Solution of simplified neutron diffusion equation by FDM in Pascal

1. 1
Core Neutronics
FDM for Neutron Transport
Syeilendra Pramuditya
Nuclear Science and Engineering
Institut Teknologi Bandung

2. 2
Energy Release & Deposition

3. 3
Calculation Model of Reactor Analysis

4. 4
Simplified…

5. 5
Energy Transport Model (LWR)

6. 6
Neutron Transport
 General neutron transport equation is
reduced to neutron diffusion equation
 Main variables
 Neutron flux (BVP / IVP)
 Multiplication factor (EVP)
 Simple system
 Neutron flux / BVP
 Numerical solution
 Iterative FDM
 Direct FDM

7. 7
Finite Difference
1 1
2
1 1
2 2 2
( ) ( )
( ) ( )
2 2
2
( ) 2 ( ) ( )
( ) ( )
( )
i i
i i i
f f
d f x x f x x
f x f x
dx x h
f f f
d f x x f x f x x
f x f x
dx x h
 
 

    
   

 
     
   

2
2
( ) 48
'' 48
( 0) 6
Boundary Conditions
( 1) 7
0.05, 0.10, 0.15, ... , 1.00
i
d
y x x
dx
y x
y x
y x
x


  

  


8. 8
Finite Difference
1 1
2
2
1 1
2
1 1
'' 48
2
48
2 48
48
2
i i i
i
i i i i
i i i
i
y x
y y y
x
h
y y y h x
y y h x
y
 
 
 

 

  
 

1 1
2
1 1
2 2
( ) ( )
2
2
( ) ( )
i i
i i i
f f
d
f x f x
dx h
f f f
d
f x f x
dx h
 
 

  
 
  

9. 9
Finite Difference
Initial Guess
2
2
( ) 48
'' 48
( 0) 6
Boundary Conditions
( 1) 7
0.05, 0.10, 0.15, ... , 1.00
i
d
y x x
dx
y x
y x
y x
x


  

  

2
1 1 48
2
old old
new i i i
i
y y h x
y  
 

Loop only the internal points,
Until:
0.0001
For all points i
new old
i i
old
i
y y
y


 

10. 10
Finite Difference
10-point Grid
old
i
y 1
old
i
y 
1
old
i
y 
old
i
y
1
old
i
y 
new
i
y
2
1 1 48
2
old old
new i i i
i
y y h x
y  
 


11. 11
Code Structure
 Define resolution & domain boundaries
 Define number of points (xi)
 Define x1 & xN
 Calculate h or x
 Define boundary conditions  f(x1) & f(xN)
 Dirichlet BC
 Define initial guess for internal points
 Double Loop
 Inner loop  Update internal points using finite
difference equation
 Outer loop  Convergence check: how much the change
of each point is?
 Loop until iteration is converged
 No significant change for all points

12. 12
Code Output

13. 13
Gnuplot Command
# Script to plot 1D dataset
reset
unset label
unset key
set key left top
set xrange [0:1]
set yrange [3:8]
set title "Plot Image"
set xlabel "X Value"
set ylabel "Y Value"
set terminal wxt size 600,400 font "Verdana,10"
plot 'output.txt' using 1:2 title "Numeric" with linespoints pointtype 6 lw 1 lc 7,
'output.txt' using 1:3 title "Analytic" with linespoints pointtype 6 lw 1 lc 6

14. 14
Gnuplot Image

15. 15
Finite Difference
2
1 1 48
2
old old
new i i i
i
y y h x
y  
 


16. 16
Finite Difference
Max error = 1E-4  ~379 iterations to converged

17. 17
Speed up the process..?
10-point Grid
old
i
y 1
old
i
y 
1
old
i
y 
old
i
y
1
old
i
y 
new
i
y
2
1 1 48
2
old old
new i i i
i
y y h x
y  
 


18. 18
Speed up the process..?
2
1 1 48
2
old old
new i i i
i
y y h x
y  
 

2
1 1 48
2
old new
new i i i
i
y y h x
y  
 

Jacobi Method
Gauss-Siedel Method
379 iterations  219 iterations
1.73X Speed up!

19. 19
Finite Difference: Example
1 1
2
1 1
2 2 2
( ) ( )
( ) ( )
2 2
2
( ) 2 ( ) ( )
( ) ( )
( )
i i
i i i
f f
d f x x f x x
f x f x
dx x h
f f f
d f x x f x f x x
f x f x
dx x h
 
 

    
  

 
     
  

5 10 10
(0) 0
( ) 100
0.0, 0.1, 0.2, ... , 1.0
i
y y y x
y
y n
x
 
  




20. 20
Physics Sample Problem
 Neutron diffusion in nuclear reactor

21. 21
Neutron Diffusion in Nuclear Reactor
laju perubahan kebocoran absorpsi neutron muncul neu
jumlah neutron neutron dari sumber
= - - + -
neutron dari sistem di grup g neutron
di grup g ( ) di grup g
leakage
       
       
       
       
       
       
tron neutron
terhambur terhambur
+
keluar dari masuk ke
grup g grup g
   
   
   
   
   
   
' '
' 1
( , )
1
( ) ( , ) ( ) ( , )
( , ) ( ) ( , ) ( ) ( , )
g
g g ag g
g
G
g
g sg g sg g g
g
eff
r t
D r r t r r t
v t
S r t r r t r r t
k

 

 


     

  

' ' '
' 1
( , ) ( ) ( , )
G
g fg g
g
S r t v r r t


 

Neutron diffusion equation

22. 22
Neutron Diffusion Equation
 Neutron diffusion equation, much much simplified...
 Steady state
 1 dimension
 1 energy group
 Without kinf calculation
 FDM Gauss-Siedel Scheme
2
2
( ) ( )
( )
Boundary conditions:
( 0) ( ) 0
( 0) ( ) 0
a
d x S x
x
dx D D
x x L
S x S x L


 

  
   
   
1 1
2
1 1
2
2
2
( )
( )
2
( )
i i i a i
i
old new
i i i
new
i
a
S
x D D
S
x D
D x
  

 

 
 
  
  









23. 23
Neutron Diffusion Equation
Rod length L 3
Number of partition 50
Absorption cross section a
1
Diffusion coef. D 1/6
Max iteration 1000
Max error  1e-6
Source (uniform) 100
Flux initial guess (interior) 50
Calculation Parameters
1 1
2
2
( )
2
( )
Boundary conditions:
( 0) ( ) 0
( 0) ( ) 0
Max error:
old new
i i i
new
i
a
new old
i i
new
i
S
x D
D x
x x L
S x S x L
 

 
 


 







   
   



24. 24
Calculation Result
 Uniform Source S(x)

25. 25
Calculation Result
 Non-Uniform Source S(x)