Space dependent considerations in the non-linear prompt neutron kinetics of a nuclear reactor

Journal of NuclearEnergy, Vol. 24.pp. 21710221. PergamonPress 1970. Printed in Northern Ireland

LETTERS TO THE EDITORS

Space dependent considerations in the non-linear prompt neutron kinetics of a nuclear reactor (First received 6 August 1969 and infinalform

15 December

1969)

THE FUCHS’model describes the space and energy independent description of the kinetic behaviour of a temperature coefficient dominated reactor (FUCHS, 1946; SCALLETAR,1963 and SMETS,1958). There have been only a relatively few attempts to calculate space dependent effects in the non-linear kinetics problem. HAPKE (1962) calculated space dependent effects using the adiabatic approximation familiar in quantum mechanics, whereas SCALLETAR(1963) used a perturbation-theoretic method for handling the space dependent effects for a bare reactor. In a previous note (GHATAK, 1968), the Fuchs’ model was derived as a well defined approximation to the general neutron balance equation and the parameters appearing in the equation appropriately defined. In this note, we will obtain corrections to the time dependent reactor power due to the excitation of higher spatial modes for a two region reactor using one group diffusion theory. We will assume that the contributions from the higher modes is small (as compared to that of the fundamental mode) and hence the calculations are based on the use of perturbation theory. Based on the model developed, higher mode contributions are calculated for a graphite moderated pulse reactor. We start with the eauations V’(Q)

+ [y(v &), - (u Cz),lp = $, in the core

and YD,P)

-

(o x:),P

3P = 3 , in the reflector

(2)

wherep = bin, n denotes the neutron density, u the neutron speed, Y the number of neutrons born per fission, E the energy released per fission, C, the macroscopic fission cross-section; Cz and C: are the macroscopic absorption cross-sections of the core and reflector respectively and D( = u/38,,) is the diffusion coefficient, C,, being the macroscopic transport cross-section. The subscripts c and r refer to a suitable average of the quantities over the core and reflector spectrum respectively. We shall assume that the fission and absorption cross-sections have strict l/v energy dependence, thus the average values of UC,, UC: and vC: will be independent of the neutron spectrum and hence independent of temperature. (The effects due to thermal expansion are, of course, neglected). We will assume D, to be a linear function of the local temperature rise T(?, t): D, = Do + qT(?, t).

(3)

Since there is no change in the reflector temperature, D, will remain the same. (This is, however, not strictly true. With the heating of the core, the core spectrum changes which will also change the neutron spectrum in the reflector. The effect of this change, which is expected to be small is wglected in the present treatment). It should be pointed out that in the core (since XI has been assumed to have a strict l/v energy dependence), p represents the local power density, but in the reflector, p is just a variable and has no physical significance other than being proportional to the local neutron density. We expand p(r’, t) in terms of the orthogonal functions Y,(F):

where Y,(F) satisfies the following differential equations: V2Y, + CZ,*Y~= 0, in the core

(5)

VaY,, + m”Y, = 0, in the reflector.

(6)

Explicit expressions for CC,,~ and yn2 will be found later. It will be found that whereas an* is always positive, Y,,~may sometimes be negative. In that case ynZ has to be replaced by -8,“. The functions Y, will satisfy the same boundary and the continuity conditions as the neutron density (orp). In one 217

218

Letters to the editors

group two region diffusion theory, the Y;s will form a complete set of orthogonal power associated with the nth mode will be given by* p,,(t) = p,(t)

functions.

I Y,,Y) dr’.

(7)

c

and the total reactor power will be xP,(t). II

Further,

cy

The

the heat capacity, C, is defined by

=p(3, r)V,

where Vis the volume of the core. If we substitute the expansion (equation 4) in equations (1) and (2), use equations (3)-(8) and carry out some manipulations we would obtain dP, = U, dt

+ 2 FxP P&(r) [/$f)dl],

(9)

kl

where 1, = Y(u&) - UC: - Doan8 = - (zq)

(104

- L&2,

W’b)

and the constants &” involve suitable integrations of they function. As is obvious from equation (9) the various modes are intercoupled through the coefficients FE,“. If we consistently neglect the higher modes (i.e. assume all F,,” to be zero except for k = I = n = 1) then equation (9) will reduce to the Fuchs’ model equation (GHATAK, 1968) as it should. If we now compare the coefficient FIX1with the corresponding coefficient in the Fuchs’ model equation we would obtain

F1” =Fkz”= - &I’

a

iE D

(lib)

LL”

(1 lc) Equations (10) and (11) plus the continuity conditions for the neutron density and the normal component of the current density enables us to calculate a,,, Y,,, 1, and F,,n. We shall now solve equation 9 by a perturbation approach. Thus, if we assume that the power associated with higher modes, (n > l), is small compared to the fundamental mode then, we have, for n > 1: P,(r) g Fll” &

I

i Plo(t ‘) e-M

[I

;‘Pl”(rs) dt”

1

dr’,

(12)

where PI”(r) corresponds to the zeroeth order approximation and hence, is the solution to the Fuchs’ equation (SMETS, 1958). In writing equation 12 we have assumed that at t = 0 only the fundamental mode is present. Further, the perturbation to the fundamental mode, API(t), due to the presence of higher modes is approximately given by (13) Approximate

evaluation of the integrals in equations (12) and (13) gives the following results:

P,(t) G

,I,* 1 i ;D,l” D,,” 0-20 07Al 0 -_-* [P,Y~)Y 1, 3 1 \ *o

(144 for t >
* The letter ‘c’ below the integral sign denotes that the integration is carried out over the core only; whereas ‘CR’ indicates the integration over the core and reflector, i.e. over the entire reactor.

Letters to the editors

219

and

(1W

4 * D,,’ D,,” 05 -+ 0

for t > t, + ; . 0

-p-1

Case (i) : Bare slab reactor For a bare slab reactor of extrapolated where

thickness 2~2, the Y functions

are of the form cos alrx

k = 1,3,5,. . . .

CQ= &, and Dk,* =

3n2ka12 [na - (k - 1)2] [n2 - (k + 1)*] ;

n,k,l = 1,3,5,7,

....

Further LyM,a - ccl*) 1 + L%,a ;i; = ’ keff - 1

42

(15)

where La(= DO/v&,) is the square of the diffusion length and kerf multiplication factor.

is the effective

Case (ii) : Rejected slab reactor For a reflected slab reactor of core thickness 2a and extrapolated reflector thickness b - a, the solution of equations (5) and (6) is quite straightforward and if the appropriate boundary conditions are applied, one would obtain the following equations: coth [q($

- I)]

=f,/>

tan k

3

4-1

(164 .

and cot[rc-I)]

=$JFtan&\/R’+/‘1

,OforA,<

-;.

(16b)

where

t=aka,

rl=Bxa;

1 = km - 1 - L,~u,~ L

f = (vz:),; 0 DO L’ = @zj, ;

I,

=

5 = ha;

(174

- 1 - L,ayk2 - 1 + L,2pb* ; I, = 4

(17b)

f = (VIZ:),; I L,‘=&,;

D

p* = [(km - 1) + ;] (;)” ; I c

k

_ v(v’f)

m

(G%

. ’

Rz=[(km-l)~+l](;)z.

0

I

Knowing the reactor parameters, the above equations can be solved to give a discrete set of values of r] and 5. The corresponding values of 5 and jl can be determined from equations (17). The integrals in Dkzn are readily evaluable. It is interesting to note that for a tixed value of ‘a’, if we tend ‘b’to infinity, the solutions of (16b) move closer until in the limit the roots are continuous. However, the roots of (16a) remain discrete. Thus a system with an infinite reflector has one or at most a small number of modes with discrete 4

220

Letters to the editors

values of & which are exponential in the reflector, plus a continuum of modes which are oscillatory in the reflector. In fact, when the reflector thickness is large, it is easier to assume it to be intinite, calculate the density of modes and obtain the total contribution from the continuum by evaluating a suitable integral. For example, for 2/L, < I < t,,,, the result is 1 [Pro(t)]* 1, * 6 -1, R fi(E)g&) P’0) G z - Pm.x OS

dE,

where km - 1 -

1

gx=2jX)

km

-

= - MEr)sin

1-

E

(;)‘pB @*El1

&-j 1 [

sin (25, + 0 + $& ~~‘-R’cos”E+;~&

E1 + sin & cos E1 + z ~0s.’E1 II,(&) = x sina E1 1 - 5 sina [I [

(1=CQU.

APPLICATIONS

sin (X1 - 0

1

sin ‘5

1’

,

TO THE

TREAT

REACTOR

TREAT (Transient Reactor Test Facility) is a small homogeneous graphite moderated reactor with an atomic ratio of graphite to highly enriched uranium close to 10,000. A description of the TREAT reactor has been given in many Argonne National Laboratory Reports (see, for example, FREUND ef al., 1959 and KIRN et al.,1960). A series of transients has been performed on TREAT with initial reactivity insertions between 0.2 and 1.9 per cent. The resulting reactor periods were between 22 and 0.075 sec. These transients were initiated by a step insertion of reactivity with the reactor critical at a power of 11 W. Thus, before the initiation of the transient the reactor was at such a low power level that all temperature effects are negligible. A detailed neutronic analysis of TREAT, including the calculation of the temperature coefficient has already been carried out (GHATAK, 1962). After carrying out suitable averages of various cross sections over calculated reactor spectrum (GHATAK, 1962), one obtains the following values of various parameters in one group two region representation of the TREAT reactor: I, = 7.3934 x 1O-4 set; Lo2 = 273 cma I, = 3.6057 x 1O-S set; Lra = 1323 cma. The reflector savings, 8, is assumed to be 36 cm (= L,) and by equating [r/2@ + S)]* to the measured value of Be (= 9 x lo-’ cm*), we get a w 165 cm. By solving the two region, one group criticality equation we got km = l-3941. For an initial reactivity insertion of 28, the values of 1, E, 71and 5, as obtained by solving equations (16) and (17), are tabulated in Table 1. It is interesting to note that, if we calculate the promptTABLE1. I, = 15.329 set-I; 2, = -730.71 set-‘;

6, = 0.03854; Ea = 0.05930;

11= 0.466 & = 0.58

& = - 1820.35 set-‘;

55 = 0.08055;

56 = 1.07

1, = - 3419.9 set-1;

& = 0.1042;

5, = 1.53

neutron lifetime, I(= Ak,/I,), then the value calculated is 9.37 x lo-& set as compared to the experimental value of 9.0 x lo-* sec. The value obtained from a detailed two dimensional computer calculation, using multigroup diffusion theory, was 9.2 x 1O-4 set (GHATAK, 1962). This shows that a one group two region representation of the TREAT reactor is not a bad approximation. Using the

Letters to the editors

221

calculated values of 1,5, 7 and 5, we get the following values for the total power associated with the third and fifth harmonic:

1 P&)tat_+;. P(r) 6 1 P(t)

pm PO <

-0.2

Pi(t) 1 6 -0.1 P(t) and

P:(t) 1

x IO-$_

p;

+
x lo-“p,,,

-0.2 -0.4

x 10-s x 10-1

1

For large reflector thicknesses, calculation of only a few harmonics will not correctly depict the higher mode effects, because contributions from many modes will be comparable. However, for such cases, it is often simpler (and fairly accurate) to assume the reflector to be intinitely thick and the modes to form a continuum. Acknowledgements-One of the authors (AKG) wishes to thank Professor Mark Nelkin for stimulating discussions on this topic and to Dr. R. Scalletar and Dr. B. Hapke for communicating the results of their unpublished work. AJOY K. GHATAK PAL SINGH

Department of Physics Indian Institute of Technology New Delhi-29 India

OM

REFERENCES FREUND G. A. (1960) Design Summary on TREAT, ANL-6025. FUCHS K. (1946) LOS Alamos Reo. LA-596. GIUTAK A: K. (1962) Neutron& Analysis of TREAT, General Atomic Rep. GA-3575. GHATAICA. K. (1968) 1. Nucl. Energy 22,699. HAPKE B. W. (1962) Problems connected with the use ofpulsed Reactors with Very High Neutron Fluxes, Ph.D. Thesis, Cornell University. KIRN F., BOLAND J., LAWROSKI H. and CIXK R. (1960), Reactor Physics Measurements in TREAT, ANL-6173. SCALLXTAR R. (1962) General Atomic Rep. GA-3400. Shrrrrs H. B. (1958) Low and High Power Nuclear Reactor Kinetics, Ph.D. Thesis, Massachusetts Institute of Technology.

Journalof NucIcarEncrt?y. Vol. 24,pi. 221to 225. PermmonPress1970. Printed in Northern Ireland

Conversion of a horizontal thermal column into a useful neutron source plane (First received 2 January 1969 and in final form 9 January 1970) A VER~CALthermal column--located above the reactor core-is frequently used as a neutron source plane for studying the reactor physics parameters of various moderators and of various configurations of fuel and moderator. A horizontal column-usually located beside the core-is not useful in such experiments because the longitudinal neutron flux at the vertical-access position (Fig. 1) of this column decreases exponentially with distance from the core while the transverse flux usually follows a cosine distribution, and mathematical treatments of such complex source conditions have not been published. Conversion of this non-symmetric flux shape into a symmetriconecan, at least in theory, be accomplished by removing a portion of the graphite from the column. The literature failed to reveal any direct information about accomplishing this task; although some related work has been reported. ANNO et al. (1958) removed graphite from the thermal column of the Batelle Memorial Institute Reactor in order to increase and to flatten the neutron flux at the horizontal access position. CLARK