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A n a l y s i s o f t h e d a t a o b t a i n e d f o r d i f f e r e n t d e p t h s o f f u e l b u r n u p s h o w t h e e x c e l l e n t a g r e e m e n t b e t w e e nt h e m e a s u r e m e n t r e s u l t s o f t h e b u r n u p a c c o r d i n g to d i f f e r e n t is o t o p e s . T h i s g i v e s a b a s i s f o r s u p p o s i n g th a tt h e c e s i u m m i g r a t i o n e f f e c t an d th e r a d i a t i v e c a p t u r e o f n e u t r o n s b y t h e f i s s i o n p r o d u c t s d o n o t d i s t o r t t h er e s u l t s o f t h e b u r n u p d e t e r m i n a t io n b y c e s i u m . C o m p a r i s o n o f t h e r e s u l t s o f t h e b u r n u p d e t e r m i n a t io n o b -t a i n e d b y t h e d i f f e r e n t m e t h o d s ( s e e T a b l e 2) s h o w s t h e i r e x c e l l e n t a g r e e m e n t .

W h e n i n v e s t i g a t i n g th e b u il d u p o f i s o t o p e s o f t h e t r a n s u r a n i u m e l e m e n t s i n i r r a d i a t e d f u e l , t h e b u r n u pc a n b e d e t e r m i n e d b y th e m e t h o d o f h e a v y a t o m s w i t h an e r r o r o f 7% . T h e b u r n u p c a n b e d e t e r m i n e d m o r ea c c u r a t e l y ( w it h an e r r o r o f 1 t o 3%) b y m e a s u r i n g t h e c o n t e n t i n t h e f u e l o f a n u m b e r o f f i s s i o n p r o d u c t s .

L I T E R A T U R E C I T E Di . R . W e b s t e r , i n : P r o g r e s s i n M a s s S p e c t r o s c o p y [ R u s s i a n t r a n s l a t i o n ] , I L , M o s c o w ( 1 9 6 3 ) , p . 1 0 7 .2 . N e u t r o n C r o s s S e c t i o n s , B N L - 3 2 5 , 2 r i d e d i t i o n , S u p p l . 2 , V o l . 3 ( 1 9 6 5 ) .3 . V . D . S i d e r e n k o a n d E . D . B e l y a e v a , P r e p r i n t I A E - 1 9 / 8 9 5 [ i n R u s s i a n ] , M o s c o w ( 1 9 6 6 ) .4 . G . A . S t o l y a r o v e t a l . , S e s s i o n o f t h e A c a d e m y o f S c i e n c e s o f t h e U S S R o n t h e P e a c e f u l A p p l i c a t i o n o f

N u c l e a r E n e r g y [ i n R u s s i a n ] ; I z d . A k a d ; N a u k S S S R , M o s c o w ( 1 9 5 5 ) , p . 2 1 7 .5 . A . I . L e i p u n s k i i e t a l . , i n: P r o c e e d i n g s o f t h e T h i r d G e n e v a C o n f e r e n c e . R e p o r t o f S o v i e t S c i e n t i s t s l

V o l . 2 [ i n R u s s i a n ] , A t o m i z d a t , M o s c o w ( 1 9 5 9 ) , p . 3 7 7 .6 . G . Y a . R u m y a n t s e v , C a l c u l a t i o n o f a T h e r m a l N e u t r o n N u c l e a r R e a c t o r [ i n R u s s i a n ] , A t o m i z d a t , M o s c o w

( 1 9 6 7 ) .7 . J . R e i n , i n : P r o c e e d i n g s o f t h e I A E A S y m p o s i u m o n A n a l y t i c a l M e t h o d s i n t h e N u c l e a r F u e l C y c l e ,

V i e n n a , N o v . 2 9 - D e c . 3 , 1 9 7 1 , p . 4 4 9 .8 . A . F u d g e , i n : P r o c e e d i n g s o f t h e I A E A S y m p o s i u m o n R e a c t o r B u r n u p P h y s i c s , V i e n n a , J u l y 1 2 - 1 6 ,1 9 7 3 , p . 2 3 9 .9 . N e u t r o n C r o s s S e c t i o n s , B N L - 3 3 5 , 2 n d e d i t i o n , S u p p l . 2 , V o l . l I B ( 1 9 6 5 ) .

I 0 . E . C r o u c h , i n : P r o c e e d i n g s o f t h e I A E A S y m p o s i u m o n N u c l e a r D a t a i n S c i e n c e a n d T e c h n o l o g y , P a r i s ,M a r . 1 2 - 1 6 , 1 9 7 3 , V o l . I , p . 3 9 3 .

A U T O M A T I C R E A C T O R R E G U L A T O R A N D

X E N O N O S C I L L A T I O N SA . M . A f a n a s ' e v a n d B . Z . T o r l i n U D C 6 2 i . 0 3 9 . 5 1 4

A t p r e s e n t , t h e r e a r e m a n y s t u d i e s w h e r e t h e s p a t i a l in s t a b i l i t y o f a r e a c t o r t o x e n o n o s c i l l a ti o n s h a sb e e n i n v e s t i g a t e d b y a n a l y t i c a l m e t h o d s . H o w e v e r , i n t h e e a r l i e s t s t u d i e s [ 1] , a s w e l l as in t h e r e c e n t m o n o -g r a p h s [ 2, 3 ] , t h e r e a c t o r c o n t r o l s y s t e m h a s n o t b e e n t a k e n i n to c o n s i d e r a t i o n i n t h e c o m p u t a t i o n s i n a n e x -p l i c i t f o r m . A t t h e s a m e t i m e , v e r y c o n v i n c i n g a r g u m e n t s h a v e b e e n p r e s e n t e d [4] s h o w i n g t h a t i n s o m e c a s e st h e a u t o m a t i c r e g u l a t o r (A R) m a y h a v e a p o s i t i v e e f f e c t , w h i l e i n o t h e r c a s e s i t m a y h a v e a n e g a t i v e e f f e c t o nt he n a t u r e o f t he p r o c e s s . F u r t h e r m o r e , t h e e l i m i n a ti o n o f a n A R f r o m t h e i n v e s ti g a t io n m a k e s t he s y s t e m i n -c o m p a t i b l e w i th th e i n t r in s i c c o n t a i n m e n t r e q u i r e m e n t , e . g . , t h e r e a c t o r p o w e r .

W e i n t r o d u c e t h e r e g u l a t o r i n an e x p l ic i t m a n n e r a n d s h o w t h a t t h e r e a c t o r s t a b i l i ty i s e n t i r e l y d i f f e r e n td e p e n d i n g o n t h e t y p e of A R , i t s a r r a n g e m e n t , a n d h o w t h e s e n s o r s f o r m t h e c o n t r o l s ig n a l . F o r a d e t a i l e da n a l y s i s o f t h e e f f e c t o f t h e c o n t ro l s y s t e m o n t h e r e a c t o r d y n a m i c s~ a s p e c i a l m I N A p r o g r a m w a s d e v e l o p e d ,

s "w h i c h w a s w r i t t e n in F O R T R A N l a n g u a g e f o r t h e B E S M c o m p u t e r . H o w e v e r , t h e m a i n q u a l i t a ti v e f e a t u r e s c a nb e a n a l y t i c a l l y d e t e r m i n e d w i t h o u t t h e u s e o f t h e c o m p u t e r .

I t i s kn o w n f r o m [ 1, 4 - 6 ] t h a t t h e s p a t i a l s t a b i l i t y of a r e a c t o r i s h i g h e r f o r th e l a r g e r m i n i m u m e i g e n -v a l u e ~ o f t h e b o u n d a r y - v a l u e p r o b l e m

hq~ + B~(p + ~tq)= 0 (1)

T r a n s l a t e d f r o m A t o m n a y a E n e r g i y a , V o l . 4 3, N o . 4 , p p . 2 4 3 - 2 4 6 , O c t o b e r , 1 9 77 . O r i g i n a l a r t i c l e s u b -m i t t e d A p r i l 6 , 1 9 7 7 .

8 7 4 0 0 3 8 - 5 3 1 X / 7 7 / 4 3 0 4 - 0 8 7 4 5 0 7 . 5 0 9 1 9 7 8 P l e n u m P u b l i s h i n g C o r p o r a t i o n

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TABLE 1. Eigensolutions in Presen ce of Local Control~n

zsin 2an H (4n%-- 1) B~H2K = 1 sin 2~n Zp - - - - '

0

K----fi (Z-go)Zsin an Za

H - - Zsin 2~n - -H - - Z psin an Z_F-pZ0 sin an-H- Z

sin ~ n H ' ~ z t~ Z~Z 0H - - Zs i f t ~ l n H - - Z p

4n H2 i ) B~(//--Z v)//2

H2

with homogeneous boundary conditions, where, in the fir st- ord er per turbation theo ry [5], correspo nding toeach eigenvalue Pn there is an elgenmode of oscill atory pro ces s ~n and there are definite char act eris tics ofits t ime behavior (decreme nt, frequency); B 0 is the Laplace opera tor depending on the coordinates.

The consid eration of the regu lator s req uire s, besides the introduction of the new variable Pj in Eq. (1)(reactivity introduced by the j- th regu lating unit) , the use of additional equations which describ e the operatio nalgorith ms of the regulators .

We assume that the reg ulat ors are of the high-sp eed kind and we neglect the retardat ion in the sensorcircuits . The ref ore , instead of Eq. (1), we have

NAtp + B~q~+ q)o : ~ FiPJ + ~q~ = 0;" v~ p~ = f K ~ e ~ d u ; ] = I , 2 , . . . , N , ( 2 )

V

where n 0 is the stat ionary distr ibut ion of the neutron flux; Fj de scribe s the spatial localization of the reactivityintrodu ced by the j-th regul atin g unit, Kj is the weight function for the format ion of the sen sor signals fo r thej-th control unit; and vj becom es unity and zero for static (proportional) and astatic regul ators , respective ly.

Let us co nsi der how functions Fj and Kj are related to the operation algo rith m of the re gula tor.Let the )-th regulating unit introduce the reactiv ity uniformly ove r the volume (e.g., when the re tar deris contamin ated by boron); then Fj = 1. If an abso rbing bar is the j-th regula ting unit, whose endlies at the point r), then F) = 6 ( r- rj) (only sma ll di spla ceme nts of the rod are c onside red). Whenthe deviation of the reac tor power from the statio nary value is the controlling action for the )-th re-gulator, then Kj is propor tio nal to the distri butio n of the fission cro ss section. If the deviation of thecu rre nt fro m the sens or at the point r 0 is the contro lling action, then Kj = 6(r - r0). In the cas e of a regul ato roperating in the proportional static mode (v = 1), its action p is proportional to the control action formed bysens ors accord ing to the given algorithm. V~len the regulator ope rates in astatic mode the reactivi ty, whichit must int roduc e, is dete rmin ed fr om the condition of supp ress ion of the control ling action iv = 0).

Let us investigate the astatic mode of operation of AR, which is at prese nt used in almost all reac tors .We consid er the si mpl est sys te m in which AIR introd uces rea cti vit y p in the entire zone (Y = 1). In this c asethe s olutio n of Eq. (2) is of the for m

~ . = - - P'~ r n = 1 , 2 , 3 ,n " ~ 'V I r

whe re r ar e the eig ens olu tio ns of Eq. (1). The eige nval ues/ ~n of Eqs. (1) and (2) coinci de. Henc e, it followsthat the concl usions on the rea ct or stability without the co nsider ation of AR are valid only under the conditionswhen the compensation of reactiv ity is uniform over the react or. With rega rd to the eigenmodes of oscillations

875

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(3), their f orm is com pletely determi ned by the algorith m, in accor danc e with which the control signals aregenerated by the sens ors and only in the par tic ul ar c ase S KCndv = 0 they coincide with the eigenfunctions of Eq. (1)."V

We show that the consideration of the AR leads to significantly different results if the regulation is car-ried out locally. For simplicity we consider a uniform one-dimensional react or with zero boundary conditionand with astatic r egul ator imb edded at a depth Z r. In this case F = 8(Z - Zr). For the regu lato r tuned forkeeping the power constant, K = 1. When the regula tor is tuned to hold the cur ren t of the sens or constant,place d at a point Z 0, then K = 8(Z - Z0). Fo r d efin iten ess we as sum e that Z 0 lies in the fir st zone [0, Zr].The zone [Zr, H] will be called the secon d zone. The sol ution s in the fir st

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Let the sen sor be place d at the point wher e the fir st mode is equal to zero (without regu lato r); then theregu lato r has no effect on it. If at the points wher e the sen sor and the end of the regul ato r are located thefirst mode has (without regulator) one and the same sign, Re ~l dec rea ses monotonically. When the sensorand the end of the regul ator are located in the region where the first mode has (without regulator) differentsigns , then Rew 1 at fi rst inc rea ses with the K, w 0 and w t get cl ose r, and for a cert ain value Kcr w 1 = ~0. Afurt her inc rea se of K* cause s a monotonic decr ea se of Re wl. Fo r the investigat ed modes the limiting (for K -*~) are the value s of w obtained for astat ic regu lato r. The study of the deforma tion of the eigenm edes with theincre ase of K confirms the phenomena detected in the computations with the pro gra m describ ed in [8]. A regu-lator whose end lies close to the senso r quite easily blocks the field change at this point, deprivingt he sens or ofthe information about the pro ces ses occ urri ng in the reac tor. The separation of the sensor and the regul atormakes this blocking difficult. Ther efor e, such syste ms are more stable. The nonmonotonic dependence ofRew 1 on K for ap prec iabl e sepa rati on of the se nso r and the regul ato r is also explained in this way. For K ~