The Pennsylvania State University

The Graduate School

Department of Mechanical and Nuclear Engineering

IMPROVED REFLECTOR MODELING FOR LIGHT WATER REACTOR ANALYSIS

A Thesis in

Nuclear Engineering

by

David V. Colameco

2010 David V. Colameco

Submitted in Partial Fulfillment

of the Requirements

for the Degree of

Master of Science

May 2010

ii

The thesis of David Colameco was reviewed and approved* by the following:

Kostadin Ivanov

Distinguished Professor of Nuclear Engineering

Thesis Advisor

Maria Avramova

Assistant Professor of Nuclear Engineering

Jack Brenizer, Jr.

Professor of Nuclear Engineering

Chair of Nuclear Engineering Program

*Signatures are on file in the Graduate School

iii

ABSTRACT

Special treatment of the reflector in reactor analysis is required due to the drastic

differences of neutronics properties between the core and reflector regions. The strong

spectrum change observed at core/reflector boundaries combined with geometry

complexity, material and structural heterogeneity of radial and axial reflectors make such

treatment a challenging task. The correct modeling of the reflector response is important

for accurate predictions of core power distribution especially in regions next to the

reflector. For this reason special care is taken in generation of reflector homogenized

cross-sections and discontinuity factors (DFs). Historically, one-dimensional (1-D) color

set problems are used for the reflector, which are different from the unit fuel assembly

cross-section generation models. The investigations presented in this thesis are further

extension of studies performed elsewhere to achieve a more correct modeling by

introducing improved color set models for reflector cross-section and DF generation and

parameterization. These color set models more accurately capture the 2-D effects that

occur on reentrant surfaces. From the transport solution of the color set model,

discontinuity factors are calculated, which in turn preserve the transport solution in the

nodal code calculations. Two sensitivity studies have been performed. The first study

evaluates the effect of the size of the color-set model on calculated reflector constants as

compared to full 1/8 core sector of symmetry. The second sensitivity study is conducted

with the aim of determining a parameterization for the reflector discontinuity factors at

the core-reflector interface as function of the core conditions such as boron concentration,

iv

moderator temperature and density. In addition, the effects of the loading pattern next to

the reflector region on the discontinuity factors are examined through the use of color sets

that include Mixed Oxide (MOX) fuel and the traditional UO2 fuel. The prediction

improvements that are achieved in both the global eigenvalue and power distribution

from the selected optimal 2-D color sets as compared to the 1-D models currently in use

are discussed in this thesis for two nuclear power plants. The newly calculated

discontinuity factors show an improvement in predicting the global eigenvalue and power

distribution and correcting the power tilt that was previously observed.

v

TABLE OF CONTENTS

LIST OF FIGURES ................................................................................................................. vi

LIST OF TABLES ................................................................................................................... viii

ACKNOWLEDGEMENTS ..................................................................................................... ix

CHAPTER 1 Introduction ....................................................................................................... 1

1.1 Background ................................................................................................................ 1 1.2 Modeling .................................................................................................................... 3 1.3 Sensitivity Studies ...................................................................................................... 4 1.4 Parameterization of Factors and Constants ................................................................ 6

CHAPTER 2 Reflector Modeling ........................................................................................... 8

2.1 Axial Reflector Modeling .......................................................................................... 9 2.2 Radial Reflector Modeling ......................................................................................... 11

CHAPTER 3 Sensitivity Studies ............................................................................................ 22

CHAPTER 4 Parameterization ............................................................................................... 53

CHAPTER 5 Conclusion ........................................................................................................ 63

REFERENCES ........................................................................................................................ 66

APPENDIX A: MCNP Modeling ............................................................................................ 68

APPENDIX B: Sample DF Routine Input ............................................................................... 69

vi

LIST OF FIGURES

Figure 1: 1-D Assembly/Reflector Models .............................................................................. 10

Figure 2: 3x3 Mini-Core Model ............................................................................................... 13

Figure 3: 1/8th Core Results .................................................................................................... 16

Figure 4: 1/8th Core Normalized Power Absolute Error Results ............................................ 20

Figure 5: Mini-Cores Used in the Sensitivity Study ................................................................ 23

Figure 6: Reflector Node Identification ................................................................................... 25

Figure 7: Constant Boron Concentration Fast Discontinuity Factor Behavior ........................ 27

Figure 8: Constant Temperature Fast Discontinuity Behavior ................................................ 28

Figure 9: Constant Boron Thermal Discontinuity Factor Behavior ......................................... 29

Figure 10: Constant Temperature Discontinuity Factor Behavior ........................................... 30

Figure 11: Fast Flat Side Discontinuity Factor Behavior......................................................... 31

Figure 12: Thermal Flat Side Discontinuity Factor Behavior .................................................. 32

Figure 13: Fast Inner-Corner Discontinuity Factor Behavior .................................................. 33

Figure 14: Inner-Corner Thermal Discontinuity Factor Behavior ........................................... 34

Figure 15: Flat-Sides Fast Discontinuity Factor Behavior ....................................................... 35

Figure 16: Flat-Sides Thermal Discontinuity Factor Behavior ................................................ 36

Figure 17: Inner-Corner Fast Discontinuity Factor Behavior .................................................. 37

Figure 18: Inner-Corner Thermal Discontinuity Factor Behavior ........................................... 38

Figure 19: MOX Flat-Side Fast Discontinuity Factor Behavior .............................................. 39

Figure 20: MOX Flat-Side Thermal Discontinuity Factor Behavior ....................................... 40

Figure 21: MOX Inner-Corner Fast Discontinuity Factor Behavior ........................................ 41

Figure 22: MOX Inner-Corner Thermal Discontinuity Factor Behavior ................................. 42

vii

Figure 23: Flat-Sides Fast Absorption Reflector Constant Behavior ....................................... 43

Figure 24: Flat-Side Thermal Absorption Reflector Constant Behavior ................................. 44

Figure 25: Inner-Corner Fast Absorption Reflector Constant Behavior .................................. 45

Figure 26: Inner-Corner Thermal Absorption Reflector Constant Behavior ........................... 46

Figure 27: Flat Sides Fast Diffusion Reflector Constant Behavior .......................................... 47

Figure 28: Flat Side Thermal Diffusion Reflector Constant Behavior .................................... 48

Figure 29: Inner-Corner Fast Diffusion Reflector Constant Behavior ..................................... 49

Figure 30: Inner-Corner Thermal Diffusion Reflector Constant Behavior .............................. 50

Figure 31: Flat Side Removal Reflector Constant Behavior .................................................... 51

Figure 32: Inner-Corner Removal Reflector Constant Behavior ............................................. 52

viii

LIST OF TABLES

Table 1: 3x3 Mini-Core Results ............................................................................................... 15

Table 2: 1/8th Core Results ...................................................................................................... 16

Table 3: PWR Axial Absolute Percent Error of Predicted Constant and Factor Data ............. 54

Table 4: BWR Axial Factor and Constant Absolute Percent Error ......................................... 55

Table 5: Individual Side Maximum Absolute Errors ............................................................... 58

Table 6: Individual Assembly Side Maximum Absolute Errors .............................................. 58

Table 7: Flat and Inner-Corner Side Maximum Absolute Errors ............................................ 59

Table 8: Maximum Absolute Errors for Multiple Minicore Modelings .................................. 60

ix

ACKNOWLEDGEMENTS

This thesis would not be possible without the dedication of my advisors, Dr.

Kostadin Ivanov and Dr. Mohamed Ouisloumen. Their guidance has been greatly

appreciated through the development and implementation of this project.

I am also grateful to Huria Harish, Boacheng Zhang, and Larry Mayhue for their

guidance while conducting the research behind this thesis. Last but not least,

Westinghouse Electric Company made this research possible through their sponsorship

for which I am extremely grateful.

CHAPTER 1

Introduction

1.1 Background

The objectives of this research are to investigate, develop and test a reflector

model for generation of equivalent homogenous (homogenized) reflector diffusion

parameters (cross-sections, diffusion coefficients and discontinuity factors) as well to

study and determine the important feedback effects for core reflector modeling. The

investigations presented in this thesis are further extension of studies performed

elsewhere [1-12] to achieve a more correct modeling by introducing improved color set

models for LWR reflector cross-sections and DF generation and parameterization. These

color set models more accurately capture the 2-D effects that occur on reentrant surfaces

that was not in previous 1-D models.

In the past the direct treatment of the reflector region in reactor core analysis was

ignored by relying on user-adjusted albedo boundary conditions to treat the neutron

reflection at the core/reflector interface. One-Dimensional (1-D) and Two-Dimensional

(2-D) transport models were used to calculate such albedos. The major deficiencies of

albedo boundary conditions are their dependency on fuel loading and the fact that they

are not capable of modeling axial neutron transfer through the radial reflector.

The above described disadvantages of the albedo-type boundary conditions were

avoided by explicit modeling the reflector (by introducing additional ring of

2

homogenized radial nodes – for radial reflector, and layer of homogenized axial nodes –

for each bottom and top reflector), and treating reflector homogenization in an analogous

manner to the fuel assembly homogenization. The difference is that instead of the single

assembly model used for fuel assembly homogenization, a color set model including the

reflector region and the adjacent core region is utilized. This approach requires definition

of few-group (usually two-group) diffusion parameters for the reflector nodes based on

the Equivalence theory, which should preserve both volume-averaged reaction rates and

surface-averaged net currents, corresponding to multi-group heterogeneous solution. This

can be achieved for any spatial discretization method used to solve the few-group

diffusion equation but the equivalent homogenized few-group diffusion parameters are

specific for a given flux solver [1, 2].

Special treatment of the reflector in reactor analysis is required due to the drastic

differences of neutronics properties between the core and reflector regions. The strong

spectrum change observed at core/reflector boundaries combined with geometry

complexity, material and structural heterogeneity of radial and axial reflectors make such

treatment a challenging task. The correct modeling of reflector response is important for

accurate prediction of core power distributions, especially in regions next to the reflector.

The radial and axial reflectors of Light Water Reactors (LWRs) consist of heterogeneous

arrangements of materials.

The radial reflector of a Pressurized Water Reactor (PWR) contains a stainless

steel plate with an approximate thickness of 2-4 cm depending on the reactor

manufacturer. The physical nature of the reflector has an impact on neutron behavior – it

3

affects the energy-dependent flux gradient in the reflector as well as the reflection of

neutrons of different energies to the core thus affecting the power distribution in the core.

This effect is seen not only by the power distribution (assembly-wise and pin-wise) of

periphery fuel assemblies but it may result in a power distribution tilt across the core.

For this reason special care is taken in generation of reflector homogenized cross-

sections and discontinuity factors (DFs). Historically, one-dimensional (1-D) color set

problems have been used for the reflector for the reasons described above. The

investigations presented in this thesis will utilize the more accurate modeling through

color sets which capture the 2-D effects that occur on reentrant surfaces.

1.2 Modeling

Modeling consisted of the traditional 1-D and 2-D color sets. 1-D models were

utilized with the axial reflector and had been previously developed by Westinghouse.

Both PWR and BWR axial models were utilized. In using the same models developed by

Westinghouse, comparisons of the current reflector modeling methodology to the

equivalence reflector methodology can be made.

The radial modeling for PWR reflectors was performed through the use of 2-D

color sets. These color sets consisted of 3x3 and 5x5 mini-cores which represented a

variety of loading patterns with fuel of varying enrichments, uranium oxide only and

uranium and mixed oxide, and varying beginning of cycle assembly burnups. The

reflector region was explicitly modeled and included the baffle and core barrel. The

radial color sets were burned out to 40GWd/MTU in eleven burnup steps. Boron

4

concentrations were varied from 0 ppm, to 500 ppm, 1000 ppm, 2000 ppm, and 3000

ppm. The moderator temperature was also varied from 293K, to 538K, 558K, 578K,

598K, and 618K. The results from the axial and radial modeling discussed in Chapter 2,

provided data sets for the sensitivity studies and parameterization. With the models in

place sensitivity studies can be completed.

1.3 Sensitivity Studies

Two sensitivity studies have been performed. The first study evaluates the effect

of the size of the color-set model on reflector constants. Different color sets have been

introduced and examined. From the transport solution of the color set model,

discontinuity factors are calculated which in turn preserve the transport solution in the

nodal code solution. The basis for this work was developed by Ivanov [9] with the

improvements to Penn State’s Nodal Expansion Code (NEM), which was originally

developed by Bandini [16], through the discontinuity factors from Smith [2]. For this

study the discontinuity factors are calculated using the polynomial flux representation

homogenization procedure (which is consistent with the NEM solution) developed by

Ivanov [9]. This procedure is based on the Generalized Equivalence Theory [2], which

preserves both volume-averaged reaction rates and surface-averaged net currents,

corresponding to the multi-group heterogeneous solution. In this work the

homogenization procedure has been further developed to accommodate the semi-analytic

flux representation of the Westinghouse Advanced Nodal Code (ANC) solution. The

discontinuity factors calculated in this way were then used in NEM and ANC [15]

5

respectively. This semi-analytic flux representation was developed through the following

approximation to the one-dimensional transverse-integrated flux, which is the same

approximation used in ANC:

ϕg,xl x = Asinh kglx + B cosh kglx + ag,xn

l fn x nn=0 (1)

Where for two groups:

kg,l = Ag

l Dgl 12

k1,l = A1l D1

l 12

k2,l = A2l D2

l 12

Ag = rgl ;r1

l = a1 + s1→2

r2l = a2

l

(2)

The transport lattice physics code PARAGON [13] was used to generate the

homogenized cross sections, surface average currents, surface average fluxes, volume

average fluxes based on user defined regions, and the global eigenvalue for the 1-D and

2-D models in this study. In addition, PARAGON provided the reference solutions by

which the nodal calculations were compared.

This thesis first analyzes the improvements gained in power distribution and

global eigenvalue by using the discontinuity factors calculated by using 2-D color-sets

compared to factors calculated using 1-D color set models. For this study the NEM-

based polynomial flux expansion homogenization procedure is used. Next the results

using discontinuity factors, calculated by using semi-analytic flux expansion based

homogenization procedure, were compared to a generic set of discontinuity factors with

6

the Westinghouse nodal code ANC for 1/8th

core models. A second sensitivity study was

conducted with the aim of determining a parameterization for the discontinuity factors as

function of the core conditions such as boron concentration, moderator temperature and

density. In addition, the effects of loading pattern variations next to the reflector region

on the discontinuity factors were examined through the use of color sets that included

Mixed Oxide (MOX) fuel and the traditional UO2 fuel. Multiple color set models were

developed in order to determine the magnitude of the changes to the discontinuity factors

for each effect and the feasibility of a correlation to capture these discontinuity factor

changes caused by changing core conditions. The core condition changes that were

analyzed included: boron concentration, moderator density, moderator temperature, and

variations in the loading pattern next to the reflector region.

1.4 Parameterization of Factors and Constants

The discontinuity factors along with the reflector constants were gathered from

the sensitivity studies. The reflector constants are simply the ratios of the homogenized

reflector constants to the discontinuity factors. The goal of the parameterization was to

find a relationship for the discontinuity factors and reflector constants that was not

dependent upon loading pattern or fuel burnup. Ultimately this was accomplished with

two sets of parameterizations. One set for uranium oxide only loading patterns and

another for loading patterns including MOX. A generic parameterization for flat sides

and inner corners was also desired; however the study found that the most accurate

parameterization was specific to the location upon the fuel-reflector interface, meaning

7

that each individual node surface on the fuel reflector interface had an individual

parameterization for each energy group.

8

CHAPTER 2

Reflector Modeling

The axial reflector models were provided by Westinghouse in the form of two

documents. The first document was an axial offset study performed for PWRs and the

second document was F. Reitsma’s Master’s thesis, which served as the basis for BWR

axial reflector modeling. In both of these documents the modeling and dimensions were

specified. The “pin by pin” data in the PARAGON models for the axial offset study were

not changed, however homogenization of the group constants occurred over regions of

equal size for use in a discontinuity factor routine. The discontinuity factor routine was a

standalone program which took the boundary conditions from the transport reference

solution and generated discontinuity factors. Likewise for BWRs, the models in

Reitsma’s thesis were kept the same except for minor adjustments to achieve fuel sized

reflector nodes such as extending the reflector region. It should be noted that this

extension occurred in such a way as not to affect the results. When the reflector is

modeled of an adequate depth or length, as discussed by Reitsma, additional reflector or

changes in boundary conditions will not affect the results because once the neutrons

reach that distance away from the reflector core interface, they are not going to make

their way back to the reflector core interface in sufficient numbers to affect the results.

9

2.1 Axial Reflector Modeling

This section will describe the details of the axial reflector modeling for the PWR

cases first, followed by the BWR cases. The axial reflector models were based on an

axial offset study conducted by Westinghouse. The original PARAGON input files were

provided by Westinghouse and the models were not changed in their physical

dimensions, only the regions over which the reflector constants were homogenized were

changed to match the regions of the fuel. In some cases the outer reflector region was

extended to provide a consistent node size throughout the model.

The axial offset cases covered the following conditions:

1. Fresh and Burned Fuel.

2. Rodded and Unrodded Cases.

3. Cases with 0 and 900 ppm Boron.

4. Two sets of moderator temperatures.

575K and 610K – Top Axial Models

540K and 575K – Bottom Axial Models

Figure 1 below shows the 1-D assembly-reflector color-set models that were

composed with reflective and vacuum boundary conditions. Although the detail of the

reflector node is not shown, the reflector is modeled explicitly to include different vessel

internals and moderator.

10

Figure 1: 1-D Assembly/Reflector Models

There was a wide variation in the discontinuity factors produced from these

models when comparing the Equivalence DFs (consistent with the nodal method) to the

DFs (determined by analytical solution of the 1-D diffusion equation) used in the current

Westinghouse methodology. This is expected due to the differences in the

methodologies. The equivalence reflector constants generated were then utilized in the

benchmarking against an actual cycle. It was found that the reflector constants did not

affect the overall global solution much at all, but any change was in the correct direction

towards the measured values. The axial leakage is small and causes this lack of effect in

the core’s global solution.

The BWR modeling was conducted in the same way as the PWR modeling in that

the original models developed previously would not be changed unless necessary.

Changes to the models involved extending the outer reflector region, as was done in the

PWR cases. This allowed for homogenization of the group constants over regions of

equal size for use in the discontinuity factor routine. There was a single temperature

bottom reflector model that was rodded and unrodded. The top reflector model was also

single temperature with voiding from zero to ninety percent void in increments of ten

percent. As in the case of the PWR models, the BWR models also displayed a difference

between the equivalence DFs and the DFs of the current methodology.

11

The axial modeling of this project is not as extensive as the radial modeling and

this is due to core conditions being fewer in number. While the PWR axial cases could

have included more boron concentrations, the multiple other conditions were adequate.

More exact details of the upper and lower plenum were not readily available, however in

the future if a more detailed exploration of the axial models is desired more conditions

could be added.

2.2 Radial Reflector Modeling

This section will describe the details of the radial reflector modeling. The radial

reflector modeling consisted of three robust sets of models. The first set of models was

used to verify the accuracy of the discontinuity factor routine and to display the

improvements over the existing modeling methodology. The second set of radial models

was developed to benchmark the new methodology against an actual cycle. This was

accomplished using data for various types of plants and fuel loading patterns. The third

set of radial reflector models was utilized to perform the sensitivity study and consisted

of color set models. These color sets included both uranium oxide only loading patterns

and loading patterns with both uranium oxide and mixed oxide fuel assemblies.

The radial reflector modeling underwent various changes as it was being

developed over the course of the project. At first, full sized assembly nodes were used to

model both the fuel and reflector region. Then modeling of the fuel and reflector region

using quarter assembly sized nodes was deemed desirable. This evolution in the

modeling has in part led to the first set of models being completed with full sized

12

assembly nodes, while the second and third set were completed with both full assembly

sized and quarter assembly sized nodes. For the second and third sets of radial models

the quarter sized nodes will be discussed.

Traditionally the homogenized constants (cross sections and diffusion

coefficients) and the discontinuity factors for fuel assemblies are calculated in an infinite

environment (unit assembly calculations). This data combined with homogenized

constants and discontinuity factors calculated for the core-reflector interface are then

used in a nodal code for core simulation and analysis. The first comparison looks at a 2-D

3x3 mini-core. The following models were developed for PARAGON: the individual

assemblies in an infinite environment, a 1-D assembly-reflector node color-set, and the

entire 3x3 mini-core model. Using the PARAGON data from these various models,

discontinuity factors based on a polynomial flux expansion were calculated, which were

then used with NEM to model the 3x3 mini-core. This accomplishes two things. First, the

nodal code solution shows the need for including color-set models for reflector cross

section and DF generation that include 2-D effects. Second, the nodal code solution

shows that the transport reference solution can be reproduced when all nodes have side

dependent discontinuity factors generated by the above-mentioned homogenization

procedure applied to the whole 3x3 mini-core. It is important to note that the 3x3 mini-

core by its very nature will produce exaggerated results as compared to a 1/8th

core

symmetry model due to its small size.

Furthermore the discontinuity factors, calculated using the 3x3 color-set model

above, are used in conjunction with an 1/8th

core symmetry model. The nodal code

13

solutions for the 1/8th

core model show that the transport reference solution can again be

reproduced using side dependent discontinuity factors generated by the homogenization

procedure applied to the whole 1/8th

core. The nodal code solutions also show that the

3x3 color-set model performs well in providing representative discontinuity factors.

Nodal code solutions using discontinuity factors from 1-D assembly-reflector node color

sets (the standard models currently in use) are also provided for comparison. To verify

the accuracy of the equivalence methodology, the following 3x3 mini-core that is

representative of the fuel-reflector interface of a 1/8th

core sector of symmetry was

developed and is shown in Figure 2 below.

Figure 2: 3x3 Mini-Core Model

As in Figure 1, Figure 2 does not illustrate the detail of the reflector region. The

reflector region was modeled explicitly in PARAGON with a baffle and core barrel.

14

Discontinuity factors were developed for the reflector and the fuel regions. The

individual fuel assemblies were modeled in an infinite environment in PARAGON and

assembly discontinuity factors were calculated. With the discontinuity factors from the

models described above, 3x3 mini-core nodal code models were developed and compared

to the PARAGON reference solutions.

Table 1 below contains the 3x3 mini-core results. The PARAGON reference

solution is shown for keff along with the normalized assembly powers. The differences in

pcm and absolute percent difference are shown for the eigenvalue and normalized

assembly powers respectively. “No DF” means that the nodal code was executed without

any discontinuity factors, fuel assembly cross-sections are generated in infinite

environment and the reflector cross-sections are taken from 1-D color-set models. “IA”

means that the fuel assemblies have infinite environment discontinuity factors and cross

sections, while ARR are reflector discontinuity factors and cross-sections from the 1-D

assembly-reflector models. “DF” means side dependent assembly discontinuity factors

and cross sections calculated from the 3x3 mini-core while “RDF” represents side

dependent reflector discontinuity factors and cross sections calculated from the 3x3 mini-

core. A case using equivalence discontinuity factors in the fuel and reflector regions

reproduced the reference transport solution and it was performed to demonstrate that the

utilized homogenization procedure works correctly. The results in Table 1 show that the

3x3 mini-core exaggerates the effects with the 9766 pcm difference between the

reference transport solution and the nodal code solution with “No DF”. Using the

assembly discontinuity factors (ADFs) calculated in an infinite environment combined

15

with the discontinuity factors at the fuel-reflector interface from the 1-D assembly-

reflector models (case IA-ARR) shows significant improvement as expected. This is the

standard model currently in use. In this model the side dependent discontinuity factors for

all reflector-reflector interfaces were set to one.

Table 1: 3x3 Mini-Core Results

Model Eigenvalue Fuel Type A Fuel Type B Fuel Type C

Reference 0.95000 1.7531 1.0188 0.2281 No DF 9766.0 pcm -12.6068 ABS % 10.4789 ABS % 2.1283 ABS % IA-ARR -219.5 pcm 0.4832 ABS % -1.7711 ABS % 1.2883 ABS % IA-RDF 7.1 pcm 0.0922 ABS % -0.0230 ABS % -0.0688 ABS %

The use of the new discontinuity factors is shown in the “IA-RDF” model. The 2-

D color-set model used to generate the RDF discontinuity factors shows further

improvement over the 1-D ARR model used to generate discontinuity factors. The next

comparison is even more important because it represents the actual color-set models that

could be used to produce side dependent discontinuity factors for an actual 1/8th

core

symmetry model. For this comparison the side dependent discontinuity factors from

above are used.

Table 2 shows the 1/8th core results. The PARAGON reference solution is shown

along with the differences in pcm for the eigenvalue. Normalized assembly power

absolute errors are displayed in Figure 3. As in Table 1 “IA” means that the fuel

assemblies have infinite environment discontinuity factors and cross sections, “1-D” are

models with reflector discontinuity factors and cross sections from the 1-D assembly-

reflector models while “MC” means models with reflector discontinuity factors and cross

sections from the 2-D 3x3 mini-core model.

16

Table 2: 1/8th Core Results

Model Eigenvalue

Reference 0.98782 IA-1D 22.8 pcm IA-MC -0.2 pcm

The results in Table 2 show a spread in eigenvalues that is smaller than for mini-

core results of Table 1. This is due to the exaggerated effects from a smaller core. The

reflector constants and cross sections from the 3x3 mini-core are proving to be excellent

substitutes. This is seen in the eigenvalue results above and in the normalized power

absolute percent errors from the PARAGON reference solution in Figure 3.

Figure 3: 1/8th Core Results

IA-1D IA-MC

ABS ABS

% %

Error Error

0.9006 0.8530 0.5407 0.1159

0.7737 0.1824

-0.5657

0.7333 0.1414 0.4897 -0.0042 -0.5536

0.2115 -0.1036 0.1659 -0.2787

0.0015 0.4023 -0.1445 0.0749

-0.7527

-0.5134 -0.7341

-0.5652 -1.1616 -0.2095

-0.2665

0.6811

1.1018

0.4178

0.3706

1.1943

0.3392

-0.7757

-0.2607

-0.1289

0.9815

1.5318

-0.3274

0.6889

1.1999

1.2790

-1.5870

-2.1511

-0.6180

0.0700

0.2290

1.4224

-1.6975

-1.7574

-1.3460

-0.5912

17

The absolute errors above show that the mini-core, with its capturing of 2-D

effects that occur at reentrant surfaces, is a better means of modeling the reflector

response. The results in this section used side dependent discontinuity factors that were

generated from the polynomial flux expansion based homogenization procedure used in

the NEM code. This was done as an extension of the methodology developed by B.

Ivanov [9] to be applicable to larger and more complex modeling as compared to the

original study. The next section presents the results of using the semi-analytic flux

expansion based homogenization procedure to calculate side dependent discontinuity

factors, which are then used in the Westinghouse nodal code ANC [15].

With the polynomial flux expansion based homogenization procedure (consistent

with the NEM solution) of the previous set yielding the desired results in both eigenvalue

and assembly power predictions, similar comparisons were made using discontinuity

factors from a semi-analytic flux expansion based homogenization procedure (consistent

with the ANC solution). Please note that in NEM calculations one node per assembly in

the radial plane was used. In ANC PWR calculations usually four nodes per assembly

model are utilized.

In ANC the explicit representation of the reflector is utilized, i.e. the baffle and

reflector are represented by a homogenous node. The homogenization is performed using

1-D color-set calculation with PARAGON. Since ANC is using a four nodes per

assembly calculation scheme, a half-assembly size (one radial node) reflector is used

along with albedo boundary conditions on the outer surface of the reflector node. The

18

homogenized flux distribution in the reflector node is obtained by analytical solution of

1-D diffusion homogeneous boundary value problem. This analytical solution is not

consistent with the nodal solver of ANC. The generic reflector DFs obtained in this way

are used with the reflector cross-sections to generate reflector constants, which are then

utilized in ANC for core calculations.

The difference with the NEM study is that instead of using only one side–

dependent discontinuity factor per assembly side (as it is the case for NEM), two side

dependent discontinuity factors are calculated since four nodes per assembly are used.

Utilization of two assembly side dependent discontinuity factors will more accurately

model the reflector response in inner corners of the core-reflector interface and this

modeling will also match the methodology of the Westinghouse nodal code ANC.

In the ANC reflector model one row of quarter assembly reflector nodes are used

to explicitly model the reflector response while in the NEM model one row of full

assembly reflector nodes is used. In the ANC reflector model albedo boundary conditions

are used on the outer surface of reflector nodes in difference of the NEM model where

vacuum boundary conditions are used on the outer surface of the reflector nodes. For a

water reflector at room temperature a thickness of ~ 20 cm (an assembly-size reflector

node) is thick enough. The boundary conditions, applied on the outer surface of such

reflector, do not affect keff and power distribution in the core. In cases when a steel region

is also part of the reflector, or the moderator density is low, or a half-assembly size

reflector node is used, the boundary conditions on outer surface of reflector are

important. Adding additional rings of reflector nodes will affect the efficiency of core

19

nodal calculations, and the calculations of DFs for the additional reflector nodes may be

problematic. This is the reason why the ANC model uses albedo boundary conditions and

there is a need to investigate whether these albedos have to be parameterized also in

terms of feedback parameters.

Two 3-Loop PWR (Westinghouse type) plants were chosen for this study. One is

a core loaded with only UO2 assemblies while the second is loaded with a combination of

UO2 and MOX assemblies. The side dependent discontinuity factors calculated with a

semi-analytic flux expansion based homogenization procedure are utilized in addition to

a generic set of discontinuity factors. The normalized assembly powers generated using

these two sets of discontinuity factors with the nodal code ANC are compared to one

another in Figure 4 below. Discontinuity Factors using 3x3 models as described in the

Polynomial cases before are used, and additionally, larger 5x5 mini-cores were used to

determine the effect of modeling the inner fuel assemblies would have. The results in

Figure 4 are from 5x5 mini-cores, while not shown the 3x3 mini-cores generated similar

discontinuity factors, which varied by less than 1%, which in-turn had little effect on the

power distributions. It should be noted that even though more fuel was modeled with the

5x5 mini-cores, only the discontinuity factors at the fuel reflector interface were utilized.

The results in Figure 4 show a tilt correction. The generic set of discontinuity

factors suffered from a power tilt of over predicting assembly powers towards the center

of the core and under predicting assembly powers toward the periphery when compared

to measured data, which was especially pronounced for the loading pattern shown in the

right part of Figure 4. The new discontinuity factors are correcting this error in the proper

20

direction when compared to measured data. The effect of the new constants on the boron

prediction is usually negligible (few ppm) but could be in the range of 10 ppm with the

depletion for high leakage cores. More pronounced improvements are seen in the

absolute percent errors in assembly powers of Figure 4. The observed deviations in

critical boron concentration predictions were below 10 ppm.

Figure 4: 1/8th Core Normalized Power Absolute Error Results

The absolute errors above show an improvement in the power tilt problem

discussed above. The larger 5x5 mini-cores produced results that maintain good results

for UO2 loading patterns, and improve the results where the generic set struggles in the

case of loading patterns with MOX fuel.

150 MWd/MTU, 3Loop PWR Core, UO2 Fuel 150 MWd/MTU, 3Loop PWR Core, UO2 and MOX Fuel

ABS ABS

% %

Error Error

0.2

0.3

0.3

0.2

0.1

0.2

-0.1

0.2

0.2

0.3

0.2

0.2

0.2

-0.4

-0.3

0.2

0.2

-0.2

-1.00.2

0.1

-0.7

2.3

2.6 4.0

0.2

0.2

0.2

0.1

-1.3

3.9 2.5 2.0

1.6 1.8 1.1

-3.5

-0.5 -0.9 -2.8 -4.9

1.3

2.0 1.2 1.0 -0.2

-0.7 -1.2

0.6 0.1 -0.1 -2.1

21

The issue of P3 scattering in large core reflectors has been raised by J. Vidal, R.

Tellier, et. al., [17]. To investigate the effects of P3 scattering on the models of this study

an MCNP reference model was developed. Unfortunately the cross sections used with

MCNP were not developed for the 3x3 mini-core and thus generated large errors. These

results can be found in Appendix A. Likewise, the discontinuity factors were generated

using a standalone routine. A sample input is given in Appendix B.

The process of generating discontinuity factors can be a computationally costly

endeavor. Sensitivity studies are carried out in the following section in an effort to

generate a correlation (parameterization) for changing core conditions.

22

CHAPTER 3

Sensitivity Studies

Using a single generic set of discontinuity factors does not always produce the

desired results. Calculating a new set of discontinuity factors for changing core

conditions and loading patterns is also not desirable due to the computational costs

involved. Developing a correlation for changing core conditions to be used in conjunction

with a set or sets of discontinuity factors is one of the goals of the research discussed in

this thesis.

The current reflector model in ANC does not take into account feedback effects.

From a consistency point of view it will be better if the reflector parameter representation

is expressed in similar manner as for fuel assemblies if possible. One of objectives of this

study is to identify the correct feedback parameters for accurate and efficient reflector

modeling.

The parameters shown below were individually adjusted to the following values

in the sensitivity study:

Moderator Temperature: 293K, 538K, 558K, 578K, 598K, and 618K

Moderator Density: 0.40, 0.50, 0.60, and 0.72 g/cc

Moderator Boron Concentration 0, 500, 1000, 2000, and 3000 ppm

Fuel Burnup 0, 0.5, 1, 2, 3, …, 38, 39, and 40 GWd/MTU

Loading Pattern Various assemblies with different initial burnup,

enrichment as well as UO2 vs. MOX fuel

23

A total of eight loading patterns were developed. The following four in Figure 5

are discussed first.

Figure 5: Mini-Cores Used in the Sensitivity Study

The reflector region contains a baffle and core barrel that has been explicitly

modeled even though it is not depicted in the illustration of Figure 5. The same reflective

and vacuum boundaries depicted in Figure 2 are applied to the four mini cores of Figure

5. All four mini-cores also had a moderator density of 0.72 g/cc. In addition the mini-

core depicted in the top right of Figure 5 had its moderator density varied. The

discontinuity factors generated via the polynomial flux expansion based homogenization

procedure are discussed below.

24

In [10] it was concluded that the only meaningful spectrum parameter is the

core/reflector net current spectrum (e.g., J1/J2 on core/reflector interface) since this

determines the spatial shape of the flux (especially the thermal flux) throughout the

reflector. Since this spatial shape is primarily responsible for environment sensitivity (to

core loading for instance) that results from the spatial smearing, this (J1/J2) is the correct

parameter to use and not the fuel or reflector spectrum index, or even fuel temperature.

The basis of the Westinghouse improved cross section representation methodology,

implemented in the NEXUS system [11], is the modeling of both macroscopic and

microscopic cross sections primarily as a function of spectrum index (SI) (the ratio of fast

to thermal group node-average fluxes). These two spectrum representative parameters

have been investigated in this work along with some additional parameters as listed

below.

The discontinuity factors were plotted against numerous variables including:

Mini-core Burnup

Spectral Index (Fast Node Average Flux/Thermal Node Average Flux)

Incoming Current/Node Average Flux per energy group

Albedo at the Fuel/Reflector Interface per energy group

Net Current per energy group

Net Current Ratio

Delta Spectral Index (Surface SI – Node SI)

Net Current/Surface Flux at the Fuel/Reflector Interface per energy group

As will be seen in the data that follows, Net Current/Surface Flux at the

Fuel/Reflector Interface yielded the best results when visually inspecting the plots of the

25

data. In each 3x3 mini-core there are four fuel/reflector interface sides. The nodes are

labeled on an XY plane that originates in the upper left hand corner of the mini-cores as

depicted in Figure 5. The X-axis is positive to the right, and the Y-axis is positive moving

down the page. For example the central node is node X2Y2. The West side of this node

is X2Y2W. The legend in Figure 6 depicts these sides and the particular boron

concentration for the mini-core depicted in the top left of Figure 5 with a moderator

temperature of 578 K, density of 0.72 g/cc. Each point in the plot represents a burnup

step as the mini-core is depleted.

The discontinuity factor and reflector constant data was gathered based on the

nodes labeled in accordance with the following Figure:

1 2 3 4 5 6

7 8 9 10 11 12

13 14 15 16 17 18

19 20 21 22 23 24

25 26 27 28 29 30

31

32

33

34

35

36

Figure 6: Reflector Node Identification

26

Figure 6 shows the fuel assemblies in yellow, green and red. The reflector region

is in blue. The reflector nodes, depicted in white large numbers, are the nodes of interest.

Each node has the data associated with the node sides labeled as south, east, north and

west. From Figure 6 only the west and north sides of the respective reflector nodes were

gathered for this sensitivity study. Although not shown in Figure 6, the reflector

components such as the baffle and core barrel are explicitly modeled.

The following plots and discussion presented will cover 3x3 mini-cores loaded

with only UO2 followed immediately by those with MOX-UO2 in sets of fast and thermal

results. Figures 7 through 10 show the behavior of the discontinuity factors for a

particular side of node 5. Figures 11 through 18 show the behavior of the discontinuity

factors for flat and inner corner sides respectively grouped together. Like Figures 11

through 18, Figures 19 through 22 show the behavior of the discontinuity factors for flat

and inner respectively grouped together, but for a mini-core loaded with MOX-UO2.

Figures 23 through 32 show the behavior of reflector constants for both energy groups

and flat and inner-corners for a UO2 loaded mini-core. The reflector constants behave in

a similar manner for all mini-cores and the smaller subset of data shown in the Figures

below provides a basic display of parameter behavior.

The following plot in Figure 7 depicts the change in the fast discontinuity factor

with a constant boron concentration and multiple temperatures. Side 5W refers to the

west side of node 5 in Figure 6. The discontinuity factors display a linear behavior

followed by an inflection point and a leveling off. As the mini-core is burned the

discontinuity factors decrease as the Net Fast J/Surface Average Fast Flux increases. The

27

burnup effects are captured in the environmental effects of the dependent variable. The

blue diamonds represent a moderator with no boron and a temperature of 293K. When

the moderator temperature increases to 538K the data shifts to the right and downwards

to the red squares on the same plot. This shifting behavior is common throughout all

similar data sets. This also occurs when temperature is held constant and the boron

concentration is changed. Uranium only loading patterns are discussed here.

Figure 7: Constant Boron Concentration Fast Discontinuity Factor Behavior

0.934

0.935

0.935

0.936

0.936

0.937

0.937

0.938

0.938

0.939

0.939

0.1723 0.1724 0.1725 0.1726 0.1727 0.1728 0.1729 0.1730 0.1731

DF 1

Net Fast J/Surface Average Fast Flux

Side 5W DF1 vs. Net Fast J/Surface Average Fast Flux0ppm 293K

0ppm 538K

0ppm 558K

0ppm 578K

0ppm 598K

0ppm 618K

28

The following plot in Figure 8 depicts the change in the fast discontinuity factor

with a constant temperature and multiple boron concentrations. As in Figure 7, the

discontinuity factor displays a linear behavior and inflection point. For comparison the

shift to the right as boron concentration increases is much more pronounced than the shift

from a change in temperature. As in Figure 7, as the mini-core is burned, the

discontinuity factors decrease as the Net Fast J/Surface Average Fast Flux increases. The

burnup effects are again captured in the environmental effects of the dependent variable.

When the boron concentration increases, the data shifts to the right. This shifting

behavior is common throughout all data sets of changing boron concentration.

Figure 8: Constant Temperature Fast Discontinuity Behavior

0.9350

0.9355

0.9360

0.9365

0.9370

0.9375

0.9380

0.9385

0.9390

0.9395

0.1724 0.1726 0.1728 0.1730 0.1732 0.1734

DF 1

Net Fast J/Surface Average Fast Flux

Side 5W DF1 vs. Net Fast J/Surface Average Fast Flux0ppm 293K

500ppm 293K

1000ppm 293K

2000ppm 293K

3000ppm 293K

29

The next two plots are a repeat of Figures 7 and 8 but for the thermal

discontinuity factors. The following plot in Figure 9 depicts the change in the thermal

discontinuity factor with a constant boron concentration and multiple temperatures. As in

Figure 7, the discontinuity factors display a linear behavior. As the mini-core is burned,

the discontinuity factors increase with the Net Thermal J/Surface Average Thermal Flux

ratio. The burnup effects are again captured in the environmental effects of the

dependent variable. The effects are more drastic in the thermal discontinuity factor as

expected. This overall behavior is common throughout all data sets of changing

temperature on the thermal discontinuity factor.

Figure 9: Constant Boron Thermal Discontinuity Factor Behavior

0.150

0.160

0.170

0.180

0.190

0.200

0.210

0.040 0.050 0.060 0.070 0.080 0.090

DF 2

Net Thermal J/Surface Average Thermal Flux

Side 5W DF2 vs. Net Thermal J/Surface Average Thermal Flux0ppm 293K

0ppm 538K

0ppm 558K

0ppm 578K

0ppm 598K

0ppm 618K

30

The following plot in Figure 10 depicts the change in the thermal discontinuity

factor with a constant temperature and multiple boron concentrations. As in Figure 9, the

discontinuity factors display a linear behavior. Unlike in Figure 7, as the mini-core is

burned, the discontinuity factors increase with the Net Thermal J/Surface Average

Thermal Flux ratio. The burnup effects are again captured in the environmental effects of

the dependent variable. The effects are more drastic in the thermal discontinuity factor as

expected. As in the comparison of Figure 7 to Figure 8, comparing Figure 9 to Figure 10,

the change in boron concentration has a larger effect than change in temperature. This

behavior is also common throughout all plots of similar data sets.

Figure 10: Constant Temperature Discontinuity Factor Behavior

0.150

0.170

0.190

0.210

0.230

0.250

0.270

0.290

0.310

0.330

0.060 0.070 0.080 0.090 0.100 0.110

DF 2

Net Thermal J/Surface Average Thermal Flux

Side 5W DF2 vs. Net Thermal J/Surface Average Thermal Flux0ppm 293K

500ppm 293K

1000ppm 293K

2000ppm 293K

3000ppm 293K

31

Figures 7 through 10 above display the behavior of just one node within the mini-

core. Behaviors of other nodes are similar in trends but different in magnitude. The

sensitivity study revealed that the flat sides along the fuel-reflector interface (sides 5

west, 11 west, 25 north and 26 north, from Figure 6), behave in a different manner as

compared to the inner corners of sides 15 west, 15 north, 16 north and 21 west. A further

distinction will be made amongst the individual sides with the parameterization; that the

flat and inner corner sides behave based location on the fuel-reflector interface and not on

the type of fuel. Figure 11 below illustrates flat sides.

Figure 11: Fast Flat Side Discontinuity Factor Behavior

0.900

0.905

0.910

0.915

0.920

0.925

0.930

0.935

0.940

0.945

0.1660 0.1680 0.1700 0.1720 0.1740 0.1760 0.1780

DF 1

Net Fast J/Surface Average Fast Flux

Flat Sides DF1 vs. Net Fast J/Surface Average Fast Flux0ppm 293K

0ppm 538K

0ppm 558K

0ppm 578K

0ppm 598K

0ppm 618K

0ppm 293K

0ppm 538K

0ppm 558K

0ppm 578K

0ppm 598K

0ppm 618K

0ppm 293K

0ppm 538K

0ppm 558K

0ppm 578K

0ppm 598K

0ppm 618K

0ppm 293K

0ppm 538K

0ppm 558K

0ppm 578K

0ppm 598K

0ppm 618K

32

Figure 11 above shows several interesting trends. First the top right of the plot is

data for sides 5 west and 11 west (See Figure 6 for node numbering). The lower left

includes both sides 25 north and 26 north. Sides 5 west and 11 west in the upper right

hand corner are separated whereas 25 and 26 overlap. This may be due to a lack of a core

barrel in close proximity to the fuel-reflector interface being analyzed. Figure 12 shows

data for constant boron of 0 ppm and multiple temperatures. The scale used in Figure 12

adequately shows the differences in data behavior between the different sides, however

now the shift due to changes in temperature is not as visible as in Figures 7 through 10.

Likewise we can combine the thermal discontinuity factors below. Sides 5 west and 11

west overlap at the top right, and sides 25 north and 26 north at bottom left.

Figure 12: Thermal Flat Side Discontinuity Factor Behavior

0.100

0.150

0.200

0.250

0.300

0.350

0.400

0.450

0.020 0.040 0.060 0.080 0.100 0.120 0.140 0.160

DF 2

Net Thermal J/Surface Average Thermal Flux

Flat Sides DF2 vs. Net Thermal J/Surface Average Thermal Flux0ppm 293K

0ppm 538K

0ppm 558K

0ppm 578K

0ppm 598K

0ppm 618K

0ppm 293K

0ppm 538K

0ppm 558K

0ppm 578K

0ppm 598K

0ppm 618K

0ppm 293K

0ppm 538K

0ppm 558K

0ppm 578K

0ppm 598K

0ppm 618K

0ppm 293K

0ppm 538K

0ppm 558K

0ppm 578K

0ppm 598K

0ppm 618K

33

The internal corners will be examined next. Sides 15 west, 15 north, 16 north,

and 21 west from Figure 6 are the internal corners. The discontinuity factors for the

internal corners shift with changes in moderator temperature and moderator boron

concentration. Unlike the flat sides, the internal corners group themselves more based on

location. Sides 15 west and 21 west are interfacing with a fresh naturally enriched

assembly, sides 15 north and 16 north are facing a fresh assembly enriched to 3.2 w/o

without burnable absorbers. In Figure 13 below working clockwise from the left most

grouping of discontinuity factors is side 15 west, 16 north, 21 west, and 15 north. The

two sides touching the inner corner, 15 west and 15 north are grouped together to the

lower left and the two inner corner sides that are further away from the inner corner are at

the top right of the plot. Position matters, a fact reinforced with parameterization.

Figure 13: Fast Inner-Corner Discontinuity Factor Behavior

0.890

0.900

0.910

0.920

0.930

0.940

0.950

0.960

0.970

0.980

0.990

0.0500 0.0700 0.0900 0.1100 0.1300 0.1500 0.1700

DF 1

Net Fast J/Surface Average Fast Flux

Inner-Corner DF1 vs. Net Fast J/Surface Average Fast Flux0ppm 293K

0ppm 538K

0ppm 558K

0ppm 578K

0ppm 598K

0ppm 618K

0ppm 293K

0ppm 538K

0ppm 558K

0ppm 578K

0ppm 598K

0ppm 618K

0ppm 293K

0ppm 538K

0ppm 558K

0ppm 578K

0ppm 598K

0ppm 618K

0ppm 293K

0ppm 538K

0ppm 558K

0ppm 578K

0ppm 598K

0ppm 618K

34

The inner corner thermal discontinuity factors display a grouping behavior and

are shown in Figure 14 below. The grouping behavior for the thermal discontinuity

factors is slightly different than that seen above with the fast discontinuity factors.

Looking closely at Figure 14 there is overlap between the four sides. The data is

arranged clockwise from the top right as 15 west, 21 west, 16 north, and 15 north. Both

data sets of 15 north, 15 west and 16 north, 21 west overlap each other in a continuing

manner; meaning that the light blue diamonds at the bottom of the plot for 16 north 293K

form a line that continues up to the darker blue diamonds of 21 west 293K. The same

occurs for other temperatures between 16 north and 21 west. Likewise the inner-inner

corner sides of 15 west and 15 north enjoy some overlap as well. This again shows that

location affects the data more so than fuel type.

Figure 14: Inner-Corner Thermal Discontinuity Factor Behavior

0.100

0.150

0.200

0.250

0.300

0.350

0.400

0.020 0.040 0.060 0.080 0.100 0.120 0.140

DF 2

Net Thermal J/Surface Average Thermal Flux

Inner-Corner DF2 vs. Net Thermal J/Surface Average Thermal Flux0ppm 293K

0ppm 538K

0ppm 558K

0ppm 578K

0ppm 598K

0ppm 618K

0ppm 293K

0ppm 538K

0ppm 558K

0ppm 578K

0ppm 598K

0ppm 618K

0ppm 293K

0ppm 538K

0ppm 558K

0ppm 578K

0ppm 598K

0ppm 618K

0ppm 293K

0ppm 538K

0ppm 558K

0ppm 578K

0ppm 598K

0ppm 618K

35

The data above in Figures 7 through 14 were from a mini-core loaded with fresh

uranium oxide fuel only. This was done to provide a data set from both different

positions and different fuel assemblies at the interface. The next set of plots comes from

a mini-core model with burned assemblies at the periphery in a symmetric loading

pattern. This symmetric pattern does not offer this difference in fuel next to the interface

as seen in the first set of plots; however it represents the typical in-out loading pattern

that is used today. As with Figures 7 through 14 the data for each side is grouped in

overlapping sets of data of different temperatures. Looking at Figure 15 below and

working from left to right in the figure below are: 25 north, 26 north, 5 west, and 11 west.

The sides closest to the reflective boundary behave similarly as do the two sides which

are one node away from the reflective boundary. Their magnitudes are of course shifted.

Figure 15: Flat-Sides Fast Discontinuity Factor Behavior

0.900

0.905

0.910

0.915

0.920

0.925

0.930

0.935

0.940

0.1670 0.1680 0.1690 0.1700 0.1710 0.1720 0.1730 0.1740 0.1750 0.1760

DF 1

Net Fast J/Surface Average Fast Flux

Flat Sides DF1 vs. Net Fast J/Surface Average Fast Flux0ppm 293K

0ppm 538K

0ppm 558K

0ppm 578K

0ppm 598K

0ppm 618K

0ppm 293K

0ppm 538K

0ppm 558K

0ppm 578K

0ppm 598K

0ppm 618K

0ppm 293K

0ppm 538K

0ppm 558K

0ppm 578K

0ppm 598K

0ppm 618K

0ppm 293K

0ppm 538K

0ppm 558K

0ppm 578K

0ppm 598K

0ppm 618K

36

Analyzing the thermal discontinuity factors for the same flat sides as in Figure 15,

we see in Figure 16 below. The overlap of sides in Figure 16 is very similar to the

overlap seen in Figure 12. Sides 5 west and 11 west are overlapping each other in the

bottom left of the plot, while sides 25 north and 26 north are overlapping each other in

the top right portion of the plot. Even with identical fuel, a twice burned 3.2 w/o

assembly, in each position next to the reflector-core interface, we still see the same

groupings of sides as with different assemblies of Figure 12. This grouping of the results

is consistent with the parameterization results which will show that position is dominant

not the loading pattern.

Figure 16: Flat-Sides Thermal Discontinuity Factor Behavior

0.150

0.170

0.190

0.210

0.230

0.250

0.270

0.290

0.040 0.050 0.060 0.070 0.080 0.090

DF 2

Net Thermal J/Surface Average Thermal Flux

Flat Sides DF2 vs. Net Thermal J/Surface Average Thermal Flux0ppm 293K

0ppm 538K

0ppm 558K

0ppm 578K

0ppm 598K

0ppm 618K

0ppm 293K

0ppm 538K

0ppm 558K

0ppm 578K

0ppm 598K

0ppm 618K

0ppm 293K

0ppm 538K

0ppm 558K

0ppm 578K

0ppm 598K

0ppm 618K

0ppm 293K

0ppm 538K

0ppm 558K

0ppm 578K

0ppm 598K

0ppm 618K

37

The inner-corner fast discontinuity factors are analyzed next. Sides 15 west and

15 north overlap each other at the bottom left of Figure 17. Sides 16 north and 21 west

overlap each other at the top right of Figure 17. This grouping of data based on reflector

location is, again, consistent with the overall findings in this study. As in the other plots

the change in temperature still exerts a shift in the data, however that shift is very small in

comparison to change in magnitude due to the change in position along the fuel-reflector

interface.

Figure 17: Inner-Corner Fast Discontinuity Factor Behavior

0.900

0.910

0.920

0.930

0.940

0.950

0.960

0.1000 0.1100 0.1200 0.1300 0.1400 0.1500 0.1600

DF 1

Net Fast J/Surface Average Fast Flux

Inner-Corner DF1 vs. Net Fast J/Surface Average Fast Flux0ppm 293K

0ppm 538K

0ppm 558K

0ppm 578K

0ppm 598K

0ppm 618K

0ppm 293K

0ppm 538K

0ppm 558K

0ppm 578K

0ppm 598K

0ppm 618K

0ppm 293K

0ppm 538K

0ppm 558K

0ppm 578K

0ppm 598K

0ppm 618K

0ppm 293K

0ppm 538K

0ppm 558K

0ppm 578K

0ppm 598K

0ppm 618K

38

Looking to the inner-corner thermal discontinuity factor behavior in Figure 18

below, similar results are illustrated in terms of groupings of sides. Sides 16 north and 21

west both overlap each other in the data shown in the lower left portion of the plot. Sides

15 west and 15 north overlap each other and are seen in the top right portion of the plot.

Position again dominates in discontinuity factor behavior.

Figure 18: Inner-Corner Thermal Discontinuity Factor Behavior

Analysis of core loadings with both MOX and Uranium assemblies at the

periphery show a similar tendency to be position dependent. MOX does have an effect

on the inner-corner uranium assembly fuel-reflector interface. The harder spectrum shifts

the discontinuity factors. The following figures will show the MOX loading pattern with

0.140

0.160

0.180

0.200

0.220

0.240

0.260

0.010 0.020 0.030 0.040 0.050 0.060 0.070 0.080

DF 2

Net Thermal J/Surface Average Thermal Flux

Inner-Corner DF2 vs. Net Thermal J/Surface Average Thermal Flux0ppm 293K

0ppm 538K

0ppm 558K

0ppm 578K

0ppm 598K

0ppm 618K

0ppm 293K

0ppm 538K

0ppm 558K

0ppm 578K

0ppm 598K

0ppm 618K

0ppm 293K

0ppm 538K

0ppm 558K

0ppm 578K

0ppm 598K

0ppm 618K

0ppm 293K

0ppm 538K

0ppm 558K

0ppm 578K

0ppm 598K

0ppm 618K

39

all MOX at the periphery. Following that we will return to the above mini-core but look

at reflector constants. The discontinuity factors of the MOX fuel-reflector interface

behave more like steady points rather than shifting data as seen in the figures above. The

discontinuity factors still shift, and as a matter of fact their behavior is not as linear. In

fact, close inspection of the data reveals interesting trends with inflection points. Plotted

with a bigger scale however, as in Figure 19 we see the variation due to burnup,

moderator temperature, and moderator boron concentration is rather small compared to

the uranium loading patterns. In Figure 19 below, if we work clockwise from the top the

data is grouped by sides in the figure as follows: side 11 west, side 26 north, side 25

north, and side 5 west. Sides 5 west and 11 west are facing a fresh MOX assembly while

sides 25 north and 26 north are facing that same assembly but twice burned.

Figure 19: MOX Flat-Side Fast Discontinuity Factor Behavior

0.600

0.700

0.800

0.900

1.000

1.100

1.200

1.300

1.400

1.500

1.600

0.1500 0.1700 0.1900 0.2100 0.2300 0.2500

DF 1

Net Fast J/Surface Average Fast Flux

Flat-Side DF1 vs. Net Fast J/Surface Average Fast Flux0ppm 293K

0ppm 538K

0ppm 558K

0ppm 578K

0ppm 598K

0ppm 618K

0ppm 293K

0ppm 538K

0ppm 558K

0ppm 578K

0ppm 598K

0ppm 618K

0ppm 293K

0ppm 538K

0ppm 558K

0ppm 578K

0ppm 598K

0ppm 618K

0ppm 293K

0ppm 538K

0ppm 558K

0ppm 578K

0ppm 598K

0ppm 618K

40

Analyzing the thermal discontinuity factors for the flat-sides occurs in Figure 20

below. Side 5 west data, below a discontinuity factor value of 0.160 and 0.140 appears in

the lower left hand corner of the plot. Sides 11 west and 26 north overlap a little bit and

are seen near discontinuity values of 0.180 and above 0.160. Side 25 north is seen above

all of the other data near a discontinuity value below 0.240 and 0.220 in Figure 19. The

shift in discontinuity factors due to changes in temperature is more drastic than changes

in burnup as expected with MOX.

Figure 20: MOX Flat-Side Thermal Discontinuity Factor Behavior

0.120

0.140

0.160

0.180

0.200

0.220

0.240

0.260

0.000 0.020 0.040 0.060 0.080 0.100

DF 2

Net Thermal J/Surface Average Thermal Flux

Flat-Side DF2 vs. Net Thermal J/Surface Average Thermal Flux0ppm 293K

0ppm 538K

0ppm 558K

0ppm 578K

0ppm 598K

0ppm 618K

0ppm 293K

0ppm 538K

0ppm 558K

0ppm 578K

0ppm 598K

0ppm 618K

0ppm 293K

0ppm 538K

0ppm 558K

0ppm 578K

0ppm 598K

0ppm 618K

0ppm 293K

0ppm 538K

0ppm 558K

0ppm 578K

0ppm 598K

0ppm 618K

41

The inner-corner fast discontinuity factor behavior is shown below in Figure 21.

The sides are grouped together as in all previous plots, and here no overlap from one side

to another occurs. Clockwise from the top left of the figure are sides: 15 north, 21 west,

16 north and 15 west. As in the strictly uranium loading patterns the inner-inner corner

sides of 15 west and 15 north are to the left of the flat-inner corner sides of 16 north and

21 west. For MOX, location matters over fuel type, just as it did with uranium loading

patterns. In addition the variation in the discontinuity factors due to the change in

temperature is not noticeable at this scale compared to the change due to location on the

fuel-reflector interface.

Figure 21: MOX Inner-Corner Fast Discontinuity Factor Behavior

0.600

0.800

1.000

1.200

1.400

1.600

1.800

2.000

-0.0500 0.0000 0.0500 0.1000 0.1500 0.2000 0.2500

DF 1

Net Fast J/Surface Average Fast Flux

Inner-Corner DF1 vs. Net Fast J/Surface Average Fast Flux0ppm 293K

0ppm 538K

0ppm 558K

0ppm 578K

0ppm 598K

0ppm 618K

0ppm 293K

0ppm 538K

0ppm 558K

0ppm 578K

0ppm 598K

0ppm 618K

0ppm 293K

0ppm 538K

0ppm 558K

0ppm 578K

0ppm 598K

0ppm 618K

0ppm 293K

0ppm 538K

0ppm 558K

0ppm 578K

0ppm 598K

0ppm 618K

42

The inner-corner thermal discontinuity factors are displayed in Figure 22 below.

The discontinuity factors have grouped themselves according to location along the fuel-

reflector interface. Side 15 north is located in the top left portion of the figure below.

Side 15 west is located just below the discontinuity factor value of 0.300 in the figure

below. Sides 21 west and 16 north are in close proximity to each other. Side 21 west is

elongated more along the x-axis and appears above the 16 north side data which is

directly below and more tightly located along the x-axis.

Figure 22: MOX Inner-Corner Thermal Discontinuity Factor Behavior

0.100

0.200

0.300

0.400

0.500

0.600

0.700

-0.150 -0.100 -0.050 0.000 0.050 0.100 0.150

DF 2

Net Thermal J/Surface Average Thermal Flux

Inner-Corner DF2 vs. Net Thermal J/Surface Average Thermal Flux0ppm 293K

0ppm 538K

0ppm 558K

0ppm 578K

0ppm 598K

0ppm 618K

0ppm 293K

0ppm 538K

0ppm 558K

0ppm 578K

0ppm 598K

0ppm 618K

0ppm 293K

0ppm 538K

0ppm 558K

0ppm 578K

0ppm 598K

0ppm 618K

0ppm 293K

0ppm 538K

0ppm 558K

0ppm 578K

0ppm 598K

0ppm 618K

43

The discontinuity factors were the initial step in calculating the reflector constants

which are the ratios of the homogenized nodal constants to the discontinuity factors. The

plots in the next set of Figures will show the reflector constant behavior of the mini-core

which most closely matches actual core loading patterns at the fuel-reflector interface

with burned fuel at the periphery. This pattern matches the in-out loading patterns used

today. Figures 7 through 14 above showed data for a loading pattern of a non-symmetric

nature, the following plots have a loading pattern that is symmetric. Starting from the

data at the top of the figure below and working clockwise are sides: 26 north, 11 west, 5

west, and 25 north. The reflector constants group themselves as the discontinuity factors

did.

Figure 23: Flat-Sides Fast Absorption Reflector Constant Behavior

0.0023

0.0024

0.0025

0.0026

0.0027

0.0028

0.0029

0.0030

0.1670 0.1680 0.1690 0.1700 0.1710 0.1720 0.1730 0.1740 0.1750 0.1760

a1

/DF 1

[c

m-1

]

Net Fast J/Surface Average Fast Flux

Flat Sides a1/DF1 vs. Net Fast J/Surface Average Fast Flux0ppm 293K

0ppm 538K

0ppm 558K

0ppm 578K

0ppm 598K

0ppm 618K

0ppm 293K

0ppm 538K

0ppm 558K

0ppm 578K

0ppm 598K

0ppm 618K

0ppm 293K

0ppm 538K

0ppm 558K

0ppm 578K

0ppm 598K

0ppm 618K

0ppm 293K

0ppm 538K

0ppm 558K

0ppm 578K

0ppm 598K

0ppm 618K

44

The flat-side thermal absorption reflector constant behavior is shown in Figure 24

below. This reflector constant overlaps more so than the discontinuity factors did. In

Figure 24, sides 25 north and 26 north appear on the upper right diagonal of the figure.

Similarly the data for sides 5 west and 11 west occurs towards the bottom left of the

figure. It is important to note that the 293K data for both sides 5 west and 11 west occurs

further to the top right portion of the plot on the diagonal than the higher temperature data

for the sides 25 north and side 26 north.

Figure 24: Flat-Side Thermal Absorption Reflector Constant Behavior

0.100

0.120

0.140

0.160

0.180

0.200

0.220

0.040 0.050 0.060 0.070 0.080 0.090

a2

/DF 2

[cm

-1]

Net Thermal J/Surface Average Thermal Flux

Flat-Side a2/DF2 vs. Net Thermal J/Surface Average Thermal Flux0ppm 293K

0ppm 538K

0ppm 558K

0ppm 578K

0ppm 598K

0ppm 618K

0ppm 293K

0ppm 538K

0ppm 558K

0ppm 578K

0ppm 598K

0ppm 618K

0ppm 293K

0ppm 538K

0ppm 558K

0ppm 578K

0ppm 598K

0ppm 618K

0ppm 293K

0ppm 538K

0ppm 558K

0ppm 578K

0ppm 598K

0ppm 618K

45

The inner-corner fast absorption reflector constant behavior is shown below in

Figure 25. The data below for each side is grouped, however sides 15 west and 15 north

overlap each other in the top left portion of the plot below, while sides 16 north and 21

west overlap in the bottom right portion of the figure below. The elongation of the data is

due to burnup as described with the first plot of discontinuity factor data.

Figure 25: Inner-Corner Fast Absorption Reflector Constant Behavior

0.0020

0.0022

0.0024

0.0026

0.0028

0.0030

0.0032

0.1000 0.1100 0.1200 0.1300 0.1400 0.1500 0.1600

a1

/DF 1

[cm

-1]

Net Fast J/Surface Average Fast Flux

Inner-Corner a1/DF1 vs. Net Fast J/Surface Average Fast Flux0ppm 293K

0ppm 538K

0ppm 558K

0ppm 578K

0ppm 598K

0ppm 618K

0ppm 293K

0ppm 538K

0ppm 558K

0ppm 578K

0ppm 598K

0ppm 618K

0ppm 293K

0ppm 538K

0ppm 558K

0ppm 578K

0ppm 598K

0ppm 618K

0ppm 293K

0ppm 538K

0ppm 558K

0ppm 578K

0ppm 598K

0ppm 618K

46

The inner-corner thermal absorption reflector constant behavior is shown in

Figure 26 below. Sides 15 west and 15 north data overlap on one another and appear in

the right portion of the plot. The higher temperature data appears towards the bottom of

the plot, while the 293K data appears higher around a reflector constant value of 0.170.

The data for sides 16 north and 21 west overlap as well and appear to the left of the side

15 data. The data overlaps less than the fast data did causing the different temperature

data sets to stratify the plot area in bands of their respective data colors.

Figure 26: Inner-Corner Thermal Absorption Reflector Constant Behavior

0.100

0.110

0.120

0.130

0.140

0.150

0.160

0.170

0.180

0.000 0.010 0.020 0.030 0.040 0.050 0.060 0.070 0.080

a2

/DF 2

[cm

-1]

Net Thermal J/Surface Average Thermal Flux

Inner-Corner a2/DF2 vs. Net Thermal J/Surface Average Thermal Flux

0ppm 293K

0ppm 538K

0ppm 558K

0ppm 578K

0ppm 598K

0ppm 618K

0ppm 293K

0ppm 538K

0ppm 558K

0ppm 578K

0ppm 598K

0ppm 618K

0ppm 293K

0ppm 538K

0ppm 558K

0ppm 578K

0ppm 598K

0ppm 618K

0ppm 293K

0ppm 538K

0ppm 558K

0ppm 578K

0ppm 598K

0ppm 618K

47

The diffusion coefficient reflector constant appears in the next set of four plots.

The data is grouped by side in the plot below. Starting from the data set furthest to the

left in the figure and working to the right are the sides: 25 north, 26 north, 5 west, and 11

west. The diffusion reflector constants incur the most amount of absolute error when

they are parameterized using a linear fit. There small reliance on the independent

variable as seen below could be a possible cause.

Figure 27: Flat Sides Fast Diffusion Reflector Constant Behavior

0.805

0.810

0.815

0.820

0.825

0.830

0.1670 0.1680 0.1690 0.1700 0.1710 0.1720 0.1730 0.1740 0.1750 0.1760

D1/

DF 1

[cm

]

Net Fast J/Surface Average Fast Flux

Flat-Sides D1/DF1 vs. Net Fast J/Surface Average Fast Flux0ppm 293K

0ppm 538K

0ppm 558K

0ppm 578K

0ppm 598K

0ppm 618K

0ppm 293K

0ppm 538K

0ppm 558K

0ppm 578K

0ppm 598K

0ppm 618K

0ppm 293K

0ppm 538K

0ppm 558K

0ppm 578K

0ppm 598K

0ppm 618K

0ppm 293K

0ppm 538K

0ppm 558K

0ppm 578K

0ppm 598K

0ppm 618K

48

The flat-sides thermal diffusion reflector constant appears in the figure below.

Sides 5 west and 11 west are grouped together at the top of the plot. Their respective

293K data is to the top right of the plot, while the higher temperatures lie to the top left.

Sides 25 north and 26 north data lie below and to the right of sides 5 west and 11 west.

Similarly the 293K data is to the right of the higher temperature data sets.

Figure 28: Flat Side Thermal Diffusion Reflector Constant Behavior

0.700

0.800

0.900

1.000

1.100

1.200

1.300

0.040 0.050 0.060 0.070 0.080 0.090

D2/

DF 2

[cm

]

Net Thermal J/Surface Average Thermal Flux

Flat-Side D2/DF2 vs. Net Thermal J/Surface Average Thermal Flux0ppm 293K

0ppm 538K

0ppm 558K

0ppm 578K

0ppm 598K

0ppm 618K

0ppm 293K

0ppm 538K

0ppm 558K

0ppm 578K

0ppm 598K

0ppm 618K

0ppm 293K

0ppm 538K

0ppm 558K

0ppm 578K

0ppm 598K

0ppm 618K

0ppm 293K

0ppm 538K

0ppm 558K

0ppm 578K

0ppm 598K

0ppm 618K

49

The inner-corner fast diffusion coefficient reflector constant behavior is shown in

Figure 29 below. As with the flat-sides, here the inner-corner side data sets are

overlapping one another. Sides 15 west and 15 north are in the lower left portion of the

figure while sides 16 north and 21 west are found in the top right of the figure. Similarly

the temperature variation is not seen in each side’s data set due to the overlap. The

elongation is due to the burning of the mini-core.

Figure 29: Inner-Corner Fast Diffusion Reflector Constant Behavior

0.770

0.780

0.790

0.800

0.810

0.820

0.830

0.840

0.1000 0.1100 0.1200 0.1300 0.1400 0.1500 0.1600

D1/

DF 1

[cm

]

Net Fast J/Surface Average Fast Flux

Inner-Corner D1/DF1 vs. Net Fast J/Surface Average Fast Flux0ppm 293K

0ppm 538K

0ppm 558K

0ppm 578K

0ppm 598K

0ppm 618K

0ppm 293K

0ppm 538K

0ppm 558K

0ppm 578K

0ppm 598K

0ppm 618K

0ppm 293K

0ppm 538K

0ppm 558K

0ppm 578K

0ppm 598K

0ppm 618K

0ppm 293K

0ppm 538K

0ppm 558K

0ppm 578K

0ppm 598K

0ppm 618K

50

The inner-corner thermal diffusion reflector constant behavior is shown in Figure

30 below. As in Figure 28 above the data sets for each side are overlapping one another.

Sides 15 west and 15 north appear at the bottom of the plot with data for the 293K

conditions appearing to the right of the higher temperature data sets for the same side.

Sides 16 north and 21 west lie in the upper region of Figure 30. Similarly, the 293K data

appears to the right of the higher temperature data sets.

Figure 30: Inner-Corner Thermal Diffusion Reflector Constant Behavior

0.800

0.850

0.900

0.950

1.000

1.050

1.100

1.150

1.200

1.250

1.300

0.010 0.020 0.030 0.040 0.050 0.060 0.070 0.080

D2/D

F 2[c

m]

Net Thermal J/Surface Average Thermal Flux

Inner-Corner D2/DF2 vs. Net Thermal J/Surface Average Thermal Flux

0ppm 293K

0ppm 538K

0ppm 558K

0ppm 578K

0ppm 598K

0ppm 618K

0ppm 293K

0ppm 538K

0ppm 558K

0ppm 578K

0ppm 598K

0ppm 618K

0ppm 293K

0ppm 538K

0ppm 558K

0ppm 578K

0ppm 598K

0ppm 618K

0ppm 293K

0ppm 538K

0ppm 558K

0ppm 578K

0ppm 598K

0ppm 618K

51

The flat-side removal reflector constant data appears in Figure 31 below. The

data sets for each side do not overlap in the figure below. Working clockwise from the

data set on the far right is side: 11 west, 26 north, 25 north and 5 west. Data sets for sides

11 west and 26 north have similar behavior as do sides 25 north and 5 west. This

behavior is dependent on the location of the surface along the fuel-reflector interface.

Figure 31: Flat Side Removal Reflector Constant Behavior

0.025

0.025

0.026

0.026

0.027

0.027

0.028

0.028

0.1670 0.1680 0.1690 0.1700 0.1710 0.1720 0.1730 0.1740 0.1750 0.1760

R/D

F 1[c

m-1

]

Net Fast J/Surface Average Fast Flux

Flat-Side R/DF1 vs. Net Fast J/Surface Average Fast Flux0ppm 293K

0ppm 538K

0ppm 558K

0ppm 578K

0ppm 598K

0ppm 618K

0ppm 293K

0ppm 538K

0ppm 558K

0ppm 578K

0ppm 598K

0ppm 618K

0ppm 293K

0ppm 538K

0ppm 558K

0ppm 578K

0ppm 598K

0ppm 618K

0ppm 293K

0ppm 538K

0ppm 558K

0ppm 578K

0ppm 598K

0ppm 618K

52

The inner-corner removal reflector constant behavior is shown in Figure 32

below. Unlike Figure 31 above, the sides are grouping themselves together and

overlapping. Sides 15 west and 15 north appear as the dot in the lower left portion of

Figure 32, while sides 16 north and 21 west overlap and appear in the upper right hand

portion of the plot.

Figure 32: Inner-Corner Removal Reflector Constant Behavior

Discontinuity factor and reflector constant behavior is similar for all mini-cores

studied. The behavior as the mini-core was burned may differ, however the shifts due to

changes in temperature and boron concentration are essentially the same.

0.020

0.022

0.024

0.026

0.028

0.030

0.032

0.1000 0.1100 0.1200 0.1300 0.1400 0.1500 0.1600

R/D

F 1[c

m-1

]

Net Fast J/Surface Average Fast Flux

Inner-Corner R/DF1 vs. Net Fast J/Surface Average Fast Flux0ppm 293K

0ppm 538K

0ppm 558K

0ppm 578K

0ppm 598K

0ppm 618K

0ppm 293K

0ppm 538K

0ppm 558K

0ppm 578K

0ppm 598K

0ppm 618K

0ppm 293K

0ppm 538K

0ppm 558K

0ppm 578K

0ppm 598K

0ppm 618K

0ppm 293K

0ppm 538K

0ppm 558K

0ppm 578K

0ppm 598K

0ppm 618K

53

CHAPTER 4

Parameterization

The parameterization occurred in two steps. The axial discontinuity factors

contained a much smaller set of data and were analyzed separately from the radial factors

and constants. Due to the location along the fuel-reflector interface being most

important, the PWR axial models were parameterized separately from the BWR axial

models. This is due to the much larger difference relative difference in reflector

composition between the PWR and BWR models.

Ideally axial reflector composition drawings would have been obtained and a

detailed axial model would have been set up in PARAGON as it was done for the radial

modeling. The axial data was not readily obtainable and reproduction of previously

developed models was done. Leakage in the axial direction is typically small; however it

will be recommended for future work that axial models be given more attention.

The axial discontinuity factors and reflector constants from the sensitivity study

were gathered and analyzed. Each factor was regressed using a linear fit. The linear

equation was then used to predict the factors and constants. An absolute percent error

was calculated by taking the prediction and subtracting the actual factor from it. The

PWR axial models are discussed first followed by the BWR axial models.

54

For the PWR axial models large absolute errors appeared when using a linear

regression. These absolute errors in Table 3 below suggest that changes to increase the

data set overall to include more temperatures and conditions need to be done, modeling

of the reflector as a two-dimensional assembly rather than one row of pins, or a different

grouping of data for regression needs to occur. Table 3 has been conditionally formatted

from the largest positive magnitude in red to the largest magnitude negative in blue.

Table 3: PWR Axial Absolute Percent Error of Predicted Constant and Factor Data

Absolute Error: (Predicted - Actual)*100%

1-D Axial Model Equivalence Equivalence Reflector Constants

DF1 DF2 D1 /DF1 a1 /DF1 r /DF1 D2 /DF2 a /DF2

top_0ppm_575 11.748521 -1.040185 -6.366993 -0.146158 0.006515 4.847764 -0.277547

top_0ppm_575_CR 17.303182 3.148093 -12.928029 -0.176401 -0.022979 -6.562464 -2.407928

top_0ppm_575_fresh -6.600885 -4.586771 14.285907 0.073508 0.125216 9.642762 3.423666

top_0ppm_575_CR_fresh -30.016796 2.626068 6.336146 0.260091 -0.079357 -5.722557 -0.116571

top_0ppm_610 4.555617 -1.815705 -7.554697 -0.104450 0.021654 -0.083035 -7.580553

top_0ppm_610_CR 19.530475 3.969232 -11.405973 -0.219160 -0.019534 -4.233767 3.705561

top_0ppm_610_fresh -5.922213 -5.610432 11.064776 0.118618 0.072028 4.718047 -3.507039

top_0ppm_610_CR_fresh -10.597901 3.309699 6.568863 0.193951 -0.103543 -2.606748 6.760410

top_900ppm_575 3.778634 4.728230 -8.485686 -0.044166 -0.123997 -6.414436 3.557084

top_900ppm_575_CR 17.207546 -3.757466 -10.511164 -0.239523 0.067259 3.433773 -6.932813

top_900ppm_575_fresh -5.456249 3.093657 10.629633 0.144213 -0.003959 -3.082441 7.666406

top_900ppm_575_CR_fresh -7.963952 -4.211627 7.040187 0.128434 0.031302 3.857600 -4.912299

top_900ppm_610 -6.311541 4.090252 -8.399805 -0.002591 -0.072478 -7.259078 -2.940364

top_900ppm_610_CR 13.531308 -2.790221 -7.841617 -0.260503 0.087367 5.961572 -0.387032

top_900ppm_610_fresh -11.310267 2.195169 8.997368 0.187765 -0.012101 -3.632170 1.551470

top_900ppm_610_CR_fresh -3.475478 -3.347996 8.571085 0.086370 0.026608 7.135180 2.397547

bottom_0ppm_540_feed 25.202320 1.555384 0.441501 0.001896 0.015466 1.814191 0.671744

bottom_0ppm_575_feed 19.376897 -1.634993 -0.247166 0.000985 -0.014897 -2.607189 -0.755203

fbottom_0ppm_540_feed -48.850431 1.688185 1.002920 0.000536 0.047294 2.337967 0.732165

fbottom_0ppm_575_feed 4.271215 -1.608576 -1.197256 -0.003417 -0.047862 -1.544970 -0.648705

bottom_900ppm_540_feed -12.831517 -1.564678 -1.153449 -0.003679 -0.045480 -1.439335 -0.619238

bottom_900ppm_575_feed -38.964625 1.705564 0.948786 0.000381 0.045695 2.278997 0.731328

fbottom_900ppm_540_feed 36.479629 -1.678891 -0.290972 0.001247 -0.017279 -2.712824 -0.784671

fbottom_900ppm_575_feed 15.316513 1.538005 0.495635 0.002051 0.017064 1.873161 0.672581

Maximum 36.479629

Minimum -48.850431

55

For the BWR axial models, only the top reflector model is regressed because

bottom reflector models only contained two data points that don’t make an accurate

portrayal of the true regression. The regressions are done as a function of the Net

Current/Surface Flux and the void. The BWR regressions have a smaller absolute error

at the maximum and minimum as seen in Table 4 below, however the variety of cases

regressed was much smaller. There was one reflector configuration regressed over

multiple void fractions. The BWR data suggests that the PWR data could be regressed

more successfully if the reflector region compositions were taken into account. The

PWR bottom axial reflector data contradicts this assumption because the bottom reflector

region remains the same with a changing boron and moderator temperature. Further

strengthening of the axial methodology and plan of attack is required in future work.

Table 4: BWR Axial Factor and Constant Absolute Percent Error

Absolute Percent Error: (Predicted - Actual)*100%

1-D Axial Model Equivalence Equivalence Reflector Constants

DF1 DF2 D1/DF1 a1/DF1 r/DF1 D2/DF2 a/DF2

Bottom Control Blade

Bottom No Control Blade

Top 0 Void -0.045751 -0.021764 -0.542765 1.733446 -0.001285 0.187979 -0.003550

Top 10 Void 0.003111 0.008049 -0.016676 1.271695 -0.000037 -0.008463 0.000466

Top 20 Void 0.029303 0.018889 0.279114 0.846202 0.000778 -0.112899 0.002434

Top 30 Void 0.036635 0.012918 0.376956 0.467264 0.001171 -0.136810 0.002794

Top 40 Void 0.022750 -0.000575 0.466116 0.154192 0.000905 -0.131896 0.001730

Top 50 Void 0.003650 -0.008106 0.235655 -0.093137 -0.000024 -0.062465 -0.000188

Top 60 Void -0.020128 -0.009828 -0.203703 -0.249628 -0.000651 0.091904 -0.001971

Top 70 Void -0.038266 -0.007709 -0.572515 -0.278332 -0.001019 0.212212 -0.002884

Top 80 Void -0.029494 0.001953 -0.714296 -0.139802 -0.001054 0.196256 -0.001794

Top 90 Void 0.038190 0.006173 0.692115 0.274180 0.001215 -0.235817 0.002963

Maximum 1.733446

Minimum -0.714296

56

Next the details of the parameterization of the radial discontinuity factors and

reflector constants are discussed. The radial reflector modeling contained a vastly wider

array of fuel types, fuel burnup, moderator temperatures, boron concentrations, and

moderator densities when compared with the axial modeling. This is somewhat expected

as the radial modeling can be more complex when considering that many assemblies

contain natural blankets at both their top and bottom.

The parameterization was carried out as follows:

1. Discontinuity Factors and Reflector Constants were gathered.

2. Linear Regressions were Performed based on Moderator Temp, Boron

Concentration, Net J/Surface Flux, and Moderator Density if Applicable.

3. Predictions were made based on the Regressions.

4. Absolute Errors were Calculated: (Predicted – Actual)*100%

A linear regression was found to be adequate for both uranium loading and MOX

loading patterns. The two types of fuel cannot be regressed together for the same

reflector location without introducing excessively large errors. Two mini-cores were

voided in the fuel region. These voided cases were difficult to parameterize. It is

recommended that a relationship for uranium at the fuel-reflector interface and a separate

relationship for MOX at the fuel-reflector interface be developed. The difficulty in

parameterizing voided cases may require a separate relationship. Further investigation is

required.

57

During the parameterization process each mini-core had a relationship developed

for each individual node side along the fuel-reflector interface for both fast and thermal

energy groups. Initial discussions suggested developing relationships for the

discontinuity factors and reflector constants based on whether they were on a flat or

inner-corner side; the errors encountered is higher than acceptable. Developing constants

for the same fuel assembly along the fuel-reflector interface by grouping neighboring

node faces leads to errors smaller than those seen from grouping flat and inner-corner

sides, but much larger than individual node relationships. Using individually developed

relationships for each node face on the fuel-reflector interface for each energy group is

best. It was also found that relationships can be developed for uranium and MOX fuel

that covers a wide array of enrichments and burnups. The following tables illustrate the

magnitudes of the errors in each case through a listing of the highest absolute errors for

each mini-core or set of mini-cores.

The errors in Table 5 below occur in the thermal energy group, if it exists for that

constant. The errors for the voided cases are by far the highest. The maximum error is

determined by finding the error with the highest positive magnitude, whereas the

minimum error is determined by finding the largest negative magnitude. The errors are

located across the data set for all sides and energy groups. The maximum errors are not

indicative of the average error seen throughout the data sets. Locations of the errors will

be discussed for the combination of mini-cores that were parameterized, and absolute

errors will be located in the appendix for inspection. In addition to the individual sides,

assemblies were modeled by grouping neighboring faces in Table 6.

58

Table 5: Individual Side Maximum Absolute Errors

Individual

MiniCore DF Sig A/DF D/DF Sig R/DF

MAX MIN MAX MIN MAX MIN MAX MIN

F480_2320_2320_DENS040 1.558459 -2.070572 1.346410 -1.675945 7.443446 -11.942856 0.005918 -0.010583

F480_2320_2320_DENS050 1.686324 -1.943243 1.586682 -1.927153 6.163832 -9.913567 0.005476 -0.008953

F480_2320_2320_DENS060 1.750503 -1.829825 1.597617 -2.046164 5.593410 -8.766644 0.007577 -0.008170

F480_2320_2320_DENS072 1.814575 -1.756600 1.702681 -2.287026 5.121642 -8.543196 0.011272 -0.012595

F480_F320_2480 2.498985 -1.854561 2.154268 -2.185231 7.695234 -10.539861 0.018415 -0.013086

F480_F320_2480_REFL 2.499737 -1.855540 2.153717 -2.185944 7.697723 -10.547070 0.018437 -0.013086

F480_F320_F074 3.258230 -3.107092 2.079169 -2.187517 4.331855 -7.766403 0.023510 -0.039375

F480_F320_F074_VOID 76.035910 -11.093110 9.675415 -21.857952 66.400262 -249.641920 5.354190 -0.826182

F480_F320_2MOX 4.631051 -2.566974 2.353082 -2.170346 6.292381 -9.104068 0.095136 -0.070592

F480_FMOX_2MOX 7.777114 -7.707727 2.279489 -2.016213 5.295540 -10.044947 0.056606 -0.108233

F480_FMOX_1320 4.552449 -3.278872 2.638410 -2.402291 6.374035 -12.446458 0.021717 -0.028814

F480_FMOX_2480 2.977110 -3.270520 2.599822 -2.259979 6.729886 -12.416460 0.044663 -0.029600

F480_FMOX_F074 4.441298 -4.189975 2.976249 -2.256110 8.345223 -12.141917 0.035679 -0.039001

F480_FMOX_F074_VOID 73.348685 -10.870875 9.231620 -27.835555 63.695456 -286.272852 5.467360 -0.409471

F480_FMOX_F074_REFL 4.444354 -4.195300 2.972911 -2.254378 8.339583 -12.171421 0.035036 -0.038974

Table 6: Individual Assembly Side Maximum Absolute Errors

Assembly Sides

MiniCore DF Sig A/DF D/DF Sig R/DF

MAX MIN MAX MIN MAX MIN MAX MIN

F480_2320_2320_DENS040 1.708568 -2.100777 1.401304 -1.904103 8.790753 -15.254537 0.028991 -0.024619

F480_2320_2320_DENS050 1.691336 -1.967303 1.605338 -2.054535 6.257670 -12.014126 0.021181 -0.019309

F480_2320_2320_DENS060 1.752892 -1.832302 1.630091 -2.204527 5.390081 -9.665478 0.030306 -0.026710

F480_2320_2320_DENS072 1.814356 -1.753409 1.723860 -2.365641 4.738726 -9.110183 0.033468 -0.030751

F480_F320_2480 2.689446 -2.707301 2.962474 -2.268548 8.291249 -10.817198 0.018286 -0.022587

F480_F320_2480_REFL 2.689596 -2.707705 2.961927 -2.312471 8.296514 -10.831534 0.019386 -0.023511

F480_F320_F074 3.840963 -4.856397 3.487299 -2.915349 5.102979 -8.327818 0.048462 -0.039933

F480_F320_F074_VOID 25.034826 -10.043342 8.932102 -21.954329 61.266537 -220.343161 4.428826 -0.338352

F480_F320_2MOX 28.771553 -32.688062 3.292235 -4.076205 18.460803 -17.382684 0.576498 -0.508959

F480_FMOX_2MOX 14.409543 -13.557742 5.294981 -6.662330 31.788764 -36.481058 0.295928 -0.242544

F480_FMOX_1320 36.594435 -28.092579 11.640742 -9.072063 30.298004 -23.062474 0.506835 -0.640174

F480_FMOX_2480 12.086597 -16.099326 11.775511 -9.686356 49.494908 -31.965910 0.268906 -0.212730

F480_FMOX_F074 12.093628 -19.994265 12.073787 -9.413937 23.112128 -15.727417 0.295227 -0.276381

F480_FMOX_F074_VOID 26.691221 -9.999030 8.691459 -24.160861 59.918256 -264.370978 4.460751 -2.824725

F480_FMOX_F074_REFL 12.524468 -19.985154 12.077672 -9.406447 23.101018 -15.742229 0.294956 -0.276147

59

The voided cases are again the cases with the highest errors. Also there is an

increase in the magnitude of the maximum and minimum errors for each mini-core from

the modeling of individual sides to individual assemblies. Modeling of individual

assemblies in Table 6 was performed by combining neighboring faces on the fuel-

reflector interface. The following table has even higher magnitude errors and is the result

of combining all of the flat sides, (5 west, 11 west, 25 north, 26 north from Figure 6) in

one relationship, and combining all inner-corners, (15 west, 21 west, 15 north, 16 north

from Figure 6):

Table 7: Flat and Inner-Corner Side Maximum Absolute Errors

Flat and Inner Corner

MiniCore DF Sig A/DF D/DF Sig R/DF

MAX MIN MAX MIN MAX MIN MAX MIN

F480_2320_2320_DENS040 5.816247 -6.514283 3.094978 -3.302006 41.347819 -35.109391 0.096236 -0.099024

F480_2320_2320_DENS050 5.634528 -6.912250 3.226782 -3.389403 31.035074 -25.015197 0.114536 -0.117313

F480_2320_2320_DENS060 5.374862 -6.972131 3.383028 -3.432719 23.934402 -17.738379 0.136497 -0.140731

F480_2320_2320_DENS072 4.991794 -6.593102 3.823919 -3.684974 18.292299 -13.615393 0.157455 -0.162977

F480_F320_2480 5.494831 -7.801155 4.372796 -4.109073 23.633188 -22.653216 0.490033 -0.670180

F480_F320_2480_REFL 5.489097 -7.801430 4.369681 -4.104868 23.609301 -22.776427 0.489898 -0.669830

F480_F320_F074 7.478829 -10.078945 5.312602 -4.657267 12.585547 -16.614271 0.521297 -0.873499

F480_F320_F074_VOID 15.472410 -48.266480 6.638049 -28.831804 101.579197 -222.014497 -1.202610 -2.404721

F480_F320_2MOX 46.242935 -47.747673 9.453420 -8.451485 40.927715 -40.341798 0.718038 -0.737156

F480_FMOX_2MOX 44.397017 -50.110925 10.671342 -15.393044 46.336767 -70.567760 0.783430 -0.486387

F480_FMOX_1320 23.511508 -23.442068 15.526964 -17.424082 35.791986 -59.895967 0.399186 -0.458542

F480_FMOX_2480 24.553888 -24.981290 18.491579 -17.584854 41.566285 -60.422841 0.296508 -0.364186

F480_FMOX_F074 25.957294 -26.955643 14.936521 -18.274640 47.436681 -62.768594 0.334665 -0.746665

F480_FMOX_F074_VOID 16.789241 -26.517476 2.034958 -32.654679 49.994132 -273.678184 -0.647004 -3.623273

F480_FMOX_F074_REFL 25.975058 -26.956181 14.933590 -18.241219 47.323140 -62.536680 0.334260 -0.746014

From the tables and inspection of the set of absolute errors, it is seen that the

lowest errors occur when each individual half-node side along the fuel-reflector interface

is modeled individually. The next table combines several mini-cores together to

60

determine if the parameterization is dependent on the loading pattern or if the Net

J/Surface Flux parameter adequately captures the effect of the environment. It was found

that is does; however MOX and uranium fuel behave differently enough to warrant

separate parameterization for each.

Table 8: Maximum Absolute Errors for Multiple Minicore Modelings

Combination

MiniCore DF Sig A/DF D/DF Sig R/DF

MAX MIN MAX MIN MAX MIN MAX MIN

F480_2320_2320_DENS040

F480_2320_2320_DENS050

F480_2320_2320_DENS060

F480_2320_2320_DENS072 2.7986036 -8.2285967 5.5491636 -1.2409698 17.634372 -20.0518686 0.06038 -0.072902

F480_F320_2480

F480_F320_2480_REFL

F480_F320_F074

F480_F320_F074_VOID

F480_F320_2MOX

F480_FMOX_2MOX

F480_FMOX_F074_VOID

F480_FMOX_1320

F480_FMOX_2480 6.1559926 -6.4273296 3.9044655 -3.1922584 10.365159 -13.3646958 0.0451 -0.056218

F480_FMOX_F074

F480_FMOX_F074_REFL

Table 8 above shows that the different mini-cores can be combined together in

one parameterization and adequately model the discontinuity factors and reflector

constants. The Voided cases were not combined in the uranium and MOX combinations

due to their large errors in Table 5. The cases with MOX in a minority location as

compared to the other MOX cases was not combined with MOX cases due to the high

61

errors associated with mixing uranium and MOX loadings for the same location on the

fuel-reflector interface. These errors were upwards of forty absolute percent. The

maximum and minimum errors occur under specific conditions to be discussed next.

The maximum errors for the uranium combination of mini-cores occur primarily

in the mini-cores with the lower moderator density, high boron concentration and lower

temperatures as a whole. Some of these high errors occur at other temperatures from

293K but the cases are 2000 ppm and 3000 ppm boron. Errors above 5 absolute percent

in the absorption reflector constant occurred in the thermal group, 3000 ppm, and 293K

for the moderator density case of 0.60 g/cc. For the diffusion reflector constant, absolute

errors occur throughout the mini-cores and conditions for the thermal energy group.

Absolute errors for the fast diffusion reflector constant are typically under one percent.

Errors for the removal reflector constant are acceptable and below a fraction of an

absolute percent.

For the MOX loading patterns that were combined the maximum absolute errors

for the discontinuity factors are localized in two mini-cores under high the highest burnup

for the MOX assembly, side 11W in the fast group. The absorption reflector constant had

errors of less than 4 absolute percent. The diffusion reflector constant for the MOX

parameterization showed more of a clustering of high absolute errors, however they still

occurred over a wide range of conditions. The highest errors both positive and negative

appear to occur at cases of 293K. Errors for the removal reflector constant are acceptable

and below a fraction of an absolute percent.

62

The parameterization performed in this study is just the first step in developing

the relationships of the discontinuity factors and reflector constants to the varying core

conditions. Further development is necessary, however the initial analysis shows that

parameterization is possible using the fuel-reflector interface condition of the net current

to the surface flux.

63

CHAPTER 5

Conclusion

The discontinuity factor routine developed based on equivalence theory has been

shown to be properly implemented through its reproduction of the PARAGON reference

solution in Chapter 2. In addition, the routine provides smaller improvements where the

generic set of Westinghouse reflector constants currently in use generates good results,

and more importantly the routine provides a much larger improvement where the generic

constants do not perform as well. These improvements are in both the global eigenvalue

and local power distribution as noted in Chapter 2.

Chapter 3 discussed the behavior of the various factors and constants. Most

behavior was shown to be in a linear fashion, although some was not. This non-linear

behavior was mostly seen in the loading patterns containing MOX. Chapter 4

demonstrated that a linear regression can be used to accurately predict most factors and

constants. The diffusion reflector constant is proving to be more difficult to regress,

although the errors may be deemed acceptable based on the conditions in which they

occur and their magnitude relative to previous studies conducted by Westinghouse on the

sensitivity of the global eigenvalue and local power distribution to changes in the

reflector constants of up to fifteen percent.

The voided color-sets could not be parameterized based on a linear regression.

Separate parameterizations for voided conditions could be made, however a general

parameterization to cover all core conditions was desired. Parameterizing the voided

64

conditions separately may have to be done just as with MOX. The axial voiding cases

were parameterized with much smaller errors. The axial PWR cases proved to be

difficult and require more investigation and development.

It is recommended that color-sets be used to calculate the reflector constants as

they capture the two dimensional effects in the inner-corners of the fuel-reflector

interface. Due to the behavior of the errors seen in the previous chapter, it is

recommended that the boron concentrations and temperatures used in this study be used.

Burnup steps can be reduced further if necessary. Originally the color-sets were burned

out to 40 GWd/MTU using 42 burnup steps. This was reduced to 11 with no noticeable

degradation in the results. Reduction to 6 burnup steps may be possible in strictly

uranium cases, however the behavior of MOX is best captured with the 11 burnup steps

used.

Changing the albedo boundary conditions one fuel pitch away from the fuel-

reflector interface had little to no noticeable effect on the factors and constants. A half of

an assembly pitch in distance from the interface may result in noticeable errors. The core

barrel that is located in Node 25 (Figure 6) resulted in noticeable changes in the factors

and constants when comparing them to the factors and constants generated in Node 5

with a symmetric loading pattern.

Care must be taken when modeling voiding in axial models, as noted by F.

Reitsma [12]. The axially voided models did result in parameterization with acceptable

errors, however the smeared reflector region and single row of pins may not accurately

capture the two dimensional nature it possesses. It is recommended that the axial models

include more than one row of pins and move to an assembly pitch size. This

65

recommendation is the result of the difficulties seen in parameterizing the axial constants

and from the benefits the two-dimensional color sets brought to the radial models.

66

REFERENCES

1. K. Koebke, “Advances in Homogenization and Dehomogenization,” Proc. Conf.

Advances in Mathematical Methods for Solution of Nuclear Engineering Problems,

Volume II, Munich, American Nuclear Society. p. 59, April 1981.

2. K. Smith, “Assembly Homogenization Techniques for Light Water Reactor Analysis,”

Prog. In Nuclear Energy, Vol. 17, No. 3, pp. 303-335, 1986.

3. L. Hetzelt, H-J Winter, “Generalization of the Equaivalent Reflector Model for the

Siemens Standard Core Design Procedure”, Proc. of M&C 99 Conference, Madrid, Spain

(1999).

4. E. Muller, “Environment-Insensitive Equivalence Diffusion Theory Group Constants for

Pressurized Water Reactor Radial Reflector Regions”, Nuclear Science and Engineering,

103, 359-376 (1989).

5. E. Muller, “Improved Pressurized Water Reactor Radial Reflector Modeling in Nodal

Analysis”, Nuclear Science and Engineering, 109, 200-214 (1991).

6. Y. Takara, T. Kanagawa, and H. Sekimoto, “Two-dimensional Baffle/Reflector

Constants for Nodal Code in PWR Core Design”, Journal of Nuclear Science and

Technology, Vol. 37, p. 986-995 (2000).

7. Y. Takara, H. Sekimoto, “Two-dimensional Baffle/Reflector Constants Based on

Transport Equaivalent Diffusion Parameters”, Proc. of PHYSOR-2002, Seoul, Korea

(2002).

8. Y. Takara, H. Sekimoto, “Transport Equivalent Diffusion Constants for Reflector Region

in PWRs”, Journal of Nuclear Science and Technology, Vol. 39, p. 716-728 (2002).

9. B. Ivanov, “Methoddology for Embedded Transport Core Calculation”, PhD Thesis, PSU

2007.

67

10. E. Muller, “Development of an Environment-Insensitive PWR Radial Reflector Model

Applicable to Modern Nodal Reactor Analysis Methods”, PhD Thesis, Potchefstroom

University for CHO, Potchefstroom, South Africa (1989).

11. L. Mayhue, B. Zhang, D. Sato, and D. Jung, “PWR Core Modeling using the NEXUS

Once-Through Cross Section Model”, PHYSOR 2006 Conference, Vancouver, Canada,

(2006).

12. F. Reitsma, “The Application of Advanced Homogenization Models to the Reflector

Regions of Boiling Water Reactors”, Potchfstroom University for CHE, 1989.

13. “PARAGON User Manual”, PARAGON 1.2.0 Release, Westinghouse Electric Company

(2005).

14. T. Beam, K. Ivanov, A. Baratta, H. Finnemann, “Nodal Kinetics Model Upgrade in the

Penn State Coupled TRAC/NEM Codes”, Annals of Nuclear Energy, 26, 1205-1219

(1999).

15. “ANC User Manual”, Westinghouse Electric Company, ANC 8.7.3 Release (2001).

16. B. Bandini, “A Three-Dimensional Transient Neutronics Routine for the TRAC-PF1

Reactor Thermal Hydraulic Computer Code,” PhD Thesis, The Pennsylvania State

University, May 1990.

17. J.F. Vidal, P.B. Tellier, et. al., “Analysis of the FLOULE Experiment for the APOLLO2

Validation of PWR Core Reflectors”, PHYSOR 2008 Conference, Interlaken,

Switzerland, (2008).

68

APPENDIX A: MCNP Modeling

The effects of P3 scattering on thick reflectors in PWRs were raised by J. Vidal,

R. Tellier, et. al. [17]. To investigate this effect on our modeling methodology an MCNP

reference was developed. Unfortunately the libraries used were not specific to this

model, and introduced large errors. In the near future 70 group libraries will be

developed for MCNP so that a benchmark of this type can be carried out. The details of

the MCNP modeling performed in this project are given here as a reference:

1. 3980 Total Cycles with 3890 Active Cycles.

2. Source of 30,000 Particles per Cycle.

3. K-effective 0.95691.

4. 1 Standard Deviation = 0.00007.

5. 95% CI (0.95678 0.95704).

6. PARAGON k-effective 0.95000

The 691 pcm difference was the result of non model specific multi-group libraries being

utilized.

69

APPENDIX B: Sample DF Routine Input

The Discontinuity Factors generated in this study were done so using a routine

described in Chapter 1. The routine takes the boundary conditions from a transport

reference solution, calculates the flux based on the approximation selected (polynomial or

semi-analytic) and then calculated the discontinuity factors. In the case where reflector

constants were analyzed, the homogenized constants from the transport reference solution

were divided by the discontinuity factors from the routine.

The routine was a standalone program and a sample input for a 3x3 mini-core is

given below:

&NEM

NEM_Option = 1

/

&CORE

Number_Groups = 2

Number_Regions = 36

Number_Fuel_Assemblies_Y = 6

Number_Fuel_Assemblies_X = 6

Quad_Leakage = 1

/

&DIMENSIONS

XDimension = 10.799975D+00

YDimension = 10.799975D+00

/

&COMPOSITIONS

Composition =

1 2 3 4 5 6

7 8 9 10 11 12

70

13 14 15 16 17 18

19 20 21 22 23 24

25 26 27 28 29 30

31 32 33 34 35 36

/

&ASSEMBLY_BOUNDARY_CONDITIONS

Assembly_Boundary_Condition =

0 0 2 2 0 0 2 0 0 0 2 0 0 0 2 0 0 0 2 0 0 1 2 0

0 0 0 2 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0

0 0 0 2 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0

0 0 0 2 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0

0 0 0 2 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0

1 0 0 2 1 0 0 0 1 0 0 0 1 0 0 0 1 0 0 0 1 1 0 0

/

&AVERAGE_NET_CURRENTS

Average_Net_Current =

-3.123797870780E+13 -3.132302031400E+13 0.000000000000E+00 0.000000000000E+00

-2.153099236660E+12 -2.154747180180E+12 0.000000000000E+00 0.000000000000E+00

-2.632435303210E+13 -5.972966118160E+13 0.000000000000E+00 3.233382727200E+13

-1.866672294930E+12 -4.546289151800E+11 0.000000000000E+00 -8.821482642000E+11

-1.783715156220E+13 -3.816599275924E+13 0.000000000000E+00 5.972966118160E+13

-1.053596245671E+12 -1.430417749909E+12 0.000000000000E+00 4.546289151800E+11

-1.069733616340E+13 -2.930473572770E+13 0.000000000000E+00 4.121596149955E+13

-6.728286652190E+11 -4.488944936280E+11 0.000000000000E+00 3.196233267450E+11

-3.490096946440E+12 -1.083515559744E+13 0.000000000000E+00 2.930473572770E+13

-4.477720282000E+11 -1.518294845130E+12 0.000000000000E+00 4.488944936280E+11

-1.428575795142E+12 -4.420803210809E+12 0.000000000000E+00 1.239935935217E+13

-3.461199175900E+11 -3.097809012438E+12 0.000000000000E+00 9.999785722700E+11

-5.959973124340E+13 -2.640829763300E+13 3.225785152970E+13 0.000000000000E+00

-4.501168188800E+11 -1.868855241140E+12 -8.839922952100E+11 0.000000000000E+00

-5.012744277800E+13 -5.026084322030E+13 2.712670314860E+13 2.720380093150E+13

-4.180287664840E+11 -4.222176810890E+11 -7.967153718900E+11 -7.954056657100E+11

71

-3.316263602890E+13 -3.104410386924E+13 1.746476218980E+13 5.026084322030E+13

-8.673252462520E+11 -1.200197957746E+12 -3.271423296190E+11 4.222176810890E+11

-2.114557932749E+13 -2.289919591371E+13 1.041258177387E+13 3.376069531738E+13

1.153255368620E+11 -3.379570620540E+11 -2.169594520570E+11 2.399878203980E+11

-6.047802686405E+12 -8.291191481917E+12 3.127113669970E+12 2.289919591371E+13

1.087720217600E+11 -1.268768676790E+12 4.119597828000E+11 3.379570620540E+11

-1.954893773351E+12 -3.596032795807E+12 1.294611994554E+12 9.445035250622E+12

-3.785468735530E+11 -2.564127949224E+12 3.191083582900E+11 8.449212630600E+11

-3.794345064583E+13 -1.792217670420E+13 5.959973124340E+13 0.000000000000E+00

-1.426779734955E+12 -1.057986763618E+12 4.501168188800E+11 0.000000000000E+00

-3.084677855441E+13 -3.329315323044E+13 5.012744277800E+13 1.754599221830E+13

-1.198271324361E+12 -8.744106480600E+11 4.180287664840E+11 -3.276653963730E+11

-1.796079314022E+13 -1.819421951469E+13 3.316263602890E+13 3.329315323044E+13

-1.184878988520E+12 -1.112549379750E+12 8.673252462520E+11 8.744106480600E+11

-1.094147117052E+13 -1.058048544161E+13 2.114557932749E+13 1.986968754655E+13

-1.006271713830E+12 -1.127672588050E+12 -1.153255368620E+11 6.464392126400E+11

-3.683155396035E+12 -4.438314645206E+12 6.047802686405E+12 1.058048544161E+13

-9.361053426300E+11 -1.335552605003E+12 -1.087720217600E+11 1.127672588050E+12

-1.278424203317E+12 -2.415957781220E+12 1.954893773351E+12 4.908533075783E+12

-7.210135581270E+11 -1.672634163963E+12 3.785468735530E+11 1.231018411700E+12

-2.849922393797E+13 -1.080025801385E+13 4.101281858734E+13 0.000000000000E+00

-7.305643328820E+11 -6.807344584510E+11 3.090858677110E+11 0.000000000000E+00

-2.209980442141E+13 -2.124519364339E+13 3.357961427121E+13 1.051187361483E+13

-6.491577564350E+11 6.207653510500E+10 2.318932729960E+11 -2.167924318580E+11

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/

&ASSEMBLY_DATA

Assembly_kinf =

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8.072011150720E-01

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8.072011150720E-01

Assembly_Removal_XSEC =

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2.044380648505E+13 1.856092956885E+13 4.544654639861E+13 3.150834275616E+13

4.596062270641E+12 4.568063164355E+12 1.061573389643E+13 1.015339891125E+13

1.153213401722E+13 9.861570021216E+12 1.436903171767E+13 1.856092956885E+13

2.235151528907E+12 1.609551931039E+12 4.231580001675E+12 5.002223984467E+12

4.579442083607E+12 8.148429431892E+11 6.805704824910E+12 1.086512525095E+13

8.287552037702E+12 1.639641587529E+13 4.250181455880E+13 1.860094353371E+13

4.514519966316E+12 2.621004820368E+13 3.091933146517E+13 2.468586279154E+13

6.759081299368E+12 1.163660569918E+13 3.269684783461E+13 1.687323237866E+13

3.930396191124E+12 2.168935131441E+13 3.218459582544E+13 2.652352457851E+13

4.371969550328E+12 6.779561453421E+12 1.783605520930E+13 1.163660569918E+13

2.778988816176E+12 1.553429396053E+13 2.826407723375E+13 2.168935131441E+13

2.801018831812E+12 4.115559966561E+12 1.004814293490E+13 7.211358381060E+12

1.820153831450E+12 9.941736058858E+12 2.185202055500E+13 1.623718398375E+13

1.636254511261E+12 2.217418665777E+12 5.037194889657E+12 4.115559966561E+12

79

9.768037974662E+11 4.695185856161E+12 1.252885395775E+13 9.941736058858E+12

9.126889300238E+11 9.035428364076E+11 2.411355948538E+12 2.390184676083E+12

3.840398075142E+11 3.718517775112E+11 5.002436158674E+12 5.221145263494E+12

/