The Pennsylvania State University. The Graduate School. Department of Mechanical and Nuclear Engineering

Transcription

1 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

2 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

3 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,

4 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 UO 2 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.

5 v TABLE OF CONTENTS LIST OF FIGURES... vi LIST OF TABLES... viii ACKNOWLEDGEMENTS... ix CHAPTER 1 Introduction Background Modeling Sensitivity Studies Parameterization of Factors and Constants... 6 CHAPTER 2 Reflector Modeling Axial Reflector Modeling Radial Reflector Modeling CHAPTER 3 Sensitivity Studies CHAPTER 4 Parameterization CHAPTER 5 Conclusion REFERENCES APPENDIX A: MCNP Modeling APPENDIX B: Sample DF Routine Input... 69

6 vi LIST OF FIGURES Figure 1: 1-D Assembly/Reflector Models Figure 2: 3x3 Mini-Core Model Figure 3: 1/8th Core Results Figure 4: 1/8th Core Normalized Power Absolute Error Results Figure 5: Mini-Cores Used in the Sensitivity Study Figure 6: Reflector Node Identification Figure 7: Constant Boron Concentration Fast Discontinuity Factor Behavior Figure 8: Constant Temperature Fast Discontinuity Behavior Figure 9: Constant Boron Thermal Discontinuity Factor Behavior Figure 10: Constant Temperature Discontinuity Factor Behavior Figure 11: Fast Flat Side Discontinuity Factor Behavior Figure 12: Thermal Flat Side Discontinuity Factor Behavior Figure 13: Fast Inner-Corner Discontinuity Factor Behavior Figure 14: Inner-Corner Thermal Discontinuity Factor Behavior Figure 15: Flat-Sides Fast Discontinuity Factor Behavior Figure 16: Flat-Sides Thermal Discontinuity Factor Behavior Figure 17: Inner-Corner Fast Discontinuity Factor Behavior Figure 18: Inner-Corner Thermal Discontinuity Factor Behavior Figure 19: MOX Flat-Side Fast Discontinuity Factor Behavior Figure 20: MOX Flat-Side Thermal Discontinuity Factor Behavior Figure 21: MOX Inner-Corner Fast Discontinuity Factor Behavior Figure 22: MOX Inner-Corner Thermal Discontinuity Factor Behavior... 42

7 vii Figure 23: Flat-Sides Fast Absorption Reflector Constant Behavior Figure 24: Flat-Side Thermal Absorption Reflector Constant Behavior Figure 25: Inner-Corner Fast Absorption Reflector Constant Behavior Figure 26: Inner-Corner Thermal Absorption Reflector Constant Behavior Figure 27: Flat Sides Fast Diffusion Reflector Constant Behavior Figure 28: Flat Side Thermal Diffusion Reflector Constant Behavior Figure 29: Inner-Corner Fast Diffusion Reflector Constant Behavior Figure 30: Inner-Corner Thermal Diffusion Reflector Constant Behavior Figure 31: Flat Side Removal Reflector Constant Behavior Figure 32: Inner-Corner Removal Reflector Constant Behavior... 52

8 viii LIST OF TABLES Table 1: 3x3 Mini-Core Results Table 2: 1/8th Core Results Table 3: PWR Axial Absolute Percent Error of Predicted Constant and Factor Data Table 4: BWR Axial Factor and Constant Absolute Percent Error Table 5: Individual Side Maximum Absolute Errors Table 6: Individual Assembly Side Maximum Absolute Errors Table 7: Flat and Inner-Corner Side Maximum Absolute Errors Table 8: Maximum Absolute Errors for Multiple Minicore Modelings... 60

9 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.

10 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

11 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

12 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 crosssections 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

13 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]

14 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: l ϕ g,x x = A sinh k gl x + B cosh k gl x + n l n=0 a g,xn f n x (1) Where for two groups: k g,l = A g l D g l 1 2 k 1,l = A 1 l D 1 l 1 2 k 2,l = A 2 l D 2 l 1 2 l l A g = rg ; r1 = a1 + s1 2 l r2 l = a2 (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 NEMbased 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

15 6 the Westinghouse nodal code ANC for 1/8 th 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 UO 2 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

16 that each individual node surface on the fuel reflector interface had an individual parameterization for each energy group. 7

17 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.

18 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.

19 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.

20 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

21 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 minicore by its very nature will produce exaggerated results as compared to a 1/8 th 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 1/8 th core symmetry model. The nodal code

22 13 solutions for the 1/8 th 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 1/8 th 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 1/8 th 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.

23 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 k eff 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 minicore. 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

24 15 with the discontinuity factors at the fuel-reflector interface from the 1-D assemblyreflector 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 No DF pcm ABS % ABS % ABS % IA-ARR pcm ABS % ABS % ABS % IA-RDF 7.1 pcm ABS % ABS % 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 1/8 th 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 assemblyreflector models while MC means models with reflector discontinuity factors and cross sections from the 2-D 3x3 mini-core model.

25 16 Table 2: 1/8th Core Results Model Eigenvalue Reference 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 minicore 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. IA-1D IA-MC ABS ABS % % Error Error Figure 3: 1/8th Core Results

26 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

27 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 k eff 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

28 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 UO 2 assemblies while the second is loaded with a combination of UO 2 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

29 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. 150 MWd/MTU, 3Loop PWR Core, UO 2 Fuel 150 MWd/MTU, 3Loop PWR Core, UO 2 and MOX Fuel 0.2 ABS ABS % 2.3 % Error Error 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 UO 2 loading patterns, and improve the results where the generic set struggles in the case of loading patterns with MOX fuel.

30 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.

31 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 UO 2 vs. MOX fuel

32 A total of eight loading patterns were developed. The following four in Figure 5 are discussed first. 23 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 minicore 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.

33 24 In [10] it was concluded that the only meaningful spectrum parameter is the core/reflector net current spectrum (e.g., J 1 /J 2 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 (J 1 /J 2 ) 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

34 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: Figure 6: Reflector Node Identification

35 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 UO 2 followed immediately by those with MOX-UO 2 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

36 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. DF 1 Side 5W DF 1 vs. Net Fast J/Surface Average Fast Flux Net Fast J/Surface Average Fast Flux Figure 7: Constant Boron Concentration Fast Discontinuity Factor Behavior

37 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 Side 5W DF 1 vs. Net Fast J/Surface Average Fast Flux 50 DF Net Fast J/Surface Average Fast Flux Figure 8: Constant Temperature Fast Discontinuity Behavior

38 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 Side 5W DF 2 vs. Net Thermal J/Surface Average Thermal Flux DF Net Thermal J/Surface Average Thermal Flux Figure 9: Constant Boron Thermal Discontinuity Factor Behavior

39 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 Side 5W DF 2 vs. Net Thermal J/Surface Average Thermal Flux DF Net Thermal J/Surface Average Thermal Flux Figure 10: Constant Temperature Discontinuity Factor Behavior

40 31 Figures 7 through 10 above display the behavior of just one node within the minicore. 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. DF 1 Flat Sides DF 1 vs. Net Fast J/Surface Average Fast Flux Net Fast J/Surface Average Fast Flux Figure 11: Fast Flat Side Discontinuity Factor Behavior

41 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. DF 2 Flat Sides DF 2 vs. Net Thermal J/Surface Average Thermal Flux Net Thermal J/Surface Average Thermal Flux Figure 12: Thermal Flat Side Discontinuity Factor Behavior

42 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. DF 1 Inner-Corner DF 1 vs. Net Fast J/Surface Average Fast Flux Net Fast J/Surface Average Fast Flux Figure 13: Fast Inner-Corner Discontinuity Factor Behavior

43 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. DF 2 Inner-Corner DF 2 vs. Net Thermal J/Surface Average Thermal Flux Net Thermal J/Surface Average Thermal Flux Figure 14: Inner-Corner Thermal Discontinuity Factor Behavior