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Neutron Transport Theory: The diffusion approximation · Neutron Transport - Introduction •...
Transcript of Neutron Transport Theory: The diffusion approximation · Neutron Transport - Introduction •...
Neutron Transport Theory: The diffusion approximation
Jonathan Evans
Department of Mathematical SciencesUniversity of Bath, UK
March 26 2015
Web pages: people.bath.ac.uk/masjde/teaching people.bath.ac.uk/masigg/ITT2
Outline
1. Modelling Process
2. Neutron Transport
1. Problem Formulation
2. Non-dimensionalisation
3. Asymptotic Analysis - (limit of small mean free path)
Modelling Process: Overview
Physical Description Mathematical Description
Mathematical ModelMathematical Problem
Statement
Physical ModelIdentification & Statement
of physical problem/process
Answer Original Questions of Physical Problem
Mathematical Results
Interpretation
Interpretation
AnalysisRefinement
Modelling Process: Analysis
Mathematical Statementof Problem
Dimensionless Model +
Dimensionless Parameters
Perturbation Analysis
Numerical Analysis on Full Problem
ModelRefinement
Numerical Analysison Reduced Problem
ProblemSolution
Modelling Process: Research Problems
Research Problems
Type 1 Novel Mathematics
New Techniques
TechnologyTransfer
Type 2Known Mathematics
Novel Aspects
TechnologyTransfer
MathematicalModel
Application
Area 1
Application
Area 2Application
Area N
Neutron Transport - Introduction
• References:
G.J. Habetler & B.J. Matkowsky, Uniform asymptotic expansions in tranport theory with small mean free paths and the diffustion approximation, J. Math. Phys. 16 (1975) 846-854.
K.M. Case and P.F. Zweifel, Linear Transport Theory, Addison-Wesley, (1967)
• Neutron Motion - Boltzmann Transport Equation - Linear Integro-differential Equation
• Diffusion Theory - Common Approximation - Neutron Flux = Fick’s law - Good working results
• Objective: Derive Diffusion Equation + B.C.s
Neutron Transport - Problem Formulation
• Neutron conservation statement: @N
@t+r ·H = q
1
4⇡
Z
S2v�s(r,⌦ ·⌦0)N(r,⌦0, t) dS(⌦0)+
⌫(r)
4⇡v�f (r)
Z
S2N(r,⌦0, t) dS(⌦0) Q(r, t)++
N(r,⌦, t) (angular) neutron density
(number per unit volume)
•
H = Nv v = v⌦ v speed (constant)
⌦ direction
Flux
r ·H = v⌦ ·rN
•
�(r) = �f (r) + �c(r) +1
4⇡
Z
S2�s(r,⌦ ·⌦0) dS(⌦0)
q = �v�(r)N(r,⌦, t)Source•
Total macroscopic cross-section
Neutron Transport - Problem Formulation
c(r) =�s(r) + ⌫�f (r)
�(r)
Isotropic scattering �s(r,⌦.⌦0) = �s(r)
mean number secondary neutrons
1
v
@ (r,⌦, t)
@t+⌦ ·r (r,⌦, t) + �(r) (r,⌦, t)
=�(r)c(r)
4⇡
Z
S2 (r,⌦0, t) dS(⌦0) Q(r, t)+
•
•
•
Neutron Transport - Problem Formulation
•
•
• 1-D problem
1 space coordinate x
1 direction coordinate
µ = ⌦ · i = cos ✓
= (x, µ, t)✓ 2 [0,⇡]
1
v
@ (x, µ, t)
@t
+ µ
@ (x, µ, t)
@x
+ �(x) (x, µ, t)
�(x)c(x)
2
Z 1
�1 (x, µ0
, t) dµ0 +Q(x, t)=
0 < x < d
�1 µ 1
slab geometry
x = 0
x = d
= f1(µ, t)
= f2(µ, t)
µ > 0
µ < 0
at t = 0 = g(x, µ) for 0 x d,�1 µ 1
Neutron Transport - Problem Formulation
x
µµ = 1
µ = �1x = d
= f1(µ, t)
= f2(µ, t)
Neutron Transport - Non-dimensionalisation
•
•
•
x = dx
t =d
vt = 0 Q = 0�(x)Q
f1 = 0f1 f2 = 0f2 g = 0g
� = �0�(x) c = c(x)
Non-dimensionalise:
Neutron Transport - Non-dimensionalisation
•
•
•
•
✏
�(x)
@
@ t
+✏µ
�(x)
@ (x, µ, t)
@x
+ (x, µ, t)
c(x)
2
Z 1
�1 (x, µ0
, t) dµ0 + Q(x, t)=
0 < x < 1,�1 µ 1,
on x = 0
= f1(µ, t)
= f2(µ, t)
= g(x, µ)
on x = 1
at t = 0
µ > 0
µ < 0
for 0 x 1,�1 µ 1
(d, v, 0,�0)
4 dimensional
parameters✏
1 dimensionless
parameter
Dimensionless Problem:
Dimensionless Parameter
✏ =1
�0d⇠ 10�9 � 10�3
Mean free path
Neutron Transport - Matched Asymptotics
•
• Matched Asymptotic Expansions ✏ ! 0
=
x
µ µ = 1
µ = �1
= f1(µ, t)
= f2(µ, t)
x = 1
Outer0 < x < 1
Inner InnerO(✏) O(✏)
c(x)
2
Z 1
�1 (x, µ0
, t) dµ0 +Q(x, t)
✏
�(x)
@
@t
+✏µ
�(x)
@ (x, µ, t)
@x
+ (x, µ, t)
Neutron Transport - Outer Expansion
•
•
• Outer region 0 < x < 1
= 0(x, µ, t) + ✏ 1(x, µ, t) + ✏
2 2(x, µ, t) + . . .
as ✏ ! 0
Pose
c(x) = c0(x) + ✏c1(x) + ✏
2c2(x) + . . .
Q(x, t) = Q0(x, t) + ✏Q1(x, t) + ✏
2Q2(x, t) + . . .
t =✏
�⌧ � = K0 +K1✏+K2✏
2 + . . .
•
Neutron Transport - Outer Expansion
•
•
K0
�(x)
@ 0
@⌧
+ 0 =c0(x)
2
Z 1
�1 0(x, µ
0, t)dµ0 +Q0(x, t)
• 0 = 0(x, t)K0
�(x)
@ 0
@⌧
+ (1� c0(x)) 0 = Q0(x, ⌧)
K0 = 0 c0 = 1 Q0 = 0
At O(✏0):
Neutron Transport - Outer Expansion
•
•
•
K1
�(x)
@ 0
@⌧
+µ
�(x)
@ 0
@x
+ 1 = c0(x)
2
Z 1
�1 1(x, µ
0, ⌧)dµ0
+c1(x)
2
Z 1
�1 0(x, ⌧)dµ
0 +Q1(x, ⌧)
1 = 10(x, ⌧) + µ 11(x, ⌧)
• K1 = 0 c1 = 0 Q1 = 0
1
�(x)
@ 0
@x
+ 11 = 0
K1
�(x)
@ 0
@⌧
= c1(x) 0 +Q1(x, ⌧)
At O(✏):
Neutron Transport - Outer Expansion
•
•
•
At O(✏2): K2
�(x)
@ 0
@⌧
+µ
�(x)
@ 1
@x
+ 2 = 1
2
Z 1
�1 2(x, µ
0, ⌧)dµ0
+c2(x)
2
Z 1
�1 0(x, ⌧)dµ
0 +Q2(x, ⌧)
2 = 20(x, ⌧) + µ 21(x, ⌧) + µ
2 22(x, ⌧)
K2
�(x)
@ 0
@⌧
=1
3 22 + c2(x) 0 +Q2(x, ⌧)
1
�(x)
@ 10
@x
+ 21 = 0
1
�(x)
@ 11
@x
+ 22 = 0
K2
�(x)
@ 0
@⌧
=1
3�(x)
@
@x
✓1
�(x)
@ 0
@x
◆+ c2(x) 0 +Q2
• Q(x, t) = ✏
2Q2(x, ⌧)t =
⌧
✏c(x) = 1 + ✏
2c2(x)
Neutron Transport - Inner Expansion
•
•
•
•
•
•
x = ✏y =
c(✏y) = 1 + ✏2c2(0) +O(✏3)
�(✏y) = �(0) Q(✏y, t) = ✏2Q2(0, ⌧)
=
✏2
�(0)
@
@⌧+
µ
�(0)
@ (y, µ, ⌧)
@y+ (y, µ, ⌧)
c(✏y)
2
Z 1
�1 (y, µ0, ⌧) dµ0 +✏2Q2(0, ⌧)
0 < y < 1,�1 µ 1Domain:
Inner region at x = 0
Neutron Transport - Inner Expansion
•
•
•
•
Pose:
= 0(y, µ, ⌧) + ✏ 1(y, µ, ⌧) + ✏2 2(y, µ, ⌧) + . . .
as ✏ ! 0
At O(✏0): µ
�(0)
@ 0
@y+ 0 =
1
2
Z 1
�1 0(y, µ
0, ⌧)dµ0
0(0, µ, ⌧) = f1(µ, ⌧)
�⌫(µ) =⌫
2P
1
⌫ � µ+ �(⌫)�(⌫ � µ) �(⌫) = 1� ⌫ tanh�1 ⌫
A0(⌫, ⌧) = 0 ⌫ < 0
General solution:
0(y, µ, ⌧) = a0(⌧) + b0(⌧)(�(0)y � µ)
+
Z 1
�1A0(⌫, ⌧)�⌫(µ)e
�y�(0)/⌫ d⌫
•
Neutron Transport - Inner Expansion
•
•
•
•
•
Z 1
0�⌫(µ) �(µ) dµ = 0
Z 1
0�⌫(µ)�⌫0(µ) �(µ) dµ =
�(⌫)
⌫N(⌫)�(⌫ � ⌫0)
�(µ) =3µ
2X(�µ)N(⌫) = ⌫
✓�(⌫)2 +
⇡2⌫2
4
◆
X(z) =1
1� zexp
✓1
⇡
Z 1
0
1
(µ0 � z)tan
�1
⇡µ0
2�(µ0)
�◆dµ0
Orthogonality Conditions:
Neutron Transport - Inner Expansion
•
•
• a0(⌧) =�1�0
b0(⌧) +1
�0
Z 1
0f1(µ, ⌧)�(µ)dµ
A0(⌫, ⌧) = � ⌫2�0b0(⌧)
2�(⌫)N(⌫)+
⌫
�(⌫)N(⌫)
Z 1
0f1(µ, ⌧)�⌫(µ) �(µ) dµ
b0(⌧) determined by matching to outer
�i =
Z 1
0µi�(µ)dµ for i = 0, 1
Neutron Transport - Matching
• •
•
•
✏ ⌧ x = ✏y ⌧ 1Overlap domain:
Outer in Inner variables:
= 0(✏y, µ, ⌧) + ✏ 1(✏y, µ, ⌧) + ✏2 2(✏y, µ, ⌧) + . . .
= 0(0, ⌧) + ✏
y
@ 0(0, µ, ⌧)
@x
+ 1(0, µ, ⌧)
�
+✏2y
2
2
@
2 0(0, µ, ⌧)
@x
2+ y
@ 1(0, µ, ⌧)
@x
+ 2(0, µ, ⌧)
�+ . . .
Outer limit of Inner:
= 0(y, µ, ⌧) + ✏ 1(y, µ, ⌧) + ✏2 2(y, µ, ⌧) + . . .
as y ! 1• At O(✏0): lim
y!1 0(y, µ, ⌧) = 0(0, ⌧)
Neutron Transport - Matching
•
• 0(y, µ, ⌧) = b0(⌧)�(0)y + (a0(⌧)� b0(⌧)µ) + o(1)
as y ! 1
b0(⌧) = 0 a0(⌧) = 0(0, ⌧)
Neutron Transport - Summary
•
•
•
K2
�(x)
@ 0
@⌧
=1
3�(x)
@
@x
✓1
�(x)
@ 0
@x
◆+ c2(x) 0 +Q2
in 0 < x < 1, ⌧ > 0
Boundary Conditions:
0(0, ⌧) =
Z 1
0f1(µ, ⌧)�(µ)dµ
0(1, ⌧) =
Z 1
0f2(µ, ⌧)�(µ)dµ
Initial Condition:
0(x, 0) = G(x) ??
Governing Equation:
Thank You !
The End