Annals of Nuclear Energy, Vol. 2, pp. 11 to 16o Pergamon Press 1975. Printed in Northern Ireland

SOLUTION OF GROUP DIFFUSION EQUATION FOR X - Y GEOMETRY BY FINITE FOURIER TRANSFORMATION

KEL.qUKE KOBAYASHI Department of Nuclear Engineering, Kyoto University, Kyoto, Japan

(Received 13 June 1974; in revlsed form 9 September 1974)

Abstract--A new difference equation to the two dimensional diffusion equation for x -y geometry is derived by using the finite Fourier transformation. This difference equation has a form of a coupled equation of the 3 point difference equations for each coordinate, and can be easily solved by the iterative method of the alternative direction implicit method. Group diffusion equations are solved using this difference equation and sample calculations show that accurate results can be obtained with less mesh points than the usual 5 points difference equation.

1. I N T R O D U C T I O N

For one dimensional diffusion equations, exact 3 point difference equations were derived by Kobay- ashi et al. (1967) and they were used to solve group diffusion equations using source iteration method. Since the source term contains the integration over source distribution in those difference equations, appropriate form for the source distribution must be assumed to perform integration analytically. By assuming the source distribution in a quadratic form, it was shown that accurate results could be obtained with fewer mesh point than the usual difference equations.

Aoki and Tsuiki (1973) also used similar difference equations to solve one dimensional group diffusion equations and showed that accurate results could be obtained with less mesh point than the usual difference equations, even if the source term is assumed to be linear over a mesh interval.

Recently, monoenergetic diffusion equation of two dimension is shown to be solved by the finite Fourier transformation by Kobayashi et al. (1974). In this method, applying Fourier transformation to the two dimensional diffusion equation for each finite region of constant cross sections, integral equations for the flux and current at the boundary are derived. Ex- panding the boundary values into Fourier series and choosing appropriate discrete values for the trans- formation variable, the integral equations are re- duced to a system of linear equations for the Fourier coefficients of the boundary values. Using the boundary condition that the flux and current at the material interface are continuous and a given bound- ary condition for the outermost boundary, they can be solved and all boundary values are determined.

If the region in which Fourier transformation is applied is chosen to be small so that the variation

of flux and current along the boundary is not signifi- cant, it may be enough to use first few terms of Fourier series for the boundary values.

In this paper, taking regions so small that use is made of only the first term of Fourier series for boundary values and assuming the source is piece- wise linear in each mesh box, an approximate difference equation is derived for x - y geometry. The difference equations thus obtained take a form of a coupled equation of 3 point difference equations for each coordinate and can be easily solved by the iterative method of the alternative direction im- plicit method. If the flux is constant with respect to one coordinate, say, y-coordinate, the difference equation is reduced to the similar form of the exact one dimensional 3 point difference equation for x- coordinate as is obtained using Green's function by Kobayashi et al. (1967) or Aoki et al. (1973).

Using this difference equation, group diffusion equation is solved and comparison is made with the usual 5 point difference equation. In Section 2, out- line of the finite Fourier transformation method is given briefly and the difference equation is derived and in Section 3, numerical examples are given and comparison with the usual difference method is made.

2. D E R I V A T I O N O F D I F F E R E N C E E Q U A T I O N

Group diffusion equation is

VSgg(r) - ,%z~%(r) = - ag ( r ) , where

1 as(r) = -A- Sg(r), L,g

11

(z)

(2a) a--1 G

~g a~zvZ1o. ~,.(r). SAr) = ~ r ' . ( z~g ' l~ , ( r ) + T

(2b)

12 K.mSUKrZ

where 9o(r), x o and k are the flux, the reciprocal of the diffusion length of g-th group and criticality factor respectively. Other notations have usual meaning. For simplicity, we drop the suffix g here- after.

Let V be the region in which cross sections are constant. Multiplying equation (1) by e -~k'~ and integrating over/I , we obtain

B(k) + ~(k) 93(k) = k~ + ~. , (3)

where

k . r = k ~ + kvy , k 2 = kx 2 + kv 2, (4)

f~ dr e~k"q~(r), (5) ¢(k)

fo dr eqk'~Q(r) (6)

B(k) =/~(k,, k~) = Io ds e-~k"(n • V~(r)

+ in .kg(r) ) . (7)

The integral in equation (7) is the surface integral over surface S of region V and n is the unit out- word vector normal to surface S.

Since ~(k) should not have singular point for any finite values ofkx and k¢ by definition (5), the follow- ing equation must hold from equation (3);

BCk+, kv) + Q(kx, kv) = 0, (8a)

when k~ and k u satisfy

k : / + kv 2 + ,~ = 0. (8b)

Equation (8a) is the Fredholm integral equation of the first kind for the flux and its normal derivative at the boundary, and boundary values may be determined by solving equation (8a).

From equation (1), the average flux in region V can he obtained using the boundary values as,

93 = ~. drg(r) = ~ dsn- V~(r)

+ f drQ(r)). (9)

Now, let V be a rectangular region of x ~ [x 0, x d and y ~ [Yo, Yx], and expand flux and current at boundary into Fourier series,

[ 2~rm Y') = [ cos ( x - • )

2rrm ) + ~s~(Y~) sin ~ (x - ~) , i = 0, 1, (10a)

KOBAYASHI

q~(x~, y) = ~.(x~) cos ~ (y - ~)

2rrn ) + ~ (x~ ) sin ~ (y -- p) , i = O , 1, (lOb)

~o( 2,~m J.(x~ yj) = = ~',~(y~) cos ~ (x - ~)

• 21rm ) + J,~@~) sm ~ (x - .~) , j = 0, 1, (1 la)

~ - o ~ 2rrn Sx(x~, y) = . .(x3 cos ~ O' - Y)

--2rrn ) +Jsn(x,)sin--~-, (y - P ) , i = 0 , 1, ( l lb)

where ff = ½(xl + Xo), P = ½(Yl + Yo), l~ = x I - x o, lu =) '1 - Yo and Ju(x , y~) = -D(Oqo(x , y)/ay)l,=,, etc.

Substituting equations (10) and (11) into equation (Sa) and setting kx = 2rrm/lx or k v = 2rrn[l v, a system of linear algebraic equations for the Fourier coefficients of the boundary values is obtained. Applying this procedure to each region of constant cross section, a system of linear equations for Fourier coefficients of fluxes and currents at the material boundaries is derived. It can be solved using the boundary conditions that the flux and the current are continuous across the material interface and a given boundary condition for the outermost bound- ary.

If region Vis taken to be small such that the varia- tion of the flux and current on surface S is small, the 0-th term of Fourier series may give accurate results. Substituting equations (10) and (11) with M = N = 0 into equation (8a) and setting k~ = 0 or kv = 0, we obtain

:q: ~I u e-+-k~j2 q~o(X x) ± ~l v e~k~,/2 9~o(x o)

lv e~./~j,o(XO + lv - -6 -3 e~m/~s~°(x°) 2l~ KI~

+ ~--~sinh~-Sx ± = 0 , (12)

:F Klx e:~t, /2 9~o(yl) ± Klx eT'a*/2 q%0, o)

lx t 2 l~ - - ~ s l ~ / 2 r . . -- -~ e ±'C v/ Jeo(Yl) + D e Jco(Yo)

2l. % + ~)--~xsinh-~--Su± = O, (13)

where ( 2 t~ h Kt~ ~Tx) s~ S~ + = S 0 + y \ c o t 2

1 1~ (J,o(Yl) - J,o(Yo)) 04)

Solution of group diffusion equation for x - y geometry

and

S ~ = S O 5: ~ \cot 2

1 - ~ (.roo(XO - S,o(Xo)). (15)

In the above equations, double signs should be taken in the same order. In deriving equations (12) and (13), the source term is assumed to be linear in v;

S ( x , y ) = So + (x - $) s , + o , - y ) S ~. (16)

By substituting equation (11) into equation (9), the average flux in region V is obtained;

1 S - ¢ = / ' ~ = T~ (S0oCx0

' 1 -- Jco(Xo)) - ~ (Jeo(Yx) - Jto(Yo)) • (17)

Taking mesh points as shown in Fig. 1 and using equations (12) and (13) for each mesh box, we can derive a difference equation. Now, we will derive a difference equation for the ease where the following boundary condition is used;

J , (xx , y ) = O at x = x t = 0 , (18a)

and

~0(xi+ ~, y) = 0 at the outermost boundary.

(18b)

Eliminating Jeo(Xl) in equation (12), we obtain

Yeo(Xi) = bxi'~%o(Xi) - cxi~%o(X~+a) - dxi', (19)

J+.l

J

I I 1 1 i I

- I I

I I I I

I I t it'll 2 i i + l

I 1 I I+I

Fig. 1. Mesh points for x - y coordinates.

13

j + l

J

where a ~ = c~i-x (20e)

bxl = b ' i_ l + b~i' (20t")

dxi = d~t_l + dzi' (20g)

Similarly, we can obtain the difference equation for y coordinate from equation (13). It takes the same form as equations (22) and (20) in which the suffixes i and x are replaced b y j and y respectively, if the same boundary conditions as equations (18a) and (18b) are used for y coordinate.

I f the flux is constant along y coordinate direction, difference equation (22) is reduced to the similar form as those of one dimensional case obtained by Kobayashi et al. (1967) or Aoki et aL (1973) and this one dimensional difference equation is exact in the case of monoenergetic problem in which the external source is given to be piecewise linear.

where (xo)~ = x i , (xl) ~ = xi+l

bxi' = (DK cot h Klx)i (20a)

C~4 ~ i

, ,,:IS I Jco(Yo))] dxi = (~ tan h - f L So - g (J~o(Y,) -

Similarly, eliminating J¢o(Xo) in equation (12), we obtain

fco(Xi+t) = cztq~©o(xi) -- b~,'q~eo(x~+t) + dttt", (21)

where

[1 ,d7 _ j,o(Yo))-Ij a, ,, ~ = t a n h y L S ° 1 - ?~ (Joo(yO

lz [ Klz

Using equation (19) and boundary condition (18a) at the first mesh point, we obtain

-b~i'~co(Xx) + oxiq~co(x~) = --d~t'. (22a)

Using equation (21) for the (i - 1)-th region and equation (19) for the i-th region, and using the boundary condition that the flux and current are continuous at x = xi , we obtain

a~%o(x~_O - b~O~o(X~) + ¢~i~o(Xt+x) = - d ~ ,

(22b) for i = 2, 3 . . . . . I,

14 KEI~KE KOBAYASHI

I f source S(r) of equation (16) is given and the current to y coordinate direction in equations (20c) and (20d) is assumed, we can obtain all coefficients (20) and can solve equation (22) as in one dimen- sional case. Using the flux thus obtained, the current to x coordinate direction can be obtained from equations (19) and (21). I f we use the current to x coordinate direction just obtained, the difference equation of y coordinate can be solved. Thus, solving the coupled system of linear equations of x and y coordinates alternatively, we obtain the solu- tion of equations (12) and (13).

In order to solve group diffusion equation by the source iteration method, it is necessary to calculate the flux distribution inside region V to compute the source term of equations (2) and (16) for next group and for next outer iteration. The flux inside V can be obtained by solving the similar equation as equa- tion (8) that is obtained by choosing appropriate smaller region V ' in V and by repeating the same procedure to V'. Here, we assume that the flux in- side V is expressed by the linear function of x and

Y; ~ ( x , y ) = ~o + (x - Yc) ~% + (y - y ) ~ u i n V . (23)

Integrating equation (23) along boundaries at x = xi or y = y~., we obtain

1 cPx = 7 (C&o(X*) - ~,o(Xo), (24a)

1 ~Pu = ~ (e&oY~) - CPco(Yo)), (24b)

~o = ½(~°o(x~) + ~oo(xo)) or (25)

~o = ½(~oo(y3 + ~oo(yo)).

9o can be also obtained by substituting equation (23) into equation (17);

~o = D K-----~ So -- ~ (4o(X~) -- Jco(Xo)

t~ (4o(y~) - 4o(yo)) • (26)

Using equation (23) in equation (2b), we obtain the source term of equation (16) for next iteration.

3. NUMERICAL RESULTS

Sample calculations were performed for U-H~O thermal reactor and for ~ U - T h fast reactor, and comparisons were made with the usual finite differ- ence method to study the numerical accuracy of the present method.

Two group calculations are performed for U - H 2 0 thermal reactor using the group constant and the reactor configuration given by Garabedian and Thomas (1962), which is also reproduced in Table 1 and in Fig. 3 by Kobayashi (1968). The reactor has a square core surrounded by a square water re- flector. Half of the core dimension is 15 cm and the thickness of the water reflector is 20 cm on all sides. Calculations are also performed for U-HzO thermal reactor using the 3 group constants given in Table 1 by Kobayashi et al. (1967). In this case, the reactor has a square core of half core dimension of 25 cm which is surrounded by a water reflector of 25 cm thicknes~ on all sides. Fast reactor studied has a square core of half core dimension of 50 cm loaded by 2aaU and surrounded by a thorium blanket of 50 cm thickness on all sides. Four group con- stants are taken from Table 2 of Kobayashi et aL (1967).

Table 1. Criticality factor and average flux in the core and reflector for a thermal reactor by two groups

Number Criticality Average flux* Number C.P.U. of factor Core Reflector of time

meshes k ~1 ~ ~x ¢~ iteration (sec)

Present Method 4 x 4 1.3415 4.705 1.301 0"4646 0.77•7 18 0.3 8 × 8 1.3445 4-703 1.301 0"4614 0"7601 25 1-1

16 × 16 1.3442 4.701 1-301 0"4621 0.7601 23 4 32 × 32 1.3441 4.700 1-301 0"4625 0.7603 28 16 64 x 64 1.3440 4.700 1.301 0.4626 0.7604 34 75

EXTERMINATQR-2 4 × 4 1-3862 3.834 1-302 0.5799 0.5781 12 0.6 8 × 8 1.3615 4.399 1.302 0.5023 0.6940 16 1-1

16 × 16 1.3492 4-613 1.302 0.4741 0-7411 32 6 32 × 32 1.3454 4.677 1-302 0.4656 0.7554 60 35

* Average flux is multiplied by l0 S.

Solution of group diffusion equation for x-y geometry 15

Table 2. Criticality factor and average flux in the core and reflector for a thermal reactor by three groups

Number Criticality Average flux* Number C.P.U. of factor Core Reflector of time

meshes k ~1 ~2 ~3 ~ ~ ~g3 iteration (sec)

Present method 4 x 4 1.1299 21.49 9-551 8'484 1.157 0 . 6 2 5 9 0.3944 19 0"5 8 x 8 1"1399 21"47 9-567 8"483 1.107 0 - 5 9 4 3 0.3725 22 1"4

16 x 16 1.1410 21.46 9.568 8"483 1"102 0 . 5 9 1 4 0.3702 22 5 32 × 32 1.1409 21"46 9.567 8.483 1.102 0 - 5 9 1 6 0"3702 24 21 64 x 64 1"1409 21"46 9.566 8.483 1.103 0 . 5 9 1 7 0.3703 25 86

EXTERMINATOR-2 4 x 4 1"1912 16.35 7"425 8.697 2.040 0 . 9 8 3 8 0.3198 17 0.8 8 x 8 1.1617 19.53 8.776 8.562 1.451 0 " 7 3 3 7 0.3498 25 2"1

16 × 16 1.1473 20.87 9.328 8"507 1.210 0 " 6 3 4 4 0.3640 74 18 32 × 32 1"1326 21"30 9-502 8"490 1.132 0 - 6 0 3 4 0.3687 118 100 64 x 64 1.1413 21-42 9.550 8"485 1.I10 0 . 5 9 4 8 0.3699 96 383

* Average flux is multiplied by 103.

Iteration calculation is started assuming a con- stant fission source and zero currents in the whole reactor which are used in equations (20c) and (20d), and use is made of the alternative direction iteration method described in the preceding section. For every outer iteration, the inner iteration is performed once for each energy group. The iteration is termin- ated, when the relative change of flux becomes less than 10-4. Any acceleration method is not used for the inner and outer iterations. Computational results* for the criticality factor and average fluxes in the core and the reflector or in the blanket are given in Tables 1, 2 and 3. Flux in the Table is normalized such that

°L ~, drvZla(r)q~a(r) = 1. (27) ¢~1 eactor

* Computations are performed by the FACOM 230-75 of the Data Processing Center of Kyoto University.

In the present method, equation (26) is used to calculate ~%, because equations (25) are found to give less accuracy.

For comparison, the results of the usual difference equations obtained by the EXTERMINATOR-2 code (Fowler, Tobias and Vondy, 1967), are also shown in the Tables.

In the computation by the EXTERMINATOR-2 code, computation is started with the initial guess of fiat flux and is terminated by the error criterion of 10 -4 for the relative change of flux and criticality factor. The option of "exponential-B" is used for the over relaxation coefficient of acceleration calcu- lation. In both methods, the criticality factor con- verges at least with six significant figules.

As seen in the Tables, the criticality factor and the average flux by the present method converge to the limiting value more rapidly than those by the usual difference method, as the number of mesh points increases. For example, the results by the present

Table 3. Criticality factor and average flux in the core and blanket for a fast reactor by four groups

Number Criticality Average flux* Number C.P.U. of factor Core Blanket of time

meshes k ~1 4g~ ~3 ~, ,~x ~2 ~3 ~4 iteration (see)

Present method 4 × 4 1 .0491 5"155 2 0 . 8 6 16 .88 5 .581 0"1903 2-278 2.666 1-218 15 0"5 8 x 8 1"0460 5"163 20.86 16 .88 5 .580 0.1924 2"297 2 - 676 1-219 33 3.1

16 x 16 1 .0452 5'164 20.86 16 .87 5 .580 0.1932 2.304 2"683 1.221 71 24 32 x 32 1"0450 5"165 20 -86 16 .87 5 .580 0.1935 2"306 2 -685 1.222 205 262

EXTERMINATOR-2 4 X 4 1"0190 4"794 2 0 . 7 5 17 .15 5 .815 0.3063 2"522 2 . 7 3 0 1.176 12 0.8 8 X 8 1"0369 5 - 0 3 7 20.84 16 .95 5 .645 0"2314 2"373 2 -705 1.212 18 2"1

16 x 16 1 .0428 5"128 20.86 16"89 5 .596 0"2041 2-323 2 .691 1.220 41 15 32 x 32 1 .0444 5"155 2 0 . 8 6 16 .88 5 .584 0.1963 2 .311 2"687 1.221 84 100 64 x 64 1 .0448 5"162 20"86 16 .87 5 .581 0'1942 2 .308 2 . 6 8 6 1.222 85 459

* Average flux is multiplied by 10 a.

16 K~ISUKE KOBAYASHI

method for mesh points of 8 × 8 of Table 1 have higher accuracy than those by EXTERMINATOR-2 for mesh points of 32 x 32, and the results by the present method for mesh points of 16 x 16 of Tables 2 and 3 have similar accuracy to those by EXTER- MINATOR-2 for mesh points of 64 × 64.

Although C.P.U. (Central Processing Uni0 time shown in the Tables is only rough estimate on account of its dependence on the coding method and on the amount of the output printing, we can expect that the present method economize the C.P.U. time and the volume of core memories.

For the case of fast reactor, a large number of iterations is needed for the present method. If some effective acceleration methods can be developed for the present method, computation time may be farther reduced.

An extension of the present method to the three dimensional geometries of x, y and z coordinates may be straight forward and this method can be further applied to other geometries of two and three dimensions. Some applications will be reported later somewhere.

REFERENCES

Aoki K. and Tsuiki M. (1973) J. nucl. Sci. Technol. 10 275.

Fowler T. B., Tobias M. L. and Vondy D. R. (1967) ORNL-4078.

Garabedian H. L. and Thomas D. H. (1962) Nucl. Sci. Engng 14, 266.

Kobayashi K. and Nishihara H. (1967) Nucl. ScL Engng 28, 93.

Kobayashi K. (1968) Nucl. Sci. Engng 31, 91. Kobayashi K., Ohtani N. and Jung J. (1974) Nucl. Sci.

Engng. 55, 320.