The Pennsylvania State University - ETDA - Penn State

The Pennsylvania State University The Graduate School College of Engineering

METHODOLOGY FOR EMBEDDED TRANSPORT CORE CALCULATION

A Thesis in Nuclear Engineering by Boyan D. Ivanov

© 2007 Boyan D. Ivanov

Submitted in Partial Fulfillment of the Requirements for the Degree of

Doctor of Philosophy

December 2007

The thesis of Boyan D. Ivanov was reviewed and approved* by the following:

Kostadin N. Ivanov Professor of Nuclear Engineering Thesis Adviser Chair of Committee

Yousry Y. Azmy Professor of Nuclear Engineering

John H. Mahaffy Associate Professor of Nuclear Engineering

Victor Nistor Professor of Mathematics

Mohamed Ouisloumen Principal Engineer Westinghouse Electric Company Special Member

Jack Brenizer Professor of Mechanical and Nuclear Engineering Chair of Nuclear Engineering Program

*

Signatures are on file in the Graduate School

ii

ABSTRACT

The progress in the Nuclear Engineering field leads to developing new generations of Nuclear Power Plants (NPP) with complex rector core designs, such as cores loaded partially with mixed-oxide (MOX) fuel, high burn-up loadings, and cores with advanced designs of fuel assemblies and control rods. Such heterogeneous cores introduce challenges for the diffusion theory that has been used for several decades for calculations of the current Pressurized Water Rector (PWR) cores. To address the difficulties the diffusion approximation encounters new core calculation methodologies need to be developed by improving accuracy, while preserving efficiency of the current reactor core calculations.

In this thesis, an advanced core calculation methodology is introduced, based on embedded transport calculations. Two different approaches are investigated. The first approach is based on embedded finite element (FEM), simplified P3 approximation (SP3), fuel assembly (FA) homogenization calculation within the framework of the diffusion core calculation with NEM code (Nodal Expansion Method). The second approach involves embedded FA lattice physics eigenvalue calculation based on collision probability method (CPM) again within the framework of the NEM diffusion core calculation. The second approach is superior to the first because most of the uncertainties introduced by the off-line cross-section generation are eliminated.

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TABLE OF CONTENTS

LIST OF ABBREVIATIONS............................................................................................ vi LIST OF FIGURES ......................................................................................................... viii LIST OF TABLES...............................................................................................................x ACKNOWLEDGMENTS ............................................................................................... xiv CHAPTER 1 INTRODUCTION AND BACKGROUND ..................................................1 1.1 Introduction............................................................................................................... 1 1.2 Background information ........................................................................................... 2 1.3 Research objectives................................................................................................... 7 1.4 Thesis outline ............................................................................................................ 9 CHAPTER 2 STANDARD METHODOLOGY ...............................................................11 2.1 Description of the standard methodology for core calculations ............................. 11 2.1.1 Cross-section generation and modeling .................................................... 11 2.1.2 Reactor core calculations .......................................................................... 14 2.2 Improvements to the standard methodology........................................................... 16 2.2.1 Introduction............................................................................................... 16 2.2.2 Master and Transient libraries development............................................. 17 2.2.3 Verification of the TLG code.................................................................... 19 2.2.4 Sensitivity studies ..................................................................................... 21 2.2.4.1 Cross-section dependence on fuel temperature......................................... 22 2.2.4.2 Cross-section dependence on moderator density...................................... 26 2.2.4.3 Cross-section dependence on boron concentration................................... 28 2.2.4.4 Conclusions of the sensitivity study ......................................................... 31 2.2.4.5 Application of the extended methodology................................................ 31 2.3 Concluding remarks ................................................................................................ 34 CHAPTER 3 IMPROVEMENTS AND MODIFICATIONS IN NEM ............................35 3.1 Introduction............................................................................................................. 35 3.2 Homogeneous partial current formulation in NEM ................................................ 36 3.3 Implementation of the heterogeneous partial current formulation in NEM ........... 41 3.4 Equivalence procedure for discontinuity factors generation .................................. 44 iv

3.5 Verification ............................................................................................................. 46 3.6 Concluding remarks ................................................................................................ 47 CHAPTER 4 DEVELOPMENT OF THE EMBEDDED SP3 TRANSPORT METHODOLOGY ......................................................................................................49 4.1 Introduction............................................................................................................. 49 4.2 Development of the embedded SP3 methodology .................................................. 51 4.3 The SP3 approximation ........................................................................................... 54 4.4 Development of PSU-FEM diffusion and SP3 code ............................................... 56 4.4.1 Derivation of the multi-group diffusion and SP3 FEM discretization ...... 56 4.4.2 Verification of PSU-FEM ......................................................................... 59 4.5 Application of the embedded SP3 methodology ..................................................... 64 4.6 Concluding remarks ................................................................................................ 68 CHAPTER 5 DEVELOPMENT OF THE EMBEDDED LATTICE TRANSPORT METHODOLOGY ..............................................................................70 5.1 Introduction............................................................................................................. 70 5.2 Development of the embedded lattice transport methodology ............................... 72 5.2.1 Method description ................................................................................... 72 5.2.2 Reconstruction of the global boundary conditions ................................... 76 5.3 Application of the embedded lattice transport methodology.................................. 80 5.3.1 Sensitivity studies using the C3 and C5 problems reference solutions .... 80 5.3.2 PRG-NEM solution to the C3 benchmark problem.................................. 91 5.3.3 PRG-NEM solution to the C5 benchmark problem................................ 109 5.3.4 Sensitivity study on the utilization of the DFs........................................ 122 5.4 Concluding remarks .............................................................................................. 125 CHAPTER 6 CONCLUSIONS AND FUTURE WORK................................................128 6.1 Conclusions........................................................................................................... 128 6.2 Recommendations for future work ....................................................................... 131 REFERENCES ................................................................................................................133 APPENDIX A. DESCRIPTION OF C3 AND C5 PROBLEMS....................................138 APPENDIX B. C3 AND C5 PIN-POWERS AND COMPARISONS ...........................142

v

LIST OF ABBREVIATIONS

1-,2- or 3-D ADF BCs BEM

One-, Two- or Three-Dimensional Assembly Discontinuity Factor Boundary Conditions Boundary Element Method

BWR CEA CPM DF FA FEM HFP HZP LWR ML MOX NEM NFI N-K NPA NPP NRC ORNL PSU PWR RDFMG SP3

Boiling Water Reactor Commissariat à l’Energie Atomique Collision Probability Method Discontinuity Factor Fuel Assembly Finite Element Method Hot Full Power Hot Zero Power Light Water Reactor Master Library Mixed-Oxide Nodal Expansion Method Nuclear Fuel Industries Neutron-Kinetics Nodes Per Assembly Nuclear Power Plant Nuclear Regulatory Commission Oak Ridge National Laboratory The Pennsylvania State University Pressurized Water Reactor Reactor Dynamics and Fuel Management Group

TL TLG

Simplified P3 Transient Library Transient Library Generator vi

T-H UOX V1000CT

Thermal-Hydraulic Uranium-Oxide VVER-1000 Coolant Transient Benchmark

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LIST OF FIGURES

Figure 2.1.1: Axial coupling scheme .............................................................................. 15 Figure 2.1.2: Radial coupling scheme............................................................................. 16 Figure 2.2.1: Σa1 dependence on fuel temperature.......................................................... 24 Figure 2.2.2: Σa2 dependence on fuel temperature.......................................................... 25 Figure 2.2.3: Σa1 dependence on moderator density ....................................................... 27 Figure 2.2.4: Σa2 dependence on moderator density ....................................................... 28 Figure 2.2.5: Σa1 dependence on boron concentration .................................................... 30 Figure 2.2.6: Σa2 dependence on boron concentration .................................................... 30 Figure 2.2.7: Comparison of VVER-440 normalized axial power distribution.............. 32 Figure 2.2.8: Comparison of VVER-440 normalized radial power distribution ............ 33 Figure 4.2.1: FEM-SP3-NEM coupling algorithm.......................................................... 52 Figure 4.4.1: Geometry of the 1-D test problem used for verification of PSUFEM (by Capilla) .......................................................................................... 60 Figure 4.4.2: Geometry of the test problem 2 used for verification of PSU-NEM (by Brantley) ................................................................................................. 61 Figure 4.4.3: Spatial flux distribution obtained with diffusion approximation .............. 62 Figure 4.4.4: Spatial flux distribution obtained with SP3 approximation ....................... 62 Figure 5.2.1: PARAGON-NEM coupling algorithm...................................................... 74 Figure 5.2.2: FA-1/FA-2 interface .................................................................................. 76 Figure 5.3.1: South side albedo energy and spatial dependence (C3, NW-UOX).......... 88 Figure 5.3.2: North side albedo energy and spatial dependence (C3, SW-MOX).......... 88 Figure 5.3.3: South side albedo energy and spatial dependence (C5, NE-MOX) .......... 89 Figure 5.3.4: North side albedo energy and spatial dependence (C5, SE-UOX)............ 90 Figure 5.3.5: East side albedo energy and spatial dependence (C5, NE-MOX)............. 90 viii

Figure 5.3.6: East side albedo energy and spatial dependence (C5, SE-UOX) .............. 91 Figure 5.3.7: Results after iteration # 2........................................................................... 95 Figure 5.3.8: Results after iteration # 3........................................................................... 95 Figure 5.3.9: NEM albedo (17x17 NPA) versus reference (SW-MOX South) ............. 98 Figure 5.3.10: Albedo for the North side of the SW-MOX assembly .......................... 100 Figure 5.3.11: Convergence through the iterations....................................................... 104 Figure 5.3.12: keff through the iterations....................................................................... 105 Figure 5.3.13: Convergence through the iterations for Case 4 (4-group)..................... 107 Figure 5.3.14: Convergence through the iterations for Case 4 (7-group)..................... 109 Figure 5.3.15: NEM 17x17 albedo versus reference albedo (NW-UOX South) .......... 111 Figure 5.3.16: NEM 17x17 albedo versus reference albedo (NE-MOX South)........... 111 Figure 5.3.17: NEM 17x17 albedo versus reference albedo (NE-MOX East) ............. 111 Figure 5.3.18: NEM 17x17 albedo versus reference albedo (SE-UOX East) .............. 112 Figure 5.3.19: keff through the iterations....................................................................... 113 Figure 5.3.20: Convergence through the iterations....................................................... 113 Figure 5.3.21: keff through the iterations....................................................................... 118 Figure 5.3.22: Convergence through the iterations....................................................... 118 Figure 5.3.23: Albedo through the iterations (South side of NW-UOX)...................... 119 Figure 5.3.24: Albedo through the iterations (South side of NE-MOX) ...................... 120 Figure 5.3.25: Albedo through the iterations (East side of NE-MOX)......................... 120 Figure 5.3.26: Albedo through the iterations (East side of SE-UOX) .......................... 120 Figure A.1: C3 core configuration ................................................................................ 139 Figure A.2: C5 core configuration ................................................................................ 139 Figure A.3: Fuel pin compositions and numbering scheme ......................................... 140

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LIST OF TABLES

Table 2.2.1: Cases definition .......................................................................................... 19 Table 2.2.2: TLG and HELIOS generated cross-sections differences for Case #1 ........ 20 Table 2.2.3: TLG and HELIOS generated cross-sections differences for Case #2 ........ 20 Table 2.2.4: TLG and HELIOS generated cross-sections differences for Case #3 ........ 21 Table 2.2.5: TLG and HELIOS generated cross-sections differences for Case #4 ........ 21 Table 2.2.6: Σa1 and Σa2 interpolation error (fuel temperature)....................................... 24 Table 2.2.7: Σa1 and Σa2 interpolation ( Tfuel ) error (fuel temperature) ...................... 25 Table 2.2.8: Σa1 and Σa2 interpolation error (moderator density).................................... 27 Table 2.2.9: Σa1 and Σa2 interpolation error (boron concentration)................................. 29 Table 2.2.10: Maximum interpolation errors, %............................................................. 31 Table 3.5.1: keff and normalized FA power comparisons (C3)....................................... 47 Table 3.5.2: keff and normalized FA power comparisons (C5)....................................... 47 Table 4.4.1: Comparison of FEM results to the analytical solution and the S96 reference........................................................................................................ 60 Table 4.4.2: Comparison of FEM results to the analytical solution and the S16 reference........................................................................................................ 61 Table 4.4.3: Comparison of PSU-FEM results to NEM results for C3 MOX benchmark test case ...................................................................................... 63 Table 4.5.1: Reference two-group pin-by-pin PSU-FEM SP3 solution to the C3 benchmark problem ...................................................................................... 65 Table 4.5.2: Reference seven-group pin-by-pin PSU-FEM SP3 solution to the C3 benchmark problem................................................................................. 65 Table 4.5.3: Comparison of FEM-NEM results to reference (Case 1, iteration 1)......... 66 Table 4.5.4: Comparison of FEM-NEM results to reference (Case 1, iteration 5)......... 66 x

Table 4.5.5: Comparison of FEM-NEM results to reference (Case 2, iteration 1)......... 67 Table 4.5.6: Comparison of FEM-NEM results to reference (Case 2, iteration 4)......... 67 Table 4.5.7: Run time statistics for PSU-FEM execution............................................... 68 Table 5.3.1: Reference PARAGON results for C3 test problem .................................... 81 Table 5.3.2: Reference PARAGON results for C5 test problem .................................... 81 Table 5.3.3: Case 1 pin-power statistics ......................................................................... 83 Table 5.3.4: Case 2 pin-power statistics ......................................................................... 84 Table 5.3.5: Case 3 pin-power statistics ......................................................................... 85 Table 5.3.6: Case 4 pin-power statistics ......................................................................... 85 Table 5.3.7: Standard methodology results compared to reference................................ 92 Table 5.3.8: Embedded lattice results compared to reference (loc-0) ............................ 92 Table 5.3.9: Embedded lattice pin-power statistics ........................................................ 93 Table 5.3.10: keff and normalized FA power comparisons (2-group) ........................... 101 Table 5.3.11: Pin-power statistics (2-group)................................................................. 102 Table 5.3.12: keff and normalized FA power comparisons (Case 5)............................. 103 Table 5.3.13: Pin-power statistics (Case 5) .................................................................. 103 Table 5.3.14: Four energy group structure.................................................................... 106 Table 5.3.15: keff and normalized FA power comparisons (Case 4-4) ......................... 106 Table 5.3.16: Pin-power statistics (Case 4-4) ............................................................... 107 Table 5.3.17: Seven energy group structure ................................................................. 108 Table 5.3.18: keff and normalized FA power comparisons (Case 4-7) ......................... 108 Table 5.3.19: Pin-power statistics (Case 4-7) ............................................................... 108 Table 5.3.20: keff and normalized FA power comparisons ........................................... 113 Table 5.3.21: Pin-power statistics................................................................................. 114 Table 5.3.22: keff and normalized FA power comparisons ........................................... 115 Table 5.3.23: Pin-power statistics................................................................................. 116 Table 5.3.24: Embedded vs reference albedo (South side of NW-UOX)..................... 121 Table 5.3.25: Embedded vs reference albedo (South side of NE-MOX) ..................... 121 Table 5.3.26: Embedded vs reference albedo (East side of NE-MOX)........................ 121 Table 5.3.27: Embedded vs reference albedo (East side of SE-UOX) ......................... 121 Table 5.3.28: keff and normalized FA power comparisons ........................................... 123 xi

Table 5.3.29: Pin-power statistics................................................................................. 123 Table 5.3.30: Eigenvalues comparisons (Case 1) ......................................................... 124 Table 5.3.31: Eigenvalues comparisons (Case 2) ......................................................... 124 Table 5.3.32: Eigenvalues comparisons (Case 3) ......................................................... 124 Table A.1: Fuel cell dimensions ................................................................................... 141 Table A.2: Guide-Tube cell dimensions ....................................................................... 141 Table A.3: Isotopic Number Densities for each medium ............................................. 141 Table A.4: Isotopic Number Densities for the moderator and the clad ........................ 141 Table B.1: C3 NW-UOX assembly reference normalized pin-powers......................... 142 Table B.2: C3 NE-MOX assembly reference normalized pin-powers ......................... 142 Table B.3: C3 SW-MOX assembly reference normalized pin-powers......................... 143 Table B.4: C3 SE-UOX assembly reference normalized pin-powers........................... 143 Table B.5: C5 NW-UOX assembly reference normalized pin-powers......................... 144 Table B.6: C5 NE-MOX assembly reference normalized pin-powers ......................... 144 Table B.7: C5 SW-MOX assembly reference normalized pin-powers......................... 145 Table B.8: C5 SE-UOX assembly reference normalized pin-powers........................... 145 Table B.9: C3 NW-UOX pin-power differences multiplied by 100 (Case 1) .............. 146 Table B.10: C3 NE-MOX pin-power differences multiplied by 100 (Case 1)............. 146 Table B.11: C3 SW-MOX pin-power differences multiplied by 100 (Case 1) ............ 146 Table B.12: C3 SE-UOX pin-power differences multiplied by 100 (Case 1) .............. 147 Table B.13: C5 NW-UOX pin-power differences multiplied by 100 (Case 2) ............ 147 Table B.14: C5 NE-MOX pin-power differences multiplied by 100 (Case 2)............. 147 Table B.15: C5 SW-MOX pin-power differences multiplied by 100 (Case 2) ............ 148 Table B.16: C5 SE-UOX pin-power differences multiplied by 100 (Case 2) .............. 148 Table B.17: C3 NW-UOX pin-power differences multiplied by 100 (Case 3) ............ 148 Table B.18: C3 NE-MOX pin-power differences multiplied by 100 (Case 3)............. 149 Table B.19: C3 SW-MOX pin-power differences multiplied by 100 (Case 3) ............ 149 Table B.20: C3 SE-UOX pin-power differences multiplied by 100 (Case 3) .............. 149 Table B.21: C5 NW-UOX pin-power differences multiplied by 100 (Case 4) ............ 150 Table B.22: C5 NE-MOX pin-power differences multiplied by 100 (Case 4)............. 150 Table B.23: C5 SW-MOX pin-power differences multiplied by 100 (Case 4) ............ 150 xii

Table B.24: C5 SE-UOX pin-power differences multiplied by 100 (Case 4) .............. 151

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ACKNOWLEDGMENTS

I would like to thank my advisor, Dr. Kostadin Ivanov, for his support, encouragement and assistance during the course of this work.

I would like also to thank Dr. Mohamed Ouisloumen, Dr. Erwin Müller, and Dr. Ludmil Zikatanov for the guidance and suggestions they have given me during my thesis research.

Further, I would like to thank the committee members Dr. Yousry Azmy, Dr. John Mahaffy, Dr. Mohamed Ouisloumen, and Dr. Victor Nistor for reading and making additional suggestions to improve this thesis.

I also wish to thank the Westinghouse Electric Company, the U.S. Department of Energy and the Pennsylvania State University for providing funding during the course of my graduate studies.

Finally, I wish to thank my son Damian and my wife Nadejda, and also my parents and the rest of my family for their lasting love, support, and encouragement.

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CHAPTER 1 INTRODUCTION AND BACKGROUND

1.1 Introduction

Nowadays, reactor core analyses and design are performed using nodal (coarse-mesh) two-group diffusion methods. These methods are based on precomputed assembly homogenized cross-sections and assembly discontinuity factors obtained by single assembly calculation with reflective boundary conditions (infinite lattice). These methods are very efficient and accurate when applied to the current Light Water Reactor (LWR) cores. With the progress in the Nuclear Engineering field, new generations of Nuclear Power Plants are considered with more complicated rector core designs, such as cores loaded partially with mixed-oxide fuel, high burn-up loadings, and cores with advanced fuel assembly and control rod designs. Such heterogeneous cores have much more pronounced leakage and spatial thermal flux gradients between the unlike assemblies, which introduce challenges to the current methods for core calculation. First, the use of precomputed fuel-assembly homogenized cross-section could lead to significant errors in the coarse-mesh solution. On the other hand, the application of the two-group diffusion theory for such heterogeneous cores could also be source for large errors because of the fundamental approximations of the diffusion theory itself. 1

New core calculation methodologies need to be developed to improve the accuracy of the current methodology for core calculation, while preserving the efficiency. These new core calculation methodologies should be based on fine-mesh (pin-by-pin), higher-order transport theory, multi-group methods to address the difficulties the diffusion approximation encounters.

In this thesis, an advanced core calculation methodology is proposed, based on embedded transport calculations. This advanced methodology includes on-line cross-section homogenization, which accounts for the environment of each FA in the core; consistent generation of NEM-specific side-dependent assembly discontinuity factors; and utilization of an efficient and accurate local calculation, based on FEM-SP3 or CPM.

1.2 Background information

The current (standard) methodology for performing reactor calculations is based on a two step off-line process. First, a lattice physics calculation is performed on a fuel assembly basis using reflective boundary conditions (infinite lattice calculation). The resultant neutron spectrum is used to collapse and homogenize the cross-sections to few-group cross-sections over the entire FA domain. Such a lattice physics calculation has to be performed for each FA type in the core. In order to cover all possible core conditions numerous sets of cross-sections for each FA type at different thermal-hydraulic and control feedback parameters (such as fuel temperature, moderator temperature, moderator density, etc) and different burnup conditions must be generated. The generation of such cross-section sets is achieved by performing numerous so called “base” and “branch” 2

lattice calculations. At the end of this step all cross-sections must be tabulated in a huge parameterized cross-section library. The second step in the process is to utilize this crosssection library by using different methods in order to analyze the reactor core. Traditionally, in the second step, the two-group diffusion theory is used for performing reactor core calculations. This approach was proven to be accurate for the current Pressurized Water Reactor (PWR) and Boiling Water Reactor (BWR) plants but its accuracy is questionable when applied to more heterogeneous cores, such as cores partially loaded with MOX fuel [Downar, 2000].

The errors in the calculation of cores partially loaded with MOX fuel, high burn-up loadings, and cores with advanced fuel assembly and control rod designs can be attributed to the following simplifications used in the diffusion theory: -

Transport approximation to the linear Boltzmann equation. The presence of different materials in the core, such as MOX and UOX, leads to significant neutron streaming. Such significant transport effects are difficult to be modeled accurately with the diffusion approximation;

-

Spatial homogenization and group collapsing. In the cross-section generation process, infinite lattice physics calculations are performed, which do not account for the actual environment of the assembly in the core. In the presence of MOX and UOX fuel assemblies, a strong thermal flux gradient exists at the interface, which may lead to significant errors in the homogenized group constants generated in the conventional manner with the infinite lattice spectrum. The group collapsing problem is related to the spatial 3

homogenization problem because the group constants are collapsed to few groups using the same infinite lattice energy spectra. MOX cross-sections are collapsed using infinite lattice MOX spectra and UOX cross-sections are collapsed using infinite lattice UOX spectra. However, in the reactor core the energy spectrum is neither MOX or UOX, which can lead to significant errors in the group constants generated by the single assembly calculations ; -

Spatial discretization. The strong thermal flux gradients at the interface of the unlike assemblies can cause difficulties to the current nodal methods to model this phenomena in an accurate manner;

-

Approximations associated with the conventional cross-section representation. This problem is not only associated with the calculation of the mixed UOX and MOX core loadings but also with the current core loadings. The current cross-section representation uses polynomial fitting or tables and cannot represent all core state conditions exactly because of the finite number of state points.

The best way to address all of the above mentioned shortcomings of the current core calculation methodology is to solve the exact Boltzmann equation in many energy groups in a heterogeneous domain using exact cross-sections. However, such a problem is very computationally demanding and is unfeasible to be accomplished with the present day computers.

Many researchers dedicated their work on solving the above mentioned problems. Some of them concentrated their research on solving full core problems in 2-D or 3-D using 4

integral transport methods [Cho, 2000], [Smith, 2002], [Joo, 2002], while others used differential transport methods [Langenbuch, 2005], [Kriangchaiporn, 2005]. Such calculations can be useful for obtaining reference solutions but they are not practical for transient and depletion calculations because of the limited computer power.

One of the most innovative methods for performing core calculations is the Simplified PN (SPN) approximation to the neutron transport equation. SPN was first proposed by Gelbard in early 1960s. In the beginning, this method was rarely used due to the weak theoretical basis (Gelbard simply replaced the second derivatives in the one-dimensional PN equation with general three-dimensional Laplacian operators). However, several researchers reported significant improvements in the results when SPN is used compared to the diffusion approximation [Lemanska, 1981], [Smith, 1986a]. Larsen provided the first theoretical basis for the SPN equations in multidimensional applications [Larsen, 1993], [Larsen, 1996]. Application of the SPN approximation followed with increased confidence as the theoretical basis of the method became more firmly established [Brantley, 2000].

The SP3 method is currently used by different companies and universities, such as Purdue University [Lee, 2001 and 2004], Studsvik [Bahadir, 2005], NFI Japan [Tatsumi, 2002], and CEA, Saclay [Ragusa, 2003]. The method was implemented by Purdue University into the U.S. NRC neutronics code PARCS, by Studsvik into SIMULATE-4, by NFI into SCOPE2 code, and by CEA in CRONOS2 code. The SP3 method was also implemented in the Pennsylvania State University (PSU) core simulator NEM by Dianna Hahn [Hahn, 2003]. Hahn implemented the SP3 transport model within the framework of NEM for 3-D 5

pin-by-in multi-group calculations. To improve further the computational efficiency and performance of the transport model to be applied for practical transient analysis, Hahn coupled the local pin-by-pin A2-BEM (Boundary Element Method) solution with the global core NEM diffusion (assembly–by-assembly) calculation and entitled this scheme as Hybrid NEM/BEM calculations. The Hybrid NEM/BEM uses the incoming partial currents from the NEM global diffusion calculation and separate pin-wise cross-sections in the A2-BEM calculation. The A2-BEM calculates the partial currents that are then used to update the global NEM incoming partial currents. Finally, NEM uses the updated partial currents and its own assembly-wise cross-sections to perform new global calculation. Hahn did not provide sufficient verification of both NEM-SP3 and Hybrid NEM/BEM in her work. Consequently, attempt to verify the SP3 transient option in NEM was made by Mathieu Boydens [Boydens, 2004] during his research visit at PSU.

Today’s reactor analysts consider the homogenization error as the most significant shortcoming of the standard methodology for core calculations. To homogenize the crosssections, infinite lattice calculations (single FA with reflective boundary conditions) are performed because the exact boundary conditions are unknown. If the exact boundary conditions were used, the homogenized constants will be different than the ones obtained under the reflective boundary condition assumption. In order to avoid this assumption several remedies were proposed. Mondot and Sanchez [Mondot, 2003] proposed an iterative homogenization technique that preserves assembly core exchanges. The methodology is based on embedded, fixed source and eigenvalue, transport calculations that preserves macro-group assembly exchanges in the core. Another approach used by Clarno and Adams [Clarno, 2005] focused on accurate approximation of the effects that 6

neighboring assemblies have on the few-group cross-sections, assembly discontinuity factors, and form functions by tabulating these effects. Rahnema and Nichita [Rahnema, 1997] proposed another improvement of the homogenization process by introducing parameterization of the leakage effects on the homogenized parameters. Another method proposed by Smith [Smith, 1994] and used by Palmtag [Palmtag, 1997] is the crosssection re-homogenization. In this iterative method the homogeneous cross-sections are recalculated using the “actual” flux shape in the reactor. This flux shape is obtained by the assumption that the heterogeneous flux shape is equal to the superposition of the flux shape from the diffusion calculation and the form functions from the heterogeneous single assembly calculation.

1.3 Research objectives

The main objective of this thesis is the development, implementation, and qualification of an advanced core calculation methodology based on embedded transport calculations. Two different approaches are followed. The first approach is based on embedded SP3 fuel assembly calculation within the framework of the NEM diffusion core calculation. In order to accomplish this step, a new FEM-based code has been developed to enable, performing both diffusion and SP3 calculations in 2-D. The new code is used in the framework of the NEM code in order to perform embedded pin-by-pin SP3 calculations on FA basis. The SP3 approximation was chosen due to its accuracy and efficiency in terms of computing time. This approach is used to evaluate the superiority of the SP3 approximation compared to the diffusion approximation of the Boltzmann equation. In contrast to Hahn’s work, the embedded SP3 calculations are coupled to the NEM global 7

calculation through on-line generation of Discontinuity Factors (DF) [Smith, 1986b] and cross-section homogenization (non-linear application of the equivalence theory) and the albedos are used as boundary conditions for the local SP3 calculation.

In the second approach, the SP3 calculation is replaced with lattice physics transport eigenvalue calculation. This involves embedded FA lattice physics eigenvalue calculation based on collision probability method, used in the PARAGON code, again within the framework of the NEM diffusion core calculation. PARAGON is a lattice physics code in development at The Westinghouse Electric Company [Westinghouse, 2005]. The second methodology is superior to the first one due to the elimination of most of the uncertainties introduced by the off-line approach of cross-section generation. In this step new code has been developed, which allows the coupling of NEM and PARAGON.

The embedded SP3 approach aims to improve the shortcomings of the diffusion approximation to the Boltzmann equation by using embedded simplified P3 calculations on pin-by-pin basis and taking into account the assembly environment on the level of already collapsed few-group pin-wise cross-sections. It is also used to demonstrate the feasibility of the embedded approach on a simple framework. The embedded lattice approach is more sophisticated and addresses all of the above mentioned shortcomings of the currently used methodology for reactor core analysis. This method is used to generate the few-group cross-sections using embedded lattice calculations avoiding the standard two step off-line methodology. The work presented here shows that the new embedded lattice methodology improves the current two step off-line methodology by: •

avoiding all the approximations associated with the conventional cross8

section parameterization; •

avoiding the preparation of the huge cross-section library;

•

accounting for the environment of each FA in the core in the entire crosssection generation process avoiding, by that, the infinite lattice calculations;

•

significantly improves the two-group nodal diffusion theory results and in some cases even reproduces the higher order transport results.

During the course of this research it was realized that the NEM code needs to be modified in order to be able to use this code for the purpose of the embedded transport calculations. The existing discontinuity factor utilization procedure in NEM was implemented based on the homogeneous partial current formulation following the Lawrence’s approach [Lawrence, 1986] of replacing the continuity of the partial currents by introducing discontinuity on the interface boundary conditions. This implementation is not suitable for computing partial currents or albedo boundary conditions that are required in the embedded transport calculation. Computed in this way partial currents or albedos will not be physical and it will introduce challenges for the methods used in the embedded local calculations. A separate chapter of this thesis is devoted to solving this problem in NEM.

1.4 Thesis outline

Chapter 2 presents an overview of the current (standard) two step off-line methodology for performing core calculations. In addition, improvements of the current off-line methodology made by the author are also described. 9

Chapter 3 presents the modifications done in the NEM code required by the proposed embedded calculation methodology. The development of the DF generation procedure, which generates NEM specific side-dependent DFs is also described.

Chapter 4 focuses on the development of the embedded SP3 method. The first part presents, the philosophy behind the embedded SP3 calculations. The second part describes the new diffusion and SP3 FEM based code and provides verification of the code. At the end, the application of this method to the C3 benchmark problem is presented.

Chapter 5 describes the development, implementation, and qualification of the advanced core calculation methodology based on embedded lattice calculations followed by its application to the C3 and C5 benchmark problems.

Chapter 6 summarizes this thesis and its unique contributions and discusses recommendations for future work.

10

CHAPTER 2 STANDARD METHODOLOGY

2.1 Description of the standard methodology for core calculations

This chapter presents the standard two step off-line methodology for performing core calculations. In addition, some improvements of the current off-line methodology used at PSU are also described.

The current standard off-line methodology for performing core calculations is based on two main steps. The first step is cross-section generation and modeling described in subsection 2.1.1. The second step is to employ these cross-sections by using different methods in order to analyze the reactor core. Traditionally, in the second step, two-group diffusion theory is used for performing reactor core calculations. The second step is presented in subsection 2.1.2. 2.1.1

Cross-section generation and modeling

The development of efficient and accurate procedures for generation and modeling of nuclear cross-sections for coupled three-dimensional neutronics/thermal-hydraulic calculations of current and next generation nuclear power plants is becoming an

11

increasingly important issue. Studies performed at PSU [Watson, 2002] and elsewhere over the last several years have revealed the advantages of using multi-dimensional table cross-section representation for accurate modeling of both initial conditions and transients. In the Penn State transient cross-section modeling methodology, the entire range of changes of the major feedback parameters are covered using multi-dimensional table interpolation, thus avoiding simplified polynomial fitting and allowing for extrapolation over the parameter limits. This methodology has been applied in several international benchmarks on coupled 3-D neutron kinetics thermal-hydraulics calculations [Ivanov K., 1999a], [Solis, 2001], and [Ivanov B., 2002].

The reactor core is assembled from fuel assemblies with different enrichment and burnup, i.e. with different compositions. For computational purposes the reactor core is divided in nodes. Typically, in radial direction each numerical node is of the size of the fuel assembly and axially the nodalization depends on the material characteristics in axial direction. Therefore, cross-sections for the composition of each node in the core have to be generated and modeled. The cross-sections are generated by single-assembly lattice physics calculation with reflective boundary conditions (infinite lattice calculation). The resultant infinite-medium neutron spectrum is used to collapse and homogenize the crosssections to few-group cross-sections over the entire FA. Such a lattice physics calculation has to be performed for each composition in the core.

There are two main groups of parameters on which the cross-sections greatly depend: instantaneous parameters and history parameters. The history dependence is a burnup dependence and the history parameters include the exposure, control history, and spectral 12

history. The control history in a PWR is a burnable poison history since the control rod history can be neglected as the control rods are rarely used during the fuel cycle. The spectral history effect is the moderator density effect, which is very important as shown in [Watson, 2002] and must be carefully modeled. This effect is based mainly on the axial moderator density distribution in the core. Different moderator density causes different neutron spectrum. Fuel depleted at different moderator density will have different isotopic content, which will significantly affect the core calculation. The methodology used at PSU incorporates separate depletion calculations for each axial layer in the core nodalization. In this way the methodology takes into account the different neutron spectra during the depletion process. Usually the cross-section library is created for a specific transient, which is a small fraction of the fuel cycle. Therefore, the depletion is done up to the point in the fuel cycle where the transient of interest starts and all the branch-off calculations (lattice physics calculations at different combinations of the conditions) are performed at this point.

Most of the instantaneous parameters are the local thermal-hydraulic parameters. For a PWR the usual instantaneous parameters are the moderator temperature and density, the fuel temperature, and the boron concentration. Boron concentration is to be modeled only if there is change in the concentration during the transient to be analyzed. As mentioned earlier, branch-off calculations are performed with a lattice code for all of the possible combinations of the instantaneous parameters. The cross-sections generated by these branch-offs are then tabulated in tables. Such a modeling of the cross-sections allows taking into account not only the dependence of the cross-sections on the instantaneous parameters, but also the cross-term effects (when two or more parameters change 13

simultaneously). This is actually one of the best features of the current PSU methodology in contrast to the standard polynomial fitting.

In summary, the cross-section library consists of cross-sections for each node in the core known as a cross-section set (cross-sections for given composition). Each cross-section set is generated by separate lattice code depletion calculations at the moderator density corresponding to the location of the node in the core. Once the fuel is depleted to the required burnup, branch-off calculations are performed for all combinations of the instantaneous parameters. At the end, all cross-sections are assembled in a single file named cross-section library. 2.1.2

Reactor core calculations

The current state-of-the-art methodology for performing core calculations is the coupled 3-D neutron kinetic/thermal-hydraulics scheme. At PSU we use the nodal diffusion code NEM [Bandini, 1990], which is coupled to the system code TRAC-PF1 [Schnurr, 1992] in order to be able to perform coupled 3-D neutron kinetics/thermal-hydraulics calculations. NEM is a 3-D multi-group nodal code developed at PSU for modeling both steady-state and transient core conditions [Ivanov K., 1999b]. The code utilizes a transverse integration procedure and is based on the partial current formulation of the nodal balance equations. The TRAC-PF1 code calculates the local thermal-hydraulic parameters that are needed for appropriate extraction of the necessary cross-sections from the cross-section library. Further, the cross-sections are used by NEM to calculate the core power distribution that TRAC uses to evaluate the new local thermal-hydraulic parameters. 14

Important components of the coupled 3-D neutron kinetics/thermal-hydraulics calculation are the axial and the radial coupling schemes. These coupling schemes define the parameter exchanges between the neutron kinetics (N-K) code and the thermal hydraulic (T-H) code. Figure 2.1.1 shows an example of an axial coupling scheme used in the V1000CT benchmark [Ivanov B., 2006]. Axially, there is a one-to-one correspondence between the T-H model and the neutron kinetics model as seen in the figure. The radial coupling scheme is shown in Figure 2.1.2. This scheme is obtained by combining several neutronics nodes into a single T-H cell. Dashed lines represent T-H cells, while the thick line represents the neutronics nodes that are lumped together into one T-H cell. The number of the neutronics nodes lumped in one T-H cell is not the same, because the area of the T-H cells is different in different rings. Each number shows to which T-H channel the neutronics node is mapped. N-K model

T-H model

Figure 2.1.1: Axial coupling scheme

15

19 20 20 20 20 20 20 20 21 21

14

15

21

15 21

9

15

9 15

15 21

10

16 22 22

16 16

22

10

10 10

16 16

22

5

16 16

22

11

22

11

16 16

17

22

17

24

17

17

24

17

17

17

23

18

17

17

24 18

17

17

17

16

12

11

24 18

18

11

11

11

18

12

11

24 18

12

11

11

10

12

6

24 18

18

12

5 5

18

12

6

24

18

12

6

5

18

12

5

24 18

12

6

6

4

10

7

6

5 4

10

16 16

6

4

10

1

19 13

7 1

1

4 4

7

1

19 13

13

7

1

3

4

10

7

2

3 3

9

15 21

3

9

15

2

19 3

13

7 1

2

3 9

7

2

19 13

13

7

2

3

9

7

2

9

15

21

8

19 13

13

8 8

8

9 15

8

8

19 13

13

8

8

9 15

15

14

8

19 13

14

14 14

14

15 21

4

14

14

14

14

14

20

23 23

23 23

23 23

23

22

Figure 2.1.2: Radial coupling scheme

2.2 Improvements to the standard methodology 2.2.1

Introduction

The improvements to the standard core calculation methodology are mostly in the first step, namely the cross-section generation and modeling procedure. These improvements were developed, implemented, and verified during a project sponsored by the Oak Ridge National Laboratory (ORNL) [Ivanov B., 2004a]. These improvements were later used successfully in a second ORNL-sponsored project in support of the Westinghouse IRIS project [Ivanov B., 2005] and in the second phase of the V1000CT International Benchmark [Kolev, 2006], as well as in a study for improvement of the current RBMK analysis methodology [Parisi, 2006].

16

The PSU cross-section generation methodology described in the previous section performs very well when a single transient have to be analyzed and is suitable for use in benchmark problems.

However, for a real application this procedure is very time

consuming and not practical.

For example, in the V1000CT-1 benchmark problem

[Ivanov B., 2002] the cross-section library consists of 283 unrodded cross-section sets (numerical node without control rod) and 110 rodded cross-section sets (numerical node with control rod inserted). This means that 393 lattice code depletion calculations had to be performed, which are very time consuming. After each depletion step a number of branch-off calculations have to be performed. In the case of V1000CT-1 benchmark, only five moderator densities and four fuel temperatures were used as instantaneous parameters due to the nature of the transient to be analyzed. Therefore, after each depletion step, another twenty branch-off calculations had to be performed. It is obvious that such a process could be very time consuming and the outcome of it is a cross-section library that can be used only for the given transient. One can imagine how many lattice code calculations could be involved if more transients have to be analyzed, which is usually the case at the current NPPs. 2.2.2

Master and Transient libraries development

Until recently, at the PSU we have been developing cross-section libraries for each material composition contained in the core and for given point of the fuel cycle, i.e. for specific burn-up [Watson, 2002]. For the first ORNL sponsored project “Methodology for nuclear cross-sections generation and implementation” [Ivanov B., 2004a] the existing methodology was extended for multi-transient cross-section generation. The 17

extended methodology incorporates two types of libraries. The first one named Master Library (ML) is generated once by lattice calculations and includes a full set of crosssections for each fuel type and possible core state for different points of the fuel cycle. Each state is defined by exposure, spectral history, and instantaneous parameters. All cross-sections are tabulated in multi-dimensional tables similar to the previously used cross-section libraries. The difference is that these multi-dimensional tables are arranged in cross-section sets based also on exposure and spectral history. Therefore, the Master Library consists of cross-sections for the entire fuel cycle, not only for a given transient as in the previously used libraries. A new code has been developed – Transient Library Generator (TLG), which reads the ML and generates a new cross-section library called Transient Library (TL), which is the same as the previously used cross-section libraries. Therefore, the new methodology does not require changes in the current coupled codes. The TLG code reads the cross-sections from the ML and based on the user input creates a TL for specific transient at a certain point of the cycle. TLG linearly interpolates the burnup and the moderator history in the master library and then saves the transient library in the format used in numerous international benchmarks for coupled code calculations [Ivanov K., 1999a], [Solis, 2001], and [Ivanov B., 2002]. In this methodology, only the generation of the Master Library is time consuming task, while the generation of the Transient Library by the TLG code is done in seconds once the user data is supplied. Consequently, using this extended methodology numerous Transient Libraries could be generated in very efficient manner. The only remaining question is how accurate this methodology is. The verification of the methodology is presented in the next subsection.

18

2.2.3

Verification of the TLG code

The TLG code is verified against HELIOS [Studsvik, 2000] calculations. HELIOS is a two-dimensional (2-D) current-coupling collision-probability code that is typically used for lattice burnup calculations with a 45 or 190 energy group nuclear data library based on ENDF-B/VI. HELIOS code was also used for generation of the Master Library.

Table 2.2.1 shows four cases that were calculated with the HELIOS package in order to obtain the exact cross-sections. The cross-sections generated by HELIOS (reference cross-sections) were compared to the cross-sections in the Transient Library obtained by the Transient Library Generator code. These cases were chosen in order to compare the cross-sections for four different points of the fuel cycle. Moderator temperature shown in Table 2.2.1 is the spectral history used for the depletion calculations with HELIOS.

Table 2.2.1: Cases definition Case #

Burnup

Tmod

1 2 3 4

2325 7200 27500 4900

565.00 569.00 582.00 569.00

For each case the cross-sections in the following branch-off states: Branch1, Branch2, Branch3, and Branch4 (different Tfuel, Tmod, and boron concentration) were compared. The following four tables (Tables 2.2.2 to 2.2.5) show the comparison for the four different cases for each branch-off state. The tables show that the relative error for Cases 1, 3, and 4 is less then 0.1%. However, the relative error for Case 2 reaches almost 0.4%. The increase in the relative error for this case could be explained with the size of the 19

depletion steps in the Master Library. There are five depletion steps at the beginning from 0 to 5000 MWd/tU and from 5000 to 30000 MWd/tU there are another five steps. Case 2 was calculated at a burnup 7200 MWd/tU, which falls into the first large depletion step. Most probably at this exposure the burnup dependence of the cross-sections is still nonlinear and smaller burnup steps must be used for generating the master library.

Table 2.2.2: TLG and HELIOS generated cross-sections differences for Case #1

D1 D2 ∑a1 ∑a2 ∑s ∑f1 ∑f2 ν∑f1 ν∑f2

Branch1 Relative error, %




-0.003 -0.014 -0.019 0.053 -0.003 0.002 0.052 0.007 0.068

0.001 -0.006 -0.016 0.061 -0.004 0.006 0.060 0.009 0.076

0.003 -0.005 -0.016 0.072 0.005 0.008 0.069 0.010 0.086

0.000 -0.011 -0.017 0.080 0.003 0.004 0.062 0.009 0.080


D1 D2 ∑a1 ∑a2 ∑s ∑f1 ∑f2 ν∑f1 ν∑f2





0.003 -0.083 0.090 0.275 -0.052 -0.086 0.242 -0.057 0.335

0.004 -0.077 0.093 0.284 -0.057 -0.086 0.248 -0.056 0.344

0.007 -0.086 0.098 0.337 -0.077 -0.094 0.290 -0.057 0.385

0.009 -0.088 0.101 0.381 -0.058 -0.087 0.286 -0.056 0.385

20


D1 D2 ∑a1 ∑a2 ∑s ∑f1 ∑f2 ν∑f1 ν∑f2





-0.001 -0.030 0.072 -0.003 -0.027 -0.114 -0.065 -0.100 -0.042

0.003 -0.030 0.068 -0.004 -0.031 -0.113 -0.062 -0.099 -0.036

0.004 -0.025 0.069 0.012 -0.022 -0.116 -0.050 -0.099 -0.025

0.003 -0.030 0.074 0.007 -0.036 -0.116 -0.054 -0.099 -0.031


D1 D2 ∑a1 ∑a2 ∑s ∑f1 ∑f2 ν∑f1 ν∑f2

2.2.4





-0.005 -0.006 -0.001 0.005 0.000 -0.002 0.008 -0.002 0.013

-0.001 0.000 0.002 0.011 -0.004 0.000 0.011 0.001 0.012

0.001 -0.004 0.002 0.012 -0.005 -0.001 0.010 0.001 0.013

0.001 -0.004 0.002 0.014 -0.002 0.000 0.007 0.001 0.011

Sensitivity studies

The generation of Master Library by lattice calculations could be very CPU demanding and time consuming task. The following issues were identified as possible reasons for the increased CPU time: •

The number of branch-off calculations in some cases could be more than 200 21

•

Extended range of the burnup

•

Small burnup steps – at the beginning less than or approximately 1000 MWd/t intervals have to be used and after 6000 MWd/t the steps are 2000 or 3000 MWd/t

While nothing can be done for the second and third points, as was shown in the TLG verification, the first point can be reviewed and eventually lead to significant decrease of the computation time. The number of the branch-offs could be decreased by using fewer fuel temperatures and boron concentrations, which will lead to significant decrease of CPU time. The basis for such a decision is presented in this subsection.

The goal of this study is to show that the interpolation error is acceptable when decreased number of independent variables’ reference points (grid points) were used to cover the expected range of the core conditions. The data used in this work was taken from HELIOS-1.7 calculation of 2.10 w/o

235

U enriched composition without burnable

absorbers and 2.10 w/o 235U enriched composition with burnable absorber using 45 group cross-section library adjusted for

238

U resonances. The differences in the interpolation

error for the two different cases – with burnable absorber and without burnable absorber are very small. Therefore, selected results only from the calculation of the composition with burnable absorber are shown. Further, detailed results only for the absorption cross-section as a representative will be shown while at the end the overall results will be summarized. The following three subsections present the cross-section dependence on the independent instantaneous variables and their associated interpolation error. 2.2.4.1 Cross-section dependence on fuel temperature

The calculation of the cross-section dependence on fuel temperature was performed at the 22

following core conditions: •

Boron concentration, ppm – 0

•

Moderator density, kg/m3 – 690

•

Moderator temperature, K – 589.47

•

Fuel temperature, K – 300, 550, 850, 1000, 1500, 2000

The results for the absorption cross-section are shown in Tables 2.2.6 and 2.2.7. The results are presented for three different burnups – 0 MWd/t, 6000 MWd/t, and 50000 MWd/t in order to see if there is change in the behavior of the cross-section dependence on fuel temperature with the change in the burnup. Table 2.2.6 shows the relative difference of the exact cross-section value calculated by HELIOS compared to the one obtained by interpolation using the neighboring points. Table 2.2.7 shows the same results except that they were obtained by interpolating on the square root of the fuel temperature instead of the fuel temperature itself. These comparisons show that using interpolation on the square root of the fuel temperature gives better results compared to the one using fuel temperature since the square root dependence is much more linear as compared to the direct dependence.

Absorption cross-section dependence on fuel temperature is shown also as plots on Fig. 2.2.1-2. Presented results show that the maximum difference from the square root of the fuel temperature interpolation for the diffusion coefficients is 0.017 %, for the absorption cross-section is 0.099 %, for the fission cross-sections is 0.108 %, and for the downscattering cross-section is 0.032 %. Please note that these errors represent a case in which the intervals for obtaining the cross-section by interpolation are twice as big as the

23

original ones. For example to obtain a cross-section at 550 K, the cross-sections at 300 K and 850 K were used. This means that these errors bound the possible errors in case when the number of fuel temperatures is decreased to 3. Table 2.2.6: Σa1 and Σa2 interpolation error (fuel temperature) Σa1 calculated by HELIOS:

Σa1 obtained by interpolation

Interpolation error, %

Tfuel

0 MWd/t

6000 MWd/t

50000 MWd/t

0 MWd/t

6000 MWd/t

50000 MWd/t

0 MWd/t

6000 MWd/t

50000 MWd/t

300.00

9.119E-03

8.916E-03

1.051E-02

-

-

-

-

-

-

550.00

9.319E-03

9.117E-03

1.074E-02

9.297E-03

9.093E-03

1.071E-02

-0.239

-0.263

-0.223

850.00

9.511E-03

9.307E-03

1.096E-02

9.502E-03

9.298E-03

1.095E-02

-0.090

-0.090

-0.082

1000.00

9.594E-03

9.389E-03

1.106E-02

9.587E-03

9.381E-03

1.105E-02

-0.064

-0.083

-0.086

1500.00

9.844E-03

9.629E-03

1.134E-02

9.824E-03

9.613E-03

1.132E-02

-0.196

-0.173

-0.181

2000.00

1.006E-02

9.836E-03

1.158E-02

-

-

-

-

-

-

Σa2 calculated by HELIOS:

Σa2 obtained by interpolation 0 MWd/t

6000 MWd/t

50000 MWd/t


Tfuel

0 MWd/t

6000 MWd/t

50000 MWd/t

0 MWd/t

6000 MWd/t

50000 MWd/t

300.00

8.614E-02

8.283E-02

7.008E-02

-

-

-

-

-

-

550.00

8.585E-02

8.259E-02

6.996E-02

8.589E-02

8.265E-02

7.004E-02

0.052

0.079

0.107

850.00

8.560E-02

8.244E-02

6.999E-02

8.564E-02

8.249E-02

7.003E-02

0.046

0.053

0.054

1000.00

8.553E-02

8.244E-02

7.006E-02

8.557E-02

8.245E-02

7.005E-02

0.046

0.013

-0.019

1500.00

8.548E-02

8.246E-02

7.024E-02

8.549E-02

8.246E-02

7.023E-02

0.016

0.002

-0.010

2000.00

8.545E-02

8.248E-02

7.040E-02

-

-

-

-

-

-

1.200E-02 1.150E-02

Sig A 1, cm -1

1.100E-02 A1 at 0 MWd/t

1.050E-02

A1 at 6000 MWd/t 1.000E-02

A1 at 50000 MWd/t

9.500E-03 9.000E-03 8.500E-03 300

550

800

1050

1300

1550

1800

Fuel temperature, K

Figure 2.2.1: Σa1 dependence on fuel temperature 24

Table 2.2.7: Σa1 and Σa2 interpolation ( Tfuel ) error (fuel temperature) Σa1 calculated by HELIOS:



Tfuel

0 MWd/t

6000 MWd/t

50000 MWd/t

0 MWd/t

6000 MWd/t

50000 MWd/t

0 MWd/t

6000 MWd/t

50000 MWd/t

17.32

9.119E-03

8.916E-03

1.051E-02

-

-

-

-

-

-

23.45

9.319E-03

9.117E-03

1.074E-02

9.322E-03

9.118E-03

1.074E-02

0.028

0.010

0.043

29.15

9.511E-03

9.307E-03

1.096E-02

9.511E-03

9.307E-03

1.096E-02

0.001

0.002

0.009

31.62

9.594E-03

9.389E-03

1.106E-02

9.596E-03

9.390E-03

1.106E-02

0.030

0.010

0.006

38.73

9.844E-03

9.629E-03

1.134E-02

44.72

1.006E-02

9.836E-03

1.158E-02

9.844E-03 -

9.632E-03 -

1.134E-02 -

0.004 -

0.024 -

0.016 -

Σa2 calculated by HELIOS:


6000 MWd/t

50000 MWd/t


Tfuel

0 MWd/t

6000 MWd/t

50000 MWd/t

0 MWd/t

6000 MWd/t

50000 MWd/t

17.32

8.614E-02

8.283E-02

7.008E-02

-

-

-

-

-

-

23.45

8.585E-02

8.259E-02

6.996E-02

8.586E-02

8.263E-02

7.003E-02

0.012

0.049

0.099

29.15

8.560E-02

8.244E-02

6.999E-02

8.563E-02

8.248E-02

7.003E-02

0.034

0.047

0.058

31.62

8.553E-02

8.244E-02

7.006E-02

8.556E-02

8.245E-02

7.005E-02

0.042

0.014

-0.009

38.73

8.548E-02

8.246E-02

7.024E-02

44.72

8.545E-02

8.248E-02

7.040E-02

8.549E-02 -

8.246E-02 -

7.025E-02 -

0.012 -

0.004 -

0.011 -

9.000E-02

Sig A 2, cm -1

8.500E-02

8.000E-02

A2 at 0 MW d/t A2 at 6000 MW d/t A2 at 50000 MW d/t

7.500E-02

7.000E-02

6.500E-02 300

550

800

1050

1300

1550

1800

Fuel temperature, K

Figure 2.2.2: Σa2 dependence on fuel temperature

25

2.2.4.2 Cross-section dependence on moderator density

The calculation of the cross-section dependence on moderator density was performed at the following core conditions: •

Boron concentration, ppm – 0

•

Fuel temperature, K – 850

•

Moderator temperature, K – 618.05, 616.83, 589.47, 561.29, 531.17, 449.66

•

Moderator density, kg/m3 – 500, 600, 690, 750, 800,900

The results for the absorption cross-section are shown in Table 2.2.8. The results are given again for three different burnups – 0 MWd/t, 6000 MWd/t, and 50000 MWd/t in order to see if there is change in the behavior of the cross-section dependence on moderator density with the change in burnup. This table shows the relative difference of the cross-section calculated by HELIOS compared to the one obtained by interpolation by using the neighboring points. The results are presented also as plots on Fig. 2.2.3-4.

Presented results show that the maximum difference from the interpolation for the diffusion coefficients is 1.462 %, for the absorption cross-section is 0.607 %, for the fission cross-sections is 0.709 %, and for the down-scattering cross-section is less than 0.233 %. Again the errors represent a case in which the intervals for obtaining the crosssection by interpolation are twice as big as the original ones. This means that these errors bound the possible errors in the case when the number of moderator densities’ reference points is decreased.

26

Table 2.2.8: Σa1 and Σa2 interpolation error (moderator density) Σa1 calculated by HELIOS



Mod. Density

0 MWd/t

6000 MWd/t

50000 MWd/t

0 MWd/t

6000 MWd/t

50000 MWd/t

0 MWd/t

6000 MWd/t

50000 MWd/t

500.00

9.072E-03

8.869E-03

1.033E-02

-

-

-

-

-

-

600.00

9.329E-03

9.131E-03

1.069E-02

9.303E-03

9.099E-03

1.066E-02

-0.283

-0.345

-0.265

690.00

9.511E-03

9.307E-03

1.096E-02

9.498E-03

9.302E-03

1.095E-02

-0.135

-0.047

-0.097

750.00

9.610E-03

9.417E-03

1.112E-02

9.608E-03

9.411E-03

1.111E-02

-0.023

-0.059

-0.058

800.00

9.689E-03

9.498E-03

1.125E-02

900.00

9.805E-03

9.630E-03

1.146E-02

9.675E-03 -

9.488E-03 -

1.123E-02 -

-0.143 -

-0.111 -

-0.101 -

Σa2 calculated by HELIOS


6000 MWd/t


Mod. Density

0 MWd/t

6000 MWd/t

50000 MWd/t

50000 MWd/t

0 MWd/t

6000 MWd/t

50000 MWd/t

500.00

8.501E-02

8.130E-02

6.837E-02

-

-

-

-

-

-

600.00

8.499E-02

8.169E-02

6.921E-02

8.532E-02

8.190E-02

6.922E-02

0.392

0.262

0.026

690.00

8.560E-02

8.244E-02

6.999E-02

8.586E-02

8.263E-02

7.005E-02

0.314

0.223

0.088

750.00

8.645E-02

8.325E-02

7.062E-02

8.664E-02

8.339E-02

7.067E-02

0.217

0.163

0.078

800.00

8.751E-02

8.418E-02

7.124E-02

900.00

9.122E-02

8.725E-02

7.316E-02

8.804E-02 -

8.459E-02 -

7.147E-02 -

0.607 -

0.485 -

0.316 -

1.150E-02

Sig A 1, cm -1

1.100E-02 1.050E-02 A1 at 0 MWd/t 1.000E-02

A1 at 6000 MWd/t A1 at 50000 MWd/t

9.500E-03 9.000E-03 8.500E-03 500

600

700

800

900

Moderator density, kg/m3

Figure 2.2.3: Σa1 dependence on moderator density

27

9.500E-02

Sig A 2, cm -1

9.000E-02 8.500E-02 A2 at 0 MWd/t A2 at 6000 MWd/t

8.000E-02

A2 at 50000 MWd/t 7.500E-02 7.000E-02 6.500E-02 500

600

700

800

900

Moderator density, kg/m3

Figure 2.2.4: Σa2 dependence on moderator density

2.2.4.3 Cross-section dependence on boron concentration

The calculation of the cross-section dependence on boron concentration was performed at the following core conditions: •

Boron concentration, ppm – 0, 500, 1000, 1500, 2000

•

Fuel temperature, K – 850

•

Moderator temperature, K –589.47

•

Moderator density, kg/m3 – 690

The results for the absorption cross-section are shown in Table 2.2.9. Again three different burnup values were used. This table shows the relative difference of the calculated by HELIOS exact cross-section value compared to the one obtained by

28

interpolation by using the neighboring points. The results are presented also as plots on Fig. 2.2.5-6.

Presented results show that the maximum difference from the interpolation for the diffusion coefficients when the neighboring points were used is 0.066 %, for the absorption cross-section is 0.057 %, for the fission cross-sections is 0.025 %, and for the down-scattering cross-section is less than 0.103 %. Similar to the previous two cases the errors represent a case in which the intervals for obtaining the cross-section by interpolation are twice as big as the original ones. For example to obtain a cross-section at 500 ppm, the cross-sections at 0 ppm and 1000 ppm were used. This means that these errors bound the possible errors in case when the number of boron concentrations’ reference points is decreased. Table 2.2.9: Σa1 and Σa2 interpolation error (boron concentration) Σa1 calculated by HELIOS



Boron Concentration

0 MWd/t

6000 MWd/t

50000 MWd/t

0 MWd/t

6000 MWd/t

50000 MWd/t

0 MWd/t

6000 MWd/t

50000 MWd/t

0.00

9.511E-03

9.307E-03

1.096E-02

-

-

-

-

-

-

500.00

9.676E-03

9.477E-03

1.113E-02

9.672E-03

9.474E-03

1.112E-02

-0.046

-0.029

-0.027

1000.00

9.833E-03

9.642E-03

1.129E-02

9.833E-03

9.641E-03

1.128E-02

0.002

-0.007

-0.031

1500.00

9.989E-03

9.805E-03

1.144E-02

2000.00

1.014E-02

9.961E-03

1.159E-02

9.987E-03 -

9.802E-03 -

1.144E-02 -

-0.021 -

-0.038 -

-0.013 -

Σa2 calculated by HELIOS



Boron Concentration

0 MWd/t

6000 MWd/t

50000 MWd/t

0 MWd/t

6000 MWd/t

50000 MWd/t

0 MWd/t

6000 MWd/t

50000 MWd/t

0.00

8.560E-02

8.244E-02

6.999E-02

-

-

-

-

-

-

500.00

9.110E-02

8.772E-02

7.521E-02

9.106E-02

8.769E-02

7.517E-02

-0.042

-0.044

-0.057

1000.00

9.653E-02

9.293E-02

8.035E-02

9.650E-02

9.289E-02

8.031E-02

-0.035

-0.038

-0.050

1500.00

1.019E-01

9.806E-02

8.540E-02

1.018E-01

9.802E-02

8.536E-02

-0.040

-0.037

-0.042

2000.00

1.072E-01

1.031E-01

9.038E-02

-

-

-

-

-

-

29

1.200E-02

Sig A 1, cm -1

1.150E-02 1.100E-02 A1 at 0 MWd/t

1.050E-02


1.000E-02 9.500E-03 9.000E-03 0

250

500

750

1000 1250 1500 1750 2000

Boron concentration, ppm

Figure 2.2.5: Σa1 dependence on boron concentration

1.100E-01 1.050E-01

Sig A 2, cm -1

1.000E-01 9.500E-02

A2 at 0 MWd/t

9.000E-02


8.500E-02 8.000E-02 7.500E-02 7.000E-02 0

250

500

750

1000

1250

1500 1750

2000

Boron concentration, ppm

Figure 2.2.6: Σa2 dependence on boron concentration

30

2.2.4.4 Conclusions of the sensitivity study

Table 2.2.10 shows the maximum interpolation relative differences for the different independent variables.

Table 2.2.10: Maximum interpolation errors, % D Σa Σf Σs12

Fuel temp. 0.017 0.099 0.108 0.032

Mod. Density 1.462 0.607 0.709 0.233

Boron 0.066 0.057 0.025 0.103

The results presented in Table 2.2.10 show that the interpolation error for fuel temperature (using the square root of the fuel temperature) and boron concentration is relatively small compared to the interpolation error for moderator density. Therefore, it could be concluded that using fewer number of boron concentrations and fuel temperatures will not increase significantly the interpolation error. By employing fewer number of instantaneous parameters’ reference points, the CPU time for performing lattice calculations will be significantly reduced because of the lower number of branchoff calculations. Another interesting observation is that the behavior of the cross-section dependence on the discussed instantaneous parameters does not change with the change in burnup. 2.2.4.5 Application of the extended methodology

In order to verify the entire performance of the extended methodology for cross-section generation and modeling a previously analyzed problem using the original cross-section generation and modeling procedure was analyzed again using the new extended 31

methodology. The problem chosen is a VVER-440 core at hot zero power (HZP) conditions [Ivanov B., 2004], originally proposed and analyzed in the framework of the International Nuclear Safety Program (INSP) established by the United States Department of Energy (DOE) and managed by Battelle, Pacific Northwest National Laboratory (PNNL). This problem was selected for performing the comparisons because the burnup of this core loading ranges from approximately 3 MWd/tU to 40 MWd/tU.

The HZP conditions were intentionally chosen because the thermal-hydraulic feedback is frozen, which allows consistent comparison to be performed. The same computer code TRAC-PF/NEM was used in both calculations. Figures 2.2.7 and 2.2.8 show the comparison of the normalized axial power distribution and the normalized radial power distribution obtained using the original cross-section library and the new cross-section library.

1.800 1.600

Normalized Axial Power

1.400 1.200 1.000

New Lib

0.800

Orig lib

0.600 0.400 0.200 0.000 0

2

4

6

8

10

12

Axial level #

Figure 2.2.7: Comparison of VVER-440 normalized axial power distribution

32

52 0.7281 0.7259 0.22

FA number New library Original library Abs difference, %

52 0.7281 0.7259 0.22 49 1.6971 1.6922 0.49 45 1.4831 1.4770 0.61 40 1.1027 1.0984 0.43 34 0.9880 0.9857 0.23 27 1.0829 1.0793 0.36

19 0.7061 0.7026 0.35 11 0.8911 0.8844 0.67 1 0.1149 0.1074 0.75

2 0.6223 0.6161 0.62

20 0.8484 0.8465 0.19

12 0.7048 0.7014 0.34

3 0.7038 0.6990 0.48

35 1.5959 1.5964 -0.05

28 1.0044 1.0044 0.00

4 0.0000 0.0000 0.00

29 1.1668 1.1723 -0.55

14 0.9622 0.9620 0.02

5 0.8420 0.8395 0.25

6 1.0261 1.0214 0.47

39 0.3805 0.3818 -0.13

32 0.0000 0.0000 0.00

24 1.1434 1.1461 -0.27

16 1.2736 1.2706 0.30

7 0.2887 0.2711 1.76

44 0.8126 0.8148 -0.22

38 1.2375 1.2436 -0.61

31 1.2616 1.2715 -0.99

23 1.2399 1.2468 -0.69

15 1.0139 1.0131 0.08

48 0.6286 0.6294 -0.08

43 1.5620 1.5669 -0.49

37 1.5142 1.5240 -0.98

30 1.2351 1.2397 -0.46

22 1.4889 1.4964 -0.75

51 0.5850 0.5846 0.04

47 1.5148 1.5158 -0.10

42 1.6883 1.6923 -0.40

36 1.3034 1.3107 -0.73

21 0.9867 0.9880 -0.13

13 1.0716 1.0688 0.28

50 1.4298 1.4273 0.25

46 1.6537 1.6528 0.09

41 1.4858 1.4854 0.04

53 0.4357 0.4342 0.15

25 0.8774 0.8767 0.07

17 1.3957 1.3928 0.29

8 1.4138 1.4076 0.62

33 0.2776 0.2776 0.00 26 0.2774 0.2767 0.07

18 0.5941 0.5926 0.15

9 1.3749 1.3704 0.45

10 0.5768 0.5741 0.27

Figure 2.2.8: Comparison of VVER-440 normalized radial power distribution The comparison of the normalized axial and radial distribution shows reasonable agreement. Generally the error in the normalized radial power distribution is less than 1 % except in FA number 7 where the error is 1.76 %. In FA number 7 the control assembly is partially inserted which most probably enhances the discrepancy. The originally calculated keff is 1.00078 while the one obtained with the new library is 0.99701. This difference is most probably due to the different type of depletion calculations. The original cross-section library was generated using 25 equally spaced 33

depletion steps for all of the fuel assemblies because of the requirements of the automatic process used for the depletion calculations. This means that the low burned assemblies were depleted in detail while for the assemblies with high burnup the case is opposite. As mentioned earlier, in the new methodology for cross-section generation predefined burnup steps were used, which are smaller in the range from 0 to 5000 MWd/tU and getting larger after that, which is very different depletion process then the one used for generation of the old library. The difference in the keff could be decreased if similar depletion processes were used in both calculations.

2.3 Concluding remarks

This chapter presented the standard two-step off-line methodology for performing core calculations as well as some improvements of the current off-line methodology. However, even with the implementation of the improvements to the standard methodology, the methodology is limited and needs to be further improved in order to be applied for analyses of the next generation nuclear power plants.

34

CHAPTER 3 IMPROVEMENTS AND MODIFICATIONS IN NEM

3.1 Introduction

The course of this research led the author to the conclusion that the NEM code needs to be modified in order to be able to use this code for the purpose of the embedded transport calculations. The existing discontinuity factor utilization procedure in NEM has been implemented based on the homogeneous partial current formulation following the Lawrence’s approach [Lawrence, 1986] of replacing the continuity of the partial currents by introducing discontinuity. This implementation proved to be unsuitable for computing partial currents or albedo boundary conditions that the embedded transport calculation requires. The partial currents or albedos computed in this way will not be physical and will introduce challenges for the methods used in the embedded local calculations. This chapter of the thesis describes the original DFs implementation in NEM, as well as the development and implementation of the new formulation of the DFs in NEM.

This chapter also gives details on the development of the equivalence procedure used to generate NEM-specific side-dependent DFs required in the embedded transport calculations. It was Koebke, who first realized that additional degrees of freedom are needed into the homogeneous problem to reproduce the reaction rates and the surface35

averaged currents from the heterogeneous problem and introduced heterogeneity factors [Koebke, 1978, 1981]. Smith generalized Koebke’s method by introducing discontinuity factors for each surface of the node [Smith, 1985]. The work presented in this thesis is based on Smith’s theory for generation of NEM-specific side-dependent DFs using the equivalence procedure.

3.2 Homogeneous partial current formulation in NEM

The existing DFs procedure is implemented in NEM based on the homogeneous partial current formulation following the Lawrence’s approach [Lawrence, 1986] by replacing continuity of the partial currents with: ⎛ 1 ⎞ hom,out ,l in , k out , k J gxhom, =⎜ + α J gxhom, + + ⎟ J gx − α 1 − ⎝ ⎠ where

(

1⎛ 2⎝

α ≡ ⎜1 − ⎜

f f

k gx + l gx −

) (3.1)

⎞ ⎟⎟ ⎠

where f gxk + and f gxl − are the discontinuity factors at the plus and minus boundary of two neighboring nodes k and l. Standard notation is used for the remainder of the parameters.

In NEM the partial current equations, which, in each node, express the outgoing partial current as function of the incoming partial currents and intra-node sources/sinks, are derived from the Fick’s Law [Bandini, 1990]. On the two faces of node l normal to the xaxis, Fick’s Law takes on the following form:

36

d l φgx ( x ) |x =∆x / 2 dx d l + Dgl φgx ( x ) |x =− ∆x / 2 dx

out ,l in ,l = J gxhom, − Dgl J gxhom, + + out ,l in ,l = J gxhom, J gxhom, − −

(3.2) (3.3)

Replacing the fluxes in Eqs. (3.2) and (3.3) with the polynomial expansion results in:

out ,l in ,l J gxhom, = J gxhom, − + +

J

hom, out ,l gx −

=J

hom,in ,l gx −

Dgl

(16φ ∆x

Dgl

+

∆x

( −4φ

)

(3.4)

)

(3.5)

l gx +

+ 4φgxl − − 20φ gl − 60φ gxl 1 − 140φ gxl 2

l gx +

− 16φgxl − + 20φ gl − 60φ gxl 1 + 140φ gxl 2

where φgxl + and φgxl − are group g transverse integrated surface fluxes on the two faces of node l normal to the x-axis; φgl is the node volume average flux; and φgxl 1 and φgxl 2 are group g first and second flux moments. Standard notation is used for the rest.

The flux moment equations can be expressed as:

φ

φ

l gx 2

l gx1

⎛ 1 Dgl l l J gx + + J gx − + 2 φgxl + − φgxl − ⎜⎜ ∆x ⎝ 2∆x 1 l 1 l ⎞ l −Qgx Lgyx1 + Lgzx1 ⎟ 1+ ∆y ∆z ⎠

(

1 =− l Ag

)

(

)

⎛ 1 3Dgl l l l J gx + − J gx − + 2 φgx + + φgxl − − 2φgl ⎜⎜ ∆ ∆x x 2 ⎝ 1 l 1 l ⎞ − Qgxl 2 + Lgyx 2 + Lgzx 2 ⎟ ∆y ∆z ⎠

(

1 =− l Ag

)

(

(3.6)

) (3.7)

l l where J gx + and J gx − are group g face averaged net currents on the two faces of node l l

l

l

normal to the x-axis; Q gx1 and Q gx 2 are group g first and second source moments; L gyx1 l

l

and L gyx 2 are group g first and second moments of y-direction transfers leakages; L gzx1 37

l

and L gzx 2 are group g first and second moments of z-direction transfers leakages, and Agl are the following coefficients:

Agl = Σlag +

N

∑Σ

l sgg ′ g ′ =1, g ′ ≠ g

Substituting the node average fluxes and the flux moments in Eqs. (3.4) and (3.5) using the neutron balance equation and the flux moments equations, Eqs. (3.6) and (3.7), the response matrix equations can be obtained:

J

hom,out ,l gx +

⎛ l + J gx ⎜ + ⎜ ⎝

(

+ J

l gy +

⎛ ⎛ Dgl ⎛ 30 Dgl ⎞ 90 Dgl ⎞ l l =J − ⎜16φgx + ⎜ 1 + l 2 ⎟ + 4φgx − ⎜ 1 + l 2 ⎟ ⎜ ⎜ ∆x ⎜⎝ Ag ∆x ⎟⎠ Ag ∆x ⎟⎠ ⎝ ⎝ ⎛ −60 ⎛ 7 Dgl ⎞ ⎞ 14 Dgl ⎞ ⎞ 120 ⎛ l ⎜ 1 + l 2 ⎟⎟ ⎟ + J gx − ⎜ l ⎜⎜ 1 + l 2 ⎟⎟ ⎟ ⎜ Ag ∆x Agl ∆x ⎜⎝ Ag ∆x ⎠ ⎟⎠ Ag ∆x ⎠ ⎟⎠ ⎝ ⎝

−J

hom,in ,l gx +

l gy −

)

⎛ 20 ⎛ 42 Dgl ⎞ ⎞ ⎜ l ⎜ 1 + l 2 ⎟ ⎟ + J gzl + − J gzl − ⎜ Ag ∆y ⎜ Ag ∆x ⎟⎠ ⎟⎠ ⎝ ⎝

(

)

⎛ 20 ⎛ 42 Dgl ⎞ ⎞ ⎜ l ⎜1 + l 2 ⎟ ⎟ ⎜ Ag ∆z ⎜ Ag ∆x ⎟⎠ ⎟⎠ ⎝ ⎝

⎛ −20 ⎛ ⎛ −60 ⎞ ⎛ −140 ⎞ l ⎛ 60 ⎞ 42 Dgl ⎞ ⎞ l l Q +Q ⎜ l ⎜1 + l 2 ⎟ ⎟ + Qgx + +L ⎜ ⎟ 1⎜ gx 2 ⎜ l ⎟ ⎜ Agl ⎟⎟ gyx1 ⎜⎜ Agl ∆y ⎟⎟ ⎜ Ag ⎜ Ag ∆x ⎟ ⎟ A g ⎝ ⎠ ⎝ ⎠ ⎝ ⎠ ⎝ ⎠ ⎝ ⎠ ⎛ 60 ⎞ ⎛ 140 ⎞ ⎞ ⎛ 140 ⎞ + Llgyx 2 ⎜ l ⎟ + Llgzx1 ⎜ l ⎟ + Llgzx 2 ⎜ l ⎟ ⎟ ⎜ Ag ∆z ⎟ ⎜ Ag ∆z ⎟ ⎟ ⎜ Ag ∆y ⎟ ⎝ ⎠ ⎝ ⎠⎠ ⎝ ⎠ l g

38

(3.8)

out ,l in ,l J gxhom, = J gxhom, + − −

⎛ ⎛ Dgl ⎛ 90 Dgl ⎞ 30 Dgl ⎞ ⎜ −4φgxl + ⎜1 + l 2 ⎟ − 16φgxl − ⎜1 + l 2 ⎟ ⎜ ⎜ ∆x ⎜⎝ Ag ∆x ⎟⎠ Ag ∆x ⎟⎠ ⎝ ⎝

⎛ −60 ⎛ ⎛ 120 ⎛ 14 Dgl ⎞ ⎞ 7 Dgl ⎞ ⎞ l l + J gx + + + J 1 1 ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ + 2 ⎟ 2 ⎟ l l ⎟⎟ ⎜ Agl ∆x ⎜ ⎟ gx − ⎜ Agl ∆x ⎜ ∆ ∆ A x A x g g ⎝ ⎠⎠ ⎝ ⎠⎠ ⎝ ⎝ ⎛ −20 ⎛ ⎛ −20 ⎛ 42 Dgl ⎞ ⎞ 42 Dgl ⎞ ⎞ l l l l + J gy − + J J J + − + 1 1 ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ gz + gz − + gy − l l 2 ⎟ 2 ⎟ ⎟⎟ ⎜ Agl ∆y ⎜ ⎟ ⎜ Agl ∆z ⎜ ∆ ∆ A x A x g g ⎝ ⎠⎠ ⎝ ⎠⎠ ⎝ ⎝

(

)

(

)

(3.9)

⎛ 20 ⎛ ⎛ −60 ⎞ ⎛ 140 ⎞ l ⎛ 60 ⎞ 42 Dgl ⎞ ⎞ l l +Qgl ⎜ l ⎜1 + l 2 ⎟ ⎟ + Qgx + Qgx ⎜ ⎟ ⎜⎜ l ⎟⎟ + Lgyx1 ⎜⎜ l ⎟⎟ 1 2 l ⎟⎟ ⎜ Ag ⎟ ⎜ Ag ⎜ ∆ A x g ⎠⎠ ⎝ ⎠ ⎝ Ag ⎠ ⎝ Ag ∆y ⎠ ⎝ ⎝ ⎛ −140 ⎞ ⎛ 60 ⎞ ⎛ −140 ⎞ ⎞ + Llgyx 2 ⎜ l ⎟ + Llgzx1 ⎜ l ⎟ + Llgzx 2 ⎜ l ⎟ ⎟ ⎜ Ag ∆y ⎟ ⎜ Ag ∆z ⎟ ⎜ Ag ∆z ⎟ ⎟ ⎝ ⎠ ⎝ ⎠ ⎝ ⎠⎠ If the relations of the incoming and outgoing partial currents to the surface averaged flux and net currents are substituted in Eqs. (3.8) and (3.9), these two partial current equations and their y and z direction analogues may be written in matrix form as:

[ A] ⋅ J ghom,out ,l = [C ] ⋅ J ghom,in,l + [ B1 ] ⋅ Qgl + [ B2 ] ⋅ Llg where

⎛ a1 ⎜ ⎜ a2 ⎜a A=⎜ 7 ⎜ a7 ⎜ a11 ⎜⎜ ⎝ a11

a2

a3

a3

a4

a1 a7

a3 a5

a3 a6

a4 a8

a7

a6

a5

a8

a11 a11

a12 a12

a12 a12

a9 a10

a4 ⎞ ⎟ a4 ⎟ a8 ⎟ ⎟ a8 ⎟ a10 ⎟ ⎟ a9 ⎟⎠

(

out ,l out ,l out ,l out ,l out ,l out ,l J gout ,l = J gx + , J gx − , J gy + , J gy − , J gz + , J gz −

39

)

T

(3.10)

⎛ c1 c2 c3 ⎜ ⎜ c2 c1 c3 ⎜ c c7 c5 C =⎜ 7 ⎜ c7 c7 c6 ⎜ c11 c11 c12 ⎜⎜ ⎝ c11 c11 c12

c3

c4

c3 c6

c4 c8

c5

c8

c12 c12

c9 c10

c4 ⎞ ⎟ c4 ⎟ c8 ⎟ ⎟ c8 ⎟ c10 ⎟ ⎟ c9 ⎟⎠

(

in ,l in ,l in ,l in ,l in ,l in ,l J gin ,l = J gx + , J gx − , J gy + , J gy − , J gz + , J gz −

⎛ b1 ⎜ ⎜ b1 ⎜b B1 = ⎜ 8 ⎜ b8 ⎜ b13 ⎜⎜ ⎝ b13

b2

b3

0

0

0

−b2 0

b3 0

0 b9

0 b10

0 0

0

0

−b9

b10

0

0 0

0 0

0 0

0 0

b14 −b14

(

)

T

0⎞ ⎟ 0⎟ 0⎟ ⎟ 0⎟ b15 ⎟ ⎟ b15 ⎟⎠

l l l l l l Qgl = Qgl , Qgx 1 , Qgx 2 , Qgy1 , Qgy 2 , Qgz1 , Qgz 2

⎛ −b4 ⎜ ⎜ b1 ⎜ 0 B2 = ⎜ ⎜ 0 ⎜ 0 ⎜⎜ ⎝ 0

b5

−b6

b7

0

0

b5 0

b6 0

b7 0

0 −b4

0

0

0

0 0

0 0

0 0

)

0

T

0

0

0

0

0 b5

0 0 −b11 b12

0 0

0 0

0 0

b4

b5

b11

b12

0

0

0

0 0

0 0

0 0

0 0

−b6 b6

b7 b7

−b11 b11

0⎞ ⎟ 0⎟ 0⎟ ⎟ 0⎟ b12 ⎟ ⎟ b12 ⎟⎠

(

Llg = Llgyx1 , Llgyx 2 , Llgzx1 , Llgzx 2 , Llgxy1 , Llgxy 2 , Llgzy1 , Llgxy 2 , Llgxz1 , Llgxz 2 , Llgyz1 , Llgyz 2

)

T

.

The coefficients a’s, b’s, and c’s can be expressed in terms of known quantities [Bandini, 1990].

The Response Matrix Equation, Eq. (3.10), can be written in simplified notation as: 40

[ A] ⋅ J = [b]

(3.11)

where ⎛ a1 ⎜ ⎜ a2 ⎜a A=⎜ 7 ⎜ a7 ⎜ a11 ⎜⎜ ⎝ a11

a2

a3

a3

a4

a1 a7

a3 a5

a3 a6

a4 a8

a7

a6

a5

a8

a11 a11

a12 a12

a12 a12

a9 a10

a4 ⎞ ⎟ a4 ⎟ a8 ⎟ ⎟ a8 ⎟ a10 ⎟ ⎟ a9 ⎟⎠

(

out ,l out ,l out ,l out ,l out ,l out ,l J = J gx + , J gx − , J gy + , J gy − , J gz + , J gz −

)

T

b = ( b1 , b2 , b3 , b4 , b5 , b6 )

T

Here the entire right hand side of Eq. (3.10) has been replaced by the b vector in Eq. (3.11). After series of row operations, the 6x6 matrix in (3.11) can be easily transformed into two 3x3 matrices, which require less computational effort to be solved in comparison with the original 6x6 response matrix equation given in Eq. (3.11). Further, if the discontinuity factors need to be used, Eq. (3.1) and the 3x3 matrices are alternatively solved in order to obtain the partial currents.

3.3 Implementation of the heterogeneous partial current formulation in NEM

The heterogeneous partial current formulation (physical response matrix method) is originally proposed by Yamamoto [Yamamoto, 2003]. This method is based on treating

41

the partial currents of the heterogeneous system (heterogeneous partial currents) as unknowns instead of the partial currents of the homogeneous system (homogeneous partial currents) in the response matrix formulation. He implemented his idea into an analytical nodal method code (1-D calculations) and proved through eigenvalue analysis of the red-black iteration that, when heterogeneous partial currents are used the response matrix method converges even with very large or small DFs, which could be the case in the embedded transport calculations. Further he verified his method by using more realistic problem, i.e. a 2-D PWR core calculation. For the PWR core calculation he utilized the discontinuity factors only in the reflector node. Instead of modifying the Advanced Nodal Method (ANM) code he simply divided the reflector’s cross-sections by the DF (he did not use side dependant DFs). Please note that above treatment of the DF is identical to the heterogeneous partial current formulation when single DF per node is used. Both homogeneous and heterogeneous partial current formulations obtained identical solutions except for the cases with extremely large or small DFs where the homogeneous partial current formulation did not converge while the heterogeneous partial current formulation provided converged solution [Yamamoto, 2003].

In the heterogeneous partial current formulation the continuity of the net current Eq. (3.12) and the discontinuity of the homogeneous surface fluxes Eq. (3.13) are used like in the homogeneous partial current formulation. However, in contrast to the homogeneous partial current formulation in the heterogeneous partial current formulation the continuity of the heterogeneous partial currents is also preserved Eq. (3.14) as opposed to the discontinuity of the homogeneous partial currents observed in Eq. (3.1).

42

J gxhet+,out ,k − J gxhet+,in,k = J gxhet−,in,l − J gxhet−,out ,l

(

)

(

(3.12)

)

k l f gxk + Φ hom, = 2 J gxhet+,out ,k + J gxhet+,in ,k = 2 J gxhet−,out ,l + J gxhet−,in ,l = f gxl − Φ hom, gx + gx −

(3.13)

J gxhet+,out ,k = J gxhet−,in,l

(3.14)

The heterogeneous partial current formulation was added in NEM as a new option so the existing homogeneous partial current formulation still can be used. If the heterogeneous partial current option is turned on the NEM algorithm will skip Eq. (3.1) and it will use Eq. (3.14) making by that the partial currents continuous. In order to implement the heterogeneous partial current formulation in NEM the relation of the incoming and outgoing partial currents to the surface averaged flux, Eq. (3.13), had to be substituted in Eqs. (3.8) and (3.9) resulting in slightly different [A] and [C] matrices in Eq. (3.10):

⎛ a1 ⎜ ⎜ a13 ⎜a A=⎜ 7 ⎜ a7 ⎜ a11 ⎜⎜ ⎝ a11

a2

a3

a3

a4

a14

a3

a3

a4

a7

a5

a6

a8

a7 a11

a15 a12

a16 a12

a8 a9

a11

a12

a12

a10

⎛ c1 c2 ⎜ ⎜ c13 c14 ⎜c c7 C =⎜ 7 ⎜ c7 c7 ⎜ c11 c11 ⎜⎜ ⎝ c11 c11

c3

c3

c4

c3 c5

c3 c6

c4 c8

c15

c16

c8

c12 c12

c12 c12

c9 c10

a4 ⎞ ⎟ a4 ⎟ a8 ⎟ ⎟ a8 ⎟ a10 ⎟ ⎟ a9 ⎟⎠ c4 ⎞ ⎟ c4 ⎟ c8 ⎟ ⎟ c8 ⎟ c10 ⎟ ⎟ c9 ⎟⎠

As it can be seen from these new [A] and [C] matrices, four new coefficients (a13, a14, a15, 43

a16, and c13, c14, c15, c16) for each of the matrices replaced the existing coefficients (a2, a1, a6, a5, and c2, c1, c6, c5) respectively. These changes were implemented because of the use of side dependent DFs. In the case of only one DF per node the changes in the [A] and [C] matrices would not be necessary. Another change had to be introduced in flux moment equations, Eqs. (3.6) and (3.7). The surface averaged fluxes had to be divided by the DFs as shown below:

φ

l gx1

1 =− l Ag

⎛ 1 Dgl ⎛ φgxhet+,l φgxhet−,l l l J + J gx − + 2 ⎜ l − l ⎜ ⎜ 2∆x gx + f gx − ∆x ⎜⎝ f gx + ⎝

(

l −Qgx 1+

φ

l gx 2

)

⎞ ⎟⎟ ⎠

1 l 1 l ⎞ Lgyx1 + Lgzx1 ⎟ ∆y ∆z ⎠

⎛ 1 ⎞ 3Dgl ⎛ φgxhet+,l φgxhet−,l l l J gx + − J gx − + 2 ⎜ l + l − 2φgl ⎟ ⎜ ⎟ ⎜ 2∆x ∆x ⎜⎝ f gx + f gx − ⎠ ⎝ 1 l 1 l ⎞ − Qgxl 2 + Lgyx 2 + Lgzx 2 ⎟ ∆y ∆z ⎠

1 =− l Ag

(

(3.15)

)

(3.16)

All these changes into the NEM code required also changes in the NEM solver. As mentioned earlier NEM solves two 3x3 matrices instead of the original 6x6 matrix. Because of the new coefficients in the [A] and [C] matrices the transformation of the 6x6 matrix into two 3x3 matrices is no longer possible. Therefore, in the new version of NEM, which has the heterogeneous partial current formulation option, the 6x6 matrix is solved instead of the two 3x3 matrices. New matrix solver from LAPACK was introduced in NEM for solving the 6x6 matrix.

3.4 Equivalence procedure for discontinuity factors generation

In the conventional methodology, the ADFs are calculated in the following way based on 44

the infinite lattice fuel assembly calculation only:

ADFg =

(Φ g ) SHet (Φ g ) SHom

(3.17)

where

(Φ g ) SHet - heterogeneous surface flux from the lattice calculation (Φ g ) SHom - homogeneous surface flux, which in the case of infinite lattice Het calculation is equal to the (Φ g )VOL , which is also available from the lattice calculation

In order to evaluate the discontinuity factors [Smith, 1986b], the 1-D boundary value problem for the 1-D homogenized flux distribution must be solved in every node in x and y directions. The approximation method used for the core calculations (e.g. NEM) has to be employed also for the evaluation of the homogeneous surface fluxes in order to obtain the exact discontinuity factors. The homogenized surface fluxes on each node surface can be evaluated by using Eqs (3.4) to (3.7). The transverse leakage moments (needed to solve Eqs (3.4) to (3.7)) are calculated again by using the method utilized in the NEM code. They are calculated either by assuming flat leakage or by approximating the leakage using quadratic polynomials using the information not only from the current node but also from the two surrounding neighbors.

In two energy groups these four equations form a linear system of eight equations with eight unknowns that can readily be solved with the available information from the lattice calculation. The obtained surface fluxes are then used to calculate the discontinuity factors using Eq. (3.17). A special subroutine is developed to solve the above linear 45

system and to provide the necessary surface fluxes for the evaluation of the discontinuity factors. In the case of multi-group calculations the four equations, Eqs (3.4) to (3.7) are solved iteratively in each group in order to obtain the surface fluxes and the flux moments. Using the flux moments from the previous iteration the source moments are updated and the linear system of four equations is solved again. This procedure is repeated until convergence on the flux moments is attained. The convergence criteria are set to be 1E-6 on the relative difference of the flux moments.

3.5 Verification

The equivalence procedure and the implementations in NEM were verified against the C3 and C5 benchmark problems, which are described in the Appendix A of this thesis. These problems were first computed with the lattice physics code PARAGON [Westinghouse, 2005] in order to obtain the reference solution. Mini-core color-set calculation with PARAGON was performed for both problems. PARAGON was also used to generate the reference four-group cross-sections from the mini-core calculation. If we use the reference four-group cross-sections and the reference four-group DFs, generated by the equivalence procedure using the average flux, net currents, and the heterogeneous surface fluxes from the reference mini-core PARAGON solution, the NEM code must by definition reproduce exactly the PARAGON mini-core reference solution. The comparison of the results of the NEM code to the reference results is shown in Table 3.5.1 for the C3 problem and Table 3.5.2 for the C5 problem. We can observe that NEM basically reproduces the reference solution. The small differences can be attributed to round-off error. 46

Table 3.5.1: keff and normalized FA power comparisons (C3) NEM

1.25495

keff

Reference

1.25496 -1

2D Power Distribution

Diff, pcm NEM Reference Abs Diff, % NEM Reference Abs Diff, %

1.1241 1.1238 0.03 0.8761 0.8764 -0.03

0.8761 0.8764 -0.03 1.1237 1.1234 0.03

Table 3.5.2: keff and normalized FA power comparisons (C5) NEM

1.17439

keff

Reference

1.17442 -3


Diff, pcm NEM Reference Abs Diff, % NEM Reference Abs Diff, %

1.7849 1.7847 0.02 0.8354 0.8354 0.00

0.8354 0.8354 0.00 0.5443 0.5444 -0.01

These calculations were repeated also with two- and seven-group cross-sections and similar agreement between NEM and PARAGON reference results was observed demonstrating the multi-group performance of the equivalence procedure.


A new DFs utilization procedure for NEM was developed and implemented based on the physical response matrix method. In comparison to the old procedure, the new one uses the heterogeneous partial current formulation instead of the homogeneous partial current formulation. The old formulation of the DF procedure was not suitable for computing partial currents or albedo boundary conditions to be passed to the embedded transport 47

calculation. The partial currents or albedos computed in the old way are unphysical and introduce challenges for the methods used in the embedded local calculations. The new procedure is improvement in the application of DFs in NEM.

This chapter also presented the development of the NEM-dependent DFs generation procedure. The verifications of the DF generation procedure and the new implementations in NEM were also presented.

48

CHAPTER 4 DEVELOPMENT OF THE EMBEDDED SP3 TRANSPORT METHODOLOGY

4.1 Introduction

The SP3 method has gained popularity in the last ten years as an advanced method for neutronics calculations. The SP3 method is currently used by different companies and universities such as Purdue University [Lee, 2001 and 2004], Studsvik [Bahadir, 2005], NFI Japan [Tatsumi, 2002], and CEA Saclay [Ragusa, 2003]. The method has been implemented by Purdue University into the U.S. NRC neutronics code PARCS, by Studsvik into SIMULATE-4, by NFI into SCOPE2 code, and by CEA in CRONOS2. The scheme utilized in PARCS code is to solve the multi-group NEM-SP3 equations for pinby-pin whole core calculations. SP3 has been also developed within a framework of the “one-node” scheme that updates the global diffusion solution of PARCS by correcting the partial currents and homogenizing the cross-sections. In contrast to PARCS code, the SP3 method built into SCOPE2 and CRONOS2 allows performing only whole core pin-bypin calculations. Interesting but complicated scheme is implemented into SIMULATE-4 code. The analytical nodal code SIMULATE-4 solves the 3-D multi-group SP3 or diffusion equations in tandem with axial homogenization model and radial sub-mesh 49

cross-section model. The axial homogenization model computes average cross-sections and axial discontinuity factors for each node. The radial sub-mesh cross-section model divides the core into a set of 2-D radial slices and solves the equations using a two-step scheme. First, full 2-D slice is solved at a time and homogenized assembly cross-sections and discontinuity factors are generated. In the second step, the code solves the diffusion or SP3 equations in each assembly with reflective boundary conditions by using submesh, non-corrected, cross-sections conventionally computed by a lattice code. The outcome of this step is a set of homogenized assembly cross-sections and side averaged discontinuity factors. The cross-sections and the discontinuity factors generated in these two steps are then used to correct the node-average cross-sections, conventionally generated by a lattice code. Corrected cross-sections are then used to solve the global core problem using either diffusion or SP3 equations.

Recently, PSU started a project for improving the standard core calculation methodology [Hahn, 2003] and [Boydens, 2004]. Hahn implemented the SP3 transport model within the framework of the NEM for 3-D pin-by-pin multi-group calculations. To improve further the computational efficiency and performance of the transport model, in order to be applied for practical transient analysis, Hahn coupled the local A2-BEM (pin-by-pin) procedure with the global core NEM diffusion (assembly–by-assembly) calculation. The so-called AN (A2) equations are equivalent to the SP2N–1 (SP3) equations. The Hybrid NEM/BEM is based on local A2-BEM calculation using the incoming partial currents from the NEM global diffusion calculation and the separate pin-wise cross-sections. The partial currents calculated by A2-BEM are then used to update the global NEM incoming partial currents. NEM then uses updated partial currents and its own assembly-wise 50

cross-sections to perform new global calculation. Hahn did not provide sufficient verification of both NEM-SP3 and Hybrid NEM/BEM in her work. Consequently, Mathieu Boydens made an attempt to verify the SP3 transient option in NEM during his research visit at PSU [Boydens, 2004].

4.2 Development of the embedded SP3 methodology

The scheme proposed in this thesis is based on embedded SP3 pin-by-pin local fuel assembly calculation within the framework of the NEM diffusion core calculation. This is a non-linear iteration process that involves cross-section homogenization, on-line discontinuity factors generation, and albedo boundary conditions evaluation by the global solution send to the local calculation. In order to accomplish this step, the author developed a new code based on FEM, capable of performing both diffusion and SP3 calculations. The new code is used in the framework of the NEM code in order to perform embedded pin-by-pin SP3 calculations on fuel assembly basis. The development of this code is described in subsection 4.4.

The initial step of the proposed NEM-FEM iteration scheme is an execution of embedded FEM-SP3 pin-by-pin calculation for each fuel assembly in the core with reflective boundary conditions using multi-group pin-wise cross-sections conventionally generated with a lattice code. The FEM-SP3 then uses the resultant neutron spectrum to collapse and homogenize the multi-group pin-wise cross-sections to few-group cross-sections over the entire fuel-assembly. The output of this execution is the conventional set of assembly homogenized cross-sections and assembly discontinuity factors (ADF) [Smith, 1986b] for 51

each fuel assembly present in the core. The iteration process starts in the next step. NEM uses the cross-sections and the ADFs to perform global core calculation and obtain new albedo boundary conditions for the local FEM-SP3 calculation. The new albedo boundary conditions are then send to the FEM-SP3 solver for each fuel assembly and then new updated cross-sections and side-dependent discontinuity factors are generated. Subsequently, NEM performs new global core calculation with the new data and then convergence test is performed with respect to the local albedo boundary conditions. The iteration process continues until convergence criteria are satisfied. The algorithm for this calculation is depicted in Figure 4.2.1.

FEM-SP3

NEM

FA pin-by-pin calculation

Solves for partial currents, flux, and eigenvalue

Homogenize and collapse the pin-wise cross-sections and compute DF

FA XS and DF

Prepare and send the albedo boundary conditions

Y

NEM converged?

N NEM-FEM-SP3 converged?

Y END

Figure 4.2.1: FEM-SP3-NEM coupling algorithm

52

N

The embedded SP3 methodology improves the standard two-step off-line methodology in the following three aspects: -

The transport effect is treated through the local assembly-based SP3 transport calculations. Using the discontinuity factors, the low-order diffusion calculation exactly reproduces the higher-order SP3 solution since in this case one can consider that the higher-order solution is known a priori;

-

The homogenization and the group collapsing are improved due to the homogenization of the cross-sections using the “true” boundary conditions from the global diffusion solution;

-

The spatial resolution is improved through the pin-based local transport calculation, which is used to correct the nodal diffusion solution.

In contrast to Hahn’s work, the embedded SP3 calculation is coupled to the NEM global calculation through on-line generation of discontinuity factors and homogenization of cross-section, and the albedos are used as boundary conditions for the local SP3 calculation. Using consistent cross-sections on pin and FA level makes the new methodology more refined compared to the methodology used in the Hahn’s work. The proposed methodology is also different in comparison to the methodology within SIMULATE-4 with the proposed non-linear iteration scheme, which accounts for the environment of each assembly in the core, during the homogenization process, by using albedo boundary conditions.

The embedded SP3 calculation is somehow similar to the methodology used by Purdue 53

University due to the consistent utilization of the cross-section, i.e. the homogenization of the cross-sections on fuel assembly basis. However, in the embedded SP3 calculation the homogenization is improved because of the use of the NEM consistent side-dependent discontinuity factors. Another difference is that the coupling of the local and global solutions is done through the discontinuity factors, the homogenization of the crosssections, and the albedo boundary conditions at each iteration step.

The embedded SP3 methodology is used to demonstrate the feasibility of the embedded approach on a simple framework.

4.3 The SP3 approximation

The SPN approximation was first proposed by Gelbard in early 1960s. In the beginning, this method was rarely used because of the weak theoretical basis (Gelbard simply replaced the second derivatives in the one-dimensional planar geometry PN equation with general three-dimensional Laplacian operators). However, several researchers reported significant improvements in the results when SPN is used compared to the diffusion approximation [Lemanska, 1981], [Smith, 1986a]. Larsen provided the first theoretical basis for the SPN equations in multidimensional applications [Larsen, 1993], [Larsen, 1996]. Application of the SPN approximation followed with increased confidence as the theoretical basis of the method became more firmly established [Brantley, 2000].

The embedded transport methodology uses the SP3 approximation because of its accuracy and efficiency in terms of computing time. The derivation of the SP3 equations from the 54

PN equations in slab geometry has been shown in Hahn’s work [Hahn, 2003] as well as in several other publications. The steady-state SP3 equations for group g, assuming isotropic sources and isotropic scattering, are given below [Stacey, 2001]: − D0, g ∇ 2 F0 g + (Σt , g − Σ s 0, g ) F0 g = S0 g + 2(Σ t , g − Σ s 0, g ) F1g (4.1) 4 2 ⎡5 ⎤ − D1, g ∇ 2 F1g + ⎢ (Σt , g − Σ s 2, g ) + (Σt , g − Σ s 0, g ) ⎥ F1, g = ⎡⎣(Σ t , g − Σ s 0, g ) F0 g − S0 g ⎤⎦ 3 3 ⎣3 ⎦

where the following change of variables is applied

F0 g = 2Φ 2 g + Φ 0 g F1g = Φ 2 g

(4.2)

and

D 0g =

S0 =

1 3 ; D1 g = 3 ( Σ tg − Σ s 0 g ) 7 ( Σ tg − Σ s 3 g )

χg keff

G

∑ν g'

G

g'

Σ f , g 'Φ 0 g ' + ∑ Σ s , g '→ g ,0 Φ 0 g ' g '≠ g

G

Σ r , g ≡ Σ t , g − ∑ Σ s , g '→ g = Group g removal cross section g '= g

G

Σ tr ,n=odd = Σ t − µ 0 ∑ Σ s , g '→ g ,n=odd g '= g

The SP3 approximation is more accurate when applied to transport problems compared to

55

the diffusion approximation with considerably less computation expense than the discrete ordinates (SN) or spherical harmonics (PN) approximations. Another advantage of the use of SP3 equations is that they can be solved by straightforward extensions of the common nodal diffusion methods with little computation resources overhead. Tatsumi [Tatsumi, 2002] also showed that the multi-group SP3 pin-by-pin calculations can be used for practical core depletion and transient analyses.

4.4 Development of PSU-FEM diffusion and SP3 code 4.4.1

Derivation of the multi-group diffusion and SP3 FEM discretization

The FEM was chosen for solving the SP3 equations because of its geometrical flexibility and computation efficiency. It has been successfully utilized for diffusion and SP3 direct core calculations in the solvers within the framework of the CRONOS2 code [Ragusa, 2003]. The FEM-based code named temporarily PSU-FEM is designed for both – direct reference pin-by-pin core calculations and embedded assembly pin-by-pin calculation within the NEM core calculations. PSU-FEM is currently applied to rectangular structured mesh but new development allows application also to curvilinear unstructured mesh in order to be able to perform embedded heterogeneous calculations.

The FEM method is applied to the numerical solution of the time independent multigroup neutron diffusion and SP3 equations. To describe the FEM discretization in more detail, the lowest order angular approximation to the linear Boltzmann equation is used, which is given by the following diffusion equation. 56

−∇ ( Dg ∇Φ g ) + Σ Rg Φ g = Fg

r ∈ Ω ; Fg ∈ L 2 (Ω )

(4.3)

subject to the following boundary conditions Φg = 0 Dg

∂Φ g ∂n

r ∈ Γ1 + γΦ g = 0

r ∈ Γ2

Where: Fg =

1 Qg + SS g = Source k G

Qg = χ g ∑νΣ fg 'Φ g ' = Fission source g '=1

SS g =

G

∑

g '=1, g '≠ g

Σ sg '→ g Φ g ' = Scattering source

Here the source term is square integrable Fg ∈ L2 ( Ω ) and ∂Ω = Γ1 ∪ Γ 2 . In the following paragraphs the group index g will be omitted since the derivation below is analogous for all groups.

The FE discretization is based on the weak form of Eq. (4.3), which amounts to finding a

(

)

(

solution Φ ∈ V = H 01 Ω; Γ1 . The Sobolev space H 01 Ω; Γ1

)

is the space of functions,

which together with their first derivatives are square integrable and their traces on Γ1 vanish, that is, they satisfy the Dirichlet boundary condition on Γ1 . Multiplying by a

test function Ψ ∈ V , integrating over Ω , applying the Green’s formula and the boundary conditions on Γ2 we arrive at the following weak formulation of Eq. (4.3): Find Φ ∈V , 57

such that the following is satisfied for all Ψ ∈ V :

∫ D∇Φ.∇Ψdr + ∫ γΦΨds + ∫ Σ

Ω

Γ2

Ω

R

ΦΨ dr = ∫ F Ψ dr Ω

(4.4)

Since the left side of Eq. (4.4) is linear in both Ψ and Φ and the right hand side is linear in Ψ we can write the weak form in the following way: Find Φ ∈V such that a (Φ , Ψ ) = F ( Ψ ), for all Ψ ∈ V

(4.5)

where the bilinear form a(.,.) and the linear form F(.) are defined as follows

a ( Φ , Ψ ) := ∫ D∇Φ.∇Ψ dr + Ω

∫ γΦΨds + ∫ Σ

Γ2

R

ΦΨ dr

Ω

(4.6)

F ( Ψ ) = ∫ F Ψ dr Ω

The FE discretization then is obtained by triangulating Ω and restricting Eq. (4.5) to a finite dimensional subspace Vs of V , consisting of piece-wise polynomials, continuous functions. The finite element solution then is a function Ψ s ∈ Vs satisfying the following equations a (Φ s , Ψ s ) = F (Ψ s ), for all Ψ s ∈ Vs

(4.7)

In the calculations linear and quadratic polynomials to define Vs were used. Writing the N

solution Φ s = ∑ U iϕi , where ϕi is the canonical basis for Vs , we have the following i =1

linear system of equations for the coefficients U i : ΚU = F

58

(4.8)

The entries of the matrix Κ and the right hand side F are obtained using Eq. (4.7), and are given by Kij = a(ϕi , ϕ j )

(4.9)

Fi = ∫ Fϕi

(4.10)

Ω

The weak form of the SP3 equations (4.1) are derived in a completely analogous fashion using the following boundary conditions on Γ2 :

∂F0 − γ 1 F0 + γ 2 F1 = 0 ∂n ∂F D 1 − β1 F1 + β 2 F0 = 0 ∂n

D

r ∈ Γ2 (4.11)

r ∈ Γ2

For different values of the constants γ and β the boundary conditions can include vacuum, zero-flux, reflective and albedo.

The diffusion and SP3 equations are solved by the common source iterations. 4.4.2

Verification of PSU-FEM

The first problem considered for verification is a simple 1-D slab used by Capilla [Capilla, 2005]. The geometry for this problem is shown in Fig. 4.4.1. The slab is 2 cm long and has the following one-group cross-sections: Σt = 1 cm-1, Σs = 0.9 cm-1 and υΣf = 0.25 cm-1.

59

Figure 4.4.1: Geometry of the 1-D test problem used for verification of PSU-FEM (by

Capilla) Table 4.4.1 shows results using 40 meshes in the PSU-FEM calculation. The solutions of the PSU-FEM code for both P1 and P3 match exactly the reported analytical solutions. These results also show that SP3 solution is closer to the reference S96 solution, as expected since in 1-D SP3 is equivalent to P3. Table 4.4.1: Comparison of FEM results to the analytical solution and the S96 reference

P-1 P-3 S96

Analytical solution

PSUFEM

0.587489 0.652956 0.662951

0.58749 0.65296 -

Difference to analytical solution, pcm 0 0 -

Difference to S96, pcm

-7881 -1308 0

The second problem chosen is a one-group, isotropic scattering eigenvalue problem used by Brantley [Brantley, 2000] in the system shown in Fig 4.4.2. This problem has onedimensional character along y ≈ 0 plane and becomes increasingly two-dimensional as y increases.

60

Vacuum 10

M

F

F

M

M

F

M

Vacuum

Reflecting

9

0 0

1

2

4

7

5

8

10

Reflecting

Figure 4.4.2: Geometry of the test problem 2 used for verification of PSU-NEM (by

Brantley) The eigenvalue results are shown in Table 4.4.2 and the diffusion and SP3 spatial flux solutions are shown in Figures 4.4.3 and 4.4.4. These results were obtained with 60x60 meshes and with the refining of the mesh it is observed that the FEM code converges to the provided in Brantley’s paper solution.

Table 4.4.2: Comparison of FEM results to the analytical solution and the S16 reference

P-1 SP3 S16

Brantley

PSU-FEM

0.776534 0.798617 0.806132

0.77680 0.79904 -

61

Difference, pcm 27 42 -

Difference to S16, pcm -2933 -709 0

5 4 3 2

Y

1 0 -1 -2 -3 -4 -5 -5

0

5

X

Figure 4.4.3: Spatial flux distribution obtained with diffusion approximation

5 4 3 2

Y

1 0 -1 -2 -3 -4 -5 -5

0

5

X

Figure 4.4.4: Spatial flux distribution obtained with SP3 approximation

62

In order to verify the multi-group performance of the PSU-FEM code the C3 MOX benchmark problem, specified by Cavarec, et. al., [Cavarec, 1994], was computed in two, four and seven energy groups by PSU-FEM and NEM. As mentioned earlier, NEM is a multi-group nodal diffusion code developed, maintained, and continuously improved at PSU. The C3 MOX benchmark specification is provided in the Appendix A. Both codes, PSU-FEM and NEM, used homogenized FA cross-sections in two, four, and seven energy groups generated by the lattice physics code PARAGON [Westinghouse, 2005]. The results of the PSU-FEM code were compared to the results of the NEM code in Table 4.4.3. In order to make a consistent comparison the NEM code must be spatially converged. The NEM calculations were performed with 1, 4 and 9 nodes per assembly (NPA) in order to obtain spatially converged results.

Table 4.4.3: Comparison of PSU-FEM results to NEM results for C3 MOX benchmark test case

NEM 2G

NEM 4G

NEM 7G

NPA

keff Diff, pcm

UOX Abs. diff, %

MOX Abs. diff, %

1 4 9 1 4 9 1 4

30.1 4.1 1.1 68.0 13.0 4.0 61.6 10.6

1.68 0.41 0.14 1.61 0.36 0.12 1.60 0.35

-1.68 -0.41 -0.15 -1.61 -0.35 -0.12 -1.60 -0.34

9

3.6

0.12

-0.11

As the comparison in Table 4.4.3 shows, the PSU-FEM results agree very well with the NEM results obtained with 9 NPA which demonstrates the correct multi-group performance of the PSU-FEM code. The column under “keff” shows the difference in the keff in pcm. As it can be seen the difference is in the order of few pcm. The next two 63

columns show the differences for the normalized assembly power distribution. The comparison of the normalized FA powers shows that the difference in the normalized FA distribution for both the uranium and the mixed oxide fuel assemblies is less than 0.2 %.

4.5 Application of the embedded SP3 methodology

This section presents the preliminary results of the C3 MOX benchmark, specified by Cavarec, et. al., [Cavarec, 1994]. The number densities as well as the dimensions of the fuel, cladding, guide tubes, and the fission chambers for this problem were taken from Cathalau, et. al. [Cathalau, 1996] and used with minor modifications for generating the pin cross-sections. The benchmark problem is described in detail in the Appendix A.

The results for the C3 MOX benchmark were obtained with the coupled code FEMSP3/NEM. FEM-SP3 was used for the local pin-based calculations on assembly basis while NEM was applied for the global core calculation. These results were obtained using assembly-wise albedo boundary conditions. The DFs were generated as described in subsection 3.4 of this thesis and continuously updated through the iteration process.

Two different calculations were performed. In the first case, the local SP3 calculation uses two-group pin-wise cross-sections and provides two-group assembly homogenized cross-sections to the NEM code to perform the global core-calculation. Therefore, in this case there is no cross-section collapsing. In the second case, the local SP3 calculation uses seven-group pin-wise cross-sections and supplies the NEM code with collapsed to twogroup assembly homogenized cross-sections. The results of the embedded SP3 64

calculations were compared to the reference pin-by-pin whole core PSU-FEM SP3 calculation. For the first case the reference calculation was performed using two-group pin-wise cross-sections, while for the second case the seven-group pin-wise crosssections were used. Both sets of cross-sections were generated by the lattice physics code PARAGON [Westinghouse, 2005]. The reference solutions are provided in Tables 4.5.1 and 4.5.2.

Table 4.5.1: Reference two-group pin-by-pin PSU-FEM SP3 solution to the C3 benchmark problem

1.26072 keff 2-D 1.1376 0.8624 Normalized 0.8624 1.1376 Power

Table 4.5.2: Reference seven-group pin-by-pin PSU-FEM SP3 solution to the C3 benchmark problem

1.26550 keff 2-D 1.1137 0.8863 Normalized 0.8863 1.1137 Power For the first case the embedded calculations converged in 5 iterations, while for the second case the embedded calculations converged in 4 iterations. The convergence criterion was specified on the absolute change of the boundary conditions through the iterations and it was set to be 1.0E-04.

The results of the first case after the first iteration and after the last converged iteration are shown in Tables 4.5.3 and 4.5.4 respectively.

65

Table 4.5.3: Comparison of FEM-NEM results to reference (Case 1, iteration 1) FEM-NEM

1.26239

keff

Reference

1.26072


Diff, pcm FEM-NEM Reference Abs Diff, % FEM-NEM Reference Abs Diff, %

167 1.1615 0.8385 1.1376 0.8624 2.39 -2.39 0.8385 1.1615 0.8624 1.1376 -2.39 2.39


1.26186

keff

Reference

1.26072



114 1.1550 1.1376 1.74 0.8450 0.8624 -1.74

0.8450 0.8624 -1.74 1.1550 1.1376 1.74

The results from the first iteration can be considered as results of the standard methodology used for core calculations as in the first iteration the cross-sections are obtained by single-assembly calculation with reflective boundary conditions. In other words, the effect of the environment of each assembly in the core is not taken into consideration. From these results one can observe an improvement of keff with 53 pcm, in comparison to the solution in the first iteration, and reduction of the difference in the normalized powers from the reference from 2.39 % to 1.74 %.

66

The results of the second case after the first iteration and after the last converged iteration are shown in Tables 4.5.5 and 4.5.6, respectively.


1.26826

keff

Reference

1.26550



276 1.1406 0.8594 1.1137 0.8863 2.69 -2.69 0.8594 1.1406 0.8863 1.1137 -2.69 2.69

In the second case, which uses seven-group cross-sections for the local calculation, we observed improvement of keff with 82 pcm in comparison to the solution in the first iteration and the difference in the normalized power from the reference decreased from 2.69 % in the first iteration to 1.94 % in the last iteration.


1.26744

keff

Reference

1.26550 194



1.1331 1.1137 1.94 0.8669 0.8863 -1.94

0.8669 0.8863 -1.94 1.1331 1.1137 1.94

The same calculations were performed using the diffusion option of the PSU-FEM code for the embedded local calculations instead of the SP3 option and the results were compared to the PSU-FEM reference solutions over whole core pin-by-pin. The differences of the results of these calculations from the reference have similar trend 67

compared to the differences of the results with SP3 option reported above.

All calculations were performed on a four processor workstation and the embedded calculations were performed in parallel. The run time statistics are provided in Table 4.5.7.

Table 4.5.7: Run time statistics for PSU-FEM execution Embedded Reference 70.7 124.7 Case 1 307.1 822.96 Case 2

The run time statistics show that the embedded calculations are 2-3 times faster than the reference pin-by-pin whole core calculations. Having in mind that the embedded calculations were performed in parallel on 4 processors, one can conclude that actually there is no gain in CPU time. However, with increase of the problem size the embedded calculations will become much more efficient in comparison to the whole core pin-by-pin calculations.


New methodology for improving the standard calculation scheme for core analysis was developed. This methodology is based on embedded SP3 pin-by-pin calculation as well as on-line cross-section and discontinuity factors generation. The embedded SP3 calculations improves the standard methodology because of the improved cross-section homogenization and also because of the increased spatial resolution due to the SP3 pinby-pin calculation. 68

A new FEM diffusion and SP3 code (PSU-FEM) was developed. The results of both diffusion and SP3 options of the code agree well with other published results. The PSUFEM code was designed for both direct reference pin-by-pin core calculations and embedded assembly pin-by-pin calculation within the NEM core calculations. The PSUFEM code was coupled to the NEM code for performing embedded SP3 and diffusion calculations. The current version of the coupled code showed that the embedded calculations are feasible in terms of fast convergence. The coupled PSU-FEM/NEM code also produced reasonable results in comparison to direct pin-by-pin solution, having in mind that currently the code uses only assembly wise albedo boundary conditions. We expect that with the implementation of the pin-wise boundary conditions the accuracy of the results from the embedded SP3 calculations will significantly improve.

Despite all the improvements to the current methodology, embedded SP3 calculations have limited accuracy because of the use of pre-calculated few-group pin-wise homogenized cross-sections. The use of SP3 also imposes limitations in the improvement of the angular representation. Nevertheless, the development and application of this methodology confirms that the embedded approach is practical in terms of improved results and fast convergence. Substituting the SP3 calculation with higher order method may further improve the accuracy of the embedded SP3 methodology, proposed in this chapter. Such a calculation scheme is developed and described in the next chapter of this thesis.

69

CHAPTER 5 DEVELOPMENT OF THE EMBEDDED LATTICE TRANSPORT METHODOLOGY

5.1 Introduction

The reactor analysts consider the homogenization error the most significant deficiency of the current standard methodology for core calculations. Therefore, special efforts were devoted to improve the homogenization procedure. Mondot and Sanchez [Mondot, 2003] proposed an iterative homogenization technique that preserves assembly core exchanges. This iterative methodology is based on embedded, fixed source and eigenvalue, transport calculations that computes fine multi-group transport fluxes, which preserve macro-group assembly exchanges in the core. Mondot and Sanchez tested this methodology against 1D problem and obtained unsatisfactory results. In contrast, Clarno and Adams [Clarno, 2005] focused on approximating accurately the effects that neighboring assemblies have on the few-group cross-sections, the assembly discontinuity factors, and the form functions by tabulating these effects. They used core-level algorithm to interpolate in these tables in order to account for the environment of each fuel assembly in the core. This methodology shows promising results but is connected with cumbersome evaluation and parameterization of the detailed angle- and energy-dependent albedo used as a 70

boundary condition. Another approach associated again with parameterization of the leakage effects on the homogenized parameters was proposed by Rahnema and Nichita [Rahnema, 1997].

The embedded lattice methodology proposed in this thesis is a non-linear iteration process based on global NEM core calculations and embedded (local) fuel assembly based

eigenvalue

calculations

that

significantly

improve

the

cross-section

homogenization because of the use of boundary conditions that account for the environment of each FA in the core. The lattice code PARAGON [Westinghouse, 2005] performs the embedded calculations. During the non-linear iteration process NEM sends local (pin-wise) albedo boundary conditions to PARAGON from the global core solution. PARAGON then performs new set of single assembly calculations and recalculates the cross-sections using the local boundary conditions. NEM uses the updated cross-sections and discontinuity factors in order to perform new global calculation. The iterations continue until convergence on the albedo boundary conditions is attained.

In contrast with the embedded SP3 methodology, described in the previous chapter, the embedded lattice methodology addresses all of the earlier described shortcomings of the present core calculation methodology. The improvements in comparison with the present methodology are as follows: -

The transport effect can be considered treated even though the global solution is still diffusion based. Using the discontinuity factors, the low-order diffusion calculation exactly reproduces the higher-order collision probability heterogeneous transport

71

solution since in this case one can consider that the higher-order solution is known a priori; -

The homogenization and group collapsing are improved owing to the fact that the local higher order heterogeneous calculations are performed not with reflective boundary conditions but with the “true” boundary conditions from the nodal code global solution. In contrast to the embedded SP3 calculations, where conventionally computed, few-group pin-wise, homogenized cross-sections are used, in this method the cross-sections are generated on-line by the lattice code multi-group heterogeneous calculations accounting for the environment of each assembly in the core;

-

The spatial discretization effect is also taken into account through the pin-based embedded heterogeneous transport calculation, which is used to correct the nodal solution;

-

There is no need to perform branch calculations or cross-section parameterization. The PARAGON calculations for each node/assembly are done on-line at the exact assembly (node) averaged thermal-hydraulic conditions in the core.

5.2 Development of the embedded lattice transport methodology 5.2.1

Method description

The embedded lattice transport methodology is a non-linear iteration process based on global NEM core calculations and embedded, FA-based, heterogeneous calculations. The method uses on-line cross-section and discontinuity factors generation. The initial step of 72

the proposed iteration process is execution of the embedded heterogeneous lattice calculation for each fuel assembly in the conventional way using reflective boundary conditions (infinite lattice calculation). The outcome of this execution is the conventional cross-sections and assembly discontinuity factors (ADF) [Smith, 1986b] for each fuel assembly present in the core. The iteration process starts in the next step. NEM uses the cross-sections and the ADFs to perform global core calculations in order to obtain new albedo boundary conditions for the local lattice calculations. Until this point everything is the same as in the conventional methodology for core analysis and therefore the NEM solution in the first iteration can be considered as the solution of the conventional methodology for core analysis currently used. The albedo boundary conditions are evaluated in the global core NEM calculation and then are passed to the lattice code for each fuel assembly. PARAGON performs new set of single-assembly calculations and then uses the resultant neutron spectrum to collapse and homogenize again the crosssections to few-group cross-sections over the entire FA domain. The equivalence procedure generates new discontinuity factors using the updated by PARAGON surface averaged net currents and fluxes, the average flux, and the eigenvalue from the embedded calculation. In the second iteration NEM performs global core calculation with the new data and then convergence test on the local albedo boundary conditions is performed. The iteration process continues until convergence criteria are satisfied. If depletion calculation is desired, PARAGON can use the final boundary conditions from the iteration process in order to perform a depletion step. The algorithm for this calculation is depicted in Figure 5.2.1.

A new code, called PRG-NEM was developed to perform NEM and PARAGON 73

executions in parallel, and to exchange the information between the two codes. As Figure 5.2.1 shows, PARAGON performs all steps in the blue box while NEM performs all steps in the orange box. Everything outside the two boxes is performed by the PRG-NEM code.

PRG-NEM

PARAGON

Prepare set of PARAGON inputs

FA based transport lattice calculation

Update BC (Calculate Albedos)

Generation of fewgroup homogenized cross-sections

Expand the few-group global currents j g = jG × f

PARAGON Depletion Step

NEM Calculate DF Solves for partial currents, flux, and eigenvalue

N

Y

NEM-PRGN converged?

NEM converged?

N

Y Last Depletion Step?

N

Y END

Figure 5.2.1: PARAGON-NEM coupling algorithm

74

The boundary conditions used in the iteration process could be either albedo or incoming partial currents. In this thesis the albedo boundary conditions were used because the PARAGON code currently does not accept incoming partial currents as boundary conditions. In the initial step, when the first sets of cross-sections are being generated, reflective boundary conditions are used in the embedded PARAGON fuel assembly calculation. Once the global NEM solution is performed, the reflective boundary conditions are replaced for the next iteration with the “real” ones from the global solution. In the case when the incoming partial currents are used as boundary conditions, the boundary conditions can be re-constructed in space, energy and angle in order to provide local boundary condition for each pin-cell in the fuel assembly. The reconstruction can be done by using the spatial, energy and angular shapes of the outgoing partial currents from the neighboring assembly from the previous iteration step, which will be incoming partial currents for the assembly of interest. However, in the case when the albedos are used as boundary conditions the problem is a little bit different. The albedo’s spatial, energy, and angular shapes are fixed since they are used as boundary conditions and can not be updated in a simple way as in the case of the partial incoming currents. Therefore, another way had to be found in order to be able to reconstruct the albedos. Even though the albedos are fixed for a given assembly the idea of using the partial currents can be utilized in order to reconstruct the albedos. The new idea of computing the local boundary conditions for the embedded calculations is to obtain the spatial, spectral and angular shapes of the outgoing currents for each assembly from the PARAGON single-assembly calculations from the previous iteration step. Using these shapes, the outgoing and incoming currents from the NEM global calculation can be 75

expanded in space, energy, and angle followed by a computation of the local albedos.

5.2.2

Reconstruction of the global boundary conditions

Figure 5.2.2 shows two neighboring pin-cells i and i + 1 . For example, let’s imagine that pin-cell i is located on the East border of fuel assembly 1 (FA-1) and pin-cell i + 1 is located on the West border of the fuel assembly 2 (FA-2). Therefore, we will need the outgoing partial currents on the East side of pin-cell i and the West side of pin-cell i + 1 .

FA-1

FA-2

i

i+1

J iout +1, k , g ′

J iout ,k , g ′

Figure 5.2.2: FA-1/FA-2 interface

For example, having this information the local (pin-wise) spatial, spectral, and angular dependant albedos for pin cells i and i + 1 can then be calculated using Eqs. (5.1) and (5.2).

76

α L ,i , k , g ′ =

J iout +1, k , g ′

α L ,i +1,k , g ′ =

J iout ,k , g ′ J iout ,k , g ′ J iout +1, k , g ′

(5.1)

(5.2)

where, i is the pin-cell number, “L” means “local”, “k” represents the direction, g ′ is the fine group number (used in PARAGON), and α is the albedo.

In order to expand the NEM albedos, first the spatial, energy, and angular shapes of the outgoing currents will be computed from the information generated by the embedded PARAGON calculation in the previous iteration step, then the incoming and outgoing NEM partial currents will be expanded in space, energy, and angle using these shapes and finally the expanded NEM incoming and outgoing currents will be used to compute the local albedos. In order to better understand the importance of each of the spatial, energy, and angular reconstructions, the spatial, energy, and angular shapes are computed separately allowing in this way their separate application in the embedded calculation. The PRG-NEM code was first modified to be able to use only the spatial shape. This was done by first integrating over the directions of the outgoing partial currents for each fine group and pin-cell on FA periphery from the single-assembly PARAGON calculation as shown in Eq. (5.3). out ,l −1

K

out ,l −1

J L ,i , g ′ = ∑ J L ,i , k , g ′

(5.3)

k =1

where, g ′ is the fine group number, i is the pin-cell number, k represents the direction, L means “local”, and l is the iteration number.

77

out ,l −1

Secondly, the integrated over the directions outgoing partial currents J L ,i , g ′ were integrated over the energy within each coarse energy group g in order to obtain the coarse group outgoing partial currents for each pin-cell as shown in Eq. (5.4).

out ,l −1 J ave ,i , g =

Gg

∑J

out ,l −1 L ,i , g ′∈g

(5.4)

g ′∈g

out ,l −1 These currents J ave ,i , g were then divided by the average outgoing current per FA side per out ,l −1 (evaluated in Eq. 5.5) in order to obtain the spatial shape as shown coarse group J ave ,g

in Eq. (5.6). Gg

out ,l −1 J ave ,g =

out ,l −1

I

∑ ∑ J L,i, g′∈g × si

g ′∈g i =1

I

∑s i =1

f

space ,l i,g

=

(5.5)

i

out ,l −1 J ave ,i , g out ,l −1 J ave ,g

(5.6)

where, si is the surface of the pin-cell, Gg is the number of fine groups within a coarse group g , i is the pin-cell number, I is the total number of pins per fuel assembly side ,l and fi ,space is the spatial shape. g

The spatially reconstructed local albedo is then obtained by multiplying the global (assembly-wise) NEM coarse-group partial average currents per FA side by the corresponding spatial shape functions and then by dividing the expanded in space global NEM incoming current to the expanded in space global NEM outgoing current for each 78

FA side and each FA in the core.

The PRG-NEM code was also modified to be able to use only the energy shape. This was done by first integrating over all directions the outgoing partial currents for each fine group and pin-cell as shown in Eq. (5.3). Then the spatially averaged outgoing current per fine group and per assembly side is obtained by averaging the currents within each fine group for each assembly side as shown in Eq. (5.7). out ,l −1

I

out ,l −1 J ave ,g′ =

∑ J L,i, g′∈g × si i =1

(5.7)

I

∑s i =1

i

,l The energy shape f genergy is than obtained by Eq. (5.8). ′

f

=

energy ,l g′

out ,l −1 J ave , g ′∈g

(5.8)

out ,l −1 J ave ,g

The energy-reconstructed local albedo is then obtained by multiplying the global NEM coarse-group partial average currents per FA side by the corresponding energy shape functions and then by dividing the expanded in energy global NEM incoming current to the expanded in energy global NEM outgoing current for each FA side and each FA in the core.

out ,l −1

Using the average outgoing partial currents per pin-cell side J L ,i ,k , g ′ and the integrated out ,l −1

,l over the directions outgoing partial current J L ,i , g ′ the angular shape fi ,angle for each pink , g′

cell and each group can be obtained by: 79

out ,l −1

f

angle ,l i ,k , g ′

=

J L ,i , k , g ′ out ,l −1

(5.9)

J L ,i , g ′

The following five options for the albedo reconstruction in the PRG-NEM code were created: •

No reconstruction – global albedo is used (loc-0)

•

Reconstruction only in space (loc-1)

•

Reconstruction only in energy (loc-2)

•

Reconstruction in space and energy (loc-3)

•

Reconstruction in space, energy, and angle (loc-4)

Global albedo means one average value per FA side per coarse-energy group. Using the above options, the NEM albedo obtained from the global core calculation can be reconstructed in a different ways or not reconstructed and applied as a boundary condition to the embedded single-assembly PARAGON calculation.

5.3 Application of the embedded lattice transport methodology 5.3.1

Sensitivity studies using the C3 and C5 problems reference solutions

The embedded lattice transport methodology is applied to the C3 and C5 benchmark problems, specified by Cavarec, et. al., [Cavarec, 1994]. The number densities, the dimensions of the fuel, cladding, guide tubes and the fission chambers for these problems were taken from Cathalau, et. al. [Cathalau, 1996] and used with slight modifications. The benchmark problems description is provided in detail in Appendix A. 80

In order to evaluate the performance of the embedded lattice methodology, reference solutions obtained by higher order methods are needed. In this case the reference solutions of both problems were computed by PARAGON mini-core heterogeneous lattice calculation (for C3 and C5). We use PARAGON mini-core solutions as reference because the best possible solution of the embedded methodology is to reproduce the PARAGON mini-core solution.

The reference PARAGON mini-core calculations were performed using current coupling order 3 for both problems. All PARAGON calculations (embedded and reference) were performed with the following convergence criteria: • • •

Eigenvalue: 1.0E-6 Inner (current) iterations: 1.0E-4 Source iterations: 1.0E-5

Tables 5.3.1 and 5.3.2 show the reference PARAGON mini-core results for the FA normalized power distributions and keff for the C3 and C5 benchmark problems respectively.

Table 5.3.1: Reference PARAGON results for C3 test problem

1.25496 keff 2-D 1.1238 0.8764 Normalized 0.8764 1.1234 Power

Table 5.3.2: Reference PARAGON results for C5 test problem

1.17442 keff 2-D 1.7847 0.8354 Normalized 0.8354 0.5444 Power 81

The normalized pin-powers obtained from the reference PARAGON mini-core calculations are provided in Appendix B in Tables B.1 to B.4 for the C3 problem and Tables B.5 to B.8 for the C5 problem. The abbreviations used in the Table’s captions are as follows: Table B.1 shows the results for the North-West Uranium assembly (NWUOX), Table B.2 shows the results for the North-East Mixed-oxide assembly (NEMOX), Table B.3 shows the results for the South-West Mixed-oxide assembly (SWMOX) and Table B.4 shows the results for the South-East Uranium assembly (SE-UOX). The same abbreviations are used in the rest of the Tables and Figures, which show results for the different assemblies.

In order to investigate the applicability of the albedos for the embedded lattice calculations, single-assembly PARAGON calculations were performed using albedo boundary conditions extracted from the reference PARAGON solution. These calculations demonstrate if the local multi-group albedo boundary conditions are sufficient to reproduce the reference mini-core solution. The single assembly PARAGON calculations with reference pin-wise albedos in 70 energy groups from the reference mini-core PARAGON calculations were performed for the following cases:

•

Case 1: Single FA calculations for all of the 4 assemblies of the C3 problem.

Albedos from the reference mini-core calculation were used as a boundary condition.

•


82

Albedos from the reference mini-core calculation were used as a boundary condition.

The pin-powers obtained by the single-assembly calculations are normalized on assembly basis while the reference pin-powers obtained by the mini-core PARAGON calculation are normalized on core basis and therefore there is inconsistency of the power normalization. In order to perform consistent comparison of the single-assembly pinpowers to the reference mini-core pin-powers the reference mini-core pin-powers were re-normalized on assembly basis. The pin-power differences (multiplied by 100) from the reference for the two cases above for each of the fuel assemblies are provided in the Appendix B in Tables B.9 to B.12 for Case 1 and B.13 to B.16 for Case 2. These results show that, when albedo is used as a boundary condition, the maximum pin-power differences are approximately 2.5 % for the C3 problem and 2.9 % for the C5 problem all located in the point where the four assemblies meet.

Tables 5.3.3 and 5.3.4 summarized the results provided in Appendix B for Case 1 and Case 2. These tables provide the minimum and maximum differences (multiplied by 100), the average difference (multiplied by 100), and the root mean square (RMS).

Table 5.3.3: Case 1 pin-power statistics MIN MAX AVG RMS

NW-UOX -0.236 0.794 0.089 0.122

NE-MOX -2.494 0.511 0.141 0.243

83

SW-MOX -2.493 0.512 0.142 0.244

SE-UOX -0.207 0.872 0.100 0.133


NW-UOX -0.175 0.354 0.049 0.063

NE-MOX -2.864 1.035 0.256 0.389

SW-MOX -2.865 1.035 0.253 0.386

SE-UOX -0.594 2.025 0.157 0.240

The only possible reason for these differences was identified to be the angular representation, which is not taken into account in the albedo boundary conditions. Therefore, to investigate the reason for these differences, the PARAGON code developers at Westinghouse introduced a new option for boundary conditions in PARAGON. The new option was named “Partial_albedos” and allows the user to specify direction dependent albedos. The direction dependant albedos were calculated again using the information from the reference mini-core PARAGON solution. The directional albedos for cell i were calculated by dividing the incoming current in direction K for cell i by the incoming current in the mirror image of direction K of the neighboring cell

i +1 .

The same cases, as described above, were calculated again with the new “Partial_albedo” option in PARAGON:

•


Directional albedos from the reference mini-core calculation were used as a boundary condition.

•


Directional albedos from the reference mini-core calculation were used as a 84

boundary condition.

The pin-power differences (multiplied by 100) from the reference for the two cases above for each of the fuel assemblies are provided in the Appendix B in Tables B.17 to B.20 for Case 3 and B.21 to B.24 for Case 4. The same results are summarized also in Tables 5.3.5 for Case 3 and Table 5.3.6 for Case 4. These results show that when directional dependent albedos are used as boundary conditions, the maximum pin-power differences decreased from approximately 2.5% to less than 0.2% for the C3 problem and from 2.9% to less than 0.3% for the C5 problem.


NW-UOX -0.089 0.088 0.030 0.038

NE-MOX -0.142 0.181 0.044 0.056

SW-MOX -0.140 0.183 0.044 0.056

SE-UOX -0.099 0.078 0.036 0.042


NW-UOX -0.077 0.070 0.029 0.035

NE-MOX -0.167 0.202 0.056 0.070

SW-MOX -0.168 0.188 0.055 0.069

SE-UOX -0.158 0.168 0.061 0.073

The average pin-power difference per assembly was calculated using Eq. (5.10) and the RMS was calculated by using Eq. (5.11). In the calculations of the AVG and RMS the guide tubes were excluded. The N is the number of pin-cells in the assembly and en is pin-power difference multiplied by 100. 85

AVG =

RMS =

∑e

n

N

(5.10)

N

∑e

2 n

N

N

(5.11)

The AVG provides a point of reference for the RMS measure. The RMS gives an estimate of the severity of the pin-power differences distribution. If the RMS is significantly larger than the AVG it will mean that there are pin-cells with substantial pin-power difference. The results in Tables 5.3.3 to 5.3.6 show that for the C3 problem, when non-directional albedo is used as a boundary condition, the differences in the UOX assemblies are smaller than the differences in the MOX assemblies. Looking at the RMS results one can conclude that the pin-power differences distribution is uneven, which was actually observed earlier – the highest errors are in the point where the four assemblies meet.

The results of Cases 3 and 4, which uses directional albedo as boundary

conditions, indicate that the use of angular dependant boundary conditions further improve the results. The average difference for the C3 problem for the UOX assemblies decreased from 0.1% to 0.036% and for the MOX assembly from 0.142% to 0.044%. The average difference for the C5 problem for the UOX assemblies decreased from 0.157% to 0.061% and for the MOX assembly from 0.256% to 0.056%. Overall, one can observe that the pin-power differences are slightly higher for the C5 problem than for the C3 problem and also slightly higher for the MOX assemblies than for the UOX assemblies.

The reference albedos used for the C3 problem are plotted in Figures 5.3.1 and 5.3.2 in order to analyze their energy and spatial behavior. These figures show the albedo energy

86

group dependence for all pin cells located at the south side of the NW-UOX assembly and the north side of the SW-MOX assembly for the C3 configuration. The order of the pin cells used in these two figures as well as the rest of the figures is from left to right for the North and South sides and from top to bottom for the East and West sides of the assembly. This means that in Figures 5.3.1 and 5.3.2 cell # 1 is located at the periphery while cell # 17 is located in the center of the core. As these figures illustrate, the spectrum dependence of the albedo is very important from group 29 to 70 (0.0- 3.0 eV) and it seems unimportant for energies above 3.0 eV, except in the high energy spectrum where slight energy dependence can be also observed. This behavior of the albedos can be explained with the resonances of the

240

Pu,

241

Pu and

242

Pu isotopes at approximately

0.3 eV (group 52) and 1 eV (group 29). The increased thermal neutron absorption at these energies in MOX FAs leads to significant neutron streaming from the UOX assemblies to the MOX assemblies. Therefore, we observe albedos significantly higher than 1 for the MOX assemblies for the above mentioned energies and albedos less than 1 for the UOX assemblies. The results in these figures actually suggest that more energy groups in the NEM calculations should be used in the energy range 0.0 – 3.0 eV. Figures 5.3.1 and 5.3.2 also show the spatial dependence of the albedo. In both figures one can notice that the albedos for pin-cells 16 and 17 are different, which actually suggest that the spatial dependence of the albedos is most important in the vicinity where the four different assemblies meet (center of the core for the C3 configuration).

87

1

1.20

2 1.10

3 4

1.00

5 6

Albedo

0.90

7 0.80

8 9

0.70

10 11

0.60

12 0.50

13 14

0.40 0

10

20

30

40

50

60

70

Enrgy group #

15 16 17

Figure 5.3.1: South side albedo energy and spatial dependence (C3, NW-UOX)

1

2.20

2 3

2.00

4 5

1.80

Albedo

6 7

1.60

8 1.40

9 10

1.20

11 12

1.00

13 14

0.80 0

10

20

30

40

Enrgy group #

50

60

70

15 16 17

Figure 5.3.2: North side albedo energy and spatial dependence (C3, SW-MOX)

Figures 5.3.3 and 5.3.4 show the albedo energy group dependence for all pin cells located in the south side of the NE-MOX assembly and the north side of the SE-UOX assembly for the C5 configuration. The order of the pin cells, as explained earlier, is from left to 88

right for the North and South sides and from top to bottom for the East and West sides of the assemblies. This means that in Figures 5.3.3 and 5.3.4 cell # 1 is located in the point where the four assemblies meet while cell # 17 is located in the periphery of the core. Figures 5.3.3 and 5.3.4 show that the albedo energy dependence is similar to the one observed for the C3 problem on the UOX/MOX interface. The only difference is in the spatial distribution of the albedos. In this case, the albedos are different not only for pin cells 1 and 2 but also for pin cells 16 and 17 showing that the albedos are spatially dependent also at the periphery of the core at the fuel/reflector interface.

1

2.20

2 2.00

3

1.80

4 5

Albedo

1.60

6 7

1.40

8 1.20

9 10

1.00

11 0.80

12 13

0.60

14

0.40 0

10

20

30

40

Enrgy group #

50

60

70

15 16 17

Figure 5.3.3: South side albedo energy and spatial dependence (C5, NE-MOX)

89

1

1.60

2 3

1.40

4 5

Albedo

1.20

6 7

1.00

8 9 10

0.80

11 12

0.60

13 14

0.40 0

10

20

30

40

50

60

15

70

16

Enrgy group #

17

Figure 5.3.4: North side albedo energy and spatial dependence (C5, SE-UOX)

1

2.50

2 3 2.00

4 5 6

1.50 Albedo

7 8 9

1.00

10 11 0.50

12 13 14

0.00 0

10

20

30

40

Enrgy group #

50

60

70

15 16 17

Figure 5.3.5: East side albedo energy and spatial dependence (C5, NE-MOX)

90

1

2.00

2 1.80

3

1.60

4

1.40

5 6

Albedo

1.20

7

1.00

8

0.80

9 10

0.60

11

0.40

12

0.20

13 14

0.00 0

10

20

30

40

50

Enrgy group #

60

70

15 16 17

Figure 5.3.6: East side albedo energy and spatial dependence (C5, SE-UOX)

The albedo energy dependence at the fuel/reflector interface is provided in Figures 5.3.5 and 5.3.6. These figures demonstrate that the albedos are very small in the high energy region and steadily increase with the decrease of the energy. This is due to the fact that the fast neutrons, which leak from the core, are thermalized in the reflector (moderator) and some of them reenter the core as thermal neutrons. This means that for the C5 problem more energy groups will need to be used in the NEM calculation not only for the thermal energy range but for the entire energy spectrum.

5.3.2

PRG-NEM solution to the C3 benchmark problem

The C3 problem was first computed with the global albedos (without any reconstruction) in two energy groups to investigate the applicability and the convergence of the embedded lattice methodology. Option “loc-0” of the PRG-NEM code was used. The results in the fist iteration for normalized FA power and keff were compared to the 91

PARAGON mini-core reference solution. The comparisons are shown in Table 5.3.7. As explained earlier, the results after the first iteration can be considered as results of the conventional methodology for core calculations to which from now on I will refer as “standard methodology”. The results of the embedded calculation (after the last converged iteration) are also compared to the PARAGON reference results for normalized FA power, normalized pin-powers and keff and are shown in Tables 5.3.8 and 5.3.9.

Table 5.3.7: Standard methodology results compared to reference keff


NEM

1.25955

Reference

1.25496

Diff, pcm NEM Reference Abs Diff, % PRGN-NEM Reference Abs Diff, %

1.165 1.1238 4.12 0.8348 0.8764 -4.16

459 0.8347 0.8764 -4.17 1.1655 1.1234 4.21

Table 5.3.8: Embedded lattice results compared to reference (loc-0) keff


PRGN-NEM

1.25813

Reference

1.25496

Diff, pcm PRGN-NEM Reference Abs Diff, % PRGN-NEM Reference Abs Diff, %

92

317 1.1411 0.8589 1.1238 0.8764 1.73 -1.75 0.8589 1.1411 0.8764 1.1234 -1.75 1.77

Table 5.3.9: Embedded lattice pin-power statistics MIN MAX AVG RMS

NW-UOX -9.11 2.92 1.05 1.47

NE-MOX -6.30 44.11 2.30 4.47

SW-MOX -6.30 44.11 2.30 4.47

SE-UOX -9.13 2.98 1.05 1.48

The convergence criteria for the embedded lattice iterations were set to be 1.0E-4 on the absolute change of the albedo boundary conditions. The embedded lattice methodology converged for 12 iterations. As it can be seen from these results, there is improvement in keff and the normalized FA powers. The eigenvalue improved with 142 pcm and the differences in the normalized FA power decreased from 4.2 % to 1.8 %. However, as Table 5.3.9 shows, the pin power differences from the reference are unacceptably high. The pin-power statistic shows that the average error is in the range of 1-2 % but in some pin cells the difference reaches 44 %. Even though high pin-power errors were observed, these results showed that the embedded lattice calculations converge and improve the results of the standard methodology. Please keep in mind that the non-linear iteration scheme is not completely implemented in PRG-NEM as we don’t have access to the PARAGON’s source. It is expected that with its full implementation the embedded calculations will converge faster.

The C3 problem was next computed using reconstructed in space, energy and angle albedo boundary conditions. The reconstruction of the boundary conditions is expected to significantly improve the pin-powers because this type of boundary conditions will supply more detailed information to the embedded single-assembly heterogeneous calculation. This calculation was performed by using two energy groups in the global core calculation while for the single-assembly PARAGON heterogeneous calculation 70 93

energy groups were used. The PARAGON calculations are always performed in 70 energy groups because the available PARAGON cross-section library is a 70-group energy library. Surprisingly, the single-assembly PARAGON calculations did not converge in the 4th iteration. In order to investigate this problem the reconstruction of the boundary conditions was separated. Using only expansion in space of the albedos, new calculation with the PRG-NEM code was performed again with two energy groups in the global core calculation. Similar to the previous calculation, PARAGON did not converge in the 4th iteration. Figures 5.3.7 and 5.3.8 show detailed results after the second and third iterations. Each figure shows the spatial shapes of the outgoing currents, for the North side of the SW-MOX assembly and for the South side of the NW-UOX assembly, used for computing the local albedos on the North side of the SW-MOX assembly. These figures also show the division of the spatial shape 2 by spatial shape 1 results and the computed albedo for the North side of the SW-MOX assembly. In Fig. 5.3.8 the reference spatial shape from the reference PARAGON calculation in two energy groups is also added for comparison purposes. In this figure “E.” stands for Embedded and “R.” stands for Reference.

94

Spatial shape-1 (SW-MOX-North)

Spatial shape-2 (NW-UOX-South)

1.25

1.10

1.20 Current's spatial shape

Current's spatial shape

1.05 1.15 1.10 1.05

1

1.00

2

0.95 0.90

1.00 1

0.95

2

0.90 0.85

0.85 0.80

0.80 0

2

4

6

8

10

12

14

16

18

0

2

4

6

Pin-cell #

8

10

12

14

16

18

Pin-cell #

Shape-2 / Shape-1

Albedo (SW-MOX-North)

1.20

1.60 1.50

1.10

1.40 1.00

Albedo

1.30 1

0.90

2

1

1.20

2

1.10

0.80

1.00 0.70

0.90 0.60

0.80 0

2

4

6

8

10

12

14

16

18

0

2

4

6

Pin-cell #

8

10

12

14

16

18

Pin-cell #

Figure 5.3.7: Results after iteration # 2

Spatial shape-1 (SW-MOX-North)

Spatial shape-2 (NW-UOX-South)

1.40

1.80

1.30

1.60 Current's spatial shape

Current's spatial shape

1.20 1.10 E. Group 1

1.00

E. Group 2

0.90

R. Group 1

0.80

R. Group 2

0.70 0.60 0.50

1.40

E. Group 1 E. Group 2

1.20

R. Group 1 R. Group 2

1.00 0.80

0.40 0

5

10

15

20

0.60 0

Pin-cell #

5

10

15

20

Pin-cell #

Shape-2 / Shape-1

Albedo (SW-MOX-North)

3.50

5.00

3.00

4.50 4.00

2.50

3.50

E. Group 1 2.00

Albedo

E. Group 2

R. Group 1

1.50

R. Group 2

E. Group 1

3.00

E. Group 2

2.50

R. Group 1

2.00

R. Group 2

1.00

1.50 0.50

1.00 0.50

0.00 0

5

10

15

20

0.00

Pin-cell #

0

5

10 Pin-cell #

Figure 5.3.8: Results after iteration # 3

95

15

20

One can see from Fig. 5.3.7 after the second iteration the outgoing currents on the North side of the SW-MOX assembly are increasing from the periphery of the core to the center of the core (pin-cell # 1 is at the periphery and pin-cell # 17 is at the center of the core) and for the South side of the NW-UOX assembly the outgoing currents shape is opposite. From Fig. 5.3.8 it can be seen that after iteration 3 the shape of the outgoing currents and the shape of the albedos are exactly opposite to the ones after the second iteration shown in Fig. 5.3.7. In other words, the spatial shape of the outgoing currents as computed by PARAGON significantly changes its shape from iteration to iteration instead of converging. The comparison of the albedos after the 3rd iteration to the reference albedos (Fig. 5.3.8) also shows that the albedos after the third iteration (used as a boundary condition in the 4th iteration) are completely different than the reference albedos. The same situation was observed in the previous calculations when the combined expansion in space, energy and angle was used. These results suggested that the iterations for computing the spatial shape of the outgoing currents are non-convergent process when applying albedo boundary conditions, and that the spatial shape of the outgoing currents from PARAGON cannot be used in order to expand the albedos from the global calculation. Finally, the reason for the observed PARAGON non-convergence in the 4th iteration is the oscillation of the spatial shape, which produced non-physical albedo boundary conditions for PARAGON. The oscillations observed in the embedded calculations are most probably because albedos are used as boundary conditions instead of incoming currents. The albedos fix only the ratio between the partial currents and not the currents themselves which gives PARAGON the freedom to redistribute spatially the currents but still preserve the albedos. Most probably if incoming partial currents were 96

used as boundary conditions the oscillations of the spatial shape would not have been observed. However, since PARAGON does not accept incoming currents as boundary conditions we had to continue the investigations using the albedos. The only possible way to continue this study with the albedo boundary conditions was either to obtain the spatial shape of the albedos from the global NEM calculation by using more than one node per assembly nodalization in NEM (increase the spatial resolution) or to use the reference spatial shape from the PARAGON mini-core solution. Using more than one node per assembly NEM will generate several values for the albedo (instead of only one global albedo) on each side of the fuel assemblies. These several albedo (partial currents) values could be used to approximate the spatial shape. In the case of 17x17 nodes per assembly the exact spatial shape from the global calculation can be obtained for this particular problem.

In order to test if the NEM code will be able to obtain reasonable spatial shape for the reconstruction of the albedo boundary conditions the NEM code was run with 17x17 nodes per assembly. Fig. 5.3.9 shows the albedos obtained by NEM (17x17 NPA) after the first iteration compared to the reference albedos from the mini-core PARAGON calculation for the North side of the SW-MOX assembly. For group one, we observe very good agreement between the reference albedo and the NEM obtained albedo. For group two, one can see that the agreement is also reasonable. More calculations with NEM were performed using different number of nodes per assembly in order to investigate how many nodes per FA side are needed in order to accurately approximate the boundary conditions.

97

1.5000 1.4000

Albedo

1.3000

NEM gr 1 NEM gr 2

1.2000

Reference gr 1

1.1000

Reference gr 2

1.0000 0.9000 0

5

10

15

20

Pin-cell #

Figure 5.3.9: NEM albedo (17x17 NPA) versus reference (SW-MOX South)

Embedded calculations with NEM using 17x17 NPA were performed. In these calculations the spatial shape is obtained from the NEM (17x17 NPA) calculation (the spatial shape from the previous single-assembly calculation was not used), while the energy and angular shapes were computed from the previous single-assembly PARAGON heterogeneous calculation and used to reconstruct the NEM obtained incoming and outgoing currents (17 per FA side) in energy and angle. Surprisingly, in the 5th iteration PARGON did not converge for the MOX fuel assemblies. After careful examination of the results of the embedded calculations prior to iteration 5 it was observed that the boundary conditions as calculated by NEM are not converging. It was observed that the albedos through the iterations continuously increase or decrease. In contrast to the previously observed problems with the spatial shape in these calculations the shape is not changing, only the amplitude of the albedo is changing. In order to investigate the reason for the observed problems the same embedded calculations but

98

without using DFs was performed. In this calculation PARAGON converged but the final results were not very close to the reference because in this calculation the spatial shape and the values of the albedos are not as close to the reference as in the case with the DFs. The reason for the increasing/decreasing values of the albedos was identified to be in the equivalence procedure for DF generation. When more than one node per assembly in the NEM calculation is used, NEM becomes higher order method and the consistency between the equivalence procedure and NEM is no longer valid. In order to be able to use NEM in the embedded calculations with more than one node per assembly, new equivalence procedure will need to be developed. The development of such equivalence procedure is left for future studies. Nevertheless,we observed that the spatial shapes of the albedo boundary conditions from NEM are stable and not changing significantly through the iterations when the DFs were not used, which is an indication that the embedded calculations with NEM calculated spatial shape are feasible.

The next step was to investigate the convergence of the energy shape. Using the energy reconstruction option for expansion of the albedo boundary conditions only in energy (no spatial or angular), new calculations with the PRG-NEM code were performed using two energy groups for the global calculation. Contrary to the calculations with only spatial expansion of the BCs embedded calculations did not experience any convergence problems. Fig. 5.3.10 shows the albedo for the North side of the SW-MOX assembly after each of the first 5 iterations in 70 energy groups compared to the albedos from the reference mini-core calculation. The albedos from the reference mini-core calculation were obtained by spatially averaging the currents per FA side in each fine group. As it can be seen from the figure the highest difference is observed in the thermal region after 99

the first two iterations (group 55 to 70). However, one can see that from iteration to iteration the albedos in the thermal region are getting closer to the reference which means that the energy shape is converging to the reference one.

2.50E+00

2.00E+00

Albedo

Iteration 1 Iteration 2

1.50E+00

Iteration 3 Iteration 4

1.00E+00

Iteration 5 Reference

5.00E-01

0.00E+00 0

10

20

30

40

50

60

70

Energy group #

Figure 5.3.10: Albedo for the North side of the SW-MOX assembly

The angular reconstruction can not be used separately and have to be done in combination with the spatial and energy reconstruction. Obviously, because of the problems with the oscillation of the spatial shape, the reconstruction in space using the PARAGON results from the single-assembly calculation from the previous iteration is not feasible and it can not be used to investigate the behavior of the angular reconstruction. Therefore, either the reference spatial shape of the outgoing currents from the reference PARAGON mini-core calculation will have to be used or the NEM (17x17) spatial shape obtained after the first iteration. The reference spatial shape of the outgoing currents from the reference PARAGON mini-core calculation was selected to be used for the purposes of continuing this study and investigating the angular reconstruction. The reference spatial shape was fixed through the embedded iterations while the energy and

100

angular shapes were updated in each iteration step using the information from the singleassembly embedded calculation from the previous iteration step.

The following calculations were performed with threefold purpose: first to investigate the convergence of the embedded iterations when a fixed spatial shape is used; second to examine the convergence of the angular shape, namely to see if the reconstruction in angle will improve the results; and finally to see what the contribution is of each of the reconstructions to the overall improvement of the results. The following four cases were considered: •

Case 1: Local albedo – reconstruction in space (loc-1)

•

Case 2: Local albedo – reconstruction in energy (loc-2)

•

Case 3: Local albedo – reconstruction in space and energy (loc-3)

•

Case 4: Local albedo – reconstruction in space, energy and angle (loc-4)

In order also to investigate the convergence of the embedded calculations the convergence criterion was tighten to 1.0E-4 absolute difference. All of the cases were run with only two energy groups in the global core calculation. The comparisons of the normalized FA power distributions and keff to the reference PARAGON results for all of the cases are provided in Table 5.3.10 and pin-power statistics is shown in Table 5.3.11.

Table 5.3.10: keff and normalized FA power comparisons (2-group)

Case 1 Case 2 Case 3 Case 4

keff Diff, pcm

NW-UOX Abs. diff. %

NE-MOX Abs. diff. %

SW-MOX Abs. diff. %

SE-UOX Abs. diff. %

198 162 38 50

0.01 1.56 -0.14 -0.04

-0.03 -1.58 0.12 0.01

-0.03 -1.58 0.12 0.01

0.05 1.60 -0.10 0.01

101

The results in Tables 5.3.10 and 5.3.11 show that the spatial reconstruction of the albedo boundary conditions (Case 1) significantly improved not only the assembly normalized powers and keff but also the pin-powers in comparison to the results when only global albedo is used (Tables 5.3.8 and 5.3.9). The reconstruction in energy (Case 2) led to significant improvement of the keff while the differences in the normalized FA powers stayed almost the same as in the case of embedded with only global albedo used as boundary condition.

Table 5.3.11: Pin-power statistics (2-group)

Case 1

Case 2

Case 3

Case 4

NW-UOX

NE-MOX

SW-MOX

SE-UOX

MIN Diff, % MAX Diff, % AVG

-0.96 3.12 0.79

-4.09 0.94 0.58

-4.09 0.94 0.57

-1.00 3.09 0.80

RMS

0.97

0.81

0.81

0.98

-10.81 2.22 1.05

-5.60 42.91 2.41

-5.60 42.81 2.41

-10.83 2.28 1.05

RMS

1.62

4.52

4.52

1.63


-0.76 1.09 0.20

-1.59 1.74 0.33

-1.59 1.64 0.33

-0.69 1.07 0.20

RMS

0.29

0.51

0.50

0.28


-0.66 0.41 0.14

-0.90 1.71 0.25

-0.90 1.71 0.25

-0.59 0.42 0.14

RMS

0.20

0.37

0.37

0.19


The application of both reconstructions in space and energy (Case 3) further improved the results. The keff difference decreased to 38 pcm, the difference in the normalized FA power also decreased to less than 0.2 % and the maximum pin-power errors decreased to less than 2 %. In Case 4, the keff and the normalized FA power stayed approximately the same as in Case 3 but the application of the angular reconstruction of the albedo 102

boundary conditions further decreased the differences in the pin-powers. Overall these results show that the increase of the complexity of the boundary conditions leads to increase of the accuracy of the results as expected. These results actually prove that the embedded calculations are feasible as long as the correct spatial shape is provided. Currently the only practical way of computing the spatial shape is by performing NEM calculation with 17 by 17 nodes per assembly. Since the equivalence procedure for DF generation will not be consistent when NEM is used with more than one node per assembly the embedded calculations will need to be performed with one node per assembly and use the precomputed by NEM (17x17 NPA) spatial shape which will be fixed through the embedded iterations. Such calculation was performed again in two energy groups (Case 5). The results of this calculation are provided in Tables 5.3.12 and 5.3.l3.

Table 5.3.12: keff and normalized FA power comparisons (Case 5) PRG-NEM

1.25541

keff

Reference

1.25496


Diff, pcm PRG-NEM Reference Abs Diff, % PRG-NEM Reference Abs Diff, %

1.1234 1.1238 -0.04 0.8767 0.8764 0.03

45 0.8767 0.8764 0.03 1.1233 1.1234 -0.01

Table 5.3.13: Pin-power statistics (Case 5) MIN Diff, % MAX Diff, % AVG RMS

NW-UOX -1.51 0.57 0.15 0.23

NE-MOX -1.35 6.81 0.27 0.60

103

SW-MOX -1.24 6.91 0.27 0.60

SE-UOX -1.53 0.55 0.15 0.22

As it can be seen from the results in Table 5.3.12 the keff and the normalized FA powers of Case 5 agree very well with the reference results and are comparable to the results of Case 4 when the reference spatial shape was used. However, higher pin-power differences are observed in comparison to Case 4, which was expected since the NEM generated shape of the albedos is close to the reference but is not exactly the same (see Fig. 5.3.9) and also because the precomputed NEM spatial shape is fixed and it is not updated through the embedded iterations.

The convergence of all 5 cases and also when only the global albedo is used are provided in Fig. 5.3.11. This figure shows the absolute change in the albedo boundary conditions through the iterations. Figure 5.3.12 shows the change of the keff through the iterations again for all of the cases including the global albedo case.

1.0E+00

BC absolute differnce

1.0E-01 Global albedo Case 1

1.0E-02

Case 2 Case 3

1.0E-03

Case 4 Case 5

1.0E-04

1.0E-05 0

5

10

15

20

25

30

35

Iteration #

Figure 5.3.11: Convergence through the iterations

104

1.26000 1.25900 Global albedo

1.25800

Case 1

kinf

Case 2

1.25700

Case 3 Case 4

1.25600

Case 5 Reference

1.25500 1.25400 0

5

10

15

20

25

30

35

Iteration #

Figure 5.3.12: keff through the iterations

Fig. 5.3.12 shows that basically the largest change in keff is in the 2nd iteration and in the rest of the iterations keff does not change significantly. The change in the 2nd iteration is associated with the first application of the boundary conditions obtained from the global core calculation which accounts for the environment of each assembly in the core. Fig. 5.3.11 shows that the convergence is faster for the case which uses global albedo as a boundary condition and slower for the cases which use more complex boundary conditions. One can observe that all cases are converged to 1E-3 in 6 to 8 iterations but to converge to 1E-4 is taking approximately 30 iterations. However, the results after the 6th iteration do not significantly change which means that the convergence criteria may be relaxed to 1E-2 or 1E-3. It is expected that with the full implementation of the non-linear iteration scheme, the embedded calculations will converge faster.

The analyses of the reference albedo boundary conditions, discussed previously in

105

subsection 5.3.1, suggested that using more energy groups in the global calculation will eventually lead to better accuracy of the embedded calculation results. Therefore, Case 4 was computed again by using four energy groups in the global calculation instead of two energy groups. Selected four energy group structure is shown in Table 5.3.14. As suggested by the albedo energy dependence, one fast group was created to isolate the high energy dependence of the albedo, while the rest three groups are in the thermal and epithermal regions. The results of Case 4 with four energy groups are shown in Tables 5.3.15 and 5.3.16.

Table 5.3.14: Four energy group structure Group #

1 2 3 4

Energy cut-off, eV 9.118E+3 4.000E+0 0.625E+0 0.000E+0

PARAGON’s library group # 14 27 46 70

Table 5.3.15: keff and normalized FA power comparisons (Case 4-4) PRG-NEM

1.25544

keff

Reference

1.25496



106

1.1238 1.1238 0.00 0.8762 0.8764 -0.02

48 0.8762 0.8764 -0.02 1.1238 1.1234 0.04

Table 5.3.16: Pin-power statistics (Case 4-4) NW-UOX -0.28 0.29 0.08 0.10

MIN Diff, % MAX Diff, % AVG RMS

NE-MOX -0.24 0.91 0.12 0.16

SW-MOX -0.24 0.91 0.12 0.16

SE-UOX -0.26 0.27 0.07 0.09

Tables 5.3.15 and 5.3.16 show that the results further improve with the increase of the number of energy groups in the global calculation. The pin-power differences decreased to less than 1 % while the differences in the keff and the normalized FA power remained approximately the same as in the two group calculations. Figure 5.3.13 shows the convergence through the iterations for Case 4 with four energy groups. One can observe that the convergence process is also improving when the number of energy groups in the global calculation is increased. Using four energy groups Case 4 converged to 1E-4 in 15 iterations while when 2 groups were used Case 4 converged to 1E-4 in 30 iterations.

1.0E+00

Convergence

1.0E-01

1.0E-02

1.0E-03

1.0E-04

1.0E-05 0

2

4

6

8

10

12

14

16

Iteration #

Figure 5.3.13: Convergence through the iterations for Case 4 (4-group)

Case 4 was calculated also by using seven groups in the global calculation. Selected seven group energy structure is shown in Table 5.3.17. Again in this structure more 107

groups were added in the thermal and epithermal regions as suggested by the study on the albedo energy dependence.

Table 5.3.17: Seven energy group structure Energy cut-off, eV 9.118E+3 2.770E+1 4.000E+0 2.100E+0 0.625E+0 0.100E+0 0.000E+0

Group #

1 2 3 4 5 6 7

PARAGON’s library group # 14 24 27 31 46 57 70

Using the seven group structure Case 4 was computed again and its results are shown in Tables 5.3.18 and 5.3.19. Similar to the four group calculation, further decrease in the pin-power errors can be observed from these results while the keff and the normalized FA powers differences remained the same as in the previous Case 4 calculations.

Table 5.3.18: keff and normalized FA power comparisons (Case 4-7) PRG-NEM

1.25546

keff

Reference

1.25496



1.1237 1.1238 -0.01 0.8762 0.8764 -0.02

50 0.8762 0.8764 -0.02 1.1238 1.1234 0.04

Table 5.3.19: Pin-power statistics (Case 4-7) MIN Diff, % MAX Diff, % AVG RMS

NW-UOX -0.26 0.19 0.08 0.10

NE-MOX -0.24 0.54 0.13 0.16

108

SW-MOX -0.24 0.54 0.13 0.16

SE-UOX -0.26 0.19 0.07 0.09

The convergence of Case 4 with seven group cross-sections is shown in Figure 5.3.14. As the figure demonstrates, using seven energy groups slightly improved the convergence in comparison to the four group calculation.

1.0E+00

Convergence

1.0E-01

1.0E-02

1.0E-03

1.0E-04

1.0E-05 0

2

4

6

8

10

12

14

Iteration #

Figure 5.3.14: Convergence through the iterations for Case 4 (7-group)

5.3.3

PRG-NEM solution to the C5 benchmark problem

This subsection of the thesis discusses the C5 problem results obtained with the embedded lattice methodology. The methodology for the embedded lattice calculation was slightly modified in order to be applied to the C5 problem. The most important differences had to be introduced because of the reflector assemblies present in the C5 problem. The reflector cross-sections and the data needed for the generation of the discontinuity factors cannot be updated through the embedded iterations using the current methodology for embedded lattice calculations due to the PARAGON eigenvalue calculations. Therefore the reflector data was fixed in the embedded process and it was 109

not updated. Another difference is that in order to compute and reconstruct the albedos on the Fuel/Reflector interfaces the spatial, energy and angular shapes of the outgoing from the reflector currents were extracted from the mini-core reference calculation. The shapes of the outgoing from the reflector currents were also not updated during the embedded iterations. Only the energy shape of the outgoing currents on the Fuel/Fuel interfaces and the energy shape of the outgoing currents from the fuel region to the reflector were updated through the embedded iterations. Such treatment of the reflector is not perfect because it requires the existence of the reference mini-core solution. Other treatments of the reflector are possible, for example 1-D reflector, but are left for future studies. Even though this treatment of the reflector is not perfect it was decided that this treatment is sufficient for the purposes of this feasibility study.

The first performed computation used two energy groups with 17x17 nodes per assembly in the NEM model. This computation was performed in order to investigate if the albedos calculated by NEM for the C5 problem will be close to the reference albedos as for the C3 problem. Comparison of the albedos for South side of NW-UOX assembly, South side of the NE-MOX assembly, East side of the NE-MOX assembly and East side of the SE-UOX assembly is provided in Figures 5.3.15 to 5.3.18 respectively. The first two comparisons show the albedos at the UOX/MOX interface while the last two comparisons show the albedos at the MOX/Reflector interface and UOX/Reflector interface. One can observe from these comparisons that the NEM 17x17 solution reasonably well predicts the albedos on all possible interfaces in comparison to the reference albedos. Therefore, the NEM obtained spatial shapes of the partial currents can be used to reconstruct spatially the boundary conditions in the embedded calculations. 110

0.95 0.90 0.85 Albedo

NEM gr1 0.80

NEM gr2 REF gr1

0.75

REF gr2 0.70 0.65 0.60 0

2

4

6

8

10

12

14

16

18

Pin-cell #

Figure 5.3.15: NEM 17x17 albedo versus reference albedo (NW-UOX South)

1.60 1.40 1.20

Albedo

1.00

NEM gr1 NEM gr2

0.80

REF gr1 REF gr2

0.60 0.40 0.20 0.00 0

2

4

6

8

10

12

14

16

18

Pin-cell #

Figure 5.3.16: NEM 17x17 albedo versus reference albedo (NE-MOX South)

2.50

Albedo

2.00

NEM gr1

1.50

NEM gr2 REF gr1 1.00

REF gr2

0.50

0.00 0

2

4

6

8

10

12

14

16

18

Pin-cell #

Figure 5.3.17: NEM 17x17 albedo versus reference albedo (NE-MOX East)

111

1.80 1.60 1.40

Albedo

1.20

NEM gr1

1.00

NEM gr2

0.80

REF gr1 REF gr2

0.60 0.40 0.20 0.00 0

2

4

6

8

10

12

14

16

18

Pin-cell #

Figure 5.3.18: NEM 17x17 albedo versus reference albedo (SE-UOX East)

The cases used for the C3 problem were run also for the C5 problem. Surprisingly, for the C5 problem the embedded iterations converged to a state different than the reference. One case was selected and run for 100 iterations in order to analyze the reasons for this behavior of the embedded non-linear iterations. The case selected is the four-group calculation using albedo boundary conditions expanded in space and energy (Case 3 in the C3 problem calculations). The reconstruction in space of the albedo boundary conditions was done by using the outgoing partial currents spatial shape from the reference mini-core PARAGON solution. As explained earlier, the reflector crosssections and discontinuity factors were not updated during the embedded iterations. The change of keff and the convergence through the iterations is shown in Figures 5.3.19 and 5.3.20. The comparisons of the embedded lattice methodology solution after 100 iteration steps is provided in Tables 5.3.20 and 5.3.21 for the normalized FA powers and the normalized pin-powers to the reference mini-core PARAGON solution.

112

1.18600 1.18400 1.18200 1.18000 Keff

1.17600

Reference Keff

1.17400 1.17200 1.17000 1.16800 1.16600 0

20

40

60

80

100

Iteration #


1.0E+00

1.0E-01

Convergence

Keff

1.17800

1.0E-02

1.0E-03

1.0E-04

1.0E-05 0

20

40

60

80

100

120

Iteration #

Figure 5.3.20: Convergence through the iterations Table 5.3.20: keff and normalized FA power comparisons PRG-NEM

1.17164

keff

Reference

1.17442



-278 1.8362 0.8137 1.7847 0.8354 5.15 -2.17 0.8138 0.5363 0.8354 0.5444 -2.16 -0.81

113

Table 5.3.21: Pin-power statistics MIN Diff, % MAX Diff, % AVG RMS

NW-UOX -1.45 0.91 0.56 0.66

NE-MOX -41.72 19.22 9.17 13.36

SW-MOX -41.72 19.22 9.16 13.36

SE-UOX -1.12 3.21 0.32 0.45

Fig. 5.3.19 shows that the embedded calculations converge to a different state. The comparison of the results given in Tables 5.3.20 and 5.3.21 show once more that the embedded calculation converges to a different state. However, one can see from Fig. 5.3.19 that the embedded calculations initially converge to a close proximity of the reference keff but after iteration 17 the solution deteriorates. In order to investigate this problem several cases were computed with relaxed convergence criteria for the embedded calculations from 1E-4 to 1E-2. Fig. 5.3.20 shows that using this convergence criteria the embedded calculations are expected to stop approximately between 6th and 9th iteration and the solution is expected to be much closer to the reference one. The following six cases were run: •

Case 1: Global albedo (2-group)

•

Case 2:Global albedo (4-group)

•

Case 3: Local albedo – reconstruction in space and energy (2-group)

•

Case 4: Local albedo – reconstruction in space (NEM) and energy (2-group)

•

Case 5: Local albedo – reconstruction in space and energy (4-group)

•

Case 6: Local albedo – reconstruction in space, energy and angle (4-group)

Only one case using NEM obtained spatial shape is calculated (Case 4) just to show that the results are similar to the ones obtained with the reference spatial shape. Further, only 114

one case with angular reconstruction of the boundary conditions is used because the PARAGON’s inner iteration convergence criteria had to be significantly lowered to 1E-2 in order for PARAGON to converge. This case is shown just to give an idea of the importance of the angular expansion in the C5 problem even though the inner iteration convergence criteria in PARAGON had to be significantly relaxed. The comparisons of the normalized power distributions and keff to the reference PARAGON results for all of the cases as well as the results after the first iteration are provided in Table 5.3.22 and pin-power statistics is shown in Table 5.3.23. The convergence criteria for the embedded iterations used is 1E-2.

Table 5.3.22: keff and normalized FA power comparisons

Iteration 1 Case 1 Case 2 Case 3 Case 4 Case 5 Case 6

keff Diff, pcm

NW-UOX Abs. diff. %

NE-MOX Abs. diff. %

SW-MOX Abs. diff. %

SE-UOX Abs. diff. %

1881 342 326 -67 -65 32 -13

14.46 4.03 1.46 0.16 0.29 -0.14 -0.11

-6.32 -1.89 -0.82 0.02 -0.14 0.25 0.12

-6.32 -1.89 -0.82 0.03 -0.15 0.25 0.12

-1.81 -0.24 0.20 -0.21 0.01 -0.36 -0.13

Using the relaxed convergence criteria the embedded calculations obtained significantly better results than the ones with convergence criteria 1E-4. From these results, similar to the C3 problem, it can be observed that with the increase of the complexity of the boundary conditions and the number of groups in the global calculations the results improve. However, in contrast to the C3 problem, the pin-power errors are significantly higher than the ones for the C3 problem, which actually suggest that there are problems in the embedded calculations. Most probably the non-linear embedded iteration does not converge to the reference PARAGON solution because of the way the reflector data is 115

used. As mentioned earlier, the reflector data is not updated during the non-linear embedded iterations which most probably causes problems and makes the non-linear iterations to converge to a different state.

Table 5.3.23: Pin-power statistics NW-UOX

Case 1

Case 2

Case 3

Case 4

Case 5

Case 6

NE-MOX

SW-MOX

SE-UOX


-7.25 0.68 0.67

-43.62 66.04 10.88

-43.62 66.04 10.88

-22.71 13.31 6.11

RMS

0.98

15.62

15.61

7.46


-6.35 0.72 0.47

-19.19 47.84 4.78

-19.19 48.04 4.78

-15.88 19.61 2.11

RMS

0.76

7.39

7.40

3.38


-2.05 0.75 0.54

-25.43 18.42 5.88

-25.53 18.42 5.88

-0.85 3.02 0.52

RMS

0.66

8.67

8.68

0.69


-2.65 0.65 0.48

-25.59 19.42 5.94

-25.69 19.52 5.94

-2.49 2.53 0.59

RMS

0.61

8.72

8.73

0.81


-1.35 0.46 0.39

-18.43 13.26 4.21

-18.43 13.24 4.20

-0.52 4.51 0.22

RMS

0.49

6.27

6.27

0.46


-1.55 0.41 0.32

-14.93 10.14 3.45

-14.93 10.14 3.45

-1.10 8.31 0.59

RMS

0.42

5.05

5.05

0.97

To investigate the observed convergence problems of the embedded methodology when applied to the C5 problem a new calculation was performed using fixed reference spatial and energy shapes and 4 energy groups in the global NEM calculation (the same case as case 5 above and the case which was run for 100 iterations except that fixed reference 116

shapes were used). The reference shapes were obtained from the PARAGON mini-core calculation. The goal of this calculation is to show the impact of the energy shape convergence on the embedded methodology convergence. Figures 5.3.21 and 5.3.22 show the keff and the embedded convergence through the iterations. This calculation was run only for 25 iterations as the phenomenon that needs to be analyzed is taking place from iteration 15 to 25. As it can be seen from these figures the results with the fixed reference energy shape are very close to the ones previously obtained for case 5 (Fig. 5.3.19). The same deterioration of the results from iteration 15 to 20 is observed as in Fig. 5.3.19, which actually suggests that the convergence problems of the embedded methodology for the C5 problem are not caused by the energy reconstruction of the boundary conditions. The only significant difference caused by the use of fixed energy shape is in iterations 2 to 5. When the energy shape is obtained from the previous iteration we see in Fig. 5.3.19 that the embedded calculations need several iterations in order to converge to the reference eigenvalue (time needed for converging the energy shape) while when the reference energy shape is used it can be seen that there are no fluctuations in the eigenvalue for iteration 2 to 5 (Fig. 5.3.21).

117

1.18600 1.18400 1.18200

Keff

1.18000 1.17800

Keff

1.17600

Reference Keff

1.17400 1.17200 1.17000 1.16800 0

5

10

15

20

25

30

Iteration #


1.0E+00

Convergence

1.0E-01

1.0E-02

1.0E-03 0

5

10

15

20

25

30

Iteration #

Figure 5.3.22: Convergence through the iterations

To further investigate the observed convergence problems of the embedded methodology, when applied to the C5 problem, the albedos calculated by NEM for case 5 (space and energy reconstruction) were extracted from the outputs in order to investigate the albedo change through the iterations. The albedo change through the iterations is plotted in Figures 5.3.23 to 5.3.26 for the South side of the NW-UOX, South side of the NE-MOX,

118

East side of the NE-MOX and East side of the SE-UOX assemblies respectively. All these interfaces can be considered as representatives of all of the interfaces encountered in the C5 problem due to the symmetry of the problem. The albedos are plotted for 32 iterations as the phenomenon that we would like to analyze is taking place up to iteration 25. The values of the albedos do not change significantly after iteration 30 (Fig. 5.3.20). The final albedo values are compared to the reference albedos obtained from the minicore PARAGON calculation in Tables 5.3.24 to 5.3.27. One can see from these results that the most significant change in the albedos is observed on the MOX/Reflector interface (East side of the NE-MOX) in the thermal group. Looking into Table 5.3.26 and Figure 5.3.27 it can be seen that initially the thermal albedo on the MOX/Reflector interface increases and attempts to converge to the reference albedo (1.9823) but after iteration 5 starts to decrease and finally converges to a very different value - 1.0270. Slightly high errors can be observed also for some of the other albedos but the error in the thermal albedo on the MOX/Reflector interface is the most significant one which is actually the reason for the convergence of the embedded calculations to a state different than the reference one.

0.95 0.90 0.85

Albedo

0.80

Group 1

0.75

Group 2

0.70

Group 3 Group 4

0.65 0.60 0.55 0.50 0

5

10

15

20

25

30

35

Iteration #

Figure 5.3.23: Albedo through the iterations (South side of NW-UOX)

119

1.50 1.40 1.30

Albedo

1.20 Group 1

1.10

Group 2

1.00

Group 3

0.90

Group 4

0.80 0.70 0.60 0.50 0

5

10

15

20

25

30

35

Iteration #

Figure 5.3.24: Albedo through the iterations (South side of NE-MOX) 2.00 1.80 1.60

Albedo

1.40 Group 1

1.20

Group 2

1.00

Group 3

0.80

Group 4

0.60 0.40 0.20 0.00 0

5

10

15

20

25

30

35

Iteration #

Figure 5.3.25: Albedo through the iterations (East side of NE-MOX) 1.60 1.40 1.20

Albedo

1.00

Group 1 Group 2

0.80

Group 3 Group 4

0.60 0.40 0.20 0.00 0

5

10

15

20

25

30

35

Iteration #

Figure 5.3.26: Albedo through the iterations (East side of SE-UOX)

120

Table 5.3.24: Embedded vs reference albedo (South side of NW-UOX) Groups 1 2 3 4

Reference 0.8702 0.9084 0.7770 0.6745

Embedded 0.8688 0.9044 0.7733 0.6485

Rel. Diff, % -0.16 -0.43 -0.48 -3.85

Table 5.3.25: Embedded vs reference albedo (South side of NE-MOX) Groups 1 2 3 4

Reference 0.8374 0.9438 1.1017 1.3436

Embedded 0.8387 0.9442 1.1040 1.3580

Rel. Diff, % 0.16 0.04 0.21 1.07

Table 5.3.26: Embedded vs reference albedo (East side of NE-MOX) Groups 1 2 3 4

Reference 0.2529 0.7513 1.0273 1.9823

Embedded 0.2535 0.7347 0.9929 1.0270

Rel. Diff, % 0.24 -2.21 -3.35 -48.19

Table 5.3.27: Embedded vs reference albedo (East side of SE-UOX) Groups 1 2 3 4

Reference 0.2530 0.7447 0.8521 1.4923

Embedded 0.2518 0.7454 0.8487 1.4880

Rel. Diff, % -0.47 0.09 -0.40 -0.29

These comparisons once more show that most probably the reason for the observed 121

divergence of the embedded calculations is the fixed reflector data through the iterations. The albedo boundary conditions also enhance the observed divergence as they fix only ratio of two physical quantities instead of the quantities themselves. One possible solution for this problem could be to use partial currents as boundary conditions for the embedded calculations instead of the albedos. If partial currents are used along with PARAGON fixed source calculations it will be possible to update not only the fuel assembly’s data but also the reflector’s data during the embedded iterations. Investigation of such treatment of the reflector data is left for future studies as currently PARAGON does not accept incoming currents as boundary conditions.

5.3.4

Sensitivity study on the utilization of the DFs

Several calculations for the C3 problem were performed with the PRG-NEM code in order to investigate the importance of the physical response matrix method implementation for the utilization of the DFs in NEM. Case 4 from the previous C3 problem calculations was selected (spatial, energy and angular reconstruction of the boundary conditions) in order to conduct this investigation. The following three cases were considered:

•

Case 1: The DFs were not use in the global NEM calculation;

•

Case 2: The homogeneous formulation of the DFs in NEM was used;

•

Case 3: The physical formulation of the DFs in NEM was used.

122

Case 3 is actually exactly the same case used in subsection 5.3.2 (Case 4) of this Chapter. Even though Case’s 3 results were shown earlier, they will be shown also here for clarity. Tables 5.3.28 and 5.3.29 show the results of all cases.

Table 5.3.28: keff and normalized FA power comparisons

Case 1 Case 2 Case 3

keff Diff, pcm 143 46 50

NW-UOX Abs. diff. % 2.01 -0.02 -0.04

NE-MOX Abs. diff. % -2.03 0.01 0.01

SW-MOX Abs. diff. % -2.03 0.01 0.01

SE-UOX Abs. diff. % 2.06 0.01 0.01

Table 5.3.29: Pin-power statistics NW-UOX

Case 1

Case 2

Case 3


-0.68 4.29 0.64

NE-MOX -8.49 1.70 1.76

SW-MOX -8.49 1.70 1.75

SE-UOX -0.67 4.37 0.65

RMS

0.85

2.29

2.29

0.86


-3.51 1.14 0.70

-5.79 1.80 1.50

-5.79 1.80 1.49

-3.53 1.10 0.69

RMS

0.87

1.91

1.91

0.86


-0.66 0.41 0.14

-0.90 1.71 0.25

-0.90 1.71 0.25

-0.59 0.42 0.14

RMS

0.20

0.37

0.37

0.19

Case’s 1 results show that when the DFs are not used the results of the embedded lattice calculation methodology are significantly worse in comparison to the cases when the DFs are used. The results of Cases 2 and 3 are comparable for keff and the normalized FA power but the pin-power differences for Case 3 are smaller than the ones for Case 2. In order to further investigate the differences between all of the cases additional results were compared. The eigenvalues of the single-assembly PARAGON embedded calculations after the last converged iteration are compared to the reference mini-core PARAGON 123

eigenvalue in Tables 5.3.30 to 5.3.32. In the best case scenario, the eigenvalues from the single-assembly calculations should reproduce the eigenvalue of the mini-core reference solution because during the embedded iterations the single-assembly calculation accounts for the environment of each assembly in the core.

Table 5.3.30: Eigenvalues comparisons (Case 1) Eigenvalues 1.25496 Reference 1.25398 NW-UOX 1.25011 NE-MOX 1.25011 SW-MOX 1.25398 SE-UOX

Difference, pcm -98 -485 -485 -98


Difference, pcm -405 -1015 -1017 -405


Difference, pcm 48 59 59 48

One can see from the results of these tables the worst comparison of the eigenvalues can be observed for Case 2. Because of the homogeneous implementation of the DFs, NEM provides unphysical boundary conditions due to the discontinuity of the partial currents. Therefore, PARAGON converges to some state different from the reference. The best comparison is observed for Case 3 as expected. The comparison of the results for Cases 1 124

and 3 shows the importance of the utilization of the DFs in the embedded lattice calculation. The eigenvalue difference for Case 3 is decreased from 142 pcm to 50 pcm and the differences in the normalized FA power is decreased from approximately 2 % to 0.04 %. The pin-power differences also decreased from about 9 % to less than 2 %.


Innovative embedded lattice calculation methodology for core calculations was developed and tested. This methodology is based on embedded lattice physics heterogeneous calculation as well as on-line cross-section and discontinuity factors generation. This methodology is superior to the standard two-step off-line methodology and to the SP3 embedded methodology due to the elimination of most of the uncertainties introduced by the off-line approach of cross-section generation and the introduction of higher order transport method for the embedded calculations.

A detailed investigation showed that the embedded lattice methodology have the potential to significantly improve the results of the core calculation, and even for some cases to basically reproduce the solution of higher order transport methods, but this investigation also revealed the limitations of this methodology. In principle, this method produced its best results only if the albedo boundary conditions were reconstructed in space, energy and angle. However, difficulties were observed in the spatial reconstruction of the boundary conditions. It was discovered that the spatial shape obtained from the embedded single-assembly heterogeneous solution is oscillating when albedos are used as boundary conditions and can not be used for the spatial reconstruction. Remedies to 125

partially overcome this problem were proposed in order to be able to continue this study. First the spatial resolution of the global core calculation was increased in order to be able to obtain the albedo boundary conditions with enough spatial resolution and therefore not to perform spatial reconstruction of the boundary conditions. The second approach was to utilize the spatial shape from the reference mini-core calculation. The second approach is not practical as the first approach but allowed the analysis of the best possible performance of this methodology.

The other problem observed was revealed when this methodology was applied for solving the C5 benchmark problem. The introduction of the reflector caused convergence problems to the embedded lattice methodology. It was observed that if the convergence criteria are tight enough the embedded lattice methodology converges to a state different than the reference. When convergence criteria were relaxed the embedded lattice methodology obtained significantly improved results for keff and normalized FA powers in comparison to the standard methodology. The pin-powers were also improved but pin-power errors still remained higher than expected. This problem was attributed to the fact that the reflector data is not updated through the non-linear iterations. It is expected that if the boundary conditions for the embedded single-assembly calculations are changed from albedos to incoming partial currents this problem will be overcome. The current version of PARAGON code does not accept incoming partial currents as boundary conditions and therefore this investigation is left for future studies.

One of the observations from this study was that the increase of the number of energy groups in the global core calculation improves not only the accuracy of the results but 126

also improves the convergence of the non-linear iterations. The complete implementation of the non-linear iteration scheme is expected to further improve the convergence of the embedded lattice methodology.

This study also showed the importance of the on-line application of the equivalence theory. The on-line application of the equivalence theory improved the global solution, which in turn led to improved boundary conditions for the embedded single-assembly heterogeneous calculation.

Last but not least, this methodology is easy to be implemented since it is an extension of the current methodology. The licensing procedure will be also easier since any licensed lattice code and core simulator can be used within this methodology.

127

CHAPTER 6 CONCLUSIONS AND FUTURE WORK

6.1 Conclusions

Current two-step off-line standard methodology for performing core calculations gives accurate results in an efficient manner despite the fact that infinite lattice calculations and two-group coarse mesh diffusion approximation for the core level calculations are used. While the current methodology produces reasonably accurate pin-power distributions for majority of utilized core loadings it is challenged for some of the core loadings used nowadays in the current reactors. In addition, recently new advanced core configurations were proposed that will be much more heterogeneous compared to the current core loadings. Such advanced cores will introduce challenges for the current two-step off-line methodology and therefore new methodologies for core analysis will have to be developed.

In this thesis two innovative methodologies for core calculations were developed and described. The first methodology is based on embedded SP3 pin-by-pin calculation within the framework of the NEM diffusion core calculation as well as on-line cross-section and discontinuity factors generation. A new FEM diffusion and SP3 code (PSU-FEM) was developed to be used for the embedded SP3 and diffusion pin-by-pin calculations. The 128

PSU-FEM code was designed for both – direct reference pin-by-pin core calculation and embedded assembly pin-by-pin calculation within the NEM core calculation. The verification of the code showed that the results of both diffusion and SP3 options of the PSU-FEM code agree well with other published results. The PSU-FEM code was then coupled to the NEM code for performing embedded SP3 and diffusion calculations. The current version of the coupled code showed that the embedded calculations are feasible in terms of fast convergence. The coupled PSU-FEM/NEM code also obtained reasonable results in comparison to direct pin-by-pin solution, having in mind that currently the code uses assembly wise albedo boundary conditions. The embedded SP3 calculations improved the standard methodology results because of the improved cross-section homogenization and also because of the spatial discretization due to the SP3 pin-by-pin calculation. Despite of all the improvements to the current methodology, the embedded SP3 calculations have limited accuracy due to the use of pre-calculated few-group pinwise homogenized cross-sections. Nevertheless, this methodology was developed in order to demonstrate the feasibility of the embedded approach on a simpler framework.

In the second method, the embedded SP3 calculation was substituted by embedded heterogeneous lattice calculations. In this thesis, the collision probability method was used but any other method can be used. This method is superior to the standard and the SP3 embedded methodologies due to the elimination of most of the uncertainties introduced by the off-line approach of cross-section generation. A detailed investigation showed that the embedded lattice methodology have the potential to significantly improve the results of the core calculation, and even for some cases to basically reproduce the solution of higher order transport methods, but this investigation also 129

revealed the limitations of the approach based on albedo boundary conditions. In principle, this method produced its best results only if the albedo boundary conditions were reconstructed in space, energy and angle. However, difficulties were observed in the spatial reconstruction of the boundary conditions. In order to overcome this problem two remedies were proposed, which partially solved the problem and allowed the continuation of this study. First, the spatial resolution of the global core calculation was increased in order to be able to obtain the albedo boundary conditions with sufficient spatial resolution. The second approach was to utilize the spatial shape from the reference minicore calculation. The second approach is not practical as the first approach but allowed the analysis of the best possible performance of this methodology.

The other problem observed in the application of the embedded lattice methodology was revealed when this methodology was applied for solving the C5 benchmark problem. The introduction of the reflector caused convergence problems to the embedded lattice methodology. It was observed that if the convergence criteria are tight enough the embedded lattice methodology converges to a state different than the reference. When convergence criteria were relaxed the embedded lattice methodology obtained significantly improved results for keff and normalized FA powers in comparison to the standard methodology. The pin-powers were also improved but the pin-power errors still remained higher than expected. This problem was attributed to the fact that the reflector data is not updated through the non-linear iterations. It is expected that if the boundary conditions for the embedded single-assembly calculations are changed from albedos to incoming partial currents this problem will be overcome. The current version of PARAGON code does not accept incoming partial currents as boundary conditions and is 130

not able to perform fixed source calculations and therefore this investigation is left for future studies.

Another important original contribution of this work is the implementation of the physical response matrix method for DF utilization in the NEM code and also the development of the NEM-consistent side-dependant DFs generation procedure. Both contributions allowed the on-line application of the equivalence theory in the embedded calculation methodologies. It was also shown that the on-line application of the equivalence theory is actually one of the biggest contributors to the accuracy of the embedded lattice calculations. The on-line application of the equivalence theory improved the global solution due to the consistent homogenization, which in turn led to improved boundary conditions for the embedded single-assembly heterogeneous calculation.

Last but not least, this methodology is easy to be implemented since it is an extension of the current methodology. If licensing of the embedded methodology is needed in the future, the licensing procedure will be easier than usual since any licensed lattice code and core simulator can be used within this methodology.

6.2 Recommendations for future work

One of the areas where this research could be continued is the improvement of the embedded SP3 methodology. This methodology can be improved by implementation of the pin-wise boundary conditions instead of the global assembly-wise albedos. The 131

embedded SP3 methodology can also be improved by the change of the boundary conditions from albedos to incoming partial currents.

The two problems observed in the embedded lattice methodology should be further investigated as their solution is fundamental prerequisite for the successful application of this methodology to practical problems. An initial investigation of the problem with the spatial reconstruction of the boundary conditions was performed. This investigation showed that one possible solution to this problem is the use of higher spatial resolution in the global core calculation. However, we expect that the utilization of the partial incoming currents as boundary conditions will solve both problems: the spatial reconstruction of the boundary conditions and the treatment of the reflector during the non-linear embedded iterations. Therefore, it is suggested that the further studies in this area should investigate the performance of the embedded methodologies when partial incoming currents are used as boundary conditions.

Another important area for future work is the complete implementation of the non-linear iterations scheme into the embedded lattice methodology. It is expected that the complete implementation of the non-linear scheme will notably improve the convergence performance of this methodology.

132

REFERENCES

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137

APPENDIX A. DESCRIPTION OF C3 AND C5 PROBLEMS

The C3 and C5 MOX fuel assembly problems are specified by Cavarec, et al. [Cavarec, 1994]. Later the C5 problem has been extended to C5G7 MOX Benchmark specified by Lewis, et al. [Lewis, 2001]. The C3 and C5 benchmark problems are used as a basis to investigate the performance and validity of the proposed embedded transport calculation methodologies. The advantages of these benchmarks are: •

Well defined – consist all of the needed information

•

Small problem (It could be calculated with PARAGON and obtained solution could be used as a reference)

•

Challenging configuration (MOX/UOX core loading with or without reflector)

•

Internationally well accepted benchmarks with a lot of published different solutions

•

The C5 problem has 3-D extension that can be used for further studies

The problem to be solved is a small core made of MOX and UO2 fuel assemblies surrounded by a reflector in the C5 case and the same core configuration but with reflective boundary conditions substituting the reflector in the C3 case. One-quarter or one-eight of the core symmetry could be used for solving the problems. The C3 problem 138

core layout is shown in Fig. A.1. One-quarter of the C5 2-D core configuration as well as the boundary conditions is shown in Figure A.2. The overall dimensions of each fuel assembly are 21.421394×21.421394 cm, which are slightly different due to performing the calculations at Hot Zero Power (HZP) conditions instead of cold conditions as specified in the original benchmark specification. The core is surrounded by additional 21.421394 cm radial water reflector.

UO2

MOX

MOX

UO2

Reflective BC

Reflective BC

Reflective BC

Reflective BC

Figure A.1: C3 core configuration

Reflective BC

MOX

MOX Vacuum BC

Reflective BC

UO2

UO2

Moderator Vacuum BC

Figure A.2: C5 core configuration

139

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

1 2 3 4 5 6 7 8

- UO2 Fuel

9 10 11

- 4.3% MOX Fuel

12 13 14

- 7.0% MOX Fuel

15 16

- 8.7% MOX Fuel

17 18 19

- Guide Tube

20 21 22

- Fission Chamber

23 24 25 26 27 28 29 30 31 32 33 34

Figure A.3: Fuel pin compositions and numbering scheme

Each fuel assembly is made up of a 17x17 lattice of square fuel-pin cells, as seen in Figure A.3. The side length of every fuel-pin cell is 1.260082 cm (square lattice pitch). Table A.1 shows the fuel cells dimensions (MOX – 4.3%, 7.0%, 8.7%, and UO2) while Table A.2 shows the guide tube cell dimensions. Tables A.3 and A.4 list the number densities for each medium. The number densities, the dimensions of the fuel, cladding, guide tubes and the fission chambers were taken originally from [Cathalau, 1996] and they were modified to comply with ALPHA code’s results at Hot Zero Power (HZP) conditions. The fission chamber is defined the same as the guide tube.

140

Table A.1: Fuel cell dimensions Medium Fuel Zirconium Clad Moderator

External Radius 0.409501 cm 0.540000 cm Square lattice pitch = 1.260082 cm

Table A.2: Guide-Tube cell dimensions Medium Moderator Aluminum clad Moderator

External Radius 0.340000 cm 0.540000 cm Square lattice pitch = 1.26 cm

Table A.3: Isotopic Number Densities for each medium MOX 4.3% 92235 92238 92234 92236 94238 94239 94240 94241 94242 95241 8016

5.0017595E-05 2.2096719E-02 1.0000000E-09 1.0000000E-09 1.5042693E-05 5.8165081E-04 2.4068309E-04 9.8278930E-05 5.4153696E-05 1.3037001E-05 4.6299170E-02

MOX 7.0%

MOX 8.7%

5.0020146E-05 2.2097846E-02 1.0000000E-09 1.0000000E-09 2.4025892E-05 9.3100330E-04 3.9042074E-04 1.5216398E-04 8.4090620E-05 2.0021576E-05 4.7499189E-02

4.9986268E-05 2.2082880E-02 1.0000000E-09 1.0000000E-09 3.0016944E-05 1.1606552E-03 4.9027675E-04 1.9010731E-04 1.0505930E-04 2.5014120E-05 4.8267995E-02

UOX 8.6557939E-04 2.2243897E-02 1.0000000E-09 1.0000000E-09

4.6218958E-02

Table A.4: Isotopic Number Densities for the moderator and the clad Moderator 1001 8016 5010 40000 13027

Zr clad

Al clad

6.7118404E-02 3.3559202E-02 5.5252484E-06 3.6967821E-02 6.0198146E-02

141

APPENDIX B. C3 AND C5 PIN-POWERS AND COMPARISONS

Table B.1: C3 NW-UOX assembly reference normalized pin-powers

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17

1 1.158 1.162 1.171 1.182 1.193 1.200 1.189 1.181 1.176 1.158 1.144 1.130 1.097 1.055 1.000 0.925 0.812

2 1.162 1.171 1.187 1.211 1.234 1.280 1.228 1.217 1.251 1.195 1.180 1.206 1.136 1.082 1.015 0.932 0.814

3 1.171 1.187 1.228 1.305 1.327 0.000 1.300 1.287 0.000 1.264 1.250 0.000 1.223 1.167 1.053 0.947 0.819

4 1.182 1.211 1.305 0.000 1.348 1.332 1.259 1.245 1.283 1.222 1.212 1.256 1.242 0.000 1.122 0.969 0.828

5 1.193 1.234 1.327 1.348 1.304 1.327 1.259 1.246 1.285 1.224 1.213 1.253 1.204 1.209 1.143 0.991 0.838

6 1.200 1.280 0.000 1.332 1.327 0.000 1.305 1.293 0.000 1.271 1.258 0.000 1.228 1.196 0.000 1.033 0.847

7 1.189 1.228 1.300 1.259 1.259 1.305 1.246 1.236 1.276 1.216 1.203 1.237 1.168 1.134 1.125 0.992 0.842

8 1.181 1.217 1.287 1.245 1.246 1.293 1.236 1.228 1.268 1.209 1.196 1.229 1.160 1.126 1.119 0.989 0.841

9 1.176 1.251 0.000 1.283 1.285 0.000 1.276 1.268 0.000 1.250 1.237 0.000 1.201 1.165 0.000 1.025 0.845

10 1.158 1.195 1.264 1.222 1.224 1.271 1.216 1.209 1.250 1.192 1.181 1.215 1.148 1.117 1.112 0.984 0.839

11 1.144 1.180 1.250 1.212 1.213 1.258 1.203 1.196 1.237 1.181 1.172 1.207 1.144 1.115 1.109 0.981 0.837

12 1.130 1.206 0.000 1.256 1.253 0.000 1.237 1.229 0.000 1.215 1.207 0.000 1.189 1.163 0.000 1.016 0.837

13 1.097 1.136 1.223 1.242 1.204 1.228 1.168 1.160 1.201 1.148 1.144 1.189 1.148 1.160 1.106 0.966 0.824

14 1.055 1.082 1.167 0.000 1.209 1.196 1.134 1.126 1.165 1.117 1.115 1.163 1.160 0.000 1.069 0.934 0.809

15 1.000 1.015 1.053 1.122 1.143 0.000 1.125 1.119 0.000 1.112 1.109 0.000 1.106 1.069 0.977 0.895 0.791

16 0.925 0.932 0.947 0.969 0.991 1.033 0.992 0.989 1.025 0.984 0.981 1.016 0.966 0.934 0.895 0.846 0.770

17 0.812 0.814 0.819 0.828 0.838 0.847 0.842 0.841 0.845 0.839 0.837 0.837 0.824 0.809 0.791 0.770 0.735

33 0.595 0.757 0.748 0.762 0.786 0.843 0.795 0.800 0.858 0.817 0.830 0.899 0.858 0.855 0.879 0.969 0.902

34 0.605 0.595 0.592 0.596 0.605 0.614 0.616 0.621 0.630 0.635 0.643 0.655 0.661 0.671 0.697 0.761 0.912

Table B.2: C3 NE-MOX assembly reference normalized pin-powers

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17

18 0.913 0.903 0.901 0.908 0.917 0.925 0.922 0.922 0.925 0.921 0.920 0.921 0.911 0.901 0.894 0.904 0.957

19 0.761 0.969 0.957 0.970 0.996 1.057 0.997 0.995 1.053 0.996 1.000 1.063 1.003 0.981 0.978 1.030 0.904

20 0.697 0.879 0.886 0.959 0.982 0.000 0.950 0.947 0.000 0.950 0.958 0.000 1.001 0.987 0.930 0.978 0.894

21 0.672 0.856 0.924 0.000 0.937 1.022 0.930 0.926 0.995 0.932 0.942 1.042 0.964 0.000 0.987 0.980 0.901

22 0.661 0.858 0.920 0.915 0.948 0.964 0.887 0.884 0.950 0.892 0.902 0.990 0.984 0.964 1.000 1.002 0.911

23 0.655 0.899 0.000 0.975 0.948 0.000 0.928 0.928 0.000 0.936 0.949 0.000 0.990 1.042 0.000 1.063 0.921

24 0.644 0.830 0.860 0.875 0.857 0.914 0.850 0.852 0.919 0.862 0.871 0.948 0.902 0.942 0.958 1.000 0.920

25 0.635 0.817 0.843 0.857 0.841 0.897 0.838 0.842 0.907 0.853 0.862 0.936 0.892 0.931 0.950 0.996 0.920

26 0.630 0.858 0.000 0.906 0.891 0.000 0.890 0.895 0.000 0.907 0.919 0.000 0.949 0.994 0.000 1.052 0.924

142

27 0.622 0.800 0.827 0.840 0.825 0.882 0.824 0.828 0.895 0.841 0.852 0.928 0.884 0.926 0.947 0.995 0.921

28 0.617 0.795 0.823 0.839 0.822 0.878 0.820 0.824 0.890 0.838 0.850 0.928 0.886 0.930 0.950 0.996 0.922

29 0.614 0.843 0.000 0.915 0.892 0.000 0.878 0.882 0.000 0.897 0.914 0.000 0.964 1.022 0.000 1.057 0.925

30 0.605 0.786 0.845 0.838 0.872 0.892 0.822 0.824 0.891 0.840 0.856 0.947 0.948 0.936 0.982 0.995 0.916

31 0.597 0.762 0.825 0.000 0.838 0.915 0.839 0.840 0.906 0.857 0.875 0.975 0.914 0.000 0.959 0.970 0.907

32 0.592 0.748 0.758 0.825 0.845 0.000 0.823 0.826 0.000 0.842 0.860 0.000 0.920 0.924 0.886 0.956 0.901

Table B.3: C3 SW-MOX assembly reference normalized pin-powers

18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34

1 0.913 0.761 0.697 0.672 0.661 0.655 0.644 0.635 0.630 0.622 0.617 0.614 0.605 0.597 0.592 0.595 0.605

2 0.903 0.969 0.879 0.856 0.858 0.899 0.830 0.817 0.858 0.800 0.795 0.843 0.786 0.762 0.748 0.757 0.595

3 0.901 0.957 0.886 0.924 0.920 0.000 0.860 0.843 0.000 0.827 0.823 0.000 0.845 0.825 0.758 0.748 0.592

4 0.908 0.970 0.959 0.000 0.915 0.975 0.875 0.857 0.906 0.840 0.839 0.915 0.838 0.000 0.825 0.762 0.596

5 0.917 0.996 0.982 0.937 0.948 0.948 0.857 0.841 0.891 0.825 0.822 0.892 0.872 0.838 0.845 0.786 0.605

6 0.925 1.057 0.000 1.022 0.964 0.000 0.914 0.897 0.000 0.882 0.878 0.000 0.892 0.915 0.000 0.843 0.614

7 0.922 0.997 0.950 0.930 0.887 0.928 0.850 0.838 0.890 0.824 0.820 0.878 0.822 0.839 0.823 0.795 0.616

8 0.922 0.995 0.947 0.926 0.885 0.928 0.852 0.842 0.895 0.828 0.824 0.882 0.824 0.840 0.826 0.800 0.621

9 0.925 1.053 0.000 0.995 0.950 0.000 0.919 0.907 0.000 0.895 0.890 0.000 0.891 0.906 0.000 0.858 0.630

10 0.921 0.996 0.950 0.932 0.892 0.936 0.862 0.853 0.907 0.841 0.838 0.897 0.840 0.857 0.842 0.817 0.635

11 0.920 1.000 0.958 0.942 0.902 0.949 0.871 0.862 0.919 0.852 0.850 0.914 0.856 0.875 0.860 0.830 0.643

12 0.921 1.063 0.000 1.042 0.990 0.000 0.949 0.936 0.000 0.928 0.928 0.000 0.947 0.975 0.000 0.899 0.655

13 0.911 1.003 1.001 0.964 0.984 0.990 0.902 0.892 0.950 0.884 0.886 0.964 0.948 0.914 0.920 0.858 0.661

14 0.901 0.981 0.987 0.000 0.964 1.042 0.942 0.931 0.994 0.926 0.930 1.022 0.936 0.000 0.924 0.855 0.671

15 0.894 0.978 0.930 0.987 1.000 0.000 0.958 0.950 0.000 0.947 0.950 0.000 0.982 0.959 0.886 0.879 0.697

16 0.904 1.030 0.978 0.980 1.002 1.063 1.000 0.996 1.052 0.995 0.996 1.057 0.995 0.970 0.956 0.969 0.761

17 0.957 0.904 0.894 0.901 0.911 0.921 0.920 0.920 0.924 0.921 0.922 0.925 0.916 0.907 0.901 0.902 0.912

33 0.813 0.932 1.015 1.081 1.136 1.206 1.180 1.194 1.251 1.217 1.227 1.280 1.234 1.212 1.187 1.171 1.162

34 0.811 0.924 0.999 1.054 1.096 1.130 1.143 1.158 1.176 1.181 1.189 1.200 1.193 1.182 1.171 1.162 1.158

Table B.4: C3 SE-UOX assembly reference normalized pin-powers

18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34

18 0.735 0.770 0.791 0.809 0.824 0.837 0.836 0.838 0.844 0.841 0.842 0.846 0.837 0.828 0.818 0.813 0.811

19 0.770 0.846 0.894 0.934 0.966 1.015 0.981 0.984 1.025 0.989 0.992 1.033 0.990 0.969 0.946 0.932 0.924

20 0.791 0.894 0.977 1.069 1.105 0.000 1.108 1.111 0.000 1.119 1.124 0.000 1.142 1.122 1.052 1.015 0.999

21 0.809 0.934 1.069 0.000 1.160 1.163 1.114 1.116 1.165 1.126 1.134 1.196 1.208 0.000 1.166 1.081 1.054

22 0.824 0.966 1.105 1.160 1.148 1.188 1.143 1.148 1.200 1.159 1.167 1.228 1.203 1.241 1.223 1.136 1.096

23 0.837 1.015 0.000 1.163 1.188 0.000 1.207 1.215 0.000 1.228 1.236 0.000 1.253 1.256 0.000 1.206 1.130

24 0.836 0.981 1.108 1.114 1.143 1.207 1.171 1.181 1.236 1.196 1.203 1.257 1.212 1.211 1.250 1.180 1.143

25 0.838 0.984 1.111 1.116 1.148 1.215 1.181 1.192 1.250 1.208 1.215 1.270 1.223 1.222 1.263 1.194 1.158

26 0.844 1.025 0.000 1.165 1.200 0.000 1.236 1.250 0.000 1.268 1.276 0.000 1.284 1.283 0.000 1.251 1.176

143

27 0.841 0.989 1.119 1.126 1.159 1.228 1.196 1.208 1.268 1.228 1.236 1.292 1.246 1.245 1.287 1.217 1.181

28 0.842 0.992 1.124 1.134 1.167 1.236 1.203 1.215 1.276 1.236 1.245 1.304 1.259 1.259 1.300 1.227 1.189

29 0.846 1.033 0.000 1.196 1.228 0.000 1.257 1.270 0.000 1.292 1.304 0.000 1.327 1.332 0.000 1.280 1.200

30 0.837 0.990 1.142 1.208 1.203 1.253 1.212 1.223 1.284 1.246 1.259 1.327 1.304 1.348 1.327 1.234 1.193

31 0.828 0.969 1.122 0.000 1.241 1.256 1.211 1.222 1.283 1.245 1.259 1.332 1.348 0.000 1.305 1.212 1.182

32 0.818 0.946 1.052 1.166 1.223 0.000 1.250 1.263 0.000 1.287 1.300 0.000 1.327 1.305 1.228 1.187 1.171

Table B.5: C5 NW-UOX assembly reference normalized pin-powers

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17

1 2.078 2.083 2.093 2.105 2.113 2.113 2.078 2.044 2.015 1.961 1.910 1.860 1.777 1.679 1.562 1.417 1.218

2 2.083 2.096 2.119 2.154 2.184 2.251 2.142 2.104 2.139 2.019 1.968 1.981 1.837 1.720 1.584 1.426 1.219

3 2.093 2.119 2.187 2.313 2.342 0.000 2.262 2.218 0.000 2.130 2.076 0.000 1.972 1.849 1.638 1.445 1.223

4 2.105 2.154 2.313 0.000 2.371 2.328 2.182 2.137 2.179 2.051 2.005 2.047 1.991 0.000 1.739 1.472 1.231

5 2.113 2.184 2.342 2.371 2.281 2.305 2.170 2.127 2.169 2.041 1.994 2.029 1.918 1.893 1.757 1.495 1.237

6 2.113 2.251 0.000 2.328 2.305 0.000 2.234 2.190 0.000 2.105 2.052 0.000 1.943 1.857 0.000 1.547 1.240

7 2.078 2.142 2.262 2.182 2.170 2.234 2.116 2.079 2.122 1.996 1.947 1.970 1.830 1.745 1.698 1.469 1.220

8 2.044 2.104 2.218 2.137 2.127 2.190 2.079 2.044 2.086 1.964 1.914 1.934 1.796 1.711 1.667 1.446 1.203

9 2.015 2.139 0.000 2.179 2.169 0.000 2.122 2.086 0.000 2.007 1.954 0.000 1.836 1.746 0.000 1.478 1.190

10 1.961 2.019 2.130 2.051 2.041 2.105 1.996 1.964 2.007 1.888 1.841 1.864 1.730 1.649 1.610 1.396 1.162

11 1.910 1.968 2.076 2.005 1.994 2.052 1.947 1.914 1.954 1.841 1.797 1.820 1.693 1.617 1.575 1.366 1.138

12 1.860 1.981 0.000 2.047 2.029 0.000 1.970 1.934 0.000 1.864 1.820 0.000 1.729 1.655 0.000 1.387 1.116

13 1.777 1.837 1.972 1.991 1.918 1.943 1.830 1.796 1.836 1.730 1.693 1.729 1.636 1.619 1.513 1.292 1.074

14 1.679 1.720 1.849 0.000 1.893 1.857 1.745 1.711 1.746 1.649 1.617 1.655 1.619 0.000 1.431 1.219 1.027

15 1.562 1.584 1.638 1.739 1.757 0.000 1.698 1.667 0.000 1.610 1.575 0.000 1.513 1.431 1.276 1.136 0.975

16 1.417 1.426 1.445 1.472 1.495 1.547 1.469 1.446 1.478 1.396 1.366 1.387 1.292 1.219 1.136 1.043 0.918

17 1.218 1.219 1.223 1.231 1.237 1.240 1.220 1.203 1.190 1.162 1.138 1.116 1.074 1.027 0.975 0.918 0.845

33 0.452 0.573 0.563 0.567 0.577 0.607 0.567 0.561 0.586 0.549 0.544 0.570 0.533 0.516 0.509 0.533 0.466

34 0.656 0.648 0.644 0.643 0.642 0.642 0.636 0.630 0.625 0.617 0.610 0.604 0.594 0.585 0.581 0.591 0.636

Table B.6: C5 NE-MOX assembly reference normalized pin-powers

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17

18 1.334 1.318 1.312 1.314 1.317 1.317 1.299 1.281 1.266 1.239 1.214 1.190 1.150 1.106 1.065 1.037 1.049

19 1.075 1.367 1.344 1.354 1.379 1.448 1.354 1.333 1.385 1.291 1.268 1.314 1.213 1.153 1.114 1.126 0.942

20 0.948 1.192 1.195 1.283 1.308 0.000 1.244 1.219 0.000 1.185 1.165 0.000 1.159 1.105 1.010 1.018 0.888

21 0.877 1.114 1.197 0.000 1.203 1.304 1.171 1.147 1.215 1.114 1.100 1.190 1.066 0.000 1.026 0.974 0.853

22 0.827 1.071 1.143 1.138 1.167 1.170 1.065 1.045 1.100 1.016 1.004 1.072 1.041 0.991 0.987 0.949 0.822

23 0.784 1.075 0.000 1.158 1.114 0.000 1.066 1.045 0.000 1.019 1.005 0.000 1.000 1.014 0.000 0.961 0.791

24 0.734 0.946 0.981 0.989 0.959 1.016 0.929 0.915 0.973 0.892 0.879 0.937 0.864 0.871 0.857 0.855 0.748

25 0.686 0.880 0.901 0.914 0.888 0.934 0.864 0.853 0.899 0.831 0.820 0.864 0.803 0.810 0.793 0.801 0.706

26 0.642 0.872 0.000 0.913 0.887 0.000 0.866 0.854 0.000 0.835 0.822 0.000 0.806 0.811 0.000 0.798 0.666

144

27 0.594 0.764 0.789 0.794 0.772 0.820 0.752 0.743 0.791 0.726 0.716 0.762 0.703 0.709 0.702 0.705 0.621

28 0.549 0.705 0.723 0.735 0.713 0.749 0.693 0.685 0.722 0.669 0.661 0.699 0.651 0.659 0.646 0.654 0.578

29 0.508 0.694 0.000 0.742 0.715 0.000 0.689 0.679 0.000 0.665 0.657 0.000 0.656 0.667 0.000 0.645 0.538

30 0.463 0.601 0.643 0.625 0.647 0.659 0.596 0.587 0.626 0.574 0.570 0.616 0.593 0.564 0.580 0.562 0.493

31 0.426 0.540 0.581 0.000 0.577 0.619 0.561 0.551 0.581 0.540 0.536 0.580 0.530 0.000 0.525 0.508 0.455

32 0.409 0.513 0.515 0.557 0.562 0.000 0.531 0.522 0.000 0.512 0.508 0.000 0.518 0.506 0.467 0.482 0.435

Table B.7: C5 SW-MOX assembly reference normalized pin-powers

18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34

1 1.334 1.075 0.948 0.877 0.827 0.784 0.734 0.686 0.642 0.594 0.549 0.508 0.463 0.426 0.409 0.452 0.656

2 1.318 1.367 1.192 1.114 1.071 1.075 0.946 0.880 0.872 0.764 0.705 0.694 0.601 0.540 0.513 0.573 0.648

3 1.312 1.344 1.195 1.197 1.143 0.000 0.980 0.901 0.000 0.789 0.723 0.000 0.643 0.581 0.515 0.563 0.644

4 1.314 1.354 1.283 0.000 1.138 1.158 0.989 0.914 0.913 0.794 0.735 0.742 0.625 0.000 0.557 0.567 0.643

5 1.317 1.379 1.308 1.203 1.167 1.114 0.959 0.888 0.887 0.772 0.713 0.715 0.647 0.577 0.562 0.577 0.642

6 1.317 1.448 0.000 1.304 1.170 0.000 1.016 0.934 0.000 0.820 0.749 0.000 0.659 0.619 0.000 0.607 0.642

7 1.299 1.354 1.244 1.171 1.065 1.066 0.929 0.864 0.866 0.752 0.693 0.689 0.596 0.561 0.531 0.567 0.636

8 1.281 1.333 1.219 1.147 1.045 1.045 0.915 0.853 0.854 0.743 0.685 0.679 0.587 0.551 0.522 0.561 0.630

9 1.266 1.385 0.000 1.215 1.100 0.000 0.973 0.899 0.000 0.791 0.722 0.000 0.626 0.581 0.000 0.586 0.625

10 1.239 1.291 1.185 1.114 1.016 1.019 0.892 0.831 0.835 0.726 0.669 0.665 0.574 0.540 0.512 0.549 0.617

11 1.214 1.268 1.165 1.100 1.004 1.005 0.879 0.820 0.822 0.716 0.661 0.657 0.570 0.536 0.508 0.544 0.610

12 1.190 1.314 0.000 1.190 1.072 0.000 0.937 0.864 0.000 0.762 0.698 0.000 0.616 0.580 0.000 0.570 0.604

13 1.150 1.213 1.159 1.066 1.041 1.000 0.864 0.803 0.806 0.703 0.651 0.656 0.593 0.530 0.518 0.533 0.594

14 1.106 1.153 1.105 0.000 0.991 1.014 0.871 0.810 0.811 0.709 0.659 0.667 0.564 0.000 0.506 0.516 0.585

15 1.065 1.114 1.010 1.026 0.987 0.000 0.857 0.793 0.000 0.702 0.646 0.000 0.580 0.525 0.467 0.509 0.581

16 1.037 1.126 1.018 0.974 0.949 0.961 0.855 0.801 0.798 0.705 0.654 0.645 0.562 0.508 0.482 0.533 0.591

17 1.049 0.942 0.888 0.853 0.822 0.791 0.748 0.706 0.666 0.621 0.578 0.538 0.493 0.455 0.435 0.466 0.636

33 0.413 0.449 0.466 0.472 0.471 0.474 0.440 0.420 0.413 0.377 0.354 0.344 0.310 0.285 0.268 0.270 0.320

34 0.539 0.567 0.575 0.572 0.563 0.549 0.526 0.503 0.480 0.452 0.425 0.399 0.370 0.343 0.323 0.320 0.356

Table B.8: C5 SE-UOX assembly reference normalized pin-powers

18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34

18 0.768 0.768 0.754 0.735 0.714 0.689 0.653 0.617 0.584 0.544 0.507 0.473 0.435 0.403 0.387 0.413 0.539

19 0.768 0.809 0.818 0.816 0.804 0.804 0.738 0.697 0.681 0.616 0.574 0.554 0.495 0.454 0.430 0.449 0.567

20 0.754 0.818 0.855 0.894 0.884 0.000 0.803 0.757 0.000 0.672 0.625 0.000 0.550 0.505 0.459 0.466 0.575

21 0.735 0.816 0.894 0.000 0.894 0.853 0.774 0.730 0.717 0.648 0.605 0.593 0.555 0.000 0.487 0.472 0.572

22 0.714 0.804 0.884 0.894 0.844 0.830 0.757 0.716 0.702 0.636 0.594 0.580 0.529 0.512 0.485 0.471 0.563

23 0.689 0.804 0.000 0.853 0.830 0.000 0.762 0.720 0.000 0.642 0.598 0.000 0.526 0.491 0.000 0.474 0.549

24 0.653 0.738 0.803 0.774 0.757 0.762 0.700 0.664 0.654 0.592 0.553 0.538 0.482 0.450 0.448 0.440 0.526

25 0.617 0.697 0.757 0.730 0.716 0.720 0.664 0.631 0.621 0.563 0.526 0.510 0.458 0.427 0.425 0.420 0.503

26 0.584 0.681 0.000 0.717 0.702 0.000 0.654 0.621 0.000 0.556 0.518 0.000 0.452 0.421 0.000 0.413 0.480

145

27 0.544 0.616 0.672 0.648 0.636 0.642 0.592 0.563 0.556 0.504 0.471 0.458 0.410 0.383 0.383 0.377 0.452

28 0.507 0.574 0.625 0.605 0.594 0.598 0.553 0.526 0.518 0.471 0.441 0.428 0.385 0.360 0.359 0.354 0.425

29 0.473 0.554 0.000 0.593 0.580 0.000 0.538 0.510 0.000 0.458 0.428 0.000 0.378 0.354 0.000 0.344 0.398

30 0.435 0.495 0.550 0.555 0.529 0.526 0.482 0.458 0.452 0.410 0.385 0.378 0.345 0.333 0.319 0.310 0.370

31 0.403 0.454 0.505 0.000 0.512 0.491 0.450 0.427 0.421 0.383 0.360 0.354 0.333 0.000 0.295 0.285 0.343

32 0.387 0.430 0.459 0.487 0.485 0.000 0.448 0.425 0.000 0.383 0.359 0.000 0.319 0.295 0.267 0.268 0.323

Table B.9: C3 NW-UOX pin-power differences multiplied by 100 (Case 1) 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17

1 -0.047 -0.003 -0.104 0.017 -0.062 0.015 -0.006 -0.094 -0.049 -0.047 -0.102 -0.056 -0.119 -0.082 -0.187 -0.113 -0.158

2 -0.003 -0.104 -0.028 0.036 -0.011 -0.004 -0.077 0.002 -0.123 -0.140 -0.005 -0.119 -0.090 -0.084 -0.122 -0.136 -0.236

3 -0.104 -0.028 0.023 -0.029 0.014 0.000 -0.084 -0.027 0.000 -0.080 -0.034 0.000 -0.032 -0.048 -0.104 -0.171 -0.181

4 0.017 0.036 -0.029 0.000 0.045 -0.031 -0.035 -0.089 -0.071 -0.043 -0.053 -0.068 -0.122 0.000 -0.044 -0.129 -0.182

5 -0.062 -0.011 0.014 0.045 -0.040 0.014 -0.035 -0.078 -0.049 -0.021 -0.042 -0.001 -0.041 -0.086 -0.113 -0.087 -0.171

6 0.015 -0.004 0.000 -0.031 0.014 0.000 -0.029 -0.061 0.000 -0.003 -0.046 0.000 0.023 -0.029 0.000 -0.024 -0.172

7 -0.006 -0.077 -0.084 -0.035 -0.035 -0.029 -0.078 0.011 0.052 -0.009 0.048 -0.077 -0.037 0.088 -0.011 0.024 -0.027

8 -0.094 0.002 -0.027 -0.089 -0.078 -0.061 0.011 0.023 0.064 0.014 0.071 0.034 -0.025 0.100 0.023 0.091 -0.038

9 -0.049 -0.123 0.000 -0.071 -0.049 0.000 0.052 0.064 0.000 0.066 0.023 0.000 0.026 0.130 0.000 0.088 0.006

10 -0.047 -0.140 -0.080 -0.043 -0.021 -0.003 -0.009 0.014 0.066 0.127 0.106 0.080 0.142 0.101 0.046 0.136 -0.060

11 -0.102 -0.005 -0.034 -0.053 -0.042 -0.046 0.048 0.071 0.023 0.106 0.107 0.192 0.098 0.079 0.113 0.203 0.017

12 -0.056 -0.119 0.000 -0.068 -0.001 0.000 -0.077 0.034 0.000 0.080 0.192 0.000 0.194 0.208 0.000 0.189 0.117

13 -0.119 -0.090 -0.032 -0.122 -0.041 0.023 -0.037 -0.025 0.026 0.142 0.098 0.194 0.242 0.275 0.180 0.238 0.174

14 -0.082 -0.084 -0.048 0.000 -0.086 -0.029 0.088 0.100 0.130 0.101 0.079 0.208 0.275 0.000 0.272 0.286 0.109

15 -0.187 -0.122 -0.104 -0.044 -0.113 0.000 -0.011 0.023 0.000 0.046 0.113 0.000 0.180 0.272 0.259 0.256 0.111

16 -0.113 -0.136 -0.171 -0.129 -0.087 -0.024 0.024 0.091 0.088 0.136 0.203 0.189 0.238 0.286 0.256 0.317 0.280

17 -0.158 -0.236 -0.181 -0.182 -0.171 -0.172 -0.027 -0.038 0.006 -0.060 0.017 0.117 0.174 0.109 0.111 0.280 0.794

Table B.10: C3 NE-MOX pin-power differences multiplied by 100 (Case 1) 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17

18 0.326 0.367 0.495 0.397 0.370 0.357 0.299 0.199 0.157 0.013 0.028 0.013 -0.046 -0.105 -0.206 -0.747 -2.494

18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34

1 0.328 0.271 0.173 0.125 0.081 0.165 0.020 0.047 0.118 0.030 0.001 0.043 -0.030 -0.017 -0.047 0.011 -0.030

19 0.270 0.337 0.206 0.223 0.156 0.196 0.042 0.070 -0.048 -0.044 -0.200 -0.289 -0.343 -0.433 -0.390 -0.624 -0.747

20 0.172 0.106 0.207 0.178 0.153 0.000 0.005 -0.053 0.000 -0.095 -0.208 0.000 -0.415 -0.417 -0.413 -0.390 -0.206

21 0.124 0.130 0.171 0.000 0.088 -0.011 -0.013 -0.057 -0.130 -0.242 -0.283 -0.293 -0.293 0.000 -0.417 -0.318 -0.105

22 0.080 0.102 0.128 0.098 0.033 0.007 -0.107 -0.065 -0.195 -0.178 -0.219 -0.259 -0.375 -0.293 -0.300 -0.229 -0.046

23 0.164 0.124 0.000 0.052 0.033 0.000 0.015 -0.085 0.000 -0.098 -0.281 0.000 -0.259 -0.293 0.000 -0.289 0.013

24 0.019 0.097 0.074 0.062 -0.084 0.012 0.015 -0.114 -0.058 -0.155 -0.082 -0.167 -0.219 -0.283 -0.208 -0.200 0.028

25 0.046 0.080 0.013 0.016 -0.059 0.052 -0.016 -0.073 0.011 -0.128 -0.155 -0.098 -0.178 -0.128 -0.095 -0.044 0.128

26 0.117 0.102 0.000 0.025 0.036 0.000 -0.049 -0.020 0.000 0.011 -0.058 0.000 -0.081 -0.016 0.000 0.066 0.271

27 0.029 0.020 0.039 0.056 -0.033 -0.037 -0.119 -0.075 -0.020 0.041 -0.114 -0.085 -0.065 -0.057 -0.053 0.070 0.313

28 0.000 0.090 -0.005 -0.030 0.009 0.020 -0.062 -0.119 -0.049 -0.016 0.015 0.015 0.007 -0.013 0.105 0.156 0.299

29 0.042 0.013 0.000 -0.002 -0.078 0.000 0.020 -0.037 0.000 0.052 0.012 0.000 0.007 -0.011 0.000 0.196 0.357

30 -0.031 0.017 0.085 -0.016 0.004 -0.078 0.009 0.081 0.036 0.056 0.030 0.047 0.033 0.202 0.153 0.270 0.484

31 -0.018 -0.045 -0.033 0.000 -0.016 -0.002 -0.030 0.056 0.025 0.016 0.062 0.052 0.212 0.000 0.178 0.223 0.511

32 -0.047 0.053 0.012 -0.033 0.085 0.000 -0.005 0.153 0.000 0.127 0.074 0.000 0.128 0.171 0.207 0.320 0.495

33 0.010 0.026 0.053 -0.045 0.017 0.013 -0.010 0.020 0.102 0.080 0.097 0.124 0.102 0.244 0.106 0.337 0.481

34 -0.031 0.010 -0.047 0.096 -0.031 0.042 0.114 0.044 0.017 0.046 0.133 0.164 0.080 0.239 0.172 0.270 0.440

Table B.11: C3 SW-MOX pin-power differences multiplied by 100 (Case 1) 2 0.369 0.338 0.107 0.131 0.103 0.125 0.098 0.081 0.103 0.021 0.091 0.015 0.018 -0.043 0.054 0.027 0.011

3 0.497 0.207 0.208 0.173 0.129 0.000 0.075 0.015 0.000 0.040 -0.003 0.000 0.086 -0.032 0.013 0.054 -0.047

4 0.398 0.224 0.179 0.000 0.099 0.053 0.063 0.017 0.026 0.057 -0.029 -0.001 -0.015 0.000 -0.032 -0.043 0.097

5 0.371 0.157 0.155 0.089 0.034 0.034 -0.083 -0.057 0.038 -0.032 0.011 -0.076 0.006 -0.015 0.086 0.018 -0.030

6 0.358 0.197 0.000 -0.009 0.009 0.000 0.014 0.053 0.000 -0.035 0.021 0.000 -0.076 -0.001 0.000 0.015 0.043

7 0.301 0.043 0.006 -0.012 -0.106 0.016 0.016 -0.015 -0.048 -0.118 -0.061 0.021 0.011 -0.029 -0.003 -0.009 0.115

8 0.201 0.071 -0.052 -0.056 -0.178 -0.084 -0.112 -0.071 -0.019 -0.074 -0.118 -0.035 0.082 0.057 0.154 0.021 0.045

9 0.158 -0.046 0.000 -0.129 -0.194 0.000 -0.057 0.012 0.000 -0.019 -0.048 0.000 0.038 0.026 0.000 0.103 0.018

146

10 0.015 -0.043 -0.094 -0.240 -0.176 -0.097 -0.153 -0.126 0.012 0.043 -0.015 0.053 0.057 0.017 0.129 0.081 0.047

11 0.029 -0.199 -0.207 -0.281 -0.217 -0.280 -0.080 -0.153 -0.057 -0.112 0.016 0.014 0.031 0.063 0.075 0.098 0.134

12 0.015 -0.287 0.000 -0.291 -0.258 0.000 -0.280 -0.097 0.000 -0.084 0.016 0.000 0.048 0.053 0.000 0.125 0.165

13 -0.044 -0.341 -0.413 -0.291 -0.373 -0.258 -0.217 -0.176 -0.194 -0.064 0.008 0.009 0.034 0.214 0.129 0.103 0.081

14 -0.103 -0.431 -0.416 0.000 -0.291 -0.291 -0.281 -0.126 -0.014 -0.056 -0.012 -0.009 0.203 0.000 0.173 0.245 0.240

15 -0.205 -0.389 -0.412 -0.416 -0.299 0.000 -0.207 -0.094 0.000 -0.052 0.006 0.000 0.155 0.179 0.208 0.107 0.173

16 -0.746 -0.622 -0.389 -0.317 -0.227 -0.287 -0.199 -0.043 0.068 0.071 0.157 0.197 0.271 0.224 0.321 0.338 0.271

17 -2.493 -0.746 -0.205 -0.103 -0.044 0.015 0.029 0.129 0.273 0.315 0.301 0.358 0.485 0.512 0.497 0.483 0.442

Table B.12: C3 SE-UOX pin-power differences multiplied by 100 (Case 1) 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34

18 0.872 0.356 0.187 0.184 0.149 0.092 0.081 0.003 0.069 -0.064 -0.053 -0.109 -0.108 -0.207 -0.117 -0.172 -0.094

19 0.356 0.291 0.318 0.257 0.209 0.247 0.173 0.106 0.056 0.061 -0.006 -0.056 -0.028 -0.159 -0.111 -0.165 -0.053

20 0.187 0.318 0.329 0.240 0.335 0.000 0.168 0.201 0.000 -0.011 0.044 0.000 -0.059 -0.078 -0.047 -0.153 -0.129

21 0.184 0.257 0.240 0.000 0.239 0.272 0.134 0.156 0.094 0.066 -0.046 -0.066 -0.034 0.000 -0.095 -0.029 -0.125

22 0.149 0.209 0.335 0.239 0.207 0.247 0.152 0.107 0.078 0.028 0.016 -0.014 -0.089 -0.071 -0.169 -0.125 -0.064

23 0.092 0.247 0.000 0.272 0.247 0.000 0.155 0.043 0.000 0.086 -0.026 0.000 -0.040 -0.107 0.000 -0.156 -0.090

24 0.081 0.173 0.168 0.134 0.152 0.155 0.160 0.070 0.074 0.034 0.011 0.004 0.010 -0.101 -0.173 -0.041 -0.048

25 0.003 0.106 0.201 0.156 0.107 0.043 0.070 -0.010 0.027 0.066 0.043 -0.053 -0.069 -0.080 -0.030 -0.088 -0.083

26 0.069 0.056 0.000 0.094 0.078 0.000 0.074 0.027 0.000 0.025 0.013 0.000 0.001 -0.110 0.000 -0.162 -0.085

27 -0.064 0.061 -0.011 0.066 0.028 0.086 0.034 0.066 0.025 -0.014 -0.026 -0.011 -0.117 -0.127 -0.066 -0.035 -0.130

28 -0.053 -0.006 0.044 -0.046 0.016 -0.026 0.011 0.043 0.013 -0.026 -0.027 -0.080 -0.074 -0.074 -0.123 -0.025 -0.042

29 -0.109 -0.056 0.000 -0.066 -0.014 0.000 0.004 -0.053 0.000 -0.011 -0.080 0.000 -0.127 -0.072 0.000 -0.043 -0.022

30 -0.108 -0.028 -0.059 -0.034 -0.089 -0.040 0.010 -0.069 0.001 -0.117 -0.074 -0.127 -0.080 0.004 -0.027 -0.048 -0.099

31 -0.207 -0.159 -0.078 0.000 -0.071 -0.107 -0.101 -0.080 -0.110 -0.127 -0.074 -0.072 0.004 0.000 -0.069 -0.090 -0.019

32 -0.117 -0.111 -0.047 -0.095 -0.169 0.000 -0.173 -0.030 0.000 -0.066 -0.123 0.000 -0.027 -0.069 -0.014 -0.064 -0.140

33 -0.172 -0.165 -0.153 -0.029 -0.125 -0.156 -0.041 -0.088 -0.162 -0.035 -0.025 -0.043 -0.048 -0.090 -0.064 -0.140 -0.039

34 -0.094 -0.053 -0.129 -0.125 -0.064 -0.090 -0.048 -0.083 -0.085 -0.130 -0.042 -0.022 -0.099 -0.019 -0.140 -0.039 -0.083


1 -0.033 -0.013 -0.074 0.054 0.006 0.006 -0.033 -0.028 -0.003 -0.077 -0.020 -0.018 -0.068 -0.077 -0.021 -0.096 -0.046

2 -0.013 -0.042 -0.030 0.009 -0.072 -0.027 -0.019 0.010 -0.051 -0.027 -0.070 0.002 -0.030 -0.074 -0.054 -0.101 -0.102

3 -0.074 -0.030 -0.041 0.000 -0.025 0.000 0.057 -0.077 0.000 -0.047 -0.021 0.000 0.006 -0.002 -0.079 -0.065 -0.126

4 0.054 0.009 0.000 0.000 -0.050 -0.041 0.040 -0.039 0.008 -0.020 -0.043 0.004 -0.058 0.000 -0.038 -0.078 -0.175

5 0.006 -0.072 -0.025 -0.050 -0.007 0.048 0.012 0.021 -0.032 0.040 -0.026 0.012 0.032 0.033 -0.047 -0.067 -0.111

6 0.006 -0.027 0.000 -0.041 0.048 0.000 0.026 -0.009 0.000 -0.046 0.024 0.000 0.031 0.050 0.000 -0.080 -0.079

7 -0.033 -0.019 0.057 0.040 0.012 0.026 -0.062 0.011 0.002 0.061 0.007 0.018 -0.037 0.025 0.059 -0.010 -0.058

8 -0.028 0.010 -0.077 -0.039 0.021 -0.009 0.011 -0.028 0.019 -0.046 0.056 0.035 0.068 0.030 -0.004 -0.021 -0.106

9 -0.003 -0.051 0.000 0.008 -0.032 0.000 0.002 0.019 0.000 0.045 0.015 0.000 0.026 0.069 0.000 -0.014 -0.077

10 -0.077 -0.027 -0.047 -0.020 0.040 -0.046 0.061 -0.046 0.045 0.013 0.046 0.058 0.066 0.104 -0.010 -0.020 -0.008

11 -0.020 -0.070 -0.021 -0.043 -0.026 0.024 0.007 0.056 0.015 0.046 0.112 0.023 0.039 0.097 0.051 0.061 -0.064

12 -0.018 0.002 0.000 0.004 0.012 0.000 0.018 0.035 0.000 0.058 0.023 0.000 0.122 0.168 0.000 0.085 -0.031

13 -0.068 -0.030 0.006 -0.058 0.032 0.031 -0.037 0.068 0.026 0.066 0.039 0.122 0.133 0.085 0.125 0.108 0.022

14 -0.077 -0.074 -0.002 0.000 0.033 0.050 0.025 0.030 0.069 0.104 0.097 0.168 0.085 0.000 0.119 0.098 -0.044

15 -0.021 -0.054 -0.079 -0.038 -0.047 0.000 0.059 -0.004 0.000 0.090 0.051 0.000 0.125 0.119 0.104 0.048 -0.031

16 -0.096 -0.101 -0.065 -0.078 -0.067 -0.080 -0.010 -0.021 -0.014 -0.020 0.061 0.085 0.108 0.098 0.048 0.059 0.063

17 -0.046 -0.102 -0.126 -0.175 -0.111 -0.079 -0.058 -0.106 -0.077 -0.008 -0.064 -0.031 0.022 -0.044 -0.031 0.063 0.354


18 0.722 0.837 0.855 0.816 0.757 0.657 0.612 0.566 0.362 0.293 0.186 0.059 -0.053 0.013 -0.179 -0.827 -2.864

19 0.324 0.472 0.425 0.428 0.336 0.276 0.228 0.142 0.017 -0.031 -0.178 -0.284 -0.394 -0.312 -0.544 -0.781 -0.956

20 0.126 0.219 0.260 0.227 0.134 0.000 -0.005 -0.013 0.000 -0.243 -0.349 0.000 -0.531 -0.567 -0.596 -0.553 -0.392

21 0.124 0.156 0.221 0.000 0.103 0.013 -0.167 -0.094 -0.234 -0.244 -0.368 -0.441 -0.399 0.000 -0.611 -0.486 -0.303

22 0.109 0.003 0.085 0.083 0.012 -0.047 -0.079 -0.185 -0.268 -0.314 -0.377 -0.417 -0.506 -0.421 -0.542 -0.494 -0.292

23 0.056 0.024 0.000 0.089 -0.044 0.000 -0.099 -0.085 0.000 -0.273 -0.397 0.000 -0.399 -0.474 0.000 -0.430 -0.282

24 0.041 0.065 -0.024 -0.082 0.009 -0.114 -0.100 -0.124 -0.167 -0.271 -0.215 -0.358 -0.420 -0.457 -0.382 -0.442 -0.235

25 -0.013 -0.035 0.052 -0.004 -0.092 -0.098 -0.120 -0.203 -0.209 -0.170 -0.353 -0.320 -0.318 -0.356 -0.321 -0.279 -0.207

26 0.054 0.023 0.000 -0.085 0.027 0.000 -0.059 -0.123 0.000 -0.248 -0.292 0.000 -0.177 -0.276 0.000 -0.319 -0.119

147

27 -0.001 0.050 0.058 -0.041 -0.107 -0.053 -0.113 -0.136 -0.082 -0.201 -0.204 -0.210 -0.248 -0.166 -0.228 -0.287 -0.133

28 0.085 0.013 -0.042 -0.078 0.055 0.046 -0.051 -0.094 -0.122 -0.078 -0.121 -0.169 -0.124 -0.181 -0.125 -0.183 -0.086

29 -0.007 0.029 0.000 -0.016 0.016 0.000 0.028 -0.075 0.000 -0.100 -0.042 0.000 -0.122 -0.039 0.000 -0.106 -0.098

30 0.080 -0.039 0.134 -0.012 -0.045 0.019 0.060 0.037 -0.031 -0.007 -0.128 -0.034 -0.081 -0.010 -0.125 -0.071 0.089

31 0.008 0.163 0.055 0.000 0.034 0.107 -0.051 0.046 0.055 -0.037 0.042 -0.025 -0.040 0.000 0.058 -0.007 0.037

32 0.143 0.095 0.155 0.228 0.129 0.000 0.140 0.117 0.000 0.114 0.093 0.000 0.096 0.033 0.101 0.105 0.131

33 0.196 0.313 0.310 0.331 0.334 0.343 0.331 0.249 0.357 0.285 0.284 0.372 0.201 0.236 0.273 0.201 0.120

34 0.978 1.035 1.014 0.934 0.954 0.854 0.872 0.890 0.988 0.946 0.884 0.902 0.799 0.776 0.755 0.558 -0.328

Table B.15: C5 SW-MOX pin-power differences multiplied by 100 (Case 2) 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34

1 0.721 0.323 0.225 0.123 0.108 0.055 0.040 -0.014 0.053 -0.002 0.085 -0.007 0.079 0.008 0.143 0.196 0.977

2 0.836 0.471 0.218 0.155 0.002 0.023 0.064 -0.036 0.022 0.049 0.012 0.028 -0.039 0.062 0.094 0.312 1.035

3 0.854 0.424 0.259 0.220 0.083 0.000 0.094 0.051 0.000 0.057 -0.043 0.000 0.133 0.054 0.155 0.309 1.013

4 0.815 0.427 0.225 0.000 0.082 0.088 -0.083 -0.005 -0.086 -0.042 -0.079 -0.017 -0.012 0.000 0.227 0.330 0.933

5 0.756 0.334 0.133 0.101 0.010 -0.045 0.008 -0.093 0.026 -0.108 0.054 0.015 -0.046 0.033 0.129 0.333 0.953

6 0.656 0.275 0.000 0.012 -0.049 0.000 -0.115 -0.099 0.000 -0.054 0.045 0.000 0.018 0.106 0.000 0.342 0.853

7 0.610 0.227 -0.006 -0.168 -0.080 -0.100 -0.101 -0.121 -0.060 -0.114 -0.052 0.027 0.059 -0.052 0.139 0.330 0.871

8 0.565 0.140 -0.014 -0.096 -0.186 -0.086 -0.125 -0.204 -0.124 -0.137 -0.094 -0.076 -0.064 0.045 0.117 0.248 0.889

9 0.360 0.016 0.000 -0.235 -0.270 0.000 -0.168 -0.210 0.000 -0.082 -0.123 0.000 -0.032 0.054 0.000 0.356 0.988

10 0.292 -0.032 -0.244 -0.245 -0.315 -0.274 -0.272 -0.170 -0.249 -0.202 -0.079 -0.100 -0.008 -0.038 0.114 0.285 0.845

11 0.185 -0.179 -0.350 -0.370 -0.378 -0.398 -0.216 -0.354 -0.293 -0.205 -0.121 -0.043 -0.129 0.041 0.093 0.283 0.883

12 0.057 -0.285 0.000 -0.443 -0.418 0.000 -0.359 -0.321 0.000 -0.211 -0.050 0.000 -0.035 -0.026 0.000 0.371 0.801

13 0.045 -0.296 -0.432 -0.400 -0.507 -0.400 -0.421 -0.319 -0.178 -0.249 -0.124 -0.123 -0.082 -0.041 0.096 0.200 0.798

14 0.012 -0.314 -0.568 0.000 -0.422 -0.475 -0.458 -0.357 -0.276 -0.167 -0.182 -0.040 -0.011 0.000 0.032 0.235 0.776

15 -0.180 -0.545 -0.597 -0.612 -0.544 0.000 -0.383 -0.322 0.000 -0.229 -0.126 0.000 -0.126 0.058 0.100 0.273 0.754

16 -0.829 -0.782 -0.554 -0.487 -0.495 -0.431 -0.443 -0.279 -0.220 -0.288 -0.184 -0.106 -0.071 -0.007 0.005 0.200 0.457

17 -2.865 -0.957 -0.393 -0.304 -0.293 -0.282 -0.235 -0.208 -0.120 -0.133 -0.086 -0.098 0.088 0.037 0.131 0.120 -0.329


18 2.025 0.825 0.296 0.387 0.244 0.236 0.149 0.162 0.024 -0.028 -0.032 -0.086 -0.206 -0.228 -0.089 -0.065 0.890

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17

1 -0.047 -0.003 -0.004 0.017 -0.062 0.015 -0.006 0.006 0.051 0.053 -0.002 0.044 -0.019 0.018 -0.087 -0.013 -0.058

19 0.825 0.693 0.540 0.308 0.312 0.212 0.135 0.067 0.106 0.046 -0.039 0.035 -0.027 -0.096 -0.087 0.022 0.347

20 0.296 0.540 0.544 0.480 0.316 0.000 0.295 0.045 0.000 -0.041 -0.007 0.000 -0.130 -0.264 -0.215 -0.200 0.077

21 0.387 0.308 0.480 0.000 0.280 0.311 0.223 0.105 -0.007 -0.132 -0.034 -0.129 -0.149 0.000 -0.258 -0.203 0.128

22 0.244 0.312 0.316 0.280 0.164 0.136 0.145 -0.023 0.048 -0.128 -0.013 -0.141 -0.173 -0.250 -0.091 -0.219 -0.118

23 0.236 0.212 0.000 0.311 0.136 0.000 0.027 -0.058 0.000 -0.130 -0.048 0.000 -0.222 -0.093 0.000 -0.170 -0.147

24 0.149 0.135 0.295 0.223 0.145 0.027 0.016 0.029 -0.034 -0.146 -0.182 -0.226 -0.239 -0.161 -0.194 -0.124 -0.022

25 0.162 0.067 0.045 0.105 -0.023 -0.058 0.029 -0.010 -0.073 -0.118 -0.122 -0.083 -0.231 -0.136 -0.169 -0.151 -0.097

26 0.024 0.106 0.000 -0.007 0.048 0.000 -0.034 -0.073 0.000 -0.033 -0.052 0.000 -0.029 -0.134 0.000 -0.165 -0.072

27 -0.028 0.046 -0.041 -0.132 -0.128 -0.130 -0.146 -0.118 -0.033 -0.181 -0.119 -0.031 -0.114 -0.154 -0.154 -0.052 -0.029

28 -0.032 -0.039 -0.007 -0.034 -0.013 -0.048 -0.182 -0.122 -0.052 -0.119 -0.108 -0.120 -0.121 -0.129 -0.245 -0.027 -0.069

29 -0.086 0.035 0.000 -0.129 -0.141 0.000 -0.226 -0.083 0.000 -0.031 -0.120 0.000 -0.135 -0.127 0.000 -0.190 0.091

30 -0.206 -0.027 -0.130 -0.149 -0.173 -0.222 -0.239 -0.231 -0.029 -0.114 -0.121 -0.135 -0.174 -0.069 -0.098 -0.144 0.034

31 -0.228 -0.096 -0.164 0.000 -0.250 -0.093 -0.161 -0.136 -0.134 -0.154 -0.129 -0.127 -0.069 0.000 -0.089 -0.052 0.194

32 -0.089 -0.087 -0.215 -0.158 -0.091 0.000 -0.194 -0.169 0.000 -0.154 -0.245 0.000 -0.098 -0.089 0.054 -0.029 0.268

33 0.035 0.022 -0.200 -0.203 -0.219 -0.170 -0.124 -0.151 -0.165 -0.052 -0.027 -0.190 -0.144 -0.052 -0.029 0.003 0.019

34 0.890 0.347 0.077 0.128 -0.118 -0.047 -0.022 0.003 -0.072 -0.029 -0.069 0.007 0.034 0.194 0.268 0.019 -0.594

Table B.17: C3 NW-UOX pin-power differences multiplied by 100 (Case 3) 2 -0.003 -0.004 -0.028 0.036 -0.011 -0.004 -0.077 0.002 -0.023 -0.040 -0.005 -0.019 0.010 0.016 -0.022 -0.036 -0.036

3 -0.004 -0.028 0.023 -0.029 0.014 0.000 0.016 -0.027 0.000 0.020 -0.034 0.000 -0.032 -0.048 -0.004 -0.071 -0.081

4 0.017 0.036 -0.029 0.000 0.045 0.069 -0.035 0.011 0.029 0.057 -0.053 0.032 -0.022 0.000 0.056 -0.029 0.018

5 -0.062 -0.011 0.014 0.045 0.060 0.014 -0.035 0.022 -0.049 -0.021 -0.042 -0.001 -0.041 0.014 -0.013 0.013 -0.071

6 0.015 -0.004 0.000 0.069 0.014 0.000 -0.029 -0.061 0.000 -0.003 -0.046 0.000 0.023 -0.029 0.000 -0.024 -0.072

7 -0.006 -0.077 0.016 -0.035 -0.035 -0.029 0.022 0.011 0.052 -0.009 0.048 0.023 -0.037 0.088 -0.011 0.024 -0.027

8 0.006 0.002 -0.027 0.011 0.022 -0.061 0.011 0.023 0.064 0.014 0.071 -0.066 -0.025 0.000 0.023 -0.009 0.062

9 0.051 -0.023 0.000 0.029 -0.049 0.000 0.052 0.064 0.000 -0.034 0.023 0.000 0.026 0.030 0.000 -0.012 0.006

148

10 0.053 -0.040 0.020 0.057 -0.021 -0.003 -0.009 0.014 -0.034 0.027 0.006 -0.020 0.042 0.001 -0.054 0.036 -0.060

11 -0.002 -0.005 -0.034 -0.053 -0.042 -0.046 0.048 0.071 0.023 0.006 0.007 -0.008 -0.002 -0.021 0.013 0.003 -0.083

12 0.044 -0.019 0.000 0.032 -0.001 0.000 0.023 -0.066 0.000 -0.020 -0.008 0.000 -0.006 0.008 0.000 -0.011 0.017

13 -0.019 0.010 -0.032 -0.022 -0.041 0.023 -0.037 -0.025 0.026 0.042 -0.002 -0.006 0.042 0.075 -0.020 0.038 -0.026

14 0.018 0.016 -0.048 0.000 0.014 -0.029 0.088 0.000 0.030 0.001 -0.021 0.008 0.075 0.000 -0.028 -0.014 0.009

15 -0.087 -0.022 -0.004 0.056 -0.013 0.000 -0.011 0.023 0.000 -0.054 0.013 0.000 -0.020 -0.028 0.059 -0.044 -0.089

16 -0.013 -0.036 -0.071 -0.029 0.013 -0.024 0.024 -0.009 -0.012 0.036 0.003 -0.011 0.038 -0.014 -0.044 0.017 -0.020

17 -0.058 -0.036 -0.081 0.018 -0.071 -0.072 -0.027 0.062 0.006 -0.060 -0.083 0.017 -0.026 0.009 -0.089 -0.020 -0.006


18 0.026 0.067 0.095 -0.003 -0.030 0.057 0.099 -0.001 0.057 0.013 0.028 0.113 0.054 -0.005 0.094 -0.047 0.106

19 0.070 0.037 -0.094 0.023 -0.044 0.096 -0.058 -0.030 -0.048 0.056 0.000 0.011 -0.043 -0.033 0.010 -0.024 -0.047

20 0.072 0.006 0.007 -0.022 0.053 0.000 0.005 -0.053 0.000 0.005 -0.008 0.000 -0.015 -0.017 -0.013 0.010 0.094

21 -0.076 -0.070 -0.029 0.000 -0.012 -0.011 -0.013 -0.057 -0.130 -0.142 -0.083 -0.093 0.007 0.000 -0.017 0.082 -0.005

22 -0.020 0.002 0.028 -0.002 -0.067 0.007 -0.107 0.035 -0.095 -0.078 -0.019 -0.059 -0.075 0.007 0.100 0.071 0.054

23 0.064 0.024 0.000 -0.048 -0.067 0.000 0.015 -0.085 0.000 0.002 -0.081 0.000 -0.059 -0.093 0.000 0.011 0.113

24 -0.081 -0.003 0.074 -0.038 -0.084 0.012 0.015 -0.014 -0.058 -0.055 0.018 0.033 -0.019 -0.083 -0.008 0.000 0.028

25 -0.054 -0.020 -0.087 0.016 -0.059 -0.048 -0.016 -0.073 0.011 -0.028 -0.055 0.002 -0.078 0.072 0.005 0.056 0.128

26 0.017 0.002 0.000 0.025 -0.064 0.000 -0.049 -0.020 0.000 0.011 -0.058 0.000 0.019 -0.016 0.000 0.066 0.171

27 -0.071 0.020 -0.061 -0.044 -0.133 -0.037 -0.119 -0.075 -0.020 0.041 -0.014 -0.085 0.035 -0.057 -0.053 -0.030 0.113

28 0.000 -0.010 -0.005 -0.030 0.009 0.020 -0.062 -0.119 -0.049 -0.016 0.015 0.015 0.007 -0.013 0.005 0.056 0.099

29 0.042 0.013 0.000 -0.002 -0.078 0.000 0.020 -0.037 0.000 -0.048 0.012 0.000 0.007 -0.011 0.000 0.096 0.057

30 -0.031 0.017 -0.015 -0.016 -0.096 -0.078 0.009 -0.019 -0.064 0.056 0.030 0.047 -0.067 0.102 0.053 0.070 0.084

31 -0.018 -0.045 -0.033 0.000 -0.016 -0.002 -0.030 -0.044 0.025 0.016 -0.038 -0.048 0.112 0.000 -0.022 0.023 0.111

32 -0.047 0.053 -0.088 -0.033 -0.015 0.000 -0.005 0.053 0.000 0.027 0.074 0.000 0.028 -0.029 0.007 0.020 0.095

33 0.010 0.026 0.053 -0.045 0.017 0.013 -0.010 0.020 0.002 -0.020 -0.003 0.024 0.002 0.044 0.006 0.037 0.181

34 -0.031 0.010 -0.047 0.096 -0.031 0.042 0.014 0.044 0.017 -0.054 0.033 0.064 -0.020 0.039 0.072 0.070 0.140


1 0.028 0.071 0.073 -0.075 -0.019 0.065 -0.080 0.047 0.018 -0.070 0.001 0.043 -0.030 -0.017 -0.047 0.011 -0.030

2 0.069 0.038 0.007 -0.069 0.003 0.025 -0.002 -0.019 0.003 0.021 -0.009 0.015 0.018 -0.043 0.054 0.027 0.011

3 0.097 -0.093 0.008 -0.027 0.029 0.000 0.075 -0.085 0.000 -0.060 -0.003 0.000 -0.014 -0.032 -0.087 0.054 -0.047

4 -0.002 0.024 -0.021 0.000 -0.001 -0.047 -0.037 0.017 0.026 -0.043 -0.029 -0.001 -0.015 0.000 -0.032 -0.043 0.097

5 -0.029 -0.043 0.055 -0.011 -0.066 -0.066 -0.083 -0.057 -0.062 -0.032 0.011 -0.076 -0.094 -0.015 -0.014 0.018 -0.030

6 0.058 0.097 0.000 -0.009 0.009 0.000 0.014 -0.047 0.000 -0.035 0.021 0.000 -0.076 -0.001 0.000 0.015 0.043

7 0.101 -0.057 0.006 -0.012 -0.106 0.016 0.016 -0.015 -0.048 -0.118 -0.061 0.021 0.011 -0.029 -0.003 -0.009 0.015

8 0.001 -0.029 -0.052 -0.056 -0.078 -0.084 -0.012 -0.071 -0.019 -0.074 -0.118 -0.035 -0.018 -0.043 0.054 0.021 0.045

9 0.058 -0.046 0.000 -0.129 -0.094 0.000 -0.057 0.012 0.000 -0.019 -0.048 0.000 -0.062 0.026 0.000 0.003 0.018

10 0.015 0.057 0.006 -0.140 -0.076 0.003 -0.053 -0.026 0.012 0.043 -0.015 -0.047 0.057 0.017 0.029 -0.019 -0.053

11 0.029 0.001 -0.007 -0.081 -0.017 -0.080 0.020 -0.053 -0.057 -0.012 0.016 0.014 0.031 -0.037 0.075 -0.002 0.034

12 0.115 0.013 0.000 -0.091 -0.058 0.000 -0.080 0.003 0.000 -0.084 0.016 0.000 0.048 -0.047 0.000 0.025 0.065

13 0.056 -0.041 -0.013 0.009 -0.073 -0.058 -0.017 -0.076 -0.094 0.036 0.008 0.009 -0.066 0.114 0.029 0.003 -0.019

14 -0.003 -0.031 -0.016 0.000 0.009 -0.091 -0.081 0.074 -0.014 -0.056 -0.012 -0.009 0.103 0.000 -0.027 0.045 0.040

15 0.095 0.011 -0.012 -0.016 0.101 0.000 -0.007 0.006 0.000 -0.052 0.006 0.000 0.055 -0.021 0.008 0.007 0.073

16 -0.046 -0.022 0.011 0.083 0.073 0.013 0.001 0.057 0.068 -0.029 0.057 0.097 0.071 0.024 0.021 0.038 0.071

17 0.107 -0.046 0.095 -0.003 0.056 0.115 0.029 0.129 0.173 0.115 0.101 0.058 0.085 0.112 0.097 0.183 0.142


18 0.072 -0.044 -0.013 -0.016 0.049 -0.008 -0.019 0.003 0.069 0.036 0.047 -0.009 -0.008 -0.007 -0.017 0.028 0.006

19 -0.044 -0.009 0.018 0.057 0.009 0.047 0.073 0.006 -0.044 -0.039 -0.006 -0.056 -0.028 -0.059 -0.011 -0.065 0.047

20 -0.013 0.018 0.029 0.040 0.035 0.000 0.068 0.001 0.000 -0.011 0.044 0.000 0.041 -0.078 0.053 -0.053 -0.029

21 -0.016 0.057 0.040 0.000 0.039 0.072 0.034 0.056 -0.006 -0.034 -0.046 -0.066 0.066 0.000 0.005 0.071 -0.025

22 0.049 0.009 0.035 0.039 0.007 0.047 0.052 0.007 0.078 0.028 0.016 -0.014 0.011 0.029 -0.069 -0.025 0.036

23 -0.008 0.047 0.000 0.072 0.047 0.000 -0.045 -0.057 0.000 -0.014 -0.026 0.000 -0.040 -0.007 0.000 -0.056 0.010

24 -0.019 0.073 0.068 0.034 0.052 -0.045 0.060 -0.030 0.074 -0.066 0.011 0.004 0.010 -0.001 -0.073 -0.041 0.052

25 0.003 0.006 0.001 0.056 0.007 -0.057 -0.030 -0.010 -0.073 0.066 0.043 0.047 0.031 0.020 -0.030 0.012 0.017

26 0.069 -0.044 0.000 -0.006 0.078 0.000 0.074 -0.073 0.000 0.025 0.013 0.000 0.001 -0.010 0.000 -0.062 0.015

149

27 0.036 -0.039 -0.011 -0.034 0.028 -0.014 -0.066 0.066 0.025 -0.014 -0.026 -0.011 -0.017 -0.027 -0.066 -0.035 -0.030

28 0.047 -0.006 0.044 -0.046 0.016 -0.026 0.011 0.043 0.013 -0.026 0.073 0.020 -0.074 -0.074 -0.023 -0.025 -0.042

29 -0.009 -0.056 0.000 -0.066 -0.014 0.000 0.004 0.047 0.000 -0.011 0.020 0.000 -0.027 0.028 0.000 -0.043 -0.022

30 -0.008 -0.028 0.041 0.066 0.011 -0.040 0.010 0.031 0.001 -0.017 -0.074 -0.027 0.020 0.004 -0.027 0.052 -0.099

31 -0.007 -0.059 -0.078 0.000 0.029 -0.007 -0.001 0.020 -0.010 -0.027 -0.074 0.028 0.004 0.000 -0.069 -0.090 -0.019

32 -0.017 -0.011 0.053 0.005 -0.069 0.000 -0.073 -0.030 0.000 -0.066 -0.023 0.000 -0.027 -0.069 -0.014 0.036 -0.040

33 0.028 -0.065 -0.053 0.071 -0.025 -0.056 -0.041 0.012 -0.062 -0.035 -0.025 -0.043 0.052 -0.090 0.036 -0.040 -0.039

34 0.006 0.047 -0.029 -0.025 0.036 0.010 0.052 0.017 0.015 -0.030 -0.042 -0.022 -0.099 -0.019 -0.040 -0.039 -0.083


1 -0.033 -0.013 -0.074 0.054 0.006 0.006 -0.033 -0.028 -0.003 -0.077 -0.020 -0.018 -0.068 0.023 -0.021 0.004 0.054

2 -0.013 -0.042 -0.030 0.009 -0.072 -0.027 -0.019 0.010 0.049 -0.027 0.030 0.002 0.070 0.026 0.046 -0.001 -0.002

3 -0.074 -0.030 -0.041 0.000 -0.025 0.000 0.057 0.023 0.000 0.053 -0.021 0.000 0.006 -0.002 0.021 0.035 -0.026

4 0.054 0.009 0.000 0.000 -0.050 -0.041 0.040 -0.039 0.008 -0.020 -0.043 0.004 -0.058 0.000 -0.038 0.022 0.025

5 0.006 -0.072 -0.025 -0.050 -0.007 0.048 0.012 0.021 -0.032 0.040 -0.026 0.012 0.032 0.033 0.053 0.033 -0.011

6 0.006 -0.027 0.000 -0.041 0.048 0.000 0.026 -0.009 0.000 -0.046 0.024 0.000 0.031 0.050 0.000 0.020 0.021

7 -0.033 -0.019 0.057 0.040 0.012 0.026 -0.062 0.011 0.002 0.061 0.007 0.018 -0.037 0.025 0.059 -0.010 0.042

8 -0.028 0.010 0.023 -0.039 0.021 -0.009 0.011 -0.028 0.019 -0.046 -0.044 0.035 -0.032 0.030 -0.004 -0.021 -0.006

9 -0.003 0.049 0.000 0.008 -0.032 0.000 0.002 0.019 0.000 -0.055 0.015 0.000 0.026 0.069 0.000 -0.014 0.023

10 -0.077 -0.027 0.053 -0.020 0.040 -0.046 0.061 -0.046 -0.055 0.013 0.046 -0.042 -0.034 0.004 -0.010 -0.020 -0.008

11 -0.020 0.030 -0.021 -0.043 -0.026 0.024 0.007 -0.044 0.015 0.046 0.012 0.023 0.039 -0.003 0.051 0.061 -0.064

12 -0.018 0.002 0.000 0.004 0.012 0.000 0.018 0.035 0.000 -0.042 0.023 0.000 0.022 0.068 0.000 -0.015 -0.031

13 -0.068 0.070 0.006 -0.058 0.032 0.031 -0.037 -0.032 0.026 -0.034 0.039 0.022 0.033 -0.015 0.025 0.008 0.022

14 0.023 0.026 -0.002 0.000 0.033 0.050 0.025 0.030 0.069 0.004 -0.003 0.068 -0.015 0.000 0.019 -0.002 -0.044

15 -0.021 0.046 0.021 -0.038 0.053 0.000 0.059 -0.004 0.000 -0.010 0.051 0.000 0.025 0.019 0.004 -0.052 -0.031

16 0.004 -0.001 0.035 0.022 0.033 0.020 -0.010 -0.021 -0.014 -0.020 0.061 -0.015 0.008 -0.002 -0.052 -0.041 -0.037

17 0.054 -0.002 -0.026 0.025 -0.011 0.021 0.042 -0.006 0.023 -0.008 -0.064 -0.031 0.022 -0.044 -0.031 -0.037 -0.046

Table B.22: C5 NE-MOX pin-power differences multiplied by 100 (Case 4) 18 19 20 21 22 23 24 25 1 0.122 0.024 -0.074 -0.076 0.009 0.056 -0.059 -0.013 2 0.137 0.072 -0.081 -0.044 -0.097 -0.076 -0.035 -0.035 3 0.055 0.025 -0.040 0.021 -0.115 0.000 -0.124 -0.048 4 0.116 0.028 -0.073 0.000 -0.017 -0.011 -0.082 -0.004 5 0.057 0.036 -0.066 0.003 -0.088 -0.144 -0.091 -0.092 6 0.057 -0.024 0.000 0.013 -0.047 0.000 -0.014 -0.098 7 0.112 0.028 -0.005 -0.167 -0.079 -0.099 0.000 -0.120 8 0.066 -0.058 -0.013 0.006 -0.085 0.015 -0.024 -0.103 9 0.062 -0.083 0.000 -0.034 -0.068 0.000 -0.067 -0.009 10 -0.007 -0.031 -0.043 -0.044 -0.014 -0.073 -0.071 0.030 11 0.086 -0.078 -0.049 -0.068 -0.077 -0.097 -0.015 -0.153 12 0.059 0.016 0.000 -0.041 -0.117 0.000 -0.058 -0.020 13 -0.053 0.006 -0.031 0.001 -0.106 0.001 -0.120 -0.018 14 0.113 -0.012 -0.067 0.000 -0.021 0.026 -0.057 -0.056 15 0.021 -0.044 -0.096 -0.011 -0.042 0.000 0.018 -0.021 16 -0.027 0.019 0.047 0.014 0.006 -0.030 -0.042 0.021 17 0.036 0.044 0.108 0.097 0.108 0.018 0.065 0.093

26 -0.046 0.023 0.000 -0.085 0.027 0.000 -0.059 -0.023 0.000 -0.048 -0.092 0.000 0.023 0.024 0.000 -0.019 0.081

27 -0.001 -0.050 0.058 -0.041 -0.107 -0.053 -0.013 -0.036 0.018 -0.101 -0.104 -0.010 -0.048 0.034 -0.028 -0.087 0.067

28 -0.015 0.013 -0.042 -0.078 -0.045 0.046 0.049 -0.094 -0.022 -0.078 -0.021 -0.069 -0.024 -0.081 0.075 0.017 0.014

29 -0.007 -0.071 0.000 -0.016 0.016 0.000 0.028 -0.075 0.000 -0.100 -0.042 0.000 -0.022 0.061 0.000 -0.006 0.002

30 -0.020 -0.039 0.034 -0.012 -0.045 -0.081 0.060 -0.063 -0.031 -0.007 -0.028 -0.034 -0.081 -0.010 -0.025 0.029 0.089

31 0.008 0.063 -0.045 0.000 0.034 0.007 -0.051 0.046 0.055 -0.037 0.042 -0.025 -0.040 0.000 0.058 -0.007 0.037

32 0.043 -0.005 0.055 0.128 0.029 0.000 0.040 0.117 0.000 0.014 -0.007 0.000 -0.004 0.033 0.001 0.005 0.131

33 -0.004 0.013 0.010 0.131 0.034 0.043 0.131 0.049 0.057 0.085 0.084 0.172 0.001 0.036 0.073 0.101 0.120

34 0.178 0.135 0.114 0.134 0.154 0.154 0.072 0.090 0.188 0.146 0.184 0.202 0.099 0.076 0.055 0.158 0.072


1 0.121 0.023 -0.075 -0.077 0.008 0.055 -0.060 -0.014 -0.047 -0.002 -0.015 -0.007 -0.021 0.008 0.043 -0.004 0.177

2 0.136 0.071 -0.082 -0.045 -0.098 -0.077 -0.036 -0.036 0.022 -0.051 0.012 -0.072 -0.039 0.062 -0.006 0.012 0.135

3 0.054 0.024 -0.041 0.020 -0.117 0.000 -0.006 -0.049 0.000 0.057 -0.043 0.000 0.033 -0.046 0.055 0.009 0.113

4 0.115 0.027 -0.075 0.000 -0.018 -0.012 -0.083 -0.005 -0.086 -0.042 -0.079 -0.017 -0.012 0.000 0.127 0.030 0.133

5 0.056 0.034 -0.067 0.001 -0.090 -0.145 -0.092 -0.093 0.026 -0.108 -0.046 0.015 -0.046 0.033 0.029 0.033 0.153

6 0.056 0.075 0.000 0.012 -0.049 0.000 -0.015 -0.099 0.000 -0.054 0.045 0.000 -0.082 0.006 0.000 0.042 0.053

7 0.110 0.027 -0.006 -0.168 -0.080 -0.100 -0.001 -0.121 -0.060 -0.014 0.048 0.027 0.059 -0.052 0.039 0.130 0.071

8 0.065 -0.060 -0.014 0.004 -0.086 0.014 -0.025 -0.104 -0.024 -0.037 -0.094 -0.076 -0.064 0.045 0.117 0.048 0.089

9 0.060 -0.084 0.000 -0.035 -0.070 0.000 -0.068 -0.010 0.000 0.018 -0.023 0.000 -0.032 0.054 0.000 0.056 0.188

150

10 0.092 -0.032 -0.044 -0.045 -0.015 -0.074 -0.072 0.030 -0.049 -0.102 -0.079 -0.100 -0.008 -0.038 0.014 0.085 0.145

11 0.085 -0.079 -0.050 -0.070 -0.078 -0.098 -0.016 -0.154 -0.093 -0.105 -0.021 -0.043 -0.029 0.041 -0.007 0.083 0.183

12 0.057 0.015 0.000 -0.043 -0.118 0.000 -0.059 -0.021 0.000 -0.011 0.050 0.000 -0.035 -0.026 0.000 0.171 0.101

13 0.045 0.004 -0.032 0.000 -0.107 0.000 -0.121 -0.019 0.022 -0.049 -0.024 -0.023 -0.082 -0.041 -0.004 0.000 0.098

14 0.112 -0.014 -0.068 0.000 0.078 0.025 -0.058 -0.057 0.024 0.033 -0.082 0.060 -0.011 0.000 0.032 0.035 0.076

15 0.020 -0.045 -0.097 -0.012 -0.044 0.000 0.017 -0.022 0.000 -0.029 0.074 0.000 -0.026 0.058 0.000 0.073 0.054

16 -0.029 0.018 0.046 0.013 0.005 -0.031 -0.043 0.021 -0.020 -0.088 0.016 -0.006 0.029 -0.007 0.005 0.100 0.157

17 0.135 0.043 0.107 0.096 0.107 0.018 0.065 0.092 0.080 0.067 0.014 0.002 0.088 0.037 0.131 0.120 0.071


18 0.025 0.025 -0.104 -0.013 -0.056 0.036 -0.051 0.062 0.024 0.072 -0.032 -0.086 -0.106 -0.128 0.011 0.035 0.090

19 0.025 -0.007 -0.060 -0.092 0.012 -0.088 -0.065 -0.033 0.106 0.046 -0.039 0.035 -0.027 0.004 0.113 0.122 0.147

20 -0.104 -0.060 0.044 -0.020 0.016 0.000 0.095 -0.155 0.000 -0.041 -0.007 0.000 -0.030 -0.064 -0.015 -0.100 0.077

21 -0.013 -0.092 -0.020 0.000 -0.120 0.011 0.023 0.005 -0.107 -0.132 0.066 -0.029 0.051 0.000 -0.058 -0.003 0.128

22 -0.056 0.012 0.016 -0.120 -0.036 -0.064 0.045 -0.123 0.048 -0.128 -0.013 -0.041 0.027 -0.050 0.109 -0.019 -0.018

23 0.036 -0.088 0.000 0.111 -0.064 0.000 -0.073 -0.158 0.000 -0.030 -0.048 0.000 -0.122 0.107 0.000 0.030 -0.047

24 -0.051 -0.065 0.095 0.023 0.045 -0.073 -0.084 -0.071 0.066 -0.046 -0.082 -0.126 -0.139 -0.061 0.006 0.076 0.078

25 0.062 -0.033 -0.155 0.005 -0.123 -0.158 -0.071 -0.010 -0.073 -0.018 -0.022 0.017 -0.131 0.064 0.031 0.049 0.003

26 0.024 0.106 0.000 -0.107 0.048 0.000 0.066 -0.073 0.000 0.067 0.048 0.000 0.071 -0.034 0.000 0.035 -0.072

151

27 0.072 0.046 -0.041 -0.132 -0.128 -0.030 -0.046 -0.018 0.067 -0.081 -0.019 0.069 0.086 0.046 -0.054 0.148 -0.029

28 -0.032 -0.039 -0.007 0.066 -0.013 -0.048 -0.082 -0.022 0.048 -0.019 -0.008 0.080 -0.021 0.071 -0.045 0.073 -0.069

29 -0.086 0.035 0.000 -0.029 -0.041 0.000 -0.126 0.017 0.000 0.069 0.080 0.000 -0.035 -0.027 0.000 0.010 0.091

30 -0.106 -0.027 -0.030 0.051 0.027 -0.122 -0.139 -0.131 0.071 0.086 -0.021 -0.035 -0.074 0.131 0.102 -0.044 -0.066

31 -0.128 0.004 -0.064 0.000 -0.050 0.107 -0.061 0.064 -0.034 0.046 0.071 -0.027 0.131 0.000 0.011 0.048 0.094

32 0.011 0.113 -0.015 0.042 0.109 0.000 0.006 0.031 0.000 -0.054 -0.045 0.000 0.102 0.011 0.154 0.071 0.168

33 0.035 0.122 0.000 -0.003 -0.019 0.030 0.076 0.049 0.035 0.148 0.073 0.010 -0.044 0.048 0.071 0.103 0.019

34 0.090 0.147 0.077 0.128 0.082 0.053 0.078 0.003 0.028 -0.029 0.031 0.007 -0.066 0.094 0.168 0.019 0.006

Vita

Boyan Ivanov was born in Pleven, Bulgaria on April 8, 1972. Boyan received his M.S. degree in Nuclear Power Engineering from Technical University, Sofia, Bulgaria in July of 1996. At the end of 1996, he started his job as a Senior Engineer Operator in the Nuclear Power Plant in Kozloduy, Bulgaria. He began post graduate studies in Nuclear Engineering at the Pennsylvania State University in August, 2000, where he received his M.S. degree in August, 2003. He continued his study at the Pennsylvania State University and earned the Ph.D. degree in Nuclear Engineering in December, 2007.