Nodal Methods for Core Neutron Diffusion Calculations

Nodal Methods for Core Neutron Diffusion Calculations

May 8, 2008

Prof. Joo Han-gyu

Department of Nuclear Engineering

Reactor Numerical Analysis and Design

1st Semester of 2008

Lecture Note 10

Contents

Transverse Integration and Resulting One-Dimensional Neutron Diffusion Equation

Treatment of Transverse leakage

Nodal Expansion Method with One-Node Formulation

 Polynomial Intra-nodal Flux Expansion

 Response Matrix Formulation

 Iterative Solution Sequence

Analytic Nodal Method with Two-Node Formulation

 Two-Node Problem

 Analytic Solution of Two-Group, One-D Neutron Diffusion Eqn.

 Implementation with the CMFD Framework

Semi-Analytic Nodal Method

 Polynomial Intra-nodal Source Expansion

 Analytic Solution for One Node

Introduction

3-D Steady-State Multigroup Neutron Diffusion Equation

 Fick's Law of Diffusion for Current out of Flux

Computational Node in 3-D Space

 Property assumed constant within each homogenized node

 FDM accurate only if the node size is sufficiently small (~1cm)

 Nodal methods to achieve high accuracy with large nodes (20 cm)

 

 

















G

g

G

g

g g sg g

fg g

g rg

g r E r E r E r E

J

1

' ' 1

' ' '

' ( , ) ( , )

) , ( )

,

(   

     

) , ( ) , ( )

,

(r E D r E r E

J_g  _g  _g 









x

y z

0

) , , (h_x h_y h_z

Volume Averaging of Diffusion Equation for a Node

 Integrate over the node volume then divide by volume

 Volume Average Flux

 Integration of the Divergence Term using Gauss Theorem

 Surface Average Current

Nodal Balance Equation for Average Quantities of Interest (Nodal Power)

Nodal Balance Equation (NBE)

   

  



 



 





 



 

 



 



 







 

 

 

  

x y

x z

y z

x y z

h h

z z z

h h

y y

y

h h

x x

x

h h h

g

dxdy y

x J h y x J

dxdz z

x J z h x J dydz

z y J z y h J dxdydz

J

0 0

0 0

0 0

0 0 0

) 0 , , ( ) , , (

) , 0 , ( ) , , ( )

, , 0 ( ) , , (



 

  ^

 ^{y z hy hz} _{x x}

z y gxr

h h

x z

y

gxl J h y z dydz

h h J

dydz z y J

h h J

0 0

0 0

) , , 1 (

, )

, , 0 1 (

  

 ^{hx hy hz}

z y x

dxdydz z

y x h

h

h 1 ^{0 0 0} ( , , )









  













  ^G

g

g g sg G

g

g fg g

g rg z

y x

u u

gul gur

h J J

1 '

' ' 1

'

' ' ,

,











NBE Solution Consideration

 Information on 6 surface average currents only required for obtaining the node average flux which will determine the nodal power

 Surface Average Currents

- Average of Flux Derivative on a Surface

- Equals to Derivative of Average Flux at the Surface

 Better to work with the neutron diffusion equation for average flux rather than one for the point wise flux (3-D)

Transverse Integration

 Set a direction of interest (e.g. x)

 Perform integration within node over 2-D plane normal to the direction, then divide by plane area

Need for Transverse Integration

 

  ^{  }

 ^{hz hy} _{g rg g}

z y

dydz E

r E

r J h

h LHS

0 0

) , ( )

,

1  ( 

 x

y

z

0

Normalized Independent Variables

Transformation of Integration and Derivative Operator

Simplified Averaging

Normalized 3-D Diffusion Equation

Normalization of Variables

z z y

y u

x h

z h

y h

x  

  

 , ,

z y x u d

h d du

d d h du

u u u

u

, ,

,

,  

  

  

 

 







1 0

1 0

1 0

1 0

1 0 1

0 1 0

) , , (

, )

, , 1 ( ,

) , , 0 (

z y x z y x

z y z y x gxr

z y z y x gxl

d d d

d d J

J d

d J

J

































) , , ( )

, , (

1

' ' 1

' '

, ,

2 2

2 g x y z

G

g

G

g

g sg fg

g z

y x g rg z

y x

u u u

g

h

D           

 ^_







   

 











 



    

 



Transverse Integration of Leakage Term

Plane Average One-Dimensional Flux

Line Average Surface Current at Arbitrary Position x Transverse Integrated Quantities





 

 

 

 











 





 



 



 

 

 



 



 





1

0 1

0 1

0 1

2 0 2

1

0 1

0 2

2

2 1

0 1

0 1

0 1

0

)) 0 , , ( ) 1 , , ( 1 (

)) , 0 , ( ) , 1 , ( 1 (

) , , (

1 ) ( 1

1 ) 1

( 1

y y

x z y

x z z z z x y z x y y

z y z y x x

x g

z y z z

z y y

y x x

g z

y z z

z y y

y x x

x z y

d J

J h d J

J h

d d h

D

d J d

h J h h

D d

J d h J h J h

d d J































 



 





 

 







 

z

x zl x

zr

y

x l y x

r y

x x x

x g

h J J

h J J

h

D ( ) ( ) ( ) ( ) ( )

2 2



 







 

 

 

 



 

 ¹

0 1 0

) , , ( )

( _{x x y z y z}

x      d d





 ^

 ^l _{y x z z yl x} ^l _{y x z z}

x

yr J d J J d

J

0 0

) , 0 , ( )

( ,

) , 1 , ( )

(       



 ^

 ^l _{z x y y zl x} ^l _{z x y y}

x

zr J d J J d

J

0 0

) ,0 , ( )

( ,

) 1 , , ( )

(       

x y

z

0

) ( _x Jyl 

) ( _x Jzl 

) ( _x Jyr  )

( _x Jzr 

Transverse Integration of 3-D Neutron Diffusion Equation

Define Transverse Leakage to Move to RHS

Transverse Integrated One-Dimensional Neutron Diffusion Equation (Final Form)

 Diffusion Equivalent Group Constant

Transverse Integrated One-Dimensional Neutron Diffusion Equation





  













  ^G

g

x x g g sg G

g

x x g fg g

x gx rg x

gx z x

y

u u x

x gul x

gur

d d h

D h

J J

1 '

' ' 1

'

' 2 '

2

,

2 ( ) ( ) ( ) ( )

) ( )

(          







^{J J}  ^{u y z}

h

L _{gur x gul x}

u x

gu 1 ( ) ( ) , ,

)

(     

) ( )

( )

( )

( )

( )

(

1 '

' ' 1

'

' 2 '

2

x gz x

gy G

g

x x g g sg G

g

x x g fg g

x gx rg x

gx x x

Dg L L

d

d            

 ^{       }



  





2 x x

Dg h

 D



Set of 3 Directional 1-D Neutron Diffusion Equations

 3-D Partial Differential Equation

→ Three 1-D Ordinary Differential Equations

 Coupled through average transverse leakage term

- Exact if the proper transverse leakages are used

Approximation on Transverse Leakage

 Quadratic Shape (2^nd order polynomial)

 based on observation that change of flux distribution is not sensitive to change of transverse leakage

 Iteratively update transverse leakage

Transverse Integrated One-dimensional Neutron Diffusion Equations

) ( )

( )

( )

( )

( )

(

1 '

' ' 1

'

' 2 '

2

y gx y

gz G

g

y y g g sg G

g

y y g fg g

y gy rg y

gy y y

Dg L L

d

d            

 ^{       }



  





) ( )

( )

( )

( )

( )

(

1 '

' ' 1

'

' 2 '

2

x gz x

gy G

g

x x g g sg G

g

x x g fg g

x gx rg x

gx x x

Dg L L

d

d            

 ^{       }



  





) ( )

( )

( )

( )

( )

(

1 '

' ' 1

'

' 2 '

2

z gy z

gx G

g

z z g g sg G

g

z z g fg g

z gz rg z

gz z z

Dg L L

d

d            

 ^{       }



  





Transverse Leakage Approximation

Quadratic Approximation in Each Node

Average TL Conservation Scheme to Determine l

and l

 Use three node average transverse leakages

- Values of own node and two adjacent nodes

 Impose constraint of conserving the averages of two adjacent nodes

) ( _x Lz 

Lleft ^center

L

right

L

) ( )

( )

( L l₁P₁  l₂P₂ 

L   

Intranodal Flux Expansion of 1-D Flux

 Approximate 1-D Flux by 4^th Order Polynomial

 Basis Functions

- Not Orthogonal Function

- Integration in Range [0,1] results 0.

2

Order Transverse Leakage

Nodal Expansion Method

4

0

( ) _{i i}( )

i

  a P 



 

1 )

0(  P

1 2 )

1(   

P

1 ) 1 ( 6 )

2(    

P

) 1 2 )(

1 ( 6 )

3(     

P

) 1 5 5

)(

1 ( 6 )

( ²

4        

P

0.2 0.4 0.6 0.8 1

-1 -0.5

0.5 1

)

2( P )

4(

P P₃()

)

1( P







2

0

) ( )

(

i i iP l

L  

Given Conditions

 Incoming Partial Currents at Both Boundaries

 Quartic Intranodal Variation of Source

Aim

 Solve for flux expansion

 Then update the outgoing partial current and source polynomial

One Node Formulation

m

g _m1

g

1 m

g

 right

Jg_,

 left

Jg_,

 left

Jg_{, Jg}^_,right

0 1

) 0

, (

m g

x ₍₁₎

, m

g

x

)

, (

_x^m_g

) (

) ( )

( )

( )

( )

( ) ( )

(

4

0

' 1

'

' '

1 '































i

i i

g g

G

g

g g g

G

g

f g

P q

L L

s Q



























Three Physical Constraints

 2 Incoming Current Boundary Conditions

 1 Nodal Balance

 Two-Additional Conditions Required to Determine 5 Coeff.

Weighted Residual Method for 1-D Neutron Diff. Eqn.

1

Moment of Neutron Diffusion Equation

- contains a₁ which is unknown in principle

2

Moment of Neutron Diffusion Equation

- contains a₂ which is unknown in principle

Weighted Residual Method

^{   }  ^











 

 d w s L d

d

w( ) _D d ( ) _r ( ) ( ) ( ) ( ) ( )

1 0 1

0 2 

 ^_^{    }_^{   }



 







    

0¹ 1^{( )} 2 ⁽⁾ ⁽^{) (} ⁾  ⁰

  Q d

d

P _D d _r

) 60

( 3

5 3

5 _{1 3 1}

3

r D

a r

q a q











 



 







    

0¹ 2^{( )} 2 ⁽⁾ ⁽ ^{) (}⁾  ⁰

  Q d

d

P _D d _r

) 3 (

420

7 3

7 _{2 4 2}

4

r D

a r

q a q













 

One-Node NEM Iterative Solution Sequence

For a given group

 Determine sequentially

- Source expansion coeff.

- a₁ and a₂ from previous surface fluxes

- a₃ and a₄ using source moments and a₁ and a₂ - node average flux

- outgoing current

 Move to next group

 Move to next node once all groups are done

Group sweep and node sweep can be reversed (node sweep then group sweep) Update eigenvalue

Source

) , , , ,

1(J_r^ J_l^ J_r^ J_r^  a

) , , , ,

2(Jr^ Jl^ Jr^ Jl^ a

) , , ( _{1 1 3}

3 a q q

a

) , , ( _{2 2 4}

4a q q a

) , , , ,

(J J a₃ a₄ J_r^ _r^ _l^ 

) , , , ,

(J J a₃ a₄ J_l^ _r^ _l^ 

4 3 2 1

; i , , , q_i 

Converge?

End Condition Initial

No

Yes

. .

( , C B

1,2,3,4) i

a Update

i 



Analytic Nodal Method for 2-G Problem

1D, Two-Group Diffusion Equation

 All source terms except transverse leakage now on LHS

Analytic Solution: Homogeneous + Particular Sol.

Trial Homogeneous Solution

( ) ^H ( ) ^P( )

g x g x g x

   

( ) ˆ

H H iBx

g x g e

  

 

2 1

1 2 1 1 1 1 2 2 1

2 2

2 2 2 1 2

( ) ( ) ( ) ( ) ( )

( ) ( ) ( )

r f f

r

d x

D x x x L x

dx

d x

D x L x

dx

      

 

        

    

2

2 2

2

( ) ˆ ( )

H

g H iB x H

g g

d x

B e B x

dx

      

Determination of Buckling Eigenvalues

Characteristic Equation

 For Nontrivial Solution

Eigen-Buckling (Roots of Characteristic Equation)

 Fundamental Mode

 Second Harmonics Mode

2

1 1 1 2 1

2

1 2 2 2 2

ˆ 0

ˆ 0

H

r f f

H r

D B

D B

  



 

         

  

         

   

0 )

)(

( 0

)

(A   D₁B²  _r1   _f1 D₂B²  _r2   _f2₁₂ 

Det  

 

 

2 2

1 2 1

1 1

2 2

1 2 1

2       











 

 







 

 

 

 ^ B b B c

D D k

B k D

D D

B ^{r r}

eff r f

r 























  







eff eff

k k

k k

b b c

B 0 ,

, 0 1

1 2

2 1

0 1

1 2

2

2  









  



b b c

B

1 1

2

E ig V ec ˆ

ˆ 1

H

H

 r



   

    

   

 

1 2

ˆ

ˆ 1

H

H

 r



   

    

 

 

 

2

2 2 2

1 2 _m 2 _{m r}2 0 _m D B^{m r}

r D B r  

      



Homogeneous Solutions

Each Group Homogenous Solution

 Fundamental Mode

 Second-Harmonics Mode

Combined Homogenous Solution

 Linearly Dependent Group 1 and Group 2 Equations

 Fast-to-Thermal Flux Ratio

1 1 2 1

1

1 1 2 1

sin ( ) co s( ) ,

( )

sin h ( ) co sh ( ) ,

g g eff

H g

g g eff

a B x a B x k k

x

a B x a B x k k

 ^



 

 

 



2( ) 3sinh( 2 ) 4 cosh( 2 )

H

g x ag B x ag B x

  

1 1 2 21 1 22 1

23 2 24 2

2

( ) sin( ) cos( )

sinh( ) cosh( )

1 1

( )

H g

H g

x r r a B x a B x

a B x a B x

x





     

      

     

 

2

11 12 2 1 2

1

21 22 12

a a D B r

r

a a

    



2

13 14 2 2 2

2

23 24 12

a a D B r

r

a a

    

