ADVANCED REACTOR PHYSICS METHODS FOR HETEROGENEOUS REACTOR ...

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The Pennsylvania State University The Graduate School College of Engineering ADVANCED REACTOR PHYSICS METHODS FOR HETEROGENEOUS REACTOR CORES A Dissertation in Nuclear Engineering by Steven A. Thompson 2014 STEVEN ANDREW THOMPSON Submitted in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy December 2014

Transcript of ADVANCED REACTOR PHYSICS METHODS FOR HETEROGENEOUS REACTOR ...

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The Pennsylvania State University

The Graduate School

College of Engineering

ADVANCED REACTOR PHYSICS METHODS FOR

HETEROGENEOUS REACTOR CORES

A Dissertation in

Nuclear Engineering

by

Steven A. Thompson

2014 STEVEN ANDREW THOMPSON

Submitted in Partial Fulfillment

of the Requirements

for the Degree of

Doctor of Philosophy

December 2014

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The dissertation of Steven A. Thompson was reviewed and approved* by the following:

Kostadin Ivanov Distinguished Professor of Nuclear Engineering Dissertation Advisor Chair of Committee

Maria Avramova Assistant Professor of Nuclear Engineering

Igor Jovanovic Associate Professor of Nuclear Engineering Ludmil Zikatanov Professor of Mathematics

Daniel Haworth Professor of Mechanical Engineering Professor-In-Charge of Mechanical and Nuclear Engineering Graduate Programs

*Signatures are on file in the Graduate School

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ABSTRACT

To maintain the economic viability of nuclear power the industry has begun to emphasize

maximizing the efficiency and output of existing nuclear power plants by using longer

fuel cycles, stretch power uprates, shorter outage lengths, mixed-oxide (MOX) fuel and

more aggressive operating strategies. In order to accommodate these changes, while still

satisfying the peaking factor and power envelope requirements necessary to maintain safe

operation, more complexity in commercial core designs have been implemented, such as

an increase in the number of sub-batches and an increase in the use of both discrete and

integral burnable poisons. A consequence of the increased complexity of core designs, as

well as the use of MOX fuel, is an increase in the neutronic heterogeneity of the core.

Such heterogeneous cores introduce challenges for the current methods that are used for

reactor analysis. New methods must be developed to address these deficiencies while still

maintaining the computational efficiency of existing reactor analysis methods.

In this thesis, advanced core design methodologies are developed to be able to adequately

analyze the highly heterogeneous core designs which are currently in use in commercial

power reactors. These methodological improvements are being pursued with the goal of

not sacrificing the computational efficiency which core designers require. More

specifically, the PSU nodal code NEM is being updated to include an SP3 solution option,

an advanced transverse leakage option, and a semi-analytical NEM solution option.

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TABLE OF CONTENTS

LIST OF ABBREVIATIONS…………………………………………….……………………...vii

LIST OF FIGURES…………………………………………………………………...……........viii

LIST OF TABLES………………………………………………………………………………....x

ACKNOWLEDGMENTS…………………….………………………………..………….…….xiii

CHAPTER 1 INTRODUCTION AND BACKGROUND…..…….…………………..…………..1

1.1 Introduction…………………………………………….……………..………………1

1.2 Statement of Objectives………………………………….…………………………...4

1.3 Thesis Outline………………………………………..….……………………………7

CHAPTER 2 LITERATURE REVIEW………………………………………………………….10

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

2.2 The SP3

2.3 The Semi-Analytical Nodal Expansion Method……………………………..………13

Approximation………………………………..…………………………….10

2.4 Transverse Leakage Approximation.………………..…………….…………………14

2.5 Discontinuity Factors………………………………………………………………...17

CHAPTER 3 THE SIMPLIFIED P3

3.1 Introduction……………………………………..……………….…………………...22

METHODOLOGY………..……….………………………22

3.2 The Simplified PN

3.2.1 The SP

Approximation……………..………....………………………….23

N

3.2.2 Marshak Boundary Conditions……………..….……………..………………..28

Equations…….……………..…………….…………………….……23

3.2.3 Response Matrix Equations for SP3

3.3 Benchmarking of the SP

Solution in NEM….……...……………...29

3 Solution in NEM……….…..…….…………..…….…….32

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3.3.1 OECD/NEA 2-D C5G7 MOX Benchmark………..….……………………….32

3.3.2 OECD/NEA 3-D C5G7 MOX Benchmarks (Rodded)…….……....………….39

3.3.3 PWR MOX/UO2

3.4 Final Remarks on SP

Core Transient Benchmark……………..…..………………48

3

CHAPTER 4 THE SEMI-ANALYTICAL NODAL EXPANSION METHOD………....………64

Nodal Expansion Method…………………..………………..60

4.1 Introduction……………………………………………………..………….………...64

4.2 The Semi-Analytical Nodal Expansion Method……………..…………….………...65

4.3 Benchmarking of SA-NEM Solution……..………………..………………………..72

4.3.1 OECD/NEA 2-D C5G7 MOX Benchmark……….…..….……………………72

4.3.2 C3 2x2 Mini-Core Benchmark…………………..….………..………………..76

4.3.3 C5 2x2 Mini-Core Benchmark……………..…..…….………………………..77

4.3.4 Mesh Width Sensitivity………………………..……..………………………..78

4.4 Concluding Remarks on the SA-NEM……………..……….....….…………………80

CHAPTER 5 THE ANALYTIC BASIS FUNCTION

TRANSVERSE LEAKAGE METHOD………………………..…………………….…………..83

5.1 Introduction…………………………….…………..……………………...….……...83

5.2 Analytic Basis Function Transverse Leakage……..…………………….…....……...85

5.2.1 Representation of Intra-Nodal Flux by Analytic Basis Functions…..…..….....86

5.2.2 Analytic Basis Function Transverse Leakage Method….…..…..….………….88

5.3 C3 Benchmarking of the ABFTL Method…………..….………..….……………….90

5.4 C5 Benchmarking of the ABFTL Method…………..….………..….……………….91

5.5 Mesh Width Sensitivity…………………………..…….…...…….…………………92

5.6 Discussion and Final Remarks on ABFTL Method…………...…….....……………94

CHAPTER 6 DISCONTINUITY FACTORS FOR SP3

EQUATIONS…….…….……….…….97

6.1 Introduction…………………………………………………………………………..97

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6.2 Expression of Angular Flux in SPN

6.3 Calculation of Discontinuity Factors for SP

Approximation………….……………………..98

3

CHAPTER 7 CONCLUSIONS AND FUTURE WORK………………..……………….……..103

Method…………..………….………101

7.1 Conclusions…………..………………………………………………………...…..103

7.2 Recommendations for Future Work…………………………………………....…..107

REFERENCES………………………………………………………………………………….109

APPENDIX A. DESCRIPTION OF THE BENCHMARKS…………………………………...115

APPENDIX B. BENCHMARK CROSS SECTIONS……………………..……………………120

APPENDIX C. 2-D C5G7 Benchmark Pin Powers and % Error Comparison………..………..127

APPENDIX D. SP3

APPENDIX E. NEM INPUT and NEMTAB INPUT for SP

RESPONSE MATRIX EQUATIONS…………….……..…….………….137

3 OPTION…………..…….……..146

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LIST OF ABBREVIATIONS

ABFTL Analytic Basis Function Transverse Leakage Approximation ADF Assembly Discontinuity Factor AFEM Analytical Function Expansion Method ANM Analytical Nodal Method ARI All Rods In ARO All Rods Out BWR Boiling Water Reactor CMFD Coarse-Mesh Finite Difference CQLA Consistent Quadratic Leakage Approximation DF Discontinuity Factor ET Equivalence Theory GET Generalized Equivalence Theory IFBA Integral Fuel Burnable Absorber MOX Mixed-Oxide NEA Nuclear Energy Agency NEM Nodal Expansion Method NFI Nuclear Fuel Industries NPA Nodes Per Assembly NRC Nuclear Regulatory Commission OECD Organization for Economic Cooperation and Development P-NEM Polynomial Nodal Expansion Method PSU The Pennsylvania State University PWR Pressurized Water Reactor QLA Quadratic Leakage Approximation RM Response Matrix SA-NEM Semi-Analytical Nodal Expansion Method SP3 Simplified P3 SPH Super-homogenization SSS Scattered Source Subtraction TLA Transverse Leakage Approximation TL Transverse Leakage WABA Wet Annular Burnable Absorber

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LIST OF FIGURES

Figure 2.1- Thermal Flux Distribution in Two Neighboring Fuel Assemblies………….……18

Figure 3.1- P-NEM Diffusion Theory Pin Power

Distribution for 2-D C5G7 MOX Benchmark……………...………..…………....37

Figure 3.2- SP3 Pin Power Distribution for 2-D C5G7 MOX Benchmark……………..……..38

Figure 3.3- Geometry for the Unrodded Benchmark Case…....………………………..……...40

Figure 3.4- Geometry Configuration for the Upper Axial Water Reflector……….….………41

Figure 3.5- Geometry for the Rodded A Benchmark Case…………………..………………..45

Figure 3.6- Quarter-Core Configuration of MOX/UO2 Core Transient Benchmark…….…...50

Figure 3.7- Pin Layout of UO2 and MOX Fuel Assemblies…………………….……..………51

Figure 3.8- Percent Deviations in Assembly Powers for Three Solution Methodologies vs.

Reference DeCART Solution for ARO Configuration……………..………….....53

Figure 3.9- Deviation of Pin Powers for Assembly A1 at ARO Conditions…………..……...55

Figure 3.10- Percent Deviations in Assembly Powers for Two Solution Methodologies vs.

Reference DeCART Solution for ARI Configuration…………………..……….57

Figure 3.11- Deviation of Pin Powers for Assembly F6 at ARI Conditions…………..……...59

Figure 4.1- SA-NEM Pin Power Distribution for 2-D C5G7 MOX Benchmark………….….75

Figure 4.2- Assembly Power RMS % Error vs. Mesh Size

for P-NEM and SA-NEM Solutions………………………………..…………..….80

Figure 5.1- Assembly Power RMS % Error vs. Mesh Size……………..……….…………….94

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Figure A.1- OECD/NEA 2-D C5G7 MOX Benchmark Core Configuration………...…..…116

Figure A.2- Pin-by-Pin Layout of 2D C5G7 MOX Benchmark Fuel Assemblies……….....117

Figure A.3- C3 Core Configuration……………………………………………………….…..118

Figure A.4- C5 Core Configuration…………………………………………………….……..119

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LIST OF TABLES

Table 3.1- Calculated Pin Powers from NEM Compared

with MCNP Reference Solution (SP3 2-D C5G7 Benchmark)…………………….....35

Table 3.2- Calculated Assembly Powers from NEM Compared

with MCNP Reference Solution (SP3

Table 3.3- Calculated Pin Powers from NEM Compared with MCNP

2-D C5G7 Benchmark).......….……………....35

Reference Solution (C5G7 unrodded Benchmark)…..………………..…………...43

Table 3.4- Calculated Pin Powers from NEM Compared with MCNP

Reference Solution (C5G7 Rodded A Benchmark)…………………………..……46

Table 3.5- Eigenvalue keff and Assembly Power Deviation for ARO Configuration………...53

Table 3.6- Pin Power PWE (%) for ARO Configuration……….………..……………………54

Table 3.7- Eigenvalue keff and Assembly Power

Deviation for ARI Configuration…...………………………………………………57

Table 3.8- Pin Power PWE (%) for ARI Configuration…………………………..….….…….58

Table 4.1 Calculated Pin Powers from NEM Compared

with MCNP Reference Solution (SA-NEM 2-D C5G7 Benchmark)……..……….74

Table 4.2 Calculated Assembly Powers from NEM Compared

with MCNP Reference Solution (SA-NEM 2-D C5G7 Benchmark)…………………74

Table 5.1- 2D Power Distribution for C3 Benchmark………………..………..……………....91

Table 5.2- 2D Power Distribution for C5 Benchmark…………..……….…………………….92

Table B.1- C5G7 Control Rod Cross Sections………………………….…………………….120

Table B.2- C5G7 Moderator Cross Sections………………………………………………….121

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Table B.3- C5G7 Guide Tube Cross Sections……………………….………………………..121

Table B.4- C5G7 UO2 Fuel-Clad Cross Sections…………………….….……………………122

Table B.5- C5G7 4.3% MOX Fuel-Clad Cross Sections………………………….…………122

Table B.6- C5G7 7.0% MOX Fuel-Clad Cross Sections…………………………………….123

Table B.7- C5G7 8.7% MOX Fuel-Clad Cross Sections…………………………………….123

Table B.8- C5G7 Fission Chamber Cross Sections………………………..………….…..….124

Table B.9- C5G7 Fission Spectrum…………………………………………………….……….124

Table B.10- C3 Cross Sections (UO2)………………………………………………………...125

Table B.11- C3 Cross Sections (MOX)……………………………………………………….125

Table B.12- C5 Cross Sections (UO2)……………..……………………………………….…126

Table B.13- C5 Cross Sections (MOX)……………..…………………………………….…..126

Table B.14- C5 Cross Sections (Water Moderator)…………….…………………………….126

Table C1- C5G7 Distribution of % Errors in UO2 Assembly Near Axis of Symmetry

(NEM with P-NEM Diffusion Theory Solution)…………………………..……..127

Table C2- C5G7 Distribution of % Errors in UO2 Assembly Near Axis of Symmetry

(NEM with SP3 Solution)………………………………………………….………127

Table C3- C5G7 Distribution of % Errors in MOX Assembly

(NEM with P-NEM Diffusion Theory Solution)………………………………....128

Table C4- C5G7 Distribution of % Errors in MOX Assembly

(NEM with SP3 Solution)………………………………………………………….128

Table C5- C5G7 Distribution of % Errors in UO2 Assembly Near Water Reflector

(NEM with P-NEM Diffusion Theory Solution)…………………………………129

Table C6- C5G7 Distribution of % Errors in UO2 Assembly Near Water Reflector

(NEM with SP3 Solution)………………………………………………………….129

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Table C7- C5G7 Distribution of % Errors in UO2 Assembly Near Axis of Symmetry

(NEM with SA-NEM Diffusion Theory Solution)…………………….…………130

Table C8- C5G7 Distribution of % Errors in MOX Assembly

(NEM with SA-NEM Diffusion Theory Solution)……………………………….130

Table C9- C5G7 Distribution of % Errors in UO2 Assembly Near Water Reflector

(NEM with SA-NEM Diffusion Theory Solution)……………………………….131

Table C10- C5G7 Distribution of Pin Powers in UO2 Assembly Near Axis of Symmetry

(NEM with P-NEM Diffusion Theory Solution)………………………………...132

Table C11- C5G7 Distribution of Pin Powers in UO2 Assembly Near Axis of Symmetry

(NEM with SP3 Solution)…………………………………………………………132

Table C12- C5G7 Distribution of Pin Powers in UO2 Assembly Near Axis of Symmetry

(NEM with SA-NEM Diffusion Theory Solution)………………………..…….133

Table C13- C5G7 Distribution of Pin Powers in MOX Assembly

(NEM with P-NEM Diffusion Theory Solution)…………………………….….133

Table C14- C5G7 Distribution of Pin Powers in MOX Assembly

(NEM with SP3 Solution)…………………………………………………………134

Table C15- C5G7 Distribution of Pin Powers in MOX Assembly

(NEM with SA-NEM Diffusion Theory Solution)………………………………134

Table C16- C5G7 Distribution of Pin Powers in UO2 Assembly Near Water Reflector

(NEM with P-NEM Diffusion Theory Solution)………………………………..135

Table C17- C5G7 Distribution of Pin Powers in UO2 Assembly Near Water Reflector

(NEM with SP3 Solution)…………………………………………………………135

Table C18- C5G7 Distribution of Pin Powers in UO2 Assembly Near Water Reflector

(NEM with SA-NEM Diffusion Theory Solution)………………………….…..136

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ACKNOWLEDGMENTS

I would like to thank my advisor, Dr. Kostadin Ivanov, for his advice and assistance during

the course of this thesis work.

I would also like to thank Dr. Maria Avramova, Dr. Igor Jovanovic, and Dr. Ludmil

Zikatanov for the guidance and suggestions they have given me during my thesis research.

Finally and foremost I would like to thank my wife Meredith for her love, support, and

encouragement during this long and difficult process.

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CHAPTER 1

INTRODUCTION AND BACKGROUND

1.1 Introduction

In order to maintain the economic viability of nuclear power the industry has begun to

emphasize maximizing the efficiency and power output of existing nuclear power plants in

lieu of building newer more expensive next generation plants. Longer fuel cycles, stretch

power uprates, shorter outage lengths, and more aggressive operating strategies have been

implemented to maximize the output of existing nuclear units. Furthermore, in Europe about

30 reactors in Belgium, Switzerland, France and Germany use mixed-oxide (MOX) fuel and

ten reactors in Japan are licensed to use MOX [1]. MOX fuel allows the plant operator to

significantly increase the amount of excess reactivity needed for the longer operating cycles

without requiring an increase in fuel enrichment. However, the cost associated with the

manufacturing of MOX fuel has limited its use in commercial power reactors. Thus far,

MOX fuel has only been used in US reactors as demonstration projects in single fuel batches.

Most recently, in 2005 four MOX test assemblies manufactured in France were burned at

Catawba [1].

An additional change in the way in which nuclear reactor cores are designed, in the United

States in particular, is related to the ongoing re-licensure process. The original 40 year

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operating license is due to expire in the near future at most US nuclear power plants. As part

of the re-licensure application, the US Nuclear Regulatory Commission (NRC) has required

that utilities demonstrate the ability of their reactor pressure vessels, embrittled by 40 years

of fast neutron fluence, to still maintain adequate integrity, as well as ductility, to be able to

avoid brittle fracture during the re-pressurization stage of a loss of coolant accident [2]. As a

result of this requirement, which is primarily concerned with the amount of fluence

accumulated by the pressure vessel welds, core designs in US commercial power reactors are

now almost exclusively of the low leakage design, meaning that they concentrate the new

fuel assemblies on the core interior, and place the higher burnup and therefore lower

reactivity fuel assemblies on the core periphery near the core baffle. Low leakage core

designs reduce the neutron leakage and therefore reduce the fluence to the pressure vessel

welds. The use of low leakage cores, however, increase peaking factors due to the

concentration of the higher reactivity fuel in the core interior, where the neutron flux is at its

highest.

In order to accommodate these changes, while still satisfying the peaking factor and power

envelope requirements necessary to maintain safe reactor operation, more complexity in

commercial reactor core designs has been implemented, such as an increase in the number of

sub-batches, increase in fresh feed enrichment, and an increase in the use of both discrete and

integral burnable poisons. In addition, in Europe, where it is economically advantageous for

nuclear power plants to load follow (operate at full power during peak electricity demand

times and operate at reduced power at lower electricity demand times), there is significant

operation with the reactor in deeply rodded conditions, adding even more complexity to the

design and analysis of the reactor core due to the presence of the strong absorbers (control

rods) in the reactor core.

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A consequence of the increased complexity of reactor core designs, as well as the use of

MOX fuel, is an increase in the neutronic heterogeneity of the nuclear core. These

heterogeneous cores have much more pronounced leakage and thermal flux gradients, which

introduce challenges to the current generation of reactor analysis codes based upon diffusion

theory. One of the fundamental assumptions of diffusion theory is that neutrons behave

essentially as an inert gas, diffusing from areas of high concentration to areas of low

concentration according to the description of Fick’s Law. When the medium under

consideration is dominated by linearly anisotropic scattering (or isotropic scattering) and has

little absorption, this approximation is quite valid. For the modern heterogeneous cores which

contain discrete and integral burnable poisons of varying strengths and poison materials

(gadolinium, erbium, boron-carbide, zirconium diboride, etc.), MOX fuel, and many sub-

batches of varying enrichments and burnups, the diffusion theory assumptions are invalid.

High absorption (such as is the case when control rods are present or when discrete or

integral burnable poisons are present) leads to a rapid spatial variation in the neutron flux and

invalidates assumptions made in the derivation of the diffusion theory. Therefore, analysis of

modern heterogeneous reactor cores with the diffusion theory may produce less than

satisfactory results.

The current generation of nodal codes is still based upon methods which were in large part

developed for the relatively homogeneous cores used in the past, and are generally

inadequate for the complex reactor core designs which are currently in use. The significant

amount of neutron streaming which is introduced in a mixed MOX/UO2 reactor core, as well

as the use of multiple types of burnable absorbers and multiple sub-batches is difficult to

accurately model with the diffusion approximation using current nodal methods (polynomial

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nodal expansion method with quadratic leakage approximation). The polynomial nodal

expansion method has been found to be inaccurate in areas where steep flux gradients occur

at assembly interfaces, such as would be found in MOX or high-burnup cores, near material

boundaries and near control rods. Furthermore, the complex leakage shapes associated with

these modern heterogeneous reactor cores introduce additional challenges that the current

methods, namely the quadratic leakage approximation, cannot adequately satisfy.

More advanced core design methodologies need to be developed in order to improve the

accuracy of reactor core design codes, while preserving the computational efficiency which

core designers require. These newer core design methodologies should address the

inadequacies of the methods commonly used in most nodal codes currently in use: namely,

the inadequacy of the diffusion approximation in highly heterogeneous cores and near

material boundaries and strong absorbers such as control rods and burnable poisons, the

limitations of the polynomial nodal expansion method in larger spatial nodes with more

complicated flux shapes and in areas where steep flux gradients occur, as well as the

inadequacy of the quadratic leakage approximation to be able to model complex leakage

shapes.

1.2 Statement of Objectives

The research documented in this thesis is being undertaken to develop more advanced core

design methodologies which will have the ability to be able to adequately analyze the highly

heterogeneous core designs which are currently in use in commercial power reactors. These

methodological improvements are being pursued with the goal of not sacrificing the

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computational efficiency which core designers require. More specifically, the Pennsylvania

State University (PSU) Nodal Expansion Method (NEM) code is being updated with the

following advanced features:

1) A simplified P3 (SP3) option has been added to NEM. This option will allow for some

transport capability, while not introducing prohibitively high computation times

which a full nodal transport solution, such as the discrete ordinates or spherical

harmonics approximations, would introduce. The SP3 approximation is more accurate

than the diffusion approximation with a considerably lower runtime than a full

transport theory solution. Another advantage of the use of the SP3 equations is that

they can be solved by straightforward extensions of the common nodal diffusion

theory methods with little computation resources overhead. Therefore, there are very

few changes to the basic structure of the NEM code itself in order to implement the

SP3 solution option.

2) An advanced transverse leakage capability based upon a direct calculation of the

intra-nodal flux using analytic basis functions has been added to NEM. This

methodology is an improvement over the existing quadratic leakage approximation

(QLA). This capability uses the existing information from the response matrix and

flux solution (flux moments and surface currents) to solve for the basis function

coefficients, and is therefore fully integratable into existing nodal codes without the

need to introduce more variables, which could introduce prohibitively longer run

times.

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3) A semi-analytical nodal expansion method (SA-NEM) solution option, which is

based upon the within group form of the neutron diffusion equation, has been

completed in NEM. The SA-NEM option uses hyperbolic functions to solve for the

homogenous portion of the neutron diffusion equation and uses polynomial basis

functions for the inhomogeneous portion (source moments and transverse leakage

term) of the neutron diffusion equation.

4) A method for the incorporation of discontinuity factors (DFs) into the SP3 solution

methodology is described. The method for the incorporation of DFs makes use of an

angularly symmetric (with respect to the net current vector) expansion of angular flux

up to order P2 which is inserted into the neutron transport equation. Integration is

performed over the angular space which results in a relationship between the partial

currents and the first and second flux moments. These relationships can then be used

to calculate surface discontinuity factors which can be used in the SP3 solution.

Each of these features has been incorporated into PSU’s NEM nodal code. It is expected that

each feature will result in improved accuracy compared with the standard diffusion theory

utilizing the existing nodal solution methods (polynomial nodal expansion method and the

QLA). Each new feature has been individually tested against benchmarks which are designed

to test the ability of codes to model heterogeneous cores. Namely, the OECD/NEA C5G7

MOX benchmarks [3, 4], which have mini-cores of MOX fuel assemblies mixed with UO2

fuel assemblies, are utilized along with the 2x2 C3 and C5 Benchmarks [44]. The

OECD/NEA and U.S. NRC PWR MOX/UO2 Core Transient Benchmark [42] was also used

for benchmarking. These benchmarks, due to their use of mixed MOX/UO2 fuel as well as

both discrete and integral burnable poisons (for OECD/NEA and U.S. NRC PWR MOX/UO2

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Core Transient Benchmark only), will produce the sharp thermal flux gradients, complicated

leakage shapes, and increased rod to rod streaming that the present work is being pursued in

order to accurately model. These benchmarks are, therefore, ideal test platforms for the newly

implemented methodologies in PSU’s NEM code which have been added in this thesis work.

The unique contributions of this thesis arise primarily from the transverse leakage work,

which provides a new, explicit description of the transverse leakage in terms of the intra-

nodal flux, which is described in terms of analytic basis functions. This work provides a

unique way of handling the transverse leakage term which combines the eigenfunction

description of the intra-nodal flux from the analytical function expansion method with the

more traditional transversely integrated diffusion theory method used in most currently

available nodal codes.

1.3 Thesis Outline

This thesis is divided into seven chapters. The chapters each contain their own self-contained

numbering scheme for the equations, figures, and tables. The chapters are arranged as

follows:

Chapter 1 provides some brief background information on the research topic and a summary

of the research objectives. It provides the reader with a brief discussion of the problems

which the present work is seeking to address and discusses the benefits which will be derived

from the present work.

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Chapter 2 summarizes the results of the literature review performed for this thesis work. The

literature related to the historical background of the SP3 method is reviewed and summarized,

along with the papers which aided in its theoretical basis being more firmly solidified. The

work performed by others attempting to improve the transverse leakage treatment is also

summarized. The literature related to the development of the semi-analytical nodal expansion

method is also included in this chapter. A review of literature related to the equivalence

theory, generalized equivalence theory, super-homogenization theory, and surface

discontinuity factors (DFs) and assembly discontinuity factors (ADFs), including DFs for the

SPN method, is summarized.

Chapter 3 presents the SP3 equations as they are implemented into NEM. The derivation of

the response matrix equations, the nodal expansion method used to solve for the flux

moments, and the Marshak boundary conditions as they are used in NEM is also presented.

Benchmarking of the SP3 solution is also included. Some final remarks and a discussion of

the benchmarking results conclude the chapter.

Chapter 4 presents the SA-NEM solution. The SA-NEM solution procedure is developed

analytically, followed by a discussion of its implementation into NEM. Benchmarking of the

SA-NEM solution is also provided along with a mesh size sensitivity study. Some final

remarks on the SA-NEM and a discussion of the benchmarking results conclude the chapter.

Chapter 5 presents the advanced TL method that has been developed as an improvement to

the QLA. The TL method is based upon an exact representation of the intra-nodal flux in

terms of analytic basis functions. Benchmarking of the new TL treatment using the 2x2 C3

and C5 benchmarks is included in this chapter along with a mesh size sensitivity study. Some

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final remarks on the advanced TL method and a discussion of the benchmarking results

conclude the chapter.

Chapter 6 presents the methodology for the incorporation of DFs into the SP3 solution. This

work, at present, stands to be completed at a future time; however, the theoretical

background, originally presented by Yamamoto and Chao [38], is presented in this chapter.

The coding and future benchmarking of the method is left for future work.

Chapter 7 provides a summary of the work which was performed in this thesis, discusses the

conclusions which were reached regarding each method, and lays out some suggestions for

future work.

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CHAPTER 2

LITERATURE REVIEW

2.1 Introduction

This chapter documents the literature research that has been performed for this thesis.

Literature related to the development of the SP3 methodology, the semi-analytical nodal

expansion method, the equivalence theory and the use of discontinuity factors in nodal codes

and the various transverse leakage treatments which have been used in nodal codes were

reviewed and are discussed in this chapter. Although this chapter is by no means an

exhaustive literature review, the majority of the seminal works for each of the pertinent

topics is presented and summarized in this chapter.

2.2 The SP3 Approximation

The Simplified PN (SPN) approximation to the neutron transport equation was first proposed

by Gelbard in three papers in the early 1960s [5-7]. His aim was to add additional transport

effects into the standard P1 equations, without introducing the complexities and undesired

increase in runtime that a full transport theory solution would entail. The PN equations in slab

geometry, in an optically thick medium dominated by scattering, may be written as a system

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of planar diffusion problems in each direction, which can be solved by Fick’s Law as is done

in the diffusion theory approximation. Gelbard then generalized the equations to 3-D to

create the Simplified PN equations. The approximation that he introduced, according to

Gamino [8-9], is capable of producing “greater than 80%” of the transport correction to

diffusion theory. The salient feature of the SPN approximation is that in a truly planar

problem, which is the applicable domain of Fick’s Law which is used in its derivation, the

SPN approximation is equivalent to the PN equations, as well as the SN+1 equations.

Due to a mathematical prestidigitation the SPN approximation was slow to catch on in the

nuclear community due to the somewhat axiomatic approach taken by Gelbard. In his

derivation, Gelbard replaced the odd Legendre moments of the angular flux with vectors and

the even Legendre moments of the angular flux with scalars. The first order derivatives of

the even moments were then replaced with gradient operators, and the first order derivatives

of the odd order Legendre moments were replaced by divergence operators. These

substitutions were not adequately supported in the works, but rather were axioms upon which

the approximation was based. The theoretical underpinnings of the method were not fully

understood until work done by Larsen and Pomraning [10], and later by Brantley and Larsen

[11].

In their 1995 paper [12], E.W. Larsen et. al. demonstrated using Big O notation for their

asymptotic analysis, that the P1 equations are the leading-order asymptotic approximation of

the transport equation with an error of O(ε3), where ε is a dimensionless parameter. They

further demonstrate, using the same approach, that the SP2 and SP3 equations are asymptotic

approximations to the transport equation with errors of O(ε5) and O(ε7) respectively.

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Therefore, the contribution of Reference 12 to fortifying the theoretical basis of the SPN

equations is to demonstrate that the simplified PN equations are asymptotic corrections to the

P1 equations.

Pomraning [10] further fortified the theoretical basis of the SPN equations by demonstrating

that the SPN equations are a leading order asymptotic limit of the transport equation,

corresponding to nearly planar transport with a phase space which has a highly forward

peaked scattering kernel. He also showed that this asymptotic limit exists for time-dependent

transport in heterogeneous mediums. The most significant conclusion from Pomraning [10],

however, was his variational characterization of the SPN equations. The paper shows that the

SPN equations may be derived from the use of trial functions in the self-adjoint variational

characterization of the even parity transport equation.

Once the SPN equations were validated from a theoretical standpoint, due in large part to the

previously discussed papers, the method obtained more widespread use. One example [13]

reported that the SPN equations produced results which were significantly more accurate than

diffusion theory, which concurs with the asymptotic limit conclusions of Pomraning and

Larsen.

At present the SP3

approximation is being used by Studsvik in its SIMULATE-5 nodal code

[14], NFI’s SCOPE2 code [15], the Forschungszentrum Dresden-Rossendorf Institute of

Safety Research in its DYN3D code [16], and the PARCS code developed by Purdue

University for the US NRC [53].

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2.3 The Semi-Analytical Nodal Expansion Method

The SA-NEM is, in effect, a hybrid method in which the homogenous solution to the

transversely integrated diffusion equation is obtained analytically while the inhomogeneous

solution, namely the source and leakage terms, are expanded in polynomial basis functions as

in the polynomial NEM. It therefore uses the analytical nodal method [17], as well as the

conventional polynomial NEM. The general idea of the method goes back to work performed

by Fischer and Finnemann [43], Wagner [55] and Rajic and Ougouag [56] in the 1980s.

There are several ways in which the SA-NEM has been implemented by various researchers.

In the semi-analytical two-group nodal method [18], the analytical solution is used only for

the thermal group flux, while the fast group flux, which does not have the sharp spatial

gradients like the thermal flux, is analyzed using a fourth-order polynomial.

Kim et. al. [19] introduced an SA-NEM method which involves the analytical solution

consisting of two exponential functions and a fourth-order polynomial. Kim’s method can be

applied to multigroup problems, unlike that of Esser and Smith [18] which is strictly a two-

group method.

One of the more recent works is that by Yamamoto and Tatsumi [20]. This paper describes

the scattered source subtraction (SSS) method which is used to reduce the spatial

discretization error in the SCOPE2 code. In the SSS method, the scattered source is

subtracted from both sides of the diffusion equation, which reduces the spatial variation of

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the source term and thereby reduces the error associated with the flat-source approximation

which is used in SCOPE2.

In the work performed by Fu and Cho [21], a nonlinear semi-analytical method was

developed using the coarse-mesh finite difference (CMFD) scheme. In CMFD, the nodal

equations are used for the local two-node current equations. The global solution is obtained

with the CMFD equations, whose coupling coefficients are obtained by requiring that the

finite difference equations produce the same surface-averaged neutron currents as the nodal

solution.

Han, Joo and Kim [54] developed a two-group CMFD which accelerates the semi-analytical

nodal method kernel. In this method, a quartic expansion of the source terms is used in the

semi-analytical nodal method. In their method, the three transverse integrated equations are

solved simultaneously for the outgoing currents and the node average flux, using the

incoming currents as boundary conditions. The two-group CMFD is used for the global

calculation meaning that the multigroup calculation needs to be performed only at the local

one-node level.

2.4 Transverse Leakage Approximation

The transverse leakage (TL) term arises in nodal codes which use the transverse integration

procedure to integrate the three-dimensional neutron diffusion equation over the directions

which are transverse to the one being analyzed. This results in a simplification of the phase

space from three spatial dimensions (in Cartesian geometry) to one spatial dimension. The

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TL term is the leakage from the directions transverse to the one under consideration. This

term also serves to maintain the coupling between the three one-dimensional equations. An

accurate treatment of this term is of paramount importance if one is to capture strong spatial

variations in pin and assembly power, which could possibly have quite complicated leakage

shapes. A few of the more significant approaches to the treatment of the TL term are

provided below.

In a series of papers in the late 70s, the flat leakage approximation to the transverse leakage

term was proposed [22-23]. The flat leakage approximation, as the name implies, treats the

leakage as being flat across the node. The leakage is simply set equal to its average value,

which can be expressed in terms of side average currents. As one would expect, this approach

is perhaps the simplest of any published approximation to the TL. However, its accuracy,

particularly when sharp flux gradients are present, leaves something to be desired and as a

result this method is no longer used.

If one assumes that the TL shape assumes the same shape as the one-dimensional flux, then

the buckling approach may be used to calculate the TL term [23]. This approach is likewise

quite simplistic. It relates the transverse leakage to the one-dimensional flux via the buckling,

by requiring that the average transverse leakage from the two transverse directions is

conserved.

The TL approximation approach which is the most widely used at present is the quadratic

leakage approximation (QLA). Current nodal codes such as ANC and SIMULATE utilize

this approximation. The QLA was first proposed as a means of calculating the TL by

Bennewitz et. al. [24]. In the QLA, a quadratic polynomial is used to approximate the shape

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of the TL. The three coefficients for the polynomial are obtained by forcing agreement with

the average leakage values of the node of interest, along with its two neighboring nodes. The

obtained shape is applied only to the central node. It has been well documented that the QLA,

while adequate for checkerboard loading patterns, is inadequate near boundaries, in cores

loaded with mixed-oxide fuel, and near strong absorbers such as control rods.

An advanced TL method developed by Prinsloo et al. [28-30], the consistent quadratic

leakage approximation (CQLA), is based upon weighted transverse integration. The weighted

transverse integration is performed by expressing the intra-nodal flux distribution as a multi-

variate expansion, with the solution projected onto Legendre polynomials. In a 3D

benchmark [29], the CQLA led to a decrease in the maximum assembly power error from

1.48% (with QLA) to 0.64%, with an increase in the computational time of 1.7

(computational cost factor). The CQLA method, at present, is one of the more promising

options for the replacement of the QLA. However, the increase in computational time leaves

some further work to do to obtain a more efficient solution.

A few methods which do not use the transverse integration procedure have also been

developed. These methods have not gained widespread popularity but are nonetheless

included. It is one such method, the analytical function expansion method (AFEM) that the

present work expands upon to formulate a more precise TL treatment for a transversely

integrated nodal code. The AFEM, as it appears in the literature, is briefly summarized

below.

The AFEM uses analytic basis functions to directly solve for the intra-nodal flux. Thus, there

is no transverse integration performed. The combinations of trigonometric and hyperbolic

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basis functions are each individual eigensolutions of the diffusion equation at any point in the

node. In their original work, Noh and Cho [25-26] used nine analytic basis functions to

describe the homogeneous intra-nodal flux. The coefficients were expressed in terms of the

corner-point fluxes, side-averaged fluxes, and the node-average flux (nine variables in all).

In a refinement to the AFEM, Woo and Cho [27] introduced transverse gradient basis

functions. These additional basis functions are the original one-dimensional eigensolutions

from the original method [25-26] multiplied by linear functions transverse to the one-

dimensional solution. The additional terms and continuity conditions are satisfied by the

introduction of flux moments. These interface flux moments are defined by the interface-

averaged fluxes and currents, which have been weighted by some independent functions

which are parallel to the direction of the interface.

2.5 Discontinuity Factors

The traditional nodal code methodology relies upon the discretization of the reactor phase

space into individual, materially homogenous regions referred to as nodes. Each node utilizes

a homogenized cross section which represents the flux weighted contribution of each material

in the region. However, flux-weighted constants do not preserve the keff, nodal reaction rates

or nodal surface currents of the respective heterogeneous problem. This is due to the

continuity of homogenized flux interface condition which causes the homogenized currents

to be different than the reference heterogeneous currents. This can be seen in Figure 2.1

below, which also shows qualitatively the relatively smooth behavior of the homogeneous

flux at an assembly boundary and the comparatively more complex behavior of the

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heterogeneous flux at an assembly boundary. Smith [31] also provides an excellent

description and visualization of this effect.

Figure 2.1- Thermal Flux Distribution in Two Neighboring Fuel Assemblies

To remedy this problem, Koebke proposed the Equivalence Theory [32-35]. In the

Equivalence Theory (ET), the homogeneous flux is allowed to be discontinuous across the

nodal boundary. In doing so, the heterogeneous flux distribution can be preserved when the

two-node boundary value problem is solved. In the ET there are equivalence factors which

represent the relationship between the surface homogeneous and surface heterogeneous flux.

When the homogenized two-node problem is solved, the homogeneous flux is made

discontinuous by the ratio of these equivalence factors between the two nodes; in so doing,

the homogeneous flux, while discontinuous, still results in the preservation of the interface

currents as well as the preservation of the surface heterogeneous flux. The novel thing about

the ET is that these equivalence factors can be defined directly from information from the

Heterogeneous Flux

Homogeneous Flux

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reference solution, and can therefore be considered as homogenization parameters. They

provide additional degrees of freedom so that the surface currents and reaction rates are

preserved in the homogeneous problem.

A variation of Koebke’s ET, termed the Generalized Equivalence Theory (GET), takes into

consideration the fact that equivalence factors on either face of a node will be different. In the

GET, Smith [36] introduced assembly discontinuity factors (ADFs) which are the ratio of the

surface-averaged fluxes to the cell-averaged fluxes in the heterogeneous assembly

calculation. In Reference 37, it is demonstrated that the use of ADFs can reduce the assembly

power % error by as much as 5-8%.

The success of nodal methods using multigroup diffusion theory is based largely upon the

ability to, via the use of ADFs, reproduce the reference transport theory solution. By

multiplying the homogeneous surface fluxes by the ADFs and allowing the homogeneous

flux to be discontinuous across the assembly boundary, the reference heterogeneous flux can

be obtained which is the sought after parameter in nodal calculations.

Similar in concept to the ADFs are surface DFs. Surface DFs are used to represent the ratio

of the surface-averaged fluxes to the cell-averaged fluxes. Surface DFs are explicitly defined

on each surface of the assembly, meaning that a traditional square-lattice assembly has four

DFs, hexagonal-lattice assembly has six DFs, and so forth. This allows for a more descriptive

reconstruction of the reference heterogeneous flux distribution than using the single ADFs for

each assembly. Surface dependent DFs should especially be used for assemblies which

border the reflector/baffle region, control rods or strong absorbers, and when a UO2 assembly

borders a MOX assembly on one side and not on the other side. In these environments, the

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single ADF cannot successfully reproduce the heterogeneous flux distribution on each of the

four sides of the assembly due to its inability to be able to capture differing relationships

between the homogeneous and heterogeneous fluxes on the differing assembly faces.

In the SPN approximation to the transport equation, the angular flux cannot be explicitly

reconstructed from the SPN solution. Furthermore, the reference transport theory solution

cannot be explicitly reproduced in the process of solving the SPN equations. As a result,

discontinuity factors cannot be defined to force agreement between the homogenous nodal

solution and the reference heterogeneous problem as is done in the traditional diffusion

theory method.

To remedy this problem, Chao and Yamamoto [38] proposed an SPN formulation that

provides for an explicit angular flux solution such that surface discontinuity factors can be

utilized. The explicit angular flux representation makes it possible to reconstruct the angular

flux from the SPN solution and to extract from a given transport solution the corresponding

SPN solution. The angular flux representation follows from the basic assumption of the SPN

physics model of being one dimensional locally in space and also that the even parity angular

flux is cylindrically symmetric in the angular space with respect to the net current direction.

With this angular flux representation, it is possible to define and calculate surface dependent

discontinuity factors to compensate for the SPN approximation to the transport solution. By

enabling the use of DFs for the SPN approximation, the superiority of the SPN approximation

over the diffusion theory is maintained.

Although the Yamamoto method seems quite promising, it should be pointed out that

experience with the SP3 method has shown that the primary benefit with SP3 is in pin-by-pin

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geometry. Therefore the ADFs or surface DFs described previously would be of limited

practical benefit. In pin-by-pin geometry, the most beneficial method to be used is the super-

homogenization (SPH) method, originally proposed and developed by Hébert [52] and

Kavenoky [57] in order to improve homogenized cross sections and improve the accuracy of

reactor core calculations. The SPH method, just like the ET and GET, seeks to preserve the

reaction rates and surface currents of the heterogeneous region in the homogenized region.

The primary two differences between the ET/GET and the SPH method is that the primary

homogenization region for the SPH is the individual pin cell as opposed to the assembly, and

also that the equivalence parameter is the cross section itself instead of a modification of the

interface condition in the global solution as is the case in the ET and GET.

The SPH method, as mentioned previously, was developed by Hébert [52] and Kavenoky

[57]. In Reference 57, Kavenoky demonstrated that the reaction rates of the homogeneous

environment and the reaction rates of the heterogeneous environment can be forced to agree

with one another if the cross sections are modified by some constant multiplier, which is

termed an SPH factor. The SPH factor is determined in an iterative manner and is directly

applied to the cross sections in order to calculate accurate outgoing currents of the individual

pin cells.

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CHAPTER 3

THE SIMPLIFIED P3 METHODOLOGY

3.1 Introduction

As mentioned in Chapter 2, the Simplified PN (SPN) approximation to the neutron transport

equation was first proposed by Gelbard in three papers in the early 1960s [5-7]. The aim of

the SPN approximation is to introduce additional transport effects into the standard P1

equations, without introducing the complexities and undesired increase in runtime that a full

transport theory solution, such as the discrete ordinates or spherical harmonics methods,

would entail. The SP3 approximation has been demonstrated to be more accurate than the

diffusion approximation when applied to neutron transport problems but with a significantly

less computational burden than either the discrete ordinates or spherical harmonics

approximations.

This chapter provides a discussion of the SP3 equations as they are implemented in NEM. The

chapter is organized as follows: firstly, the SP3 equations are derived; secondly, the Marshak

boundary conditions are developed and discussed; and thirdly, the nodal expansion method

used to solve for the flux moments and to derive the response matrix (RM) equations is

presented. Following this, the method is tested against both 2-D and 3-D versions of the

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OECD/NEA C5G7 MOX benchmarks (both rodded and unrodded benchmarks are

considered) and the OECD/NEA and U.S. NRC PWR MOX/UO2 Core Transient Benchmark

(rodded and unrodded and with the inclusion of SPH factors). In Appendix E an example of

the NEM input and NEMTAB input for the SP3 option in NEM is provided.

3.2 The Simplified PN Approximation

The SPN equations are a simplification of the PN equations, which are themselves

simplifications of the spherical harmonic equations. The assumption of planar transport with

azimuthal symmetry and material isotropy allows one to neglect the azimuthal dependence of

the angular flux and the azimuthal dependence of the differential scattering cross section, and

thereby obtain a planar problem which can be solved by the diffusion approximation via

Fick’s Law. As a result of this assumption, the SPN equations are most accurate for problems

that have strong transport regions in which the solution behaves nearly one-dimensionally

and have weak tangential derivatives at material interfaces. The mathematical development

of the SPN equations follows, followed by a development of the Marshak boundary

conditions, followed by a development of the RM equations which are solved in the NEM

code.

3.2.1 The SPN Equations

The spherical harmonic approximation (PN) is developed by expanding the angular

dependence of the neutron flux and the differential scattering cross section in orthogonal

Legendre polynomials up to order N. The simplification from spherical harmonics to

Legendre polynomials comes from assuming azimuthal symmetry and material isotropy of

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the scattering medium. More specifically, it is assumed that the scattering medium is

invariant under rotation in the phase space R3 and therefore only depends on the cosine of the

scattering angle. This amounts to assuming that the problem under consideration may be

approximated as planar transport with a highly forward-peaked scattering kernel. This allows

the PN equations to be written in one-dimension (for arbitrary node # n) as

)r(S)r()r()r(drd

1l21l)r(

drd

1l2l n

g,lG

1'g

ng'g,l,s

n'g,l

ng,l

ng,t

ng,1l

ng,1l +φ=φΣ+φ

++

+φ+

∑ ∑=

→+− (3.1)

with φl being the flux moments, l = 0, 1, …, N, r is an arbitrary spatial coordinate r: r = x, y,

z and r ∈ Vnode and G = energy group. As is typically assumed in the PN equations, for l = 0

and l = N, φΝ-1 and φN+1 are assumed to be 0, ngt ,Σ is the group g total macroscopic cross

section, nggls →Σ ',, is the lth moment of the macroscopic scattering cross section from group g’

into group g.

The isotropic source is defined as follows,

)(,0 rS ng = )()(1

,1'

',0', rSrk

ngex

G

g

ng

ngf

ng

eff+Σ∑

=

φνχ (3.2)

where χg is the fission spectrum, keff is the neutron multiplication factor, ngf ',Σ is the

macroscopic fission cross section for group g’, n is the node number, and )(, rS ngex is the

isotropic external source (if present).

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The P3 equations are obtained from the PN equations above by inputting l = 0, 1, 2, 3 into

Equation 3.1, which produces four linear differential equations for the four flux moments.

Using the same assumption as Brantley and Larsen [11], that there is no anisotropic group to

group scattering, eliminates all group-to-group scattering terms higher than l = 0. However,

Beckert and Grundmann [16] have reported fairly large errors in pin-by-pin SP3 calculations

in which anisotropic group-to-group scattering was completely neglected. Due to these

conclusions from Beckert and Grudmann [16], in which first-order anisotropic group-to-

group scattering was considered in the DYN3D code and found to provide significantly better

results, only scattering orders higher than l = 1 were eliminated from the P3 equations (for

group-to-group scattering). Therefore, linearly anisotropic group-to-group scattering is

considered. This is shown in Equation 3.3 below.

∑=

→ +φΣ=φΣ+φG

1'g

ng,0

n'g,0

ng'g,0,s

ng,0

ng,t

ng,1 )r(S)r()r()r(

drd

∑=

→ φΣ=φΣ+φ+φG

1'g

n'g,1

ng'g,1,s

ng,1

ng,t

ng2

ng,0 )r()r()r(

drd

32)r(

drd

31 (3.3)

∑=

→ φΣ=φΣ+φ+φG

1'g

n'g,2

ng'g,2,s

ng,2

ng,t

ng3

ng,1 )r()r()r(

drd

53)r(

drd

52

∑=

→ φΣ=φΣ+φG

1'g

n'g,3

ng'g,3,s

ng,3

ng,t

ng2 )r()r()r(

drd

73

where,

0ng'g,l,s =Σ → for g’ ≠ g , l = 2, 3

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To arrive at the simplified P3 equations from Equation 3.3, a few simplifying

assumptions/substitutions must be made:

1) The total macroscopic cross section minus the first scattering moment is replaced with the

transport cross section in the second equation above.

−Σ=Σ )r()r( ng,t

ng,tr ∑

=→Σ

G

g

nggs

1'',1, = −Σ )(, rn

gt )()( ,0,,0 rr ngsg Σµ (3.4)

where )(,0 rgµ is the average cosine of the scattering angle.

This assumption is equivalent to the transport correction of the diffusion theory and is

based upon the assumption:

)r()r( n'g,1

G

1'g

ng'g,1,s

ng,1

G

1'g

n'gg,1,s ΦΣ≈ΦΣ ∑∑

=→

=→

This approximation has been shown to be fairly accurate in diffusive environments with

weak absorption.

2) The even flux moments are assumed to be scalars, while the odd flux moments are

assumed to be vectors. Pomraning discusses this in more detail is his paper [10] and this

assumption was utilized by Brantley and Larsen [11] as well. From a more mathematically

rigorous standpoint, the higher order flux moments are in fact higher order tensors, but

nonetheless the convention is retained. This assumption is made in order to extend the

utility of the SP3 equations from 1-D planar problems to 3-D equations. The results of this

assumption is that the d/dr terms in Equation 3.4 are replaced by a divergence operator for

the odd moments and a gradient operator for the even order moments.

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3) The removal cross section is introduced, which is equal to the total cross section minus the

within group scattering cross section as follows:

n

gg,l,sn

g,tn

g,l,r →Σ−Σ=Σ for l = 0, 2, 3 (3.5)

4) The synthesized flux approximation is implemented for the scalar flux and second flux

moments as follows:

)r(2)r()r( g,2g,0g,0 φ+φ=Φ (3.6)

Using the aforementioned approximations and Equation 3.3, the next step in the derivation is

to solve for the odd flux moments in terms of the spatial derivatives of the even flux

moments. The following is then obtained:

)r(drd

31)r( g,0

g,trg,1 Φ

Σ−=φ (3.7)

)(7

3)( ,2,3,

,3 rdrdr g

grg φφ

Σ−=

These two diffusion equations are then inserted into the first and third equations of

Equation 3.3 above. The approximation made by Brantley and Larsen [11] to extend the

utility of the P3 equations to three dimensions was then applied. Brantley and Larsen

replaced the second derivatives in the PN equations with the Laplacian operator. After

doing this, and performing some simplifications, the SP3 equations as they are

implemented in NEM can be obtained. Equations 3.8 and 3.9 below show the final SP3

equations as they are implemented in NEM.

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)r(S)r(2)r()r(D g,0g,2g,0,remg,0g,0,remg,02

g,0 =φΣ−ΦΣ+Φ∇− (3.8)

(3.9)

with,

)r(S g,0 = [ ]∑ φ−ΦΣ≠=

G

g'g,1'gg,2g,0g'g,0,s )r(2)r( + [ ])(2)( ,2,0

1'', rr

k gg

G

ggf

eff

g φνχ

−ΦΣ∑=

(3.10)

g,trg,0 3

1DΣ

= and gr

gD,3,

,2 359

Σ=

3.2.2 Marshak Boundary Conditions

The only remaining parameter to be determined for the SP3 equations is the boundary

conditions. The exact boundary condition (using the x dimension as an example), as

described by Marchuk and Lebedev [39], is

0),x( right =µφ for 0<µ (3.11)

0),x( left =µφ for 0>µ

Since this exact boundary condition cannot be exactly satisfied based on the continuous finite

expansion of angular flux using the Legendre polynomials, Marshak [40] proposed to use the

same Legendre polynomial expansion for the angular flux as in the PN equations, but

substitute it into Equation 3.11, use only the odd Legendre polynomials as weighting

functions, and integrate from 0 1≤µ≤ (left) and -1 0≤≤ µ (right) separately. The use of

)r(S52)r(

52)r(

54)r(D g,0g,0g,0,remg,2g,0,remg,2,remg,2

2g,2

−=ΦΣ−φ

Σ+Σ+φ∇−

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only the odd Legendre polynomials is predicated on the fact that only the odd Legendre

polynomials represent direction since they attain different values for µ and -µ. This produces

(L+1)/2 boundary conditions for each boundary. In his seminal work on the mathematics of

neutron transport [41], Vladimirov demonstrates that the Marshak boundary conditions are

optimal approximations of the exact boundary conditions. The Marshak boundary conditions

ensure that the exact inward partial current at the boundary is incorporated into the solution.

Furthermore, the Marshak boundary conditions lead to spatial continuity of all of the flux

moments across the interfaces, including the continuity of real scalar flux and net currents.

After performing the aforementioned integrations, the Marshak boundary conditions in terms

of surface fluxes, as used in NEM, are as follows:

)(

1532)(

258

)(58)(

2556

33112

33110

inoutinouts

inoutinouts

jjjj

jjjj

+++=

+++=Φ

φ (3.12)

3.2.3 Response Matrix Equations for SP3 Solution in NEM

After performing transverse integration to Equations 3.8 and 3.9 to obtain two one-

dimensional equations for each of the three nodal directions, the intra-nodal flux shape is

expanded in series within each node using fourth order polynomial basis functions as

follows:

∑=

+Φ=ΦN

nnn rfar

100 )()( (3.13)

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∑=

+=N

nnn rfbr

122 )()( φφ (3.14)

where,

0Φ = nodal volume averaged flux

2φ = nodal volume averaged second flux moment

In the fourth-order approximation, the series is truncated after the first four basis functions

(given below for the x direction), which are given by

( )1

xf xx

=∆

(3.15)

( )2

2134

xf xx

= − ∆ (3.16)

( )3

31 1 12 2 4

x x x x xf xx x x x x

= − + = − ∆ ∆ ∆ ∆ ∆ (3.17)

( )2 4 2

41 1 1 3 120 2 2 10 80

x x x x xf xx x x x x

= − − + = − + ∆ ∆ ∆ ∆ ∆ (3.18)

The first two expansion coefficients for Equations 3.13 and 3.14 can be obtained by

evaluating the intra-nodal flux expansions at the endpoints of the node. The remaining two

expansion coefficients are determined by a weighted residual procedure. The weight

functions used are Equations 3.15 and 3.16 from the basis functions above. This moment

weighting method allows the final two expansion coefficients for Equations 3.13 and 3.14 to

be obtained. In NEM, a moments weighting method, using the same two basis functions, is

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applied to obtain the flux moments and source moments needed to derive the response matrix

equations.

Sufficient information now exists to derive two RM equations for each nodal direction, using

Fick’s Law expressions for the partial currents on the node boundaries. As in traditional

nodal codes, the outgoing partial currents are expressed as a function of incoming partial

currents as well as intra-nodal sources and sinks. Using the x direction as an example, Fick’s

Law takes the following form:

2

0011 )( xxxinr

outr x

dxdDjj ∆

=Φ−= (3.19)

2

0011 )( xxxinl

outl x

dxdDjj ∆

−=Φ+= (3.20)

2

2233 )( xxxinr

outr x

dxdDjj ∆

=−= φ (3.21)

2

2233 )( xxxinl

outl x

dxdDjj ∆

−=+= φ (3.22)

To produce the final current matrices which are solved in NEM, the fluxes from Equations

3.19 – 3.22 are replaced with the polynomial expansions from Equations 3.13 and 3.14. The

differentiation is performed, and the polynomial expansion coefficients are substituted for in

the resulting expression. This results in four RM equations for each nodal direction.

Appendix D provides the actual response matrix equations as used in NEM.

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The RM equations are solved sequentially by NEM using a traditional inner/outer iteration

scheme. The RM equations (Equations 3.19 and 3.20) are solved first, along with the scalar

flux and flux moments. The solutions are then used to solve for Equations 3.21 and 3.22,

which update the currents and scalar fluxes and flux moments. The incoming currents are

determined from the outgoing currents of the neighboring nodes.

3.3 Benchmarking of the SP3 Solution in NEM

The SP3 solution option in NEM was tested against four different, well documented,

benchmarks. These benchmarks, which include both MOX and UO2 fuel assemblies, are

designed so as to challenge the code’s ability to accurately predict pin and assembly power in

a highly heterogeneous core environment. The presence of the MOX fuel assemblies leads to

steep thermal flux gradients between neighboring fuel pins; furthermore, the water reflector

challenges the current diffusion theory method. For three of the benchmarks, control rods are

inserted into the core, providing an even greater challenge to the code by providing a gross

distortion in the flux distribution, which challenges the initial assumptions of the diffusion

theory as mentioned in Chapter 1. These benchmarks will be an ideal setting to demonstrate

the superior performance of the newly implemented SP3 method over the existing polynomial

NEM diffusion theory solution. Each benchmark and its results are presented individually

below followed by a general discussion of the results of the benchmarks.

All of the benchmarking runs were performed using an Intel Core i7-2620M CPU with a 2.7

GHz processor with the exception of the 16 group (with SPH factors) OECD/NEA and U.S.

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NRC PWR MOX/UO2 Core Transient Benchmark run, which was run on a RedHat LINUX

platform.

3.3.1 OECD/NEA 2-D C5G7 MOX Benchmark

The OECD/NEA 2-D C5G7 MOX benchmark was used to benchmark the newly

implemented SP3 solution methodology in the NEM code. The benchmark is described in

detail in Appendix A. A more thorough description of the benchmark may be found in

Reference 3.

The seven-group cross sections provided in Reference 3 for each of the seven materials (four

fuel compositions, guide tube material, fission chamber material, and water moderator

material) were used in this benchmarking. These cross sections are provided in Appendix B.

Unfortunately, P1 scattering cross sections were not available for use in the benchmark; the

second diffusion coefficient and second removal cross section was approximated using the

total cross section. Up-scattering was explicitly considered for groups 5-7. A convergence of

1E-05 was used for keff and 1E-06 was used for point-wise flux. NEM was run in pin-by-pin

geometry.

The NEM results for the SP3 and polynomial diffusion theory methods were compared to the

reference MCNP solution [3]. The value of keff calculated by MCNP is 1.18655 (±0.003 %)

and the value of keff calculated by NEM with the SP3 solution is 1.18699. Therefore, the error

in keff for NEM with the SP3 solution option for this benchmark equals 0.037 %. The value of

keff calculated by NEM with the diffusion theory solution is 1.18707, which corresponds to an

error of 0.044%.

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Table 3.1 below shows comparisons of the pin powers calculated by NEM with both the

diffusion theory and SP3 solutions and the reference MCNP solution. The results of the

benchmark for the AVG (average on module pin power percent distinction), RMS (root mean

square of the percent distinction) and MRE (mean relative pin power percent error) were

calculated by the following formulas from Reference 3.

AVG = ∑=

N

nne

N 1

1

RMS = ∑=

N

nne

N 1

21

MRE = avg

n

N

nn

Np

pe∑=1

where,

N = total number of pins

n = pin #

en = pin power error for pin n

pavg = average pin power

pn = pin power for pin n

Table 3.2 below shows comparisons of the assembly powers calculated by NEM with both

the diffusion theory and SP3 solutions and the reference MCNP solutions. The terminology

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from Reference 3 was retained to refer to the fuel assemblies. “Inner UO2” refers to the upper

left assembly in Figure A.1 with reflective boundary conditions on its left and top faces;

“Outer UO2” refers to the bottom right assembly in Figure A.1 which borders the water

reflector on its right and bottom faces; and MOX refers to the MOX fuel assemblies.

Table 3.1- Calculated Pin Powers from NEM Compared with MCNP Reference Solution (SP3 2-D C5G7 Benchmark)

Evaluated Parameter MCNP (Reference) NEM-SP3 NEM-DIFF CPU Time (seconds) - 1768 893 Maximum Pin power 2.498 2.573 2.574 Minimum Pin power 0.232 0.233 0.233

AVG (in percent) - 2.204 2.227 RMS (in percent) - 2.627 2.653 MRE (in percent) - 2.021 2.041

Number of pins within 99.9% confidence interval of MCNP

-

102

97

Average pin power in inner UO2 assembly (% error)

1.867

1.894

(1.472)

1.894

(1.475) Average pin power in outer UO2 assembly

(% error)

0.529

0.524 (-1.066)

0.524

(-1.111) Average pin power in FA-MOX

(% error)

0.802

0.791 (-1.362)

0.791

(-1.349)

Table 3.2- Calculated Assembly Powers from NEM Compared with MCNP Reference Solution

(SP3 2-D C5G7 Benchmark)

Fuel Assembly MCNP (Reference) NEM-SP3 NEM-DIFF Inner UO2 492.8 ±0.10 500.1 500.1

MOX 211.7 ±0.18 208.8 208.9 Outer UO2 139.8 ±0.20 138.3 138.2

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As can be observed from the results of the benchmarking, both codes calculate keff to within

less than 0.05% of the reference value. Both the SP3 and the diffusion theory solutions for keff

are within the range of values for the other codes submitted to the C5G7 benchmark, as can

be seen in Table 17 of Reference 3.

With regard to the pin powers, the newly implemented SP3 method resulted in small

improvements in the AVG, RMS, and MRE values. As expected, the most pronounced

improvements in the pin power agreement between the NEM code and the reference MCNP

solution was near the material boundaries between the UO2 and MOX fuel assemblies, where

the ratio of 2φ / 0Φ is greatest. The calculated pin powers for both the diffusion theory and

SP3 solutions are approximately equal away from material boundaries. Appendix C provides

the individual pin power and the individual pin power error results for both the SP3 and

diffusion theory solutions. The individual pin powers can be seen for both the P-NEM

diffusion theory and SP3 solutions in Figures 3.1 and 3.2 below. The similarity between the

two solutions is readily apparent from the figures.

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Figure 3.1- P-NEM Diffusion Theory Pin Power Distribution for 2-D C5G7 MOX Benchmark

1

5

9 13 17

21 25

29 33

0.10 0.30 0.50 0.70 0.90 1.10 1.30 1.50 1.70 1.90 2.10 2.30 2.50

1

8

15

22

29

2.50-2.60 2.30-2.50 2.10-2.30 1.90-2.10 1.70-1.90 1.50-1.70 1.30-1.50 1.10-1.30 0.90-1.10 0.70-0.90 0.50-0.70 0.30-0.50 0.10-0.30

Water Moderator

Pin Power

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Figure 3.2- SP3 Pin Power Distribution for 2-D C5G7 MOX Benchmark

1

5

9 13 17

21 25

29 33

0.10 0.30 0.50 0.70 0.90 1.10 1.30 1.50 1.70 1.90 2.10 2.30 2.50

1

8

15

22

29

2.50-2.60 2.30-2.50 2.10-2.30 1.90-2.10 1.70-1.90 1.50-1.70 1.30-1.50 1.10-1.30 0.90-1.10 0.70-0.90 0.50-0.70 0.30-0.50 0.10-0.30

Water Moderator

Pin Power

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The reason that the improvement afforded by the SP3 solution was not larger is likely due to

the absence of P1 scattering cross sections in the benchmark. The approximation of the

second diffusion coefficient and the second removal cross section using the total cross

section reduced the overall transport contribution, causing the solution to be more similar to

the diffusion theory. This effect is discussed in more detail in Section 3.4.

3.3.2 OECD/NEA 3-D C5G7 MOX Benchmarks (Rodded)

The previous benchmark demonstrated a small improvement in the prediction of pin powers

in 2-D geometry. Reference 4 provides two more configurations which provide even greater

heterogeneity than the previous case by the insertion of control rods into the 2x2 cores,

causing sharp flux distortions in the surrounding areas. As previously mentioned, the

diffusion approximation is not valid in the presence of strong absorbers such as control rods.

Therefore, the rodded benchmarks in Reference 4 provide a measure of the improvement in

the transport approximation afforded by the SP3 equations. The rodded benchmarks should

provide an ideal setting to demonstrate the improvement afforded by the SP3 solution in a

highly absorbing environment.

The rodded benchmarks are identical to the 3-dimensional 2x2 core in Reference 3 with three

exceptions:

1) The height of the geometry is reduced from 192.78 cm to 64.26 cm.

2) The control rod guide tubes and fission chamber are defined in the upper axial

reflector.

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3) A control rod macroscopic cross section definition is introduced and is used to

replace the control rod guide tube composition in certain parts of the reactor.

The two benchmarks from Reference 4 which were modelled, referred to as unrodded and

Rodded A, vary only by the amount and location of control rod insertion. The unrodded

configuration, detailed by Figure 3.3, has control rod clusters (one cluster for each assembly)

inserted into the upper axial water reflector as indicated by the shading in Figure 3.3. Figure

3.4 shows a slice in the radial direction through the upper axial reflector and should more

clearly show the layout of the control rod clusters and fission chamber in the axial reflector

region. It should be noted that all four assemblies have control rods present in the upper

water reflector region.

Figure 3.3- Geometry for the Unrodded Benchmark Case

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Figure 3.4- Geometry Configuration for the Upper Axial Water Reflector

The axial length of the assemblies were partitioned into three axial nodes each 14.28 cm in

height; the water reflector, which is 21.42 cm thick, was modeled as two axial nodes. The

seven-group cross sections provided in Reference 4 for each of the eight materials (four fuel

compositions, guide tube material, fission chamber material, control rod material and water

moderator material) were used in this benchmarking. These cross sections are provided for

each material in Appendix B. As in the 2-D version of the benchmark, P1 scattering cross

sections were not available and so the second diffusion coefficients and second removal cross

sections were approximated using the total cross section. Up-scattering was explicitly

considered for groups 5-7. A convergence of 1E-05 was used for keff and 1E-06 was used for

point-wise flux. NEM was run in pin-by-pin geometry.

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For the unrodded benchmark, the value of keff calculated by MCNP is 1.143080 (±0.0026 %)

and the value of keff calculated by NEM with the SP3 solution is 1.14304. Therefore, the error

of keff for NEM with the SP3 solution option for the unrodded benchmark equals -0.0020 %.

The value of keff calculated by NEM with the diffusion theory solution is 1.14310, which

corresponds to an error of 0.0017 %.

Table 3.3 below shows comparisons of pin powers for each of the three axial slices calculated

by NEM with both the diffusion theory and SP3 solutions and the reference MCNP solution.

Note that Slice #3 is the top of the core which is closest to the reflector and control rods.

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Table 3.3- Calculated Pin Powers from NEM Compared with MCNP Reference Solution (C5G7 unrodded Benchmark)

Evaluated Parameter MCNP

(Reference) NEM-SP3 NEM-DIFF

CPU Time (seconds) - 6419 3188 Slice #1

AVG (in percent) - 2.150 2.137 RMS (in percent) - 2.612 2.599 MRE (in percent) - 0.878 0.900

Assembly power in inner UO2 assembly (% error)

219.04

222.09 (1.394)

222.55 (1.601)

Assembly power in outer UO2 assembly (% error)

62.12

61.66 (-0.731)

61.72 (-0.647)

Assembly power in FA-MOX (% error)

94.53

93.22 (-1.387)

93.47 (-1.127)

Slice #2 AVG (in percent) - 2.210 2.271 RMS (in percent) - 2.680 2.739 MRE (in percent) - 0.721 0.731

Assembly power in inner UO2 assembly (% error)

174.24

176.80 (1.472)

176.68 (1.399)

Assembly power in outer UO2 assembly (% error)

49.45

49.13 (-0.650)

49.01 (-0.893)

Assembly power in FA-MOX (% error)

75.25

74.14 (-1.477)

74.10 (-1.531)

Slice #3 AVG (in percent) - 2.053 2.267 RMS (in percent) - 2.510 2.699 MRE (in percent) - 0.377 0.404

Assembly power in inner UO2 assembly (% error)

97.93 99.19 (1.280)

99.08 (1.168)

Assembly power in outer UO2 assembly (% error)

27.82

27.60 (-0.802)

27.44 (-1.361)

Assembly power in FA-MOX (% error)

42.92

42.41 (-1.198)

42.20 (-1.669)

Overall AVG (in percent) - 2.145 2.198 RMS (in percent) - 2.609 2.653 MRE (in percent) - 1.972 2.025

Assembly power in inner UO2 assembly (% error)

491.21

498.08 (1.399)

498.30 (1.443)

Assembly power in outer UO2 assembly (% error)

139.39 138.39 (-0.716)

138.16 (-0.877)

Assembly power in FA-MOX (% error)

212.70

209.76 (-1.381)

209.77 (-1.379)

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For the Rodded A benchmark, a control rod is inserted into the UO2 assembly on the reflected

boundary. As with the unrodded case, the control rods are still inserted into the upper axial

water reflector. Figure 3.5 below shows the control rod insertion for the Rodded A

configuration. The same seven group cross sections as were used for the unrodded case were

used in the Rodded A case. The same axial node structure as was used in the unrodded case

(three axial nodes for the fuel region and two axial nodes for the upper reflector region) was

again used in the Rodded A benchmark case.

For the Rodded A benchmark, the value of keff calculated by MCNP is 1.128060 (±0.0027 %)

and the value of keff calculated by NEM with the SP3 solution is 1.127470. Therefore, the

error of keff for NEM with the SP3 solution option for the Rodded A benchmark equals -

0.0523 %. The value of keff calculated by NEM with the diffusion theory solution is

1.127390, which corresponds to an error of -0.0594 %.

Table 3.4 below shows comparisons of pin powers for each of the three axial slices calculated

by NEM with both the diffusion theory and SP3 solutions along with the reference MCNP

solution.

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Figure 3.5- Geometry for the Rodded A Benchmark Case

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Table 3.4- Calculated Pin Powers from NEM Compared with MCNP Reference Solution (C5G7 Rodded A Benchmark)

Evaluated Parameter MCNP

(Reference) NEM-SP3 NEM-DIFF

CPU Time (Seconds) - 6205 3066 Slice #1

AVG (in percent) - 2.106 2.141 RMS (in percent) - 2.615 2.668 MRE (in percent) - 1.108 1.149

Assembly power in inner UO2 assembly (% error)

237.41

243.59 (2.603)

244.05 (2.799)

Assembly power in outer UO2 assembly (% error)

69.80 70.03 (0.326)

69.95 (0.211)

Assembly power in FA-MOX (% error)

104.48 104.12 (-0.341)

104.34 (-0.136)

Slice #2 AVG (in percent) - 2.264 2.332 RMS (in percent) - 2.770 2.802 MRE (in percent) - 0.823 0.862

Assembly power in inner UO2 assembly (% error)

167.51

171.44 (2.342)

171.78 (2.550)

Assembly power in outer UO2 assembly (% error)

53.39

53.41 (0.049)

53.19 (-0.375)

Assembly power in FA-MOX (% error)

78.01

77.18 (-1.064)

77.19 (-1.042)

Slice #3 AVG (in percent) - 4.742 5.162 RMS (in percent) - 5.820 6.128 MRE (in percent) - 0.831 0.897

Assembly power in inner UO2 assembly (% error)

56.26

50.89 (-9.544)

50.68 (-9.915)

Assembly power in outer UO2 assembly (% error)

28.21

27.95 (-0.895)

27.69 (-1.821)

Assembly power in FA-MOX (% error)

39.23

38.05 (-3.012)

37.80 (-3.649)

Overall AVG (in percent) - 1.946 1.993 RMS (in percent) - 2.489 2.495 MRE (in percent) - 1.757 1.830

Assembly power in inner UO2 assembly (% error)

461.18

465.91 (1.026)

466.52 (1.158)

Assembly power in outer UO2 assembly (% error)

151.39

151.40 (0.001)

150.83 (-0.374)

Assembly power in FA-MOX (% error)

221.71

219.34 (-1.068)

219.33 (-1.076)

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For both the unrodded and the Rodded A configurations, the SP3 solution generally produced

a small improvement in the pin and assembly power results in comparison with the diffusion

theory solution with polynomial NEM. The degree of improvement in the predicted pin and

assembly powers, for both the unrodded and the Rodded A configurations, was greatest in the

areas nearest to the control rods (node 3). As was discussed previously, this is as would be

expected due to the assumptions used in the derivation of the diffusion theory being

invalidated in the vicinity of strong absorbers such as control rods. Furthermore, node 3 is

closest to the water reflector (nodes 4 and 5). Therefore, as discussed for the 2-D benchmark,

the ratio of 2φ / 0Φ is greatest in the vicinity of the reflector, and thus the improvement

afforded by the transport effect introduced by the SP3 approximation is demonstrated near the

reflector.

As previously mentioned, the improvement with the SP3 solution for the unrodded and

Rodded A configurations was relatively small. The degree of improvement with the SP3

solution, in comparison to the diffusion theory solution, was likely inhibited by the absence

of higher order scattering cross sections in the benchmark. The use of the total cross section

as an approximation for the second diffusion coefficient and the second removal cross section

reduces the contribution of the second and third flux moments, which are responsible for the

improvement afforded by the SP3 method. Further benchmarking is necessary to demonstrate

the improvement afforded by the SP3 solution in NEM. A benchmark with higher order

scattering cross sections, which allow the full improvement of the SP3 solution to be realized,

is required in order to adequately evaluate the new method.

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3.3.3 PWR MOX/UO2 Core Transient Benchmark

The OECD/NEA and U.S. NRC PWR MOX/UO2 Core Transient Benchmark [42], which

was designed specifically to assess core simulators, was used to benchmark the SP3 solution

methodology in NEM. The core used in the simulation is a four-loop Westinghouse PWR

core. A quarter-core loading pattern is specified due to core symmetry. This quarter-core

configuration has uniform fuel composition in the axial direction and consists of 49 MOX

fuel assemblies with 4.0 and 4.3 wt% Pu-fissile and 144 UO2 assemblies with enrichments of

4.2 and 4.5 wt%. The burnup of the assemblies is different, as shown in Figure 3.6 below, but

uniform in the axial direction. The MOX fuel assemblies contain wet annular burnable

absorbers (WABAs) while the UO2 assemblies contain integral fuel burnable absorbers

(IFBA). The core is surrounded by a single row of reflector assemblies of the same width as

the fuel assembly pitch. Each reflector assembly contains a 2.52-cm thick baffle and has

fixed moderator at the same condition as the core inlet [42]. As shown in Figure 3.6 below, a

quarter of the core was modeled with reflective boundary conditions on all faces (axial and

radial), making the problem a 2-D benchmark. Figure 3.7 below shows the pin arrangements

of the UO2 and MOX fuel assemblies, which consist of a 17×17 array of pins. A more

thorough description of the benchmark as well as some published results may be found in

Reference 42.

Pin-wise calculations were performed with eight energy groups (for diffusion theory runs and

for SP3 runs without SPH factors) and 16 energy groups (for SP3 run with SPH factors). SPH

(super-homogenization) factors were initially introduced by Hébert [52] in order to improve

homogenized cross sections and improve the accuracy of reactor core calculations. The SPH

factors are generated in an iterative way to preserve the reaction rates in the pin cells, using

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the homogenized cross sections of the heterogeneous lattice calculation. The SPH factors

used in this run were obtained from GRS; later they were utilized in the DYN3D SP3 code

calculations [50]. The SPH factors are incorporated into the 16-group cross section library.

No pin discontinuity factors or SPH factors are utilized in the eight energy group

calculations. The eight group pin-by-pin cross sections used in this benchmarking were

generated by HELIOS. HELIOS yields only PN data up to the order N = 1. It is therefore

assumed that scattering moments higher than l=1 are equal to zero. Up-scattering was

explicitly considered for groups 7-8 for the no-SPH cases and groups 13-16 for the SPH case

which uses 16 energy groups. A convergence of 1E-05 was used for keff and 1E-05 was used

for point-wise flux. Benchmark cases are performed for an all rods out (ARO) configuration

and an all rods in (ARI) configuration. Unfortunately, SPH factors are only available for the

ARO configuration. The results for each of these configurations are presented below.

The reference solution for this benchmark was generated with the DeCART code, which uses

the method of characteristics. The heterogeneous structure of the individual pin cells was

modeled in the reference solution. A 47-group cross section library generated with HELIOS

was used in the generation of the reference solution. The results for the reference solution

were taken from Reference 42.

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Figure 3.6- Quarter-Core Configuration of MOX/UO2 Core Transient Benchmark

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Figure 3.7- Pin Layout of UO2 and MOX Fuel Assemblies

Three different NEM calculations were performed for the ARO benchmark: a diffusion

theory calculation in eight-groups using a polynomial NEM, an SP3 solution in eight-groups

with no SPH factors and an SP3 solution in 16-groups using SPH factors. As stated

previously, a quarter of the core was modeled with reflective boundary conditions on all

faces (axial and radial), making the problem a 2-D benchmark. Table 3.5 shows the results of

the eigenvalues of the three NEM calculations compared with the reference solution of the

multigroup transport code DeCART as well as the mean deviations of the fuel-assembly

powers from the reference DeCART solution. For additional comparison, MCNP results from

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Reference 42 are also provided as well as pin-by-pin results from the diffusion theory code

COBAYA3 and the SP3 code DYNSUB from Reference 58. The measurements of the error

in the fuel assembly powers use the two parameters from Reference 42, the PWE and the

EWE. PWE is the mean weighted deviation and EWE is the mean quadratic deviation. The

calculation for each of these parameters is shown below. Note that PWE and EWE can also

be calculated for the pin powers. Figure 3.8 below shows the deviations in the assembly

power results for NEM with the three solution options compared to the reference DeCART

solution.

PWE = 100P

Pe

i

refi

refi

ii

•∑

EWE = 100e

e

ii

i

2i

•∑

where,

ei = 100P

PPrefi

refii •

Pi = NEM assembly (or pin) power

refiP = DeCART reference assembly (or pin) power

The CPU requirements for the three NEM runs for the ARO benchmark are 12 hours 14

minutes for the diffusion theory run, 25 hours 46 minutes for the SP3 with no SPH factors run

and 74 hours 16 minutes for the SP3 with SPH factors run.

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Table 3.5- Eigenvalue keff and Assembly Power Deviation for ARO Configuration

Code / Method keff PWE (%) EWE (%)

MCNP / Monte Carlo 1.05699 - - DeCART / Method of

Characteristics (Reference Solution)

1.05852 - -

NEM / Diffusion Theory 1.06128 0.78 1.31 NEM / SP3 No SPH 1.06024 0.49 0.64

NEM / SP3 with SPH 1.06011 0.41 0.49 COBAYA3 / Diffusion Theory 1.0636 1.47 1.81

DYNSUB / SP3 1.05888 0.70 1.09

Figure 3.8- Percent Deviations in Assembly Powers for Three Solution Methodologies vs. Reference DeCART Solution

for ARO Configuration

0.754 0.281-0.58 -2.28-0.44 0.50-0.73 -0.64

0.904 1.067 0.5851.65 0.62 -3.061.37 0.46 0.360.73 0.48 -0.32

1.076 1.308 1.143 0.892 0.3410.99 0.48 0.38 -0.47 0.730.46 0.34 0.24 -0.54 0.230.32 0.34 -0.45 -0.83 0.38

1.325 1.446 1.247 1.114 0.991 0.393-0.60 0.19 0.63 -0.66 0.66 1.12-0.50 0.13 0.52 -0.51 0.53 0.560.29 0.26 0.61 -0.62 -0.46 -0.43

1.563 1.245 1.277 1.349 0.918 0.978 0.491-1.38 -0.88 -0.24 0.30 0.77 -0.42 0.57-1.07 -0.67 -0.11 0.12 0.63 -0.20 0.26-0.31 -0.33 0.29 0.30 0.32 -0.52 -0.39

1.374 1.735 1.418 1.525 1.035 1.032 0.997 0.413 DeCART Reference RPD-1.30 -1.44 -1.16 0.98 0.34 0.35 0.58 1.26 % Difference Diffusion Theory-0.57 -0.92 -1.04 -0.43 0.43 -0.06 0.42 0.85 % Difference SP3 No SPH0.33 0.28 -0.27 0.30 0.43 0.36 0.43 -0.68 % Difference SP3 with SPH

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The pin powers were also examined for comparison with the reference solution. The six fuel

assemblies along the diagonal quadrant of the core were examined. Referring to Figure 3.6

for the numbering scheme, assemblies A1, B2, C3, D4, E5, and F6 were analyzed for pin

power comparison. The PWE for the pin powers were calculated using the previously

described equation. The results are presented in Table 3.6 below. The DYNSUB results from

Reference 58 are also included. A representative assembly (assembly A1) with the pin power

% error for the three analytical methods is shown in Figure 3.9 below.

Table 3.6- Pin Power PWE (%) for ARO Configuration

Code / Method A1 B2 C3 D4 E5 F6

NEM / Diffusion Theory

2.08 1.12 1.29 2.02 2.31 2.19

NEM / SP3 No SPH 1.59 0.79 1.12 1.44 1.52 1.49 NEM / SP3 with SPH 1.48

0.72 1.03 1.41 1.48 1.42

DYNSUB / SP3 1.66 0.71 0.97 1.58 1.81 1.72

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Figure 3.9- Deviation of Pin Powers for Assembly A1 at ARO Conditions

1.28 1.30 1.31 1.33 1.34 1.34 1.34 1.34 1.34 1.34 1.34 1.34 1.34 1.33 1.31 1.30 1.28-2.28 0.92 0.83 0.83 0.99 0.55 1.08 1.10 0.84 1.10 1.08 0.55 0.99 0.83 0.83 0.92 -2.28-1.96 0.78 0.71 0.71 0.92 0.58 1.03 1.04 0.62 1.04 1.03 0.58 0.92 0.71 0.71 0.78 -1.96-0.72 1.80 1.76 1.76 1.89 1.50 1.78 1.90 1.55 1.90 1.78 1.49 1.89 1.76 1.76 1.80 -0.721.30 1.30 1.32 1.34 1.36 1.40 1.36 1.36 1.40 1.36 1.36 1.40 1.36 1.34 1.32 1.30 1.300.84 0.69 0.70 1.06 1.32 -2.72 2.11 1.92 -2.68 1.92 2.11 -2.72 1.32 1.06 0.70 0.69 0.840.70 0.42 0.43 0.38 1.16 -2.30 1.01 0.92 -2.40 0.92 1.01 -2.30 1.16 0.38 0.43 0.42 0.701.78 1.67 1.67 1.54 2.09 -0.68 1.91 2.01 1.96 2.01 1.91 -0.68 2.09 1.54 1.68 1.67 1.781.21 1.32 1.35 1.42 1.44 1.42 1.41 1.41 1.42 1.44 1.42 1.35 1.32 1.211.88 1.87 1.88 -2.74 -1.68 -2.89 -2.88 -2.88 -2.89 -1.68 -2.74 1.88 1.87 1.880.84 0.81 0.83 -2.71 -1.72 -2.63 -2.62 -2.62 -2.63 -1.72 -2.71 0.83 0.81 0.841.76 1.75 1.76 -1.19 -0.74 -1.41 -1.42 -1.42 -1.41 -0.74 -1.19 1.76 1.75 1.761.33 1.34 1.42 1.46 1.44 1.38 1.37 1.41 1.37 1.38 1.44 1.46 1.42 1.34 1.332.12 1.80 -2.53 -2.04 -2.98 1.72 1.43 -2.78 1.43 1.72 -2.98 -2.04 -2.53 1.80 2.120.84 0.68 -2.38 -1.92 -2.74 0.98 1.00 -2.66 1.00 0.97 -2.74 -1.92 -2.38 0.68 0.841.80 1.37 -1.11 -0.68 -1.55 2.10 2.08 -1.62 2.08 2.10 -1.55 -0.68 -1.11 1.37 1.801.34 1.36 1.44 1.46 1.42 1.43 1.38 1.37 1.41 1.37 1.38 1.43 1.42 1.46 1.44 1.36 1.341.22 2.06 -1.97 -1.94 -1.70 -2.80 2.12 2.14 -1.97 2.14 2.12 -2.80 -1.70 -1.97 -1.98 2.06 1.220.84 0.88 -1.76 -1.70 1.42 -2.56 0.80 1.08 -2.55 1.08 0.80 -2.56 1.42 -1.70 -1.76 0.88 0.841.88 2.04 -0.47 -0.53 2.09 -1.24 1.99 2.06 -1.18 2.06 1.99 -1.24 2.09 -0.53 -0.47 2.04 1.881.34 1.40 1.44 1.43 1.41 1.40 1.40 1.41 1.43 1.44 1.40 1.341.68 -2.93 -2.88 -2.99 -2.99 -2.92 -2.92 -2.99 -2.99 -2.88 -2.93 1.680.32 -2.11 -2.78 -2.59 -2.59 -2.60 -2.60 -2.59 -2.59 -2.78 -2.11 0.321.56 -1.01 -1.33 -1.37 -1.47 -1.51 -1.51 -1.47 -1.37 -1.33 -1.01 1.561.34 1.36 1.42 1.38 1.38 1.41 1.37 1.36 1.40 1.36 1.37 1.41 1.38 1.38 1.42 1.36 1.341.71 1.72 -1.78 1.18 1.18 -2.97 1.52 1.77 -2.44 1.77 1.52 -2.97 1.18 1.18 -1.78 1.72 1.710.98 1.18 -2.58 1.02 1.00 -2.61 1.01 0.99 -2.70 0.99 1.01 -2.61 1.00 1.02 -2.58 1.18 0.981.89 2.11 -1.31 1.98 2.04 -1.29 2.01 1.96 -1.57 1.96 2.01 -1.29 2.04 1.98 -1.31 2.11 1.891.34 1.36 1.41 1.37 1.37 1.40 1.36 1.36 1.40 1.36 1.36 1.40 1.37 1.37 1.41 1.36 1.341.78 1.02 1.66 1.47 1.48 -2.88 1.08 1.07 -2.66 1.07 1.08 -2.88 1.48 1.47 1.66 1.02 1.781.03 0.91 -2.54 1.06 1.08 -2.66 0.99 0.98 -2.38 0.98 0.99 -2.66 1.08 1.06 -2.54 0.91 1.031.89 1.90 -1.39 2.05 2.00 -1.22 1.85 1.80 -1.33 1.80 1.85 -1.22 2.00 2.05 -1.39 1.90 1.891.34 1.40 1.41 1.41 1.40 1.40 1.40 1.40 1.41 1.41 1.40 1.341.68 -2.53 -2.88 -2.91 -2.91 -2.92 -2.92 -2.91 -2.91 -2.88 -2.53 1.680.42 -2.33 -2.40 -2.31 -2.40 -2.39 -2.39 -2.40 -2.31 -2.40 -2.33 0.421.51 -1.02 -1.30 -1.32 -1.30 -1.40 -1.40 -1.30 -1.32 -1.30 -1.02 1.511.34 1.36 1.41 1.37 1.37 1.40 1.36 1.36 1.40 1.36 1.36 1.40 1.37 1.37 1.41 1.36 1.341.78 1.02 1.66 1.47 1.48 -2.88 1.08 1.07 -2.66 1.07 1.08 -2.88 1.48 1.47 1.66 1.02 1.781.03 0.91 -2.54 1.06 1.08 -2.66 0.99 0.98 -2.38 0.98 0.99 -2.66 1.08 1.06 -2.54 0.91 1.031.89 1.90 -1.39 2.05 2.00 -1.22 1.85 1.80 -1.33 1.80 1.85 -1.22 2.00 2.05 -1.39 1.90 1.891.34 1.36 1.42 1.38 1.38 1.41 1.37 1.36 1.40 1.36 1.37 1.41 1.38 1.38 1.42 1.36 1.341.71 1.72 -1.78 1.18 1.18 -2.97 1.52 1.77 -2.44 1.77 1.52 -2.97 1.18 1.18 -1.78 1.72 1.710.98 1.18 -2.58 1.02 1.00 -2.61 1.01 0.99 -2.70 0.99 1.01 -2.61 1.00 1.02 -2.58 1.18 0.981.89 2.11 -1.31 1.98 2.04 -1.29 2.01 1.96 -1.57 1.96 2.01 -1.29 2.04 1.98 -1.31 2.11 1.891.34 1.40 1.44 1.43 1.41 1.40 1.40 1.41 1.43 1.44 1.40 1.341.68 -2.93 -2.88 -2.99 -2.99 -2.70 -2.70 -2.99 -2.99 -2.88 -2.93 1.680.32 -2.11 -2.78 -2.59 -2.59 -2.60 -2.60 -2.59 -2.59 -2.78 -2.11 0.321.56 -1.01 -1.33 -1.37 -1.47 -1.51 -1.51 -1.47 -1.37 -1.33 -1.01 1.561.34 1.36 1.44 1.46 1.42 1.43 1.38 1.37 1.41 1.37 1.38 1.43 1.42 1.46 1.44 1.36 1.341.22 2.06 -1.97 -1.94 -1.70 -2.80 2.12 2.14 -1.97 2.14 2.12 -2.80 -1.70 -1.97 -1.98 2.06 1.220.84 0.88 -1.76 -1.70 1.42 -2.56 0.80 1.08 -2.55 1.08 0.80 -2.56 1.42 -1.70 -1.76 0.88 0.841.88 2.04 -0.47 -0.53 2.09 -1.24 1.99 2.06 -1.18 2.06 1.99 -1.24 2.09 -0.53 -0.47 2.04 1.881.33 1.34 1.42 1.46 1.44 1.38 1.37 1.41 1.37 1.38 1.44 1.46 1.42 1.34 1.332.12 1.80 -2.53 -2.04 -2.98 1.72 1.43 -2.78 1.43 1.72 -2.98 -2.04 -2.53 1.80 2.120.84 0.68 -2.38 -1.92 -2.74 0.98 1.00 -2.66 1.00 0.97 -2.74 -1.92 -2.38 0.68 0.841.80 1.37 -1.11 -0.68 -1.55 2.10 2.08 -1.62 2.08 2.10 -1.55 -0.68 -1.11 1.37 1.801.21 1.32 1.35 1.42 1.44 1.42 1.41 1.41 1.42 1.44 1.42 1.35 1.32 1.211.88 1.87 1.88 -2.74 -1.68 -2.89 -2.88 -2.88 -2.89 -1.68 -2.74 1.88 1.87 1.880.84 0.81 0.83 -2.71 -1.72 -2.63 -2.62 -2.62 -2.63 -1.72 -2.71 0.83 0.81 0.841.76 1.75 1.76 -1.19 -0.74 -1.41 -1.42 -1.42 -1.41 -0.74 -1.19 1.76 1.75 1.761.30 1.30 1.32 1.34 1.36 1.40 1.36 1.36 1.40 1.36 1.36 1.40 1.36 1.34 1.32 1.30 1.300.84 0.69 0.70 1.06 1.32 -2.72 2.11 1.92 -2.68 1.92 2.11 -2.72 1.32 1.06 0.70 0.69 0.840.70 0.42 0.43 0.38 1.16 -2.30 1.01 0.92 -2.40 0.92 1.01 -2.30 1.16 0.38 0.43 0.42 0.701.78 1.67 1.67 1.54 2.09 -0.68 1.91 2.01 1.96 2.01 1.91 -0.68 2.09 1.54 1.68 1.67 1.781.28 1.30 1.31 1.33 1.34 1.34 1.34 1.34 1.34 1.34 1.34 1.34 1.34 1.33 1.31 1.30 1.28 DeCART Reference Pin Power-2.28 0.92 0.83 0.83 0.99 0.55 1.08 1.10 0.84 1.10 1.08 0.55 0.99 0.83 0.83 0.92 -2.28 % Difference Diffusion Theory-1.96 0.78 0.71 0.71 0.92 0.58 1.03 1.04 0.62 1.04 1.03 0.58 0.92 0.71 0.71 0.78 -1.96 % Difference SP3 No SPH-0.72 1.8 1.76 1.76 1.89 1.5 1.78 1.9 1.55 1.9 1.78 1.49 1.72 1.66 1.65 1.8 -0.72 % Difference SP3 with SPH

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The benchmark also included the calculation of control rod worths and the power distribution

at rodded conditions. Heavily rodded conditions are significantly more difficult for a code to

predict and so the ARI power distribution is a good condition to benchmark the improvement

afforded by the SP3 methodology over the current diffusion theory NEM. As in the ARO

benchmark, in the ARI benchmark, a quarter of the core was modeled with reflective

boundary conditions on all faces (axial and radial), making the problem a 2-D benchmark.

Eight-group rodded cross sections were generated by HELIOS with scattering moments up to

l=1. The same assumptions used in the unrodded cases with regard to the l=2 and l=3

scattering cross sections was used in the rodded cases. Unfortunately, SPH factors for rodded

conditions are not available. Since the rodded condition represents the most deviation from

the reference condition, SPH factors should be very important and result in a substantial

improvement.

Table 3.7 below shows the results of the eigenvalues of the two NEM rodded calculations

compared with the reference DeCART solution. The total rod worth is also provided. For

additional comparison, MCNP results from Reference 42 are provided as well as pin-by-pin

results from the diffusion theory code COBAYA3 and the SP3 code DYNSUB from

Reference 58. The EWE and PWE of the relative differences in the assembly powers is also

provided in Table 3.7. Figure 3.10 below shows the deviations in the assembly power results

for NEM with the two solution options compared to the reference DeCART solution. The

rodded assemblies are shaded in gray for clarity. The CPU requirements for the two NEM

runs for the ARI benchmark are 15 hours 22 minutes for the diffusion theory run and 33

hours 19 minutes for the SP3 with no SPH factors run

.

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Table 3.7- Eigenvalue keff and Assembly Power Deviation for ARI Configuration

Code / Method keff PWE (%) EWE (%) Total Control

Rod Worth (∆k/k) MCNP / Monte Carlo 0.98540 - 6873 DeCART / Method of

Characteristics (Reference Solution)

0.98743

-

6801

NEM / Diffusion Theory 0.99339 3.86 6.98 6440 NEM / SP3 No SPH Factors 0.99008 3.03 5.87 6684

COBAYA3 / Diffusion Theory 0.98900 1.68 3.02 7092 DYNSUB / SP3 0.98878 1.05 2.43 6695

Figure 3.10- Percent Deviations in Assembly Powers for

Two Solution Methodologies vs. Reference DeCART Solution for ARI Configuration

0.562 0.186-4.27 -3.66-3.56 -3.42

0.508 0.696 0.190-3.35 -3.88 -10.05-2.36 -2.59 -8.79

1.823 1.675 0.531 0.450 0.1863.51 2.09 -6.03 -6.87 -6.514.06 2.99 -4.52 -7.02 -7.47

1.198 2.452 1.944 0.985 0.329 0.198-3.67 4.53 4.84 -1.52 -12.16 -5.05-2.92 3.67 3.55 -0.41 -9.12 -6.57

2.459 1.812 2.103 1.832 0.449 0.489 0.268-1.18 -1.71 -3.57 2.73 -4.32 -7.77 -8.21-0.98 -0.72 -2.90 2.29 -4.45 -6.75 -6.23

1.209 2.533 1.202 2.196 0.742 0.669 0.300 0.205-5.54 -2.37 -5.07 -2.23 4.45 -7.62 -11.67 -10.73-4.05 -1.50 -4.08 -1.55 1.75 -4.19 -9.67 -7.32

DeCART Reference RPD% Difference Diffusion Theory% Difference SP3 No SPH

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The same six fuel assemblies along the diagonal quadrant of the core were examined for pin

power comparison to the reference solution. Assemblies A1, B2, C3, D4, E5, and F6 were

analyzed for pin power comparison. The PWE for the pin powers were calculated using the

previously described equation. The results are presented in Table 3.8 below. A representative

assembly (assembly F6) with the pin power % error for the two analytical methods is shown

in Figure 3.11 below.

Table 3.8- Pin Power PWE (%) for ARI Configuration

Code / Method A1 B2 C3 D4 E5 F6

NEM / Diffusion Theory

1.64 1.84 1.89 2.28 2.50 3.69

NEM / SP3 No SPH 0.91 1.14 1.18 1.72 1.94 3.18 DYNSUB / SP3 0.83 1.02 1.08 1.87 2.18 3.02

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Figure 3.11- Deviation of Pin Powers for Assembly F6 at ARI Conditions

As can be observed from the results in Table 3.5, the SP3 ARO solutions resulted in an

approximately 100 pcm improvement over the diffusion theory solution in the prediction of

keff. The PWE and EWE assembly power and pin power errors are reduced with the SP3

solutions (with and without SPH factors) when compared to the diffusion theory NEM

solution. This is discussed in more detail in Section 3.4.

0.49 0.45 0.48 0.46 0.45 0.44 0.43 0.42 0.41 0.40 0.38 0.37 0.35 0.34 0.33 0.28 0.27-5.58 -5.02 -5.17 -5.23 -5.19 -5.22 -5.18 -5.12 -5.10 -5.08 -4.42 -4.41 -4.82 -4.78 -4.55 -3.81 -4.02-5.06 -4.41 -4.77 -4.85 -4.92 -4.97 -4.90 -4.88 -4.62 -4.58 -4.60 -4.61 -4.58 -4.60 -4.42 -4.18 -4.560.50 0.50 0.45 0.57 0.55 0.40 0.52 0.51 0.37 0.48 0.46 0.33 0.43 0.41 0.31 0.31 0.28-5.05 -4.19 -4.17 -3.87 -4.06 -4.55 -4.25 -4.24 -4.77 -4.50 -4.44 -4.71 -4.23 -4.06 -3.99 -4.06 -5.02-4.64 -4.08 -4.05 -3.50 -3.76 -3.99 -3.72 -3.70 -4.08 -4.06 -3.92 -4.21 -3.23 -3.19 -3.67 -3.89 -4.120.58 0.49 0.59 0.55 0.53 0.50 0.49 0.46 0.45 0.41 0.40 0.40 0.31 0.33-5.03 -4.02 -4.08 -3.98 -3.75 -3.75 -3.79 -3.79 -3.80 -3.85 -3.74 -4.08 -4.28 -5.15-4.63 -3.78 -3.41 -3.29 -3.33 -3.22 -3.21 -3.08 -3.01 -3.19 -3.12 -3.56 -3.60 -4.550.59 0.66 0.58 0.53 0.52 0.51 0.50 0.49 0.47 0.46 0.44 0.42 0.40 0.41 0.34-4.92 -3.78 -3.56 -3.82 -3.80 -4.03 -4.08 -3.90 -3.98 -3.88 -3.54 -3.60 -3.58 -3.71 -4.91-4.60 -3.22 -3.03 -2.92 -2.94 -3.30 -3.32 -3.12 -3.27 -3.38 -3.09 -3.14 -3.12 -3.30 -4.600.61 0.68 0.60 0.57 0.56 0.54 0.53 0.52 0.50 0.49 0.47 0.45 0.44 0.42 0.41 0.43 0.35-4.62 -3.42 -3.38 -3.40 -3.68 -3.48 -3.60 -3.59 -3.47 -3.57 -3.60 -3.39 -3.58 -3.61 -3.64 -3.60 -4.71-4.42 -2.88 -2.77 -3.03 -3.33 -3.14 -3.29 -3.27 -3.22 -3.08 -3.41 -2.99 -3.20 -3.16 -3.19 -3.44 -4.320.64 0.52 0.59 0.57 0.55 0.54 0.51 0.49 0.45 0.44 0.33 0.37-4.58 -3.59 -3.38 -3.44 -3.53 -3.71 -3.77 -3.59 -3.63 -3.50 -4.12 -4.78-3.98 -3.44 -2.99 -3.06 -3.16 -3.28 -3.18 -3.02 -3.19 -3.12 -3.70 -4.610.66 0.73 0.64 0.61 0.60 0.58 0.58 0.57 0.56 0.54 0.51 0.49 0.47 0.46 0.45 0.46 0.38-4.57 -3.08 -2.96 -3.32 -3.41 -3.38 -3.68 -3.92 -4.06 -3.97 -3.74 -3.88 -3.90 -3.69 -3.40 -4.02 -5.12-4.22 -2.90 -2.77 -3.11 -3.24 -2.92 -3.31 -3.63 -3.78 -3.71 -3.34 -3.22 -3.41 -3.39 -2.97 -3.47 -4.490.68 0.75 0.66 0.63 0.62 0.60 0.60 0.62 0.64 0.59 0.54 0.51 0.49 0.47 0.46 0.48 0.40-4.48 -3.18 -3.05 -3.59 -3.63 -3.50 -3.88 -4.07 -4.19 -3.98 -3.70 -3.49 -3.66 -3.66 -3.48 -4.08 -4.991.03 0.91 -2.54 1.06 1.08 -2.66 0.99 0.98 -2.38 0.98 0.99 -2.66 1.08 1.06 -2.54 0.91 1.030.70 0.57 0.65 0.63 0.62 0.67 0.64 0.56 0.50 0.49 0.37 0.41-4.74 -4.08 -3.77 -3.72 -4.12 -4.88 -4.96 -4.12 -3.88 -3.97 -4.28 -5.07-4.09 -3.39 -2.88 -2.83 -3.22 -4.18 -4.12 -3.68 -3.21 -3.33 -4.18 -4.710.71 0.79 0.70 0.67 0.65 0.64 0.63 0.65 0.67 0.62 0.57 0.54 0.52 0.50 0.49 0.51 0.42-4.67 -3.18 -3.12 -3.66 -3.71 -3.68 -3.92 -4.22 -4.44 -4.01 -3.89 -3.65 -3.96 -4.00 -4.41 -4.67 -5.12-3.90 -2.90 -2.66 -2.99 -3.08 -3.02 -3.40 -3.66 -4.08 -3.77 -3.29 -3.07 -3.24 -3.43 -3.09 -3.68 -4.600.72 0.80 0.71 0.68 0.66 0.65 0.64 0.63 0.62 0.60 0.58 0.55 0.53 0.51 0.50 0.52 0.43-4.62 -3.11 -2.87 -2.98 -3.67 -3.51 -3.77 -3.88 -3.98 -3.97 -3.79 -3.58 -3.84 -3.81 -3.50 -4.18 -5.34-4.11 -2.44 -2.49 -3.02 -3.12 -2.88 -3.12 -3.30 -3.44 -3.42 -3.37 -2.94 -2.34 -2.49 -3.14 -3.62 -4.870.73 0.61 0.69 0.67 0.65 0.64 0.60 0.58 0.54 0.52 0.40 0.44-3.88 -3.55 -3.18 -3.28 -3.33 -3.40 -3.51 -3.50 -3.56 -3.71 -4.68 -4.97-3.80 -2.91 -2.71 -2.88 -2.82 -3.02 -3.18 -3.13 -3.24 -3.16 -4.00 -4.580.74 0.83 0.74 0.70 0.69 0.67 0.66 0.65 0.63 0.62 0.60 0.57 0.56 0.53 0.53 0.55 0.45-4.16 -3.47 -3.38 -3.42 -3.55 -3.40 -3.38 -3.30 -3.11 -4.08 -4.19 -3.88 -3.99 -4.18 -4.31 -4.67 -5.19-3.72 -2.77 -2.33 -2.32 -2.98 -2.83 -3.08 -2.91 -2.82 -3.12 -3.32 -3.10 -3.38 -3.19 -3.22 -3.57 -4.820.75 0.85 0.76 0.70 0.69 0.68 0.67 0.65 0.63 0.61 0.59 0.57 0.55 0.57 0.46-4.18 -3.40 -3.22 -3.39 -3.44 -3.56 -3.62 -3.57 -3.74 -3.74 -3.46 -3.49 -3.66 -3.90 -5.22-3.89 -2.68 -2.49 -2.32 -2.33 -2.99 -3.02 -2.93 -3.03 -3.04 -2.96 -2.99 -3.19 -3.55 -4.620.77 0.66 0.81 0.76 0.74 0.71 0.70 0.66 0.64 0.60 0.58 0.59 0.45 0.48-4.41 -3.72 -3.55 -3.29 -3.47 -3.47 -3.42 -3.53 -3.61 -3.78 -3.83 -4.39 -4.77 -5.01-3.89 -3.10 -2.91 -2.44 -2.49 -2.65 -2.63 -2.42 -2.47 -2.95 -2.97 -3.66 -4.01 -4.830.71 0.72 0.66 0.85 0.83 0.61 0.80 0.79 0.57 0.75 0.73 0.52 0.68 0.66 0.49 0.50 0.45-4.40 -4.16 -4.02 -3.12 -3.48 -3.90 -3.55 -3.68 -4.06 -3.44 -3.39 -3.97 -3.66 -3.71 -4.30 -4.51 -4.97-3.63 -3.12 -3.08 -2.78 -2.73 -3.11 -2.88 -2.94 -3.33 -3.00 -2.92 -3.44 -3.09 -3.13 -3.85 -4.09 -4.280.75 0.71 0.77 0.75 0.74 0.73 0.72 0.71 0.70 0.68 0.66 0.64 0.61 0.59 0.58 0.50 0.49 DeCART Reference RPD-4.92 -4.56 -4.69 -4.78 -4.75 -4.81 -4.92 -4.72 -4.67 -4.88 -4.90 -5.04 -5.15 -5.22 -5.19 -5.16 -5.55 % Difference Diffusion Theory-4.18 -3.57 -3.89 -4.02 -4.02 -4.02 -4.18 -3.90 -3.78 -3.91 -3.88 -4.57 -4.45 -4.68 -4.72 -4.40 -4.83 % Difference SP3 No SPH

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For the rodded benchmark, the SP3 solution resulted in an over 300 pcm improvement in the

predicted keff when compared to the diffusion theory model. The SP3 solution also resulted in

a reduction of EWE and PWE assembly power deviations and pin power deviations in

comparison to that of the diffusion theory solution. For the calculation of the total control rod

worth, which is also a measure of the power distribution, the SP3 solution produced results

which were better than the diffusion theory solution by over 200 ∆k/k.

As expected, the most pronounced improvement in the assembly power prediction with the

SP3 solution was near the material boundaries between the UO2 and MOX fuel assemblies

and in the rodded assemblies, where the ratio of 2φ / 0Φ is the greatest. For both the ARO and

ARI benchmarks, as can be visualized in Figures 3.8 and 3.10, there was improvement with

the SP3 solution in the predicted assembly powers in the peripheral fuel assemblies in contact

with the reflector. This is very likely due to the benchmark’s utilization of P1 scattering

moments. Previous benchmarking of NEM-SP3 using the C5G7 benchmarks has revealed that

there exists a limitation on the improvement afforded by the SP3 solution when only P0

scattering is included. It is likely that the inclusion of the P1 scattering moments, particularly

with regard to the second removal cross section, results in an improved ability to be able to

model the complicated scattering that occurs in the baffle region of reactor cores.

3.4 Final Remarks on SP3 Nodal Expansion Method

The SP3 method which has been implemented in the NEM code was benchmarked against

five well known mixed UO2/MOX benchmarks. The results from the three C5G7 benchmarks

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(Tables 3.1 - 3.4) for the SP3 method were shown to produce a minor improvement (a few

tenths of a percent) in the prediction of the pin and assembly powers. However, the amount

of runtime for the SP3 method for these benchmarks is approximately twice that of the

runtime for the diffusion theory polynomial NEM. It can therefore be concluded that, with

the absence of P1 scattering cross sections, as was the case in all three of the C5G7

benchmarks, the small increase in accuracy in using the SP3 method is not worth the doubling

of the runtime in comparison with current methods.

The results of the OECD/NEA and U.S. NRC PWR MOX/UO2 Core Transient Benchmark,

which included P1 scattering cross sections, demonstrated a more significant improvement

than the C5G7 benchmarks when the SP3 solution was used. The prediction of reactivity

improved with the SP3 method by approximately 100 and 300 pcm for the ARO and ARI

benchmarks respectively in comparison with the diffusion theory solution. In addition, the

error in the assembly power, as measured by the EWE and PWE, was also reduced by the SP3

method in both rodded and unrodded configurations.

For the ARO benchmark, the SP3 method resulted in a small improvement in the prediction

of assembly power and pin power in comparison to the diffusion theory solution. As can be

observed from Figure 3.8, the most substantial improvement is in the peripheral assembly

locations (i.e. near the baffle reflector). This is very likely the contribution of the P1

scattering moments which are included in the benchmark. The scattering in the reflector

region should be expected to be more anisotropic due to the heavier materials present (steel,

inconel, etc.). Therefore, the ratio of 2φ / 0Φ is greatest in these anisotropic regions, so the

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improvement afforded by the SP3 method should be more apparent in this region, which was

the observed result in this benchmark.

Another observation in the ARO results is that the SPH factors appeared to not have a

significant effect in ether the assembly power or pin power calculations. There was a small

improvement, but not nearly enough to justify the increase in computational burden

associated with a 16 energy group run, which involved an increase in runtime of about six

times the diffusion theory result. This is consistent with the conclusions reached in Reference

50 regarding SPH factors which states “The results show that the SPH correction is not

relevant for the considered PWR fuel assembly configurations without control rods”. In

Reference 50 the SPH factors were found to be relevant only for the rodded configuration.

For the rodded benchmark, the SP3 method likewise resulted in an improvement in the

predicted assembly powers and pin powers, but larger than desired errors still remain. As can

be seen from Figure 3.10, the most substantial improvement with the SP3 method is in the

rodded assemblies, which are shaded gray. These are the most heterogeneous portions of the

core due to the sharp flux distortions caused by the strong absorber being present.

Furthermore, it is obvious that these regions have a strong absorber present, which should be

expected to present problems for the diffusion theory. However, the amount of improvement

obtained with the SP3 method, as previously mentioned, was likely hindered by the lack of

SPH factors for the rodded cross sections. Reference 50 shows these factors to be very

important in rodded environments for the SP3 solution.

The larger improvement obtained with the OECD/NEA and U.S. NRC PWR MOX/UO2 Core

Transient Benchmark in comparison to the C5G7 benchmark is almost certainly due to the

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inclusion of the P1 scattering cross sections. The primary contribution of the P1 scattering

cross sections is in the definition of the removal cross sections. As can be seen from Equation

3.5, the higher order scattering cross sections are subtracted from the total cross sections in

the SP3 method. When the P1 scattering cross sections are absent, as in the C5G7

benchmarks, the second removal cross section is incorrect. When the P1 cross sections are

present, as in the OECD/NEA and U.S. NRC PWR MOX/UO2 Core Transient benchmark,

the second removal cross section can be adequately represented which leads to the improved

results with the SP3 method in that benchmark.

It is therefore concluded from the results of the benchmarking that it is only advantageous to

use the SP3 solution when P1 scattering cross sections are present. The marginal improvement

afforded by the SP3 method when the P1 scattering cross sections are not present does not

justify the nearly doubling of the computational time.

This benchmarking work also agrees with the conclusions of Reference 50, namely that the

improvement obtained with the SP3 method in heavily rodded configurations requires SPH

factors to obtain an acceptable result. This suggests that the primary source of code error in

problems with heavily rodded conditions is in fact the cross sections. The form of nodal

solution (diffusion theory, SP3, etc.) has a limited effect on the calculation of the overall

power distribution.

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CHAPTER 4

THE SEMI-ANALYTICAL NODAL EXPANSION METHOD

4.1 Introduction

The polynomial nodal expansion method (P-NEM) solution which is currently available in

PSU’s NEM code has been demonstrated to produce acceptable results in standard PWR and

BWR geometries. The P-NEM solution, which is based on the transverse integration

procedure and a one-dimensional fourth-order polynomial flux expansion, has been observed

to be less accurate and potentially inefficient in areas where steep flux gradients occur at

assembly interfaces, such as would be found in MOX cores, near material boundaries and

near control rods or burnable absorbers. It is, therefore, a less than desirable solution option

for the modern heterogeneous reactor cores that the present work is being performed in order

to more accurately analyze.

The semi-analytical nodal expansion method (SA-NEM) solution was developed and

implemented into the NEM code in order to improve the capability of the code to be able to

model the steep flux gradients which are commonly present in modern heterogeneous reactor

cores. The salient feature of the SA-NEM solution implemented in NEM is that it solves the

within-group diffusion equation using a standard linear response matrix procedure without

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the flat-source approximation which is sometimes implemented to eliminate the non-linear

iteration [20].

This chapter provides a discussion of the SA-NEM solution as implemented in the NEM

code. The chapter is organized as follows: firstly, the theoretical/mathematical background of

the SA-NEM solution is presented; secondly, the SA-NEM solution is benchmarked against

three international benchmarks and the results are presented; and thirdly, some final remarks

on the SA-NEM solution and the results of the benchmarks are presented. The work

documented in this chapter is a continuation of the SA-NEM work initiated by Reed [46] and

Beam [49].

4.2 The Semi-Analytical Nodal Expansion Method

The SA-NEM, similar to the polynomial NEM, is based upon the transverse integrated

diffusion equation. Unlike the analytical nodal method (ANM) or the nodal integration

method [43], whose only approximation is the quadratic representation of the transverse

leakage term, the SA-NEM utilizes an additional approximation by using polynomial basis

functions to represent the spatial dependence of the source terms. However, the primary

advantage of the SA-NEM in comparison to the ANM is that the latter is known to encounter

stability problems when the buckling matrix is singular (i.e. contains a zero eigenvalue). This

happens when k∞ = keff, meaning that the neutron production from fission equals the loss by

absorption and therefore there is no net leakage from the node [21]. The SA-NEM avoids this

singularity by using the within group form of the diffusion equation, which has two separate

operators for the gain and loss terms. This can be seen in Equation 4.1 below.

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The derivation of the SA-NEM begins with the within group diffusion equation, written in 1-

D in matrix form below for an arbitrary direction u.

)u(L)u(A)u(k)u(dud

2

2

−φ=φ+φ− (4.1)

where,

k = g

rem

A =

∑ Σ+Σχν

≠=→

G

g'g,1'gg'g,sf

effg k1

D1

L(u) = transverse leakage

The general solution of Equation 4.1 is obtained analytically by solving for the eigenvalues

and eigenvectors of the k matrix. The particular solution of Equation 4.1 is obtained by

expanding the flux in fourth-order polynomial basis functions.

To obtain the general solution of Equation 4.1, the leakage and source terms are transformed

into a form more amenable to analytical solution procedures. In each case, this is done using

a polynomial expansion approximation. Thus, the total leakage for the x-direction (equivalent

expressions are obtained for the y and z directions) in node l is defined as:

( ) ( ) ( )1 1l l l

g gy gzL x L x L xy z

= +∆ ∆

(4.2)

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and is approximated by a quadratic polynomial fn(x) as follows:

( ) ( ) ( ) ( )2

1 1 2 20

xll l l lgg gx gx gxn n

nL x L p f x p f x p f x

=

= + + = ∑ (4.3)

In Equation 4.3, xlgL is the average x-directed leakage in node l and l

gxnp are expansion

coefficients for energy group g, direction x, and node l.

The source term in Equation 4.1 is approximated by a fourth-order polynomial as follows:

)x(fs)x(Q n4

0nlgxn

lgx ∑ == (4.4)

where,

l l lgxn gxn gxns Q p= − (4.5)

' ' ' '' 1 ' 1

lG Ggl l l l l

gxn g g g xn fg g xng g

Qk

χφ ν φ→

= =

= Σ + Σ∑ ∑ (4.6)

Equations 4.5 and 4.6 introduce the parameters lgxnQ and l

gxnφ , which are the source moments

and flux moments, respectively. The basis functions for Equations 4.3 and 4.4 are the same as

those used in the polynomial NEM expansions, namely

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( )0 1f x =

( )1xf xx

=∆

( )2

2134

xf xx

= − ∆

( )3

31 1 12 2 4

x x x x xf xx x x x x

= − + = − ∆ ∆ ∆ ∆ ∆

( )2 4 2

41 1 1 3 120 2 2 10 80

x x x x xf xx x x x x

= − − + = − + ∆ ∆ ∆ ∆ ∆

Equation 4.1, with the simplifications presented in Equations 4.2 - 4.6, may now be written in

multigroup form (using the x direction for arbitrary node l) as follows:

)(1)(4

0

22

2

xfbD

xkdx

dn

n

lgxnl

g

lgxgl

lgx ∑

=−=− φ

φ (4.7)

where,

lg

remgl D

=

2 2x xx−∆ ∆≤ ≤

l l l

gxn gxn gxnb p s= − +

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The first step in solving Equation 4.7 is to find the general solution of the homogeneous

eigenvalue problem. The homogenous solution to Equation 4.7 is found in the space of

hyperbolic functions. The sinh and cosh functions, which are analytic over the full problem

domain, are the appropriate eigenfunctions.

The general solution for Equation 4.7 is:

( ) ( ) ∑=

++=4

0)(coshsinh)(

n nlgxnglgl

lgx xfaxkBxkAxφ (4.8)

where the coefficients lgxna (n=1,4) are:

∆+−

Σ= l

3gx2gl

2l

1gxl

1gxrem

l1gx Q

kx6pQ1a

∆+−

Σ= l

4gx2gl

2l

2gxl

2gxrem

l2gx Q

kx4pQ1a

l3gx

rem

l3gx Q1a

Σ=

l4gx

rem

l4gx Q1a

Σ=

Reference 46 provides a complete derivation of these four coefficients.

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The remaining coefficient, 0lgxa , follows from the consistency condition:

( )2

2

1 x llgx gx

x dxx

φ φ∆

−∆=

∆ ∫

where,

gl

φ = node volume averaged flux

This leads to the following value for coefficient 0lgxa :

02 sinh

2ll

gx glggl

xa B kxk

φ ∆ = − ∆

The general solution to Equation 4.1 may now be written as follows:

( ) ( ) ∑=

+

∆−++=

4

0)(

2sinh2coshsinh)(

n nlgxngl

glglgl

lg

lgx xfaxk

xkxkBxkAx φφ (4.9)

The coefficients A and B are determined by the continuity (discontinuity) boundary

conditions at the nodal interfaces [46, 49]. These coefficients are derived in detail in

Reference 46.

At this point, one has enough information to begin producing the final partial current

equations, the 6 x 6 matrix of equations that describe, for each node, the outgoing partial

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currents as a function of incoming partial currents and intra-nodal sources/sinks. The

expressions for these partial current matrices are derived in a manner similar to that for the

polynomial NEM, from Fick’s law expressions for the partial currents on the node

boundaries. On the two faces of node l normal to the x-axis, as stated previously,

( ), ,

2

out l in l l lgx gx g gx

xx

dJ J D xdx

φ+ +∆=

= − (4.10)

( ), ,

2

out l in l l lgx gx g gx

xx

dJ J D xdx

φ− −−∆=

= + (4.11)

The SA-NEM expansion (Equation 4.9) must now be substituted into Equations 4.10 and

4.11, yielding,

∆+

∆+

∆+

∆+

∆+

−= ++ xa

xa

xa

xaxk

BkxkAkDJJlgx

lgx

lgx

lgxgl

glglgllg

lingx

loutgx 52

32

sinh2

cosh 4321,, (4.12)

∆−

∆+

∆−

∆+

∆−

+= −− xa

xa

xa

xaxk

BkxkAkDJJlgx

lgx

lgx

lgxgl

glglgllg

lingx

loutgx 52

32

sinh2

cosh 4321,, (4.13)

Equations 4.12 - 4.13, with similar equations for the y and z directions, are solved by NEM

using a traditional inner/outer iteration scheme. For each group, inner iterations or multiple

sweeps through the mesh with a known internal source are performed. Outer fission source

iterations are then performed around the inner iterations to calculate values for the problem

multiplication eigenvalue (keff) and the space and energy dependent fission neutron source

distribution.

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4.3 Benchmarking of SA-NEM Solution

The SA-NEM solution was tested using the OECD/NEA 2-D C5G7 MOX benchmark [3] and

the C3 and C5 benchmarks [44]. All three of these benchmarks are 2x2 mini-cores with both

MOX and UO2 fuel assemblies. The benchmarks are designed to challenge the code’s ability

to accurately predict pin power and assembly power in a highly heterogeneous core

environment. The presence of the MOX fuel assemblies leads to steep thermal flux gradients

between neighboring fuel assemblies; this allows the improvement afforded by the SA-NEM

solution to be demonstrated. In addition, a sensitivity study is performed to study the impact

of varying mesh size on the performance of the SA-NEM solution.

All of the benchmarking runs were performed using an Intel Core i7-2620M CPU with a 2.7

GHz processor.

4.3.1 OECD/NEA 2-D C5G7 MOX Benchmark

The OECD/NEA 2-D C5G7 MOX benchmark [3] was used to benchmark the newly

completed SA-NEM solution methodology in NEM. This benchmark is described in detail in

Appendix A. The seven energy group cross sections from Reference 3, which are provided in

Appendix B, were used in this benchmark. A convergence of 1E-05 was used for keff and 1E-

06 was used for the point-wise flux. NEM was run in pin-by-pin geometry using a 34 x 34

lattice for the 2x2 configuration. For comparison, the polynomial NEM solution is also

generated.

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The NEM results for both methods were compared to the reference MCNP solution [3]. The

value of keff calculated by MCNP is 1.18655 (±0.003 %) and the value of keff calculated by

NEM with the SA-NEM solution is 1.18737. Therefore, the keff error for NEM with the SA-

NEM solution option for this benchmark equals 0.069 %. The value of keff calculated by

NEM with the polynomial NEM solution is 1.18707, which corresponds to an error of

0.044%.

Table 4.1 below shows the pin powers calculated by NEM with both the SA-NEM and

polynomial NEM (P-NEM) solutions as well as the reference MCNP solution. The results of

the benchmark for the AVG (average on module pin power percent distinction), RMS (root

mean square of the percent distinction) and MRE (mean relative pin power percent error)

were calculated by the following formulas from Reference 3.

AVG = ∑=

N

nne

N 1

1

RMS = ∑=

N

nne

N 1

21

MRE = avg

n

N

nn

Np

pe∑=1

where,

N = total number of pins

n = pin #

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en = pin power error for pin n

pavg = average pin power

pn = pin power for pin n

The assembly power results are presented in Table 4.2. Figure 4.1 below shows the pin power

distribution of the 2x2 array calculated with the SA-NEM solution option. The individual pin

power results for this benchmark for both the SA-NEM solution as well as the P-NEM

solution are located in Appendix C.

Table 4.1 Calculated Pin Powers from NEM Compared with MCNP Reference Solution (SA-NEM 2-D C5G7Benchmark)

Evaluated Parameter MCNP (Reference) SA-NEM P-NEM CPU Time (seconds) - 917 893 Maximum Pin power 2.498 2.576 2.574 Minimum Pin power 0.232 0.232 0.233

AVG (in percent) - 2.366 2.227 RMS (in percent) - 2.832 2.653 MRE (in percent) - 2.149 2.041

Average pin power in inner UO2 assembly

(% error)

1.867

1.895

(1.522)

1.894

(1.475) Average pin power in outer UO2 assembly

(% error)

0.529

0.523

(-1.216)

0.524

(-1.111) Average pin power in

FA-MOX (% error)

0.802

0.791

(-1.370)

0.791

(-1.349)

Table 4.2 Calculated Assembly Powers from NEM Compared with MCNP Reference Solution

(SA-NEM 2-D C5G7 Benchmark)

Fuel Assembly MCNP (Reference) SA-NEM P-NEM Inner UO2 492.8 ±0.10 500.3 500.1

MOX 211.7 ±0.18 208.8 208.9 Outer UO2 139.8 ±0.20 138.1 138.2

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Figure 4.1- SA-NEM Pin Power Distribution for 2-D C5G7 MOX Benchmark

As can be seen from the results in Tables 4.1 and 4.2, the P-NEM and SA-NEM solutions

give similar results in pin-by-pin geometry. This is to be expected at the pin cell level since

the spatial variation of the flux is quite small. However, as the mesh size is increased to the

size of an assembly node, the disparity between the two methods should be more pronounced.

At the nodal level, the spatial variation of the flux within the mesh is increased, and therefore

the deficiency of approximating the flux using polynomials, as in the P-NEM, is more

apparent. To see this effect, the C3 and C5 benchmarks, which model full assembly

geometry, are modeled next.

1

5

9 13 17

21 25

29 33

0.10 0.30 0.50 0.70 0.90 1.10 1.30 1.50 1.70 1.90 2.10 2.30 2.50

1 5

9 13

17 21

25 29

33

2.50-2.60

2.30-2.50

2.10-2.30

1.90-2.10

1.70-1.90

1.50-1.70

1.30-1.50

1.10-1.30

0.90-1.10

0.70-0.90

0.50-0.70

0.30-0.50

0.10-0.30

Pin Powers

Water

Reflector

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4.3.2 C3 2x2 Mini-Core Benchmark

The C3 benchmark [44] was modeled to compare the results of NEM using the semi-

analytical NEM solution to the results with the P-NEM solution. The C3 benchmark is

described in detail in Appendix A. The two-group assembly-homogenized cross sections

(Appendix B) and assembly discontinuity factors, which were generated using PARAGON,

were taken from Reference 45. The reference solution for the C3 benchmark was generated

using NEM in pin-by-pin geometry. Both the P-NEM and SA-NEM solutions were run in

pin-by-pin geometry to obtain the reference solution.

The value of keff calculated by NEM in pin-by-pin geometry is 1.25883 (for both P-NEM and

SA-NEM solutions) and the value of keff calculated by NEM in full assembly geometry with

the SA-NEM solution is 1.25714. The value of keff calculated by NEM in full assembly

geometry with the P-NEM solution is 1.25949.

The assembly powers calculated by NEM with the semi-analytical NEM and polynomial

NEM solutions were compared with the reference solution calculated by NEM in pin-by-pin

geometry (Note: the reference pin-by-pin results with the polynomial NEM and the SA-NEM

were essential equal so only one reference value is provided below). The comparison is

showed below. The CPU time for the SA-NEM run was 0.36 seconds compared with 0.31

seconds for the P-NEM run.

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2D Assembly Powers for C3 Benchmark

SA-NEM

REFERENCE (pin-by-pin geometry)

P-NEM

1.1258 0.8742

1.1382 0.8618

1.1600 0.8400

0.8742 1.1258

0.8618 1.1382

0.8400 1.1600

As can be seen from the above results, there was observed to be an improvement in the

prediction of the 2D assembly powers with the use of the SA-NEM solution in comparison

with the P-NEM solution. The ability of the SA-NEM solution to be able to capture the sharp

flux distortions caused by the mixed MOX/UO2 core was demonstrated in this benchmark.

4.3.3 C5 2x2 Mini-Core Benchmark

The C5 benchmark [44] is a 2x2 mini-core with two MOX assemblies and two UO2

assemblies, but with a water reflector on two faces. The C5 benchmark is described in detail

in Appendix A. The two-group assembly-homogenized cross sections (Appendix B) and

assembly discontinuity factors, which were generated using PARAGON, were taken from

Reference 45.

The value of keff calculated by NEM in pin-by-pin geometry is 0.944162 and the value of keff

calculated by NEM in full assembly geometry with the SA-NEM solution is 0.941820. The

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value of keff calculated by NEM in full assembly geometry with the P-NEM solution is

0.941716.

The assembly powers calculated by NEM with the SA-NEM and polynomial NEM solutions

were compared with the reference solution calculated by NEM in pin-by-pin geometry (Note:

the reference pin-by-pin results with the polynomial NEM and the SA-NEM were essentially

equal so only one reference value is provided below). The comparison is showed below. The

CPU time for the SA-NEM run was 0.41 seconds compared with 0.38 seconds for the P-

NEM run.

2D Assembly Powers for C5 Benchmark

SA-NEM

REFERENCE (pin-by-pin geometry)

P-NEM

1.7891 0.7744

1.7611 0.7887

1.7960 0.7929

0.7744 0.6621

0.7887 0.6614

0.7929 0.6182

As can be seen from the above results, the SA-NEM solution produced an improvement in

the calculation of assembly power for the two UO2 assemblies. For the two MOX assemblies,

both the P-NEM and SA-NEM solutions produced acceptable results (about 1% error) when

compared with the pin-by-pin reference solution.

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4.3.4 Mesh Width Sensitivity

The results of the C3 and C5 benchmarks demonstrate that the SA-NEM solution generally

produces improved results in the prediction of assembly power in comparison with the P-

NEM solution when full assembly geometry is used. The sensitivity of the SA-NEM to the

mesh size used was analyzed. In the sensitivity study, smaller mesh sizes are used to see the

effect on assembly power prediction. It should be the case that as the mesh size is made

smaller, the solution converges to the NEM pin-by-pin reference solution.

The C3 benchmark was used for the sensitivity study. The 2x2 configuration was run using

four radial nodes (in a 2x2 configuration) per assembly (four npa model) and nine radial

nodes (in a 3x3 configuration) per assembly (nine npa model). Both the P-NEM and SA-

NEM solutions were considered.

The root mean squared (RMS) % error (in comparison to the NEM pin-by-pin reference

solution) of the assembly powers was plotted versus mesh size and is presented in Figure 4.2.

As can be seen from the results in Figure 4.2, the % error of the assembly powers decrease

with decreasing mesh size in an approximately linear fashion. As expected, the solution

appears to converge to the reference pin-by-pin solution as the mesh size is made smaller.

Furthermore, the deviation between the SA-NEM and P-NEM solutions decreases with

decreasing mesh size. The reason that the two solutions converge at smaller mesh sizes is due

to the approximation made to describe the intra-nodal flux shape in the P-NEM solution. At

smaller mesh sizes (i.e. pin cell), the flux distribution across the mesh is essentially flat; thus,

the spatial distribution of the flux may be well approximated by a low-order polynomial. Put

another way, the polynomial well approximates the analytical solution to the homogeneous

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eigenvalue problem which is solved analytically in the SA-NEM solution procedure. For this

reason, the two solutions should be expected to converge at smaller mesh sizes, which was

the observed result in this study.

Figure 4.2- Assembly Power RMS % Error vs. Mesh Size for P-NEM and SA-NEM Solutions

4.4 Concluding Remarks on the SA-NEM

The SA-NEM solution was completed in the NEM code and was benchmarked against three

well known mixed UO2/MOX benchmarks. For the C5G7 benchmark, which tested the

solution in pin-by-pin geometry, the SA-NEM solution was demonstrated to produce

comparable results to the P-NEM solution. The runtimes were nearly identical as well. This

0

0.5

1

1.5

2

2.5

3

0 5 10 15 20 25

Ass

embl

y Po

wer

RM

S %

Err

or

Mesh Size (cm)

MOX Assembly (SA-NEM)

UO2 Assembly (SA-NEM)

MOX Assembly (P-NEM)

UO2 Assembly (P-NEM)

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81

was the expected result due to the fact that the spatial variation in the flux across a pin cell is

quite small. The advantage that the SA-NEM solution affords is in its ability to be able to

model more complex flux distributions due to its analytical solution of the homogeneous

eigenvalue problem (Equation 4.7). The quadratic polynomial approximation proves to be

inadequate in larger node volumes where the spatial flux shape is more complex.

To demonstrate this, the SA-NEM solution was benchmarked against the C3 and C5

benchmarks [44] which are modeled in full assembly geometry. The SA-NEM results for the

C3 and C5 benchmarks are quite promising. For the C3 benchmark, the % error of the

assembly powers was reduced by nearly one half with the SA-NEM solution in comparison

with the P-NEM solution. Similar improvements in the calculation of assembly power were

observed for the UO2 assemblies in the C5 benchmark.

The reason for this improvement, as mentioned previously, is the ability of the SA-NEM to

be able to better model the complicated flux shape in a larger node via the analytical solution

of the homogeneous eigenvalue problem (Equation 4.7). In the P-NEM solution, the flux

shape within a node is approximated using a quadratic polynomial. For smaller nodes where

the shape of the flux within the node is relatively flat the P-NEM gives approximately the

same solution as the SA-NEM. This can be seen in Figure 4.2, where the two solutions (SA-

NEM and P-NEM) converge to the pin-by-pin reference solution at the smaller mesh sizes.

However, in full assembly geometry, where the spatial variation of the flux within a node

may be much more complicated, the approximation using a quadratic polynomial is no longer

adequate in many situations. In these situations, the analytical solution of the homogeneous

eigenvalue problem allows the SA-NEM solution to be able to capture much more

complicated flux shapes within a node than the P-NEM solution can capture. This is the

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element which allows the SA-NEM solution to better calculate assembly powers in full

assembly geometry when compared to the P-NEM solution.

It is uncertain at this time as to why exactly the results for the C3 benchmark are better than

the results for the C5 benchmark with respect to the assembly powers. For the two UO2

assemblies, both the C3 benchmark and the C5 benchmark show improvement with the SA-

NEM solution. However, the MOX assemblies have slightly worse results with the SA-NEM

solution in comparison to the P-NEM solution for the C5 benchmark. Part of the issue could

be related to the convergence of the two methods. During the benchmarking of the SA-NEM

solution it has been observed that the convergence behavior is different than that of the P-

NEM solution. Specifically, the source tends to converge slower than the P-NEM solution.

This should be expected due to the fact that the SA-NEM has four source moments compared

with only two source moments with the P-NEM. Therefore, there are four moments which

must converge, leading to a somewhat slower convergence. Using slightly looser

convergence criteria than that used in the P-NEM solution enables the assembly powers to be

improved somewhat compared to that previously reported for the C5 benchmark. However,

when the convergence criteria are the same for the SA-NEM and P-NEM solutions, there

remains a disparity in the improvement between the C3 and C5 benchmarks for the two

MOX assemblies.

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CHAPTER 5

THE ANALYTIC BASIS FUNCTION

TRANSVERSE LEAKAGE METHOD

5.1 Introduction

Current nodal reactor analysis codes typically rely on the transverse integration procedure to

convert the initial three-dimensional problem into three auxiliary one-dimensional problems.

This conversion is accomplished by integrating the nodal diffusion equation over the two

directions transverse to the direction under consideration, which produces the three auxiliary

one-dimensional equations. The parameter which links the solutions of the flux/current

equations in the two transverse directions to the direction under consideration is the

transverse leakage term. Accurate representation of the transverse leakage term remains the

major shortcoming of nodal codes because explicit representation of the intra-nodal flux

shape, which is required to express the transverse leakage explicitly, is lost when transverse

integration is used. The intra-nodal flux must therefore be approximated or else the transverse

leakage term must be approximated in some other manner.

The most popular method at present for approximating the transverse leakage term is the

quadratic leakage approximation (QLA). In the QLA, a quadratic polynomial is used to

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84

approximate the spatial variation of the leakage. This polynomial is applied to three

consecutive nodes and the coefficients of the polynomial are obtained by assuming that the

average transverse leakages for each of the three nodes are preserved by the quadratic

polynomial. The average transverse leakages may be obtained, without knowing the intra-

nodal flux shape, using the average net currents from the node of interest to the transverse

directions. After obtaining the coefficients, the quadratic polynomial is then applied only to

the middle node. The advantage of the QLA is that all quantities are directly available from

the nodal solution itself (namely, the average leakages for both the node of interest as well as

the neighboring nodes). However, the disadvantage of the QLA is that the true leakage shape

cannot always be well approximated by a quadratic polynomial fit. The QLA generally

performs well in checkerboard cores, but has been found to be less accurate near core

boundaries, in cores loaded with MOX fuel, at material boundaries, and near strong absorbers

such as control rods or burnable absorber rods. The QLA, therefore, would be expected to

perform less than satisfactorily in the modern heterogeneous reactor cores which are

presently in use in commercial power reactors.

An explicit method is presented in this chapter for calculating the transverse leakage based

upon the use of analytic basis functions (ABFs), which represent individual eigenfunctions of

the neutron diffusion equation. The intra-nodal flux solution, which is required to explicitly

describe the transverse leakage term, may be expressed as a linear combination of

eigenfunctions. Its coefficients may be solved for using the already calculated surface

currents and flux moments as boundary conditions. The salient feature of the method,

therefore, is that no ad hoc presumptions are made with regard to the leakage shape, such as

is done with the QLA method. The individual eigenfunctions are calculated based upon

already available parameters from the flux solution and response matrix solution, and

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therefore no additional parameters are introduced into the problem which could lead to an

unwanted increase in computational time, as is the case in other advanced transverse leakage

methods.

5.2. Analytic Basis Function Transverse Leakage

An expression for the intra-nodal flux shape is sought which may be utilized to calculate a

more accurate representation of the transverse leakage term of the transversely integrated

neutron diffusion equation. Previous attempts to accomplish this have used variational

methods [47] or self-consistent methods based upon weighted transverse integration with

Legendre polynomials [48]. These advanced methods, while a definite improvement over the

QLA in terms of accuracy, do so with increased computational time and also introduce more

variables to be solved for in the nodal flux solution. The analytic basis function transverse

leakage (ABFTL) method solves for the intra-nodal flux needed to explicitly calculate the

transverse leakage using available parameters from the response matrix and flux solutions

(surface currents and flux moments). By using already calculated parameters, the ABFTL

method improves the treatment of the transverse leakage term, without introducing additional

variables into the solution, and therefore does not introduce an increase in computational time

as in other advanced transverse leakage methods.

In this section, the concept of utilizing analytic basis functions to represent the intra-nodal

flux is developed mathematically. Following this is a description of the ABFTL method, as

implemented in NEM.

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5.2.1 Representation of Intra-Nodal Flux by Analytic Basis Functions

Considering a three-dimensional phase space (node) in standard Cartesian geometry (x, y, z

∈ Vnode and Vnode 3R⊂ ), the steady-state multigroup neutron diffusion equation (NDE) may

be written as follows:

)r(S)r()r(D g,0g,0g,0,remg,02

g,0 =ΦΣ+Φ∇− (5.1)

with,

)(,0 rS g = [ ]∑≠=

→ ΦΣG

g'g,1'gg,0g'g,0,s )r( + ∑

=Σν

χ G

1'g'g,f

eff

g

k[ ])r(g,0Φ

Written in a more compact operator notation, the multigroup NDE with down-scattering only

may be written as follows:

Φ=ΦeffkFM (5.2)

where,

φφ

φφφ

χνχνχνχνχνχν

φφφ

ΣΣΣΣΣΣ

=

Σ+∇⋅∇−Σ−Σ−Σ+∇⋅∇−Σ−

Σ+∇⋅∇−

3

2

1

3231

2221

1211

3

2

1

33

22

1

21

21

21

2313

12

1

1000

F

ff

ff

ff

M

Rss

Rs

R

kDD

D

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It can therefore easily be seen from Equation 5.2 that the multigroup NDE is an eigenvalue

problem, whose eigenfunctions correspond to the neutron flux.

Returning to Equation 5.1, the solution to this elliptic partial differential equation is sought.

The solution space for Equation 5.1 is the space of all analytic functions infinitely

differentiable on x, y, z ∈ Vnode and Vnode 3R⊂ . For the ABFTL method, the solution for

Equation 5.1 is expanded (for arbitrary node n) as a linear series of harmonic functions as

follows:

)y,x,z()x,z,y()z,y,x()z,y,x( 321

n ϕ+ϕ+ϕ=φ (5.3)

where,

=ϕ )z,y,x(1 [ ] [ ]x2

x1

x0

nx2

x1

x0

n zByBB)xcosh(zAyAA)xsinh( ++γ+++γ

=ϕ )x,z,y(2 [ ] [ ]y2

y1

y0

ny2

y1

y0

n xBzBB)ycosh(xAzAA)ysinh( ++γ+++γ

=ϕ )y,x,z(3 [ ] [ ]z2

z1

z0

nz2

z1

z0

n yBxBB)zcosh(yAxAA)zsinh( ++γ+++γ

nγ = n

eff

gRg

1g k

FD

−Σ−

where,

φeff

g

kF

= ∑ Σνχ

=

G

1'g'g,f

eff

g

k)r(g,0φ + ∑ φΣ

≠=→

G

g'g,1'gg,0g'g,0,s )r(

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Each of the 18 terms in Equation 5.3 are separate eigensolutions of Equation 5.1. Since the

NDE is a linear equation, any linear combination of solutions is also a solution. This property

can be utilized to build the general solution as a finite linear combination of the individual

eigensolutions of the intra-nodal flux. In this manner, the eigenspace containing the eighteen

eigensolutions of Equation 5.1 can be used to construct the transverse leakage term of the

NDE.

In Equation 5.3, each direction (x, y, and z) has six separate solutions. Two are general

analytic solutions and the other four are general analytic solutions multiplied by linear

functions transverse to the direction under consideration. These transverse basis functions are

also solutions of the NDE.

5.2.2 Analytic Basis Function Transverse Leakage Method

The coefficients in the three expansions of Equation 5.3 may be expressed in terms of

parameters which are already known from the response matrix and flux solutions from the

previous iteration. Specifically, in Cartesian geometry, there are six partial current values and

six interface flux moments, as well as an average nodal flux value. The first and second flux

moments as well as the surface currents may be used to obtain expressions for the

coefficients in Equation 5.3. To facilitate this, transverse integration is performed on

Equation 5.3 to obtain an expression for the transverse integrated flux. The transverse

integrated flux is then used to derive Fick’s law expressions for the surface currents, which

are then set equal to the surface currents from the response matrix solution. Moments

weighting, using Equations 3.15 and 3.16, is then performed on the transverse integrated flux

to obtain expressions for the first and second flux moments. These flux moments are then set

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89

equal to the flux moments calculated in the nodal flux solution. Once obtained, these three

equations are used to solve for the coefficients for each direction.

When calculated using the aforementioned procedure, the coefficients are combined with the

analytic basis functions to form a single eigensolution, which describes the intra-nodal flux

shape which may be used to solve for the transverse leakage term explicitly. This intra-nodal

flux may be utilized to calculate the transverse leakage without the unwanted assumptions of

the quadratic leakage approximation. To avoid having to calculate any additional parameters

in the nodal flux and response matrix solutions, only the cosh terms from Equation 5.3 are

implemented into NEM for the ABFTL method. The introduction of additional parameters

into the nodal flux and response matrix solutions, which would be necessary to solve for all

eighteen coefficients in Equation 5.3, would likely result in an increase in computational time

which the present work sought to avoid.

Once obtained, the intra-nodal flux solution may be utilized to explicitly calculate the

transverse leakage as demonstrated for the x-direction in Equation 5.4 below. Similar

equations describe the transverse leakage for the y and z directions.

φ

∂∂

−+

φ

∂∂

−= ∫∫−−

dz)z,y,x(y

Dh1

h1dy)z,y,x(

zD

h1

h1)x(L g

ng

n2

h

2hzy

gn

gn

2h

2hyz

x,ng

z

z

y

y

(5.4)

where,

hy = length of node in y-coordinate

hz =length of node in z-coordinate

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Using Equation 5.3 for the flux, Equation 5.4 may now be written for the x-direction as

follows:

=)x(L x,ng ( ) ( )z

1z

0n

gn

gny

2y

0n

gn

gn xBB

2zsinh

zD2

xBB2ysinh

yD2

+

γ

∆∆

γ−+

γ

∆∆

γ− (5.5)

The method used in the derivation of Equation 5.5 does not require as many assumptions

about the leakage shape as the QLA does in order to solve for the coefficients. The long

range coupling that the QLA introduces over three consecutive nodes is not present in the

ABFTL method. Equation 5.5 accounts for the exact leakage shape in terms of individual

eigenfunctions of the NDE. As a result, it is expected that the ABFTL method will result in a

more accurate calculation of the transverse leakage term in NEM.

5.3 C3 Benchmarking of the ABFTL Method

The C3 benchmark [44], which is described in detail in Appendix A, was used to benchmark

the ABFTL method in NEM. The two-group assembly-homogenized cross sections (included

in Appendix B) and ADFs for the benchmark, which were generated using PARAGON, were

taken from Reference 45.

The reference solution for the C3 benchmark is NEM run in pin-by-pin geometry using the

same two-group assembly-homogenized cross sections. NEM was also run using the

conventional QLA in nodal (full assembly) geometry. A run was also performed in nodal

geometry with no transverse leakage for comparison (IQL = 0 in NEM). The NEM

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assembly-wise results for all three benchmark runs (ABFTL, QLA and no transverse

leakage) were generated using a one node per assembly (1 npa) model.

The reference keff calculated by NEM in pin-by-pin geometry is 1.25789; the keff calculated

by NEM using the ABFTL and QLA methods is 1.25890 for the ABFTL and 1.25948 for the

QLA method. With no transverse leakage the keff calculated by NEM is 1.26061. For the C3

benchmark, both transverse leakage methods (QLA and ABFTL) required 22 outer source

iterations and ran in less than half of a second of CPU time. Table 5.1 below provides the

results of the C3 benchmark for the normalized assembly powers.

Table 5.1- 2D Power Distribution for C3 Benchmark

Assembly Position

ABFTL Method

Reference Solution (pin-by-pin)

QLA

No Transverse Leakage

NW UO2 1.1513 1.1239 1.1600 1.1758 SW MOX 0.8487 0.8761 0.8400 0.8242

SE UO2 1.1513 1.1239 1.1600 1.1758 NE MOX 0.8487 0.8761 0.8400 0.8242

5.4 C5 Benchmarking of the ABFTL Method

The C5 benchmark [44], which is described in detail in Appendix A, was used as an

additional benchmark to test the ABFTL method in NEM. The primary difference with the

C5 benchmark, in comparison with the C3 benchmark, is the inclusion of the water reflector

on two sides of the 2x2 array. This can be seen in Figure A.4. The two-group assembly-

homogenized cross sections (Appendix B) and ADFs, which were generated using

PARAGON, were taken from Reference 45.

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The reference solution for the C5 benchmark, like the C3 benchmark, is NEM run in pin-by-

pin geometry using the same two-group assembly-homogenized cross sections. NEM was

also run using the conventional QLA in nodal (full assembly) geometry and a run was

performed in nodal geometry with no transverse leakage for comparison (IQL = 0 in NEM).

The NEM assembly-wise results are generated using a one node per assembly (1 npa)

model.

The reference keff calculated by NEM in pin-by-pin geometry is 0.944162; the keff calculated

by NEM using the ABFTL and QLA methods is 0.941716 for the ABFTL method and

0.940968 for the QLA method. The ABFTL method required 29 outer source iterations while

the QLA method required 30 outer source iterations. Both TL methods ran in approximately

0.9 seconds of CPU time. Table 5.2 below provides the results of the C5 benchmark for the

normalized assembly powers.

Table 5.2- 2D Power Distribution for C5 Benchmark

Assembly Position

ABFTL Method

Reference Solution

(pin-by-pin)

QLA

No Transverse

Leakage NW UO2 1.7933 1.7611 1.7960 1.6932 SW MOX 0.7657 0.7887 0.7929 0.8428 SE UO2 0.6754 0.6614 0.6182 0.6212

NE MOX 0.7657 0.7887 0.7929 0.8428

5.5 Mesh Width Sensitivity

The results of the C3 and C5 benchmarks, as presented in Tables 5.1 and 5.2 respectively,

demonstrate that the ABFTL method generally produces improved results in the prediction of

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assembly power in comparison with the QLA method for full assembly geometry (1 npa). In

addition, it was determined from both of the benchmarks that there was no increase in

computational runtime with the implementation of the ABFTL method, which was one of the

goals of this work.

In addition to the benchmarking, the sensitivity of the ABFTL method to the mesh size used

was analyzed. In the previously described benchmark results for the C3 and C5 benchmarks,

an assembly-sized mesh (21.42 cm) was used, as is done in current generation nodal codes. In

the sensitivity study, smaller mesh sizes are used to see the effect on assembly power

prediction.

The C3 benchmark was used for the mesh-width sensitivity study. The 2x2 configuration

(shown in Figure A.3) was run using four radial nodes (in a 2x2 configuration) per assembly

(four npa model) and nine radial nodes (in a 3x3 configuration) per assembly (nine npa

model). NEM cases were performed using both the ABFTL method and the QLA method for

comparison.

The root mean squared (RMS) % error (in comparison to the reference pin-by-pin solution)

of the assembly powers was plotted versus mesh size and is presented in Figure 5.1 below.

As can be seen from Figure 5.1, the RMS % error of the assembly powers for both methods

(QLA and ABFTL) decrease in an approximately linear fashion with decreasing mesh size. It

can also be seen that the ABFTL method approximately converges to the QLA method for

smaller mesh sizes. For larger mesh sizes, such as in nodal assembly geometry, the ABFTL

method provides an improvement in the calculation of assembly power.

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Figure 5.1- Assembly Power RMS % Error vs. Mesh Size

5.6 Discussion and Final Remarks on ABFTL Method

A method which utilizes analytic basis functions to explicitly model the transverse leakage

shape has been implemented in NEM. The intra-nodal flux solution is expressed as a linear

combination of eigenfunctions of the NDE. The coefficients of the eigenfunctions are solved

for using the already calculated surface currents and flux moments as boundary conditions.

The method, therefore, does not require additional parameters to be calculated which could

result in an unwanted increase in computational time. Other very successful advanced

transverse leakage methods have been devised; however, thus far, only the ABFTL method

has been able to improve the transverse leakage treatment without an increase in the

computational runtime. This was able to be accomplished due to the method utilizing already

calculated parameters from the flux and RM solutions (surface currents and flux moments) as

0

0.5

1

1.5

2

2.5

3

3.5

4

4.5

0 5 10 15 20 25

Ass

embl

y Po

wer

RM

S %

Err

or

Mesh Size (cm)

MOX Assembly (ABFTL)

UO2 Assembly (ABFTL)

MOX Assembly (QLA)

UO2 Assembly (QLA)

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boundary conditions, eliminating the need to calculate additional parameters, which is the

thing which causes other advanced transverse leakage methods to require additional

computational resources.

The ABFTL method was tested against two well-known international benchmarks consisting

of mixed UO2/MOX mini-cores. These two benchmarks were selected to provide

environments in which the QLA method is known to suffer from inaccuracies, and therefore

provide ideal platforms with which to test the improvement afforded by the ABFTL method.

Specifically, both benchmarks (C3 and C5) contain MOX fuel which, when bordered by UO2

fuel, would be expected to produce a somewhat complicated leakage shape due to the

significantly different material properties of UO2 and MOX fuel. This complicated leakage

shape would not be well approximated by a quadratic polynomial. Also, for the C5

benchmark, a core boundary/reflector is present, which again will provide a challenge for the

traditional QLA method.

The results for the two benchmarks demonstrated that the ABFTL method generally

improves the accuracy of NEM in calculating the assembly (nodal) power distribution. A

small improvement in the calculation of keff was also obtained with the ABFTL method in

comparison with the QLA method. The reason for the improvement with the ABFTL method

compared with the QLA method, as mentioned in Section 5.2, lies within the basic

assumptions of the methods. The QLA assumes the transverse leakage shape to be well

approximated by a quadratic polynomial, which is fitted across three nodes, with the

coefficients for the polynomial being obtained by forcing agreement with the average leakage

values of the nodes of interest (which are calculated using the average net current values),

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along with its two neighboring nodes. In cores with complicated leakage shapes, the

transverse leakage shape is too complicated to be well approximated by the quadratic

polynomial. The ABFTL method, however, improves upon the transverse leakage treatment

by incorporating information about the transverse leakage shape into the nodal solution itself.

This is done by using the eigenfunctions of the NDE, described in terms of a linear series of

harmonic functions, to describe the intra-nodal flux shape explicitly; this is then used to solve

for the transverse leakage term, which is subsequently used in the flux solution and RM

equations. By directly incorporating information from the transverse leakage shape into the

nodal solution itself, a more accurate representation of the transverse leakage term is obtained

without the long range coupling and a priori leakage shape assumption of the QLA.

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CHAPTER 6

DISCONTINUITY FACTORS

FOR SP3 EQUATIONS

6.1 Introduction

The success of nodal methods based upon diffusion theory is due in large part to the

reconstruction of the reference transport theory solution via the use of surface discontinuity

factors (DFs). However, the SP3 solution does not provide an explicit representation of the

angular flux. Furthermore, it is not possible to obtain from a reference transport solution the

corresponding SP3 solution, making it impossible to calculate DFs for the SP3 nodal solution.

A consequence of this is that a diffusion theory solution obtained using DFs may be more

accurate than a corresponding SP3 solution without DFs [38].

To remedy this problem, the SPN equations can be written in a way so that the angular flux

can be reconstructed from the SPN solution [38]. The SPN formulation is derived using the

physical assumption that neutron transport is approximately one-dimensional in any local

point in space. Using this assumption, it may be argued that the singular characteristic

angular direction must coincide with the direction of the net current vector, allowing for the

expression of a unique angular flux representation. From this angular flux representation,

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surface discontinuity factors can be calculated to ensure that the SP3 solution remains

superior to the diffusion theory solution.

The method of Chao and Yamamoto [38] will be introduced into NEM to allow the SP3

solution (discussed in Chapter 3) to be able to obtain consistency with the reference transport

theory solution by the use of surface DFs. This work, at present, remains unfinished. The

theory will be discussed below. Following this will be suggestions for the completion of this

work in the future. The completion of the SP3 DF work for NEM will be documented in

future work.

6.2 Expression of Angular Flux in SPN Approximation

In order to be able to utilize DFs, and therefore to be able to reconstruct from the

homogenized nodal solution the corresponding heterogeneous transport solution, the angular

flux must be able to be reconstructed from the SPN solution. To express the angular flux in

the SPN approximation, it is most convenient to begin with the physical picture of the SPN

approximation; namely, that neutron transport behaves as a continuously varying 1-D

problem whose direction changes from point to point. Chao and Yamamoto [38] explain in

greater detail that this local direction must be the direction of the net current vector. The

mathematical ramifications of this assumption can be shown using the even and odd parity

portions of the angular neutron flux (shown in Equations 6.1 and 6.2 below).

π

=φΣ+φ∇•Ω4So

eto (6.1)

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et

o1

φ∇•ΩΣ

−=φ (6.2)

where,

φο = odd parity flux

φe = even parity flux

Σt = total cross section

So = isotropic source

In a truly 1-D problem the angular distribution of the neutron flux is azimuthally symmetric

about the characteristic direction, which is the direction of the net current. From Equations

6.1 and 6.2 above, it can be seen that the only instance in which the angular flux is

azimuthally symmetric with respect to the net current is either in a 1-D problem or in the

diffusion approximation, in which the gradient of the even parity flux is parallel to the current

[38]. In a 1-D (planar) problem, the even parity flux may be expanded by the complete set of

basis functions of Legendre polynomials as shown in Equation 6.3 below.

)r()(P4

1n2)r,( nneven

e φµπ+

=µφ ∑ (6.3)

where,

r = 1-D direction

µ = projection of Ω in the direction of the net current

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If Equation 6.3 is inserted into Equation 6.2, Equation 6.4 is the result.

)r()(P4

1n2)r,( nneven t

o φ∇•ΩµΣπ+

=µφ ∑ (6.4)

If one combines Equations 6.3 and 6.4, the following is obtained for the angular flux

representation:

∇•Ω

Σ−

+=Ω ∑ )(1)()(

412),( rrPnr n

tnn φφµ

πφ (6.5)

Using Equation 6.5 the angular flux is then expanded up to P2, which is angularly symmetric

with respect to the net current vector. This is shown in Equation 6.6 below.

[ ])r()(P5J3)r(41)r,( 2J20 φµ+•Ω+φπ

=Ωφ (6.6)

By multiplying Equation 6.6 by the angular vector Ω and performing integration over half of

the angular space (this is performed in detailed in Appendix A of Reference 38), an

expression for the partial currents in terms of the first and second surface fluxes may be

obtained. This relationship is expressed in Equation 6.7 below.

20inout 85

21JJ φ+φ=+ (6.7)

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Using the relationship derived in Equation 6.7, the reference value for φ2 can be easily

derived in terms of parameters which are available from the reference transport solution;

namely, the partial currents and the reference surface flux φ0

. This is demonstrated in

Equation 6.8 below.

[ ]0inout2 J2J254

φ−+=φ (6.8)

6.3 Calculation of Discontinuity Factors for SP3

Method

In current diffusion theory codes, the DFs are obtained from the ratio of the surface fluxes

between the transport solution and the diffusion solution. The surface fluxes of the transport

solution and the diffusion solution are allowed to be discontinuous in order to maintain the

requirement of continuity of surface currents between the two methods. This is necessary to

maintain continuity of the reaction rates between the transport solution (heterogeneous) and

the diffusion solution (homogeneous). The DFs are obtained from a reference case (lattice

calculation) and applied in the nodal solution when implementing the interface boundary

condition between nodes. The same basic procedure should be able to be used in an SP3

calculation in order to obtain DFs. However, for obvious reasons, there are more variables

which must be accounted for in the reference solution.

In the reference transport solution, which is used to collapse the cross sections from the fine

group structure to the few group structure as well as to homogenize the heterogeneous flux

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into a homogenous flux, the surface flux and surface currents are obtained in the ordinary

solution procedure. As stated in the previous section, the second surface flux moment may

be obtained from the partial currents and the reference surface flux φ0 using Equation 6.8.

The ratio of the SP3 values of φ0 and φ2

to their reference values provides the DF to be used

to reproduce the reference transport solution [38].

In the method of Yamamoto and Chao [38], it is stated that the reference values of ∇ φ0 and

φ2 should be used as the nodal boundary conditions to solve for the surface flux moments

φ0 and φ2

. In this way the nodal reaction rates should be preserved. They suggest that the

reference value for φ0

be calculated in the transport solution, then use the fact that the net

current is a linear combination of φ0 and ∇ φ2 to find the reference value for ∇ φ2

.

However, Yu et. al. [51] express pragmatic concerns with this approach, stating that it is

frequently difficult to calculate this gradient ( φ0) effectively. To remedy this problem, they

present an approach which should be given consideration in the implementation of the SP3

DFs in NEM. In their paper, they suggest a method in which the SP3 method is viewed as an

improvement on the diffusion theory. Thus, they calculate the DF for φ0

(surface flux

moment) the same as in the diffusion theory calculation.

An interesting proposition from Reference 51 is the fact that, for small values of φ2 the

discontinuity factor calculated for φ2 may be too large, giving a non-realistic result. In the

limit of indefinitely small φ2 it is known that the SP3 should, in fact must, converge to

diffusion theory. Thus, Yu et. al. propose that, instead of having a DF for φ2

200 2~φφφ +=

, a DF for the

synthesized flux which they call is instead used.

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CHAPTER 7

CONCLUSIONS AND FUTURE WORK

7.1 Conclusions

Modern reactor core designs, due to the use of MOX fuel, low leakage core designs, and

aggressive operating strategies, have led to an increased demand on the analysis codes used

to analyze their behavior due to the increased complexity/heterogeneity of such designs. New

methodologies will have to be developed to enable the analysis of these challenging core

configurations. The work documented herein was undertaken with the objective of improving

the analytical capability of the PSU NEM code to be able to analyze the highly

heterogeneous core designs which are currently in use in commercial power reactors. These

methodological improvements were pursued with the goal of not sacrificing the

computational efficiency which is required by modern reactor analysis codes.

A transport solution based upon the SP3 approximation was implemented into the NEM code.

The SP3 solution was benchmarked against the OECD/NEA C5G7 MOX benchmarks in both

2-D pin-by-pin geometry and 3-D pin-by-pin geometry with control rods inserted. The

diffusion theory approximation in NEM was also benchmarked for comparison. Results for

keff, pin power and assembly power were presented in Chapter 3. The SP3 results from the

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C5G7 benchmarks were shown to generally produce a minor improvement (a few tenths of a

percent) in the prediction of the pin and assembly powers. The improvement in the results

with the SP3 method, however, was limited by the absence of P1 scattering cross sections in

the C5G7 benchmarks, which is one of the primary parameters in which the SP3 method

provides improvement over the diffusion theory solution. It can therefore be concluded that,

with the absence of P1 scattering cross sections, the small increase in accuracy obtained with

the SP3 method is not worth the increase in runtime that it entails (the SP3 runtime

approximately doubles in comparison to the diffusion theory solution due to the solution of

two RM equations for the SP3 solution compared to one RM equation for the diffusion theory

solution).

The SP3 results of the OECD/NEA and U.S. NRC PWR MOX/UO2 Core Transient

Benchmark, which included P1 scattering cross sections, demonstrated a more significant

improvement than the C5G7 benchmarks. The prediction of reactivity improved with the SP3

method by approximately 100 and 300 pcm for the ARO and ARI benchmarks respectively.

The error in the pin and assembly powers was reduced with the SP3 solution in comparison

with the diffusion theory solution in both the ARO and ARI benchmarks.

The OECD/NEA and U.S. NRC PWR MOX/UO2 Core Transient Benchmark results for

NEM with the SP3 solution demonstrated a small improvement in the results with the SPH

factors included in comparison to the SP3 solution with no SPH factors. However, for the

ARO configuration, which is the only configuration in which SPH factors were available, the

amount of improvement with the SPH factors was found to be small. This conclusion concurs

with the results presented in Reference 50 for the ARO configuration of this benchmark.

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For the ARI configuration of the benchmark, the SP3 solution was found to provide an

improvement over the diffusion theory solution in the core power distribution calculation, as

well as the calculation of the control rod worths, which are themselves a measurement of core

power distribution. However, even with the SP3 solution, the errors in core power distribution

were still found to be larger than desired. This is almost certainly due to the absence of SPH

factors, which were unfortunately only available for the ARO benchmark case. This

benchmarking work agrees with the conclusions of Reference 50, namely that the

improvement obtained with the SP3 method in heavily rodded configurations requires SPH

factors to obtain an acceptable result. This suggests that the primary source of code error in

problems with heavily rodded conditions is in fact the cross sections. The form of nodal

solution (diffusion theory, SP3, etc.) has a limited effect on the calculation of the overall

power distribution.

The larger improvement obtained with the OECD/NEA and U.S. NRC PWR MOX/UO2 Core

Transient benchmark in comparison with the C5G7 benchmark is almost certainly due to the

inclusion of the P1 scattering cross sections. It is therefore concluded from the results of the

benchmarking that it is only advantageous to use the SP3 solution when P1 scattering cross

sections are present.

The SA-NEM solution was implemented into the NEM code and was benchmarked against

three well known mixed UO2/MOX benchmarks. For the OECD/NEA C5G7 MOX

benchmark, which tested the code in 2-D using pin-by-pin geometry, the SA-NEM solution

was demonstrated to produce comparable results in comparison with the P-NEM solution.

This was the expected result due to the fact that the spatial variation in the flux across a pin

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cell is quite small. Therefore, the quadratic polynomial well approximates the spatial

variation of the flux within the node. This effect is demonstrated in the mesh-size sensitivity

study which was performed in which it was demonstrated that the SA-NEM and P-NEM

solutions converge to one another in the limit of decreasing mesh-size, which explains the

similar results for the two methods for the C5G7 benchmark. The SA-NEM solution was also

benchmarked against the C3 and C5 benchmarks which were modeled in full assembly

geometry. The SA-NEM results for the C3 and C5 benchmarks are quite promising. For the

C3 benchmark, the % error of the assembly powers was reduced by nearly one half with the

SA-NEM solution in comparison with the P-NEM solution. Similar improvements in the

calculation of assembly power were observed for the UO2 assemblies in the C5 benchmark.

An explicit method for calculating the transverse leakage has been developed which is based

upon the use of analytic basis functions, which represent individual eigenfunctions of the

neutron diffusion equation. The intra-nodal flux solution is expressed as an eigenspace, and

may be solved for using the already calculated surface currents and flux moments as

boundary conditions. The salient feature of the method, therefore, is that no ad hoc

presumptions are made with regard to the leakage shape. The individual eigenfunctions are

calculated based upon already calculated parameters from the flux solution and response

matrix solution, and therefore no additional parameters are introduced into the problem

which could lead to an unwanted increase in computation time as with other advanced TL

methods.

The ABFTL method was benchmarked using the C3 and C5 mixed UO2/MOX mini-cores.

The ABFTL method was demonstrated to produce improved results in the calculation of

assembly power in comparison with the standard QLA treatment. Equally important, the

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ABFTL method did not result in an increase in runtime as is the case with other advanced

transverse leakage treatments. In a mesh-size sensitivity study, the ABFTL method was

shown to converge to the QLA solution with decreasing mesh size.

7.2 Recommendations for Future Work

One of the areas where this research could be expanded on is the implementation of the DFs

for the SP3 solution which was discussed in Chapter 6. As stated previously, the success of

nodal methods is due in large part to the reconstruction of the reference transport theory

solution via the use of surface DFs. However, the SP3 solution does not provide an explicit

representation of the angular flux; furthermore, the SP3 solution cannot be obtained from the

reference transport solution, making it impossible to calculate surface DFs for the SP3 nodal

solution. A consequence of this is that a diffusion theory solution obtained using DFs may be

more accurate than a corresponding SP3 solution without DFs. Chao and Yamamoto [38]

have presented a very promising remedy for this problem which should be implemented into

the SP3 solution in NEM. The concerns and recommendations of Yu et. al in Reference 51

should also be addressed. A summary of the theoretical background of the method is

provided in Chapter 6. The coding and testing of the method remains for future development.

An additional benchmark for the SP3 solution should also be performed. As demonstrated in

Chapter 3, the SP3 solution with SPH factors provides the best results for the OECD/NEA

and U.S. NRC PWR MOX/UO2 Core Transient Benchmark at ARO conditions. However, for

the relatively homogeneous ARO case, the difference between the SP3 runs with SPH factors

and without SPH factors concurs with the authors of Reference 50, which state that the SPH

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factors make very little difference for an unrodded configuration. Their true benefit comes in

rodded configurations, where the environment in the core can be expected to be significantly

different than the environment in the cross section generating code. Therefore, rodded SPH

factors should be obtained for the U.S. NRC PWR MOX/UO2 Core Transient Benchmark and

the rodded case (ARI case) should be run with SPH factors. It is expected that the NEM

results using the SP3 method with SPH factors should result in a significant improvement

over the current results with regard to the assembly power and pin power. This is what was

observed in Reference 50.

Due to the larger than desired runtime for the SP3 pin-by-pin calculations, it is desirable for

the SP3 solution to be parallelized. It may be possible, due to the construction of the NEM

code, to solve the response matrix equations in parallel, which should lead to a decrease in

runtime of roughly one half.

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Technical Committee Meeting on Homogenization Methods in Reactor Physics, Lugano,

November 13-15, 1978.

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Meeting on Advances in Mathematical Methods for the Solution of Nuclear Engineering

Problems, Volume 2, p.59, Munich, April 27-29, 1981.

35. Wagner M.R. and Koebke K., Progress in Nodal Reactor Analysis, A Topical Meeting on

Advances in Reactor Computations, Volume 2, p.941, Salt Lake City, March 28-30,

1983.

36. K.S. Smith, Spatial Homogenization Methods for Light Water Reactor Analysis, nuclear

engineering thesis, MIT, 1980.

37. G. Gomes, Importance of Assembly Discontinuity Factors In Simulating Reactor Cores

Containing Highly Heterogeneous Fuel Assemblies, Excerpt from the Proceedings of the

COMSOL Conference 2009 Boston.

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38. Y. Chao and A. Yamamoto, The Explicit Representation for the Angular Flux Solution in

the Simplified PN (SPN) Theory, PHYSOR 2012, Knoxville, Tennessee, April 15-20,

2012.

39. G.I. Marchuk, V.I. Lebedev, Numerical Methods in the Theory of Neutron Transport,

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40. R. Marchuk, Theory of the Slowing-Down of Neutrons by Elastic Collision with Atomic

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Report, NEA/NSC/DOC(2006)20, January 2007.

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Neutron Diffusion Problems, Atomkernenergie-Kerntechnik 1981.

44. C. Cavarec, et al., The OECD/NEA Benchmark Calculations of Power Distributions

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45. B.D. Ivanov, Methodology for Embedded Transport Core Calculation, PhD Thesis, The

Pennsylvania State University, December 2007.

46. James Reed, Semi-Analytical Nodal Expansion Method, Honors Thesis, The Pennsylvania

State University, Spring 2010.

47. D.V. Altiparmakov and D.I. Tomasevic, Variational Formulation of a Higher Order

Nodal Diffusion Method, Nuclear Science and Engineering: 105, 256-270, 1990.

48. A.M. Ougouag and H.L. Rajic, ILLICO-HO: A Self-Consistent Higher Order Coarse-

Mesh Nodal Method, Nuclear Science and Engineering: 100, 332-341, 1988.

49. T.M. Beam et. al., Nodal Kinetics Model Upgrade in the Penn State Coupled TRAC/NEM

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Codes, Annals of Nuclear Energy: 26, pp. 1205-1219, 1999.

50. U. Grundmann and S. Mittag, Super-Homogenisation Factors in Pinwise Calculations by

the Reactor Dynamics Code DYN3D, Annals of Nuclear Energy: 38, pp. 2111-2119,

2011.

51. L. Yu, et. al., The Calculation Method for SP3 Discontinuity Factor and its Application,

Annals of Nuclear Energy: 69, pp. 14-24, 2014.

52. Hébert, A., A Consistent Technique for the pin-by-pin Homogenization of a Pressurized

Water Reactor Assembly, Nuclear Science and Engineering: 113, pp. 227-238, 1993.

53. Tomasz Kozlowski, Chang-Ho Lee, and Thomas Downar, Benchmarking of the

Multigroup, Fine Mesh, SP3 Methods in PARCS with the VENUS-2 MOX Critical

Experiments, PHYSOR 2002, October 7-10, 2002.

54. Tae Young Han, Han Gyu Joo, and Chang Hyo Kim, Two-Group CMFD Accelerated

Multi-group Calculation with a Semi-Analytic Nodal Kernel, PHYSOR-2006,

Vancouver, BC, Canada, September 10-14, 2006.

55. M. R. Wagner, Three-dimensional Nodal Diffusion and Transport Theory Methods for

Hexagonal-z geometry, Nuclear Science and Engineering: Volume 103, 1989.

56. H. L. Rajic, A. M. Ougouag, ILLICO: A Nodal Neutron Diffusion Method for Modern

Computer Architectures, Nuclear Science Engineering: Volume 103, 1989.

57. Kavenoky, A., The SPH Homogenization Method, Proceedings of the Specialists’

Meeting on Homogenization Methods in Reactor Physics, Lugano, Switzerland, 1978.

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the Extended Coupled Code System DYNSUB, Accepted for Publication in Annals of

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APPENDIX A. DESCRIPTION OF THE BENCHMARKS

2-D OECD/NEA C5G7 MOX Benchmark

The 2-D OECD/NEA C5G7 MOX benchmark is described in Reference 3. This benchmark

was used to test the performance of the SP3 and semi-analytical nodal expansion method

solution. This benchmark was selected so as to provide a challenging configuration for the

methods to be adequately exercised.

The 2-D benchmark is a 2x2 representation of a mini-core with UO2 and MOX fuel

assemblies, and a water reflector on the bottom and right faces. The overall dimensions of the

2-D configuration are 64.26 x 64.26 cm, with each assembly being 21.42 x 21.42 cm. Each of

the fuel assemblies is a 17 x 17 lattice, with each pin cell having a side length of 1.26 cm.

The thickness of the water moderator reflector is the length of one assembly (21.42 cm).

The layout of the 2x2 configuration of the assemblies is shown in Figure A.1 below, followed

by a pin-by-pin layout of the UO2 and MOX fuel assemblies shown in Figure A.2. As can be

seen in Figure A.1, vacuum boundary conditions are imposed on the water reflector

boundaries and mirror reflective boundary conditions are imposed on the assembly

boundaries.

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Figure A.1- OECD/NEA 2-D C5G7 MOX Benchmark Core Configuration

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Figure A.2- Pin-by-Pin Layout of 2D C5G7 MOX Benchmark Fuel Assemblies

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34123456789

10111213141516171819202122232425262728293031323334

UO2 Fuel Guide Tube 4.3% MOX Fuel8.7% MOX Fuel Fission Chamber 7.0% MOX Fuel

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C3 and C5 Benchmarks

The C3 and C5 UO2/MOX benchmarks are described in Reference 44. The C3 and C5

benchmark problems are used to investigate the performance of the semi-analytical nodal

expansion method solution and the analytic basis function transverse leakage approximation.

Each one is described below.

The C3 benchmark is a 2x2 mini-core with two MOX assemblies and two UO2

assemblies

and reflected boundary conditions on all four faces. Each assembly has a side length of

approximately 21.42 cm on each face. The C3 problem core layout is shown in Figure A.3

below.

Figure A.3- C3 Core Configuration

The C5 benchmark is a 2x2 mini-core with two MOX assemblies and two UO2 assemblies,

but with a water reflector on two faces. The C5 benchmark has vacuum boundary conditions

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on the water reflector faces and mirror reflective boundary conditions on the assembly faces.

Each assembly has a side length of approximately 21.42 cm on each face. The water reflector

is 21.42 cm thick. The C5 problem core layout is shown in Figure A.4 below.

Figure A.4- C5 Core Configuration

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APPENDIX B. BENCHMARK CROSS SECTIONS

Table B.1- C5G7 Control Rod Cross Sections

Transport Absorption Capture Group Cross Section Cross Section Cross Section

1 2.16768E-01 1.70490E-03 1.70490E-03 2 4.80098E-01 8.36224E-03 8.36224E-03 3 8.86369E-01 8.37901E-02 8.37901E-02 4 9.70009E-01 3.97797E-01 3.97797E-01 5 9.10482E-01 6.98763E-01 6.98763E-01 6 1.13775E+00 9.29508E-01 9.29508E-01 7 1.84048E+00 1.17836E+00 1.17836E+00

Scattering Block to Group 1 to Group 2 to Group 3 to Group 4 to Group 5 to Group 6 to Group 7

Group 1 1.70563E-01 0.00000E+00 0.00000E+00 0.00000E+00 0.00000E+00 0.00000E+00 0.00000E+00 Group 2 4.44012E-02 4.71050E-01 0.00000E+00 0.00000E+00 0.00000E+00 0.00000E+00 0.00000E+00 Group 3 9.83670E-05 6.85480E-04 8.01859E-01 0.00000E+00 0.00000E+00 0.00000E+00 0.00000E+00 Group 4 1.27786E-07 3.91395E-10 7.20132E-04 5.70752E-01 6.55562E-05 0.00000E+00 0.00000E+00 Group 5 0.00000E+00 0.00000E+00 0.00000E+00 1.46015E-03 2.07838E-01 1.02427E-03 0.00000E+00 Group 6 0.00000E+00 0.00000E+00 0.00000E+00 0.00000E+00 3.81486E-03 2.02465E-01 3.53043E-03 Group 7 0.00000E+00 0.00000E+00 0.00000E+00 0.00000E+00 3.69760E-09 4.75290E-03 6.58597E-01

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Table B.2- C5G7 Moderator Cross Sections

Transport Absorption Capture

Group Cross Section Cross Section Cross Section 1 1.59206E-01 6.01050E-04 6.01050E-04 2 4.12970E-01 1.57930E-05 1.57930E-05 3 5.90310E-01 3.37160E-04 3.37160E-04 4 5.84350E-01 1.94060E-03 1.94060E-03 5 7.18000E-01 5.74160E-03 5.74160E-03 6 1.25445E+00 1.50010E-02 1.50010E-02 7 2.65038E+00 3.72390E-02 3.72390E-02

Scattering Block to Group 1 to Group 2 to Group 3 to Group 4 to Group 5 to Group 6 to Group 7

Group 1 4.44777E-02 0.00000E+00 0.00000E+00 0.00000E+00 0.00000E+00 0.00000E+00 0.00000E+00 Group 2 1.13400E-01 2.82334E-01 0.00000E+00 0.00000E+00 0.00000E+00 0.00000E+00 0.00000E+00 Group 3 7.23470E-04 1.29940E-01 3.45256E-01 0.00000E+00 0.00000E+00 0.00000E+00 0.00000E+00 Group 4 3.74990E-06 6.23400E-04 2.24570E-01 9.10284E-02 7.14370E-05 0.00000E+00 0.00000E+00 Group 5 5.31840E-08 4.80020E-05 1.69990E-02 4.15510E-01 1.39138E-01 2.21570E-03 0.00000E+00 Group 6 0.00000E+00 7.44860E-06 2.64430E-03 6.37320E-02 5.11820E-01 6.99913E-01 1.32440E-01 Group 7 0.00000E+00 1.04550E-06 5.03440E-04 1.21390E-02 6.12290E-02 5.37320E-01 2.48070E+00

Table B.3- C5G7 Guide Tube Cross Sections

Transport Absorption Capture Group Cross Section Cross Section Cross Section

1 1.26032E-01 5.11320E-04 5.11320E-04 2 2.93160E-01 7.58010E-05 7.58010E-05 3 2.84240E-01 3.15720E-04 3.15720E-04 4 2.80960E-01 1.15820E-03 1.15820E-03 5 3.34440E-01 3.39750E-03 3.39750E-03 6 5.65640E-01 9.18780E-03 9.18780E-03 7 1.17215E+00 2.32420E-02 2.32420E-02

Scattering Block to Group 1 to Group 2 to Group 3 to Group 4 to Group 5 to Group 6 to Group 7

Group 1 6.61659E-02 0.00000E+00 0.00000E+00 0.00000E+00 0.00000E+00 0.00000E+00 0.00000E+00 Group 2 5.90700E-02 2.40377E-01 0.00000E+00 0.00000E+00 0.00000E+00 0.00000E+00 0.00000E+00 Group 3 2.83340E-04 5.24350E-02 1.83297E-01 0.00000E+00 0.00000E+00 0.00000E+00 0.00000E+00 Group 4 1.46220E-06 2.49900E-04 9.23970E-02 7.88511E-02 3.73330E-05 0.00000E+00 0.00000E+00 Group 5 2.06420E-08 1.92390E-05 6.94460E-03 1.70140E-01 9.97372E-02 9.17260E-04 0.00000E+00 Group 6 0.00000E+00 2.98750E-06 1.08030E-03 2.58810E-02 2.06790E-01 3.16765E-01 4.97920E-02 Group 7 0.00000E+00 4.21400E-07 2.05670E-04 4.92970E-03 2.44780E-02 2.38770E-01 1.09912E+00

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Table B.4- C5G7 UO2 Fuel-Clad Cross Sections

Transport Absorption Capture Fission Group Cross Section Cross Section Cross Section Cross Section ν

1 1.77949E-01 8.02480E-03 8.12740E-04 7.21206E-03 2.78145E+00 2 3.29805E-01 3.71740E-03 2.89810E-03 8.19301E-04 2.47443E+00 3 4.80388E-01 2.67690E-02 2.03158E-02 6.45320E-03 2.43383E+00 4 5.54367E-01 9.62360E-02 7.76712E-02 1.85648E-02 2.43380E+00 5 3.11801E-01 3.00200E-02 1.22116E-02 1.78084E-02 2.43380E+00 6 3.95168E-01 1.11260E-01 2.82252E-02 8.30348E-02 2.43380E+00 7 5.64406E-01 2.82780E-01 6.67760E-02 2.16004E-01 2.43380E+00

Scattering Block to Group 1 to Group 2 to Group 3 to Group 4 to Group 5 to Group 6 to Group 7

Group 1 1.27537E-01 0.00000E+00 0.00000E+00 0.00000E+00 0.00000E+00 0.00000E+00 0.00000E+00 Group 2 4.23780E-02 3.24456E-01 0.00000E+00 0.00000E+00 0.00000E+00 0.00000E+00 0.00000E+00 Group 3 9.43740E-06 1.63140E-03 4.50940E-01 0.00000E+00 0.00000E+00 0.00000E+00 0.00000E+00 Group 4 5.51630E-09 3.14270E-09 2.67920E-03 4.52565E-01 1.25250E-04 0.00000E+00 0.00000E+00 Group 5 0.00000E+00 0.00000E+00 0.00000E+00 5.56640E-03 2.71401E-01 1.29680E-03 0.00000E+00 Group 6 0.00000E+00 0.00000E+00 0.00000E+00 0.00000E+00 1.02550E-02 2.65802E-01 8.54580E-03 Group 7 0.00000E+00 0.00000E+00 0.00000E+00 0.00000E+00 1.00210E-08 1.68090E-02 2.73080E-01

Table B.5- C5G7 4.3% MOX Fuel-Clad Cross Sections

Transport Absorption Capture Fission Group Cross Section Cross Section Cross Section Cross Section ν

1 1.78731E-01 8.43390E-03 8.06860E-04 7.62704E-03 2.85209E+00 2 3.30849E-01 3.75770E-03 2.88080E-03 8.76898E-04 2.89099E+00 3 4.83772E-01 2.79700E-02 2.22717E-02 5.69835E-03 2.85486E+00 4 5.66922E-01 1.04210E-01 8.13228E-02 2.28872E-02 2.86073E+00 5 4.26227E-01 1.39940E-01 1.29177E-01 1.07635E-02 2.85447E+00 6 6.78997E-01 4.09180E-01 1.76423E-01 2.32757E-01 2.86415E+00 7 6.82852E-01 4.09350E-01 1.60382E-01 2.48968E-01 2.86780E+00

Scattering Block to Group 1 to Group 2 to Group 3 to Group 4 to Group 5 to Group 6 to Group 7

Group 1 1.28876E-01 0.00000E+00 0.00000E+00 0.00000E+00 0.00000E+00 0.00000E+00 0.00000E+00 Group 2 4.14130E-02 3.25452E-01 0.00000E+00 0.00000E+00 0.00000E+00 0.00000E+00 0.00000E+00 Group 3 8.22900E-06 1.63950E-03 4.53188E-01 0.00000E+00 0.00000E+00 0.00000E+00 0.00000E+00 Group 4 5.04050E-09 1.59820E-09 2.61420E-03 4.57173E-01 1.60460E-04 0.00000E+00 0.00000E+00 Group 5 0.00000E+00 0.00000E+00 0.00000E+00 5.53940E-03 2.76814E-01 2.00510E-03 0.00000E+00 Group 6 0.00000E+00 0.00000E+00 0.00000E+00 0.00000E+00 9.31270E-03 2.52962E-01 8.49480E-03 Group 7 0.00000E+00 0.00000E+00 0.00000E+00 0.00000E+00 9.16560E-09 1.48500E-02 2.65007E-01

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Table B.6- C5G7 7.0% MOX Fuel-Clad Cross Sections

Transport Absorption Capture Fission Group Cross Section Cross Section Cross Section Cross Section ν

1 1.81323E-01 9.06570E-03 8.11240E-04 8.25446E-03 2.88498E+00 2 3.34368E-01 4.29670E-03 2.97105E-03 1.32565E-03 2.91079E+00 3 4.93785E-01 3.28810E-02 2.44594E-02 8.42156E-03 2.86574E+00 4 5.91216E-01 1.22030E-01 8.91570E-02 3.28730E-02 2.87063E+00 5 4.74198E-01 1.82980E-01 1.67016E-01 1.59636E-02 2.86714E+00 6 8.33601E-01 5.68460E-01 2.44666E-01 3.23794E-01 2.86658E+00 7 8.53603E-01 5.85210E-01 2.22407E-01 3.62803E-01 2.87539E+00

Scattering Block to Group 1 to Group 2 to Group 3 to Group 4 to Group 5 to Group 6 to Group 7

Group 1 1.30457E-01 0.00000E+00 0.00000E+00 0.00000E+00 0.00000E+00 0.00000E+00 0.00000E+00 Group 2 4.17920E-02 3.28428E-01 0.00000E+00 0.00000E+00 0.00000E+00 0.00000E+00 0.00000E+00 Group 3 8.51050E-06 1.64360E-03 4.58371E-01 0.00000E+00 0.00000E+00 0.00000E+00 0.00000E+00 Group 4 5.13290E-09 2.20170E-09 2.53310E-03 4.63709E-01 1.76190E-04 0.00000E+00 0.00000E+00 Group 5 0.00000E+00 0.00000E+00 0.00000E+00 5.47660E-03 2.82313E-01 2.27600E-03 0.00000E+00 Group 6 0.00000E+00 0.00000E+00 0.00000E+00 0.00000E+00 8.72890E-03 2.49751E-01 8.86450E-03 Group 7 0.00000E+00 0.00000E+00 0.00000E+00 0.00000E+00 9.00160E-09 1.31140E-02 2.59529E-01

Table B.7- C5G7 8.7% MOX Fuel-Clad Cross Sections

Transport Absorption Capture Fission

Group Cross Section Cross Section Cross Section Cross Section ν 1 1.83045E-01 9.48620E-03 8.14110E-04 8.67209E-03 2.90426E+00 2 3.36705E-01 4.65560E-03 3.03134E-03 1.62426E-03 2.91795E+00 3 5.00507E-01 3.62400E-02 2.59684E-02 1.02716E-02 2.86986E+00 4 6.06174E-01 1.32720E-01 9.36753E-02 3.90447E-02 2.87491E+00 5 5.02754E-01 2.08400E-01 1.89142E-01 1.92576E-02 2.87175E+00 6 9.21028E-01 6.58700E-01 2.83812E-01 3.74888E-01 2.86752E+00 7 9.55231E-01 6.90170E-01 2.59571E-01 4.30599E-01 2.87808E+00

Scattering Block to Group 1 to Group 2 to Group 3 to Group 4 to Group 5 to Group 6 to Group 7

Group 1 1.31504E-01 0.00000E+00 0.00000E+00 0.00000E+00 0.00000E+00 0.00000E+00 0.00000E+00 Group 2 4.20460E-02 3.30403E-01 0.00000E+00 0.00000E+00 0.00000E+00 0.00000E+00 0.00000E+00 Group 3 8.69720E-06 1.64630E-03 4.61792E-01 0.00000E+00 0.00000E+00 0.00000E+00 0.00000E+00 Group 4 5.19380E-09 2.60060E-09 2.47490E-03 4.68021E-01 1.85970E-04 0.00000E+00 0.00000E+00 Group 5 0.00000E+00 0.00000E+00 0.00000E+00 5.43300E-03 2.85771E-01 2.39160E-03 0.00000E+00 Group 6 0.00000E+00 0.00000E+00 0.00000E+00 0.00000E+00 8.39730E-03 2.47614E-01 8.96810E-03 Group 7 0.00000E+00 0.00000E+00 0.00000E+00 0.00000E+00 8.92800E-09 1.23220E-02 2.56093E-01

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Table B.8- C5G7 Fission Chamber Cross Sections

Transport Absorption Capture Fission Group Cross Section Cross Section Cross Section Cross Section ν

1 1.26032E-01 5.11320E-04 5.11315E-04 4.79002E-09 2.76283E+00 2 2.93160E-01 7.58130E-05 7.58072E-05 5.82564E-09 2.46239E+00 3 2.84250E-01 3.16430E-04 3.15966E-04 4.63719E-07 2.43380E+00 4 2.81020E-01 1.16750E-03 1.16226E-03 5.24406E-06 2.43380E+00 5 3.34460E-01 3.39770E-03 3.39755E-03 1.45390E-07 2.43380E+00 6 5.65640E-01 9.18860E-03 9.18789E-03 7.14972E-07 2.43380E+00 7 1.17214E+00 2.32440E-02 2.32419E-02 2.08041E-06 2.43380E+00

Scattering Block to Group 1 to Group 2 to Group 3 to Group 4 to Group 5 to Group 6 to Group 7

Group 1 6.61659E-02 0.00000E+00 0.00000E+00 0.00000E+00 0.00000E+00 0.00000E+00 0.00000E+00 Group 2 5.90700E-02 2.40377E-01 0.00000E+00 0.00000E+00 0.00000E+00 0.00000E+00 0.00000E+00 Group 3 2.83340E-04 5.24350E-02 1.83425E-01 0.00000E+00 0.00000E+00 0.00000E+00 0.00000E+00 Group 4 1.46220E-06 2.49900E-04 9.22880E-02 7.90769E-02 3.73400E-05 0.00000E+00 0.00000E+00 Group 5 2.06420E-08 1.92390E-05 6.93650E-03 1.69990E-01 9.97570E-02 9.17420E-04 0.00000E+00 Group 6 0.00000E+00 2.98750E-06 1.07900E-03 2.58600E-02 2.06790E-01 3.16774E-01 4.97930E-02 Group 7 0.00000E+00 4.21400E-07 2.05430E-04 4.92560E-03 2.44780E-02 2.38760E-01 1.09910E+00

Table B.9- C5G7 Fission Spectrum

Group χ(E)

1 5.87910E-01 2 4.11760E-01 3 3.39060E-04 4 1.17610E-07 5 0.00000E+00 6 0.00000E+00 7 0.00000E+00

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Table B.10- C3 Cross Sections (UO2)

Group Diffusion Coefficient

Absorption Cross Section

Fission Cross Section Σfν

Scattering Cross Section

1 9.76800E-01 9.9180E-03 9.6897E-14 7.5455E-03 1.8066E-02

2 2.6867E-01 1.1459E-01 2.4876E-12 1.8755E-01 -

Table B.11- C3 Cross Sections (MOX)

Group Diffusion Coefficient

Absorption Cross Section

Fission Cross Section Σfν

Scattering Cross Section

1 9.6660E-01 1.4565E-02 1.2854E-13 1.0991E-02 1.4173E-02

2 2.4264E-01 2.6453E-01 5.0439E-12 4.2883E-01 -

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Table B.12- C5 Cross Sections (UO2)

Group Diffusion Coefficient

Absorption Cross Section

Fission Cross Section Σfν

Scattering Cross Section

1 1.200 9.22597E-03 4.569983E-03 4.569983E-03 2.043E-02

2 4.000E-01

9.265912E-02 1.135299E-01 1.135299E-01 -

Table B.13- C5 Cross Sections (MOX)

Group Diffusion Coefficient

Absorption Cross Section

Fission Cross Section Σfν

Scattering Cross Section

1 1.200 1.379115E-02 6.85240E-03 6.85240E-03 1.58635E-02

2 4.000E-01

2.3163960E-01 3.44500E-01 3.44500E-01 -

Table B.14- C5 Cross Sections (Water Moderator)

Group Diffusion Coefficient

Absorption Cross Section

Fission Cross Section Σfν

Scattering Cross Section

1 1.200 1.0000E-03 0 0 5.000E-02

2 4.000E-01

4.000E-02 0 0 -

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APPENDIX C. 2-D C5G7 Benchmark Pin Powers and

% Error Comparison

Table C1- C5G7 Distribution of % Errors in UO2 Assembly Near Axis of Symmetry (NEM with P-NEM Diffusion Theory Solution)

Table C2- C5G7 Distribution of % Errors in UO2 Assembly Near Axis of Symmetry (NEM with SP3 Solution)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 171 -1.9990 -1.9817 -1.9405 -1.9679 -2.0206 -1.8966 -2.2793 -2.1479 -1.6189 -2.0510 -2.0260 -1.5591 -1.7715 -1.3978 -0.8986 -0.1244 2.29622 -2.1649 -2.1298 -2.1071 -3.0273 -2.0097 -2.9187 -2.8212 -1.7597 -2.7295 -2.5748 -1.5655 -2.5815 -1.2926 -0.9228 -0.0205 2.21533 -2.7241 -1.8431 -2.7402 -1.7386 -1.6821 -1.3769 -1.5018 -2.2849 -1.1680 -1.6369 -0.1622 2.19474 -3.0294 -1.8031 -3.0687 -3.1208 -1.4627 -2.8404 -2.7571 -1.3069 -2.5026 -0.6953 0.1332 2.27965 -4.0500 -1.9938 -3.1793 -3.1426 -1.6615 -2.9007 -2.8000 -1.5261 -3.4825 -2.2229 -1.7136 -0.9849 2.04036 -1.8935 -1.7767 -1.5040 -1.3266 -1.3236 -0.9179 0.2336 2.26417 -3.1825 -3.2055 -1.5645 -2.9623 -2.9653 -1.3679 -2.7646 -2.3661 -0.7167 -1.0327 2.09168 -2.9382 -1.5233 -2.9530 -2.8137 -1.3735 -2.6569 -2.3797 -0.6904 -0.9089 1.96709 -1.3593 -1.4454 -1.3290 -1.1128 0.3731 2.5391

10 -2.7394 -2.5778 -1.1076 -2.4895 -2.3399 -0.3827 -0.7561 2.242211 -2.5612 -1.0360 -2.4695 -2.1924 -0.4597 -0.6916 2.314012 -1.0200 -0.6026 0.5272 2.965913 -3.0017 -1.8915 -1.1398 -0.5030 2.566014 0.0451 0.5210 2.761815 -0.6851 0.6826 3.175816 1.7106 3.633317 4.5145

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 171 -1.9898 -1.9735 -1.9249 -1.9499 -1.9996 -1.8743 -2.2550 -2.1254 -1.5985 -2.0290 -2.0023 -1.5381 -1.7525 -1.3824 -0.8948 -0.1460 2.20672 -2.1544 -2.1151 -2.0852 -3.0053 -2.0125 -2.8964 -2.8000 -1.7584 -2.7130 -2.5516 -1.5645 -2.5634 -1.2701 -0.9066 -0.0310 2.12573 -2.7071 -1.8446 -2.7374 -1.7383 -1.6778 -1.3729 -1.4962 -2.2888 -1.1679 -1.6247 -0.1711 2.11214 -3.0236 -1.7937 -3.0413 -3.0927 -1.4551 -2.8118 -2.7322 -1.2946 -2.4967 -0.7020 0.1244 2.19545 -4.0194 -1.9808 -3.1490 -3.1120 -1.6560 -2.8744 -2.7725 -1.5151 -3.4546 -2.2201 -1.7170 -0.9899 1.95446 -1.8830 -1.7666 -1.4988 -1.3200 -1.3182 -0.9088 0.2023 2.18647 -3.1538 -3.1779 -1.5584 -2.9352 -2.9413 -1.3580 -2.7334 -2.3352 -0.7192 -1.0366 2.01038 -2.9122 -1.5145 -2.9232 -2.7821 -1.3659 -2.6278 -2.3569 -0.6913 -0.9129 1.88229 -1.3525 -1.4371 -1.3247 -1.1046 0.3399 2.4589

10 -2.7169 -2.5491 -1.1042 -2.4653 -2.3120 -0.3865 -0.7660 2.160311 -2.5373 -1.0279 -2.4470 -2.1672 -0.4670 -0.6997 2.230512 -1.0121 -0.5997 0.4977 2.889713 -2.9781 -1.8889 -1.1460 -0.5105 2.483114 0.0314 0.5087 2.674515 -0.6798 0.6639 3.084916 1.6710 3.519417 4.3574

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Table C3- C5G7 Distribution of % Errors in MOX Assembly (NEM with P-NEM Diffusion Theory Solution)

Table C4- C5G7 Distribution of % Errors in MOX Assembly (NEM with SP3 Solution)

18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 341 -1.8356 1.3010 3.0565 3.3616 3.6739 3.8463 3.6186 4.0398 4.5110 4.3040 4.8362 5.2023 5.1394 5.6636 5.9393 3.7701 -4.85222 -1.5279 -1.9069 -0.0538 0.5807 -0.1212 1.2644 0.2073 0.5089 1.8295 0.7315 1.0129 2.5539 1.5726 2.8606 2.8873 -0.0784 -4.90383 -1.2675 -1.3148 0.2215 2.1250 1.5793 3.0544 3.6211 3.8549 3.9550 3.3937 4.4421 3.5146 0.6254 -4.75724 -1.1055 -0.7332 1.8239 2.3051 1.8444 0.8445 1.1583 2.9591 1.3849 1.5760 3.0530 3.9987 5.0390 0.9700 -4.39275 -1.1931 -1.7680 1.1424 1.7616 -0.6749 2.2975 1.2372 1.8390 3.5052 2.1076 2.2444 3.4041 1.4802 4.5360 4.6668 -0.1880 -4.73996 -1.0132 -0.5348 1.8075 2.1893 2.8936 3.2791 3.7161 3.6765 3.9428 4.2030 1.0036 -4.09557 -1.1886 -1.5319 2.7592 0.3443 1.3090 2.9695 1.6437 1.9426 3.6150 2.7239 2.7611 4.2765 3.1018 2.9591 5.5613 0.0695 -4.77968 -1.1805 -1.4416 2.6575 0.4924 1.4050 3.0339 2.0110 2.0615 3.6466 2.4661 2.8754 4.3541 2.8624 3.2139 5.5994 0.5905 -4.54249 -0.6342 -0.4396 1.9289 2.7155 3.2725 3.7543 3.7547 4.2205 4.5060 4.4199 1.3354 -4.2992

10 -1.1043 -1.4315 2.7628 0.5364 1.3636 2.9702 1.8454 2.2046 3.7697 2.7423 2.8962 4.2148 3.1408 2.9549 5.6032 0.0589 -4.566511 -0.9619 -1.3313 2.9228 0.3438 1.4068 2.8354 1.6871 2.1314 3.7035 2.1809 2.8420 4.2598 3.0173 3.0702 5.8383 0.2286 -4.338412 -0.2654 0.0170 1.7925 2.5899 3.3235 3.3582 3.6494 4.2227 4.3767 4.1751 1.5584 -3.916813 -0.7299 -1.3431 1.6338 2.4522 0.0685 2.7674 1.7995 1.9617 3.4759 2.5855 2.6866 4.1785 1.8667 4.9554 4.2571 0.0703 -4.564214 -0.6918 -0.4043 2.0900 2.8634 2.6339 1.0698 1.3283 2.8312 1.5113 2.1463 3.8371 4.3538 5.2522 1.4067 -4.319515 -0.5423 -0.5275 0.7781 2.8529 2.1769 3.6834 3.7175 4.1692 4.3482 3.8532 4.6540 3.6734 0.6859 -4.754716 -0.5429 -1.8267 -0.1460 0.6436 -0.3763 1.1122 -0.1980 0.1114 1.5065 0.3370 0.6062 2.1004 1.2717 2.4742 2.1724 -1.1250 -4.884017 -0.4892 0.1765 0.3637 0.7308 0.6484 0.9786 0.5987 0.7343 1.3603 1.0707 1.3021 2.3497 2.2970 2.4463 2.0031 0.0697 -6.8184

18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 341 -1.7565 1.2663 2.9877 3.3077 3.6156 3.7939 3.5617 3.9767 4.4545 4.2391 4.7793 5.1315 5.0536 5.6071 5.8738 3.7693 -4.76752 -1.4573 -1.8392 -0.0231 0.6152 -0.0814 1.2820 0.2529 0.5427 1.8386 0.7644 1.0392 2.5492 1.6045 2.8888 2.9229 -0.0134 -4.84563 -1.2012 -1.2667 0.2347 2.1196 1.5719 3.0241 3.5883 3.8345 3.9105 3.3734 4.4115 3.5266 0.6816 -4.71164 -1.0473 -0.6879 1.8082 2.2848 1.8798 0.8857 1.1920 2.9666 1.4193 1.6042 3.0555 3.9397 5.0479 1.0230 -4.34635 -1.1283 -1.7118 1.1325 1.7368 -0.6184 2.2963 1.2621 1.8525 3.4903 2.1160 2.2580 3.3782 1.5325 4.4894 4.6504 -0.1271 -4.69366 -0.9399 -0.5154 1.8331 2.1861 2.8816 3.2561 3.6949 3.6341 3.9336 4.1855 1.0480 -4.04857 -1.1243 -1.4739 2.7334 0.3852 1.3298 2.9584 1.6599 1.9554 3.5853 2.7287 2.7558 4.2498 3.1081 2.9742 5.5103 0.1385 -4.72808 -1.1174 -1.3915 2.6306 0.5248 1.4241 3.0101 2.0222 2.0675 3.6218 2.4617 2.8749 4.3171 2.8568 3.2167 5.5537 0.6450 -4.48689 -0.5651 -0.4228 1.9360 2.7010 3.2661 3.7337 3.7331 4.1906 4.4832 4.3901 1.3757 -4.2576

10 -1.0412 -1.3820 2.7297 0.5726 1.3847 2.9459 1.8557 2.2083 3.7409 2.7470 2.8889 4.1850 3.1435 2.9459 5.5639 0.1027 -4.519111 -0.8973 -1.2809 2.8872 0.3766 1.4230 2.8166 1.6918 2.1411 3.6677 2.1767 2.8395 4.2182 3.0047 3.0635 5.7804 0.2769 -4.303812 -0.1996 0.0264 1.8080 2.5847 3.3104 3.3286 3.6280 4.1900 4.3587 4.1820 1.5914 -3.877613 -0.6656 -1.2887 1.6135 2.4181 0.1132 2.7599 1.8235 1.9799 3.4469 2.5885 2.6899 4.1530 1.8876 4.8979 4.2293 0.1273 -4.517114 -0.6362 -0.3669 2.0720 2.8269 2.6505 1.1007 1.3415 2.8116 1.5238 2.1593 3.8356 4.2918 5.2358 1.4608 -4.302215 -0.4977 -0.4839 0.7783 2.8248 2.1496 3.6335 3.6682 4.1203 4.2954 3.8182 4.6195 3.6567 0.7237 -4.734316 -0.4862 -1.7716 -0.1093 0.6696 -0.3310 1.1163 -0.1627 0.1377 1.5016 0.3627 0.6177 2.0752 1.2950 2.4646 2.1875 -1.0483 -4.872617 -0.4070 0.2265 0.3960 0.7572 0.6781 1.0133 0.6305 0.7624 1.3985 1.0896 1.3158 2.3751 2.2957 2.4567 2.0093 0.1639 -6.7390

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Table C5- C5G7 Distribution of % Errors in UO2 Assembly Near Water Reflector (NEM with P-NEM Diffusion Theory Solution)

Table C6- C5G7 Distribution of % Errors in UO2 Assembly Near Water Reflector (NEM with SP3 Solution)

18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 3418 4.9060 4.3389 4.2219 4.0729 4.0959 4.4632 4.2984 4.3847 4.7179 4.5287 4.8076 5.5610 5.5455 5.7526 5.9540 3.8172 -2.599319 2.5902 1.9313 2.1055 1.0556 1.9174 1.1615 1.2614 2.4982 1.5944 2.2888 3.0665 2.1809 3.5271 2.9842 1.0381 -4.121120 0.7872 1.4739 0.4235 1.3023 1.5749 1.9101 2.2333 1.7436 2.8997 1.5550 -0.0026 -4.509821 -0.1519 1.2073 -0.2556 -0.1190 1.3197 0.2988 0.4987 2.1751 1.1638 2.0423 -0.0551 -4.805722 -1.1546 0.6964 -0.1415 0.1631 1.2046 0.0816 0.4897 2.0644 0.2036 1.4107 0.9842 -0.6919 -5.329223 1.1777 1.5404 1.5136 1.6864 2.0776 2.3608 0.3584 -4.273424 -0.0307 0.1412 1.6292 0.3048 0.2074 2.1865 0.9566 1.0072 2.0632 -0.6848 -4.858725 0.4588 1.6123 0.5737 0.6704 2.3454 0.9996 1.5725 2.3040 -0.4780 -4.950426 1.9732 2.2233 2.4834 2.8095 0.6295 -4.538027 0.5245 0.8704 2.6212 1.0539 1.3125 2.4137 -0.4484 -4.926228 1.1194 2.6316 1.4515 1.5777 2.1214 -0.2870 -4.776829 2.5104 2.7731 0.6644 -4.193230 0.6191 1.4363 1.3938 -0.0920 -4.481731 2.7342 1.2692 -4.012932 2.1414 1.1621 -4.057233 -0.9049 -4.945334 -7.4120

18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 3418 4.7341 4.2039 4.1046 3.9623 3.9798 4.3435 4.1774 4.2629 4.6117 4.4051 4.6638 5.4359 5.4193 5.6228 5.8101 3.7076 -2.610519 2.5354 1.8964 2.0727 1.0254 1.8553 1.1246 1.2172 2.4227 1.5476 2.2332 2.9912 2.1186 3.4769 2.9301 0.9928 -4.081520 0.7692 1.4442 0.3840 1.2657 1.5337 1.8693 2.1863 1.6689 2.8371 1.5221 0.0065 -4.495721 -0.1837 1.1851 -0.2578 -0.1246 1.2845 0.2815 0.4750 2.1309 1.1030 1.9716 -0.0737 -4.770422 -1.1566 0.6722 -0.1477 0.1520 1.1626 0.0558 0.4561 2.0276 0.1559 1.3605 0.9325 -0.6898 -5.305023 1.1485 1.5041 1.4741 1.6383 2.0203 2.3072 0.3396 -4.256524 -0.0474 0.1148 1.5828 0.2725 0.1826 2.1402 0.9245 0.9564 1.9972 -0.6785 -4.844125 0.4359 1.5512 0.5419 0.6451 2.2970 0.9640 1.5401 2.2548 -0.4770 -4.936826 1.9146 2.1689 2.4117 2.7378 0.5900 -4.547527 0.4951 0.8450 2.5673 1.0092 1.2671 2.3482 -0.4864 -4.910528 1.0730 2.5526 1.4032 1.5260 2.0478 -0.3309 -4.783729 2.4468 2.7035 0.6051 -4.198130 0.5498 1.3512 1.3256 -0.1163 -4.481931 2.6246 1.2065 -4.074832 2.0998 1.1666 -4.126933 -0.8123 -5.004234 -7.4349

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Table C7- C5G7 Distribution of % Errors in UO2 Assembly Near Axis of Symmetry (NEM with SA-NEM Diffusion Theory Solution)

Table C8- C5G7 Distribution of % Errors in MOX Assembly (NEM with SA-NEM Diffusion Theory Solution)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 171 -2.0399 -2.0316 -1.9947 -2.0262 -2.0877 -1.9772 -2.3433 -2.2129 -1.6942 -2.1093 -2.0859 -1.6360 -1.8307 -1.4321 -0.8804 0.0101 2.59142 -2.2145 -2.1877 -2.1773 -3.0577 -2.1821 -2.9586 -2.8529 -1.9238 -2.7672 -2.5991 -1.7290 -2.6179 -1.3482 -0.9048 0.1129 2.51083 -2.7544 -2.0226 -2.8572 -1.9268 -1.8400 -1.5549 -1.6529 -2.4108 -1.3230 -1.6253 -0.0567 2.48174 -3.1372 -1.9737 -3.0992 -3.1476 -1.6277 -2.8683 -2.7762 -1.4649 -2.5794 -0.8548 0.2171 2.54935 -4.0418 -2.1540 -3.2100 -3.1695 -1.8279 -2.9288 -2.8192 -1.6717 -3.4527 -2.2987 -1.7898 -0.8893 2.27856 -2.0849 -1.9415 -1.6893 -1.4842 -1.5055 -1.0207 0.1533 2.48667 -3.2186 -3.2331 -1.7438 -2.9958 -2.9949 -1.5522 -2.7909 -2.3607 -0.8408 -0.9416 2.32568 -2.9616 -1.6784 -2.9774 -2.8338 -1.5317 -2.6784 -2.3798 -0.7882 -0.8098 2.19719 -1.5400 -1.5971 -1.5173 -1.2117 0.2953 2.7631

10 -2.7699 -2.5987 -1.2927 -2.5174 -2.3341 -0.5020 -0.6670 2.472211 -2.5880 -1.1888 -2.4924 -2.1865 -0.5635 -0.5934 2.532412 -1.1926 -0.7129 0.4440 3.179413 -2.9665 -1.9452 -1.2227 -0.4362 2.788814 -0.1100 0.5920 3.004515 -0.6625 0.8098 3.450816 1.9225 3.935817 4.8432

18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 341 -1.5926 1.4324 3.2477 3.6036 3.8817 4.0016 3.7722 4.1595 4.5910 4.4086 4.9496 5.3254 5.2984 5.8634 5.9392 3.1415 -6.35942 -1.2819 -1.8848 0.0482 0.6899 0.1446 1.3976 0.4345 0.6843 1.9006 0.9352 1.1758 2.6597 1.8356 2.9592 2.8449 -0.8870 -6.45163 -1.0511 -1.3073 0.4333 2.2350 1.7924 3.1278 3.5638 3.9339 3.8681 3.7054 4.5522 3.6400 -0.1587 -6.35174 -0.8973 -0.7632 1.8475 2.6359 1.8266 1.0417 1.3051 2.9137 1.5553 1.7044 2.9962 4.1842 5.1549 0.1940 -5.96815 -1.0007 -1.6287 1.3665 2.0825 -0.3171 2.3342 1.4521 2.0018 3.4935 2.2965 2.3915 3.4188 1.8390 4.8315 4.8195 -0.7800 -6.32016 -0.8515 -0.5349 1.8618 2.1979 2.9948 3.2238 3.8175 3.5925 4.1195 4.0996 0.4764 -5.65237 -1.0086 -1.4049 2.8160 0.6049 1.5392 3.0366 1.8983 2.1340 3.6387 2.9444 2.9419 4.3221 3.3853 3.1879 5.5205 -0.4931 -6.32718 -0.9978 -1.3202 2.7073 0.7584 1.6495 3.0827 2.2575 2.2432 3.6465 2.6760 3.0585 4.4003 3.1513 3.4464 5.5786 0.0043 -6.08349 -0.4820 -0.4397 2.0295 2.7340 3.4097 3.7199 3.8598 4.1627 4.7087 4.3830 0.8288 -5.8490

10 -0.9232 -1.3058 2.8398 0.8108 1.6157 3.0606 2.1108 2.4034 3.8068 2.9711 3.0837 4.2937 3.4539 3.2130 5.6243 -0.5027 -6.087511 -0.7854 -1.2192 2.9489 0.6040 1.6418 2.8865 1.9454 2.3336 3.7160 2.4145 3.0320 4.3077 3.3150 3.3296 5.8169 -0.3177 -5.856412 -0.1288 0.0092 1.8439 2.6184 3.4444 3.3341 3.7864 4.1927 4.6168 4.1381 1.0959 -5.409113 -0.5611 -1.2088 1.8796 2.7483 0.4413 2.8598 2.0506 2.1687 3.5404 2.8376 2.8961 4.2906 2.3114 5.2972 4.5282 -0.4261 -6.036814 -0.5158 -0.4487 2.1085 3.1835 2.5933 1.2837 1.5082 2.7925 1.7330 2.3374 3.8370 4.6532 5.4655 0.7672 -5.796015 -0.3786 -0.5736 0.9300 2.9325 2.3531 3.7316 3.6650 4.2437 4.3156 4.2360 4.8778 3.9047 0.0342 -6.229716 -0.3943 -1.9641 -0.2473 0.5484 -0.3111 1.1013 -0.1373 0.1373 1.4933 0.4262 0.6867 2.1659 1.4979 2.5585 2.1723 -1.7710 -6.272917 -0.4206 0.1654 0.3520 0.7186 0.6103 0.9258 0.5705 0.7043 1.3286 1.0876 1.3388 2.4284 2.4483 2.6593 2.1284 -0.1896 -7.9388

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Table C9- C5G7 Distribution of % Errors in UO2 Assembly Near Water Reflector (NEM with SA-NEM Diffusion Theory Solution)

18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 3418 5.0687 4.4900 4.3506 4.1922 4.1779 4.5199 4.3886 4.4645 4.8025 4.6564 4.9648 5.7507 5.8225 6.0799 6.1922 3.7662 -3.074819 2.7588 2.0271 2.1535 1.1537 1.8309 1.2018 1.3328 2.4249 1.6918 2.4455 3.1210 2.4062 3.7964 3.2485 1.1084 -4.422320 0.8561 1.3523 0.3560 1.1278 1.4823 1.8050 2.2332 1.8547 3.0213 1.8035 0.0883 -4.770121 -0.2526 1.0434 -0.2817 -0.1053 1.2211 0.3456 0.6160 2.2093 1.3293 2.1492 0.0571 -5.011922 -1.1665 0.5390 -0.1681 0.1630 1.1037 0.1294 0.6094 2.1520 0.4941 1.6523 1.1990 -0.5120 -5.501923 0.9790 1.4423 1.4185 1.7203 2.1557 2.5709 0.4701 -4.390824 -0.0452 0.1716 1.5207 0.3908 0.3741 2.2625 1.2558 1.3765 2.2736 -0.4192 -4.920425 0.5389 1.5959 0.7184 0.8841 2.5056 1.3379 2.0088 2.6485 -0.1484 -4.972126 1.9731 2.3808 2.6888 3.1543 0.8878 -4.538227 0.7485 1.1757 2.8230 1.4842 1.8309 2.7720 -0.0501 -4.853728 1.4932 2.9686 1.9645 2.1889 2.5954 0.2001 -4.673329 2.9260 3.3092 1.0752 -4.026730 1.3234 2.1388 2.0026 0.4740 -4.269431 3.3241 1.8510 -3.814332 2.8796 1.7916 -3.841733 -0.2166 -4.720334 -7.1685

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Table C10- C5G7 Distribution of Pin Powers in UO2 Assembly Near Axis of Symmetry (NEM with P-NEM Diffusion Theory Solution)

Table C11- C5G7 Distribution of Pin Powers in UO2 Assembly Near Axis of Symmetry (NEM with SP3 Solution)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 171 2.2417 2.2469 2.2566 2.2684 2.2770 2.2700 2.2339 2.1943 2.1546 2.0970 2.0382 1.9760 1.8860 1.7754 1.6457 1.4826 1.25242 2.2603 2.2875 2.3218 2.3723 2.4182 2.3206 2.2764 2.2875 2.1765 2.1160 2.1048 1.9666 1.8189 1.6705 1.4924 1.25253 2.3755 2.4889 2.5401 2.4252 2.3744 2.2710 2.2102 2.1055 1.9519 1.7392 1.5129 1.25594 2.5736 2.4980 2.3662 2.3134 2.3283 2.2122 2.1568 2.1727 2.1308 1.8260 1.5393 1.26305 2.4970 2.4755 2.3508 2.2998 2.3144 2.1995 2.1437 2.1545 2.0689 2.0170 1.8632 1.5768 1.26986 2.3883 2.3393 2.2386 2.1787 2.0548 1.9580 1.6105 1.26867 2.2859 2.2424 2.2590 2.1461 2.0876 2.0835 1.9537 1.8571 1.7795 1.5457 1.24988 2.2014 2.2181 2.1078 2.0497 2.0436 1.9143 1.8187 1.7454 1.5194 1.23069 2.1253 2.0663 1.9304 1.8340 1.5328 1.2132

10 2.0199 1.9654 1.9610 1.8376 1.7473 1.6786 1.4623 1.185011 1.9139 1.9118 1.7952 1.7091 1.6405 1.4285 1.158012 1.8091 1.7280 1.4310 1.131213 1.7410 1.7032 1.5819 1.3453 1.088514 1.4779 1.2542 1.037115 1.3270 1.1659 0.981716 1.0626 0.920117 0.8390

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 171 2.2415 2.2467 2.2562 2.2680 2.2765 2.2695 2.2334 2.1938 2.1542 2.0965 2.0377 1.9756 1.8856 1.7751 1.6456 1.4829 1.25352 2.2601 2.2872 2.3213 2.3718 2.4183 2.3201 2.2759 2.2875 2.1761 2.1155 2.1048 1.9662 1.8185 1.6702 1.4926 1.25363 2.3751 2.4889 2.5400 2.4252 2.3743 2.2709 2.2101 2.1056 1.9519 1.7390 1.5130 1.25704 2.5734 2.4978 2.3656 2.3128 2.3281 2.2116 2.1563 2.1724 2.1307 1.8261 1.5394 1.26415 2.4963 2.4752 2.3501 2.2991 2.3143 2.1989 2.1431 2.1543 2.0683 2.0169 1.8633 1.5769 1.27096 2.3880 2.3391 2.2385 2.1786 2.0547 1.9578 1.6110 1.26967 2.2853 2.2418 2.2589 2.1455 2.0871 2.0833 1.9531 1.8565 1.7795 1.5458 1.25088 2.2008 2.2179 2.1072 2.0491 2.0434 1.9138 1.8183 1.7454 1.5195 1.23179 2.1251 2.0661 1.9303 1.8338 1.5333 1.2142

10 2.0195 1.9648 1.9609 1.8372 1.7468 1.6787 1.4624 1.186011 1.9134 1.9116 1.7948 1.7087 1.6406 1.4286 1.159012 1.8090 1.7279 1.4314 1.132113 1.7406 1.7032 1.5820 1.3454 1.089414 1.4781 1.2544 1.038015 1.3269 1.1661 0.982616 1.0630 0.921217 0.8404

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Table C12- C5G7 Distribution of Pin Powers in UO2 Assembly Near Axis of Symmetry (NEM with SA-NEM Diffusion Theory Solution)

Table C13- C5G7 Distribution of Pin Powers in MOX Assembly (NEM with P-NEM Diffusion Theory Solution)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 171 2.2426 2.2480 2.2578 2.2697 2.2785 2.2718 2.2353 2.1957 2.1562 2.0982 2.0394 1.9775 1.8871 1.7760 1.6454 1.4806 1.24862 2.2614 2.2888 2.3234 2.3730 2.4223 2.3215 2.2771 2.2912 2.1773 2.1165 2.1082 1.9673 1.8199 1.6702 1.4905 1.24873 2.3762 2.4933 2.5430 2.4297 2.3781 2.2750 2.2135 2.1081 1.9549 1.7390 1.5113 1.25234 2.5762 2.5022 2.3669 2.3140 2.3321 2.2128 2.1572 2.1761 2.1324 1.8289 1.5380 1.25955 2.4968 2.4794 2.3515 2.3004 2.3182 2.2001 2.1441 2.1576 2.0683 2.0185 1.8646 1.5753 1.26676 2.3927 2.3431 2.2427 2.1821 2.0585 1.9600 1.6118 1.26577 2.2867 2.2430 2.2630 2.1468 2.0882 2.0873 1.9542 1.8570 1.7817 1.5444 1.24688 2.2019 2.2215 2.1083 2.0501 2.0468 1.9148 1.8187 1.7471 1.5180 1.22789 2.1290 2.0694 1.9340 1.8357 1.5340 1.2104

10 2.0205 1.9658 1.9646 1.8381 1.7472 1.6806 1.4610 1.182211 1.9144 1.9147 1.7956 1.7090 1.6422 1.4271 1.155412 1.8122 1.7299 1.4322 1.128713 1.7404 1.7041 1.5832 1.3444 1.086014 1.4802 1.2534 1.034515 1.3267 1.1644 0.978916 1.0603 0.917217 0.8361

18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 341 1.3355 1.0469 0.9089 0.8352 0.7848 0.7395 0.6874 0.6384 0.5940 0.5460 0.5012 0.4600 0.4157 0.3760 0.3555 0.3966 0.63072 1.3153 1.3691 1.1715 1.0876 1.0506 1.0332 0.9181 0.8474 0.8240 0.7278 0.6657 0.6413 0.5589 0.4900 0.4571 0.5179 0.62123 1.3048 1.3363 1.1728 1.1510 1.1036 0.9198 0.8404 0.7265 0.6611 0.5865 0.5186 0.4598 0.5051 0.61534 1.3055 1.3443 1.2397 1.0885 1.1080 0.9509 0.8714 0.8533 0.7488 0.6867 0.6812 0.5667 0.4891 0.5086 0.61405 1.3091 1.3815 1.2735 1.1586 1.1489 1.0588 0.9151 0.8400 0.8250 0.7223 0.6616 0.6530 0.6016 0.5148 0.4974 0.5228 0.61426 1.3065 1.4308 1.2588 1.1175 0.9545 0.8726 0.7563 0.6865 0.5953 0.5542 0.5425 0.61297 1.2878 1.3529 1.1912 1.1425 1.0246 1.0066 0.8846 0.8161 0.8039 0.7031 0.6424 0.6265 0.5446 0.5068 0.4616 0.5133 0.60718 1.2687 1.3304 1.1666 1.1176 1.0039 0.9872 0.8707 0.8048 0.7924 0.6941 0.6340 0.6169 0.5357 0.4975 0.4537 0.5070 0.60159 1.2509 1.3643 1.1647 1.0436 0.9128 0.8388 0.7291 0.6612 0.5632 0.5174 0.5239 0.5967

10 1.2231 1.2854 1.1304 1.0830 0.9740 0.9604 0.8470 0.7836 0.7734 0.6772 0.6191 0.6035 0.5238 0.4868 0.4445 0.4965 0.589111 1.1963 1.2592 1.1098 1.0681 0.9609 0.9451 0.8337 0.7714 0.7605 0.6672 0.6108 0.5959 0.5191 0.4838 0.4408 0.4913 0.582112 1.1695 1.2871 1.1397 1.0166 0.8753 0.8036 0.7005 0.6379 0.5551 0.5177 0.5090 0.575813 1.1288 1.2016 1.1160 1.0170 1.0146 0.9424 0.8179 0.7545 0.7451 0.6541 0.6013 0.5959 0.5495 0.4709 0.4572 0.4815 0.566014 1.0824 1.1289 1.0537 0.9370 0.9591 0.8287 0.7644 0.7522 0.6636 0.6118 0.6093 0.5091 0.4422 0.4609 0.556115 1.0390 1.0889 0.9755 0.9710 0.9396 0.7943 0.7319 0.6403 0.5866 0.5253 0.4666 0.4147 0.4555 0.551916 1.0104 1.1081 0.9862 0.9376 0.9194 0.9133 0.8205 0.7647 0.7486 0.6668 0.6143 0.5943 0.5215 0.4605 0.4309 0.4836 0.564317 1.0198 0.9087 0.8506 0.8136 0.7823 0.7486 0.7054 0.6627 0.6223 0.5775 0.5344 0.4937 0.4500 0.4105 0.3892 0.4225 0.6175

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Table C14- C5G7 Distribution of Pin Powers in MOX Assembly (NEM with SP3 Solution)

Table C15- C5G7 Distribution of Pin Powers in MOX Assembly (NEM with SA-NEM Diffusion Theory Solution)

1.3345 1.0473 0.9095 0.8357 0.7853 0.7399 0.6878 0.6388 0.5943 0.5464 0.5015 0.4603 0.4161 0.3762 0.3557 0.3966 0.63021.3144 1.3682 1.1711 1.0872 1.0502 1.0330 0.9177 0.8471 0.8239 0.7276 0.6655 0.6413 0.5587 0.4899 0.4569 0.5176 0.62091.3039 1.3357 1.1726 1.1511 1.1037 0.9201 0.8407 0.7267 0.6614 0.5866 0.5188 0.4597 0.5048 0.61501.3047 1.3437 1.2399 1.0887 1.1076 0.9505 0.8711 0.8532 0.7485 0.6865 0.6812 0.5670 0.4891 0.5083 0.61371.3083 1.3807 1.2736 1.1589 1.1483 1.0588 0.9149 0.8399 0.8251 0.7222 0.6615 0.6532 0.6013 0.5150 0.4975 0.5225 0.61391.3055 1.4305 1.2585 1.1175 0.9546 0.8728 0.7565 0.6868 0.5954 0.5543 0.5423 0.61261.2870 1.3521 1.1915 1.1420 1.0244 1.0067 0.8845 0.8160 0.8041 0.7031 0.6424 0.6267 0.5446 0.5067 0.4618 0.5129 0.60681.2679 1.3297 1.1669 1.1172 1.0037 0.9874 0.8706 0.8047 0.7926 0.6941 0.6340 0.6171 0.5357 0.4975 0.4539 0.5067 0.60121.2500 1.3641 1.1646 1.0438 0.9129 0.8390 0.7293 0.6614 0.5633 0.5176 0.5237 0.59651.2223 1.2848 1.1308 1.0826 0.9738 0.9606 0.8469 0.7836 0.7736 0.6772 0.6191 0.6037 0.5238 0.4868 0.4447 0.4963 0.58881.1955 1.2586 1.1102 1.0677 0.9607 0.9453 0.8337 0.7713 0.7608 0.6672 0.6108 0.5962 0.5192 0.4838 0.4411 0.4911 0.58191.1687 1.2870 1.1395 1.0167 0.8754 0.8038 0.7007 0.6381 0.5552 0.5177 0.5088 0.57561.1281 1.2010 1.1162 1.0174 1.0141 0.9425 0.8177 0.7544 0.7453 0.6541 0.6013 0.5961 0.5494 0.4712 0.4573 0.4812 0.56571.0818 1.1285 1.0539 0.9373 0.9589 0.8284 0.7643 0.7523 0.6635 0.6117 0.6093 0.5094 0.4423 0.4606 0.55601.0385 1.0884 0.9755 0.9713 0.9399 0.7947 0.7323 0.6406 0.5869 0.5255 0.4668 0.4148 0.4553 0.55181.0098 1.1075 0.9858 0.9374 0.9190 0.9133 0.8202 0.7645 0.7486 0.6666 0.6142 0.5945 0.5214 0.4605 0.4308 0.4832 0.56421.0190 0.9082 0.8503 0.8134 0.7821 0.7483 0.7052 0.6625 0.6221 0.5774 0.5343 0.4936 0.4500 0.4105 0.3892 0.4221 0.6170

18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 341 1.3324 1.0455 0.9071 0.8331 0.7831 0.7383 0.6863 0.6376 0.5935 0.5454 0.5006 0.4594 0.4150 0.3752 0.3555 0.3992 0.63982 1.3121 1.3688 1.1703 1.0864 1.0478 1.0318 0.9160 0.8459 0.8234 0.7263 0.6646 0.6406 0.5574 0.4895 0.4573 0.5221 0.63043 1.3020 1.3362 1.1703 1.1497 1.1012 0.9191 0.8409 0.7260 0.6617 0.5846 0.5180 0.4592 0.5091 0.62464 1.3028 1.3447 1.2394 1.0848 1.1082 0.9490 0.8701 0.8537 0.7475 0.6858 0.6816 0.5656 0.4886 0.5126 0.62325 1.3067 1.3796 1.2706 1.1548 1.1449 1.0584 0.9131 0.8386 0.8251 0.7209 0.6606 0.6529 0.5994 0.5132 0.4966 0.5259 0.62346 1.3044 1.4308 1.2581 1.1174 0.9535 0.8731 0.7555 0.6871 0.5942 0.5548 0.5454 0.62207 1.2855 1.3512 1.1905 1.1395 1.0222 1.0059 0.8824 0.8145 0.8037 0.7015 0.6412 0.6262 0.5430 0.5056 0.4618 0.5161 0.61618 1.2664 1.3288 1.1660 1.1146 1.0014 0.9867 0.8685 0.8033 0.7924 0.6926 0.6328 0.6166 0.5341 0.4963 0.4538 0.5100 0.61049 1.2490 1.3643 1.1635 1.0434 0.9115 0.8391 0.7283 0.6616 0.5620 0.5176 0.5266 0.6056

10 1.2209 1.2838 1.1295 1.0800 0.9715 0.9595 0.8447 0.7820 0.7731 0.6756 0.6179 0.6030 0.5221 0.4855 0.4444 0.4993 0.597611 1.1942 1.2578 1.1095 1.0653 0.9586 0.9446 0.8316 0.7698 0.7604 0.6656 0.6096 0.5956 0.5175 0.4825 0.4409 0.4940 0.590612 1.1679 1.2872 1.1391 1.0164 0.8742 0.8038 0.6996 0.6381 0.5537 0.5179 0.5114 0.584113 1.1269 1.2001 1.1132 1.0140 1.0108 0.9415 0.8158 0.7529 0.7446 0.6524 0.6000 0.5952 0.5470 0.4692 0.4559 0.4839 0.573914 1.0805 1.1294 1.0535 0.9339 0.9595 0.8269 0.7630 0.7525 0.6621 0.6106 0.6093 0.5075 0.4412 0.4638 0.564015 1.0373 1.0894 0.9740 0.9702 0.9379 0.7939 0.7323 0.6398 0.5868 0.5232 0.4655 0.4137 0.4585 0.559716 1.0089 1.1096 0.9872 0.9385 0.9188 0.9134 0.8200 0.7645 0.7487 0.6662 0.6138 0.5940 0.5203 0.4601 0.4309 0.4867 0.571717 1.0191 0.9088 0.8507 0.8137 0.7826 0.7490 0.7056 0.6629 0.6225 0.5774 0.5342 0.4933 0.4493 0.4096 0.3887 0.4236 0.6239

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Table C16- C5G7 Distribution of Pin Powers in UO2 Assembly Near Water Reflector (NEM with P-NEM Diffusion Theory Solution)

Table C17- C5G7 Distribution of Pin Powers in UO2 Assembly Near Water Reflector (NEM with SP3 Solution)

18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 3418 0.7565 0.7566 0.7405 0.7204 0.6977 0.6700 0.6332 0.5960 0.5604 0.5207 0.4824 0.4461 0.4075 0.3728 0.3539 0.3772 0.516019 0.8053 0.8151 0.8101 0.8034 0.7918 0.7315 0.6881 0.6630 0.6032 0.5588 0.5309 0.4756 0.4280 0.4021 0.4200 0.551320 0.8589 0.8883 0.8839 0.7895 0.7415 0.6518 0.6042 0.5280 0.4767 0.4339 0.4379 0.560421 0.8926 0.8412 0.7695 0.7229 0.6989 0.6359 0.5912 0.5690 0.5353 0.4562 0.4438 0.557322 0.8487 0.8175 0.7510 0.7066 0.6834 0.6227 0.5792 0.5565 0.5132 0.4877 0.4589 0.4460 0.547223 0.7438 0.7012 0.6197 0.5755 0.4992 0.4627 0.4442 0.531324 0.6900 0.6523 0.6328 0.5770 0.5364 0.5125 0.4617 0.4272 0.4171 0.4152 0.508925 0.6179 0.5995 0.5474 0.5090 0.4857 0.4372 0.4045 0.3954 0.3946 0.484626 0.5324 0.4947 0.4255 0.3930 0.3831 0.459527 0.4864 0.4526 0.4325 0.3893 0.3602 0.3526 0.3517 0.431428 0.4215 0.4029 0.3635 0.3368 0.3291 0.3282 0.402729 0.3505 0.3252 0.3131 0.374030 0.3233 0.3074 0.2904 0.2818 0.343131 0.2628 0.2535 0.312932 0.2377 0.2346 0.288433 0.2336 0.278634 0.3075

18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 3418 0.7579 0.7577 0.7414 0.7212 0.6985 0.6708 0.6340 0.5968 0.5610 0.5214 0.4831 0.4467 0.4080 0.3733 0.3544 0.3776 0.516119 0.8057 0.8154 0.8104 0.8036 0.7923 0.7318 0.6884 0.6635 0.6035 0.5591 0.5313 0.4759 0.4282 0.4023 0.4202 0.551120 0.8591 0.8886 0.8842 0.7898 0.7418 0.6521 0.6045 0.5284 0.4770 0.4340 0.4379 0.560321 0.8929 0.8414 0.7695 0.7229 0.6991 0.6360 0.5913 0.5693 0.5356 0.4565 0.4439 0.557122 0.8487 0.8177 0.7510 0.7067 0.6837 0.6229 0.5794 0.5567 0.5134 0.4879 0.4591 0.4460 0.547123 0.7440 0.7015 0.6199 0.5758 0.4995 0.4630 0.4443 0.531224 0.6901 0.6525 0.6331 0.5772 0.5365 0.5127 0.4618 0.4274 0.4174 0.4152 0.508825 0.6180 0.5999 0.5476 0.5091 0.4859 0.4374 0.4046 0.3956 0.3946 0.484526 0.5327 0.4950 0.4258 0.3933 0.3833 0.459527 0.4865 0.4527 0.4327 0.3895 0.3604 0.3528 0.3518 0.431328 0.4217 0.4032 0.3637 0.3370 0.3293 0.3283 0.402729 0.3507 0.3254 0.3133 0.374030 0.3235 0.3077 0.2906 0.2819 0.343131 0.2631 0.2537 0.313132 0.2378 0.2346 0.288633 0.2334 0.278834 0.3076

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Table C18- C5G7 Distribution of Pin Powers in UO2 Assembly Near Water Reflector (NEM with SA-NEM Diffusion Theory Solution)

18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 3418 0.7552 0.7554 0.7395 0.7195 0.6971 0.6696 0.6326 0.5955 0.5599 0.5200 0.4816 0.4452 0.4063 0.3715 0.3530 0.3774 0.518419 0.8039 0.8143 0.8097 0.8026 0.7925 0.7312 0.6876 0.6635 0.6026 0.5579 0.5306 0.4745 0.4268 0.4010 0.4197 0.552920 0.8583 0.8894 0.8845 0.7909 0.7422 0.6525 0.6042 0.5274 0.4761 0.4328 0.4375 0.561821 0.8935 0.8426 0.7697 0.7228 0.6996 0.6356 0.5905 0.5688 0.5344 0.4557 0.4433 0.558422 0.8488 0.8188 0.7512 0.7066 0.6841 0.6224 0.5785 0.5560 0.5117 0.4865 0.4579 0.4452 0.548123 0.7453 0.7019 0.6203 0.5753 0.4988 0.4618 0.4437 0.531924 0.6901 0.6521 0.6335 0.5765 0.5355 0.5121 0.4603 0.4256 0.4162 0.4141 0.509225 0.6174 0.5996 0.5466 0.5079 0.4849 0.4358 0.4027 0.3940 0.3933 0.484726 0.5324 0.4939 0.4246 0.3916 0.3822 0.459527 0.4853 0.4512 0.4316 0.3876 0.3583 0.3513 0.3503 0.431128 0.4199 0.4015 0.3616 0.3347 0.3275 0.3266 0.402329 0.3490 0.3234 0.3118 0.373430 0.3210 0.3052 0.2886 0.2802 0.342431 0.2612 0.2520 0.312332 0.2359 0.2331 0.287833 0.2320 0.278034 0.3068

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137

APPENDIX D. SP3 RESPONSE MATRIX EQUATIONS

00,rem20,rem2,rem22

2

02,rem00,rem02

1

S52)r(

52)r(

54)r(D

S)r(2)r()r(D

−=ΦΣ−φ

Σ+Σ+φ∇−

=φΣ−ΦΣ+Φ∇−

With,

)(,0 rS g = [ ]∑≠=

→ −ΦΣG

gggggggs rr

',1',2,0',0, )(2)( φ + ∑

=

ΣG

ggf

eff

g

k 1'',ν

χ [ ])(2)( ,2,0 rr gg φ−Φ

g,trg,0 3

1DΣ

= and g,3,r

g,2 359D

Σ=

)(2)()( ,2,0,0 rrr ggg φφ +=Φ

Let, α = 2,remΣ + 0,rem54

Σ

The Marshak boundary conditions, as used in NEM, are as follows

)(2532)(

258

)(2524)(

2556

33112

33110

−+−+

−+−+

+++=

+++=Φ

jjjj

jjjj

s

s

φ

Expansion Coefficients, from polynomial expansion currently in NEM,

a1 = )(2524)(

2556

3333111100inL

outL

inR

outR

inL

outL

inR

outRLR jjjjjjjj −−++−−+=Φ−Φ

a2 = =Φ−Φ+Φ 000 2LR 033331111 2)(2524)(

2556

Φ−+++++++ inL

outL

inR

outR

inL

outL

inR

outR jjjjjjjj

a3 = 10a1 - 120 10uΦ = 1033331111 120 -)(25

240)(25

560u

inL

outL

inR

outR

inL

outL

inR

outR jjjjjjjj Φ−−++−−+

a4 = 35a2 - 700 20uΦ = 20033331111 700 -70)(25

840)(25

1960u

inL

outL

inR

outR

inL

outL

inR

outR jjjjjjjj ΦΦ−+++++++

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138

for the second equation the expansion coefficients,

b1 = )(2532)(

258

3333111100inL

outL

inR

outR

inL

outL

inR

outRLR jjjjjjjj −−++−−+=− φφ

b2 = =−+ 200 2φφφ LR 233331111 2)(2532)(

258 φ−+++++++ in

LoutL

inR

outR

inL

outL

inR

outR jjjjjjjj

b3 = 10b1 - 120 12uφ = 1233331111 120 -)(25

320)(2580

uinL

outL

inR

outR

inL

outL

inR

outR jjjjjjjj φ−−++−−+

b4 = 35b2 - 700 22uφ = 22033331111 700 -70)(25

1120)(25

280u

inL

outL

inR

outR

inL

outL

inR

outR jjjjjjjj φφ−+++++++

After performing transverse integration, the flux moments for u,v,w ∈R3 are as follows:

10uΦ =

( )

∆+

∆+−Σ−

−−++−−+

∆++−−

∆Σ

111120,

3333111121

1111

0, 112

)(2524)(

2556

21

1

uwuvuurem

inL

outL

inR

outR

inL

outL

inR

outR

inL

outL

inR

outR

rem Lw

Lv

Q

jjjjjjjju

Djjjj

u

φ

2u0Φ =

( )

Φ∆

−∆

+∆

+−Σ−

+++++++

∆+−+−

∆Σ

021

222220,

3333111121

1111

0, 6112

)(2524)(

25563

21

1

uD

Lw

Lv

Q

jjjjjjjjuD

jjjju

uwuvuurem

inL

outL

inR

outR

inL

outL

inR

outR

inL

outL

inR

outR

rem φ

12uφ =

( )

∆+

∆++ΦΣ−

−−++−−+

∆++−−

∆−

12121100,

3333111122

3333

1152

52

)(2532)(

258

21

1

uwuvuurem

inL

outL

inR

outR

inL

outL

inR

outR

inL

outL

inR

outR

Lw

Lv

Q

jjjjjjjju

Djjjj

2u2φ =

( )

∆−

∆+

∆++ΦΣ−

+++++++

∆+−+−

∆−

222

22222200,

3333111122

3333

61152

52

)(2532)(

2583

21

1

φα

uD

Lw

Lv

Q

jjjjjjjjuD

jjjju

uwuvuurem

inL

outL

inR

outR

inL

outL

inR

outR

inL

outL

inR

outR

Page 152: ADVANCED REACTOR PHYSICS METHODS FOR HETEROGENEOUS REACTOR ...

139

Response Matrix Equations (1st Currents)

20011 )( uuu

inR

outR u

dudDjj ∆

=Φ−=

20011 )( uuu

inL

outL u

dudDjj ∆

−=Φ+=

)(0 udud

uΦ = u

au

au

au

a∆

+∆

+∆

+∆ 52

3 4321 2uu ∆

=

)(0 udud

uΦ = u

au

au

au

a∆

−∆

+∆

−∆ 52

3 4321 2uu ∆

−=

( )

( ) ( )

( )

( ) ( )

++

+++++++

∆+

Φ∆

−−Σ−+−−∆

Σ+

++

−−++−−+

∆+

−Σ−−+−∆

Σ+

Φ−+++++++

+−−++−−+

∆−=

22

3333111120

020

2220,1111

0,

11

3333111120

1210,1111

0,

033331111

33331111

011

112572

25168

6)(2

21

140

112524

2556

)(22

1

60

20)(25

240)(25

560

)(25

144)(25

336

uwuv

inL

outL

inR

outR

inL

outL

inR

outR

uremoutL

inL

inR

outR

rem

uwuv

inL

outL

inR

outR

inL

outL

inR

outR

uremoutL

inL

inR

outR

rem

inL

outL

inR

outR

inL

outL

inR

outR

inL

outL

inR

outR

inL

outL

inR

outR

inR

outR

Lw

Lv

jjjjjjjju

DuD

Qujjjju

Lw

Lv

jjjjjjjju

D

Qujjjju

jjjjjjjj

jjjjjjjj

uD

jj

φ

φ

Page 153: ADVANCED REACTOR PHYSICS METHODS FOR HETEROGENEOUS REACTOR ...

140

Σ+

Σ+

Σ+

Σ+Σ−Σ−

Σ−

Σ−

∆Σ+

Σ−

∆Σ+

+

∆Σ++

∆Σ++

∆Σ+

+

∆∆Σ−

∆Σ−+

∆∆Σ−

∆Σ−

+

∆∆Σ−

∆Σ−+

∆∆Σ−

∆Σ−

+

∆∆Σ+

∆Σ+

∆∆Σ+

∆Σ

+

∆∆Σ+

∆Σ+

∆∆Σ+

∆Σ

+

∆Σ−

∆Σ+

∆Σ−

+

∆Σ+

∆Σ+

∆Σ+

+

∆Σ−

∆Σ+

∆Σ−

+

∆Σ+

∆Σ+

∆Σ+

∆−=

20,

20,

10,

10,

220,210,

20,

10,

220,

0

0,32

0,

0

320,

032

0,

032

0,

0

1220,

0

0,122

0,

0

0,

1220,

0

0,122

0,

0

0,

1220,

0

0,122

0,

0

0,

1220,

0

0,122

0,

0

0,

1320,

02

0,

0

0,

1320,

02

0,

0

0,

1320,

02

0,

0

0,

1320,

02

0,

0

0,

011

1140

1140160160)(280)(120

140608402025

86402596

258640

2596

2511520

25384

2511520

25384

8402084020

8402084020

8402084020

8402084020

84025

201606025

224

84025

201606025

224

84025

2688012025

896

84025

2688012025

896

uwrem

uvrem

uwrem

uvrem

remrem

urem

uremremrem

outL

rem

inL

rem

inuR

rem

outuR

rem

inwL

remrem

invL

remrem

inwR

remrem

invR

remrem

outwL

remrem

outvL

remrem

outwR

remrem

outvR

remrem

inuL

remremrem

outuL

remremrem

inuR

remremrem

outuR

remremrem

inuR

outuR

Lw

Lv

Lw

Lv

uu

QQQu

Dj

uD

ju

Dj

uD

ju

D

jwu

Dw

jvu

Dv

jwu

Dw

jvu

Dv

jwu

Dw

jvu

Dv

jwu

Dw

jvu

Dv

ju

Du

Du

ju

Du

Du

ju

Du

Du

ju

Du

Du

uD

jj

φφ

Page 154: ADVANCED REACTOR PHYSICS METHODS FOR HETEROGENEOUS REACTOR ...

141

( )

( ) ( )

( )

( ) ( )

++

+++++++

∆+

Φ∆

−−Σ−+−−∆

Σ−

++

−−++−−+

∆+

−Σ−−+−∆

Σ+

Φ++++−+++

−−−++−−+

∆+=

22

3333111120

020

2220,1111

0,

11

3333111120

1210,1111

0,

033331111

33331111

011

112572

25168

6)(2

21

140

112524

2556

)(22

1

60

20)(25

240)(25

560

)(25

144)(25

336

uwuv

inL

outL

inR

outR

inL

outL

inR

outR

uremoutL

inL

inR

outR

rem

uwuv

inL

outL

inR

outR

inL

outL

inR

outR

uremoutL

inL

inR

outR

rem

inL

outL

inR

outR

inL

outL

inR

outR

inL

outL

inR

outR

inL

outL

inR

outR

inuL

outuL

Lw

Lv

jjjjjjjju

DuD

Qujjjju

Lw

Lv

jjjjjjjju

D

Qujjjju

jjjjjjjj

jjjjjjjj

uD

jj

φ

φ

Page 155: ADVANCED REACTOR PHYSICS METHODS FOR HETEROGENEOUS REACTOR ...

142

Σ−

Σ−

Σ+

Σ+Σ+Σ−

Σ+

Σ−

∆Σ+

Σ+

∆Σ−−

+

∆Σ−−+

∆Σ−−+

∆Σ−−

+

∆∆Σ+

∆Σ+

∆∆Σ+

∆Σ

+

∆∆Σ+

∆Σ+

∆∆Σ+

∆Σ

+

∆∆Σ−

∆Σ−+

∆∆Σ−

∆Σ−

+

∆∆Σ−

∆Σ−+

∆∆Σ−

∆Σ−

+

∆Σ+

∆Σ+

∆Σ+−

+

∆Σ−

∆Σ−

∆Σ−−

+

∆Σ+

∆Σ−

∆Σ+−

+

∆Σ−

∆Σ−

∆Σ−−

∆+=

20,

20,

10,

10,

220,210,

20,

10,

220,

0

0,32

0,

0

320,

032

0,

032

0,

0

1220,

0

0,122

0,

0

0,

1220,

0

0,122

0,

0

0,

1220,

0

0,122

0,

0

0,

1220,

0

0,122

0,

0

0,

1320,

02

0,

0

0,

1320,

02

0,

0

0,

1320,

02

0,

0

0,

1320,

02

0,

0

0,

011

11401140

160160)(280)(120

140608402025

1152025

384

2511520

25384

258640

2596

258640

2596

8402084020

8402084020

8402084020

8402084020

84025

2688012025

896

84025

2688012025

896

84025

201606025

224

84025

201606025

224

uwrem

uvrem

uwrem

uvrem

remrem

urem

uremremrem

outL

rem

inL

rem

inuR

rem

outuR

rem

inwL

remrem

invL

remrem

inwR

remrem

invR

remrem

outwL

remrem

outvL

remrem

outwR

remrem

outvR

remrem

inuL

remremrem

outuL

remremrem

inuR

remremrem

outuR

remremrem

inuL

outuL

Lw

Lv

Lw

Lv

uu

QQQu

Dj

uD

ju

Dj

uD

ju

D

jwu

Dw

jvu

Dv

jwu

Dw

jvu

Dv

jwu

Dw

jvu

Dv

jwu

Dw

jvu

Dv

ju

Du

Du

ju

Du

Du

ju

Du

Du

ju

Du

Du

uD

jj

φφ

Page 156: ADVANCED REACTOR PHYSICS METHODS FOR HETEROGENEOUS REACTOR ...

143

Response Matrix Equations (2st Currents)

22233 )( uuu

inR

outR u

dudDjj ∆

=−= φ

22233 )( uuu

inL

outL u

dudDjj ∆

−=+= φ

)(2 udud

uφ = u

bu

bu

bu

b∆

+∆

+∆

+∆ 52

3 4321 2uu ∆

=

)(2 udud

uφ = u

bu

bu

bu

b∆

−∆

+∆

−∆ 52

3 4321 2uu ∆

−=

( )

( )

∆−

∆+

∆++ΦΣ−

++++

+++

∆+−+−

∆+

∆+

∆++ΦΣ−

−−++

−−+

∆++−−

∆+

−+++++++

+−−++−−+

∆−=

222

22222200,

3333

1111

22

3333

12121100,

3333

1111

22

3333

233331111

33331111

233

61152

52

)(2532

)(258

32

1140

1152

52

)(2532

)(258

21

60

20)(25

320)(2580

)(25

192)(2548

φ

α

α

φ

uD

Lw

Lv

Q

jjjj

jjjj

uD

jjjju

Lw

Lv

Q

jjjj

jjjj

uD

jjjju

jjjjjjjj

jjjjjjjj

uD

jj

uwuvuurem

inL

outL

inR

outR

inL

outL

inR

outR

inL

outL

inR

outR

uwuvuurem

inL

outL

inR

outR

inL

outL

inR

outR

inL

outL

inR

outR

inL

outL

inR

outR

inL

outL

inR

outR

inL

outL

inR

outR

inL

outL

inR

outR

inR

outR

Page 157: ADVANCED REACTOR PHYSICS METHODS FOR HETEROGENEOUS REACTOR ...

144

∆+

∆++Φ

Σ−

∆+

∆++Φ

Σ−

−−+

+

+

++

++

++

+∆

+

+∆

++

+∆

−+

+∆

+

∆−=

22222200,

12121100,

222

122

122

122

122

322

322

322

322

233

1401405656

6060242484020

252880

2532

252880

2532

253840

25128

253840

25128

251152040

25128

251152040

25128

2515360100

25512

2515360100

25512

uwuvuurem

uwuvuuremin

L

outL

inR

outR

inL

outL

inR

outR

inR

outR

Lw

Lv

Q

Lw

Lv

QuD

juD

juD

juD

juD

juD

u

juD

u

juD

uj

uD

u

uD

jj

αααα

ααααφ

αα

αα

ααα

αα

αααα

( )

( )

∆−

∆+

∆++ΦΣ−

++++

+++

∆+−+−

∆−

∆+

∆++ΦΣ−

−−++

−−+

∆++−−

∆+

++++−+++

−−−++−−+

∆+=

222

22222200,

3333

1111

22

3333

12121100,

3333

1111

22

3333

233331111

33331111

233

61152

52

)(2532

)(258

32

1140

1152

52

)(2532

)(258

21

60

20)(25

320)(2580

)(25

192)(2548

φ

α

α

φ

uD

Lw

Lv

Q

jjjj

jjjj

uD

jjjju

Lw

Lv

Q

jjjj

jjjj

uD

jjjju

jjjjjjjj

jjjjjjjj

uD

jj

uwuvuurem

inL

outL

inR

outR

inL

outL

inR

outR

inL

outL

inR

outR

uwuvuurem

inL

outL

inR

outR

inL

outL

inR

outR

inL

outL

inR

outR

inL

outL

inR

outR

inL

outL

inR

outR

inL

outL

inR

outR

inL

outL

inR

outR

inL

outL

Page 158: ADVANCED REACTOR PHYSICS METHODS FOR HETEROGENEOUS REACTOR ...

145

∆−

∆−+Φ

Σ+

+∆

++ΦΣ

++

−−+

−−

+

−−+

−−+

−∆

+−

+

−∆

−−+

−∆

+−+

−∆

−−

∆+=

22222200,

12

121100,

222

122

122

122

122

322

322

322

322

233

140140565660

602424

84020

253840

25128

253840

25128

252880

2532

252880

2532

2515360100

25512

2515360100

25512

251152040

25128

251152040

25128

uwuvuurem

uw

uvuurem

inL

outL

inR

outR

inL

outL

inR

outR

inL

outL

Lw

Lv

QLw

Lv

Q

uD

juD

juD

juD

juD

juD

u

juD

u

juD

uj

uD

u

uD

jj

ααααα

ααα

φααα

αααα

αα

αααα

Page 159: ADVANCED REACTOR PHYSICS METHODS FOR HETEROGENEOUS REACTOR ...

146

APPENDIX E. NEM INPUT and NEMTAB INPUT for SP3 OPTION

NEM Input !--------------------------------------------------------------! ! Card 1 ! !--------------------------------------------------------------! !Type of calc !Number of!Geometry !Number of!0=steady !0= no DF !XS File Option !TH option ! !0 = Pol !delayed !type !energy !1=transie!1=single DF(hom. partial currents) !0=use values here !0 = fixed ! !1 = SA !neutron !0=cartes !groups ! !2=directional DF(hom.partial currents) !1=use nemtab ! temp ! !2 = SP3 !groups !1=cylind ! ! !3=single DF(het. partial currents) ! !1 = coupled! ! ! !2=hex-z ! ! !4=directional DF(het. partial currents)! ! ! !NEMTYPE !NDGR !IGEOM !NG !ITRANS !IADF !EXTXS !COUPTH ! !--------------------------------------------------------------! 2 6 0 7 0 0 1 0 !--------------------------------------------------------------! ! Card 1a ! !--------------------------------------------------------------! !Fuel Temp !Mod Dens !Boron !Mod Temp ! ! ! ! ! ! !TMP_F !MOD_D !MOD_B !MOD_T ! !--------------------------------------------------------------! 550.0 650.0 0.0 0.0 !--------------------------------------------------------------! ! Card 4 ! !--------------------------------------------------------------! !Number !Number !Number !Number !Number !Number !Number ! !nodes in !nodes in !nodes in !cross !asemblies!asemblies!asemblies! !x or r !y or O !z !section !x or r !y or O !z ! !direction!direction!direction!sets !direction!direction!direction! !NXNDS !NYNDS !NZNDS !NXSETS !NASMX !NASMY !NASMZ ! !--------------------------------------------------------------! 51 51 1 7 51 51 1 !-------------------------------------------------! ! Card 5 ! !-------------------------------------------------! !Number !Number !Number !Number !Number ! !coarse !coarse !coarse !material !inner ! !mesh !mesh !mesh !cards or !iteration! !nodes !nodes !nodes !Number !per outer! !x or r !y or O !z !Assembly !iteration! !direction!direction!direction! ! ! !NXCDS !NYCDS !NZCDS !NXSCDS !NINNER ! !-------------------------------------------------- 51 51 1 7 350 !--------------------------------------------------------------! ! Card 5b - Output options ! !--------------------------------------------------------------! !Flag !Flag !Print !Print !Echo !Echo !Echo !

Page 160: ADVANCED REACTOR PHYSICS METHODS FOR HETEROGENEOUS REACTOR ...

147

!indicatin!indicatin!albedos !converg. !input? !material !X-Sec ! !output !input !data? !data? !0=no !maps? !data? ! !0=short !0=short !0=no !0=no !1=yes !0=no !0=no ! !1=long !1=long !1=yes !1=yes ! !1=yes !1=yes ! ! ! ! ! ! ! ! ! !ILNG !IIN !IALB !ICONV !IIECHO !IMAPS !IXSD ! !-------------------!----------!----------!---------!---------!------! 1 0 1 1 1 1 1 !--------------------------------------------------------------! ! Card 8 ! !--------------------------------------------------------------! !Maximum !Use !Use !quadratic!Point or !Average !convergenc! !number !coarse !asymptoti!leakage !L4 norm !or L2norm !criterion ! !outer !mesh !extrapola!option !convergen !convergen !for Keff ! !iteration !rebalance !0=NO !0=NO !criterion !criterion ! ! ! !0=NO !1=YES !1=YES !on nodal !on nodal ! ! ! !1=YES ! ! !fission !fission ! ! ! ! ! ! !source !source ! ! !MOUTER !IREB !IAEX !IQL !APTCVG !AAVCVG !AKCVG ! !--------------------------------------------------------------! 1000 0 0 1 5.000D-06 5.000D-06 5.000D-06 !--------------------------------------------------------! ! Card 9 ! !--------------------------------------------------------! !Upscatter !MAximum !Weilant !Weilant !steady ! !option !number !shift !shift !state ! !0=NO !coarse !parameter !option !relaxati! !1=YES !mesh ! !1=YES !paramete! ! !iteration! !0=NO !Jacobi ! ! ! ! ! !iteratio! !IUSCAT !MXCMIT !CMEMD !IWEIL !OMEGOR ! !--------------------------------------------------------! 1 100 0.000D+00 0 1.200D+00 5 3

Cards 12-42 remain unchanged from current version of NEM.

Page 161: ADVANCED REACTOR PHYSICS METHODS FOR HETEROGENEOUS REACTOR ...

148

NEMTAB Input * * NEM-Cross Section Table Input * * T Fuel Rho Mod. Boron ppm. T Mod. 0 0 0 0 * ******* 1 UO2 Cross Section 1 * * Group No. 1 * *************** Diffusion Coefficient Table * 1.96054D+00 * *************** 2nd Diffusion Coefficient Table * 1.16935D+00 * *************** sigR0 X-Section Table * 1.27498D-01 * *************** sigR1 X-Section Table * 2.19903D-01 * *************** sigTOT X-Section Table * 2.19903D-01 * *************** Capture X-Section Table * 4.88476D-03 * *************** Fission X-Section Table * 4.16155D-03 * *************** Nu-Fission X-Section Table * 1.15752D-02 * *************** Scattering X-Section Table * **** 7.24184E-02 3.11454E-04 1.58929E-06 2.24954E-08 0.00000E+00 0.00000E+00 *

Page 162: ADVANCED REACTOR PHYSICS METHODS FOR HETEROGENEOUS REACTOR ...

Vita

Steven Thompson was born in Tampa, Florida on November 21, 1983. Steven received his

Bachelor of Science degree in nuclear engineering from the University of Florida in 2006. In

May of 2007 he began work as a nuclear core design engineer for Dominion Virginia Power.

While there, he has authored over 50 calculations and technical reports. In 2007 he began his

post graduate studies in nuclear engineering at the Pennsylvania State University and received his

Master of Engineering degree in 2010. He continued his study at the

Pennsylvania State University

and earned his PhD in nuclear engineering in December of 2014. His research interests include

numerical methods in radiation transport, reactor physics, and reactor noise analysis.