Title Page

Abstract

Contents

I. Introduction 12

II. Overestimation of 238U Cross Sections(이미지참조) 14

2.1. Equivalence Theory 14

2.2. Numerical Test with Equivalence Theory 19

2.3. Pointwise Energy Approach 21

2.4. Numerical Test with Pointwise Energy Approach 23

2.5. Numerical Test without Resonance Scattering Cross Section 27

2.6. Contemporary Spatially Dependent Self-shielding Method 28

III. Pin-based Pointwise Energy Slowing-down Method 31

3.1. General Derivation 31

3.2. Collision Probability Calculation: First step - Isolated Fuel Pellet 34

3.3. Collision Probability Calculation: Second step - Fuel Pin in Lattice 37

3.4. Resonance Upscattering Treatment 41

3.5. Techniques to Achieve High Performance 42

3.6. Calculation Flow 45

IV. Numerical Result 48

4.1. Sensitivity Test for Calculation Option in PSM 51

4.2. Sensitivity Test for Ratio of Fuel Diameter to Pin-pitch 55

4.3. Base Pin-cell Problem 57

4.4. Pin-cell with Uniform Material Composition and Temperature Profile 59

4.5. Pin-cell with Non-uniform Material Composition and Uniform Temperature Profile 80

4.6. Pin-cell with Non-uniform Material Composition and Temperature Profile 87

4.7. SNU Non-uniform Temperature Pin-cell Benchmark 94

4.8. VERA 17x17 Fuel Assembly Problem 101

4.9. 2x2 Multi-assembly Problem 104

4.10. 17x17 Fuel Assembly Depletion Benchmark 110

4.11. Test for Computing Time 113

V. Discussion 116

VI. Conclusions 118

References 120

Journal Publications 123

Table 1. Material composition of base pin-cell problem. 19

Table 2. Coefficients of three-term rational equation. 23

Table 3. Grouping of scattering nuclides. 44

Table 4. Summary of test cases. 48

Table 5. Elapsed time in resonance treatment with PSM. 52

Table 6. Nuclide-wise contribution to k-inf difference (Mosteller benchmark 5 wt.% UO₂ pin-cell). 63

Table 7. Nuclide-wise contribution to k-inf difference (Mosteller benchmark 8 wt.% PuO₂ reactor-... 73

Table 8. Geometry information of the burned pin-cell problem. 80

Table 9. k-inf and difference (60 MWd/㎏ burned fuel pin-cell). 81

Table 10. Nuclide-wise contribution to k-inf difference (60 MWd/㎏ burned fuel pin-cell). 82

Table 11. Parameters for TH feedback calculation. 88

Table 12. k-inf and difference (60 MWd/㎏ burned fuel pin-cell with TH feedback). 89

Table 13. Nuclide-wise contribution to k-inf difference (60 MWd/㎏ burned fuel pin-cell with TH... 89

Table 14. Fuel temperature coefficients (SNU benchmark). 96

Table 15. Description for fuel assembly problem. 102

Table 16. k-inf and pin power distribution results - SDDM. 103

Table 17. k-inf and pin power distribution results - PSM. 104

Table 18. Results for k-inf and pin power distribution (2x2 multi-assembly problem). 107

Table 19. Nuclide-wise contribution to k-inf difference (Pin-1 in mutli-assembly problem). 109

Table 20. Nuclide-wise contribution to k-inf difference (Pin-2 in mutli-assembly problem). 109

Table 21. Description for fuel assembly depletion problems. 110

Table 22. Comparison for elapsed time 114

Fig. 1. Comparison of 238U absorption XSs with the equivalence theory (base pin-cell problem).(이미지참조) 20

Fig. 2. Flat source divisions for pointwise energy MOC calculation. 24

Fig. 3. Comparison of fuel region-averaged 238U absorption XSs with pointwise energy approaches...(이미지참조) 24

Fig. 4. Comparison of fuel-to-fuel collision probability (base pin-cell problem). 25

Fig. 5. Scattering source distribution in fuel pellet with 15-mesh-PW-MOC. 26

Fig. 6. XSs of 238U and fictitious 238U.(이미지참조) 27

Fig. 7. Comparison of fictitious 238U absorption XSs with pointwise energy approaches (base pin-cell...(이미지참조) 28

Fig. 8. Example for fuel collision probability of neutron born in sub-region 3. 35

Fig. 9. Example for fuel escape probability of fuel lump and ratio. 39

Fig. 10. Energy integration range for current and previous scattering source calculations. 44

Fig. 11. Flowchart of the pin-based pointwise energy slowing-down solution method (PSM). 46

Fig. 12. Flowchart of the pin-based pointwise energy slowing-down solution method with CPM... 47

Fig. 13. Comparison of k-inf from PSM with different number of energy points in the XS libraries. 52

Fig. 14. Comparison of fuel region-averaged 238U absorption XS for PSM sub-region sensitivity test...(이미지참조) 54

Fig. 15. Geometries of pin-cells with different ratios of fuel diameter to pin-pitch. 55

Fig. 16. Results of sensitivity test for ratio of fuel diameter to pin-pitch. 56

Fig. 17. Comparison of fuel region-averaged 238U absorption XS with spatially dependent resonance...(이미지참조) 57

Fig. 18. Comparison of region-wise 238U absorption XS of Groups 21, 25 and 26 (base pin-cell...(이미지참조) 58

Fig. 19. Results for Mosteller benchmark UO₂ fuel problems. 60

Fig. 20. Results for Mosteller benchmark reactor-recycle MOX fuel problems. 61

Fig. 21. Results for Mosteller benchmark weapons-grade MOX fuel problems. 61

Fig. 22. Doppler coefficients for Mosteller benchmark with SVT and DBRC scattering kernels. 62

Fig. 23. Contribution to k-inf difference for 238U in all regions (Mosteller benchmark 5 wt.% UO₂ pin-...(이미지참조) 64

Fig. 24. Comparison of absorption and nu*fission reaction rates for 238U in all energy groups...(이미지참조) 64

Fig. 25. Comparison of absorption and nu*fission reaction rates for 238U in fast energy groups...(이미지참조) 65

Fig. 26. Comparison of absorption and nu*fission reaction rates for 238U in resonance energy groups...(이미지참조) 65

Fig. 27. Comparison of absorption and nu*fission reaction rates for 238U in thermal energy groups...(이미지참조) 66

Fig. 28. Comparison of absorption XS and reaction rate for 238U in Group 26 (Mosteller benchmark 5...(이미지참조) 67

Fig. 29. Comparison of absorption XS and reaction rate for 238U in Group 27 (Mosteller benchmark 5...(이미지참조) 67

Fig. 30. Comparison of absorption XS and reaction rate for 238U in Group 29 (Mosteller benchmark 5...(이미지참조) 68

Fig. 31. Contribution to k-inf difference for 235U in all regions (Mosteller benchmark 5 wt.% UO₂ pin-...(이미지참조) 69

Fig. 32. Comparison of absorption and nu*fission reaction rates for 235U in all energy groups...(이미지참조) 69

Fig. 33. Comparison of absorption and nu*fission reaction rates for 235U in resonance energy groups...(이미지참조) 70

Fig. 34. Comparison of absorption and nu*fission reaction rates for 235U in thermal energy groups...(이미지참조) 70

Fig. 35. Comparison of absorption XS and reaction rate for 235U in Group 29 (Mosteller benchmark 5...(이미지참조) 72

Fig. 36. Comparison of nu*fission XS and reaction rate for 235U in Group 29 (Mosteller benchmark 5...(이미지참조) 72

Fig. 37. Contribution to k-inf difference for 238U in all regions (Mosteller benchmark 8 wt.% PuO₂...(이미지참조) 73

Fig. 38. Comparison of absorption XS and reaction rate for 238U in Group 27 (Mosteller benchmark 8...(이미지참조) 74

Fig. 39. Comparison of absorption XS and reaction rate for 238U in Group 29 (Mosteller benchmark 8...(이미지참조) 75

Fig. 40. Contribution to k-inf difference for 239Pu in all regions (Mosteller benchmark 8 wt.% PuO₂...(이미지참조) 75

Fig. 41. Comparison of absorption XS and reaction rate for 239Pu in Group 25 (Mosteller benchmark 8...(이미지참조) 76

Fig. 42. Comparison of nu*fission XS and reaction rate for 239Pu in Group 25 (Mosteller benchmark 8...(이미지참조) 77

Fig. 43. Comparison of absorption XS and reaction rate for 239Pu in Group 29 (Mosteller benchmark 8...(이미지참조) 77

Fig. 44. Comparison of nu*fission XS and reaction rate for 239Pu in Group 29 (Mosteller benchmark 8...(이미지참조) 78

Fig. 45. Contribution to k-inf difference for 242Pu in all regions (Mosteller benchmark 8 wt.% PuO₂...(이미지참조) 78

Fig. 46. Comparison of absorption XS and reaction rate for 242Pu in Group 31 (Mosteller benchmark 8...(이미지참조) 79

Fig. 47. Temperature profile and number densities (60 MWd/㎏ burned fuel pin-cell). 81

Fig. 48. Contribution to k-inf difference for 239Pu in all regions (Burned UO₂ pin-cell).(이미지참조) 82

Fig. 49. Comparison of absorption and nu*fission reaction rates for 239Pu in resonance energy groups...(이미지참조) 83

Fig. 50. Comparison of absorption XS and reaction rate for 239Pu in Group 29 (Burned UO₂ pin-cell).(이미지참조) 83

Fig. 51. Comparison of nu*fission XS and reaction rate for 239Pu in Group 29 (Burned UO₂ pin-cell).(이미지참조) 84

Fig. 52. Contribution to k-inf difference for 238U in all regions (Burned UO₂ pin-cell).(이미지참조) 85

Fig. 53. Comparison of absorption XS and reaction rate for 238U in Group 27 (Burned UO₂ pin-cell).(이미지참조) 85

Fig. 54. Contribution to k-inf difference for 150Sm in all regions (Burned UO₂ pin-cell).(이미지참조) 86

Fig. 55. Comparison of absorption XS and reaction rate for 150Sm in Group 27 (Burned UO₂ pin-cell).(이미지참조) 86

Fig. 56. Temperature profile and number densities (60 MWd/㎏ burned fuel pin-cell with TH...(이미지참조) 88

Fig. 57. Contribution to k-inf difference for 239Pu in all regions (Burned UO₂ pin-cell with TH...(이미지참조) 90

Fig. 58. Comparison of absorption XS and reaction rate for 239Pu in Group 29 (Burned UO₂ pin-cell...(이미지참조) 90

Fig. 59. Comparison of nu*fission XS and reaction rate for 239Pu in Group 29 (Burned UO₂ pin-cell...(이미지참조) 91

Fig. 60. Contribution to k-inf difference for 238U in all regions (Burned UO₂ pin-cell with TH...(이미지참조) 92

Fig. 61. Comparison of absorption XS and reaction rate for 238U in Group 27 (Burned UO₂ pin-cell...(이미지참조) 92

Fig. 62. Comparison of absorption XS and reaction rate for 238U in Group 29 (Burned UO₂ pin-cell...(이미지참조) 93

Fig. 63. Temperature profiles of uniform temperature cases (SNU benchmark). 94

Fig. 64. Temperature profiles of non-uniform temperature cases (SNU benchmark). 94

Fig. 65. Comparison of reactivity (SNU benchmark). 95

Fig. 66. Contribution to k-inf difference for 238U in all regions (200% power non-uniform temperature...(이미지참조) 96

Fig. 67. Comparison of absorption XS for 238U in Group 27 (200% power non-uniform temperature...(이미지참조) 97

Fig. 68. Comparison of absorption XS for 238U in Group 29 (200% power non-uniform temperature...(이미지참조) 97

Fig. 69. Macroscopic total XSs in fuel pellet (200% power non-uniform temperature case). 98

Fig. 70. Contribution to k-inf difference for 235U in all regions (200% power non-uniform temperature...(이미지참조) 99

Fig. 71. Comparison of absorption XS for 235U in Group 29 (200% power non-uniform temperature...(이미지참조) 99

Fig. 72. Comparison of nu*fission XS for 235U in Group 29 (200% power non-uniform temperature...(이미지참조) 100

Fig. 73. Configuration of rods in 17x17 Fuel assembly problem. 101

Fig. 74. Fuel assembly configuration of 2x2 multi-assembly problem. 105

Fig. 75. Temperature profile of fuel pellets. 106

Fig. 76. Pin power distribution from MCNP6 (2x2 multi-assembly problem). 106

Fig. 77. Relative difference in pin power distribution with PSM 107

Fig. 78. Relative difference in pin power distribution with PSM-CPM 108

Fig. 79. Comparison of reaction rates of 235U and 238U in resonance energy groups (Pin-1 in mutli-...(이미지참조) 109

Fig. 80. Comparison of reaction rates of 235U and 238U in resonance energy groups (Pin-2 in mutli...(이미지참조) 110

Fig. 81. Analysis result of 17x17 fuel assembly without poison. 111

Fig. 82. Analysis result of 17x17 fuel assembly with 24 Pyrex. 112

Fig. 83. Analysis result of 17x17 fuel assembly with 24 Gadolinia. 112

Fig. 84. Elapsed time as a function of the number of radial meshes. 115

 A new resonance self-shielding method using a pointwise energy solution has been developed to overcome the drawbacks in the equivalence theory. In reactor physics, the equivalence theory has been widely used in calculating the effective multi-group cross sections for the neutron transport analyses. The neutron transport codes adopting the equivalence theory give reasonable solutions within short computation time. However, there are still a lot of limitations in the equivalence theory even though many modified and improved equivalence theories have been published over the past several decades. The significant drawbacks in the equivalence theory are newly figured out in this work, and the new method is proposed to overcome the problems. The equivalence theory uses the intermediate resonance approximation on the resonance scattering source and the multi-term rational approximation to represent the fuel escape probability. With these approximations, the effective multi-group cross section is derived with the asymptotic equations. However, there is a gap between the derivation and the practical usage in the lattice physics code. In addition, the equivalence theory assumes that the constant distribution of the scattering sources in the fuel pellet even though the source distribution is quite important in view point of the fuel escape probability. These methods and approximations cause significant errors, in that they overestimate the effective multi-group cross sections, especially for 238U. The new resonance self-shielding method solves pointwise energy slowing-down equations which are derived for a sub-divided fuel pellet and a non-fuel region. A two-step method is developed to efficiently calculate the collision probabilities of the sub-divided fuel pin-cell. In the first step, the collision probabilities of the sub-divided fuel pellet are calculated assuming that the fuel pellet is isolated. In the second step, a shadowing effect correction factor is derived based on the equivalence theory to consider the global self-shielding effect. In addition, a fictitious moderator material is generated to model realistic scattering source from the moderator. The slowing-down solutions are used to generate the multi-group cross sections of the sub-divided fuel pellet. Various techniques and assumptions are incorporated to maximize calculation efficiency in solving the pointwise energy slowing-down equations. Especially, the new method significantly reduces the number of MOC fixed-source calculations which is one of major time consuming calculations in the resonance self-shielding calculation. Although the new method performs the pointwise energy slowing-down calculations, the computational cost is not expensive even compared to that with the conventional equivalence theory. With various light water reactor problems, it is demonstrated that the new resonance self-shielding method successfully overcomes the limitations of the equivalence theory and shows great accuracy in calculating the multiplication factor, the multi-group cross section, the reaction rate, and the power distribution with no compromise in computation time.