A nnals of Nuclear Ener,tty, Vol. 8. pp. 145 to 146 0306-6549/81/030145-02 $02.0(),0 Pergamon Press Ltd 1981. Printed in Great Britain

T E C H N I C A L N O T E

DISTRIBUTED TIME LAG IN A STOCHASTIC POINT REACTOR MODEL

K ARMESH U *

University of Waterloo, Department of Physics, Waterloo, Ontario N2L 3G1, Canada

(Rece ived 1 S e p t e m b e r 1980)

Al~tract--The effects of distributed time lag in reactivity feedback are investigated in a point reactor model with random reactivity insertion. The time-delay kernel is assumed to be of the exponential type and is found to affect the mean neutron density in the stationary state.

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

Recently, Dutr6 and Debosscher (1977) have presented an exact statistical analysis of neutron density fluctuations resulting from Gaussian, white-noise reactivity insertion in a point reactor model (without delayed neutrons) with proportional power feedback. This stochastic model works under the assumption that the reactivity feedback is instantaneous, i.e. neglecting slowly responding processes such as moderator-temperature reactivity feedback.

In this note we attempt to generalize the model of Dutr6 and Debosscher to account for the time lag or delay introduced as a result of slowly responding processes. We, therefore, study a point reactor model (without delayed neutrons) with stochastic reactivity perturbations and distributed time lag in reactivity feedback proportional to power level. The Stratonovich calculus is employed for the stochastic analysis of the problem (see Karmeshu, 1978).

Z R E A C T O R SYSTEM E Q U A T I O N A N D S T O C H A S T I C ANALYSIS

The reactor system is represented by the following equation

I dt = P + Air) - k( t - r)n(r)dr n(t) (1)

where n(t), l and p are neutron density, effective neutron life-time and constant external reactivity respectively and A(tt is Gaussian, white-noise reactivity with

(Aft)) = 0. (A(t)A(t + z)) = a]5(z). (2)

In equation (l) k ( t ) i s the time-delay kernel and

f l k(t - dr ~)n(T)

* On leave of absence from the University of Delhi, Delhi 110007, India.

introduces distributed time lag. When k( t ) is proportional to a delta function, i.e. instantaneous reactivity feedback, we find equation (1) reduces to the one considered by Dutr6 and Debosscher. Equation (1) is a nonlinear stochastic integro-differential equation and consequently the variable n(t) is a non-Markov process. Whilst we see that there is no analytic way to solve equation (1), it seems useful to consider a special case k(t) = ye -~', which qualitatively implies that past densities have decreasing influence. The mathematical advantage of this choice is that the augmented process In(t), x(t)] constitutes a Markov process, where

x(t) = ~' e -~I' '~n(r)dz (31

Equations (1) and (3) can be rewritten as a pair of coupled nonlinear stochastic differential equations

dt = [p + A(t)]n - nx (4)

d x = 7n -- 2x, 0 ~ x , n < or+ (5)

dt

The Fokker Planck equation corresponding to (41 and (5) is

~ ,~,, L \ t + 2 F - i "p

# 6.2 •2 (n2p}

- ~ x [ ( 7 n - ~x)p] + 21 ~ On 2 (6)

Introducing a new set of variables u = ?nl'~tx and v = ,;/p and defining

.f(u, v, t) du dv = p(n, x , t) dn d x (7)

145

146 KARMESHU

equation (6) reduces to

2120t = ~ U ~uu + ~ V + ~i~ -

+ ~(1 - u) ~ i g v ~ v + T v + ~ i 7 - f

(8)

This form of equation becomes amenable to the forward approach due to Liu (1968). We obtain the stationary density with normalization constant

f~t(u,v) = JV~l e-"~-"Vu"-lv "-1 (9)

The constants are

a = 212~/o2a, rl = 2pl/a 2 (I0)

The stationary probability density in terms of the original variables n and x is

I 2yl 2 n 21 ] p~t(n,x) = ~ n " - l x " - " - I exp a~ x a~ x (11)

where ~22 is a normalization constant. Using equation (1) we get the probability density of n in terms of modified

Bessel function K v(n) i.e.

fo Psi(n) = psi(n, x) dx

a+ "2 a /412 = Cn " - K . _ , ~ 7x~l) (12)

The constant C is given by

2a+ n + 1 yla+ 7)/213ta + hi~2 c = 0 3 )

a ~" + "~r(a)r(n)

The average neutron density turns out to be

( n ) = (~tp/7) (14)

which increases or decreases depending upon whether :t is greater or less than 7. Similarly, the higher moments of n can be obtained from (12).

It is interesting to note that the first moment ( n ) is independent of a~. This result is due to the particular nature of the nonlinearity of the problem. Similarly, the higher moments of n can be obtained from (12) and are found to depend on an as well.

REFERENCES

Dutr6 W. L. and Debosscher A. F. (1977) Nucl. Sci. Engng 62, 355.

Karmeshu (1978) Ann. nucl. Energy 5, 21. Liu S. C. (1968) Bell System Tech. J. 48, 2031.