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HETEROGENEOUS REACTOR CORES
A Dissertation in
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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 ProfessorInCharge 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, mixedoxide (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 subbatches 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 semianalytical NEM solution option.
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1.1 Introduction…………………………………………….……………..………………1
1.3 Thesis Outline………………………………………..….……………………………7
2.1 Introduction……………………………………..…………………….……………...10
2.2 The SP3
Approximation………………………………..…………………………….10
2.5 Discontinuity Factors………………………………………………………………...17
3.1 Introduction……………………………………..……………….…………………...22
Equations…….……………..…………….…………………….……23
3.3 Benchmarking of the SP
Solution in NEM….……...……………...29
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3.3.3 PWR MOX/UO2
Core Transient Benchmark……………..…..………………48
Nodal Expansion Method…………………..………………..60
4.3 Benchmarking of SANEM Solution……..………………..………………………..72
4.3.1 OECD/NEA 2D C5G7 MOX Benchmark……….…..….……………………72
4.3.2 C3 2x2 MiniCore Benchmark…………………..….………..………………..76
4.3.3 C5 2x2 MiniCore Benchmark……………..…..…….………………………..77
4.3.4 Mesh Width Sensitivity………………………..……..………………………..78
4.4 Concluding Remarks on the SANEM……………..……….....….…………………80
CHAPTER 5 THE ANALYTIC BASIS FUNCTION
TRANSVERSE LEAKAGE METHOD………………………..…………………….…………..83
5.2.1 Representation of IntraNodal 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
Approximation………….……………………..98
Method…………..………….………101
REFERENCES………………………………………………………………………………….109
APPENDIX B. BENCHMARK CROSS SECTIONS……………………..……………………120
APPENDIX C. 2D C5G7 Benchmark Pin Powers and % Error Comparison………..………..127
APPENDIX D. SP3
RESPONSE MATRIX EQUATIONS…………….……..…….………….137
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 CoarseMesh Finite Difference CQLA Consistent Quadratic Leakage Approximation DF Discontinuity Factor ET Equivalence Theory GET Generalized Equivalence Theory IFBA Integral Fuel Burnable Absorber MOX MixedOxide 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 PNEM Polynomial Nodal Expansion Method PSU The Pennsylvania State University PWR Pressurized Water Reactor QLA Quadratic Leakage Approximation RM Response Matrix SANEM SemiAnalytical Nodal Expansion Method SP3 Simplified P3 SPH Superhomogenization 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 PNEM Diffusion Theory Pin Power
Distribution for 2D C5G7 MOX Benchmark……………...………..…………....37
Figure 3.2 SP3 Pin Power Distribution for 2D 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 QuarterCore 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 SANEM Pin Power Distribution for 2D C5G7 MOX Benchmark………….….75
Figure 4.2 Assembly Power RMS % Error vs. Mesh Size
for PNEM and SANEM Solutions………………………………..…………..….80
Figure 5.1 Assembly Power RMS % Error vs. Mesh Size……………..……….…………….94
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Figure A.1 OECD/NEA 2D C5G7 MOX Benchmark Core Configuration………...…..…116
Figure A.2 PinbyPin 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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with MCNP Reference Solution (SP3
Table 3.3 Calculated Pin Powers from NEM Compared with MCNP
2D C5G7 Benchmark).......….……………....35
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 (SANEM 2D C5G7 Benchmark)……..……….74
Table 4.2 Calculated Assembly Powers from NEM Compared
with MCNP Reference Solution (SANEM 2D 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.5 C5G7 4.3% MOX FuelClad Cross Sections………………………….…………122
Table B.6 C5G7 7.0% MOX FuelClad Cross Sections…………………………………….123
Table B.7 C5G7 8.7% MOX FuelClad 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 PNEM 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 PNEM 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 PNEM 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 SANEM Diffusion Theory Solution)…………………….…………130
Table C8 C5G7 Distribution of % Errors in MOX Assembly
(NEM with SANEM Diffusion Theory Solution)……………………………….130
Table C9 C5G7 Distribution of % Errors in UO2 Assembly Near Water Reflector
(NEM with SANEM Diffusion Theory Solution)……………………………….131
Table C10 C5G7 Distribution of Pin Powers in UO2 Assembly Near Axis of Symmetry
(NEM with PNEM 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 SANEM Diffusion Theory Solution)………………………..…….133
Table C13 C5G7 Distribution of Pin Powers in MOX Assembly
(NEM with PNEM 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 SANEM Diffusion Theory Solution)………………………………134
Table C16 C5G7 Distribution of Pin Powers in UO2 Assembly Near Water Reflector
(NEM with PNEM 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 SANEM 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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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 mixedoxide (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 relicensure 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 relicensure 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 repressurization 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
subbatches, 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, boroncarbide, 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 subbatches 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 highburnup 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
intranodal 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 semianalytical nodal expansion method (SANEM) solution option, which is
based upon the within group form of the neutron diffusion equation, has been
completed in NEM. The SANEM 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 minicores 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 intranodal 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 selfcontained
numbering scheme for the equations, figures, and tables. The chapters are arranged as
follows:
Chapter 1 provides some brief background information on the research topic and a summary
of the research objectives. It provides the reader with a brief discussion of the problems
which the present work is seeking to address and discusses the benefits which will be derived
from the present work.
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Chapter 2 summarizes the results of the literature review performed for this thesis work. The
literature related to the historical background of the SP3 method is reviewed and summarized,
along with the papers which aided in its theoretical basis being more firmly solidified. The
work performed by others attempting to improve the transverse leakage treatment is also
summarized. The literature related to the development of the semianalytical nodal expansion
method is also included in this chapter. A review of literature related to the equivalence
theory, generalized equivalence theory, superhomogenization 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 SANEM solution. The SANEM solution procedure is developed
analytically, followed by a discussion of its implementation into NEM. Benchmarking of the
SANEM solution is also provided along with a mesh size sensitivity study. Some final
remarks on the SANEM 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 intranodal 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.
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 semianalytical 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 [57]. 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 3D to
create the Simplified PN equations. The approximation that he introduced, according to
Gamino [89], 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 leadingorder asymptotic approximation of
the transport equation with an error of O(ε3), where ε is a dimensionless parameter. They
further demonstrate, using the same approach, that the SP2 and SP3 equations are asymptotic
approximations to the transport equation with errors of O(ε5) and O(ε7) respectively.
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Therefore, the contribution of Reference 12 to fortifying the theoretical basis of the SPN
equations is to demonstrate that the simplified PN equations are asymptotic corrections to the
P1 equations.
Pomraning [10] further fortified the theoretical basis of the SPN equations by demonstrating
that the SPN equations are a leading order asymptotic limit of the transport equation,
corresponding to nearly planar transport with a phase space which has a highly forward
peaked scattering kernel. He also showed that this asymptotic limit exists for timedependent
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 selfadjoint 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 SIMULATE5 nodal code
[14], NFI’s SCOPE2 code [15], the Forschungszentrum DresdenRossendorf Institute of
Safety Research in its DYN3D code [16], and the PARCS code developed by Purdue
University for the US NRC [53].
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2.3 The SemiAnalytical Nodal Expansion Method
The SANEM 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 SANEM has been implemented by various researchers.
In the semianalytical twogroup 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 fourthorder polynomial.
Kim et. al. [19] introduced an SANEM method which involves the analytical solution
consisting of two exponential functions and a fourthorder polynomial. Kim’s method can be
applied to multigroup problems, unlike that of Esser and Smith [18] which is strictly a two
group method.
One of the more recent works is that by Yamamoto and Tatsumi [20]. This paper describes
the scattered source subtraction (SSS) method which is used to reduce the spatial
discretization error in the SCOPE2 code. In the SSS method, the scattered source is
subtracted from both sides of the diffusion equation, which reduces the spatial variation of
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the source term and thereby reduces the error associated with the flatsource approximation
which is used in SCOPE2.
In the work performed by Fu and Cho [21], a nonlinear semianalytical method was
developed using the coarsemesh finite difference (CMFD) scheme. In CMFD, the nodal
equations are used for the local twonode 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 surfaceaveraged neutron currents as the nodal
solution.
Han, Joo and Kim [54] developed a twogroup CMFD which accelerates the semianalytical
nodal method kernel. In this method, a quartic expansion of the source terms is used in the
semianalytical 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 twogroup CMFD is used for the global
calculation meaning that the multigroup calculation needs to be performed only at the local
onenode level.
2.4 Transverse Leakage Approximation
The transverse leakage (TL) term arises in nodal codes which use the transverse integration
procedure to integrate the threedimensional 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 onedimensional 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 [2223]. 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 onedimensional 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 onedimensional 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 mixedoxide fuel, and near strong absorbers such as control rods.
An advanced TL method developed by Prinsloo et al. [2830], the consistent quadratic
leakage approximation (CQLA), is based upon weighted transverse integration. The weighted
transverse integration is performed by expressing the intranodal 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 intranodal 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 [2526] used nine analytic basis functions to
describe the homogeneous intranodal flux. The coefficients were expressed in terms of the
cornerpoint fluxes, sideaveraged fluxes, and the nodeaverage 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 onedimensional eigensolutions
from the original method [2526] 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, fluxweighted 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 [3235]. 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
twonode 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 twonode 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
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
surfaceaveraged fluxes to the cellaveraged 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 58%.
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 surfaceaveraged fluxes to the cellaveraged fluxes. Surface DFs are explicitly defined
on each surface of the assembly, meaning that a traditional squarelattice assembly has four
DFs, hexagonallattice 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 pinbypin
21
geometry. Therefore the ADFs or surface DFs described previously would be of limited
practical benefit. In pinbypin 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.
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 [57]. 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 2D and 3D 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 onedimensionally
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 forwardpeaked scattering kernel. This allows
the PN equations to be written in onedimension (for arbitrary node # n) as
)r(S)r()r()r( dr d
+ +
→+− (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, φΝ1 and φN+1 are assumed to be 0, n gt ,Σ is the group g total macroscopic cross
section, n ggls →Σ ',, is the lth moment of the macroscopic scattering cross section from group g’
into group g.
)(,0 rS n g = )()(1
, 1'
φνχ (3.2)
where χg is the fission spectrum, keff is the neutron multiplication factor, n gf ',Σ is the
macroscopic fission cross section for group g’, n is the node number, and )(, rS n gex 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 grouptogroup scattering terms higher than l = 0. However,
Beckert and Grundmann [16] have reported fairly large errors in pinbypin SP3 calculations
in which anisotropic grouptogroup scattering was completely neglected. Due to these
conclusions from Beckert and Grudmann [16], in which firstorder anisotropic groupto
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
grouptogroup scattering). Therefore, linearly anisotropic grouptogroup scattering is
considered. This is shown in Equation 3.3 below.
∑ =
1'g
1'g
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( n g,t
n g,tr ∑
gt )()( ,0,,0 rr n gsg 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:
= →
= →
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 1D planar problems to 3D 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
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( dr d
3 1)r( g,0
g,tr g,1 Φ
Σ −=φ (3.7)
gr g 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
(3.9)
with,
1' ', rr
k gg
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(S 5 2)r(
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
)(
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 intranodal flux shape is
expanded in series within each node using fourth order polynomial basis functions as
follows:
2φ = nodal volume averaged second flux moment
In the fourthorder approximation, the series is truncated after the first four basis functions
(given below for the x direction), which are given by
( )1
3 1 1 1 2 2 4
x x x x xf x x x x x x
= − + = − (3.17)
( ) 2 4 2
4 1 1 1 3 1 20 2 2 10 80
x x x x xf x x x x x x
= − − + = − + (3.18)
The first two expansion coefficients for Equations 3.13 and 3.14 can be obtained by
evaluating the intranodal 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 intranodal sources and sinks. Using the x direction as an example, Fick’s
Law takes the following form:
2
dx dDjj
−= += 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 i72620M 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 2D C5G7 MOX Benchmark
The OECD/NEA 2D 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 sevengroup 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. Upscattering was explicitly considered for groups 57. A convergence of
1E05 was used for keff and 1E06 was used for pointwise flux. NEM was run in pinbypin
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
n = pin #
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 2D C5G7 Benchmark)
Evaluated Parameter MCNP (Reference) NEMSP3 NEMDIFF 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

1.867
1.894
(1.472)
1.894
(% error)
0.529
(% error)
0.802
0.791
(1.349)
Table 3.2 Calculated Assembly Powers from NEM Compared with MCNP Reference Solution
(SP3 2D C5G7 Benchmark)
Fuel Assembly MCNP (Reference) NEMSP3 NEMDIFF 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 PNEM
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 PNEM Diffusion Theory Pin Power Distribution for 2D C5G7 MOX Benchmark
1
5
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
Water Moderator
Pin Power
38
Figure 3.2 SP3 Pin Power Distribution for 2D C5G7 MOX Benchmark
1
5
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
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 3D C5G7 MOX Benchmarks (Rodded)
The previous benchmark demonstrated a small improvement in the prediction of pin powers
in 2D 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 3dimensional 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.
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
sevengroup 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 2D 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. Upscattering was explicitly
considered for groups 57. A convergence of 1E05 was used for keff and 1E06 was used for
pointwise flux. NEM was run in pinbypin 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) NEMSP3 NEMDIFF
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
62.12
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
49.45
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)
27.82
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
139.39 138.39 (0.716)
212.70
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
46
Table 3.4 Calculated Pin Powers from NEM Compared with MCNP Reference Solution (C5G7 Rodded A Benchmark)
Evaluated Parameter MCNP
(Reference) NEMSP3 NEMDIFF
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
69.80 70.03 (0.326)
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
53.39
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
28.21
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
151.39
221.71
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 2D 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 fourloop Westinghouse PWR
core. A quartercore loading pattern is specified due to core symmetry. This quartercore
configuration has uniform fuel composition in the axial direction and consists of 49 MOX
fuel assemblies with 4.0 and 4.3 wt% Pufissile 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 2.52cm 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 2D 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.
Pinwise 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
(superhomogenization) 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 16group cross section library.
No pin discontinuity factors or SPH factors are utilized in the eight energy group
calculations. The eight group pinbypin 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. Upscattering was
explicitly considered for groups 78 for the noSPH cases and groups 1316 for the SPH case
which uses 16 energy groups. A convergence of 1E05 was used for keff and 1E05 was used
for pointwise 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 47group cross
A Dissertation in
ii
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 ProfessorInCharge of Mechanical and Nuclear Engineering Graduate Programs
*Signatures are on file in the Graduate School
iii
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, mixedoxide (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 subbatches 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 semianalytical NEM solution option.
iv
1.1 Introduction…………………………………………….……………..………………1
1.3 Thesis Outline………………………………………..….……………………………7
2.1 Introduction……………………………………..…………………….……………...10
2.2 The SP3
Approximation………………………………..…………………………….10
2.5 Discontinuity Factors………………………………………………………………...17
3.1 Introduction……………………………………..……………….…………………...22
Equations…….……………..…………….…………………….……23
3.3 Benchmarking of the SP
Solution in NEM….……...……………...29
v
3.3.3 PWR MOX/UO2
Core Transient Benchmark……………..…..………………48
Nodal Expansion Method…………………..………………..60
4.3 Benchmarking of SANEM Solution……..………………..………………………..72
4.3.1 OECD/NEA 2D C5G7 MOX Benchmark……….…..….……………………72
4.3.2 C3 2x2 MiniCore Benchmark…………………..….………..………………..76
4.3.3 C5 2x2 MiniCore Benchmark……………..…..…….………………………..77
4.3.4 Mesh Width Sensitivity………………………..……..………………………..78
4.4 Concluding Remarks on the SANEM……………..……….....….…………………80
CHAPTER 5 THE ANALYTIC BASIS FUNCTION
TRANSVERSE LEAKAGE METHOD………………………..…………………….…………..83
5.2.1 Representation of IntraNodal 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
Approximation………….……………………..98
Method…………..………….………101
REFERENCES………………………………………………………………………………….109
APPENDIX B. BENCHMARK CROSS SECTIONS……………………..……………………120
APPENDIX C. 2D C5G7 Benchmark Pin Powers and % Error Comparison………..………..127
APPENDIX D. SP3
RESPONSE MATRIX EQUATIONS…………….……..…….………….137
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 CoarseMesh Finite Difference CQLA Consistent Quadratic Leakage Approximation DF Discontinuity Factor ET Equivalence Theory GET Generalized Equivalence Theory IFBA Integral Fuel Burnable Absorber MOX MixedOxide 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 PNEM Polynomial Nodal Expansion Method PSU The Pennsylvania State University PWR Pressurized Water Reactor QLA Quadratic Leakage Approximation RM Response Matrix SANEM SemiAnalytical Nodal Expansion Method SP3 Simplified P3 SPH Superhomogenization SSS Scattered Source Subtraction TLA Transverse Leakage Approximation TL Transverse Leakage WABA Wet Annular Burnable Absorber
viii
LIST OF FIGURES
Figure 2.1 Thermal Flux Distribution in Two Neighboring Fuel Assemblies………….……18
Figure 3.1 PNEM Diffusion Theory Pin Power
Distribution for 2D C5G7 MOX Benchmark……………...………..…………....37
Figure 3.2 SP3 Pin Power Distribution for 2D 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 QuarterCore 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 SANEM Pin Power Distribution for 2D C5G7 MOX Benchmark………….….75
Figure 4.2 Assembly Power RMS % Error vs. Mesh Size
for PNEM and SANEM Solutions………………………………..…………..….80
Figure 5.1 Assembly Power RMS % Error vs. Mesh Size……………..……….…………….94
ix
Figure A.1 OECD/NEA 2D C5G7 MOX Benchmark Core Configuration………...…..…116
Figure A.2 PinbyPin Layout of 2D C5G7 MOX Benchmark Fuel Assemblies……….....117
Figure A.3 C3 Core Configuration……………………………………………………….…..118
Figure A.4 C5 Core Configuration…………………………………………………….……..119
x
with MCNP Reference Solution (SP3
Table 3.3 Calculated Pin Powers from NEM Compared with MCNP
2D C5G7 Benchmark).......….……………....35
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 (SANEM 2D C5G7 Benchmark)……..……….74
Table 4.2 Calculated Assembly Powers from NEM Compared
with MCNP Reference Solution (SANEM 2D 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
xi
Table B.5 C5G7 4.3% MOX FuelClad Cross Sections………………………….…………122
Table B.6 C5G7 7.0% MOX FuelClad Cross Sections…………………………………….123
Table B.7 C5G7 8.7% MOX FuelClad 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 PNEM 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 PNEM 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 PNEM Diffusion Theory Solution)…………………………………129
Table C6 C5G7 Distribution of % Errors in UO2 Assembly Near Water Reflector
(NEM with SP3 Solution)………………………………………………………….129
xii
Table C7 C5G7 Distribution of % Errors in UO2 Assembly Near Axis of Symmetry
(NEM with SANEM Diffusion Theory Solution)…………………….…………130
Table C8 C5G7 Distribution of % Errors in MOX Assembly
(NEM with SANEM Diffusion Theory Solution)……………………………….130
Table C9 C5G7 Distribution of % Errors in UO2 Assembly Near Water Reflector
(NEM with SANEM Diffusion Theory Solution)……………………………….131
Table C10 C5G7 Distribution of Pin Powers in UO2 Assembly Near Axis of Symmetry
(NEM with PNEM 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 SANEM Diffusion Theory Solution)………………………..…….133
Table C13 C5G7 Distribution of Pin Powers in MOX Assembly
(NEM with PNEM 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 SANEM Diffusion Theory Solution)………………………………134
Table C16 C5G7 Distribution of Pin Powers in UO2 Assembly Near Water Reflector
(NEM with PNEM 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 SANEM Diffusion Theory Solution)………………………….…..136
xiii
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.
1
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 mixedoxide (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 relicensure process. The original 40 year
2
operating license is due to expire in the near future at most US nuclear power plants. As part
of the relicensure 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 repressurization 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
subbatches, 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.
3
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, boroncarbide, 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 subbatches is difficult to
accurately model with the diffusion approximation using current nodal methods (polynomial
4
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 highburnup 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
5
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
intranodal 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.
6
3) A semianalytical nodal expansion method (SANEM) solution option, which is
based upon the within group form of the neutron diffusion equation, has been
completed in NEM. The SANEM 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 minicores 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
7
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 intranodal 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 selfcontained
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 semianalytical nodal expansion
method is also included in this chapter. A review of literature related to the equivalence
theory, generalized equivalence theory, superhomogenization 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 SANEM solution. The SANEM solution procedure is developed
analytically, followed by a discussion of its implementation into NEM. Benchmarking of the
SANEM solution is also provided along with a mesh size sensitivity study. Some final
remarks on the SANEM 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 intranodal 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
9
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.
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 semianalytical 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 [57]. 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
11
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 3D to
create the Simplified PN equations. The approximation that he introduced, according to
Gamino [89], 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 leadingorder 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 timedependent
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 selfadjoint 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 SIMULATE5 nodal code
[14], NFI’s SCOPE2 code [15], the Forschungszentrum DresdenRossendorf 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 SemiAnalytical Nodal Expansion Method
The SANEM 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 SANEM has been implemented by various researchers.
In the semianalytical twogroup 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 fourthorder polynomial.
Kim et. al. [19] introduced an SANEM method which involves the analytical solution
consisting of two exponential functions and a fourthorder 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 flatsource approximation
which is used in SCOPE2.
In the work performed by Fu and Cho [21], a nonlinear semianalytical method was
developed using the coarsemesh finite difference (CMFD) scheme. In CMFD, the nodal
equations are used for the local twonode 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 surfaceaveraged neutron currents as the nodal
solution.
Han, Joo and Kim [54] developed a twogroup CMFD which accelerates the semianalytical
nodal method kernel. In this method, a quartic expansion of the source terms is used in the
semianalytical 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 twogroup CMFD is used for the global
calculation meaning that the multigroup calculation needs to be performed only at the local
onenode level.
2.4 Transverse Leakage Approximation
The transverse leakage (TL) term arises in nodal codes which use the transverse integration
procedure to integrate the threedimensional 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 onedimensional 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 [2223]. 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 onedimensional 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 onedimensional 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 mixedoxide fuel, and near strong absorbers such as control rods.
An advanced TL method developed by Prinsloo et al. [2830], the consistent quadratic
leakage approximation (CQLA), is based upon weighted transverse integration. The weighted
transverse integration is performed by expressing the intranodal 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 intranodal 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 [2526] used nine analytic basis functions to
describe the homogeneous intranodal flux. The coefficients were expressed in terms of the
cornerpoint fluxes, sideaveraged fluxes, and the nodeaverage 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 onedimensional eigensolutions
from the original method [2526] 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, fluxweighted 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 [3235]. 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
twonode 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 twonode 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
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
surfaceaveraged fluxes to the cellaveraged 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 58%.
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 surfaceaveraged fluxes to the cellaveraged fluxes. Surface DFs are explicitly defined
on each surface of the assembly, meaning that a traditional squarelattice assembly has four
DFs, hexagonallattice 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 pinbypin
21
geometry. Therefore the ADFs or surface DFs described previously would be of limited
practical benefit. In pinbypin 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.
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 [57]. 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 2D and 3D 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 onedimensionally
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 forwardpeaked scattering kernel. This allows
the PN equations to be written in onedimension (for arbitrary node # n) as
)r(S)r()r()r( dr d
+ +
→+− (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, φΝ1 and φN+1 are assumed to be 0, n gt ,Σ is the group g total macroscopic cross
section, n ggls →Σ ',, is the lth moment of the macroscopic scattering cross section from group g’
into group g.
)(,0 rS n g = )()(1
, 1'
φνχ (3.2)
where χg is the fission spectrum, keff is the neutron multiplication factor, n gf ',Σ is the
macroscopic fission cross section for group g’, n is the node number, and )(, rS n gex 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 grouptogroup scattering terms higher than l = 0. However,
Beckert and Grundmann [16] have reported fairly large errors in pinbypin SP3 calculations
in which anisotropic grouptogroup scattering was completely neglected. Due to these
conclusions from Beckert and Grudmann [16], in which firstorder anisotropic groupto
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
grouptogroup scattering). Therefore, linearly anisotropic grouptogroup scattering is
considered. This is shown in Equation 3.3 below.
∑ =
1'g
1'g
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( n g,t
n g,tr ∑
gt )()( ,0,,0 rr n gsg 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:
= →
= →
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 1D planar problems to 3D 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
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( dr d
3 1)r( g,0
g,tr g,1 Φ
Σ −=φ (3.7)
gr g 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
(3.9)
with,
1' ', rr
k gg
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(S 5 2)r(
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
)(
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 intranodal flux shape is
expanded in series within each node using fourth order polynomial basis functions as
follows:
2φ = nodal volume averaged second flux moment
In the fourthorder approximation, the series is truncated after the first four basis functions
(given below for the x direction), which are given by
( )1
3 1 1 1 2 2 4
x x x x xf x x x x x x
= − + = − (3.17)
( ) 2 4 2
4 1 1 1 3 1 20 2 2 10 80
x x x x xf x x x x x x
= − − + = − + (3.18)
The first two expansion coefficients for Equations 3.13 and 3.14 can be obtained by
evaluating the intranodal 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 intranodal sources and sinks. Using the x direction as an example, Fick’s
Law takes the following form:
2
dx dDjj
−= += 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 i72620M 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 2D C5G7 MOX Benchmark
The OECD/NEA 2D 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 sevengroup 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. Upscattering was explicitly considered for groups 57. A convergence of
1E05 was used for keff and 1E06 was used for pointwise flux. NEM was run in pinbypin
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
n = pin #
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 2D C5G7 Benchmark)
Evaluated Parameter MCNP (Reference) NEMSP3 NEMDIFF 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

1.867
1.894
(1.472)
1.894
(% error)
0.529
(% error)
0.802
0.791
(1.349)
Table 3.2 Calculated Assembly Powers from NEM Compared with MCNP Reference Solution
(SP3 2D C5G7 Benchmark)
Fuel Assembly MCNP (Reference) NEMSP3 NEMDIFF 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 PNEM
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 PNEM Diffusion Theory Pin Power Distribution for 2D C5G7 MOX Benchmark
1
5
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
Water Moderator
Pin Power
38
Figure 3.2 SP3 Pin Power Distribution for 2D C5G7 MOX Benchmark
1
5
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
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 3D C5G7 MOX Benchmarks (Rodded)
The previous benchmark demonstrated a small improvement in the prediction of pin powers
in 2D 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 3dimensional 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.
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
sevengroup 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 2D 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. Upscattering was explicitly
considered for groups 57. A convergence of 1E05 was used for keff and 1E06 was used for
pointwise flux. NEM was run in pinbypin 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) NEMSP3 NEMDIFF
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
62.12
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
49.45
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)
27.82
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
139.39 138.39 (0.716)
212.70
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
46
Table 3.4 Calculated Pin Powers from NEM Compared with MCNP Reference Solution (C5G7 Rodded A Benchmark)
Evaluated Parameter MCNP
(Reference) NEMSP3 NEMDIFF
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
69.80 70.03 (0.326)
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
53.39
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
28.21
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
151.39
221.71
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 2D 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 fourloop Westinghouse PWR
core. A quartercore loading pattern is specified due to core symmetry. This quartercore
configuration has uniform fuel composition in the axial direction and consists of 49 MOX
fuel assemblies with 4.0 and 4.3 wt% Pufissile 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 2.52cm 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 2D 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.
Pinwise 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
(superhomogenization) 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 16group cross section library.
No pin discontinuity factors or SPH factors are utilized in the eight energy group
calculations. The eight group pinbypin 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. Upscattering was
explicitly considered for groups 78 for the noSPH cases and groups 1316 for the SPH case
which uses 16 energy groups. A convergence of 1E05 was used for keff and 1E05 was used
for pointwise 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 47group cross