Application of the transfer matrix for a rectangular void in a 3-D neutron diffusion nodal model

Ann. nucl. Energy, Vol. 18, No. 4, pp. 177-181, 1991 Printed in G r e a t Britain. All rights reserved

0306-4549/91 $ 3 . 0 0 + 0 . 0 0 Copyright © 1991 P e r g a m o n Press plc

APPLICATION OF THE TRANSFER MATRIX FOR A RECTANGULAR VOID IN A 3-D NEUTRON DIFFUSION NODAL MODEL A. F. ROHACH and M. BOUSSOUFI Nuclear Engineering Program, Iowa State University, Ames, IA 50011, U.S.A.

(Received for publication 18 September 1990) Abstract--A transfer matrix void model was incorporated into a diffusion nodal model. The P~ flux distributions on the nodal model interfaces are transported across the void by the transfer matrix method. The matrix results from a combination of numerical and analytical solutions for radiation transport across the void. Sample problems were developed to verify the void model. Results demonstrated that the void model is applicable to the diffusion nodal model within the accuracy of the PI flux approximation.

INTRODUCTION

group diffusion equation is :

z),[O%(x,y,z)

A transport model using the transfer matrix method for a void has been developed (Boussoufi and Rohach, 1991). In this paper we will describe an application of the method in the diffusion nodal model. In this model the spatial neutron flux is expanded in a Legendre polynomnial series over the node (Rohach, 1987). From the expansion, the flux can be described over a node surface. Therefore, the P~ determination of the angular flux can also be described over the node surface using Fick's law. Hence, the source for the void transfer matrix model is established. This transfer matrix model, in turn, will determine the inbound partial currents into the nodes surrounding the void. Finally, sample problems for the nodal model have been developed to establish the validity of the void transfer matrix model.

l

(3x2

a%(x,y,z)] +

+Ztdp~(x,y,z)+

+

Oz2

l

Zfh+

vZ h C~h(X,y,z) = 0

h=l

f o r g = 1,2 . . . . . G,

(1)

where G

= Za+ " Zs~ = Z~+ y Z2"

Z~

h=l

The fourth-order Legendre polynomial expansion for the trial functions is : 4

~O°(u,v,w) = Z

4--' 4 ,--rn

~

, = 0 m=O

~

aTm.P,(u)P,.(vlP.(w)

(21

n=O

where (Fig. 1)"

THE DIFFUSION NODAL MODEL In the diffusion nodal model in three dimensions, the multigroup diffusion flux is approximated by a series of Legendre polynomials for the three spatial variables and these trial functions are minimized in a least square sense over the node (Rohach, 1987). From the minimization conditions several independent equations are determined for the unknown polynomial coefficients. The remaining coefficients are determined from the node interface conditions. These conditions are determined from applying continuity of averaged partial currents across the nodal interface for selected regions on the interfaces. Sufficient interface conditions are chosen such that all of the nodal polynomial coefficients can be determined. The multi-

x q

U=--~

y v

V --~ --~

W ~

z /~

The minimization conditions are (Rohach, 1987): Ag g alto n q- elm" = O,

(3)

with l + r n + n <. 2; all groups 9, where:

177 AHE 18~4-A

OY2

G # = C,,.n

y~ O~gha,rnn, h

(4)

h=l

a qh = ~

vZfh + Zsgh _ 6ghZf

5ah = Kronecker delta,

(5)

178

A . F . ROHACHand M. BOUSSOUH Z

I

I

I V

,

l

.

T +

..

,

I

+.t

,'J

I

-+ .......

i

IJ

~ - - r / .-~ ,._.77 --~ Fig. 2 Fig. 1

]~3(0) =

1

1

/~4(0)

&~oo = r/2 (3a~oo + lOa~,oo)+ ~ (3ag2o + lOa~+o)

~_

2 5 2 +tv2,2v2-3)-o,(~o,-3)],

I[IQ2/'~Q2

1 7 4 ~[02 (~02 - 5022 -q- 32 ) - - 0 1 ( 2 0 17 - - 54 0

15 3 3 6~oo = r/z a~oo+ v~a~2o+ ~ a"~o2,

(6)

Similar relations can be written for all five other interfaces for a node.

(7/ INCORPORATION

35 3 3 fi~oo = ~ a g o o + ~ a ~ 2 o + ~-2 a%2, 15 15 3 a~,0 = ~ a ~ , 0 + ~-a%o + ~ a q , 2 .

(8) (9)

The equations in the other two directions can be determined by incrementing the cyclic index for the last three equations. The interface conditions are determined by matching corresponding averaged partial currents across specified interface windows. Enough windows are used to satisfy the needed additional conditions for the node coefficients. The averaged partial currents can be written for the input and output conditions at an interface, for example, where x = r/as (Fig. 2) : ( + 1)t l=0

[

-

E m=O

/%(0) = 02--0,,

#, (0)

fl~,atm,,

n=O

/~m° = /~m(0)#,(~),

&(o)

MATRIX

= ~(0~-0,%,

= 1 [ o 2 ( o ~ - l ) - O , ( O , ~ - 1)1,

(10) (11) (12) (13) (14)

OF THE TRANSFER FOR THE

VOID

The void is incorporated into the nodal model by replacing one of the nodes by the void. In this case the Legendre polynomial coefficients for a node neighboring the void are used in the transfer matrix model to determine the partial currents entering the remaining neighboring nodes. Therefore, the void transfer matrix transfers neutrons from all nodes neighboring the void into partial currents that enter, in turn, all of the nodes neighboring the void. This transfer process serves the usual calculation for diffusion for a material node. The procedure is outlined in Boussoufi and Rohach (1991). SAMPLE

|-

2 3 I -~- ~ ) ] .

(16)

1

+ ~ ( 3 a ~ o 2 + lOa~o4),

J~+ = ~

(15)

PROBLEMS

Sample problems were developed to evaluate the application of the transfer matrix for an internal void in a diffusing medium. A model of 27 nodes, each with 20 cm on a side, was arranged in the configuration of a cube with three nodes on a side (Fig. 3). The center node in this model is the void. The configuration is chosen symmetrically for evaluation purposes. The A N L benchmark cross sections (Argonne Code Center, 1977) were chosen for the problem. Albedo

1

q

N

k*l

Zg

8l

xnl~

e

o,~7

9

t~

? QI oJ

>-

J

~ /'~ ~ ~ /\

/

e

N

m

i

~'n/~.l

oA 1 7~/oJ

180

A . F . ROHACHand M. BOUSSOUFI

k ',1

1]

"" tq t.

Fig. 7

boundary conditions (~ = 0.877) were chosen for all boundaries and both groups such that the resulting geometric configuration would be critical with the void in place. The flux distribution in the void was estimated from the values of the averaged fluxes at the surfaces of the void and the coefficients of the Legendre polynomials

in the void were fitted to these fluxes. Not all of the coefficients for a fourth-order polynomial expansion were determined, but a reasonable approximation to the function was established. In order to compare the void model to another model, we developed a model in which the void was replaced with a node with adjusted cross sections so

Fig. 8

Application of the transfer matrix for a rectangular void that the ko~ for the node is unity and the two group fluxes in the node match the spectrum on the surfaces of the surrounding nodes. Therefore, if kog is unity the central node would not have any net multiplying and the overall distribution should approximately match that of the void node. Of course, the P~ distributions at the node interfaces of the central node will not be exactly the same as those of the void node. However, calculations confirmed that the scalar flux distributions and the system eigenvalues were the very nearly equal. The scalar flux distribution for the central void is shown in Fig. 4. The 2-D slice for the display is at position O on Fig. 3. The deviations at the corner of the void display are the results of the fitting errors due to a limited number of polynomial coefficients. The results for the model with a material region in the central node gave a much smoother distribution (Fig. 5). A second sample problem was developed in which the void node was placed at the overall system boundary (Fig. 6). In this case, the system boundary conditions were chosen as symmetric at three interfaces and as zero inbound partial currents at the remaining three, one of which contained the void. The results for the scalar flux distributions are shown in Figs 7 and 8. No polynomial fitting was done for the flux in the void since the nodal model does not calculate the flux at the outer boundary of the void. Sample problems were also calculated for void models with uneven dimensions. The results were

181

similar to those for the equally dimensioned models. However, in these cases nine transfer matrices must be generated for the void. Results from these calculations are not presented here.

CONCLUSIONS The transfer matrix void model was incorporated into the Legendre polynomial nodal model. The Pt flux distributions on the nodel surfaces interfacing with the void were transported across the void by the transfer matrix. This transfer resulted in windowaveraged partial currents entering the material nodes adjacent to the void. Sample calculations were performed for both uniform node sizes and nonuniform node sizes. In the case of uniform node sizes only three transfer matrices are necessary; however, nine transfer matrices are required for the nonuniform node sizes. Results for a void located in a multiplying medium compared favorably to a problem in which the void was replaced by a multiplying node with an infinite multiplication factor of unity.

REFERENCES

Argonne Code Center (1977) Report ANL-7416 (Suppl. 2). Boussoufi M. and Rohach A. F. (1991) Ann. nucl. Energy 18, 5.

Rohach A. F. (1987) Ann nucl. Energy 14, 653.