COARSE-MESH METHOD FOR TWO-DIMENSIONAL, MIXED-LATTICE DIFFUSION THEORY CALCULATIONS

COARSE-MESH METHOD FOR TWO-DIMENSIONAL, MIXED-LATTICE DIFFUSION THEORY CALCULATIONS

H.

L. Dodds, Jr.,

H.

C. Moneck, and D. E. H o s t e t l e r Savannah R i v e r Laboratory

E . I. du Pont de Nemours & Company Ai ken, South Carol i n a 29801

DP-MS-75-3

w- ^$ei?ba+

7 , --

in New Orleans, Louisiana, on

" " j

June 8-13,

1975

:-q

¹

1 I L

L I

MASTER

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COARSE-MESH METHOD FOR TWO-DIMENSIONAL, MIXED-LATTICE DIFFUSION THEORY CALCULATIONS b y

H. L. Dodds, Jr., H. C. Honeck, and D. E. H o s t e t l e r Savannah R i v e r L a b o r a t o r y

E. I. du Pont de Nemours 81 Co.

Aiken, South C a r o l i n a 29801

.

SUMMARY AND CONCLUSIONS ( S l i d e s 1 and 9 )

A coarse-mesh (CM) f i n i t e - d i f f e r e n c e method h a s been developed1- f o r two-dimensional d i f f u s i o n c a l c u l a t i o n s i n hexagonal geometry.

R e s u l t s o b t a i n e d with t h e CM method have been compared w i t h con- v e n t i o n a l f i n i t e - d i f f e r e n c e (CFD) r e s u l t s and with experimental

r e s u l t s . . The r e s u l t s o f t h i s work s u p p o r t . t h e f o l l o w i n g conclusions:

(1) The accuracy o f t h e ^CM method w i t h 1 p o i n t / h e x i s about t h e same a s t h e CFD method with 3 o r 6 p o i n t s / hex, depending on t h e problem b e i n g s o l v e d .

(2) The computing c o s t s , c o r e s t o r a g e , and CPU time

of

t h e CM method with 1 p o i n t / h e x a r e about t h e same as t h e CFB method w i t h 1 p o i n t / h e k .

DESCRIPTION OF THE CM METHOD

The e q u a t i o n t o be s o l v e d i s t h e two-dimensional multigroup d i f f u s i o n e q u a t i o n which i s p r e s e n t e d i n S l i d e 2 . The numerical approximation t h a t i s used f o r t h e f l u x , @, i n d e r i v i n g t h e

d i f f e r e n c e e q u a t i o n i s p r e s e n t e d i n S l i d e 3 . Consider 1 / 6 o f hex

The i n f o r m a t i o n c o n t a i n e d i n t h i s a r t i c l e was developed d u r i n g t h e c o u r s e o f work under C o n t r a e t No. AT(07-2)-1 w i t h ' t h e U . S.

Energy Research .and Development A d m i n i s t r a t i o n .

c e l l i., which i s shown a s an e q u i l a t e r a l t r i a n g l e i n S l i d e 3 . $ i s assumed t o v a r y l i n e a r l y from X = 0 t o X = h . / 2 . From X = hi/2 t o

1

X = H/2, $ i s assumed t o be c o n s t a n t where H i s t h e d i s t a n c e between mesh c e l l s , $i i s c a l l e d t h e p o i n t f l u x i n c e l l i, and $in i s t h e f l u x on t h e i n t e r f a c e between c e l l s i and n .

A d i f f e r e n t l i n e a r e x p r e s s i o n . i s used f o r each t r i a n g l e i n t h e hex because t h e a r e d i f f e r e n t . N i s t h e number of neighbors,and $. i s i n v a r i a n t w i t h r e s p e c t t o N . Using t h i s

1

approximation f o r t h e f l u x and t h e i n t e g r a l method t o d e r i v e t h e d i f f e r e n c e e q u a t i o n , t h e d i f f e r e n c e e q u a t i o n shown i n S l i d e 4 may b e o b t a i n e d .

I n S l i d e 4, .$n i s t h e p o i n t f l u x i n neighbor n . Cin i s t h e

, .

l e a k a g e c o e f f i c i e n t between c e l l i and n e i g h b o r i n g c e l l n . Note t h a t Cin depends on pi and pn where pi = hi/H, whereas t h e

f a c t o r f i depends o n l y on t h e p r o p e r t i e s o f c e l l i . If f . ⁼ 1

1

and p ' s = 1, t h i s e q u a t i o n r e d u c e s t o t h e c o n v e n t i o n a l f i n i t e - d i f f e r e n c e e q u a t i o n f o r mesh-centered mesh p o i n t s . Thus, t h e

d i f f e r e n c e e q u a t i o n i n S l i d e 4 has t h e same form as t h e conventional f i n i t e - d i f f e r e n c e e q u a t i o n and, hence, i s e a s i l y implemented i n e x i s t i n g codes w i t h o n l y minor code m o d i f i c a t i o n s . Qi and a l l r e a c t i o n r a t e s a r e computed u s i n g t h e e x p r e s s i o n f o r t h e average f l u x ,

$i,

which i s a l s o d e f i n e d i n S l i d e 4 .

The q u a n t i t y pi i s computed by a r u l e of thumb t h a t guaran- t e e s f . > O f o r numerical s t a b i l i t y . The r u l e o f thumb, which i s v e r y

1

crude b u t h a s proved s a t i s f a c t o r y t h u s f a r , i s d e s c r i b e d i n S l i d e s 5 and 6 .

The d i f f e r e n c e e q u a t i o n f o r a homogeneous r e a c t o r u s i n g 1 p o i n t / hex w i t h Qi = 0 i s shown i n S l i d e 5. T h i s e q u a t i o n may b e r e -

a r r a n g e d t o o b t a i n a t h i r d o r d e r polynomial i n p a s shown i n i

'

S l i d e 5. Note t h a t pi i s e q u a l t o t h e mesh s i z e measured i n d i f - f u s i o n l e n g t h s . The q u a n t i t y Gi i s determined a n a l y t i c a l l y from t h e s o l u t i o r l o f t h e model problem shown i n t h e n e x t s l i d e .

I n S l i d e 6 , t h e two-dimensional problem o f a s q u a r e , homo- geneous r e a c t o r w i t h a s o u r c e on two o f t h e f o u r s i d e s and z e r o f l u x on t h e o t h e r two s i d e s i s p r e s e n t e d . Gi, determined from t h e a n a l y t i c a l s o l u t i o n o f t h i s model problem, i s d e f i n e d i n t h e s l i d e . Note t h a t i f pi>O, t h e n Gi>O. Using t h i s e x p r e s s i o n f o r G t h e

i

'

t h i r d o r d e r polynomial i n S l i d e 5 i s s o l v e d f o r t h e a l g e b r a i c a l l y s m a l l e s t r o o t t h a t s a t i s f i e s 0 < p 4 1. S i n c e p p;, and G . a r e

i i

'

^{1 .}

a l l p o s i t i v e , f . i s a l s o p o s i t i v e , which i s t h e d e s i r e d n e c e s s a r y

1

c o n d i t i o n f o r s t a b i l i t y .

RESULTS

R e s u l t s o b t a i n e d with t h e CM method were compared w i t h r e s u l t s o b t a i n e d w i t h t h e CFn m'ethnd and w i t h e x p e r i m e n t a l r e s u l t s . A

schematic diagram of two s e c t o r s o f t h e r e a c t o r used i n t h e compari- sons i s p r e s e n t e d i n S l i d e 7. The r e a c t o r l a t t i c e , which i s a very heterogeneous (mixed) l a t t i c e w i t h a D20 r e f l e c t o r , c o n s i s t e d p r i - m a r i l y o f f u e l ( d r i v e r ) , t a r g e t , a n d c o n t r o l a s s e m b l i e s . The

moderator i s a l s o D20. A t o t a l o f 4 0 . d i f f e r e n t m a t e r i a l s a r e r e p r e s e n t e d . The r e a c t o r i s c r i t i c a l and t h e c o n t r o l a s s e m b l i e s a r e p o s i t i o n e d uniformly. The r e l a t i v e power produced i n each assembly was determined e x p e r i m e n t a l l y by flow *AT measurements.

-

3

-

R e s u l t s o f t h e c a l c u l a t i o n s and experimental r e s u l t s a r e shown i n S l i d e 8 .

R e s u l t s of two-group k c a l c u l a t i o n s u s i n g t h e CFD method e f f

with 1, 3, and 6 p o i n t s / h e x and t h e CM method with 1 p o i n t / h e x a r e p r e s e n t e d i n S l i d e ' 8 along w i t h e x p e r i m e n t a l r e s u l t s . The q u a n t i t i e s t h a t a r e p r e s e n t e d f o r each c a s e a r e :

(1) keff'

(2) R a t i o o f average t a r g e t power t o average d r i v e r power.

(3) Average power i n d r i v e r rods ( r a t i o o f c a l c u l a t e d v a l u e t o experimental v a l u e ) .

(4) CPU ( C e n t r a l

-

- P r o c e s s i n g

-

U n i t ) time.

(5) Core s t o r a g e r e q u i r e m e n t s .

A s may be observed i n S l i d e 8, t h e 1 p o i n t / h e x CM method and t h e 6 p o i n t s / h e x CFD method a r e . e s s e n t i a l l y ' e q u i v a l e n t with r e s p e c t t o accuracy, b u t t h e CM method r e q u i r e s l e s s CPU t i m e ( a f a c t o r of 2 . 4 l e s s ) a d l e s s c o r e s t o r a g e ( a f a c t o r o f 1 . 4 l e s s ) . I n f a c t , 1 p o i n t / h e x CM i s more a c c u r a t e t h a n 3 p o i n t s / l ~ e x CFD c a s e , whereas, 1 p o i n t / h e x CFD has c o n s i d e r a b l e e r r o r when compared with 6 p o i n t s / hex CFD imd w i t h t h e experimental r e s u l t s . 'I'he CPU

time

and core s t o r a g e requirements f o r t h e CM method u s i n g 1 p o i n t / h e x a r e e s s e r l t i a l l y t h e same a s r e q u i r e d by t h e CFD method u s i n g 1 p o i n t / hex.

S l i d e s 10 and 11 a r e a d d i t i o n a l s l i d e s t h a t w i l l n o t b e d i s c u s s e d u n l e s s r e l a t i v e q u e s t i o n s a r e asked by someone i n t h e audience.

REFERENCES

1. H. C . Honeck, J . E . S u i c h , J . C . J e n s e n , C . E . B a i l e y , and

J . W . S t e w a r t , "JOSHUA

-

^A Reactor P h y s i c s Computational System,"

Proceedings of the Conference on &he E f f e c t i v e Use o f Computers i n the Nuclear Industry,

Knoxville, Tennessee, A p r i l 21-23, 1969.

CONF-690401 (1969).

2 . S. Borresen, "A S i m p l i f i e d , Coarse-Mesh, Three Dimensional D i f f u s i o n Scheme f o r C a l c u l a t i n g t h e Gross Power D i s t r i b u t i o n i n a B o i l i n g Water Reactor

," NucZ. Sci.

Engr., 44, 37 (1971)

.

3. A. B i r k h o f e r , S. ~ a n ~ e n b u c h , and W. Werner, "Coarse-Mesh Method f o r Space-Time K i n e t i c s , "

T r m s .

Am.

'Nucl. Soc.

18,

153 (1974.).

4 . A . F. Henry, "Refinements i n Accuracy o f Coarse-Mesh F i n i t e - D i f f e r e n c e S o l u t i o n o f t h e Group-Diffusion Equations,"

Proceedings of

a

Seminar on Nwnerical Reactor CaZcuZations,

Vienna, A u s t r i a , J a n u a r y 17-21, 1972.

5. H . C. Honeck and J . W. S t e w a r t , "Simultaneous Line Over- Relaxation (SLOR) i n Hexagonal L a t t i c e s

," Proceedings of National Topical Meeting on

N e w

Developments i n Reactor Physics and Shielding,

Kiamesha Lake, New York, September

12-15, 1972, CONF-720901.

SUMMARY

Coarse-mesh (CM) f i n i te - d i f f e r e n c e method has been developed f o r two-dimensional d i f f u s i o n c a l c u l a t i o n s i n hexagonal geometry.

R e s u l t s obtained w i t h t h e CM method have been compared w i t h t h e conventional f i n i te - d i f f e r e n c e (CFD) method and w i t h experiment.

Accuracy o f CM method w i t h 1 p o i n t l h e x i s about t h e same as t h e CFD method w i t h 3 o r 6 p o i n t s l h e x , de- pending on t h e problem being solved.

e Computing c o s t s ( c o r e s t o r a g e and r u n times) o f CM method w i t h 1 p o i n t l h e x a r e about t h e same as t h e CFD method w i t h 1 p o i n t l h e x .

S L I D E 1

For Each Energy Group,'

D = D i f f u s i o n c o e f f i c i e n t

R = Removal cross s e c t i o n ( a b s o r p t i o n and o u t - s c a t t e r i ng)

Q = Source term due t o f i s s i o n s , i n - s c a t t e r i n g , and e x t e r n a l sources

@ = Neutron f l u x , @ = $(X,Y)

S L I D E 2

f o r n = 1;

...,

^{N .}

.

Hex C e l l i

S L I D E 3

where

and

qi i s computed using t h e a u x i l i a r y expression f o r the average f 1 ux,

Ti,

where

S L I D E 4

RULE OF THUMB

a 20, d i f f e r e n c e e q u a t i o n f o r 1 p o i n t / h e x w i t h Qi = 0

P i P i 2 H2Ri 6

o r , fi =

-

^{where =}

^{- ,}

^{Gi = 2}

Gi . D ~

3 2 36vi

o r , ^{p 1}

-

^3pi

- -

⁺

% =

⁰

i P i

S L I D E 5

$ = O

MATERIAL

SOURCE

-

CELL i

\L-

SOURCE

a S o l v e eq. 2 f o r t h e s m a l l e s t v a l u e o f pi t h a t s a t i s f i e s 0 < pi ( 1.

Pi P i 2

P i s i n c e fi =

-

G~ >

o

Gi

S L I D E 6

S L I D E 7

TWO-DIMENSIONAL kef CALCULATIONS

R a t i o o f Average

Target Power t o Average Power Go Core

Average D r i v e r In D r i v e r rod^,^ CPU ~ i m e ' , d

Method Points/Hexa k e f f b owe$ Cal c/Exp Sec Bytes

C FD 1 1.04614 0.258 1.081 39 41 8K

CFD 3 1.02106 0.285 , 1.021 49 . 476K

a . Reactor model c o n s i s t s o f 1183 assemblies on a 7-inch p i t c h .

b. Experimentally, k e f f = 1 and t h e r a t i o o f average t a r g e t power t o average d r i v e r power = 0.310.

C. R a t i o o f c a l c u l a t e d value t o experimental value.

d. I B M 360 Model 195.

S L I D E 8

CONCLUSIONS

1 . Accuracy o f CM method w i t h 1 p o i n t l h e x i s about t h e same as t h e CFD method w i t h 3 o r 6 points/hex, de- pending on t h e problem being solved.

2. Computing c o s t s ( c o r e storage and r u n t i m e s ) o f CM method w i t h 1 p o i n t l h e x a're about t h e same as t h e .CFD method w i t h 1. p o i n t / h e x .

S L I D E 9

RADIAL FLUX DISTRIBUTION

0 1 I I I I I I

0 14 28 42 56 70 8 4 ' . 9 8 112

$

^' Radius, inches

S L I D E 10

R A D I A L POWER DISTRIBUTION

Normolizotion

- _- 8

Q) >

.- ^a CFD lpt/hex

I

CFD 6pts/hex

0 I I b--be-b-so ^I

J

8

^{14 28 4 2 98 112}

Radius, inches

S L I D E 11