ADVANCED REACTOR PHYSICS METHODS FOR HETEROGENEOUS REACTOR ...
Transcript of ADVANCED REACTOR PHYSICS METHODS FOR HETEROGENEOUS REACTOR ...
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
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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
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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.
8
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.
12
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].
13
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
14
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
15
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
16
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
17
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
18
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
19
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
20
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
21
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.
22
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
23
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
24
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).
25
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
26
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.
27
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.
28
)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∇−
29
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)
30
∑=
+=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
31
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.
32
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.
33
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%.
34
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
35
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
36
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.
37
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
38
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
39
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.
40
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
41
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.
42
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.
43
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)
44
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.
45
Figure 3.5- Geometry for the Rodded A Benchmark Case
46
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)
47
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.
48
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
49
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.
50
Figure 3.6- Quarter-Core Configuration of MOX/UO2 Core Transient Benchmark
51
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
52
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.
53
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
54
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
55
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
56
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
.
57
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
58
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
59
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
60
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
61
(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
62
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
63
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.
64
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
65
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.
66
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
DΣ
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)
67
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
68
( )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
kΣ
=
2 2x xx−∆ ∆≤ ≤
l l l
gxn gxn gxnb p s= − +
69
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.
70
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
71
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.
72
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.
73
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 #
74
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
75
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
76
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.
77
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
78
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.
79
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
80
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)
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
82
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.
83
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
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
85
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.
86
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
87
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(
88
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
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
90
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
91
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.
92
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
93
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.
94
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)
95
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),
96
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.
97
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,
98
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)
99
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
100
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)
101
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
102
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.
103
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
104
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.
105
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
106
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
107
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
108
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.
109
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115
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.
116
Figure A.1- OECD/NEA 2-D C5G7 MOX Benchmark Core Configuration
117
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
118
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
119
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
120
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
121
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
122
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
123
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
124
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
125
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 -
126
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 -
127
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
128
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
129
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
130
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
131
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
132
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
133
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
134
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
135
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
136
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
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−+++++++
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
uα
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
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
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
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
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
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
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
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
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 !
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.
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 *
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.