EXAMINATION AND IMPROVEMENT OF THE SHEM ENERGY …

The Pennsylvania State University

The Graduate School

College of Engineering

EXAMINATION AND IMPROVEMENT OF THE SHEM ENERGY GROUP STRUCTURE

FOR HTR AND DEEP BURN HTR DESIGN AND ANALYSIS

A Dissertation in

Nuclear Engineering

by

Tholakele Prisca Ngeleka

© 2012 Tholakele Prisca Ngeleka

Submitted in Partial Fulfillment

of the Requirements

for the Degree of

Doctor of Philosophy

August 2012

ii

The dissertation of Tholakele Prisca Ngeleka was reviewed and approved* by the following:

Kostadin N. Ivanov

Distinguished Professor of Nuclear Engineering

Dissertation Advisor

Chair of Committee

Maria Avramova

Assistant Professor of Nuclear Engineering

Samuel Levine

Professor Emeritus of Nuclear Engineering

Chimay J. Anumba

Professor of Architectural Engineering

Arthur Motta

Chair of Nuclear Engineering Program

*Signatures are on file in the Graduate School.

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Abstract

The purpose of this study was to study and improve the SHEM energy group structures (281 and

361) and General Atomics-193 energy group structure utilizing the more systematic, consistent,

and sophisticated energy group selection method referred to as contributon and point-wise cross-

section driven (CPXSD) method. The SHEM-281 and -361 energy group structures were

developed for LWR and General Atomics energy group structure was developed for the fast

reactors. Pebble bed and Prismatic hexagonal block type fuel are used for cell analysis.

DRAGON transport code was used for this task taking advantage of its capability to compute

adjoint fluxes for reactor analysis. MCNP5 was used for generation of the reference solution

selected due to its accuracy of neutron transport calculations. Comparisons with DRAGON

calculations are presented. Pebble fuel element and Prismatic hexagonal block models were

created for both codes. In the DRAGON code, analysis are conducted for the starting energy

group structure by computing both forward and adjoint fluxes as well as the reaction rates and k-

effective. Then forward and adjoint fluxes were used in computing the importance function of

the groups, and the groups with high importance function are subdivided accordingly. The whole

energy group interval of interest was divided into fast, epithermal and thermal regions. Firstly,

the improvement was done for fast region and a new library was created and applied in the fuel

cell analysis until the selected target criteria’s were met (10 pcm relative deviation of /k k∆ and

1 percent deviation of reaction rate of interest). Then similar procedure was repeated for

epithermal and thermal regions. The dominant parameters for each energy region were

considered as required such as the fission cross section for fast region, absorption and scattering

iv

cross sections for epithermal region and absorption cross section for thermal region and k-

effective applied for all energy regions. Pebble fuel element and the Prismatic hexagonal block

were analyzed for depletion based on the improved energy group structure SHEM_TPN-531.

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Table of Contents

List of Figures ........................................................................................................................... ix

List of Tables ............................................................................................................................ xi

Acknowledgements ................................................................................................................ xvii

Chapter 1 ....................................................................................................................................1

Introduction ................................................................................................................................1

1.1 Background .......................................................................................................................1

1.2 Motivation .........................................................................................................................5

1.3 Research Objectives ...........................................................................................................7

1.4 Scope of Research ..............................................................................................................8

Chapter 2 .................................................................................................................................. 10

Literature Review ...................................................................................................................... 10

2.1 Introduction ..................................................................................................................... 10

2.2 Nuclear Energy Group Structures ..................................................................................... 10

2.2.1 SHEM Energy Group Structure ................................................................................. 11

2.2.2 Ultra Fine Energy Group Structure ............................................................................ 16

2.2.3 Contributon and Point-Wise Cross Section Driven Method ........................................ 17

2.2.4 HTR Energy Group Structure Selection Method ........................................................ 18

2.2.5 An Adaptive Energy Group Constructor .................................................................... 19

Chapter 3 .................................................................................................................................. 21

Nuclear Physics Theory ............................................................................................................. 21

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3.1 Introduction ..................................................................................................................... 21

3.2 Transport Equation .......................................................................................................... 21

3.2.1 Angular Discretization ............................................................................................... 25

3.2.2 Energy Discretization ................................................................................................ 28

3.3 Adjoint Transport Equation .............................................................................................. 30

3.4 Neutron Energy Regions .................................................................................................. 31

3.4.1 Neutron Slowing Down ............................................................................................. 32

3.4.2 Resonance Absorption ............................................................................................... 34

3.4.3 Neutron Thermalization ............................................................................................. 39

Chapter 4 .................................................................................................................................. 40

Monte Carlo Reference Results ................................................................................................. 40

4.1 Introduction ..................................................................................................................... 40

4.2 MCNP5 Code Description ............................................................................................... 40

4.3 HTR Fuel Specifications .................................................................................................. 41

4.4 MCNP5 Results ............................................................................................................... 46

Chapter 5 .................................................................................................................................. 61

Multi-group Structure Analysis ................................................................................................. 61

5.1 Introduction ..................................................................................................................... 61

5.2 DRAGON Code Description ............................................................................................ 61

5.3 Cross Section Library Generation .................................................................................... 67

5.4 Sensitivity Analysis Results ............................................................................................. 72

5.4.1 DRAGON Results used in the Sensitivity Study ........................................................ 72

5.4.2 Sensitivity Analysis Comparison with MCNP5 Results ............................................. 75

Chapter 6 .................................................................................................................................. 82

Multi-group Energy Structure Improvement .............................................................................. 82

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6.1 Introduction ..................................................................................................................... 82

6.2 Contributon and Point-Wise Cross Section Driven Method .............................................. 83

6.3 Improvement of SHEM-281 Energy Group Structure ....................................................... 85

6.3.1 Fast Energy Region Improvement .............................................................................. 85

6.3.2 Epithermal Energy Region Improvement ................................................................... 90

6.3.3 Thermal Energy Region Improvement ....................................................................... 93

6.3.4 Improved SHEM-281 Energy Group Structure for all Regions .................................. 98

6.4 SHEM-361 Energy Group Structure Results .................................................................. 104

6.4.1 Fast Energy Region Improvement ............................................................................ 105

6.4.2 Epithermal Energy Region Improvement ................................................................. 110

6.4.3 Thermal Energy Region Improvement ..................................................................... 114

6.4.4 Improved Energy Group Structure for all Regions (SHEM-361) .............................. 119

6.5 General Atomics-193 Energy Group Structure ............................................................... 125

6.5.1 Fast Energy Region Improvement ............................................................................ 125

6.5.2 Epithermal Energy Region Improvement ................................................................. 130

6.5.3 Thermal Energy Region Improvement ..................................................................... 135

6.5.4 Improved GA-193 Energy Group Structure for all Regions ...................................... 140

Chapter 7 ................................................................................................................................ 147

Comparative Analysis of ENDF Data Files ............................................................................. 147

7.1 Introduction ................................................................................................................... 147

7.2 Comparison Results ....................................................................................................... 147

7.3 Nuclear Data Advancements .......................................................................................... 153

Chapter 8 ................................................................................................................................ 156

Depletion Analysis .................................................................................................................. 156

8.1 Introduction ................................................................................................................... 156

8.2 Pebble Fuel Element ...................................................................................................... 158

8.3 Prismatic Hexagonal Block ............................................................................................ 163

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Chapter 9 ................................................................................................................................ 169

Conclusions and Recommendations ........................................................................................ 169

9.1 Conclusions ................................................................................................................... 169

9.2 Recommendation ........................................................................................................... 174

References .............................................................................................................................. 176

Appendices ............................................................................................................................. 181

A1 MCNP5 Input Decks ...................................................................................................... 181

A1.1 Pebble Fuel Element ................................................................................................ 181

A1.2 Prismatic Hexagonal Block ...................................................................................... 187

A2 DRAGON Input Decks .................................................................................................. 200

A2.1 Pebble Fuel Element ................................................................................................ 200

A2.2 Prismatic Hexagonal Block ...................................................................................... 203

A3 Energy Group Structures ................................................................................................ 208

A3.1 SHEM-361 .............................................................................................................. 208

A3.2 SHEM-281 .............................................................................................................. 211

A3.3 GA-193 ................................................................................................................... 213

A3.4 SHEM_TPN-407 ..................................................................................................... 215

A3.5 SHEM_TPN-531 ..................................................................................................... 218

A3.6 GA_TPN-537 .......................................................................................................... 222

A4 Depletion Data Analysis ................................................................................................ 226

A4.1 Pebble Fuel Element Nuclides Concentrations ......................................................... 226

A4.2 Pebble Fuel Element Fission Products Concentrations ............................................. 227

A4.3 Prismatic Hexagonal Block Nuclides Concentrations ............................................... 229

A4.4 Prismatic Hexagonal Block Fission Products Concentrations ................................... 230

A4.5 Pebble Fuel Element Criticality Data per Burnup Step ............................................. 232

A4.6 Prismatic Hexagonal Block Criticality Data per Burnup Step ................................... 232

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List of Figures

Figure 3.1: The position and direction characterizing a neutron ................................................. 22

Figure 3.2: Low energy resonances for U-238 ........................................................................... 34

Figure 3.3: Low energy resonances for Th-232 .......................................................................... 35

Figure 3.4: Flux depreciation under the resonance region .......................................................... 36

Figure 3.5: Unresolved resonances for U-235 (t2.lanl.gov) ........................................................ 37

Figure 3.6: Unresolved resonances for U-238 (t2.lanl.gov) ........................................................ 37

Figure 4.1: MCNP5 Pebble model ............................................................................................. 48

Figure 4.2: MCNP5 Prismatic block model ............................................................................... 49

Figure 4.3: Pebble keffective and source convergence ............................................................... 54

Figure 4.4: Prismatic keffective and source convergence ........................................................... 54

Figure 5.1: DRAGON flow chart [11] ....................................................................................... 66

Figure 5.2: Flow chart for DRAGON library production ........................................................... 71

Figure 6.1: Importance function for fast energy region for 281, 309 1nd 323 groups. ................ 87

Figure 6.2: Importance function for epithermal energy region for 281 and 333 groups .............. 91

Figure 6.3: Importance function for thermal energy region for 281, 297 and 313 groups ........... 95

Figure 6.4: Importance function for fast energy region for 361,389 and 403 groups ................. 106

Figure 6.5: Importance function for epithermal energy region for 361 and 455 groups ............ 111

Figure 6.6: Importance function for thermal energy region for 361, 393 and 395 groups ......... 115

Figure 6.7: Importance function for fast energy region for 193 and 227 groups ....................... 127

Figure 6.8: Importance function for epithermal energy region ................................................. 132

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for 193, 235, 319, 355, 399 and 485 groups ............................................................................. 132

Figure 6.9: Importance function for thermal energy region for 193, 205 and 211 groups ......... 137

Figure 7.1: Neutron capture on natural carbon for ENDF/B-VII.1 ........................................... 149

and ENDF/B-VII.0 (Chadwick et al, 2011) .............................................................................. 149

Figure 8.1: K-effective versus burnup ..................................................................................... 159

Figure 8.2: Neutron flux spectrum at the beginning of life ....................................................... 160

Figure 8.3: Neutron flux spectrum at the end of life ................................................................. 160

Figure 8.4: Nuclides concentration in atom/barm.cm ............................................................... 162

Figure 8.5: Fission products buildup in a Pebble fuel element ................................................. 163

Figure 8.6: Prismatic block K-effective versus burnup ............................................................ 164

Figure 8.7 Neutron flux spectrum at the beginning of life ........................................................ 164

Figure 8.8: Neutron flux spectrum at the end of life ................................................................. 165

Figure 8.9: Nuclides concentration in atom/barn.cm ................................................................ 167

Figure 8.10: Fission products buildup in a Prismatic hexagonal block ..................................... 168

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List of Tables

Table 4.1: Coated Particle specifications (common for all types of fuel) .................................... 43

Table 4.2: Material specification (common for all fuel types) .................................................... 44

Table 4.3: Pebbles representative of PBMR fuel ........................................................................ 45

Table 4.4: Prismatic fuel lattice data .......................................................................................... 46

Table 4.5: Pebble fuel element results calculated using ENDF/B-VII.0 ..................................... 50

Table 4.6: Pebble fuel element results calculated using ENDF/B-VI.8 ....................................... 50

Table 4.7: Percent deviation of pebble FE results for ENDF/B-VII.0 and ENDF/B-VI.8 ........... 51

Table 4.8: Prismatic block results calculated using ENDF/B-VII.0 ............................................ 52

Table 4.9: Prismatic block results calculated using ENDF/B-VI.8 ............................................. 52

Table 4.10 Percent deviation of prismatic block results for ENDF/B-VII.0 and ENDF/B-VI.8 .. 53

Table 4.11: Pebble fuel element results calculated using ENDF/B-VII.0 ................................... 55

Table 4.12: Pebble fuel element results calculated using ENDF/B-VI.8 ..................................... 56

Table 4.13: Percent deviation of pebble FE results for ENDF/B-VII.0 and ENDF/B-VI.8 ......... 56

Table 4.14: Prismatic block results calculated using ENDF/B-VII.0 .......................................... 57

Table 4.15: Prismatic block results calculated using ENDF/B-VI.8 ........................................... 57

Table 4.16 Percent deviation of prismatic block results for ENDF/B-VII.0 and ENDF/B-VI.8 .. 58

Table 4.17: ENDF/B-VII.0 and ENDF/B-VI.8 data files improvement comparisons. ................. 60

Table 5.1: Pebble reaction rates and criticality for SHEM-281 energy group structure ............... 72

Table 5.2: Prismatic block reaction rates and criticality for SHEM-281 energy group structure . 73

Table 5.3: Pebble reaction rates and criticality for SHEM-361 energy group structure ............... 73

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Table 5.4: Prismatic block reaction rates and criticality for SHEM-361energy group structure .. 74

Table 5.5: Pebble reaction rates and criticality for GA-193 energy group structure .................... 74

Table 5.6: Prismatic block reaction rates and criticality for GA-193 energy group structure ...... 75

Table 5.7: Criticality calculation comparisons for ENDF/B-VII.0 .............................................. 76

Table 5.8: Criticality calculation comparisons for ENDF/B-VII.0 .............................................. 76

Table 5.9: Pebble FE reaction rates calculation comparisons for ENDF/B-VII.0 ........................ 77

Table 5.10: Prismatic block reaction rates calculation comparisons for ENDF/B-VII.0 .............. 78

Table 5.11: Pebble FE reaction rates calculation comparisons for ENDF/B-VII.0 ...................... 80

Table 5.12: Prismatic block reaction rates calculation comparisons for ENDF/B-VII.0 .............. 81

Table 6.1: Fast group selected in the fast range .......................................................................... 86

Table 6.2: Eigen-value results for fast energy group structure improvement .............................. 87

Table 6.4: Prismatic block results .............................................................................................. 89

Table 6.5: Epithermal energy groups selected in the epithermal range ....................................... 90

Table 6.6: Eigen-value resulted for epithermal energy group structure improvement ................. 90

Table 6.7: Pebble FE results ...................................................................................................... 92

Table 6.8: Prismatic block results .............................................................................................. 93

Table 6.9: Thermal groups selected in the thermal region .......................................................... 94

Table 6.10: Eigen-values resulted for thermal energy group structure improvement .................. 94

Table 6.11: Pebble FE results .................................................................................................... 96

Table 6.12: Prismatic block results ........................................................................................... 97

Table 6.13: Energy group structure improved from SHEM-281 to 407 ...................................... 98

Table 6.14: Reaction for SHEM-281 and 407 energy group structures ....................................... 99

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Table 6.15: Reaction for SHEM-281 and 407 energy group structures ..................................... 100

Table 6.16: Comparisons for SHEM_TPN-407 with SHEM-281 energy group structure for the pebble FE ................................................................................................................................ 101

Table 6.17: Comparisons for SHEM_TPN-407 with SHEM-281 energy group structure for the prismatic block ........................................................................................................................ 101

Table 6.18: MCNP5 results for the pebble ............................................................................... 102

Table 6.19: Comparisons of SHEM_TPN-407 energy group structure to MCNP5 results for the pebble FE ................................................................................................................................ 103

Table 6.20: MCNP5 results for the prismatic block ................................................................. 103

Table 6.21: Comparisons of SHEM_TPN-407 energy group structure to MCNP5 results for the prismatic block ........................................................................................................................ 104

Table 6.22: Fast group selected in the fast range ...................................................................... 105

Table 6.23: Eigen-value results for fast energy group structure improvement .......................... 107

Table 6.24: Pebble FE results .................................................................................................. 108

Table 6.25: Prismatic block results .......................................................................................... 109

Table 6.26: Epithermal energy groups selected in the epithermal range ................................... 110

Table 6.27: Eigen-value resulted for epithermal energy group structure improvement ............. 111



Table 6.30: Thermal groups selected in the thermal region ...................................................... 114

Table 6.31: Eigen-values resulted for thermal energy group structure improvement ................ 116



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Table 6.34: Energy group structure improved from SHEM-361 to SHEM_TPN-531 ............... 119

Table 6.35: Reaction rates for SHEM-361 and SHEM_TPN-531 energy group structures ....... 120

Table 6.36: Reaction rates for SHEM-361 and SHEM_TPN-531 energy group structures ....... 121

Table 6.37: Comparisons for SHEM_TPN-531 with SHEM-361 energy group structure for the pebble FE ................................................................................................................................ 122

Table 6.38: Comparisons for SHEM_TPN-531 with SHEM-361 energy group structure for the prismatic block ........................................................................................................................ 122

Table 6.39: MCNP5 results for the pebble FE ......................................................................... 123

Table 6.40: Comparisons of SHEM_TPN-531 energy group structures to MCNP5 results for the pebble FE ................................................................................................................................ 123

Table 6.41: MCNP5 results for the prismatic block ................................................................. 124

Table 6.42: Comparisons of SHEM_TPN-531 energy group structures to MCNP5 results for the pebble FE ................................................................................................................................ 124

Table 6.43: Fast group selected in the fast range ...................................................................... 126

Table 6.44: Eigen-values resulted for fast energy group structure improvement ....................... 126

Table 6.45: Reaction rates for GA-193 and 227 energy group structures .................................. 128

Table 6.46: Reaction rates for GA-193 and 227 energy group structures .................................. 129

Table 6.47: Epithermal group selected in the epithermal range ................................................ 131

Table 6.48: Eigen-values resulted for epithermal energy group structure improvement ............ 131

Table 6.49: Reaction rates for GA-193, 235, 319, 355, 399, and 485 energy group structures .. 133

Table 6.50: Reaction rates for GA-193, 235, 319, 355, 399, 485 energy group structures ........ 134

Table 6.51: Thermal groups selected in the thermal region ...................................................... 135

Table 6.52: Eigen-values resulted for thermal energy group improvement ............................... 136

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Table 6.55: Energy group structure improved from GA-193 to GA_TPN-537 ......................... 140

Table 6.56: Reaction rates for GA-193 and GA_TPN-537 energy group structures .................. 141

Table 6.57: Reaction rates for GA-193 and GA_TPN-537 energy group structures .................. 142

Table 6.58: Comparisons for 537 with GA-193 energy group structure for the pebble FE. ....... 143

Table 6.59: Comparisons for GA_TPN-537 with GA-193 energy group structure for the prismatic block. ....................................................................................................................... 143

Table 6.60: MCNP5 results for the pebble FE ......................................................................... 144

Table 6.61: Comparisons of GA_TPN-537 energy group structures to MCNP5 results for the pebble FE ................................................................................................................................ 145

Table 6.62: MCNP5 results for the prismatic block. ................................................................ 145

Table 6.63: Comparisons of GA_TPN-537 energy group structures to MCNP5 results for the prismatic block. ....................................................................................................................... 146

Table 7.1: Pebble FE results using ENDF/B-VII.0 and ENDF.B-VII.1 comparisons ................ 150

Table 7.2: END/B-VII.0 and ENDF/B-VII.1 % deviations for the pebble reaction rates and k-effective in pcm ..................................................................................................................... 151

Table 7.3: Prismatic block results using ENDF/B-VII.0 and ENDF.B-VII.1 comparisons ........ 152

Table 7.4: END/B-VII.0 and ENDF/B-VII.1 % deviation for the prismatic block reaction rates and k-effective in pcm ........................................................................................................... 153

Table 7.5: ENDF/B-VII.1, ENDF/B-VII.0 and ENDF/B-VI.8 data file advancements ............. 155

Table 8.1: Nuclides concentration in atom/barm.cm ................................................................ 161

Table 8.2: Fission products in atom/barn.cm ........................................................................... 162

Table 8.3: Nuclides concentration in atom/barn.cm ................................................................. 166

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Table 8.4: Fission products in atom/barn.cm ........................................................................... 168

Table 9.1: Pebble fuel element deviations ................................................................................ 170

Table 9.2: Prismatic hexagonal block deviations ..................................................................... 171

xvii

Acknowledgements

I give great honor to God Almighty for giving me the strength, confidence and patience for

undertaking the study. Thank you for providing a great team to work with and all opportunities

accompanied the study. A conducive environment and all the campus community and friendly

student organizations, worth mentioning is the International Christian Fellowship.

My sincere gratitude goes to Dr Kostadin N. Ivanov for his supervision and assistance

throughout the study. I thank him for giving me this opportunity to be part of his research group,

Fuel Dynamics and Management Research Group. I am thankful to Dr Samuel Levine, for his

contribution throughout the study.

My special thanks to the Department of Mechanical and Nuclear Engineering of the

Pennsylvania State University, and the members of the Reactor Dynamics and Fuel Management

Research Group for the assistance and support.

I greatly appreciate help rendered by the Idaho National Laboratory (INL) for funding this study.

Thank you to Abderrafi Ougouag, Hans Gougar for being part of this work.

xviii

Not forgetting the technical support obtained from Alan Hebert and Vincent Descotes of the

Ecole Polytechnique de Montreal, Michael Pope of INL, Volkan Seker of the University of

Michigan.

Thanking parents, family and friends for their support, encouragements and lot of prayers.

1

Chapter 1

Introduction

1.1 Background

Nuclear energy is presently contributing around 14 % of the world’s electric energy needs and

produces vast amount of energy from a small amount of fuel. Its advantages lie on the fact that it

does not have green gas emissions (CO2, SO2 and NO2

34

) since it is a major problem with the

current fossil fuel power generation industries. Since the advent of nuclear reactor industry,

many power reactors have been designed and operated such as Light water reactors (Pressurized

and Boiling water reactors), Heavy water reactors (CANDU) and Magnox gas cooled reactor

(named after its cladding material magnesium non-oxidizing alloy) [ ]. Gas cooled type

reactors were also designed and operated such as German Arbeitsgemeinschaft Versuchs Reaktor

(AVR) for testing of fuel and other reactor components; however, it was shut down after high

temperature fuel instabilities. A demonstration plant (Peach Bottom) and the fuel development

plants, the DRAGON reactors were built in the United States of America (USA) but were shut

down following technical challenges. Therefore, gas cooled nuclear reactor research and

development was not performed for quite some time.

2

In spite of the challenges that gas cooled reactors encountered, the inherent safety features and

high outlet temperatures have caused recent interest throughout the world for initiating research

to improve their design. They are currently among the proposed generation IV nuclear reactors

(NGNP) that are under development, namely gas cooled fast reactor (GFR), very high

temperature reactor (VHTR), all grouped under high temperature reactors (HTRs). These

reactors promise to provide a secure nuclear system using materials with improved management

of nuclear waste, effective utilization of fuel, economically sound characteristics, and a well

developed safety performance. Their high temperature outlet can be utilized in other processes

such as the hydrogen production and thus make these reactors more useful [9]. There is a need to

improve the accuracy of HTR analysis to make HTRs more economically viable. Improvement

can be made by reducing the computer analysis uncertainties and using more sophisticated

methods.

Due to recent interests on these concepts, research facilities were constructed. At this stage, the

research (experimental) facilities in operation are a 10 megawatt (MW) high temperature reactor

(HTR-10), a high temperature engineering test reactor (HTTR) and the ASTRA critical facility

[33] to obtain detailed understanding of HTRs. All these facilities use helium gas as a coolant

and are graphite moderated with high temperature outlet of approximately 850°C to 950°C.

HTR-10 is a pebble bed type reactor that has been in operation at the Institute of Nuclear Energy

in China (INET) since December 2002. HTTR is a 30 MW thermal power reactor using

hexagonal assemblies (pin in block) type of fuel, operated in Oarai research reactor of Japan

Atomic Research Institute.

3

Whereas ASTRA is a Russian zero power critical facility at the Kurchatov Institute built for the

investigation of neutronics of high temperature reactors such as the GT-MHR reactor under

development at General Atomics in USA, [30].

Major developments are now focusing on General Atomics design of the GT-MHR and on the

VHTR under development at the Idaho National Laboratory (INL). Both GT-MHR and VHTR

use a prismatic block fuel type. South Africa had major contributions into the HTR research and

developments through the Pebble Bed Modular Reactor that was designed to use pebbles, an

extension of the German AVR. The similarities of the Pebble fuel element and the prismatic

hexagonal block fuel types are that they are both embedded with tri-structural isotopic coated

particles. These coated particles consists of a kernel (UO2

), low density carbon (buffer layer),

pyrolytic carbon (inner and outer layers) and silicon carbide. Consequently, these coated

particles make an HTR design to be a double heterogeneous system, firstly between the particles

in a graphite matrix and then between the fuel element and other structural materials like

moderator and a reflector. This results in a significant change in the physics of neutron slowing

down, absorption and scattering processes in the high temperature reactors as compared to that in

the light water reactors. Hence, advanced ways of treating double heterogeneity in graphite

moderators and neutron behavior in these reactors is of utmost importance.

4

One of the tasks in meeting the research objectives of the development program is to obtain

excellent fine energy group structures that allow accurate calculation of neutron cross sections

for reactor analysis. Neutron cross section is the probability that the various types of nuclear

reactions (fission, scattering and absorption) will occur. The most important dependent

parameters during the reactions for the equilibrium neutron conditions are energy, direction, and

space position. Neutron group cross sections play a very important role in reactor core analysis

and design calculations [8]. Neutron group cross sections are generated from continuous cross

section data files that have been collected for the past decades by experimental measurements

and theoretical calculations (in the United States of America). These data files contain all of

various types of neutron cross sections, e.g. scattering, absorption etc. for all the important

nuclides resulting in a vast volume of data stored in standardized format in the Evaluated

Nuclear Data File (ENDF). The ENDF data files contain both neutron and photon cross sections.

Its latest version is ENDF/B, which contains the complete and evaluated data ready for use by

the nuclear designers and nuclear analysts. Some nuclear data files also exist in other countries

such as Evaluated Nuclear Data Library of the Lawrence Livermore National Laboratory

(ENDL), United Kingdom Nuclear Data Library (UKNDL), Japanese Evaluated Nuclear Data

Library (JENDL), Karlsruhe Nuclear Data File (KEDAK), Russian Evaluated Nuclear Data File

(BROND) and Joint Evaluated Fission File of NEA Countries (JEFF), etc.

5

1.2 Motivation

To advance the development of the accurate analysis of the HTR and ongoing relevant research,

INL, in collaboration with Pennsylvania State University (PSU), have been engaged on

sensitivity studies of energy group structures for the analysis of HTRs [10]. Their previous study

addressed the broad energy group structure used in analyzing graphite moderated high

temperature gas cooled reactors. It is likely that one of the largest sources of error encountered in

that study [10] stems from the method through which nodal leakage was incorporated into the

spectrum calculation.

In these analysis, tables of neutron cross sections were generated for a few selected values of fast

and thermal buckling. Neutron cross sections for the core simulation were interpolated from

these tables using two group internodal currents, omitting much of the information contained in

the fine group energy spectrum. Furthermore, the fine energy group structure hardwired within

INL’s COMBINE-6 code that was used to generate broad group constants is fixed for light water

reactors and has not been optimized for HTR analysis. If the fine energy group structure is not

sufficiently refined in energy regions of importance, such a group structure may prevent the

flexibility needed for more accurate cell level energy collapsing. COMBINE-6 has also some

limitations that must be addressed explicitly in their own right.

As a result, the examination of a new fine energy group structure that is optimized for the

compositions expected in the (Next Generation Nuclear Plant) NGNP was necessary. Hence the

6

goal was to study the currently available energy group structures that are in use like SHEM

energy group structures of Santamarina and Hfaiedh [17] that were developed for LWRs as well

as General Atomics energy group structure, which was developed for the fast reactors.

The SHEM energy group structure has been optimized for LWR applications, addressing their

fuel components as well as the structural materials expected to be present in them. It was verified

to be accurate for both uranium and mixed oxide fuels in LWRs. In addition, it does address

actinides extensively in that their resonant reactions are well covered by the structure. This may

imply that the SHEM energy group structure might be applicable to NGNP and deep burn (DB)

applications without further modifications. However, it is uncertain whether it addresses all the

actinides that arise in the case of very high burnup applications, such as those contemplated for

NGNP and DB-NGNP. Furthermore, the SHEM energy group structure does not give any

particular attention to graphite and its slowing down properties. Although the graphite neutron

cross sections appear very smooth (with no resonance structures) at all energies below about 2

MeV, the absence of special consideration of graphite may imply an inadequate coverage of the

NGNP and deep burn physical situations. Therefore it was necessary to examine this potential

shortcoming of the SHEM energy group structure, and be modified to cover all NGNP and deep

burn related physical phenomena.

General Atomics energy group structure developed for fast energy reactors has been repeatedly

used for the analysis of HTRs without consideration of the effects that may be caused by the

7

underlying purpose of its development. At this point there are no details on the nuclear reactions,

isotopes or any materials addressed for this energy group structure. Pelloni et al. [29] noted that

the GA-193 energy group structure has 92 fast energy groups below 14.92 MeV that are equally

spaced. GA-193 energy group structure is incorporated in MICROX-2 code system that creates

the broad group cross sections with resonance interference and self-shielding from fine group

and point-wise cross section. Therefore, it is important to give a thorough investigation on its

performance towards the thermal reactor physical phenomena.

1.3 Research Objectives

The purpose of this PhD research was to study and improve the SHEM energy group structures

(281 and 361) as well as GA-193 energy group structure utilizing the more systematic,

consistent, and sophisticated energy group selection method referred to as contributon and point-

wise cross-section driven (CPXSD) [19], a method developed by PSU. The sensitivity analysis

on these energy group structures were conducted for the pebble fuel element and prismatic

hexagonal block based on PBR and VHTR respectively. Reference solution was developed with

the MCNP5 code, where three energy boundaries were selected for three regions that correspond

to the SHEM-281, SHEM-361 and GA-193 energy group structures. This study was based on

ENDF/B-VII.0 and ENDF/B-VI.8 nuclear data libraries. Then the energy group structures were

improved using the CPXSD method mentioned above. The best performing energy group

structure was selected for the depletion calculations on both pebble FE and Prismatic hexagonal

block.

8

1.4 Scope of Research

The outline is as follows:

Chapter 1 presents an overview of the background, motivation of the study and the

description of the research objectives, as well as the research scope.

Literature review is presented in chapter 2. The currently available energy group

structures and methods used during their development are summarized.

In chapter 3, nuclear physics theory and neutron transport equation theory is

presented. Included is the Boltzmann transport equation accompanied by methods

that can be used in solving it. Basic information is provided on neutron slowing

down, resonance absorption and thermalization processes.

Chapter 4 presents the description of the continuous energy Monte Carlo Neutron

Particle Transport (MCNP) code used to develop a reference solution for this study. A

detailed description of used HTR fuel specifications is given. Then, the results

(reference solution) along with the sensitivity analysis for the evaluated data files

(ENDF/B-VI.8 and ENDF/B-VII.0) are provided.

A description of DRAGON, a multi-group deterministic transport code used for the

energy group structure improvement and burnup calculations is given in Chapter 5.

Methods utilized for cross section generation is also given in details. Multi-group

energy structure sensitivity analysis are shown.

9

The energy group structure improvement for SHEM-281, SHEM-361 and GA-193

are shown in chapter 6. Each energy group structure was improved for three different

energy regions (fast, epithermal and thermal) separately. These were then compared

to the reference solution (MCNP5 results in chapter 4).

Chapter 7 presents the sensitivity analysis for the evaluated data files (ENDF/B-VII.1

and NDF/B-VII.0) that were conducted in DRAGON for selected energy group

structures. The differences in both evaluated data files are also discussed.

Depletion analysis and the results are discussed in chapter 8.

Conclusions drawn from the study and recommendations made based on findings are

given in chapter 9.

10

Chapter 2

Literature Review

2.1 Introduction

This section gives an overview of the work conducted in developing energy group structures and

the method used by other researchers [1] [10] [17] [18] [24] [25]. It also highlights the

considerations made during those developments to account for important phenomena such as

self-shielding, resonance absorption and overlapping and other nuclear reactor physics

assumptions made.

2.2 Nuclear Energy Group Structures

The distribution of neutron production, slowing down and interaction in a core of a reactor

depends on the neutron energy. At high energy, neutron energy dependence is dominated by

fission spectrum. At intermediate energies, it is dominated by both neutron slowing down and

resonance absorption. While at the low energies it is dominated by thermalization process [8].

Since neutronic calculations form the basis of the research and development of the nuclear

industry, energy group structures have been studied to reduce the errors in these calculations.

However, physics related challenges such as self-shielding are the major concerns and many

researchers are involved in ongoing improvements of the currently used energy group structures

to eliminate all the errors experienced. These energy group structures are used to generate

11

neutron cross section libraries utilized by different transport and diffusion codes for reactor

simulations, analysis and design. Some codes e.g. Monte Carlo utilizes continuous energies and

some utilize multi-group energies e.g. DRAGON, COMBINE etc. Due to the current advances in

both computing and programming languages it has became feasible to develop and use finer

energy group structures for improved core calculations and simulations [18], so finer energy

group structures has been developed and in some places developments are still in progress to

reduce the errors in reactor calculations.

2.2.1 SHEM Energy Group Structure

One of the latest multi-group energy structure developed by Hfaiedh and Santamarina is the

optimized SHEM energy group structure that promised to eliminate simplified self-shielding

calculations of thermal resonances [17]. SHEM energy group structure determination was meant

to correct the setback that was encountered in the XMAS-172 neutron energy group structure

used and developed in CEA-UKEA. The intention was also to elude the use of self-shielding

model in U-238 first large resonances and also in those for LWR absorbers, such as Silver (Ag),

Indium (In), Cadmium (Cd) and Hafnium (Hf) claiming that the assumptions involved in XMAS

were to crude. Reason for elimination of self-shielding model was due to their hypotheses that

were made:

(i) Pure slowing down,

(ii) Asymptotic flux and fine structure uncoupling ( ).ψ ϕΦ ≅

(iii) Wide resonance assumption,

12

(iv) Pij

Work conducted by Mosca et al, [

(inaccurate probability on a neutron interaction) from interface current method for

Dancoff calculations

26] for fast reactors also showed similar problems related to

the self-shielding model approximations as applied in APPOLO-2. SHEM energy group

structure’s expectations were to account for the mutual-shielding effects (resonance

overlapping). The isotopes and resonances that were accounted for included mostly actinides,

fission products and absorbers, moderators/coolants and the structural materials. The resonances

of the isotopes were considered up to 23 eV . The actinides involved are U-234, U-235, U-236,

U-238, Np-237, Pu-238, Pu-239, Pu-240, Pu-241, Pu-242, Am-241, Am-243, Cm-243 and Cm-

244 whom resonances possibly appear at different energies within the range of interest. Whereas,

fission product resonances involved were for Xe-135, Sm-149, Rh-103, Xe-131, Cs-133, Pm-

147, Sm-152, Tc-99, Nd-145, Eu-153, Mo-95, Eu-155 and Eu-154. Absorber isotopes considered

were Ag-107, Ag-109, Cd-113, In-115, Hf-176, Hf-177, Hf-178 and Hf-179, and the burnable

poisons were Gd-155, Gd-157 and Er-167. For structural materials and coolants, the isotopes

considered were Mn-55, Ni-58, Fe-56, Al-27, Na-23 and O-16. The multi-group resonance

reaction rate (Equation 2.1) was compared to point-wise mesh reaction rate (Equation 2.2),

which was taken as a reference.

mg g g

gR σ φ=∑ 2.1

( ) ( )max

min

uref

u

R du u uσ φ= ∫ 2.2

13

0log EuE

=

2.3

( )1g

g

du uu

σ σ=∆ ∫ 2.4

Where:

( )uσ - is the cross section at lethargy u

( )uφ - is the neutron flux at lethargy u

gσ - is the group g cross section

g - present a group number with flat slowing down flux assumption

gφ - is the group g flux

Resonant absorption approximation and fine structure is presented in equation 2.5 where

subscript c and m represents fuel and moderator respectively.

c c c cc m m m mc c c cV R P V R P V Rϕ ϕ ϕ+ = 2.5

Then a reciprocity theorem was applied, which details are found in Nasr and Roushdy [27]. The

flux was simplified to be .ϕ ψ φ≅ , where m m

m

R ϕψ =∑

- is the macroscopic slowing down flux. The

discretization technique of energy group structure was used with small group numbers

consequently reaching the required target of accuracy (about 1%) on reaction rate. Point-wise

14

resonances were calculated using Breit and Wigner multilevel approximation while the flux was

calculated based on heavy nuclide slowing down models such as wide resonance (WR) and

narrow resonance (NR) models. The NR model was used for high energy resonances (actinides)

and it assumes that the neutrons absorbed in the resonance have been scattered at energies higher

than that of the resonance. WR model was also used for low energy resonances with an

assumption that the resonance is very wide compared to the energy change of the scattered

nuclide. Though wide resonance assumptions has reported to cause problems for self-shielding

model, in this work it was used on resonant nuclide slowing down operator shown to give

accurate results. An algorithm was developed for the groups inside the resonance consisting of

discretization of energy variable for equal distribution of reaction rate errors between energy

groups of the resonance. The central peak group was determined and the central group

boundaries fixed as the nearest point-wise energy values around the peak energy value.

Then the group was extended from both ends of the resonance resulting in an increase in a group

error. The calculation was stopped when the total group error reached the target accuracy, which

was fixed in the external iteration. Group widths under the resonance were selected as wide as

possible without violating the established error.

The findings concluded that SHEM energy group structure (281 groups) has proven its accuracy

in calculating the resonance absorption up to 23 eV without using self-shielding approximation

method and also it managed to account for actinides, fission products and absorbers. It also

15

contributed in solving the mutual-shielding effects between resonant isotopes. The SHEM energy

group structure reproduced the structural material neutron cross sections and accounted for U-

238 threshold reactions, oxygen and sodium coolant resonances, which is well suited for fast

spectra calculations (e.g. LMFBR). The overall contribution of the SHEM energy group structure

optimization task was the introduction of an innovative energy scheme that will eliminate self-

shielding model. Its contribution to calculation of reaction rates more accurately will be

advantageous for the nuclear industry, because it allows better design, safety, and control of the

reactor.

The SHEM energy group structure was later refined from 281 to 361 energy groups by Hebert

and Santamarina, [13] by expanding the number of groups in the ranges between 22.5 eV and

11.4 keV using subgroup projection method (SPM). This took advantage of the computing

resource of DRAGON code version 4.02. This work increased the number of groups in the above

mentioned energy ranges from 38 to 118 groups. The idea was to remove the slowing down

correlation model used before, and it is assumed that below 22.5 eV the effects were taken care

of in the previous optimization work and above 11.4 keV the correlation effects vanishes. SPM is

a subgroup approach based on the CALENDF type probability tables. Detailed description of this

method is published by Hebert and Santamarina [13].

16

2.2.2 Ultra Fine Energy Group Structure

Hurai and Ouislomen, 2008 [18] also developed an optimized ultra fine energy group structure

with the intention of avoiding the entire self-shielding problem, and also expected to eliminate

the errors that are caused by resonance interference and overlapping effects. This broadens their

applications to different fuel assemblies. Major concern was the simplifying approximations that

are used in PARAGON and WIMS for self-shielding treatment. The ultra fine energy group

structure has 6064 groups and it was developed based on SHEM energy group structure as

discussed above. It is also developed for light water reactors.

The group boundaries of the SHEM energy group structure were refined keeping in mind the

importance of the location and the practical widths of the resonances. A certain number of

iterations were performed before attaining the final ultra fine energy group structure. This work

used MCNP5 for comparisons of the group boundary selection method. The equation below

presents the iteration method used, wherein the cell iterates until equation 2.6 is satisfied.

, ,g g gP M g MMax σ σ ε σ ξ− ≤ ∀ ∀ ≥ 2.6

Where ε was set to 3%, ξ was chosen to be equal to 10 barns, gPσ is the multi-group cross

section, gMσ is the Monte Carlo derived cross sections from the reaction rates and flux

calculations and ∀ represents all groups. Then, the resulting broad group distribution for their

proposed ultra fine energy group structure was: fast energy range is 20 MeV to 9.118 keV (about

64 groups), resonance energy range is 9.118 keV to 4.0 eV (5877 groups) and the thermal energy

17

range is 4.0 eV to 0.0eV (123 groups). The proposed energy group structure was then tested and

compared to MCNP calculations where the results showed deviations in an acceptable range

resulting in confidence that their use will lead to the reduction of neutronic calculation errors.

2.2.3 Contributon and Point-Wise Cross Section Driven Method

Alpan and Haghighat, 2003 [1] developed a method for the selection of energy group structure

based on contributon theory (William, 1991) [35]. Contributon and Point-wise Cross Section

Driven (CPXSD) method focus on the self-shielding problem as well. CPXSD method refined

the selected initial arbitrary group structure by using the importance of energy groups in the

initial energy group structure and point-wise cross section of an important isotope/material in the

problem. Cross sections were processed for the initial energy group structures using NJOY. Then

self-shielding calculations were performed. Adjoint and forward fluxes were calculated and

further used in the calculation of the importance function. Where maximum importance is

identified, the relevant group structures are refined considering the point-wise cross section of

the important material. If there is a resonance structure, resonance and non-resonance parts are

determined and their areas are computed. Those resonances with larger areas are enclosed in a

sub-group and the remaining resonances and non-resonances are combined about the size of the

largest resonance area. Thus sub-groups are placed within the important groups. If that is not the

case, the group is partitioned evenly by the user’s judgement.

18

The number of subdivisions in other groups is set based on their importance compared to the

maximum importance. The iteration is performed until the subdivisions set for that group are

closely matched. Then a new multi-group energy structure is constructed and ready for use in

generation of the new library.

2.2.4 HTR Energy Group Structure Selection Method

For HTR analysis, Mkhabela et al, 2007 [24] developed a neutron energy group structure

selection method based on systematic survey of the group structures that can predict the

modeling objectives such as effective multiplication factor, reaction rates, and flux distributions.

COMBINE6 code was used for cross section generation, Nodal Expansion Method (NEM) for

core calculations and Monte Carlo Neutron Particle (MCNP) code for the reference solution.

The objectives produced by the group structure were compared to the group structure of the

reference solution. Later, Han, 2008 continued with similar work in determining the best broad

energy group structures consisting of 5, 6, 7, 8 or 9 groups. The starting group structure was

COMBINE6 fine group structure developed at INL. It consisted of 166 energy groups.

Mphahlele, 2008 [25] also conducted a broad group structure selection for PBR analysis. The

starting energy group structure was a recommended best energy group structures by Han, 2008.

It was improved by adding the suitable group boundaries for point-wise resonance calculations

using MICROX-2 code.

19

The new group structure boundaries were examined based on the boundaries of General

Atomics-193 energy group structure. K-effective for the pebble bed reactor was used as the

parameter for these tests and where the minimum deviation between k-effective was observed,

those boundaries were selected as good boundaries. The k-effective for the reference solution

was calculated using Monte Carlo methods. This selection was done for three different

temperatures (800K, 1000K and 1200K). The best boundaries from all three temperatures were

combined into one energy group structure. Also the effectiveness of the new energy group

structure was tested. The iterative method was repeated until 8-group, 10-group and 13-group

structures were obtained.

2.2.5 An Adaptive Energy Group Constructor

Mosca et al, 2011 [26] developed another method to construct a multi-group energy structure

referred to as an adaptive energy mesh constructor (AEMC) and integrated it into nuclear data

processing project GALILEE. AEMC is used to optimize the energy group over infinite

homogeneous medium neutron problems characterizing a given reactor, in this case a fast

reactor. Parameters that were considered are medium temperature, isotopic concentration, cross

section and slowing down of neutrons.

20

The optimization was carried out in two steps, namely, the investigation of the energy group

structure that have results close to the Monte Carlo solutions used as references and followed by

the selection of the appropriate self-shielding model. However, the use of a shielding model later

revealed that it underestimates the U-238 fission production rates as it was pointed out by

Hfaiedh and Santamarina, 2005 [17].

Nonetheless, three self-shielding models were used, that is sub-group self-shielding, Livolant-

Jeanpierre approach and the extended Livolant-Jeanpierre approach. Then, the fine energy group

structure was collapsed into broad energy group structure. AEMC tool searches for the optimal

boundaries with minimum errors during multi-group transport calculations for a specified

problem using swarm algorithm. At the end, the optimization technique was tested with

heterogeneous medium for accuracy where the neutron flux is known to be anisotropic. Details

on the technique used and the outcomes for using different shielding models are published on

Mosca et al, 2011 [26].

21

Chapter 3

Nuclear Physics Theory

3.1 Introduction

This chapter gives a brief summary of forward and adjoint transport equation used to solve

neutron problems. Other neutron physics concepts such as neutron slowing down, resonance

absorption, and thermalization process are highlighted since they form part of the problem

statement of this work.

3.2 Transport Equation

The rate at which nuclear reactions occur is determined by the neutron flux distribution and the

corresponding material distribution in the reactor. Neutron’s behavior and population as occurs

in the reaction rates is the key element in the reactor and chain reaction control. Neutron

distribution has been studied by investigating their motion as they stream inside the reactor [8].

Diffusion theory has been used to analyze nuclear reactors using neutron multi-group theory.

However, this technique has limitations because it is not valid between regions of different

compositions. Therefore, the Boltzmann transport equation adopted from gas dynamics (equation

3.1) is used to study neutrons transport in the reactors in developing the neutron energy group

spectrum.

22

( ) ( )

( ) ( ) ( )4 0

ˆ ˆ ˆ, , , , , ,

ˆ ˆ ˆ ˆ ˆ' ' ' ' , ' , ', ', , , ,

t

s

n n r E t n r E tt

d d E E E n r E t s r E tπ

ν ν

ν∞

∂+ Ω⋅∇ Ω + ∑ Ω =

∂

Ω ∑ → Ω →Ω Ω + Ω∫ ∫ 3.1

where: nt

∂∂

- is the rate of change of neutrons of neutron density

The omega in equation 3.1 is given by ˆ ˆ ˆ ˆx y zi j kΩ = Ω + Ω + Ω = angular distribution of neutrons.

where ˆ sin cosx θ θΩ = , ˆ sin siny θ θΩ = and ˆ cosz θΩ = as shown in Figure 3.1

Z

Y

X

Ω

ZΩ

XΩ

YΩ

φ

θ

Figure 3.1: The position and direction characterizing a neutron

23

( )ˆ ˆ, , ,n r E tνΩ⋅∇ Ω

Streaming term

This term defines the flow of neutrons in a volume 3d r about r at energies

between E and E dE+ traveling in a direction ˆdΩ about Ω . Angular flux is presented by

( )ˆ, ,nv r E Ω and Ω is the unit vector that gives a direction of a particle.

( )ˆ, , ,t n r E tν ∑ Ω

Collision term

- is the rate at which neutron suffer collisions at point r in a volume 3d r at

energies between E and E dE+ traveling in a direction ˆdΩ about Ω . The collision types

involved are elastic, fission, inelastic scattering, and absorption. ( ),t r E∑ , is the total

macroscopic cross section at point r and energy E giving the probability that a neutron will

interact per unit length.

( ) ( )4 0ˆ ˆ ˆ ˆ' ' ' ' , ' , ', ',sd dE E E n r E t

πν

∞Ω ∑ → Ω →Ω Ω∫ ∫

Scattering term

is the neutron gain at time t due to

scattering in a volume 3d r about r at energies between 'E and 'E dE+ to energies between E

and E dE+ traveling in direction ' 'dΩ + Ω and scattered in direction dΩ+ Ω .

24

^, , ,s r E t Ω

Source term

This is the source of neutrons in a volume 3d r . Rate of neutron are produced at

position r having energy E , moving in direction Ω at time t . The analysis in this program deal

with a reactor in equilibrium where 0nt

∂=

∂. Therefore the transport equation becomes:

( ) ( )

( ) ( ) ( )04

ˆ , , , ,

ˆ ˆ ˆ ˆ' ' ' ' , ' , , , ,

t

s

v n r E v n r E

d d Ev E E n r E s r Eπ

∞

Ω ⋅∇ Ω + ∑ Ω =

Ω ∑ → Ω →Ω Ω + Ω∫ ∫ 3.2

It is complicated to solve the transport equation due to space, energy and angular dependence of

neutrons. There are methods to simplify transport calculations to obtain solutions. In this

program the numerical discretization methods are applied to solve the transport equation. Here,

( )ˆ, ,r Eφ Ω is expanded as explained in the next section.

25

3.2.1 Angular Discretization

The angular discretization method is most appropriate for obtaining solution in transport theory

codes where the energy spectrum is initially represented by many energy groups. NP method

uses a flux expansion into a series of Ω (angle), which results in an accurate flux presentation at

position r and energy E in direction Ω . The reaction rate accuracy then depends on the accuracy

of the cross section as given in ENDF data files. Discrete Ordinates method results in SN

4π

equations. The Discrete Ordinates divides the solid angle into segments n∆Ω and

approximates each segment by a linear expression or weights defined by its value, w within the

segment. The value of n determines the order of the approximation. The 2S approximation is

better than diffusion theory and the 4S approximation is adequate for many practical purposes.

Normally, the n∆Ω is expressed as units of 4π or that 1nw =∑ and

( ) ( )1

, , , 1,N

n mn

r E w r E n Nφ φ=

= =∑

The transport equation reduces to:

( ) ( )

( ) ( ) ( )'

1

ˆ ˆ, , ,

ˆ ˆ' ' ' , , ' ,

n t n

N

n s nn n n nn

r E r E

w dE E E r E s r E

φ φ

φ=

Ω⋅∇ Ω +∑ =

∑ − Ω →Ω +∑ ∫

3.3

26

Thus a treatment of the neutron directional dependence is summarized. Equation 3.3 is solved to

determine neutron flux in the NS method where:

( ) ( ) ( )14

ˆ ˆ, , , ,N

m mn

r E r E d w r Eπ

φ φ φ=

= Ω Ω =∑∫ 3.4

The first step in the NP method is to expand ( )ˆ, ,r Eφ Ω in terms of spherical harmonics:

( ) ( ) ( )0

ˆ ˆ, , .N l

lm lml m l

r E r E Yφ ϕ+

= =−

Ω = Ω∑ ∑ 3.5

where: ( )ˆlmY Ω

represents the spherical harmonics. Here ( ) ( ) ( )

4

ˆ, , ,lmr E d Y r Eπ

φ φ= Ω Ω Ω∫ .

Considering a one dimensional equation to simplify a problem,

r z= , 0x y= =

( ) ( )0

2 1ˆ4

N

lm ll

lY P µπ=

+Ω =∑

where Cosµ θ=

( )lP µ is a Legendre polynomial

27

In one dimension, the flux expands into:

( ) ( ) ( )0

2 1, , ,4

N

n ll

lz E z E Pφ µ φ µπ=

+ =

∑ 3.6

( )lP µ = Legendre polynomial

The general form of the equation becomes:

( )( ) ( ) ( )

( ) ( ) ( )

1 1

0

1,

2 1 2 1

' ' , , ,

l lt l

sl l

l l z El z l z

dE E E z E s z E

ϕ ϕ ϕ

ϕ µ

+ −

∞

+ ∂ ∂+ +∑ =

+ ∂ + ∂

∑ → +∫ 3.7

1

1

1

0

l mPP du if l m

if l m

+

−

= =

= ≠

∫

The angular components in the NP expansion of the differential scattering cross section is

( ) ( ) ( )1

0 0 01' 2 ' ,sl s lE E d E E Pπ µ µ µ

+

−∑ → = ∑ →∫ 3.8

0ˆ ˆ'µ = Ω ⋅Ω

28

Here ( ) ( ) ( ) ( )( ) ( ) ( ) ( )0

1

!' 2 ' cos '

!

lm m

l l l l lm

n mP P P P P m

n mµ µ µ µ µ φ φ

=

−= + −

+∑ 3.9

where 0 0cosµ θ= , cosµ θ= and ' cos 'µ θ= . The orthorgonality of the Legendre equations

eliminates the last term in the above equation.

3.2.2 Energy Discretization

Both the spherical harmonic formulation ( )NP of the transport equation as given in equation 3.7

and discrete ordinates ( )NS formulation use the multi-group discretization of the energy variable

E . The discretization of energy in equation 3.7 will provide adequate results, particularly for the

higher NP equations and NS equations. Of importance in solving the multi-group equations are

the different reaction rates that occur over the overall energy interval from 310− eV to 710 eV in

a nuclear reactor. The neutron spectrum is quite different in each of the three broad energy

ranges, fast, epithermal and thermal. In the fast region, the energy spectrum is dominated by the

fission, in the epithermal region by the slowing down and resonance absorption, and in the

thermal region by thermalization. The solutions to NP and NS equations are handled in energy

groups of three broad regions. The energy spectrum is divided into many energy groups, maybe

up to several hundred groups, forming a large matrix. In each group g , the group cross sections

are determined as shown in equation 3.10.

29

( ) ( )

( )

1

1

g

g

g g

g

E

xEx E

E

dE E E

dE E

ϕ

ϕ

−

−

∑∑ =

∫∫

3.10

where xg∑ = macroscopic cross section for reaction x .

x = f (fission), a (absorption), etc.

For example, the actual NS equation then reduces to:

'

^' '

' '' '

1g n n

gg g g g g gn

n n t n n s n nn gg

w Stϕ ϕ ϕ ϕ

ν →

→∂+Ω ⋅∇ +∑ = ∑ +

∂ ∑ ∑ 3.11

1,...............,n N= and 1................,g G= and ( )1g

g

E

gE

E dEϕ ϕ−

= ∫

The equation 3.11 is the NS solved to obtain gϕ .

30

3.3 Adjoint Transport Equation

The adjoint operator is defined as:

1 1g g

g g

E E

E E

M d M dψ ψ ψ ψ ψ ψ− −

+ + + +=∫ ∫ 3.12

where, ψ + is the adjoint flux and M + is the adjoint operator matrix.

'

' ''

''

n n

g g g g g gn tg n g s n n

gM w Sψ ψ ψ

→

→= Ω⋅∇ +∑ − ∑ +∑ ∑ 3.13

M + = transposed of M when the matrix ijM M= and jiM M+ = formed by interchanging rows

and columns. The Eigen - functions of ψ + are then orthogonal to those of M , so that

( )Mψ λψ=

( )M ψ ηψ+ + +=

The matrix multiplication of the NS multi-group equation is

( ) 0M Mψ ψ ψ ψ η λ ψψ+ + + +− = − = 3.14

since ψ and ψ + are orthogonal, thus

'

' '' 'n n

g g g g g g gn n tg n g g n gM w Sψ ψ ψ ψ

→

+ + + → + += Ω⋅∇ +∑ − ∑ +∑ ∑ 3.15

31

These equations can be shown to define the adjoint function to be the importance function where

gψ + is the probability that a neutron born in energy group g will fission. Therefore, the

importance of the group ( )g is given by:

( ) ( )3

4, ,g g

g VC d r d r r

πψ ψ += Ω Ω Ω∫ ∫ 3.16

Adjoint function/flux is the response of a detector in the core to a unit point source inserted at

position ( )0 0 0, ,r EΩ , such as it is a measure of the importance of neutron events contributing to

the response of a detector [8] [3]. So it does not perform the neutron flux calculation for neutron

source or energy. An important application of adjoint flux is in the perturbation theory where it is

used to analyze the small changes that occur in the reactor [3] [15] [21]. The neutron importance

and its properties are discussed thoroughly by Henry, 1975 [16].

3.4 Neutron Energy Regions

As stated previously, neutron energy regions in nuclear reactors are estimated to be in the range

of 10-3 to 107 electron volts. This large energy range is conveniently subdivided into three

regions, fast, epithermal and thermal. The fast region is dominated by slowing down process

through elastic and inelastic scattering processes. Epithermal also known as 1/E region is

dominated by resonance absorption and scattering processes in which slowing down of neutrons

also occurs. In the thermal region both up scattering and down scattering occur as the neutrons

are thermalized, and subsequently are absorbed.

32

3.4.1 Neutron Slowing Down

Neutron slowing down is governed by two processes, elastic and inelastic scattering where

elastic scattering can easily occur with neutron of any energy while the inelastic scattering

requires a sufficiently high energy to excite the target nucleus to a higher energy level. It occurs

in the fast energy region. The slowing down process also depends on the atomic mass of a

moderator, the lighter the nucleus the better the moderating ability. However, a moderator must

exhibit low neutron absorption.

Graphite (carbon) has the highest atomic mass, MC=12 of any moderator whereas the hydrogen

in water has the lowest mass MH

1E

=1. Thus, during the slowing down of neutrons from the fast to

thermal energies due to mainly elastic scattering, neutrons lose less energy per collision with

graphite than hydrogen. During elastic scattering, the initial neutron energy is related to the

scattered energy LE by:

( ) ( )12 1 1

2EE Cosα α θ= + + − 3.17

where 21

1uu

α − = + and θ = scattered angle of the neutron

For , 0.716C α = whereas 0α = for H .

33

Of importance is the average energy lost per collision in a moderator. It is convenient to

calculate the energy loss as a function of lethargy u . The average gain in lethargy per collision

is independent of the neutron energy.

0ln EuE

=

and dEduE

= −

where, ( )− indicates a decrease in E , as u increases.

( )

( )

11

2111

2

1

lnln

E d CosEE

Ed Cos

θξ

θ

+

−+

−

= =

∫

∫ 3.18

1 ln1αξ αα

= +−

3.19

Lethargy is the convenient variable used to perform fast spectrum calculations. The average

lethargy change per collision ξ is used to calculate the average number of collisions needed to

thermalize neutrons starting at 2 MeV . It takes 118 collisions for neutrons to thermalize in

graphite whereas only 18 collisions are needed in H . The large energy changes that take place in

light water reactor allow the neutron to escape the resonances relatively often while slowing

down in the epithermal energy range.

It also allows the total energy spectrum to be divided into two groups, fast and thermal, for

successful analysis of the light water reactors. This is not the case for graphite moderated

34

reactors. Due to the extra number of scattering required for a neutron to slow down through the

epithermal region, there is greater chance for the neutrons to be captured by the resonance cross

sections. For this reason, more energy groups are required in analyzing graphite moderated

reactors.

3.4.2 Resonance Absorption

Resonance absorption is dominant in the epithermal energy region of the nuclear reactor because

neutrons with low energies have a high possibility to collide during slowing down thereby have

high probability to be absorbed [8]. The significant resonance absorption in thermal reactors is in

the lower energy resonances (resolved) such as those resonances of heavy nuclides such as U-

238 and Th-232 (see Figure 3.2 and 3.3).

Figure 3.2: Low energy resonances for U-238

35

Figure 3.3: Low energy resonances for Th-232

This resonance absorption cannot be ignored since it affects the multiplication factor, fuel

burnup, breeding and reactor control characteristics (reactivity, control rods, period, prompt and

delayed neutrons) [8]. Generally, in strong resonance regions there will be a depression of

neutron flux (see figure 3.4), and this need to be treated thoroughly to ensure the accurate kinetic

behavior of a reactor. This is known as energy self-shielding. There is also a related spatial self-

shielding encountered in some cases but it is not a major concern in this study.

36

Resonance

( )Eφσφ

E

Figure 3.4: Flux depreciation under the resonance region

At energies of keV, the resonance structure becomes fine such that they cannot be treated as

individual resonances (unresolved resonances) and in this case they are solved by means of

models. Examples of unresolved resonances are given in Figure 3.5 and 3.6.

37

Figure 3.5: Unresolved resonances for U-235 (t2.lanl.gov)

Figure 3.6: Unresolved resonances for U-238 (t2.lanl.gov)

38

The models are narrow resonance (NR) and wide resonance (WR). Narrow resonance assumes

that the practical width of the resonance is small compared to the average energy lost in the

collision with a moderator. Wide resonance assumes that the practical width of the resonance is

very wide compared to the energy lost in the collision with a moderator. Details assumptions and

equations of these models are given in literature [8] [3] [31].

In heterogeneous cases, which is true for all nuclear reactors flux now depends on both lethargy

and position. So, they can be computed by considering average values per region by assuming

first collision probabilities. This considered that the probability that a neutron born in the fuel

region will undergo its first collision in the moderator region (equation 3.20). Resonant

absorption and Wigner and Bell-Wigner approximations are used for proper considerations of

heterogeneity.

( ) ( )0exp

' 'c eff

S f S m

V N Ip

V Vξ ξ

= − ∑ + ∑

3.20

( ),eff a effI u duσ= ∫ 3.21

Subscript f and m represents the fuel and moderator regions respectively. V is the volume of the

fuel region and moderator, N is the neutron density and ( )ξ ∑ is the moderating power. Equation

3.21 is the effective resonance integral.

39

3.4.3 Neutron Thermalization

In this region, the thermal neutron cross section depends on temperature and the physical state of

the scattering medium. This complicates the nuclear reactor analysis in this region since there is

a neutron up-scattering during the collisions. The equivalence of the thermal neutron energy and

the binding energy of the scattering nucleus also make the analysis difficult when determining

the change in neutron energy and angle. In heterogeneous reactor cores such as HTGRs where

considerable absorption occurs at thermal energies a detail modeling is required. In pure

graphite, the thermalization process is described by the Maxwell-Boltzmann distribution

(equation 3.22).

Neutron energy ( TE ) and its speed ( Tv ) as functions of temperature (T ) are expressed in

equation 3.23 and 3.24 respectively. Subscript ( m ) is the neutron mass.

( )( )

32

2 exp EM E EkTkT

π

π

= −

3.22

58.62 10 ( )TE kT T eV−= = × 3.23

( )42 1.28 10 / secTkTv T cmm

= = × 3.24

However in reality for an actual reactor, the thermal spectrum is modified by the presence of

absorption, in-scattering, leakages etc. There are methods available for use to model such

conditions [8] [20].

40

Chapter 4

Monte Carlo Reference Results 4.1 Introduction

This chapter presents the MCNP5 work as a reference solution to the study. An overview of

Monte Carlo code, MCNP (version 5.1.5) is given in section 4.2. MCNP5 is used to give a

reference detailed accurate solution to be compared with the DRAGON code results. The

DRAGON code is a deterministic code having inherent approximations resulting in some

inaccuracy in the solution; techniques are available to reduce these inaccuracies. The MCNP5

code gives accurate solutions to the same problem, which can be used to guide technique to

increase the accuracy of the DRAGON code. HTR fuel specifications that are used for the

analysis are tabulated in section 4.3

4.2 MCNP5 Code Description

Monte Carlo Neutron Particle Transport code system (MCNP) is a continuous energy statistical

code developed at Los Alamos National Laboratory [23]. For this study, the used MCNP5

version 5.1.5 was supplied by the Radiation Safety Information Computational Center (RSICC)

organization of USA that conducts and regulates the use of nuclear computational tools. The

MCNP5 code models exactly the geometry in the calculation and then tracks individual neutron

particles as they traverse nuclear region based on accurate reactor physics laws and stores all

41

reactions of the neutron as it travels through the medium as the history of that particle. The

results are therefore statistical and the more histories the more accurate the solution.

Running thousands of histories gives an accurate solution that is a reference solution to be

compared with other codes that are less accurate such as deterministic codes. MCNP5 can handle

both simple and complex geometries and uses continuous energy neutron cross section data. It is

the best choice for analysis that involves complex geometries since it statistically can evaluate

the system without geometric approximations. However it is time consuming, especially when

many histories are being sampled to obtain solutions having small uncertainties. A random

number generation method is used to determine the particle interactions (absorption, scattering,

fission) as well as energy losses, new scattered direction, and the number of neutrons created in a

fission process. For criticality calculations, k-effective cycle (multiplication factor is a ratio of

the number of neutrons generated at the end of the cycle to those created during previous cycle)

is estimated by averaging over the events in the life time cycle. The first few cycles are discarded

to eliminate bad initial source distributions. MCNP5 is chosen to give reference solution to this

study.

4.3 HTR Fuel Specifications

There are two forms of HTR fuel used in this research work. Pebbles (fuel element of 6 cm

diameter) are used for pebbled bed reactors and prismatic (hexagonal blocks) are used for

modular high temperature gas cooled reactors. Both fuel elements are composed of TRISO

42

coated particles (CP) that are dispersed or embedded in a graphite matrix. The current design for

pebble fuel element has 15000 of CP’s and each prismatic cylinder in the hexagonal block has

about 3000 CP’s each. The fuel specifications used in this study were taken from the benchmark

specification by DeHart [6] for the HTGR fuel element under the US Department of Energy

(DOE)’s NGNP development program.

Table 4.1 presents the CP’s specifications, and the lattice data and material specification is

presented in Table 4.2. Table 4.3 and Table 4.4 give the specifications for the pebble bed and

prismatic lattice data respectively.

43

Table 4.1: Coated Particle specifications (common for all types of fuel)

Coated Particle specifications

Item Units Value

UO2 g/cm fuel density 10.4 3

Uranium enrichment (by

mass 235U/(235U+238U)

% 8.2

Fuel natural boron impurity

by mass

ppm 1

Outer coated particle radius mm 0.455

Fuel kernel radius mm 0.25

Coated material - C/C/SiC/C

Coated thickness mm 0.09/0.04/0.035/0.04

Coated densities g/cm 1.05/1.9/3.18/1.9 3

Coated Particle lattice data

Item Units Value

Unit cell grain square array

pitch (cubical outer

boundary)

cm 0.16341

Unit cell grain outer radius

(spherical outer boundary

cm 0.10137

Grain outer radius cm 0.0455

Packing fraction of coated

particles

% 9.043

Graphite matrix density g/cm 1.75 3

Graphite matrix natural

boron impurity by mass

ppm 0.5

UO2 g fuel mass 6.806e-04

44

Table 4.2: Material specification (common for all fuel types)

Material Nuclide Atoms per barn cm

( concentrations)

UO2 U-238 fuel 2.12877e-02

U-235 1.92585e-03

O 4.64272e-02

B-10 1.14694e-07

B-11 4.64570e-07

Inner low density carbon

kernel coating

C (natural) 5.26449e-02

Pyrolytic carbon kernel

coating (inner and outer)

C (natural) 9.52621e-02

Silicon carbide kernel coating C (natural) 4.77240e-02

S (natural) 4.77240e-02

Pebble/Compact carbon

matrix

C (natural) 8.77414e-02

B-10 9.64977e-09

B-11 3.90864e-08

Pebble outer

coating/Prismatic block (note:

fuel grain has the same

packing)

C (natural) 8.77414e-02

B-10 9.64977e-09

B-11 3.90864e-08

Helium Coolant He-3 3.71220e-11

He-4 2.65156e-05

45

Table 4.3: Pebbles representative of PBMR fuel

Item Units Value

Unit cell coolant outer radius

(spherical outer boundary)

cm 3.53735

Pebble radius cm 3.0

Radius of fuel zone cm 2.5

Pebble outer carbon coating

thickness

cm 0.5

Pebble outer carbon natural

boron impurity by mass

ppm 0.5

Number of coated particles per

pebble

- 15,000

Packing fraction of coated

particles

% 9.043


Graphite matrix natural boron

impurity by mass

ppm 0.5

Pebble outer carbon density g/cm 1.75 3

UO2 g fuel mass per pebble 10.210

46

Table 4.4: Prismatic fuel lattice data

Item Units Value

Triangular pitch (coolant channel – rod

channel and rod channel – rod channel)

cm 1.880

Fuel channel diameter cm 1.270

Coolant channel diameter cm 1.588

Fuel compact (centered in fuel channel)

diameter

cm 1.245

Compact height cm 4.93

Number of coated particles per compact - 3000

Packing fraction of coated particles % 19.723


Graphite matrix natural boron impurity by

mass

ppm 0.5

UO2 g fuel mass per compact 2.042

4.4 MCNP5 Results

The HTR models developed in MCNP5 for both the Pebble fuel element and Prismatic

hexagonal block fuel are shown in Figures 4.1 and 4.2 below. The corresponding results are

given in Table 4.5 to 4.16. The reaction rates were sampled for three energy regions, fast,

epithermal and thermal regions. The energy ranges were selected as 10 61.100 10 3.142 10− −× → ×

MeV for thermal region, 6 13.142 10 1.156 10− −× → × MeV for epithermal region and

1 11.156 10 1.964 10− +× → × MeV for fast region and these values corresponds to the SHEM

energy group structures that form the foundation of this work. The pebble absorption rates, nu-

47

fission rates and the average flux for all energy regions are presented in Table 4.5 that were

calculated using the ENDF/B-VII.0 version of the evaluated Nuclear Data files and Table 4.6

presents results for ENDF/B-VI.8 which was succeeded by ENDF/B-VII.0. The percent

deviation of the pebble reaction rates is presented in Table 4.7 where thermal region is shown to

be higher than 1 percent, but epithermal and fast regions have percent deviations below 1

percent. The eigenvalue relative deviation in pcm of /k k∆ between the data files is below the

acceptable 500 pcm difference. The average flux from the ENDF/B-VI.8 calculations is higher

compared to the ENDF/B-VII.0 results. It is observed that the difference was in the helium

region of the pebble. This is thought to be due to the updates that are in ENDF/B-VII.0 for He-3

(see Table 4.17).

48

Figure 4.1: MCNP5 Pebble model

49

Figure 4.2: MCNP5 Prismatic block model

50

Table 4.5: Pebble fuel element results calculated using ENDF/B-VII.0

PEBBLE ENDF/B-VII.0

Energy Range

Reaction Rates and criticality Thermal Epithermal Fast

Absorption (collisions/cm3 7.24977E-01 -s) 2.51396E-01 5.33901E-03

Nu-Fission (fissions/cm3 1.40447E+00 -s) 1.01377E-01 8.65859E-03

Average Flux (particles/cm2 1.09878E+00 -s) 1.32725E+00 5.97252E-01

K-effective and deviation 1.52881± 0.00046

Table 4.6: Pebble fuel element results calculated using ENDF/B-VI.8

PEBBLE ENDF/B-VI.8

Energy Range






51

Table 4.7: Percent deviation of pebble FE results for ENDF/B-VII.0 and ENDF/B-VI.8

PEBBLE ENDF/B-VII.0

Energy Range


Absorption -1.28991 0.36327 -0.35949

Nu-Fission -1.28361 0.35469 -0.29800

Average flux -17.74941 8.00124 15.32708

K-effective- Relative Deviation in pcm of /k k∆

with previous version of ENDF

23

Table 4.8 and 4.9 presents the reaction rates for the prismatic block fuel element in the same

energy ranges. In this case the percent deviation (Table 4.10) for all reactions is below 1 percent

except for the nu-fission in the thermal region. The average fluxes deviation percent is also

below 1. The criticality relative deviation in pcm of /k k∆ is also below the acceptable 500

pcm difference.

52

Table 4.8: Prismatic block results calculated using ENDF/B-VII.0

PRISMATIC ENDF/B-VII.0

Energy Range






Table 4.9: Prismatic block results calculated using ENDF/B-VI.8

PRISMATIC ENDF/B-VI.8

Energy Range






53

Table 4.10 Percent deviation of prismatic block results for ENDF/B-VII.0 and ENDF/B-VI.8

PRISMATIC ENDF/B-VII.0 and ENDF/B-VI.8 % deviation

Reaction Rates and criticality

Energy Range

Thermal Epithermal Fast

Absorption 0.64366 0.011907 -0.97457

Nu-Fission 3.05850 0.26963 0.69606

Average Flux 0.50197 0.074997 -0.19075

K-effective- Relative Deviation in pcm of /k k∆ with previous version of ENDF

23

Figure 4.3 and 4.2 below presents the keffective and source convergence for the Pebble and

Prismatic block respectively. For the pebble fuel element the runs had 2000 cycles (150 cycles

skipped) and 15000 histories per cycle. While for the Prismatic block the runs had 1000 cycles

(150 were skipped) and 200 histories per cycle. The results were normalized to the mean of the

active cycles.

54

Figure 4.3: Pebble keffective and source convergence

Figure 4.4: Prismatic keffective and source convergence

0.98

0.985

0.99

0.995

1

1.005

1.01

1.015

0 500 1000 1500 2000 2500

Kef

fect

ive a

nd so

urce

_ent

ropy

Number of cycles

Source_entropyKeffective

0.8

0.85

0.9

0.95

1

1.05

1.1

1.15

0 200 400 600 800 1000 1200

Kef

fect

ive a

nd so

urce

_ent

ropy

Number of cycles

Source_entropy

Keffective

55

The General Atomics energy group structure was also studied in this work. The energy ranges

for GA-193 group structure were selected to be between 10 65.000 10 3.059 10− −× → × MeV for

thermal, 6 13.059 10 1.111 10− −× → × MeV for epithermal, and 1 11.111 10 1.49182 10− +× → × MeV

for fast region. This cutoff energies are the closest possible to the cutoffs used for the SHEM

energy group structures. GA-193 energy group structure was developed for fast reactors, the fast

region consists of 49 groups while the slowing down (epithermal) region consists of 42 groups

and the thermal region consists of 102 groups. The reference solution was computed for the

corresponding cutoff points in MCNP5 to ensure accuracy during DRAGON results

comparisons. Table 4.11 and 4.12 presents Pebble fuel elements reaction rates calculated using

ENDF/B-VII.0 and ENDF/B-VI.8 respectively. The comparisons are shown in Table 4.13.

Similarly, the prismatic hexagonal block results are shown in Table 4.14 and 4.15, with their

comparisons in Table 4.16.

Table 4.11: Pebble fuel element results calculated using ENDF/B-VII.0

PEBBLE ENDF/B-VII.0

Energy Range

Reaction Rates Thermal Epithermal Fast

Absorption (collisions/cm3 0.72463533 -s) 0.251696718 0.005378067

Nu-Fission (fissions/cm3 1.403901224 -s) 0.101907034 0.008697482

Average Flux (particles/cm2 1.085554073 -s) 1.321156429 0.616512002

56

Table 4.12: Pebble fuel element results calculated using ENDF/B-VI.8

PEBBLE ENDF/B-VI.8

Energy Range





Table 4.13: Percent deviation of pebble FE results for ENDF/B-VII.0 and ENDF/B-VI.8

PEBBLE ENDF/B-VII.0 and ENDF/B-VI.8 results % deviation

Energy Range


Absorption -1.29242 0.36798 -0.32085

Nu-Fission -1.28596 0.37223 -0.27531

Average Flux -19.00287 7.74801 17.28592

57

Table 4.14: Prismatic block results calculated using ENDF/B-VII.0

PRISMATIC ENDF/B-VII.0

Energy Range





Table 4.15: Prismatic block results calculated using ENDF/B-VI.8

PRISMATIC ENDF/B-VI.8

Energy Range





58

Table 4.16 Percent deviation of prismatic block results for ENDF/B-VII.0 and ENDF/B-VI.8

PRISMATIC ENDF/B-VII.0 and ENDF/B-VI.8 results % deviation

Energy Range


Absorption 0.59461 0.00919 -0.77860

Nu-Fission 0.64337 0.47077 -0.21039

Average Flux -0.13837 0.06438 -0.16895

The Evaluated Nuclear Data Files (ENDF/B-VII.0) version contains data with reactions with

incident neutrons, protons and photons of approximately 400 isotopes. Its advancements

compared to ENDF/B-VI.8 are that it has new cross sections for U , Pu ,Th , Np and Am

actinides isotopes resulting in the improved performance in integral validation criticality and

neutron transmission [5]. It also has more precise standard cross sections for neutron reactions on

H , 6Li , 10B , Au and for 235U and 238U fission. The thermal neutron scattering was improved

including a set of neutron cross sections on fission products. Neutron and proton induced

evaluations were extended up to 150 MeV and new light nucleus neutron proton reactions were

added. Among these developments are large suites of photonuclear reactions, post fission beta

delayed photon decay spectra, new radioactive decay beta, new method for uncertainties and

covariance’s with covariance evaluations, new actinides fission energy deposition [4]. These

advances attribute to the reaction rates deviations for ENDF/B-VII.0 and ENDF/B-VI.8 results

shown above for the pebble fuel element and the prismatic hexagonal block fuel. Table 4.17

59

below presents the comparisons for ENDF/B-VII.0 and ENDF/B-VI.8 libraries. NSUB is the

sub-library identification number in ENDF/B-VI format. The last two columns present the

number of materials (or isotopes) for specified libraries.

60

Table 4.17: ENDF/B-VII.0 and ENDF/B-VI.8 data files improvement comparisons.

No. NSUB Sub-library name Short name ENDF/B-VII.0 ENDF/B-VI.8

1 0 Photonuclear g 163 -

2 3 Photo-atomic photo 100 100

3 4 Radioactive decay decay 3838 979

4 5 Spontaneous

fission yields

s/fpy 9 9

5 6 Atomic relaxation ard 100 100

6 10 Neutron n 393 328

7 11 Neutron fission

yields

n/fpy 31 31

8 12 Thermal scattering tsl 20 15

9 19 Standards std 8 8

10 113 Electro-atomic e 100 100

11 10010 Proton p 48 35

12 10020 Deuteron d 5 2

13 10030 Triton t 3 1

14 20030 3He He-3 2 1

61

Chapter 5

Multi-group Structure Analysis

5.1 Introduction

Sensitivity analysis were peformed on the currently available energy group structures, SHEM-

281, SHEM-361 and GA-193. The DRAGON transport code is used for the analysis. Section 5.2

describes the DRAGON code. The method used for the library generation is discussed in section

5.3. DRAGON libraries were generated using ENDF/B-VII.0 data files and used to analyze the

pebble fuel element and prismatic hexagonal block models for reaction rates and criticality. The

results and the observations are discussed in section 5.4.

5.2 DRAGON Code Description

DRAGON code is a lattice physics code, which is divided into many different calculation

modules linked together using a GAN generalized driver. A GAN generalized driver is used to

call sequential series of modules sharing a common calling convention and template to build

FORTRAN applications through linking independent modules. The modules exchange the

information through defined data structures. These data structures are memory resident or

persistent [22]. A simplified flow chart in Figure 5.1 presents the data structure and sequence of

modules and its brief description is given below.

MACROLIB

62

MICROLIB

GEOMETRY

TRACKING

ASMPIJ

FLUXUNK

EDITION

BURNUP

DRAGLIB

MULTICOMPO

MACROLIB is a standard data structure used to transfer group ordered macroscopic cross

sections between its modules. It can either be used on its own or included into MICROLIB or

EDITION structures. It can be created by MAC (macro library generation), LIB (micro library

generation) and EDI (editing) modules or modified by SHI (perform self-shielding using

generalized Stammler’s method), USS (perform self-shielding using subgroup method) and EVO

(burnup) modules. It is required for a successful execution of assembly (ASM) and flux (FLU)

modules.

MICROLIB is used to transfer microscopic and macroscopic cross sections between modules. It

follows the MACROLIB substructure. It can be created by LIB and EDI modules or modified by

MAC, SHI, USS and EVO modules.

63

GEOMETRY is used to transfer the geometry between modules. It is created by the GEO

(generate a geometry) module. It is required for a successful execution of the tracking modules

SYBILT, EXCELT and MCCGT modules.

TRACKING is used to transfer the general tracking information between its modules. It is a

stand-alone structure. It is created by either: SYBILT, EXCELT or MCCGT modules. It is

required for a successful execution of the ASM (assembly) module.

ASMPIJ is used to transfer multi-group response and collision probability matrices between

modules. It is a stand-alone structure. It can be created by the ASM module. It is required for a

successful execution of the FLUX module.

FLUXUNK is used to transfer fluxes between modules. It is a stand-alone structure. It can be

created by the FLUX module. It is required for a successful execution of the EDI and EVO

modules.

EDITION is used to store condensed and merged microscopic and macroscopic cross sections. It

is a stand-alone structure but can contain MICROLIB and MACROLIB sub-structures. It can be

created by EDI module. It is required for a successful execution of the COMPO (database

construction) module.

64

BURNUP is used to store burnup information. It is created by the EVO module. It is required to

deplete the fuel assembly.

DRAGLIB is used to recover isotopic, dilution and temperature dependent information including

multi-group microscopic cross sections and burnup data. It is a stand-alone structure. It is created

by the DRAGR module which is in NJOY.

MULTICOMPO is used to store reactor related information and to classify it using immutable

list of local and global parameters. It is a stand-alone structure. It is created by the COMPO

module. For details in specific modules refer to [22].

The DRAGON code is designed to simulate the neutronic behavior of a unit cell, a fuel assembly

and multi-assembly arrays of the nuclear reactor. It has a number of lattice code functional

characteristics such as the interpolation of microscopic cross sections supplied by the standard

library. It computes the resonance self-shielding in multi-dimensional geometries and the multi-

group and multi-dimensional neutron flux that can take into account the neutron leakage. It

considers transport-transport and transport-diffusion equivalence calculations and can edit the

condensed and homogenized properties of nuclear reactor calculations. It also performs the

depletions analysis. DRAGON has two main components, one solves the multi-group flux and

the other calculates the collision probabilities (CP). It contains a multi-group iterator to control

the different algorithms to solve the neutron transport equation. These algorithms are SYBIL

65

option that solves the integral transport equation using the collision probability method (for 1-D)

and the interface current method (for 2-D) Cartesian or hexagonal assemblies. EXCELL option

solves the integral transport equation using collision probability method for 2-D and 3-D

geometries. There is also the MCCG option that uses a long characteristics method for 2D and

3D geometries. An option is also available for treating specular boundary conditions in 2D

rectangular geometry. In solving of the integral transport equations, the results include reaction

rates, neutron flux and the multiplication factor (k-effective).

66

MCCGTSHI

TRIVACT

SNT

FLU

LIB

BIVACT

USSEVO

EDI

NXT

GEO

COMPO

SYBILT

ASM

END

EXCELT

edition

asmpij

Microlib (Macrolib)2

compo

burnup

flux

Microlib (Macrolib)

2

Draglib

Track

Geometry

Microlib (Macrolib)1

Track

Figure 5.1: DRAGON flow chart [11]

67

5.3 Cross Section Library Generation

The DRAGON library is referred to as DRAGLIB and is generated using a Python Script [14].

The ENDF/B data files are processed using NJOY through an object oriented Python Script

named PyNjoy. NJOY is a nuclear data processing system that converts the evaluated nuclear

data files ENDF/B format into usable libraries for nuclear reactor analysis. It transforms

continuous cross section data stored in data files like ENDF/B format for use with Monte Carlo

codes or lattice physics transport codes. It handles all nuclear reactor parameters such as

resonances, Doppler broadening, heating, radiation damage, scattering, gas production, neutrons

and charged particles, photo-atomic interactions, self-shielding, probability tables, photon

production and high energy interactions. NJOY consists of a set of modules from which several

can be selected to perform specific calculations. For DRAGON library generation, the following

modules are used:

RECONR

BROADR

GROUPR

UNRESR

PURR

THERMR

GROUPR

68

The RECONR module reads ENDF/B data and produces a common energy grid for all reactions

to obtain all cross sections within specified tolerance by linear interpolation. It then reconstructs

resonance cross sections from resonance parameters and cross sections from ENDF/B using

nonlinear interpolation schemes. The output is written in point-wise- ENDF (PENDF) file with

all cross sections kept on a unionized energy grid where they are improved for accuracy and

usability.

BROADR generates Doppler broadened cross sections from the PENDF module, the output file

of RECONR. It uses the SIGMA1 module, also called the kernel broadening because it uses a

detailed integration of the integral transport equation defining the cross section. It is known to be

accurate since it treats all resonances and non-resonance cross section including multilevel

effects. The output is written on the PENDF file.

The GROUPR produces self-shielded multi-group cross sections, anisotropic group to group

scattering matrices as well as anisotropic photon production matrices. It presents fission as a

group to group matrix. If necessary, self-shielding for scattering matrices and the photon

production matrices may be modeled. It uses the Bondarenko narrow resonance weighting

scheme. It also allows an option for computing a weighted flux for various mixtures of heavy

absorbers with light moderators. GROUPR uses an accurate point-wise solution of the integral

slowing down equation to account for intermediate resonance effects in the epithermal range. It

uses the PENDF module for input.

69

The UNRESR module produces the effective self-shielded cross sections for resonance reactions

in the unresolved energy range. The unresolved resonance range begins at the energy where it is

difficult to measure individual resonances. It extends to energies where the effects of fluctuations

in the resonance cross section become unimportant for analysis. Resonance information is

averaged for resonance widths and spacing. This is then converted to the effective cross sections.

The UNRESR module uses PENDF module and the ENDF data from BROADR as input. The

effective cross sections are then written into the PENDF module.

The PURR module produces the probability tables that are used to treat unresolved resonance

self-shielding cross section. The unresolved self-shielding data generated by UNRESR is suitable

for use in multi-group calculation after processing by GROUPR. This is done by the generation

of resonances using statistical analysis. Then, cross sections sampling is random at selected

energies and are accumulated into probability tables and Bondarenko moments. These will not be

used for Monte Carlo codes where the probability tables are used. The probability tables are

written on an output file (PENDF module).

The THERMR module generates point-wise neutron scattering cross sections in the thermal

range. It generates elastic cross sections for crystalline materials and non crystalline materials. It

also generates inelastic cross sections and energy to energy transfer matrices for gas free atoms

or for a bound scatters. Since in thermal energy range, neutron scatterings are temperature

dependent, THERMR module considers this temperature. Therefore it expects the requested

70

temperature (T) to be of the temperature included in the ENDF/B thermal file or within few

degrees of the value such that it can pick the closest possible value if necessary. It uses ENDF

and PENDF module from the RECONR as input.

DRAGR is an interface module developed to perform advanced lattice code functions. These are

self-shielding models that have capabilities of presenting distributed and mutual resonance self-

shielding effects, leakage models with space dependent isotropic or anisotropic streaming

effects, availability of the characteristics method and burnup calculations featuring energy

production reactions [12]. The capability of radiative transport of gamma energy produced by

decay or any nuclear reaction in the lattice is among the essential features. DRAGR produces a

DRAGLIB which is a direct access cross section library in a format compatible for DRAGON

code.

The list of modules discussed are the ones specifically used for the generation of library for this

study, Please note that NJOY has a various modules to select depending on the user’s need. For

NJOY details can review the NJOY, 99.90 manual [28].

71

GROUPR

ENDFB/VI or VII

NJOY RECONR

BROADR PURR

THERMR

PENDF

GENDF

DRAGR

DRAGLIB

Figure 5.2: Flow chart for DRAGON library production

72

5.4 Sensitivity Analysis Results

5.4.1 DRAGON Results used in the Sensitivity Study

In this section, DRAGON analysis results are given, which are used in the sensitivity analysis for

optimization of multi-group libraries. The cross section libraries were generated for the available

group structures, SHEM-281, SHEM-361 and GA-193 using ENDF/B-VII.0. These libraries

were used to compute the reaction rates and criticality of pebble fuel element and prismatic

hexagonal block with specifications given in Chapter 4. The results are shown in Tables 5.1

through 5.6.

Table 5.1: Pebble reaction rates and criticality for SHEM-281 energy group structure

PEBBLE

Group Structure

Energy Range


SHEM-281




K-effective 1.51692 (convergence = 2.79E-09)

73

Table 5.2: Prismatic block reaction rates and criticality for SHEM-281 energy group structure

PRISMASTIC

Group Structure

Energy Range


SHEM-281



Average Flux (particles/cm2 1.46823E+00 -s) 2.25204E+00 1.03317E+00

K-effective 1.46094 (convergence =1.53E-07)

Table 5.3: Pebble reaction rates and criticality for SHEM-361 energy group structure

PEBBLE

Group Structure

Energy Range


SHEM-361





74

Table 5.4: Prismatic block reaction rates and criticality for SHEM-361energy group structure

PRISMATIC

Group Structure

Energy Range


SHEM-361





PEBBLE

Table 5.5: Pebble reaction rates and criticality for GA-193 energy group structure

Group Structure

Energy Range


GA-193 Absorption (collisions/cm3 6.25219E-01 -s) 3.68398E-01 6.36363E-03


Average Flux (particles/cm2 8.98270E-01 -s) 1.30071E+00 6.10101E-01


75

PRISMATIC

Table 5.6: Prismatic block reaction rates and criticality for GA-193 energy group structure

Group Structure

Energy Range






5.4.2 Sensitivity Analysis Comparison with MCNP5 Results

The reaction rates for Pebble FE and prismatic hexagonal block computed with DRAGON using

the SHEM-281, SHEM-361 and GA-193 generated libraries are compared to the reference

solution calculations from MCNP5 (chapter 4). K-effective relative percent deviation in pcm is

given in Table 5.7 and 5.8 for each model. Table 5.9 and 5.10 shows the relative percent

deviation for the reaction rates comparisons.

76

PEBBLE

Table 5.7: Criticality calculation comparisons for ENDF/B-VII.0

Group Structure K-effective deviation in pcm

MCNP5 - 1.52881

DRAGON

SHEM-281 1.51692 1189

SHEM-361 1.517046 1176.4

GA-193 1.307130 22168.5

Table 5.8: Criticality calculation comparisons for ENDF/B-VII.0

PRISMATIC

Codes Group Structure K-effective deviation in pcm

MCNP5 - 1.46946

DRAGON

SHEM-281 1.46094E+00 852.4

SHEM-361 1.461067E+00 839.3

GA-193 1.21880E+00 25066.1

77

Table 5.9: Pebble FE reaction rates calculation comparisons for ENDF/B-VII.0

PEBBLE

Energy Range

MCNP5 Reaction rates Thermal Epithermal Fast




DRAGON

SHEM-281

Absorption (collisions/cm3-s) 7.350480E-01 2.586430E-01 6.26975E-03

Nu-Fission (fissions/cm3-s) 1.40377E+00 1.04519E-01 8.63650E-03

Average Flux (particles/cm2-s) 1.07132E+00 1.32069E+00 5.97364E-01

% deviation

Absorption -1.389212277 -2.882545451 -17.43271822

Nu-Fission 0.05013748 -3.099762817 0.255135027

Average Flux 2.499473073 0.494391026 -0.018703628

SHEM-361

Absorption (collisions/cm3-s) 7.350480E-01 2.586860E-01 6.261710E-03

Nu-Fission (fissions/cm3-s) 1.403680E+00 1.047250E-01 8.636350E-03

Average Flux (particles/cm2-s) 1.071270E+00 1.320840E+00 5.973630E-01

% deviation

Absorption -1.389212277 -2.899649913 -17.28212864

Nu-Fission 0.056545572 -3.302965595 0.25686741

Average Flux 2.504023558 0.483089478 -0.018536195

78

Table 5.10: Prismatic block reaction rates calculation comparisons for ENDF/B-VII.0

PRISMATIC

Energy Range





DRAGON

SHEM-281 Absorption (collisions/cm3 6.869630E-01 -s) 3.053760E-01 7.65777E-03



% deviation Absorption 1.521522941 -2.520222601 -3.753748785

Nu-Fission 3.807588733 -0.699405992 -2.290771967

Average Flux -0.972081048 -2.807681093 -9.778257997





Nu-Fission 3.818534366 -0.922847458 -2.290771967

Average Flux -0.959702214 -2.820463348 -9.778257997

79

Since the cutoff points for GA-193 energy group structure were slightly different from the

SHEM energy group structures, their reference solution was generated separately. Though there

might not be much difference in the results due to the fact that the cutoff points were selected to

be as close as possible to the SHEM energy group structures. The comparisons are therefore

presented in Table 5.11 and 5.12 for both models. The observation is that the reaction rates

percent deviation is very high in some cases for the GA-193 group structure, thus the energy

group structure should not be used or either trusted for simulation for other reactors other than

fast reactors for which it was developed. Note that the errors in k-effective are greater than 1000

pcm for the SHEM energy group structure for both the Pebble fuel element and the Prismatic

hexagonal block. This large difference in k-effective is probably due to the large deviations of 17

% in the fast absorption reaction rate in the Pebble fuel element. However, the large difference in

k-effective for the Prismatic hexagonal block appears to be due to the 3.8 % deviation in the

thermal nu-fission reaction rate. The k-effective deviation is more sensitive to deviations in

thermal nu-fission reaction rate than fast absorption rate reaction rates.

80

PEBBLE

Table 5.11: Pebble FE reaction rates calculation comparisons for ENDF/B-VII.0

Energy Range





DRAGON





Nu-Fission 14.932049 -2.186273 -0.215900

Average Flux 17.252395 1.065463 1.039883

81

PRISMATIC

Table 5.12: Prismatic block reaction rates calculation comparisons for ENDF/B-VII.0

Energy Range





DRAGON





Nu-Fission 19.4233870 0.6311153 -3.7209620

Average Flux 18.2443740 -1.6283556 -10.9788243

82

Chapter 6

Multi-group Energy Structure Improvement

6.1 Introduction

The SHEM multi-group energy structure has been optimized for LWRs. The objective of this

dissertation was to optimize it for graphite moderated reactors, both the Pebble bed and the

Prismatic type reactors by altering the energy group structure. The DRAGON-4 code was used

for the optimization process where the currently available energy group structure (SHEM-281,

SHEM-361 and GA-193) are improved, and the obtained results are compared to the MCNP5

results wherein the MCNP5 results are assumed to be reference results. The advantages of using

the DRAGON code are its capabilities to calculate the flux and the adjoint flux for each of its

energy groups. The adjoint flux allows computing the neutron importance function of each

energy group, which is used to improve the energy group structure. The group that has higher

importance function was subdivided into two groups or more and a new energy group structure

was developed from which a new library was created. The new library is applied in the

DRAGON code to analyze a pebble fuel element and prismatic hexagonal block. Then, a pebble

fuel element and prismatic hexagonal block were analyzed for k-effective, flux spectrum and

reaction rates (absorption, scattering, production, fission cross sections). These parameters were

analyzed for three macro energy groups, fast, epithermal and thermal regions. A pebble fuel

element comprises of 15000 TRISO coated particles consisting of fuel kernel of UO2 which is

surrounded by low density carbon, inner pyrolitic carbon, silicon carbide and outer pyrolitic

83

carbon. It has a diameter of 6 cm. Each prismatic cylinder in a hexagonal block has 3000 coated

particles of the same specifications. A fuel rod of 1.245 cm diameter is placed in a fuel channel

diameter of 1.270 cm and the compact height is 4.93 cm.

6.2 Contributon and Point-Wise Cross Section Driven Method

The improvement of the energy group structures was conducted using the contributon and point-

wise cross section driven method (CPXSD) developed by Alpan and Haghighat in 2005 initially

for shielding problems [2]. Nateekol [19] in her PhD extended the method to TRIGA (Penn

State) reactor. In this study the method was applied for first time to HTR problems. In the former

two applications the method was applied using NS formulation. In this study the method was

applied in Collision Probability. The CPXSD method was derived based on the product of the

forward and adjoint angular fluxes, and the point-wise cross section of important

isotope/material of interest. It is an iterative method that selects effective fine and broad group

energy structures for a problem of interest. The energy dependent response flux referred to as

contributon is expressed by equation 6.1:

( ) ( ) ( )4

ˆ ˆ, , , ,v

C E d r d r E r Eπ

+= ΩΨ Ω Ψ Ω∫ ∫ 6.1

In equation 6.1, ( )C E is the importance function, ( )ˆ, ,r EΨ Ω

is the forward flux and

( )ˆ, ,r E+Ψ Ω is the adjoint flux dependent on position ( r ), energy (E) and direction ( Ω ). When

84

considering spherical harmonics expansion of forward flux and its adjoint, and using

orthogonality, the group-dependent contributon ( )C E is given by:

,, , , ,

0 0

2 14

L lm m

g s l g s l g ss D l m

lC Vπ

+

∈ = =

+= Ψ Ψ∑ ∑ ∑ 6.2

In equation 6.2, s presents a uniform material region in D , sV is the volume of the sub-domain.

Where l and m are polar and azimuthal indices for the spherical harmonic polynomial, and g is

the energy group [1]. Also, , ,ml g sΨ is the flux moment and ,

, ,ml g s

+Ψ is the adjoint flux moment.

Therefore, the fine energy group structure improvement followed in this study is as follows:

(i) An initial multi-group energy structure was selected (SHEM-281 and SHEM-361 groups)

and GA-193 group structures.

(ii) Cross sections were generated for the initial multi-group energy structure with the

established procedure of cross section generation relevant to DRAGON transport code as

discussed in chapter 5.

(iii) The forward and adjoint flux calculations were performed to determine the importance

function of the groups in the initial energy group structure of interest.

(iv) After identifying the energy groups with higher importance, this energy group structure is

improved by the resonance structure of a spectrum representing the unit cell (fuel) by

dividing the energy group into two or more energy groups.

85

(v) When the improvement process was complete for all energy groups, the new energy group

structure was used for cross section generation. The new cross section library was used to

calculate the reaction rates and k-effective of the problem of interest.

(vi) The reaction rates and k-effective calculated using the new library are compared with the

results obtained from the previous library analysis. If the results are within a specified

tolerance, the procedure ends; otherwise, steps (iii) through (v) were repeated.

6.3 Improvement of SHEM-281 Energy Group Structure

In this section the SHEM-281 energy group structure was improved for three different energy

regions (Fast, Epithermal and Thermal) separately. After each region has met the target criteria,

they are combined to make up a whole SHEM-281 improved energy group structure. This was

done to study the effect of each region behavior during the energy group structure improvement

and to ensure that there was no under estimations in the procedure.

6.3.1 Fast Energy Region Improvement

The fast energy region was selected to be between 6 11.156235 10 1.964030 10− +× → × MeV . The

starting energy group structure is SHEM-281. The two selected criteria for determining fine

group structure were 10 pcm relative deviation of /k k∆ and 1% relative deviation of the reaction

rates. The reaction rate used for the fast region was the total neutron production (Nu-fission)

( )fν ∑ and the k-effective. During the fast region energy group structure improvement the

86

epithermal (163 groups) and thermal (81 groups) regions were kept constant. Table 6.1 shows the

number of groups obtained during fast energy group structure improvement.

Table 6.1: Fast group selected in the fast range

SHEM 281-group structure improvement

Group structure number

Number of Groups in different energy ranges

Fast Epithermal Thermal Total

1 37 163 81 281

2 65 163 81 309

3 79 163 81 323

The importance function for the starting and improved energy group structure is presented in

Figure 6.1. It was observed that as the energy groups with high importance function are split as

defined previously their importance was reduced. The relative deviation differences of /k k∆

and the percent relative deviation for selected reaction rates are shown in Table 6.2. The 1%

relative deviation in the reaction rate was met in the 309 energy group structure, however, the

relative deviation for /k k∆ was slightly higher than that of the target criteria. Then further

improvement to 323 energy groups resulted in the achievement of both target criteria’s, thus, the

fast range of the energy derived from 323 energy group structure was selected to be used for

further improvement in the final energy group structure.

87

Figure 6.1: Importance function for fast energy region for 281, 309 1nd 323 groups.

Table 6.2: Eigen-value results for fast energy group structure improvement

Eigen-values results of fine energy group structure

Group structure K-effective

Relative Deviation in pcm

of /k k∆ with previous group

Nu-fission rate for Fast region

( )fν ∑

% Relative Deviation with previous group

SHEM-281 1.516920E+00 - 8.63650E-03 -

309 1.516816E+00 10.4 8.60509E-03 0.363688994

323 1.516797E+00 1.9 8.59872E-03 0.074025954

0

0.02

0.04

0.06

0.08

0.1

1.00E-01 1.00E+00 1.00E+01

Impo

rtan

ce

Energy (MeV)

281-groups

309 groups

323 groups

88

Table 6.3 and 6.4 gives the data for each energy group structure for the Pebble FE and Prismatic

hexagonal block, respectively. It was anticipated that the reaction rates in each energy group

would increase with the energy group structure improvement, however, this did not happen in the

groups wherein reaction rates decreased and the k-effective also decreased. Note that the changes

are very small.

Table 6.3: Pebble FE results

PEBBLE

Group Structure Reaction Rates

Energy Range





K-effective 1.51692 (convergence =2.79E-09)

309 Absorption (collisions/cm3 7.35102E-01 -s) 2.58647E-01 6.24979E-03








89

Table 6.4: Prismatic block results

PRISMATIC


Energy Range





K-effective 1.46094+1.53E-07









90

6.3.2 Epithermal Energy Region Improvement

Similarly, starting from the SHEM-281 energy group structure the epithermal energy region

( )6 13.14 10 1.156 10− −× → × MeV was improved. The absorption reaction rate was used for

comparisons in this energy range as well as the k-effective. The energy group numbers per

region are given in Table 6.5 where both fast (37 groups) and thermal (81 groups) regions were

kept constant.

Table 6.5: Epithermal energy groups selected in the epithermal range


Group structure number Number of Groups in Different energy ranges


1 37 163 81 281

2 37 215 81 333

Table 6.6: Eigen-value resulted for epithermal energy group structure improvement




of /k k∆ with previous group Absorption


SHEM-281 1.51692E+00 - 2.58643E-01 -

333 1.51699E+00 6.6 2.58696E-01 0.02049157

91

The importance function plot is given in Figure 6.2 showing the reduction of energy group

importance as the group structure was increased. The target criteria of 10 pcm relative deviation

of /k k∆ and the 1 percent relative deviations of the objective function were met with the 333

energy group structure. The absorption reaction rates and k-effective comparisons results for

energy groups are presented in Table 6.6. The additional reaction rates data for all regions for

the energy group structure of interest are given in Table 6.7 and 6.8 for pebble FE and Prismatic

hexagonal block respectively.

Figure 6.2: Importance function for epithermal energy region for 281 and 333 groups

0

0.02

0.04

0.06

0.08

1.00E-06 1.00E-04 1.00E-02 1.00E+00

Impo

rtan

ce

Energy (MeV)

281-groups

333-groups

92


PEBBLE


Energy Range










93


PRISMATIC


Energy Range










6.3.3 Thermal Energy Region Improvement

The thermal region energy range is between 10 61.10 10 3.14 10− −× − × MeV . The epithermal (163

groups) and fast (37 groups) region were kept constant as the starting energy group structure

(SHEM-281) was improved for the thermal region. The number of energy groups in each energy

range is presented in Table 6.9 and the related importance function plot in Figure 6.3. Table 6.10

presents the comparisons for the selected reaction rates (absorption, neutron production) and the

k-effective. The selected energy group structure for use in the final group structure for all regions

is 313.

94

Table 6.9: Thermal groups selected in the thermal region




1 37 163 81 281

2 37 163 97 297

3 37 163 113 313

Table 6.10: Eigen-values resulted for thermal energy group structure improvement



Relative Deviation in

pcm of

/k k∆ with previous

group ( )a∑

Absorption


( )fν ∑

Nu-Fission


SHEM-281 1.516920E+00 - 7.35093E-01 - 1.40377E+00 -

297 1.516752E+00 16.8 7.35875E-01 0.106381097 1.40488E+00 0.079072783

313 1.51675E+00 0.6 7.35578E-01 0.040360116 1.40438E+00 0.035590228

95

Figure 6.3: Importance function for thermal energy region for 281, 297 and 313 groups

0

0.05

0.1

0.15

0.2

0.25

0.3

1.00E-09 1.00E-08 1.00E-07 1.00E-06 1.00E-05

Impo

rtan

ce

Energy (MeV)

281-groups297-groups313-groups

96


PEBBLE


Energy Range














97


PRISMATIC


Energy Range














98

6.3.4 Improved SHEM-281 Energy Group Structure for all Regions

This section combines all the individually improved fast, epithermal and thermal energy regions

to 407 energy group structure as shown in Table 6.13 and the new group structure is referred to

as SHEM_TPN-407. The neutron cross sections were generated for this energy group structure

and Pebble fuel element and Prismatic hexagonal block analysis were conducted as the same as

previous sections. Then, the objective functions were compared to the SHEM-281 as the starting

group structure, see Table 6.14 and Table 6.15.

Table 6.13: Energy group structure improved from SHEM-281 to 407




1 37 163 81 281

2 79 215 113 407

99

Table 6.14: Results for SHEM-281 and 407 energy group structures

PEBBLE


Energy Range






SHEM_TPN-407 Absorption (collisions/cm3 7.35662E-01 -s) 2.58207E-01 6.13159E-03




100

Table 6.15: Results for SHEM-281 and 407 energy group structures

PRISMATIC


Energy Range










The Pebble fuel element results are compared in Table 6.16 below. The comparisons are based

on the relative deviation of /k k∆ in pcm for k-effective and percent relative deviation for

reaction rates. Similarly, Table 6.17 shows the comparisons for the prismatic hexagonal block. It

was observed that the fast region has higher reaction relative deviation in the fast region which is

believed to be the effect from other changes related to the improvement in epithermal and

thermal energy regions since during its improvement the relative deviations were very low.

Epithermal and thermal energy regions relative deviation are below the selected target criteria of

1 percent. The relative deviation of /k k∆ also met the 10 pcm target criteria.

101

Table 6.16: Comparisons for SHEM_TPN-407 with SHEM-281 energy group structure for the

pebble FE

% relative deviation for three regions

Reaction rates Thermal Epithermal Fast

Relative Deviation in pcm of /k k∆ with previous

group

Absorption 0.077345303 -0.16885677 -2.253249157

k-effective -1.9 Nu-Fission 0.054822219 -0.626757038 -1.746531971

Average Flux 0.956853754 0.6686322 -2.765936217


prismatic block

% relative deviation



group

Absorption 0.099179082 -0.166629492 -2.24511

k-effective

1

Nu-Fission 0.073529412 -0.583950532 -1.76614

Average Flux 1.025319528 0.278524396 -2.77026

102

Table 6.18 and 6.20 are the MCNP5 reference solution recalled from chapter 4 and are compared

with the results for a new energy group structure (407 groups) in table 6.19 and 6.21 for both fuel

types. The improvement is observed in the reaction rates relative deviations with no significant

trend.

Table 6.18: MCNP5 results for the pebble

Revise to use MCNP5 values given in Chapter 4

Reaction Rates

Energy Range






103

Table 6.19: Comparisons of SHEM_TPN-407 energy group structure to MCNP5 results for the

pebble FE

% relative deviation criticality


Relative Deviation in pcm of /k k∆with previous

group

Absorption 1.452496263 2.637657019 12.9260987

K-effective -1190.9 Nu-Fission 0.00468709 2.398653219 -2.006787014

Average Flux -1.582165109 0.175106886 -2.746718853

Table 6.20: MCNP5 results for the prismatic block


Reaction Rates

Energy Range






104

Table 6.21: Comparisons of SHEM_TPN-407 energy group structure to MCNP5 results for the

prismatic block




group

Absorption -1.44432 2.29573 1.45406

K-effective

-851.4 Nu-Fission -3.88186 0.11465 0.51288

Average Flux 1.97817 3.00192 6.38377

6.4 SHEM-361 Energy Group Structure Results

Similarly, the SHEM-361 energy group structure was improved for fast, epithermal and thermal

regions separately. After each region improvements, the energy group structures selected for

different regions are combined to make a complete improved energy group structure. Thus the

effect of each region is not underestimated.

105


The fast energy region as previously selected is 6 11.156 10 1.964 10− +× → × MeV . The starting

energy group structure is SHEM-361. The same target criteria of 10 pcm relative deviation of

/k k∆ and 1 percent relative deviation of the objective function are used. The reaction rate used

for the fast energy region was the total neutron production (Nu-Fission) and the k-effective. As

the fast region was being improved, the epithermal (243 groups) and thermal (81 groups) regions

remained constant. The number of energy groups obtained during fast energy region

improvement is shown in Table 6.22. The importance function for all energy groups is presented

in figure 6.4 showing a significant reduction of the importance function as the energy groups are

being improved.





1 37 243 81 361

2 65 243 81 389

3 79 243 81 403

106

Figure 6.4: Importance function for fast energy region for 361,389 and 403 groups

The relative deviation differences for /k k∆ and percent relative deviation for the selected

reaction rates are given in Table 6.23. Both target criteria’s were met by improving the SHEM-

361 to 389 groups, however 9.9 was relatively too close to the 10 pcm , so the decision was made

to continue with another improvement that led to 403 energy group structures. Therefore 403

energy group structure was selected for further use. Table 6.24 and 6.25 present additional

information of the three regions reaction rates obtained during the energy group structure

improvement for the pebble fuel element and prismatic hexagonal block respectively.

0

0.02

0.04

0.06

0.08

0.1

1.00E-01 1.00E+00 1.00E+01

Impo

rtan

ce

Energy (MeV)

361 groups

389 groups

403 groups

107

Table 6.23: Eigen-value results for fast energy group structure improvement




group


( )fν ∑


361 1.51705E+00 - 8.63635E-03

389 1.516947E+00 9.9 8.60508E-03 0.36207

403 1.516928E+00 1.9 8.598720E-03 0.07391

108


PEBBLE


Energy Range














109


PRISMATIC


Energy Range














110


During the improvement of epithermal energy region ( )6 13.142 10 1.156 10 MeV− −× → × for

SHEM-361 energy group structure the target criteria’s were met at 455 energy group structure

(Table 6.26). The fast (37 groups) and thermal (81 groups) regions were kept constant during the

epithermal region improvement. Significant improvement in the importance function is shown in

Figure 6.5. The relative deviation in pcm of /k k∆ and percent relative deviation of the selected

reaction rates (absorption rate) are given in Table 6.27. Pebble fuel element and prismatic

hexagonal block results also given in Table 6.28 and Table 6.29 respectively.

Table 6.26: Epithermal energy groups selected in the epithermal range


Group structure number Number of Groups in Different Energy Ranges


1 37 243 81 361

2 37 337 81 455

111

Figure 6.5: Importance function for epithermal energy region for 361 and 455 groups

Table 6.27: Eigen-value resulted for epithermal energy group structure improvement




group

Absorption

( )a∑


361 1.51705E+00 - 2.58686E-01 -

455 1.516965E+00 8.1 2.58849E-01 -0.06301

0

0.02

0.04

0.06

0.08

0.1

1.00E-06 1.00E-04 1.00E-02 1.00E+00

Impo

rtan

ce

Energy (MeV)

361 groups

455-groups

112


PEBBLE


Energy Range









K-effective and precision 1.51697 (convergence = 5.78E-08)

113


PRISMATIC


Energy Range










114


In thermal region ( )60 3.14 10 MeV−→ × , the selected reaction rates for improving the energy

group structure were absorption and nu-fission and then k-effective. The fast and epithermal

energy regions were kept constant as well. The number of energy group structures obtained

during the improvement is given in Table 6.38. Improving the SHEM-361 to 393 did not meet

the 10 pcm target criteria of /k k∆ , while both absorption rate and nu-fission rate were met.

Then the energy group structure was further improved to 395 groups, it is also importance to

note that the difference between 393 and 395 groups is not just two groups, but some energy

group structures with less importance were taken out while some energy group structure were

subdivided into two or more energy groups leading to 395 energy groups. Then all the target

criteria’s were met and 395 energy group structures were selected for further use. The

importance function is shown in Figure 6.6.



Group structure number Number of Groups in Different Energy Ranges


1 37 243 81 361

2 37 243 113 393

3 37 243 115 395

115


The relative deviation in pcm of /k k∆ and percent relative deviation of selected reaction rates is

given in Table 6.31. The results for all regions of both pebble fuel element and prismatic

hexagonal block are given in Table 6.32 and Table 6.33.

0

0.05

0.1

0.15

0.2

0.25

0.3

1.00E-09 1.00E-08 1.00E-07 1.00E-06 1.00E-05

Impo

rtan

ce

Energy (MeV)

361 groups

393-groups

395 groups

116

Table 6.31: Eigen-values resulted for thermal energy group structure improvement



Relative Deviation in pcm of

/k k∆ with

previous group

Absorption

( )a∑

% Relative Deviation

with previous group

Nu-Fission

( )fν ∑


361 1.517046E+00 - 7.35048E-01 - 1.40368E+00 -

393 1.516874E+00 17.2 7.35535E-01 -0.066254177 1.40430E+00 -0.04417

395 1.516877E+00 -0.3 7.355350E-01 0 1.404300E+00 0

117


PEBBLE


Energy Range














118


PRISMATIC


Energy Range














119

6.4.4 Improved Energy Group Structure for all Regions (SHEM-361)

The energy group structure selected during the specific regions improvement are combined

leading to 531 energy groups referred to as SHEM_TPN-531. The library was generated for this

group structure and used for the analysis of the pebble FE and prismatic hexagonal block

benchmark problems. Reaction rates and k-effetive were compared with the ones from the

original SHEM-361 energy group structure as shown in Table 6.37 and 6.38. The MCNP5 results

were again recalled (Table 6.39 and 6.41) and they were compared with the results obtained from

the new energy group structure analysis (see Table 6.40 and 6.42).

Table 6.34: Energy group structure improved from SHEM-361 to SHEM_TPN-531




1 37 243 81 361

2 79 337 115 531

120

Table 6.35: Reaction rates for SHEM-361 and SHEM_TPN-531 energy group structures

PEBBLE


Energy Range










121

Table 6.36: Reaction rates for SHEM-361 and SHEM_TPN-531 energy group structures

PRISMATIC


Energy Range










122


pebble FE




of /k k∆ with previous group

Absorption -0.06884 0.12563 2.89665

k-effective 16.3 Nu-Fission -0.04701 0.59203 2.30167

Average Flux -0.96241 -1.18864 3.87553

Table 6.38: Comparisons for SHEM_TPN-531 with SHEM-361 energy group structure for the prismatic block



Relative Deviation in pcm of /k k∆

with previous group

Absorption -0.08910 0.12375 3.08219

k-effective 15.8 Nu-Fission -0.06373 0.56184 2.32899

Average Flux -1.03334 -1.24316 3.88165

123

Table 6.39: MCNP5 results for the pebble FE


Reaction Rates

Energy Range






Table 6.40: Comparisons of SHEM_TPN-531 energy group structures to MCNP5 results for the

pebble FE

% relative deviation Criticality



group

Absorption 1.43803 2.69569 12.19203

k-effective -1192.7 Nu-Fission -0.00955 2.62085 -2.61949

Average Flux -1.59062 0.69494 -4.01251

It is important to compare the deviation in Table 6.37 with the corresponding deviation for the

Pebble fuel element in Table 6.40. Only the thermal nu-fission reaction rate deviation was

124

decreased by increasing the number of energy groups to 531. However, the fast absorption

reaction rate remained high at 12 %. The increase in energy groups to 531 showed no

improvement in the k-effective.

Table 6.41: MCNP5 results for the prismatic block


Reaction rates

Energy Range






Table 6.42: Comparisons of SHEM_TPN-531 energy group structures to MCNP5 results for the

pebble FE

% relative deviation Criticality



group

Absorption -1.46675 2.36299 0.56668

k-effective -855.1 Nu-Fission -3.90392 0.35456 -0.09166

Average Flux 1.96363 3.93731 5.22859

125

Similar comparison can be made for the Prismatic hexagonal block by comparing Table 6.38 to

Table 6.42. Again the k-effective was not improved using the 531 energy group structure

whereas the nu-fission reaction rate remained relatively high at 3.9 %. In fact, the 531 energy

group structure increases the deviation.

6.5 General Atomics-193 Energy Group Structure

General Atomics energy group structure was divided into three energy regions (fast, epithermal

and thermal) and each region was improved separately. Similar target criteria’s are used for the

improvement of GA-193 energy group structure, which are 10 pcm relative deviation of /k k∆

and 1 percent deviation of reaction rates.


The fast energy region of GA-193 which is between 1 11.111 10 1.492 10− +× → × MeV has 49

groups as shown in Table 6.43. This was increased to 83 groups while keeping the epithermal

and thermal regions constant. The relative deviations for k-effective and percent relative

deviation for nu-fission rate are presented in Table 6.44. These were met with the group

structure improvement to 227 energy groups.

126

GA 193-group structure improvement




1 49 42 102 193

2 83 42 102 227


Table 6.44: Eigen-values resulted for fast energy group structure improvement


Relative Deviation in pcm of /k k∆

with previous group


( )fν ∑


193 1.30713E+00 - 8.71626E-03

227 1.30712E+00 0.5 8.70836E-03 0.090635

The importance function plotted in Figure 6.13. The data for the pebble fuel element and

prismatic hexagonal block results for this region are given in Table 6.45 and 6.46, where a slight

decrease in both k-effective and nu-fission rate was observed.

127

Figure 6.7: Importance function for fast energy region for 193 and 227 groups

0

0.002

0.004

0.006

0.008

0.01

0.012

0.014

0.016

1.00E-01 1.00E+00 1.00E+01 1.00E+02

Impo

rtan

ce

Energy (MeV)

GA-193GA-227

128

PEBBLE

Table 6.45: Reaction rates for GA-193 and 227 energy group structures


Energy Range










129

PRISMATIC

Table 6.46: Reaction rates for GA-193 and 227 energy group structures


Energy Range










130


The epithermal energy region of 6 13.059 10 1.111 10− −× → × MeV was improved from 42 groups

to 334 groups. The energy group structures selected in this region is given in Table 6.47, while

the relative deviation for the k-effective and the percent relative deviation for the absorption

reaction rate are given in Table 6.56. The epithermal region improvement of GA-193 was

challenging due to the instability of the reaction rates behavior. The reaction rates vary with no

trend as the energy group structures are increased. The k-effective showed a high sensitivity on

the energy group structure changes. This is attributed to the fact that GA-193 was developed for

fast energy reactors, which use thorium fuel cycle therefore it is obvious that the resonances and

other consideration made were based on the thorium fuel cycle physics and also there are

breeding properties involved in such reactors. The target criteria of 1 percent deviation for

reaction rates was met at 235 groups, however the k-effective was higher than the targeted value.

Therefore the improvement continued with intention to reach the target 10 pcm . But, the k-

effective high sensitivity observed as the work progresses leading into accepting the closest

possible value to 10 pcm target criteria, which is 17.9 pcm . Therefore 485 energy groups were

selected for this region.

131

GA-193 group structure improvement

Table 6.47: Epithermal group selected in the epithermal range



1 49 42 102 193

2 49 84 102 235

3 49 168 102 319

4 49 204 102 355

5 49 334 102 485


Table 6.48: Eigen-values resulted for epithermal energy group structure improvement



/k k∆ with previous group

Absorption

( )a∑


193 1.30713E+00

3.68398E-01

235 1.30675E+00 37.8 3.68842E-01 0.120376747

319 1.32884E+00 2208.9 3.56242E-01 3.416096865

355 1.32166E+00 717.4 3.59880E-01 1.01089252

399 1.36102E+00 3935.4 3.39782E-01 5.584639324

485 1.36120E+00 17.9 3.39685E-01 0.028547716

132

Figure 6.8: Importance function for epithermal energy region

for 193, 235, 319, 355, 399 and 485 groups

0

0.005

0.01

0.015

0.02

0.025

0.03

1.00E-06 1.00E-04 1.00E-02 1.00E+00

Impo

rtan

ce

Energy (MeV)

GA-193 GA-235 GA-319

GA-355 GA-399 GA-485

133

Table 6.49: Reaction rates for GA-193, 235, 319, 355, 399, and 485 energy group structures

PEBBLE


Energy Range





K-effective and precision 1.30713E+00 (precision = 4.22E-08)




K-effective and precision 1.30675+1.36E-07

















134

Table 6.50: Reaction rates for GA-193, 235, 319, 355, 399, 485 energy group structures

PRISMATIC


Energy Range


























135


Thermal energy region ( )10 65.000 10 3.059 10− −× → × MeV has 102 energy groups. The fast (49

groups) and epithermal (42 groups) are kept constant during thermal region improvement.

During the improvement of the GA-193 to 205 groups, the selected reaction rates (absorption)

and the k-effective met the target criteria of 1 percent deviation and the 10 pcm relative

deviation of /k k∆ . However, the nu-fission did not meet the target criteria. Therefore the 205

energy group structure was further improved to 211 where all reaction rates and the k-effective

target criteria’s were met. A slight increase in the absorption rate and nu-fission rate was

observed from GA-193 energy group to 205 energy group, which is a positive improvement

when compared to the reference solution from the MCNP5. The k-effective is decreasing, which

is different from what was expected (Table 6.52).

GA193-group structure improvement




1 49 42 102 193

2 49 42 114 205

3 49 42 120 211

136

Table 6.52: Eigen-values resulted for thermal energy group improvement




pcm of /k k∆ with

previous group

( )a∑Absorption


with previous group

( )fν ∑

Nu-Fission


with previous group

193 1.307125E+00 - 6.25219E-01 - 1.11943E+00 -

205 1.307042E+00 8.3 6.25886E-01 -0.1066826 1.19502E+00 -6.7528298

211 1.30703E+00 0.9 6.25885E-01 0.0001598 1.19501E+00 0.0008368

The importance function plotted in Figure 6.9 shows a high importance in the energy ranges of

8 75.50 10 2.18 10− −× → × MeV and that of 6 62.38 10 3.06 10− −× → × MeV . This was reduced as the

energy group structure was divided into two or more groups thus improving the reaction rates

results. Table 6.53 and 6.54 presents the reaction rates of all regions for the Pebble fuel element

and prismatic hexagonal block respectively. The changing of one region results in the changes in

the reaction rates for all regions showing that all regions energy group structure has effect from

any changes that are happening in the neighboring regions.

137


0

0.02

0.04

0.06

0.08

1.00E-09 1.00E-08 1.00E-07 1.00E-06 1.00E-05

impo

rtan

ce

Energy (MeV)

GA-193

GA-205

GA-211

138

PEBBLE



Energy Range














139

PRISMATIC


Group Structure

Reaction Rates

Energy Range














140

6.5.4 Improved GA-193 Energy Group Structure for all Regions

Three energy group structures selected during each region’s improvement are combined and

totaling to 537 and is referred to as GA_TPN-537. The library was generated for this final energy

group structure and used for the analysis of the pebble FE and prismatic hexagonal block. Table

6.55 gives the number of the starting and final energy groups. The reaction rates for the initial

GA-193 and 537 are given in Table 6.56 and 6.57 for the pebble FE and prismatic hexagonal

block. The reaction rates were compared with the ones from the original GA-193 energy group

structure as shown in Table 6.58 and 6.59.

GA 193-group structure improvement

Table 6.55: Energy group structure improved from GA-193 to GA_TPN-537

Group structure number Number of Groups in different energy ranges


1 49 42 102 193

2 83 334 120 537

141

PEBBLE

Table 6.56: Reaction rates for GA-193 and GA_TPN-537 energy group structures


Energy Range










142

PRISMATIC

Table 6.57: Reaction rates for GA-193 and GA_TPN-537 energy group structures


Energy Range






537 Absorption (collisions/cm3 6.003060E-01 -s) 3.922280E-01 7.460500E+00




143

Table 6.58: Comparisons for 537 with GA-193 energy group structure for the pebble FE


.



pcm of /k k∆ with

previous group

Absorption 4.53273 -8.67775 -4.16351

k-effective 5398.4 Nu-Fission 10.47592 -1.88039 -2.84042

Average Flux 6.28759 1.71542 -5.58395


Table 6.59: Comparisons for GA_TPN-537 with GA-193 energy group structure for the

prismatic block.



group

Absorption 6.55432 -9.94269 99.89558 k-effective

7212.2

Nu-Fission 6.50332 -1.92044 -2.86855

Average Flux 8.56961 1.46818 -5.58969

144

The MCNP5 results for both pebble fuel element and prismatic hexagonal block are given in

Table 6.60 and 6.62. They are then compared with the improved energy group structure results in

Table 6.61 and 6.63. A significant change in the reaction rates deviations is observed. However,

it is believed that there are some underlying issues about the energy group structure. The k-

effective was improved significantly about 5000 pcm from initial deviations.

Table 6.60: MCNP5 results for the pebble FE


Reaction Rates

Energy Range






145


Table 6.61: Comparisons of GA_TPN-537 energy group structures to MCNP5 results for the

pebble FE



pcm of /k k∆ with

previous group

Absorption 9.66549 -34.83965 -14.42692

k-effective

-16770.1 Nu-Fission 10.96881 -0.82508 2.11433

Average Flux 12.76364 0.28946 3.25110

Table 6.62: MCNP5 results for the prismatic block.


Reaction Rates

Energy Range






146


Table 6.63: Comparisons of GA_TPN-537 energy group structures to MCNP5 results for the

prismatic block.



group

Absorption 13.94410 -31.67800 -100980.973

k-effective 17853.9 Nu-Fission 15.95797 1.82043 -0.36076

Average Flux 11.41982 -2.88848 -6.18167

147

Chapter 7

Comparative Analysis of ENDF Data Files

7.1 Introduction

The purpose of this section was to analyze the latest released ENDF/B-VII.1. Two multi-group

energy structures (SHEM-361 and SHEM_TPN-531) were selected for the analysis. Section 7.2

compares the Pebble fuel element and Prismatic hexagonal block analysis based on the ENDF/B-

VII.0 and ENDF/B-VII.1. The discussion on the differences of the evaluation data files is given

in section 7.3.

7.2 Comparison Results

Table 7.1 presents the Pebble fuel element results for ENDF/B-VII.0 and ENDF/B-VII.1. This

was computed for SHEM-361 and the improved energy group structure SHEM_TPN-531. Their

percent relative deviations are shown in Table 7.2. The fission reaction rate percent relative

deviation is below 1 percent. The absorption reaction rate for the thermal and epithermal energy

regions for both energy groups are also below 1 percent, however it is observed that the

absorption reaction rate at fast energy region is higher. The relative percent deviation of /k k∆ is

higher than the accepted 500 pcm for the pebble FE. Similar behavior is observed for the

prismatic hexagonal block results with the higher percent relative deviation at the fast energy

region (Table 7.3 and 7.4). As seen that the reaction rate percent relative deviations between the

148

ENDF/B-VII.1 and ENDF/B-VII.0 data files is insignificant (Table 7.2 and 7.4), this may be due

to the fact important nuclides like U-235, U-238 were not changed during data advancements,

noting their importance (fuel kernel). The relative deviation for /k k∆ is below 500 pcm for the

prismatic hexagonal block. It is noticeable that the k-effective sensitivity for the pebble fuel

element is higher compared to that of the prismatic hexagonal block, which can be attributed to

the geometry. This is in agreement with the MCNP5 results (Chapter 4) when comparing

ENDF/B-VI.8 and ENDF/B-VII.0.

It is pointed out in the Nuclear Data Sheets by Chadwick et al, 2011[4] that the validation testing

for ENDF/B-VII.1 maintained a good performance compared to that of ENDF/B-VII.0 for

nuclear criticality of which the improved performance is attributed to the new structural

materials evaluations. Figure 7.1 shows the neutron cross section for natural carbon as an

example. Concerns were raised on the shortfall of major actinides (U-235, U-238 and Pu-239)

for the previous ENDF/B-VII.0 data files, for example poor performance of Pu-239 in thermal

ranges and the cross section capture for U-235 that is reported to be 25% higher in the 1 keV

region. Therefore, U-235, U-238, Pu-239 were changed back to the ENDF/B-VI.8 data neutron

parameters since there are no new advancements on these nuclides as yet.

149

Figure 7.1: Neutron capture on natural carbon for ENDF/B-VII.1

and ENDF/B-VII.0 (Chadwick et al, 2011)

150

PEBBLE

Table 7.1: Pebble FE results using ENDF/B-VII.0 and ENDF.B-VII.1 comparisons

ENDF/B-VII.0


Energy Range










ENDF/B-VII.1









151

pcm

Table 7.2: END/B-VII.0 and ENDF/B-VII.1 % deviations for the pebble reaction rates and k-

effective in

% Deviation


SHEM-361 Absorption 0.025032 -0.034018 -1.522907

Nu-Fission 0.408213 0.021962 0.024200

Average Flux 0.007468 0.014385 0.010546

K-effective ( pcm ) 575.6

SHEM_TPN-531 Absorption 0.024471 -0.034061 -1.504195

Nu-Fission 0.318228 0.023054 0.025837

Average Flux 0.272749 0.014964 0.010797

K-effective ( pcm ) 565.6

152

PRISMATIC

Table 7.3: Prismatic block results using ENDF/B-VII.0 and ENDF.B-VII.1 comparisons

ENDF/B-VII.0


Energy Range










ENDF/B-VII.1









153

pcm

Table 7.4: END/B-VII.0 and ENDF/B-VII.1 % deviation for the prismatic block reaction rates

and k-effective in

% Deviation


SHEM-361 Absorption 0.025186 -0.024881 -1.242094

Nu-Fission 0.327749 0.022746 0.024317

Average Flux -0.000681 0.014652 0.010647

K-effective (pcm) 434.5

SHEM_TPN-531 Reaction Rates Thermal Epithermal Fast

Absorption 0.024728 -0.025239 -1.239697

Nu-Fission 0.320717 0.022875 0.025819

Average Flux 0.211027 0.015349 0.010875

K-effective (pcm) 425.8

7.3 Nuclear Data Advancements

The most important advances of the new evaluated data files (ENDF/B-VII.1) include an

increase in neutron reaction cross section coverage from 393 to 423 nuclides, and the covariance

uncertainty data for 190 of the most important nuclides. New R-matrix evaluations on light

nuclides (He, Li, and Be) are give as well as resonance parameter analysis at lower energies and

statistical high energy reactions for isotopes (Cl, K, Ti, V, Mn, Cr, Ni, Zr and W). Also, changes

154

on thermal neutron reactions on fission products (Mo, Tc, Rh, Ag, Cs, Nd, Sm, Eu) and neutron

absorbers (Cd, Gd). Minor actinides evaluations for isotopes (U, Np, Pu, and Am) were

improved, but there were no improvements on the major actinides (U-235, U-238, and Pu-239)

as discussed above. They then adopted JENDL-4.0 evaluations for Cm, Bk, Cf, Es, Fm, isotopes

and other minor actinides. New data for fission energy release evaluations and a new decay data

sub-library was created. Fission product yield advances for fission spectrum neutrons and 14

MeV neutrons incident on Pu-239.

Table 7.5 presents the comparisons for the ENDF/B-VII.0, ENDF/B-VII.1 libraries and

ENDF/B-VI.8. NSUB is the sub-library number in ENDF/B-VI format. The last two columns

give the materials (or isotopes) for the libraries.

155

Table 7.5: ENDF/B-VII.1, ENDF/B-VII.0 and ENDF/B-VI.8 data file advancements

No. NSUB Sub-library name Short name ENDF/B-

VII.1

ENDF/B-

VII.0

ENDF/B-

VI.8

1 0 Photonuclear g 163 163 -

2 3 Photo-atomic photo 100 100 100

3 4 Radioactive decay decay 3817 3838 979

4 5 Spontaneous

fission yields

s/fpy 9 9 9

5 6 Atomic relaxation ard 100 100 100

6 10 Neutron n 423 393 328

7 11 Neutron fission

yields

n/fpy 31 31 31

8 12 Thermal scattering tsl 20 20 15

9 19 Standards std 8 8 8

10 113 Electro-atomic e 100 100 100

11 10010 Proton p 48 48 35

12 10020 Deuteron d 5 5 2

13 10030 Triton t 3 3 1

14 20030 3He 3he 2 2 1

156

Chapter 8

Depletion Analysis

8.1 Introduction

Depletion is an important aspect of reactor analysis for the reactor safety and for the prediction

of the economic performance of the reactor. It includes various nuclear reactions, and is

described by the isotopic depletion rate equations where isotopic concentrations are solved as a

function of time and position. Secondly, the multi-group transport/diffusion equations are solved

for the neutron flux. The depletion calculations in this study were performed using DRAGON

code. It uses an EVO module wherein depletion equations for different isotopes in the library are

solved using burnup chains as available in the generated library. Depletion can be performed at

constant flux or constant power in MW/tonne of initial uranium using burnup time steps (in

Days), assuming linear flux/power changes. The burnup mixtures of the unit cell are solved using

the rate equations describing the isotopic changes in the core composition during the reactor

operation. The rate equations are expressed as shown below, Marleau, et al, 2010 [22].

( ) ( ) ( ) ,kk k k

dN N t t S tdt

+ Λ = 1,k K= 8.1

( ) ( ) ( ), ,k k a kt t tλ σ φΛ = + 8.2

( ) ( ) ( ) ( ) ( ) ( ),1 1

,L K

k kl f l l kl ll l

S t Y t t N t m t N tσ φ= =

= +∑ ∑ 8.3

157

( ) ( ) ( ) ( ), ,0,x l x lt t u t u duσ φ σ φ

∞= ∫ 8.4

( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )( ), , 0 0

, , 0 0 00

, , , ,, , , , x k f f x k

x k x kf

t u t u t u t ut u t u t u t u t t

t tσ φ σ φ

σ φ σ φ−

= + −−

8.5

Where

K = number of depleting isotopes

L = number of fissile isotopes

( )kN t = time dependent number density for thk isotope

kλ = radioactive decay constant for thk isotope

( ), ,x k t uσ = time and lethargy dependent microscopic cross section for nuclear reaction x on thk

isotope, x can be absorption fission and radiative capture cross sections

( ),t uφ = time and lethargy dependent neutron flux

klY = fission yield for production of fission product k by fissile l

klm = radioactive decay constant or ( ) ( ),x l t tσ φ term for the production of isotope k by isotope

l

The advantage of the DRAGON code in depletion analysis is that it allows the calculations of

rate equations for as many isotopes as required per type of the nuclear reactor of interest. Thus,

158

there is sufficient capability for monitoring the nuclides concentrations, fission products buildup,

and burnable poison concentrations. Similar HTGR models as in previous chapters are used

(Pebble fuel element and Prismatic hexagonal blocks) for depletion analysis. The depletion

analysis were performed using input multi-group cross section library based on selected

SHEM_TPN-531 multi-group structure at a constant power of 62 MW/tonne and for burnup

steps of 0.5, 5, 10, 20, 30, 40, 50, 60, 70, 80, 100 and 120 GWD/tonne of initial uranium.

8.2 Pebble Fuel Element

Figure 8.1 presents the criticality behavior of the Pebble fuel element during the depletion

process per burnup steps in GWD/tonne. The effect of the xenon fission product buildup and k-

effective is observed at the beginning of the depletion process. The xenon causes a sharp drop in

the k-effective during the first few days of core depletion. The overall criticality curve

presentation produce good results as expected for assumed constant power. The k-effective

results are in agreement with the work reported by Dehart and Goluoglu [7], where depletion

calculations were conducted using Serpent and TRITON/NEWT and TRITON/KENO codes.

The neutron flux spectrum for the beginning of life and end of life are given in Figures 8.2 and

8.3 respectively, showing an increase in neutron flux peak for thermal regions for the end of life.

The neutron flux spectrum increase observed in the end of life (Figure 8.3) as compared to the

beginning of life (Figure 8.2) is due to the number of nuclides and fission products with thermal

energy that are produced during burnup process. These nuclides and fission products then

159

compete with the control rods resulting in a need to remove control rods from the system in order

to keep the reactor at constant power. The nuclide concentration changes, for example of U-235

as it fissions has been reduced by 98 percent from the beginning of life to the end of life. More

nuclides and fission products were produced (Table 8.1) and their concentration increased with

an increase in burnup time step. Pu-239 increased by 98 percent from its initial production

(beginning of life) to the end of life. The summary of the nuclides concentrations in atom/barn-

cm are given in Table 8.1 and Figure 8.4 for selected burnup steps. Fission products are shown in

Table 8.2 and Figure 8.5. The xenon-135 buildup is about 66 percent and that of Sm149 is 10

percent from beginning of life to the end of life.

Figure 8.1: K-effective versus burnup

0.7

0.8

0.9

1

1.1

1.2

1.3

1.4

1.5

1.6

0 20 40 60 80 100 120 140

K-e

ffec

tive

Burnup (GWD/t)

K-effective

160

Figure 8.2: Neutron flux spectrum at the beginning of life

Figure 8.3: Neutron flux spectrum at the end of life

0

0.1

0.2

0.3

0.4

0.5

1.00E-09 1.00E-07 1.00E-05 1.00E-03 1.00E-01 1.00E+01

Flux

/leth

argy

wid

th

Energy (MeV)

Pebble_BOL

0

0.2

0.4

0.6

0.8

1.00E-09 1.00E-07 1.00E-05 1.00E-03 1.00E-01 1.00E+01

Flux

/leth

argy

wid

th

Energy (MeV)

Pebble_EOL

161

Table 8.1: Nuclides concentration in atom/barm.cm

Nuclides

Burnup (GWD/t)

10 20 40 80 120

U-235 1.64E-03 1.38E-03 9.27E-04 2.98E-04 4.27E-05

U-236 4.84E-05 9.15E-05 1.63E-04 2.48E-04 2.58E-04

U-238 2.12E-02 2.11E-02 2.09E-02 2.03E-02 1.96E-02

Np-237 3.69E-07 1.47E-06 5.53E-06 1.81E-05 2.95E-05

Pu-238 8.61E-09 7.39E-08 6.21E-07 4.87E-06 1.19E-05

Pu-239 7.61E-05 1.30E-04 1.88E-04 1.98E-04 1.70E-04

Pu-240 5.02E-06 1.61E-05 4.15E-05 7.46E-05 7.90E-05

Pu-241 7.40E-07 4.68E-06 2.14E-05 5.10E-05 5.06E-05

Pu-242 2.12E-08 2.95E-07 3.40E-06 2.78E-05 6.95E-05

Am-241 3.89E-09 4.90E-08 4.41E-07 1.75E-06 1.58E-06

Am-242m 1.71E-11 3.51E-10 4.42E-09 1.98E-08 1.74E-08

Am-243 2.61E-10 7.80E-09 2.02E-07 4.02E-06 1.77E-05

Cm-242 1.43E-10 3.58E-09 6.57E-08 6.41E-07 1.20E-06

Cm-244 4.47E-12 2.86E-10 1.69E-08 8.86E-07 8.07E-06

Cm-245 2.11E-14 2.72E-12 3.13E-10 2.73E-08 2.74E-07

162

Figure 8.4: Nuclides concentration in atom/barm.cm

Table 8.2: Fission products in atom/barn.cm

Fission Products

Burnup (GWD/t)

10 20 40 80 120

Kr-85 6.22E-07 1.19E-06 2.18E-06 3.62E-06 4.37E-06

Sr-90 1.40E-05 2.70E-05 5.03E-05 8.66E-05 1.09E-04

Ag-110m 2.71E-10 1.46E-09 8.30E-09 5.15E-08 1.58E-07

Cs-137 1.55E-05 3.08E-05 6.09E-05 1.19E-04 1.73E-04

Xe-135 1.62E-08 1.54E-08 1.34E-08 8.74E-09 5.57E-09

Sm-149 1.43E-07 1.41E-07 1.32E-07 9.65E-08 6.76E-08

Sm-151 4.12E-07 4.72E-07 5.04E-07 4.98E-07 4.77E-07

1.0E-24

1.0E-22

1.0E-20

1.0E-18

1.0E-16

1.0E-14

1.0E-12

1.0E-10

1.0E-08

1.0E-06

1.0E-04

1.0E-02

1.0E+00

0 20 40 60 80 100 120 140

Con

cent

ratio

n (a

tom

/bar

m.c

m)

Burnup (GWD/t)

U235 U236 U238 Np237

Pu238 Pu239 Pu240 Pu241

Pu242 Am241 Am242m Am243

Cm242 Cm244 Cm245

163

Figure 8.5: Fission products buildup in a Pebble fuel element

8.3 Prismatic Hexagonal Block

Similar depletion analysis were conducted for the prismatic hexagonal block and the criticality

behavior is presented in Figure 8.6. The neutron flux spectrum at the beginning and end of life

are shown in Figure 8.7 and 8.8 with significant neutron flux spectrum peak in thermal region as

a result more nuclides and fission products are produced and the U-235 is reduced. A significant

decrease in U-235 concentration of about 95 percent is observed. Pu-239 concentration had

increased by 98 percent during the burnup process. The xenon-135 buildup change is about 51

percent. This is in agreement with prismatic hexagonal blocks depletion analysis results reported

by Rohde et al, 2011[32], where BGCore and Helios codes were used. The nuclides

1.0E-13

1.0E-12

1.0E-11

1.0E-10

1.0E-09

1.0E-08

1.0E-07

1.0E-06

1.0E-05

1.0E-04

1.0E-03

0 20 40 60 80 100 120 140

Con

cent

ratio

n (a

tom

/bar

m.c

m)

Burnup (GWD/t)

Kr85 Sr90 Ag110m

Cs137 Xe135 Sm149

Sm151

164

concentration per selected burnup steps is given in Table 8.3 and Figure 8.9. The fission products

are given in Table 8.4 and Figure 8.10.

Figure 8.6: Prismatic block K-effective versus burnup

Figure 8.7 Neutron flux spectrum at the beginning of life

0.7

0.8

0.9

1

1.1

1.2

1.3

1.4

1.5

0 20 40 60 80 100 120 140

K-e

ffec

tive

Burnup (GWD/t)

K-effective

0

0.1

0.2

0.3

0.4

0.5

0.6

1.00E-09 1.00E-07 1.00E-05 1.00E-03 1.00E-01 1.00E+01

Flux

/leth

argy

wid

th

Energy (MeV)

Prismatic_BOL

165

Figure 8.8: Neutron flux spectrum at the end of life

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

1.00E-09 1.00E-07 1.00E-05 1.00E-03 1.00E-01 1.00E+01

Flux

/leth

argy

wid

th

Energy (MeV)

Prismatic_EOL

166

Table 8.3: Nuclides concentration in atom/barn.cm

Nuclides

Burnup (GWD/t)

10 20 40 80 120

U-235 1.64E-03 1.39E-03 9.55E-04 3.67E-04 9.32E-05

U-236 5.02E-05 9.39E-05 1.64E-04 2.45E-04 2.58E-04

U-238 2.12E-02 2.11E-02 2.08E-02 2.02E-02 1.94E-02

Np-237 4.78E-07 1.88E-06 6.93E-06 2.19E-05 3.56E-05

Pu-238 1.26E-08 1.07E-07 8.74E-07 6.51E-06 1.64E-05

Pu-239 8.74E-05 1.48E-04 2.16E-04 2.45E-04 2.27E-04

Pu-240 6.09E-06 1.87E-05 4.52E-05 7.77E-05 8.48E-05

Pu-241 1.09E-06 6.60E-06 2.83E-05 6.65E-05 7.24E-05

Pu-242 3.21E-08 4.24E-07 4.40E-06 3.05E-05 6.99E-05

Am-241 5.77E-09 6.98E-08 5.91E-07 2.38E-06 2.71E-06

Am-242m 2.81E-11 5.52E-10 6.60E-09 3.06E-08 3.45E-08

Am-243 4.92E-10 1.39E-08 3.28E-07 5.34E-06 2.00E-05

Cm-242 2.32E-10 5.55E-09 9.28E-08 7.91E-07 1.53E-06

Cm-244 1.03E-11 6.22E-10 3.35E-08 1.40E-06 9.88E-06

Cm-245 6.13E-14 7.51E-12 8.14E-10 6.11E-08 5.23E-07

167

Figure 8.9: Nuclides concentration in atom/barn.cm

3.0E-23

3.0E-21

3.0E-19

3.0E-17

3.0E-15

3.0E-13

3.0E-11

3.0E-09

3.0E-07

3.0E-05

3.0E-03

3.0E-01

0 20 40 60 80 100 120 140

Con

cent

ratio

n (a

tom

/bar

m.c

m)

Burnup (GWD/t)

U235 U236 U238 Np237

Pu238 Pu239 Pu240 Pu241

Pu242 Am241 Am243 Cm242

Cm244 Cm245

168

Table 8.4: Fission products in atom/barn.cm

Fission Products

Burnup (GWD/t)

10 20 40 80 120

Kr-85 6.17E-07 1.17E-06 2.13E-06 3.51E-06 4.30E-06

Sr-90 1.39E-05 2.66E-05 4.91E-05 8.38E-05 1.07E-04

Ag-110m 3.56E-10 1.95E-09 1.10E-08 6.25E-08 1.68E-07

Cs-137 1.54E-05 3.07E-05 6.07E-05 1.19E-04 1.74E-04

Xe-135 1.69E-08 1.64E-08 1.50E-08 1.13E-08 8.37E-09

Sm-149 1.48E-07 1.51E-07 1.49E-07 1.25E-07 9.83E-08

Sm-151 4.40E-07 5.28E-07 6.00E-07 6.60E-07 6.74E-07

Figure 8.10: Fission products buildup in a Prismatic hexagonal block

1.0E-13

1.0E-12

1.0E-11

1.0E-10

1.0E-09

1.0E-08

1.0E-07

1.0E-06

1.0E-05

1.0E-04

0 20 40 60 80 100 120 140

Con

cent

ratio

n (a

tom

/bar

m.c

m)

Burnup (GWD/t)

Kr85 Sr90 Ag110m

Cs137 Xe135 Sm149

Sm151

169

Chapter 9

Conclusions and Recommendations

9.1 Conclusions

The objective of this study was to investigate the applicability of the SHEM energy group

structures (281 and 361) and GA-193 for HTR applications and improve them utilizing the more

systematic, consistent, and sophisticated energy group selection method referred to as

contributon and point-wise cross-section driven (CPXSD). The DRAGON code was used for the

energy group structure improvement and the MCNP5 as reference for comparing the results to

determine the magnitude of the improvement.

MCNP5 code was selected to provide a reference solution to the study. Therefore both the pebble

fuel element and prismatic hexagonal block are the basic components of the PBR and VHTR

analyzed by DRAGON and MCNP5 for comparing results. Then, energy boundaries were

selected to divide the energy spectrum into three regions (fast, epithermal and thermal) that

correspond to the SHEM-281, SHEM-361 and GA-193 energy group structures. These analysis

were based on ENDF/B-VII.0 and ENDF/B-VI.8 data files. Additionally, the sensitivity caused

by the data files was studied. When comparing reaction rates from the three regions, the thermal

region had higher percent deviations and the epithermal and fast regions had their percent

deviations below 1 percent. The major changes were observed to be due to helium advancement

170

from ENDF/B -VI.8 to ENDF/B -VII.0 data files. Also neutron sub-library was increased from

328 to 393 as well as radioactive decay (from 979 to 3838), and a new 163 photonuclear sub-

library was added. Similar Pebble fuel element and Prismatic hexagonal block models were

created for the DRAGON code analysis. The neutron cross sections were generated using

SHEM-361, SHEM-281 and GA-193 energy group structures. Then, the pebble fuel element and

prismatic hexagonal block fuel analysis were subjected to these libraries and the results were

compared to the MCNP5 (reference solution) as shown in Table 9.1 and 9.2.

Table 9.1: Pebble fuel element deviations

K-effective deviation

Group Structure K-effective deviation in pcm MCNP5 - 1.528750

DRAGON SHEM-281 1.516920 1189 SHEM-361 1.517046 1176.4 GA-193 1.307130 22168.5


% deviation SHEM-281

Absorption -1.389212277 -2.882545451 -17.43271822 Nu-Fission 0.05013748 -3.099762817 0.255135027

Average Flux 2.499473073 0.494391026 -0.018703628 Reaction rates Thermal Epithermal Fast


Absorption -1.389212277 -2.899649913 -17.28212864 Nu-Fission 0.056545572 -3.302965595 0.25686741

Average Flux 2.504023558 0.483089478 -0.018536195

171

Table 9.2: Prismatic hexagonal block deviations

K-effective deviation

Group Structure K-effective deviation in pcm MCNP5 - 1.46946

DRAGON SHEM-281 1.46094E+00 852.4 SHEM-361 1.461067E+00 839.3 GA-193 1.21880E+00 25066.1



Absorption 1.521522941 -2.520222601 -3.753748785 Nu-Fission 3.807588733 -0.699405992 -2.290771967

Average Flux -0.972081048 -2.807681093 -9.778257997 Thermal Epithermal Fast


Absorption 1.53327792 -2.547080042 -3.768246022 Nu-Fission 3.818534366 -0.922847458 -2.290771967

Average Flux -0.959702214 -2.820463348 -9.778257997

The k-effective difference between the MCNP5 calculation and DRAGON are approximately

1.19 % for the pebble fuel element and 0.85 % for the prismatic hexagonal fuel block. For the

DRAGON code to become reliable for analyzing Pebble bed and Prismatic reactors, the k-

effective deviation should be less than 500 pcm . Further examination of Table 9.1 and 9.2 show

that for the Pebble fuel element the major reaction rates deviation is -17 % for the fast absorption

rate and for the Prismatic block is 3.8 % for the thermal nu-fission reaction rates. This could

explain the major errors that occur in the DRAGON calculation. A small error of 3.83 in the

thermal nu-fission reaction rate for the Pebble fuel element could propagate into an error greater

than 1 % in k-effective whereas a -17 % error in fast absorption rate for the Prismatic hexagonal

block would propagate also into an error greater than 1 % for k-effective because k-effective is

less sensitive to errors in the fast absorption rate. Note that an increase in the MCNP5 thermal

172

nu-fission and a decrease in MCNP5 fast absorption rate both cause the MCNP5 k-effective to be

greater than the corresponding DRAGON k-effective. It is interesting to note that the errors are

different for reaction rates for the Pebble fuel and Prismatic block. This might indicate that

geometry plays some part in the way DRAGON calculates reaction rates (see Table 5.7, 5.8 and

5.9 and 5.10).

Optimizing the SHEM-281 and SHEM-361multi-group structures to 407 and 531 groups caused

relatively small changes in the k-effective and reaction rates. The small changes in the k-

effective and reaction rates due to the increase in the multi-groups (from 281 to 407 and from

361 to 531) groups were insignificant compared to the significant deviation in k-effective caused

by the 17 % deviation in the fast absorption reaction rate in the Pebble fuel element and the 3.8

% deviation in the nu-fission reaction rate for the Prismatic hexagonal block. Thus, using

SHEM_TPN-407 and SHEM_TPN-531 group structures in the DRAGON calculation does not

correct for the relatively large deviation in the k-effective for either Pebble fuel element or

Prismatic hexagonal block. Nor, will it correct the deviation in the fast absorption reaction rate in

the Pebble fuel element or the nu-fission reaction rate in the Prismatic hexagonal block. The

DRAGON code calculation method of the Pebble cell fast absorption reaction rate and the

Prismatic block nu-fission reaction rate should be corrected so that the DRAGON code

calculates these reaction rates more accurately.

173

It is also observed that SHEM-281 energy group structure development had solved sufficient

nuclear physics problems that were introduced by the assumptions considered for XMAS-172

energy group structure. This was done by eliminating the self-shielding calculation of thermal

resonances, removing self-shielding models for U-238 resonances, 6.7rE = and 20.9 eV .

Sufficient nuclides and fission products were taken in to account. Then, SHEM-281 was

improved to SHEM-361 using a subgroup model in the regions of 22.5 eV and 11.14 keV

(epithermal region). However, there is little effect on the reaction rates and k-effective results as

shown in chapter 5 when comparing the two energy group structures. This is in agreement with

the results obtained in this study, the improvement of SHEM energy group structures had little

effect on the reaction rates and k-effective as the pebble fuel element and prismatic hexagonal

block models were analyzed.

GA-193 was improved to GA_TPN-537 energy group structure. The fast energy region was

improved from 49 to 83 groups, epithermal region from 42 to 334 and thermal region from 102

to 120. When comparing the results obtained using GA-193 and GA_TPN-537 energy group

structures, the relative percent deviation for k-effective was 5398.4 pcm for the pebble fuel

element and 7212.2 pcm for the prismatic hexagonal block. By improving GA-193 to GA_TPN-

537, the relative percent deviation is reduced from 22168.5 pcm to 16770.1 pcm to the MCNP5

results for the pebble fuel element. For the prismatic hexagonal block, the relative percent

deviation is reduced from 25066.1 pcm to 17853.9 pcm to the MCNP5 results. There was a

significant improvement in reaction rates as well but the deviations remain very large.

174

There was also recently published ENDF/B-VII.1 that was decided to be tested in this study and

be compared to ENDF/B-VII.0. The neutron cross sections were generated for both data files and

then pebble fuel element and prismatic hexagonal block models were analyzed using these

libraries. There were no significant changes in the reaction rates, the reason being that there were

no significant changes in the important nuclides (U-235, U-238). The observed changes were in

k-effective, which were about 500 pcm in difference. This is attributed to the advancement of

structural materials, for example natural carbon cross sections (Figure 7.1), which is important

for HTR since graphite is the moderator.

Lastly, the SHEM_TPN-531 energy group structure was used for the depletion analysis. The

objective was to test its ability for the HTR deep burn analysis. It is interesting to note that the

change in k-effective with increase in burnup, Figure 8.1 is a straight line once the sharp drop in

k-effective due to xenon buildup is passed. No such straight line occurs for the change in k-

effective as a function of burnup in light water reactors. Then, the conclusion is that the energy

group structure is applicable for HTR deep burn analysis as the results are showing a good

agreement with other researchers [7] [32].

9.2 Recommendation

The main recommendation based on the results of this study is to determine why there are large

deviations, -17 % in the fast absorption reaction rate for the Pebble fuel element and 3.8 % in the

thermal nu-fission reaction rate, when DRAGON calculates the reaction rates. Once these

175

problems are fixed DRAGON should be an excellent code to generate multi-group cross section

for graphite moderated reactors. In addition, although DRAGON does create broad group cross

sections, these cross sections cannot be reprocessed within DRAGON for further cross section

improvement. This should be corrected.

Next step will be to utilize the SHEM_TPN-531 multi-group structure with DRAGON and

develop an optimized broad group cross-section structure for core analysis using again the

contributon method. Such study would require coupling DRAGON with core analysis code,

which has also forward and adjoint flux calculation capability.

176

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181

Appendices

A1 MCNP5 Input Decks

A1.1 Pebble Fuel Element Pebble 1 1 -10.4 -13 u=5 imp:n=1 $ kernel UO2 2 2 -1.05 13 -14 u=5 imp:n=1 $ inner low density carbon 3 3 -1.9 14 -15 u=5 imp:n=1 $ inner pyro carbon layer 4 4 -3.18 15 -16 u=5 imp:n=1 $ Silicon carbide layer 5 3 -1.90 16 -17 u=5 imp:n=1 $ Outer pyro carbon layer 6 5 -1.75 17 u=5 imp:n=1 $ graphite matrix 7 5 -1.75 -18 19 -20 21 -22 23 lat=1 u=8 imp:n=1 $ cube fill=-14:14 -14:14 -14:14 c (121) 8 260r 8 8r 5 10r 8 8r 8 8r 5 10r 8 8r 8 8r 5 10r 8 8r 8 8r 5 10r 8 8r 8 8r 5 10r 8 8r 8 8r 5 10r 8 8r 8 8r 5 10r 8 8r 8 8r 5 10r 8 8r 8 8r 5 10r 8 8r 8 8r 5 10r 8 8r 8 8r 5 10r 8 8r 8 260r c (169) 8 231r 8 7r 5 12r 8 7r 8 7r 5 12r 8 7r 8 7r 5 12r 8 7r 8 7r 5 12r 8 7r 8 7r 5 12r 8 7r 8 7r 5 12r 8 7r 8 7r 5 12r 8 7r 8 7r 5 12r 8 7r 8 7r 5 12r 8 7r 8 7r 5 12r 8 7r 8 7r 5 12r 8 7r 8 7r 5 12r 8 7r 8 7r 5 12r 8 7r 8 231r c (285) 8 173r 8 6r 5 14r 8 6r 8 5r 5 16r 8 5r 8 5r 5 16r 8 5r 8 5r 5 16r 8 5r 8 5r 5 16r 8 5r 8 5r 5 16r 8 5r 8 5r 5 16r 8 5r 8 5r 5 16r 8 5r 8 5r 5 16r 8 5r 8 5r 5 16r 8 5r 8 5r 5 16r 8 5r 8 5r 5 16r 8 5r 8 5r 5 16r 8 5r 8 5r 5 16r 8 5r 8 5r 5 16r 8 5r 8 5r 5 16r 8 5r 8 6r 5 14r 8 6r 8 173r c (401) 8 115r 8 7r 5 12r 8 7r 8 6r 5 14r 8 6r 8 5r 5 16r 8 5r 8 4r 5 18r 8 4r 8 3r 5 20r 8 3r 8 3r 5 20r 8 3r 8 3r 5 20r 8 3r 8 3r 5 20r 8 3r 8 3r 5 20r 8 3r 8 3r 5 20r 8 3r 8 3r 5 20r 8 3r 8 3r 5 20r 8 3r 8 3r 5 20r 8 3r 8 3r 5 20r 8 3r 8 3r 5 20r 8 3r 8 3r 5 20r 8 3r 8 3r 5 20r 8 3r 8 4r 5 18r 8 4r 8 5r 5 16r 8 5r 8 6r 5 14r 8 6r 8 7r 5 12r 8 7r 8 115r c (469) 8 86r 8 7r 5 12r 8 7r 8 6r 5 14r 8 6r 8 5r 5 16r 8 5r 8 4r 5 18r 8 4r 8 3r 5 20r 8 3r 8 2r 5 22r 8 2r 8 2r 5 22r 8 2r 8 2r 5 22r 8 2r 8 2r 5 22r 8 2r 8 2r 5 22r 8 2r 8 2r 5 22r 8 2r 8 2r 5 22r 8 2r 8 2r 5 22r 8 2r 8 2r 5 22r 8 2r 8 2r 5 22r 8 2r 8 2r 5 22r 8 2r 8 2r 5 22r 8 2r 8 2r 5 22r 8 2r 8 3r 5 20r 8 3r 8 4r 5 18r 8 4r 8 5r 5 16r 8 5r 8 6r 5 14r 8 6r 8 7r 5 12r 8 7r 8 86r c (469) 8 86r 8 7r 5 12r 8 7r 8 6r 5 14r 8 6r 8 5r 5 16r 8 5r 8 4r 5 18r 8 4r 8 3r 5 20r 8 3r 8 2r 5 22r 8 2r 8 2r 5 22r 8 2r 8 2r 5 22r 8 2r 8 2r 5 22r 8 2r 8 2r 5 22r 8 2r 8 2r 5 22r 8 2r 8 2r 5 22r 8 2r 8 2r 5 22r 8 2r 8 2r 5 22r 8 2r 8 2r 5 22r 8 2r 8 2r 5 22r 8 2r 8 2r 5 22r 8 2r 8 2r 5 22r 8 2r 8 3r 5 20r 8 3r 8 4r 5 18r 8 4r 8 5r 5 16r 8 5r 8 6r 5 14r 8 6r 8 7r 5 12r 8 7r 8 86r c (549) 8 57r 8 7r 5 12r 8 7r 8 5r 5 16r 8 5r 8 4r 5 18r 8 4r 8 4r 5 18r 8 4r 8 3r 5 20r 8 3r 8 2r 5 22r 8 2r 8 8 5 24r 8 8 8 8 5 24r 8 8

182

8 8 5 24r 8 8 8 8 5 24r 8 8 8 8 5 24r 8 8 8 8 5 24r 8 8 8 8 5 24r 8 8 8 8 5 24r 8 8 8 8 5 24r 8 8 8 8 5 24r 8 8 8 8 5 24r 8 8 8 8 5 24r 8 8 8 8 5 24r 8 8 8 2r 5 22r 8 2r 8 3r 5 20r 8 3r 8 4r 5 18r 8 4r 8 4r 5 18r 8 4r 8 5r 5 16r 8 5r 8 7r 5 12r 8 7r 8 57r c (625) 8 28r 8 7r 5 12r 8 7r 8 6r 5 14r 8 6r 8 4r 5 18r 8 4r 8 3r 5 20r 8 3r 8 3r 5 20r 8 3r 8 2r 5 22r 8 2r 8 8 5 24r 8 8 8 5 26r 8 8 5 26r 8 8 5 26r 8 8 5 26r 8 8 5 26r 8 8 5 26r 8 8 5 26r 8 8 5 26r 8 8 5 26r 8 8 5 26r 8 8 5 26r 8 8 5 26r 8 8 5 26r 8 8 8 5 24r 8 8 8 2r 5 22r 8 2r 8 3r 5 20r 8 3r 8 3r 5 20r 8 3r 8 4r 5 18r 8 4r 8 6r 5 14r 8 6r 8 7r 5 12r 8 7r 8 28r c (625) 8 28r 8 7r 5 12r 8 7r 8 6r 5 14r 8 6r 8 4r 5 18r 8 4r 8 3r 5 20r 8 3r 8 3r 5 20r 8 3r 8 2r 5 22r 8 2r 8 8 5 24r 8 8 8 5 26r 8 8 5 26r 8 8 5 26r 8 8 5 26r 8 8 5 26r 8 8 5 26r 8 8 5 26r 8 8 5 26r 8 8 5 26r 8 8 5 26r 8 8 5 26r 8 8 5 26r 8 8 5 26r 8 8 8 5 24r 8 8 8 2r 5 22r 8 2r 8 3r 5 20r 8 3r 8 3r 5 20r 8 3r 8 4r 5 18r 8 4r 8 6r 5 14r 8 6r 8 7r 5 12r 8 7r 8 28r c (697) 8 8r 5 10r 8 8r 8 6r 5 14r 8 6r 8 5r 5 16r 8 5r 8 3r 5 20r 8 3r 8 2r 5 22r 8 2r 8 2r 5 22r 8 2r 8 8 5 24r 8 8 8 5 26r 8 8 5 26r 8 5 318r 8 5 26r 8 8 5 26r 8 8 8 5 24r 8 8 8 2r 5 22r 8 2r 8 2r 5 22r 8 2r 8 3r 5 20r 8 3r 8 5r 5 16r 8 5r 8 6r 5 14r 8 6r 8 8r 5 10r 8 8r c (697) 8 8r 5 10r 8 8r 8 6r 5 14r 8 6r 8 5r 5 16r 8 5r 8 3r 5 20r 8 3r 8 2r 5 22r 8 2r 8 2r 5 22r 8 2r 8 8 5 24r 8 8 8 5 26r 8 8 5 26r 8 5 318r 8 5 26r 8 8 5 26r 8 8 8 5 24r 8 8 8 2r 5 22r 8 2r 8 2r 5 22r 8 2r 8 3r 5 20r 8 3r 8 5r 5 16r 8 5r 8 6r 5 14r 8 6r 8 8r 5 10r 8 8r c (697) 8 8r 5 10r 8 8r 8 6r 5 14r 8 6r 8 5r 5 16r 8 5r 8 3r 5 20r 8 3r 8 2r 5 22r 8 2r 8 2r 5 22r 8 2r 8 8 5 24r 8 8 8 5 26r 8 8 5 26r 8 5 318r 8 5 26r 8 8 5 26r 8 8 8 5 24r 8 8 8 2r 5 22r 8 2r 8 2r 5 22r 8 2r 8 3r 5 20r 8 3r 8 5r 5 16r 8 5r 8 6r 5 14r 8 6r 8 8r 5 10r 8 8r c (697) 8 8r 5 10r 8 8r 8 6r 5 14r 8 6r 8 5r 5 16r 8 5r 8 3r 5 20r 8 3r 8 2r 5 22r 8 2r 8 2r 5 22r 8 2r 8 8 5 24r 8 8 8 5 26r 8 8 5 26r 8 5 318r 8 5 26r 8 8 5 26r 8 8 8 5 24r 8 8 8 2r 5 22r 8 2r 8 2r 5 22r 8 2r 8 3r 5 20r 8 3r 8 5r 5 16r 8 5r 8 6r 5 14r 8 6r 8 8r 5 10r 8 8r c (697) 8 8r 5 10r 8 8r 8 6r 5 14r 8 6r 8 5r 5 16r 8 5r 8 3r 5 20r 8 3r 8 2r 5 22r 8 2r 8 2r 5 22r 8 2r 8 8 5 24r 8 8 8 5 26r 8 8 5 26r 8 5 318r 8 5 26r 8 8 5 26r 8 8 8 5 24r 8 8 8 2r 5 22r 8 2r 8 2r 5 22r 8 2r 8 3r 5 20r 8 3r 8 5r 5 16r 8 5r 8 6r 5 14r 8 6r 8 8r 5 10r 8 8r c (697) 8 8r 5 10r 8 8r 8 6r 5 14r 8 6r 8 5r 5 16r 8 5r 8 3r 5 20r 8 3r 8 2r 5 22r 8 2r 8 2r 5 22r 8 2r 8 8 5 24r 8 8 8 5 26r 8 8 5 26r 8 5 318r 8 5 26r 8 8 5 26r 8 8 8 5 24r 8 8 8 2r 5 22r 8 2r 8 2r 5 22r 8 2r 8 3r 5 20r 8 3r 8 5r 5 16r 8 5r 8 6r 5 14r 8 6r 8 8r 5 10r 8 8r c (697) 8 8r 5 10r 8 8r 8 6r 5 14r 8 6r 8 5r 5 16r 8 5r 8 3r 5 20r 8 3r 8 2r 5 22r 8 2r 8 2r 5 22r 8 2r 8 8 5 24r 8 8 8 5 26r 8 8 5 26r 8 5 318r 8 5 26r 8 8 5 26r 8 8 8 5 24r 8 8 8 2r 5 22r 8 2r 8 2r

183

5 22r 8 2r 8 3r 5 20r 8 3r 8 5r 5 16r 8 5r 8 6r 5 14r 8 6r 8 8r 5 10r 8 8r c (697) 8 8r 5 10r 8 8r 8 6r 5 14r 8 6r 8 5r 5 16r 8 5r 8 3r 5 20r 8 3r 8 2r 5 22r 8 2r 8 2r 5 22r 8 2r 8 8 5 24r 8 8 8 5 26r 8 8 5 26r 8 5 318r 8 5 26r 8 8 5 26r 8 8 8 5 24r 8 8 8 2r 5 22r 8 2r 8 2r 5 22r 8 2r 8 3r 5 20r 8 3r 8 5r 5 16r 8 5r 8 6r 5 14r 8 6r 8 8r 5 10r 8 8r c (697) 8 8r 5 10r 8 8r 8 6r 5 14r 8 6r 8 5r 5 16r 8 5r 8 3r 5 20r 8 3r 8 2r 5 22r 8 2r 8 2r 5 22r 8 2r 8 8 5 24r 8 8 8 5 26r 8 8 5 26r 8 5 318r 8 5 26r 8 8 5 26r 8 8 8 5 24r 8 8 8 2r 5 22r 8 2r 8 2r 5 22r 8 2r 8 3r 5 20r 8 3r 8 5r 5 16r 8 5r 8 6r 5 14r 8 6r 8 8r 5 10r 8 8r c (697) 8 8r 5 10r 8 8r 8 6r 5 14r 8 6r 8 5r 5 16r 8 5r 8 3r 5 20r 8 3r 8 2r 5 22r 8 2r 8 2r 5 22r 8 2r 8 8 5 24r 8 8 8 5 26r 8 8 5 26r 8 5 318r 8 5 26r 8 8 5 26r 8 8 8 5 24r 8 8 8 2r 5 22r 8 2r 8 2r 5 22r 8 2r 8 3r 5 20r 8 3r 8 5r 5 16r 8 5r 8 6r 5 14r 8 6r 8 8r 5 10r 8 8r c (697) 8 8r 5 10r 8 8r 8 6r 5 14r 8 6r 8 5r 5 16r 8 5r 8 3r 5 20r 8 3r 8 2r 5 22r 8 2r 8 2r 5 22r 8 2r 8 8 5 24r 8 8 8 5 26r 8 8 5 26r 8 5 318r 8 5 26r 8 8 5 26r 8 8 8 5 24r 8 8 8 2r 5 22r 8 2r 8 2r 5 22r 8 2r 8 3r 5 20r 8 3r 8 5r 5 16r 8 5r 8 6r 5 14r 8 6r 8 8r 5 10r 8 8r c (625) 8 28r 8 7r 5 12r 8 7r 8 6r 5 14r 8 6r 8 4r 5 18r 8 4r 8 3r 5 20r 8 3r 8 3r 5 20r 8 3r 8 2r 5 22r 8 2r 8 8 5 24r 8 8 8 5 26r 8 8 5 26r 8 8 5 26r 8 8 5 26r 8 8 5 26r 8 8 5 26r 8 8 5 26r 8 8 5 26r 8 8 5 26r 8 8 5 26r 8 8 5 26r 8 8 5 26r 8 8 5 26r 8 8 8 5 24r 8 8 8 2r 5 22r 8 2r 8 3r 5 20r 8 3r 8 3r 5 20r 8 3r 8 4r 5 18r 8 4r 8 6r 5 14r 8 6r 8 7r 5 12r 8 7r 8 28r c (625) 8 28r 8 7r 5 12r 8 7r 8 6r 5 14r 8 6r 8 4r 5 18r 8 4r 8 3r 5 20r 8 3r 8 3r 5 20r 8 3r 8 2r 5 22r 8 2r 8 8 5 24r 8 8 8 5 26r 8 8 5 26r 8 8 5 26r 8 8 5 26r 8 8 5 26r 8 8 5 26r 8 8 5 26r 8 8 5 26r 8 8 5 26r 8 8 5 26r 8 8 5 26r 8 8 5 26r 8 8 5 26r 8 8 8 5 24r 8 8 8 2r 5 22r 8 2r 8 3r 5 20r 8 3r 8 3r 5 20r 8 3r 8 4r 5 18r 8 4r 8 6r 5 14r 8 6r 8 7r 5 12r 8 7r 8 28r c (549) 8 57r 8 7r 5 12r 8 7r 8 5r 5 16r 8 5r 8 4r 5 18r 8 4r 8 4r 5 18r 8 4r 8 3r 5 20r 8 3r 8 2r 5 22r 8 2r 8 8 5 24r 8 8 8 8 5 24r 8 8 8 8 5 24r 8 8 8 8 5 24r 8 8 8 8 5 24r 8 8 8 8 5 24r 8 8 8 8 5 24r 8 8 8 8 5 24r 8 8 8 8 5 24r 8 8 8 8 5 24r 8 8 8 8 5 24r 8 8 8 8 5 24r 8 8 8 8 5 24r 8 8 8 2r 5 22r 8 2r 8 3r 5 20r 8 3r 8 4r 5 18r 8 4r 8 4r 5 18r 8 4r 8 5r 5 16r 8 5r 8 7r 5 12r 8 7r 8 57r c (469) 8 86r 8 7r 5 12r 8 7r 8 6r 5 14r 8 6r 8 5r 5 16r 8 5r 8 4r 5 18r 8 4r 8 3r 5 20r 8 3r 8 2r 5 22r 8 2r 8 2r 5 22r 8 2r 8 2r 5 22r 8 2r 8 2r 5 22r 8 2r 8 2r 5 22r 8 2r 8 2r 5 22r 8 2r 8 2r 5 22r 8 2r 8 2r 5 22r 8 2r 8 2r 5 22r 8 2r 8 2r 5 22r 8 2r 8 2r 5 22r 8 2r 8 2r 5 22r 8 2r 8 2r 5 22r 8 2r 8 3r 5 20r 8 3r 8 4r 5 18r 8 4r 8 5r 5 16r 8 5r 8 6r 5 14r 8 6r 8 7r 5 12r 8 7r 8 86r c (469) 8 86r 8 7r 5 12r 8 7r 8 6r 5 14r 8 6r 8 5r 5 16r 8 5r 8 4r 5 18r 8 4r 8 3r 5 20r 8 3r 8 2r 5 22r 8 2r 8 2r 5 22r 8 2r 8 2r 5 22r 8 2r 8 2r 5 22r 8 2r 8 2r 5 22r 8 2r 8 2r 5 22r 8 2r 8 2r 5 22r

184

8 2r 8 2r 5 22r 8 2r 8 2r 5 22r 8 2r 8 2r 5 22r 8 2r 8 2r 5 22r 8 2r 8 2r 5 22r 8 2r 8 2r 5 22r 8 2r 8 3r 5 20r 8 3r 8 4r 5 18r 8 4r 8 5r 5 16r 8 5r 8 6r 5 14r 8 6r 8 7r 5 12r 8 7r 8 86r c (401) 8 115r 8 7r 5 12r 8 7r 8 6r 5 14r 8 6r 8 5r 5 16r 8 5r 8 4r 5 18r 8 4r 8 3r 5 20r 8 3r 8 3r 5 20r 8 3r 8 3r 5 20r 8 3r 8 3r 5 20r 8 3r 8 3r 5 20r 8 3r 8 3r 5 20r 8 3r 8 3r 5 20r 8 3r 8 3r 5 20r 8 3r 8 3r 5 20r 8 3r 8 3r 5 20r 8 3r 8 3r 5 20r 8 3r 8 3r 5 20r 8 3r 8 3r 5 20r 8 3r 8 4r 5 18r 8 4r 8 5r 5 16r 8 5r 8 6r 5 14r 8 6r 8 7r 5 12r 8 7r 8 115r c (285) 8 173r 8 6r 5 14r 8 6r 8 5r 5 16r 8 5r 8 5r 5 16r 8 5r 8 5r 5 16r 8 5r 8 5r 5 16r 8 5r 8 5r 5 16r 8 5r 8 5r 5 16r 8 5r 8 5r 5 16r 8 5r 8 5r 5 16r 8 5r 8 5r 5 16r 8 5r 8 5r 5 16r 8 5r 8 5r 5 16r 8 5r 8 5r 5 16r 8 5r 8 5r 5 16r 8 5r 8 5r 5 16r 8 5r 8 5r 5 16r 8 5r 8 6r 5 14r 8 6r 8 173r c (169) 8 231r 8 7r 5 12r 8 7r 8 7r 5 12r 8 7r 8 7r 5 12r 8 7r 8 7r 5 12r 8 7r 8 7r 5 12r 8 7r 8 7r 5 12r 8 7r 8 7r 5 12r 8 7r 8 7r 5 12r 8 7r 8 7r 5 12r 8 7r 8 7r 5 12r 8 7r 8 7r 5 12r 8 7r 8 7r 5 12r 8 7r 8 7r 5 12r 8 7r 8 231r c (121) 8 260r 8 8r 5 10r 8 8r 8 8r 5 10r 8 8r 8 8r 5 10r 8 8r 8 8r 5 10r 8 8r 8 8r 5 10r 8 8r 8 8r 5 10r 8 8r 8 8r 5 10r 8 8r 8 8r 5 10r 8 8r 8 8r 5 10r 8 8r 8 8r 5 10r 8 8r 8 8r 5 10r 8 8r 8 260r 8 0 -40 41 -42 43 -44 45 u=52 fill=8 imp:n=1 9 5 -1.75 40:-41:42:-43:44:-45 u=52 imp:n=1 10 0 -24 u=53 fill=52 imp:n=1 11 5 -1.75 24 u=53 imp:n=1 12 0 -25 u=54 fill=53 imp:n=1 13 13 -0.000176 25 u=54 imp:n=1 14 0 -26 fill=54 imp:n=1 15 0 26 imp:n=0 c Surfaces 13 so 0.025 14 so 0.034 15 so 0.038 16 so 0.0415 17 so 0.0455 18 px 0.081705 19 px -0.081705 20 py 0.081705 21 py -0.081705 22 pz 0.081705 23 pz -0.081705 24 so 2.5 25 so 3.0 *26 so 3.53735 40 px 2.369445 41 px -2.369445 42 py 2.369445 43 py -2.369445 44 pz 2.369445 45 pz -2.369445 c imp:n 1 13r 0 c

185

c Material data mode n m1 92238.66c 2.12877E-02 $ U-238 293.6 1993 92235.66c 1.92585E-03 $ U-235 293.6K 1997 8016.62c 4.64272E-02 $ Oxygen 293.6K 2000 5010.66c 1.14694E-07 $ Boron-10 5011.66c 4.64570E-07 $ Boron-11 m2 6000.24c 5.26449E-02 $ Natural Carbon (buffer) mt2 grph.60t m3 6000.24c 9.52621E-02 $ Natural Carbon mt3 grph.60t m4 6000.24c 4.77240E-02 $ Natural Carbon 14000.60c 4.77240E-02 $ Silicon 293.6K 1976 mt4 grph.60t m5 6000.24c 8.77414E-02 $ Natural Carbon 5010.66c 9.64977E-09 $ Boron-10 5011.66c 3.90864E-08 $ Boron-11Fuel mt5 grph.60t m13 2003.66c 3.71220E-11 $ Helium gas outside the FE 293.6K 1990 2004.60c 2.65156E-05 $ Helium gas outside the FE 293.6K 1973 ksrc 0 0 0 kcode 2000 1.0 200 1200 prdmp j 20 0 2 c c Tallies: Fluxes and Reaction rates c Energy bins c E0 3.14200E-06 1.15600E-01 1.96400E+01 $ 3 broad groups E0 3.05902E-06 1.11090E-01 1.49182E+01 c c Thermal, epithermal and fast fluxes F4:N 9 $ graphite cube 11 $ graphite layer 13 $ heliumlayer SD4 65.44984695 47.64748858 72.30820487 c c Coated Particle flux F14:N (1 < (7 [-14:14 -14:14 -14:14])<10) $ fuel kernel (2 < (7 [-14:14 -14:14 -14:14])<10) $ buffer (3 < (7 [-14:14 -14:14 -14:14])<10) $ IPyC (4 < (7 [-14:14 -14:14 -14:14])<10) $ SiC (5 < (7 [-14:14 -14:14 -14:14])<10) $ OPyC SD14 0.987835 1.49702 0.984231 1.049562 1.43658 c c reaction rates c F24:N 9 FM24 -1.0 5 (-1) (-2) (-6) (-7) (-6 -7) SD24 0.044085965 c F44:N 11 FM44 -1.0 5 (-1) (-2) (-6) (-7) (-6 -7) SD44 47.64748858

186

c F54:N 13 FM54 -1.0 13 (-1) (-2) (-6) (-7) (-6 -7) SD54 72.30820487 c F64:N (1 < (7 [-14:14 -14:14 -14:14])<10) FM64 -1.0 1 (-1) (-2) (-6) (-7) (-6 -7) SD64 0.987835 c F74:N (2 < (7 [-14:14 -14:14 -14:14])<10) FM74 -1.0 2 (-1) (-2) (-6) (-7) (-6 -7) SD74 1.49702 c F84:N (3 < (7 [-14:14 -14:14 -14:14])<10) FM84 -1.0 3 (-1) (-2) (-6) (-7) (-6 -7) SD84 0.984231 c F94:N (4 < (7 [-14:14 -14:14 -14:14])<10) FM94 -1.0 4 (-1) (-2) (-6) (-7) (-6 -7) SD94 1.049562 c F104:N (5 < (7 [-14:14 -14:14 -14:14])<10) FM104 -1.0 5 (-1) (-2) (-6) (-7) (-6 -7) SD104 1.43658 c

187

A1.2 Prismatic Hexagonal Block Prismatic Assembly lattice pattern c Hexagonal Lattice c -------------------------------- note --------------------------------------- c Preserve the coated particle packing fraction c Removed the particles sitting at the boundary of the fuel compact c ----------------------------------------------------------------------------- c ----------------------------------------------------------------------------- c Cells 10 10 -0.000176 -10 u=10 imp:n=1 $ coolant channel 20 0 -20 fill=1(1) u=10 imp:n=1 $ fuel channel 30 0 -30 fill=1(2) u=10 imp:n=1 40 0 -40 fill=1(3) u=10 imp:n=1 50 0 -50 fill=1(4) u=10 imp:n=1 60 0 -60 fill=1(5) u=10 imp:n=1 70 0 -70 fill=1(6) u=10 imp:n=1 80 80 -1.75 10 20 30 40 50 60 70 u=10 imp:n=1 c lattice 90 21 -10.4 -12 u=20 imp:n=1 $ kernel 91 22 -1.05 12 -13 u=20 imp:n=1 $ Inner low density kernel coating layer 92 23 -1.90 13 -14 u=20 imp:n=1 $ Inner Pyro Carbon kernel coating layer 93 24 -3.18 14 -15 u=20 imp:n=1 $ Silicon carbide layer 94 25 -1.90 15 -16 u=20 imp:n=1 $ Outer pyro Carbon coating layer 95 26 -1.75 16 u=20 imp:n=1 $ outside of the coating layer 85 26 -1.75 -16 u=50 imp:n=1 $ graphite medium interior 86 26 -1.75 16 u=50 imp:n=1 $ graphite medium exterior 96 0 -90 91 -92 93 -94 95 lat=1 u=21 imp:n=1 $ cube fill=-6:6 -6:6 -20:20 50 50 50 50 50 50 50 50 50 50 50 50 50 $row *** 50 50 50 50 50 50 20 50 50 50 50 50 50 $row 50 50 50 50 20 20 20 20 20 50 50 50 50 $row 50 50 50 20 20 20 20 20 20 20 50 50 50 $row 50 50 20 20 20 20 20 20 20 20 20 50 50 $row 50 50 20 20 20 20 20 20 20 20 20 50 50 $row 50 20 20 20 20 20 20 20 20 20 20 20 50 $row 50 50 20 20 20 20 20 20 20 20 20 50 50 $row 50 50 20 20 20 20 20 20 20 20 20 50 50 $row 50 50 50 20 20 20 20 20 20 20 50 50 50 $row 50 50 50 50 20 20 20 20 20 50 50 50 50 $row 50 50 50 50 50 50 20 50 50 50 50 50 50 $row 50 50 50 50 50 50 50 50 50 50 50 50 50 $row 50 50 50 50 50 50 50 50 50 50 50 50 50 $row *** 50 50 50 50 50 50 20 50 50 50 50 50 50 $row 50 50 50 50 20 20 20 20 20 50 50 50 50 $row 50 50 50 20 20 20 20 20 20 20 50 50 50 $row 50 50 20 20 20 20 20 20 20 20 20 50 50 $row 50 50 20 20 20 20 20 20 20 20 20 50 50 $row 50 20 20 20 20 20 20 20 20 20 20 20 50 $row 50 50 20 20 20 20 20 20 20 20 20 50 50 $row 50 50 20 20 20 20 20 20 20 20 20 50 50 $row 50 50 50 20 20 20 20 20 20 20 50 50 50 $row 50 50 50 50 20 20 20 20 20 50 50 50 50 $row 50 50 50 50 50 50 20 50 50 50 50 50 50 $row 50 50 50 50 50 50 50 50 50 50 50 50 50 $row 50 50 50 50 50 50 50 50 50 50 50 50 50 $row *** 50 50 50 50 50 50 20 50 50 50 50 50 50 $row 50 50 50 50 20 20 20 20 20 50 50 50 50 $row

188

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50 50 50 50 50 50 50 50 50 50 50 50 50 $row 50 50 50 50 50 50 50 50 50 50 50 50 50 $row *** 50 50 50 50 50 50 20 50 50 50 50 50 50 $row 50 50 50 50 20 20 20 20 20 50 50 50 50 $row 50 50 50 20 20 20 20 20 20 20 50 50 50 $row 50 50 20 20 20 20 20 20 20 20 20 50 50 $row 50 50 20 20 20 20 20 20 20 20 20 50 50 $row 50 20 20 20 20 20 20 20 20 20 20 20 50 $row 50 50 20 20 20 20 20 20 20 20 20 50 50 $row 50 50 20 20 20 20 20 20 20 20 20 50 50 $row 50 50 50 20 20 20 20 20 20 20 50 50 50 $row 50 50 50 50 20 20 20 20 20 50 50 50 50 $row 50 50 50 50 50 50 20 50 50 50 50 50 50 $row 50 50 50 50 50 50 50 50 50 50 50 50 50 $row 50 50 50 50 50 50 50 50 50 50 50 50 50 $row *** 50 50 50 50 50 50 20 50 50 50 50 50 50 $row 50 50 50 50 20 20 20 20 20 50 50 50 50 $row 50 50 50 20 20 20 20 20 20 20 50 50 50 $row 50 50 20 20 20 20 20 20 20 20 20 50 50 $row 50 50 20 20 20 20 20 20 20 20 20 50 50 $row 50 20 20 20 20 20 20 20 20 20 20 20 50 $row 50 50 20 20 20 20 20 20 20 20 20 50 50 $row 50 50 20 20 20 20 20 20 20 20 20 50 50 $row 50 50 50 20 20 20 20 20 20 20 50 50 50 $row 50 50 50 50 20 20 20 20 20 50 50 50 50 $row 50 50 50 50 50 50 20 50 50 50 50 50 50 $row 50 50 50 50 50 50 50 50 50 50 50 50 50 $row 50 50 50 50 50 50 50 50 50 50 50 50 50 $row *** 50 50 50 50 50 50 20 50 50 50 50 50 50 $row 50 50 50 50 20 20 20 20 20 50 50 50 50 $row 50 50 50 20 20 20 20 20 20 20 50 50 50 $row 50 50 20 20 20 20 20 20 20 20 20 50 50 $row 50 50 20 20 20 20 20 20 20 20 20 50 50 $row 50 20 20 20 20 20 20 20 20 20 20 20 50 $row 50 50 20 20 20 20 20 20 20 20 20 50 50 $row 50 50 20 20 20 20 20 20 20 20 20 50 50 $row 50 50 50 20 20 20 20 20 20 20 50 50 50 $row 50 50 50 50 20 20 20 20 20 50 50 50 50 $row 50 50 50 50 50 50 20 50 50 50 50 50 50 $row 50 50 50 50 50 50 50 50 50 50 50 50 50 $row c 87 0 90 -91 92 -93 94 -95 u=21 imp:n=1 $ fuel cube exterior 97 0 -17 fill=21 u=1 imp:n=1 $ inside He gap 98 27 -0.000176 17 u=1 imp:n=1 $ outside c Hexagonal cell 100 0 -100 200 -300 400 -500 600 u=30 fill=10 lat=2 imp:n=1 $ Hexagon c c Sample cell 200 0 -700 710 -720 730 -740 750 fill=30 imp:n=1 250 0 (700:-710:720:-730:740:-750) imp:n=0 c c 300 0 -700 710 -720 730 -740 750 20 30 40 50 60 70 imp:n=1 c ----------------------------------------------------------------------------- c ----------------------------------------------------------------------------- c SURFACE CARDS c Cylinder surfaces 10 cz 0.79400 20 c/z 1.628127759 0.94 0.63500

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30 c/z 0 1.88 0.63500 40 c/z -1.628127759 0.94 0.63500 50 c/z -1.628127759 -0.94 0.63500 60 c/z 0 -1.88 0.63500 70 c/z 1.628127759 -0.94 0.63500 c c Hex cell surfaces 100 px 1.628127759 200 px -1.628127759 300 p 0.577350269 1 0 1.88 400 p 0.577350269 1 0 -1.88 500 p -0.577350269 1 0 1.88 600 p -0.577350269 1 0 -1.88 c c Boundary Cell (Assembly) *700 px 3.256 *710 px -3.256 *720 pz 2.465 *730 pz -2.465 *740 py 2.820 *750 py -2.820 c surface 17 cz 0.6225 12 so 0.025 13 so 0.034 14 so 0.038 15 so 0.0415 16 so 0.0455 90 px 0.0560 91 px -0.0560 92 py 0.0560 93 py -0.0560 94 pz 0.0601220 95 pz -0.0601220 c ----------------------------------------------------------------------------- c ----------------------------------------------------------------------------- c DATA CARDS tr1 1.628127759 0.94 0 tr2 0 1.88 0 tr3 -1.628127759 0.94 0 tr4 -1.628127759 -0.94 0 tr5 0 -1.88 0 tr6 1.628127759 -0.94 0 mode n c MATERIAL m10 2003.66c 1.40000e-06 $ Helium gas outside the FE 293.6K 1990 2004.60c 9.99999e-01 $ Helium gas outside the FE 293.6K 1973 m21 92238.66c 3.05676e-01 $ U-238 293.6 1993 92235.66c 2.76538e-02 $ U-235 293.6K 1997 8016.62c 6.66662e-01 $ Oxygen 293.6K 2000 5010.66c 1.64692e-06 $ Boron-10 5011.66c 6.67090e-06 $ Boron-11 m22 6000.24c 1.00000e+00 $ Natural Carbon (buffer) mt22 grph.60t m23 6000.24c 1.00000e+00 $ Natural Carbon mt23 grph.60t m24 6000.24c 5.00000e-01 $ Natural Carbon 14000.60c 5.00000e-01 $ Silicon 293.6K 1976

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mt24 grph.60t m25 6000.24c 1.00000e+00 $ Natural Carbon mt25 grph.60t m26 6000.24c 9.99999e-01 $ Natural Carbon 5010.66c 1.09980e-07 $ Boron-10 5011.66c 4.45472e-07 $ Boron-11 mt26 grph.60t m27 2003.66c 1.40000e-06 $ Helium gas outside the FE 293.6K 1990 2004.60c 9.99999e-01 $ Helium gas outside the FE 293.6K 1973 m80 6000.24c 9.99999e-01 $ Natural Carbon 5010.66c 1.09980e-07 $ Boron-10 5011.66c 4.45472e-07 $ Boron-11 mt80 grph.60t c c CRITICALITY CONTROL CARDS kcode 200 1.0 200 1200 ksrc 0.000 0.000 0.000 c c TALLY CARDS c E0 3.14200E-06 1.15600E-01 1.96400E+01 E0 3.05902E-06 1.11090E-01 1.49182E+01 c -------------------- FLUX IN SYSTEM ------------------------- FC4 Broad-group FLUX in CELLS (6-group) F4:n 10 $ Coolant channel 98 $ Helium gap 80 $ Moderator SD4 9.7642 2.4345e-01 23.0159 c FC14 Broad-group FLUX in FUEL (6-group) F14:N (90 < (96 [-6:6 -6:6 -20:20]) < 97) $ Fuel kernel (91 < (96 [-6:6 -6:6 -20:20]) < 97) $ 1 layer coating (92 < (96 [-6:6 -6:6 -20:20]) < 97) $ 2 layer coating (93 < (96 [-6:6 -6:6 -20:20]) < 97) $ 3 layer coating (94 < (96 [-6:6 -6:6 -20:20]) < 97) $ 4 layer coating SD14 0.1963 0.2976 0.1956 0.2086 0.2855 FC24 Broad-group FLUX outside FUEL (6-group) F24:N ( (95 85 86) < (96 [-6:6 -6:6 -20:20]) < 97) $ graphite matrix SD24 4.8180 c -------------------------------------------------------------- c c ---------------- REACTION RATE IN SYSTEM --------------------- F34:N 10 $ Coolant channel FM34 -1.0 10 (-1) (-2) (-6) (-7) SD34 9.7642 c F44:N 98 $ Helium gap FM44 -1.0 27 (-1) (-2) (-6) (-6 -7) SD44 2.4345e-01 c F54:N 80 $ Moderator FM54 -1.0 80 (-1) (-2) (-6) (-6 -7) SD54 23.0159

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c F64:N (90 < (96 [-6:6 -6:6 -20:20]) < 97) $ Fuel kernel FM64 -1.0 21 (-1) (-2) (-6) (-6 -7) SD64 0.1963 c F74:N (91 < (96 [-6:6 -6:6 -20:20]) < 97) $ 1 layer coating FM74 -1.0 22 (-1) (-2) (-6) (-6 -7) SD74 0.2976 c F84:N (92 < (96 [-6:6 -6:6 -20:20]) < 97) $ 2 layer coating FM84 -1.0 23 (-1) (-2) (-6) (-6 -7) SD84 0.1956 c F94:N (93 < (96 [-6:6 -6:6 -20:20]) < 97) $ 3 layer coating FM94 -1.0 24 (-1) (-2) (-6) (-6 -7) SD94 0.2086 c F104:N (94 < (96 [-6:6 -6:6 -20:20]) < 97) $ 4 layer coating FM104 -1.0 25 (-1) (-2) (-6) (-6 -7) SD104 0.2855 c F114:N ( (95 85 86) < (96 [-6:6 -6:6 -20:20]) < 97) $ graphite matrix FM114 -1.0 26 (-1) (-2) (-6) (-6 -7) SD114 4.8180 c --------------------------------------------------------------

200

A2 DRAGON Input Decks

A2.1 Pebble Fuel Element *----- * HTR FUEL MODELING * Pebble1 * Surrounded by Helium * ENDF7_361 * Author: Tholakele Prisca Ngeleka *----- * Define STRUCTURES and MODULES *------ LINKED_LIST PEBBLE DISCR DISCR1 LIBRARY LIBRARY1 LIBRARY2 MACRO1 MACRO2 CP CPAM CALC CALCB OUT OUTA OUTB OUTC OUTD OUTE OUTF OUTG OUTH OUT1 OUT2 OUT3 OUT4 OUT5 OUT6 OUT7 OUT8 COMPO FLUX DATABASE DATABASE1 ; SEQ_ASCII calcFl1 calcFl2 calcFl3 calcFl4 calcFl5 calcFl6 calcFl7 calcFl8 calcFlA calcFlB calcFlC calcFlD calcFlE calcFlF calcFlG calcFlH multicompo ; MODULE LIB: GEO: SYBILT: USS: SHI: ASM: FLU: EDI: T: COMPO: END: ; * *------ * Microscopic cross section *------ * LIBRARY := LIB: :: NMIX 8 CTRA NONE SUBG MACR MIXS LIB: DRAGON FIL: ENDF7_361 * * Mixtures * 1 = fuel kernel * 2 = Inner low density carbon kernel coating (Buffer) * 3 = Inner Pyro carbon kernel coating * 4 = Silicon Carbide kernel coating * 5 = Outer Pyro carbon kernel coating * 6 = Compact carbon matrix * 7 = Helium coolant * 8 = Pebble outer coating * * Fuel kernel MIX 1 293.6 U238 = 'U238' 2.12877E-02 1 U235 = 'U235' 1.92585E-03 1 O16 = 'O16' 4.64272E-02 B10 = 'B10' 1.14694E-07 B11 = 'B11' 4.64570E-07 * Inner low density carbon kernel (buffer) MIX 2 293.6

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Cnat = 'C12_GRA' 5.26449E-02 * Inner Pyro carbon kernel coating MIX 3 293.6 Cnat = 'C12_GRA' 9.52621E-02 * Silicon carbide kernel coating MIX 4 293.6 Si28 = 'Si28' 4.402E-02 Si29 = 'Si29' 2.235E-03 Si30 = 'Si30' 1.473E-03 C0 = 'C12' 4.772E-02 * Outer Pyro carbon kernel coating MIX 5 293.6 Cnat = 'C12_GRA' 9.52621E-02 * Compact carbon matrix MIX 6 293.6 Cnat = 'C12_GRA' 8.77414E-02 B10 = 'B10' 9.64977E-09 B11 = 'B11' 3.90864E-08 * Helium coolant MIX 7 293.6 He3 = 'He3' 3.71220E-11 He4 = 'He4' 2.65156E-05 * Pebble outer coating MIX 8 293.6 Cnat = 'C12_GRA' 8.77414E-02 B10 = 'B10' 9.64977E-09 B11 = 'B11' 3.90864E-08 ; *------- * GEOMETRY PEBBLE : PEBBLE := GEO: :: SPHERE 3 EDIT 10000 R+ REFL RADIUS 0.0 2.5 3.0 3.53735 MIX 9 8 7 * Coated Particles * NMISTR = Number of microstructures/coated particles types in region * NMILG = Number of microstructure/coated particles regions * NS = ARRAY OF SUB REGIONS IN THPebble1.x2mE COATED PATICLE/MICROSTRUCTURES; LEN=NMILG * RS = RADIUS OF COATED PARTICLES/MICROSTRUCTURES, LEN=NS(I); I=1; NMISTR; * milie = COMPOSITION OF EACH COATED PARTICLE/MICROSTRUCTURE, LEN=NMISTR; * "NOTE: MILIE NO'S ARE GREATER THAN MIX NO'S" * mixdil = Base composition of each region, LEN=NMILG * fract = Microstructure type volume fraction in region LEN=NMILG * mixgr = LIBRARY MIXTURES FOR EACH COATED PARTICLE/MICROSTRUCTURE LAYER; LEN=NS(I) *NMISTR, NMILG BIHET SPHE 1 1 (* NS *) 5 (* RS *) 0.0 0.025 0.034 0.038 0.0415 0.0455 (* milie *) 9 (* mixdil *) 6 (* fract *) 0.09042852 (* mixgr *) 1 2 3 4 5 ; *----- * * TRACKING MODULES * SYBILT -used for 1D geometries (plane, cyl or spherical) * EXCELT-perform full cell collision probability tracking with isotropic/specular surface current

202

* NXT -extension of excelt for more complex geometries * MCCGT-implementation of open characteristics method * SNT -implementation of the discrete ordinates method (1D/2D/3D geom) * BIVACT-perform a finite element (Diff/SPn) 1D/2D tracking * TRIVACT-perform a finite element (Diff/SPn) 1D/2D/3D trackikng * *SYBILT (CASE 1) DISCR := SYBILT: PEBBLE :: TITLE 'Pebble1: HTR FUEL MODELING (SYBIL /SYBIL)' MAXR 1000 ; LIBRARY1 := USS: LIBRARY DISCR :: EDIT 1 ; CP := ASM: LIBRARY1 DISCR :: EDIT 0 ; CALC := FLU: CP LIBRARY1 DISCR :: EDIT 1 TYPE K ; OUTA := EDI: CALC LIBRARY1 DISCR :: EDIT 2 MERG COMP COND 38 281 361 MICR ALL SAVE ; *SYBILT (CASE 1) DISCR1 := SYBILT: PEBBLE :: TITLE 'Pebble1: HTR FUEL MODELING (SYBIL /SYBIL)' MAXR 1000 ; LIBRARY2 := USS: LIBRARY DISCR1 :: EDIT 1 ; MACRO1 := LIBRARY2 :: STEP UP MACROLIB ; MACRO2 := T: MACRO1 ; CPAM := ASM: MACRO2 DISCR1 :: EDIT 0 ; CALCB := FLU: CPAM MACRO2 DISCR1 :: EDIT 4 TYPE K ; OUTB := EDI: CALCB MACRO2 DISCR1 :: EDIT 2 MERGE COMP MICR ALL SAVE ; END: ;

203

A2.2 Prismatic Hexagonal Block *----- * HTR FUEL MODELING * Prismatic block * Surrounded by Helium * ENDF7 SHEM 361 * Author:Tholakele Prisca Ngeleka *----- * Define STRUCTURES and MODULES *------ LINKED_LIST PRISM DISCR DISCR1 LIBRARY LIBRARY1 LIBRARY2 CPAM OUTK OUTL OUTM OUTN OUTP OUTQ OUTR OUTS COMPO CALC FLUX DATABASE3 ; * SEQ_BINARY TRACK_F1 TRACK_F2 ; SEQ_ASCII calcFl1 calcFl2 calcFl3 calcFl4 calcFl5 calcFl6 calcFl7 calcFl8 multicompo ; MODULE LIB: GEO: SYBILT: USS: ASM: SHI: FLU: EDI: COMPO: END: ; * * Microscopic cross section *------ LIBRARY := LIB: :: NMIX 8 CTRA NONE SUBG MIXS LIB: DRAGON FIL: ENDF7_361 * * Mixtures * 1 = fuel kernel * 2 = Inner low density carbon kernel coating (Buffer) * 3 = Inner Pyro carbon kernel coating * 4 = Silicon Carbide kernel coating * 5 = Outer Pyro carbon kernel coating * 6 = Compact carbon matrix * 7 = Helium coolant * 8 = Compact carbon matrix * * Fuel kernel MIX 1 293.6 U238 = 'U238' 2.12877E-02 1 U235 = 'U235' 1.92585E-03 1 O16 = 'O16' 4.64272E-02 B10 = 'B10' 1.14694E-07 B11 = 'B11' 4.64570E-07 * * Inner low density carbon kernel (buffer) MIX 2 293.6 Cnat = 'C12_GRA' 5.26449E-02 * * Inner Pyro carbon kernel coating MIX 3 293.6 Cnat = 'C12_GRA' 9.52621E-02

204

* * Silicon carbide kernel coating MIX 4 293.6 Si28 = 'Si28' 4.402E-02 Si29 = 'Si29' 2.235E-03 Si30 = 'Si30' 1.473E-03 C0 = 'C12' 4.772E-02 * * Outer Pyro carbon kernel coating MIX 5 293.6 Cnat = 'C12_GRA' 9.52621E-02 * * Compact carbon matrix MIX 6 293.6 Cnat = 'C12_GRA' 8.77414E-02 B10 = 'B10' 9.64977E-09 B11 = 'B11' 3.90864E-08 * * Helium coolant MIX 7 293.6 He3 = 'He3' 3.71220E-11 He4 = 'He4' 2.65156E-05 * * Pebble outer coating MIX 8 293.6 Cnat = 'C12_GRA' 8.77414E-02 B10 = 'B10' 9.64977E-09 B11 = 'B11' 3.90864E-08 ; *------- * * GEOMETRY PRISMATIC : * PRISM := GEO: :: HEX 36 HBC S30 REFL CELL * Ring 0 (center cell) C * Ring 1 F * Ring 2 C F * Ring 3 F C * Ring 4 C F F * Ring 5 F C F * Ring 6 C F F C * Ring 7 F C F F * Ring 8 C F F C F * Ring 9 F C F F C * Ring 10 C F F C F F *

205

* Coated Particles * NMISTR = Number of microstructures/coated particles types in region * NMILG = Number of microstructure/coated particles regions * NS = ARRAY OF SUB REGIONS IN THE COATED PATICLE/MICROSTRUCTURES; LEN=NMILG * RS = RADIUS OF COATED PARTICLES/MICROSTRUCTURES, LEN=NS(I); I=1; NMISTR; * milie = COMPOSITION OF EACH COATED PARTICLE/MICROSTRUCTURE, LEN=NMISTR; * "NOTE: MILIE NO'S ARE GREATER THAN MIX NO'S" * misdil = Base composition of each region, LEN=NMILG * fract = Microstructure type volume fraction in region LEN=NMILG * mixgr = LIBRARY MIXTURES FOR EACH COATED PARTICLE/MICROSTRUCTURE LAYER; LEN=NS(I) * *NMISTR, NMILG BIHET SPHE 1 1 (* NS *) 5 (* RS *) 0.0 0.025 0.034 0.038 0.0415 0.0455 (* milie *) 9 (* mixdil *) 6 (* fract *) 0.19723 (* mixgr *) 1 2 3 4 5 ::: C := GEO: HEXCEL 1 EDIT 1000 SIDE 1.085 RADIUS 0.0 0.794 MIX 7 8 ; ::: F := GEO: HEXCEL 2 EDIT 1000 SIDE 1.085 RADIUS 0.0 0.6225 0.635 MIX 9 7 8 ; ; * *PSPPLOT (Graphical presentation) *----- * TRACKING MODULES * SYBILT -used for 1D geometries (plane, cyl or spherical) * EXCELT-perform full cell collision probability tracking with isotropic/specular surface * current * NXT -extension of excelt for more complex geometries * MCCGT-implementation of open characteristics method * SNT -implementation of the discrete ordinates method (1D/2D/3D geom) * BIVACT-perform a finite element (Diff/SPn) 1D/2D trackikng * TRIVACT-perform a finite element (Diff/SPn) 1D/2D/3D tracking * *SYBILT (CASE 1) * TRACKING MODULES * SYBILT -used for 1D geometries (plane, cyl or spherical) * EXCELT-perform full cell collision probability tracking with isotropic/specular surface current * NXT -extension of excelt for more complex geometries * MCCGT-implementation of open characteristics method * SNT -implementation of the discrete ordinates method (1D/2D/3D geom) * BIVACT-perform a finite element (Diff/SPn) 1D/2D tracking * TRIVACT-perform a finite element (Diff/SPn) 1D/2D/3D trackikng * * SYBILT *---------------------------------------------- * Tracking calculation for self-shielding *----------------------------------------------

206

* TITLE 'Prismatic: HTR FUEL MODELING (SYBILT)' DISCR := SYBILT: PRISM :: EDIT 1 MAXR 1000 ; LIBRARY1 := USS: LIBRARY DISCR :: EDIT 1 ; CPAM := ASM: LIBRARY1 DISCR :: EDIT 1 ; CALC := FLU: CPAM LIBRARY1 DISCR :: EDIT 2 TYPE K ; * 1 = fuel kernel OUTK := EDI: CALC LIBRARY1 DISCR :: EDIT 2 MERGE MIX 1 0 0 0 0 0 0 0 MICR ALL SAVE ; calcFl1 := OUTK ; * 2 = Inner low density carbon kernel coating (Buffer) OUTL := EDI: CALC LIBRARY1 DISCR :: EDIT 2 MERGE MIX 0 1 0 0 0 0 0 0 MICR ALL SAVE ; calcFl2 := OUTL ; * 3 = Inner Pyro carbon kernel coating OUTM := EDI: CALC LIBRARY1 DISCR :: EDIT 2 MERGE MIX 0 0 1 0 0 0 0 0 MICR ALL SAVE ; calcFl3 := OUTM ; * 4 = Silicon Carbide kernel coating OUTN := EDI: CALC LIBRARY1 DISCR :: EDIT 2 MERGE MIX 0 0 0 1 0 0 0 0 MICR ALL SAVE ; calcFl4 := OUTN ; * 5 = Outer Pyro carbon kernel coating OUTP := EDI: CALC LIBRARY1 DISCR :: EDIT 2 MERGE MIX 0 0 0 0 1 0 0 0 MICR ALL SAVE ;

207

calcFl5 := OUTP ; * 6 = Compact carbon matrix OUTQ := EDI: CALC LIBRARY1 DISCR :: EDIT 2 MERGE MIX 0 0 0 0 0 1 0 0 MICR ALL SAVE ; calcFl6 := OUTQ ; * 7 = Helium coolant OUTR := EDI: CALC LIBRARY1 DISCR :: EDIT 2 MERGE MIX 0 0 0 0 0 0 1 0 MICR ALL SAVE ; calcFl7 := OUTR ; * 8 = Pebble outer coating OUTS := EDI: CALC LIBRARY1 DISCR :: EDIT 2 MERGE MIX 0 0 0 0 0 0 0 1 MICR ALL SAVE ; calcFl8 := OUTS ; * COMPO (FLUX SAMPLING) DATABASE3 := COMPO: :: EDIT 5 COMM 'Multi-parameter reactor database' ENDC INIT ; DATABASE3 := COMPO: DATABASE3 OUTK OUTL OUTM OUTN OUTP OUTQ OUTR OUTS :: EDIT 3 STEP UP 'default' ; multicompo := DATABASE3 ; END: ;

208

A3 Energy Group Structures A3.1 SHEM-361 Group number Energy (eV) Group number Energy (eV) Group number Energy (eV)

1 1.96403E+07 51 2.26994E+04 101 2.00958E+02 2 1.49182E+07 52 1.85847E+04 102 1.95996E+02 3 1.38403E+07 53 1.62005E+04 103 1.93078E+02 4 1.16183E+07 54 1.48997E+04 104 1.90204E+02 5 9.99999E+06 55 1.36037E+04 105 1.88877E+02 6 9.04836E+06 56 1.11377E+04 106 1.87559E+02 7 8.18730E+06 57 9.11881E+03 107 1.86251E+02 8 7.40817E+06 58 7.46585E+03 108 1.84952E+02 9 6.70319E+06 59 6.11252E+03 109 1.83295E+02

10 6.06530E+06 60 5.00451E+03 110 1.75229E+02 11 4.96585E+06 61 4.09735E+03 111 1.67519E+02 12 4.06569E+06 62 3.48107E+03 112 1.63056E+02 13 3.32871E+06 63 2.99618E+03 113 1.54176E+02 14 2.72531E+06 64 2.70024E+03 114 1.46657E+02 15 2.23130E+06 65 2.39729E+03 115 1.39504E+02 16 1.90139E+06 66 2.08410E+03 116 1.32701E+02 17 1.63654E+06 67 1.81183E+03 117 1.26229E+02 18 1.40577E+06 68 1.58620E+03 118 1.20554E+02 19 1.33694E+06 69 1.34358E+03 119 1.17577E+02 20 1.28696E+06 70 1.13467E+03 120 1.16524E+02 21 1.16205E+06 71 1.06432E+03 121 1.15480E+02 22 1.05115E+06 72 9.82494E+02 122 1.12854E+02 23 9.51119E+05 73 9.09681E+02 123 1.10288E+02 24 8.60006E+05 74 8.32218E+02 124 1.05646E+02 25 7.06511E+05 75 7.48517E+02 125 1.03038E+02 26 5.78443E+05 76 6.77287E+02 126 1.02115E+02 27 4.94002E+05 77 6.46837E+02 127 1.01605E+02 28 4.56021E+05 78 6.12834E+02 128 1.01098E+02 29 4.12501E+05 79 6.00099E+02 129 1.00594E+02 30 3.83884E+05 80 5.92941E+02 130 9.73287E+01 31 3.20646E+05 81 5.77146E+02 131 9.33256E+01 32 2.67826E+05 82 5.39204E+02 132 8.87741E+01 33 2.30014E+05 83 5.01746E+02 133 8.39393E+01 34 1.95008E+05 84 4.53999E+02 134 7.93679E+01 35 1.64999E+05 85 4.19094E+02 135 7.63322E+01 36 1.40000E+05 86 3.90760E+02 136 7.35595E+01 37 1.22773E+05 87 3.71703E+02 137 7.18869E+01 38 1.15624E+05 88 3.53575E+02 138 6.90682E+01 39 9.46645E+04 89 3.35323E+02 139 6.68261E+01 40 8.22974E+04 90 3.19928E+02 140 6.64929E+01 41 6.73794E+04 91 2.95922E+02 141 6.61612E+01 42 5.51656E+04 92 2.88327E+02 142 6.58312E+01 43 4.99159E+04 93 2.84888E+02 143 6.55029E+01 44 4.08677E+04 94 2.76468E+02 144 6.50460E+01 45 3.69786E+04 95 2.68297E+02 145 6.45923E+01 46 3.34596E+04 96 2.56748E+02 146 6.36306E+01 47 2.92810E+04 97 2.41796E+02 147 6.23083E+01 48 2.73944E+04 98 2.35590E+02 148 5.99250E+01 49 2.61001E+04 99 2.24325E+02 149 5.70595E+01 50 2.49991E+04 100 2.12108E+02 150 5.40600E+01

209

Group number Energy (eV) Group number Energy (eV) Group number Energy (eV)

151 5.29895E+01 201 1.65501E+01 251 6.57184E+00 152 5.17847E+01 202 1.60498E+01 252 6.55609E+00 153 4.92591E+01 203 1.57792E+01 253 6.53907E+00 154 4.75173E+01 204 1.48662E+01 254 6.51492E+00 155 4.62053E+01 205 1.47301E+01 255 6.48178E+00 156 4.52904E+01 206 1.45952E+01 256 6.43206E+00 157 4.41721E+01 207 1.44702E+01 257 6.35978E+00 158 4.31246E+01 208 1.42505E+01 258 6.28016E+00 159 4.21441E+01 209 1.40496E+01 259 6.16011E+00 160 4.12270E+01 210 1.35460E+01 260 6.05991E+00 161 3.97295E+01 211 1.33297E+01 261 5.96014E+00 162 3.87874E+01 212 1.26000E+01 262 5.80021E+00 163 3.77919E+01 213 1.24721E+01 263 5.72015E+00 164 3.73038E+01 214 1.23086E+01 264 5.61979E+00 165 3.68588E+01 215 1.21302E+01 265 5.53004E+00 166 3.64191E+01 216 1.19795E+01 266 5.48817E+00 167 3.60568E+01 217 1.18153E+01 267 5.41025E+00 168 3.56980E+01 218 1.17094E+01 268 5.38003E+00 169 3.45392E+01 219 1.15894E+01 269 5.32011E+00 170 3.30855E+01 220 1.12694E+01 270 5.21008E+00 171 3.16930E+01 221 1.10529E+01 271 5.10997E+00 172 2.78852E+01 222 1.08038E+01 272 4.93323E+00 173 2.46578E+01 223 1.05793E+01 273 4.76785E+00 174 2.25356E+01 224 9.50002E+00 274 4.41980E+00 175 2.23788E+01 225 9.14031E+00 275 4.30981E+00 176 2.21557E+01 226 8.97995E+00 276 4.21983E+00 177 2.20011E+01 227 8.80038E+00 277 4.00000E+00 178 2.17018E+01 228 8.67369E+00 278 3.88217E+00 179 2.14859E+01 229 8.52407E+00 279 3.71209E+00 180 2.13360E+01 230 8.30032E+00 280 3.54307E+00 181 2.12296E+01 231 8.13027E+00 281 3.14211E+00 182 2.11448E+01 232 7.97008E+00 282 2.88405E+00 183 2.10604E+01 233 7.83965E+00 283 2.77512E+00 184 2.09763E+01 234 7.73994E+00 284 2.74092E+00 185 2.07676E+01 235 7.60035E+00 285 2.71990E+00 186 2.06847E+01 236 7.38015E+00 286 2.70012E+00 187 2.06021E+01 237 7.13987E+00 287 2.64004E+00 188 2.05199E+01 238 6.99429E+00 288 2.62005E+00 189 2.04175E+01 239 6.91778E+00 289 2.59009E+00 190 2.02751E+01 240 6.87021E+00 290 2.55000E+00 191 2.00734E+01 241 6.83526E+00 291 2.46994E+00 192 1.95974E+01 242 6.81070E+00 292 2.33006E+00 193 1.93927E+01 243 6.79165E+00 293 2.27299E+00 194 1.91997E+01 244 6.77605E+00 294 2.21709E+00 195 1.90848E+01 245 6.75981E+00 295 2.15695E+00 196 1.79591E+01 246 6.74225E+00 296 2.07010E+00 197 1.77590E+01 247 6.71668E+00 297 1.98992E+00 198 1.75648E+01 248 6.63126E+00 298 1.90008E+00 199 1.74457E+01 249 6.60611E+00 299 1.77997E+00 200 1.68305E+01 250 6.58829E+00 300 1.66895E+00

210

Group number Energy (eV) Group number Energy (eV)

301 1.58803E+00 351 4.73019E-02 302 1.51998E+00 352 4.02999E-02 303 1.44397E+00 353 3.43998E-02 304 1.41001E+00 354 2.92989E-02 305 1.38098E+00 355 2.49394E-02 306 1.33095E+00 356 2.00104E-02 307 1.29304E+00 357 1.48300E-02 308 1.25094E+00 358 1.04505E-02 309 1.21397E+00 359 7.14526E-03 310 1.16999E+00 360 4.55602E-03 311 1.14797E+00 361 2.49990E-03 312 1.12997E+00 313 1.11605E+00 314 1.10395E+00 315 1.09198E+00 316 1.07799E+00 317 1.03499E+00 318 1.02101E+00 319 1.00904E+00 320 9.96501E-01 321 9.81959E-01 322 9.63960E-01 323 9.44022E-01 324 9.19978E-01 325 8.80024E-01 326 8.00371E-01 327 7.19999E-01 328 6.24999E-01 329 5.94993E-01 330 5.54990E-01 331 5.20011E-01 332 4.75017E-01 333 4.31579E-01 334 3.90001E-01 335 3.52994E-01 336 3.25008E-01 337 3.05012E-01 338 2.79989E-01 339 2.54997E-01 340 2.31192E-01 341 2.09610E-01 342 1.90005E-01 343 1.61895E-01 344 1.37999E-01 345 1.19995E-01 346 1.04298E-01 347 8.97968E-02 348 7.64969E-02 349 6.51999E-02 350 5.54982E-02

211

A3.2 SHEM-281 Group number Energy (eV) Group number Energy (eV) Group number Energy (eV)


212


151 8.13027E+00 201 3.14211E+00 251 5.20011E-01 152 7.97008E+00 202 2.88405E+00 252 4.75017E-01 153 7.83965E+00 203 2.77512E+00 253 4.31579E-01 154 7.73994E+00 204 2.74092E+00 254 3.90001E-01 155 7.60035E+00 205 2.71990E+00 255 3.52994E-01 156 7.38015E+00 206 2.70012E+00 256 3.25008E-01 157 7.13987E+00 207 2.64004E+00 257 3.05012E-01 158 6.99429E+00 208 2.62005E+00 258 2.79989E-01 159 6.91778E+00 209 2.59009E+00 259 2.54997E-01 160 6.87021E+00 210 2.55000E+00 260 2.31192E-01 161 6.83526E+00 211 2.46994E+00 261 2.09610E-01 162 6.81070E+00 212 2.33006E+00 262 1.90005E-01 163 6.79165E+00 213 2.27299E+00 263 1.61895E-01 164 6.77605E+00 214 2.21709E+00 264 1.37999E-01 165 6.75981E+00 215 2.15695E+00 265 1.19995E-01 166 6.74225E+00 216 2.07010E+00 266 1.04298E-01 167 6.71668E+00 217 1.98992E+00 267 8.97968E-02 168 6.63126E+00 218 1.90008E+00 268 7.64969E-02 169 6.60611E+00 219 1.77997E+00 269 6.51999E-02 170 6.58829E+00 220 1.66895E+00 270 5.54982E-02 171 6.57184E+00 221 1.58803E+00 271 4.73019E-02 172 6.55609E+00 222 1.51998E+00 272 4.02999E-02 173 6.53907E+00 223 1.44397E+00 273 3.43998E-02 174 6.51492E+00 224 1.41001E+00 274 2.92989E-02 175 6.48178E+00 225 1.38098E+00 275 2.49394E-02 176 6.43206E+00 226 1.33095E+00 276 2.00104E-02 177 6.35978E+00 227 1.29304E+00 277 1.48300E-02 178 6.28015E+00 228 1.25094E+00 278 1.04505E-02 179 6.16011E+00 229 1.21397E+00 279 7.14526E-03 180 6.05991E+00 230 1.16999E+00 280 4.55602E-03 181 5.96014E+00 231 1.14797E+00 281 2.49990E-03 182 5.80021E+00 232 1.12997E+00 183 5.72015E+00 233 1.11605E+00 184 5.61979E+00 234 1.10395E+00 185 5.53004E+00 235 1.09198E+00 186 5.48817E+00 236 1.07799E+00 187 5.41025E+00 237 1.03499E+00 188 5.38003E+00 238 1.02101E+00 189 5.32011E+00 239 1.00904E+00 190 5.21008E+00 240 9.96501E-01 191 5.10997E+00 241 9.81959E-01 192 4.93323E+00 242 9.63960E-01 193 4.76785E+00 243 9.44022E-01 194 4.41980E+00 244 9.19978E-01 195 4.30981E+00 245 8.80024E-01 196 4.21983E+00 246 8.00371E-01 197 4.00000E+00 247 7.19999E-01 198 3.88217E+00 248 6.24999E-01 199 3.71209E+00 249 5.94993E-01 200 3.54307E+00 250 5.54990E-01

213

A3.3 GA-193 Group number Energy (eV) Group number Energy (eV)

1 1.491820E+07 51 8.651700E+04 2 1.349860E+07 52 6.738000E+04 3 1.221400E+07 53 5.247500E+04 4 1.105170E+07 54 4.086800E+04 5 1.000000E+07 55 3.182800E+04 6 9.048370E+06 56 2.478800E+04 7 8.187310E+06 57 1.930500E+04 8 7.408180E+06 58 1.503400E+04 9 6.703200E+06 59 1.170900E+04 10 6.065310E+06 60 9.118800E+03 11 5.488120E+06 61 7.101700E+03 12 4.965850E+06 62 5.530800E+03 13 4.493300E+06 63 4.307400E+03 14 4.065700E+06 64 3.354600E+03 15 3.678800E+06 65 2.612600E+03 16 3.328700E+06 66 2.034700E+03 17 3.011900E+06 67 1.584600E+03 18 2.725300E+06 68 1.234100E+03 19 2.466000E+06 69 9.611200E+02 20 2.231300E+06 70 7.485200E+02 21 2.019000E+06 71 5.829500E+02 22 1.826800E+06 72 4.540000E+02 23 1.653000E+06 73 3.535800E+02 24 1.495700E+06 74 2.753600E+02 25 1.353400E+06 75 2.144500E+02 26 1.224600E+06 76 1.670200E+02 27 1.108000E+06 77 1.300700E+02 28 1.002600E+06 78 1.013000E+02 29 9.071800E+05 79 7.889300E+01 30 8.208500E+05 80 6.144200E+01 31 7.427400E+05 81 4.785100E+01 32 6.720600E+05 82 3.726600E+01 33 6.081000E+05 83 2.902300E+01 34 5.502300E+05 84 2.260300E+01 35 4.978700E+05 85 1.760400E+01 36 4.504900E+05 86 1.371000E+01 37 4.076200E+05 87 1.067700E+01 38 3.688300E+05 88 8.315300E+00 39 3.337300E+05 89 6.475900E+00 40 3.019700E+05 90 5.043500E+00 41 2.732400E+05 91 3.927900E+00 42 2.472400E+05 92 3.059000E+00 43 2.237100E+05 93 2.382400E+00 44 2.024200E+05 94 2.355000E+00 45 1.831600E+05 95 2.310000E+00 46 1.657300E+05 96 2.245000E+00 47 1.499600E+05 97 2.150000E+00 48 1.356900E+05 98 2.050000E+00 49 1.227700E+05 99 1.950000E+00 50 1.110900E+05 100 1.880000E+00

214


101 1.820000E+00 151 3.970000E-01 102 1.740000E+00 152 3.700000E-01 103 1.650000E+00 153 3.550000E-01 104 1.550000E+00 154 3.450000E-01 105 1.470000E+00 155 3.350000E-01 106 1.395000E+00 156 3.250000E-01 107 1.325000E+00 157 3.150000E-01 108 1.275000E+00 158 3.050000E-01 109 1.225000E+00 159 2.950000E-01 110 1.175000E+00 160 2.850000E-01 111 1.140000E+00 161 2.750000E-01 112 1.127500E+00 162 2.650000E-01 113 1.117500E+00 163 2.550000E-01 114 1.100000E+00 164 2.450000E-01 115 1.085000E+00 165 2.350000E-01 116 1.075000E+00 166 2.250000E-01 117 1.065000E+00 167 2.100000E-01 118 1.055000E+00 168 1.900000E-01 119 1.037500E+00 169 1.700000E-01 120 1.012500E+00 170 1.500000E-01 121 9.950000E-01 171 1.300000E-01 122 9.850000E-01 172 1.100000E-01 123 9.750000E-01 173 9.750000E-02 124 9.600000E-01 174 9.250000E-02 125 9.400000E-01 175 8.750000E-02 126 9.200000E-01 176 8.250000E-02 127 9.000000E-01 177 7.750000E-02 128 8.830000E-01 178 7.250000E-02 129 8.630000E-01 179 6.750000E-02 130 8.250000E-01 180 6.250000E-02 131 7.750000E-01 181 5.500000E-02 132 7.250000E-01 182 4.500000E-02 133 6.915000E-01 183 3.500000E-02 134 6.665000E-01 184 2.765000E-02 135 6.375000E-01 185 2.265000E-02 136 6.125000E-01 186 1.750000E-02 137 5.950000E-01 187 1.250000E-02 138 5.825000E-01 188 9.000000E-03 139 5.625000E-01 189 7.500000E-03 140 5.410000E-01 190 6.000000E-03 141 5.160000E-01 191 4.500000E-03 142 4.950000E-01 192 3.000000E-03 143 4.850000E-01 193 1.500000E-03 144 4.775000E-01 145 4.725000E-01 146 4.650000E-01 147 4.550000E-01 148 4.400000E-01 149 4.250000E-01 150 4.170000E-01

215

A3.4 SHEM_TPN-407 Group number Energy (eV) Group number Energy (eV) Group number Energy (eV)


216



217


301 2.77512E+00 351 7.70018E-01 401 2.49394E-02 302 2.74092E+00 352 7.45008E-01 402 2.00104E-02 303 2.71990E+00 353 7.19999E-01 403 1.48300E-02 304 2.70012E+00 354 6.96249E-01 404 1.04505E-02 305 2.64004E+00 355 6.72499E-01 405 7.14526E-03 306 2.62005E+00 356 6.48749E-01 406 4.55602E-03 307 2.59009E+00 357 6.24999E-01 407 2.49990E-03 308 2.55000E+00 358 6.09996E-01 309 2.46994E+00 359 5.94993E-01 310 2.40000E+00 360 5.74991E-01 311 2.33006E+00 361 5.54990E-01 312 2.27299E+00 362 5.37500E-01 313 2.21709E+00 363 5.20011E-01 314 2.15695E+00 364 4.97514E-01 315 2.07010E+00 365 4.75017E-01 316 1.98992E+00 366 4.53298E-01 317 1.90008E+00 367 4.31579E-01 318 1.84002E+00 368 4.10790E-01 319 1.77997E+00 369 3.90001E-01 320 1.72446E+00 370 3.71497E-01 321 1.66895E+00 371 3.52994E-01 322 1.62849E+00 372 3.39001E-01 323 1.58803E+00 373 3.32004E-01 324 1.51998E+00 374 3.25008E-01 325 1.44397E+00 375 3.15010E-01 326 1.41001E+00 376 3.05012E-01 327 1.38098E+00 377 2.92500E-01 328 1.33095E+00 378 2.79989E-01 329 1.29304E+00 379 2.67493E-01 330 1.25094E+00 380 2.54997E-01 331 1.21397E+00 381 2.43094E-01 332 1.16999E+00 382 2.31192E-01 333 1.14797E+00 383 2.20401E-01 334 1.12997E+00 384 2.09610E-01 335 1.11605E+00 385 1.99808E-01 336 1.10395E+00 386 1.90005E-01 337 1.09198E+00 387 1.75950E-01 338 1.07799E+00 388 1.61895E-01 339 1.05649E+00 389 1.49947E-01 340 1.03499E+00 390 1.37999E-01 341 1.02101E+00 391 1.19995E-01 342 1.00904E+00 392 1.04298E-01 343 9.96501E-01 393 8.97968E-02 344 9.81959E-01 394 7.64969E-02 345 9.63960E-01 395 6.51994E-02 346 9.44022E-01 396 5.54982E-02 347 9.19978E-01 397 4.73019E-02 348 8.80024E-01 398 4.02999E-02 349 8.50031E-01 399 3.43998E-02 350 8.20037E-01 400 2.95460E-02

218

A3.5 SHEM_TPN-531 Group number Energy (eV) Group number Energy (eV) Group number Energy (eV)


219



220



221


451 1.29304E+00 501 2.92500E-01 452 1.25094E+00 502 2.79989E-01 453 1.21397E+00 503 2.67493E-01 454 1.16999E+00 504 2.54997E-01 455 1.14797E+00 505 2.43094E-01 456 1.12997E+00 506 2.31192E-01 457 1.11605E+00 507 2.20401E-01 458 1.10395E+00 508 2.09610E-01 459 1.09198E+00 509 1.99808E-01 460 1.07799E+00 510 1.82978E-01 461 1.05649E+00 511 1.61895E-01 462 1.03499E+00 512 1.49947E-01 463 1.02101E+00 513 1.37999E-01 464 1.00904E+00 514 1.28997E-01 465 9.96501E-01 515 1.19995E-01 466 9.81959E-01 516 1.04298E-01 467 9.63960E-01 517 8.97968E-02 468 9.44022E-01 518 7.64969E-02 469 9.19978E-01 519 6.51994E-02 470 9.00001E-01 520 5.54982E-02 471 8.80024E-01 521 4.73019E-02 472 8.50031E-01 522 4.02999E-02 473 8.35034E-01 523 3.43998E-02 474 8.20037E-01 524 2.95460E-02 475 7.95028E-01 525 2.49394E-02 476 7.70018E-01 526 2.00104E-02 477 7.45008E-01 527 1.48300E-02 478 7.19999E-01 528 1.04505E-02 479 6.96249E-01 529 7.14526E-03 480 6.72499E-01 530 4.55602E-03 481 6.48749E-01 531 2.49990E-03 482 6.24999E-01 483 6.09996E-01 484 5.94993E-01 485 5.74991E-01 486 5.54990E-01 487 5.37500E-01 488 5.20011E-01 489 4.97514E-01 490 4.75017E-01 491 4.53298E-01 492 4.31579E-01 493 4.10790E-01 494 3.90001E-01 495 3.71497E-01 496 3.52994E-01 497 3.39001E-01 498 3.32004E-01 499 3.20009E-01 500 3.05012E-01

222

A3.6 GA_TPN-537 Group number Energy (eV) Group number Energy (eV) Group number Energy (eV)


223



224


301 1.26474E+02 351 2.58130E+01 401 5.22255E+00 302 1.22877E+02 352 2.50105E+01 402 5.04350E+00 303 1.19281E+02 353 2.42080E+01 403 4.90405E+00 304 1.15685E+02 354 2.33426E+01 404 4.76460E+00 305 1.12089E+02 355 2.24771E+01 405 4.62515E+00 306 1.08492E+02 356 2.18837E+01 406 4.48570E+00 307 1.04896E+02 357 2.12903E+01 407 4.34625E+00 308 1.01300E+02 358 2.01035E+01 408 4.20680E+00 309 9.84991E+01 359 1.94786E+01 409 4.06735E+00 310 9.56982E+01 360 1.88537E+01 410 3.92790E+00 311 9.28974E+01 361 1.82289E+01 411 3.81929E+00 312 9.00965E+01 362 1.76040E+01 412 3.71068E+00 313 8.72956E+01 363 1.71172E+01 413 3.60206E+00 314 8.44947E+01 364 1.66305E+01 414 3.49345E+00 315 8.16939E+01 365 1.61437E+01 415 3.38484E+00 316 7.88930E+01 366 1.56570E+01 416 3.27623E+00 317 7.67116E+01 367 1.51703E+01 417 3.16761E+00 318 7.45303E+01 368 1.46835E+01 418 3.05900E+00 319 7.23489E+01 369 1.41968E+01 419 2.88985E+00 320 7.01675E+01 370 1.37100E+01 420 2.72070E+00 321 6.79861E+01 371 1.33309E+01 421 2.55155E+00 322 6.58048E+01 372 1.29518E+01 422 2.38240E+00 323 6.36234E+01 373 1.25726E+01 423 2.35500E+00 324 6.14420E+01 374 1.21935E+01 424 2.31000E+00 325 5.97431E+01 375 1.18144E+01 425 2.24500E+00 326 5.80443E+01 376 1.14353E+01 426 2.15000E+00 327 5.63454E+01 377 1.10561E+01 427 2.05000E+00 328 5.46465E+01 378 1.06770E+01 428 1.95000E+00 329 5.29476E+01 379 1.03818E+01 429 1.88000E+00 330 5.12487E+01 380 1.00866E+01 430 1.82000E+00 331 4.95499E+01 381 9.79136E+00 431 1.74000E+00 332 4.78510E+01 382 9.49615E+00 432 1.65000E+00 333 4.65279E+01 383 9.20094E+00 433 1.55000E+00 334 4.52047E+01 384 8.90573E+00 434 1.47000E+00 335 4.38816E+01 385 8.61051E+00 435 1.39500E+00 336 4.25585E+01 386 8.31530E+00 436 1.32500E+00 337 4.12354E+01 387 8.08538E+00 437 1.27500E+00 338 3.99122E+01 388 7.85545E+00 438 1.22500E+00 339 3.82996E+01 389 7.65427E+00 439 1.17500E+00 340 3.66870E+01 390 7.45308E+00 440 1.14000E+00 341 3.49157E+01 391 7.35249E+00 441 1.12750E+00 342 3.40301E+01 392 7.25190E+00 442 1.11750E+00 343 3.31445E+01 393 7.05071E+00 443 1.10000E+00 344 3.21141E+01 394 6.47590E+00 444 1.08500E+00 345 3.10838E+01 395 6.29685E+00 445 1.07500E+00 346 3.00534E+01 396 6.11780E+00 446 1.06500E+00 347 2.90230E+01 397 5.93875E+00 447 1.05500E+00 348 2.82205E+01 398 5.75970E+00 448 1.03750E+00 349 2.74180E+01 399 5.58065E+00 449 1.01250E+00 350 2.66155E+01 400 5.40160E+00 450 9.95000E-01

225


451 9.85000E-01 501 1.90000E-01 452 9.75000E-01 502 1.80000E-01 453 9.60000E-01 503 1.70000E-01 454 9.40000E-01 504 1.60000E-01 455 9.20000E-01 505 1.55000E-01 456 9.00000E-01 506 1.50000E-01 457 8.83000E-01 507 1.45000E-01 458 8.63000E-01 508 1.40000E-01 459 8.25000E-01 509 1.35000E-01 460 7.75000E-01 510 1.30000E-01 461 7.50000E-01 511 1.25000E-01 462 7.25000E-01 512 1.20000E-01 463 6.91500E-01 513 1.15000E-01 464 6.66500E-01 514 1.10000E-01 465 6.37500E-01 515 1.06094E-01 466 6.12500E-01 516 1.02188E-01 467 5.95000E-01 517 9.50000E-02 468 5.82500E-01 518 9.25000E-02 469 5.62500E-01 519 8.75000E-02 470 5.41000E-01 520 8.25000E-02 471 5.16000E-01 521 7.75000E-02 472 4.95000E-01 522 7.25000E-02 473 4.85000E-01 523 6.75000E-02 474 4.77500E-01 524 6.25000E-02 475 4.72500E-01 525 5.50000E-02 476 4.65000E-01 526 4.50000E-02 477 4.55000E-01 527 3.50000E-02 478 4.40000E-01 528 2.76500E-02 479 4.25000E-01 529 2.26500E-02 480 4.17000E-01 530 1.75000E-02 481 3.97000E-01 531 1.25000E-02 482 3.83500E-01 532 9.00000E-03 483 3.70000E-01 533 7.50000E-03 484 3.55000E-01 534 6.00000E-03 485 3.45000E-01 535 4.50000E-03 486 3.35000E-01 536 3.00000E-03 487 3.25000E-01 537 1.50000E-03 488 3.15000E-01 489 3.05000E-01 490 2.95000E-01 491 2.85000E-01 492 2.75000E-01 493 2.65000E-01 494 2.55000E-01 495 2.45000E-01 496 2.35000E-01 497 2.25000E-01 498 2.17500E-01 499 2.10000E-01 500 2.00000E-01

226

A4 Depletion Data Analysis

A4.1 Pebble Fuel Element Nuclides Concentrations Burnup steps (GWD/t)

Nuclides 0.5 5 10 20 30 40 50 60 70 80 100 120

U-233 4.85E-14 4.72E-13 9.02E-13 1.67E-12 2.40E-12 3.14E-12 3.92E-12 4.74E-12 5.57E-12 6.34E-12 7.55E-12 8.13E-12

U-234 8.23E-11 8.21E-10 1.60E-09 3.08E-09 4.73E-09 7.03E-09 1.06E-08 1.63E-08 2.48E-08 3.69E-08 7.25E-08 1.20E-07

U-235 1.91E-03 1.78E-03 1.64E-03 1.38E-03 1.14E-03 9.27E-04 7.36E-04 5.67E-04 4.21E-04 2.98E-04 1.28E-04 4.27E-05

U-236 2.55E-06 2.49E-05 4.84E-05 9.15E-05 1.30E-04 1.63E-04 1.91E-04 2.15E-04 2.34E-04 2.48E-04 2.61E-04 2.58E-04

U-237 1.51E-09 2.29E-08 4.70E-08 9.57E-08 1.43E-07 1.91E-07 2.38E-07 2.84E-07 3.31E-07 3.76E-07 4.77E-07 5.22E-07

U-238 2.13E-02 2.12E-02 2.12E-02 2.11E-02 2.10E-02 2.09E-02 2.07E-02 2.06E-02 2.05E-02 2.03E-02 2.00E-02 1.96E-02

Np-237 5.79E-10 8.87E-08 3.69E-07 1.47E-06 3.22E-06 5.53E-06 8.29E-06 1.14E-05 1.47E-05 1.81E-05 2.48E-05 2.95E-05

Np-238 4.05E-13 1.18E-10 5.29E-10 2.31E-09 5.46E-09 1.02E-08 1.66E-08 2.51E-08 3.60E-08 4.94E-08 8.44E-08 1.20E-07

Np-239 1.72E-06 2.00E-06 2.06E-06 2.20E-06 2.35E-06 2.51E-06 2.68E-06 2.88E-06 3.11E-06 3.36E-06 4.04E-06 4.50E-06

Pu-238 3.02E-13 9.47E-10 8.61E-09 7.39E-08 2.57E-07 6.21E-07 1.23E-06 2.13E-06 3.35E-06 4.87E-06 8.57E-06 1.19E-05

Pu-239 2.76E-06 4.06E-05 7.61E-05 1.30E-04 1.66E-04 1.88E-04 2.00E-04 2.05E-04 2.03E-04 1.98E-04 1.86E-04 1.70E-04

Pu-240 9.64E-09 1.37E-06 5.02E-06 1.61E-05 2.90E-05 4.15E-05 5.27E-05 6.20E-05 6.93E-05 7.46E-05 7.87E-05 7.90E-05

Pu-241 6.38E-11 1.00E-07 7.40E-07 4.68E-06 1.20E-05 2.14E-05 3.12E-05 4.00E-05 4.67E-05 5.10E-05 5.43E-05 5.06E-05

Pu-242 7.86E-14 1.36E-09 2.12E-08 2.95E-07 1.27E-06 3.40E-06 7.00E-06 1.23E-05 1.93E-05 2.78E-05 4.81E-05 6.95E-05

Pu-243 7.10E-18 1.51E-13 2.43E-12 3.63E-11 1.67E-10 4.75E-10 1.05E-09 1.96E-09 3.29E-09 5.10E-09 1.06E-08 1.67E-08

Am-241 1.58E-14 2.61E-10 3.89E-09 4.90E-08 1.88E-07 4.41E-07 7.84E-07 1.16E-06 1.51E-06 1.75E-06 1.86E-06 1.58E-06

Am-242m 4.13E-18 6.41E-13 1.71E-11 3.51E-10 1.68E-09 4.42E-09 8.39E-09 1.29E-08 1.70E-08 1.98E-08 2.11E-08 1.74E-08

Am-242 1.38E-17 3.53E-13 5.67E-12 7.63E-11 3.14E-10 7.95E-10 1.54E-09 2.52E-09 3.64E-09 4.76E-09 6.36E-09 6.55E-09

Am-243 3.50E-17 8.00E-12 2.61E-10 7.80E-09 5.36E-08 2.02E-07 5.50E-07 1.22E-06 2.33E-06 4.02E-06 9.61E-06 1.77E-05

Cm-242 1.66E-17 4.52E-12 1.43E-10 3.58E-09 2.07E-08 6.57E-08 1.50E-07 2.79E-07 4.47E-07 6.41E-07 1.01E-06 1.20E-06

Cm-243 1.27E-21 3.92E-15 2.59E-13 1.39E-11 1.28E-10 5.72E-10 1.72E-09 4.02E-09 7.84E-09 1.33E-08 2.79E-08 4.00E-08

Cm-244 2.59E-20 6.59E-14 4.47E-12 2.86E-10 3.15E-09 1.69E-08 6.14E-08 1.74E-07 4.18E-07 8.86E-07 3.11E-06 8.07E-06

Cm-245 5.66E-24 1.54E-16 2.11E-14 2.72E-12 4.45E-11 3.13E-10 1.38E-09 4.54E-09 1.20E-08 2.73E-08 1.06E-07 2.74E-07

227

A4.2 Pebble Fuel Element Fission Products Concentrations Burnup (GWD/t)

Fission Products 0.5 5 10 20 30 40 50 60 70 80 100 120

Kr-85 3.26E-08 3.19E-07 6.22E-07 1.19E-06 1.71E-06 2.18E-06 2.61E-06 2.99E-06 3.33E-06 3.62E-06 4.07E-06 4.37E-06

Sr-90 7.27E-07 7.13E-06 1.40E-05 2.70E-05 3.91E-05 5.03E-05 6.07E-05 7.02E-05 7.88E-05 8.66E-05 9.93E-05 1.09E-04

Ag-109 4.01E-09 6.33E-08 1.76E-07 5.32E-07 1.05E-06 1.72E-06 2.54E-06 3.49E-06 4.58E-06 5.78E-06 8.36E-06 1.10E-05

Ag-110m 3.99E-13 5.40E-11 2.71E-10 1.46E-09 4.01E-09 8.30E-09 1.47E-08 2.37E-08 3.58E-08 5.15E-08 9.74E-08 1.58E-07

Cs-137 7.79E-07 7.75E-06 1.55E-05 3.08E-05 4.59E-05 6.09E-05 7.57E-05 9.03E-05 1.05E-04 1.19E-04 1.47E-04 1.73E-04

Xe-131 1.02E-07 3.08E-06 6.61E-06 1.34E-05 1.99E-05 2.58E-05 3.13E-05 3.61E-05 4.03E-05 4.38E-05 4.81E-05 4.94E-05

Xe-133 5.18E-07 7.87E-07 7.86E-07 7.84E-07 7.82E-07 7.79E-07 7.76E-07 7.72E-07 7.68E-07 7.64E-07 7.54E-07 7.45E-07

Xe-135 1.67E-08 1.65E-08 1.62E-08 1.54E-08 1.45E-08 1.34E-08 1.23E-08 1.11E-08 9.91E-09 8.74E-09 6.94E-09 5.57E-09

Xe-136 1.34E-06 1.37E-05 2.75E-05 5.55E-05 8.38E-05 1.13E-04 1.42E-04 1.71E-04 2.01E-04 2.32E-04 2.93E-04 3.55E-04

Nd-143 8.60E-08 5.36E-06 1.23E-05 2.50E-05 3.61E-05 4.56E-05 5.34E-05 5.92E-05 6.30E-05 6.46E-05 6.18E-05 5.31E-05

Nd-144 7.12E-09 7.33E-07 2.83E-06 1.03E-05 2.13E-05 3.50E-05 5.10E-05 6.89E-05 8.86E-05 1.10E-04 1.57E-04 2.07E-04

Nd-145 4.95E-07 4.88E-06 9.64E-06 1.88E-05 2.75E-05 3.57E-05 4.35E-05 5.06E-05 5.72E-05 6.32E-05 7.28E-05 7.94E-05

Nd-146 3.78E-07 3.77E-06 7.56E-06 1.52E-05 2.29E-05 3.08E-05 3.89E-05 4.73E-05 5.59E-05 6.48E-05 8.40E-05 1.05E-04

Pm-147 6.09E-08 2.13E-06 4.47E-06 8.28E-06 1.11E-05 1.31E-05 1.44E-05 1.51E-05 1.54E-05 1.53E-05 1.41E-05 1.24E-05

Pm-148 6.77E-11 7.68E-09 1.78E-08 3.61E-08 5.20E-08 6.59E-08 7.82E-08 8.91E-08 9.88E-08 1.07E-07 1.21E-07 1.22E-07

Pm-148m 4.05E-11 7.81E-09 1.98E-08 4.01E-08 5.58E-08 6.72E-08 7.50E-08 7.95E-08 8.12E-08 8.04E-08 7.49E-08 6.45E-08

Pm-149 4.94E-08 5.46E-08 5.65E-08 6.01E-08 6.34E-08 6.67E-08 6.99E-08 7.32E-08 7.64E-08 7.96E-08 8.58E-08 8.91E-08

Pm-151 1.11E-08 1.15E-08 1.18E-08 1.25E-08 1.31E-08 1.37E-08 1.43E-08 1.50E-08 1.57E-08 1.64E-08 1.78E-08 1.88E-08

Sm-147 1.24E-10 5.35E-08 2.44E-07 9.68E-07 2.02E-06 3.27E-06 4.62E-06 5.96E-06 7.23E-06 8.35E-06 9.94E-06 1.06E-05

Sm-148 2.05E-11 3.52E-08 1.90E-07 8.62E-07 1.98E-06 3.52E-06 5.43E-06 7.71E-06 1.03E-05 1.33E-05 2.01E-05 2.76E-05

Sm-149 6.12E-08 1.42E-07 1.43E-07 1.41E-07 1.38E-07 1.32E-07 1.24E-07 1.16E-07 1.06E-07 9.65E-08 8.09E-08 6.76E-08

Sm-151 3.97E-08 3.07E-07 4.12E-07 4.72E-07 4.93E-07 5.04E-07 5.09E-07 5.10E-07 5.05E-07 4.98E-07 4.96E-07 4.77E-07

228

Eu-153 1.35E-08 2.11E-07 4.86E-07 1.23E-06 2.23E-06 3.45E-06 4.83E-06 6.35E-06 7.94E-06 9.54E-06 1.26E-05 1.49E-05

Eu-154 3.36E-11 5.83E-09 2.43E-08 1.03E-07 2.43E-07 4.46E-07 7.08E-07 1.02E-06 1.36E-06 1.72E-06 2.43E-06 2.96E-06

Eu-155 3.93E-09 2.98E-08 4.85E-08 8.20E-08 1.28E-07 1.93E-07 2.80E-07 3.87E-07 5.10E-07 6.44E-07 9.12E-07 1.14E-06

Gd-155 5.57E-12 1.77E-10 3.16E-10 5.16E-10 7.44E-10 1.03E-09 1.36E-09 1.68E-09 1.98E-09 2.22E-09 2.64E-09 2.84E-09

Gd-156 3.31E-10 2.49E-08 8.26E-08 2.68E-07 5.58E-07 9.97E-07 1.66E-06 2.62E-06 4.01E-06 5.94E-06 1.20E-05 2.16E-05

Gd-157 3.46E-10 6.37E-10 8.32E-10 1.16E-09 1.42E-09 1.65E-09 1.86E-09 2.07E-09 2.29E-09 2.51E-09 3.26E-09 3.95E-09

229

A4.3 Prismatic Hexagonal Block Nuclides Concentrations Burnup (GWD/t)

Nuclides 0.5 5 10 20 30 40 50 60 70 80 100 120

U-233 6.39E-14 6.19E-13 1.19E-12 2.20E-12 3.18E-12 4.18E-12 5.24E-12 6.37E-12 7.55E-12 8.73E-12 1.09E-11 1.26E-11

U-234 1.09E-10 1.08E-09 2.10E-09 4.07E-09 6.31E-09 9.49E-09 1.45E-08 2.23E-08 3.39E-08 5.02E-08 9.89E-08 1.67E-07

U-235 1.91E-03 1.78E-03 1.64E-03 1.39E-03 1.16E-03 9.55E-04 7.75E-04 6.18E-04 4.83E-04 3.67E-04 1.97E-04 9.32E-05

U-236 2.67E-06 2.60E-05 5.02E-05 9.39E-05 1.32E-04 1.64E-04 1.92E-04 2.14E-04 2.32E-04 2.45E-04 2.58E-04 2.58E-04

U-237 1.99E-09 2.96E-08 6.06E-08 1.21E-07 1.80E-07 2.36E-07 2.89E-07 3.40E-07 3.87E-07 4.30E-07 5.15E-07 5.57E-07

U-238 2.13E-02 2.12E-02 2.12E-02 2.11E-02 2.09E-02 2.08E-02 2.07E-02 2.05E-02 2.04E-02 2.02E-02 1.98E-02 1.94E-02

Np-237 7.62E-10 1.15E-07 4.78E-07 1.88E-06 4.08E-06 6.93E-06 1.03E-05 1.40E-05 1.79E-05 2.19E-05 2.96E-05 3.56E-05

Np-238 6.04E-13 1.73E-10 7.67E-10 3.26E-09 7.55E-09 1.37E-08 2.17E-08 3.16E-08 4.33E-08 5.68E-08 8.86E-08 1.20E-07

Np-239 2.01E-06 2.31E-06 2.38E-06 2.52E-06 2.69E-06 2.86E-06 3.03E-06 3.21E-06 3.39E-06 3.58E-06 4.04E-06 4.38E-06

Pu-238 4.51E-13 1.40E-09 1.26E-08 1.07E-07 3.66E-07 8.74E-07 1.71E-06 2.91E-06 4.52E-06 6.51E-06 1.14E-05 1.64E-05

Pu-239 3.22E-06 4.68E-05 8.74E-05 1.48E-04 1.89E-04 2.16E-04 2.33E-04 2.42E-04 2.45E-04 2.45E-04 2.38E-04 2.27E-04

Pu-240 1.24E-08 1.71E-06 6.09E-06 1.87E-05 3.24E-05 4.52E-05 5.62E-05 6.53E-05 7.24E-05 7.77E-05 8.29E-05 8.48E-05

Pu-241 1.00E-10 1.52E-07 1.09E-06 6.60E-06 1.63E-05 2.83E-05 4.05E-05 5.14E-05 6.02E-05 6.65E-05 7.33E-05 7.24E-05

Pu-242 1.28E-13 2.13E-09 3.21E-08 4.24E-07 1.73E-06 4.40E-06 8.66E-06 1.46E-05 2.19E-05 3.05E-05 5.00E-05 6.99E-05

Pu-243 1.44E-17 2.91E-13 4.55E-12 6.37E-11 2.77E-10 7.50E-10 1.56E-09 2.77E-09 4.39E-09 6.40E-09 1.18E-08 1.76E-08

Am-241 2.48E-14 3.96E-10 5.77E-09 6.98E-08 2.59E-07 5.91E-07 1.03E-06 1.52E-06 1.99E-06 2.38E-06 2.77E-06 2.71E-06

Am-242m 7.20E-18 1.07E-12 2.81E-11 5.52E-10 2.56E-09 6.60E-09 1.24E-08 1.90E-08 2.54E-08 3.06E-08 3.61E-08 3.45E-08

Am-242 2.40E-17 5.87E-13 9.15E-12 1.16E-10 4.53E-10 1.09E-09 2.03E-09 3.20E-09 4.48E-09 5.75E-09 7.80E-09 8.69E-09

Am-243 7.12E-17 1.55E-11 4.92E-10 1.39E-08 9.12E-08 3.28E-07 8.49E-07 1.79E-06 3.26E-06 5.34E-06 1.16E-05 2.00E-05

Cm-242 2.89E-17 7.56E-12 2.32E-10 5.55E-09 3.06E-08 9.28E-08 2.04E-07 3.64E-07 5.65E-07 7.91E-07 1.23E-06 1.53E-06

Cm-243 2.61E-21 7.70E-15 4.96E-13 2.53E-11 2.21E-10 9.44E-10 2.71E-09 6.04E-09 1.13E-08 1.85E-08 3.72E-08 5.53E-08

Cm-244 6.40E-20 1.55E-13 1.03E-11 6.22E-10 6.55E-09 3.35E-08 1.16E-07 3.11E-07 7.02E-07 1.40E-06 4.26E-06 9.88E-06

Cm-245 1.76E-23 4.53E-16 6.13E-14 7.51E-12 1.19E-10 8.14E-10 3.47E-09 1.10E-08 2.80E-08 6.11E-08 2.14E-07 5.23E-07

230

A4.4 Prismatic Hexagonal Block Fission Products Concentrations Burnup steps (GWD/t)

Fission Products 0.5 5 10 20 30 40 50 60 70 80 100 120

Kr-85 3.25E-08 3.17E-07 6.17E-07 1.17E-06 1.68E-06 2.13E-06 2.54E-06 2.90E-06 3.23E-06 3.51E-06 3.97E-06 4.30E-06

Sr-90 7.25E-07 7.09E-06 1.39E-05 2.66E-05 3.83E-05 4.91E-05 5.90E-05 6.81E-05 7.63E-05 8.38E-05 9.66E-05 1.07E-04

Ag-109 4.06E-09 6.90E-08 1.97E-07 6.06E-07 1.19E-06 1.95E-06 2.85E-06 3.89E-06 5.04E-06 6.27E-06 8.89E-06 1.16E-05

Ag-110m 4.80E-13 6.86E-11 3.56E-10 1.95E-09 5.37E-09 1.10E-08 1.93E-08 3.05E-08 4.48E-08 6.25E-08 1.09E-07 1.68E-07

Cs-137 7.78E-07 7.74E-06 1.54E-05 3.07E-05 4.58E-05 6.07E-05 7.55E-05 9.01E-05 1.04E-04 1.19E-04 1.47E-04 1.74E-04

Xe-131 1.02E-07 3.07E-06 6.58E-06 1.33E-05 1.96E-05 2.54E-05 3.06E-05 3.52E-05 3.92E-05 4.26E-05 4.74E-05 4.99E-05

Xe-133 5.18E-07 7.85E-07 7.84E-07 7.82E-07 7.80E-07 7.77E-07 7.74E-07 7.71E-07 7.67E-07 7.64E-07 7.57E-07 7.51E-07

Xe-135 1.71E-08 1.71E-08 1.69E-08 1.64E-08 1.58E-08 1.50E-08 1.41E-08 1.32E-08 1.23E-08 1.13E-08 9.76E-09 8.37E-09

Xe-136 1.33E-06 1.36E-05 2.73E-05 5.50E-05 8.30E-05 1.11E-04 1.40E-04 1.69E-04 1.98E-04 2.28E-04 2.88E-04 3.48E-04

Nd-143 8.58E-08 5.35E-06 1.22E-05 2.49E-05 3.60E-05 4.56E-05 5.37E-05 6.03E-05 6.53E-05 6.87E-05 7.10E-05 6.82E-05

Nd-144 7.12E-09 7.30E-07 2.81E-06 1.02E-05 2.09E-05 3.41E-05 4.93E-05 6.61E-05 8.42E-05 1.04E-04 1.45E-04 1.90E-04

Nd-145 4.94E-07 4.86E-06 9.58E-06 1.86E-05 2.72E-05 3.52E-05 4.27E-05 4.97E-05 5.62E-05 6.22E-05 7.23E-05 8.03E-05

Nd-146 3.77E-07 3.77E-06 7.54E-06 1.52E-05 2.29E-05 3.08E-05 3.88E-05 4.71E-05 5.56E-05 6.44E-05 8.29E-05 1.03E-04

Pm-147 6.08E-08 2.12E-06 4.41E-06 8.07E-06 1.07E-05 1.25E-05 1.36E-05 1.43E-05 1.46E-05 1.45E-05 1.38E-05 1.28E-05

Pm-148 7.99E-11 8.95E-09 2.06E-08 4.08E-08 5.78E-08 7.19E-08 8.34E-08 9.28E-08 1.00E-07 1.06E-07 1.16E-07 1.18E-07

Pm-148m 4.77E-11 8.91E-09 2.24E-08 4.50E-08 6.25E-08 7.52E-08 8.39E-08 8.90E-08 9.13E-08 9.14E-08 8.80E-08 8.03E-08

Pm-149 4.93E-08 5.48E-08 5.69E-08 6.10E-08 6.46E-08 6.79E-08 7.09E-08 7.36E-08 7.62E-08 7.85E-08 8.27E-08 8.54E-08

Pm-151 1.11E-08 1.16E-08 1.20E-08 1.27E-08 1.34E-08 1.41E-08 1.47E-08 1.53E-08 1.60E-08 1.66E-08 1.77E-08 1.86E-08

Sm-147 1.23E-10 5.31E-08 2.41E-07 9.44E-07 1.95E-06 3.12E-06 4.36E-06 5.58E-06 6.72E-06 7.73E-06 9.26E-06 1.01E-05

Sm-148 2.42E-11 4.12E-08 2.20E-07 9.90E-07 2.26E-06 3.96E-06 6.06E-06 8.49E-06 1.12E-05 1.42E-05 2.09E-05 2.81E-05

Sm-149 6.13E-08 1.45E-07 1.48E-07 1.51E-07 1.52E-07 1.49E-07 1.45E-07 1.39E-07 1.32E-07 1.25E-07 1.11E-07 9.83E-08

Sm-151 3.97E-08 3.18E-07 4.40E-07 5.28E-07 5.69E-07 6.00E-07 6.25E-07 6.42E-07 6.54E-07 6.60E-07 6.80E-07 6.74E-07

231

Eu-153 1.35E-08 2.15E-07 5.04E-07 1.31E-06 2.40E-06 3.71E-06 5.20E-06 6.79E-06 8.44E-06 1.01E-05 1.32E-05 1.57E-05

Eu-154 3.76E-11 6.54E-09 2.76E-08 1.19E-07 2.83E-07 5.21E-07 8.26E-07 1.18E-06 1.58E-06 1.98E-06 2.78E-06 3.43E-06

Eu-155 3.92E-09 2.94E-08 4.80E-08 8.38E-08 1.34E-07 2.06E-07 3.00E-07 4.13E-07 5.40E-07 6.77E-07 9.50E-07 1.19E-06

Gd-155 5.62E-12 1.87E-10 3.44E-10 5.96E-10 9.12E-10 1.33E-09 1.82E-09 2.35E-09 2.87E-09 3.34E-09 4.23E-09 4.79E-09

Gd-156 3.34E-10 2.61E-08 8.80E-08 2.90E-07 6.10E-07 1.10E-06 1.82E-06 2.85E-06 4.27E-06 6.18E-06 1.17E-05 1.99E-05

Gd-157 3.66E-10 7.55E-10 1.03E-09 1.50E-09 1.91E-09 2.29E-09 2.68E-09 3.07E-09 3.48E-09 3.91E-09 5.13E-09 6.29E-09

232

A4.5 Pebble Fuel Element Criticality Data per Burnup Step Burnup (GWD/t) K-effective

0 1.516883

0.5 1.456028

5 1.428048

10 1.401606

20 1.344026

30 1.285172

40 1.227301

50 1.170394

60 1.11E+00

70 1.06E+00

80 9.98E-01

100 8.91E-01

120 8.03E-01

A4.6 Prismatic Hexagonal Block Criticality Data per Burnup Step Burnup (GWD/t) K-effective

0 1.460909

0.5 1.402415

5 1.372111

10 1.342032

20 1.279117

30 1.219249

40 1.164353

50 1.113787

60 1.07E+00

70 1.02E+00

80 9.78E-01

100 9.01E-01

120 8.34E-01

VITA

Tholakele Prisca Ngeleka

Education

2000 National Diploma in Chemical Engineering

Mangosuthu Technikon, Durban, South Africa

2001 Baccalaureus Technologiae in Chemical Engineering

Mangosuthu Technikon, Durban, South Africa

2006 Master of Science in Chemical Engineering

North West University, Potchefstroom, South Africa

2009 Master of Science in Nuclear Engineering

North West University, Potchefstroom, South Africa

2012 Doctor of Philosophy in Nuclear Engineering

Pennsylvania State University, Pennsylvania, United States of America

EXAMINATION AND IMPROVEMENT OF THE SHEM ENERGY …

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