The Pennsylvania State University

The Graduate School

College of Engineering

SENSITIVITY STUDY ON THE ENERGY GROUP STRUCTURE

FOR HIGH TEMPERATURE REACTOR ANALYSIS

A Thesis in

Nuclear Engineering

by

James Sanggene Han

Submitted in Partial Fulfillment of the Requirements for the Degree of

Master of Science

May 2008

The thesis of James Sanggene Han was reviewed and approved* by the following:

Kostadin N. Ivanov Distinguished Professor of Nuclear Engineering Thesis Advisor

Samuel Levine Professor Emeritus of Nuclear Engineering

Abderrafi M. Ougouag Nuclear Engineer Idaho National Laboratory

Hans Gougar Senior Consultant Pebble Bed Modular Reactor (Pty) Ltd

Jack Brenizer J. 'Lee' Everett Professor of Mechanical and Nuclear Engineering Chair of the Nuclear Engineering

*Signatures are on file in the Graduate School

ii iii ABSTRACT

The interactions of neutrons and materials in a core are strongly dependent on the energy of the incoming neutron. A nearly continuous energy dependent cross section data set makes for a challenging usage in computation. Therefore to simplify the computational problem, the cross sections are grouped into energy structures and properties dependent on energy are effectively averaged into energy groups. This thesis is focused on the selection of energy group structures for the reactor analysis of the Pebble Bed Modular Reactor. Energy group structure is simply defined as the discretization of energy into groups by the selection of energy boundaries. Traditional reactor analysis requires a two-step process. The first step utilizes a cross section generation code which takes the material composition and geometry of a particular region and its given conditions, to produce cross sections. This part is sometimes referred to as the cell calculation. The second part of the process is the reactor level calculation and utilizes the computed cross sections from the first step to calculate overall reactor properties. In this study, COMBINE6 was used for the cross section generation code and the Penn State nodal diffusion code NEM was used for the reactor analysis code. To arrive at an optimal group structure, many series of combinations of group structures were tested. Each set of results computed from the two-step process were compared with solution of a method independent of energy discretization, MCNP5, which is a Monte Carlo continuous energy code. MCNP5 is used as a reference because it generates a neutron transport solution using continuous energy cross sections and thus is a better representation of the physical system. From these numerical case studies, an optimal 5, 6, 7, 8, and 9 Group structure has been chosen based on the results that best match MNCP5’s results. In addition to evidence from simulations, the group structures determined can also be justified by arguments in reactor physics. iv TABLE OF CONTENTS

LIST OF FIGURES ...... vii

ACKNOWLEDGEMENTS...... xix

Chapter 1 Introduction to Gas Cooled Reactors ...... 1

Chapter 2 Nuclear Engineering and the Fundamentals of Reactor Physics...... 4

Chapter 3 Research Goals and Motivation ...... 9

Motivation for Study of Group Structures...... 10

Chapter 4 Theory: Selection Methodology & Review of Literature ...... 12

4.1 Group Structure General Theoretical Considerations...... 12 4.2 General Guidelines for Energy Group Structures from Literature ...... 12 4.4 Fast Energy Group Structure Selection Basis...... 18 4.5 Thermal Energy Group Selection Basis ...... 22 4.6 Thermal Cutoff Energy Selection Basis ...... 24 4.7 Important Isotopes in the Thermal Region ...... 24

Chapter 5 Computational Methods and Models ...... 26

5.1 Problem Description and Assumptions ...... 26 5.2 Models ...... 28 5.3 Model Assumptions & Errors...... 29 5.3.1 Low Buckling Assumption...... 29 5.3.2 20 Isotope Limitation Per Material...... 30 5.3.3 Uniform Temperature Distribution...... 31 5.3.4 Homogeneous versus Heterogeneous Material Compositions...... 31

Chapter 6 Sources of Error ...... 33

The Diffusion Theory Approximation...... 33

Chapter 7 Comparison Methodology...... 35

Chapter 8 Results and Discussion...... 37

8.1 Thermal Cutoff Study...... 37 8.1.1 Thermal Cut-off Study using 2-Group No Up-scattering...... 38 8.1.2 Thermal Cutoff Study using 2-Group with Up-scattering...... 41 8.1.3 Thermal Cutoff Study using Fixed 9-Group Structure...... 44 8.1.4 Thermal Cut-off Study Conclusion ...... 47 v 8.2 3-Group Study ...... 48 8.2.1 Fission Source Containment...... 48 8.2.2 3-Group Conclusion ...... 52 8.3 Thermal Group Studies...... 53 8.3.1 Group Structure Abbreviation ...... 53 8.3.2 Thermal Studies 1: 5-Group of 2 fast and 3 thermal groups ...... 54 8.3.3 Thermal Studies 2: 5-Group of 3 fast groups and 2 thermal groups ....60 8.3.4 Thermal Studies 3: 6-Group ...... 63 8.3.5 Thermal Studies 4: 7-Group ...... 66 8.3.6 Thermal Studies 5: 8-Group ...... 69 8.3.7 Thermal Studies Conclusion...... 73 8.4 Fast Group Studies...... 73 8.4.1 Preliminary Assessment of Fast Group Structure and Importance ...... 73 8.4.3 Fast Studies 2...... 81 8.4.4 Physical Explanations for Partitions: 2.04 keV, 639 keV, 1.74 MeV..83 8.4.5 Fast Studies 3...... 86 8.4.6 Physical Explanations for Partitions: 10 keV...... 89 8.4.7 Fast Studies 4...... 90 8.4.8 Fast Studies 5...... 94 8.4.9 Fast Studies 6...... 96 8.4.10 Fast Studies Conclusion ...... 99 8.4.11 Fast Studies 7: Constant Lethargy Divisions...... 99 8.4.12 Constant Lethargy of Fast Region Conclusion...... 102 8.4.13 Maximum Number of Energy Group Structures...... 103 8.5 G&S Studies of Broad Group Structures from LWR and Recommended modifications for HTGR...... 107 8.6 Recommended Best Group Structures Subdivided By Number of Groups....111 8.6.1 5 Group Selection...... 112 8.6.1.1 Spatial Shielding Effects in Diffusion Theory without Environment Feedback ...... 114 8.6.1.2 k-effective as a Good Indicator ...... 114 8.6.2 6 Group Selection...... 115 8.6.3 7 Group Selection...... 118 8.6.4 8 Group Selection...... 121 8.6.5 9+ Group Selection...... 124

Chapter 9 Conclusions ...... 127

Chapter 10 Future Work ...... 133

Bibliography ...... 135

Appendix A Normalization Issues Concerning Flux and Power Profiles between NEM and MCNP5 ...... 137

Appendix B Fine Group Energy Structure of COMBINE6...... 140 vi Appendix C Summary of Selected Best Results...... 142

Summary of Selected Best Results from Thermal Studies...... 142 Summary of Selected Best Results from Fast Studies...... 146 Summary of Selected Best Results from Popular Group Structures ...... 149

Appendix D Popular Energy Group Structures ...... 158

D.1 Gulf General Atomic Fort Saint Vrain (FSV) ...... 158 D.2 HRB 13 Group Structure ...... 161 D.3 MICROX ...... 164 D.4 JAERI Group Structure Sensitivity on Xenon Oscillation Analysis ...... 166 D.5 Group Structure Proposed by Originators of VSOP...... 168 D.6 Custom A 1000K ...... 171

vii LIST OF FIGURES

Figure 1-1: Design Schematic of Pebble Bed Modular Reactor [19] ...... 3

Figure 2-1: General Reactor Design ...... 5

Figure 2-2: Capture and Fast Fission Cross Section of 238U...... 7

Figure 2-3: A Generic Homogeneous Composition Reactor...... 8

Figure 2-4: A Generic Heterogeneous Composition Reactor...... 8

Figure 4-1: Summary of Cross Section Behavior Physical Considerations [6]...... 14

Figure 4-2: Fort Saint Vrain Fast Group Structure and Fast Spectrum[13]...... 15

Figure 4-3: Fort Saint Vrain Group Structure Spectrum at Several Burnups within Epithermal Energy Range[13]...... 15

Figure 4-4: The Fort Saint Vrain 4, 7, and 9 Group Structures [12]...... 18

Figure 4-5: Fort Saint Vrain Group Structure Spectrum at Several Burnups within Epithermal Energy Range [13]...... 19

Figure 4-6: Fast Group Structure for FSV 9-Group Structure and Summary of Selection Basis...... 19

Figure 4-7: Thorium Cycle [12]...... 20

Figure 4-8: Low Enriched U Cycle [12]...... 20

Figure 4-9: Th-232 Total Cross Section at 300K from ENDF/B-6.0 mat9040 [9].....21

Figure 4-10: U-238 Total Cross Section at 300K from ENDF/B-6.2 mat9237 [9] ....21

Figure 4-11: Average Radial Power Distribution for Peach Bottom HTGR and Effect of the Number of Thermal Groups on Power Distribution [12] ...... 23

Figure 4-12: Plutonium Isotopes and U-235 Cross Sections in the Thermal Region, GCR Thermal Spectrums and the Fort Saint Vrain Thermal Energy Group Structure [12]...... 23

Figure 4-13: Low-Energy Cross Section Behaviour of Several Important Isotopes[6] ...... 25

Figure 5-1: Energy Group Structure Study Comparison Methodology...... 27 viii Figure 5-2: OECD PBMR Benchmark Model (R-Z View) with 190 Material Sets..28

Figure 5-3: Simplified 4 Material Sets (R-Z View)...... 29

Figure 6-1: : Ratio of NEM / MCNP Fluxes on a Node by Node Basis (R-Z geometry – at the top, bottom and right boundaries of the model the zero flux boundary conditions are applied; the blue region of the upper right coincides with the approximate location of the Control Rod)...... 34

Figure 8-1: 2-Group k-effective’s at 1000K with no up-scattering ...... 38

Figure 8-2: 2-Group k-effective’s at 300K with no up-scattering ...... 38

Figure 8-3: 2-Group Average Radial Power Shape Percent Difference at 1000K ...... 39

Figure 8-4: 2-Group Average Radial Flux Shape Percent Difference at 1000K ...... 39

Figure 8-5: 2-Group Average Radial Power Shape Percent Difference at 300K ...... 39

Figure 8-6: 2-Group Average Radial Flux Shape Percent Difference at 300K ...... 39

Figure 8-7: Expected Thermal Spectrum at 300K and 1000K ...... 40

Figure 8-8: 2-Group Radial Power Profile at 1000K...... 40

Figure 8-9: 2-Group Radial Power Profile at 300K...... 40

Figure 8-10: 2-Group Radial Power Percent Difference at 1000K...... 41

Figure 8-11: 2-Group Radial Power Percent Difference at 300K...... 41

Figure 8-12: 2-Group k-effective for Cases at 1000K with up-scattering...... 42

Figure 8-13: 2-Group k-effective for Cases at 300K with up-scattering...... 42

Figure 8-14: 2-Group Improvement with Up-scattering for Percent Differences in Radial Power at 1000K...... 43

Figure 8-15: 2-Group Improvement with Up-scattering for Percent Differences in Radial Power at 300K...... 43

Figure 8-16: 2-Group Improvement with Up-scattering for Percent Differences in Radial Flux at 1000K...... 43

Figure 8-17: 2-Group Improvement with Up-scattering for Percent Differences in Radial Flux at 300K...... 43 ix Figure 8-18: 2-Group Improvement with Up-scattering for Percent Differences in Axial Power at 1000K ...... 43

Figure 8-19: 2-Group Improvement with Up-scattering for Percent Differences in Axial Power at 300K ...... 43

Figure 8-20: 2-Group Improvement with Up-scattering for Percent Differences in Axial Flux at 1000K ...... 44

Figure 8-21: 2-Group Improvement with Up-scattering for Percent Differences in Axial Flux at 300K ...... 44

Figure 8-22: Upper Energy Boundaries of 9 Group Energy Structure ...... 44

Figure 8-23: k-effective’s for 9-Group Thermal Cut-off at 1000K ...... 46

Figure 8-24: k-effective’s for 9-Group Thermal Cut-off at 300K ...... 46

Figure 8-25: Average Radial Power Shape Percent Difference at 1000K...... 46

Figure 8-26: Average Radial Power Shape Percent Difference at 300K...... 46

Figure 8-27: Radial Power Percent Difference at 1000K ...... 46

Figure 8-28: Radial Power Percent Difference at 300K ...... 46

Figure 8-29: 3 Group Spectrum Containment k-effective at 1000K ...... 49

Figure 8-30: 3 Group Spectrum Containment k-effective at 300K ...... 49

Figure 8-31: 3-Group Fission Source Containment Average Radial Power Shape Percent Difference at 1000K ...... 50

Figure 8-32: 3-Group Fission Source Containment Average Radial Power Shape Percent Difference at 300K ...... 50

Figure 8-33: 3 Group Spectrum Containment Radial Power Percent Difference at 1000K ...... 50

Figure 8-34: 3 Group Spectrum Containment Radial Power Percent Difference at 300K ...... 50

Figure 8-35: 3 Group Spectrum Containment Fission Source Fraction per Energy Group ...... 50

Figure 8-36: Average Radial Flux Shape Percent Difference at 1000K...... 51 x Figure 8-37: Average Radial Flux Shape Percent Difference at 296K...... 51

Figure 8-38: The Best and Worst Radial Power Percent Difference at 300K for all possible combinations of thermal cut-off with the variation of the fast group’s lower energy boundary...... 52

Figure 8-39: k-effective’s of (166_146_101_X_12)...... 55

Figure 8-40: Low Lying Resonances...... 55

Figure 8-41: Average Radial Power Shape Percent Difference of (166_146_101_X_12)...... 56

Figure 8-42: Average Radial Flux Shape Percent Difference of (166_146_101_X_12)...... 56

Figure 8-43: Average Axial Power Shape Percent Difference of (166_146_101_X_12)...... 56

Figure 8-44: Average Axial Flux Shape Percent Difference of (166_146_101_X_12)...... 56

Figure 8-45: Np, U, & Pu isotopes in 0.1-3.0 eV ...... 56

Figure 8-46: Legend for Figure 8-47, Figure 8-48, Figure 8-50...... 57

Figure 8-47: Radial Power Profiles of (166_146_101_X_12)...... 57

Figure 8-48: Axial Power Profiles of (166_146_101_X_12) ...... 57

Figure 8-49: Best and Worst Radial Power Percent Difference of (166_146_101_X_12)...... 58

Figure 8-50: Axial Power Percent Difference of (166_146_101_X_12)...... 58

Figure 8-51: k-effective of (166_146_101_12_X) ...... 58

Figure 8-52: Radial Power Percent Difference of (166_146_101_12_X)...... 58

Figure 8-53: U-235 Fission and Total Cross Section...... 60

Figure 8-54: k-effective’s of (166_145_128_101_X)...... 61

Figure 8-55: Comparison of k-effectives from (166_146_101_X_12) and (166_145_128_101_X)...... 61 xi Figure 8-56: Average Radial Power Shape Percent Difference of (166_145_128_101_X)...... 62

Figure 8-57: Average Axial Power Shape Percent Difference of (166_145_128_101_X)...... 62

Figure 8-58: Average Radial Flux Shape Percent Difference of (166_145_128_101_X)...... 62

Figure 8-59: Average Axial Flux Shape Percent Difference of (166_145_128_101_X)...... 62

Figure 8-60: k-effective of (166_145_128_101_|_46_|)...... 65

Figure 8-61: Radial Power Percent Difference of (166_145_128_101_|_46_|) ...... 65

Figure 8-62: Average Radial Power Shape Percent Difference of (166_145_128_101_|_46_|)...... 65

Figure 8-63: Average Radial Flux Shape Percent Difference of (166_145_128_101_|_46_|)...... 65

Figure 8-64: Average Axial Power Shape Percent Difference of (166_145_128_101_|_46_|)...... 65

Figure 8-65: Average Axial Flux Shape Percent Difference of (166_145_128_101_|_46_|)...... 65

Figure 8-66: k-effective of (166_145_128_101|46|23|)*...... 67

Figure 8-67: Deviations of K-eff from reference of 5 PCM or less for (166_145_128_101|46|23|) ...... 67

Figure 8-68: Average Radial Power Shape Percent Difference of (166_145_128_101|46|23|)* ...... 67

Figure 8-69: Average Axial Power Shape Percent Difference of (166_145_128_101|46|23|)* ...... 67

Figure 8-70: Average Radial Flux Shape Percent Difference of (166_145_128_101|46|23|)* ...... 67

Figure 8-71: Average Axial Flux Shape Percent Difference of (166_145_128_101|46|23|)* ...... 67

Figure 8-72: Radial Power Percent Difference of (166_145_128_101|46|23|) ...... 68 xii Figure 8-73: k-effective of (166_145_128_101|91|46|23|) ...... 70

Figure 8-74: k-effective of All Thermal Studies...... 70

Figure 8-75: Average Radial Power Shape Percent Difference of (166_145_128_101|91|46|23|) ...... 70

Figure 8-76: Average Radial Flux Shape Percent Difference of (166_145_128_101|91|46|23|) ...... 70

Figure 8-77: Average Axial Power Shape Percent Difference of (166_145_128_101|91|46|23|) ...... 70

Figure 8-78: Average Axial Flux Shape Percent Difference of (166_145_128_101|91|46|23|) ...... 70

Figure 8-79: Radial Power Percent Difference Worst and Best of (166_145_128_101|91|46|23|) ...... 72

Figure 8-80: K-eff results for Preliminary Fast Group Study (See Figure 8-81)...... 74

Figure 8-81: Preliminary Fast Group Study Abbreviations...... 74

Figure 8-82: k-eff results with variations of fast group structure (166_145_128_101_91_46_23)...... 74

Figure 8-83: Average Radial Power Shape Percent Difference for Preliminary Fast Group Study ...... 75

Figure 8-84: Radial Power Percent Difference for Preliminary Fast Group Study .....75

Figure 8-85: k-effective results of (166_X_101_91_46_23)...... 78

Figure 8-86: k-effective of (166_X_101_91_46_23) ...... 78

Figure 8-87: COMBINE6 Fission Source Fraction Per Energy Group of (166_X_101_91_46_23)...... 78

Figure 8-88: Average Radial Power Shape Percent Difference of (166_X_101_91_46_23)...... 79

Figure 8-89: Average Radial Flux Shape Percent Difference of (166_X_101_91_46_23)...... 79

Figure 8-90: Average Axial Power Shape Percent Difference of (166_X_101_91_46_23)...... 79 xiii Figure 8-91: Average Axial Flux Shape Percent Difference of (166_X_101_91_46_23)...... 79

Figure 8-92: Radial Power Percent Difference of (166_X_101_91_46_23); Best: 135-146 and Worst: 161-165...... 80

Figure 8-93: Axial Flux Percent Difference of (166_X_101_91_46_23) without Fuel-to-Moderator Interfacial Region Data...... 80

Figure 8-94: k-effective of (166|146|101_91_46_ 23)...... 81

Figure 8-95: K-effective Deviations less than 20 PCM from Reference and the base group structure at 1000K of (166|146|101_91_46_ 23) ...... 81

Figure 8-96: Average Radial Power Shape Percent Difference of (166|146|101_91_46_ 23) ...... 82

Figure 8-97: Average Radial Flux Shape Percent Difference of (166|146|101_91_46_ 23) ...... 82

Figure 8-98: Average Axial Power Shape Percent Difference of (166|146|101_91_46_ 23) ...... 82

Figure 8-99: Average Axial Flux Shape Percent Difference of (166|146|101_91_46_ 23) ...... 82

Figure 8-100: U-235 Fission and Total Cross Section; the fission cross section has a marked change at approximately 2 keV...... 83

Figure 8-101: U-238 Fast Fission Cross Sections and Total Absorption ...... 84

Figure 8-102: Natural Carbon’s cross section ...... 85

Figure 8-103: k-effective results of this case (166|146|101 91 46 23) and the comparison with (166|141|101 91 46 23) results...... 88

Figure 8-104: Radial Power Percent Difference for (166|141|101 91 46 23)...... 88

Figure 8-105: Radial Flux Profile Percent Difference for (166|141|101 91 46 23).....88

Figure 8-106: Radial Power Profile Percent Difference for (166|141|101 91 46 23)...... 88

Figure 8-107: Axial Flux Profile Percent Difference for (166|141|101 91 46 23) ...... 88

Figure 8-108: Axial Power Profile Percent Difference for (166|141|101 91 46 23)....89 xiv Figure 8-109: U-238 absorbtion cross section showing its transition from smooth to unresolved resonance region at approximately 10 keV...... 89

Figure 8-110: k-effective for (166|146|128|101 91 46 23)...... 92

Figure 8-111: Natural Carbon’s Elastic Scattering Cross Section for (166|146|128|101 91 46 23) ...... 91

Figure 8-112: Average Radial Power Shape Percent Difference for (166|146|128|101 91 46 23) ...... 92

Figure 8-113: Average Radial Flux Shape Percent Difference for (166|146|128|101 91 46 23) ...... 92

Figure 8-114: Average Axial Power Shape Percent Difference for (166|146|128|101 91 46 23) ...... 92

Figure 8-115: Average Axial Flux Shape Percent Difference for (166|146|128|101 91 46 23)...... 92

Figure 8-116: Radial Power Percent Difference for (166|146|128|101 91 46 23) ...... 92

Figure 8-117: k-eff comparison between the insertion of fixed partition 134 of (166 141 134 101 91 46 23)*...... 95

Figure 8-118: Average Radial Power Shape Percent Difference for (166 141 134 101 91 46 23)*...... 95

Figure 8-119: k-effective for (166|152|141|134|128|101 91 46 23)...... 97

Figure 8-120: k-effective results with Comparison to previous group structures ...... 97

Figure 8-121: Average Radial Power Shape Percent Difference for (166|152|141|134|128|101 91 46 23)...... 97

Figure 8-122: Average Radial Flux Shape Percent Difference for (166|152|141|134|128|101 91 46 23)...... 97

Figure 8-123: Radial Power Percent Difference for (166|152|141|134|128|101 91 46 23)...... 97

Figure 8-124: Radial Power Figure 8-of Merit for (166|152|141|134|128|101 91 46 23)...... 97

Figure 8-125: U-238 T=300 K from ENDF/B-6.2 mat9237 Total Cross Section...... 98

Figure 8-126: Fast Region Study using fixed ∆ Lethargy...... 100 xv Figure 8-127: k-effective results of Constant Lethargy Study*...... 101

Figure 8-128: Average Radial Power Shape Percent Difference of Constant Lethargy Study *...... 101

Figure 8-129: Radial Power Profile of Constant Lethargy Study...... 102

Figure 8-130: k-effective ...... 104

Figure 8-131: Energy Group Structures of Max Groups Case Study (Group 11 – Group 23) in COMBINE6 Upper Energy Numbers...... 104

Figure 8-132: Energy Group Structures of Max Groups Case Study (Group 31 – Group 71) in COMBINE6 Input Fine-Group Structure ...... 105

Figure 8-133: Average Radial Power Shape Percent Difference of Max Groups Case Study ...... 106

Figure 8-134: Radial Power Percent Difference...... 106

Figure 8-135: Radial Power Profile Comparison of Selected Group Numbers of Max Groups Case Study ...... 106

Figure 8-136: of LWR and Modifications for HTGR...... 109

Figure 8-137: Average Radial Power Shape Percent Difference of LWR and Modifications for HTGR ...... 109

Figure 8-138: Average Radial Flux Shape Percent Difference of LWR and Modifications for HTGR ...... 109

Figure 8-139: Average Axial Power Shape Percent Difference of LWR and Modifications for HTGR ...... 110

Figure 8-140: Average Radial Flux Shape Percent Difference of LWR and Modifications for HTGR ...... 110

Figure 8-141: of LWR and Modifications for HTGR...... 110

Figure 8-142: of LWR and Modifications for HTGR...... 111

Figure 8-143: k-eff of Best Performing 5-Group Cases ...... 113

Figure 8-144: Percent Difference of Radial Power Profile of Best Performing 5- Group Cases...... 113 xvi Figure 8-145: Percent Difference in Axial Profile of Best Performing 5-Group Cases...... 114

Figure 8-146: k-eff results of Best Performing 6-Group Cases...... 116

Figure 8-147: Percent Difference of Radial Power Profile of Best Performing 6- Group Cases...... 117

Figure 8-148: Percent Difference of Axial Power Profile of Best Performing 6- Group Cases...... 117

Figure 8-149: k-eff results of Best Performing 7-Group Cases...... 119

Figure 8-150: Percent Difference of Radial Power Profile of Best Performing 7- Group Cases...... 119

Figure 8-151: Percent Difference of Axial Power Profile of Best Performing 7- Group Cases...... 120

Figure 8-152: k-eff results of Best Performing 8-Group Cases...... 122

Figure 8-153: Percent Difference of Radial Power Profile of Best Performing 8- Group Cases...... 122

Figure 8-154: Percent Difference of Axial Power Profile of Best Performing 8- Group Cases...... 123

Figure 8-155: k-eff results of Best Performing 9 or More Group Cases ...... 125

Figure 8-156: Percent Difference of Radial Power Profile of Best Performing 9 or More Group Cases ...... 125

Figure 8-157: Percent Difference of Axial Power Profile of Best Performing 9 or More Group Cases ...... 126

Figure A-1: Initial Method: Absolute Radial Flux Comparison*...... 137

Figure A-2: Final Method: Normalized Radial Flux Comparison...... 138 xvii LIST OF TABLES

Table 5-1: Isotopes and Number Densities in the Fuel Material ...... 31

Table 8-1: Evaluation of k-effective in 3 Group Spectrum Containment at 1000K....49

Table 8-2: Evaluation of k-effective in 3 Group Spectrum Containment at 300K...... 49

Table 8-3: Thermal Study 1 Cases with Percent Differences of k-effective within ±0.2%...... 55

Table 8-4: Thermal Study 2 Cases with Percent Differences of k-effective within ±0.4%...... 61

The results of this case are shown in Figures 8-60 through 8-65 and in Table 8-5.....63

Table 8-5: k-effective within ±0.09% Percent of (166_145_128_101_|_46_|) ...... 64

Table 8-6: (166_145_128_101|91|46|23|) k-effective of cases of 30 pcm...... 71

Table 8-7: k-effective Results of less than 100 pcm for (166|141|101 91 46 23)...... 87

Table 8-8: k-eff with less than 50 PCM difference (166|146|128|101 91 46 23) ...... 91

Table 8-9: Cases with k-eff less than 50 PCM for (166 141 134 101 91 46 23) ...... 94

Table 8-10: k-eff with less than 50 pcm for (166|152|141|134|128|101 91 46 23).....96

Table 8-11: k-eff of Constant Lethargy Study...... 100

Table 8-12: : k-eff of Maximum Number of Energy Groups Case Study ...... 103

Table 8-14: k-effective results of LWR and Modifications for HTGR ...... 108

Table 8-15: Best Performing 5-Group Energy Structures ...... 112

Table 8-16:k-eff results of Best Performing 5-Group Cases ...... 112

Table 8-17: Table 8-1: Energy Structure of Best Performing 6-Group Cases...... 115

Table 8-18: Table 8-2: k-eff results of Best Performing 6-Group Cases...... 116

Table 8-19: Table 8-3: Group Structures of Best Performing 7-Group Cases ...... 118

Table 8-20: k-eff results of Best Performing 7-Group Cases ...... 118

Table 8-21: Group Structures of Best Performing 8-Group Cases...... 121 xviii Table 8-22: k-eff results of Best Performing 8-Group Cases ...... 121

Table 8-23: k-eff results of Best Performing 9 or more Group Cases...... 124

Table 8-24: k-eff results of Best Performing 9 or More Group Cases...... 124

Table 9-1: Recommended Group Structures for the PBMR...... 131

Table 9-2: Physical Explanations for Recommended Group Structures of the PBMR (Groups 5 – 7)...... 131

Table 9-3: Physical Explanations for Recommended Group Structures ...... 132 of the PBMR (Groups 8 & 10)...... 132

xix ACKNOWLEDGEMENTS

I would like to thank my advisor, Professor Kostadin N. Ivanov, for providing me with

the great opportunity to do research in reactor physics and also for his encouraging advice and moral support.

I would also like to thank the Dr. Hans Gougar and Dr. Abderrafi Ougouag of Idaho

National Laboratory for providing funding and valuable technical guidance.

I would also like to recognize the National Academy for Nuclear Training, the

Mechanical & Nuclear Engineering Department of the Pennsylvania State University,

and the members of the Reactor Dynamics & Fuel Management Research Group for their

support and research assistance. 1

Chapter 1

Introduction to Gas Cooled Reactors

The history of gas-cooled graphite moderated reactors (GCR) originates in United Kingdom with the Magnox. Magnox reactors are moderated with graphite, fueled with natural uranium, and cooled with CO2. Magnox is an acronym of the cladding material used in this reactor design; the cladding material is MAGnesium Non-OXidizable alloy. Because Magnox reactors were originally designed for the production of plutonium for military purposes, its design was not suitable for producing low cost electricity. In order to address the needs for low cost electricity, the Magnox design was revised into what a design known as the Advanced Gas-cooled Reactor (AGR). The

AGR used enriched UO2 as fuel, stainless steel as its cladding material, and CO2 as its coolant. The notable contrasting features of the AGR with the Magnox are its smaller core and its ability to be loaded with enriched fuel instead of natural fuel. The advantage of using enriched fuel is the reduction in frequency of refueling and therefore reduces the costs associated with frequent shutdowns for refueling. The AGR was also designed to output higher coolant temperatures than the Magnox and this feature provided better thermal efficiency. The next evolutionary stage in GCR technology is the High Temperature Gas- cooled Reactor (HTGR). Its main features in contrast to its predecessor designs is in its design for a higher outlet temperature primarily by removing any metal in the core so as to avoid melting and also in its design to use a more chemically inert coolant, helium. The first steps toward HTGR development began with fuel development at the Dragon reactor. This was followed by demonstration plants of Peach Bottom in the US and the AVR (Arbeitsgemeinschaft Versuchs Reaktor) in Germany. Following successful operation and experimental testing, the next stage was economic feasibility for commercial use. For this stage, the US developed the Fort Saint Vrain (FSV), while Germany developed the THTR (Thorium Hoch-Temperatur Reaktor).

2 Both plants were successfully brought online. However both plants faced many technical issues as expected with many first of a kind reactors. The issues affecting the FSV reactor ranged from power fluctuations, water ingress, control rod jams, and stress hairline fractures. The plant maintenance became too costly to operate and was closed in 1989. The THTR was also deactivated due to its cost and the increased public scrutiny and disapproval of nuclear technologies. A notable accident that occurred with the THTR reactor during its operations was a fuel pellet that became lodged in a fuel feed pipe to the core. The THTR closed after only 3 years of operation. Despite the technical difficulties in GCR technology, the promise of inherent safety features and high outlet temperatures has encouraged the further development of HTGR technology.

3

Place Figure Here Figure 1-1: Design Schematic of Pebble Bed Modular Reactor [19]

One of the designs that emerged following the HTGR experience is the Pebble Bed Modular Reactor, shown in Figure 1-2, which is currently in progress of being built in South Africa. The PBMR is based on the AVR and comprises of a steel pressure vessel, helium coolant, graphite moderation, 8-13% enriched uranium, and a direct Brayton cycle. The PBMR design and a popular method of neutronic analysis is the focus of this thesis.

Chapter 2

Nuclear Engineering and the Fundamentals of Reactor Physics

Today’s growing concern of the uncertain consequences of global warming and the economic instability associated with the international supply of petroleum resources, nuclear power is an ever increasingly important source of energy. This importance is due mainly because nuclear technologies provide a consistent, predictable, and viable energy source for today’s electricity demands. Nuclear engineering in reactor physics is primarily a study of reactor design based on a focus on safety, efficiency, and economics. Since the basic physics of nuclear engineering is sufficiently known, the challenge in nuclear engineering is to solve the open ended design problem and to achieve maximum optimization in terms of safety and economic efficiency. A general reactor schematic is shown in Figure 2-1. A reactor is a term popularly used in both chemical and nuclear engineering. It is basically a system (closed or open) where reactions occur. Nuclear reactions are reactions based on the interactions of the atom’s nucleus where chemical reactions are based on the electron fields surrounding the atom. The commercial use of nuclear reactors is typically for the production of heat. The heat is transferred to a fluid which in turn is used to power a turbine to produce electricity. The most common types of nuclear reactors consist of the following components:

5

• a moveable control rod • a fuel source • a moderator • cladding material • coolant • reflector • containment or shielding

Figure 2-1: General Reactor Design

The following is a list of nuclear material types, typically found in a reactor.: • The fuel source is a material of high density which has desirable nuclear properties. The usual desirable nuclear property is a high ratio of fission to absorption. Uranium-235 is typically a choice for the conventional reactor. • Moderators are materials that slow down neutrons. They are only present in thermal reactors because thermal reactors are designed to have neutron reactions in the lower energy range, which is called the thermal energy range. Since their main purpose is only to slow down neutrons, they are materials that typically are chosen to have low probabilities of neutron absorption. An commonly used moderator is graphite (carbon) or water. • Cladding is a material that serves as a sleeve that contains the fuel. Its purpose is to prevent the release of the fuel or fission products and isolate these materials from being other components such as the coolant or the moderator. Zirconium is a typical choice of cladding due to its low absorption cross section. • Coolant is a fluid (liquid or gas) that removes the heat that is generated in the fuel and carries it away to do work in the turbine.

6 • The control rod is a material that can absorb neutrons. The absorption of neutrons is its most important feature since it serves to balance the gain and losses of neutrons and sustain a chain reaction. • The reflector is the material that reflects neutrons back into the reactor, which has purposes of preserving neutron economy. • A blanket, which is not shown in the figure, is a material placed to absorb neutrons in order to convert an element into another. It is usually only in reactor designs that intend to use escaping neutrons to transmute certain materials. • Containment or shielding is self explanatory as it is used to contain material inside a system and shield the outside environment from radiation exposure. The following is a list of key terms typically used in nuclear engineering: • The conversion of materials is often referred to as transmutation. • Another term frequently used in nuclear engineering is burning. It usually does not actually refer an oxidation process, but refers to a consumption of a fuel. It is likely to be a term derived from other fields such as in conventional energy fields like coal, wood, or gas, where its fuels are actually “burning.” • Thermal is an energy range typically referring to energy ranges less than 3 eV where the room of the temperature affects the energy of the neutron. Merriam- Webster’s dictionary defines a thermal neutron as a neutron “having low energies of the order of those due to thermal agitation.” Often the parameters of interest for the reactor physicist are the k-effective and the space and energy dependent flux. These parameters help determine the balance of neutron production, losses, and in general, the rate of reactions. The rate of reactions in turn approximate heat generation from the energy released per fission. The reactions also determine conversion of material or utilization of fuel, which is commonly referred to as burn-up. The k-effective is a typical number in diffusion theory that represents the number of neutrons in the new generation of reactions versus the old generation. At steady state operation, the reactor has achieved a constant chain reaction and therefore a constant yielding heat source; a k-effective of one is achieved in steady-state.

7 One common approach utilized to predict the flux and k-effective is to use approximate methods such as diffusion theory. Diffusion theory uses the assumption that neutrons act like ideal gases particles and that they diffuse around the reactor. This theory is known to be generally correct, but is inaccurate near large neutron densities gradients where the assumption of weakly anisotropic neutron scattering is not valid. Regions where the assumption and use of diffusion are poor are near the fuel source, the fuel-to-moderator region, the boundaries, and near control rods. An important variable that needs to be accounted for is neutron energy. Shown in Figure 2-2 is the capture and fast fission cross section of 238U. A cross section is a reaction probability parameter. As shown, it is highly dependent on the energy of the incoming neutron.

Figure 2-2: Capture and Fast Fission Cross Section of 238U

8 The two most important variables in reactor physics are the selection of material and the geometry of the reactor design. Since every material has different properties of reactions, changing the composition of the core clearly alters the chain reaction process. The dependence on geometry is less obvious. For example, when a reactor is composed of materials of the exact composition, but are placed in different configurations, the k-effective and flux solutions are not the same. Figure 2-3 and Figure 2-4 represent a homogeneous reactor and a heterogeneous reactor of the same total material composition. Despite having exactly the same material composition, the resulting solutions would not be the same. This resulting difference is a consequence of the concept of spatial and energy self-shielding.

Figure 2-3: A Generic Homogeneous Composition Reactor

Figure 2-4: A Generic Heterogeneous Composition Reactor

9 Chapter 3

Research Goals and Motivation

As discussed in Chapter 2 the flux solution of reactor physics calculations energy and space dependent. The issue of space and energy are two independent variables. A continuous solution of a complex reactor design is computationally expensive; numerical approximations must be made for practical calculations in deterministic methods. The history of experience in reactor has established spatial and energy discretizations of certain reactor types that have been proven to produce results of reasonably accuracy. For example, the PWR is typically discretized on a pin-by-pin basis for each assembly and then on an assembly basis for the entire core. Such a method has supporting arguments in physics such as average mean free path, but more importantly calculations in the past have shown these methods to arrive at accurate solutions. Energy discretization which is important for knowing how to group the cross section terms to be able to use discrete or finite representative cross sections for energy ranges have also been previously established by experience. The well known PWR has been historically analyzed using two groups where the energy group boundary is set at 0.625 eV. This thesis explores, numerically, the dependence of energy group structures on the analysis of the PBMR. As discussed above, another problem of spatial discretization exists. Because the PBMR has spherical fuel elements embedded within a graphite matrix instead of fuel pins and pebble zones instead of assemblies, it is difficult to spatially discretize the reactor into regions as traditionally performed for LWR’s. For this study, an initial assumption is made that a general spatial discretization chosen and used in the OECD PBMR 400 Transient Benchmark [19] model is sufficient to obtain preliminary findings and dependencies on energy group structures.

10

Motivation for Study of Group Structures

Lately, HTR designs such as the Pebble Bed Modular Reactor (PBMR) have received much interest and are being studied internationally with the use of reactor physics deterministic codes. Typically, these deterministic codes treat the energy dependence by using multi-group theory with few groups (1 – 5) or many groups (6 - 20). Because the selection of broad group energy structures is unique for a reactor type[4] and can strongly affect accuracy and computational time of these reactor calculations, there is a need to find optimal broad group structures for the PBMR. While some texts and articles in public domain have discussed the basic approach to broad group energy structure selection[13], none have explicitly applied this methodology to the PBMR. Neutron energy group structures are usually selected by separating different types of nuclear reactions or other significant phenomena. This separation of groups by reaction type is beneficial in two ways. It provides better info on the effect of different reactions, and improves accuracy of modeling. Group selection has been performed to distinguish a range for the fission source, a range of unresolved and resolved resonances, and thermal energy range. For the case of high Plutonium (Pu) loading, it may be important to emphasize the low lying Pu resonances[13]. The approach of this study is to analyze group structures by comparing deterministic reactor calculations with selected group structures against a continuous energy based solution. This continuous energy solution is obtained by MCNP5[21]. The computational tools used in these investigations are COMBINE6[8] (cross section generation code), NEM[1] (3-D multi-group diffusion code), and MCNP5 (continuous energy Monte Carlo code), which is used to generate a reference solution. The key parameters for comparison are the k-effective, power, and flux distributions. The goal of this study is to arrive at optimal group structures by reviewing literature and applying the traditional approach of selecting energy group structure selection in conjunction with a brute-force numerical approach to find the minimum difference in results when compared with a reference continuous energy calculation. The

11 numerical studies will also benefit in developing a basis of understanding the effects of energy group structures. The calculations in this study are strictly with steady state analysis, but one might infer possible uses for transient analysis.

Chapter 4

Theory: Selection Methodology & Review of Literature

4.1 Group Structure General Theoretical Considerations

In theory, the best energy group structure should be the one that groups neutrons into energy regions of similar characteristics for significant reactions. By grouping neutrons in this manner, the calculated results are expected to be close to the true physical solution. Therefore, group structures that are selected by dominant physical reactions within an energy range should provide the best group structure. The challenge then becomes finding the correct energy regions and divisions (cut-ff points or boundaries) for specific material compositions.

4.2 General Guidelines for Energy Group Structures from Literature

The following is a list of general guidelines for energy group selection found in literature: • Creating energy partitions near marked changes in Cross Section. Authors Koclas[10] and Bell & Glasstone[2] recommend selecting group boundaries that correspond to neutron energies where cross sections of important isotopes, such as fissile isotopes, undergo a marked change. • Selection of group structures based on dominating physics of particular energy regions. o The condition for retaining the true k-eff (or some suitable characterisitic) of the particular reactor can be maintained with group-averaged values by retaining the essential physics typifying each group of neutrons[4]. o D&H[6] states that it is customary to divide the range of neutron energies into three general regions, each characterized by different types of interactions. The regions are described as follows:

13 ƒ Neutron Thermalization Region (approximately 0 eV to 1 eV) • Upscattering • Chemical Binding • Diffraction ƒ Neutron Moderation or Slowing Down Region (approx. 1 eV to 105 eV) • Elastic Scattering • No Upscattering • Resolved Resonance Absorption ƒ Fast Fission Region (approximately 105 eV to 107 eV) • Elastic Scattering • Inelastic Scattering • No Upscattering • Unresolved Resonance Absorption • Fission Sources o A summary of cross section behavior physical considerations referenced in D&H is reproduced in Figure 4-1. o Gulf General Atomic recommends dividing energy regions according to the type of nuclear reactions which occur in particular energy ranges. These include division of the resolved and unresolved energy ranges of the most important fertile and fissile materials. In addition, groups to contain the fission source, fast fission, inelastic thresholds, and large thermal energy resonances are recommended. o Glasstone and Sesonske[7] discusses a 4-Group structure for water moderated reactors. The basis of the 4-Group structure is as follows: ƒ The fast region (10 MeV to 0.05 MeV) • Essentially all of the fission neutrons have energies in this group. • Slowing down can occur by both inelasic and elastic scattering. ƒ Slowing-down region (0.05 MeV – 0.5 keV) • Slowing down by elastic scattering

14 • Absorption by unresolved resonances is possible. ƒ Resonance region (500 eV – 0.625 eV) • Slowing down occurs by elastic scattering. • Absorption by resolved resonances. ƒ Thermal Region (0.625 eV – 0) • Both down-scattering and up-scattering can occur. • Absorption can also occur. o Glasstone and Sesonske state that graphite-moderated reactors may require more than 4 groups and that there are at least two thermal regions up to a few eV.

Figure 4-1: Summary of Cross Section Behavior Physical Considerations [6]

• Spectrum based considerations for energy groups While Gulf General Atomic’s energy group structure for the Fort Saint Vrain is justified on the basis of marked changes of cross-sections of important isotopes and the isolation of important physics, it is also justified by the neutron spectrum as shown in Figure 4-2 for steady state and Figure 4-3 for different burn ups. Figure 4-2 shows that correlation

15 exists between a proper selected group structure and the fast spectrum. Figure 4-3 shows that the group structure in the epithermal region should pick up changes in spectrum which are caused by burnup. D&H also supports a spectrum based group structure emphasizing that “neutron energy spectrum is the key to the generation of group constants that yeild an accurate few-group description of nuclear reactor behavior.”

Figure 4-2: Fort Saint Vrain Fast Group Structure and Fast Spectrum[13]

Figure 4-3: Fort Saint Vrain Group Structure Spectrum at Several Burnups within Epithermal Energy Range[13]

16 • Energy regions of importance for reactor types.

Author Yigal Ronen[20] recommends that energy partitions should be placed in energy regions of importance where the importance can be measured based upon the magnitude of the neutron population in that energy range.

17

4.3 Gulf General Atomic’s Neutron Energy Group Structure for HTGR Analysis[13]

Of the literature documents found on group structure methodology, the Gulf General Atomic (GGA) report on nuclear design methods states was the most detailed. It is cited as a reference by both Massimo and D&H. GGA states that the choice of broad group energy boundaries is a matter involving judgement and some trial and error experimentation.

Gulf General Atomic also comments on spectral zones stating that few group (generally 1 -4 groups) calculations can accurately produce results for region-dependent cross sections, while many groups with no region-dependency (typically 6 -20 groups) will adjust itself sufficiently in the different regions of the spatial calculational model to accomplish an effective re-averaging of the cross sections over the local spectrum. This statement remarks on the interdependence of spectral zone and energy group selection. This work is likely to support the closely related joint INL, PSU and PBMR (Pty) Ltd study on spectral zones[17].

GGA used the group structure selection methodology for the design and analysis of the Fort Saint Vrain reactor. Their approach on this reactor design will be reviewed for understanding of group structure selection methodology. The FSV group structure is shown in Figure 4-4.

18

Figure 4-4: The Fort Saint Vrain 4, 7, and 9 Group Structures [12]

There are some important differences between the PBMR and FSV so they will be briefly discussed. Both are high temperature gas cooled graphite moderated reactors. Both use TRISO coated particles. The FSV reactor is fueled with a 93% uranium enrichment where PBMR is 8- 11% enriched (9.6% according to benchmark[19]). The main fissile and fertile materials of the

FSV are UC2 and ThC2, respectively. This contrasts to PBMR design which uses UO2 to serve as both the main fissile material through U-235 and the main fertile material through U-238.

4.4 Fast Energy Group Structure Selection Basis

In GGA, the fast group (Group 1) is selected to contain nearly all of the fission source. The second and third groups isolate the unresolved and resolved resonance absorption of the principle resonance absorber, Th-232. GGA prepared its broad group scattering matrix by assuming that all neutrons upscattered from thermal go into the lowest energy broad group in the fast region. Because of this, a small group (2.38 – 3.93 eV) was created to prevent thermal neutrons from reaching energies they could not physically reach. When it was more computationally economic not to have this upscattering containment group boundary of 3.93 eV, neutrons were allowed to upscatter to 17.6 eV or less because the upscattering neutrons were below the first resonance peak for Thorium, which occurs at 21.9 eV (see Figure 4-9). This is allowable for FSV, but is not for PBMR because U-238’s first resonance occurs at 6.8 eV (see

19 Figure 4-10). Partitions between the fission source and the thermal cut-off also serve to follow the spectral hardening that occurs with burnup, which is shown in Figure 4-5.

Figure 4-5: Fort Saint Vrain Group Structure Spectrum at Several Burnups within Epithermal Energy Range [13]

In addition these partitions above the thermal cut-off also assist in more accurate downscattering to the thermal region. The physical basis of group selection for the fast energy region is summarized in Figure 4-6. It should be emphasized that the selection basis and fast group structure should not be directly applied from FSV to PBMR because the FSV design utilizes the thorium cycle (Figure 4-7) while the PBMR design utilizes the low enriched uranium cycle (Figure 4-8).

Figure 4-6: Fast Group Structure for FSV 9-Group Structure and Summary of Selection Basis

20

Figure 4-7: Thorium Cycle [12]

Figure 4-8: Low Enriched U Cycle [12]

Stamm’ler notes that for light water reactors and probably for all low enriched U cycle fuelled reactors, the 0.5 MeV limit is a good energy boundary because of U-238’s fission threshold and also benefits by being roughly the average fission energy of delayed neutrons. He also notes the importance of proper treatment of the 1.05 eV Pu-240 resonance. He also suggests that in the epithermal range from 10 keV to 0.625 eV no more than three groups are needed because the diffusion coefficient is almost constant with the exception of the low lying resonances.

21

Figure 4-9: Th-232 Total Cross Section at 300K from ENDF/B-6.0 mat9040 [9]

Figure 4-10: U-238 Total Cross Section at 300K from ENDF/B-6.2 mat9237 [9]

22

4.5 Thermal Energy Group Selection Basis

GGA states that multiple thermal groups are required to treat both the change in spectrum with burnup and spatial variation of the spectrum at the core-reflector interfaces and at the boundaries of refueling regions with the core. GGA’s thermal group structure is a four thermal group structure that was originally selected by systematically studying the variations in reactivity and power peaking as the number of groups and their boundaries were varied. When placing the thermal group structures established by GGA on a figure of important thermal cross sections, one can see the relationship of the partitions around low lying resonance peaks of important isotopes (see Figure 4-12 ).

Massimo agrees with GGA and emphasizes that for accurate power distribution in regions of strong space-dependence of the neutron spectrum such as near the fuel-to-moderator interface many thermal groups are needed. This claim is supported by Figure 4-11. Massimo also states that Doppler broadening of the low lying resoances has a very small effect and that instead of Doppler broadening of these resonances, treating them with a fine group structure is sufficient.

Stamm’ler[22] studied light water energy group structures and stated that in the thermal range, the diffusion coefficient decreases with energy and the spectrum interaction becomes confined to short distances, thus the microflux (or local flux) calculation alone should be sufficient to establish the correct condensation spectrum. This leads to an arguement that the difference between the infinite medium and actual criticality spectrum is mainly evident in the ratio of total thermal flux to other group flux’s. In addition, Stamm’ler21 states that as few as three thermal groups are sufficient for light water reactors and more would be computationally inefficient.

23

Figure 4-11: Average Radial Power Distribution for Peach Bottom HTGR and Effect of the Number of Thermal Groups on Power Distribution [12]

Figure 4-12: Plutonium Isotopes and U-235 Cross Sections in the Thermal Region, GCR Thermal Spectrums and the Fort Saint Vrain Thermal Energy Group Structure [12]

The presence of thermal groups may also be dependent primarily on whether Doppler broadening is incorporated in the calculation of the thermal energy region. The CASMO manual

24 [3] states that its resonance energy region is treated only within 4 eV – 9.118 keV. The CASMO manual states that low lying resonances such as the 1 eV Pu-240 resonance and 0.3 eV Pu-239 resonances are considered to be adequately covered simply by creating a concentration of energy boundaries around these resonances.

4.6 Thermal Cutoff Energy Selection Basis

The literature suggests that HTR’s require a high thermal cut-off for the purpose of containing up-scattering. Massimo states that the range for this boundary for HTR’s is normally between 2- 4 eV and that the typical value is 2 eV. Massimo also states that the 2 eV cut-off has a negligibly low probability of neutrons scattering above it. D&H states that the boundary may be recommended to be as high as 3 eV.

4.7 Important Isotopes in the Thermal Region

Massimo states that for HTR’s 239Pu, 240Pu, 241Pu, 135Xe, 149Sm, 103Rh are the isotopes of special importance for proper treament because they have resonances which occur in an energy range where up-scattering starts to play an important role. Figure 4-13 taken from D&H shows the low lying resonances of the aforementioned isotopes with the exception of 103Rh.

25

Figure 4-13: Low-Energy Cross Section Behaviour of Several Important Isotopes[6]

Chapter 5

Computational Methods and Models

5.1 Problem Description and Assumptions

The computational tools used in these investigations are COMBINE6 (cross section generation code), NEM (3-D multi-group nodal diffusion code), and MCNP5 (continuous energy Monte Carlo code). The cross section generation code and the diffusion code were used to generate group structure dependent results, which were then compared with the reference solution provided by MCNP5. The key parameters for comparison are the k-effective, power, and flux distributions. The comparison methodology is shown in Figure 5-1.

27

Figure 5-1: Energy Group Structure Study Comparison Methodology

In this study, a whole core calculation is performed with an r-z-θ diffusion model and a 3-D Monte Carlo model. The complication of double heterogeneity that arises due to the pebble and kernel spatial effect in modeling the PBMR is avoided by assuming material compositions of uniform homogeneous distribution. Therefore, the reference solution is produced using a spatially homogenized MCNP5 model and the cross sections are generated assuming a homogeneous medium. Both models assume a constant and uniform temperature distribution at either 296 or 1000K. The materials in the model consists of the reflector, control rod region, core barrel and fuel regions. The cross sections for broad group are generated assuming a very low buckling. A low buckling input was used because a non-positive buckling is not allowed in COMBINE6.

28 5.2 Models

The models used in this study were created by Peter Mkhabela[14][15]. The equilibrium model of the OECD PBMR-400 benchmark, shown in Figure 5-2, is the basis of our model in this study. For computational efficiency the 190 material sets that originally define the OECD Benchmark Model are simplified to 4 sets. The fuel material chosen is referenced in the benchmark as Fuel (3, 15). The simplified R-Z model used in this study is shown in Figure 5-3 . In addition to these simplifications, the void regions (colored in blue in Figure 5-2) located at the top of the core and next to the barrel are replaced with reflector material to reduce any diffusion related complications.

0 10 41 73.6 80.55 92.05 100 117 134 151 168 185 192.95 204.45 211.4 225 243.6 260.6 275 287.5 292.5 -200 10 31 32.6 6.95 11.5 7.95 17 17 17 17 17 7.95 11.5 6.95 13.6 18.6 17 14.4 12.5 5 -150 50 133 133 133 133 155 116 113 113 113 113 113 135 164 144 144 152 152 152 189 190 -100 50 133 133 133 133 155 116 113 113 113 113 113 135 164 144 144 152 152 152 189 190 -50 50 133 133 133 133 155 116 112 112 112 112 112 135 164 144 144 152 152 152 189 190 0 50 133 133 133 133 155 116 111 111 111 111 111 135 165 144 144 152 152 152 189 190 50 50 134 134 134 125 156 117 1 23 45 67 89 136 166 145 145 153 153 153 189 190 100 50 134 134 134 125 156 117 2 24 46 68 90 136 167 145 145 153 153 153 189 190 150 50 134 134 134 126 157 118 3 25 47 69 91 137 168 146 146 153 153 153 189 190 200 50 134 134 134 126 157 118 4 26 48 70 92 137 169 146 146 153 153 153 189 190 250 50 134 134 134 126 157 118 5 27 49 71 93 137 170 146 146 153 153 153 189 190 300 50 134 134 134 127 158 119 6 28 50 72 94 138 171 147 147 153 153 153 189 190 350 50 134 134 134 127 158 119 7 29 51 73 95 138 172 147 147 153 153 153 189 190 400 50 134 134 134 127 158 119 8 30 52 74 96 138 173 147 147 153 153 153 189 190 450 50 134 134 134 127 158 119 9 31 53 75 97 138 174 147 147 153 153 153 189 190 500 50 134 134 134 128 159 120 10 32 54 76 98 139 175 148 148 153 153 153 189 190 550 50 134 134 134 128 159 120 11 33 55 77 99 139 176 148 148 153 153 153 189 190 600 50 134 134 134 128 159 120 12 34 56 78 100 139 177 148 148 153 153 153 189 190 650 50 134 134 134 128 159 120 13 35 57 79 101 139 178 148 148 153 153 153 189 190 700 50 134 134 134 129 160 121 14 36 58 80 102 140 179 149 149 153 153 153 189 190 750 50 134 134 134 129 160 121 15 37 59 81 103 140 180 149 149 153 153 153 189 190 800 50 134 134 134 129 160 121 16 38 60 82 104 140 181 149 149 153 153 153 189 190 850 50 134 134 134 129 160 121 17 39 61 83 105 140 182 149 149 153 153 153 189 190 900 50 134 134 134 130 161 122 18 40 62 84 106 141 183 150 150 153 153 153 189 190 950 50 134 134 134 130 161 122 19 41 63 85 107 141 184 150 150 153 153 153 189 190 1000 50 134 134 134 130 161 122 20 42 64 86 108 141 185 150 150 153 153 153 189 190 1050 50 134 134 134 131 162 123 21 43 65 87 109 142 186 151 151 153 153 153 189 190 1100 50 134 134 134 131 162 123 22 44 66 88 110 142 187 151 151 153 153 153 189 190 1150 50 132 132 132 132 163 124 114 114 114 114 114 143 188 151 151 154 154 154 189 190 1200 50 132 132 132 132 163 124 115 115 115 115 115 143 188 151 151 154 154 154 189 190 1250 50 132 132 132 132 163 124 115 115 115 115 115 143 188 151 151 154 154 154 189 190 Figure 5-2: OECD PBMR Benchmark Model (R-Z View) with 190 Material Sets

29

Figure 5-3: Simplified 4 Material Sets (R-Z View)

5.3 Model Assumptions & Errors

5.3.1 Low Buckling Assumption

Possibly the biggest assumption in this study is the assumption of a low buckling value used to generate cross sections in COMBINE6. Buckling may be interpreted as the second derivative of the flux and therefore the concavity of its flux shape. So the use of a low buckling value or near zero value would be similar to assuming that the cell of calculation is surrounded by identical cells such as itself or in other words an infinite medium of exactly the same type of cells. Therefore, our low bucking assumption loses the feedback due to regional leakage dependency, which is also known as environment effects. The lack of environmental effects is likely to have a significant effect on producing the cross sections.

30 5.3.2 20 Isotope Limitation Per Material

While the OECD PBMR Benchmark has information on 71 isotopes in the fuel material, only 20 materials were modeled due to the 20 isotope limitation per cross section material in COMBINE6. The isotopes that were included in the fuel material are shown in Table 5-1. The isotopes Pm-149, I-135, and B-10 are marked with an asterik due to inconsistency of model with benchmark. Pm-149 was not available in the given fast spectrum library and was only modelled in the thermal spectrum. I-135 was included in lieu of I-131. This iodine isotope substitution was made for purposes of a simplified Xe-135 decay chain. B-10 was cited in the benchmark with a number density of 0.0 in contrast to that listed. These errors are assumed to have a minor influence. Because the I-135 and B-10 isotopes were also consistently applied to the MCNP model, the errors are of relatively low importance. In addition to these errors, two potentially significant isotopes that are present in the benchmark are not included in the model of study; these isotopes are Pu-232 and Gd.

31

Table 5-1: Isotopes and Number Densities in the Fuel Material

* indicates a minor inconsistency with Benchmark.

5.3.3 Uniform Temperature Distribution

The reactor model assumes uniform temperature everywhere. This assumption is unrealistic since no reactor can be completely uniform in temperature except in low power testing or room temperature criticality testing. For our case, this assumption may be fine since our comparison is between methods. The non-uniformity of temperature is likely to have a more significant effect for spatial discretization studies.

5.3.4 Homogeneous versus Heterogeneous Material Compositions

The model used in this study is composed of uniformly distributed isotopes for each material set. The significance and level of impact between

32 heterogeneous and homogeneous materials on energy group structure is not completely understood for the study of group structure. Theory describes that solving the spatial discretization problem also requires simultaneously solving the energy discretization problem; space and energy discretization is a coupled problem. General nuclear engineering literature such as found in Lamarsh discuss the important difference between heterogeneous and homogeneous material distribution, but in addition to this, the Monte Carlo simulations performed by Colak & Seker[5] verifies that a homogenized fuel model of PBMR underestimates criticality of the HTR design. The same article suggests that the double heterogeneity may be the most challenging issue to resolve between improving diffusion methods. Therefore to reduce complication in this study, the materials are modeled with complete homogeneity and uniformity. This contrasts to typical HTGR cell calculations which utilize Dancoff factors to account for the double heterogeneity of the pebble and kernel during cross section generation.

Chapter 6

Sources of Error

The Diffusion Theory Approximation

Because of the methodological differences between Monte Carlo and diffusion theory, an inherent difference is expected to exist. Figure 6-1 shows a ratio of the flux shape calculated by diffusion over flux shape calculated by MCNP. The results for the figure are generated from a 2-group case with cross sections produced by infinite medium assumption. The ratios highlight where most of the differences between the two methods are spatially located. The results confirm the theoretical error associated with diffusion theory as it shows poor agreement near boundaries and near the control rod.

The figure most importantly illustrates the essential meaning of the deficiency of diffusion theory as it shows that in any regions of large anisotropic behavior for neutrons, diffusion will perform poorly. Therefore, the intelligent reactor analyst using a diffusion code will understand his need to apply for corrections in areas of sudden cross section changes such as from void regions, fuel-to-moderator interfacial regions, large changes in fuel content between spatial regions, reactor boundaries, and near strong absorbers.

34

0.13 0.11 0.11 0.11 0.12 0.11 0.12 0.12 0.12 0.12 0.12 0.12 0.12 0.13 0.13 0.13 0.14 0.14 0.14 0.15 0.16 0.18 0.22 0.29 0.39 0.62 0.82 0.85 0.77 0.64 0.64 0.54 0.51 0.48 0.55 0.58 0.53 0.62 1.22 0.33 0.34 0.32 0.32 0.33 0.33 0.33 0.34 0.35 0.35 0.35 0.36 0.36 0.37 0.37 0.38 0.39 0.40 0.41 0.44 0.47 0.52 0.63 0.79 1.08 1.85 2.45 2.27 1.91 1.88 1.90 1.75 1.64 1.58 1.62 1.62 1.37 1.55 2.33 0.52 0.51 0.50 0.49 0.51 0.52 0.54 0.54 0.53 0.53 0.55 0.56 0.58 0.59 0.59 0.59 0.62 0.64 0.65 0.69 0.73 0.79 0.94 1.17 1.63 2.85 3.92 3.86 3.33 3.20 2.99 2.87 2.50 2.37 2.41 2.49 2.42 2.29 3.80 0.68 0.69 0.64 0.65 0.65 0.66 0.69 0.71 0.73 0.73 0.73 0.75 0.74 0.75 0.75 0.75 0.78 0.81 0.83 0.88 0.94 0.99 1.13 1.40 2.01 3.31 4.94 4.72 4.25 3.93 3.44 3.14 3.01 2.74 2.84 2.80 2.69 2.76 4.64 0.74 0.78 0.83 0.79 0.77 0.80 0.81 0.81 0.84 0.85 0.85 0.86 0.85 0.85 0.86 0.90 0.91 0.91 0.95 1.01 1.08 1.15 1.25 1.54 2.15 3.48 4.64 4.72 4.34 3.77 3.38 3.31 3.01 2.98 2.61 2.48 2.75 2.66 4.31 0.90 0.88 0.89 0.89 0.88 0.90 0.90 0.92 0.92 0.95 0.97 0.95 0.94 0.93 0.95 0.98 0.98 1.00 1.02 1.07 1.13 1.22 1.36 1.59 2.19 3.58 4.90 4.59 4.19 3.78 3.51 3.03 2.76 2.65 2.63 2.64 2.45 2.71 4.80 1.03 1.01 0.97 0.95 0.97 0.97 1.00 0.98 0.98 1.00 1.02 1.01 1.00 0.98 1.01 1.03 1.03 1.04 1.04 1.09 1.15 1.23 1.37 1.63 2.17 3.36 4.41 3.93 3.73 3.46 3.29 2.90 2.60 2.39 2.41 2.38 2.38 2.36 4.09 1.03 1.03 1.02 1.00 0.99 1.03 1.03 1.02 1.03 1.04 1.04 1.05 1.05 1.04 1.05 1.05 1.03 1.04 1.05 1.11 1.14 1.20 1.29 1.49 1.91 3.00 3.88 3.47 3.17 2.89 2.72 2.44 2.31 2.16 2.17 2.19 2.20 2.28 3.65 1.09 1.10 1.09 1.04 1.02 1.03 1.03 1.04 1.05 1.05 1.07 1.06 1.06 1.05 1.06 1.05 1.05 1.07 1.08 1.08 1.14 1.18 1.23 1.37 1.69 2.45 2.91 2.77 2.51 2.49 2.38 2.13 2.04 2.04 2.09 2.16 2.26 2.57 4.55 1.24 1.12 1.08 1.04 1.04 1.05 1.07 1.06 1.06 1.07 1.08 1.05 1.05 1.05 1.05 1.06 1.06 1.08 1.08 1.09 1.12 1.15 1.19 1.28 1.41 1.73 2.04 2.04 2.02 2.00 1.98 1.84 1.80 1.82 1.87 2.00 2.27 2.50 4.33 1.06 1.05 1.07 1.08 1.09 1.11 1.09 1.07 1.09 1.08 1.07 1.05 1.05 1.05 1.05 1.05 1.06 1.07 1.08 1.09 1.12 1.11 1.14 1.19 1.30 1.47 1.70 1.70 1.72 1.74 1.72 1.74 1.77 1.79 1.79 1.89 2.06 2.39 4.06 1.04 1.03 1.07 1.10 1.08 1.08 1.08 1.08 1.08 1.08 1.05 1.06 1.07 1.06 1.05 1.04 1.05 1.06 1.06 1.07 1.10 1.10 1.13 1.15 1.24 1.54 1.76 1.68 1.67 1.63 1.64 1.69 1.66 1.73 1.76 1.81 1.97 2.27 3.82 1.05 1.06 1.07 1.08 1.07 1.06 1.05 1.05 1.05 1.05 1.05 1.05 1.05 1.05 1.06 1.06 1.06 1.07 1.08 1.10 1.11 1.12 1.13 1.15 1.20 1.33 1.41 1.40 1.39 1.40 1.42 1.42 1.45 1.49 1.52 1.57 1.67 2.01 3.45 1.04 1.04 1.05 1.06 1.06 1.06 1.06 1.05 1.05 1.05 1.05 1.05 1.05 1.07 1.09 1.11 1.11 1.11 1.13 1.13 1.14 1.16 1.17 1.19 1.21 1.26 1.30 1.29 1.30 1.34 1.34 1.34 1.39 1.38 1.42 1.47 1.60 1.88 3.15 1.02 1.02 1.04 1.04 1.04 1.05 1.05 1.05 1.04 1.04 1.04 1.04 1.05 1.06 1.09 1.10 1.11 1.11 1.12 1.13 1.14 1.16 1.17 1.18 1.21 1.25 1.29 1.28 1.30 1.29 1.28 1.32 1.34 1.36 1.41 1.46 1.56 1.84 3.01 1.02 1.04 1.05 1.06 1.06 1.05 1.05 1.04 1.04 1.05 1.04 1.05 1.05 1.06 1.08 1.10 1.11 1.11 1.12 1.13 1.14 1.15 1.16 1.17 1.20 1.25 1.28 1.28 1.27 1.29 1.28 1.30 1.30 1.32 1.37 1.43 1.55 1.83 3.04 1.04 1.05 1.05 1.05 1.06 1.06 1.05 1.04 1.04 1.04 1.03 1.03 1.04 1.05 1.07 1.09 1.10 1.11 1.12 1.13 1.13 1.14 1.15 1.16 1.18 1.23 1.27 1.25 1.25 1.25 1.24 1.25 1.26 1.28 1.31 1.39 1.48 1.72 2.82 1.03 1.04 1.04 1.03 1.04 1.05 1.03 1.03 1.03 1.03 1.03 1.03 1.03 1.05 1.07 1.09 1.10 1.10 1.11 1.12 1.12 1.13 1.14 1.14 1.16 1.21 1.25 1.22 1.21 1.21 1.20 1.20 1.21 1.24 1.26 1.31 1.41 1.66 2.76 1.04 1.04 1.04 1.04 1.04 1.03 1.03 1.03 1.03 1.02 1.03 1.03 1.03 1.05 1.07 1.08 1.08 1.09 1.09 1.10 1.11 1.12 1.12 1.12 1.12 1.15 1.17 1.15 1.14 1.14 1.14 1.15 1.16 1.18 1.21 1.26 1.36 1.60 2.65 1.05 1.04 1.04 1.03 1.03 1.04 1.03 1.03 1.03 1.02 1.02 1.02 1.02 1.04 1.05 1.07 1.07 1.08 1.08 1.08 1.08 1.09 1.09 1.08 1.07 1.06 1.06 1.06 1.06 1.07 1.08 1.09 1.11 1.13 1.16 1.20 1.29 1.53 2.54 1.03 1.03 1.04 1.03 1.03 1.03 1.03 1.02 1.02 1.02 1.01 1.01 1.02 1.03 1.04 1.06 1.06 1.06 1.07 1.07 1.07 1.07 1.06 1.05 1.04 1.03 1.03 1.03 1.04 1.05 1.05 1.06 1.08 1.09 1.12 1.17 1.26 1.48 2.46 1.05 1.04 1.03 1.03 1.02 1.02 1.02 1.02 1.01 1.01 1.00 1.01 1.01 1.02 1.04 1.05 1.06 1.06 1.06 1.06 1.06 1.06 1.06 1.05 1.03 1.02 1.02 1.02 1.03 1.04 1.04 1.05 1.06 1.08 1.11 1.16 1.23 1.46 2.43 1.02 1.01 1.02 1.02 1.02 1.02 1.01 1.01 1.01 1.01 1.00 1.00 1.01 1.02 1.04 1.05 1.05 1.06 1.06 1.06 1.05 1.05 1.05 1.04 1.02 1.01 1.01 1.01 1.02 1.02 1.03 1.05 1.06 1.08 1.10 1.14 1.22 1.44 2.40 1.00 1.00 1.01 1.01 1.01 1.01 1.00 1.00 1.00 1.00 0.99 1.00 1.00 1.01 1.03 1.05 1.05 1.05 1.05 1.05 1.05 1.05 1.04 1.03 1.02 1.01 1.00 1.01 1.01 1.02 1.02 1.04 1.05 1.07 1.10 1.14 1.22 1.43 2.37 1.00 1.00 1.00 1.00 1.01 1.00 1.00 0.99 1.00 0.99 0.99 0.99 0.99 1.01 1.02 1.04 1.04 1.04 1.05 1.04 1.04 1.04 1.04 1.03 1.01 1.00 1.00 1.00 1.01 1.01 1.02 1.03 1.04 1.06 1.09 1.13 1.21 1.43 2.37 0.99 0.99 1.00 1.00 1.00 1.00 0.99 0.99 0.99 0.99 0.99 0.99 0.99 1.00 1.02 1.03 1.04 1.04 1.04 1.04 1.04 1.04 1.03 1.02 1.00 0.99 0.99 1.00 1.00 1.01 1.02 1.03 1.04 1.06 1.09 1.13 1.21 1.43 2.39 0.97 0.98 0.99 0.99 0.99 0.99 0.99 0.99 0.99 0.98 0.98 0.98 0.98 1.00 1.01 1.03 1.03 1.03 1.03 1.03 1.03 1.03 1.03 1.02 1.00 0.99 0.99 0.99 0.99 1.00 1.01 1.02 1.04 1.05 1.08 1.12 1.20 1.42 2.36 0.97 0.98 0.98 0.99 0.99 0.99 0.99 0.99 0.99 0.98 0.98 0.98 0.98 0.99 1.01 1.03 1.03 1.03 1.03 1.03 1.03 1.03 1.02 1.01 1.00 0.99 0.98 0.99 0.99 1.00 1.01 1.02 1.03 1.05 1.08 1.12 1.20 1.41 2.35 0.99 0.99 0.98 0.98 0.99 0.99 0.99 0.98 0.98 0.98 0.97 0.97 0.98 0.99 1.01 1.02 1.03 1.02 1.03 1.03 1.02 1.02 1.02 1.01 0.99 0.98 0.98 0.98 0.99 1.00 1.01 1.02 1.03 1.05 1.08 1.12 1.20 1.41 2.35 0.98 0.98 0.98 0.98 0.98 0.99 0.99 0.98 0.98 0.98 0.97 0.97 0.97 0.98 1.00 1.02 1.02 1.02 1.03 1.02 1.02 1.02 1.02 1.01 0.99 0.98 0.98 0.98 0.99 1.00 1.00 1.01 1.03 1.04 1.07 1.12 1.20 1.41 2.35 0.98 0.98 0.98 0.98 0.98 0.98 0.98 0.98 0.98 0.97 0.97 0.97 0.97 0.98 1.00 1.01 1.02 1.02 1.02 1.02 1.02 1.02 1.01 1.01 0.99 0.98 0.98 0.98 0.98 0.99 1.00 1.01 1.02 1.04 1.07 1.11 1.20 1.41 2.34 0.98 0.98 0.99 0.98 0.98 0.98 0.98 0.97 0.97 0.97 0.96 0.97 0.97 0.98 1.00 1.01 1.01 1.01 1.02 1.02 1.02 1.02 1.01 1.00 0.99 0.98 0.97 0.98 0.98 0.99 0.99 1.00 1.02 1.03 1.06 1.11 1.19 1.41 2.32 0.98 0.98 0.98 0.98 0.98 0.98 0.97 0.97 0.97 0.97 0.96 0.96 0.96 0.98 0.99 1.01 1.01 1.01 1.02 1.02 1.01 1.01 1.01 1.00 0.98 0.97 0.97 0.98 0.98 0.99 0.99 1.00 1.02 1.03 1.06 1.11 1.18 1.40 2.32 0.98 0.98 0.98 0.98 0.98 0.98 0.97 0.97 0.97 0.96 0.96 0.96 0.96 0.97 0.99 1.00 1.01 1.01 1.01 1.01 1.01 1.01 1.01 0.99 0.98 0.97 0.97 0.97 0.97 0.98 0.99 1.00 1.01 1.03 1.06 1.10 1.18 1.40 2.32 0.97 0.97 0.98 0.98 0.98 0.98 0.98 0.97 0.97 0.96 0.96 0.96 0.96 0.98 0.99 1.00 1.01 1.01 1.01 1.01 1.01 1.01 1.00 0.99 0.98 0.97 0.96 0.97 0.97 0.98 0.99 1.00 1.01 1.03 1.06 1.10 1.18 1.39 2.31 0.98 0.98 0.98 0.98 0.98 0.98 0.98 0.98 0.97 0.97 0.96 0.96 0.96 0.98 0.99 1.00 1.01 1.01 1.01 1.01 1.01 1.01 1.00 0.99 0.98 0.97 0.96 0.97 0.97 0.98 0.98 1.00 1.01 1.03 1.06 1.10 1.18 1.39 2.31 0.99 0.99 0.98 0.98 0.98 0.98 0.97 0.97 0.97 0.96 0.96 0.96 0.96 0.98 0.99 1.00 1.01 1.00 1.01 1.01 1.00 1.01 1.00 0.99 0.97 0.96 0.96 0.97 0.97 0.98 0.98 0.99 1.01 1.03 1.05 1.09 1.17 1.38 2.29 0.99 0.99 0.98 0.98 0.98 0.98 0.97 0.97 0.97 0.96 0.96 0.96 0.96 0.97 0.99 1.00 1.00 1.00 1.01 1.00 1.00 1.00 1.00 0.99 0.97 0.96 0.96 0.96 0.97 0.97 0.98 0.99 1.01 1.02 1.05 1.09 1.17 1.39 2.32 0.97 0.98 0.98 0.98 0.97 0.97 0.97 0.97 0.96 0.96 0.96 0.96 0.96 0.97 0.99 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 0.99 0.97 0.96 0.96 0.96 0.97 0.97 0.98 0.99 1.00 1.02 1.05 1.09 1.17 1.38 2.30 0.97 0.97 0.97 0.97 0.97 0.97 0.97 0.96 0.96 0.96 0.96 0.96 0.96 0.97 0.98 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 0.99 0.97 0.96 0.96 0.96 0.96 0.97 0.98 0.99 1.00 1.02 1.05 1.09 1.17 1.38 2.30 0.96 0.96 0.96 0.97 0.97 0.97 0.97 0.96 0.96 0.96 0.95 0.95 0.95 0.96 0.98 1.00 1.00 1.00 1.00 1.00 1.00 1.00 0.99 0.98 0.97 0.96 0.96 0.96 0.96 0.97 0.97 0.99 1.00 1.02 1.04 1.09 1.16 1.37 2.27 0.96 0.97 0.96 0.96 0.96 0.96 0.96 0.96 0.96 0.95 0.95 0.95 0.95 0.96 0.98 0.99 1.00 1.00 1.00 1.00 1.00 1.00 0.99 0.98 0.97 0.96 0.95 0.96 0.96 0.97 0.98 0.99 1.00 1.02 1.04 1.08 1.16 1.37 2.29 0.97 0.96 0.96 0.96 0.96 0.96 0.96 0.95 0.95 0.95 0.95 0.95 0.95 0.96 0.98 0.99 0.99 0.99 1.00 1.00 0.99 0.99 0.99 0.98 0.96 0.95 0.95 0.96 0.96 0.97 0.97 0.99 1.00 1.01 1.04 1.08 1.16 1.36 2.27 0.95 0.96 0.96 0.96 0.96 0.96 0.96 0.95 0.95 0.94 0.94 0.94 0.94 0.96 0.97 0.99 0.99 0.99 0.99 0.99 0.99 0.99 0.99 0.98 0.96 0.95 0.95 0.95 0.95 0.96 0.97 0.98 0.99 1.01 1.03 1.08 1.15 1.36 2.27 0.96 0.96 0.95 0.95 0.95 0.96 0.95 0.95 0.95 0.94 0.94 0.94 0.94 0.95 0.97 0.98 0.98 0.99 0.99 0.99 0.99 0.99 0.98 0.97 0.96 0.95 0.95 0.95 0.95 0.96 0.97 0.98 0.99 1.01 1.03 1.07 1.15 1.36 2.26 0.94 0.95 0.94 0.94 0.95 0.95 0.95 0.94 0.94 0.94 0.94 0.94 0.94 0.95 0.97 0.98 0.98 0.98 0.99 0.99 0.99 0.99 0.98 0.97 0.95 0.95 0.94 0.94 0.95 0.96 0.96 0.97 0.99 1.00 1.03 1.07 1.14 1.35 2.24 0.94 0.94 0.94 0.94 0.95 0.94 0.94 0.94 0.94 0.94 0.93 0.94 0.94 0.95 0.97 0.98 0.98 0.98 0.99 0.99 0.98 0.98 0.98 0.97 0.95 0.94 0.94 0.94 0.94 0.95 0.96 0.97 0.98 1.00 1.02 1.06 1.14 1.34 2.24 0.94 0.94 0.94 0.94 0.95 0.95 0.94 0.94 0.94 0.93 0.93 0.93 0.93 0.95 0.96 0.98 0.98 0.98 0.98 0.98 0.98 0.98 0.97 0.96 0.95 0.94 0.93 0.94 0.94 0.95 0.95 0.96 0.98 0.99 1.02 1.06 1.13 1.33 2.23 0.94 0.95 0.95 0.94 0.95 0.95 0.94 0.94 0.93 0.93 0.93 0.93 0.93 0.94 0.96 0.97 0.97 0.97 0.98 0.98 0.97 0.97 0.97 0.96 0.94 0.93 0.93 0.93 0.94 0.94 0.95 0.96 0.97 0.99 1.02 1.06 1.13 1.34 2.23 0.94 0.95 0.95 0.94 0.94 0.94 0.94 0.94 0.93 0.93 0.93 0.93 0.93 0.94 0.95 0.97 0.97 0.97 0.97 0.97 0.97 0.97 0.96 0.95 0.94 0.93 0.93 0.93 0.93 0.94 0.95 0.96 0.97 0.99 1.01 1.05 1.13 1.33 2.21 0.96 0.95 0.95 0.94 0.93 0.93 0.93 0.93 0.92 0.92 0.92 0.92 0.92 0.93 0.95 0.96 0.96 0.96 0.96 0.96 0.96 0.96 0.96 0.95 0.93 0.92 0.92 0.92 0.93 0.93 0.94 0.95 0.96 0.98 1.00 1.04 1.11 1.31 2.19 0.93 0.93 0.93 0.92 0.92 0.92 0.92 0.92 0.91 0.91 0.91 0.91 0.91 0.92 0.94 0.95 0.95 0.96 0.96 0.96 0.95 0.96 0.95 0.94 0.93 0.91 0.91 0.92 0.92 0.93 0.93 0.94 0.96 0.97 1.00 1.04 1.11 1.31 2.17 0.93 0.92 0.91 0.91 0.91 0.91 0.91 0.91 0.91 0.90 0.90 0.90 0.90 0.92 0.93 0.95 0.95 0.95 0.95 0.95 0.95 0.95 0.95 0.94 0.92 0.91 0.91 0.91 0.91 0.92 0.93 0.94 0.95 0.96 0.99 1.03 1.10 1.30 2.16 0.90 0.89 0.90 0.90 0.91 0.91 0.90 0.90 0.90 0.89 0.89 0.89 0.89 0.91 0.92 0.94 0.94 0.94 0.94 0.94 0.94 0.94 0.93 0.93 0.91 0.90 0.90 0.90 0.90 0.91 0.92 0.93 0.94 0.96 0.98 1.02 1.09 1.29 2.14 0.88 0.88 0.88 0.88 0.89 0.89 0.89 0.89 0.89 0.88 0.88 0.88 0.88 0.90 0.91 0.92 0.93 0.92 0.93 0.93 0.93 0.93 0.92 0.91 0.90 0.89 0.89 0.89 0.89 0.90 0.91 0.92 0.93 0.94 0.97 1.01 1.08 1.28 2.13 0.88 0.87 0.87 0.88 0.88 0.88 0.88 0.88 0.88 0.87 0.87 0.87 0.87 0.88 0.90 0.91 0.91 0.91 0.91 0.91 0.91 0.91 0.91 0.90 0.89 0.88 0.88 0.88 0.88 0.89 0.90 0.91 0.92 0.94 0.96 0.99 1.06 1.26 2.09 0.87 0.85 0.86 0.87 0.87 0.87 0.87 0.86 0.86 0.86 0.85 0.85 0.85 0.86 0.88 0.89 0.89 0.89 0.90 0.89 0.89 0.90 0.89 0.89 0.88 0.87 0.86 0.87 0.87 0.88 0.88 0.89 0.91 0.92 0.94 0.98 1.05 1.25 2.09 0.83 0.84 0.85 0.86 0.86 0.85 0.85 0.85 0.84 0.84 0.84 0.84 0.84 0.84 0.85 0.85 0.85 0.85 0.86 0.86 0.86 0.86 0.86 0.86 0.85 0.85 0.86 0.86 0.87 0.87 0.88 0.89 0.90 0.91 0.93 0.97 1.04 1.23 2.03 0.84 0.84 0.85 0.85 0.85 0.84 0.83 0.83 0.83 0.83 0.83 0.84 0.84 0.83 0.84 0.83 0.83 0.83 0.84 0.84 0.84 0.84 0.84 0.84 0.84 0.85 0.85 0.85 0.86 0.86 0.87 0.88 0.88 0.90 0.92 0.96 1.03 1.22 2.01 0.82 0.82 0.82 0.83 0.83 0.83 0.82 0.82 0.82 0.82 0.82 0.83 0.83 0.83 0.82 0.82 0.82 0.82 0.83 0.83 0.83 0.83 0.83 0.83 0.84 0.84 0.84 0.84 0.85 0.85 0.86 0.86 0.87 0.89 0.91 0.94 1.02 1.19 1.98 0.81 0.81 0.80 0.80 0.80 0.80 0.81 0.81 0.81 0.80 0.80 0.81 0.81 0.81 0.81 0.81 0.81 0.81 0.82 0.82 0.82 0.82 0.82 0.83 0.83 0.82 0.82 0.82 0.83 0.83 0.84 0.85 0.86 0.87 0.89 0.93 1.00 1.19 1.95 0.78 0.77 0.77 0.77 0.77 0.77 0.77 0.77 0.77 0.78 0.78 0.78 0.78 0.78 0.79 0.79 0.79 0.79 0.79 0.80 0.80 0.79 0.80 0.80 0.80 0.80 0.80 0.80 0.81 0.81 0.82 0.83 0.83 0.85 0.86 0.90 0.97 1.15 1.93 0.73 0.73 0.73 0.72 0.72 0.73 0.73 0.73 0.73 0.74 0.74 0.74 0.74 0.75 0.75 0.75 0.75 0.76 0.76 0.76 0.76 0.76 0.76 0.76 0.76 0.76 0.77 0.77 0.77 0.77 0.78 0.78 0.79 0.81 0.82 0.86 0.93 1.09 1.82 0.68 0.67 0.66 0.67 0.67 0.67 0.67 0.68 0.68 0.69 0.69 0.69 0.70 0.69 0.70 0.70 0.70 0.70 0.70 0.71 0.72 0.71 0.71 0.71 0.71 0.72 0.72 0.72 0.72 0.72 0.72 0.72 0.73 0.75 0.76 0.79 0.85 1.00 1.68 0.60 0.60 0.59 0.59 0.60 0.59 0.60 0.60 0.61 0.62 0.62 0.62 0.62 0.62 0.63 0.63 0.63 0.63 0.64 0.64 0.64 0.64 0.64 0.64 0.65 0.65 0.65 0.64 0.65 0.64 0.65 0.65 0.65 0.66 0.68 0.71 0.76 0.90 1.53 0.49 0.49 0.49 0.50 0.50 0.50 0.51 0.51 0.51 0.52 0.52 0.52 0.52 0.53 0.53 0.53 0.53 0.54 0.54 0.54 0.54 0.54 0.55 0.55 0.55 0.54 0.55 0.54 0.55 0.54 0.55 0.55 0.56 0.56 0.58 0.60 0.65 0.77 1.28 0.36 0.37 0.37 0.37 0.38 0.38 0.38 0.39 0.39 0.39 0.40 0.40 0.40 0.40 0.40 0.41 0.41 0.41 0.42 0.41 0.42 0.42 0.42 0.42 0.42 0.41 0.42 0.41 0.42 0.42 0.42 0.42 0.43 0.43 0.44 0.46 0.50 0.59 0.96 0.24 0.24 0.24 0.24 0.24 0.24 0.24 0.24 0.24 0.25 0.25 0.25 0.25 0.25 0.25 0.26 0.26 0.26 0.26 0.26 0.26 0.26 0.26 0.26 0.26 0.26 0.26 0.26 0.26 0.26 0.27 0.27 0.27 0.27 0.28 0.29 0.31 0.37 0.61 0.08 0.08 0.08 0.08 0.08 0.08 0.08 0.08 0.08 0.08 0.08 0.08 0.09 0.09 0.09 0.09 0.09 0.09 0.09 0.09 0.09 0.09 0.09 0.09 0.09 0.09 0.09 0.09 0.09 0.09 0.09 0.09 0.09 0.09 0.10 0.10 0.11 0.13 0.45 Figure 6-1: : Ratio of NEM / MCNP Fluxes on a Node by Node Basis (R-Z geometry – at the top, bottom and right boundaries of the model the zero flux boundary conditions are applied; the blue region of the upper right coincides with the approximate location of the Control Rod).

Chapter 7

Comparison Methodology

The 2-D flux and relative power maps are reduced to axial and radial power profiles by volume averaging. Before the 2-D spatial results are reduced to 1-D profiles, the spatial results are normalized to average. Normalization is necessary because of the discrepancy in methods that exist between stochastic and deterministic. For more information, see Appendix A.

To assist in the analysis of the spatial profile, the results are compared with reference by using both percent difference and by using a Figure of Merit method.

The Figure of Merit [19] is the number of deviations based on the sample’s standard deviation. The sample standard deviation is based on the deviations of all cases against the reference in a given set of cases.

Equation 1: Definition of Figure of Merit xx− FOM = i MCNP σ

Equation 2: Definition of deviation

N 2 ∑()xxiMCNP− σ = i=1 N −1

th where xi is the diffusion code result for the i position of power or flux shape

and xMCNP is the solution for the respective position using MCNP.

36 In addition to Figure of Merit, the averages of percent difference spatial distributions are used to help evaluate which group structure is spatially better overall.

Chapter 8

Results and Discussion

The numerical results have been subdivided into case studies. Please note that because of limited graph labeling space, many results are noted with its equivalent COMBINE6 energy group number instead of the units of energy (eV). The table of energy listings for corresponding energy groups is listed in the Appendix B.

8.1 Thermal Cutoff Study

Purpose: To investigate the impact of the thermal cut-off, the energy boundary that separates the thermal and fast group. For COMBINE6, this boundary defines the thermal energy bound for the upper limit in INCITE and the lower boundary for the fast region in PHROG. Expectation: A high thermal cut-off is expected to produce better results than a low cut-off due to thermal spectrum and up-scattering containment.

In 2-Group, there is only one variable partition and it serves as both the only boundary that defines the group structure and also the thermal cut-off. The thermal cut-off is the energy at which the thermal spectrum module (INCITE) defines its upper energy boundary and also where the fast spectrum module (PHROG) defines its lower energy boundary. The 2-Group case studies have both the complication of moving the group structure and the thermal boundary and therefore for additional understanding a 9-Group study with fixed group structure is also investigated to observe only the impact of the thermal cut-off.

38 8.1.1 Thermal Cut-off Study using 2-Group No Up-scattering

Figure 8-1: 2-Group k-effective’s at 1000K Figure 8-2: 2-Group k-effective’s at 300K with no up-scattering with no up-scattering

No up-scattering modeling in this study means that the up-scattering is not modeled explicitly in the NEM but implicitly through corrected down-scattering cross-sections to include up-scattering. In terms of k-effective analysis, we see our expectation being met at 1000K. At 300K, the k-effective does not follow the same trend, but this is still consistent with our expectations because it implies that the dominating physics at high temperatures is up-scattering containment while at lower temperatures it is not. It might be speculated that dominating physics at lower temperatures are the low lying resonances of important isotopes.

Figure 8-3 - Figure 8-6 are averages of the percent difference in space and they better represent the differences between the group structures than the format shown in Figure 8-8 to Figure 8-11, which shows the details with space.

39

1000K Radial Power Shape and Comparison 300K Radial Power Shape and Comparison

Figure 8-3: 2-Group Average Radial Power Figure 8-5: 2-Group Average Radial Power Shape Percent Difference at 1000K Shape Percent Difference at 300K

Figure 8-4: 2-Group Average Radial Flux Figure 8-6: 2-Group Average Radial Flux Shape Percent Difference at 1000K Shape Percent Difference at 300K

The best results for power and flux shapes do not coincide with the best k-eff results. The trend for shapes at 300K in Figure 8-5 shows that a low cut-off is preferred for better power and flux shape. The low cut-off is also preferred for the 1000K shape results with the exception of the two lowest cut-offs 0.414 eV and 0.532 eV.

40

140 GRAPHITE 1000K Thermal Spectrum

120 GRAPHITE 300K Thermal Spectrum

0.414 eV cut‐off 100

) ‐ ( 80 Flux

60 Scaled

40

20

0 0 0.25 0.5 0.75 1 Energy (eV) Figure 8-7: Expected Thermal Spectrum at 300K and 1000K

The error seen at 1000K lowest cut-offs seems to be directly related to the error introduced by chopping off of the thermal spectrum as shown in Figure 8-7 and the message with k-eff results imply that any thermal cut-offs above 0.876 eV (66) may be sufficient at 1000K but 2 Group is shown to be too crude for analysis purposes.

1000K Radial Power Shape and Comparison 300K Radial Power Shape and Comparison

Figure 8-8: 2-Group Radial Power Profile at Figure 8-9: 2-Group Radial Power Profile at 1000K 300K

41 Figure 8-10: 2-Group Radial Power Percent Figure 8-11: 2-Group Radial Power Percent Difference at 1000K Difference at 300K

8.1.2 Thermal Cutoff Study using 2-Group with Up-scattering

The usage of up-scattering (up-scattering for the 3-D nodal code) for 2-Group produces a k-effective that is larger than its no up-scattering counterpart. The predicted k-effective with up-scattering is closer to reference for all cases.

The highest cut-off 2.38 eV does not have up-scattering because COMBINE6 forces a zero up- scattering rate when the highest cut-off (2.38 eV) is selected. Therefore, the comparison at cut-off 2.38 eV should be ignored because the results are identical for up-scatter and no up-scatter. COMBINE6 does calculate and output in long print the “above 2.38 eV” scattering rate, which can act be used for up-scattering rate, but this has not been used in this study.

42

Figure 8-12: 2-Group k-effective for Cases at Figure 8-13: 2-Group k-effective for Cases at 1000K with up-scattering 300K with up-scattering

Figure 8-14 - Figure 8-21 show the improvement in the percent differences at 1000 and 300K for power and flux shapes with use of up-scattering. The improvement of results is shown as a positive in these bar graphs. The only notable improvements in the use of the up-scatter generally are seen only at low cut-offs. The improvement of results for low cut-offs with up-scattering is clear for 1000K, especially in Figure 8-18 and Figure 8-20. It is less clear at 300K.

43

Figure 8-14: 2-Group Improvement with Up- Figure 8-15: 2-Group Improvement with Up- scattering for Percent Differences in Radial scattering for Percent Differences in Radial Power at 1000K Power at 300K

Figure 8-16: 2-Group Improvement with Up- Figure 8-17: 2-Group Improvement with Up- scattering for Percent Differences in Radial scattering for Percent Differences in Radial Flux at 1000K Flux at 300K

Figure 8-18: 2-Group Improvement with Up- Figure 8-19: 2-Group Improvement with Up- scattering for Percent Differences in Axial scattering for Percent Differences in Axial Power at 1000K Power at 300K

44

Figure 8-20: 2-Group Improvement with Up- Figure 8-21: 2-Group Improvement with Up- scattering for Percent Differences in Axial Flux scattering for Percent Differences in Axial Flux at 1000K at 300K

*Note that cut-off at COMBINE upper energy 101 (2.38 eV) is equivalent because COMBINE6 sets scattering above 2.38 eV to zero.

8.1.3 Thermal Cutoff Study using Fixed 9-Group Structure

Figure 8-22: Upper Energy Boundaries of 9 Group Energy Structure

One of the complications of the 2-Group study is the movement of the energy group boundary and the thermal cut-off. A case study of a fixed group structure with variation of only the thermal cut- off was performed. The energy group structure remained constant as that shown in Figure 8-22.

The k-effective analysis shows at low temperature, a low cut-off more accurately predicts the k- effective and as expected a high cut-off energy is required for a high temperature. So, according to k-effective analysis, the optimal cut-off for 300K is in the range of 0.414– 0.876 eV and for 1000K, it is in the range of 1.44 – 2.38 eV.

The differences between the k-effective predicted by the diffusion code at low and high cut-off’s are small. For 1000K, it is about 0.01 dk and for 300K, it is about 0.005 dk. Cutoff 82 (1.125 eV) is ignored because of its close proximity with Pu-240’s large resonance at approximately 1.06 eV.

45 The power shape shows a very small improvement at both temperatures with a higher cut-off. The power shape spatial evaluation show results with little difference compared to differences seen in 2- group. This may indicate that the group structure is more significant than the thermal cut-off. It’s also clear that thermal cut-off is more important at high temperature.

46

Figure 8-24: k-effective’s for 9-Group Figure 8-23: k-effective’s for 9-Group Thermal Cut-off at 300K Thermal Cut-off at 1000K

Figure 8-25: Average Radial Power Shape Figure 8-26: Average Radial Power Shape Percent Difference at 1000K Percent Difference at 300K

Figure 8-27: Radial Power Percent Difference Figure 8-28: Radial Power Percent at 1000K Difference at 300K

47

8.1.4 Thermal Cut-off Study Conclusion

In terms of k-effective prediction, the usage of up-scatter in 2-group produces less error. The importance of thermal cut-off is shown at high temperatures as seen in the 9-Group thermal cut-off study.

The power shape was shown to improve at higher cut-off’s in the 9-Group study, but in the 2-Group study the opposite was seen. The variation of power and flux accuracy is seen to be large in the 2- Group study, but is seen to be much less in the 9-Group thermal cut-off study. This observation of larger variation seen in the 2-Group implies that the 2-Group is more strongly influenced by the group structure than the thermal cut-off.

Because of the disagreement between the best group structure defined by k-eff and power profile in 2-Group and the evidence that shows that the group structure is more important than the thermal boundary, there is sufficient motivation to use more than 2 groups in order to obtain a better solution by means of the best thermal cut-off with the best group structure. In addition, having a good agreeing k-eff with poor spatial solution is not an acceptable group structure.

Since having a high thermal cut-off agrees with physical considerations and was shown to be the optimal case in the 9-Group radial power profile, it is recommended that future cases use the 2.38 eV cut-off.

In short, the conclusion of this study is to move to a higher number of groups with a high cut-off (2.38 eV).

48

8.2 3-Group Study

8.2.1 Fission Source Containment

Purpose: To investigate the impact of containing the fission source in a fast region. Expectation: Containing the fission source is expected to produce better results. Variation: Group 1’s lower energy boundary is varied from fine group 131 to fine group 154 (4.31 KeV – 1.74 MeV). In addition, a combinational study of varying both the thermal cut-off and Group 1’s lower energy boundary is also performed.

It’s shown that the k-effective is relatively insensitive (Table 8-1 and Table Table 8-2) when compared to the thermal cut-off study, to the choice of the 3rd partition. For all 3 group cases, the k- eff’s are all better than the corresponding 2-group k-effective. This is shown explicitly for 1000K 2.38 eV cut-off.

49

Table 8-1: Evaluation of k-effective in 3 Table 8-2: Evaluation of k-effective in 3 Group Spectrum Containment at 1000K Group Spectrum Containment at 300K

Figure 8-30: 3 Group Spectrum Containment k- Figure 8-29: 3 Group Spectrum Containment k- effective at 300K effective at 1000K

The radial power shape shows a direct relationship with the fission source (Figure 8-35). The minimum percent difference of radial power for both temperatures (Figure 8-31 and Figure 8-32) is shown to be at about 141 (52.5 keV). With the partition 141, it’s shown in Figure 8-35 that most (99%) of the fission source is contained in the top fast group.

50

Figure 8-31: 3-Group Fission Source Figure 8-32: 3-Group Fission Source Containment Average Radial Power Shape Containment Average Radial Power Shape Percent Difference at 1000K Percent Difference at 300K

Figure 8-33: 3 Group Spectrum Containment Figure 8-34: 3 Group Spectrum Containment Radial Power Percent Difference at 1000K Radial Power Percent Difference at 300K

Figure 8-35: 3 Group Spectrum Containment Fission Source Fraction per Energy Group

51

Figure 8-36: Average Radial Flux Shape Percent Figure 8-37: Average Radial Flux Shape Percent Difference at 1000K Difference at 296K

The radial flux shape seems to show a minimum percent difference at a different partition, 148-149.

When all the possible thermal cut-offs are calculated with variations of the fast group’s lower boundary from 131 to 155, it’s shown that the choice of the upper boundary for the thermal group is the more dominant effect and that the selection of the fast group’s lower boundary has a limited effect on the power distribution and the k-effective in comparison. Figure 8-38 shows this trend; the cases (166_X_44) are much better than (166_X_101) and there is little variation with boundary X in comparison with the thermal boundary.

A possible explanation for the insensitivity of the three group structure may be due to the importance of low lying resonances, which strongly varied results in the two-group studies.

52

Figure 8-38: The Best and Worst Radial Power Percent Difference at 300K for all possible combinations of thermal cut-off with the variation of the fast group’s lower energy boundary.

8.2.2 3-Group Conclusion

The 3-Group results show that if the partition is made such that it contains most of the fission source, it improves results slightly. In general, the exact partition that contains the fission source is not as important as having any partition within the range 4.31 KeV – 1.74 MeV. The results are slightly better when the partition is within the range of 139-149 (31.8 KeV – 388 KeV). Since the radial power profile at 141 (52.5 KeV) shows the best agreement and contains 99% of the fission source in Group 1, it’s recommended to set the boundary to contain fission at 141 (52.5 KeV).

53

8.3 Thermal Group Studies

Each thermal study case is performed such that there is one variable boundary with all other group boundaries fixed. The number of groups is held constant and the variable boundary is changed within the thermal region i.e. between the fine groups’ numbers 1 – 100 (0.001 eV – 2.33 eV). If a fixed boundary is placed at a given energy, the variable boundary skips over this boundary as a case. For example if the fixed boundary is added to the thermal region at 46 (0.043), the variable boundary cannot also be 46. In addition to having a variable boundary, sometimes the comparison includes a base group structure. The base group structure represents the structure without the variable boundary. To include a base group structure allows an analysis of how much improvement results have with the new group structure compared to having no additional partition.

8.3.1 Group Structure Abbreviation

The energy group structures are abbreviated in a form such as (166_146_101_X_12). The parenthesis notes that it is a group structure, the numbers are COMBINE6 input fine-group group number representing the upper energy, X is a variable energy partition, and the underscore is a space marker. In this example, the form means that the group structure boundaries are 16.905 MeV, 183 keV, 2.38 eV, a variable boundary, and 0.04 eV. The variable boundary noted “X” is placed at all energies between the energy boundaries adjacent to X. In this example case, those adjacent energies would be 2.38 and 0.04 eV. Another abbreviation of group structure is used. An example is (166_146_101_|_12_|) and it indicates that the group structure has a variable boundary X which is either between 101 and 12 or between 12 and 0.

54

8.3.2 Thermal Studies 1: 5-Group of 2 fast and 3 thermal groups

Purpose: To investigate the impact of partitioning of the thermal region and the significance of the thermal region. This energy group structure assumes 2 fast groups and 3 thermal groups. Expectation: More thermal groups should improve results. It’s also expected that partitions should be placed around resonance peaks instead of through the peaks. Variation: • The energy group structure takes on a 5-Group structure of the form (166_146_101_X_12), where X varies: 13 - 100 (0.05 – 2.38 eV). • Another study of X below 12 (.04 eV) showed little to no change in the results. • Also included in the comparison is the base 4-group structure, which is (166_146_101_12). Constants: • The “166_146” attempts to contain most of the fission source with • The “12” partition attempts to group the low energy 1/V tails of the cross sections in fissile/fissionable materials. • The “101” partition attempts to contain most of the up-scattering.

There are two energy cuts that seem to bring the k-eff very close to the reference as seen in Table 8-3. The first range is 43-53 (0.38 - 0.50 eV). The minimum difference of this group is found at cut 46 (0.43 eV) with 13 pcm difference. The second range is 75-76 (1.025 - 1.05 eV). The minimum difference is found at cut 75 (1.025 eV) with 286 pcm difference. If ignoring Np-237’s resonance, it can be said that the energy range between the resonances of Pu-239, Pu-240, Pu-241, and U-235 is of the range 0.45 eV - 0.95 eV (47 – 70). Based on just the k-effective results, there seems to be basis for the selection of a cut-off that separates these key resonances. Since cut 46

55 (0.43 eV) showed optimal k-eff, power, and flux distributions, it should be chosen as an essential energy partition for the next study.

Figure 8-39: k-effective’s of Figure 8-40: Low Lying Resonances (166_146_101_X_12) COMBINE # 46 coincides with 0.43 eV Pu-239 T=300 K from ENDF/B-6.2 mat9437 COMBINE # 77 coincides with 1.06 eV Pu-240 T=300 K from ENDF/B-6.2 mat9440 Pu-241 T=300 K from ENDF/B-6.0 mat9443

U-235 T=300 K from ENDF/B-6.2 mat9228

Table 8-3: Thermal Study 1 Cases with Percent Differences of k-effective within ±0.2%

56

Figure 8-41: Average Radial Power Shape Figure 8-42: Average Radial Flux Shape Percent Percent Difference of (166_146_101_X_12) Difference of (166_146_101_X_12)

Figure 8-43: Average Axial Power Shape Figure 8-44: Average Axial Flux Shape Percent Percent Difference of (166_146_101_X_12) Difference of (166_146_101_X_12)

Figure 8-45: Np, U, & Pu isotopes in 0.1-3.0 eV

In carefully analyzing the cross sections, it’s found that the range of 0.38 - 0.5 eV coincides with the gap between Pu-239, Pu-241, U-235 and Pu-240 resonances. It is especially focused on the minima of resonances Pu-240 and U-235. The region of 1.025 - 1.05 eV is also within the same gap, but its

57 minimum is focused more between Pu-239, Pu-241, and U-235 at the slight expense of Pu-240, which has a sharp peak at approximately 1.06 eV.

Figure 8-46: Legend for Figure 8-47, Figure 8-48, Figure 8-50

Figure 8-47: Radial Power Profiles of Figure 8-48: Axial Power Profiles of (166_146_101_X_12) (166_146_101_X_12)

58

Figure 8-50: Axial Power Percent Difference of (166_146_101_X_12) Best Cases: 46 – 75 (0.38 eV – 1.025 eV) Figure 8-49: Best and Worst Radial Power Worst Cases: 14 – 28 Percent Difference of (166_146_101_X_12) Best Cases: 43-60 (0.38– 0.65 eV) Worst Cases: X<21 & 99, 100 , Base4Grp

A hand inspection of the spatial distribution confirms the averaging of spatial distribution results shown in Figure 8-41.

The case inspection of (166_146_101_12_X) where X varies: 1 - 11 (0.001 – 0.03 eV) showed little change from the base group: (166_146_101_12).

Figure 8-51: k-effective of (166_146_101_12_X) Figure 8-52: Radial Power Percent Difference of (166_146_101_12_X)

Case Conclusion:

59 These case studies show that having three thermal groups can significantly improve the radial power distribution. The maximum average power percent difference can be in a range from about 4.5 to 2.75 %. The k-eff, radial power, and radial flux all show that any three thermal group case (166_146_101_X_12) is better than the base case of only two thermal groups (166_146_101_ 12). The radial power profile of these studies is also a much improvement from the 3-group structure of (166_141_101) which has a maximum error ranging about 7.5 – 8 % (see Figure 8-38). The best group structures of this case are also better than the best of the 3-group 166_141_44 radial power distribution (see Figure 8-38).

60

8.3.3 Thermal Studies 2: 5-Group of 3 fast groups and 2 thermal groups

Purpose: This 5-Group energy group structure assumes 3 fast groups and 2 thermal groups in contrast to Thermal Studies 1’s 2 fast groups and 3 thermal groups. It investigates the impact of partitioning of the thermal region without a lower boundary energy region which was used in Thermal Studies 1. One of the major purposes is to compare results with Thermal Studies 1 and see if similar results arise. Expectation: More thermal groups should improve results. It’s also expected that partitions should be placed around low lying resonances (resonances lying in the low energy ranges). Variation: The energy group structure takes on a 5-Group structure of the form: (166_145_128_101_X) in contrast to Thermal Studies 1’s (166_146_101_X_12). With X varying: 1 - 100 (0.001 – 2.33 eV). Constants: • The group structure (166_145_128_101_X) attempts to contain most of the fission source with “166_145”. • The “128” (2.04 keV) partition attempts to partition the marked change of U-235’s cross section where the unresolved resonance region begins as shown in Figure 8-53. • The “101” partition attempts to contain most of the up-scattering in the thermal region.

Figure 8-53: U-235 Fission and Total Cross Section

61

Table 8-4: Thermal Study 2 Cases with Percent Differences of k-effective within ±0.4%

Figure 8-54: k-effective’s of Figure 8-55: Comparison of k-effectives from (166_145_128_101_X) (166_146_101_X_12) and (166_145_128_101_X)

The comparison results are shown in Table 8-4 and Figures 8-54 through 8-55. The conclusions drawn in Thermal Studies 1 are identical in this study. The results found are similar to Thermal Studies 1 (see Figure 8-57 & Figure 8-59).

62

Figure 8-56: Average Radial Power Shape Figure 8-57: Average Axial Power Shape Percent Percent Difference of (166_145_128_101_X) Difference of (166_145_128_101_X)

Figure 8-58: Average Radial Flux Shape Percent Figure 8-59: Average Axial Flux Shape Percent Difference of (166_145_128_101_X) Difference of (166_145_128_101_X)

Case Conclusion:

The lack of change in this case study’s results as compared with thermal studies 1 indicates that there is a significant effect of partitions in the region of 25 – 95 (0.16 eV – 1.9 eV) and that this energy range is more important than the partition of the lower energy range “12” (0.04 eV) from thermal studies 1 and “128” (2.04 keV) in the fast region. Both studies seem to imply the importance of the large Pu-240 resonance which peaks at about 1.06 eV, but has a spread ranging from 0.4 – 1.6 eV. Similar to Thermal Studies 1, the best result is at cut-off 46 (0.43 eV).

63

8.3.4 Thermal Studies 3: 6-Group

Purpose: This 6-Group energy group structure study continues Thermal Studies 2 by the addition of the partition 46. It assumes 3 fast groups and 3 thermal groups. Partition “46” lies between resonance peaks. Variation: The energy group structure takes on a 6-Group structure of the form (See Table 8-5): if X < 46, then (166_145_128_101_46_X) else if 46 < X < 101, then (166_145_128_101_X_46) Constants: • The group structure “166_145” serves to contain most of the fission source. • The “128” (2.04 keV) partition is used to partition the marked change of U-235’s cross section where the unresolved resonance region begins. • The “101” partition attempts to contain most of the up-scattering in the thermal region. • The “101_46” partitions the Pu-240 and small U-235 resonance peaks.

The results of this case are shown in Figures 8-60 through 8-65 and in Table 8-5

64

Table 8-5: k-effective within ±0.09% Percent of (166_145_128_101_|_46_|)

65

Figure 8-60: k-effective of Figure 8-61: Radial Power Percent Difference (166_145_128_101_|_46_|) of (166_145_128_101_|_46_|)

Figure 8-62: Average Radial Power Shape Figure 8-63: Average Radial Flux Shape Percent Difference of Percent Difference of (166_145_128_101_|_46_|) (166_145_128_101_|_46_|)

Figure 8-64: Average Axial Power Shape Figure 8-65: Average Axial Flux Shape Percent Percent Difference of Difference of (166_145_128_101_|_46_|) (166_145_128_101_|_46_|)

Case Conclusion:

66 While the results do not indicate a major improvement in results, the k-effective is shown to have less spread of results with the change of X compared to previous Thermal Studies. In Thermal Studies 2 the k-effective varies with a range of about 0.02 dk while in this study it varies by 0.01 dk. The radial flux shape percent difference shows a minima around 21 -30 (0.095 - 0.24 eV). Because partition 26 has an optimal radial flux profile and best agrees with physical basis of a partition between the 1/V tails and the lowest lying resonance from Pu-241, Pu-239, and U-235, it has been chosen for the next fixed partition.

8.3.5 Thermal Studies 4: 7-Group

Purpose: 7-Group at 1000K energy group structure continues Thermal Studies 3 by the addition of the partition 23. It has 3 fast groups and 4 thermal groups. Expectation: Partition “46” lies around the resonances and its results with the varying partition should find better results in the addition of another between-resonance energy partition. Variation: The energy group structure takes on a 7-Group structure of the form: if 23 > X, then (166_145_128_101_46_23_X)

else if 46 > X > 23, then (166_145_128_101_46_X_23) else if 101 > X > 46, then (166_145_128_101_X_46_23) Constants: • The group structure “166_146” serves to contain most of the fission source. • The “128” (2.04 keV) partition is used to partition the marked change of U-235’s cross section where the unresolved resonance region begins. • The “101” partition attempts to contain most of the up-scattering in the thermal region. • The “101_46” partitions the Pu-240 and small U-235 resonance peaks. • The “46_23” partitions the Pu-241, Pu-239, and U-235 resonance peaks. • “23” serves as the partition dividing the lower 1/V tail. The results of this case are shown in Figures 8-66 through 8-72.

67

Figure 8-67: Deviations of K-eff from reference of 5 PCM or less for (166_145_128_101|46|23|) Figure 8-66: k-effective of (166_145_128_101|46|23|)*

Figure 8-68: Average Radial Power Shape Figure 8-69: Average Axial Power Shape Percent Percent Difference of Difference of (166_145_128_101|46|23|)* (166_145_128_101|46|23|)*

Figure 8-70: Average Radial Flux Shape Figure 8-71: Average Axial Flux Shape Percent Percent Difference of Difference of (166_145_128_101|46|23|)* (166_145_128_101|46|23|)*

68 *The erroneous results of 49 and 54 are due to a non-converged solution. Please ignore these two cases.

Figure 8-72 shows that the radial power distribution of all cases is very similar.

Figure 8-72: Radial Power Percent Difference of (166_145_128_101|46|23|)

Case Conclusion:

The radial power percent differences show no significant improvement from Thermal Studies 3; no significant improvement with 4 thermal groups over 3 thermal groups.

Despite the lack of improvement in power and flux shape, the k-effective showed a slight improvement at around 91 (See Figure 8-67). 91 (1.6 eV) is chosen for investigation because it coincides with the further containment of the Pu-240 isotope from the upper energy.

69 8.3.6 Thermal Studies 5: 8-Group

Purpose: 8-Group at 1000K energy group structure continues Thermal Studies 4 by the addition of the partition 91. It has 3 fast groups and 5 thermal groups. 166_145_128_101_91_46_23 with a varying boundary X in the region below 101. Expectation: The results are not expected to change from Thermal Studies 4.

The results of this case are shown in Figures 8-73 through 8-76 and in Table 8-6

70

Figure 8-73: k-effective of (166_145_128_101|91|46|23|) Figure 8-74: k-effective of All Thermal Studies

Figure 8-75: Average Radial Power Shape Figure 8-76: Average Radial Flux Shape Percent Percent Difference of Difference of (166_145_128_101|91|46|23|) (166_145_128_101|91|46|23|)

Figure 8-77: Average Axial Power Shape Figure 8-78: Average Axial Flux Shape Percent Percent Difference of Difference of (166_145_128_101|91|46|23|) (166_145_128_101|91|46|23|)

71

Table 8-6: (166_145_128_101|91|46|23|) k-effective of cases of 30 pcm

72

Figure 8-79: Radial Power Percent Difference Worst and Best of (166_145_128_101|91|46|23|)

Case Conclusion:

The radial and power shapes comparison results have to be magnified to a small scale to show very small differences. Figure 8-74 shows the k-effective results of all thermal studies.

73

8.3.7 Thermal Studies Conclusion

The agreement of results shows that placing partitions around resonance peaks improve the results. The recommended partitions found in this study are (in order of importance): • 46 (0.43 eV) • 23 (0.12 eV) • 91 (1.6 eV)

The above numerical findings agree with literature in respect to: • Using group partitions in the thermal range to compensate for the lack of Doppler feedback in the thermal region • Using more than 3 thermal groups showed little improvement. This point is in contrast to opinions in Massimo, but agrees with Stamm’ler. However, Stamm’ler’s discussion was specifically for a LWR design and not for a HTGR.

8.4 Fast Group Studies

8.4.1 Preliminary Assessment of Fast Group Structure and Importance

Purpose: To analyze the role and significance of the fast group structure in the base group structure of thermal studies 5. Constant: The thermal group structure was held constant as (101_91_46_23) and is abbreviated as “Th” for “Thermal.” Variation: The fast group structure of ThermalStudies5 is 166_145_128_101. To observe the significance of the fast group structure, the base 7 group structure is compared to cases with less fast groups. • Base 7 Group: (166_145_128_101_91_46_23) is the optimal group structure from thermal studies 5. • 166_128_Th: (166_128_101_91_46_23) has two fast groups with a partition at a marked change in U-235’s cross section. • 166_145_Th: (166_145_101_91_46_23) has two fast groups with a partition to contain source fission.

74 • 166_Th: (166_101_91_46_23) has only one fast group.

The results of this case are shown in Figures 8-80 through 8-84.

Figure 8-80: K-eff results for Preliminary Fast Figure 8-81: Preliminary Fast Group Study Group Study (See Figure 8-81) Abbreviations

Figure 8-82: k-eff results with variations of fast group structure (166_145_128_101_91_46_23)

75

Figure 8-83: Average Radial Power Shape Percent Difference for Preliminary Fast Group Study

Figure 8-84: Radial Power Percent Difference for Preliminary Fast Group Study

In terms of k-eff, the best ranking fast group structure is as follows: 1. (166_145_128_101_91_46_23), base group structure 2. (166_145_101_91_46_23), fission source containment 3. (166_128_101_91_46_23), division of marked change of U-235 4. (166_101_91_46_23), one fast group

76

Observations: The ranking of best power profile is not the same as the best k-eff. An unexpected occurrence is that the error of the 1-fast-group case is smaller than the 2-fast-group cases of 166_128 and 166_145 as shown in Figure 8-84 at the fuel region’s center.

Case Conclusion: The results shown conclude that there is still dependence in the fast group structure and therefore warrants more study.

77 8.4.2 Fast Studies 1

Purpose: Fast Studies 1 repeats a similar study of the fast region as the 3-Group study, but instead of just one thermal group, this study uses the optimized thermal group structure found in thermal studies 5. Variation: The 6-Group at 1000K energy group structure has 2 fast group and 4 thermal groups. (166_X_101_91_46_23) is the group structure with a varying boundary X for all possible energy partitions above upper energy partition 101 (2.38 eV).

The results of this case are shown in Figures 8-85 through 8-95. As seen in the 3-Group studies, the containment of fission produces a more accurate k-effective as well as power and flux shape. The k- eff results between cut 141 and 151 (52.5 KeV and 639 KeV) have the closest matching k-eff with the reference. The power shape is shown to improve for any cut in the range of 120 to 153. The flux shape is shown to improve for regions in 133 to 155.

78

Figure 8-85: k-effective results of (166_X_101_91_46_23)

Figure 8-86: k-effective of Figure 8-87: COMBINE6 Fission Source (166_X_101_91_46_23) Fraction Per Energy Group of (166_X_101_91_46_23)

79

Figure 8-88: Average Radial Power Shape

Percent Difference of (166_X_101_91_46_23) Figure 8-89: Average Radial Flux Shape Percent Difference of (166_X_101_91_46_23)

Figure 8-90: Average Axial Power Shape Figure 8-91: Average Axial Flux Shape Percent Percent Difference of (166_X_101_91_46_23) Difference of (166_X_101_91_46_23)

80

Figure 8-92: Radial Power Percent Difference of Figure 8-93: Axial Flux Percent Difference of (166_X_101_91_46_23); Best: 135-146 and (166_X_101_91_46_23) without Fuel-to- Worst: 161-165 Moderator Interfacial Region Data

Figure 8-92 shows that the best radial power shape agreeing results are partitions in the range of 135-146. The worst are 161-165. It’s interesting to see that having 1-fast groups is just as good as having the best of the 2-fast group cases.

Case Conclusion:

Partitioning the fast region by the fission spectrum improves the k-eff, the flux, and power profiles. The energy region that expects improvement for all 3 results is 133 – 155 or 7.1 KeV to 1.74 MeV. Since 95% of the fission is contained at 146 and the k-eff deviation is the smallest, it’s recommended that the partition be in either 141 or 146.

81

8.4.3 Fast Studies 2

Purpose: Fast Studies 2 continue Fast Studies 1 with a search for a second optimal partition in the fast region. Variation: • The 7-Group structure has 3 fast group and 4 thermal groups. • Base Group Structure (6 groups) Upper Energy Partitions: 166 146 101 91 46 23 • Sensitivity Cases (7 groups): if 166 > X > 146, then: 166 X 146 101 91 46 23 or if 146 > X > 101, then: 166 146 X 101 91 46 23 Where X is a varying boundary for all possible energy partitions above upper energy partition 101 (2.38 eV).

The results of this case are shown in Figures 8-94 through 8-95.

Figure 8-94: k-effective of (166|146|101_91_46_ Figure 8-95: K-effective Deviations less than 20 23) PCM from Reference and the base group structure at 1000K of (166|146|101_91_46_ 23)

82

Figure 8-96: Average Radial Power Shape Figure 8-97: Average Radial Flux Shape Percent Percent Difference of (166|146|101_91_46_ 23) Difference of (166|146|101_91_46_ 23)

Figure 8-98: Average Axial Power Shape Percent Figure 8-99: Average Axial Flux Shape Percent Difference of (166|146|101_91_46_ 23) Difference of (166|146|101_91_46_ 23)

Case Conclusion:

The results of this study have shown 3 potentially good boundaries. These boundaries are 128 (2.04 KeV), 151 (639 KeV), and 155 (1.74 MeV) and can be explained by being caused by marked changes in important isotopes (See Section 0).

83

8.4.4 Physical Explanations for Partitions: 2.04 keV, 639 keV, 1.74 MeV

The upper energy cut 128 (2.04 KeV) can be physically explained by the marked change of U-235’s unresolved resonance region shown in Figure 8-100.

Figure 8-100: U-235 Fission and Total Cross Section; the fission cross section has a marked change at approximately 2 keV.

The upper energy cut 151 (639 keV) can be physically explained by the sudden increase in U-238’s fast fission cross section see Figure 8-102, which is shown in Figure 8-101 by the red line.

The partition of 155 (1.74 MeV) can be explained by the marked change in C-12’s total and elastic cross section. At this energy C-12’s potential scattering region becomes the resonance region.

84

Figure 8-101: U-238 Fast Fission Cross Sections and Total Absorption U-238 T=300 K from ENDF/B-6.2 mat9237(first chance fission) U-238 T=300 K from ENDF/B-6.2 mat9237 (second chance fission) U-238 T=300 K from ENDF/B-6.2 mat9237 (third chance fission) U-238 T=300 K from ENDF/B-6.2 mat9237 (total absorption)

85

Figure 8-102: Natural Carbon’s cross section mcnp/C:2 C-nat. T=300 K from ENDF/B-6.1 mat 600 (Elastic) mcnp/C:1 C-nat. T=300 K from ENDF/B-6.1 mat 600 (Total XS)

86

8.4.5 Fast Studies 3

Purpose: This study is similar to Fast Studies 2 except it replaces the 146 as its boundary to contain fission with 141. The point of this study is to find a second fast group partition and also to compare its results against those with a fission containment energy boundary of 146. Variation: • The 7-Group structure has 3 fast group and 4 thermal groups. • Base Group Structure (6 groups) Upper Energy Partitions: 166 141 101 91 46 23 • Sensitivity Cases (7 groups): if 166 > X > 141, then: 166 X 141 101 91 46 23 or if 141 > X > 101, then: 166 141 X 101 91 46 23 Where X is a varying boundary for all possible energy partitions above upper energy partition 101 (2.38 eV).

The results of this case are shown in Figures 8-103 through 8-108 and in Table 8-7.

87

Table 8-7: k-effective Results of less than 100 pcm for (166|141|101 91 46 23)

88

Figure 8-103: k-effective results of this case Figure 8-104: Radial Power Percent Difference (166|146|101 91 46 23) and the comparison with for (166|141|101 91 46 23)

(166|141|101 91 46 23) results. All cases have better than or equal error in radial power shape when this 7-group is compared to the 6-group base (2 fast groups). Ignoring the boundary region, the partitions 119-129 have the best power profiles.

Figure 8-105: Radial Flux Profile Percent Figure 8-106: Radial Power Profile Percent Difference for (166|141|101 91 46 23) Difference for (166|141|101 91 46 23)

Figure 8-107: Axial Flux Profile Percent

89 Difference for (166|141|101 91 46 23) Figure 8-108: Axial Power Profile Percent Difference for (166|141|101 91 46 23)

The results in this case are similar to Thermal Studies 2. In addition to the 3 partitions found in Thermal Studies 2. The k-eff results of this case also show a good result in partition 134-135 (9.12 KeV – 11.7 KeV). This partition can be explained to be the region between the smooth and unresolved resonance region of U-238 as shown in Figure 8-109.

8.4.6 Physical Explanations for Partitions: 10 keV

Figure 8-109: U-238 absorbtion cross section showing its transition from smooth to unresolved resonance region at approximately 10 keV. 7U-238 T=300 K from ENDF/B-6.2 mat9237

90

8.4.7 Fast Studies 4

Purpose: This study is a progression of Fast Studies 2 with an insertion of a fixed boundary “128” in addition to the already present “146.” Variation: • The 8-Group structure has 4 fast group and 4 thermal groups. • Base Group Structure (7 groups) Upper Energy Partitions: 166 146 128 101 91 46 23 • Sensitivity Cases (8 groups): if 128 > X > 101, then: 166 146 128 X 101 91 46 23 if 146 > X > 128, then: 166 146 X 128 101 91 46 23 if 166 > X > 146, then: 166 X 146 128 101 91 46 23

The results of this case are shown in Figures 8-110 through 8-116 and in Table 8-8. The circled k- eff values in Figure 8-110 produces a shape that mimics C-12’s scattering properties. This reveals the dominating physics in the high energy region. It might be important then to isolate this region. It’s suggested to separate the physical dominating regions of scattering and resonance absorption further by an insertion at 134-135. This also coincides with the partition found in Fast Studies 3.

91

Table 8-8: k-eff with less than 50 PCM difference (166|146|128|101 91 46 23)

Figure 8-111: Natural Carbon’s Elastic

92 Figure 8-110: k-effective for (166|146|128|101 91 Scattering Cross Section for 46 23) (166|146|128|101 91 46 23)

Figure 8-112: Average Radial Power Shape Figure 8-113: Average Radial Flux Shape Percent Difference for (166|146|128|101 91 46 Percent Difference for (166|146|128|101 91 46 23) 23)

Figure 8-114: Average Axial Power Shape Figure 8-115: Average Axial Flux Shape Percent Percent Difference for (166|146|128|101 91 46 Difference for (166|146|128|101 91 46 23) 23)

Figure 8-116: Radial Power Percent Difference for (166|146|128|101 91 46 23)

93

Case Conclusion:

The differences in radial power percent shape for these case scenarios are seen to be very small as seen in Figure 8-116. But there is motivation to select the partition 134 for investigative purposes; partition 134 is chosen because of a slight improvement over all the radial power shape as well as the physical purpose of partition U-238’s smooth resonance region from its overlapping resonances.

94

8.4.8 Fast Studies 5

Purpose: Search for a 5 fast group structure using: • 166 141 134 101 91 46 23 with variable boundary X > 101.

The results of this case are shown in Figures 8-117 through 8-118 and in Table 8-9

Table 8-9: Cases with k-eff less than 50 PCM for (166 141 134 101 91 46 23)

95

Figure 8-117: k-eff comparison between the insertion of fixed partition 134 of (166 141 134 101 91 46 23)*

Figure 8-118: Average Radial Power Shape Percent Difference for (166 141 134 101 91 46 23)* *Partition Case 120 has been omitted due to non-convergence.

The differences between cases are becoming very small such that the only direction provided for choosing group structures is now only the average of the radial power profile and k-eff.

96 Case Conclusion:

Results show little improvement, but partition at 152 (821 KeV) is chosen for further investigation due to slight improvement in radial power shape.

8.4.9 Fast Studies 6

Search for a five fast group structure with 4 fixed thermal groups. • Base Group Structure (9 groups) Upper Energy Partitions: 166 152 141 34 128 101 91 46 23

• Sensitivity Cases (10 groups): if 128 > X > 101, then: 166 152 141 134 128 X 101 91 46 23 if 134 > X > 128, then: 166 152 141 134 X 128 101 91 46 23 if 141 > X > 134, then: 166 152 141 X 134 128 101 91 46 23 if 152 > X > 141, then: 166 152 X 141 134 128 101 91 46 23 if 166 > X > 152, then: 166 X 152 141 134 128 101 91 46 23

The results of this case are shown in Figures 8-119 through 8-124 and in Table 8-10.

Table 8-10: k-eff with less than 50 pcm for (166|152|141|134|128|101 91 46 23)

97

Figure 8-119: k-effective for Figure 8-120: k-effective results with (166|152|141|134|128|101 91 46 23) Comparison to previous group structures

Figure 8-121: Average Radial Power Shape Figure 8-122: Average Radial Flux Shape Percent Difference for Percent Difference for (166|152|141|134|128|101 91 46 23) (166|152|141|134|128|101 91 46 23)

Figure 8-123: Radial Power Percent Difference Figure 8-124: Radial Power Figure 8-of Merit for (166|152|141|134|128|101 91 46 23) for (166|152|141|134|128|101 91 46 23)

98 From Figure 8-120 it can be seen that the new partition “134” raises the curve of k-eff results for all different partitions of X.

The minimum difference with reference in the radial power profile and the k-eff seems to indicate that a cut at 120 (275.4 eV) produces the best result. 120 is the cut-off that separates U-238’s unresolved and resolved resonance regions. Note that the differences are so small that even the Figure 8-of Merit for the radial power profile shows very little differences.

Alternative to the selection of partition 120 are the partitions at 102, 103, or 104. These partitions have similar effects on k-eff; they increase the capture of neutrons because they isolate separated resonances of U-238 as shown in Figure 8-125.

Figure 8-125: U-238 T=300 K from ENDF/B-6.2 mat9237 Total Cross Section

Case Conclusion:

99 In general, the numerical studies seem to give the impression that for k-eff to be predicted accurately requires a proper balance of partitioning energy regions that contribute to both capture and production. Therefore, refinement of energy regions should be applied uniformly to energy regions that accurately represent the actual weighting of certain effects to an energy region. For example, the refinement should not be heavily concentrated in only production or only capture as this would skew the k-effective more towards a higher or lower value depending on which energy range has been given more focus for discretization. Energy refinement should be performed in a manner that uniformly improves the calculation of neutron production and capture rates. To determine the weighted importance of an energy range is also somewhat ambiguous.

8.4.10 Fast Studies Conclusion

The studies in the fast region have shown that the separation of fission source, smooth, unresolved, and isolated resonances for the most important fissile (U-235) and fissionable isotope (U-238) improves results. Generally, the little improvement is seen in the spatial profiles past 4 fast groups.

8.4.11 Fast Studies 7: Constant Lethargy Divisions

Purpose: To investigate if scattering is the dominating physics for energy group selections by creating group structures of equal lethargy.

Fast region group structures were created by first calculating the constraint. The total ∆ lethargy between the maximum cutoff (16.905 MeV) and thermal cut-off of choice (2.38 eV) is 15.78. The variable of study is the number of fast groups. By dividing the total ∆ lethargy of the fast region with the desired number of groups, we can find the energy partitions. These energy partitions are shown in Figure 8-126 for both COMBINE input fine-group structure partition and energy cut-offs (boundaries).

100

Figure 8-126: Fast Region Study using fixed ∆ Lethargy.

The results of this case are shown in Figures 8-127 through 8-128 and in Table Table 8-11.

Table 8-11: k-eff of Constant Lethargy Study

101

Figure 8-127: k-effective results of Constant Figure 8-128: Average Radial Power Shape Lethargy Study* Percent Difference of Constant Lethargy Study * *Base indicates group structure: (166 146 128 101 91 46 23)

The k-eff did not necessarily improve with more groups, but the average radial power shape shows improvement with more groups. Figure 8-129 shows the error of the radial power profile. The center of the fuel region decreases in error with more groups, the fuel-to-moderator interfacial region gets worse with increased number of groups. This may be attributed to the low-buckling assumption, which neglects true environmental feedback.

Applying the idea of uniform energy refinement to reactions of production and capture to k-eff results of this case, we can infer that grouping the entire fast region and subdividing by equal lethargy appears to under predict the k-eff. This might imply that the physics of neutron capture plays a more dominant role than scattering in the fast region.

102

Figure 8-129: Radial Power Profile of Constant Lethargy Study

8.4.12 Constant Lethargy of Fast Region Conclusion

Using more groups shows improved radial power profile at the fuel’s center region, but also contributes to a worsening calculation of the power profile at the fuel-to-moderator interfacial region. This error might be associated with the error caused by diffusion theory and the lack of environment feedback which can be compensated with discontinuity factors or online cross section generation.

The k-eff prediction is shown to be poor when using only equal lethargy as the basis for fast group structure. But using constant lethargy produces better k-eff results than selecting groups randomly as seen in the case study: Maximum Number of Energy Group Structures.

103

8.4.13 Maximum Number of Energy Group Structures

Purpose: To see the effect of having a large number of groups with energy structures selected arbitrarily. The thermal group structure was set fixed for cases 11 - 47 groups.

The results of this case are shown in Figures 8-130 through 8-135 and in Table 8-12 .

Table 8-12: : k-eff of Maximum Number of Energy Groups Case Study

104

Figure 8-131: Energy Group Structures of Max Figure 8-130: k-effective Groups Case Study (Group 11 – Group 23) in COMBINE6 Upper Energy Numbers

105

Figure 8-132: Energy Group Structures of Max Groups Case Study (Group 31 – Group 71) in COMBINE6 Input Fine-Group Structure

106

The k-effective results show that using more energy groups does not guarantee a more accurate k- effective. It does seem to indicate that the radial power profile continues to improve, but with marginal effect per additional group. The computational efficiency is therefore decreasing for a small increase in accuracy.

Figure 8-133: Average Radial Power Shape Percent Difference of Max Groups Case Study Figure 8-134: Radial Power Percent Difference of Max Groups Case Study

Figure 8-135: Radial Power Profile Comparison of Selected Group Numbers of Max Groups Case Study

Case Conclusion:

107

Increasing the number of energy groups, improves the radial power profile, but only slightly. The k- effective does not improve with random group structure selection.

This numerical study agrees with Massimo’s statement that more thermal groups (i.e. 10) produce a better power profile, but the improvement is very little.

8.5 G&S Studies of Broad Group Structures from LWR and Recommended modifications for HTGR

The following study is a numerical investigation of recommended broad group structure by Glasstone and Sesonske (see Table 8-13).

• The first two cases (LWR_4A and LWR_4B) explores using a 4-Group structure with the same physical basis used for LWR’s with the only modification of raising the thermal cut-off from 0.625 eV to either 2.38 eV or 1.86 eV. • The second two cases (LWR_5A and LWR_5B) perform the recommended operation for HTGR which is to use more than one thermal group and maintain a thermal cut-off in the eV range. • The cases LWR_5C and LWR_5D move the thermal boundary of 0.625 eV to 0.43 eV which was found in thermal studies 1 as an important thermal boundary. According to Massimo11, 0.625 eV may the popular thermal cut-off for LWR’s because of Cd’s threshold. • The last two cases investigate an additional boundary at 0.18 eV which is a partition separating the last group of low energy resonances with the 1/V sloped cross sections.

The results of this case are shown in Figures 8-136 through 8-142 and in Table 8-13 and Table 8- 14

108 Table 8-13: Energy Group Structures of LWR and Modifications for HTGR

Table 8-14: k-effective results of LWR and Modifications for HTGR

109

Figure 8-136: of LWR and Modifications for HTGR

Figure 8-137: Average Radial Power Shape Figure 8-138: Average Radial Flux Shape Percent Difference of LWR and Percent Difference of LWR and Modifications for HTGR Modifications for HTGR

110

Figure 8-139: Average Axial Power Shape Figure 8-140: Average Radial Flux Shape Percent Difference of LWR and Percent Difference of LWR and Modifications for HTGR Modifications for HTGR

Figure 8-141: of LWR and Modifications for HTGR

111

Figure 8-142: of LWR and Modifications for HTGR Comment:

As Glasstone & Sesonske recommends at least 2 thermal groups and thus more than 4 groups in total, we see a significant improvement with the 5-Group energy structure. The flux shows a continuing improvement with each additional group. The 2.38 eV cut-off is shown to have a better predicted k-eff value compared to the 1.86 eV cut-off cases. Therefore, there should be at least 2 thermal groups.

8.6 Recommended Best Group Structures Subdivided By Number of Groups

The following is a summary of best cases seen in previous runs. The purpose of this analysis is to compare best group structure cases from different studies to find the best performing group

112 structure. The analysis has been broken into categories by number of groups: 5,6,7,8, and 9 or more groups.

8.6.1 5 Group Selection

The best performing results of the 5-Group cases are shown in Figures 8-143 through 8-145 and in Table 8-15 and Table 8-16

Table 8-15: Best Performing 5-Group Energy Structures

Table 8-16:k-eff results of Best Performing 5-Group Cases

113

Figure 8-143: k-eff of Best Performing 5-Group Cases

Figure 8-144: Percent Difference of Radial Power Profile of Best Performing 5-Group Cases

114

Figure 8-145: Percent Difference in Axial Profile of Best Performing 5-Group Cases

8.6.1.1 Spatial Shielding Effects in Diffusion Theory without Environment Feedback

It may be that diffusion theory without buckling or spatial environment feedback, through preparation of cross sections, may not accurately perform spatial shielding compared to Monte Carlo method. This might be inferred from Figure 8-144 because the radial power profile usually shows that when the outer boundary of the fuel region has a larger error, the inner fuel region will have less error and vice versa.

8.6.1.2 k-effective as a Good Indicator

Initial studies implied that k-eff was a poor measure of a good group structure. However, it appears from results seen in Figure 8-143 and Figure 8-144 that it might be a good indicator because it’s seen that cases with larger k-eff deviation from reference is correlated to cases with larger power profile’s errors in certain regions (ie. center or edge). The worst k-eff deviation from reference is from the cases where radial power profile error is higher on the fuel boundaries (LWR_5D, Th_1B).

115 The next worst cases in terms of k-eff deviation are the cases where the power profile has the least error at the boundary and the most error at the fuel midsection (LWR_5A and LWR_5B).

It’s clear that the four cases cited have the worst agreement in k-eff (Figure 8-143). This might imply then that when k-eff is close to reference, the fission reaction rate error at the fuel-to- moderator interface is cancelling out with the error at the fuel’s center.

According to the justifications discussed above, it is recommended that of the 5 Group structures Th2_46, LWR_5C, and Th_1A are adequate. Because the radial and axial power profiles are slightly better in LWR_5C, it is the recommended choice.

8.6.2 6 Group Selection

The best performing results of the 6-Group cases are shown in Figures 8-146 through 8-148 and in Table 8-17 and Table 8-18 .

Table 8-17: Table 8-1: Energy Structure of Best Performing 6-Group Cases

116

Table 8-18: Table 8-2: k-eff results of Best Performing 6-Group Cases

Figure 8-146: k-eff results of Best Performing 6-Group Cases

117

Figure 8-147: Percent Difference of Radial Power Profile of Best Performing 6-Group Cases

Figure 8-148: Percent Difference of Axial Power Profile of Best Performing 6-Group Cases

For similar justifications as the 5-Group case, the Fa1_6G_146 is recommended as the best 6-Group case.

118 8.6.3 7 Group Selection

The best performing results of the 7-Group cases are shown in Figures 8-149 through 8-151 and in Table 8-19 and Table 8-20.

Table 8-19: Table 8-3: Group Structures of Best Performing 7-Group Cases

Table 8-20: k-eff results of Best Performing 7-Group Cases

119

Figure 8-149: k-eff results of Best Performing 7-Group Cases

Figure 8-150: Percent Difference of Radial Power Profile of Best Performing 7-Group Cases

120

Figure 8-151: Percent Difference of Axial Power Profile of Best Performing 7-Group Cases

More evidence supporting the correlation of worst agreeing k-eff and issues arising from spatial shielding is further seen in the radial power profile of these 7-Group cases (Figure 8-150). It’s clearly shown that two largest deviations of k-effective have the largest and smallest error in the fuel-to-moderator interfaces and fuel’s center.

It appears in the power profiles that Th_3A may be the best 7-Group structure.

121

8.6.4 8 Group Selection

The best performing results of the 8-Group cases are shown in Figures 8-152 through 8-154 and in Table 8-21 and Table 8-22

Table 8-21: Group Structures of Best Performing 8-Group Cases

Table 8-22: k-eff results of Best Performing 8-Group Cases

122

Figure 8-152: k-eff results of Best Performing 8-Group Cases

Figure 8-153: Percent Difference of Radial Power Profile of Best Performing 8-Group Cases

123

Figure 8-154: Percent Difference of Axial Power Profile of Best Performing 8-Group Cases

Fast2A and Fa4_8G_166146128_134 are recommended. Since the two group structures are nearly identical except for the lowest group, the structure Fa4_8G_166146128_134 is recommended because of its better physical justification of having a boundary at 0.12 eV (COMBINE #23).

124

8.6.5 9+ Group Selection

The best performing results of the 9 or more-Group cases are shown in Figures 8-155 through 8-157 and in Table 8-23 and Table 8-24

Table 8-23: k-eff results of Best Performing 9 or more Group Cases

Table 8-24: k-eff results of Best Performing 9 or More Group Cases

125

Figure 8-155: k-eff results of Best Performing 9 or More Group Cases

Figure 8-156: Percent Difference of Radial Power Profile of Best Performing 9 or More Group Cases

126

Figure 8-157: Percent Difference of Axial Power Profile of Best Performing 9 or More Group Cases

Fast4B shows it has relatively low percent error and also is well agreeing in k-effective.

Chapter 9

Conclusions

The traditional energy group selection methodology based on the energy regions of physical importance and the cross section dependence of energy for important isotopes is assessed in this study. The results of the fast and thermal studies have shown that such physical importance approach is valid. The underlying theme of this study agrees with GGA’s statement that the choice of broad energy group boundaries using this approach is a matter involving judgement and trial and error experimentation.

For the thermal spectrum, the group structures should be selected based on the low energy resonances (low lying resonances) of fuel material. The optimal partitions found are 46 (0.43 eV), 23 (0.12 eV), and 91 (1.6 eV) in order of importance; partition 46 (0.43 eV) is the most important because it bounds the lower end of the large Pu-240 resonance peak at 1.06 eV.

The fast spectrum has many physical considerations and because of the numerous physical interactions, the insertion of each boundary should be placed in the order of the dominating physics. The first containment should be the fission source; the optimal partition found in this study is at 141 (52.5 KeV) or 146 (183 KeV). The second partition should be the separation of the continuous or smooth resonances for the main fissile isotope; the optimal partition for this case study corresponds to U-235 and at 128 (2.04 KeV). The third partition should be for the same physical reasoning as the second partition, but for the principle neutron capturing isotope; in this study it was found to be U-238 and the optimal partition was found to be 134 (9.12 KeV).

Two additional fast partitions can be added, but if the two additional partitions are included, it must be done together. The reason they must be added together is because

128 one increases k-eff by improving fast fission calculations and the other reduces k-eff by improving resonance capture calculations. The first partition that improves fast fission corresponds with the fast fission threshold for U-238, the principle fissionable isotope; the numerical studies indicate this partition to be 152 (821 KeV). The second partition is the improvement of the neutron capture rate by isolating separated resonances for the same principle fissionable isotopes (U-238) from either the lower energy end or from the upper energy end. The lower energy partition separates the isolated resonances from the overlapping resonances; the numerical studies indicate the partition at the boundary 103. The upper energy partition that separates the resolved and unresolved resonances was found based on numerical results in this study to be 120 (275.4 eV).

It was found from this numerical study that there should be at least 2 thermal groups, but the use of more than 3 thermal groups produces very little improvements. This finding is in contrast to Massimo’s statement that many thermal groups are needed for an accurate power distribution. It, however, agrees with Stamm’ler findings, but Stamm’ler’s study was designed for LWR group structures. It is possible that accuracy with more thermal groups can still be achieved when performing online-cross section generation instead of the separate two-step process.

The following summarizes the group structure selection method (in order of importance):

1. Thermal spectrum and up-scatter containment. 2. Fission source containment. 3. Isolation of low lying resonances in thermal region (at least 2 thermal groups) 4. Separation of continuum (smooth) resonances from the unresolved resonances for the most important fissile isotope. 5. Separation of continuum (smooth) resonances from the unresolved resonances for the most important capture isotope. 6. (Optional) Fast fission improvement for the main fissionable isotope.*

129 7. (Optional) Improvement of the isolated resonance region by partition below or above this energy region.* * 6 and 7 should be used together.

The selection of the thermal cut-off should be carefully made to contain up- scattering with respect to the temperature of graphite moderator. For the temperature of 1000K, it is shown through the 9-Group structure study that the cut-off’s 1.44, 1.86, and 2.38 eV have similar results. Perhaps the only consideration between these 3 cut-offs are the low lying resonances. The 2-Group study can only be performed by moving both the thermal cut-off and the group structure and because of this two simultaneous variable movement; it is difficult to make any real conclusions on whether the thermal cutoff or the group boundary is the important contributor. It might be inferred, however, that 2.38 eV (the highest possible thermal cut-off) is the best selection based on the increasing dominant effect of up-scattering shown between the comparison of 2-Group k-eff results of 300K and 1000K.

It seems that by increasing the number of groups, there is a decrease in importance of the placement of each boundary. Effectively, it reduces the risk of choosing a boundary that produces more error. However, in the selection of broad thermal groups, partitions for thermal spectrum should not be selected at energies that intersect resonances.

To achieve the optimal increase in performance of each boundary, partitions should be selected based on the containment of important physical considerations. Using a large number of groups will improve the power profile, but each additional group only slightly improves results and is therefore not computational efficient. This conclusion may change when using an online cross section generation method.

The fast studies have indicated that k-eff becomes more challenging to accurately predict with a large number of groups. The cause of this may be from refining an energy

130 region that shifts k-eff up or down. For instance group structures that improve the neutron’s resonance escape probability without improving the resonance capture rates will see an over prediction of k-eff from reference if its value is already fairly accurate.

131 The recommended group structures for use on the PBMR are shown in Table 9-1. The physical explanations are shown in Table 9-2 and Table 9-3.

Table 9-1: Recommended Group Structures for the PBMR

Table 9-2: Physical Explanations for Recommended Group Structures of the PBMR (Groups 5 – 7)

132

Table 9-3: Physical Explanations for Recommended Group Structures of the PBMR (Groups 8 & 10)

Chapter 10

Future Work

It is likely that one of the largest errors introduced in this phase of the study is the generation of cross sections with the use of low buckling assumption instead of the generation of cross sections on-line. In addition, this study was constrained to using COMBINE6’s fine-group cross section library. If the fine group structure is not sufficiently refined in energy regions of importance for the local or micro-flux, it may prevent the flexibility needed for more accurate cell-level energy collapsing. COMBINE6 has also some limitations and the impact of those on the results of this study can be assessed with the improved COMBINE7. The goal of the next phase in the energy group structure study is to find optimal group structures utilizing the more systematic, consistent, and sophisticated energy group selection methodology called CPXSD (Contribution and Point-wise Cross-Section Driven) methodology [11]. This method will be applied first to fine–group (cell analysis) and then for broad group (core analysis) structures for PBMR analysis. It’s also recommended that this numerical study be repeated using online cross section generation and with a fine-group structure found using the CPXSD method. Since the research in this thesis assumed a simple spectral zone assignment, when a better spectral zone assignment is developed for the equilibrium cycle of the PBMR, it’s suggested to perform more sensitivity studies on energy group structures to experiment and observe any significant new findings. It is the author’s opinion that the end goal for this project should be to achieve an self-checking adaptive energy and spatial discretization method. The adaptive discretization assignment should employ a method that does not use brute force techniques to search for a better pattern due to the costly and inefficient manner of this type of analysis. A numerical method such as Chebyshev Polynomials might be one method of approach to attack the discretization problem. It could potentially be

134 employed for both energy and space, but complications seem to appear in which should be first or if it can be done simultaneously. This adapative method may be an overly ambitious idea, but would prove to be a significant tool for transient analysis if successful. And in the words of the famous Hungarian mathematician G. Polya, “the more ambitious plan may have more chances of success.”

Bibliography

1. Bandini, B., “A Three-dimensional Transient Neutronics Routine for the TRAC-PF1 Reactor Thermal Hydraulic Computer Code”, Ph.D. Thesis, The Pennsylvania State University (1990). 2. Bell, G.I. & Glasstone, S. Nuclear Reactor Theory, Van Nostrand, Princeton, NJ (1970). Pgs. 182, 439. 3. CASMO-3 User’s Manual. Version 4.8. Edenius, Malte & Forssen, Bengt H., STUDSVIK/SOA-94/9. Nov 1994. Section 2. Methodology. Pp: 2-1. 4. Bondarenko, I.I. Group Constants for Nuclear Reactor Calculations. Constants Bureau. NY, 1964. 5. Colak, U. & Seker, V., “Monte Carlo Criticality Calculations for a Pebble Bed Reactor with MCNP“, Nuclear Scence. and Engineering. 6. Duderstadt, J. & Hamilton, L., Nuclear Reactor Analysis, John Wiley & Sons. Canada (1976). Chapter 2, 7-10. 7. Glasstone, S. & Sesonske, A., Nuclear Reactor Engineering, 4th Ed. Vol. 1. Chapman and Hall. NY (1994), Few-Group Constants: 211. 8. Grimesey, R.A. et al., “COMBINE/PC-A Portable ENDF/B Version 6 Neutron Spectrum and Cross Section Generation Program,” Idaho National Laboratory (1994). 9. KAERI, Korea Atomic Energy Research Institute. ENDFPLOT 2.0. 2007 . 10. Koclas, Jean. Nuclear Analysis of Reactors. (1998). Chapter 4: Energy Condensation. 11. Kriangchaiporn, N., “Transport Model Based on 3-D Cross Section Generation for TRIGA Core Analysis“, Ph.D. Thesis, The Pennsylvania State University, 2006. 12. Massimo, L., Physics of High Temperature Reactors, pp. 92–94, Pergamon Press, NY (1976) 13. Merrill, M., “Nuclear Design Methods and Experimental Data in Use at Gulf General Atomic”, pp. 52-55, 129-136, Gulf-GA-A12652 (GA-LTR-2), Gulf Oil Company, San Diego, CA (1973) 14. Mkhabela, P., et al., “Systematic Method for Neutron Energy Group Structure Selection for HTR Analysis,” ANS 2007 Annual Meeting, Boston, Massachusetts, June 24-28, 2007, American Nuclear Society (2007). 15. Mkhabela, P.T. “Sensitivity Study on the Energy Group Structure for HTR analysis,” Report to INL, August 2006.

136 16. Mphahlele, R., “Cross section generation for pebble bed reactors”, M.S. Thesis. The Pennsylvania State University, 2003. 17. Mphahlele, R., et al, “Spectral Zones Selection Methodology for Pebble Bed Reactors”, Submitted to PHYSOR-2008 Conference 18. Pelloni, S. & Giesser, W. Nuclear Data Related to HTR’s. “Uncertainties in physics calculations for gas cooled reactor cores”, Proceedings of a specialist's meeting held in Villigen, Switzerland, 9-11 May 1990. International Atomic Energy Agency, Vienna (Austria). International Working Group on Gas-Cooled Reactors. IWGGCR--24, pp: 98- 103. 19. Reitsma, F., et al, “PBMR Coupled Neutronics/Thermal Hydraulics Transient Benchmark - The PBMR-400 Core Design, Benchmark Definition”, Draft V03, to be published by OECD. 20. Ronen, Y., High Converting Water Reactors, CRC Press, Boca Raton, Florida (1990), pp: 90-91. 21. RSIC Computer Code Collection, “MCNP5 Monte Carlo N-Particle Transport Code System,” Los Alamos National Laboratory (2003). 22. Stamm’ler, R. J. and Abate, M. Methods of Steady State Reactor Physics in Nuclear Design, Academic Press, 1983, pp: 100, 346. 23. Yamasita, K. et al, “Effects of Core Models and Neutron Energy Group Structures on Xenon Oscillation in Large Graphite-Moderated Reactors,” Journal of Nuclear Science and Technology, 30[3], March 1993: 249-260.

Appendix A

Normalization Issues Concerning Flux and Power Profiles between NEM and MCNP5

Initial computations of flux and power profiles showed a problematic discrepancy between NEM and MCNP5. The scaling methods used in NEM and MCNP were thoroughly reviewed and it was concluded that the simplest solution to the discrepancy issue to use a normalized flux and power profile for comparison. The normalization procedure is the same method as computing a maximum-to-average flux and power but for all flux values instead of just the maximum flux (Lamarsh, 3rd Ed., p. 281). Figure A-1 and Figure A-2 show the comparison with using absolute flux and normalized flux, respectively.

Radial Tot al Flux ‐ No Upscatter ‐ 1000K 3.5E+14 REF 0.414 3.0E+14 0.532 0.683 2.5E+14 0.876 cm2) ‐ 2.0E+14 1.125 1.44 (n/sec

1.5E+14 1.86 Flux 2.38

Total 1.0E+14 Fuel Zone C.R. Zone 5.0E+13

0.0E+00 0 50 100 150 200 250 300 350 Radial Distance from Center (CM) Figure A-1: Initial Method: Absolute Radial Flux Comparison*

*The flux output was approximately two times magnitude of difference. This result incorporates this multiple.

138

Figure A-2: Final Method: Normalized Radial Flux Comparison

Absolute Flux: fisslu φφ(l,k,n,lg)'= (l,k,n,lg)* fissot fisslu fisslu Normalized φφ(l,k,n,lg)* (l,k,n,lg)* Flux: fissot fissot φmod (l,k,n,lg)'==all_ nodes * NumOfNodes ⎛⎞fisslu fisslu average⎜⎟φ(l,k,n,lg)* ∑ φ(l,k,n,lg)* ⎝⎠fissot node fissot

φ(l,k,n,lg) φmod (l,k,n,lg)'= all_ nodes * Number of Nodes ∑ φ(l,k,n,lg) node

The calculation shown above reveals that in the normalized flux method, fisslu and fissot cancels out and therefore the resulting value is independent of the internally summed fission reaction rate and therefore, independent of reactor power level source strength. The same method occurs for MCNP.

Absolute Flux: φ = ϕ *S where S = Source Strength Normalized Flux: ϕ *S ϕ φ ==* NumOfNodes mod average(*)ϕ S all_ nodes ∑ ()ϕ node=1

139 The normalization procedure reveals by deductive reasoning that the discrepancy is either caused by the source strength multiplier fisslu/S and/or the equivalent fissot term between the two codes. A similar approach is taken for the power comparison can be shown.

The following is for NEM Relative Normalized Power at a Node:

⎛⎞⎛⎞⎛⎞fissoulk(, )' fvol fissoulk (, )' ⎜⎟⎜⎟⎜⎟ ⎝⎠⎝⎠⎝⎠nvol(, l k ) fissot ' nvol (, l k ) Asmpnmod ==* NumOfNodes ⎛⎞⎛⎞⎛⎞⎛⎞fissou(, l k )' fvol fissou(, l k )' average⎜⎟⎜⎟⎜⎟⎜⎟∑ ⎝⎠⎝⎠⎝⎠nvol(, l k ) fissot ' ⎝⎠nvol(, l k ) ⎛⎞fissou(, l k )* S ⎛⎞ fissou (, l k ) ⎜⎟nvol(, l k ) ⎜⎟ nvol (, l k ) Asmpn==⎝⎠** NumOfNodes ⎝⎠ NumOfNodes mod ⎛⎞fissou(, l k )* S ⎛⎞ fissou (, l k ) ∑∑⎜⎟ ⎜⎟ ⎝⎠nvol(, l k ) ⎝⎠ nvol (, l k )

The following is for MCNP Relative Normalized Power at a Cell

FissonEnergyDepositCELL Source Hf ** mass= NumOfNodes ∑ FissonEnergyDepositCELL Source

FissonEnergyDepositCELL Hf ** mass= NumOfNodes ∑ FissonEnergyDepositCELL

Appendix B

Fine Group Energy Structure of COMBINE6

Table 2-1: Fine Group Energy Structure of COMBINE6

141 Table 2-2: Fine Group Energy Structure of COMBINE6

Appendix C Summary of Selected Best Results

Summary of Selected Best Results from Thermal Studies

Table C-1: Energy Groups in the Best of Thermal Studies

Table C-2: k-eff in the Best of Thermal Studies

143

Figure C-1: k-eff in the Best of Thermal Studies

Figure C-2: Average Radial Power Shape Percent Difference in the Best of Thermal Studies

Figure C-3: Average Radial Flux Shape Percent Difference in the Best of Thermal Studies

144

Figure C-4: Average Axial Power Shape Percent Difference in the Best of Thermal Studies

Figure C-5: Average Axial Flux Shape Percent Difference in the Best of Thermal Studies

Figure C-6: Percent Difference of Radial Power Profile of Best of Thermal Studies

145

Figure C-7: Percent Difference of Axial Power Profile of Best of Thermal Studies

146

Summary of Selected Best Results from Fast Studies

Table C-3: Energy Groups in the Best of Fast Studies (Part 1)

Table C-4: Energy Groups in the Best of Fast Studies (Part 2)

147

Figure C-8: : k-effective in the Best of Fast Studies (graphical)

Table C-5: k-effective in the Best of Fast Studies (tabular)

Table C-6: k-effective in the Best of Fast Studies

Figure C-9: Average Radial Power Shape Percent Difference in the Best of Fast Studies

148

Table C-7: Legend in the Best of Fast Studies

Figure C-10: Radial Power Percent Difference in the Best of Fast Studies

Figure C-11: Axial Power Percent Difference in the Best of Fast Studies

149

Summary of Selected Best Results from Popular Group Structures

Figure C-12: k-effective in the Best of Popular Group Structure Studies

Table C-8: k-effective in the Best of Popular Group Structure Studies

150

Figure C-13: Average Radial Power Shape Percent Difference in the Best of Popular Group Structure Studies

Figure C-14: Average Radial Flux Shape Percent Difference in the Best of Popular Group Structure Studies

151

Figure C-15: Average Axial Power Shape Percent Difference in the Best of Popular Group Structure Studies

Figure C-16: Average Axial Flux Shape Percent Difference in the Best of Popular Group Structure Studies

152

Figure C-17: Radial Power Shape Percent Difference in the Best of Popular Group Structure Studies

Figure C-18: Axial Power Shape Percent Difference in the Best of Popular Group Structure Studies

153 Table C-9: Group Structures in Variations of the Best Thermal and Fast Group Studies

Table C-10: Group Structures in Variations of the Best Thermal and Fast Group Studies

154

Figure C-19: k-effective in Variations of the Best Thermal and Fast Group Studies

Figure C-20: Average Radial Power Shape Percent Difference of the Variations in the Best Thermal and Fast Group Studies

155

Figure C-21: Average Axial Power Shape Percent Difference of the Variations in the Best Thermal and Fast Group Studies

Table C-11: Legend of the Variations in the Best Thermal and Fast Group Studies

156

Figure C-22: Radial Power Shape Percent Difference of the Variations in the Best Thermal and Fast Group Studies

157

Figure C-23: Axial Power Shape Percent Difference of the Variations in the Best Thermal and Fast Group Studies

Appendix D

Popular Energy Group Structures

The following studies were performed to analyze particular energy group structures found in literary sources. For clarity and convenience, the reference location from which the energy group structure was found is listed at the top of each section.

D.1 Gulf General Atomic Fort Saint Vrain (FSV)

Reference: Merrill, M., “Nuclear Design Methods and Experimental Data in Use at Gulf General Atomic”, pp. 52-55, 129-136, Gulf-GA-A12652 (GA-LTR-2), Gulf Oil Company, San Diego, CA (1973).

Table D-1: k-effective in the FSV Study

159 Table D-2: Energy Group Structures of Fort Saint Vrain

Figure D-1: : k-effective of FSV

Figure D-2: Average Radial Power Shape Figure D-3: Average Axial Power Shape Percent Difference of FSV Percent Difference of FSV

Figure D-4: Average Radial Flux Shape Figure D-5: Average Axial Flux Shape Percent Difference of FSV Percent Difference of FSV

160

Figure D-6: Radial Power Percent Difference of FSV

Comment:

The FSV energy group structures are not expected to produce the best group structure because of its reliance on partitions in the fast region which were selected based on Th-232’s resonances. However, many of the other partitions are expected to be re-usable for the PBMR. These boundaries include the thermal structure and the fission source containment.

161

D.2 HRB 13 Group Structure

Case Study: Group Structures of the Hochtemperatur Reaktorbau (HRB) Reference: Pelloni, S. & Giesser, W. Nuclear Data Related to HTR’s. “Uncertainties in physics calculations for gas cooled reactor cores”, Proceedings of a specialist's meeting held in Villigen, Switzerland, 9-11 May 1990. International Atomic Energy Agency, Vienna (Austria). International Working Group on Gas-Cooled Reactors. IWGGCR--24, pp: 98-103.

Table D-3: Original HRB 13 Group Structure

162 Table D-4: k-eff of HRB Group Structure

Table D-5: Energy Group Structures of HRB

Figure D-7: k-effective of HRB Group Structure

Figure D-8: Average Radial Power Shape Figure D-9: Average Radial Flux Shape Percent Difference of HRB Group Structure Percent Difference of HRB Group Structure

163

Figure D-10: Radial Power Percent Difference of HRB Group Structure

164

D.3 MICROX

Reference: Mphahlele, R., “Cross section generation for pebble bed reactors”, M.S. Thesis. The Pennsylvania State University, 2003.

The MICROX group structure is a 7-Group structure that has been previously used in the study MICROX code for the analysis of HTGR’s.

Table D-6: k-effective of MICROX 7-Group Case Study

Table D-7: Energy Group Structures of MICROX 7-Group Case Study

Figure D-11: k-effective of MICROX 7-Group Case Study

165

Figure D-12: Average Radial Power Shape Figure D-13: Average Axial Power Shape Percent Difference of MICROX 7-Group Percent Difference of MICROX 7-Group Case Study Case Study

Figure D-14: Average Radial Flux Shape Figure D-15: Average Axial Flux Shape Percent Difference of MICROX 7-Group Percent Difference of MICROX 7-Group Case Study Case Study

Figure D-16: Radial Power Percent Difference of MICROX 7-Group Case Study

166

D.4 JAERI Group Structure Sensitivity on Xenon Oscillation Analysis

Purpose: To investigate a group structure studied at JAERI (Japan Atomic Energy Research Institute) for HTGR’s in the study of Xe oscillation effects. Reference: Yamasita, K. et al, “Effects of Core Models and Neutron Energy Group Structures on Xenon Oscillation in Large Graphite-Moderated Reactors,” Journal of Nuclear Science and Technology, 30[3], March 1993: 249-260.

Table D-8: k-eff results of JAERI Case Study

Table D-9: Energy Group Structures of JAERI Case Study

Figure D-17: k-effective of JAERI Case Study

167

Figure D-18: Average Radial Power Shape Figure D-19: Average Axial Power Shape Percent Difference of JAERI Case Study Percent Difference of JAERI Case Study

Figure D-20: Average Radial Flux Shape Figure D-21: Average Axial Flux Shape Percent Difference of JAERI Case Study Percent Difference of JAERI Case Study

Figure D-22: Radial Power Percent Figure D-23: Axial Power Percent Difference of JAERI Case Study Difference of JAERI Case Study

168

D.5 Group Structure Proposed by Originators of VSOP

Purpose: To inspect the performance of the group structure suggested by originators of VSOP (Very Superior Old Programs), variations studied at PBMR Company, and a few variations that are expected to improve results. Reference: Naicker, Vishnu. PBMR. VSOP99 PBMR400 DP3: Energy Group Analysis. T000236 Rev. B Draft. August 25, 2006.

Table D-10: k-effective of VSOP Case Study

Figure D-24: k-effective of VSOP Case Study

169 Table D-11: Energy Group Structures of VSOP Case Study

Figure D-25: Average Radial Power Shape Figure D-26: Average Axial Power Shape Percent Difference of VSOP Case Study Percent Difference of VSOP Case Study

Figure D-27: Radial Power Percent Figure D-28: Axial Power Percent Difference of VSOP Case Study Difference of VSOP Case Study

170

Figure D-29: Average Radial Flux Shape Figure D-30: Average Axial Flux Shape Percent Difference of VSOP Case Study Percent Difference of VSOP Case Study

Figure D-31: Radial Flux Percent Figure D-32: Axial Flux Percent Difference Difference of VSOP Case Study of VSOP Case Study

Comment:

The addition of one thermal partition at 0.43 eV shows a significant improvement in both the power profile and k-eff. It’s recommended that at the very least there are two thermal groups.

The addition of one thermal group might not enhance results when using on-line buckling feedback. This may arise because the thermal energy region may sufficiently be incorporated within the micro-flux spectrum of the lattice-physics calculation. Further numerical studies using on-line buckling feedback must be performed to verify this.

171

D.6 Custom A 1000K

Purpose: To test a group structure selected based on observed marked changes in the cross sections of important isotopes.

Table D-12: k-effective results of Custom Groups Study A

172 Table D-13: Energy Group Structures of Custom Selection A

Figure D-33: k-effective of Custom Groups Study A

Figure D-34: Average Radial Power Shape Figure D-35: Average Radial Flux Shape Percent Difference of Custom Groups Study A Percent Difference of Custom Groups Study A

173

Figure D-36: Radial Power Percent Difference of Custom Groups Study A

Comment:

The thermal region is shown to have a significant impact on both the k-eff and the power profile.