4.1.DiffusionEq

7/17/2019 4.1.DiffusionEq

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Diffusion Equation

f S i iept. of Energy System Engineering

Shim Hyung Jin



Reactor Theory

Neutron Current

Net flow

- -

- in 3-D

ˆ

j

j

normal componentto - plane

x

y z

ˆ ˆ ˆ ˆ ˆ

net as rec on

- in 2-D

( , s n cos s n s n cos x y z

ˆ ˆ( , , ) ( , , ) ( )r E d n r E d v E

Need for net inflowor outflowto exmine balance

4

ˆ( , ) ( , , )ˆ J r E r E d

- vec or sum o angu ar ux

net flow formed toward a direction

ˆ ˆ ˆ ˆ yˆ z

in a volume

- scalar sum

x y z

ˆ x( , )

( , ) J r E

n r E v E

ˆ( , ) ( ) ( , ) ( , ) ( , )n r E v E J r E n r E r E

r E

v

v

2 SNU Monte Carlo Lab.

4 ˆ( , ) ( , , )r E r E d

scalar flux important for reaction rate



Reactor Theory

Normal Component of Current

- Number of neutrons passing through unit area of a surface per unit time in a net flow field

plane A

after t v

ˆunit normal vector ( )n

: travel distance forv t t

n

moving with toward directionv

neutrons per unit volumen height of parallel pipe = cosv t

total volume = cosv t A

total number ofneutrons passed

= cosnv t A

ˆˆ ˆ ˆ J J x J y J z n v

- normal component to a surface to parallel to y-z plane

: number of neutrons assin throu h unit area on -z lane er unit time J nv x const




Reactor Theory

Neutron Balance in a Volume Element for given Energy

0 0 0 ,( , , ) J x x y y E

t

z

omi

z

0Inflow through surface at y z x x

z , , J x z

, , x J x y z y z

0 0

0( , , ) z z y y L

x x L J x y z dydz

at a surface point

x

A y z

V x z

x

z y

y

( , , ) x y z 0 0 z y

0( , , ) x J x y z y z 0 0 0

0 0 0

( )

( )

y y x y y

z z x z z

x mean va ue t eorem

0Outflow through surface at y z x x x Net outflow or leakage through all six surfaces

0 0

0 00( , , )

z z y y R

x x z y

L J x x y z dydz

0( , , ) x J x x y z y z

R L R L R L

x x y y z z L L L L L L L

0 0( , , ) ( , , ) x x J x x y z J x y z y z

Net outflow or leakage through surface y z

R L

x x x L L L

0 0

0 0

( , , ) ( , , )

( , , ) ( , , )

y y

z z

J x y y z J x y z z x

J x y z z J x y z x y


0 0( , , ) ( , , ) x x J x x y z J x y z y z



Reactor Theory

Neutron Balance in a Volume Element

Loss within the volume by collision

z z x x

0 0 0

( , , , )V t z y x

C x y z E dxdydz

( , , , )t x y z E V

in small volume

Production within the volume by source

s iS S S S 0

( ) ( , , , ) f E x y z E dE

0 0 0 z z y y x x

0 0 0

0 0 0

( ) ( , , ) z z y y x x

z y x E x y z dxdydz

0 0 0

0 0 0

0 0 0 0,

(

,

,

,

, , ) z z y y x x

s z y x

z y x

s x y z E dxdydz

0

( ) ( ) ( , , , ) f E E x y z E dE V


0, , , , , ,s x y z s x y z



Reactor Theory

Neutron Balance in a Volume

Overall balance

0 0 0 0( , , ) ( , , ) ( , , ) ( , , ) x x y y J x x y z J x y z y z J x y y z J x y z z x

s i

0 0

0 0

, , , , , ,

( ) ( ) ( , , , ) ( ) ( , , , ) ( , , , )

z z t

f s E E x y z E dE V E E x y z E dE V s x y z E V

a ance equa on or vo ume e emen

divide by V x y z 0 00 0

( , , ) ( , , )( , , ) ( , , ) y y x x J x y y z J x y z J x x y z J x y z

0 0( , , ) ( , , )( , , , ) z z

t

x y

J x y z z J x y z x y z E V

z

0 0( ) ( ) ( , , , ) ( ) ( , , , ) ( , , , ) f s E E x y z E dE E E x y z E dE s x y z E

0 0( , , ) ( , , )

lim x x x J x x y z J x y z J


, , , , ,0 0 0

0, ,

x x y z x x



Reactor Theory

Neuton Balance Equation

Leakage term

y x z

x y z

limV V

: outflow per unit volumedivergence J

Balance equation

( , ) ( , ) ( , ) ( ) ( , ) ( , ) ( , ) ( , )

( , )

t f s E E

J r E r E r E E r E r E dE r E E r E dE

s r E

for approximationFic of k's law cur tren

4

ˆ ˆ- exact if ( , ) ( , , ) J r E r E d

( , ) ( , ) ( , ) J r E D r E r E

- - -

net flow

- net flow occurs due to by either and/ordifference

densityveloc in two reity gions

dense re ion


or faster neutrons- proportional to the gradient of flux ( )nv



Reactor Theory

Diffusion Equation

( , ) ( , ) ( , ) ( , )t D r E r E r E r E

, , , , , f s E E

E r E r E E r E E r E E s r E

scalar flux on- contains as unknown functly ion!

- in each constant property region

2( ) ( , ) ( ) ( , ) ( ) ( ) ( , ) ( ) ( , )

( , )

t f s E E D E r E E r E E E r E dE E E r E dE

s r E

* ( ), ( ) andetc. are region dependent constantt D E E




Reactor Theory

One Group Formulation

2( ) ( , ) ( ) ( , ) ( ) ( ) ( , ) ( ) ( , ) ( , )t f s

E E D E r E E r E E E r E dE E E r E dE s r E

- define group flux and average cross sections and

0 ( ) ( )( ) ( , ) ; obtained from infinite lattice calculation x

x E E dE r r E dE

0( ) E dE

- integrate over energy

0 0

, , ,t t a sr r r

0 0( ) ( ) ( , ) = ( ) ( , ) ( ) f f f

E E E r E dE dE E r E dE r

0 0 0 0 0( ) ( , ) = ( ) ( , ) = ( ) ( , ) ( )s s s s E E r E dE dE E E dE r E dE E r E dE r

2 2 0

,

- one group diffusion equation


( ) ( ) ( ) ( )a f D r r r s r



Reactor Theory

Plane Source in Non-Multiplying Infinite Medium

2

Diffusion Equation in 1-D 0 for nonmultiplying material ( 0) f y z

20a D

dx

0: (0) , ( ) 0 BC

2neutrons er unit cm secs 2 1d D

Solution x x

x

2 2dx L

w ere : us on eng , cma

1 1

( )

L L

x Ae Ce

0( ) x

Ld D

J x D e

0

( ) L x e 0 0 0 0

4 2

s

0 0

1 Ds

x

0(0) D

J

L

0 0Relation ship between and ? J s

0 0

1

1 s D

Interpretation of diffusion length

1 36.7%e 1

Relaxation to after traveling L

0 4 2 L


x L



Reactor Theory

Uniform Source in Non-Multiplying Finite Medium

2d

: ( ) 0, ( ) 0 BC a a 1D Diffusion Equation with Independent Source2

01d s 02 a

dx

General Solution

2 2dx L D

0s

( ) ( ) ( ) H P x x x

a a- Homogeneous Solution

2

2

1

L

use cosh sinh for finite s stems x x

2 21 0 H H

d

dx L ( ) cosh sinh H x A x C x

- Particular Solution

2

22 H

H d dx

cosh x

cosh a

20 0 0

2

1P P

a

s s s L

L D D

0 after subtractionC

0 1 after additions A a-

0( ) cosh sinh 0a

sa A a C a

s

cosha a

0 cosh( ) 1

s x x


cos s n

a

a a a

a

0 cosh coshcosha

s a xa



Reactor Theory

Point Source in Infinite Non-multiplying Region

2 01 sr r

Balance Equation

L D

2 2

2

1 d d r

r dr dr Symmetry consideration (no polar and azimuthal dependence)

0

21 10

d d r

Equation at locations other than origin

12 0r r

2 10

d r r

2 10

r dr dr L L dr L

r r

r r

L L

Solution

r L

2

22

d r d r r

dr dr

L Lr Ae Ce

( )r A C r r

BC1: finite as r 0;

r

L

e A r C

r

0BC2: as r 0 J Area s r

Ld d e

1 1 1 1r r


= J D D DC

dr dr r

2 2

L L DCe DCe

r rL r rL



Reactor Theory

Point Source in Infinite Non-multiplying Region

24 4 (1 )r

L r

J r DCe L

2

00

lim 4 4r

J r DC s

0

4

sC

D

0

( ) 4

r

Ls e

r D r

Interpretations

- As 0 oint sourcer

a- If 0, L 1r

Le

0s

0( )4

sr

D r

2

4 r

- Superposition of multiple sources 0( )

ir r

Ls er


i ir r



Reactor Theory

Linear Anisotropy in Angular Flux Distribution

( , ) ( , ) ( , ) J r E D r E r E

J cos J

- exact if angular flux is linear in cos J J

0 1( ) a a axisymmetric (no dependence in )

0 1, ?a a

0 ( )2 sin d

ˆ: angle between vector and J J

1 ( )2 d

04 a

0a

1

1( )2 J J d

1

0 11

2 a a d

1 1

2 2

1 11 0

2 4a d a d

1

4

3

a

1 3

14

a J

1 3


4 4

2 2



Reactor Theory

Partial Currents and Albedos

1 3( ) 2 ( )

2 2 J

1 0 1

1 1 0( ) ( ) ( ) J d d d

1 1 1 3 1 1

0 0: par a curren o pos ve rec on

2 2 4 2 out

0 0 1 3 1 1

1 1

< <

2 2 4 2in

0 1 1: artial current to ne ative direction J d J cos J

1

: from the core (system) to surroundings

in

p

J

J

J

J ou

: from the surroundings to the core (system)out

Alternatively

J


in



Reactor Theory

Anisotropic Scattering

Isotropic in CMS : low energy, light nucleus

C12 U238

1

1( ) 0 p d




Reactor Theory

Transport Cross Section

Migration of a Source Neutron Emitted toward a Direction (source by either fission or scat.)

-

in an absorption free medium

another three vectors toocate on a erent p ane

s

escr e success ve trave

with two vectors forming a plane

and the ro ection of 3rd vectors

on the plane

trans ort mean free ath

1 s x 2

2 s x n

n s x

211

str s

1

1

with bigger , longer travel from the source point toward the initial direction

easier diffusion




Reactor Theory

Transport Cross Section and Diffusion Coefficient

Transport corrected scattering cross section

1

1tr

s

1tr

s s reduced scattering cross section to consider anisotropic scattering

tr

tr a s transport cross section, later to be used to define diffusion coefficient

a s s t s

Diffusion coefficient

1 D tr 1

- Under linear anisotropy in angular distribution

3 tr s , ,s tr 3




Reactor Theory

Boundary Conditions for Diffusion Equation

Boundary condition for angular flux at vacuum boundary

ˆ ˆ, ,v in

r invacuum

vr

Interface condition1a 2a

ˆ ˆ, , , ,

I I r E r E

I r

I r

Boundary condition for diffusion equationoutward direction

1 32 ( , ) ( ) ( )

2 2 z z J z

- 1

1. 5

2

2. 5

transport

diffusionshape

2

1 3 2

2 2 J - 1 - 0. 5 0. 5 1

0. 5s ape


pos t ve angu ar unp ys cashaded areas not the same 0in j



Reactor Theory

BC for Diffusion Equation

Adjustment for zero net incoming current

0 0

1 12 ( , )

2 2

net

in J z d J d

4 2 3 4 2

J J 0

2 J 2 Ddz

2dz D

22 0

3 3tr

tr

2( ) 0tr z

Zero flux at extrapolated location

2

( 0.711 ) 0tr z more realistic condition

3

1: albedo boundary condition

D d J or

* = : Partial current albedoin

p

J


out



Reactor Theory

Interface Condition for Diffusion Equation

ˆ ˆ

Transport interface condition

, , , , I I r E r E

ˆ ˆ

Flux continuity

4 4, , , , I I

, , I I r E r E

Current continuity

4 4

ˆ ˆ ˆ ˆ, , , , I I

r E d r E d

diffusion coefficients are different

1 2, , I I


R Th



Reactor Theory

Validity of Fick's law

Fick's law valid if free back-and-forth collisions with nuclei

net flow

from dense region or faster neutron region

Limitations of Fick's law and diffusion equation

- cases ac ng ree ac an ort co s ons w t nuc e

1. near strong absorber

. near oun ary

3. near source

1 . near interface of very different materials4. in a medium of low density

homo enize cell first usin trans ort calc. result


4.1.DiffusionEq

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