1. DIFFUSION OF NEUTRONS OVERVIEW Basic Physical Assumptions
Generic Transport Equation
Diffusion Equation
Fermi’s Age Equation
Solutions to Reactor Equation

2. Basic Physical Assumptions
Neutrons are dimensionless points
Neutron – neutron interactions are neglected
Neutrons travel in straight lines
Collisions are instantaneous
Background material properties are isotropic
Properties of background material are known and time-independent
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3. T10: Diffusion of Neutrons
Physical Model
(a)
(b)
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4. T10: Diffusion of Neutrons
Collision Model θ rm b rc χm v v´
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5. T10: Diffusion of Neutrons
Initial Definitions j r W ex x y z ey v q
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6. T10: Diffusion of Neutrons
Neutron Density
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7. Angular Flux and Current DensityJ dS
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8. Generic Transport Equation
Arbitrary volume V
We ignore macroscopic forces
Asteroid Eros
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9. Generic Transport Equation
Gauss Theorem:
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10. Substantial Derivativez r y x
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11. Transport (Boltzmann) Equation
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12. T10: Diffusion of Neutrons
Collision Term z r y x
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13. Neutron Transport Equation
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14. T10: Diffusion of Neutrons
Boundary Condition
Outgoing direction
Outward normal Ω
Incoming direction ns r
Volume V x y z Ω Rs
Surface S V
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15. T10: Diffusion of Neutrons
Difficulties
Mathematical structure is too involved
Mixed type equation (integro-differential), no way to reduce it to a differential equation
Boundary conditions are given only for a halve of the values
Too many variables (7 in general)
Angular variable
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16. T10: Diffusion of Neutrons
Angular Measures
180 Solar disks
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17. T10: Diffusion of Neutrons
Plane Angles R φ
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18. T10: Diffusion of Neutrons
Solid Angles
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19. One-Group Diffusion Model
Infinite homogeneous and isotropic medium
Neutron scattering is isotropic in Lab-system
Weak absorption Σa << Σs
All neutrons have the same velosity v. (One-Speed Approximation)
The neutron flux is slowly varying function of position
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20. T10: Diffusion of Neutrons
Derivation
Isotropic scattering z y x
r = 0 is most important
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21. T10: Diffusion of Neutrons
Derivation II
Taylor’s series at the origin:
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22. T10: Diffusion of Neutrons
Derivation III
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23. T10: Diffusion of Neutrons
Fick’s Law
CM-System → Lab-System:
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24. T10: Diffusion of Neutrons
Transport Mean Free Path
Information about the original direction is lost Y
Transport correction =
A number of anisotropic collisions is replaced by one isotropic Y Y ls
lscosY
lscosY2
ltr
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25. T10: Diffusion of Neutrons
Diffusion Equation
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26. T10: Diffusion of Neutrons
Leakage Rate z Jz dz dx
(x,y,z) dy y x
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27. T10: Diffusion of Neutrons
Diffusion Equation
Time-dependent:
Time-independent:
Time-independent from a steady source
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28. T10: Diffusion of Neutrons
Laplace’s Operator
Cartesian geometry
Cylindrical geometry
Spherical geometry
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29. T10: Diffusion of Neutrons
Symmetries
Slab geometry
Spherical geometry
Cylindrical geometry z y x z
n = 0 for slab
n = 1 for cylindrical
n = 2 for spherical
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30. T10: Diffusion of Neutrons
General Properties
Flux is finite and non-negative
Flux preserves the symmetry
No return from a free surface
Flux and current are continues
Diffusion equation describes the balance of neutrons
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31. T10: Diffusion of Neutrons
Interface Conditions A B z
for +z - direction:
for -z - direction:
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32. T10: Diffusion of Neutrons
Boundary Condition
Transport equation
Free surface
Diffusion eq.
Straight line extrapolation from x = 0 towards vacuum:
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33. Plane Infinite Source in Infinite Medium
Transport equation Q0
x = 0
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34. T10: Diffusion of Neutrons
Point Source in Infinite Medium r
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35. T10: Diffusion of Neutrons
Plane Infinite Source in Slab Medium Q0
Infinite:
Slab:
x = -a/2
x = 0
x = a/2
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36. T10: Diffusion of Neutrons
Plane Infinite Source with Reflector 2 1 Q0 1 2
Reflector
Reflector a
Bare slab
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37. T10: Diffusion of Neutrons
Age of Neutrons
Energy
q(E) - number of neutrons, which per cubic-centimeter and second pass energy E.
q(E) = [n×cm-3 s-1]
X-sections depend on E: D(E),Σs(E),... E0 Q E
q(E)
Slowing down medium:
Mean Total Slowing down distance
Can be shown
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38. T10: Diffusion of Neutrons
Fermi’s Age Equation
q(E+dE)
E+dE E
q(E)
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39. Fermi’s Age Equation II
τ ~ time
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40. Solutions to the Age Equation
No absorption
x = 0 r
No absorption
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41. Slowing Down Density for Different Fermi’s Ages
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42. T10: Diffusion of Neutrons
Migration Area (Length)
Thermal neutron absorbed
Fast neutron borne r rs
rth
Fast neutron thermalized
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43. Diffusion and Slowing Down Parameters for Various Moderators
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44. Neutrons in Multiplying Medium
Assumption:
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45. Principles of a Nuclear Reactor
Leakage N2
2 MeV N1
Fast fission
Energy
n n/fission
Slowing down
Resonance abs.
ν ≈ 2.5
Non-fissile abs.
Non-fuel abs.
1 eV
Fission
200 MeV/fission
Leakage
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46. T10: Diffusion of Neutrons
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47. T10: Diffusion of Neutrons
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48. T10: Diffusion of Neutrons
Buckling as Curvature
Large core
Small core
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49. Criticality Condition
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50. T10: Diffusion of Neutrons
Eigenvalues
Transport operator
Differential operator
Matrix
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51. T10: Diffusion of Neutrons
Eigenfunctions
Only one is physically meaningful a
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52. Solution of a Reactor Equation
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53. T10: Diffusion of Neutrons
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54. T10: Diffusion of Neutrons
Rectangular
Cylinder
Sphere
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55. Critical Size of a Reactor
We assume bare homogenous reactor
For thermal neutrons we get:
Slowing down neutrons:
Assumption:
Reactor is sufficiently big to treat neutron spectrum independently of space variables
At the beginning slowing down density is t=0
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56. T10: Diffusion of Neutrons
For t > 0 one has to take into account resonance
capture through p – resonance passage factor.
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57. Non-Leakage Probability
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58. T10: Diffusion of Neutrons
Volume of an cylindrical reactor with buckling derived from a critical equation – the smallest critical size:
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59. T10: Diffusion of Neutrons
Minimum Volume L
V = V(R)
L = L(R) L R
D = 1.08 L
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60. Optimum Core Dimensions
Core shape
Optimum dimensions
Minimal volume
Cube
Cylinder
Sphere
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61. T10: Diffusion of Neutrons
Migration Area
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62. T10: Diffusion of Neutrons
Improved Diffusion
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63. T10: Diffusion of Neutrons
The END
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64. CRITICALITY EQUATION - physical interpretation
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65. Thermal non - leakage factor:
Thermal leakage:
Thermal non - leakage factor:
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66. T10: Diffusion of Neutrons
Derivation
Number of collisions in dV
Neutrons scattered towards dA
Neutrons through dA per 1 second
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67. T10: Diffusion of Neutrons
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68. T10: Diffusion of Neutrons
Delayed Neutrons
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69. T10: Diffusion of Neutrons
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