102_diffusionequation

DIFFUSION EQUATION

Vasily ArzhanovReactor Physics, KTH

HT2008 Diffusion equation 2

Overview

• Neutron current• Continuity equation• Fick’s law• Diffusion coefficient• One-speed diffusion equation• Boundary condition• General properties of diffusion equation


Neutron Flux and Current2

,2mvv E= =v Ω

( ) ( ) ( )

( ) ( ) ( )4

4

, , , , ,

, , , , ,

ii

i ii

E vn E vn E d

E n E n E dπ

π

φ ≡ =

≡ =

∑ ∫

∑ ∫

r r Ω r Ω Ω

J r v r Ω v r Ω Ω

x xR φ= Σ

dn d= ⋅J A

J

dA


Balance of NeutronsChange rate in number Production Absorption Leakage of neutrons in V rate in V rate in V rate from V⎡ ⎤ ⎡ ⎤ ⎡ ⎤ ⎡ ⎤

= − −⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥⎣ ⎦ ⎣ ⎦ ⎣ ⎦ ⎣ ⎦

Change rate in number( , , )

of neutrons in V V V

d nn E t dV dVdt t

⎡ ⎤ ∂= =⎢ ⎥ ∂⎣ ⎦

∫ ∫r

Production rate in V tot

V

s dV⎡ ⎤=⎢ ⎥

⎣ ⎦∫

Absorption rate in V a

V

dVφ⎡ ⎤

= Σ⎢ ⎥⎣ ⎦

∫

Leakagediv

rate from V A V

d dV⎡ ⎤= ⋅ =⎢ ⎥

⎣ ⎦∫ ∫J A J


Continuity Equation

div 0tot aV

n s dVt

φ∂⎛ ⎞− + Σ + =⎜ ⎟∂⎝ ⎠∫ J

1 divφ ν φ φ∂= + Σ − Σ −

∂Jf as

v t

gradD Dφ φ= − = − ∇JDiffusion theory: (Fick’s law)

21 φ ν φ φ φ∂= + Σ − Σ + ∇

∂ f as Dv t

Diffusion equation:

Exact equation, two unknowns


Example

Sr

( )4

r LSerDr

φπ

−

=

(a) Find the neutron current at distance r(b) Net number of neutrons flowing out through a sphere of radius r


Solution

Sr

2

1 1( )4 4

r Lr L

r rd Se Sr D edr Dr r Lrπ π

−−⎛ ⎞ ⎛ ⎞= − = +⎜ ⎟ ⎜ ⎟

⎝ ⎠⎝ ⎠J e e

r rddr

∇ = e

(1)

2( ) 4 1 r LrN r r J S eL

π −⎛ ⎞= ⋅ = +⎜ ⎟⎝ ⎠

re

(2)


• 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

One-Group Diffusion Model


φR

CR

ϕ =

Plane Angles

d ds≡s n

nθr

cos r ddsdr rθϕ ⋅

= =e s

re

Full plane angle φ = 2π


2

AR

Ω =

Solid Anglesd dA≡A n

nr ≡e Ω

θ

r

2 2

cosdA ddr r

θ ⋅= =

Ω AΩ

Full solid angle Ω = 4π


Spherical Coordinates

x

y

z

θ

dθ

ψ

dψ

r

dr

rsinθdψ

rdθ

rsinθdψ

2 sinθ θ ψ=dV r d d dr


Net Current

x

y

z

+ −= −zJ J J

+J

−JdA

ψ

θ

00 20 2

θ πψ π

≤ < ∞≤ ≤≤ ≤

rr

Upper semi-space

= + +J e e ex x y y z zJ J J

Purpose is to relate J and φ


Current from dV

x

y

z

dA

θ

r

ψ

2

cosθΩ =

dAdr

Solid angle φ= Σcoll sdn dV

(1) Number of collisions in dV:

(2) Fraction of neutrons scattered towards dA: 4π

Ωd

(3) Fraction of neutrons survived while traveling r: −Σsre

(4) Number of neutrons

crossing dA from above

4φ

π−ΣΩ

= Σ ⋅ ⋅ srs

ddn dV e


Current from Upper Semi-Space

2

cos4

θφπ

−Σ− = = Σ sr

sdndJ e dVdA r

2 22

20 0 0

cos( ) sin4

π π θφ θ θ ψπ

∞−Σ

− −= = Σ∫ ∫ ∫ ∫ r srs

upper

J dJ e r d d drr

r = 0 is most important


Taylor’s series at the origin: 00 00

...x y zx y zφ φ φφ φ

⎛ ⎞∂ ∂ ∂⎛ ⎞ ⎛ ⎞= + + + +⎜ ⎟⎜ ⎟ ⎜ ⎟∂ ∂ ∂⎝ ⎠ ⎝ ⎠⎝ ⎠

sin cos ; sin sin ; cosx r y r z rθ ψ θ ψ θ= = =

00 00

( ) sin cos sin sin cosr r rx y zφ φ φφ φ θ ψ θ ψ θ

⎛ ⎞∂ ∂ ∂⎛ ⎞ ⎛ ⎞= + + +⎜ ⎟⎜ ⎟ ⎜ ⎟∂ ∂ ∂⎝ ⎠ ⎝ ⎠⎝ ⎠r

2 2

000 0 0

cos sin4

srs

r

J r e d dr dz

ππ

θ

φφ θ θ θ ψπ

∞−Σ

−Ψ= = =

⎡ ⎤Σ ∂⎛ ⎞= + ⎜ ⎟⎢ ⎥∂⎝ ⎠⎣ ⎦∫ ∫ ∫

Slowly Varying Flux


0

0

0

0

0

1(0)4 6

1(0)4 6

1(0)3

s

s

zs

Jz

Jz

Jz

φ φ

φ φ

φ

−

+

∂⎛ ⎞= + ⎜ ⎟Σ ∂⎝ ⎠

∂⎛ ⎞= − ⎜ ⎟Σ ∂⎝ ⎠

∂⎛ ⎞= − ⎜ ⎟Σ ∂⎝ ⎠

Net Current

0 0 0

1 1 1; ; 3 3 3z x y

s s s

J J Jz x yφ φ φ⎛ ⎞∂ ∂ ∂⎛ ⎞ ⎛ ⎞= − = − = − ⎜ ⎟⎜ ⎟ ⎜ ⎟Σ ∂ Σ ∂ Σ ∂⎝ ⎠ ⎝ ⎠ ⎝ ⎠

13x x y y z z x y z

s

J J Jx y z

φ

φ φ φ

∇

⎛ ⎞∂ ∂ ∂≡ + + = − + +⎜ ⎟Σ ∂ ∂ ∂⎝ ⎠

J e e e e e e

1( , ) ( , )3 ( )

φ= − ∇Σ

J r rrs

t t


Fick’s Law

1( ) ( ); ( )3 x y z

s x y zφ φ φφ φ ∂ ∂ ∂

= − ∇ ∇ = + +Σ ∂ ∂ ∂

J r r r e e e

( ) ( )φ= − ∇J r rD

It is essentially based on the hypothesis of isotropic scattering in the Lab system

The essence of diffusion theory

13 3

λ≡ =

Σs

s

D

Fick’s law:

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



Transport Approximation

13 3

λ= =

Σs

s

D

Zero order approximation:

( ) ( )1, , ,4

φπ

Φ =r Ω rt t Elementary diffusion

First order approximation:

( ) ( ) ( )1 3, , , ,4 4

t t tφπ π

Φ = + ⋅r Ω r Ω J r 13 3

tr

tr

D λ= =

ΣTransport approximation

; 1tr t s tr trμ λΣ ≡ Σ − ⋅Σ ≡ Σ

( ) ( ) ( )1 3, , , ,4 4

t t tφπ π

Φ = + ⋅ +r Ω r Ω J r …

23A

μ =

NTE

NTE


Improved Diffusion Coefficient

1 1λμ

≡ =Σ Σ − ⋅Σtrtr t s

13 3

λ= =

Σtr

tr

D

( ) ( ) ( )1 1 1 1

3 3 3 3 1μ μ μ= = ≈ =

Σ Σ − ⋅Σ Σ − ⋅Σ Σ −tr t s s s s

DWeak absorption:

( ) ( )1 1 1

3 1 3 1 2 3 3μ= = ≈

Σ − Σ − Σs s s

DA

Scattering from heavy nuclides:


Making Scattering Isotropic

3λ

= sD

1λ = Σs s

ψNeutron remembers its original direction

ψ Neutron does not remember its original direction

One combined collision


λλ λ λ μ λ μ λ μμ

= + + + + =−

… str s s s s

2 3

1

Transport Mean Free Path

λtr

Transport correction =

A number of anisotropic collisions is replaced by one isotropic

Information about the original direction is lost

ψ

ψ

ψ

λs λ μ⋅s2λ μ⋅s

Initial direction

No absorption:


2 3

1s

tr s s s sλλ λ λ μ λ μ λ μμ

= + + + + =−

…

Transport mfp with Absorption

No absorption:

( )1 1

1 1λλ

μ μ μ≡ = =Σ − ⋅Σ Σ − ⋅Σ Σ − ⋅Σ Σ

ttr

t s t s t s t

With absorption:


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


SolutionThe scattering X-section of carbon at 1 ev is 4.8 b. Estimate the diffusion coefficient.The atom density of graphite is 0.08023x1024

( )1

3 1s

Dμ

=Σ −

2 2 0.5553 36A

μ = = =

24 244.8 10 0.08023 10 0.385s s CNσ −Σ = × = × × × =

0.916D =


Interpretation of Fick’s Law

( ) ( )φ= − ∇J r rD

Neutron density increases

Gradient( )φ∇ r( )J r

Current


Two Kinds of TransportRandom walk

(self-diffusion)

Collective

transport

Examples: Neutrons in reactor Gas molecules

Neutrons do not collide with each other

Molecules do collide with each other

Equation: NTE Boltzmannn ~ 108

NB ~ 1022

(a) (b)


Warning

( ) ( )φ= − ∇J r rDFick’s law: is valid only for completely chaotic movement (random walk) with weak absorption

Collimated beam of neutrons:( )( )

φφ

= == =

rJ r v Ω

vn constn

No collisions only absorption

x

( ) (0)( )

axx eφ φφ

−Σ==J r Ω


One-Speed Diffusion Equation( ) ( ) ( ) ( ) ( ),1 , , , div ,f a

ts t t t t

v tφ

ν φ φ∂

= + Σ −Σ −∂r

r r r J r

( ) ( ) ( ), grad ,t D D tφ φ= − = − ∇J r r rDiffusion theory: (Fick’s law)

( )1f as D

v tφ ν φ φ φ∂= + Σ −Σ +∇ ∇

∂

( ) ( ) ( ) ( ) ( )1 ,3 tr t s

tr

D μ= Σ = Σ − ⋅ΣΣ

r r r rr

Balance of neutrons:

( ) 2L D Dφ φ= −∇ ∇ = − ∇Leakage from a unit volume:

Diffusion coefficient:

Diffusion equation:


Net and Partial Currents

( )4 2φ φ∂⎛ ⎞= ⋅ −⎜ ⎟∂⎝ ⎠

+nJ r n Dn

Arbitrary direction n:

–n+n

x

y

z

r( )

4 2φ φ

−∂⎛ ⎞= − ⋅ +⎜ ⎟∂⎝ ⎠

nJ r n Dn ( ) ( )φ= − ∇J r rD


4 2 4 2A A A B B BD D

x xφ φ φ φ∂ ∂

− = −∂ ∂

4 2 4 2A A A B B BD D

x xφ φ φ φ∂ ∂

+ = +∂ ∂ 0 0

( 0) ( 0)φ φφ φ

− +

− = +

∂ ∂=

∂ ∂A Bx x

x x

D Dx x

A Bφ

xAφ

Bφ

Interface Conditions

( ) ( )+ +=J A J B

4 2φ φ

+∂

= −∂

DJx4 2

φ φ−

∂= +

∂DJx

( ) ( )− −=J A J B

( 0)( 0)

φ φφ φ

≡ −≡ +

A

B

xx

x


Boundary Condition

Vacu

um

x

4 2φ φ

+∂

= −∂

DJx4 2

φ φ−

∂= +

∂DJx

Diffusion theory predicts:

Diffusion theory is not accurate near:

• Sources

• Sinks

• Interfaces

• Boundaries

04 6φ λ φ

−∂

= + =∂

trJx

First step:( )

2 3φφλ

∂⎛ ⎞ = −⎜ ⎟∂⎝ ⎠B

B trx

B


Extrapolated Length

Vacu

um

x

φ

( ) ( )( )

2 3φ

φ φλ

= − ⋅Bext B

tr

x x

B

( )φ BLinearly extrapolated flux:

Neutron flux in reality

extd

2 0.663λ λ= =ext tr trdExtrapolated length:

0

( )BB extx d

φφ∂⎛ ⎞ = −⎜ ⎟∂⎝ ⎠

( )2 3φφλ

∂⎛ ⎞ = −⎜ ⎟∂⎝ ⎠B

B trxBC:


Improved Boundary Condition

x

Transport solution

Elementary diffusion

0.66λtr 0.71λtr

Improved diffusion

( )BB extx d

φφ∂⎛ ⎞ = −⎜ ⎟∂⎝ ⎠


Practical Boundary Condition

x

Transport solution

0.71ext trd λ=

Improved diffusion

( ) 0B extx dφ + =

Natural curvature

Bx

x = 0


General Properties

• 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


The END

102_diffusionequation

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