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Cumulative Migration Method for Computing

Multi-Group Transport Cross Sections and Diffusion

Coefficients with Monte Carlo Calculations

by

Zhaoyuan Liu

B.S., Engineering Physics, Tsinghua University (2011)

M.S., Nuclear Science and Engineering, Tsinghua University (2014)

Submitted to the Department of Nuclear Science and Engineering

in partial fulfillment of the requirements for the degree of

Doctor of Philosophy in Nuclear Science and Engineering

at the

MASSACHUSETTS INSTITUTE OF TECHNOLOGY

Feburuary 2020

c

Massachusetts Institute of Technology 2020. All rights reserved.

Author . . . .

Department of Nuclear Science and Engineering

December 1, 2019

Certified by . . . .

Kord S. Smith

KEPCO Professor of the Practice of Nuclear Science and Engineering

Thesis Supervisor

Certified by . . . .

Benoit Forget

Professor of Nuclear Science and Engineering

Associate Department Head

Thesis Supervisor

Accepted by . . . .

Ju Li

Battelle Energy Alliance Professor of Nuclear Science and Engineering

Professor of Materials Science and Engineering

Chairman, Department Committee on Graduate Theses

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Cumulative Migration Method for Computing Multi-Group

Transport Cross Sections and Diffusion Coefficients with Monte

Carlo Calculations

by

Zhaoyuan Liu

Submitted to the Department of Nuclear Science and Engineering on December 1, 2019, in partial fulfillment of the

requirements for the degree of

Doctor of Philosophy in Nuclear Science and Engineering

Abstract

In nuclear reactor physics analysis, fast accurate deterministic methods are needed for the many full-core calculations required for safe and efficient operation of nuclear power plants. Multi-group diffusion coefficients and transport cross sections are the crucial parameters that balance efficiency and accuracy in full-core simulations. However, it is not clear what definition of diffusion coefficients and transport cross sections should be employed or what “transport properties” are preserved by the numerous approximations available in the literature.

Among the sources of error associated with efficient deterministic simulations of nuclear reactors, whether diffusion or transport theory, the anisotropy of neutron scatter-ing introduces one major challenge for achievscatter-ing highly accurate eigenvalues and power distributions. Anisotropic scattering has a significant impact on the neutron spatial migration, which is an important transport property in nuclear reactor systems. It is well known that the scattering is highly forward-peaking when neutrons collide with light nuclides such as hydrogen in water, but how anisotropic scattering contributes to neutron migration has not been thoroughly studied.

The Cumulative Migration Method (CMM) is developed in this thesis as a new method for computing multi-group diffusion coefficients and transport cross sections using Monte Carlo methods which preserves migration area. Thus, CMM is able to overcome the shortcomings of commonly-applied transport approximations. CMM is directly applicable to lattice calculations performed by Monte Carlo and is capable of producing rigorous homogenized diffusion coefficients and transport cross sections for arbitrarily heterogeneous lattices. By preserving neutron migration area, CMM also improves the accuracy of heterogeneous transport cross sections in multi-group transport calculations.

The advantage of CMM in achieving higher accuracy in full-core calculations is demonstrated on a series of 2D benchmark problems with both water and graphite moderators. The transport correction using CMM significantly improved agreement in

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full-core simulation results compared with other approximations. Consistent improve-ment is shown in reducing the error of eigenvalue and migration area. By employing pre-computed continuous energy correction tables for light nuclides, CMM offers a po-tential pathway to improve tally capabilities of existing Monte Carlo codes in generating transport cross sections.

Thesis Supervisor: Kord S. Smith

Title: KEPCO Professor of the Practice of Nuclear Science and Engineering Thesis Supervisor: Benoit Forget

Title: Professor of Nuclear Science and Engineering Associate Department Head

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Acknowledgments

I would like to express my deepest gratitude to my thesis co-advisors Professor Kord Smith and Professor Benoit Forget for all their guidance and support during my PhD study at MIT. They were always patient and supportive in helping me confront and solve challenges in research, and their personal and professional guidance were invaluable for me. Professor Kord Smith always welcome questions and challenges in any interesting topics, and I am very grateful to his generosity in spending time with me and sharing his rich experience and knowledge. Professor Benoit Forget encouraged me to think out of the box and expanded my scope of knowledge by exploring innovative ideas.

I would like to acknowledge the funding support for my PhD program. This research was supported by the China Scholarship Council and the U.S. Department of Energy Nuclear Energy University Program contract DE-NE0008578.

I would like to thank my thesis committee member, Professor Koroush Shirvan. His advice in my research is very helpful and inspiring. It was my great pleasure to have been working as officemate with Professor Shirvan for my first few years at MIT, and he is very generous in helping me. I also would like to thank Professor Mingda Li. It was my great pleasure to have the chance to learn from and develop friendship with Professor Li over the past few years. His passion in science and technology greatly inspired me and I am very grateful to all his help.

I am also grateful to Professor Edward Larsen and Eshed Magali in the University of Michigan. I would like to thank them for their interest and input in my research work. The discussions and cooperation with them provided valuable inspiration to my PhD project.

Many of the group members and alumni in the Computational Reactor Physics Group (CRPG) provided significant help to me and I am sincerely grateful to them. I would like to thank Lulu Li, Will Boyd, Jingang Liang, Jilang Miao for their tremendous help to me over these years. My fellow graduate students Pablo Ducru, Derek Gaston and John Tramm have been great support and I learned a lot from discussions with them. I also would like to thank Sterling Harper and Guillaume Giudicelli for their help especially in

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code development, and thank Wenbin Wu, Jiankai Yu, Xingjie Peng, Zhuoran Han and Qiudong Wang for their input in my PhD research. I am very grateful to Lisa Magnano Bleheen, Brandy Baker and Heather Barry for their kind help in administrative support over these years.

I would like to thank all my friends and fellow students and at MIT. Especially I would like to thank Lingbo Zhang, Yinan Cai, Xingang Zhao, Weiyue Zhou and Rui Sun. The experience together and friendship among us means a lot to me. I am grateful to Kaichao Sun and Yang Yang for their generous help to me in many ways. And I also want to thank my friends at MIT including: Jinyong Feng, Xu Wu, Yixiang Liu, Miaomiao Jin, Chuteng Zhou, Angxiu Ni, Zhenyu Liu, Jiayue Wang, Yifeng Che, Kieran Dolan and many others for our friendship over these years at MIT.

Lastly, I would like to thank my family for their support and understanding during these years. In particular, I owe my deepest gratitude to my wife Yang Liu. Her love and accompany is the biggest reason and motivation for me to make it to the end of my PhD journey.

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Contents

1 Introduction 19

1.1 Motivation . . . 19

1.2 Objective and Outline . . . 22

2 Background 25 2.1 Overview . . . 25

2.1.1 Standard Multi-level Approach for Steady-state Core Analysis . . 25

2.1.2 MGXS Generation Using Monte Carlo Codes . . . 27

2.2 Anisotropy in Neutron Scattering . . . 29

2.3 Transport Cross Section in Mono-energetic Models . . . 31

3 Diffusion coefficients and transport cross sections 37 3.1 The PL Method. . . 37

3.1.1 Expansion of the Angular Flux . . . 38

3.1.2 Expansion of the Scattering Kernel and Source . . . 40

3.1.3 General PL Equations . . . 41

3.2 The P1 Equations . . . 43

3.2.1 The First P1 Equation . . . 43

3.2.2 The Second P1 Equation . . . 47

3.2.3 Diffusion Equation and Diffusion Coefficients . . . 50

3.3 Transport Cross Sections from the P1 Approximation . . . 51

3.3.1 P1 Approximation of the Angular Flux . . . 52

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3.3.3 Transport Cross Section. . . 56

3.4 Analytical Transport Correction Ratio. . . 58

3.4.1 General Derivation. . . 58

3.4.2 Analytical TCR of Hydrogen . . . 62

3.5 Literature Review for Methods of Computing Multi-group Transport Cross Sections . . . 68

3.5.1 The In-scatter Method . . . 68

3.5.2 The Out-scatter Approximation . . . 69

3.5.3 The Flux-limited Approximation . . . 70

3.5.4 The Extended Transport Approximation. . . 71

3.5.5 The Consistent-P and Inconsistent-P Approximation . . . 71

3.5.6 The Diagonal Transport Approximation . . . 72

3.5.7 The B1 Method . . . 73

3.6 Test Problems of Hydrogen . . . 74

3.6.1 1D Homogeneous Hydrogen Slab. . . 74

3.6.2 Infinite Homogeneous Medium of Hydrogen . . . 80

4 The Cumulative Migration Method 87 4.1 Theory of CMM . . . 87

4.1.1 One-group Migration Area . . . 87

4.1.2 Multi-group Form and Cumulative Group Quantities . . . 88

4.2 Cumulative-group Implementation Scheme . . . 91

4.2.1 Tally Scheme for Cumulative Migration Area. . . 91

4.2.2 Phantom Tracking in Finite Medium . . . 94

4.2.3 Directional Cumulative Migration Area and Diffusion Coefficients 95 4.3 Group-wise Implementation Scheme . . . 96

4.3.1 Incremental Migration Area . . . 96

4.3.2 Equivalence between Cumulative and Incremental Tallies. . . 97

4.3.3 Tally of Incremental Migration Area . . . 98

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4.4 Application of CMM for Generating Homogenized Dg andΣt r,g . . . 103

4.4.1 Infinite Medium of Hydrogen . . . 103

4.4.2 2-group Diffusion Coefficients for a BEAVRS Assembly. . . 104

4.4.3 TREAT Fuel Assembly . . . 106

4.5 Summary . . . 110

5 Calculation of Multi-group Migration Areas in Deterministic Transport Sim-ulations 113 5.1 Mean Square Displacement . . . 113

5.1.1 Scattering Angle . . . 114

5.1.2 Analytical Mean Square Displacement. . . 116

5.1.3 Example with Hydrogen . . . 120

5.2 Rigorous Migration Area for One-group Infinite Homogeneous Problems 122 5.2.1 Mean Square Displacement in One-group Neutron Transport. . . 122

5.2.2 Verification with One-group Monte Carlo . . . 125

5.3 Migration Area for Multi-group Infinite Homogeneous Problems . . . . 127

5.3.1 Neutron Removal in Each Group . . . 127

5.3.2 Incremental Migration Area in Each Group . . . 128

5.3.3 Group Condensation for Incremental Migration Area in Different Groups . . . 129

5.3.4 Verification with Example Problem . . . 132

5.3.5 Advantage of the New Calculation Method for Multi-group Migra-tion Areas . . . 132

5.4 Understanding the Physical Meaning of Multi-group TCR from CMM . . 135

5.4.1 The Asymptotic TCR of 1/3 for Hydrogen . . . 136

5.4.2 The TCR Limit of 1.0 for High Energy Region . . . 138

5.4.3 The TCR Limit of 0.5 with Fixed-energy Source . . . 139

5.5 Multi-group Weighted Migration Areas for Heterogeneous Problems. . . 141

5.5.1 Approximations by Spatial Homogenization with Small Leakage 142 5.5.2 Test Problem: 2D C5G7 MOX Benchmark Problem . . . 143

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5.6 Summary . . . 149

6 Application of CMM on Transport Cross Sections in Heterogeneous Geome-tries 151 6.1 Challenges in applying CMM on generating heterogeneous transport cross sections . . . 152

6.2 Correction Using Infinite Homogeneous CMM TCR . . . 153

6.2.1 The B&W 1484 Benchmark problems . . . 154

6.2.2 Infinite Homogeneous TCR using CMM . . . 154

6.2.3 Transport Correction Using Pre-computed CMM TCR for the Mod-erator. . . 156

6.3 Full-core Results of the Test Problems. . . 158

6.3.1 B&W 1484 Core 1 and Core 2 . . . 160

6.3.2 2D BEAVRS Benchmark Problem . . . 168

6.3.3 Simplified model of 2D TREAT MCM. . . 172

6.4 The Improved Flux-limited Approximation. . . 177

6.4.1 Correction for Hydrogen . . . 178

6.4.2 Improving the Flux-limited Approximation Using CMM TCR for Hydrogen . . . 179

6.4.3 Test results . . . 181

7 Conclusions 189 7.1 Summary of Contributions . . . 189

7.2 Future Work . . . 191

7.2.1 CMM TCR for More Nuclides . . . 191

7.2.2 Test of 3D Problems . . . 191

7.2.3 Transport Correction for High-Order Anisotropic Scattering . . . 191

A Analytical derivations for neutron slowing down with hydrogen 193 A.1 Analytical Flux. . . 193

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B Energy Group Structures 201

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List of Figures

1-1 Traditional LWR core analysis factorizations. . . 19

2-1 Traditional multi-level calculation flow for nuclear reactor core analysis 26 2-2 Angular-dependent capture MGXS. . . 30

2-3 Probability density function of the cosine of scattering angle . . . 32

2-4 Illustration of transport effect. . . 33

3-1 Illustration of solid angle and projections in 3D Cartesian coordinate system. 39 3-2 Energy distribution of Watt fission spectrum of U-235. . . 65

3-3 Analytical TCR for hydrogen with constant cross section . . . 66

3-4 Analytical TCR for hydrogen with constant cross section (zoomed in) . . 67

3-5 Analytical TCR for hydrogen with real cross section . . . 68

3-6 Configuration of the 1D homogeneous hydrogen slab problem. . . 75

3-7 Flux profiles for the 100-cm slab . . . 77

3-8 Illustration of a cosine-shape flux profile. . . 78

3-9 Group-dependent bulking for the 100-cm slab. . . 79

3-10 70-group TCRs for 1D hydrogen slab with different widths. . . 81

3-11 Comparison of 70-group hydrogen TCRs with the 1D slab model . . . 82

3-12 Comparison of 70-group TCRs for infinite homogeneous medium hydrogen 83 3-13 Comparison of few-group TCRs for infinite homogeneous medium hydrogen 84 3-14 Comparison of few-group diffusion coefficients . . . 85

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4-2 Comparison of cumulative energy groups with conventional individual

energy groups . . . 89

4-3 Illustration of the concept of cumulative group . . . 90

4-4 Flowchart of cumulative tally scheme for CMM. . . 92

4-5 Scenario of energy transition for an example neutron . . . 93

4-6 Illustration of phantom position in a reflection.. . . 94

4-7 Illustration of directional projection of crow flight vector. . . 96

4-8 Illustration of the spatial and energetic variation of two example neutrons101 4-9 Comparison of 70-group TCRs computed using CMM vs. in-scatter . . . 104

4-10 Configuration of the BEAVRS assembly . . . 105

4-11 The 2D geometrical layout and material composition of TREAT fuel assembly.106 4-12 The axial geometrical layout and material composition of the simplified TREAT fuel assembly model. . . 107

4-13 Error distribution of flux for the TREAT fuel assembly problem . . . 109

5-1 Comparison of microscopic total cross section between H-1 and O-16. . 115

5-2 Probability density function of the cosine of scattering angle in COM and LAB system for hydrogen . . . 115

5-3 Illustration of flight vectors for a neutron born at point O. . . . 117

5-4 Mean square displacement (MSD) vs. number of neutron flights.. . . 121

5-5 CMM TCRs for hydrogen with constant scattering cross sections . . . 137

5-6 CMM TCRs for hydrogen in the high energy range . . . 139

5-7 CMM TCRs for hydrogen with fixed-energy source. . . 140

5-8 Illustration of the first two neutron flights with a small scattering angle. 140 5-9 2D layout of fuel assemblies of the C5G7 MOX benchmark problem . . . 143

5-10 Core configuration of the 2D C5G7 MOX benchmark problem. . . 144

5-11 Multi-group cumulative migration area for isolated fuel assemblies . . . 147

5-12 Multi-group cumulative migration area for the full core of the 2D C5G7 MOX benchmark problem.. . . 149

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6-1 Illustration of a neutron traveling through the interface of two homoge-neous material regions. . . 152

6-2 Illustration of a neutron traveling through the void region between two separated homogeneous material regions. . . 153

6-3 2D layout of the B&W 1484 Core 1 problem. . . 154

6-4 2D layout of the B&W 1484 Core 2 problem. . . 155

6-5 70-group TCRs using CMM for infinite homogeneous medium of the moderators of B&W 1484 Core 1 and Core 2. . . 156

6-6 70-group cross sections of infinite homogeneous medium of the modera-tors of B&W 1484 Core 1 and Core 2.. . . 157

6-7 Cumulative migration areas in infinite homogeneous medium of modera-tors of B&W 1484 Core 1 and Core 2.. . . 158

6-8 The OpenMC geometry and the OpenMOC FSRs for the fuel pin in B&W 1484 cores . . . 161

6-9 Example of FSR division for a local zone in B&W 1484 Core 1. . . 161

6-10 70-group cumulative migration areas for B&W 1484 Core 1. . . 163

6-11 The reference pin power distribution of B&W 1484 Core 1 computed from the OpenMC simulation. . . 164

6-12 Relative error distributions of pin powers for B&W 1484 Core 1 problem 165

6-13 70-group cumulative migration areas for B&W 1484 Core 2. . . 167

6-14 The reference pin power distribution of B&W Core 2 computed from the OpenMC simulation. . . 167

6-15 Relative error distributions of pin powers for B&W 1484 Core 2 problem 168

6-16 70-group cumulative migration areas for the 2D BEAVRS benchmark problem. . . 170

6-17 The reference pin power distribution of 2D BEAVRS computed from the OpenMC simulation. . . 170

6-18 Relative error distributions of pin powers for 2D BEAVRS problem . . . . 171

6-19 The 2D geometrical layout and material composition of reflector assembly.172

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6-21 11-group cumulative migration areas for the simplified 2D model of TREAT MCM.. . . 174

6-22 The normalized reference assembly power distribution of the simplified 2D model of TREAT MCM computed from continuous-energy OpenMC simulation. . . 175

6-23 Relative error distributions of assembly powers for the simplified 2D model of TREAT MCM . . . 176

6-24 Fine-grid transport correction ratios for pure hydrogen in infinite homo-geneous medium with fixed source of U-235 fission spectrum. . . 181

6-25 The transport cross sections and TCRs of moderator of B&W 1484 Core 1 184

6-26 Relative error distributions of pin powers for B&W 1484 Core 1 problem 185

6-27 Relative error distributions of pin powers for B&W 1484 Core 2 problem 185

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List of Tables

4.1 The cumulative-group and group-wise tallies for the first example neutron101

4.2 The cumulative-group and group-wise tallies for the second example

neutron . . . 102

4.3 Comparison of D1 of the 2-group constants for the BEAVRS assembly . . 105

4.4 Comparison of eigenvalues computed by different approaches for the TREAT fuel assembly problem. . . 108

4.5 Comparison of 11-group diffusion coefficients computed by different methods . . . 110

5.1 One-group cross sections of the test problem . . . 126

5.2 Results of the one-group test problem. . . 126

5.3 Three-group cross sections of infinite homogeneous test problem. . . 132

5.4 Results of the three-group test problem. . . 133

5.5 Results of condensed one-group total cross section and migration area. . 134

5.6 Results of isolated fuel assemblies with reflective boundary conditions. . 146

5.7 Results of the full core of the 2D C5G7 MOX benchmark problem. . . 148

6.1 Comparison of OpenMC and OpenMOC results with different transport correction methods for the 2D B&W 1484 Core 1 problem. . . 162

6.2 Comparison of OpenMC and OpenMOC results with different transport correction methods for the 2D B&W 1484 Core 2 problem. . . 166

6.3 Comparison of OpenMC and OpenMOC results with different transport correction methods for the 2D BEAVRS benchmark problem. . . 169

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6.4 Comparison of continuous-energy OpenMC and 11-group results with different transport correction methods for the simplified 2D TREAT MCM

problem . . . 174

6.5 Comparison of errors with different group structures . . . 177

6.6 Comparison of OpenMC and OpenMOC results with different transport correction methods for the 2D B&W 1484 Core 1 problem. . . 183

6.7 Comparison of OpenMC and OpenMOC results with different transport correction methods for the 2D B&W 1484 Core 2 problem. . . 183

6.8 Comparison of OpenMC and OpenMOC results with different transport correction methods for the 2D BEAVRS benchmark problem. . . 186

B.1 1-group energy boundaries. . . 201

B.2 2-group energy boundaries. . . 201

B.3 4-group energy boundaries. . . 201

B.4 11-group energy boundaries. . . 202

B.5 25-group energy boundaries. . . 202

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Chapter 1

Introduction

1.1

Motivation

Deterministic light water reactor (LWR) core analysis is usually performed by employing a series of factorizations of the desired pin-wise spatial and continuous-energy spectral details into a sequence of fine-to-coarse transport calculations, as depicted in Figure1-1.

Figure 1-1: Traditional LWR core analysis factorizations. Image courtesy of W. Boyd[1].

Many deterministic nuclear reactor calculations utilize transport-corrected-P0

trans-port or diffusion theory to model neutron transtrans-port within fuel assemblies and reflectors. The accuracy of such core models is inherently tied to approximations made in

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obtain-ing multi-group transport cross sections or diffusion coefficients. While classic reactor physics textbooks [2, 3] offer insights and plausible arguments for computing trans-port cross sections and diffusion coefficients, there appears to be no rigorous theory for, nor quantification of errors introduced by, these approximations. Consequently, the computational accuracy of both heterogeneous (e.g. explicit fuel pin) and nodal (e.g. homogenized fuel assemblies) core calculations is often seriously compromised by inaccurate transport approximations, and there is a lack of available guidance in the literature to assist code developers and analysts in choosing the appropriate transport approximation.

The generation of multi-group cross section data for LWR analysis usually starts by identifying some characteristic “lattice” — be it a pin-cell, a fuel assembly, or a collection of fuel assemblies. For each such lattice, a very-fine-group transport calculation (e.g., 50 – 10,000 groups) is performed to obtain the neutron flux and reaction rate distributions within the lattice. Unless this transport calculation explicitly models anisotropic scattering, an approximation for transport-corrected-P0 cross sections for

each nuclide must be introduced before the multi-group lattice transport calculation can be performed.

In addition, lattice reaction rates and fluxes are used to compute energy-condensed and/or spatially-homogenized transport cross sections (or diffusion coefficients) for use in downstream multi-group (e.g., 2 – 100 groups) core calculations. Here, additional approximations are required to compute the appropriate transport cross section that preserves some selected characteristic of the lattice calculation.

All production lattice physics codes[4–7] make such approximations, often with-out substantial justification. Moreover, the most useful of these approximations are often considered to be proprietary, and the literature remains largely silent on useful methods. One example might be that of the transport-corrected-P0 methods that have

been employed in CASMO for more than 40 years. In 2013 Herman et al.[8] published details of the method used in CASMO to generate transport-corrected-P0 cross sections

for hydrogen in LWR lattices. Herman was able to compute CASMO’s “exact” transport cross section that matched continuous-energy Monte Carlo neutron leakages (integrated

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into 70 fine energy groups) for a slab of pure hydrogen. This transport correction is markedly different from that computed using the “micro-balance” argument[9] which produces the classic “out-scatter” approximation — with its transport-to-total ratio of 1/3 for purely isotropic center-of-mass (COM) neutron scattering with hydrogen in a free gas model. Developers recognized long ago that the CASMO definition of transport cross section produced excellent eigenvalues for small LWR critical assemblies with large neutron leakages, while the classic out-scatter approximation produced errors in eigenvalue as large as 1000 pcm. In addition, SIMULATE-3 nodal code developers ob-served (more than 30 years ago) that the CASMO transport cross section also produced two-group diffusion coefficients that eliminated radial power tilts observed in large 4-loop pressurized water reactor (PWR) cores when using the out-scatter approximation.

In recent years, Monte Carlo methods are in widespread use for detailed investiga-tions of nuclear reactor core physics models and quantification of errors/uncertainties in deterministic analysis methods. Monte Carlo methods have advantages in the capability of high-fidelity modeling in geometry, material composition, continuous-energy repre-sentation and real physics simulation. However, it is abundantly clear that core design, safety analysis, and regulatory licensing of commercial power reactors will continue to rely on deterministic methods for years to come — simply because such analysis requires 10,000s, 100,000s, and even 1,000,000s of core calculations for the broad-spectrum analysis required to support each cycle of reactor operation.

At the same time, Monte Carlo neutronics calculations are becoming more and more popular in generating multi-group cross sections and diffusion coefficients in both academic research and industrial practice. But the principle challenge remains in that the multi-group diffusion coefficients and transport cross sections still depend on various approximations, and the accurate definition remains a missing link. As the “upstream” step for “downstream” deterministic simulations, the generation of accurate multi-group cross sections has a crucial role in achieving results with high precision.

Among the many approximations in spatial discretization, energy condensation and angular discretization, the anisotropy in neutron scattering reactions has a major impact on neutron migration area, which also affect the calculation accuracy in effective neutron

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multiplication factors ke f f and pin power distributions. Failure to properly model the anisotropy of scattering can cause substantial errors, especially when the scattering is with light atoms such as hydrogen in water [10–12]. To account for the anisotropy in neutron scattering reactions, one option in deterministic simulations is to model high-order anisotropic scattering moments in the solver, but this approach incurs a substantial increase of the computational cost in both time and memory. Another popular solution is to use transport-corrected cross sections (transport cross sections) in P0 calculations

using the isotropic-in-LAB approximation. Replacing total cross sections with transport cross sections allows to approximate the neutron scattering source term as isotropic, but it remains an open question on how to generate accurate transport cross sections with Monte Carlo codes.

1.2

Objective and Outline

The objective in this thesis is to generate accurate multi-group diffusion coefficients and transport cross sections using Monte Carlo codes. Based on the conservation of migration area, the Cumulative Migration Method (CMM) is developed and verified in this thesis. By conserving migration area, CMM can generate rigorous multi-group diffusion coefficients and transport cross sections using Monte Carlo codes for infinite-medium homogenized problems, and it can also achieve improved accuracy in heterogeneous problems.

Chapter 2 presents a discussion about the background of this research work, including the anisotropy in neutron scattering and the traditional transport cross section in mono-energetic models. Chapter 3 derives the diffusion coefficients and transport cross sections based on the P1 equations. An analytical solution of the transport correction is provided

in this chapter along with numerical results. The commonly used approximation methods are reviewed and compared using test problems with hydrogen.

Chapter 4 introduces the methodology of CMM starting from the concept of cumulative-group quantities, and it is generalized to the cumulative-group-wise tally scheme. The application of CMM for generating homogenized diffusion coefficients and transport cross sections are tested on a series of problems. The idea of CMM comes from conservation of neutron

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migration area, but there is little information in literature on how to calculate migration area in deterministic simulations. Through an analytical derivation of mean square displacement in infinite homogeneous medium, Chapter 5 develops a rigorous method for computing migration area in infinite homogeneous medium and shows the impact of anisotropic scattering on migration area. Chapter 5 also introduces a method for computing core-wide multi-group migration area and corresponding weighting factors in deterministic calculations. The application of CMM on generating heterogeneous transport cross sections are discussed in Chapter 6. Using this method of computing migration area in deterministic simulations, the results of ke f f, power distribution and cumulative migration area for several 2D core benchmark problems are compared with CMM and other approximation methods. Results demonstrate the advantage of CMM in improving results accuracy by conserving migration area.

Lastly, the contributions of this thesis are summarized in Chapter 7, along with remaining challenges and potential future work.

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Chapter 2

Background

This chapter reviews the background for generation of multi-group diffusion coefficients and transport cross sections using Monte Carlo codes. The Monte Carlo method is in widespread use for generating multi-group cross sections (MGXS) due to its advantage in high-fidelity modeling of complex configurations and neutron transport physics, but the ultimate full core simulation accuracy remains limited by the approximations in multi-group deterministic calculations.

2.1

Overview

2.1.1

Standard Multi-level Approach for Steady-state Core Analysis

The steady-state neutronics analysis of nuclear reactor cores is an extremely broad and challenging computational task. The computational challenges include, but are not limited to: the detailed heterogeneity in geometry and material configuration, the continuous-energy nuclear cross section library, the enormous neutron interaction physics, the spatial and energy self-shielding effect, etc.

To make the simulations computationally tractable, the mainstream of steady-state light water reactor (LWR) core analysis takes a multi-level approach as illustrated in Figure2-1. The simulation usually starts from the pin cell calculation in a very detailed manner (e.g., full heterogeneity in geometry, ultra-fine energy representation), to the

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assembly calculation with intermediate details (e.g., homogenized pin cells, 100s energy groups), and the full core calculation is conducted using homogenized assemblies with only a few energy groups. Although the advancement in computer capabilities and parallel calculation technology have enabled to use fewer approximations and incorporate more details along this process, replacing the three-step procedure by two-step or one-step, the generation of accurate multi-group cross sections for the “down-stream” full core calculation remains a challenging task.

Figure 2-1: Traditional multi-level calculation flow for nuclear reactor core analysis (from the Handbook of Nuclear Engineering[13]).

Traditionally, the generation of multi-group cross section library often uses deter-ministic lattice codes[4–7, 14, 15]. The primary challenge in using deterministic lattice codes is to accurately incorporate the energy and spatial self-shielding effects, which are often caused by the resonance interactions with certain isotopes, such as U-238. Approximations made for modeling the self-shielding can lead to errors in the calculation of flux spectrum, thus impacting the accuracy of multi-group cross sections.

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2.1.2

MGXS Generation Using Monte Carlo Codes

With the advantageous features in high-fidelity modeling, the Monte Carlo method can overcome challenges of accuracy loss due to approximations used in deterministic lattice calculations. Monte Carlo provides an alternative pathway in the MGXS generation process and there has been growing interest for this approach in the past decades [1,16–22]. This approach replaces the separate resonance self-shielding approximations and deterministic lattice physics calculation steps with detailed Monte Carlo simulations.

The generation of MGXS by Monte Carlo codes is based on tallies with specific scores and filters. Generally, a tally estimator for reaction rate integration of reaction type x over a given volume V can be represented as

〈Σx,ψ〉g= Z V Z S Z Eg−1 Eg Σx(r, E)ψ(r, E, Ω)dEdΩdr (2.1)

where the angle bracket notation〈·, ·〉 represents inner products in phase-space (incoming and/or outgoing energy, space and angle), and the physical meanings of the symbols are as follows:

Σx: macroscopic cross sections of reaction type x;

ψ: angular neutron flux;

V: integration bounds in space r; S: integration bounds in solid angleΩ;

Eg and Eg−1: lower and upper integration bounds in energy group g.

The tally for scalar flux (represented by symbolφ) integration over a given volume V is represented as〈ψ〉 and it is essentially the inner product of angular flux with unity:

〈ψ〉g≡ 〈ψ, 1〉g= Z V Z S Z Eg−1 Eg ψ(r, E, Ω)dEdΩdr (2.2)

The general spatially homogenized (over volume V ) macroscopic MGXS for reaction x and energy group g can be computed as

Σx,g=

〈Σx,ψ〉g

〈ψ〉g

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There are special cases for cross section types such as the scattering matrix and fission production cross section, and more details can be found in[22].

As the “gold standard” in reactor neutronics analysis, continuous-energy Monte Carlo can also be used to run full core simulations to produce reference results for deterministic calculations. It remains a challenging task to achieve accuracy of pin power root mean square (RMS) error within 1% in the deterministic calculations. The discrepancy between the full core Monte Carlo and multi-group deterministic calculation results is mainly caused by the approximations in multi-group deterministic calculations, including the energy discretization, spatial homogenization and approximations in angle. For criticality calculations, the continuous-energy Monte Carlos method is configured to solve the steady-state Boltzmann transport equation[3] as

Ω · ∇ψ(r, Ω, E) + Σt(r, E)ψ(r, Ω, E) = ∞ Z 0 d E0 Z 4π d0Σs(r, Ω0→ Ω, E0→ E)ψ(r, Ω0, E0) + χ(r, E) 4πke f f ∞ Z 0 d E0 Z 4π d0ν(r, E0f(r, E0)ψ(r, Ω0, E0) (2.4) where

Σt: macroscopic total cross section;

Σs: macroscopic scattering cross section;

χ: fission spectrum;

ke f f: effective neutron multiplication factor;

ν: number of neutrons emitted per fission reaction; Σf: macroscopic fission cross section.

However, the deterministic calculations are solving a simplified version of Equation (2.4) with approximations in energy, space and angle, which are contributors to the errors in deterministic calculation results. Much effort has been made to investigate

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and improve the errors caused by spatial homogenization and energy condensation with MGXS generated using Monte Carlo codes[1,23–27], but the accuracy improvement for angular approximations remains a challenging task and open question.

Most Monte Carlo-based MGXS generation applications aim to improve the accuracy of full core diffusion calculations and are focused on generating few-group cross sections and diffusion coefficients. The recent study by Boyd[1] has investigated the accuracy by directly using full core continuous energy Monte Carlo simulations to compute MGXS for full core deterministic multi-group transport calculations. But due to the difficulty caused by the anisotropy in scattering reactions, the study in[1] adopted the isotropic-in-LAB (isotropic in the laboratory system) approximation in both the Monte Carlo and deterministic simulations.

The challenges in angular approximations consist of two primary factors. The first is the angular dependence of cross sections, especially for the resonance capture cross sections in the fuel. In the study by Gibson [28] the capture cross sections of a local region in the fuel collapsed with angular flux for the 6.67 eV U-238 resonance group are shown to be strongly dependent on the direction of the angular flux, as shown in Figure 2-2. This issue is usually corrected by equivalence methods such as the SPH factor method[29] and the generalized equivalence method [30].

The second factor is the anisotropy in neutron scattering reactions, which is discussed in more details in the next section. In this thesis work, we focus on reducing errors in multi-group deterministic calculations caused by anisotropy in scattering, and we seek to generate more accurate multi-group diffusion coefficients and transport cross sections using Monte Carlo.

2.2

Anisotropy in Neutron Scattering

For elastic neutron scattering with target atom of relative atomic mass A, from the conservation of energy and momentum we can get

µ = Aµc+ 1

pA2+ 2Aµ

c+ 1

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Figure 2-2: Angular-dependent capture MGXS for the 6.67 eV resonance group as a function of azimuthal angle for the fuel region in dark gray color. (The radial axis is given in units of barns and the azimuthal axis in units of degrees.) Image courtesy of N. Gibson[28].

in whichµcis the cosine of the scattering angle in the center-of-mass (COM) system, andµ is the cosine of the scattering angle in laboratory (LAB) system.

From Equation (2.5), we can see that if A= 1 (the case of hydrogen), µ is always non-negative with the range of[0, 1]. For A larger than 1, the range of µ is [−1, 1]. Solving forµc, we get

µc=

µ2− 1 + µp

µ2+ A2− 1

A (2.6)

The scattering laws are relatively complicated for inelastic neutron scattering, where the scattering angles are sampled from distributions based on experimental data. But elastic scattering dominates neutron moderation in LWRs and the energy and momentum are conserved between the neutron and the target atom.

The elastic neutron scattering is nearly isotropic in the COM system [31], so for target atoms with any A value, µc is uniformly distributed between [−1, 1], and the

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probability density function is

p(µc) = 1

2 (2.7)

Then, the probability density function of µ can be derived base on the fact that p(µ)dµ = p(µc)dµc, and the expression of p(µ) is

p(µ) = p(µc)dµc = µ A+ 2µ2+ A2− 1 2Apµ2+ A2− 1 (2.8)

As we can see from Equation (2.8), the probability distribution ofµ is dependent on the value of A. Specially, when the target is hydrogen (A= 1),

p(µ) =    0, forµ in [−1, 0) 2µ, for µ in [0, 1] (2.9)

µ has zero probability in [−1, 0) and is always non-negative, which means the neutron can only go forward in the LAB system after each elastic scattering reaction with hydrogen.

The comparison of p(µc) and p(µ) for different A values is shown in Figure2-3. In the COM system, the probability density function ofµc is a uniform distribution independent

of A, while in the LAB system we can see the forward-peaking effect as the probability density is higher whenµ is positive. As A increases, the distribution becomes closer to the uniform distribution, so elastic scattering with heavy nuclei is close to isotropic in the LAB system.

2.3

Transport Cross Section in Mono-energetic Models

To account for the effect of anisotropic neutron scattering, transport cross sections (transport-corrected cross sections) were introduced, like those of Lamarsh[2] for an

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-1 -0.5 0 0.5 1

Cosine of scattering angle

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 Probability density c (A = 1) (A = 2) (A = 4) (A = 16) (A = 100)

Figure 2-3: Probability density function of the cosine of scattering angle in COM system (µc) and lab system (µ).

infinite homogeneous scattering medium in a mono-energetic model.

Consider neutrons moving in an infinite homogeneous medium that scatters but does not absorb neutrons. Assume the energy of neutrons does not change (mono-energetic) and the scattering cross section of the medium is constant. With the scattering cross section of the medium represented byΣs, the mean free path of neutrons in it is λs= 1/Σs. Neutrons are emitted along the x axis, as shown in Figure2-4. Each neutron

flight is denoted by a vector ln, in which the subscript n is the sequence number of each flight.

As shown in in Figure 2-4, the first flight l1is aligned with the x axis. The angle between l1and l2isθ1, which is the scattering angle of the first collision. The vectors l1 and l2are presented to be on the same plane of the paper plane, so that the azimuthal angle of the first collision (ϕ1) is 0 (not shown in the figure). For the next neutron flight

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Figure 2-4: Illustration of transport effect.

angle of the second collision with respect to the plane defined by l1and l2(ϕ2) is 0.

Now we want to calculate the distance the neutron travels along its original direction, which is equivalent to the projection of neutron location to the x axis. For the first flight, the distance is the length of l1, so x1 = |l1| and the average of it is 〈x1〉 = λs.

The distance the neutron travels along x axis by the second flight is the projection of l2, which can be computed as x2 = |l2|cosθ1. If the average cosine of the scattering angle is

¯

µ, then we can get 〈x2〉 = λsµ.¯

For the third flight the situation is more complicated, because l3is not on the same plane as the one defined by l1and l2unlessϕ2= 0. As shown in Figure2-4, to computed x3 we need to calculate the cosine of the angle between l3and the x axis (set as cosα). With trigonometry it can be shown that

cosα = cosθ1cosθ2+ sinθ1sinθ2cosϕ2 (2.10)

and x3 = |l3|cosα. Since the azimuthal angle is uniformly distributed in the range of [−π, π], the average of cosϕ2 is zero. So the average of x3 is simply

〈x3〉 = λs〈cosα〉 = λs〈cosθ1cosθ2〉 = λsµ¯

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For the next neutron flights, in a similar way it can be demonstrated that the distance the neutron travels along the x axis by the nth flight is〈xn〉 = λsµ¯n−1. So after n flights

the total distance along x axis is

dn = n X i=1 〈xi= λs+ λsµ + λ¯ ¯ 2+ · · · + λ ¯n−1 = λs 1− ¯µn−1 1− ¯µ (2.12)

As n is large enough, the limit of dn is

lim

n→∞dn= λs

1− ¯µ (2.13)

This limit is known as the “transport mean free path”. It is denoted by λt r and

computed as

λt r=

λs

1− ¯µ (2.14)

For neutron scattering with light atoms, it has the forward-peaking property, in which case ¯µ is positive and (1 − ¯µ) is smaller than 1. So the transport mean free path λt r is larger than the scattering mean free path λs and neutrons tend to propagate in their original direction of motion.

Based on the inverse relation between cross section and mean free path, the transport cross section for this mono-energetic model is

Σt r= 1 λt r = 1 λs (1 − ¯µ) = Σs(1 − ¯µ) (2.15)

The transport cross section defined as in Equation (2.15) is the traditional way of calculatingΣt r, but it should be mentioned that this result only considers the neutron penetration distance along its emitted direction, and it is only applicable to the mono-energetic model.

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In addition, the physical meaning of mean free path is usually the mean distance of neutron movement by one flight (before its next collision), but in fact the transport mean free path computed in Equation (2.14) is a cumulative result from an infinite number of neutron flights.

The analysis of transport correction in the mono-energetic model helps to illustrate the impact of anisotropic scattering. However, the mono-energetic model is inapplicable to practical neutronics calculations and the definition of transport cross section with this model is not directly applicable to the continuous-energy case. The transport cross sections in continuous-energy and multi-group cases will be discussed in the next chapter.

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Chapter 3

Diffusion coefficients and transport

cross sections

Multi-group diffusion and transport theory are the primary approaches for deterministic nuclear reactor full core calculations, in which the diffusion coefficients and transport cross sections are crucial parameters. To get high accuracy in deterministic calculations, the angular dependence in neutron scattering reactions need to be taken into account for generating diffusion coefficients and transport cross sections. This chapter starts from the general PL method and derives the diffusion coefficients and transport cross

sections based on the P1 equations. The approximation methods for computing multi-group transport cross sections are reviewed and compared using test problems of pure hydrogen.

3.1

The P

L

Method

In deterministic simulations, the angular dependence of the flux and scattering reactions is often modeled using expansions of basis functions. The angular flux can be expanded using the spherical harmonic functions. The azimuthal angle of scattering reactions can be seen as uniformly distributed, so the scattering kernel is often expanded using the Legendre polynomials (the reduced version of spherical harmonic functions). Based on these two expansions, we can transform the Boltzmann neutron transport equation into

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PL (or PN) equations. The derivation of PN equations can be found in many textbooks, but many of them use simplifications such as one-group or one-dimension[3,32–34] in the derivations. This section follows the general derivations in[9].

3.1.1

Expansion of the Angular Flux

The spherical harmonics are basis functions defined on the surface of a sphere. There are several different normalizations used for the spherical harmonics in different fields. In the field of reactor physics, the common version in use is the Schmidt semi-normalized harmonics[35] defined as Y`m(θ, ϕ) = Y`m(Ω) = v t(` − m)! (` + m)!P m ` (cosθ)eimϕ (3.1)

which have the normalization Z π θ=0 Z 2π ϕ=0 Y`m(θ, ϕ)Y`m00∗(θ, ϕ)dΩ = 4π 2` + 1δ``mm0 (3.2) The solid angleΩ can be represented using a vector (Ωx,y,z) in a 3D Cartesian system as shown in Figure3-1, and the projections on each axis are computed using the polar angleθ and azimuthal angle ϕ as

Ωx = sinθ cosϕ

Ωy = sinθsinϕ

Ωz = cosθ

(3.3)

In the definition in Equation (3.1),` is any non-negative integer and m can be any integer in the range of−` ≤ m ≤ `. Pm

` is the associated Legendre function, and the

definition with the variableµ (µ = cosθ) is

P`m(µ) = (−1)m(1 − µ2)m/2 d

m

dµmP`(µ) (3.4)

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Figure 3-1: Illustration of solid angle and projections in 3D Cartesian coordinate system. functions: ψ(r, Ω, E) = ∞ X `=0 2` + 1 4π ` X m=−` φm ` (r, E)Y`m(θ, ϕ) (3.5)

and the flux moments can be computed as

φm `(r, E) = Z 2π 0 Z π 0 ψ(r, Ω, E)Ym` (θ, ϕ)sinθ dθ dϕ = Z 4π ψ(r, Ω, E)Ym` (Ω)dΩ (3.6) where Ym

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3.1.2

Expansion of the Scattering Kernel and Source

Based on the steady-state Boltzmann transport equation in Equation (2.4), the scattering source term is Qs(r, Ω, E) = ∞ Z 0 d E0 Z 4π d0Σs(r, Ω0→ Ω, E0→ E)ψ(r, Ω0, E0) (3.7)

For neutron scattering in nuclear reactor physics, the angular dependence relies only on the magnitude of the scattering angle, which is usually represented as the polar angle θ (in the range of [0, π]). The azimuthal angle ϕ is uniformly distributed in [0, 2π] and it reduces the expression of scattering source to

Qs(r, Ω, E) = ∞ Z 0 d E0 Z 4π d0Σs(r, Ω · Ω0, E0→ E)ψ(r, Ω0, E0) = 1 2π ∞ Z 0 d E0 Z 1 −1 dµΣs(r, µ, E0→ E)ψ(r, Ω0, E0) (3.8)

The term Ω · Ω0is just the cosine of the scattering angle, thusµ = Ω · Ω0. Notice that the range ofµ is [−1, 1], so the scattering kernel is usually expressed as an expansion of Legendre polynomials.

The differential scattering cross sectionΣs(r, µ, E0→ E) in the scattering kernel can be expressed as an infinite sum of Legendre polynomials:

Σs(r, µ, E0→ E) = ∞ X `=0 2` + 1 2 Σs,` r, E 0→ E P`(µ) (3.9)

in whichΣs,`(r, E0→ E) is the expansion coefficient (angular-independent) for order `,

and P`(µ) is the `-th order Legendre polynomial.

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of the scattering kernel can be computed as

Σs,` r, E0→ E =

Z 1

−1

dµ Σs(r, µ, E0→ E)P`(µ) (3.10)

Next, with the expansion of the angular flux in Equation (3.5) and scattering cross section in Equation (3.9), the whole scattering source can be expanded using the combination of these two expansions as

Qs(r, Ω, E) = ∞ Z 0 d E0 Z 4π d0Σs(r, Ω0→ Ω, E0→ E)ψ(r, Ω0, E0) = ∞ Z 0 d E0 ∞ X `=0 2` + 1 4π Σs,` r, E 0→ E ` X m=−` φm `(r, E0)Y`m(θ, ϕ) (3.11)

3.1.3

General P

L

Equations

Recall the steady-state Boltzmann transport equation[3]:

Ω · ∇ψ(r, Ω, E) + Σt(r, E)ψ(r, Ω, E) = ∞ Z 0 d E0 Z 4π d0Σs(r, Ω0→ Ω, E0→ E)ψ(r, Ω0, E0) + χ(r, E) 4πke f f ∞ Z 0 d E0 Z 4π d0νΣf(r, E0)ψ(r, Ω0, E0) (3.12)

It should be mentioned here that in the fission term of Equation (3.12)ν is combined withΣf as one term, which is a common practice in deterministic calculations.

To transform the neutron transport equation into PL equations, first we need to insert the expansions of the angular flux of Equation (3.5) and the scattering source of Equation (3.11) into Equation (3.12). Then, by multiplying the resulting Equation by

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Y`m(Ω) and integrating over Ω, one can get the general PL equations[9] as 1 2` + 1 ”1 2Æ(` + 2 + m)(` + 1 + m)(− ∂∂ x − i ∂ y)φ m+1 `+1 (r, E) +1 2Æ(` + 1 − m)(` + 2 − m)( ∂∂ x − i ∂ y)φ m−1 `+1 (r, E) +1 2Æ(` − 1 − m)(` − m)( ∂∂ x + i ∂ y)φ m+1 `−1 (r, E) +1 2Æ(` + m)(` − 1 + m)(− ∂∂ x + i ∂ y)φ m−1 `−1 (r, E) +Æ(` + 1 + m)(` + 1 − m) ∂ ∂ zφ m `+1(r, E) +Æ(` + m)(` − m) ∂ ∂ zφ m `−1(r, E)— + Σt(r, E)φlm(r, E) = ∞ Z 0 Σs,` r, E0→ E φ`m(r, E0)dE0+ δl0 χ(r, E) ke f f ∞ Z 0 d E0νΣf(r, E000(r, E0) (3.13)

in which δ`0 is the Kronecker delta function and the fission source only exist when ` = 0. In this equation, the range of the moment order ` is {0, 1, 2, ...} and m can be any integer from(−`) to `. If the order of any term in Equation (3.13) is out of its range, the term does not exit. For instance, when` = 0, in fact the terms having φm−1

`−1 (r, E)

andφm+1

`−1 (r, E) do not exist.

This general from of PL equations is rigorous and equivalent to the original transport

equation, and there are two important properties about the general PL equations[9]. The first is that each equation ofφm

` (r, E) only contains the scattering source with the

same moment (in both` and m), so the coupling between different moments is entirely caused by the streaming term (the terms in the bracket in Equation (3.13)). The second property is that the`-th order moment of the scattering cross section shows up together with the flux moment with the same order `.

However, as pointed in[9], there is little practical use for the general PL equations without simplifications. Normally the higher-order components are small and the com-mon simplification is to decouple the spherical harcom-monic moments with ` ≤ L from

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those with` > L by setting ∂ xφ m±1 L+1 = 0 ∂ yφ m±1 L+1 = 0 ∂ zφ m L+1= 0 (3.14)

and the resulting truncated equations are the PL approximation. The most commonly

used approximation is the P1 approximation, which is discussed in the next section.

3.2

The P

1

Equations

The P1 approximation contains the` = 0 and ` = 1 components of the PL equations.

Due to the range of m (from−` to `), the ` = 0 component contains only one equation and the` = 1 component contains three equations.

3.2.1

The First P

1

Equation

From the general PL equation in Equation (3.13), the first P1 equation (` = 0, m = 0)

can be shown as ” p 2 2 (− ∂ x − i ∂ y)φ 1 1(r, E) + p 2 2 ( ∂ x − i ∂ y)φ1−1(r, E) + ∂ zφ 0 1(r, E)— + Σt(r, E)φ 0 0(r, E) = ∞ Z 0 Σs0 r, E0→ E φ00(r, E0)dE0+ χ(r, E) ke f f ∞ Z 0 d E0νΣf(r, E000(r, E0) (3.15) It should be mentioned that the components ofφlm with` = −1 do not exist.

The flux moments can be computed using Equation (3.6). To get a clear understand-ing of the first few moments, next we can show the expressions of flux moments with ` = 0 and ` = 1.

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To facilitate the following derivation steps, the expression of the spherical harmonic functions and associate Legendre polynomials for` = 0 and ` = 1 are listed in Equation (3.16) and (3.17).

P00(cosθ) = P0(cosθ) = 1 P10(cosθ) = P1(cosθ) = cosθ P11(cosθ) = −sinθ P1−1(cosθ) = 1 2sinθ (3.16) Y00(θ, ϕ) = P00(cosθ) = 1 Y10(θ, ϕ) = P10(cosθ) = cosθ Y11(θ, ϕ) = 1 2P 1 1(cosθ) e−iϕ= − p 2 2 sinθ e Y1−1(θ, ϕ) =Æ(2) P1−1(cosθ) e−iϕ= p 2 2 sinθ e −iϕ (3.17)

Thus the flux moment with` = 0 and m = 0 is

φ0 0(r, E) = Z 4π ψ(r, Ω, E)Y0∗ 0 (Ω)dΩ = Z 4π ψ(r, Ω, E)dΩ = φ(r, E) (3.18) which is the scalar flux.

And the flux moment with` = 1 and m = 0 is

φ0 1(r, E) = Z 4π ψ(r, Ω, E)Y0∗ 1 (Ω)dΩ = Z 4π ψ(r, Ω, E)cosθ dΩ (3.19)

Using the relation ofΩz= cosθ as shown in Equation (3.3), it can be further shown

that φ0 1(r, E) = Z 4π ψ(r, Ω, E)ΩzdΩ = Jz(r, E) (3.20)

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So the physical meaning of the flux moment φ10(r, E) is the neutron current in z direction.

For the flux moment with (` = 1, m = −1) and (` = 1, m = 1), the flux moments are

φ1 1(r, E) = Z 4π ψ(r, Ω, E)Y1∗ 1 (Ω)dΩ = Z 4π ψ(r, Ω, E) − p 2 2 sinθ e −iϕ d φ−1 1 (r, E) = Z 4π ψ(r, Ω, E)Y−1∗ 1 (Ω)dΩ = Z 4π ψ(r, Ω, E) p 2 2 sinθ e  d (3.21)

It is unclear as to the physical meaning of φ−1

1 (r, E) and φ 1

1(r, E) from Equation

(3.21), but the combination of these two moments can be related to the partial currents by recalling that

eiϕ− e−iϕ= 2i sinϕ eiϕ+ e−iϕ= 2 cosϕ

(3.22)

So it implies that we can combineφ1−1(r, E) and φ11(r, E) by computing their differ-ence as φ−1 1 (r, E) − φ 1 1(r, E) = Z 4π ψ(r, Ω, E) p 2 2 sinθ e + p 2 2 sinθ e −iϕ d =p2 Z 4π ψ(r, Ω, E) sinθ cosϕ d =p2 Z 4π ψ(r, Ω, E)Ωxd =p2 Jx(r, E) (3.23)

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And it is straightforward to get Jx(r, E) =p1 2 φ −1 1 (r, E) − φ 1 1(r, E)  = p 2 2 φ −1 1 (r, E) − φ 1 1(r, E)  (3.24)

Following the similar step to compute the sum ofφ1−1(r, E) and φ11(r, E), we can get

φ−1 1 (r, E) + φ 1 1(r, E) = Z 4π ψ(r, Ω, E) p 2 2 sinθ e p 2 2 sinθ e −iϕ d =p2i Z 4π ψ(r, Ω, E) sinθsinϕ d =p2i Z 4π ψ(r, Ω, E)Ωyd =p2i Jy(r, E) (3.25)

Thus it is straightforward to get

Jy(r, E) = p1 2i φ −1 1 (r, E) + φ 1 1(r, E)  = − p 2i 2 φ −1 1 (r, E) + φ 1 1(r, E)  (3.26)

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(3.24), (3.26) and (3.20), the term in the bracket of Equation (3.15) becomes p 2 2 (− ∂ x − i ∂ y)φ 1 1(r, E) + p 2 2 ( ∂ x − i ∂ y)φ1−1(r, E) + ∂ zφ 0 1(r, E) = ∂ x ” p 2 2 −1 1 (r, E)) − φ 1 1(r, E)— + ∂∂ y ” − p 2 2 −1 1 (r, E)) + φ 1 1(r, E) — + ∂ zφ 0 1(r, E) = ∂ xJx(r, E) + ∂ yJy(r, E) + ∂ zJz(r, E) =∇ · J(r, E) (3.27)

in which J(r, E) is the vector of neutron current and J(r, E) = Jx(r, E), Jy(r, E), Jz(r, E). So the first P1 equation in Equation (3.15) can be rewritten as

∇ · J(r, E) + Σt(r, E)φ(r, E) = ∞ Z 0 Σs0 r, E0→ E φ(r, E0)dE0+ χ(r, E) ke f f ∞ Z 0 νΣf(r, E0)φ(r, E0)dE0 (3.28) Note that the term of ∇ · J(r, E) in this equation is the divergence of the neutron current and it is a scalar term.

3.2.2

The Second P

1

Equation

The second P1 equation consists of three separate equations. From the general PL

equation in Equation (3.13) with the approximations in Equation (3.14) by setting the spatial derivative of all flux moments with` = 2 as 0, for ` = 1 and m = 0, one can get

1 3 ∂ zφ 0 0(r, E) + Σt(r, E)φ10(r, E) = ∞ Z 0 Σs1 r, E0→ E φ10(r, E 0)dE0 (3.29)

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Plugging in the flux moments as derived in the last section, it becomes 1 3 ∂ zφ(r, E) + Σt(r, E)Jz(r, E) = ∞ Z 0 Σs1 r, E0→ E  Jz(r, E0)dE0 (3.30)

Similar to the derivation for the partial currents, the equation for (` = 1 and m = −1) needs to be combined together with the moment of (` = 1 and m = 1). The two equations for these two moments are:

1 3 p 2 2 ( ∂ x + i ∂ y)φ 0 0(r, E) + Σt(r, E)φ1−1(r, E) = ∞ Z 0 Σs1 r, E0→ E φ1−1(r, E 0)dE0 1 3 p 2 2 (− ∂ x + i ∂ y)φ 0 0(r, E) + Σt(r, E)φ11(r, E) = ∞ Z 0 Σs1 r, E0→ E φ11(r, E0)dE0 (3.31) By subtracting the two equations in Equation (3.31) and multiplying it byp2/2, one obtains 1 3 ∂ xφ 0 0(r, E) + Σt(r, E) ” p 2 2 φ −1 1 (r, E) − φ 1 1(r, E)— = ∞ Z 0 Σs1 r, E0→ E  ” p 2 2 φ −1 1 (r, E) − φ 1 1(r, E)d E0 (3.32)

By noticing the expression of Jx in Equation (3.24), it turns into

1 3 ∂ xφ(r, E) + Σt(r, E)Jx(r, E) = ∞ Z 0 Σs1 r, E0→ E  Jx(r, E0)dE0 (3.33)

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(−p2i/2), one obtains 1 3 ∂ yφ 0 0(r, E) + Σt(r, E) ” − p 2 2 i φ −1 1 (r, E) + φ 1 1(r, E)— = ∞ Z 0 Σs1 r, E0→ E ” − p 2 2 i φ −1 1 (r, E) + φ 1 1(r, E)d E0 (3.34)

Similarly, by noticing the expression of Jy in Equation (3.26), it becomes

1 3 ∂ yφ(r, E) + Σt(r, E)Jy(r, E) = ∞ Z 0 Σs1 r, E0→ E  Jy(r, E0)dE0 (3.35)

Lastly, we combine the three equations for ` = 1 together as shown below:

1 3 ∂ xφ(r, E) + Σt(r, E)Jx(r, E) = ∞ Z 0 Σs1 r, E0→ E  Jx(r, E0)dE0 1 3 ∂ yφ(r, E) + Σt(r, E)Jy(r, E) = ∞ Z 0 Σs1 r, E0→ E  Jy(r, E0)dE0 1 3 ∂ zφ(r, E) + Σt(r, E)Jz(r, E) = ∞ Z 0 Σs1 r, E0→ E  Jz(r, E0)dE0 (3.36)

Using the vector notation, we can combine these three equations into one, as:

1 3∇φ(r, E) + Σt(r, E)J(r, E) = ∞ Z 0 Σs1 r, E0→ E J(r, E0)dE0 (3.37)

Note that every term in Equation (3.37) is in vector form and is thus essentially a combination of the three expressions of Equation (3.36).

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To summarize, the P1 equations are ∇ · J(r, E)+Σt(r, E)φ(r, E) = ∞ Z 0 Σs0 r, E0→ E φ(r, E0)dE0+ χ(r, E) ke f f ∞ Z 0 d E0νΣf(r, E0)φ(r, E0) 1 3∇φ(r, E) + Σt(r, E)J(r, E) = ∞ Z 0 Σs1 r, E0→ E J(r, E0)dE0 (3.38)

3.2.3

Diffusion Equation and Diffusion Coefficients

The neutron diffusion equation can be obtained from the P1 equations. From the second P1 equation we can get the relation between neutron current J(r, E) and the gradient of scalar flux∇φ(r, E), as

J(r, E) = − 1 3[Σt(r, E) − 1 J(r,E) R∞ 0 Σs1(r, E0→ E)J(r, E0)dE0] ∇φ(r, E) (3.39)

in which J(r, E) on the right hand side of the equation is the magnitude of J(r, E). It is assumed here that the spatial components of J(r, E) have the same energy dependence, which is true in an infinite homogeneous medium.

Equation (3.39) is in the same form as Fick’s law of diffusion theory, which postulates that the current is proportional to the gradient of the flux as

J(r, E) = −D(r, E)∇φ(r, E) (3.40)

where D(r, E) is the diffusion coefficient.

Figure

Figure 2-1: Traditional multi-level calculation flow for nuclear reactor core analysis (from the Handbook of Nuclear Engineering [ 13 ] ).
Figure 2-3: Probability density function of the cosine of scattering angle in COM system ( µ c ) and lab system ( µ ).
Figure 3-1: Illustration of solid angle and projections in 3D Cartesian coordinate system
Figure 3-4: Analytical transport correction ratio for hydrogen in pure down-scatter reactions with constant cross section and box-energy sources (zoomed in).
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