Reactor Numerical Analysis and Design
1st Semester of 2010
Lecture Note 1
I. Solution of One-Dimensional, OneGroup Neutron Diffusion Equation
Prof. Joo Han-gyu
Department of Nuclear Engineering
1
SNURPL
1D 1G Neutron Diffusion Equation
 Neutron Diffusion Equation
dJ ( x)
dx
  a ( x ) ( x )      f ( x ) ( x ) ,
Fick's Law : J ( x )   D ( x )
1
=
k eff
d ( x)

dx
 Boundary condition with Albedo
 Right ( x  x R )
R
1)  R 
J

2) J   J
R
x  xR
 D
d
dx
x  xR
 Left ( x  x L )
L
1)  L 
J

2) J    J
L
x  xL
d 
d

  D
D
dx 
dx

x  xL
J : positivie
J L
J R
Left
Right
 Homogeneous D.E. with Homogeneous B.C  Homogeneous proble m , Eigenvalue p roblem
2
SNURPL
Albedos for Various Boundary Conditions
 

J
J : positivie

J OUT
J IN
J R
  0.5   Big
 0
c)   0.5
 ( ) 
 J IN  0
1

2
J IN  0
3
1
1
1

J


(

)

d




J
 IN
0
4
2
 
1
1
1
J


(

)

d




J
0
 O U T
4
2

J
2

: Zero-incom in g current B C

 cos 


1
4

1
J 0

2
3
 
J

 0 .5
SNURPL
Positiveness of Eigenvalue
 Neutron Diffusion Equation
 Multiply  ( x )

d  d 
D
   a      f 
dx  dx 

d  d 
D
 
dx  dx 
  a        f   
 Integrate above eqn. over [ x L , x R ]
A
B 

xR
xL
xR
 
xL
C 

xR
xL
2
 d  d  
D
   dx

d
x
d
x

 

a
    dx
  f      dx


  R   L 
2
R
2
L

xR
xL
 d 
D
 dx  0
 dx 
0
0
physically λ is an adjustment factor for nontrivial solution
4
SNURPL
1D Problem in Other Coordinates
 E xtension of 1-D to other coordinates
  J    D 
 P olar

  D   if D  const .
2
1 d 
d 
d  d  1 d
 rD

D
 D
r dr 
dr 
dr 
dr  r
dr
r 
2


0
r
dJ

dr
J
r
1 d  d 
r

r dr  dr 
 Spherical

1 d  2 d 
d  d  2 d
r
D




D
 D
2
dr 
dr 
dr  r
dr
r dr 
r 
2

dJ
dr
2
J
r
1 d  2 d 
r

2
dr 
r dr 
5
SNURPL
Discetization
 P oint schem e
i 1
1
0
i
i 1
ia ia1
N
Di Di 1
xi 1 xi
 B ox schem e schem e
xi 1
i
i 1
1
i 1
ia ia1
Di Di 1
xi 1 xi
N
xi 1
6
SNURPL
Discretization with Point Scheme

d 
d ( x) 
 D ( x)
   a ( x ) ( x )      f ( x ) ( x )
dx 
dx 
dJ
dx
dJ

dx
b )  a ,i 

   i 1 
   Di i

hi  1
hi


1
1
hi  hi  1
2
2
J
i
L
i
 a ,i Di
 L
 J i   Di
hi


 J R   D  i 1   i
i 1
 i
hi  1
 D i 1
R
Ji
i 1
  a      f 
 i   i 1
a)
i 1
 a ,i 1 Di 1
xi 1
xi
hi
2
xi 1
hi 1
2
 i 1   i

2
hi  hi  1

 i 1   i
   i 1 
 Di i
  D i 1

h
h
i 1
i


hi  a , i  hi  1  a , i  1
c )   f ,i 
hi  hi  1
hi  f , i  hi 1  f , i 1
hi  hi  1
7
SNURPL
Discretization with Point Scheme
 if hi  hi 1  h
dJ
dx
a)
dJ

dx
2
hi  hi  1
 i 1   i
 i   i 1 
1

D

D
i 1
i


h
h
h


D i 1
h
2
  i 1   i  
Di
h
2
   i 1   i 1   i    i
D
D
  
D
i 1 i 1
b )  a ,i 

 
D
i 1

D
i

 i 1   i
   i 1 
 Di i
  D i 1

hi  1
hi


i 1
i
i 1
  i   i 1 
ia Di
  i   i 1 
ia1 Di 1
xi 1
 i   
D
i 1 i 1
hi
2
 a , i   a , i 1
xi 1
xi
hi 1
2
2
c )   f ,i 
  f ,i    f ,i 1
2
 At x  x i
dJ
dx
  a      f     i  1 i  1    i  1   i
D
D
D
8

  i  1 i  1   a , i  i     f , i  i
D
i
SNURPL
Boundary Condition with Point Scheme
 A t left boundary  x  0 
J 0   L  0
dJ
dx
J0  J0
J 0   D1
R
 D1
R

h1
2
R

1   0
h1
h1
L 
1   0
J
J0
0

1a D1
h1
x1
x0
   L 0 
 2D
2 L
  21 
h1
 h1

2 D1
  0  2 1
h1

1
h1
2
2
 1
2 L 
D
  a      f    2  D 
  0  2  1 1   a ,1 0     f ,1 0
dx
h1 

dJ
9
SNURPL
Boundary Condition with Point Scheme
 At right boundary  x  H
J N   R N

N 1
J N   DN
L
 N   N 1
dJ
dx
JN  JN
L

hN
2


hN
hN
R 

 2DN
2 R
   2DN 



N

1
2
2
hN
hN
 hN

N

J

xN
xN 1
 N   N 1 

N
 a , N DN
hN
 R N    D N
L
JN
h
2
2

2 R
N
N
  2  D  N 1   2  D 
hN


N


2 L
D
cf :  2  1 
h1


2 R
N
N
  a      f    2  D  N  1   2  D 
dx
hN

dJ
 N 
1
N
  D  N 1    D  R   a , N
hN
2


D
  0  2  1 1  sam e as left


  N   a , N  N     f , N  N


1
 N     f ,N N
2

10
SNURPL
Linear System for Point Scheme
 P oint schem e  R esult
 D L 1

D
  a ,1   0   1 1
 1 
h1
2


  i  1 i  1
D

1
     f ,1 0
2

  i 1   i   a ,i  i
D
D
  i  1 i  1
D
 N 
1
N
  D  N 1    D  R   a , N
hN
2

11
    f , i  i

1
 N     f ,N N
2

 x  0
 x1
x 
~ x N 1 
H

SNURPL
Matrix Form of Tridiagonal Linear System
 D L 1
  1  h  2  a ,1
1

D

 1

0







0


1
 2   f ,1












1
0
 1   2   a ,1
1
D
D
D
0
D
0
0
0
0
0
0
0
0
0
 1
 N 1   N   a , N 1
0
 N
D
  f ,1
  f , N 1
1
2
  f ,N
D
 N
D
D
D
D 
N
R
hN

1
2
 a ,N
  0 


  1 
















   N 1 


   N 
  0 
  
 1 





















 N 1 
  
 N 
12
SNURPL
Box Scheme
 T o derive conservation form
dJ
dx
  a      f 
i
s , J i
(1)
i 1
 Integrate (1) over [ xi 1 , xi ]
A
B 
C 



xi
dJ
x i 1
dx
xi
x i 1
xi
x i 1
 
a,i 1 Di 1
 a ,i D i
dx  J i  J i  1
xi 1
 a , i  ( x ) dx   a , i   i  hi
xi
xi 1
hi
   f , i  ( x ) dx     f , i   i  hi
 drop the overbar from
now on 
13
SNURPL
Current in Box Scheme
i
 C urrent in term s of surface flux
s , J i
J i   Di
R
 s  i
J i 1   D i 1
L
hi
 i 1   s
R
hi  1
L
Ji
2
2
JJi ,LR
i 1
 a ,i Di
J
 Di

R
i
 s  i
hi
J
L
i 1
  Di
Di
s 
hi
a,i 1 Di 1
xi
xi 1
hi
hi  1
2
i 
Di
hi

D i 1
hi  1
 i 1
D i 1
  i  i  (1   i )  i 1
hi  1
J i 1  J i   D i
L
xi 1
 i 1   s
2
i 1
J i ,iL
R

 i  i  (1   i )  i 1   i
2  i  i 1
 i   i 1
hi / 2
  2  i (1   i ) (  i 1   i )
  i 1   i    D i   i 1   i 
14
SNURPL
Nodal Balance Equation
 In the i-th node, xi 1  x  xi
J i   D i ( i  1   i )

2  i  i 1 
D

 i




i
i 1 

J i  1   D i  1 ( i   i  1 )

2  i 1  i 
D

 i 1

 i 1   i 

J i  J i 1   a ,i   i  hi     f ,i   i  hi
 D i ( i  1   i )  D i 1 ( i   i 1 )   a   i  hi     f   i  hi
15
SNURPL
Boundary Condition with Box Scheme
 At left boundary
D1
J 0    L  s   D1
J 0   L
1
 0  1
1   s
→
s 
h1
h1
D1
2
h1
L
2
1
1
1 
1
L
 0  1
1 
2
0
2
1  

1 
L
1
2
1   D 01
 0  1
J 1  J 0   a  1  h1     f  1  h1
 D1 ( 2  1 )  D 0 1   a ,1  1  h1     f  h1  1
 A t Left B oundary  x  0 
 At R ight Boundary  x  H
16

SNURPL
Tridiagonal Linear System for Box Scheme
 B ox schem e  R esult
( D 0  D1   a ,1  h1 )1
 D i  1 i  1
 D1 2
 ( D i  1  D i   a  h i ) i
 D N  1 N  1
 d1

l
 2
0





0

u1
0
d2
u2
0
0
0
0
li
di
0
ll  1
0
ui
0
l N 1
d N 1
0
lN
 D i i 1
 ( D N  1  D N   a , N  h N ) N
0   0 
  f ,1



0
1
















0 



u N 1   N 1 

 0
d N    N 

    f ,1  h1  1
 x0 
     f  hi   i
 x1
    f , N  h N   N
 xN 
0
  f ,2
0
0
0
0
0
0
0
0
0
0
0
  f , N 1
0
0
 f ,N
~ x N 1 
  0 



 1 










  N 1 


  N 
M   F 
1. G eneralized F orm
 S am e form every w here
2. C onservation F orm
 N odal balance assured because of startin g
from integration
17
SNURPL
Analytic Solution for Matrix Eigenvalue Problem
 For a single region
d 
2
D
dx
2
  a      f 
 Point scheme with  0  0,  N  0
 if
 i   i 1 
D

D
D
h
2
D
h
2
,
 a ,i   a ,i 1   a ,
  k 1  2 k   k  1    a  k
 k 1  2 k   k  1 
h
2
D
  f ,i    f ,i 1    f
    f  k
1) 
h
2
D

a
    f   k  0

a
    f

 f 
a 
 h
1   

D 
a 
2
18
2) k eff 
B 
2
k
1 L B
2
2

1  k

1

0
2

L  k eff

k   k eff
SNURPL
Analytic Solution for Matrix Eigenvalue Problem
 k 1  2 k   k 1  B  k  0
2
e
e
m ( k  1)
m
 2e
mk
m ( k  1)
e
B e
2
mk
0
2e B  0
m
e e
m
m
 1
2
2
B
2
2
C osh m (  1)  1 
B
2
(  1)
N ot possible
2
 Let  k  e
e
im
e
2
imk
 im
1
B
2
2
k  e
19
 imk
as well
SNURPL
Buckling Eigenvalue
 k  C1 cos m k  C 2 sin m k
 0  C 1  0

B .C 
  N  C 2 sin m N  0,

m N  l  ( l  1,..., n  1)
 m 
l
N
 cos m  1 
B
2
2


  
B  2 1  cos 

2
1 

N





2
2
4


1 
1    
1



....





 
2
N
4
!
N







2
2
6
1   
  
 

 1  

 O

1 2  N  
 N  
N 
N 
2
2
1    
   2
6

h    O (h )
 h  1 

12  H  
H 

2
2
1   
  
4
 B 
h    O (h )
  1 


12  H  
H  
H
, B  hB
h
2
20
SNURPL
Eigenvalue
B 
2
2
2
1    
  
4

h    O (h )
  1 

12  H  
H  
   f   a
D
2
    2 

1 
1   
4 
 a  D 
  
h    O (h )  
  1 

 f 
H
12
H



 

 


 Analytic Solution for differential eqn.
2
2
4
1    2
  
  
B 
vs





 h
12  H 
H 
H 
2
2
1 
   
 
 a  D 
 
  f 
 H  
*
E rror   =
1
 f
4
D    2
2


 h  h
12  H 
21
SNURPL