DIFFUSION EQUATION
Vasily Arzhanov
Reactor Physics, KTH
Overview
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Neutron current
Continuity equation
Fick’s law
Diffusion coefficient
One-speed diffusion equation
Boundary condition
General properties of diffusion equation
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Diffusion equation
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Neutron Flux and Current
v = vΩ,
φ ( r, E ) ≡ ∑ vn ( r, Ωi , E ) =
i
∫π vn ( r, Ω, E ) dΩ
Rx = Σ xφ
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J ( r, E ) ≡ ∑ v i n ( r, Ωi , E ) =
i
mv 2
E=
2
∫π vn ( r, Ω, E ) dΩ
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dn = J ⋅ dA
J
dA
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Diffusion equation
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Balance of Neutrons
⎡Change rate in number ⎤ ⎡ Production ⎤ ⎡ Absorption ⎤ ⎡ Leakage ⎤
⎢ of neutrons in V ⎥ = ⎢ rate in V ⎥ − ⎢ rate in V ⎥ − ⎢ rate from V ⎥
⎣
⎦ ⎣
⎦ ⎣
⎦ ⎣
⎦
⎡Change rate in number ⎤ d
∂n
⎢ of neutrons in V ⎥ = dt ∫ n(r, E, t )dV = ∫ ∂t dV
⎣
⎦
V
V
⎡ Production ⎤
⎢ rate in V ⎥ = ∫ stot dV
⎣
⎦ V
⎡ Absorption ⎤
⎢ rate in V ⎥ = ∫ Σ aφ dV
⎣
⎦ V
⎡ Leakage ⎤
⎢ rate from V ⎥ = ∫ J ⋅ dA = ∫ divJ dV
⎣
⎦ A
V
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Diffusion equation
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Continuity Equation
⎛ ∂n
⎞
s
−
+
Σ
φ
+
div
J
⎟ dV = 0
∫V ⎜⎝ ∂t tot a
⎠
Exact equation,
two unknowns
Diffusion theory:
Diffusion equation:
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1 ∂φ
= s + νΣ f φ − Σ aφ − divJ
v ∂t
J = − Dgradφ = − D∇φ
(Fick’s law)
1 ∂φ
= s + νΣ f φ − Σ aφ + D∇2φ
v ∂t
Diffusion equation
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Example
Se −r L
φ (r ) =
4π Dr
r
S
(a) Find the neutron current at distance r
(b) Net number of neutrons flowing out through a sphere of radius r
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Diffusion equation
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Solution
d
∇ r = er
dr
er
r
S
(1)
(2)
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d ⎛ Se −r L ⎞
S ⎛ 1 1 ⎞ −r L
=
J (r ) = − Der ⎜
e
⎟ r
⎜ 2 + ⎟e
dr ⎝ 4π Dr ⎠
4π ⎝ r
Lr ⎠
⎛ r⎞
N (r ) = 4π r 2 ⋅ J = S ⎜1 + ⎟ e −r L
⎝ L⎠
Diffusion equation
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One-Group Diffusion Model
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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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Diffusion equation
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Plane Angles
ds ≡ nds
er
θ
n
r
ϕ=
C
R
φ
R
dϕ =
ds cos θ er ⋅ ds
=
r
r
Full plane angle φ = 2π
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Diffusion equation
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Solid Angles
dA ≡ ndA
er ≡ Ω
θ n
r
dΩ =
A
Ω= 2
R
dA cos θ Ω ⋅ dA
=
2
r
r2
Full solid angle Ω = 4π
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Diffusion equation
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Spherical Coordinates
z
dr
r
rdθ
dV = r 2 sin θ dθ dψ dr
rsinθdψ
θ
dθ
dψ
x
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y
ψ
rsinθdψ
Diffusion equation
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Net Current
z
Upper semi-space
J z = J+ − J−
0≤r <∞
0 ≤θ ≤π 2
0 ≤ ψ ≤ 2π
r
θ
J−
dA
y
x
J+
ψ
J = J xe x + J ye y + J z e z
Purpose is to relate J and
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Diffusion equation
φ
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Current from dV
(1) Number of collisions in dV:
z
dncoll = Σ sφ dV
Solid angle
dΩ =
dA cos θ
r2
r
(2) Fraction of neutrons
scattered towards dA:
θ
(3) Fraction of neutrons
survived while traveling r:
dA
y
ψ
e−Σs r
(4) Number of neutrons
crossing dA from above
x
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dΩ
4π
d Ω −Σs r
⋅e
dn = Σ sφ dV ⋅
4π
Diffusion equation
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Current from Upper Semi-Space
dJ − =
J− =
∫
upper
dn
cos θ −Σs r
= Σ sφ
e dV
2
dA
4π r
∞ π 2 2π
dJ − = ∫
∫
0 0
cos θ −Σs r 2
∫0 Σsφ (r) 4π r 2 e r sin θ dθ dψ dr
r = 0 is most important
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Diffusion equation
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Slowly Varying Flux
⎛ ∂φ ⎞
⎛ ∂φ ⎞
⎛ ∂φ ⎞
+
y
+
z
⎜ ⎟
⎟
⎜ ⎟ +...
⎝ ∂x ⎠0 ⎝ ∂y ⎠0 ⎝ ∂z ⎠0
φ = φ0 + x⎜
Taylor’s series at the origin:
x = r sin θ cosψ ; y = r sin θ sinψ ; z = r cos θ
⎛ ∂φ ⎞
⎛ ∂φ ⎞
⎛ ∂φ ⎞
φ (r ) = φ0 + r sin θ cosψ ⎜ ⎟ + r sin θ sinψ ⎜ ⎟ + r cos θ ⎜ ⎟
⎝ ∂x ⎠0
⎝ ∂z ⎠0
⎝ ∂y ⎠0
Σ
J− = s
4π
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2π
π
∞
⎡
⎛ ∂φ ⎞ ⎤ −Σsr
+
φ
r
cos
θ
⎜ ⎟ ⎥ e sin θ dθ dr dψ
⎢ 0
∫
∫
∫
⎝ ∂z ⎠0 ⎦
Ψ =0 θ =0 r =0 ⎣
2
Diffusion equation
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Net Current
φ0
1 ⎛ ∂φ ⎞
J − (0) = +
⎜ ⎟
4 6Σ s ⎝ ∂z ⎠0
J + (0) =
φ0
4
J z (0) = −
−
1 ⎛ ∂φ ⎞
⎜ ⎟
6Σ s ⎝ ∂z ⎠0
1 ⎛ ∂φ ⎞
⎜ ⎟
3Σ s ⎝ ∂z ⎠0
Jz = −
1 ⎛ ∂φ ⎞
1 ⎛ ∂φ ⎞
1 ⎛ ∂φ ⎞
J
J
=
−
=
−
;
;
⎜ ⎟
x
y
⎜ ⎟
⎜ ⎟
3Σ s ⎝ ∂z ⎠0
3Σ s ⎝ ∂x ⎠0
3Σ s ⎝ ∂y ⎠0
1 ⎛ ∂φ
∂φ
∂φ ⎞
+ ey
+ ez
J ≡ ex Jx + ey Jy + ez Jz = −
⎜ ex
⎟
3Σ s ⎝ ∂x
∂y
∂z ⎠
∇φ
J (r , t ) = −
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1
∇φ ( r , t )
3Σ s (r )
Diffusion equation
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Fick’s Law
1
∂φ
∂φ
∂φ
J (r ) = −
∇φ (r ); ∇φ (r ) = e x
+ ey
+ ez
3Σ s
∂x
∂y
∂z
Fick’s law:
D≡
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J (r ) = − D∇φ (r )
1
λ
= s
3Σ s 3
The essence of diffusion theory
It is essentially based on the hypothesis
of isotropic scattering in the Lab system
Diffusion equation
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Fick’s Law Limitations
Fick’s law is not accurate
• When scattering is strongly anisotropic
• In medium with strong absorption
• Within about 3 mfp of neutron sources,
sinks, or boundaries
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Diffusion equation
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Transport Approximation
Φ ( r, Ω, t ) =
1
3
φ ( r, t ) +
Ω ⋅ J ( r, t ) + …
4π
4π
Zero order approximation:
Φ ( r, Ω, t ) =
1
φ ( r, t )
4π
1
λs
D=
=
3Σ s 3
NTE
Elementary
diffusion
First order approximation:
Φ ( r, Ω, t ) =
NTE
1
3
φ ( r, t ) +
Ω ⋅ J ( r, t )
4π
4π
Σtr ≡ Σt − μ ⋅ Σ s ;
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D=
λtr ≡ 1 Σtr
Diffusion equation
λ
1
= tr
3Σtr
3
Transport
approximation
2
μ=
3A
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Improved Diffusion
Coefficient
1
1
λtr ≡
=
Σtr Σt − μ ⋅ Σ s
Weak
absorption:
1
1
1
1
D=
=
≈
=
3Σtr 3 ( Σt − μ ⋅ Σ s ) 3 ( Σ s − μ ⋅ Σ s ) 3Σ s (1 − μ )
Scattering from
heavy nuclides:
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1
λtr
D=
=
3Σtr
3
D=
1
1
1
=
≈
3Σ s (1 − μ ) 3Σ s (1 − 2 3A ) 3Σ s
Diffusion equation
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Making Scattering Isotropic
D=
λs
3
Neutron remembers
its original direction
ψ
λs = 1 Σ s
One combined collision
ψ
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Diffusion equation
Neutron does not remember
its original direction
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Transport Mean Free Path
Information about the
original direction is lost
ψ
Transport correction =
A number of anisotropic
collisions is replaced by
one isotropic
ψ
ψ
λs
λs ⋅ μ
λs ⋅ μ
Initial direction
2
λtr
No absorption:
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λtr = λs + λs μ + λs μ + λs μ + … =
2
Diffusion equation
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λs
1− μ
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Transport mfp with Absorption
λs
1− μ
No absorption:
λtr = λs + λs μ + λs μ 2 + λs μ 3 + … =
With absorption:
1
λt
λtr ≡
=
=
Σt − μ ⋅ Σ s Σt (1 − μ ⋅ Σ s Σt ) 1 − μ ⋅ Σ s Σt
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1
Diffusion equation
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Example
The scattering X-section of carbon at 1 ev is 4.8 b.
Estimate the diffusion coefficient.
The atom density of graphite is 0.08023x1024
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Diffusion equation
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Solution
The scattering X-section of carbon at 1 ev is 4.8 b.
Estimate the diffusion coefficient.
The atom density of graphite is 0.08023x1024
D=
1
3Σ s (1 − μ )
2
2
μ=
=
= 0.555
3 A 36
Σ s = σ s × NC = 4.8 ×10−24 × 0.08023 ×1024 = 0.385
D = 0.916
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Diffusion equation
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Interpretation of Fick’s Law
Neutron density increases
J (r ) = − D∇φ (r )
Current
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J (r )
Diffusion equation
∇φ (r )
Gradient
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Two Kinds of Transport
Random walk
Collective
(self-diffusion)
transport
(a)
(b)
Molecules do
collide with
each other
Neutrons do
not collide with
each other
Examples:
Neutrons in reactor
Gas molecules
Equation:
NTE
Boltzmann
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Diffusion equation
n ~ 108
NB ~ 1022
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Warning
Fick’s law:
J (r ) = − D∇φ (r )
is valid only for completely chaotic movement
(random walk) with weak absorption
φ (r ) = vn = const
J (r ) = vn = Ωφ
Collimated beam of neutrons:
φ ( x) = φ (0)e −Σ x
J (r ) = Ωφ
No collisions
only absorption
a
x
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Diffusion equation
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One-Speed Diffusion Equation
1 ∂φ ( r, t )
= s ( r, t ) +νΣ f φ ( r, t ) − Σ aφ ( r, t ) − divJ ( r, t )
Balance of neutrons:
v ∂t
Diffusion theory: J ( r, t ) = − Dgradφ = − D ( r ) ∇φ ( r, t ) (Fick’s law)
Diffusion coefficient: D ( r ) =
Diffusion equation:
Leakage from
a unit volume:
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1
3Σtr ( r )
,
Σtr ( r ) = Σt ( r ) − μ ⋅ Σ s ( r )
1 ∂φ
= s +νΣ f φ − Σ aφ + ∇ ( D∇φ )
v ∂t
L = −∇ ( D∇φ ) = − D∇ 2φ
Diffusion equation
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Net and Partial Currents
Arbitrary direction n:
+n
–n
⎛ φ D ∂φ ⎞
J − n (r ) = − n ⋅ ⎜ +
⎟
⎝ 4 2 ∂n ⎠
r
⎛ φ D ∂φ ⎞
J +n (r ) = n ⋅ ⎜ −
⎟
⎝ 4 2 ∂n ⎠
J ( r ) = − D∇φ ( r )
z
x
y
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Diffusion equation
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Interface Conditions
x
φ
D ∂φ
J− = +
4 2 ∂x
φ
A
φ
D ∂φ
J+ = −
4 2 ∂x
B
φB
φA
x
φ A ≡ φ ( x − 0)
φB ≡ φ ( x + 0)
J + ( A) = J + ( B)
J − ( A) = J − ( B)
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φA
D A ∂φA φB DB ∂φB
=
−
4
2 ∂x
4
2 ∂x
φA D A ∂φA φB DB ∂φB
+
=
+
4
2 ∂x
4
2 ∂x
−
Diffusion equation
φ ( x − 0) = φ ( x + 0)
∂φ
∂φ
DA
∂x
= DB
x −0
∂x
x+0
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Boundary Condition
J− =
φ
4
+
D ∂φ
2 ∂x
J+ =
φ
4
−
D ∂φ
2 ∂x
Vacuum
Diffusion theory predicts:
Diffusion theory is
not accurate near:
• Sources
• Sinks
B
x
• Interfaces
• Boundaries
First step:
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J− =
φ
4
+
λtr ∂φ
6 ∂x
=0
Diffusion equation
(φ ) B
⎛ ∂φ ⎞
⎜ ⎟ =−
2 3 λtr
⎝ ∂x ⎠ B
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Extrapolated Length
BC:
Vacuum
φ
(φ ) B
⎛ ∂φ ⎞
⎜ ⎟ =−
2 3 λtr
⎝ ∂x ⎠ B
Neutron flux in reality
(φ ) B
Linearly extrapolated flux:
φext ( x) = (φ ) B −
0
B
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2 3 λtr
⋅x
x
dext
Extrapolated length:
(φ ) B
dext
2
= λtr = 0.66λtr
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Diffusion equation
(φ ) B
⎛ ∂φ ⎞
⎜ ⎟ =−
dext
⎝ ∂x ⎠ B
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Improved Boundary Condition
Transport solution
Improved diffusion
Elementary diffusion
x
0.66λtr
0.71λtr
(φ ) B
⎛ ∂φ ⎞
⎜ ⎟ =−
dext
⎝ ∂x ⎠ B
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Diffusion equation
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Practical Boundary Condition
Transport solution
Natural curvature
Improved diffusion
x=0
dext = 0.71λtr
xB
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Diffusion equation
x
φ ( xB + dext ) = 0
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General Properties
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Flux is finite and non-negative
Flux preserves the symmetry
No return from the free surface
Flux and current are continues
Diffusion equation describes the balance
of neutrons
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Diffusion equation
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The END
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Diffusion equation
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