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Diffusion Equation
f S i iept. of Energy System Engineering
Shim Hyung Jin
7/17/2019 4.1.DiffusionEq
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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
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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
3 SNU Monte Carlo Lab.
7/17/2019 4.1.DiffusionEq
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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
4 SNU Monte Carlo Lab.
0 0( , , ) ( , , ) x x J x x y z J x y z y z
7/17/2019 4.1.DiffusionEq
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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
5 SNU Monte Carlo Lab.
0, , , , , ,s x y z s x y z
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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
6 SNU Monte Carlo Lab.
, , , , ,0 0 0
0, ,
x x y z x x
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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
7 SNU Monte Carlo Lab.
or faster neutrons- proportional to the gradient of flux ( )nv
7/17/2019 4.1.DiffusionEq
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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
8 SNU Monte Carlo Lab.
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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
9 SNU Monte Carlo Lab.
( ) ( ) ( ) ( )a f D r r r s r
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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
10 SNU Monte Carlo Lab.
x L
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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
11 SNU Monte Carlo Lab.
cos s n
a
a a a
a
0 cosh coshcosha
s a xa
7/17/2019 4.1.DiffusionEq
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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
12 SNU Monte Carlo Lab.
= J D D DC
dr dr r
2 2
L L DCe DCe
r rL r rL
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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
13 SNU Monte Carlo Lab.
i ir r
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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
14 SNU Monte Carlo Lab.
4 4
2 2
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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
15 SNU Monte Carlo Lab.
in
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Reactor Theory
Anisotropic Scattering
Isotropic in CMS : low energy, light nucleus
C12 U238
1
1( ) 0 p d
16 SNU Monte Carlo Lab.
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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
17 SNU Monte Carlo Lab.
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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
18 SNU Monte Carlo Lab.
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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
19 SNU Monte Carlo Lab.
pos t ve angu ar unp ys cashaded areas not the same 0in j
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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
20 SNU Monte Carlo Lab.
out
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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
21 SNU Monte Carlo Lab.
R Th
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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
22 SNU Monte Carlo Lab.
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