Neutron Diffusion Theory is a complete lesson

1. Reactor Physics
3. Neutron Diffusion Theory
Xiang Wang
March 24, 2021
哈 尔 滨 工 程 大 学 核 科 学 与 技 术 学 院
H A R B I N E N G I N E E R I N G U N I V E R S I T Y COLLEGE OF NUCLEAR SCIENCE & TECHNOLOGY

2. Review
Nuclear Reaction
Cross Section & Yields
Fission
After Fission
Chain Reaction
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3. Outlines
One-Velocity Diffusion Equation
Two-Group Diffusion Equation
Applications to Reactor
Reflected Reactors
Two-Group Model of Reflected Reactor
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4. One-Velocity Diffusion
Equation
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5. General Introduction
• Problem
Neutron Source Neutron Sink
Diffusion
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6. General Introduction
• Basic Variables
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7. General Introduction
• Target
Neutron density determination
• General Problem
Finite, inhomogeneous, anisotropic medium
A function of space, energy, time
• What we deal with (now)
Uniform, homogeneous, isotropic medium
Steady state
• Assumptions
Finite system of various shapes
Homogeneous, isotropic medium
Uniform neutron source and sinks
Multiplying/Non-multiplying media
Space dependence
Energy independence – one velocity
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8. Equation of Continuity
• Neutron Balance
Flux
Integral form
8
Rate of change in
number of neutrons
in dV
Rate of
production of
neutrons in V
Rate of
absorption of
neutrons in V
Rate of leakage of
neutrons from V
= - -
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9. Equation of Continuity
Differential form
Complete equation
Steady-state equation of continuity
9
Rate of change of
the neutron density
in about
Number of
neutrons
produced
in per
Number of
neutrons
absorbed
in per
Net number of
neutrons escaping
from per
= - -
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10. Fick’ s Law
• Chemical Approach
Increase Area
Increase Diffusion Coeff.
Decrease Thickness
Increase Particles
10
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11. Fick’ s Law
• Physical Approach
Number of collisions in dV:
Number of neutron crossing dA from above
For slow varying flux, Taylor’s series at the origin
11
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12. Fick’ s Law
‒ let
‒ is then expressed as
The current from both sides of dA
‒ The net current on z
The total current
12
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13. Fick’ s Law
• Diffusion Coefficient
• Transport Mean Free Path
For moderator,
13
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14. Fick’ s Law
• Limitations
14
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15. Diffusion Equation
• With conclusion of Fick’s Law
General form
Steady-state diffusion equation
‒ Diffusion length
• Unit:
‒ Diffusion area
• Unit:
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16. Diffusion Equation
16
• Diffusion Length
 Definition
 From point source
 Absorbed in
 Average distance
 Meanings
‒ larger->neutron moves further
‒ more diffusive, less absorptive
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17. Diffusion Equation
• Boundary Conditions
Normal conditions
‒ Finite & nonnegative in all regions …
‒ Except in certain source distributions
‒ At an interface flux must be continuous
‒ At vacuum interface no neutrons return
Actual conditions
‒ Finite & nonnegative in all regions …
‒ except in certain source distributions
‒ Which diffusion equation applies
‒ At an interface flux must be continuous
‒ Partial current must be continuous
‒ At vacuum interface no neutrons return
‒ At vacuum interface flux must satisfy:
• Partial current from vacuum must be zero
• Flux term must vanish at some finite outside
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18. Diffusion Equation
• Expressions
I
II
III
IV
‒ Set flux at outside is zero
‒ Set close to
Extrapolation length
18
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19. Solutions
• Infinite Planar Source
Diffusion equation
Assumptions
‒ Infinite, homogeneous isotropic medium
‒ A infinite plane of isotropic neutron source
‒ neutrons per per
‒ Source region not special
‒ Steady state
Simplification
‒ normal to infinite plane source
‒ For
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20. Solutions
Solution
‒ Consider
‒ Direction of one-half the neutrons
‒ From Fick’s Law,
‒ Substitution
‒ Final solution for both sides
20
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21. Solutions
• Point Sources
Diffusion equation
Assumptions
‒ Isotropic point source
‒ >Spherically symmetric problem
‒ >Polar coordinate
Simplification
‒ independent of and
21
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22. Solutions
Solution
‒ Introduce new variable
• New equation of
• Give general solution the same as for the planar source
‒ Source condition
‒ Final solution
22
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23. Solutions
• Line Source
Assumptions
‒ Isotropic infinite line form neutron source
‒ Axial cylindrical coordinates
Simplification
23
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24. Solutions
Solution
‒ Define
‒ General solution
‒ Modified Bessel Function
‒ Boundary condition
‒ Final solution
24
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25. Solutions
• Modified Bessel Function
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26. Solutions
• Bare Slab
Assumptions
‒ Infinite slab
‒ Infinite planer source
Simplification
‒ For
Solution
‒ Boundary conditions
‒ For right-half plane
26
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27. Solutions
‒ Using source condition
‒ For all
27
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28. Two-Group Diffusion
Equation
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29. Energy Groups
• N Groups
Principle
‒ Neutron has initially energy spectrum
‒ Losing energy by collisions
Total absorption
Group transfer
29
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30. Energy Groups
Total transfer rate
Updated diffusion equation
Group diffusion coefficient
30
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31. Thermal Neutron Diffusion
• Maxwellian Function
Thermal neutron energy distribution
• Thermal Flux
One-group thermal flux
‒ Using usual formula for speed
31
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32. Thermal Neutron Diffusion
Simplification
‒ Introduce
‒ Distinguish
• 2200m/s flux : pseudoflux
• Assume all thermal neutrons at
• Yield the same using at
• Appropriate when using a reactor
• Thermal flux
• Appropriate when design a reactor
‒ For
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33. Thermal Neutron Diffusion
• One-Group Diffusion Equation
Thermal
One-Group
Relationship with density
33
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34. Two-Group Calculation
• Two-Group Calculation
Assumptions
‒ Fast group included
‒
‒
‒ Neutron scattered out of fast must enter thermal
‒ Point source
Fast Diffusion Equation
‒ Neutron age
Thermal Diffusion Equation
34
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35. Two-Group Calculation
Solution
‒ Fast Term
‒ Thermal Term
‒ Notes
• NOT an appropriate model for slowing down process
• ONLY for rough, first-order calculation for thermal reactors
35
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36. Applications to
Reactor
36
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37. Criticality in Infinite Medium
• Neutron Balance
contain infinite fissile material
homogeneous & isotropic composition
monoenergetic neutron (same speed v)
material has only and
• Four Factor Formula
Reproduction Factor
Thermal Utilization Factor
Resonance Escape Probability
Fast Fission Factor
Multiplication Factor
, ,
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38. Criticality in Infinite Medium
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39. Criticality in Finite Medium
39
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40. Criticality in Finite Medium
• Critical Fuel Concentration
 M for moderator, F for fuel
 Volume fraction
• Six Factor Formula
 Reproduction Factor
 Thermal Utilization Factor
 Resonance Escape Probability
 Fast Fission Factor
 Fast Non-leakage Probability
 Thermal Non-leakage Probability
 Multiplication Factor
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41. One-Group Reactor Equation
• Bare Reactor
Assumptions
‒ Critical fast reactor
‒ Homogeneous mixture of fuel & coolant
‒ Only one region, no blanket nor reflector
‒ Only one energy group
One-group diffusion equation
‒ Considering only fission source
‒ Time dependent
‒ Time independent
41
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42. One-Group Reactor Equation
Geometric Buckling
‒ Definition
‒ Eigenvalue equation
‒ One-group reactor equation
Multiplication factor
‒ One-group
42
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43. Slab Reactor
• Equation
Assumptions
‒ Critical system
‒ Infinite bare slab
Boundary condition
• Solution
General solution
43
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44. Slab Reactor
Approach
Power of the reactor
Final solution
44
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45. Spherical Reactor
• Equation
• Solution
General solution
Special solution
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46. Infinite Cylindrical Reactor
46
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• Equation
• Solution
General solution
Special solution

47. Infinite Cylindrical Reactor
• Bessel Functions and
47
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48. Finite Cylindrical Reactor
48
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• Equation
• Solution
General solution
Special solution

49. Max/Average Flux
• Bare Spherical Reactor
Max flux
Average flux
49
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50. Reflected Reactor
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51. General Introduction
• Benefit from reflectors
Better neutron economics
‒ Neutron escape reduced
‒ Neutron moderated in the reflector
‒ Serves as the radiation shield
‒ Flat the flux
Reactor savings
‒ Decrease size of critical core
‒ Decrease materials of critical core
• Assumptions
Central core of multiplying material
Reflector around core of non-multiplying material
Internally homogeneous core & reflector
Time-independent
51

52. General Introduction
• Boundary conditions
Symmetry in core, non-singularity of flux at center
Continuity of flux at all points of interface
Continuity of net current at all points of interface
Flux vanish on all extrapolated outer surface
• Equation Set
Separately for core and reflector
Reduced expression
52

53. Simple Approach
• Spherical Geometry (Simplest)
Special conditions
‒ Spherical core of
‒ Surrounded by an infinite reflector
Simplified equation set
Solution
‒ General solution
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54. Simple Approach
‒ Boundary condition
• Remain finite as r goes infinity & r=0
• Interface boundary condition
‒ Substitution
‒ Criticality condition (Transcendental Equation)
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55. Simple Approach
• Solution of transcendental equation
For
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56. Application to Reactors
• Spherical Geometry
Specific conditions
‒ Spherical core of
‒ Spherical shell reflector of
Solution
‒ General solution
56

57. Application to Reactors
‒ Continuity condition
‒ Introduce new critical radius
‒ Determination of critical radius
57

58. Application to Reactors
For thin reflector
‒ Simplification
58

59. Application to Reactors
For thick reflector
59

60. Application to Reactors
Coefficients
‒ Power
‒ Coefficients
60

61. Application to Reactors
• Cylindrical Geometry
With Reflector on the Curved Surface
‒ Complete Form
‒ Separation
61

62. Application to Reactors
‒ Independent constants
Solution
‒ General solution
‒ Symmetry (BC1)
62

63. Application to Reactors
‒ Continuity of flux at interface (BC2)
‒ Flux vanishes at extrapolation (BC4)
63

64. Application to Reactors
‒ Continuity of net current at interface (BC3)
‒ Other parameters
64

65. Application to Reactors
• Cylindrical Geometry
With End Reflectors Only
‒ Complete Form
‒ Separation
65

66. Application to Reactors
‒ Independent constants
Solution
‒ General solution
‒ Symmetry (BC1)
66

67. Application to Reactors
‒ Continuity of flux at interface (BC2)
‒ Flux vanishes at extrapolation (BC4)
‒ Other parameters
67

68. Application to Reactors
• Rectangular Geometry
With End Reflectors Only
‒ Complete Form
‒ Separation
68

69. Application to Reactors
‒ Independent constants
Solution
‒ General solution
69

70. Application to Reactors
‒ Continuity at interface (BC2)
‒ Flux vanishes at extrapolation (BC4)
‒ Other parameters
70

71. Two-Group Model of
Reflected Reactor
71

72. Reactor Equations
• Two-Group Diffusion Equations
Considerations
‒ chosen to approx. slowing-down density of fast out
of fast group at
‒ Source of fast is fission caused by thermal
‒ Non-multiplying reflector
‒ Introduce and
72

73. Reactor Equations
• Group Constants
Infinite multiplying core
Consider a spatial separated term
Diffusion length
73

74. Reactor Equations
Non-leakage Probability
• Solution of Core Functions
 , and independent
>Coefficients are 0 74

75. Reactor Equations
Function of B
75

76. Reactor Equations
• Solution of Reflector Equations
Same extrapolation distance
76

77. Reactor Equations
• Criticality Condition
At core-reflector interface
77

78. Summary Of Functions
• Spherical Geometry
78

79. Summary Of Functions
• Cylinder Geometry (side reflector)
79

80. Summary Of Functions
• Cylinder Geometry (end reflector)
80

81. Summary Of Functions
• Block Geometry
81

82. Critical Radius
82

83. Two-Group Approach
• Flux in a reflected thermal reactor
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84. Two-Group Approach
• Flux of a spherical reactor in one- and two -group
84

85. Heterogeneous
Reactor
85

86. Introduction
86

87. Introduction
87

88. Thermal Disadvantage Factor
• Assumptions
Unit cell
‒ f/m ratio equals to entire core
‒ Net transport of =0
Slab reactor
‒ Fuel region (2a)
‒ Moderator region (2(b-a))
Diffusion for both material
‒ Fuel region: strong absorption for
‒ Dimensions of both regions
88

89. Thermal Disadvantage Factor
• Assumptions
Edge effect however not important
Uniform source of over moderator
No neutron thermalized in fuel region
89

90. Thermal Disadvantage Factor
• Real geometry
90

91. Thermal Disadvantage Factor
• Analytical Model
Problem description
91

92. Thermal Disadvantage Factor
Definition
For infinitesimally thin
92

93. Thermal Disadvantage Factor
Visualization of definition
93

94. Thermal Disadvantage Factor
For convenience
‒ Cell has unit height
‒
‒
94

95. Thermal Disadvantage Factor
Lattice functions
Suitable for
‒ Cylindrical fuel rods in square lattice
‒ Spherical lumps of fuel in cubic lattice
95

96. Thermal Disadvantage Factor
• E and F for Various Cell Geometries
96

97. Thermal Utilization Factor
• Definition
Features of
‒ Decrease into heterogeneous arrangement
‒ Decrease with larger size for given V ratio
97

98. Thermal Utilization Factor
• Definition
For heterogeneous system
Comparison
98

99. Thank you！
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