approximation: a new method for solving integral transport equations

123 Pages·2016·1.71 MB·English

by  Zekeriya Altaç

Iowa State University Capstones, Theses and
Retrospective Theses and Dissertations
Dissertations
1989
The SK(N) approximation: a new method for
solving integral transport equations
Zekeriya Altaç
Iowa State University
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Recommended Citation
Altaç, Zekeriya, "The SK(N) approximation: a new method for solving integral transport equations " (1989).Retrospective Theses and
Dissertations. 9262.
https://lib.dr.iastate.edu/rtd/9262
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The SKn approximation: A new method for solving integral
transport equations
Altaç, Zekeriya, Ph.D.
Iowa State University, 1989
U M I
SOON.ZeebRd.
Ann Arbor, MI 48106

The SKj^ approximation: A new method for
solving integral transport equations
by
Zekeriya Altaç
A Dissertation Submitted to the
Graduate Faculty in Partial Fulfillment of the
Requirements for the Degree of
DOCTOR OF PHILOSOPHY
Major: Nuclear Engineering
Approved;
Signature was redacted for privacy.
In Charge of Majpr Work
Signature was redacted for privacy.
FFoorr tthhee MMaajjoorr DDeeppaarrttmmeenntt
Signature was redacted for privacy.
raduate College
Iowa State University
Ames, Iowa
1989
ii
TABLE OF CONTENTS
ACKNOWLEDGEMENTS ix
DEDICATION x
1 INTRODUCTION 1
2 LITERATURE REVIEW 4
3 THEORY 15
3.1 Derivation of Neutron Transport Equation 15
3.2 Derivation of the Integral Transport Equations 18
3.3 Derivation of Integral Transport Equations for 1-D Geometries ... 22
3.3.1 Slab geometry 23
3.3.2 Cylindrical geometry 26
3.3.3 Spherical geometry 30
3.4 Derivation of SK^ Equations for 1-D Geometries 32
3.4.1 Slab geometry 32
3.4.2 Cylindrical geometry 34
3.4.3 Spherical geometry 38
3.5 Derivation of the SKp^ Equations in Two Dimensions 40
iii
4 NUMERICAL SOLUTION OF THE SKj^ EQUATIONS ... 43
4.1 One Dimensional Geometries 43
4.2 Two Dimensional X-Y Geometry 49
5 RESULTS AND COMPARISONS FOR 1-D GEOMETRIES . 55
5.1 Results for Homogeneous Systems 58
5.1.1 Slab geometry 59
5.1.2 Cylindrical geometry 62
5.1.3 Spherical geometry 64
5.2 Results for Heterogeneous Systems 66
5.2.1 Slab geometry 66
5.2.2 Cylindrical geometry 68
5.2.3 Spherical geometry 69
6 RESULTS AND COMPARISONS FOR X-Y GEOMETRY . . 72
7 A POSSIBLE REMEDY FOR HETEROGENEOUS PROB­
LEMS 89
8 CONCLUSIONS 96
9 SUGGESTIONS FOR FUTURE WORK 99
10 BIBLIOGRAPHY 101
iv
11 APPENDIX A: THE EXPONENTIAL INTEGRAL FUNC­
TIONS 104
12 APPENDIX B: THE BICKLEY-NAYLOR FUNCTIONS ... 106
V
LIST OF TABLES
Table 2.1: Quadrature set-2 for various systems 12
Table 2.2: Quadrature set-3 values for various systems 13
Table 5.1: The six group macroscopic cross sections for BMPs 2 and 3 56
Table 5.2: Solution of BMP 1 for Slab Geometry 59
Table 5.3: ODSKN solutions of BMP 1 for slab geometry 60
Table 5.4: Results of BMP 2 and BMP 3 for slab 61
Table 5.5: Solution of BMP 1 for the cylinder . 62
Table 5.6: Results of BMP 2 and BMP 3 for cylinder 63
Table 5.7: The results of BMP 1 for sphere 64
Table 5.8: The results of BMP 2 for sphere 65
Table 5.9: The results of BMP 3 for sphere 65
Table 5.10: Flux profile of BMP 4 for slab with vacuum BC 67
Table 5.11: Flux profile of BMP 4 for slab with reflecting BC 67
Table 5.12: Flux profile of BMP 4 for cylinder with vacuum BC 68
Table 5.13: Flux profile of BMP 4 for cylinder with reflecting BC .... 69
Table 5.14: Flux profile of BMP 4 for sphere with vacuum BC 70
Table 5.15: Flux profile of BMP 4 for sphere with reflecting BC 71