J. Nucl. Energy. Part A: Reactor Science. 1960. Vol 12. pp. 26-31. Pcrgamon Rcss Ltd. Printed in Northem Ireland

RESONANCE ESCAPE PROBABILITIES IN CIRCULAR CYLINDRICAL CELL SYSTEMS

HIROSHI TAKAHASHI* Japan Atomic Energy Research Institute, Tokai-mura, lbaragi-ken, Japan

(Received I July 1959)

Abstract-An investigation is made of the resonance escape probability in a circular cylindrical cell system, which is an approximation to an actual rod lattice. The analysis is based upon the integral form of the Boltzmann transport equation appropriate to this system. The resonance escape probability is evaluated and compared with the value for the slab lattice model obtained by CORNGOLD. We iind that the resonance absorption in the cylindrical cell is lower than in the slab system for a particular uranium-light water lattice. Accordingly, we conclude that the resonance absorption given by the slab lattice approximation usually used for an actual rod lattice is an over-estimate.

INTRODUCTION

THE equation which describes the time development of the neutron population in phase space is called ‘Boltzmann’s transport equation’. There are two commonly used formulations of this equation, the differential form and the integral form. The latter is useful for certain neutron transport problems because it incorporated automatically the boundary condition on the neutron flux. In the first study along these lines for heterogeneous systems, CHERNICK (1955) discussed neutron transport in a slab lattice by using the first flight collision probability, which is the simplest approximation to the integral form. CORNGOLD (1957) discussed further the same problem by using directly the integral Boltzmann transport equation.

Many studies have followed upon this work. However, for simplicity, most authors have investi- gated the problem of slab lattices. In other words, the actual rod lattice, which consists of circular cylin- drical fuel embedded in moderator, is approximated by an equivalent slab lattice, which has the same surface to volume ratio of fuel, and the same moder- ator to fuel volume ratio.

As explained in the previous paper (TAKAHASHI 1960), we consider that a circular cylindrical lattice cell, which has the same volume ratio as the actual rod lattice, is a better approximation than the slab model. In this paper we discuss the integral form of the Boltzmann transport equation for this circular cylindrical cell, and apply it to the calculation of resonance escape probability. We compare numeri- cally the resonance absorption so obtained with the value from the slab lattice approximation, evaluated

* Present address: Brookhaven National Laboratory, Upton, Long Island, New York, U.S.A.

by CORNGOLD for a typical uranium-light water lattice. We find that our result is about 10 per cent lower than the latter value, and conclude that the resonance absorption does depend on the geometrical shape of the fuel.

BOLTZMANN’S TRANSPORT EQUATION

As described in the Introduction, we consider the neutron flux in a circular cylindrical cell, in which the neutron suffers perfect reflection at the boundary. According to this approximation, the neutron flux moving in any direction is periodic with respect to each cell, in the same way as for the slab lattice discussed by CHERNICK. We can now easily derive the integral form of Boltzmann’s transport equation for the neutron in the cell. In the co-ordinate system shown in Fig. 1, the differential form of the Boltzmann transport equation is expressed by the following equations. In the fuel region 0 < r < a,

sin 0 cos v a~(r,QP) sin f3 sin q ZP(r,8,~,u) -

+&wwr, 6 Ql, u) = &(r,fA%~)

+/Oti&t/:sina:

r aP

& / 2n

I I

0 d~‘pO(e,~,ule’,gl’,u’)

~sOW)Wr,%~‘,t4 ( 14

In the moderator region a < r < b,

sin e cos ~ aQ(r,e,V,4 sin 8 sin fp X%e,fp,d - & r %

+ WPW,w4

~dIW~(r,~‘,~‘,u’) W 26

Z -0xis x

Resonance escape probabilities in circular cylindrical cell systems 27

Also,

M, + 1 X,(w) = 2 e _ew12 %c - 1 ,r,.> _------_ - 2 (4)

We shall represent the integral operators appearing on the right hand sides of equations (I) by To and T,, respectively.

-8 FIG. I.-Circular cylindrical lattice cell and neutron path.

Y(r, 8, y, u), @(r, 8, v, u) and S(r, 8, v, U) are respec- tively the neutron flux in fuel and moderator, and the neutron source directed in the LI (0, v) direction at position r with lethargy u : X is the macroscopic total cross-section : Xc, is the macroscopic scattering cross- section ; P is the scattering probability; and the suffices 0 and 1 stand for the fuel and moderator regions, respectively. If the neutron-nucleus scat- tering is isotropic in the centre of mass system, the operator Pk is as follows,

(0 < u - u’ d q_d~ = 0 (u - 24’ > qd (2)

where Mk is the mass number of the moderation nucleus, and qM = ln[(M, + l)/(M, - ])I2 for Mk > 1. For Mk = 1 (hydrogenous moderator) the condition 0 5 u - U’ 5 q_4f is replaced by 0 < u - u’. For Mk-+ co (non-moderating material) the scat- tering integral term in equation (1 a) becomes

Next, let us consider the transformation from the differential form of Boltzmann’s transport equation to the integral form. The differential form expresses the neutron balance in an infinitesimally small volume, whereas the integral form expresses the history of a neutron. So we can formulate the latter as follows. Referring to Figs. 1 and 2, let us consider the neutrons directed from -B to B on the line -B, -A, A, B. By considering the history of a neutron on the path in the moderator, we obtain the integral form of Boltz- mann’s equation as follows. For

-jn-sin-‘:) >g,>7r, j7r-sin1~)<~<7r

e-+i - 2’) T,w,e,~~,4

+ rl s A’ dx’

--d sin e- -$ (A - f’)

(Toy + s)v,e,~,w

’ dx’ + s

_e-&pt’) d sin e T+w’A~‘,4

z dx’ +

s _ e- &,z - z’)

--H sin 8 T,@(r’,f4p’,4 @a)

Z-oxis

FIG. 2.-Actual neutron path (-i, --A, 5, E) and its projected ._- ..-. . . . .-. path (-B, -A, A, 8) on a plane perpenalcular to the L-axis.

28

For a -sin-l! < tp < sin-l- 9

r r

O(r,B,gyi) = e- sin 0 Sz - d) j, (l?Jla)”

x r&l ( s * dx’ _ L(B _ z’)

Gz3 e sin 0 T+Wr’,&p’,u)

+ r0 $_I: & e- 8lFi+(-’ - “) Tl 0(r’,tl,$,u)

-4 dx’ + I_, sin e- 233 (A - “) (T, Y”+ S)(r’,f3,q+,u))

+ s = dx’ _ =I - e

A Sin 8 sin@ - “) T@(r’,B,q’,u). (W

For neutrons traversing the path -AA in the fuel, we find, for -v < v < rr,

+ r1 s

* dx’ A &?j

e- & (* - “) Tl@(r’,Q’,u)

+ - s

-A dx’ Sin 8

e- se+_4 - 2’) vWY,p’,u)

-B 1

+ s

2 dx’ _ Ho - e -A sin 8 aB(’ - “) (TOY! + S)(r’$‘,p’,u)

(54 Furthermore, for neutrons which do not intersect the fuel region, that is for the path -B’, B’ in Fig. 2, we obtain for

77 - sin-l a a ; > v > sin-’ ;

and

-((n-sin-l:) <p,<-sir+:

@(r, e,v,u) = e - se@ + *‘) *sO r;2n

* dx’ X

(1 --B’sin e- $,,w - Z’)

WW%‘,~) 1

= +

I

dx’ _ 5 (z _ St) - e aln e -hne W(r’,%p’,u) (54

In equations (5a-d), the notation is as follows,

r. L e - $ead, - @%I*%

rl = e- .& (bd - @2skPp -ad - (+rPp) (6)

-3z,gl/* - (pi”% rl’ = e he

B = B’ = fb r 2 J 0 1 - ‘i; sin29

A=fa r 2

J 0 l- a sin2p,

x=rcosfp

x’=r’ r 2 J 0

l- 7 sin2 q r 1 (8)

~‘=sir+(:;sinf+J) J

We take plus or minus signs according to the sign of v. The equations as they stand determine \r(r, 0, q, u) and cO(r, 8,q, u). In calculating the resonance absorption in the fuel, however, we are concerned with

Y000(u)=/~wdy[sinf3d0~rdrY(r,Qp,u) (9)

and we should prefer to work with an equation that contains Y,,OO(u) as an explicit unknown, since Y,,, is a ‘zero space angle moment’ it is natural to transform our equations further by expanding Y and Q, in terms of their respective space and angle moments. Accordingly, we write first

where Ynrn are orthogonal functions with respect to 8 and V,

01) and E, is the Neumann factor, e. = 1, E, = 2 (n = 1,2, 3,. . . ). Also, O,(r) and Z,(r) are defined as follows,

~O,(r)O,(r)r dr = rq (1%

O,(r) = 1

s ‘Zt(r)Z,,,(r) 0

dr = f 6,, 1 Wb)

ZOO = 1 ] From these functional properties, c$,,+Ju) and ~~,~,r(u) of equations (10) are

+tS,%l04

Resonance escape probabilities in circular cylindrical cell systems 29

Thus, by the transformation, we have reduced our system from one of four equations to one of two.

After substituting equations (10) into equation (5a-d), we consider the effect of the scattering integral operation T, on functions of u, ~9 and Q, by the usual technique of representing the delta function equation (2) in terms of Legendre polynomials

fi . 6’ and X,(u - u’), thus

S[6 . sit - X,(u - u’)] m co

in in

One can show that the operator Tk is diagonal in the Y,m(& p) representation, i.e.

J J

= ,)‘dd C,(u’) 4 M s (1 + MkY k

x e-(” - u”PB(~k(u - u’))6,,d,& (15) For the special cases Mk = 1 and Mk+ co, the term on the right is

dufe-(U - U’) p,(e- c” - “‘““)C,(u’)S,,S,,,

74 nz = 0 =

I (16)

n=O

s B dx’ Xl X -e-G(B -2’)

A sin 13

s -A dx’

+ - e - s0 (-A - 2’) 0,(/j -B sin 8 I

x y,‘~m’(bP’) (184

UW

30 H~ROSHI TAKAHASHI

s A dx' X

-Asin

e- -&(A - 2’) Its (r’) Y,,T’*‘(O,cp’)

(S -(n - sin-‘(a/r))

+ --n +I_ s,,a,,,) dvpn 0 de

b x s r drO,(r) Ynm(O,y) e- sin e

a

s d dx’ _ =o x -

-d sin 8 e sinB(d - “)&,(r’) Y,?‘h’(8,#) (1 &I)

APPROXIMATE SOLUTION FOR THE RESONANCE ESCAPE

Following CORNGOLD’S discussion for the slab lattice system, we proceed as follows. We assume that if the moderator is a non-capturing medium and its volume is not great, the flux density in the interior of a cell of moderator will be ‘reasonably flat’. Moreover, ‘reasonably flat’ space behaviour implies ‘reasonably flat’ angular dependence. Consequently we can guess that d,,,, will be considerably larger than all other $p?,m,2. On the other hand, for sufficiently small &a, the flux in the fuel is fairly smooth. For a strong absorbing resonance, although the v,,~,& is of comparable order to yooO, its absolute value is small, and its contribution to the sum in equation (17) is consequently also small. In other words, we are neglecting the disadvantage factor, etc. The simplest set of simultaneous equations is obtained by neglecting moments of higher order than zero, in equation (17):

woo0 = A~,““00)7’,00&,, + ~jpoo”“‘(~,ooy,,, + &,,,),

(19a)

f&0o = ~~)(~)T1oo~ooo + 7i~od)~~0°)(Toooyooo + So,,)

(19b) A$)@“), etc. are calculated as follows,

I = x:,C,(@ - $)F (20a)

where F is given by

F= $~cosIpdgs”si&j&(‘--,‘“(;,,f”

h n/f =-

s 0 dw cos Y joK,(24u + 4)

_" Ki,(2n(u + U) + 2U) - Ki,(2n(U + U) + 2U)

+ J&(2@ + l)(u + m (21) y is the angle (rr - 9~) at r = a, and

24 = &a cos w 7

o==C,[b J 1 --dsin2~-Ilcos~] J

Similarly,

@!JJ)@o) = $ [I - a2z01;1 0

&jy”“’ = g [l - (62 - 4&F] 1

loo = a2Co&F

Substituting these expressions into equations (19), we get woo0 = (& - &)tT.o~Y~oo + sooo>

F + C&(62 - a”) ~04000~ CW

do00 = $3 ~~oooYooo + So,,)

+ (; - B,P(b2F_ u2) 1 p"#OOO Wb)

These equations for the equivalent circular cylindrical cell are similar to those obtained by CHERNICK and CORNGOLD for a slab lattice. We can solve the equations in the same way as CORNGOLD does. If we assume that the moderator is hydrogen M = I, and that the fuel atom has infinite mass, the resonance escape probability P is expressed by

P= (24)

where

(25)

Resonance (eV)

6.70 20.9 37.0 66.5 81.6 90.0

104.0 118 Sum

00 (barn)

23,000 3 1,000 37,000 24,000

2200 100

18,OOO 8000

-

-

-

Resonance escape probabilities in circular cylindrical cell systems 31

TABLE 1 .-NUMERICAL CALCULATIONS FOR “‘U RESONANCES

I’,mv I

r,mu 1 r./rn

1.5 24 16.0 8.4 25 2.98

32.0 29 0.906 27.0 17 0.630

1.9 2.5 13.1 0.09 25 27.8

70.0 25 0.357 14.0 25 1.79

-

NUMERICAL RESULTS FOR THE RESONANCE ESCAPE PROBABILITY

We now compare the resonance escape probability given by the circular cylindrical cell approximation with that obtained for the slab lattice approximation as discussed by CORNGOLD.

Numerical calculations have been made using published resonance parameters for the eight lowest resonances in 238U. The system under consideration is composed of 0.200 in. radius rods of 238U separated by light water moderator, and the equivalent cell radius is O-28284 in. The macroscopic total cross- section of light water is taken as C, = I.378 cm-r. This lattice corresponds to a slab lattice which is composed of 0.200 in. thick plates of saeU separated by layers of light water of optical thickness I&b = O-700, and with fuel-moderator volume ratio approxi- mately 1 : 1. (CORNGOLD calculates the resonance absorption in this slab lattice.) The temperature of the fuel was taken to be 300°K and the appropriate Doppler-broadened cross-sections, modified by a potential scattering cross-section of magnitude oSp = 10 barn, were used at each resonance.

We calculated the quantity Z in p=e-r (26a)

I&lab) I,(cylinder) 1, - I, --_

1,

0.1205 0.1151 0.1038 0.1089 0.05222 0.0488 1 0.04394 0.1108 0.047 15 0.04330 0.0395 I 0.09592 0.01418 0.01353 0.01211 0.1174 0.004357 0.004839 0.004232 0.1434 oOO1079 oOO1075 O~OOlOO8 OO6610 0.01370 0.01318 0.01195 0.1032 OOO6059 oOO5901 0.005249 0.1244 0.2952 0.2457 0.2218 0.1080

-

as given by the first order solution equation (23); namely

z = c I4I BP,F _

Jo in + ~P,F \

The results are shown in Table 1. We see that the resonance absorption integral in the

circular cylindrical cell approximation is smaller by about 10 per cent than the value given by the slab lattice approximation. CORNGOLD’S numerical value is also given in Table 1 but it differs from our value

because we have chosen -300 < E-E, - < 300

r/2 as the integration range of u in equation (26b). He has taken the larger values for the integration range of u than our values.

Acknowledgements-The author is extremely grateful for many valuable discussions with H. TSUT~IJMI, A. SUGIMOTO, J. CHERNICK, N. CORNGOLD, and W. ROTHENSTEIN.

The author wishes to thank Miss F. MIYAUCHI for carrying out the computations.

REFERENCES

CHERNICK J. (1955) Proceedings of the First international Conference on the Peacclful Uses of Atomic Energy, Geneva, P/603, United Nations, New York.

CORNGOLD N. (1957) J. Nucl. Energy 4, 293. TAKAHASHI H. (1960) J. Nucl. Enecyy. This issue, p. 1.