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\ /A POWER CALIBRATION METHOD USING THE X E N O N POISONING
C A L I B R A Ç Ã O D A P O T Ê N C I A P E L O E N V E N E N A M E N T O P O R X E N O N
MARCELLO DAMY DE SOUZA SANTOS PAULO SARAIVA DE TOLEDO
INSTITUTO DE ENERGIA ATÔMICA
Publicação I E A — N.° (5
— 1 9 5 8 —
I N S T I T U T O D E E N E R G I A A T Ô M I C A C a i x a P o s t a l 1 1 0 4 9 ( P i n h e i r o s ) CIDADE UNIVERSITÁRIA "ARMANDO DE SALLES OLIVEIRA"
SÃO PAULO - BRASIL
CONSELHO NACIONAL DE PESQUISAS Piesidente — Prcf. Dr. João Chi!,çtovriO Cardoso Vice-Presidente — Prof. Dr, A f h o 3 da Silveira Ramos
UNIVERSIDADE DE SÃO PAULO
Reitor — Pioí. Dr. Gabriel Sylvestre Teixeira de Carvalho Vice Reitor — Prof. Dr. Franci.sco Joíio Humberto Maffai
INSTITUTO DE ENERGIA ATÔMICA D]F,E'1'0R
Prof. Dr. Maicello Dajiiy de Souza Santos CONSELHO TÉCNICO-CIENTÍFICO
Kcprcscntantes do Conselho Nacional de Pesquisas Prcf. Dr. Luiz Cintra do Prado
P;oF. Dl-. Pauius Aulus Poinpéia
R.spresentanles da Universidade de São Paulo Prof. Dr. Francisco João Humberto Maffei Prof. Dr. José Moiu'a Gonçalves
CONSELHO DE PESQUISAS
Prof. Dr. Marcello Damy de Souza Santos
— Chefe da Divisão de Física Nuclear Prof. Eng. Paulo Saraiva de Toledo
— Chele da Divisão de Física de Reatores Prof. Dr. Fausto Walter Lima
— Chele da Divisão de Radiociuímica Prcf. Dr. Rómulo Ribeiio Pieroni
— Chefe da Diviíão de Radiobiología
Reprint from 2nd U N G e n e v a C o n f e r e n c e . Printed by Pergamon P r e s s , L o n d o n .
P / 2 2 7 3 B r a z i l
A Power Calibration Method Using the Xenon Poisoning
By Marcelio Damy de Souza Santos and Paulo Saraiva de T o l e d o *
In this paper a inethod is given of r e a c t o r power iTieasurement applicable to reactors in which the thermal neutron flux is of t h e order of 10'^ neutrons per square c e n t i m e t e r per second. T h e m e t h o d was developed in order to h a v e an independent method of power measurement of the I n s t i t u t o de E n e r g i a A t ô m i c a swimming pool r e a c t o r in the region of 1 Mw.
Essentially, the method consists in measuring the r e a c t i v i t y variation due to xenon poisoning after the pool t e m p e r a t u r e has reached an equilibrium value.
The power is easily calculated using an approximate quadratic expression for the t i m e variation of xenon poisoning.
Results of measurements performed at the I n s t i t u t o de E n e r g i a A t ô m i c a with the r e a c t o r at a nominal 1 Mw power are given and it is shown t h a t the method is quite reliable provided the bulk pool temperature is constant or slowly varying.
X E N O N C O N C E N T R A T I O N A S A F U N C T I O N O F T I M E O F O P E R A T I O N
The differential equation governing the growth of xenon concentration in the core of a thermal reactor is:i
dX(T.i)
dt = XJ{T,t)+y.I.i^{r)-X^X{v,t)-aé{r)X{r,t).
(1) The notation is the one of R e f . 1 with one single modification; the spacial and temporal dependence of the xenon concentration X{r,t) is written explicitly.
The flux <j>(r) is supposed t o be c o n s t a n t in time.
T h i s assumption is quite good since the xenon poison- ing is coiupensated in our r e a c t o r b y the withdrawal of a single regulating rod and the time needed to t a k e the necessary d a t a is of the order of three hours. During this time and with the reactor operating at about 1 Mw with a t h e r m a l flux of the order of lO^^ n e u t r o n s / cm'^ sec, the distortion of the thermal flux due to the withdrawal of the regulating rod is quite small.
S t r i c t l y speaking, this distortion must exist since the distributed effect of the xenon poisoning is compen- sated b y the quasi-local withdrawal of the regulating rod.
I n t e g r a t i n g E q . (1) with the condition t h a t at ^ = 0
* Instituto de Energia Atômica, São Paulo (on leave of absence from São Paulo University).
we h a v e X ( 0 ) = / ( 0 ) = 0 and A'^ being the equilibrium value of xenon poisoning, we get after rearranging the corresponding expression of R e f . 1:
X{r,t) = X,{v)\\^-
X
Yl with
(2)
(3)
Now, if we consider t h a t A , = 2.1 X 10-5 s e c - i ;
A; = 2 . 9 x 1 0 - s e c - ; = 2 . 8 8 x 10 - cm'^ (4) we can see t h a t for (/i(r)= lO^^ n / c m ^ sec:
= [A2+CT2vi(r)] = 4 . 9 8 X 1 0 - ^ s e c - i .
So, if ^ < 1 0 = sec (30 h r ) , we have:
X,t< 1.
T h e n , for times of the order of 3 to 5 hr we w i l l h a \ e a good a p p r o x i m a t e expression for A'(r,;) if we develop the exponentials in E q . (2) in a power series and retain only the terms up to t h e second power. Doing this we get:
X{rj) = A'o(r) I
71
+
72T h e definition of poisoning Po{T,t) is:
P,[r,i) = A'(r,¿)ao/Su,
(5)
(6) in which S u is the macroscopic absorption cross section of the fuel in the fuel elements.
L e t us define a mean poisoning I\{t) as:
(7)
where V is the core voluiue. T a k i n g into account tliat this mean poisoning is related to the variation of r e a c t i v i t y A,o(/) through the following equation:
PoW = ^p(t)l^m + ^n]/^n (8) with Sm the macroscopic absorption cross section of
P O W E R C A L I B R A T I O N F R O M X E N O N P O I S O N I N G 545
.450
.350
.300
,250
.200
100 ISO Tim« after start up
m lltinulm 250
Figure 1
the moderator and structural materials of the core, we get:
[Sn. + 2u]Ap(0 = ^ Yi + Yi
+ i{y,X,-Y,h)Kt^]dV. (9) R e m e m b e r i n g the expression for A'o(r) and the definition of we have:
2V {X,y,-Ky,)^(v)dV. (10)
Since X^=X2+(J.¿:¡>(T), it follows that the coefficient of ¿2 depends in a complicated way on the space varia- tion of the thermal flux. T h i s space variation is small in the swimming pool reactor due to the effect of the water reflector, and it will be a good approximation to consider as space independent. T h i s approximation for the coefficient of the l"^ term is very good indeed, since A-^ya contributes less than 5 % of the total value of the coefficient, even for a thermal flux as high as 10^^ n/c\v? sec. Considering then a space independent value for A^ and calling it AjC) we obtain from;!"
[S™-fSu]Ap(0 = [aiy.J.i^]t
with ^ , a mean thermal flux, defined as:
.800
600
/
<i ^
/
1
200 Time after start up
300 Minutes 400
Figure 2
F i n a l l y , since the power P is given by:
1 P = 3.1 X 101»^
(with F f t h e fuel volume) it follows that:
p . l x l O i O a . y ,
(12)
(13)
F , [ S m - f S u ] 3.1 X l 0 i« a 2
+
; ) ( A , y i ~ A , ( o ) y , ) p ] i ^ (14)_ 2 F f ( i : ^ + S „
W r i t i n g for the time dependence of Ap(^):
Ap(/) = a/-h/3i!2 (15) and determining experimentally t h e values o f a and jff
we can c a l c u l a t e P through any of those values. T h e coefficient of could be used b u t we would have in the determination of P through it an error due to the approximation we made when we considered X^ as c o n s t a n t . Besides, to calculate this c o n s t a n t we would need at least the mean value of the thermal flux, which is unknown also.
All these drawbacks do not exist if we use the value of a to calculate P , since now all c o n s t a n t s are known and t h e only approximations made are the ones related to the use of E q . (8); those are inherent to this method and catinot be avoided.
T h e expression to be used to calculate P will then be:
1 F , ( S „ + 2 : u
P = 3.1 X 101» 6)
T a b l e 1
Nominal
po„cr a P
[calculalcd) P calculated
Ea. 77 I E . / V - { A e x p ) 1 MAY 0 . 2 2 X 1 0 - ^ 2 . 0 9 9 X 1 0 - * 3 . 0 2 6 x IQ-' 2 . 4 0 X 1 0 - « 2 . 5 8 X 1 0 - » 1 M w
546 S E S S I O N A - U P / 2 2 7 3 M . D . D E S . S A N T O S and P. S . D E T O L E D O Since ViNu=Mu=ma.ss of fuel in tlie core (in
grams), we have, with <TU the absorption cross section of the fueh
P = 1 Mu
3.1 X 1 0 " • 1 +
life)-
a t t s ) . (17) I t is interesting to note t h a t even for ^=]0^^ n / cm^ sec the experimentally determined ratio (P/a) must agree with the theoretical value computed taking for A.^C) the value A2. Indeed the correct value for(P/a)
is:(ß/o.) =
A ^ y . - Ã . W y , 2/ 2
and this should be compared with the approximate value taking Aj*"* = A2:
Airi—Aaya
;app — 2y2
Introducing the numerical values for the c o n s t a n t s and taking (;J= lO^^n/cm^ sec we get:
( ; 3 / a ) - ( ; 8 / a ) a p p
iß/a)
= 0 . 0 3 ,that is, there is an error of 3 % , certainly inferior to the experimental errors, in the measurements of reactivity variation.
R E S U L T S O F E X P E R I M E N T S
During the preliminary steps of power reactor calibration, with the power measured by a tempera- ture differential method,^ the variation of the regulat- ing rod position as a function of time was determined after the pool temperature was almost stabilized. T h e experimental points of p{t) versus t for an experiment performed, at a nominal power of 1 Mw, are given in Fig. I .
F i t t i n g the points b y a curve of the form:
p(i) = p^+al+ßP 18) was done by a least sc^uares method. T h e values obtained for the coefficients, the power calculated b y use of E q . (17), the experimental and theoretical approximate ratio (P/a) are given in T a b l e 1 for our A experiment.
I n a second experiment (B) at the same nominal power of 1 Mw, the experimental points obtained are given in F i g . 2. I t can be seen t h a t after 200 minutes, the pool temperature being then almost stabihzed, the experimental points are very well fitted b y a curve with the same values of a and p. As a m a t t e r of fact, the value of for tlie curve of Fig. 2 is the same as the one for the curve of Fig. 1; l)ut this is merely a coin- cidence, since the only thing one could expect is t h a t
the two curves be parallel, especially since the temperatures are different in the two experiments.
I t is important to note that if one wishes t o fit the experimental points b y a curve using the m e t h o d of least squares, the number of experimental points must be cjuite large; otherwise the computed values of the coefficients will be in error, due to t h e greater weight given b y the least square method to the points corre- sponding to high values of /. In the analysis of our two ex^jeriments, we took as final values of the coefficients the mean between the values obtained putting the time origin for the application of t h e least square method at 1=0, t=T/2 and l~T. where 7' is the total time in which the measurements were taken. T h e fitting of the experimental points b y a curve of the form given b y E q . (11) is b e t t e r than the one using a form without p„ owing to fluctuations t h a t occur in t h e position of the control rod due to small statistical fluctuations in the power.
T a k i n g these precautions and especially considering only the mea.surements taken after the pool tempera- ture has attained its equilibrium value, t h e method seems quite reliable.
One final remark must be made: the experimental points to be used in the determination of t h e coeffi- cients a and /S, are the ones obtained after the pool temperature has stabilized; b u t the real t i m e origin must always be taken a t t h e time in vv-hicli t h e steady operation of the reactor has been initiated, in order t h a t the requirements for the applicability of E q . (2) be fulfilled. W i t h a reactor already ]3oisoned by :<enon E q . (2) is not applicable.
I t seems, then, t h a t the method is quite suitable for power measurements and t h a t there are good grounds to expect it to give reasonably reproducible results.
Anyway, the final decision is dependent on further experiments and its precision can only b e estimated theoretically after a careful e x a m i n a t i o n of t h e pre- cision of the constants used. T h o s e experiments will be done as soon as possible at the I n s t i t u t o de E n e r g í a Atómica, depending on the schedule for operation a t
1 Mw of the l E A R e a c t o r .
A C K N O W L E D G E M E N T S
W e t a k e the opportunity to t h a n k Miss E w a W a n d a Cybulska for doing almost all the numerical calcula- tions and to Professor I v a n Cunha Nascimento, Dr. Américo Frascino and Dr. Azor Camargo P e n t e a d o for getting the d a t a for experiments A and B .
R E F E R E N C E S
1. Glasstone and Edlinid, The Elements of Nuclear Reactor Theory, D. Van Nostrand Co., New York (1955).
2. M. Damy de Souza Santos et al., Preliminary Results of 5 Mw Operation with the Brazilian Swimming Pool Reactor, P/2279, Vol. 10, these Proceedings.
E R R A T A 1) p a g 5 4 4 , s e c o n d c o l u m n , 1 1 " ' l i n e :
i n s t e a d o f : <l)(r) = 1 0 ' - ' n / c m - s e c : r e a d : (|>(r) = l O ' - ' n / c m - s e c : 2 ) p a g 5 4 5 , f i r s t c o l u m n , 1 9 " ' l i n e :
i n s t e a d o f : f r o m ( ' " ) r e a d : f r o m ( 1 0 ) 3) p a g 545 ; T a b l e I s h o u l d b e s u b s t i t u t e d b y t h e f o l l o w i n g
T A B L E I Nominal
Expsriment p„ a li O V^ ) c x i . (/^/a),,,,,, P
power ( s e c ~ ' ) ( s e c - - ) s e c - ' calculated (sec • ' ) eq (17) IEA-(A exp) 1 MW 022x10'-^S^myJO^'SAOSxi^^^ ^^cTxTo"-' TMW~"
4 ) p a g 5 4 6 ;
T h e f i r s t f o u r l i n e s o f f i r s t c o l u m n s h o u l d b e s u b s t i t u t e d b y t h e f o l l o w i n g :
N„
" S i n c e VfN„ = M„ w h e r e M„ is t h e m a s s o f f u e l in t h e A„
c o r e , in g r a m s , N„ is t h e A v o g a d r o ' s n u m b e r a n d A „ i s t h e a t o m i c m a s s o f t h e f u e l , w e h a v e , w i t h a,, t h e a b s o r p t i o n c r o s s s e c t i o n o f t h e f u e l :
1 M„ N, P = .
3 . 1 x 1 0 " ' A 5) p a g 5 4 6 , f i r s t c o l u m n , 1 0 " ' l i n e :
~ • ( ' ^ ) ( — ( w a t t s ) ( 1 7 )
i n s t e a d o f : ( ^ / a ) = —
r e a d ; (/i*/a) =
2-r,
2-f- 6 ) p a g 5 4 6 , f i r s t c o l u m n , 1 6 " ' a n d 17"' l i n e s :
i n s t e a d o f :
(li/y.) _ (/i/a)„i,p
= 0.03 (/V«)
t h a t is, t h e r e is a n e r r o r o f 3 % , r e a d :
- 0.06
t h a t is, t h e r e is a n e r r o r o f t h e o r d e r o f 6 % ,