52E+00
Total
2.20E+00
1.07E+01 5.62E+00 1.28E+01
Reaction
Absorption
nu-fission
Total
Table 3-4: Percent deviations from MCNP
Energy range
Zr rod
Fuel meat Cladding Water
Fast
-0.21
-0.58
-1.92
-1.03
Epithermal
2.28
-7.35
-0.65
20.70
Thermal
2.20
0.21
-1.64
-2.68
Total
6.55
0.28
-1.86
-2.68
Fast
-1.66
Epithermal
-1.09
Thermal
0.12
Total
0.15
Fast
0.20
-0.25
-1.00
1.27
Epithermal
3.69
-0.38
-2.47
0.29
Thermal
-0.92
-0.40
-0.94
-1.36
Total
1.00
-0.75
-1.46
-0.73
Table 3-5 shows the reaction rates for each energy range from 238-group cross –
section library with Flux Calculator in NJOY for Zr. Table 3-6 shows the percent
deviations from the MCNP calculation.
37
Table 3-5: Reaction rates with 238-group cross-section library using FluxCalculator in NJOY
Reaction
Energy range
Zr rod
Fuel meat
Cladding
Water
Absorption
Fast
1.37E-03
3.04E-03
2.93E-03
7.59E-04
Epithermal
4.91E-03
3.95E-02
1.14E-02
7.67E-04
Thermal
1.10E-02
2.95E-01
4.69E-01
4.51E-02
Total
1.72E-02
3.37E-01
4.83E-01
4.66E-02
nu-fission
Fast
0.00E+00
4.20E-03
0.00E+00 0.00E+00
Epithermal
0.00E+00
2.23E-02
0.00E+00 0.00E+00
Thermal
0.00E+00
5.20E-01
0.00E+00 0.00E+00
Total
0.00E+00
5.47E-01
0.00E+00 0.00E+00
Total
Fast
8.30E-01
1.53E+00
8.20E-01
1.19E+00
Epithermal
7.08E-01
3.03E+00
1.84E+00 3.09E+00
Thermal
6.61E-01
6.18E+00
2.96E+00 8.52E+00
Total
2.20E+00
1.07E+01
5.62E+00 1.28E+01
Reaction
Absorption
nu-fission
Total
Table 3-6: Percent deviations from MCNP
Energy range
Zr rod
Fuel meat Cladding
Fast
-0.21
-0.58
-1.92
Epithermal
2.28
-7.35
19.72
Thermal
2.20
0.21
-1.64
Total
5.93
0.28
-1.86
Fast
-1.66
Epithermal
-1.09
Thermal
0.12
Total
0.15
Fast
0.20
-0.25
-1.00
Epithermal
3.69
-0.38
-2.47
Thermal
-0.92
-0.40
-0.94
Total
1.00
-0.75
-1.46
Water
-1.03
-0.65
-2.68
-2.68
1.27
0.29
-1.36
-0.73
In order to implement the CENTRM method in NJOY, the CENTRM code in
SCALE5 package (Ref.17) was used for the TRIGA fuel cell. ENDF/B-V data were used
for this calculation, since there is no ENDF/B-VI pointwise data available in SCALE
package. The average scalar flux spectrum in the Zr rod region was introduced in the
GROUPR module of NJOY as a weighting function to calculate multigroup cross
38
sections. Table 3-7 shows the reaction rates for each energy range from 238-group crosssection library with CENTRM method for Zr. Table 3-8 shows the percent deviations
from the MCNP calculation.
Table 3-7: Reaction rates with 238-group cross-section library using CENTRM
Reaction
Energy range
Zr rod
Fuel meat Cladding
Water
Absorption
Fast
1.37E-03
3.04E-03
2.93E-03
7.59E-04
Epithermal
4.09E-03
3.95E-02
1.14E-02
7.67E-04
Thermal
1.09E-02
2.95E-01
4.69E-01
4.51E-02
Total
1.64E-02
3.37E-01
4.83E-01
4.66E-02
nu-fission
Fast
0.00E+00
4.20E-03 0.00E+00 0.00E+00
Epithermal
0.00E+00
2.23E-02 0.00E+00 0.00E+00
Thermal
0.00E+00
5.20E-01 0.00E+00 0.00E+00
Total
0.00E+00
5.47E-01 0.00E+00 0.00E+00
Total
Fast
8.30E-01
1.53E+00
8.20E-01 1.19E+00
Epithermal
6.99E-01
3.03E+00 1.84E+00 3.09E+00
Thermal
6.61E-01
6.18E+00 2.96E+00 8.52E+00
Total
2.19E+00
1.07E+01 5.62E+00 1.28E+01
Reaction
Absorption
nu-fission
Total
Table 3-8: Percent deviations from MCNP
Energy range Zr rod Fuel meat Cladding
Fast
-0.21
-0.58
-1.92
Epithermal
2.28
-7.35
-0.27
Thermal
1.27
0.21
-1.64
Total
1.00
0.28
-1.86
Fast
-1.66
Epithermal
-1.09
Thermal
0.12
Total
0.15
Fast
0.20
-0.25
-1.00
Epithermal
2.37
-0.38
-2.47
Thermal
-0.92
-0.40
-0.94
Total
0.54
-0.75
-1.46
Water
-1.03
-0.65
-2.68
-2.68
1.27
0.29
-1.36
-0.73
39
From the analysis of the above-presented results, it is found that the deviation of
Zr absorption rate in epithermal range decreases from 21% to -0.3% when using the
CENTRM method for resonance treatment.
Now we focus on the resonance treatment of U238 in the fuel meat in the
epithermal range. Zr in the Zr rod was treated with CENTRM method. Since the
Bondarenko method was applied previously in Table 3-7 and Table 3-8, the Flux
Calculator, and CENTRM methods were utilized to calculate self-shielded cross sections
of U238. For other nuclides, the Bondarenko method was used. Table 3-9 shows the
reaction rates for each energy range from 238-group cross-section library with the Flux
Calculator method for U238. Table 3-10 shows the percent deviations from the MCNP
calculation.
Table 3-9: Reaction rates with 238-group cross-section library using Flux Calculator
in NJOY for U238
Reaction
Energy range
Zr rod
Fuel meat Cladding
Water
Absorption
Fast
1.37E-03
3.04E-03
2.93E-03
7.59E-04
Epithermal
4.09E-03
3.99E-02
1.14E-02
7.67E-04
Thermal
1.09E-02
2.94E-01
4.69E-01
4.51E-02
Total
1.64E-02
3.37E-01
4.83E-01
4.66E-02
nu-fission
Fast
0.00E+00
4.20E-03
0.00E+00 0.00E+00
Epithermal
0.00E+00
2.23E-02
0.00E+00 0.00E+00
Thermal
0.00E+00
5.20E-01
0.00E+00 0.00E+00
Total
0.00E+00
5.46E-01
0.00E+00 0.00E+00
Total
Fast
8.30E-01
1.53E+00
8.20E-01
1.19E+00
Epithermal
6.99E-01
3.03E+00 1.84E+00 3.09E+00
Thermal
6.60E-01
6.17E+00 2.96E+00 8.51E+00
Total
2.19E+00 1.07E+01 5.62E+00 1.28E+01
40
Reaction
Absorption
nu-fission
Total
Table 3-10: Percent deviations from MCNP
Energy range
Zr rod Fuel meat Cladding
Fast
-0.21
-0.58
-1.92
Epithermal
-0.27
-7.35
3.31
Thermal
1.27
-0.13
-1.64
Total
1.00
0.28
-1.86
Fast
-1.66
Epithermal
-1.09
Thermal
0.12
Total
-0.03
Fast
0.20
-0.25
-1.00
Epithermal
2.37
-0.38
-2.47
Thermal
-1.07
-0.57
-0.94
Total
0.54
-0.75
-1.46
Water
-1.03
-0.65
-2.68
-2.68
1.27
0.29
-1.48
-0.73
We applied the CENTRM method for U238 in the fuel meat region. Table 3-11
shows the reaction rates for each energy range from 238-group cross-section library with
CENTRM method for Zr Rod and fuel meat. Table 3-12 shows the percent deviations
from the MCNP calculation. The deviation decreases from 3.31% to 1.76% in the fuel
meat region.
Table 3-11: Reaction rates with 238-group cross-section library using Centrm
treatment for Zr and U238
Reaction
Energy range
Zr rod
Fuel meat Cladding
Water
Absorption
Fast
1.37E-03
3.04E-03
2.93E-03
7.59E-04
Epithermal
4.09E-03
3.93E-02
1.14E-02
7.67E-04
Thermal
1.09E-02
2.95E-01
4.69E-01
4.52E-02
Total
1.64E-02
3.37E-01
4.84E-01
4.67E-02
nu-fission
Fast
0.00E+00
4.21E-03 0.00E+00 0.00E+00
Epithermal
0.00E+00
2.23E-02 0.00E+00 0.00E+00
Thermal
0.00E+00
5.21E-01 0.00E+00 0.00E+00
Total
0.00E+00
5.47E-01 0.00E+00 0.00E+00
Total
Fast
8.30E-01
1.53E+00
8.20E-01 1.19E+00
Epithermal
6.99E-01
3.03E+00 1.84E+00 3.09E+00
Thermal
6.61E-01
6.18E+00 2.96E+00 8.52E+00
Total
2.19E+00
1.07E+01 5.62E+00 1.28E+01
41
Reaction
Absorption
nu-fission
Total
Table 3-12: Percent deviations from MCNP
Energy range
Zr rod Fuel meat Cladding
Fast
-0.21
-0.58
-1.92
Epithermal
-0.27
-7.35
1.76
Thermal
1.27
0.21
-1.64
Total
1.00
0.28
-1.65
Fast
-1.43
Epithermal
-1.09
Thermal
0.31
Total
0.15
Fast
0.20
-0.25
-1.00
Epithermal
2.37
-0.38
-2.47
Thermal
-0.92
-0.40
-0.94
Total
0.54
-0.75
-1.46
Water
-1.03
-0.65
-2.47
-2.47
1.27
0.29
-1.36
-0.73
This study illustrates that the CENTRM method treats the energy self-shielding
resonance cross section better than the Bondarenko method and Flux Calculator method
in NJOY for Zr in the Zr rod and U238 in the fuel meat. The reaction rates agree well with
MCNP results except the absorption reaction rate of cladding in the epithermal energy
range.
Different approaches are applied to solve the large deviation of absorption
reaction rate of cladding in the epithermal energy range. First, the CENTRM method is
used to treat Fe56, which is the main resonance isotope in cladding. Table 3-13 shows the
reaction rates for each energy range from 238-group cross-section library with the
CENTRM method in the Zr, U238, and Fe56. Table 3-14 shows the percent deviations
from the MCNP calculation. The deviation decreases from -7.35% to -5.86%.
42
Table 3-13: Reaction rates with 238-group cross-section library using Centrm
treatment in Zr, U238, and Fe56
Reaction
Energy range
Zr rod
Fuel meat
Cladding
Water
Absorption Fast
1.37E-03
3.04E-03
2.93E-03
7.59E-04
Epithermal
4.09E-03
3.93E-02
1.16E-02
7.67E-04
Thermal
1.09E-02
2.95E-01
4.68E-01
4.52E-02
Total
1.64E-02
3.37E-01
4.83E-01
4.67E-02
nu-fission
Fast
0.00E+00
4.21E-03 0.00E+00 0.00E+00
Epithermal
0.00E+00
2.23E-02 0.00E+00 0.00E+00
Thermal
0.00E+00
5.21E-01 0.00E+00 0.00E+00
Total
0.00E+00
5.47E-01 0.00E+00 0.00E+00
Total
Fast
8.30E-01
1.53E+00
8.19E-01 1.19E+00
Epithermal
6.99E-01
3.03E+00 1.83E+00 3.09E+00
Thermal
6.60E-01
6.18E+00 2.96E+00 8.52E+00
Total
2.19E+00
1.07E+01 5.60E+00 1.28E+01
Reaction
Absorption
nu-fission
Total
Table 3-14: Percent deviations from MCNP
Energy range
Zr rod Fuel meat Cladding
Fast
0.09
-0.70
-1.83
Epithermal
-0.24
1.72
-5.86
Thermal
1.38
0.15
-1.80
Total
0.86
0.32
-1.90
Fast
-1.46
Epithermal
-1.21
Thermal
0.26
Total
0.19
Fast
0.23
-0.46
-1.13
Epithermal
2.41
-0.24
-3.13
Thermal
-1.01
-0.44
-1.03
Total
0.53
-0.39
-1.74
Water
-1.01
-0.59
-2.55
-2.49
1.53
0.22
-1.37
-0.73
After that we increase the mesh model from 1554 cells to 24192 cells as
illustrated in Figure 3-8 to observe the physical effect. Table 3-15 shows the reaction
rates for each energy range from 238-group cross-section library with the CENTRM
method for Zr rod and Fuel meat, for the 24192-cell model. The Bondarenko method was
43
used for other nuclides. Table 3-16 shows the percent deviations from the MCNP
calculation. The deviation does not change at all.
Figure 3-8: Mesh Model from 1554 cells to 24192 cells
Table 3-15: Reaction rates with 238-group cross-section library, 24192 cells:
Reaction
Energy range
Zr rod
Fuel meat
Cladding
Water
Absorption Fast
1.37E-03
3.03E-03
2.93E-03
7.58E-04
Epithermal
4.09E-03
3.93E-02
1.14E-02
7.67E-04
Thermal
1.09E-02
2.95E-01
4.68E-01
4.51E-02
Total
1.64E-02
3.37E-01
4.83E-01
4.67E-02
nu-fission Fast
0.00E+00
4.21E-03
0.00E+00 0.00E+00
Epithermal
0.00E+00
2.23E-02
0.00E+00 0.00E+00
Thermal
0.00E+00
5.20E-01
0.00E+00 0.00E+00
Total
0.00E+00
5.47E-01
0.00E+00 0.00E+00
Total
Fast
8.30E-01
1.53E+00
8.19E-01 1.19E+00
Epithermal
6.99E-01
3.03E+00
1.84E+00 3.09E+00
Thermal
6.60E-01
6.17E+00
2.95E+00 8.51E+00
Total
2.19E+00
1.07E+01
5.61E+00 1.28E+01
44
Reaction
Absorption
nu-fission
Total
Table 3-16: Percent deviations from MCNP
Energy range
Zr rod
Fuel meat
Cladding
Fast
-0.21
-0.90
-1.92
Epithermal
-0.27
1.76
-7.35
Thermal
1.27
0.21
-1.85
Total
1.00
0.28
-1.86
Fast
-1.43
Epithermal
-1.09
Thermal
0.12
Total
0.15
Fast
0.20
-0.25
-1.12
Epithermal
2.37
-0.38
-2.47
Thermal
-1.07
-0.57
-1.28
Total
0.54
-0.75
-1.63
Water
-1.16
-0.65
-2.68
-2.47
1.27
0.29
-1.48
-0.73
Then a number of energy groups were increased from 238 to 253 by refining only
in the epithermal range. Table 3-17 shows the reaction rates for each energy range from
253-group cross-section library with the CENTRM method used for Zr and U238
treatment. The Bondarenko method was used for other nuclides. Table 3-18 shows the
percent deviations from the MCNP calculation. The deviation slightly decreases from 7.35% to -5.72%; however, this approach affects the absorption reaction rate of cladding
and water in fast energy range.
45
Table 3-17: Reaction rates with 253-group cross-section library
Reaction
Energy range
Zr rod
Fuel meat
Cladding
Water
Absorption Fast
1.37E-03
3.02E-03
2.88E-03 7.37E-04
Epithermal
4.10E-03
3.93E-02
1.16E-02 7.68E-04
Thermal
1.09E-02
2.95E-01
4.69E-01 4.52E-02
Total
1.64E-02
3.37E-01
4.84E-01 4.67E-02
nu-fission
Fast
0.00E+00
4.18E-03
0.00E+00 0.00E+00
Epithermal
0.00E+00
2.23E-02
0.00E+00 0.00E+00
Thermal
0.00E+00
5.21E-01
0.00E+00 0.00E+00
Total
0.00E+00
5.47E-01
0.00E+00 0.00E+00
Total
Fast
8.28E-01
1.52E+00
8.16E-01 1.19E+00
Epithermal
7.00E-01
3.03E+00
1.86E+00 3.09E+00
Thermal
6.60E-01
6.18E+00
2.96E+00 8.52E+00
Total
2.19E+00
1.07E+01
5.63E+00 1.28E+01
Reaction
Absorption
nu-fission
Total
Table 3-18: Percent deviations from MCNP
Energy range
Zr rod
Fuel meat Cladding
Fast
-0.21
-1.23
-3.59
Epithermal
-0.03
1.76
-5.72
Thermal
1.27
0.21
-1.64
Total
1.00
0.28
-1.65
Fast
-2.13
Epithermal
-1.09
Thermal
0.31
Total
0.15
Fast
-0.04
-0.90
-1.49
Epithermal
2.52
-0.38
-1.41
Thermal
-1.07
-0.40
-0.94
Total
0.54
-0.75
-1.28
Water
-3.90
-0.52
-2.47
-2.47
1.27
0.29
-1.36
-0.73
With the CENTRM resonance treatment, the deviation of absorption reaction rate
of cladding in epithermal energy range improves a few percent. With the mesh
refinement and energy group refinement, the deviation of absorption reaction rate of
cladding in epithermal energy range does not improve or otherwise slightly improve but
affect the reaction rate in other ranges of energy and materials. Therefore, the problem in
cladding region will be resolved using the CENTRM for resonance treatment in Fe56.
46
3.2
Fine Group Structure Selection
The CPXSD methodology developed by Alplan and Haghighat [Ref.2] is an
iterative method that selects effective fine- and broad-group structures for a problem of
interest, depending on the objectives of the problem. This methodology was derived
based on the “contribution” theory (the product of the forward and adjoint angular fluxes)
[Ref.19] to calculate the importance of groups and point-wise cross sections to obtain the
sub-group boundaries. The energy dependent response flux, i.e., the “contributon” is
given by:
v
ˆ )Ψ + (rv , E , Ω
ˆ)
C ( E ) = ∫ dr ∫ dΩΨ (r , E , Ω
v 4π
Equation 3-1
v
ˆ ) is the angular flux and Ψ + (rv, E , Ω
ˆ ) is the adjoint
In Equation 3-1, Ψ (r , E , Ω
v
function dependent on position r , energy E and direction Ω̂ . Considering spherical
harmonics expansion of flux and its adjoint, and using orthogonality, the groupdependent “contributon” is given by:
Cg =
2l + 1 m
Ψl , g ,s Ψlm, g, +, s
l =0 m =0 4π
L
l
∑V s ∑ ∑
s∈D
Equation 3-2
r )
r )
In Equation 3-2, ψ g (r , Ω) is the angular flux and ψ g+ (r , Ω) is the adjoint function
r
r
dependent on position r , and direction Ω in group g.
This CPXSD methodology was applied and validated for the shielding problem
but not yet for the criticality problem. Here, we extend this methodology to the criticality
problem based on the TRIGA cell/core. The objective is to generate a group structure to
determine an accurate eigenvalue and multigroup flux and power distributions.
47
3.2.1 Extension of the CPXSD Methodology to Criticality Problem
The procedure of the CPXSD methodology for generating fine-group structures,
which was adapted for criticality problem is as follows:
1. An initial group structure is selected. The initial group structure can be the existing
group structure or arbitrary one.
2. Cross sections are processed for the initial group structure with the established
procedure of cross section generation.
3. The importance of groups in the initial group structure is calculated by performing
forward and adjoint transport calculations to calculate the group-dependent response
flux. The adjoint function for criticality problem is obtained by setting the adjoint
source equal to production cross section (νΣ f ).
4. The group that has the maximum importance is identified.
5. The group that has the maximum importance is refined by the resonance structure of
an objective isotope with an arbitrary number
6. The number of sub-divisions in other groups is set relative to their importance to the
maximum importance.
7. After the refinement process is completed for all groups, the new group structure is
used for cross-section generation process. The new cross-section library is used to
calculate the objectives of a problem of interest.
48
8. In order to test the new library, a finer group structure is derived by repeating step 5
through 7 with higher arbitrary number to generate a finer group structure.
9. Calculated objectives are compared with the previous library. If results are within a
specified tolerance, the procedure ends; otherwise, steps 5 through 8 are repeated.
3.3
Cross-Section Collapsing and Homogenization
It is considered impractical to model a reactor core for routine repetitive design
and depletion calculations with its full geometrical detail employing multigroup neutron
transport theory. Therefore, the standard approach in core analysis is to combine
geometrical details as well as to collapse the energy group structure of cross section
library for whole core calculations. The purpose of the cross section homogenization and
collapsing is to preserve the sub-region average reaction rates and fluxes, while
improving the computation efficiency.
3.3.1 Fine- to Broad-Group Collapsing
The Procedure of the CPXSD Methodology
The procedure of the adapted CPXSD methodology for generated broad-group
libraries for criticality problem is as follows:
1. An initial broad-group structure is selected. The initial broad-group structure can
be selected in evenly partitioned.
2. Fine-group cross sections are collapsed to broad-group and a transport
calculation in performed with the broad-group library to calculate the objectives.
49
3. The group that has the maximum importance is refined by even partition using
the constructed fine-group structure with an arbitrary number.
4. The fine-group library is collapsed to the new broad-group library, and step 2 and
3 are repeated until a user-specified convergence criterion is achieved.
3.3.2 Cross-Section Homogenization
A typical fuel cell comprises of three explicit regions- fuel, clad, and coolant. It
can be reduced to an equivalent cell of simpler geometry to expedite calculations as
shown in Figure 3-9. The concept of the homogenization is to preserve all of the reaction
rates in the problem from the detailed "heterogeneous'' transport calculation. We utilize
the scalar flux weighting method to combine the material regions as shown in
Equation 3-3. With this method, the multigroup cross sections characterizing
materials in the cell are spatially averaged over the cell.
Figure 3-9 Fuel cell homogenization
nzone
Eg −1
3
Σg =
r
r
∑ ∫ dE∫ d rΣ (r , E)φ(r , E)
i =1
nzone
Eg
Vi
Eg −1
i =1
Eg
∑
i
r
3
∫ dE∫ d rφ(r , E)
Equation 3-3
Vi
50
3.4
Summary
In this chapter we determined the flux-weighting spectrum that is applied in
NJOY to generate the fine group cross-section library. The void was selected to be the
material in the corner of the cell for cross section generation process. The resonance
treatment in epithermal energy range was studied using different methods i.e.
Bondarenko, Flux Calculator and CENTRM. It was found that the CENTRM method is
the most effective technique to treat the energy resonance without further refining the
energy group structure. The CPXSD methodology was adapted to generate fine- and
broad- group structures for criticality problem.
51
CHAPTER 4
Two-Dimensional Cross Section Generation
In this chapter, the CPXSD methodology, adapted for criticality problem, is
applied to study the 2-D cross section generation in order to verify and validate the
methodology prior to application to the actual 3-D cross section generation, which will be
too costly in terms of computational time and resources. The other objective of
generating 2-D cross sections is to compare them with 3-D cross-sections in 3-D core
calculations. The 8.5% and 12% wt. TRIGA fuel cells are modeled for this study. The 2D fine group structure is constructed, and then the optimization study on the parameters
of SN method is performed. The 2-D broad group structure is established based on the 2-D
fine group structure. Other non-fuel material cross sections are generated with the same
fuel-studied structure. Finally, the cross sections are homogenized and compared with the
heterogeneous cases.
4.1
Two-Dimensional Model for Cross-Section Generation
One quarter of a hexagonal unit cell has been modeled for the 2-D cross section
generation study by taking advantage of the model symmetry as illustrated in Figure 4-1.
The void is filled in the corner of the rectangular geometry model based on the results
from the study presented in Chapter 3. The reflective boundary condition is applied to all
of the surfaces. Table 4-1 and Table 4-2 show the material compositions that have been
used in the cross-section generation calculations.
52
Figure 4-1: Cross section generation model
Table 4-1: Material density of the fuel elements
Nuclide
Density (atoms/barn-cm)
Fuel
12 wt.%
8.5 wt.%
H
0.05568
0.05689
Zr
0.03442
0.03506
U-234
0.000002915
U-235
0.0003642
0.000250520
U-236
0.000002434
U-238
0.0014538
0.001003000
Reflector
H
0.06683
O
0.03343
Zr Rod
Zr
0.042936
SS304 (Cladding)
SS304
0.08739
53
Table 4-2: Cladding composition
ISOTOPE
%Wt.
Fe54
3.996
Fe56
64.419
Fe57
1.501
Fe58
0.203
Cr50
0.793
Cr52
15.903
Cr53
1.838
Cr54
0.466
Ni58
6.234
Ni60
2.465
Ni61
0.109
Ni62
0.350
Ni64
0.092
Mn55
1.000
Si
0.500
4.2
Fine Group Structure for TRIGA
By using the procedure of the CPXSD methodology, adapted for criticality
problem, the fine group structure for TRIGA cross-section generation is obtained. The
238-group SCALE library is used as a starting group structure. The 238-group cross
sections are generated. Initially, the 238-group structure was divided into 3 major ranges
of energy: fast (0.1 MeV to 20 MeV.), epithermal (3 eV to 0.1 MeV), and thermal (1E-05
eV to 3 eV). We established two criteria for obtaining a fine group structure. The first
criterion is 10 pcm relative deviation of Δk/k and the second criterion is 1% relative
deviation of objective reaction rates. The objective reaction rates are different for each
range of energy. Using the flux and adjoint function moments computed from the
transport calculations with DORT [Ref.4], the contribution function - Cg’s are calculated.
Depending on the magnitude of the Cg’s per group, the group structure is refined for each
energy range. The groups corresponding to large Cg’s were partitioned into more groups.
54
The group with the highest Cg was subdivided by the resonance structure of an objective
isotope into a number of groups and the remaining groups were divided into fewer groups
based on the ratio of their Cg to the maximum Cg.
4.2.1 Fast Range Group Refinement
In this section a group structure in the fast energy range between 0.1 and 20 MeV
is derived. The 238-group SCALE library is used as a starting group structure with 44
groups in the fast energy range, 104 groups in the epithermal range, and 90 groups in the
thermal range. The 238-group Cg’s are calculated using the normalized production cross
sections (νΣf) as the adjoint source to perform the adjoint transport calculation. The pointwise cross section of U238(n,f) is used to select the group boundaries. The objectives are
eigenvalue and neutron production reaction rate of U238. The new group structures are
generated. Table 4-3 shows the number of groups in the fast energy range that are
obtained from the group refinement process.
Table 4-3: Fine groups selected in the fast energy range
Group Structure
Number of Groups in Different Energy Ranges
Number
Fast
Epithermal
Thermal
Total
1
44
104
90
238
2
52
104
90
246
3
80
104
90
274
55
The importance of different energy groups in the fast energy range, between 0.1
and 20 MeV, of 238-group and 246-group structures are plotted in Figure 4-2. The plot
shows that when the groups that have higher importance are refined, the importance of
those groups is decreased.
4.00E-02
3.50E-02
3.00E-02
Importance (E)
2.50E-02
2.00E-02
238 groups
246 groups
1.50E-02
1.00E-02
5.00E-03
0.00E+00
1.E-01
1.E+00
1.E+01
1.E+02
Energy (MeV)
Figure 4-2: Importance of groups of 238G and 246G libraries
The eigenvalues are calculated and compared between the group structures. For
246-group and 274-group comparison, Table 4-4 and Table 4-5 demonstrate that the
relative difference of Δk/k is less than 10 pcm and the percentage relative deviation of
U238(νΣf) reaction rate are 0.167% for 8.5% wt. and
0.165% for 12% wt. cases.
Consequently, we selected the 246-group structure, which contains 52 groups in the fast
energy range, for further group refinement in the epithermal energy range.
56
Table 4-4: Eigenvalue results of fine group energy for 8.5% wt. case
Group
kinf (S10P1)
Rel. Dev. in pcm
%Rel. Dev.
νΣf rate of
With
previous
U238
of Δk/k
group
With previous
> 0.1 MeV
group
238
1.40468
1.823
-
246
1.40464
-3
1.794
-1.591
274
1.40463
-1
1.791
-0.167
Table 4-5: Eigenvalue results of fine group energy for 12% wt. case
Group
kinf (S10P1)
Rel. Dev. in pcm
%Rel. Dev.
νΣf rate of
With previous
U238
of Δk/k
group
With previous
> 0.1 MeV
group
238
1.50079
1.822
-
246
1.50047
-21
1.815
-0.384
274
1.50061
-9
1.812
-0.165
4.2.2 Epithermal Range-Group Refinement
In this section a group structure in the epithermal energy range between 3eV and
0.1 MeV is derived. The 246-group structure from the fast group refinement is used as a
starting group structure with 52 groups in fast energy range, 104 groups in epithermal
range, and 90 groups in thermal range. The 246-group Cg’s are calculated using the
summation of the normalized νΣf and down-scattering cross section of H in ZrH from
epithermal group to thermal group as the adjoint source to perform the adjoint transport
calculation. The absorption point-wise cross section of U238 is used to select the group
boundaries. The objectives are eigenvalue, down-scattering reaction rate of H in ZrH
from epithermal energy range to thermal energy range and absorption reaction rate of
U238.
57
Table 4-6 shows the number of groups in epithermal energy range that are
obtained from the group refinement process. The importance of groups in epithermal
energy range, between 3 eV and 0.1 MeV, of the 246-group structure are plotted in
Figure 4-3.
Table 4-6: Fine groups generated in the epithermal energy range
Group Structure
Number of Groups in Different Energy Ranges
Number
Fast
Epithermal
Thermal
Total
1
52
104
90
246
2
52
152
90
294
1.40E-02
1.20E-02
Importance(E)
1.00E-02
8.00E-03
6.00E-03
4.00E-03
2.00E-03
0.00E+00
1.0E-06
1.0E-05
1.0E-04
1.0E-03
1.0E-02
1.0E-01
1.0E+00
Energy(MeV)
Figure 4-3: Importance in groups of 246G library
58
Table 4-7: Eigenvalue results of fine group energy
Group
246
294
for 8.5% wt. case
kinf (S10P1)
Rel. Dev. in pcm of
Δk/k
With previous group
1.40464
-
1.40457
-5
Table 4-8: Eigenvalue results of fine group energy
Group
246
294
Group
246
294
Group
246
294
for 12% wt. case
kinf (S10P1)
Rel. Dev. in pcm of
Δk/k
With previous group
1.50047
-
1.50058
7
Table 4-9: Reaction rate comparison for 8.5% wt. case
Down-scat.
%Rel. Dev.
U238(n,abs)
%Rel. Dev.
Of H in ZrH
With previous
With previous
group
group
3.683
19.014
-
3.683
0.00
18.991
-0.12
Table 4-10: Reaction rate comparison for 12% wt. case
Down-scat.
%Rel. Dev.
U238(n,abs)
%Rel. Dev.
Of H in ZrH
With previous
With previous
group
group
3.624
15.598
-
3.624
0.00
15.578
-0.13
The eigenvalues were calculated and compared between the 246-group and 294group structures in Table 4-7 for 8.5% wt. case and Table 4-8 for 12% wt. case. The
relative difference of eigenvalues are less than 10 pcm and the percentage relative
deviation of U238 absorption reaction rate and down-scattering reaction rate of H in ZrH
from epithermal range to thermal range are less than 1.0% as given in Table 4-9 and
59
Table 4-10. Consequently, we selected the 246-group structure, which contains 104
groups in epithermal energy range, for further group refinement in the thermal energy
range.
4.2.3 Thermal Range-Group Refinement
In this section a group structure in the thermal energy range between 1E-5 eV to 3
eV is derived. The 246-group structure from the fast and epithermal range group
refinements is used as a starting group structure with 52 groups in the fast energy range,
104 groups in the epithermal range, and 90 groups in the thermal range. The 246-group
Cg’s are calculated using the summation of the normalized νΣf and up-scattering cross
section of H in ZrH as the adjoint source to perform the adjoint transport calculation. The
inelastic scattering point-wise cross section of H in ZrH is used to select the group
boundaries. The objectives are eigenvalue, neutron production reaction rate of U235, and
up-scattering reaction rate of H in ZrH in the thermal energy range. Table 4-11 shows the
number of groups in the thermal energy range that are obtained from the group
refinement process. The importance of groups in the thermal energy range, between 1E-5
and 3 eV, of 246-group, 254-group and 280-group structures are plotted in Figure 4-4.
Table 4-11: Fine groups generated in the thermal energy range
Group Structure
Number of Groups in Different Energy Ranges
Number
Fast
Epithermal
Thermal
Total
1
52
104
90
246
2
52
104
98
254
3
52
104
124
280
4
52
104
180
336
60
4.00E-02
3.50E-02
3.00E-02
Importance(E)
2.50E-02
2.00E-02
246G
254G
280G
1.50E-02
1.00E-02
5.00E-03
0.00E+00
1.00E-10
1.00E-09
1.00E-08
1.00E-07
1.00E-06
1.00E-05
Energy(MeV)
Figure 4-4: Importance in groups of 246G, 254G and 280G libraries
Table 4-12: Eigenvalue results for fine group energy
Group
246
in thermal range 8.5% wt. case
kinf (S10P1)
Rel. Dev. in pcm of
Δk/k
With previous group
1.40464
-
254
1.40373
-64
280
1.40311
-44
336
1.40304
-5
61
Table 4-13: Eigenvalue results for fine group energy
in thermal range 12% wt. case
Group
kinf (S10P1)
Rel. Dev. in pcm of
Δk/k
With previous group
246
1.50047
-
Group
246
254:246
254
280:254
280
336:280
336
254
1.49986
-41
280
1.49941
-30
336
1.49955
9
Table 4-14: Reaction rate comparison of 8.5% wt. case
Up-scat.
%Rel. diff.
%Rel. diff.
U235(n, νΣf )
Of H in ZrH With previous group
With previous
group
2.511
2078.34
-
2.4881
2.779
2.7701
3.045
3.041
-0.92
2076.94
-0.07
2075.96
-0.05
2075.87
-0.01
-0.32
-0.13
1
Note: The reaction rate was calculated in a group-collapsing method to be compared with the previous
group structure
Group
246
254:246
254
280:254
280
336:280
336
Table 4-15: Reaction rate comparison of 12% wt. case
Up-scat.
%Rel. diff.
%Rel. diff.
U235(n, νΣf )
Of H in ZrH With previous group
With previous
group
1.802
1489.52
-
1.7881
1.996
1.9891
2.197
2.194
-0.78
1488.87
-0.04
1488.39
-0.03
1488.54
0.01
-0.35
-0.14
Note: 1The reaction rate was calculated in a group-collapsing method to be compared with the previous
group structure
62
Table 4-12 and Table 4-13 compare the eigenvalues for different energy group
structures. Table 4-14 and Table 4-15 compare the rate of up-scattering of H in ZrH and
neutron-production reaction rates of U235 for each energy group structure. The 280-group
structure is selected because its relative difference compared to the 336-group case
satisfies the set criterion.
The 280-group cross-section library was selected to be a fine group structure for
the TRIGA reactor based on the CPXSD methodology in 2-D geometry. In conclusion, a
methodology is established to generate the fine-group cross-section library and applied to
8.5% and 12% wt. TRIGA fuel cells.
4.3
Parametric Studies
Increasingly, the discrete ordinates method has become the dominant means for
obtaining numerical solutions to the integrodifferential form of the transport equation.
The discrete ordinates (SN) methods require a suitable multigroup cross section library,
and a reasonably accurate combination of spatial discretization, angular quadrature set,
and scattering order of cross sections. Assuming the cross sections are reliable, the
accuracy of an SN calculation is impacted by the aforementioned modeling factors. The
investigation of the parameters in transport calculations is performed to obtain the
effective value for each parameter in the view of minimizing computer memory and
time requirements for the problems, while maintaining the desired level of accuracy.
These effective parameters are obtained for the TRIGA cell cross-section generation
process.
63
4.3.1 Spatial Mesh, Angular Quadrature, and Scattering Order Studies
In this section, we perform sensitivity studies for spatial meshes, angular
quadrature set, and scattering order. Four 2-D fine-mesh models have been developed
with different uniform grid intervals.
1) 1554 cells: 37 x-axis, 42 y-axis
2) 6132 cells: 73 x-axis, 84 y-axis
3) 13625 cells: 109 x-axis, 125 y-axis
4) 24192 cells: 144 x-axis, 168 y-axis
Two different quadrature techniques, level (fully) symmetric and Square
Legendre-Chebyshev (SLC), are used to study this problem with various orders. The S4,
S6, S8, S10, and S16 orders are examined for fully symmetric and square LegendreChebyshev, respectively. The scattering orders that are used to perform sensitivities
studies are P1 and P3.
The 8.5% wt. fuel element cell model was used to perform this set of study. The
calculations were performed with DORT using different combinations of spatial meshes,
angular quadrature and scattering order. The obtained results were compared with
continuous energy MCNP results, which are used as a reference solution in this study.
The MCNP eigenvalue is 1.40195 ±0.00024(3σ). The MCNP calculation was performed
for 9000 cycles with 100 skipped cycles and 5000 histories/cycle. The DORT results and
relative deviations in pcm are provided in Table 4-16 through Table 4-19 for P1
scattering order and Table 4-20 through Table 4-23 for P3 scattering order with fully
symmetric quadrature order.
64
Table 4-16: DORT results with 280-energy group XS and 1554 cells, LevelSymmetric
SN Order Scattering
kinf
Conv.
Out.
Rel.
Iter.
Dev. in pcm of Δk/k
S4
P1
1.40290 -8.5E-07
35
68
S6
P1
1.40364
-5.9E-07
37
121
S8
P1
1.40281
-9.3E-07
36
61
S10
P1
1.40363
-9.3E-07
36
120
S16
P1
1.40260
-6.8E-07
37
46
Table 4-17: DORT results with 280-energy group XS and 6132 cells, LevelSymmetric
SN Order Scattering
kinf
Conv.
Out.
Rel.
Iter.
Dev. in pcm of Δk/k
S4
P1
1.40302 -8.5E-07
37
76
S6
P1
1.40374
-5.1E-07
38
128
S8
P1
1.40290
-7.6E-07
35
68
S10
P1
1.40371
-5.9E-07
38
126
S16
P1
1.40269
-8.5E-07
36
53
Table 4-18: DORT results with 280-energy group XS and 13625 cells, LevelSymmetric
SN Order Scattering
kinf
Conv.
Out.
Rel.
Iter.
Dev. in pcm of Δk/k
S4
P1
1.40301
7.6E-07
29
76
S6
P1
1.40374
-8.5E-07
36
128
S8
P1
1.40289
1.7E-07
26
67
S10
P1
1.40371
-4.2E-07
26
126
S16
P1
1.40269
-8.5E-07
37
53
65
Table 4-19: DORT results with 280-energy group XS and 24192 cells, LevelSymmetric
SN Order Scattering
kinf
Conv.
Out.
Rel.
Iter.
Dev. in pcm of Δk/k
S4
P1
1.40303
9.4E-07
31
77
S6
P1
1.40376
-6.8E-07
36
129
S8
P1
1.40291
1.7E-07
27
68
S10
P1
1.40372
9.4E-07
32
126
S16
P1
1.40270
7.7E-07
37
53
Table 4-20 DORT results with 280-energy group XS and 1554 cells, LevelSymmetric
SN Order Scattering
kinf
Conv.
Out.
Rel.
Iter.
Dev. in pcm of Δk/k
S4
P3
1.40288 -5.9E-07
36
66
S6
P3
1.40362
-7.6E-07
36
119
S8
P3
1.40278
-6.8E-07
35
59
S10
P3
1.40360
-6.8E-07
37
118
S16
P3
1.40257
-5.1E-07
36
44
Table 4-21: DORT results with 280-energy group XS and 6132 cells, LevelSymmetric
SN Order Scattering
kinf
Conv.
Out.
Rel.
Iter.
Dev. in pcm of Δk/k
S4
P3
1.40299 -5.1E-07
37
74
S6
P3
1.40371
-9.3E-07
36
126
S8
P3
1.40287
-8.5E-07
36
66
S10
P3
1.40367
5.9E-07
28
123
S16
P3
1.40266
-5.1E-07
27
51
66
Table 4-22: DORT results with 280-energy group XS and 13625 cells, LevelSymmetric
SN Order Scattering
kinf
Conv.
Out.
Rel.
Iter.
Dev. in pcm of Δk/k
S4
P3
1.40300 -9.3E-07
35
75
S6
P3
1.40370
-6.8E-07
25
125
S8
P3
1.40287
-5.9E-07
36
66
S10
P3
1.40367
2.5E-07
25
123
S16
P3
1.40265
-5.1E-07
25
50
Table 4-23: DORT results with 280-energy group XS and 24192 cells, LevelSymmetric
SN Order Scattering
kinf
Conv.
Out.
Rel.
Iter.
Dev. in pcm of Δk/k
S4
P3
1.40301
7.7E-07
34
76
S6
P3
1.40372
8.5E-07
31
126
S8
P3
1.40289
-3.4E-07
30
67
S10
P3
1.40370
-8.5E-07
38
125
S16
P3
1.40268
-9.4E-07
35
52
67
P1 scattering order
1.4040
1.4038
1.4036
Eigenvalues
1.4034
S4
S6
1.4032
S8
S10
S16
1.4030
1.4028
1.4026
1.4024
0
5000
10000
15000
20000
25000
30000
Meshes
Figure 4-5: P1 scattering order with level symmetric quadrature order
P3 scattering order
1.4040
1.4038
1.4036
Eigenvalues
1.4034
S4
S6
1.4032
S8
S10
S16
1.4030
1.4028
1.4026
1.4024
0
5000
10000
15000
20000
25000
30000
Meshes
Figure 4-6: P3 scattering order with level symmetric quadrature order
68
The results presented in Table 4-16 through Table 4-23 are summarized
graphically in Figure 4-5 and Figure 4-6. Several tendencies are observed: (i) All the
cases (different combinations of spatial meshes, angular quadrature and scattering order)
give a deviation of Δk/k less than 150 pcm compared with the MCNP solution. (ii) The
fluctuation of the eigenvalues is produced because of using different quadrature orders.
(iii) The finer meshes do not yield better results. (iv) The order of scattering anisotropy
from P1 to P3 affects the kinf by a maximum amount of 3 pcm. In conclusion, this study
shows that the quadrature order of level symmetric techniques does not converge the
results in the asymptotic region.
Table 4-24 through Table 4-27 show the results for P1 scattering order and Table
4-28 through Table 4-31 display the results for P3 scattering order with square LegendreChebyshev quadrature order.
Table 4-24: DORT results with 280-energy group XS and 1554 cells, SLC
SN Order
Scattering
kinf
Conv.
-7.6E-07
Out.
Iter.
36
Rel.
Dev. in pcm of Δk/k
88
S4
P1
1.40319
S6
P1
1.40307
-7.6E-07
37
80
S8
P1
1.40310
-9.3E-07
36
82
S10
P1
1.40311
7.6E-07
31
83
S16
P1
1.40314
5.9E-07
31
85
69
Table 4-25: DORT results with 280-energy group XS and 6132 cells, SLC
SN Order
Scattering
kinf
Conv.
3.4E-07
Out.
Iter.
35
Rel.
Dev. in pcm of Δk/k
96
S4
P1
1.40329
S6
P1
1.40315
1.7E-07
30
86
S8
P1
1.40317
8.5E-08
26
87
S10
P1
1.40321
8.5E-08
29
90
S16
P1
1.4023
8.5E-08
25
91
Table 4-26: DORT results with 280-energy group XS and 13625 cells, SLC
SN Order Scattering
kinf
Conv.
Out.
Rel.
Iter.
Dev. in pcm of Δk/k
S4
P1
1.40330 -5.1E-07
26
96
S6
P1
1.40316
-7.6E-07
27
86
S8
P1
1.40318
-4.2E-07
26
88
S10
P1
1.40321
-9.3E-07
36
90
S16
P1
1.40323
-8.5E-07
27
91
Table 4-27: DORT results with 280-energy group XS and 24192 cells, SLC
SN Order Scattering
kinf
Conv.
Out.
Rel.
Iter.
Dev. in pcm of Δk/k
S4
P1
1.40331
7.7E-07
25
97
S6
P1
1.40318
-6.0E-07
36
88
S8
P1
1.40319
8.5E-07
32
88
S10
P1
1.40322
-7.7E-07
37
91
S16
P1
1.40324
3.4E-07
27
92
70
Table 4-28: DORT results with 280-energy group XS and 1554 cells, SLC
SN Order Scattering
kinf
Conv.
Out.
Rel.
Iter.
Dev. in pcm of Δk/k
S4
P3
1.40317 -6.8E-07
38
87
S6
P3
1.40304
-8.5E-07
36
78
S8
P3
1.40307
-8.5E-07
37
80
S10
P3
1.40309
-7.6E-07
37
81
S16
P3
1.40313
-7.6E-07
38
84
Table 4-29: DORT results with 280-energy group XS and 6132 cells, SLC
SN Order
Scattering
kinf
Conv.
-8.5E-07
Out.
Iter.
35
Rel.
Dev. in pcm of Δk/k
94
S4
P3
1.40327
S6
P3
1.40312
5.1E-07
37
83
S8
P3
1.40315
-9.3E-07
36
86
S10
P3
1.40319
-8.5E-07
36
88
S16
P3
1.40322
-9.3E-07
35
91
Table 4-30: DORT results with 280-energy group XS and 13625 cells, SLC
SN Order
Scattering
kinf
Conv.
-8.5E-07
Out.
Iter.
36
Rel.
Dev. in pcm of Δk/k
95
S4
P3
1.40328
S6
P3
1.40314
-9.3E-07
35
85
S8
P3
1.40314
-2.5E-07
25
85
S10
P3
1.40317
-9.3E-07
26
87
S16
P3
1.40320
-7.6E-07
26
89
71
Table 4-31: DORT results with 280-energy group XS and 24192 cells, SLC
SN Order Scattering
kinf
Conv.
Out.
Rel.
Iter.
Dev. in pcm of Δk/k
S4
P3
1.40329
7.7E-07
33
96
S6
P3
1.40316
-7.7E-07
35
86
S8
P3
1.40316
5.1E-07
34
86
S10
P3
1.40320
-6.8E-07
38
89
S16
P3
1.40322
-6.8E-06
37
91
P1 scattering order
1.4040
1.4038
1.4036
Eigenvalues
1.4034
S4
S6
1.4032
S8
S10
S16
1.4030
1.4028
1.4026
1.4024
0
5000
10000
15000
20000
25000
30000
Meshes
Figure 4-7: P1 scattering order with Square Legendre-Chebyshev quadrature order
72
P3 scattering order
1.4040
1.4038
1.4036
Eigenvalues
1.4034
S4
S6
1.4032
S8
S10
S16
1.4030
1.4028
1.4026
1.4024
0
5000
10000
15000
20000
25000
30000
Meshes
Figure 4-8: P3 scattering order with Square Legendre-Chebyshev quadrature order
As it can be seen in Figure 4-7 and Figure 4-8, all the different combinations of
fine meshes, angular quadrature and scattering order show a positive bias as compared to
the reference MCNP solution by 80 to 97 pcm. The increasing of scattering order from P1
to P3 affects the kinf by maximum 3 pcm of Δk/k. The S4 kinf is higher than any other
quadrature order kinf. The results are converged with finer meshes.
It was found that kinf converges with the mesh refinement higher than 6132 cells.
The scattering order higher than P1 does not have a significant effect on kinf. Even
though, the square Legendre-Chebyshev Quadrature technique yields physically well
behaved results, the kinf results are not sensitive to the quadrature order beyond S6. As a
result, it would be difficult to select the appropriate order of quadrature for our problem
by considering only kinf value.
73
4.3.2 Qudrature Order Determination
From the above study, we concluded that the optimum cells for TRIGA cell are
6132 cells. This study is performed to determine the appropriate order of quadrature set.
We utilize the Square Legendre-Chebyshev quadrature type with S4, S6, S8, S10, S12,
S14, S16, S20, and S24. Table 4-32 gives the kinf results from DORT calculations and the
deviations between DORT and MCNP solutions.
Table 4-32: DORT results with 280-energy group XS
and 6132 cells, SLC
SN Order Scattering
kinf
Rel.
Dev. in pcm
S4
P1
1.40329
98
S6
P1
1.40315
88
S8
P1
1.40317
89
S10
P1
1.40321
92
S12
P1
1.40323
93
S14
P1
1.40322
93
S16
P1
1.40323
93
S20
P1
1.40324
94
S24
P1
1.40326
96
The scalar flux distributions are examined visually in Figure 4-9 and Figure 4-10
for the 23rd energy group as a representative of the fast energy range and the 242nd energy
group as a representative of the thermal energy range. We observe a nonphysical
behavior of flux distribution in the cell. This behavior is referred to as “ray effects”. It
results from the inability of low-order SN quadrature to integrate accurately over the
74
angular flux. As shown in Figure 4-9, these effects are very strong in S4 and S6
quadrature set orders.
S4
S6
S10
S12
S16
S20
S8
S14
S24
Figure 4-9: Flux distribution of group 23rd
75
S4
S10
S16
S6
S12
S20
S8
S14
S24
Figure 4-10: Flux distribution for group 242nd
Six-cell detectors were defined within the fuel region of 6132-cell model to
compare the neutron production reaction rate (local parameter) of the selected cells in
fuel region as shown in Figure 4-11. Table 4-33 and Table 4-34 give the neutron
76
production reaction rates predicted by MCNP and DORT for each cell detector and the
percentage of relative deviation as compared to the MCNP reference case, respectively.
3
2
6
5
4
1
Figure 4-11: Detector locations
77
Table 4-33: Neutron-production reaction rates from MCNP and DORT
Cell1
Cell2
Cell3
Cell4
Cell5
Cell6
S4
0.58685
0.58115
0.59743
0.50764
0.51262
0.48020
S6
0.58784
0.58014
0.59857
0.50747
0.51219
0.47996
S8
0.58810
0.57989
0.59850
0.50752
0.51217
0.48006
S10
0.58821
0.57993
0.59839
0.50752
0.51226
0.48009
S12
0.58822
0.57985
0.59847
0.50754
0.51226
0.48007
S14
0.58824
0.57985
0.59848
0.50754
0.51223
0.48008
S16
0.58826
0.57993
0.59848
0.50754
0.51223
0.48008
S20
0.58826
0.57993
0.59848
0.50755
0.51222
0.48008
S24
0.58826
0.57990
0.59850
0.50756
0.51224
0.48009
MCNP
0.58301±
0.0002σ
0.57098±
0.0002σ
0.58999±
0.0002σ
0.50377±
0.0002σ
0.50817±
0.0002σ
0.47729±
0.0002σ
Table 4-34: Percentage of relative deviation from MCNP
Cell1
Cell2
Cell3
Cell4
Cell5
Cell6
S4
0.658
1.781
1.261
0.768
0.875
0.611
S6
0.828
1.604
1.454
0.735
0.790
0.559
S8
0.873
1.560
1.442
0.744
0.787
0.581
S10
0.892
1.567
1.424
0.744
0.804
0.586
S12
0.894
1.553
1.437
0.749
0.805
0.583
S14
0.898
1.553
1.439
0.747
0.799
0.584
S16
0.900
1.568
1.439
0.748
0.800
0.585
S20
0.900
1.568
1.439
0.750
0.798
0.585
S24
0.900
1.561
1.442
0.753
0.801
0.587
From Table 4-34, it can be observed that the percentage deviation of reaction rate
for each cell changes relatively within 0.01% for the quadrature orders higher than S10.
As a result, S10 has been selected to be used for further study.
78
In this section we have performed sensitivity studies on the spatial meshing of the
unit cell, angular quadrature set, and scattering order in order to obtain the effective
values for the TRIGA problem. The calculations show that the 6132-cell model, S10
quadrature order of Square Legendre-Chebyshev technique, and P1 scattering order
constitute the appropriate model in terms of accuracy and efficiency for further crosssection collapsing and homogenization.
4.4
Cross-Section Collapsing
It is considered impractical to model a reactor core for routine repetitive design
and depletion calculations with a fine group structure employing multigroup neutron
transport theory. Therefore, the standard approach in core analysis is to collapse the
energy group structure of cross section library for whole core calculations. The purpose
of the cross section collapsing is to preserve the sub-region average reaction rates and
fluxes, while improving the computational efficiency.
In this section, the 280-group structure was collapsed into a broad group structure.
Using the same approach as the one utilized to select the fine group structure, we
established two criteria to obtain a broad group structure. The first criterion is 10 pcm
relative deviation of Δk/k and the second criterion is 1% relative deviation of objective
reaction rates. The objective reaction rates are different for each range of energy. The
U238(n,f) fission reaction rate is considered in the fast energy range, the down-scattering
reaction rates of H in ZrH and U238(n,a) absorption reaction rate are considered in the
79
epithermal energy range, and the U235(νΣf) reaction rate and the thermal up-scattering
reaction rate of H in ZrH are considered in the thermal range. The group collapsing
started with fast energies by initiating a very-broad-group structure and using the same
fine-group structure in the epithermal and thermal energies. Then, the aforementioned
“contributon” approach was used to refine the broad-group structure. This process is
repeated until the two criteria were met, and consequently a new broad-group structure
for the fast energies was obtained. With this new fast broad group structure, we continue
the same process for the epithermal and thermal energy ranges.
4.4.1 Fast Range-Group Collapsing
In fast energy range, we combined all the energy groups into one group. The new
group library contains 229 groups. The eigenvalue is calculated and compared with the
280-group library. Table 4-35 shows that relative difference of eigenvalue is less than 10
pcm and the percentage relative deviation of U238(n,f) is 0.11% for 8.5% wt. case. Table
4-36 shows that relative difference of eigenvalue is less than 10 pcm and the percentage
relative deviation of U238(n,f) is 0.17% for 12% wt. case. Consequently, we selected the
229-group structure to be collapsed in the epithermal energy range.
Table 4-35: Comparison between 229G and 280G for 8.5% wt. case
U238 (νΣf)
Above 0.1 MeV
1.40321
Rel. Dev.
In pcm of
Δk/k
-
1.794
%Rel. Dev.
Reaction rate of
U238(νΣf)
0.0
1.40321
0.0
1.796
0.11
Group
kinf (S10P1)
280
229
80
Table 4-36: Comparison between 229G and 280G for 12% wt. case
Group
kinf (S10P1)
Rel. Dev.
%Rel. Dev.
U238 (νΣf)
In pcm of
Reaction
rate of
Above 0.1 MeV
Δk/k
U238(νΣf)
280
1.49946
1.815
-
229
1.49947
0.0
1.818
0.17
4.4.2 Epithermal Range-Group Collapsing
In this step we develop the broad group structure in the epithermal energy range
(3.0 eV. to 0.1 MeV). The objective reaction rate for the epithermal energy range is the
down-scattering reaction rates of H in ZrH. We have placed two energy groups in this
range to separate resolved and unresolved regions and ended up with a 127-group
structure.
Table 4-37 and Table 4-38 demonstrate that the relative differences of
eigenvalues are less than 10 pcm. The percentage relative deviations of down-scattering
reaction rate of H in ZrH and absorption reaction rate of U238 in the epithermal range are
less than 1% as given in Table 4-39 and Table 4-40. As a result, we used 127-group
structure for further collapsing in the thermal energy range.
Table 4-37: kinf comparison between
229G and 127G for 8.5% wt. case
Group
kinf
Rel. Dev.
(S10P1)
In pcm of Δk/k
229
1.40321
-
127
1.40320
0
81
Table 4-38: kinf comparison between 229G
and 127G for 12% wt. case
Group
kinf
Rel. Dev.
(S10P1)
In pcm of Δk/k
229
1.49947
-
127
1.49945
-1
Table 4-39: Reaction rate comparison between 229G and 127G for 8.5% wt. case
Group
Down-scat.
%Rel. Dev.
U238(n,abs)
%Rel. Dev.
Of H in ZrH
With previous
With previous
group
group
229
3.683
-
19.011
-
127
3.683
0.00
19.015
0.02
Table 4-40: Reaction rate comparison between 229G and 127G for 12% wt. case
Down-scat.
%Rel. Dev.
U238(n,abs)
%Rel. Dev.
Group
Of H in ZrH
With previous
With previous
group
group
229
3.624
-
15.596
-
127
3.625
0.03
15.599
0.02
4.4.3 Thermal Range-Group Collapsing
In the last step, the broad group structure in thermal energy range (1.0E-05 eV. to
3.0 eV) is developed. The objective reaction rate of the thermal range is the neutronproduction reaction rate of U235 and the thermal up-scattering reaction rates of H in ZrH.
We initially introduced one energy group in this range and obtained a 4-group structure.
Then, we subdivided the most important group into three groups. Each time, we
generated a new broad-group structure until the result met the criteria. Table 4-41
indicates that percent relative difference of the eigenvalue of 12-group structure and 14-
82
group structure is 12 pcm and the percentage relative deviations of U235(νΣf) is 0.01%
and upscattering of H in ZrH is 0.00% for the 8.5% wt. case. Table 4-42 shows that
percent relative difference of the eigenvalue of 12-group structure and 14-group structure
is 9 pcm, and the percentage relative deviation of U235(νΣf) is 0.01% and upscattering
of H in ZrH is 0.00% for 12% wt. case. Therefore, we select the 12-group structure as the
broad group structure for this study.
Table 4-41: Result comparison in thermal energy range for 8.5% wt. case
U235 (νΣf)
reaction rate
1E-05 to 3 eV
1.40320
Rel. Dev.
In pcm of Δk/k
With
Previous group
-
2076.17
%Rel. Dev.
U235(νΣf)
reaction
rate
-
1.40898
412
2085.03
0.43
1.40677
-157
2081.61
-0.16
1.40483
-145
2078.61
-0.14
1.40368
-81
2076.82
-0.09
1.40351
-12
2076.55
-0.01
Group
kinf
(S10P1)
127
127:4
4
6:4
6
8:6
8
12:8
12
14:12
14
Up-scattering
of H in ZrH
3.045
0.000
0.000
0.000
0.571
0.570
0.910
0.910
1.562
1.562
1.721
%MaxRel.
Dev. In
upscattering
rate
-
-100.00
0.00
0.00
0.00
0.00
83
Table 4-42: Result comparison in thermal energy range for 12% wt. case
Group
127
127:4
4
6:4
6
8:6
8
12:8
12
14:12
14
Rel. Dev.
In pcm of Δk/k
With
Previous group
-
U235 (νΣf)
reaction rate
1E-05 to 3 eV
1.50472
351
1493.97
1.50242
-152
1491.54
1.50085
-104
1489.88
1.49984
-67
1488.81
1.49971
-9
1488.67
kinf
(S10P1)
1.49945
1488.32
%Rel. Dev. Up-scattering
U235(νΣf) of H in ZrH
reaction
rate
2.197
0.000
0.38
0.000
0.000
-0.16
0.405
0.404
-0.11
0.626
0.626
-0.07
1.085
1.085
-0.01
1.216
%MaxRel.
Dev. In
upscattering
rate
-
-100.00
0.00
-0.25
0.00
0.00
In order to verify that we have selected the effective broad-group structure for our
problem, MCNP with continuous cross-section library, DORT with 12-broad-group and
280-fine-group cross-section libraries for both 8.5% wt. and 12% wt. cases were
performed. The MCNP eigenvalues are 1.40195 ±0.00024(3σ) for 8.5% wt. case and
1.49838±0.00030(3σ) for 12% wt. case. The calculation was performed for 5800 cycles
with 900 skipped cycles and 5000 histories/cycle for 12%wt. case. Table 4-43 gives the
kinf results calculated by DORT for the 280-fine-group cross-section library and Table
4-44 gives the kinf results calculated by DORT for the 12-broad-group cross-section
library with different scattering and angular quadrature orders for the 8.5% wt. case. The
maximum of absolute percentage relative deviations in
Δk
compared to the 280-group
k
and continuous Monte Carlo calculations are 38 pcm and 136 pcm, respectively. Table
4-45 gives the kinf results calculated by DORT for the 280-fine-group cross-section
library, and Table 4-46 gives the kinf results calculated by DORT for the 12-broad-group
84
cross-section library with different scattering and angular quadrature orders for the 12%
wt. case. The maximum of absolute percentage relative deviations in
Δk
compared to the
k
280-group and continuous Monte Carlo calculations are 30 pcm and 111 pcm,
respectively. The deviations between the 12-group and the 280-group structures are less
than the deviation between the 12-group structure and the continuous-energy MCNP
solution. These differences are identified as the method difference between deterministic
(DORT) and statistic (MCNP) including the cross-section library between multigroup and
continuous energy. The 12-group structure was selected to be our final broad group
structure.
Table 4-43: DORT results with 280-energy group XS
and 6132 cells for 8.5% wt. case
Sn order
Scattering
kinf
Rel.Dev. in pcm
of Δk/k(MCNP)
S4
P1
1.40329
98
S6
P1
1.40315
88
S8
P1
1.40317
89
S10
P1
1.40321
92
S16
P1
1.40323
93
Table 4-44: DORT results with 12-energy group XS, 6132
cells for 8.5% wt. case
Sn
Scattering
kinf
Rel. Dev. In
Rel. Dev. In
order
(12 G)
pcm of Δk/k
pcm of Δk/k
(MCNP)
(280 G)
S4
P1
1.40382
136
38
S6
P1
1.40365
123
36
S8
P1
1.40366
124
35
S10
P1
1.40368
126
33
S16
P1
1.40370
127
33
85
Table 4-45: DORT results with 280-energy group XS
and 6132 cells for 12% wt. case
Sn order
Scattering
kinf
Rel.Dev. in
pcm of Δk/k
(MCNP)
S4
P1
1.49959
81
S6
P1
1.49942
69
S8
P1
1.49944
71
S10
P1
1.49946
72
S16
P1
1.49949
74
Table 4-46: DORT results with 12-energy group XS,
6132 cells for 12% wt. case
Sn
Scattering
kinf
Rel. Dev. Rel. Dev.
order
(12 G)
In pcm
In pcm
(MCNP)
(280 G)
S4
P1
1.50004
111
30
S6
P1
1.49982
96
27
S8
P1
1.49983
97
26
S10
P1
1.49984
97
25
S16
P1
1.49985
98
24
The absorption rate, neutron production, and total reaction rates are compared
between the two cross-section libraries: 280 groups and 12 groups, in each energy range
(fast, epithermal, and thermal) and region (Zr rod, fuel meat, cladding, and water). The
DORT-calculations are performed with S10P1. For the 8.5% wt. case, Table 4-47 and
Table 4-48 give the reaction rates from DORT for 280-group library and 12-group
library, respectively. The percentage of relative deviation between these two libraries is
presented in Table 4-49. For the 12% wt. case, Table 4-50 and Table 4-51 give the
reaction rates from DORT for 280-group library and 12-group library, respectively. The
86
percentage of relative deviation between these two libraries is presented in Table 4-52.
Overall, it is demonstrated very good agreement for these selected reaction rates in each
energy range with less than 0.4% difference.
Table 4-47: DORT calculation with 280-group cross section library for 8.5% wt. case
Reaction Rate
Absorption
Energy Range
Zr Rod
Fuel Meat
Cladding
Water
Fast
1.3707E-03
3.0192E-03
2.8725E-03
7.2810E-04
Epithermal
4.0919E-03
3.9292E-02
1.1432E-02
7.6757E-04
Thermal
1.0893E-02
2.9447E-01
4.7141E-01
4.5553E-02
Total
1.6355E-02
-
3.3678E-01
4.1664E-03
4.8571E-01
-
4.7049E-02
-
Epithermal
-
2.2275E-02
-
-
Thermal
-
5.2012E-01
-
-
Total
-
5.4657E-01
-
-
Fast
8.2750E-01
1.5227E+00
8.1547E-01
1.1898E+00
Epithermal
6.9933E-01
3.0345E+00
1.8447E+00
3.0879E+00
Thermal
6.5814E-01
6.1618E+00
2.9556E+00
8.5538E+00
Total
2.1850E+00
1.0719E+01
5.6158E+00
1.2832E+01
Fast
Neutron
production
Total
87
Table 4-48: DORT calculation with 12-group cross section library for 8.5% wt. case
Reaction Rate
Absorption
Energy Range
Zr Rod
Fuel Meat
Cladding
Water
Fast
1.3714E-03
3.0228E-03
2.8690E-03
7.2657E-04
Epithermal
4.0921E-03
3.9295E-02
1.1427E-02
7.6735E-04
Thermal
1.0932E-02
2.9456E-01
4.7045E-01
4.5449E-02
Total
1.6395E-02
-
3.3687E-01
4.1713E-03
4.8475E-01
-
4.6943E-02
-
Epithermal
-
2.2277E-02
-
-
Thermal
-
5.2029E-01
-
-
Total
-
5.4673E-01
-
-
Fast
8.2795E-01
1.5245E+00
8.1448E-01
1.1873E+00
Epithermal
6.9927E-01
3.0341E+00
1.8442E+00
3.0880E+00
Thermal
6.5930E-01
6.1631E+00
2.9506E+00
8.5364E+00
Total
2.1865E+00
1.0722E+01
5.6093E+00
1.2812E+01
Fast
Neutron
production
Total
Table 4-49: Reaction rates deviation between 280G and 12G for 8.5% wt. case
Reaction Rate
Energy Range
Fast
Epithermal
Absorption
Thermal
Total
Fast
Neutron
production
Fuel Meat
Cladding
Water
0.055
0.118
-0.122
-0.210
0.003
0.008
-0.041
-0.028
0.361
0.030
-0.203
-0.228
0.246
-
0.029
-0.199
-
-0.224
-
-
-
-
-
-
-
Epithermal
-
Thermal
-
Total
-
Fast
Epithermal
Total
Zr Rod
Thermal
Total
0.118
0.008
0.031
0.031
0.055
0.118
-0.122
-0.210
-0.009
-0.011
-0.023
0.003
0.177
0.021
-0.172
-0.203
0.071
0.026
-0.116
-0.154
88
Table 4-50: DORT calculation with 280-group cross section library for 12% wt. case
Reaction Rate
Energy Range
Fast
Epithermal
Absorption
Thermal
Total
Fast
Neutron
production
Fuel Meat
Cladding
Water
1.3903E-03
3.9143E-03
2.9149E-03
7.3744E-04
4.1258E-03
4.9493E-02
1.1431E-02
7.6638E-04
7.6817E-03
2.9523E-01
3.5838E-01
3.5363E-02
1.3198E-02
-
3.4864E-01
3.7272E-01
-
3.6867E-02
-
-
-
-
-
-
-
Epithermal
-
Thermal
-
Total
-
Fast
Epithermal
Total
Zr Rod
Thermal
Total
6.1437E-03
3.2231E-02
5.4563E-01
5.8401E-01
8.3951E-01
1.5169E+00
8.2892E-01
1.2106E+00
7.0426E-01
2.9901E+00
1.8608E+00
3.1196E+00
5.0555E-01
4.5472E+00
2.3540E+00
6.7744E+00
2.0493E+00
9.0542E+00
5.0437E+00
1.1105E+01
Table 4-51: DORT calculation with 12-group cross section library for 12% wt. case
Reaction Rate
Energy Range
Fast
Epithermal
Absorption
Thermal
Total
Fast
Neutron
production
Fuel Meat
Cladding
Water
1.3910E-03
3.9192E-03
2.9118E-03
7.3600E-04
4.1263E-03
4.9505E-02
1.1427E-02
7.6615E-04
7.7077E-03
2.9532E-01
3.5767E-01
3.5285E-02
1.3225E-02
-
3.4875E-01
3.7201E-01
-
3.6787E-02
-
-
-
-
-
-
-
Epithermal
-
Thermal
-
Total
-
Fast
Epithermal
Total
Zr Rod
Thermal
Total
6.1515E-03
3.2238E-02
5.4580E-01
5.8419E-01
8.3998E-01
1.5188E+00
8.2803E-01
1.2082E+00
7.0422E-01
2.9901E+00
1.8605E+00
3.1197E+00
5.0633E-01
4.5482E+00
2.3502E+00
6.7613E+00
2.0505E+00
9.0571E+00
5.0387E+00
1.1089E+01
89
Table 4-52: Reaction rates deviation between 280G and 12G for 12% wt. case
Reaction Rate
Energy Range
Fast
Epithermal
Absorption
Thermal
Total
Fast
Neutron
production
Cladding
Water
0.056
0.126
-0.106
-0.195
0.012
0.023
-0.040
-0.030
0.339
0.030
-0.197
-0.221
0.207
-
0.030
-0.192
-
-0.217
-
-
-
-
-
-
-
-
Thermal
-
Total
-
Epithermal
Thermal
Total
4.5
Fuel Meat
Epithermal
Fast
Total
Zr Rod
0.126
0.023
0.030
0.031
0.056
0.126
-0.106
-0.195
-0.006
0.001
-0.019
0.005
0.154
0.021
-0.161
-0.193
0.059
0.032
-0.099
-0.137
Two-Dimensional Cross Section Generation for Other Materials
For other materials, which are not fissile material, we use the color-set approach
model for cross section generation. The model contains the fissile material portion in
order to produce sources to the problem. Here, 280G (fine group) and 12G (broad group)
structures with S10 Square-Legendre-Chebyshev quadrature set are considered. The
MCNP eigenvalues and reaction rates are used as the reference.
4.5.1 Graphite
The 2-D color-set graphite model is illustrated in Figure 4-12. The total number of
cells is 148x118 cells. It is modeled with a uniform mesh distribution with 0.03 cm mesh
size. The 12G broad-group library is obtained by 280G flux distributions obtained from
280G DORT calculations.
90
Figure 4-12: 2-D model for graphite XS generation
Table 4-53 compares the eigenvalues from MCNP and DORT calculations. Table
4-54 through Table 4-61 give the reaction rates. These results demonstrate that 280G and
12G structures are in good agreement in eigenvalues and reaction-rate comparisons. The
scattering order does not have effect on the 2-D graphite cross-section model. Comparing
DORT with MCNP, DORT agrees well with MCNP in eigenvalue. Large deviations take
place in fast and epithermal energy ranges for most of regions; however, they are not the
main contribution to the problem, which are about 1 or 2 order of magnitude smaller than
the reaction rates in the thermal energy range.
91
Table 4-53: Eigenvalue results for graphite cross section generation model
REL. DEVIATION IN
CODE
KINF
PCM OF ΔK/K
MCNP
0.04825±0.00006(3σ)
DORT-280G,S10P3
0.04810
-311
DORT-280G,S10P1
0.04809
-332
DORT- 12G,S10P1
0.04810
-311
Abs_Fast
Abs_Epi
Abs_Thermal
Abs_Total
Table 4-54: MCNP reaction rates
Gra_left Clad_left Water_left Gra_top Clad_top Water_top
6.05E-05 1.36E-03 3.29E-04 6.09E-05 1.35E-03 3.27E-04
6.13E-06 7.81E-03 4.98E-04 6.13E-06 7.82E-03 4.98E-04
9.54E-04 8.83E-01 8.01E-02 9.54E-04 8.83E-01 8.01E-02
1.02E-03 8.92E-01 8.10E-02 1.02E-03 8.92E-01 8.10E-02
Tot_Fast
Tot_Epi
Tot_Thermal
Tot_Total
3.26E-01 4.26E-01
5.10E-01 1.17E+00
1.75E+00 4.78E+00
2.59E+00 6.38E+00
Abs_Fast
Abs_Epi
Abs_Thermal
Abs_Total
Table 4-55: DORT, 280GP3 reaction rates
Gra_left Clad_left Water_left Gra_top Clad_top Water_top
5.60E-05 1.25E-03 3.18E-04 6.11E-05 1.26E-03 2.93E-04
6.07E-06 6.84E-03 4.93E-04 6.07E-06 6.83E-03 4.93E-04
9.58E-04 8.85E-01 8.02E-02 9.58E-04 8.85E-01 8.01E-02
1.02E-03 8.93E-01 8.10E-02 1.02E-03 8.93E-01 8.09E-02
Tot_Fast
Tot_Epi
Tot_Thermal
Tot_Total
3.27E-01 3.87E-01
5.11E-01 1.09E+00
1.74E+00 4.83E+00
2.58E+00 6.30E+00
6.38E-01 3.24E-01 4.25E-01
1.91E+00 5.10E-01 1.17E+00
1.41E+01 1.71E+00 4.78E+00
1.66E+01 2.55E+00 6.38E+00
6.45E-01 3.29E-01 3.89E-01
1.91E+00 5.11E-01 1.09E+00
1.42E+01 1.74E+00 4.82E+00
1.68E+01 2.58E+00 6.30E+00
6.37E-01
1.91E+00
1.41E+01
1.66E+01
6.33E-01
1.91E+00
1.42E+01
1.68E+01
92
Abs_Fast
Abs_Epi
Abs_Thermal
Abs_Total
Table 4-56: DORT, 280GP1 reaction rates
Gra_left Clad_left Water_left Gra_top Clad_top Water_top
5.62E-05 1.25E-03 3.18E-04 6.12E-05 1.26E-03 2.93E-04
6.07E-06 6.84E-03 4.93E-04 6.07E-06 6.83E-03 4.93E-04
9.58E-04 8.85E-01 8.02E-02 9.58E-04 8.85E-01 8.01E-02
1.02E-03 8.93E-01 8.10E-02 1.02E-03 8.93E-01 8.09E-02
Tot_Fast
Tot_Epi
Tot_Thermal
Tot_Total
3.27E-01 3.88E-01
5.11E-01 1.09E+00
1.74E+00 4.83E+00
2.58E+00 6.31E+00
Abs_Fast
Abs_Epi
Abs_Thermal
Abs_Total
Table 4-57: DORT, 12GP1 reaction rates
Gra_left Clad_left Water_left Gra_top Clad_top Water_top
5.63E-05 1.24E-03 3.15E-04 6.05E-05 1.25E-03 2.93E-04
6.07E-06 6.84E-03 4.93E-04 6.07E-06 6.84E-03 4.93E-04
9.58E-04 8.85E-01 8.02E-02 9.58E-04 8.85E-01 8.01E-02
1.02E-03 8.93E-01 8.10E-02 1.02E-03 8.93E-01 8.09E-02
Tot_Fast
Tot_Epi
Tot_Thermal
Tot_Total
3.27E-01 3.87E-01
5.11E-01 1.09E+00
1.74E+00 4.82E+00
2.58E+00 6.30E+00
6.46E-01 3.29E-01 3.90E-01
1.91E+00 5.11E-01 1.09E+00
1.42E+01 1.74E+00 4.82E+00
1.68E+01 2.58E+00 6.30E+00
6.41E-01 3.26E-01 3.87E-01
1.91E+00 5.11E-01 1.09E+00
1.42E+01 1.74E+00 4.82E+00
1.68E+01 2.58E+00 6.30E+00
6.35E-01
1.91E+00
1.42E+01
1.68E+01
6.35E-01
1.91E+00
1.42E+01
1.68E+01
Table 4-58: Percentage deviation between DORT, 280GP3 and MCNP
Gra_left Clad_left Water_left Gra_top Clad_top Water_top
Abs_Fast
0.33
-7.53
-8.10
-3.34
-6.70
-10.65
Abs_Epi
-0.99
-0.87
-0.97
-0.89
-12.53
-12.60
Abs_Thermal
0.41
0.19
0.05
0.36
0.15
0.02
Abs_Total
-0.07
0.06
0.04
0.35
0.02
-0.03
Tot_Fast
Tot_Epi
Tot_Thermal
Tot_Total
0.26
0.21
-0.71
-0.41
-9.07
-6.82
0.86
-1.21
1.17
0.05
0.98
0.88
1.64
0.22
1.67
1.37
-8.35
-6.89
0.82
-1.21
-0.58
-0.04
0.94
0.77
93
Table 4-59: Percentage deviation between DORT, 280GP1 and MCNP
Gra_left Clad_left Water_left Gra_top Clad_top Water_top
Abs_Fast
0.42
-7.13
-7.87
-3.26
-6.49
-10.37
Abs_Epi
-0.98
-0.86
-0.96
-0.89
-12.53
-12.60
Abs_Thermal
0.41
0.19
0.06
0.37
0.15
0.02
Abs_Total
-0.05
0.06
0.04
0.36
0.02
-0.03
Tot_Fast
Tot_Epi
Tot_Thermal
Tot_Total
0.31
0.14
-0.71
-0.41
-8.88
-6.84
0.86
-1.20
1.38
0.05
0.98
0.89
1.61
0.17
1.67
1.36
-8.20
-6.90
0.82
-1.20
-0.31
-0.03
0.94
0.78
Table 4-60: Percentage deviation between DORT, 12GP1 and MCNP
Gra_left Clad_left Water_left Gra_top Clad_top Water_top
Abs_Fast
-0.66
-7.08
-8.27
-4.05
-7.27
-10.37
Abs_Epi
-0.94
-0.85
-0.92
-0.86
-12.49
-12.53
Abs_Thermal
0.42
0.19
0.03
0.38
0.15
-0.01
Abs_Total
-0.04
0.07
0.01
0.31
0.03
-0.05
Tot_Fast
Tot_Epi
Tot_Thermal
Tot_Total
0.37
0.23
-0.72
-0.39
-9.27
-6.76
0.86
-1.22
0.55
0.07
0.96
0.84
0.52
0.28
1.66
1.24
-8.97
-6.78
0.82
-1.23
-0.31
0.03
0.92
0.77
Table 4-61: Percentage deviation between DORT, 12GP1 and 280GP1
Gra_left Clad_left Water_left Gra_top Clad_top Water_top
Abs_Fast
0.05
-0.43
-0.82
-1.07
-0.84
0.00
Abs_Epi
0.03
0.05
0.01
0.05
0.08
0.03
Abs_Thermal
0.01
0.01
-0.02
0.01
0.01
-0.02
Abs_Total
0.01
0.01
-0.03
-0.05
0.01
-0.02
Tot_Fast
Tot_Epi
Tot_Thermal
Tot_Total
0.05
0.09
-0.01
0.02
-0.43
0.08
0.00
-0.02
-0.82
0.01
-0.02
-0.05
-1.07
0.11
-0.01
-0.12
-0.84
0.13
0.00
-0.03
0.00
0.06
-0.02
-0.01
94
4.5.2 Control Rod
The 2-D color-set control rod model is illustrated in Figure 4-13. The total
number of cells is 148x118 cells. It is modeled with a uniform mesh distribution with
0.03 cm mesh size. Using 280G cross section library, DORT calculated the 280G flux
spectrum. The 280G cross sections were collapsed to 12G broad group cross sections.
Figure 4-13: 2-D model for control rod XS generation
Table 4-62 shows the eigenvalues calculated from DORT and MCNP. The MCNP
calculation was performed with 3000 cycles with 100 inactive cycles and 5000
histories/cycles. Table 4-63 to Table 4-66 show reaction rates for each energy range and
comparisons for each case. The scattering order has the effect on the eigenvalue. The
differences of results from MCNP are 524 pcm with P1 scattering order comparing to 66
pcm for P3 scattering order. Thus, we use flux distribution P3 scattering case to collapse
to broad group structure.
95
Table 4-62: Eigenvalues calculated by DORT and MCNP
Rel. Dev. from MCNP in
Kinf
pcm of Δk/k
MCNP
0.74073 ±0.00057(3σ)
DORT (280G, S10,P1)
0.73685
-524
DORT (280G, S10,P3)
0.74122
66
Table 4-63: Reaction rates calculated by MCNP
Reaction Type
B4C
Clad
Water
(B4C)
(B4C)
Abs_Fast
1.01E-02 6.92E-04 1.82E-04
Abs_Epi
9.99E-02 1.80E-03 1.33E-04
Abs_Thermal
9.68E-02 2.42E-02 4.42E-03
Abs_Total
2.07E-01 2.67E-02 4.74E-03
Tot_Fast
Tot_Epi
Tot_Thermal
Tot_Total
2.58E-01
2.55E-01
9.87E-02
6.12E-01
2.01E-01 2.97E-01
3.36E-01 6.47E-01
1.73E-01 8.49E-01
7.09E-01 1.79E+00
Table 4-64: The reaction rates calculated by DORT with 280 groups, S10
quadrature order
P1
P3
Reaction
B4C
Clad
Clad
Water
B4C
Water
Type
(B4C)
(B4C)
(B4C)
(B4C)
Abs_Fast
1.02E-02 6.59E-04 1.72E-04 1.02E-02 6.55E-04 1.71E-04
Abs_Epi
1.02E-01 1.56E-03 1.32E-04 1.00E-01 1.55E-03 1.32E-04
Abs_Thermal 9.65E-02 2.27E-02 4.29E-03 9.66E-02 2.28E-02 4.33E-03
Abs_Total
2.08E-01 2.49E-02 4.60E-03 2.07E-01 2.50E-02 4.63E-03
Tot_Fast
Tot_Epi
Tot_Thermal
Tot_Total
2.62E-01
2.60E-01
9.84E-02
6.20E-01
1.93E-01 3.02E-01
3.13E-01 6.47E-01
1.64E-01 8.32E-01
6.70E-01 1.78E+00
2.60E-01
2.57E-01
9.85E-02
6.15E-01
1.92E-01 2.99E-01
3.11E-01 6.48E-01
1.65E-01 8.39E-01
6.68E-01 1.79E+00
96
Table 4-65: Percent deviation of reaction rates between DORT 280G and MCNP
P1
P3
Reaction Type
B4C
Clad
Clad
Water
B4C
Water
(B4C)
(B4C)
(B4C)
(B4C)
Abs_Fast
1.35
-4.74
-5.47
0.67
-5.33
-5.80
Abs_Epi
1.66
-13.14
-0.90
0.17
-13.75
-0.41
Abs_Thermal
-0.33
-6.24
-2.86
-0.20
-5.70
-2.13
Abs_Total
0.62
-6.70
-3.01
-0.08
-6.26
-2.33
Tot_Fast
Tot_Epi
Tot_Thermal
Tot_Total
1.42
1.96
-0.29
1.32
-3.99
-6.75
-5.13
-5.44
1.52
0.02
-1.95
-0.50
0.67
0.82
-0.17
0.54
-4.73
-7.33
-4.80
-5.84
0.69
0.08
-1.22
-0.27
Table 4-66: Percent deviation of reaction rates
between DORT 280GP1 and 280GP3
Reaction Type
B4C
Clad
Water
(B4C)
(B4C)
Abs_Fast
0.68
0.63
0.35
Abs_Epi
1.49
0.70
-0.48
Abs_Thermal
-0.13
-0.58
-0.74
Abs_Total
0.69
-0.47
-0.70
Tot_Fast
Tot_Epi
Tot_Thermal
Tot_Total
0.75
1.13
-0.12
0.77
0.77
0.62
-0.34
0.43
0.82
-0.06
-0.74
-0.23
The reaction rate comparisons between DORT 280GP1 and 280GP3 are in good
agreement except the absorption rate of B4C in epithermal energy range, which of course
affects the value of eigenvalue. The reaction rate comparisons between DORT and
MCNP calculations show considerable deviation in cladding region for the whole range
of energy and in thermal energy range of water region. This could be expected since the
mean-free-path (mfp) of absorber (B4C) is very small ~5.0E-4 cm. in 280-group structure
but the mesh size in B4C region is 0.03 cm, which is about 60 times larger than the mfp.
97
The neutrons most likely may not be able to escape from the absorber region once
entered.
The absorption rates in each energy range are plotted as a function of radius as
illustrated in Figure 4-14. The thermal absorption reaction rate drop dramatically. This
phenomenon explains the spatial self-shielding effect in absorber.
1.00E+00
1.00E-01
Absorption Reaction Rate
1.00E-02
1.00E-03
Thermal
Epithermal
Fast
1.00E-04
1.00E-05
1.00E-06
1.00E-07
1.00E-08
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
Radius (cm)
Figure 4-14: Absorption reaction rate as a function of B4C radius
In order to demonstrate what we have presumed for deviation in cladding region.
The outer bound of the B4C rod is refined. It should be note that we are not refining the
whole model in order to save memory and computational time and since the thermal flux
gets absorbed mostly in the outer bound as shown above. There are three models in this
98
study. Model#1 is a base model from previous calculations. It has a uniform mesh
distribution throughout the model. The mesh size for this model is 0.030 cm. Model#2 is
the model that cells in the outer bound of B4C rod is refined to 0.015 cm. Model#3 is the
model that cells in the outer bound of B4C rod is refined to 0.003 cm.
Table 4-67 shows kinf results predicted by DORT with 280 groups and relative
deviation from MCNP. The reaction rates for each energy range and comparisons for
Model #1 were shown previously in Table 4-64 and Table 4-65. Table 4-68 to Table 4-71
show reaction rates for each energy range and comparisons for Models #2 and #3. We
observed the improvement of results in cladding region when the cells in B4C has refined.
Table 4-67: Kinf results predicted by DORT with 280G
Kinf
(Rel. Dev. from MCNP in pcm of Δk/k)
P1
P3
Model #1
0.73685
0.74122
(148x118 cells)
(-524)
(66)
Model #2
0.73734
0.74172
(170x138 cells)
(-458)
(134)
Model #3
0.73742
0.74180
(347x304 cells)
(-447)
(144)
Table 4-68: Reaction rates calculated by DORT with 280 groups, S10 quadrature
order for Model#2
P1
P3
Reaction
B4C
Clad
Clad
Water
B4C
Water
Type
(B4C)
(B4C)
(B4C)
(B4C)
Abs_Fast
1.02E-02 6.59E-04 1.72E-04 1.02E-02 6.55E-04 1.71E-04
Abs_Epi
1.02E-01 1.57E-03 1.32E-04 1.00E-01 1.56E-03 1.33E-04
Abs_Thermal 9.64E-02 2.33E-02 4.31E-03 9.65E-02 2.34E-02 4.34E-03
Abs_Total
2.08E-01 2.55E-02 4.61E-03 2.07E-01 2.56E-02 4.64E-03
Tot_Fast
Tot_Epi
Tot_Thermal
Tot_Total
2.62E-01
2.60E-01
9.83E-02
6.20E-01
1.93E-01 3.02E-01
3.14E-01 6.47E-01
1.68E-01 8.35E-01
6.75E-01 1.78E+00
2.60E-01
2.57E-01
9.84E-02
6.15E-01
1.92E-01 2.99E-01
3.12E-01 6.48E-01
1.69E-01 8.41E-01
6.72E-01 1.79E+00
99
Table 4-69: Reaction rates calculated by DORT with 280 groups, S10 quadrature
order for Model#3
P1
P3
Reaction
B4C
Clad
Clad
Water
B4C
Water
Type
(B4C)
(B4C)
(B4C)
(B4C)
Abs_Fast
1.03E-02 6.61E-04 1.72E-04 1.02E-02 6.57E-04 1.72E-04
Abs_Epi
1.02E-01 1.58E-03 1.32E-04 1.00E-01 1.57E-03 1.33E-04
Abs_Thermal 9.66E-02 2.36E-02 4.32E-03 9.67E-02 2.37E-02 4.35E-03
Abs_Total
2.09E-01 2.58E-02 4.62E-03 2.07E-01 2.59E-02 4.66E-03
Tot_Fast
Tot_Epi
Tot_Thermal
Tot_Total
2.62E-01
2.61E-01
9.85E-02
6.21E-01
1.94E-01 3.02E-01
3.15E-01 6.49E-01
1.70E-01 8.37E-01
6.78E-01 1.79E+00
2.60E-01
2.58E-01
9.86E-02
6.17E-01
1.92E-01 3.00E-01
3.13E-01 6.49E-01
1.70E-01 8.43E-01
6.75E-01 1.79E+00
Table 4-70: Percent deviation of reaction rates between DORT 280G S10 Model #2
and MCNP
P1
P3
Reaction Type
B4C
Clad
Clad
Water
B4C
Water
(B4C)
(B4C)
(B4C)
(B4C)
Abs_Fast
1.37
-4.71
-5.44
0.69
-5.30
-5.77
Abs_Ephi
1.63
-12.73
-0.80
0.14
-13.33
-0.32
Abs_Thermal
-0.45
-3.77
-2.56
-0.31
-3.20
-1.83
Abs_Total
0.55
-4.42
-2.73
-0.14
-3.97
-2.04
Tot_Fast
Tot_Ephi
Tot_Thermal
Tot_Total
1.45
1.96
-0.41
1.31
-3.95
-6.56
-2.83
-4.78
1.55
0.07
-1.66
-0.34
0.69
0.81
-0.29
0.53
-4.69
-7.14
-2.49
-5.18
0.72
0.13
-0.93
-0.10
100
Table 4-71: Percent deviation of reaction rates between DORT 280G S10 Model #3
and MCNP
P1
P3
Reaction Type
B4C
Clad
Clad
Water
B4C
Water
(B4C)
(B4C)
(B4C)
(B4C)
Abs_Fast
1.61
-4.48
-5.25
0.93
-5.07
-5.58
Abs_Ephi
1.85
-12.42
-0.59
0.35
-13.02
-0.11
Abs_Thermal
-0.25
-2.67
-2.28
-0.12
-2.10
-1.55
Abs_Total
0.76
-3.40
-2.45
0.06
-2.94
-1.77
Tot_Fast
Tot_Ephi
Tot_Thermal
Tot_Total
1.69
2.19
-0.22
1.54
-3.72
-6.29
-1.80
-4.33
1.76
0.28
-1.38
-0.10
0.93
1.04
-0.10
0.76
-4.46
-6.87
-1.45
-4.73
0.92
0.34
-0.65
0.14
The 280G cross sections were collapsed to 12G broad group cross sections using
Model#3. DORT calculated the 280G flux spectrum. The eigenvalues from 280G-fine
groups and 12G-broad groups agree well under 100 pcm as shown in Table 4-72. Table
4-73 and Table 4-74 demonstrate reaction rates for each energy range and comparisons of
12G. They agree well with the 280GP3 case.
MCNP
Table 4-72: Eigenvalues calculated by DORT and MCNP
Kinf
Rel. Dev. from MCNP in
pcm of Δk/k
0.74073 ±0.00057(3σ)
DORT (280G, S10,P3)
0.74122
66
DORT (12G, S10,P3)
0.74056
-23
101
Table 4-73: Reaction rates calculated by DORT
with 12 groups, S10 quadrature order and P3
scattering order
Reaction
B4C
Clad
Water
Type
(B4C)
(B4C)
Abs_Fast
1.01E-02 6.51E-04 1.71E-04
Abs_Epi
1.01E-01 1.55E-03 1.32E-04
Abs_Thermal 9.60E-02 2.36E-02 4.33E-03
Abs_Total
2.07E-01 2.58E-02 4.63E-03
Tot_Fast
Tot_Epi
Tot_Thermal
Tot_Total
2.58E-01
2.61E-01
9.79E-02
6.17E-01
1.90E-01 2.97E-01
3.13E-01 6.46E-01
1.69E-01 8.39E-01
6.73E-01 1.78E+00
Table 4-74: Percent deviation of reaction
rates between DORT 12GP3 and 280GP3
Reaction
B4C
Clad
Water
Type
(B4C) (B4C)
Abs_Fast
-0.91
-0.95
-0.77
Abs_Epi
0.60
-0.69
-0.63
Abs_Thermal
-0.66
-0.54
-0.47
Abs_Total
-0.07
-0.56
-0.49
Tot_Fast
Tot_Epi
Tot_Thermal
Tot_Total
4.6
-0.91
1.48
-0.67
0.13
-0.95
0.07
-0.68
-0.41
-0.77
-0.46
-0.48
-0.52
Cross-Section Homogenization
After completing the study of broad-group cross-section library for TRIGA fuel
pin cell, studies on spatial homogenization were performed. The ideal concept of the
homogenization is to preserve all the reaction rates in the problem. We utilize the scalar
flux weighting method to combine the material regions as shown in Equation 4-1. Two
sets of homogenized cross sections are calculated: i) one which combines 3 regions (Zr
Rod, UZrH fuel, and SS304 Cladding) as shown in Figure 4-15 and ii) the other, which
102
combines 4 regions (Zr Rod, UZrH fuel, SS304 Cladding, and H2O) as shown in Figure
4-16 for 12-energy group structure.
nzone Eg −1
3
Σg =
r
r
∑ ∫ dE∫ d rΣ (r , E)φ(r , E)
i =1
Eg
i
Vi
nzone Eg −1
r
∑ ∫ dE∫ d rφ(r , E)
3
i =1
Eg
Vi
Equation 4-1
Zr+UZrH+SS304
Zr+UZrH+SS304
Figure 4-15: Three-Region Homogenization
Zr+UZrH+SS304+
H2O
Figure 4-16: Four-Region Homogenization
103
Table 4-75 and Table 4-76 give the kinf of three-region and four-region
homogenization approaches using scalar flux weighting, respectively. The differences of
results are ~60 pcm comparing the three-region homogenization with heterogeneous
geometry, and ~200 pcm comparing with MCNP solution. For the four-region
homogenization,
the
differences
are
~200
pcm
comparing
the
four-region
homogenization with heterogeneous geometry, and ~350 pcm comparing with MCNP
solution. The deviations of homogenized cross sections with MCNP are less than
heterogeneous solutions, since the negative deviation between heterogeneous and MCNP
compensate with the positive deviation between heterogeneous and homogenized
calculations. These differences are higher than three-region homogenization. Another
observation is that all quadrature orders including S4 give almost the same results for kinf.
As we combine water into the homogenized region, the moderation and scattering
properties of water are smeared with the fuel meat and cladding as one material.
Table 4-75: Kinf calculated by DORT (3–region combination)
kinf
Heterogeneous Homogenized
XS
XS
S4P1
1.40386
1.40480
Rel. Dev. in pcm
of Δk/k
(Homo. VS. Het.)
67
Rel. Dev. in pcm of
Δk/k
(Homo.VS. MCNP)
205
S6P1
1.40369
1.40461
66
192
S8P1
1.40370
1.40459
63
190
S10P1
1.40372
1.40459
62
190
S12P1
1.40373
1.40458
61
190
S14P1
1.40373
1.40458
61
190
S16P1
1.40374
1.40458
60
190
104
Table 4-76: Kinf calculated by DORT (4–region combination)
kinf
Heterogeneous Homogenized
XS
XS
S4P1
1.40386
1.40674
Rel. Dev. in pcm
of Δk/k
(Homo. VS. Het.)
205
S6P1
1.40369
1.40674
217
344
S8P1
1.40370
1.40674
217
344
S10P1
1.40372
1.40674
215
344
S12P1
1.40373
1.40675
215
345
S14P1
1.40373
1.40674
214
344
S16P1
1.40374
1.40674
214
344
4.7
Rel. Dev. in pcm of
Δk/k
(Homo.VS. MCNP)
344
Summary
In this chapter the fine–energy-group and broad-energy-group structures for 2-D
cross-section generation have been selected in fast, epithermal, and thermal energy
ranges by the CPXSD methodology using different objectives corresponding of each
energy range. The scalar flux weighting technique is utilized in collapsing fine- to broadgroup libraries. Results indicate very good agreement between 280 fine- and 12 broadgroup structures.
The studied fine- and broad- group structures were also applied in non-fissile
material, i.e., graphite and control rod. Results indicate good agreement in eigenvalues
compared with MCNP. The scattering order has a strong effect on a control rod model
but not in the graphite model.
105
For the homogenization, the results are in good agreement for three-region
homogenization approach. For four-region homogenization approach, the quadrature
order does not affect the solution.
106
CHAPTER 5
Three-Dimensional Cross Section Generation
In the previous chapter, the 2-D cross section group structure was established with
280 fine groups and 12 broad groups. In this chapter, the same CPXSD methodology,
adapted for criticality problem, will be used to study the actual 3-D cross section
generation. We start with the parametric optimization study for the SN methods and
follow with the fine- and broad-group structure selection process. The 8.5% TRIGA fuel
cell is selected for this study. First, the 3-D fine group structure will be constructed.
Finally, the 3-D broad-group structure will be established and compared with 2-D broadgroup structure.
5.1
Three-dimensional model for Cross-Section Generation for Fuel Element
One eight of a hexagonal unit cell has been modeled for the 3-D cross-section
generation study by taking the advantage of the model symmetry as illustrated in Figure
5-1. The reflective boundary is applied for all the surfaces except the top surface. Table
5-1 shows the material data that has been used in the cross-section generation
calculations. As the reference, the Monte Carlo MCNP5 calculation is performed with
5000 histories, 3000 cycles and 100 inactive cycles with continuous energy cross section
library. The predicted reference eigenvalue is 1.19272±0.00051(3σ).
107
Graphite
8.7376
Fuel
19.05
Unit: cm.
Figure 5-1: 3D cross section generation model
Table 5-1: Material density of the fuel elements
Nuclide
Density (atoms/barn-cm)
Fuel
12 wt.%
8.5 wt.%
H
0.05568
0.05689
Zr
0.03442
0.03506
U-234
0.000002915
U-235
0.0003642
0.00025052
U-236
0.000002434
U-238
0.0014538
0.001003
Reflector
H
0.06683
O
0.03343
Zr Rod
Zr
0.042936
SS304
SS304
0.08739
Graphite
C-12
0.080195
108
5.2
Parametric Studies
5.2.1 Spatial Mesh, Angular Quadrature, and Scattering Order Studies
In this section, the spatial discretization, angular quadrature, and scattering order
of cross sections are studied for the 3-D case. The 238-group structure is used
throughout this set of studies. Generally, performing this type of studies simultaneously
is cumbersome for 3-D problems. Thus, the sensitivity studies for spatial meshes,
angular quadrature set, and scattering order were done, separately.
A. Scattering Order Study
The scattering orders that are used to perform sensitivity studies are P1, P3, and
P5. The S4, S6, and S8 orders are examined for fully symmetric and Square LegendreChebyshev, respectively. The fine mesh model used in this study is 20x23x37 (x,y,z).
The TORT results and relative deviation in pcm from the reference MCNP5 results are
provided in Table 5-2 and Table 5-3 for fully symmetric and Square LegendreChebyshev quadratures, respectively. They are summarized graphically in Figure 5-2 and
Figure 5-3.
109
Table 5-2: TORT results with 238-energy group XS, Level-Symmetric
SN Order Scattering
Keff
Conv.
Out.
Rel.
Iter.
Dev. in pcm of
Δk/k
S4
P1
1.19240
3.0E-7
76
-27
S6
P1
1.19292
3.0E-7
82
17
S8
P1
1.19163
1.0E-7
68
-91
S4
P3
1.19453
-7.0E-7
74
152
S6
P3
1.19481
-2.0E-7
82
175
S8
P3
1.19353
-1.0E-7
68
68
S6
P5
1.19474
-2.0E-7
57
169
S8
P5
1.19346
-1.0E-7
53
62
Table 5-3: TORT results with 238-energy group XS, SLC
SN Order Scattering
Keff
Conv.
Out.
Rel.
Iter.
Dev. in pcm of
Δk/k
S4
P1
1.19393
-9.0E-7
78
101
S6
P1
1.19241
-3.0E-7
86
-26
S8
P1
1.19244
1.0E-6
86
-23
S4
P3
1.19588
-8.0E-7
90
265
S6
P3
1.19434
-3.0E-7
84
136
S8
P3
1.19438
1.0E-6
76
139
S6
P5
1.19428
-1.0E-7
88
131
S8
P5
1.19433
-1.0E-6
75
135
110
Level Symmetric
1.1960
P1
P3
P5
1.1950
Eigenvalue
1.1940
1.1930
1.1920
1.1910
1.1900
S4
S6
S8
Quadrature Order
Figure 5-2 Eigenvalue behavior under variation of scattering
order and level symmetric quadrature
Square Legendre Chebychev
1.1970
P1
P3
P5
1.1960
Eigenvalue
1.1950
1.1940
1.1930
1.1920
1.1910
1.1900
S4
S6
S8
Quadrature Order
Figure 5-3 Eigenvalue behavior under variation of scattering
order and Square Legendre-Chebyshev quadrature
111
For both level-symmetric and Square Legrendre-Chebyshev types of quadrature,
the results show that scattering order has the effect on eigenvalue predictions for TRIGA
cell in 3-D geometry with a systematical bias about 160 pcm between P1 and P3.
However, using scattering order higher than P3 does not provide a significant
improvement on keff predictions. The P3 and P5 solution yield almost the same value of
keff. As a result, the P3 scattering order should be used in order to get a good solution.
B. Spatial Mesh Study
In 3-D problems, not only the study in radial directions but also the study in axial
direction has to be performed. The Square Legendre-Chebyshev was used to study this
problem with S8 order. In order to save computing time, the P1 scattering order was used
to perform sensitivity studies. All the results were compared with the continuous energy
MCNP result, which is used as a reference solution in this study. First, we performed the
study on the radial directions by fixing the number of axial meshes. After we established
the radial meshes, then the axial mesh was studied. Three 3-D fine-mesh models have
been developed with different grid intervals to study for radial direction refinement.
1) 17020 cells: 20 x-axis, 23 y-axis, 37 z-axis
2) 104784 cells: 48 x-axis, 59 y-axis, 37 z-axis
3) 137862 cells: 54 x-axis, 69 y-axis, 37 z-axis
The TORT results and relative deviations as compared to the MCNP results in
pcm are provided in Table 5-4.
112
Table 5-4: TORT results with 238-energy group XS, S8 (SLC), P1
Meshing
keff
Conv.
Rel.
Dev. in pcm of Δk/k
20x23x37
1.19244
-9.0E-7
-23
48x59x37
1.19100
8.0E-7
-144
54x69x37
1.19103
-1.0E-6
-142
It was found that keff converges with the mesh refinement higher than 48x59 cells
in radial directions as shown in Figure 5-4. Thus, this radial meshing will be used further
for axial mesh study.
1.1930
1.1925
keff
1.1920
1.1915
1.1910
1.1905
1.1900
20x23x37
48x59x37
54x69x37
Mesh Model
Figure 5-4: Eigenvalue behavior with different radial-mesh model
113
Four 3-D fine-mesh models have been developed with different grid intervals in
axial direction.
1) 53808 cells: 48 x-axis, 59 y-axis, 19 z-axis
2) 104784 cells: 48 x-axis, 59 y-axis, 37 z-axis
3) 155760 cells: 48 x-axis, 59 y-axis, 55 z-axis
4) 198242 cells: 48 x-axis, 59 y-axis, 70 z-axis
The results are presented in Table 5-5. It was found that keff converges with the
mesh refinement higher than 55 cells in axial directions as shown in Figure 5-5. Thus, the
optimum mesh-model for 3-D TRIGA cell was established with 48x59x55 cells.
Table 5-5: TORT results with 238-energy group XS, S8 (SLC), P1
Meshing
keff
Conv.
Rel.
Dev. in pcm of Δk/k
48x59x19
1.18935
9.0E-7
-283
48x59x37
1.19100
8.0E-7
-144
48x59x55
1.19144
1.0E-6
-107
48x59x70
1.19160
4.0E-7
-94
114
1.1940
1.1930
1.1920
keff
1.1910
1.1900
1.1890
1.1880
1.1870
48x59x19
48x59x37
48x59x55
48x59x70
Mesh Model
Figure 5-5: Eigenvalue behavior with different axial-mesh model
5.2.2 Qudrature Order Determination
The optimum mesh model for 3-D TRIGA cell was established with 48x59x55
cells. Here, we study the optimum quadrature set for 3-D TRIGA cell. The S4, S6, S8,
and S10 with SLC were used. Table 5-6 gives TORT calculations and deviations from the
MCNP results. The results show that keff predictions are insensitive to the quadrature
order higher than S6.
Table 5-6: TORT results with 238-energy group XS
and 48x59x55 cells
SN order
keff
Rel.
Dev. in pcm of Δk/k
with MCNP
18
S4
1.19293
-107
S6
1.19144
-107
S8
1.19144
-108
S10
1.19143
115
The reaction rate comparison
Six-radial detectors were defined in each selected axial mesh within the fuel and
graphite region to compare the neutron production reaction rate for fuel region and the
absorption reaction rate for graphite region in Table 5-7. Overall, the reaction rates
change relatively less for higher quadrature order. As a result, S8 has been selected to be
used for further study.
Table 5-7: Neutron production reaction rate and percent deviations
Position
Nu-fission raction rate
X
Y
Z
S4
S6
S8
0.35 0.2
4.7 f 2.551E-02 2.546E-02 2.547E-02
1.73 0.2
4.7 f 3.210E-02 3.209E-02 3.213E-02
0.21 0.4
4.7 f 2.552E-02 2.547E-02 2.549E-02
0.73 0.7
4.7 f 2.686E-02 2.679E-02 2.679E-02
1.15 1.2
4.7 f 3.165E-02 3.145E-02 3.141E-02
0.21 1.7
4.7 f 3.234E-02 3.248E-02 3.242E-02
0.35 0.2
9.7 f 2.237E-02 2.235E-02 2.236E-02
1.73 0.2
9.7 f 2.816E-02 2.817E-02 2.821E-02
0.21 0.4
9.7 f 2.238E-02 2.235E-02 2.238E-02
0.73 0.7
9.7 f 2.357E-02 2.352E-02 2.353E-02
1.15 1.2
9.7 f 2.779E-02 2.763E-02 2.760E-02
0.21 1.7
9.7 f 2.839E-02 2.854E-02 2.850E-02
0.35 0.2 14.7 f 1.747E-02 1.746E-02 1.747E-02
1.73 0.2 14.7 f 2.202E-02 2.203E-02 2.206E-02
0.21 0.4 14.7 f 1.749E-02 1.747E-02 1.748E-02
0.73 0.7 14.7 f 1.843E-02 1.839E-02 1.840E-02
1.15 1.2 14.7 f 2.183E-02 2.170E-02 2.167E-02
0.21 1.7 14.7 f 2.232E-02 2.244E-02 2.240E-02
0.35 0.2
25g 1.991E-05 2.042E-05 1.983E-05
1.73 0.2
25g 2.055E-05 1.952E-05 2.003E-05
0.21 0.4
25g 1.940E-05 2.032E-05 1.967E-05
0.73 0.7
25g 2.044E-05 1.983E-05 2.005E-05
1.15 1.2
25g 2.144E-05 2.027E-05 2.006E-05
0.21 1.7
25g 2.100E-05 1.981E-05 2.040E-05
f
g
Note: is the fuel level, is the graphite level.
S10
2.547E-02
3.210E-02
2.548E-02
2.680E-02
3.142E-02
3.241E-02
2.236E-02
2.818E-02
2.237E-02
2.353E-02
2.761E-02
2.849E-02
1.746E-02
2.204E-02
1.747E-02
1.840E-02
2.167E-02
2.239E-02
1.975E-05
1.963E-05
2.013E-05
2.011E-05
1.991E-05
2.016E-05
S6 VS
S4
-0.17
-0.04
-0.20
-0.27
-0.63
0.45
-0.10
0.03
-0.13
-0.20
-0.56
0.54
-0.07
0.05
-0.11
-0.18
-0.58
0.56
2.57
-5.01
4.71
-3.00
-5.46
-5.65
% Deviation
S8 VS
S10 VS
S6
S8
0.04
-0.03
0.11
-0.09
0.09
-0.05
0.02
0.02
-0.14
0.04
-0.21
-0.01
0.07
-0.03
0.14
-0.10
0.11
-0.05
0.05
0.01
-0.11
0.03
-0.17
-0.02
0.04
-0.04
0.12
-0.11
0.09
-0.06
0.03
0.01
-0.15
0.02
-0.19
-0.03
-2.89
-0.41
2.59
-1.99
-3.18
2.31
1.12
0.31
-1.03
-0.74
2.99
-1.20
The following set of figures display the flux distribution for each quadrature to
observe the ray effect. We selected group 23rd to represent the fast range of energy and
116
group 212th to represent the thermal range of energy. Level 15th represents the fuel part
and level 50th represents the graphite part.
S4
1.a) zlev15g23
1.b) zlev15g212
1.c) zlev50g23
1.d) zlev50g212
2.a) zlev15g23
2.b) zlev15g212
2.c) zlev50g23
2.d) zlev50g212
3.a) zlev15g23
3.b) zlev15g212
3.c) zlev50g23
3.d) zlev50g212
4.a) zlev15g23
4.b) zlev15g212
4.c) zlev50g23
4.d) zlev50g212
S6
S8
S10
Figure 5-6: Flux distribution for each quadrature order
117
5.3
Fine- Group Structure for TRIGA
Using the procedure of the CPXSD methodology developed further for criticality
problem, the fine-group structure for TRIGA cross-section generation is obtained. The
238-group SCALE library is used as a starting group structure. The 238-group cross
sections are generated. Initially, the 238-group structure was divided into 3 major ranges
of energy: fast (0.1 MeV to 20 MeV), epithermal (3 eV to 0.1 MeV), and thermal (1E-05
eV to 3 eV). We established two criteria for obtaining a fine group structure. The first
criterion is 10 pcm relative deviation of Δk/k and the second criterion is 1% relative
deviation of objective reaction rates. The objective reaction rates are different for each
range of energy. Using the flux and adjoint function moments computed from the
transport calculations with TORT, the Cg’s are calculated. Depending on the magnitude
of the Cg’s per group, the group structure is refined for each energy range. The groups
corresponding to large Cg’s were partitioned into more groups. The group with the
highest Cg was subdivided into a number of groups, and the remaining groups were
divided into fewer groups based on the ratio of their Cg to the maximum Cg. This study
is performed using 1/8 of 8.5% fuel cell with 48x59x55 fine cells in x, y, and z direction
with S8 (SLC) quadrature order and P1 scattering order.
5.3.1 Fast Group Refinement
In this section a group structure in the fast energy range between 0.1 and 20 MeV
is derived. The 238-group SCALE library is used as a starting group structure with 44
groups in fast energy range, 104 groups in epithermal range, and 90 groups in thermal
118
range. The 238-group Cg’s are calculated using the normalized νΣf as the adjoint source
to perform the adjoint transport calculation. The point-wise cross section of U238(n,f) is
used to consider the group boundaries. The objectives are eigenvalue and neutron
production reaction rate of U238. The new group structures are generated. Table 5-8 gives
the number of groups in fast energy range that we obtained from the group refinement
process.
Table 5-8: Fine groups generated in the fast energy range
Group Structure
Number of Groups in Different Energy Ranges
Number
Fast
Epithermal
Thermal
Total
1
44
104
90
238
2
52
104
90
246
3
80
104
90
274
The importance of groups in fast energy range, between 0.1 and 20 MeV, of 238group and 246-group structures are plotted in Figure 5-7. The plot shows that when the
groups that have more importance are refined, the importance of those groups is
decreased.
119
4.00E-02
3.50E-02
3.00E-02
Importance (E)
2.50E-02
2.00E-02
238 groups
246 groups
1.50E-02
1.00E-02
5.00E-03
0.00E+00
1.E-01
1.E+00
1.E+01
1.E+02
Energy (MeV)
Figure 5-7: Importance in groups of 238G and 246G libraries
The eigenvalues are calculated and compared between the group structures. For
the 246G and 274G comparison, Table 5-9 shows that percent relative difference of
eigenvalue is less than 10 pcm and the percentage relative deviation of U238(n,νΣf) is
0.199%. Consequently, we selected the 246-group structure, which contains 52 groups in
fast energy range, for further group refinement in the epithermal energy range.
Table 5-9: Eigenvalue results of fine group energy for 8.5% wt. case
Group
kinf (S8P1)
Rel. Dev. in pcm
%Rel. Dev.
νΣf rate of U238
With previous
> 0.1 MeV
of Δk/k
group
With previous
group
238
1.19144
8.662E-2
-
246
1.19202
49
8.530E-2
-1.524
274
1.19212
8
8.513E-2
-0.199
120
5.3.2 Epithermal-Group Refinement
In this section a group structure in the epithermal energy range between 3eV and
0.1 MeV is derived. The 246-group structure from the fast group refinement is used as a
starting group structure with 52 groups in the fast energy range, 104 groups in the
epithermal range, and 90 groups in the thermal range. The 246-group Cg’s are calculated
using the summation of the normalized νΣf and down-scattering cross section of H in ZrH
from the epithermal group to the thermal group as adjoint source to perform the adjoint
transport calculation. The absorption point-wise cross-section of U238 is used to consider
the group boundaries. The objectives are eigenvalue, down-scattering reaction rate of H
in ZrH from the epithermal energy range to the thermal energy range and absorption
reaction rate of U238.
Table 5-10 shows the number of groups in the epithermal energy range that we
obtained from the group refinement process. The importance of groups in epithermal
energy range, between 3 eV and 0.1 MeV, of 246-group structure are plotted in Figure
5-8.
Table 5-10: Fine groups generated in the epithermal energy range
Group Structure
Number of Groups in Different Energy Ranges
Number
Fast
Epithermal
Thermal
Total
1
52
104
90
246
2
52
152
90
294
121
1.40E-02
1.20E-02
Importance(E)
1.00E-02
8.00E-03
6.00E-03
4.00E-03
2.00E-03
0.00E+00
1.0E-06
1.0E-05
1.0E-04
1.0E-03
1.0E-02
1.0E-01
1.0E+00
Energy(MeV)
Figure 5-8: Importance in groups of 246G libraries
The eigenvalues were calculated and compared between the group structures in
Table 5-11. For the 246G and 294G comparison, the relative difference of eigenvalues
are less than 10 pcm and the percentage relative deviation of U238(n,abs) and downscattering of H in ZrH from the epithermal range to the thermal range are less than 1.0%
as demonstrated in Table 5-12. Consequently, we selected the 246-group structure, which
contains 104 groups in epithermal energy range, for further group refinement in the
thermal energy range.
122
Table 5-11: Eigenvalue results of fine group energy
Group
246
294
for 8.5% wt. case
kinf (S8P1)
Rel. Dev. in pcm of
Δk/k
With previous group
1.19202
-
1.19203
1
Table 5-12: Reaction rate comparison for 8.5% wt. case
Down-scat.
%Rel. Dev.
U238(n,abs)
%Rel. Dev.
Group
Of H in ZrH
With previous group
With previous
group
246
0.167
-
0.868
-
294
0.167
-0.00
0.867
-0.115
5.3.3 Thermal-Group Refinement
In this section, a group structure in the thermal energy range between 1E-5 to 3
eV is derived. The 246-group structure from the fast and epithermal group refinements is
used as a starting group structure with 52 groups in the fast energy range, 104 groups in
the epithermal range, and 90 groups in the thermal range. The 246-group Cg’s are
calculated using the summation of the normalized νΣf and up-scattering cross section of
H in ZrH as the adjoint source to perform the adjoint transport calculation. The inelastic
scattering point-wise cross-section of H in ZrH is used to consider the group boundaries.
The objectives are eigenvalue, neutron production reaction rate of U235, and up-scattering
reaction rate of H in ZrH in the thermal energy range.
123
Table 5-13 shows the number of groups in thermal energy range that we obtained
from the group refinement process. The importance of groups in the thermal energy
range, between 1E-5 and 3 eV, of 246-group, 254-group and 280-group structures are
plotted in Figure 5-9.
Table 5-13: Fine groups generated in the thermal energy range
Group Structure
Number of Groups in Different Energy Ranges
Number
Fast
Epithermal
Thermal
Total
1
52
104
90
246
2
52
104
98
254
3
52
104
124
280
4
52
104
180
336
4.00E-02
3.50E-02
3.00E-02
Importance(E)
2.50E-02
2.00E-02
246G
254G
280G
1.50E-02
1.00E-02
5.00E-03
0.00E+00
1.00E-10
1.00E-09
1.00E-08
1.00E-07
1.00E-06
1.00E-05
Energy(MeV)
Figure 5-9: Importance in groups of 246G, 254G and 280G libraries
124
The eigenvalue results and comparisons for fine group energy in thermal range of
8.5% case are given in Table 5-14. Table 5-15 shows the result of up-scattering of H in
ZrH and neutron-production reaction rates from each group structure library and the
comparisons. Comparing between 280G and 336G, the percent relative difference of
eigenvalues, the U235 neutron production rate and the up-scattering of H in ZrH are within
the criteria. The 280-group structure is selected to be our final fine group structure. Table
5-16 lists energy group boundaries of the 280-group structure.
Table 5-14: Eigenvalue results for fine group energy
Group
246
Group
246
254:246
254
280:254
280
336:280
336
in thermal range 8.5% case
keff (S8P1)
Rel. Dev. in pcm of
Δk/k
With previous group
1.19202
-
254
1.19121
-68
280
1.19053
-57
336
1.19045
-7
Table 5-15: Reaction rate comparison of 8.5% case
Up-scat.
%Rel. Dev.
%Rel. Dev.
U235(n, νΣf )
Of H in ZrH With previous group
With previous
group
5.50E-2
45.520
-
5.44 E-21
6.08 E-2
6.06 E-21
6.66 E-2
6.65 E-2
-1.07
45.485
-0.08
45.469
-0.04
45.465
-0.01
-0.31
-0.14
Note: 1The reaction rate was calculated in a group-collapsing method to be compared with the previous
group
125
Table 5-16: Group structure of the 280 fine groups
Energy Group
Number
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
Upper
Energy(MeV)
2.0000E+01
1.7333E+01
1.5683E+01
1.4550E+01
1.3840E+01
1.2840E+01
1.0000E+01
8.1873E+00
6.4340E+00
4.8000E+00
4.3040E+00
3.8693E+00
3.4347E+00
3.0000E+00
2.7395E+00
2.4790E+00
2.3540E+00
2.1860E+00
2.0180E+00
1.8500E+00
1.7333E+00
1.6167E+00
1.5000E+00
1.4000E+00
1.3560E+00
1.3170E+00
1.2500E+00
1.2000E+00
1.1000E+00
1.0100E+00
9.2000E-01
9.0000E-01
8.7500E-01
8.6110E-01
8.2000E-01
7.5000E-01
6.7900E-01
6.7000E-01
6.0000E-01
5.7300E-01
Energy Group
Number
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
Upper
Energy(MeV)
5.5000E-01
4.9950E-01
4.7000E-01
4.4000E-01
4.2000E-01
4.0000E-01
3.3000E-01
2.7000E-01
2.3500E-01
2.0000E-01
1.5000E-01
1.2830E-01
1.0000E-01
8.5000E-02
8.2000E-02
7.5000E-02
7.3000E-02
6.0000E-02
5.2000E-02
5.0000E-02
4.5000E-02
3.0000E-02
2.5000E-02
1.7000E-02
1.3000E-02
9.5000E-03
8.0300E-03
6.0000E-03
3.9000E-03
3.7400E-03
3.0000E-03
2.5800E-03
2.2900E-03
2.2000E-03
1.8000E-03
1.5500E-03
1.5000E-03
1.1500E-03
9.5000E-04
6.8300E-04
Energy Group
Number
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
Upper
Energy(MeV)
6.7000E-04
5.5000E-04
3.0500E-04
2.8500E-04
2.4000E-04
2.1000E-04
2.0750E-04
1.9250E-04
1.8600E-04
1.2200E-04
1.1900E-04
1.1500E-04
1.0800E-04
1.0000E-04
9.0000E-05
8.2000E-05
8.0000E-05
7.6000E-05
7.2000E-05
6.7500E-05
6.5000E-05
6.1000E-05
5.9000E-05
5.3400E-05
5.2000E-05
5.0600E-05
4.9200E-05
4.8300E-05
4.7000E-05
4.5200E-05
4.4000E-05
4.2400E-05
4.1000E-05
3.9600E-05
3.9100E-05
3.8000E-05
3.7000E-05
3.5500E-05
3.4600E-05
3.3750E-05
126
Energy Group
Number
121
122
123
124
125
126
127
128
129
130
131
132
133
134
135
136
137
138
139
140
141
142
143
144
145
146
147
148
149
150
151
152
153
154
155
156
157
158
159
160
Upper
Energy(MeV)
3.3250E-05
3.1750E-05
3.1250E-05
3.0000E-05
2.7500E-05
2.5000E-05
2.2500E-05
2.1000E-05
2.0000E-05
1.9000E-05
1.8500E-05
1.7000E-05
1.6000E-05
1.5100E-05
1.4400E-05
1.3750E-05
1.2900E-05
1.1900E-05
1.1500E-05
1.0000E-05
9.1000E-06
8.1000E-06
7.1500E-06
7.0000E-06
6.7500E-06
6.5000E-06
6.2500E-06
6.0000E-06
5.4000E-06
5.0000E-06
4.7500E-06
4.0000E-06
3.7300E-06
3.5000E-06
3.1500E-06
3.0500E-06
3.0000E-06
2.9700E-06
2.8700E-06
2.7700E-06
Energy Group
Number
161
162
163
164
165
166
167
168
169
170
171
172
173
174
175
176
177
178
179
180
181
182
183
184
185
186
187
188
189
190
191
192
193
194
195
196
197
198
199
200
Upper
Energy(MeV)
2.6700E-06
2.5700E-06
2.4700E-06
2.3800E-06
2.3000E-06
2.2100E-06
2.1200E-06
2.0000E-06
1.9400E-06
1.8600E-06
1.7700E-06
1.6800E-06
1.5900E-06
1.5000E-06
1.4500E-06
1.4000E-06
1.3500E-06
1.3000E-06
1.2500E-06
1.2250E-06
1.2000E-06
1.1750E-06
1.1500E-06
1.1400E-06
1.1300E-06
1.1200E-06
1.1100E-06
1.1000E-06
1.0900E-06
1.0800E-06
1.0700E-06
1.0600E-06
1.0500E-06
1.0400E-06
1.0300E-06
1.0200E-06
1.0100E-06
1.0000E-06
9.7500E-07
9.5000E-07
Energy Group
Number
201
202
203
204
205
206
207
208
209
210
211
212
213
214
215
216
217
218
219
220
221
222
223
224
225
226
227
228
229
230
231
232
233
234
235
236
237
238
239
240
Upper
Energy(MeV)
9.2500E-07
9.0000E-07
8.5000E-07
8.0000E-07
7.5000E-07
7.0000E-07
6.5000E-07
6.2500E-07
6.0000E-07
5.5000E-07
5.0000E-07
4.5000E-07
4.0000E-07
3.7500E-07
3.5000E-07
3.2500E-07
3.0000E-07
2.7500E-07
2.5000E-07
2.2500E-07
2.0000E-07
1.7500E-07
1.5000E-07
1.4167E-07
1.3333E-07
1.2500E-07
1.1875E-07
1.1250E-07
1.0833E-07
1.0417E-07
1.0000E-07
9.6667E-08
9.3333E-08
9.0000E-08
8.6667E-08
8.3333E-08
8.0000E-08
7.7500E-08
7.5000E-08
7.2500E-08
127
Energy Group
Number
241
242
243
244
245
246
247
248
249
250
251
252
253
254
Upper
Energy(MeV)
7.0000E-08
6.7500E-08
6.5000E-08
6.2500E-08
6.0000E-08
5.7500E-08
5.5000E-08
5.2500E-08
5.0000E-08
4.7500E-08
4.5000E-08
4.3333E-08
4.1667E-08
4.0000E-08
Energy Group
Number
255
256
257
258
259
260
261
262
263
264
265
266
267
268
Upper
Energy(MeV)
3.83E-08
3.67E-08
3.50E-08
3.33E-08
3.17E-08
3.00E-08
2.77E-08
2.53E-08
2.28E-08
2.02E-08
1.77E-08
1.51E-08
1.26E-08
1.00E-08
Energy Group
Number
269
270
271
272
273
274
275
276
277
278
279
280
Upper
Energy(MeV)
7.50E-09
5.00E-09
4.00E-09
3.00E-09
2.50E-09
2.00E-09
1.50E-09
1.20E-09
1.00E-09
7.50E-10
5.00E-10
1.00E-10
1.00E-11
The 280-fine-group cross-section library was selected to be a fine-group structure
for the TRIGA reactor based on the CPXSD methodology. This 3-D fine-group structure
is the same as in the 2-D study. In conclusion, a methodology is established to generate
the fine-group cross-section library and applied to an example of 8.5% wt. TRIGA 3-D
fuel cell.
5.4
Cross-Section Collapsing
In this section, the 280-group structure was collapsed into a broad-group
structure. With the same approach as developing the fine-group structure, we established
two criteria to obtain a broad group structure. The first criterion is 10 pcm relative
deviation of Δk/k and the second criterion is 1% relative deviation of objective reaction
rates. The objective reaction rates are different for each range of energy. The U238(n,νΣf)
is considered in the fast energy range, the down-scattering reaction rates of H in ZrH and
U238(n,a) are considered in the epithermal energy range, and the U235(n,νΣf) and the
thermal up-scattering reaction rates of H in ZrH are considered in the thermal range. The
128
group collapsing started with fast energies by initiating a very-broad-group structure and
using the same fine-group structure in the epithermal and thermal energies. Then, the
aforementioned “contributon” approach was used to refine the broad-group structure.
This process is repeated until the two criteria were met, and consequently a new broadgroup structure for the fast energies was obtained. With this new fast broad group
structure, we continue the same process for the epithermal and thermal energy ranges.
5.4.1 Fast-Group Collapsing and Axial Nodalization Study
In fast energy range, we first combined all the energy groups into one group. A
new group library contains 229 groups. This group structure was also used to study the
axial nodalization for cross section collapsing. Three cases were performed as shown in
Figure 5-10. In Case 1, we used the fluxes of the material-wise full axial length to
collapse the cross sections. In Case 2, a 4-cm node was used to collapse the cross
sections. In Case 3, a 1-cm node was used to collapse the cross sections.
Table 5-17: Number of groups for each energy range
Group Structure
Number of Groups in Different Energy Ranges
Number
Fast
Epithermal
Thermal
Total
1
52
104
124
280
2
1
104
124
229
129
Case1
Case 2
Case 3
Figure 5-10 Axial mesh size used in nodal length collapsing study
Table 5-18 shows the eigenvalues for each case. Table 5-19 shows the minimum
and maximum of mesh-wise reaction rate deviations for each layer between case 2 and
case1. Table 5-20 shows the minimum and maximum of mesh-wise reaction rate
deviations for each layer between cases 3 and 2. We observed that the eigenvalue and
reaction rate deviations between these three cases are fairly small and less than 10 pcm.
From this study, a full axial length can be used to collapse the cross sections. Comparing
between 280G and 229G, the relative deviation is large. As a result, the one group
structure is not enough in the fast energy range. Further group refinement has to be
studied.
130
Table 5-18: Eigenvalue results for 3D, 8.5% fuel cell
Group
keff (S8P1)
Rel Dev. in pcm of
Δk/k with 280G
280
1.19053
-
229 (1)
1.19948
752
229 (2)
1.19942
747
229 (3)
1.19944
748
Note: (1) using the full length of axial direction to collapse the cross sections
(2) using 4-cm node in axial direction to collapse the cross sections
(3) using 1-cm node in axial direction to collapse the cross sections
Table 5-19:The minimum and maximum of mesh-wise reaction
rate deviations for each layer between case 2 and case1
neutronabsorbtion rate
production rate
total rate
layer
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
min
-0.02
-0.02
-0.02
-0.02
-0.02
-0.02
-0.02
-0.02
-0.02
-0.02
-0.02
-0.02
-0.02
-0.02
-0.02
-0.02
-0.02
-0.02
-0.02
-0.02
-0.02
-0.02
-0.02
-0.01
-0.01
max
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
min
-0.02
-0.02
-0.02
-0.02
-0.02
-0.02
-0.02
-0.02
-0.02
-0.02
-0.02
-0.02
-0.02
-0.02
-0.02
-0.02
-0.02
-0.02
-0.02
-0.02
-0.02
-0.02
-0.02
-0.01
-0.01
max
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
min
-0.02
-0.02
-0.02
-0.02
-0.02
-0.02
-0.02
-0.02
-0.02
-0.02
-0.02
-0.02
-0.02
-0.02
-0.02
-0.02
-0.02
-0.02
-0.02
-0.02
-0.02
-0.02
-0.02
-0.01
-0.01
max
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
131
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
-0.01
-0.01
-0.01
-0.02
-0.02
-0.02
-0.02
0.00
0.01
0.02
0.02
0.03
0.04
0.05
0.06
0.06
0.07
0.07
0.06
0.05
0.04
0.04
0.03
0.03
0.02
0.02
0.01
0.01
0.01
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.01
0.02
0.02
0.03
0.04
0.05
0.05
0.06
0.06
0.06
0.05
0.04
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
-0.01
-0.01
-0.01
-0.02
-0.02
-0.01
0.00
0.01
0.01
0.02
0.03
0.05
0.06
0.07
0.08
0.09
0.09
0.09
0.08
0.07
0.07
0.06
0.06
0.05
0.05
0.04
0.03
0.03
0.02
0.02
0.02
0.01
0.01
0.01
0.01
0.01
Table 5-20:The minimum and maximum of mesh-wise reaction
rate deviations for each layer between case 3 and case2
neutronproduction rate
total rate
absorbtion rate
layer
1
2
3
4
5
6
7
8
9
10
min
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
max
0.01
0.01
0.01
0.01
0.01
0.01
0.01
0.01
0.01
0.01
min
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
max
0.01
0.01
0.01
0.01
0.01
0.01
0.01
0.01
0.01
0.00
min
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
max
0.01
0.01
0.01
0.01
0.01
0.01
0.01
0.01
0.00
0.00
132
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
-0.01
-0.01
-0.01
-0.01
-0.01
-0.01
-0.01
-0.01
-0.01
-0.01
-0.01
-0.01
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.01
0.01
0.02
0.02
0.03
0.05
0.05
0.06
0.06
0.07
0.07
0.07
0.08
0.08
0.09
0.09
0.09
0.10
0.10
0.10
0.10
0.10
0.10
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
-0.01
-0.01
-0.01
-0.01
-0.01
-0.01
-0.01
-0.01
-0.01
-0.01
-0.01
-0.01
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.01
0.02
0.02
0.03
0.05
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
-0.01
-0.01
-0.01
-0.01
-0.01
-0.01
-0.01
-0.02
-0.02
-0.02
-0.01
-0.01
-0.01
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.01
0.01
0.02
0.03
0.04
0.05
0.06
0.06
0.07
0.07
0.08
0.08
0.09
0.09
0.10
0.10
0.10
0.10
0.10
0.10
0.10
0.10
133
A refining process was repeated until the criteria were met. Table 5-21 shows the
number of groups that we obtained in the fast energy range. Table 5-22 shows the
eigenvalue results of each group structure and U238(n, νΣf) above 0.1MeV reaction rate
and comparisons. We ended up placing 7 groups in fast energy range and obtaining a 235
group structure.
Table 5-21: Number of groups for each energy range
Group Structure
Number of Groups in Different Energy Ranges
Number
Fast
Epithermal
Thermal
Total
1
52
104
124
280
2
1
104
124
229
3
3
104
124
231
4
5
104
124
233
5
7
104
124
235
6
9
104
124
237
Table 5-22: Eigenvalue results for 3D, 8.5% fuel cell
U238 (n,νΣf)
above 0.1
MeV
%Rel.
Dev.Reaction rate
of U238(n,νΣf)
Group
keff (S8P1)
Rel Dev. in pcm
of Δk/k with
previous group
280
1.19053
-
8.529E-2
229 (1)
1.19948
752
8.617E-2
1.032
231
1.19259
-574
8.546E-2
-0.824
233
1.19115
-121
8.533E-2
-0.152
235
1.19090
-21
8.533E-2
0.000
237
1.19082
-7
8.532E-2
-0.012
Note: (1) using the full length of axial direction to collapse the cross sections
134
5.4.2 Epithermal Energy Range:
In the next step, we developed a broad group structure in the epithermal energy
range (3.0 eV to 0.1 MeV). The objective reaction rate for the epithermal energy range is
the down-scattering reaction rates of H in ZrH. We initially have placed two energy
groups in this range and ended up with 133-group structure. Table 5-23 shows the
number of groups that were studied in the ephithermal energy range. Table 5-24 shows
that relative difference of Δk/k is less than 10 pcm comparing between 135G and 137G.
The percentage relative deviations of down-scattering reaction rate of H in ZrH and
U238(n,a) are 0.0% as shown in Table 5-25. As a result, we used 135-group structure to
further collapse in the thermal energy range.
Table 5-23: Number of groups for each energy range
Group Structure
Number of Groups in Different Energy Ranges
Number
Fast
Epithermal
Thermal
Total
1
7
104
124
235
2
7
2
124
133
3
7
4
124
135
4
7
6
124
137
Table 5-24: Eigenvalue results for 3D, 8.5% fuel cell
Group
keff (S8P1)
Rel Dev. in pcm of Δk/k
with previous group
235
1.19090
-
133
1.19115
21
135
1.19096
-16
137
1.19093
-3
135
Table 5-25: Reaction rate comparison for broad group in epithermal range
U238(n,abs)
Down-scat.
%Rel. Dev.
%Rel. Dev.
Group
Of H in ZrH
With previous group
235
0.167
-
0.868
-
133
0.167
0.0
0.868
0.0
135
0.167
0.0
0.868
0.0
137
0.167
0.0
0.868
0.0
With previous group
5.4.3 Thermal Energy Range:
In the last step, we developed a broad group structure in the thermal energy range
(1.0E-05 eV to 3.0 eV). The objective reaction rates of the thermal range are fission rate
of U235 and the thermal up-scattering reaction rate of H in ZrH. We initially introduced
one energy group in this range and obtained 12-group structure. Then, we subdivided in
the most important group into three groups and each time we generated a new broadgroup structure until the result met the criteria. Table 5-26 shows the number of groups
that were refined in the thermal energy range.
Table 5-27 shows that relative difference of the eigenvalue of 26-group structure
and 28-group structure is 9 pcm. Table 5-28 demonstrates that the percentage relative
deviation of U235(n,νΣf) is 0.01% and the percentage relative deviation of upscattering
rate is 0.0%. The 26-group structure was selected to be our final broad group structure for
3-D study, which has 14 groups more than the 12G structure for 2-D study. Table 5-29
lists energy boundaries of the 26-group structure.
136
Table 5-26: Number of groups for each energy range
Group Structure
Number of Groups in Different Energy Ranges
Number
Fast
Epithermal
Thermal
Total
1
7
4
124
135
2
7
4
1
12
3
7
4
3
14
4
7
4
5
16
5
7
4
7
18
6
7
4
9
20
7
7
4
11
22
8
7
4
13
26
9
7
4
15
28
Table 5-27: Eigenvalue results for 3D, 8.5% fuel cell
Group
keff (S8P1)
Rel Dev. in pcm of Δk/k
With previous group
135
1.19096
-
12
1.20145
881
14
1.19598
-455
16
1.19359
-200
18
1.19208
-127
20
1.19157
-43
22
1.19136
-18
26
1.19122
-12
28
1.19111
-9
137
Table 5-28: Result comparison in thermal energy range
Group Up-scattering %MaxRel. Dev.
%Rel. Dev.
U235 (n,νΣf)
of H in ZrH
In upscattering
U235(n,νΣf)
reaction rate
rate
1E-05 to 3 eV reaction rate
135
45.472
6.661 E-2
135:12
0.000 E+0
0.00
45.732
0.572
12
0.000 E+0
14:12
0.000 E+0
0.00
-0.480
14
1.247 E-2
45.513
16:14
1.244 E-2
-0.24
45.419
-0.207
16
1.199 E-2
18:16
1.199 E-2
0.00
45.358
-0.133
18
3.373 E-2
20:18
3.371 E-2
-0.06
45.3380
-0.045
20
3.408 E-2
22:20
3.407 E-2
0.00
45.330
-0.017
22
3.753 E-2
26:22
3.753 E-2
0.00
45.325
-0.010
26
4.864 E-2
28:26
4.868 E-2
0.08
45.321
-0.009
28
Table 5-29: Energy boundaries of 26-group structures
Energy
Group
Number
1
2
3
4
5
6
7
8
9
10
Upper
Energy(MeV)
2.0000E+01
3.4347E+00
2.0180E+00
1.1000E+00
6.0000E-01
3.3000E-01
2.0000E-01
1.0000E-01
9.5000E-03
9.5000E-04
Energy
Group
Number
11
12
13
14
15
16
17
18
19
20
Upper
Energy(MeV)
1.0000E-04
3.0000E-06
9.7500E-07
3.0000E-07
2.0000E-07
1.3333E-07
1.0000E-07
8.6667E-08
7.7500E-08
6.7500E-08
Energy
Group
Number
21
22
23
24
25
26
Upper
Energy(MeV)
6.0000E-08
5.2500E-08
4.5000E-08
3.0000E-08
1.0000E-08
3.0000E-09
1.0000E-11
The TORT calculations were performed with S8P1 for both 280-group and 26group structures. The results are given in Table 5-30. The absolute relative deviations in
Δk/k of 26-group as compared to the 280-group and continuous Monte Carlo calculations
138
are 56 pcm and 125 pcm, respectively. The errors from the comparison between the 26group and the 280-group structure are less than the comparison between the 26-group
structure and the continuous-energy MCNP solution. These differences are identified as
the method difference between the deterministic (TORT) and statistical (MCNP) and the
multigroup and continuous energy cross-section libraries. The absorption rate, neutron
production, and total reaction rates are compared between the two cross-section libraries:
280G and 26G, in each energy range (fast, epithermal, and thermal) and region (Zr Rod,
fuel meat, clad (fuel), water(fuel), graphite, clad(gra), and water(gra)). Table 5-31
through Table 5-33 give the reaction rates from MCNP for continuous energy, TORT for
280-group library and 26-group library, and their comparisons. The percentage of relative
deviation between codes and libraries are presented in Table 5-34 through Table 5-36.
Compared to the MCNP continuous energy results, we observed large differences of
absorption reaction rate in fast energy range for graphite region ~10% and in epithermal
energy range for cladding region ~11%. We suspect that the cause of the differences may
be due to group refinement process that focused on only the neutron production of U238.
Thus, we may obtain good agreement with our objectives while finding the large errors in
other regions. However, those reaction rates are insignificant parts of total absorption
reaction rates. They are smaller by 1 or 2 orders of magnitude. For the reaction rate
comparisons between 280G- fine and 26G-broad groups, they show very good agreement
for these selected reaction rates in each energy range with less than 1% difference.
139
Table 5-30: Eigenvalues calculated by TORT and MCNP
keff
Rel. Dev. from
Rel. Dev. from
MCNP in pcm of
280G in pcm of
Δk/k
Δk/k
MCNP
1.19272
±0.00042(3σ)
TORT (280G, S8,P1)
1.19053
-183
-
TORT (26G, S8,P1)
1.19122
-125
56
Table 5-31: MCNP calculation with continuous cross section library
Reaction
Energy
Rate
Range
Fast
Epithermal
Absorption
Thermal
Total
Fast
Neutron
production
Epithermal
Thermal
Total
Fast
Epithermal
Total
Thermal
Total
Zr Rod
Fuel Meat
Clad
Water
Graphite
Clad
Water
6.54E-05
1.45E-04
1.39E-04
3.51E-05
2.44E-06
3.87E-05
1.02E-05
1.92E-04
1.77E-03
5.61E-04
3.50E-05
1.46E-07
1.90E-04
1.20E-05
4.82E-04
1.31E-02
2.12E-02
2.05E-03
1.05E-05
9.87E-03
9.16E-04
7.39E-04
1.50E-02
2.19E-02
2.12E-03
1.30E-05
1.01E-02
9.38E-04
0.00E+00
2.03E-04
0.00E+00
0.00E+00
0.00E+00
0.00E+00
0.00E+00
0.00E+00
1.03E-03
0.00E+00
0.00E+00
0.00E+00
0.00E+00
0.00E+00
0.00E+00
2.31E-02
0.00E+00
0.00E+00
0.00E+00
0.00E+00
0.00E+00
0.00E+00
2.44E-02
0.00E+00
0.00E+00
0.00E+00
0.00E+00
0.00E+00
3.94E-02
7.28E-02
3.90E-02
5.50E-02
8.34E-03
1.10E-02
1.61E-02
3.16E-02
1.40E-01
8.66E-02
1.41E-01
1.24E-02
2.85E-02
4.66E-02
2.97E-02
2.77E-01
1.33E-01
3.83E-01
2.07E-02
5.79E-02
1.67E-01
1.01E-01
4.90E-01
2.58E-01
5.79E-01
4.14E-02
9.74E-02
2.29E-01
140
Table 5-32: TORT calculation with 280-group cross section library
Reaction
Energy
Rate
Range
Fast
Epithermal
Absorption
Thermal
Total
Fast
Neutron
production
Epithermal
Thermal
Total
Fast
Epithermal
Total
Thermal
Total
Zr Rod
Fuel Meat
Clad
Water
Graphite
Clad
Water
6.52E-05
1.43E-04
1.35E-04
3.36E-05
2.19E-06
3.70E-05
9.92E-06
1.89E-04
1.80E-03
5.28E-04
3.48E-05
1.47E-07
1.68E-04
1.21E-05
4.88E-04
1.31E-02
2.10E-02
2.02E-03
1.05E-05
9.87E-03
9.13E-04
7.42E-04
1.51E-02
2.16E-02
2.09E-03
1.28E-05
1.01E-02
9.35E-04
0.00E+00
1.98E-04
0.00E+00
0.00E+00
0.00E+00
0.00E+00
0.00E+00
0.00E+00
1.02E-03
0.00E+00
0.00E+00
0.00E+00
0.00E+00
0.00E+00
0.00E+00
2.32E-02
0.00E+00
0.00E+00
0.00E+00
0.00E+00
0.00E+00
0.00E+00
2.44E-02
0.00E+00
0.00E+00
0.00E+00
0.00E+00
0.00E+00
3.93E-02
7.22E-02
3.85E-02
5.58E-02
8.56E-03
1.04E-02
1.66E-02
3.23E-02
1.40E-01
8.39E-02
1.41E-01
1.26E-02
2.71E-02
4.76E-02
2.95E-02
2.75E-01
1.32E-01
3.79E-01
2.10E-02
5.82E-02
1.68E-01
1.01E-01
4.87E-01
2.54E-01
5.77E-01
4.22E-02
9.57E-02
2.32E-01
Table 5-33: TORT calculation with 26-group cross section library
Reaction
Energy
Rate
Range
Fast
Epithermal
Absorption
Thermal
Total
Fast
Neutron
production
Epithermal
Thermal
Total
Fast
Epithermal
Total
Thermal
Total
Zr Rod
Fuel Meat
Clad
Water
Graphite
Clad
Water
6.50E-05
1.43E-04
1.35E-04
3.35E-05
2.18E-06
3.68E-05
9.85E-06
1.88E-04
1.79E-03
5.26E-04
3.47E-05
1.47E-07
1.67E-04
1.21E-05
4.87E-04
1.31E-02
2.09E-02
2.01E-03
1.04E-05
9.80E-03
9.06E-04
7.40E-04
1.50E-02
2.15E-02
2.08E-03
1.27E-05
1.00E-02
9.28E-04
0.00E+00
1.97E-04
0.00E+00
0.00E+00
0.00E+00
0.00E+00
0.00E+00
0.00E+00
1.01E-03
0.00E+00
0.00E+00
0.00E+00
0.00E+00
0.00E+00
0.00E+00
2.31E-02
0.00E+00
0.00E+00
0.00E+00
0.00E+00
0.00E+00
0.00E+00
2.43E-02
0.00E+00
0.00E+00
0.00E+00
0.00E+00
0.00E+00
3.92E-02
7.20E-02
3.84E-02
5.56E-02
8.51E-03
1.03E-02
1.65E-02
3.22E-02
1.39E-01
8.36E-02
1.41E-01
1.25E-02
2.69E-02
4.73E-02
2.94E-02
2.74E-01
1.31E-01
3.78E-01
2.09E-02
5.79E-02
1.67E-01
1.01E-01
4.86E-01
2.53E-01
5.74E-01
4.19E-02
9.51E-02
2.30E-01
141
Table 5-34: Reaction rates deviation between 280G and MCNP
Reaction
Energy
Rate
Range
Fast
Epithermal
Absorption
Thermal
Total
Fast
Neutron
production
Epithermal
Thermal
Total
Fast
Epithermal
Total
Thermal
Total
Zr Rod
Fuel Meat
Clad
Water
Graphite
Clad
Water
-0.24
-1.25
-3.07
-4.40
-10.33
-4.40
-2.35
-1.56
1.63
-5.88
-0.47
0.73
-11.52
0.97
1.34
0.16
-0.98
-1.62
0.05
-0.02
-0.34
0.45
0.31
-1.12
-1.64
-1.89
-0.25
-0.35
-
-2.44
-
-
-
-
-
-
-1.19
-
-
-
-
-
-
0.28
-
-
-
-
-
-
0.19
-
-
-
-
-
-0.18
-0.73
-1.12
1.54
2.63
-5.72
3.43
2.26
-0.20
-3.13
0.33
2.02
-5.16
2.09
-0.89
-0.57
-0.93
-0.88
1.47
0.66
0.62
0.37
-0.48
-1.70
-0.35
1.87
-1.77
1.12
Table 5-35: Reaction rates deviation between 26G and MCNP
Reaction
Energy
Rate
Range
Fast
Epithermal
Absorption
Thermal
Total
Fast
Neutron
production
Epithermal
Thermal
Total
Fast
Epithermal
Total
Thermal
Total
Zr Rod
Fuel Meat
Clad
Water
Graphite
Clad
Water
-0.52
-1.55
-3.36
-4.63
-10.92
-4.93
-2.97
-1.83
1.34
-6.21
-0.84
0.29
-11.94
0.51
1.01
-0.21
-1.41
-2.11
-0.63
-0.70
-1.07
0.14
-0.04
-1.55
-2.13
-2.55
-0.92
-1.07
-
-2.73
-
-
-
-
-
-
-1.47
-
-
-
-
-
-
-0.09
-
-
-
-
-
-
-0.17
-
-
-
-
-
-0.47
-1.03
-1.45
1.21
2.08
-6.22
2.86
1.97
-0.48
-3.46
-0.01
1.50
-5.64
1.57
-1.21
-0.92
-1.35
-1.34
0.82
0.00
-0.08
0.08
-0.81
-2.07
-0.78
1.28
-2.36
0.46
142
Table 5-36: Reaction rates deviation between 26G and 280G
Reaction
Energy
Rate
Range
Fast
Epithermal
Absorption
Thermal
Total
Fast
Neutron
production
Epithermal
Thermal
Total
Fast
Epithermal
Total
Thermal
Total
5.5
Zr Rod
Fuel Meat
Clad
Water
Graphite
Clad
Water
-0.28
-0.30
-0.30
-0.24
-0.65
-0.56
-0.64
-0.27
-0.28
-0.35
-0.37
-0.44
-0.47
-0.46
-0.32
-0.36
-0.44
-0.50
-0.68
-0.68
-0.73
-0.30
-0.35
-0.43
-0.49
-0.67
-0.67
-0.72
-
-0.30
-
-
-
-
-
-
-0.28
-
-
-
-
-
-
-0.36
-
-
-
-
-
-
-0.36
-
-
-
-
-
-0.28
-0.31
-0.33
-0.33
-0.53
-0.54
-0.55
-0.28
-0.29
-0.34
-0.34
-0.52
-0.51
-0.52
-0.32
-0.36
-0.42
-0.47
-0.64
-0.65
-0.70
-0.29
-0.33
-0.38
-0.42
-0.58
-0.60
-0.65
Three-Dimensional Cross Section Model for Materials with Non-Fissile Element
Non-fissile materials have to be modeled with the color set approach and with the
280 fine-group structure, which requires too large computational effort; hence, we
decided to apply the developed 26 broad-group structure, collapsed from 280 groups,
with 2-D flux spectrum for non-fissile material and compare with MCNP.
5.5.1 Control Rod
The 3-D color-set control rod model is illustrated in Figure 5-11. The total
number of cells is 148x118x46. It is modeled with a uniform mesh distribution with 0.03
cm for radial mesh size and 0.5 cm for axial mesh size.
143
4.00
19.05
Graphite
Fuel / B4C
Figure 5-11: 3-D model for control rod XS generation
Table 5-37 shows the eigenvalues calculated by TORT and MCNP. Table 5-38 to
Table 5-42 show reaction rates for each energy range and comparisons for each case. We
observed the same large deviation of reaction rate comparisons between TORT and
MCNP calculations in cladding region for the whole range of energy and in thermal
energy range of water region as observed in the 2-D case. The scattering order has the
effect on the eigenvalue predictions. The differences of results as compared to MCNP are
568 pcm with P1 scattering order and 7 pcm for P3 scattering order.
MCNP
Table 5-37: Eigenvalues calculated by TORT and MCNP
Kinf
Rel. Dev. from MCNP in
pcm of Δk/k
0.71429 ±0.00042(3σ)
TORT (26G, S8,P1)
0.71023
-568
TORT (26G, S8,P3)
0.71424
-7
144
Table 5-38: Reaction rates calculated by MCNP
Reaction Type
B4C
Clad
Water
(B4C)
(B4C)
Abs_Fast
4.93E-04 3.37E-05 8.87E-06
Abs_Epi
4.98E-03 8.88E-05 6.55E-06
Abs_Thermal
5.21E-03 1.23E-03 2.25E-04
Abs_Total
1.07E-02 1.36E-03 2.40E-04
Tot_Fast
Tot_Epi
Tot_Thermal
Tot_Total
1.26E-02
1.26E-02
5.31E-03
3.05E-02
9.79E-03
1.65E-02
8.73E-03
3.50E-02
1.45E-02
3.17E-02
4.30E-02
8.93E-02
Table 5-39: Reaction rates calculated by TORT with
26 groups, S8 quadrature order and P1 scattering
order
Reaction Type
B4C
Clad
Water
(B4C)
(B4C)
Abs_Fast
5.01E-04 3.16E-05 8.37E-06
Abs_Epi
5.09E-03 7.41E-05 6.49E-06
Abs_Thermal
5.21E-03 1.16E-03 2.19E-04
Abs_Total
1.08E-02 1.27E-03 2.34E-04
Tot_Fast
Tot_Epi
Tot_Thermal
Tot_Total
1.28E-02
1.29E-02
5.31E-03
3.11E-02
9.07E-03
1.53E-02
8.32E-03
3.27E-02
1.48E-02
3.18E-02
4.23E-02
8.90E-02
Table 5-40: Reaction rates calculated by TORT with
26 groups, S8 quadrature order and P3 scattering
order
Reaction Type
B4C
Clad
Water
(B4C)
(B4C)
Abs_Fast
4.98E-04 3.14E-05 8.35E-06
Abs_Epi
5.01E-03 7.37E-05 6.52E-06
Abs_Thermal
5.21E-03 1.17E-03 2.21E-04
Abs_Total
1.07E-02 1.27E-03 2.35E-04
Tot_Fast
Tot_Epi
Tot_Thermal
Tot_Total
1.27E-02
1.28E-02
5.31E-03
3.08E-02
9.00E-03
1.52E-02
8.35E-03
3.26E-02
1.47E-02
3.19E-02
4.26E-02
8.92E-02
145
Table 5-41: Percent deviation of reaction rates
between TORT 26GP1 S8 and MCNP
Reaction Type
B4C
Clad
Water
(B4C)
(B4C)
Abs_Fast
1.81
-6.39 -5.56
Abs_Epi
2.13
-16.48 -0.90
Abs_Thermal
-0.08
-5.76 -2.48
Abs_Total
1.03
-6.48 -2.55
Tot_Fast
Tot_Epi
Tot_Thermal
Tot_Total
1.94
2.35
-0.07
1.76
-7.41
-7.14
-4.62
-6.59
1.66
0.41
-1.57
-0.34
Table 5-42: Percent deviation of reaction rates
between TORT 26GP3 S8 and MCNP
Reaction Type
B4C
Clad
Water
(B4C)
(B4C)
Abs_Fast
1.19
-6.92 -5.84
Abs_Epi
0.70
-17.02 -0.43
Abs_Thermal
-0.01
-5.27 -1.81
Abs_Total
0.38
-6.08 -1.92
Tot_Fast
Tot_Epi
Tot_Thermal
Tot_Total
5.6
1.24
1.26
-0.01
1.03
-8.08
-7.67
-4.33
-6.95
0.89
0.49
-0.89
-0.11
Two-Dimensional vs. Three-Dimensional Cross Sections
5.6.1 Two-Dimensional vs. Three-Dimensional Flux Distribution Collapsing
The 26-group structure 2-D and 3-D cross sections were used to study the effect
of 2-D and 3-D flux distribution collapsing. TORT is used to perform the study using S8
Square Legendre-Chevbychev quadrature order and P1 scattering order for a 3-D pin cell
as shown in Figure 5-12 . Table 5-43 shows the eigenvalue results of 2-D vs. 3-D flux
cross-section collapsing cases. Table 5-44 shows the percentage deviation of reaction
rates between 2-D and 3-D flux distribution collapsing cases. The 2-D and 3-D cross-
146
section collapsing cases agree well with each other in eigenvalue. The difference between
the two cases is observed in the absorption rate in fast energy range of graphite. The 2-D
case has less absorption rate than 3-D case by 14.86%. This difference can be attributed
Graphite
8.7376
Fuel
19.05
to the axial flux distribution effect, which is not present in the 2-D case.
Figure 5-12: A pin cell model in axial direction
Table 5-43: Eigenvalue results 2-D vs 3-D flux distribution collapsing cases
CASE
keff
Deviation
in pcm of Δk/k
MCNP
1.19272±0.00051(3σ)
TORT- 26GS8P1,3-DXS
1.19052
-184
TORT- 26GS8P1,2-DXS
1.19031
-202
Table 5-44: Percentage deviation of reaction rates between 2-D and 3-D crosssection collapsing cases
Zr
Fuel
Clad_fuel Water_fuel Graphite Clad_gra Water_gra
Reaction Type
Abs_Fast
Abs_Epi
Abs_Thermal
Abs_Total
Nu-Fis_Fast
Nu-Fis_Epi
Nu-Fis_Thermal
Nu-Fis_Total
Tot_Fast
Tot_Epi
Tot_Thermal
Tot_Total
-0.02
-0.17
0.02
-0.03
-
-0.02
-0.23
0.01
-0.08
0.00
-0.25
0.03
0.00
0.07
-0.23
0.03
0.02
-0.03
-0.21
0.02
-0.05
0.41
-1.42
0.07
0.04
-
1.01
-0.12
0.05
0.06
-
0.06
0.72
0.03
0.26
-0.05
-0.20
0.04
-0.03
-1.20
-0.08
0.53
0.51
0.04
-0.13
0.50
0.49
-
-
-
0.02
-0.17
0.26
0.08
0.02
-0.21
0.31
0.13
0.11
-0.19
0.41
0.26
-14.86
-0.04
0.53
-2.11
147
5.6.2 Two-Dimensional vs. Three-Dimensional Group Structure
The 2-D, 12-group and 3-D, 26-group structures were used to study the effect of
2-D and 3-D group structure studies. The 3-D flux distribution was used to collapse 280group cross-section library to 12-group and 26-group structures. The number of groups
placed in each energy range is given in Table 5-45. TORT is used to perform the study
using S8 Square Legendre-Chevbychev quadrature order and P1 scattering order for a 3D pin cell as shown in Figure 5-12. Table 5-46 shows the eigenvalue results of 2-D vs. 3D group structure cases. Table 5-47 shows the percentage deviation of reaction rates. The
3-D, 26-group structure agrees better with MCNP than the 2-D, 12-group structure in
both eigenvalue and reaction rates.
Table 5-45: Number of groups placed in each energy range
Number of groups
Enery Range
26G
12G
Fast Range
7
1
Epithermal Range
4
2
Thermal Range
13
9
Table 5-46: Eigenvalue results 2-D vs 3-D group structure cases
Deviation
CASE
keff
in pcm of Δk/k
MCNP
1.19272±0.00051(3ρ)
-
TORT- 26GS8P1,3DXS
1.19052
-184
TORT- 12GS8P1,2DXS
1.20051
653
148
Table 5-47: Percentage deviation of reaction rates between 2-D and 3-D group
structure cases
Zr
Fuel
Clad_fuel Water_fuel Graphite Clad_gra Water_gra
Reaction Type
Abs_Fast
Abs_Epi
Abs_Thermal
Abs_Total
Nu-Fis_Fast
Nu-Fis_Epi
Nu-Fis_Thermal
Nu-Fis_Total
Tot_Fast
Tot_Epi
Tot_Thermal
Tot_Total
5.7
1.09
0.74
1.11
1.02
1.09
0.74
0.94
0.93
1.00
0.73
0.84
0.83
1.00
0.73
0.84
0.84
1.00
0.74
0.82
0.82
0.67
0.77
0.68
0.69
0.70
0.78
0.70
0.72
0.56
0.81
0.72
0.72
0.64
0.81
0.71
0.73
-3.59
-1.08
-0.33
-0.90
-3.71
-1.26
-0.40
-1.33
-3.95
-1.20
-0.33
-0.36
-3.97
-1.33
-0.39
-1.05
-3.91
-1.14
-0.32
-0.36
-4.00
-1.37
-0.36
-0.83
Summary
In this chapter, the parametric study has been performed for a 3-D pin cell model.
We selected the S8 (SLC) quadrature and P3 scattering order with 48x59x55 (x,y,z) to be
a model for fine group study. Then, the fine–energy-group and broad-energy-group
structures for 3-D cross-section generation have been selected in the fast, epithermal, and
thermal energy ranges by the CPXSD methodology using different objectives
corresponding to each energy range. The scalar flux weighting technique is utilized in
collapsing fine- to broad-group libraries. Results indicate very good agreement between
280 fine- and 26 broad-group structures. Also, we demonstrated that the group structure
for 3-D problem should be developed in 3-D geometry not in 2-D geometry. Comparing
previous 2-D, 12 groups and 3-D, 26 groups, the obtained results show the significant
impact of geometry on the group structure selection.
149
CHAPTER 6
Core Simulation
In this chapter, we intended to implement the developed 26-group cross-section
library to core simulations. The problem is that the use of 26 groups is still
computationally expensive for a whole 3-D core calculation. For this reason, the 26group structure is verified by a mini-core test problem. Coarse group structure has been
selected from the 26 broad-group structure in order to make our TRIGA core problem
feasible for 3-D transport calculations. The TRIGA core loading 2 is used to verify and
validate the selected effective coarse group structure. In both validation efforts,
continuous energy Monte Carlo solutions are used as the references.
6.1
Mini-Core Simulation
A mini-core test problem was set up to validate the 26-group cross-section library.
It consists of 7 fuel elements as shown in Figure 6-1. A 1/8 mini-core was modeled in
TORT to take advantage of the core symmetry. The overall size of the 3-D model is
11.5304x11.2839x35.7876 cm3. The studies were performed with S8-SLC quadrature set.
Fuel
8.00
8.7376
for TORT calculations.
19.05
Graphit Water
The flux convergence was set to 1x10-4 and the eigenvalue convergence was set to 1x10-6
Figure 6-1: Configuration of Mini-core
150
6.1.1 Mesh Size Study
In this part, we performed mesh size study for the mini-core model in both axial
and radial direction. In axial-mesh size study, five models with different mesh sizes were
examined. The first model has 0.5 cm cell thickness. The second model has 1 cm cell
thickness. The third model has 1.5 cm cell thickness. The forth model has 2 cm-cell
thickness and the fifth model has 2.5 cm cell thickness. The calculations were performed
using P1 scattering order with S8-SLC quadrature order.
Table 6-1 shows the eigenvalues calculated from TORT and pcm deviation for
each model. Table 6-2 through Table 6-5 show the percentage deviation of reaction rates
between each model and the 1st model. From the results, the 3rd model is chosen to be the
axial mesh size model and will be used further for radial-mesh size study.
Table 6-1: Eigenvalues calculated from TORT
Model-mesh size
keff
Deviation
(no. of cells)
From the 1st model in pcm
1-0.5 cm
0.50003
(117x114x71)
2-1.0 cm
0.49970
-33
(117x114x36)
3-1.5 cm
0.49945
-58
(117x114x24)
4-2.0 cm
0.49913
-90
(117x114x18)
5-2.5 cm
0.49835
-168
(117x114x14)
Time
(hr)
54
31
17
12
9
151
Table 6-2: Percentage deviation of reaction rates between 2nd model and 1st model
Zr
Fuel
Clad_fuel Water_fuel Graphite Clad_gra Water_gra Reflector
Reaction Type
Abs_Fast
Abs_Epi
Abs_Thermal
Abs_Total
Nu-Fis_Fast
Nu-Fis_Epi
Nu-Fis_Thermal
Nu-Fis_Total
Tot_Fast
Tot_Epi
Tot_Thermal
Tot_Total
-0.01
-0.04
-0.03
-0.03
-0.01
-0.04
-0.04
-0.02
-0.04
-0.07
-0.07
-0.07
-0.03
-0.07
-0.07
-0.07
-0.04
-0.07
-0.07
-0.06
0.00
-0.04
-0.05
-0.05
-0.01
-0.04
-0.05
-0.04
-0.01
-0.05
-0.06
-0.06
-0.03
-0.05
-0.06
-0.06
-0.05
0.26
0.04
0.03
0.14
0.26
0.08
0.13
0.01
0.27
0.06
0.06
0.09
0.27
0.08
0.12
-0.09
0.26
0.10
0.10
0.12
0.26
0.10
0.12
-0.01
-0.04
-0.03
-0.03
-0.01
-0.04
-0.04
-0.02
Table 6-3: Percentage deviation of reaction rates between 3rd model and 1st model
Zr
Fuel
Clad_fuel Water_fuel Graphite Clad_gra Water_gra Reflector
Reaction Type
Abs_Fast
Abs_Epi
Abs_Thermal
Abs_Total
Nu-Fis_Fast
Nu-Fis_Epi
Nu-Fis_Thermal
Nu-Fis_Total
Tot_Fast
Tot_Epi
Tot_Thermal
Tot_Total
-0.03
-0.09
-0.07
-0.07
-0.03
-0.09
-0.09
-0.06
-0.06
-0.13
-0.12
-0.12
-0.05
-0.13
-0.12
-0.12
-0.07
-0.12
-0.13
-0.11
-0.01
-0.10
-0.11
-0.11
-0.03
-0.09
-0.11
-0.09
-0.02
-0.11
-0.12
-0.12
-0.05
-0.09
-0.12
-0.11
-0.23
0.62
0.11
0.07
0.26
0.63
0.19
0.29
-0.03
0.64
0.15
0.15
0.18
0.64
0.20
0.28
-0.24
0.64
0.22
0.22
0.28
0.64
0.23
0.29
-0.19
0.01
-0.16
-0.16
-0.06
-0.03
-0.16
-0.14
Table 6-4: Percentage deviation of reaction rates between 4th model and 1st model
Zr
Fuel
Clad_fuel Water_fuel Graphite Clad_gra Water_gra Reflector
Reaction Type
Abs_Fast
Abs_Epi
Abs_Thermal
Abs_Total
Nu-Fis_Fast
Nu-Fis_Epi
Nu-Fis_Thermal
Nu-Fis_Total
Tot_Fast
Tot_Epi
Tot_Thermal
Tot_Total
-0.05
-0.15
-0.16
-0.14
-0.05
-0.14
-0.18
-0.11
-0.08
-0.20
-0.21
-0.21
-0.07
-0.20
-0.21
-0.21
-0.09
-0.18
-0.21
-0.18
-0.03
-0.17
-0.19
-0.19
-0.05
-0.15
-0.19
-0.15
-0.02
-0.18
-0.20
-0.20
-0.08
-0.15
-0.20
-0.18
-1.12
0.74
0.09
-0.04
-0.16
0.70
0.19
0.25
-0.72
0.77
0.14
0.14
-0.31
0.75
0.21
0.27
-0.24
0.00
-0.14
-0.14
-0.10
-0.05
-0.14
-0.13
-1.10
0.80
0.23
0.23
-0.08
0.74
0.26
0.30
152
Table 6-5: Percentage deviation of reaction rates between 5th model and 1st model
Zr
Fuel Clad_fuel Water_fuel Graphite Clad_gra Water_gra Reflector
Reaction Type
Abs_Fast
Abs_Epi
Abs_Thermal
Abs_Total
Nu-Fis_Fast
Nu-Fis_Epi
Nu-Fis_Thermal
Nu-Fis_Total
Tot_Fast
Tot_Epi
Tot_Thermal
Tot_Total
-0.07
-0.23
-0.31
-0.26
-0.07
-0.22
-0.33
-0.18
-0.09
-0.30
-0.37
-0.36
-0.08
-0.30
-0.37
-0.36
-0.11
-0.26
-0.37
-0.29
-0.03
-0.26
-0.36
-0.35
-0.07
-0.24
-0.35
-0.27
-0.03
-0.28
-0.36
-0.36
-0.11
-0.24
-0.36
-0.31
-2.08
0.92
-0.25
-0.44
-0.60
0.82
-0.08
0.03
-1.46
0.95
-0.16
-0.16
-0.82
0.89
-0.04
0.07
-0.25
0.02
-0.25
-0.25
-0.10
-0.03
-0.24
-0.22
-2.03
0.99
-0.04
-0.04
-0.47
0.87
0.00
0.09
In radial-mesh size study, there are three models. The first model has 0.10 cm cell
size. The second model has 0.15 cm cell size. The third model has 0.20 cm cell size. The
meshes are distributed uniformly and the uniform mesh thickness in axial direction of all
four models is 1.5 cm. The calculations were performed using P1 scattering order and S8SLC quadrature order.
Table 6-6 shows the eigenvalues calculated by TORT and pcm deviation for each
model. Table 6-7 and Table 6-8 show the percentage deviation of reaction rates between
each model and the 1st model. From the results, the 2nd radial mesh-size model, 0.15 cm,
is chosen for use in mini-core calculations.
Table 6-6: Eigenvalues calculated from TORT
keff
Model-mesh size
Deviation
(no. of cells)
From the 1st model in
pcm
1-0.10 cm
0.49945
(117x114x24)
2-0.15 cm
0.49943
-2
(96x91x24)
3-0.20 cm
0.50189
244
(83x75x24)
Time
(hr)
17
5
4
153
Table 6-7: Percentage deviation of reaction rates between 2nd model and 1st model
Zr
Fuel
Clad_fuel Water_fuel Graphite Clad_gra Water_gra Reflector
Reaction Type
Abs_Fast
Abs_Epi
Abs_Thermal
Abs_Total
Nu-Fis_Fast
Nu-Fis_Epi
Nu-Fis_Thermal
Nu-Fis_Total
Tot_Fast
Tot_Epi
Tot_Thermal
Tot_Total
0.87
0.94
-0.04
0.32
0.87
0.93
0.14
0.71
-0.24
-0.17
-0.01
-0.03
-0.25
-0.17
-0.01
-0.02
-0.24
-0.19
-0.03
-0.11
0.63
0.64
-0.58
-0.54
0.72
0.68
-0.39
0.14
0.14
-0.02
-0.15
-0.14
0.09
0.00
-0.14
-0.09
-0.10
-0.37
-0.13
-0.13
-0.18
-0.35
-0.18
-0.22
0.59
0.19
-0.37
-0.36
0.64
0.25
-0.33
-0.15
-0.02
-0.28
-0.23
-0.23
-0.08
-0.25
-0.23
-0.23
0.16
0.03
0.01
0.01
0.10
0.05
0.01
0.02
Table 6-8: Percentage deviation of reaction rates between 3rd model and 1st model
Zr
Fuel
Clad_fuel Water_fuel Graphite Clad_gra Water_gra Reflector
Reaction Type
Abs_Fast
Abs_Epi
Abs_Thermal
Abs_Total
Nu-Fis_Fast
Nu-Fis_Epi
Nu-Fis_Thermal
Nu-Fis_Total
Tot_Fast
Tot_Epi
Tot_Thermal
Tot_Total
0.61
1.03
0.75
0.80
0.61
0.99
0.77
0.76
-0.12
0.20
0.48
0.44
-0.13
0.20
0.48
0.47
-0.10
0.16
0.44
0.26
-3.09
-2.74
-2.30
-2.32
-2.72
-2.82
-2.04
-2.40
0.97
0.10
0.13
0.14
-
0.61
0.05
1.06
1.00
-
0.58
0.16
0.13
0.19
0.22
0.06
0.85
0.58
-2.29
-2.11
-2.72
-2.71
-2.05
-2.19
-2.37
-2.31
0.63
-0.07
0.74
0.74
0.28
-0.03
0.70
0.59
-0.15
-0.11
-0.10
-0.10
-0.17
-0.13
-0.10
-0.11
6.1.2 Mini-Core Results
The selected mesh sizes in previous section (0.15 cm for radial direction and 1.5
cm in axial direction) were used to perform mini-core calculations. The reference
solution was obtained by MCNP with 3000 number of histories per cycle, 1000 number
of skipped cycles and 4000 number of active cycles. The standard deviation is within 1%
for reaction rates. Table 6-9 gives the eigenvalues calculated by MCNP and TORT for P1
and P3 cases. Results indicate that scattering order has a pronouced effect on eigenvalue.
154
The P3 case agrees with MCNP better than the P1 case. Table 6-10, Table 6-11 and Table
6-12 display the reaction rates calculated by MCNP and TORT. Table 6-13 and Table
6-14 show the percentage deviations of reaction rates between TORT and MCNP for P1
and P3 cases, respectively. Using P3 scattering order improves also the agreement with
MCNP results on reaction rate prediction but much lesser extend than for the eigenvalue
prediction.
Table 6-9: Eigenvalues calculated from TORT and MCNP
keff
Deviation
Time
From MCNP
(hr)
In pcm of Δk
MCNP
0.50469
(±0.00020σ)
TORT,S8P1
0.49943
-526
5
TORT,S8P3
0.50556
87
15
Table 6-10: MCNP reaction rates
Reaction Type
Abs_Fast
Abs_Epi
Abs_Thermal
Abs_Total
Nu-Fis_Fast
Nu-Fis_Epi
Nu-Fis_Thermal
Nu-Fis_Total
Tot_Fast
Tot_Epi
Tot_Thermal
Tot_Total
Zr
5.28E-06
1.15E-05
2.79E-05
4.47E-05
3.17E-03
1.95E-03
1.69E-03
6.82E-03
Fuel
1.17E-05
9.88E-05
7.92E-04
9.02E-04
1.68E-05
5.76E-05
1.40E-03
1.47E-03
5.70E-03
8.35E-03
1.64E-02
3.05E-02
Clad_fuel
1.05E-05
3.11E-05
1.32E-03
1.37E-03
Water_fuel
2.51E-06
1.87E-06
1.32E-04
1.37E-04
Graphite
7.03E-08
4.06E-09
5.54E-07
6.28E-07
Clad_gra
1.14E-06
5.20E-06
5.16E-04
5.22E-04
Water_gra
2.83E-07
3.21E-07
4.73E-05
4.79E-05
Reflector
3.08E-07
3.16E-07
5.90E-05
5.96E-05
-
-
-
-
-
-
2.87E-03
4.97E-03
8.01E-03
1.58E-02
3.76E-03
8.02E-03
2.43E-02
3.61E-02
2.58E-04
3.51E-04
1.03E-03
1.64E-03
3.27E-04
7.88E-04
2.81E-03
3.92E-03
4.60E-04
1.26E-03
8.39E-03
1.01E-02
155
4.20E-04
1.20E-03
1.04E-02
1.20E-02
Table 6-11: TORT reaction rates for P1 case
Reaction Type
Abs_Fast
Abs_Epi
Abs_Thermal
Abs_Total
Nu-Fis_Fast
Nu-Fis_Epi
Nu-Fis_Thermal
Nu-Fis_Total
Tot_Fast
Tot_Epi
Tot_Thermal
Tot_Total
Zr
5.20E-06
1.13E-05
2.80E-05
4.45E-05
3.13E-03
1.95E-03
1.66E-03
6.73E-03
Fuel
1.13E-05
9.71E-05
7.85E-04
8.93E-04
1.60E-05
5.50E-05
1.39E-03
1.46E-03
5.52E-03
8.05E-03
1.62E-02
2.98E-02
Clad_fuel
1.02E-05
2.88E-05
1.28E-03
1.32E-03
Water_fuel
2.39E-06
1.80E-06
1.28E-04
1.32E-04
Graphite
6.76E-08
4.05E-09
5.51E-07
6.22E-07
Clad_gra
1.10E-06
4.59E-06
5.11E-04
5.17E-04
Water_gra
2.83E-07
3.20E-07
4.68E-05
4.74E-05
Reflector
3.30E-07
3.15E-07
5.83E-05
5.90E-05
-
-
-
-
-
-
2.81E-03
4.75E-03
7.80E-03
1.54E-02
3.71E-03
7.77E-03
2.36E-02
3.51E-02
2.62E-04
3.52E-04
1.02E-03
1.63E-03
3.08E-04
7.46E-04
2.82E-03
3.87E-03
4.68E-04
1.27E-03
8.32E-03
1.01E-02
4.31E-04
1.20E-03
1.03E-02
1.19E-02
Table 6-12: TORT reaction rates for P3 case
Reaction Type
Abs_Fast
Abs_Epi
Abs_Thermal
Abs_Total
Nu-Fis_Fast
Nu-Fis_Epi
Nu-Fis_Thermal
Nu-Fis_Total
Tot_Fast
Tot_Epi
Tot_Thermal
Tot_Total
Zr
5.33E-06
1.15E-05
2.84E-05
4.52E-05
3.20E-03
1.99E-03
1.68E-03
6.87E-03
Fuel
1.16E-05
9.87E-05
7.94E-04
9.04E-04
1.63E-05
5.60E-05
1.40E-03
1.48E-03
5.64E-03
8.20E-03
1.64E-02
3.02E-02
Clad_fuel
1.03E-05
2.93E-05
1.30E-03
1.34E-03
Water_fuel
2.42E-06
1.84E-06
1.29E-04
1.34E-04
Graphite
6.71E-08
4.04E-09
5.50E-07
6.21E-07
Clad_gra
1.10E-06
4.59E-06
5.11E-04
5.16E-04
Water_gra
2.81E-07
3.20E-07
4.67E-05
4.73E-05
Reflector
3.25E-07
3.12E-07
5.82E-05
5.88E-05
-
-
-
-
-
-
2.86E-03
4.83E-03
7.90E-03
1.56E-02
3.76E-03
7.90E-03
2.39E-02
3.55E-02
2.60E-04
3.51E-04
1.02E-03
1.63E-03
3.06E-04
7.45E-04
2.81E-03
3.87E-03
4.65E-04
1.26E-03
8.31E-03
1.00E-02
4.21E-04
1.19E-03
1.02E-02
1.19E-02
Table 6-13: Percentage deviation of reaction rates between Tort-P1 and MCNP
Zr
Fuel
Clad_fuel Water_fuel Graphite Clad_gra Water_gra Reflector
Reaction Type
Abs_Fast
Abs_Epi
Abs_Thermal
Abs_Total
Nu-Fis_Fast
Nu-Fis_Epi
Nu-Fis_Thermal
Nu-Fis_Total
Tot_Fast
Tot_Epi
Tot_Thermal
Tot_Total
-1.52
-1.76
0.52
-0.31
-1.51
-0.09
-2.03
-1.23
-3.44
-1.76
-0.91
-1.03
-4.32
-4.46
-0.78
-0.97
-3.15
-3.59
-1.46
-2.36
-3.14
-7.33
-3.26
-3.35
-2.06
-4.54
-2.53
-3.07
-4.96
-3.46
-3.31
-3.35
-1.39
-3.06
-3.18
-2.97
-3.72
-0.32
-0.54
-0.89
1.50
0.18
-1.13
-0.43
-3.34
-11.60
-0.83
-0.94
-5.72
-5.33
0.34
-1.30
-0.28
-0.34
-1.16
-1.15
1.71
0.47
-0.81
-0.54
7.22
-0.39
-1.20
-1.15
2.72
0.13
-0.84
-0.62
156
Table 6-14: Percentage deviation of reaction rates between Tort-P3 and MCNP
Zr
Fuel
Clad_fuel Water_fuel Graphite Clad_gra Water_gra Reflector
Reaction Type
Abs_Fast
Abs_Epi
Abs_Thermal
Abs_Total
Nu-Fis_Fast
Nu-Fis_Epi
Nu-Fis_Thermal
Nu-Fis_Total
Tot_Fast
Tot_Epi
Tot_Thermal
Tot_Total
6.2
0.88
0.22
1.81
1.29
0.88
1.95
-0.73
0.79
-1.49
-0.07
0.29
0.22
-2.56
-2.83
0.41
0.25
-1.01
-1.75
-0.26
-0.81
-1.65
-5.73
-2.09
-2.17
-0.38
-2.86
-1.30
-1.63
-3.62
-1.80
-2.20
-2.22
0.05
-1.43
-2.03
-1.68
-4.43
-0.41
-0.69
-1.11
0.82
0.08
-1.26
-0.65
-3.91
-11.63
-0.97
-1.09
-6.30
-5.40
0.21
-1.46
-0.95
-0.34
-1.33
-1.32
1.10
0.36
-0.97
-0.71
5.43
-1.20
-1.45
-1.42
0.27
-1.00
-1.11
-1.05
Coarse Group Study
As described in the previous section, a mini-core was used to test the obtained 26broad group cross section library and the solution agrees well with the MCNP results.
However, the calculation time is quite significant even for this mini-core simulation,
which is considered to be a very small model. With a problem of running time, it is not
practical to perform a whole core calculation with 26 groups. Consequently, we attempt
to develop a fewer group structure in order to solve the problem in a reasonable amount
of time within the accepted range of results in terms of accuracy requirements.
The collapsing process was done in each energy range starting with fast,
epithermal, and thermal. We started with the important distribution of 26-group structure
as shown in Figure 6-2. The groups that have the most importance were kept with their
original same energy interval and the groups that have lower importance were combined
together.
157
3.50E-03
Ephithermal
Range
Thermal Range
Fast Range
3.00E-03
2.50E-03
C(E)
2.00E-03
1.50E-03
1.00E-03
5.00E-04
E01
1.
E+
00
2.
E+
00
3.
E+
00
2.
E+
01
E01
6.
3.
E01
E01
2.
E02
1.
1.
E04
E03
1.
E06
1.
E06
3.
E07
1.
3.
E07
E07
2.
E07
1.
1.
E08
E08
9.
E08
8.
E08
7.
E08
6.
E08
5.
5.
E08
E08
3.
1.
3.
E09
0.00E+00
Energy (MeV.)
Figure 6-2: Importance distribution of 26-group structure
1) Fuel pin model
The 26-broad-group structure has been finally collapsed to 12-coarse-group
structure. Fuel pin model calculations have been performed with both 26-group ans 12group cross sections and the obtained results have benn compared. The eigenvalues are
shown in Table 6-15 and the reacton rate comparisons are shown in Table 6-16. The 12G
structure shows a good agreement with 26G structure in both eigenvalue and reaction rate
predictions. Table 6-17 lists energy boundaries of 12-group structure.
Table 6-15: Eigenvalues calculated from TORT
keff
Deviation
From 26G in pcm
26G
1.19122
-
12G
1.19212
90
158
Table 6-16: Percentage deviation of reaction rates between 12G and 26G cases
Zr
Fuel
Clad_fuel Water_fuel Graphite Clad_gra Water_gra
Reaction Type
Abs_Fast
Abs_Epi
Abs_Thermal
Abs_Total
Nu-Fis_Fast
Nu-Fis_Epi
Nu-Fis_Thermal
Nu-Fis_Total
Tot_Fast
Tot_Epi
Tot_Thermal
Tot_Total
0.15
0.03
0.41
0.29
0.16
0.03
0.34
0.17
0.09
0.06
0.14
0.13
0.09
0.06
0.14
0.13
0.10
0.06
0.13
0.10
0.02
0.07
-0.18
-0.17
0.01
0.07
-0.15
-0.06
0.04
0.05
-0.28
-0.27
-0.01
0.05
-0.25
-0.15
-0.54
0.04
-0.11
-0.18
-0.36
0.04
-0.10
-0.11
-0.47
0.03
-0.10
-0.10
-0.37
0.03
-0.10
-0.09
-0.54
0.00
-0.15
-0.15
-0.37
0.00
-0.14
-0.12
Table 6-17: Energy boundaries of 12-group structures
Energy Group
Number
1
2
3
4
5
6
7
8
9
10
11
12
Upper
Energy(MeV)
2.0000E+01
2.0180E+00
1.1000E+00
6.0000E-01
1.0000E-01
3.0000E-06
9.7500E-07
3.0000E-07
1.3333E-07
1.0000E-07
4.5000E-08
3.0000E-08
1.0000E-11
2) Control Rod model
For control rod modeling, the 26-broad-group structure has been collapsed to 13coarse-group structure. Calculations were performed with both 26-group ans 13-group
cross sections The 13G structure shows a good agreement with 26G structure in both
eigenvalue and reaction rates as demonstrated in Table 6-18 and Table 6-19. As a result,
159
the final coarse group structure for the core calculations is the 13-group structure. Table
6-20 lists energy boundaries of the 13-group structure.
Table 6-18: Eigenvalues calculated by TORT
keff
Deviation
From 26G in pcm
26G
0.71020
13G
0.71092
72
Table 6-19: Percentage deviation of reaction
rates between 13G and 26G cases
Reaction Type
B4C
Clad
Water
Abs_Fast
-0.55
-0.39
-0.26
Abs_Epi
0.12
-0.77
-0.45
Abs_Thermal
-0.85
-0.95
-0.72
Abs_Total
-0.38
-0.93
-0.70
Tot_Fast
Tot_Epi
Tot_Thermal
Tot_Total
-0.53
1.04
-0.85
0.07
-0.47
-0.17
-0.87
-0.43
-0.47
-0.54
-0.69
-0.60
Table 6-20: Energy boundaries of 13group structure
Energy Group
Number
1
2
3
4
5
6
7
8
9
10
11
12
13
Upper
Energy(MeV)
2.0000E+01
2.0180E+00
1.1000E+00
6.0000E-01
1.0000E-01
1.0000E-04
3.0000E-06
9.7500E-07
3.0000E-07
1.3333E-07
1.0000E-07
4.5000E-08
3.0000E-08
1.0000E-11
160
6.3
Core loading 2 Simulations
We selected the TRIGA core loading 2 because it is a fresh core. It consists of 72 8.5% wt. fuel elements with 4 control rods. Figure 6-3 shows the cross sectional view of
the core arrangement. The fuel elements were modeled explicitly specifying the detailed
structure of the rod to eliminate any homogenization effects.
Figure 6-3: TRIGA core loading 2
161
6.3.1 Mesh Size Study
We attempt to use TORT to simulate core model with the coarse-group cross
section library. The mesh size study was performed in order to get optimum mesh size for
coarse group structure. The first step is to study on the axial mesh size and later on the
radial mesh size.
6.3.1.1 Fuel Pin Model
A pin cell model with reflector layer on top is used for 12 coarse-group structure.
8.00
8.7376
19.05
Fuel
Graphite Water
Figure 6-4 shows the configuration in axial direction of the studied model.
Figure 6-4: Pin cell in axial direction
Four models with different mesh-sizes were used in this study as shown in Figure
6-5. The first model has 0.5 cm mesh thickness for all layers. The second model has 1 cm
162
mesh thickness for all layers. The third model has 2 cm mesh thickness for all layers. The
forth model has 2 cm mesh thickness for the fuel layer, 1 cm mesh thickness for the
1st model
3rd model
Unit:cm
Figure 6-5: The studied models
Table 6-21 shows the eigenvalues calculated by TORT and pcm deviation for
each model. Table 6-22 through Table 6-24 show the percentage deviation of reaction
rates between each model and the 1st model. Even though, the eigenvalue deviation of 2nd
model is smaller than the 4th model, the deviations of reaction rates in the 4th model are
even out through all regions. Thus, the 4th axial mesh-size model is chosen for use in core
calculations.
163
8.00
8.7376
Water
Graphite
4thmodel
19.05
2cm.-mesh
thickness
Fuel
8.00
8.7376
1cm.-mesh
thickness
19.05
Water
Graphite
8.00
8.7376
2nd model
2cm.-mesh
thickness
0.5cm.-mesh
thickness
Fuel
Water
Graphite
2cm.-mesh
thickness
19.05
1cm.-mesh
thickness
2cm.-mesh
thickness
Fuel
8.00
8.7376
1cm.-mesh
thickness
19.05
0.5cm.-mesh
thickness
Water
0.5cm.-mesh
thickness
1cm.-mesh
thickness
Fuel
0.5cm-mesh
thickness
Graphite
graphite layer, and 0.5 cm mesh thickness for the reflector layer.
Table 6-21: Eigenvalues calculated by TORT
Model (no. of cells)
keff
Deviation
Time (min)
From 1st model in pcm
1(48x59x71)
1.21454
157
2(48x59x36)
1.21397
-57
85
3 (48x59x18)
1.21189
-265
42
4 (48x59x35)
1.21381
-73
83
Table 6-22: Percentage deviation of reaction rates between 2nd model and 1st model
Zr
Fuel
Clad_fuel Water_fuel Graphite Clad_gra Water_gra Reflector
Reaction Type
Abs_Fast
Abs_Epi
Abs_Thermal
Abs_Total
Nu-Fis_Fast
Nu-Fis_Epi
Nu-Fis_Thermal
Nu-Fis_Total
Tot_Fast
Tot_Epi
Tot_Thermal
Tot_Total
-0.02
-0.05
-0.03
-0.03
-0.02
-0.05
-0.04
-0.04
-0.05
-0.08
-0.07
-0.07
-0.04
-0.08
-0.07
-0.07
-0.05
-0.08
-0.07
-0.07
-0.02
-0.05
-0.07
-0.06
-0.02
-0.05
-0.07
-0.05
-0.01
-0.04
-0.06
-0.06
-0.02
-0.04
-0.06
-0.05
-0.09
-0.03
-0.56
-0.51
-0.04
-0.03
-0.48
-0.31
-0.06
-0.02
-0.53
-0.52
-0.04
-0.02
-0.47
-0.35
-0.08
-0.02
-0.51
-0.50
-0.03
-0.02
-0.48
-0.40
0.20
0.23
-1.70
-1.69
0.27
0.23
-1.66
-1.48
Table 6-23: Percentage deviation of reaction rates between 3rd model and 1st model
Zr
Fuel
Clad_fuel Water_fuel Graphite Clad_gra Water_gra Reflector
Reaction Type
Abs_Fast
Abs_Epi
Abs_Thermal
Abs_Total
Nu-Fis_Fast
Nu-Fis_Epi
Nu-Fis_Thermal
Nu-Fis_Total
Tot_Fast
Tot_Epi
Tot_Thermal
Tot_Total
-0.06
-0.18
-0.17
-0.16
-0.07
-0.18
-0.20
-0.14
-0.08
-0.20
-0.24
-0.24
-0.07
-0.20
-0.24
-0.24
-0.09
-0.20
-0.25
-0.21
-0.06
-0.18
-0.27
-0.26
-0.07
-0.18
-0.26
-0.21
-0.04
-0.18
-0.27
-0.27
-0.08
-0.18
-0.27
-0.23
-0.82
-0.79
-2.30
-2.14
-0.72
-0.79
-2.09
-1.58
-0.78
-0.77
-2.24
-2.22
-0.73
-0.77
-2.09
-1.71
-0.82
-0.77
-2.18
-2.16
-0.70
-0.77
-2.11
-1.85
0.87
1.05
-6.15
-6.10
1.18
1.05
-5.99
-5.31
164
Table 6-24: Percentage deviation of reaction rates between 4th model and 1st model
Zr
Fuel
Clad_fuel Water_fuel Graphite Clad_gra Water_gra Reflector
Reaction Type
Abs_Fast
Abs_Epi
Abs_Thermal
Abs_Total
Nu-Fis_Fast
Nu-Fis_Epi
Nu-Fis_Thermal
Nu-Fis_Total
Tot_Fast
Tot_Epi
Tot_Thermal
Tot_Total
-0.06
-0.12
-0.01
-0.04
-0.06
-0.12
-0.06
-0.08
-0.08
-0.14
-0.08
-0.09
-0.08
-0.14
-0.08
-0.08
-0.09
-0.14
-0.09
-0.11
-0.06
-0.12
-0.10
-0.10
-0.07
-0.12
-0.11
-0.11
-0.05
-0.12
-0.10
-0.10
-0.07
-0.12
-0.10
-0.10
0.22
0.61
-0.17
-0.13
0.40
0.61
-0.04
0.18
0.30
0.61
-0.12
-0.11
0.37
0.61
-0.03
0.14
0.24
0.60
-0.06
-0.05
0.42
0.60
-0.02
0.09
0.74
0.95
0.55
0.56
0.85
0.95
0.56
0.60
After the axial mesh size for 12 groups was studied, the second step is to study the
radial mesh size. Four models with different mesh-size are used in this study as shown in
Figure 6-6. The first model has a uniform 0.03 cm mesh. The second model has 0.1 cm
mesh. The third model has 0.15 cm mesh. The forth model has 0.2 cm mesh.
165
1st model
2nd model
3rd model
4th model
Figure 6-6: The studied models
Table 6-25 shows the eigenvalues calculated by TORT and pcm deviation for
each model. Table 6-26 through Table 6-28 show the percentage deviation of reaction
rates between each model and the 1st model. From the results, the 3rd radial mesh-size
model of 0.15 cm mesh size is chosen for use in core calculations.
166
Table 6-25: Eigenvalues calculated from TORT
Model (no. of cells)
keff
Deviation
From 1st model in pcm
1 (48x59x35)
1.21381
-
Time
(min)
83
2 (22x25x35)
1.21421
40
16
3 (16x18x35)
1.21445
64
9
4 (13x14x35)
1.22340
959
7
Table 6-26: Percentage deviation of reaction rates between 2nd model and 1st model
Zr
Fuel
Clad_fuel Water_fuel Graphite Clad_gra Water_gra Reflector
Reaction Type
Abs_Fast
Abs_Epi
Abs_Thermal
Abs_Total
Nu-Fis_Fast
Nu-Fis_Epi
Nu-Fis_Thermal
Nu-Fis_Total
Tot_Fast
Tot_Epi
Tot_Thermal
Tot_Total
0.17
0.06
-0.02
0.02
0.17
0.06
0.01
0.09
0.06
0.05
0.09
0.08
0.06
0.05
0.09
0.09
0.05
0.05
0.08
0.07
0.03
0.04
0.18
0.18
0.00
0.04
0.15
0.09
-0.08
0.05
0.05
0.04
-0.03
0.05
0.05
0.04
0.04
-0.17
0.20
0.18
-0.03
-0.17
0.14
0.04
0.15
-0.16
0.26
0.26
0.05
-0.16
0.21
0.12
-0.08
-0.17
0.11
0.10
-0.10
-0.17
0.09
0.05
-0.22
-0.39
0.11
0.10
-0.29
-0.39
0.09
0.05
Table 6-27: Percentage deviation of reaction rates between 3rd model and 1st model
Zr
Fuel
Clad_fuel Water_fuel Graphite Clad_gra Water_gra Reflector
Reaction Type
Abs_Fast
Abs_Epi
Abs_Thermal
Abs_Total
Nu-Fis_Fast
Nu-Fis_Epi
Nu-Fis_Thermal
Nu-Fis_Total
Tot_Fast
Tot_Epi
Tot_Thermal
Tot_Total
0.11
0.05
-0.08
-0.03
0.11
0.05
-0.03
0.05
0.04
0.05
0.10
0.10
0.04
0.05
0.10
0.10
0.03
0.05
0.10
0.07
0.15
0.05
0.09
0.09
0.11
0.05
0.09
0.08
-0.05
0.04
-0.04
-0.04
0.00
0.04
-0.03
-0.01
0.16
-0.18
0.50
0.46
0.02
-0.18
0.40
0.20
0.13
-0.16
0.51
0.51
0.06
-0.16
0.43
0.27
-0.02
-0.16
0.31
0.31
-0.05
-0.16
0.29
0.21
-0.26
-0.44
0.56
0.55
-0.38
-0.44
0.54
0.44
167
Table 6-28: Percentage deviation of reaction rates between 4th model and 1st model
Zr
Fuel
Clad_fuel Water_fuel Graphite Clad_gra Water_gra Reflector
Reaction Type
Abs_Fast
Abs_Epi
Abs_Thermal
Abs_Total
Nu-Fis_Fast
Nu-Fis_Epi
Nu-Fis_Thermal
Nu-Fis_Total
Tot_Fast
Tot_Epi
Tot_Thermal
Tot_Total
-0.15
-0.04
0.52
0.32
-0.15
-0.04
0.41
0.05
-0.16
-0.02
0.88
0.76
-0.16
-0.02
0.88
0.84
-0.15
-0.02
0.77
0.41
-0.08
-0.05
-8.55
-8.30
0.69
-0.05
-6.74
-3.44
0.32
-0.01
0.60
0.58
0.19
-0.01
0.54
0.38
0.63
0.18
2.80
2.56
0.34
0.18
2.41
1.58
-0.97
-0.13
-4.36
-4.31
-0.01
-0.13
-3.61
-2.61
0.58
0.22
2.44
2.41
0.39
0.22
2.33
1.94
0.45
0.28
2.26
2.24
0.17
0.28
2.21
2.02
6.3.1.2 Control Rod Model
A 3-D color-set control rod model using the 13 coarse-group structure cross
sections is used for mesh size study of control rod. In axial direction, the same mesh
models as studied in the fuel pin model were used. The first model has 0.5 cm mesh
thickness for all layers. The second model has 1cm-mesh thick for all layers. The third
model has 2 cm mesh thickness for all layers. The forth model has 2 cm mesh thickness
for the fuel layer, 1 cm mesh thickness for the graphite layer.
Table 6-29 shows the eigenvalues calculated by TORT and pcm deviation for
each model. Table 6-30 through Table 6-32 show the percentage deviation of reaction
rates between each model and the 1st model. The 4th axial mesh-size model is chosen to
use in core calculations.
168
Table 6-29: Eigenvalues calculated by TORT
Model (no. of cells)
keff
Deviation
Time
From 1st model in pcm
(min)
1 (148x118x46)
0.71092
-
500
2 (148x118x23)
0.71124
32
471
3 (148x118x12)
0.71216
124
275
4 (148x118x14)
0.71211
119
209
Table 6-30: Percentage deviation of reaction
rates between 2nd model and 1st model
Reaction Type
B4C
Clad
Water
Abs_Fast
0.00
0.01
0.01
Abs_Epi
0.07
0.00
-0.01
Abs_Thermal
0.12
0.11
0.07
Abs_Total
0.09
0.10
0.06
Tot_Fast
Tot_Epi
Tot_Thermal
Tot_Total
0.00
0.05
0.12
0.04
0.00
0.01
0.08
0.02
-0.02
-0.01
0.06
0.02
Table 6-31: Percentage deviation of reaction
rates between 3rd model and 1st model
Reaction Type
B4C
Clad
Water
Abs_Fast
-0.07
-0.03
-0.02
Abs_Epi
-0.04
-0.07
-0.07
Abs_Thermal
0.15
0.29
0.24
Abs_Total
0.05
0.26
0.22
Tot_Fast
Tot_Epi
Tot_Thermal
Tot_Total
-0.06
-0.05
0.15
-0.02
-0.06
-0.07
0.20
0.00
-0.07
-0.08
0.20
0.06
169
Table 6-32: Percentage deviation of reaction
rates between 4th model and 1st model
Reaction Type
B4C
Clad
Water
Abs_Fast
-0.06
-0.04
-0.04
Abs_Epi
-0.03
-0.06
-0.06
Abs_Thermal
-0.03
0.18
0.14
Abs_Total
-0.03
0.16
0.13
Tot_Fast
Tot_Epi
Tot_Thermal
Tot_Total
-0.05
-0.04
-0.03
-0.04
-0.05
-0.06
0.12
-0.01
-0.06
-0.06
0.12
0.02
The second step is to study the radial mesh size. Three models with different
mesh-size are used in this study. The first model has 0.03cm-mesh. The second model
has 0.1cm-mesh. The third model has 0.15 cm-mesh.
Table 6-33 shows the eigenvalues calculated from TORT and pcm deviation for
each model. Table 6-34 and Table 6-35 show the percentage deviation of reaction rates
between each model and the 1st model. From the results, the 1rd radial mesh-size model,
0.03 cm. is chosen to use for control rod in core calculations.
Table 6-33: Eigenvalues calculated from TORT
Model (no. of cells)
keff
Deviation
From 1st model in pcm
1 (148x114x14)
0.71211
-
Time
(min)
209
2 (45x38x14)
0.70927
-284
15
3 (35x27x14)
0.70674
-537
10
170
Table 6-34: Percentage deviation of reaction
rates between 2nd model and 1st model
Reaction Type
B4C
Clad
Water
Abs_Fast
-0.05
-0.26
-0.06
Abs_Epi
0.33
-1.80
-0.49
Abs_Thermal
0.59
-7.94
-1.69
Abs_Total
0.44
-7.39
-1.60
Tot_Fast
Tot_Epi
Tot_Thermal
Tot_Total
-0.05
0.20
0.60
0.16
-0.19
-0.88
-7.96
-2.48
-0.06
-0.27
-1.62
-0.88
Table 6-35: Percentage deviation of reaction
rates between 3rd model and 1st model
Reaction Type
B4C
Clad
Water
Abs_Fast
-0.16
-0.49
-0.13
Abs_Epi
0.65
-3.22
-0.71
Abs_Thermal
1.13 -11.76
-2.05
Abs_Total
0.84 -10.98
-1.94
Tot_Fast
Tot_Epi
Tot_Thermal
Tot_Total
-0.17
0.40
1.14
0.29
-0.39
-1.58
-11.83
-3.85
-0.20
-0.37
-2.03
-1.13
6.3.2 Core Reflector Thickness Study
Pseudo-core loading 2 was modeled for TORT calculation as shown in Figure
6-7. Four control rods were replaced with 8.5% fuel cell. The fuel part with 19.05 cm. in
axial direction was modeled. This core configuration is used to study the radial thickness
of reflector. The mean-free-path (mfp) of water for is about 5 cm for fast group. Thus, we
have 3 models of radial thickness based on the mfp, 1 mfp-5 cm, 2 mfp-10 cm, and 3
mfp-15 cm. The reflective boundary condition is applied at the top and bottom of the
model and the vacuum boundary condition is utilized at front, back, left, and right of the
171
model. The total number of cells for the model with 5 cm reflector thickness is 1,372,410
cells. The total number of cells for the model with 10 cm reflector thickness is 1,918,620
cells.The total number of cells for the model with 15 cm reflector thickness is 2,521,640
cells. The 12G structure library is used for this study.
Figure 6-7: Core loading 2 with 15 cm reflector thickness
172
The results are presented in Table 6-36. The eigenvalues are not sensitive
comparing the 10-cm reflector thickness model with the 15-cm reflector thickness model.
The model with 5-cm reflector thickness differs from the model with 15 cm reflector
thickness by 238 pcm in the eigenvalue prediction.
Reflector
Thickness
(cm)
15
10
5
Table 6-36: Eigenvalues calculated by TORT
keff
Convergence
Rel.Deviation
(1E-4,1E-6)
in pcm with 15 cm
reflector thickness model
1.14363
1.0E-7
1.14353
-6.3E-7
-10
1.14125
-4.2E-7
-238
Time
(hr)
85
58
27
6.3.3 Core Loading 2 – ARI
A model with 5 cm reflector thickness was used to perform core calculations
because of the computational time. It is used in both TORT and MCNP. In this section,
we modeled the core with all control rods in (ARI) as shown in Figure 6-8 for radialcross-section view and Figure 6-9 for axial-cross-section view. The 13-coarse group cross
section library was used with S8-SLC quadrature set and P1 scattering order. The flux
convergence was set to 5x10-4 and the eigenvalue convergence was set to 1x10-5. The
selected mesh sizes in previous section were used. For fuel rods, the mesh sizes are 0.15
cm in radial direction and 2,1,0.5 cm mixed model in axial direction. For control rod, the
mesh sizes are 0.03 cm in radial direction and 2,1,0.5 cm mixed model in axial direction.
173
Figure 6-8: Radial-cross-section view of ARI
174
Figure 6-9: Axial-cross-section view of ARI
The comparison of the eigenvalue predictions indicates that TORT over-estimates
by 9 pcm keff as compared to the reference MCNP result as shown in Table 6-37. This
deviation is within 3σ. Figure 6-10 shows the normalized power map of MCNP and
TORT also the percentage relative difference of TORT results as compared to MCNP
results. The normalized power map calculated by MCNP has less than 1% of statistical
uncertainty. The relative differences vary in a range of ~-3% to +4%. The maximum
differences occur at the core periphery at which the power is low. The agreement in this
region can be improved by using 10 or 15 cm reflector thickness in both TORT and
MCNP models.
175
Table 6-37: Eigenvalues calculated from TORT and MCNP
keff
Deviation
Time
from MCNP
(hr)
in pcm of Δk
MCNP
0.90609
10
±0.00060(3σ)
TORT,S8P1
0.90618
9
248
0.731
0.746
2.084
0.852
0.862
1.185
0.905
0.901
-0.503
0.834
0.821
-1.514
0.583
0.572
-1.820
0.450
0.440
-2.188
1.024
1.027
0.281
0.839
0.823
-1.852
0.576
0.563
-2.285
0.938
0.958
2.094
1.043
1.053
0.924
0.816
0.800
-1.995
0.992
1.008
1.665
1.213
1.189
-2.006
0.833
0.804
-3.450
0.887
0.864
-2.613
1.004
0.980
-2.383
0.961
0.958
-0.305
1.105
1.106
0.072
1.384
1.356
-2.068
1.033
1.029
-0.437
0.774
0.772
-0.224
1.044
1.038
-0.530
1.204
1.187
-1.441
0.838
0.869
3.715
0.676
0.691
2.154
0.652
0.672
3.025
SH
1.267
1.249
-1.453
1.040
1.046
0.604
0.959
0.969
1.081
1.096
1.092
-0.367
1.211
1.220
0.742
1.170
1.162
-0.680
0.998
1.005
0.684
1.152
1.171
1.680
1.433
1.413
-1.446
1.374
1.343
-2.220
0.919
0.945
2.895
1.342
1.367
1.890
1.602
1.613
0.697
TR
0.913
0.911
-0.191
1.277
1.290
1.019
1.400
1.377
-1.625
1.303
1.269
-2.631
0.782
0.816
4.367
1.035
1.058
2.171
1.218
1.246
2.295
CT
1.032
1.015
-1.604
0.827
0.825
-0.248
1.086
1.100
1.302
1.310
1.289
-1.606
1.202
1.164
-3.190
0.877
0.895
1.991
1.101
1.125
2.193
SA
1.466
1.461
-0.370
1.028
1.008
-1.951
0.908
0.926
2.023
1.047
1.066
1.761
1.053
1.064
1.036
1.007
1.015
0.735
RR
0.835
0.855
2.429
0.822
0.802
-2.483
1.046
1.032
-1.338
1.050
1.054
0.360
0.960
0.965
0.558
0.901
0.892
-1.005
0.827
0.835
0.982
MCNP NP
0.715
0.713
-0.307
0.808
0.812
0.395
0.862
0.870
0.921
0.825
0.829
0.432
0.724
0.741
2.366
x.xxx
x.xxx
x.xxx
TORT NP
TORT − MCNP
x100%
MCNP
Figure 6-10: Normalized pin-power distribution for ARI
176
6.3.4 Core Loading 2 – ARO
In this section, we modeled the core all control rods out (ARO) as presented in
Figure 6-11 for radial-cross-section view and Figure 6-12 for axial-cross-section view.
We performed the calculation with the same parameter values as in the ARI case.
Figure 6-11: Radial-cross-section view of ARO
177
Figure 6-12: Axial-cross-section view of ARO
Comparison of the eigenvalue prediction indicates that TORT over-estimates by
91 pcm as shown in Table 6-38. Figure 6-13 shows the normalized power maps of MCNP
and TORT also the percentage relative difference of TORT results as compared to MCNP
results. The relative differences vary in a range of ~-3% to +4%. The maximum
difference occurs at the same location as in the ARI case and can be improved by using a
thicker reflector.
Table 6-38: Eigenvalues calculated from TORT and MCNP
keff
Deviation
Time
From MCNP
(hr)
In pcm of Δk
MCNP
1.01926
11
±0.00057(3σ)
TORT,S8P1
1.02017
91
238
178
0.629
0.637
1.353
0.762
0.773
1.444
0.852
0.849
-0.379
0.847
0.841
-0.717
0.763
0.756
-0.864
0.652
0.641
-1.772
1.014
1.015
0.094
1.015
0.993
-2.145
0.753
0.739
-1.855
0.867
0.881
1.654
1.065
1.072
0.640
0.741
0.721
-2.731
1.192
1.202
0.858
1.297
1.271
-1.966
0.990
0.958
-3.187
0.800
0.787
-1.625
0.941
0.928
-1.285
1.143
1.130
-1.141
0.578
0.583
0.875
0.652
0.659
1.145
0.675
0.690
2.178
0.835
0.856
2.473
0.971
0.957
-1.384
0.965
0.959
-0.619
0.882
0.889
0.770
0.770
0.775
0.611
0.642
0.643
0.099
0.916
0.919
0.275
1.185
1.180
-0.381
1.073
1.058
-1.440
0.769
0.791
2.778
0.924
0.920
-0.402
1.167
1.159
-0.684
0.871
0.882
1.330
0.689
0.694
0.666
1.240
1.252
0.968
1.337
1.358
-1.361
1.180
1.167
-1.153
0.802
0.815
1.709
1.009
1.013
0.395
1.371
1.390
1.424
1.340
1.321
-1.390
0.888
0.892
0.393
0.987
1.006
2.005
1.444
1.441
-0.185
1.602
1.562
-2.462
0.738
0.759
2.802
1.267
1.294
2.146
1.623
1.642
1.210
TR
0.797
0.801
0.546
1.180
1.189
0.735
1.646
1.636
-0.571
1.549
1.513
-2.287
0.613
0.639
4.208
0.858
0.875
1.994
1.415
1.443
2.006
CT
1.253
1.232
-1.605
0.709
0.710
0.152
1.255
1.264
0.730
1.597
1.568
-1.868
1.250
1.215
-2.809
0.709
0.720
1.538
0.964
0.988
2.470
1.207
1.196
-0.933
1.524
1.528
0.234
1.011
0.994
-1.674
0.758
0.771
1.634
0.971
0.989
1.890
1.303
1.312
0.742
1.190
1.193
0.223
0.815
0.801
-1.636
0.716
0.727
1.628
0.756
0.752
-0.466
0.648
0.657
1.476
0.554
0.563
1.570
MCNP NP
x.xxx
x.xxx
x.xxx
TORT NP
TORT − MCNP
x100%
MCNP
Figure 6-13: Normalized pin-power distribution for ARO
179
6.4
Summary
We successfully verify our 26-broad group cross section library with the MCNP
continuous cross section for the developed TRIGA mini-core model. A 13 coarse-group
library is developed for core calculation. In comparing the eigenvalue and normalized pin
power distribution of TRIGA core loading 2 for both ARI and ARO cases, the results
shows good agreement even for the coarse group structure without any spatial
homogenization.
The results can be improved by using P3 scattering order and a thicker reflector
model in radial plane. Unfortunately, such model involves significant running time and
not feasible at this moment for the completion of this thesis. For the same reason, the core
loading 2 was modeled symmetrically which does not allow to compare the TORT results
with the available measured data in addition to the MCNP reference results. However, the
MCNP core loading 2 model has been validated for TRIGA core analysis with the
available measured data [Ref.21].
180
CHAPTER 7
7.1
Conclusions and Future Research
Conclusions
Fine- and broad- group structures for the TRIGA cross-section generation in both
2-D and 3-D geometries were developed based on the CPXSD (Contributon and Pointwise Cross-Section Driven) methodology that selects effective group structure for a
problem of interest. We have implemented this method for the first time for criticality
reactor calculation and introduced specified objectives to refine the group structure in
three ranges of energy (i) Fast range, above 0.1 MeV. (ii) Epithermal range, 3eV to 0.1
MeV. , and (iii) Thermal range, 1E-5 to 3 eV. We consider two factors to be the criteria
of our problem (i) eigenvalue and (ii) objective reaction rates depending on the energy
range.
For 2-D cross section generation, the 280-fine-group structure (280G) was
developed in the energy range between 1E-5 eV. to 20 MeV based on the 8.5wt% and
12% fuel pin cells using the DORT code. The 280G contains 52 energy groups in the fast
range, 104 energy groups in the epithermal range, and 124 energy groups in the thermal
range. Utilizing the scalar flux weighting technique, a 12-broad group structure (12G)
was developed from 280G with the CPXSD methodology. The 12G structure contains 1
energy group in the fast range, 2 energy groups in the epithermal range, and 9 energy
groups in the thermal range. It was demonstrated that the broad-group library is in close
agreement with its fine-group library, within 50 pcm Δk/k. Also, comparing with the
continuous energy Monte Carlo predictions, we have demonstrated that these new
libraries yield good results, with deviations within 150 pcm Δk/k. In addition to cross-
181
section group condensation, we also performed the cross-section homogenization.
Compared to broad-group heterogeneous cross sections, the broad-group-homogeneous
cross-sections results differed by ~200 pcm of Δk/k for 4-region homogenization and ~60
pcm of Δk/k for 3-region homogenization .
For 3-D cross-section generation, the same 280-fine-group structure (280G) as
studied in 2-D cross-section generation was selected based on the 8.5wt% fuel pin cells
using the TORT code. The 26-broad group structure (26G) was obtained by collapsing
280G with the CPXSD methodology. The 26G structure contains 7 energy group in the
fast range, 4 energy group in the epithermal range, and 15 energy group in the thermal
range. It was demonstrated that the broad-group library is in close agreement with its
fine-group library, within 60 pcm Δk/k. Also, comparing with the continuous energy
Monte Carlo predictions, we have demonstrated that these new libraries yield good
results, with deviations less than 200 pcm Δk/k. Our studies also show that the effective
broad-group structures derived from 2-D and 3-D cross-section generation models are
different. In order to develop an effective group structure for 3-D problem, the 3-D crosssection model should be used.
The obtained broad-group structure was also applied for non-fissile material. The
results show good agreement of eigenvalues of color set model compared with the
continuous energy MCNP solution. Some differences appear in fast energy range;
however, they are insignificant compared to the total reaction rates.
Along with the study of cross section generation, the parametric studies for SN
calculations were performed to evaluate the impact of the spatial meshing, angular, and
scattering order variables and to obtain the suitable values for cross-section collapsing of
182
the TRIGA cell problem. The analysis shows that the scattering order has an effect only
on the 3-D problem. The difference of eigenvalue is about 300 pcm between P1 and P3.
A coarse group structure was developed using the 26 broad-group- structure to
perform core calculations. With 12 groups of fuel model and 13 groups of control rod
model, we have good agreement of eigenvalues and reaction rates between coarse group
and broad group structures. The 13 group structure was selected to use for core
calculations. Finally, the TRIGA core model was developed for SN calculations. The
results agree well with the MCNP continuous energy solutions for eigenvalue and
normalized pin power distribution.
7.2
Future Research
This research has created areas of problems to be improved as presented below.
•
Self-shielding in cladding
Even though we studied on the various methods of self-shielding treatment,
the problem still exists in cladding region. Further investigative work and
developing of new self-shielding method could be done to solve this
problem.
•
Doppler effect study
At present, we have performed our studies only at cold condition (room
temperature). The Doppler effect is important when the reactor is at
power. With high temperature, the resonances in the capture cross-section
183
of U-238 are broadening. A study could be done in order to examine the
effect on selected group structure.
•
Completing the TRIGA core model calculations
Calculations of the TRIGA loading 2 should be repeated with the
developed core model by using P3 scattering order and increase the
thickness of radial reflector. Further the real core loading 2 without
symmetric model should be studied in order to compare the results with
the available measured data.
•
Parallel computing environment study
Further studies in parallel computing environment could be done utilizing
domain decomposition methods. One can use codes which have such
capabilities such as PENTRAN. This will improve the efficiency of
TRIGA core calculations.
•
Developing the software interface for depletion model
In order to complete the entire process for reactor core analysis, the
developed core model should be coupled with a depletion calculation
model. The development of the interface software between transport code
and depletion code could be done to expand the fresh core simulation to
the depletion core simulation.
184
References
1.
A. Haghighat, Ce Yi, and G.E. Sjoden, “Accuracy of PENTRAN Criticality
Calculations based on the C5G7 MOX Benchmark”, Tran. Am. Nucl. Soc., 285288 (2003)
2.
Alpan, F. A and Alireza Haghighat, “Development of the CPXSD Methodology
for Generation of Fine-Group Libraries for Shielding Application”, Nuclear
Science and Engineering, Vol. 149, No.1, Jan 2005, Pages 51-64
3.
Alpan, F. A, Luiz C. Leal, and Arnaud Courcelle, “Effect of Energy SelfShielding Methods on U238 for criticality Safety Problems”, PHYSOR 2002,
Chicago Illinois(2004)
4.
DOORS 3.2 A: “One, Two- and Three-Dimensional Discrete Ordinates
Neutron/Photon Transport Code System” Oak Ridge National Laboratory. (Oct.
2003)
5.
E. E. Lewis, W. F. Miller, Jr., “Computational Methods of Neutron Transport”,
American Nuclear Society, La Grange Park, Illinois, U.S.A., 1993.
6.
G. Sjoden, A. Haghighat, “PENTRAN-Parallel Environment Neutral-particle
TRANsport in 3-D Cartesian Geometry,” Proc. Joint Int. Conf. Mathematical
Methods and Supercomputing for Nuclear Applications, Saratoga Springs, new
York (1997).
185
7.
G.E. Sjoden and A. Haghighat, “Advanced 3-D Parallel Discrete Ordinates
Methods for Criticality Safety Calculations,” Proceedings of the Mathematics and
Computation, Reactor Physics and Environmental Analysis in nuclear
Applications, Vol.2 1403-112, Sept. 1999.
8.
James J. Duderstadt, and Louis J. Hamilton, “Nuclear Reactor Analysis”, 1976.
9.
J. J. Klingensmith, Y.Y. Azmy, J. Gehin, and R. Orsi, “TORT Solution to the
Three-Dimensional MOX Neutron Transport Benchmarks”, Tran. Am. Nucl.
Soc., 276-278 (2003).
10.
Jon A. Dahl and Raymond E. Alcouffe, “PARTISN Results for the C5G7 MOX
Benchmark Problems”, Tran. Am. Nucl. Soc., 274-275 (2003).
11.
M.A. Smith, G. Palmiotti, T. A. Taiwo, E. E. Lewis, N. Tsoulfanidis, “Benchmark
Specification for Deterministic MOX Fuel Assembly Transport Calcualtions
without
Spatial
Homogenization
(3-D
extension
C5G7
MOX)”,
NEA/NSC/DOC(2003)6, April 30,2003
12.
Macfarlane, R. E., Muir, D.E., “NJOY94.61: Code System for Producing
Pointwise and Multigroup Neutron and Photon Cross Sections from DENDF/B
Data, “ PSR355, Los Alamos National Laboratory, Los Alamos, New Mexico,
December 1996.
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MCNP 5: “Monte Carlo N-Particle Transport Code System,” LA-12625-M, Los
Alamos National Lab. (Nov. 1997).
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14.
N. Kriangchaiporn, “Advanced fuel Management System” MS Thesis, The
Pennsylvania State University, Nuclear Engineering, 2000.
15.
Richard Doyas and Brian Koponen, “A Rigorous Collapse Procedure for
Multigroup Neutron Cross Sections”, Nuclear Science and Engineering, 47: 471475, 1972.
16.
Rudi J.J. Stamm’ler, “Methods of Steady-State Reactor Physics in Nuclear
Design”, 1983.
17.
SCALE 5: “Modular Code System for Performing Criticality and Shielding
Assessment for Licensing Evaluation”, Oak Ridge National Laboratory. (May
2004)
18.
W. F. Naughton, M. J. Cenko, S. H. Levine and W. F. Witzig, “TRIGA Core
Management Model”, Nuclear Technology Vol. 23, September 1974.
19.
Williams, M. L., “Generalized Contributon Response Theory, “ Nuclear Science
Engineering, 108: 355-383, 1991.
20.
Y. Su Kim, “ PSBR Core Monte Carlo Modeling and Analysis”, M.S. Thesis,
The Pennsylvania State University, 1995.
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of the PSU Research Reactor”, Proceedings of Monte Carlo 2005 Conference
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187
APPENDIX A. TORT INPUT SAMPLE FOR TRIGA
This sample is a 8.5% wt fuel TRIGA pin cell problem with 48x59x55-cell
model. The S8-SLC quadrature set and P1 scattering order with 12-group cross sections
are usded in this problem.
" single fuel element
61$$ 0 2 8 a5 -3
/;; xsecn unit
a8 10
e
62$$ 150 4 0 0 0
/source iter;flux
a9 1 1 a15 0 a17 1900 a19 1
e
63$$ 48 59 55 1 1 1 1 1 0 0
a11 9 0 144 228 0 e
64$$ 12
8 3 3 26
11 12
e
66** 1-4 1-4 1-3 a4 1-6
e t
t
/ order 8, 144 angles
82** / mu
-2.790041-1 -2.736433-1 -2.319836-1
5.443103-2
1.550065-1
2.319836-1
-6.044191-1 -5.928054-1 -5.025562-1
1.179163-1
3.357973-1
5.025562-1
-8.507736-1 -8.344262-1 -7.073924-1
1.659777-1
4.726645-1
7.073924-1
-9.830319-1 -9.641432-1 -8.173612-1
1.917800-1
5.461432-1
8.173612-1
-2.790041-1 -2.736433-1 -2.319836-1
5.443103-2
1.550065-1
2.319836-1
-6.044191-1 -5.928054-1 -5.025562-1
1.179163-1
3.357973-1
5.025562-1
-8.507736-1 -8.344262-1 -7.073924-1
1.659777-1
4.726645-1
7.073924-1
-9.830319-1 -9.641432-1 -8.173612-1
1.917800-1
5.461432-1
8.173612-1
q
72
83** / eta
-9.602899-1 -9.602899-1 -9.602899-1
-9.602899-1 -9.602899-1 -9.602899-1
-7.966665-1 -7.966665-1 -7.966665-1
-7.966665-1 -7.966665-1 -7.966665-1
-5.255324-1 -5.255324-1 -5.255324-1
-5.255324-1 -5.255324-1 -5.255324-1
-1.834346-1 -1.834346-1 -1.834346-1
-1.834346-1 -1.834346-1 -1.834346-1
q
36 g
72
81** / weights
0.000000+0
3.163392-3
3.163392-3
3.163392-3
3.163392-3
3.163392-3
0.000000+0
6.949405-3
6.949405-3
6.949405-3
6.949405-3
6.949405-3
0.000000+0
9.803335-3
9.803335-3
9.803335-3
9.803335-3
9.803335-3
0.000000+0
1.133387-2
1.133387-2
1.133387-2
1.133387-2
1.133387-2
iter;1-print XS
/k-search
-1.550065-1
2.736433-1
-3.357973-1
5.928054-1
-4.726645-1
8.344262-1
-5.461433-1
9.641432-1
-1.550065-1
2.736433-1
-3.357973-1
5.928054-1
-4.726645-1
8.344262-1
-5.461433-1
9.641432-1
-5.443105-2
-9.602899-1
-9.602899-1
-7.966665-1
-7.966665-1
-5.255324-1
-5.255324-1
-1.834346-1
-1.834346-1
-9.602899-1
3.163392-3
3.163392-3
6.949405-3
6.949405-3
9.803335-3
9.803335-3
1.133387-2
1.133387-2
3.163392-3
-1.179164-1
-1.659777-1
-1.917801-1
-5.443105-2
-1.179164-1
-1.659777-1
-1.917801-1
-7.966665-1
-5.255324-1
-1.834346-1
6.949405-3
9.803335-3
1.133387-2
188
3q
36
84** 0.00
0.2286 1.8224 1.8732 2.1768
f9999
85**
0.00
0.2286 1.8224 1.8732 2.51355
f9999
86**
0.00
6.000000e+00 1.200000e+01 1.905000e+01 21.05 2.778760e+01
f9999
t
2**
0.000000e+00 2.857500e-02 5.715000e-02 8.572499e-02 1.143000e-01
1.428750e-01 1.714500e-01 2.000250e-01 2.286000e-01 2.876296e-01
3.466593e-01 4.056889e-01 4.647186e-01 5.237482e-01 5.827778e-01
6.418074e-01 7.008371e-01 7.598667e-01 8.188964e-01 8.779260e-01
9.369556e-01 9.959853e-01 1.055015e+00 1.114044e+00 1.173074e+00
1.232104e+00 1.291133e+00 1.350163e+00 1.409192e+00 1.468222e+00
1.527251e+00 1.586281e+00 1.645311e+00 1.704340e+00 1.763370e+00
1.822400e+00 1.847800e+00 1.873200e+00 1.900800e+00 1.928400e+00
1.956000e+00 1.983600e+00 2.011200e+00 2.038800e+00 2.066400e+00
2.094000e+00 2.121600e+00 2.149200e+00 2.176800e+00
3**
0.000000e+00 3.265714e-02 6.531429e-02 9.797142e-02 1.306286e-01
1.632857e-01 1.959429e-01 2.286000e-01 2.890809e-01 3.495619e-01
4.100428e-01 4.705238e-01 5.310047e-01 5.914857e-01 6.519666e-01
7.124476e-01 7.729285e-01 8.334095e-01 8.938904e-01 9.543714e-01
1.014852e+00 1.075333e+00 1.135814e+00 1.196295e+00 1.256776e+00
1.319623e+00 1.382470e+00 1.445317e+00 1.508165e+00 1.571012e+00
1.633859e+00 1.696706e+00 1.759553e+00 1.822400e+00 1.847800e+00
1.873200e+00 1.899881e+00 1.926563e+00 1.953244e+00 1.979925e+00
2.006607e+00 2.033288e+00 2.059969e+00 2.086650e+00 2.113331e+00
2.140013e+00 2.166694e+00 2.193375e+00 2.220056e+00 2.246737e+00
2.273418e+00 2.300100e+00 2.326781e+00 2.353462e+00 2.380143e+00
2.406824e+00 2.433506e+00 2.460187e+00 2.486868e+00 2.513550e+00
4**
0.000000e+00 5.000000e-01 1.000000e+00 1.500000e+00 2.000000e+00
2.500000e+00 3.000000e+00 3.500000e+00 4.000000e+00 4.500000e+00
5.000000e+00 5.500000e+00 6.000000e+00 6.500000e+00 7.000000e+00
7.500000e+00 8.000000e+00 8.500000e+00 9.000000e+00 9.500000e+00
1.000000e+01 1.050000e+01 1.100000e+01 1.150000e+01 1.200000e+01
1.250357e+01 1.300714e+01 1.351071e+01 1.401429e+01 1.451786e+01
1.502143e+01 1.552500e+01 1.602857e+01 1.653214e+01 1.703571e+01
1.753928e+01 1.804285e+01 1.854642e+01 1.905000e+01 1.955000e+01
2.005000e+01 2.055000e+01 2.105000e+01 2.156828e+01 2.208655e+01
2.260483e+01 2.312310e+01 2.364138e+01 2.415966e+01 2.467793e+01
2.519621e+01 2.571449e+01 2.623276e+01 2.675104e+01 2.726931e+01
2.778760e+01
/14** Body Left Boundaries
14**
7r0.000000e+00
4r2.857500e-02
4r5.715000e-02
2r8.572499e-02
4r1.143000e-01
2r1.428750e-01
6r1.714500e-01
6r2.000250e-01
7r2.286000e-01
4r2.876296e-01
8r3.466593e-01
4r4.056889e-01
4r4.647186e-01
4r5.237482e-01
8r5.827778e-01
4r6.418074e-01
4r7.008371e-01
8r7.598667e-01
4r8.188964e-01
8r8.779260e-01
4r9.369556e-01
8r9.959853e-01
8r1.055015e+00
4r1.114044e+00
189
8r1.173074e+00
8r1.350163e+00
6r1.527251e+00
8r1.704340e+00
4r1.847800e+00
4r2.011200e+00
8r1.232104e+00
6r1.409192e+00
8r1.586281e+00
6r1.763370e+00
2r1.873200e+00
4r2.121600e+00
6r1.291133e+00
8r1.468222e+00
6r1.645311e+00
6r1.822400e+00
4r1.900800e+00
8.572499e-02
2r2.286000e-01
4r1.143000e-01
6r2.000250e-01
4r2.876296e-01
2r5.237482e-01
4r5.237482e-01
2r7.008371e-01
4r7.598667e-01
4r8.779260e-01
4r9.959853e-01
4r1.114044e+00
6r1.291133e+00
8r1.468222e+00
6r1.645311e+00
8r1.822400e+00
4r1.873200e+00
4r2.121600e+00
3.466593e-01
2r2.857500e-02
2r1.428750e-01
6r2.286000e-01
4r3.466593e-01
4r4.056889e-01
4r5.827778e-01
4r6.418074e-01
4r8.779260e-01
4r9.959853e-01
8r1.055015e+00
4r1.173074e+00
8r1.350163e+00
6r1.527251e+00
8r1.704340e+00
4r1.847800e+00
2r1.900800e+00
4r2.176800e+00
8.572499e-02
4r5.715000e-02
6r1.714500e-01
3r3.466593e-01
2r5.827778e-01
4r4.647186e-01
2r7.598667e-01
4r7.008371e-01
4r8.188964e-01
4r9.369556e-01
4r1.173074e+00
8r1.232104e+00
6r1.409192e+00
8r1.586281e+00
6r1.763370e+00
2r1.900800e+00
4r2.011200e+00
2r0.000000e+00
4r1.873200e+00
2r2.460187e+00
2r1.873200e+00
1.632857e-01
2r1.873200e+00
6.531429e-02
0.000000e+00
2r2.353462e+00
2r0.000000e+00
2r2.300100e+00
2r1.822400e+00
2r2.193375e+00
2r1.759553e+00
2r2.113331e+00
2r0.000000e+00
2r2.059969e+00
2r0.000000e+00
2r1.979925e+00
2r0.000000e+00
2r1.926563e+00
2r1.508165e+00
2r1.847800e+00
2r1.445317e+00
2r1.319623e+00
2r0.000000e+00
2r0.000000e+00
2r1.696706e+00
2r1.196295e+00
2r1.135814e+00
2r9.543714e-01
2.286000e-01
2r2.486868e+00
0.000000e+00
2r2.433506e+00
0.000000e+00
2r2.406824e+00
2r1.873200e+00
2r1.822400e+00
2r1.847800e+00
2r1.759553e+00
2r1.822400e+00
2r2.220056e+00
2r0.000000e+00
2r2.166694e+00
2r1.696706e+00
2r1.633859e+00
2r1.696706e+00
2r1.571012e+00
2r1.633859e+00
2r1.508165e+00
2r0.000000e+00
2r1.899881e+00
2r0.000000e+00
2r1.822400e+00
2r1.382470e+00
2r1.256776e+00
2r1.196295e+00
2r0.000000e+00
2r0.000000e+00
2r1.633859e+00
2r1.014852e+00
2r1.822400e+00
2r1.873200e+00
1.959429e-01
0.000000e+00
1.306286e-01
0.000000e+00
2r2.380143e+00
2r1.847800e+00
2r2.326781e+00
2r1.822400e+00
2r2.273418e+00
2r1.759553e+00
2r1.696706e+00
2r1.759553e+00
2r2.086650e+00
2r1.696706e+00
2r2.033288e+00
2r1.633859e+00
2r1.953244e+00
2r1.571012e+00
2r1.445317e+00
2r1.508165e+00
2r1.382470e+00
2r0.000000e+00
2r1.759553e+00
2r1.319623e+00
2r1.256776e+00
2r1.135814e+00
2r1.014852e+00
2r0.000000e+00
2r0.000000e+00
15**
16**
190
2r8.334095e-01
2r0.000000e+00
2r0.000000e+00
2r1.508165e+00
2r5.310047e-01
2r1.445317e+00
4r0.000000e+00
2r1.319623e+00
2r9.543714e-01
2r7.124476e-01
2r5.310047e-01
2r0.000000e+00
2r0.000000e+00
2r0.000000e+00
2r1.382470e+00
2r0.000000e+00
2r1.571012e+00
2r8.334095e-01
2r7.124476e-01
2r3.495619e-01
2r3.495619e-01
2r2.286000e-01
2r0.000000e+00
2r1.256776e+00
2.286000e-01
2r2.513550e+00
2r2.460187e+00
1.822400e+00
1.632857e-01
1.822400e+00
6.531429e-02
2r2.513550e+00
2r2.353462e+00
2r2.513550e+00
2r2.300100e+00
2r2.513550e+00
2r2.193375e+00
2r1.759553e+00
2r2.113331e+00
2r2.513550e+00
2r2.059969e+00
2r2.513550e+00
2r1.979925e+00
2r2.513550e+00
2r1.926563e+00
2r1.508165e+00
2r1.847800e+00
2r1.445317e+00
2r1.319623e+00
2r2.513550e+00
2r1.759553e+00
2r1.696706e+00
2r1.196295e+00
2r1.135814e+00
2r9.543714e-01
2r8.334095e-01
2r2.513550e+00
2r1.571012e+00
2r1.508165e+00
2r5.310047e-01
2r1.445317e+00
4r1.445317e+00
2r1.319623e+00
2r2.513550e+00
2r1.822400e+00
2r2.486868e+00
2r2.513550e+00
2r2.433506e+00
1.822400e+00
2r2.406824e+00
1.822400e+00
1.822400e+00
2r2.513550e+00
2r1.759553e+00
2r2.513550e+00
2r2.220056e+00
2r2.513550e+00
2r2.166694e+00
2r2.513550e+00
2r1.633859e+00
2r2.513550e+00
2r1.571012e+00
2r2.513550e+00
2r1.508165e+00
2r2.513550e+00
2r1.899881e+00
2r2.513550e+00
2r1.822400e+00
2r1.382470e+00
2r1.256776e+00
2r1.196295e+00
2r2.513550e+00
2r1.696706e+00
2r1.633859e+00
2r1.014852e+00
2r9.543714e-01
2r7.124476e-01
2r5.310047e-01
2r2.513550e+00
2r1.508165e+00
2r2.513550e+00
2r1.382470e+00
2r2.513550e+00
2r1.873200e+00
2r2.513550e+00
1.959429e-01
2r2.513550e+00
1.306286e-01
2r2.513550e+00
2r2.380143e+00
2r1.847800e+00
2r2.326781e+00
2r1.822400e+00
2r2.273418e+00
2r2.513550e+00
2r1.696706e+00
2r2.513550e+00
2r2.086650e+00
2r1.696706e+00
2r2.033288e+00
2r1.633859e+00
2r1.953244e+00
2r1.571012e+00
2r1.445317e+00
2r2.513550e+00
2r1.382470e+00
2r2.513550e+00
2r1.759553e+00
2r1.319623e+00
2r1.256776e+00
2r1.135814e+00
2r1.014852e+00
2r2.513550e+00
2r1.633859e+00
2r1.571012e+00
2r8.334095e-01
2r7.124476e-01
2r3.495619e-01
2r3.495619e-01
2r2.286000e-01
2r2.513550e+00
2r1.256776e+00
0.000000e+00
1.905000e+01
0.000000e+00
1.905000e+01
0.000000e+00
1.905000e+01
5r0.000000e+00
1.905000e+01
0.000000e+00
1.905000e+01
0.000000e+00
1.905000e+01
0.000000e+00
1.905000e+01
2r0.000000e+00
1.905000e+01
0.000000e+00
1.905000e+01
3r0.000000e+00
1.905000e+01
0.000000e+00
17**
18**
191
1.905000e+01
0.000000e+00
1.905000e+01
0.000000e+00
1.905000e+01
0.000000e+00
1.905000e+01
0.000000e+00
1.905000e+01
0.000000e+00
1.905000e+01
0.000000e+00
1.905000e+01
0.000000e+00
1.905000e+01
0.000000e+00
1.905000e+01
0.000000e+00
1.905000e+01
0.000000e+00
1.905000e+01
0.000000e+00
1.905000e+01
0.000000e+00
1.905000e+01
0.000000e+00
1.905000e+01
0.000000e+00
1.905000e+01
0.000000e+00
1.905000e+01
0.000000e+00
1.905000e+01
0.000000e+00
1.905000e+01
0.000000e+00
1.905000e+01
0.000000e+00
1.905000e+01
0.000000e+00
1.905000e+01
0.000000e+00
1.905000e+01
0.000000e+00
1.905000e+01
0.000000e+00
1.905000e+01
0.000000e+00
1.905000e+01
0.000000e+00
1.905000e+01
0.000000e+00
1.905000e+01
0.000000e+00
1.905000e+01
0.000000e+00
1.905000e+01
3r0.000000e+00
1.905000e+01
0.000000e+00
1.905000e+01
0.000000e+00
1.905000e+01
0.000000e+00
1.905000e+01
0.000000e+00
1.905000e+01
0.000000e+00
1.905000e+01
0.000000e+00
1.905000e+01
0.000000e+00
1.905000e+01
0.000000e+00
1.905000e+01
0.000000e+00
1.905000e+01
0.000000e+00
1.905000e+01
0.000000e+00
1.905000e+01
0.000000e+00
1.905000e+01
0.000000e+00
1.905000e+01
0.000000e+00
1.905000e+01
0.000000e+00
1.905000e+01
0.000000e+00
1.905000e+01
0.000000e+00
1.905000e+01
0.000000e+00
1.905000e+01
0.000000e+00
1.905000e+01
0.000000e+00
1.905000e+01
0.000000e+00
1.905000e+01
0.000000e+00
1.905000e+01
0.000000e+00
1.905000e+01
0.000000e+00
1.905000e+01
0.000000e+00
1.905000e+01
0.000000e+00
1.905000e+01
0.000000e+00
1.905000e+01
0.000000e+00
1.905000e+01
2r0.000000e+00
1.905000e+01
0.000000e+00
1.905000e+01
0.000000e+00
1.905000e+01
0.000000e+00
1.905000e+01
0.000000e+00
1.905000e+01
0.000000e+00
1.905000e+01
0.000000e+00
1.905000e+01
0.000000e+00
1.905000e+01
0.000000e+00
1.905000e+01
0.000000e+00
1.905000e+01
0.000000e+00
1.905000e+01
0.000000e+00
1.905000e+01
0.000000e+00
1.905000e+01
0.000000e+00
1.905000e+01
0.000000e+00
1.905000e+01
0.000000e+00
1.905000e+01
0.000000e+00
1.905000e+01
0.000000e+00
1.905000e+01
0.000000e+00
1.905000e+01
0.000000e+00
1.905000e+01
0.000000e+00
1.905000e+01
0.000000e+00
1.905000e+01
0.000000e+00
1.905000e+01
0.000000e+00
1.905000e+01
0.000000e+00
1.905000e+01
0.000000e+00
1.905000e+01
0.000000e+00
1.905000e+01
0.000000e+00
1.905000e+01
192
0.000000e+00
1.905000e+01
0.000000e+00
1.905000e+01
0.000000e+00
1.905000e+01
0.000000e+00
1.905000e+01
0.000000e+00
1.905000e+01
0.000000e+00
1.905000e+01
0.000000e+00
1.905000e+01
0.000000e+00
1.905000e+01
0.000000e+00
1.905000e+01
0.000000e+00
1.905000e+01
0.000000e+00
1.905000e+01
0.000000e+00
1.905000e+01
0.000000e+00
1.905000e+01
1.905000e+01
2.778760e+01
1.905000e+01
2.778760e+01
1.905000e+01
2.778760e+01
5r1.905000e+01
2.778760e+01
1.905000e+01
2.778760e+01
1.905000e+01
2.778760e+01
1.905000e+01
2.778760e+01
1.905000e+01
2.778760e+01
1.905000e+01
2.778760e+01
1.905000e+01
2.778760e+01
1.905000e+01
2.778760e+01
1.905000e+01
2.778760e+01
1.905000e+01
2.778760e+01
1.905000e+01
2.778760e+01
1.905000e+01
2.778760e+01
1.905000e+01
2.778760e+01
1.905000e+01
2.778760e+01
1.905000e+01
2.778760e+01
1.905000e+01
2.778760e+01
1.905000e+01
2.778760e+01
1.905000e+01
2.778760e+01
1.905000e+01
2.778760e+01
1.905000e+01
2.778760e+01
1.905000e+01
2.778760e+01
1.905000e+01
2.778760e+01
1.905000e+01
2.778760e+01
1.905000e+01
2.778760e+01
3r1.905000e+01
2.778760e+01
1.905000e+01
2.778760e+01
1.905000e+01
2.778760e+01
1.905000e+01
2.778760e+01
1.905000e+01
2.778760e+01
1.905000e+01
2.778760e+01
1.905000e+01
2.778760e+01
1.905000e+01
2.778760e+01
1.905000e+01
2.778760e+01
1.905000e+01
2.778760e+01
1.905000e+01
2.778760e+01
1.905000e+01
2.778760e+01
1.905000e+01
2.778760e+01
1.905000e+01
2.778760e+01
1.905000e+01
2.778760e+01
1.905000e+01
2.778760e+01
1.905000e+01
2.778760e+01
1.905000e+01
2.778760e+01
1.905000e+01
2.778760e+01
1.905000e+01
2.778760e+01
2r1.905000e+01
2.778760e+01
1.905000e+01
2.778760e+01
3r1.905000e+01
2.778760e+01
1.905000e+01
2.778760e+01
2r1.905000e+01
2.778760e+01
1.905000e+01
2.778760e+01
1.905000e+01
2.778760e+01
1.905000e+01
2.778760e+01
1.905000e+01
2.778760e+01
1.905000e+01
2.778760e+01
1.905000e+01
2.778760e+01
1.905000e+01
2.778760e+01
1.905000e+01
2.778760e+01
1.905000e+01
2.778760e+01
1.905000e+01
2.778760e+01
1.905000e+01
2.778760e+01
1.905000e+01
2.778760e+01
1.905000e+01
2.778760e+01
1.905000e+01
2.778760e+01
1.905000e+01
2.778760e+01
1.905000e+01
2.778760e+01
1.905000e+01
2.778760e+01
1.905000e+01
2.778760e+01
1.905000e+01
19**
193
2.778760e+01
1.905000e+01
2.778760e+01
1.905000e+01
2.778760e+01
1.905000e+01
2.778760e+01
1.905000e+01
2.778760e+01
1.905000e+01
2.778760e+01
1.905000e+01
2.778760e+01
1.905000e+01
2.778760e+01
1.905000e+01
2.778760e+01
1.905000e+01
2.778760e+01
1.905000e+01
2.778760e+01
1.905000e+01
2.778760e+01
1.905000e+01
2.778760e+01
1.905000e+01
8$$
1
8
2
2
8
5
3
5
5
5
5
5
3
5
5
5
3
3
3
4
4
5
2
2
3
3
5
4
4
6
5
4
4
5
9
7
9
9
9
9
9
7
9
9
9
7
7
7
8
8
9
6
6
7
7
9
8
8
2
9
8
8
9
4
4
4
2
4
2
4
4
2
2
4
4
4
4
5
2
2
3
3
4
4
3
5
5
3
4
5
5
2
8
8
8
6
8
6
8
8
6
6
8
8
8
8
9
6
6
7
7
8
8
7
9
9
1.905000e+01
2.778760e+01
1.905000e+01
2.778760e+01
1.905000e+01
2.778760e+01
1.905000e+01
2.778760e+01
1.905000e+01
2.778760e+01
1.905000e+01
2.778760e+01
1.905000e+01
2.778760e+01
1.905000e+01
2.778760e+01
1.905000e+01
2.778760e+01
1.905000e+01
2.778760e+01
1.905000e+01
2.778760e+01
1.905000e+01
2.778760e+01
1.905000e+01
2.778760e+01
7
8
9
9
3
5
5
5
3
5
3
5
5
3
3
5
5
5
2
2
3
3
4
4
5
3
4
4
4
5
1
1
7
9
9
9
7
9
7
9
9
7
7
9
9
9
6
6
7
7
8
8
9
7
8
8
8
9
2
2
4
2
4
4
4
4
4
2
4
4
4
2
2
2
3
3
4
4
5
2
2
4
4
5
2.778760e+01
1.905000e+01
2.778760e+01
1.905000e+01
2.778760e+01
1.905000e+01
2.778760e+01
1.905000e+01
2.778760e+01
1.905000e+01
2.778760e+01
1.905000e+01
2.778760e+01
1.905000e+01
2.778760e+01
1.905000e+01
2.778760e+01
1.905000e+01
2.778760e+01
1.905000e+01
2.778760e+01
1.905000e+01
2.778760e+01
1.905000e+01
2.778760e+01
4
1
1
4
8
6
8
8
8
8
8
6
8
8
8
6
6
6
7
7
8
8
9
6
6
8
8
9
194
/material number by zone
9$$
1 2 3 4 8 5 6 7 8
2
7**
1.0 0.9973 1.0651 0.9985 1.0 0.9973 1.0651 0.9985 1.0
1** / fission spectrum
9.86807e-01
1.31924e-02
4.29826e-07
1.82920e-09
3.44986e-10
5.73352e-11
6.08626e-12
3.45283e-12
1.87948e-12
1.81310e-12
3.61087e-13
7.09854e-14
t
93** 59r1
94** 55r1
95** f1.0
t
195
APPENDIX B. TRIGA FISSION SPECTRUM
6.0E-02
5.0E-02
Chi(E)
4.0E-02
3.0E-02
2.0E-02
1.0E-02
0.0E+00
1.00E-10 1.00E-09 1.00E-08 1.00E-07 1.00E-06 1.00E-05 1.00E-04 1.00E-03 1.00E-02 1.00E-01 1.00E+00 1.00E+01 1.00E+02
Neutron energy (MeV)
Figure B-1: 8.5% wt. fuel fission spectrum of 280 groups
3.0E-01
2.5E-01
Chi(E)
2.0E-01
1.5E-01
1.0E-01
5.0E-02
0.0E+00
1.0E-09
1.0E-08
1.0E-07
1.0E-06
1.0E-05
1.0E-04
1.0E-03
1.0E-02
1.0E-01
1.0E+00
1.0E+01
1.0E+02
Neutron energy (MeV)
Figure B-2: 8.5% wt. fuel fission spectrum of 26 groups
196
4.50E-01
4.00E-01
3.50E-01
3.00E-01
Chi (E)
2.50E-01
2.00E-01
1.50E-01
1.00E-01
5.00E-02
0.00E+00
1.00E-08
1.00E-07
1.00E-06
1.00E-05
1.00E-04
1.00E-03
1.00E-02
1.00E-01
1.00E+00
1.00E+01
1.00E+02
Neutron energy (MeV)
Figure B-3: 8.5% wt. fuel fission spectrum of 13 groups
6.0E-02
5.0E-02
Chi(E)
4.0E-02
3.0E-02
2.0E-02
1.0E-02
0.0E+00
1.00E-10 1.00E-09 1.00E-08 1.00E-07 1.00E-06 1.00E-05 1.00E-04 1.00E-03 1.00E-02 1.00E-01 1.00E+00 1.00E+01 1.00E+02
Neutron energy (MeV)
Figure B-4: 12% wt. fuel fission spectrum of 280 groups
197
3.0E-01
2.5E-01
Chi(E)
2.0E-01
1.5E-01
1.0E-01
5.0E-02
0.0E+00
1.0E-09
1.0E-08
1.0E-07
1.0E-06
1.0E-05
1.0E-04
1.0E-03
1.0E-02
Neutron energy (MeV)
1.0E-01
1.0E+00
1.0E+01
1.0E+02
Figure B-5: 12% wt. fuel fission spectrum of 26 groups
4.50E-01
4.00E-01
3.50E-01
3.00E-01
Chi (E)
2.50E-01
2.00E-01
1.50E-01
1.00E-01
5.00E-02
0.00E+00
1.00E-08
1.00E-07
1.00E-06
1.00E-05
1.00E-04
1.00E-03
1.00E-02
1.00E-01
1.00E+00
1.00E+01
1.00E+02
Neutron energy (MeV)
Figure B-6: 12% wt. fuel fission spectrum of 12 groups
198
Vita
Nateekool Kriangchaiporn was born in Bangkok, Thailand on September 10,
1975. Nateekool received her Bachelor degree in Electrical Engineering from Kasetsart
University in Bangkok, Thailand in May of 1997. In 1998, she entered a competitive
examination arranged by the Royal Thai Government and was granted the scholarship for
pursuit of Master’s degree in United States. In August 2001, Nateekool received the
Master of Science in Nuclear Engineering from Pennsylvania State University. She
continued her study in Nuclear Engineering at Pennsylvania State University and earned
the Ph.D. degree in May 2006.