MATHEMATICAL MODEL FOR CALCULATING FISSION PRODUCTS CONCENTRATION AND ENERGY RELEASE IN CIRCULATING NUCLEAR FUEL UDC 621.039.55

L . I. M e d v e d o v s k i i D E . S. S t a r i z n y i , V. A . C h e r k a s h i n , V. A . R u d o i , a n d K . I. S t e p a n o v a

The solution of c e r t a i n problems associated with the design and application of u r a n i u m radioactive loops r e q u i r e s calculation of fis s ion products concentration and of the three-dimensional distribution of 7 - a n d flr a d i a t i o n energy r e l e a s e and its s p e c t r a l composition in Circulating nuclear fuel. Methods and r e s u l t s for c a l culating t h e s e c h a r a c t e r i s t i c s in nuclear r e a c t o r s with noncirculating fuel have been published in s e v e r a l w o r k s [1-3]. Design works, as a r u l e , do not take into account i s o m e r i c transitions, d i r e c t generation of nuclides in chains, and burnup of active nuclides. We have designed a mathematical model of the accumulation of fission products in circulating nuclear fuel and devised a method for calculating the radiation c h a r a c t e r i s t i c s of u r a nium r a d i o a c t i v e loops (distribution of the power of ~/and ~ radiation and their s p e c t r a l composition in the u r a n i u m r a d i o a c t i v e channel). 2his mathematical model can also be employed for calculating the radiation c h a r a c t e r i s t i c s of fission products in pulsed r e a c t o r s , in which burnup of fission products is especially high, and in nuclear r e a c t o r s with fixed fuel. Consider all possible transmutations of fission product nuclei. A nucleus (including isomers) can be gene r a t e d d i r e c t l y in fission, as a r e s u l t of neutron capture by an isotope of lower m a s s , as a r e s u l t of decay of its

I

j-z

/

2

~5

\ 8

7

6

Fig. 1. Initial (a) and linearized (b) chains (O, i s o m e r ; O, ground state): a: 1) (1-~,~_q-fl;_~j)X,~_ljAi_lj; 2) ~_lj+tX~_lj+lAl_l]+l; 3) a~_jde@Al~_j+2;4) oi-lj+z@Ai_lj+2; 5) (1--c~ij+l)~ij+lAij+l; 6) czij+ 1 ~

Xiy~Aij+1, 7) fl~j)~jA~1, 8) yD_1~f@, b. i) ~i_ti_2Ai_tj_2, 2) ~i-lj-t" Ai_j_i; 3) ~li_jAi_j; 4) cri_lj+t @Ai_ij+i; 5) Yij_l~f@.

Translated f r o m Atomnaya l~aergiya, Vol. 47, No. 3, pp. 184-186, ~ p t e m b e r , 1979. Original a r t i c l e submitred July 17, 1978; r e v i s i o n submitted January 25, 1979.

0038-531X/79/4703-0737507.50

9 1980 Plenum Publishing Corporation

737

104

P~ 0

0

tO4 lO4

2 tO-~

i0~

10z

10~

i0"~

10 5

Time after instataneous fission

Fig. 2. C o m p a r i s o n of r e s u l t s obtained by the p r o p o s e d method with e x p e r i m e n t a l data: [~ data [2], (3) data [4], V} data [5], O} data [6], ) our r e s u l t s . i s o m e r (if the nucleus is not an i s o m e r } , and as a r e s u l t of d e c a y of an i s o b a r and of an i s o m e r of i s o b a r with s m a l l e r nuclear c h a r g e s . The g e n e r a t e d nucleus can c a p t u r e a neutron~ p a s s into a ground s t a t e (if the nucleus is a n i s o m e r } , into a n i s o b a r next in the chain, and into a n i s o m e r of the i s o b a r next in the chain. All t h e s e p r o c e s s e s a r e r e f l e c t e d in the following s y s t e m of d i f f e r e n t i a l equations : [ dAtI/dt=ytJ~q~kuAu~(rllq)AU+cql-xht1-1AtI-~ "4-

i aA~flat=y~i~t~--X'tcA'ii--~'~r

i-

w h e r e A~ is the c o n c e n t r a t i o n of the j-~h nucleus in the i-th chain; Yij, p r o b a b i l i t y of output in fission; Zf, n u c l e a r reel f i s s i o n m a c r o c r o s s section; # , t h e r m a l neutron flux density; hij, d e c a y constant; oij , neutron c a p t u r e m i c r o c r o s s section; G and ~, p r o b a b i l i t y coefficients (Fig. l a ) ; quantities m a r k e d with s t r o k e s c o r r e spond to i s o m e r s . To s i m p l i f y the solution let us a r r a n g e i s o m e r s with ground s t a t e s in one chain n u m e r a t i n g t h e m so that a nucleus with a higher n u m b e r cannot a t r n into a nucleus with a lower n u m b e r ; t r a n s i t i o n s take p l a c e between nuclei with n u m b e r s differing by not m o r e than t h r e e . The c o n c e n t r a t i o n Aij of isotope (i, j) in the m i x t u r e c a n then be d e s c r i b e d by a single equation (see Fig. lb) : dAu/dt~ Y u ~ A u A u - } - p I _ u + A t _ u + ~,su_sAu_s q- Z~u-~Au-2 + ~,xIj_xAII_G

&j (0) = Bu,

(2)

whe r e Y U = yUEI~; pU=~U~; AU-~-~U-E-pIj; 3 ~'kU-k ~---~'iJ-k~IJ-kJ; ~ ~'iJj+k = t,

H e r e Gijj+k is the p r o b a b i l i t y of t r a n s i t i o n of nucleus (i, j) into the nucleus (i, j + k). Since nuclear fuel in a u r a n i u m r a d i o a c t i v e loop e n t e r s the neutron field p e r i o d i c a l l y , for each e l e m e n t a r y v o l u m e we h a v e if 0~
738

J " | I if

nAu(t)~

[~

n

kl

-

-

k==t I ~ t O~t
t Ci~exp(--~i~t)

if tp~t
where j.f

0 tj + ~siJ-sA~j-3 0.1 -]Aij0 = (Yi J + Pt-IlA~-

0

.

0

-]- ~2iJ-2Aij - 2-~ ~ttJ-1Aij

i-i

~

k=t

I~1

. 9

- t)/AtJ '

J-t nA~:;

~A hl IEt

(k, I)sA(t, 1): "~ij=Wt-lJ ~ - t j ~ s~j-s ij-a~ hl nAhZ -{-~'2~Y-~n Aij-2"+L1~J-x ii-l)/(At$~A/~/) ., i 5 nAhl exp(--Akztp); = o_ o

%-A j+y E h~l

(5)

/=t j-t

h ic- -~n -O . _ _c u

~, ~c~; l~t

(t
~

nC~: exp (--~,tttt,,).

l=i

The specific power of g a m m a and beta r a d i a t i o n of f i s s i o n products a r r a n g e d into r chains of length s, each amounts a t the instant t in the n - ~ c y c l e to r

$

i P+ 7, 7, ,~ : w fO=

X

"

x

k~l/=1 oxp

(--Agtl)

"Au (0 auEu = I '. (inside co~)i

L X exp(--Z,klt) (outside). Here

I"=

i==lj = t

~.Eu;

~ukz= ~, ~,. ~

9

AiJ~:JEo'

i=k j ~ l

"

nVht= ~ nC~j~MEM' j=l

where Eij is the 7 ({~) radiation energy of isotope (i, j). The r e l i a b i l i t y of the results thus obtained depends to a large extent on the fullness and accuracy of i n i t i a l data. The l a c k of c o m p r e h e n s i v e data about short-lived f i s s i o n p r o d u c t s c o n s i d e r a b l y i m p a i r s the a c c u r a c y of calculations for s h o r t c i r c u l a t i o n p e r i o d s (on the o r d e r of minutes). Using the above p r o c e d u r e we h a v e computed the dependence of 7 r a d i a t i o n power of the products of instantaneous f i s s i o n on the holding t i m e . The r e s u l t s obtained a r e in good a g r e e m e n t with known e x p e r i m e n t a l data [2, 4-6] f o r holding t i m e s exceeding ~ 1000 sec (Fig. 2). LITERATURE 1.

2, 3. 4.

5. 6.

CITED

J. P e r k i n s and R. King, Nucl. Sci. Eng., 3_, 726 (1958). M~ R a m a k r i s h n a and A. Ganguly, in: l>coc. 3rd Int. Geneva Conf., No. 28/P~92 (India) (1964). N. G. Gusev et al., Radiation C h a r a c t e r i s t i c s of F i s s i o n Products (Handbook) [in Russian], Atomizdat, Moscow (1974). F. Meienscheir~ in: P r o c . 2nd Geneva Conf., Vol. 2, Atomizdat, Moscow (1959), p. 297. L. Bunnej and D. Sam, Nucl. Sci. Eng., 39, 81 (1970). P. F i s h e r and L. Engle, Phys. Rev., 134, B796 (1964).

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