C A L C U L A T I O N OF W A T E R - M O D E R A T E D W A T E R - C O O L E D R E A C T O R S

(UDC 621.039.51)

E. A. G a r u s o v a n d Y u . V. P e t r o v

Translated from Atomnaya gnergiya, Vol. 17, No. 5, pp. 378-379, November, 1964

Original article submitted September 23, 1963

An expression is obtained for the probability of avoiding leakage in reactors with uniform energy evo-

lution. From an analysis of this probability, together with the Fourier components of the experimental

slowing-down function and the slowing-down functions of various diffusion models for ordinary water, it is concluded that the single group model is fairly accurate forthe calculat ion of active zones con- raining not less than 10 liters of water. Increasing the number of energy groups without going over to

nondiffusion theory lowers the precision.

The calculat ion of a nuclear reactor of given composition and dim~mions should primarily give the correct value of the effective neutron breeding coefficient (Keff). If the reactor operates mainly with thermaI neutrons, the value of Keff will depend substantially on the probability of avoiding neutron leakage from the active zone during

the slowing-down process. This probability depends on the distribution of fast neutron sources in the active zone and the geometry of the reactor,andalso on the slowing-down function, i .e . , the probability that a neutron generated at a point r 0 will becomes thermal at the point r - O ( r , g ) . Since the exact calcuiat ion of the slowing-down function is a complex problem, it is customary to use various computing models (age, theory, multigroup, approximations, etc.). In the particular case of an infinite moderating medium, the slowing-down function is usually known from experiment.

In using one or other of the moderating models, we need to be sure that it gives the correct value of neutron leakage, especially for smallish reactors, in which the leakage is considerable. It is most difficult to select moder- ating models for reactors with hydrogen containing moderators. In view of the sharp variation of the scattering cross

sect ionof hydrogen with energy and the small number of collisions needed for slowing down, it is not a priori clear whether it will be possible to use the diffusion approximation. It is thus interesting for reactors with ordinary water to compare the probability of avoiding neutron leakage calculated from the experimental slowing-down function with that calculated from the diffusion model of the process.

1. The function G(r,r0) was determined experimental ly in [1], where the distribution of neutrons with indium resonance energy from a point source of U 235 fission slowed in ordinary water was measured at distances up to 92 era. This distribution led to the neutron age 7" = 30.7 cm 2. It was later shown, however, that there were some systematic

errors in [1], mainly caused by the great thickness of the U 235 target, which gave rise to "corrosion" of the resonance neutrons near the source [2]. A new value given in [2, 3] was f = 27.3 • cm 2. The value of G(r,r0) was obtained for a very thin U 235 target up to distances of 40 cm in [3]. This distribution, however, contains a large error at great distances owing to the smali dimensions of the water tank and the small number of counts. In order to obtain values of the function G(r,r0) correct for both small and great distances from the source, the curves of [1] and [3] were

nsewn together" in the range 11.5-23.5 cm, in which one would expect the two curves coincide. The so-obtained slowing-down function Gexp(r,r0) was used in subsequent calculations. The values of the moments of the slowing- down function

c o

.~ aex p (r) r '~+2 dr (rn}__ 0

o o

!" Gex p (r) r2 dr 0

(1)

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TABLE 1. Experimental and Calculated Values of the Moments of Slowing-Down Functions

Means of determinat ion

Experiment. single group ap'pr'oxi/nation 2: : : : : : : : : Two-group approximation (a = 0.216) . . . . .

Age approximation . . . . . . . . . . . . . . . Flight -age approximation . . . . . . . . . . .

1 T~-~- <r2>,

cm 2

27,3-/-1,0 27,3 27,3 27,3 27,3

<r>~cm

10,6 10,5 10,9 t t ,8 10,&

< rs> , c m s

3,6.t03 3,4. t03 3,2.t03 2,6.t08 3,7.108

<r4>, am 4

1,07.105 0,89.105 O, 74.1(15 0,45.105 1, t2. t05

<7 6>,ClTI 6

1,95. t0s t ,03.t08 0,68.10s 0 , t 7 . t 0 s t , 98. t08

are given in Table 1. For comparison the table also shows the moments for several frequently used diffusion slowing- down functions as well as for the f l ight -age function. In calculat ing the moments of the slowing-down functions, the constants in them were chosen in such a way that the second moment should coincide with the exper imenta l value.

2. A. Weinbert and E. Wigner [4] formulated the second basic theorem in reactor theory, according to which the probabil i ty of avoiding the leakage of neutron s out of a reactor without reflector (P) during the slowing-do,. 'n process equals the Fourier component of the slowing-down function of neutrons from a point source in an infinite medium

P = G (B) ------ I G (r) e iBr dr. (2)

Figure 1 shows the Fourier components for the exper imenta l slowing-down function and for various theoret ical functions in the diffusion approximation, We see from Fig. 1 that the one-group theory of decelerat ing neutrons agrees best with ~exp (B). For values of B < 0.23 cm -1, which corresponds to a spherical reactor with an act ive zone

~ o r .

volume of around 10 liters, the difference between Gex p (B) and G I (B) lzes within the l imits of exper imenta l error. The ~II (B) curve passes below the ~I (B), and the difference between ~exp (B) and ~II (B) for spherical act ive zones less than 50 liters in volume lies outside exper imental error.

From the general form of the Fourier component for the N-group approximation,

N N

l i t ( l @ a ~ . B 2 ) , ~ j {2 i = I (3) i----I i = l

it follows that the breaking up of any energy group of neutrons into several groups can only reduce ~ (B), i . e . , we always have ~N+K (B) < gN (B). For smal l B, a l l the functions G N (B) are close to each other; with increasing B,the difference between them rises. Hence, increasing the number of groups can only lead to an increase in the leakage of neutrons out of the reactor, especial ly for smallish act ive zones. The maximum error occurs for N~.o, i ,e . , on

using the age theory (see Fig. 1).

The second fundamental theorem assumes that O(r, r0) = O(Ir -rob. This assumption is approximate ly valid for fair ly large reactors, It is shown in [5, 6] that , for a direct-fl ight function with P ~ 0.5, the difference between P and ~ (B) in the case of a plane reactor is approximately 20%, increasing with increasing anisotropy of scattering. It is thus desirable to obtain the probabil i ty of avoiding neutron leakage from small act ive zones without using the

second fundamental theorem.

3. The probabil i ty of avoiding neutron leakage during the slowing-down process for a reactor with an infinite reflector may be ca lcula ted quite simply if the slowing-down properties of the reflector and the act ive zone are the same. In this case G(r,r0) = G(Ir, to]). Let the fuel in the reactor be so disposed that the number of fissions at each point of the reactor is the same. Then for a spherical act ive zone of d iameter D the formula is given in the Appen- dix [see (A-3)]. Figure 2 shows P (G,D) for light water as a function of the inverse d iameter , both for the exper i - mental function Gex p and for various theore t ica l functions in the diffusion approximation. As in the previous case, right up to volumes of the order of 10 liters (D ~27 cm), the deviat ion between P(Gex p, D) and P(GI, D) c l ea r ly l i e s within the l imits of exper imenta l error. The two-group and age theories give increased values of neutron leakage out of the active zone. The same result follows by substituting the values of <r> and <ra> from the table given into

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0,6

l \ , 0 0,08 0,Is 0,2~ B, cm -t

Fig. 1. Fourier component of slowing down functions in ordinary water: 1) ~exp (B), the result of analyzing e x -

p e r i m e n t a l data; 2) G I (B), one-group approximation;

3) ~II (B), two-group approximation (a = 0.216); 4)~B (B), age approximation.

0,~

0,s

\ I

N

"-4

0 0,G2 0,0~ tl06 D'I, c m - ~

Fig. 2. Probabili ty of avoiding neutron leakage from a spherical act ive zone: 1) P(Gexp,D), the result of ana - lyzing exper imenta l data; 2) P(GI,D), one-group approx- imat ion; 3) P(GII , D), two-group approximation (ct= 0.216); 4) P(G B, D), age approximation.

the asymptotic formula [see (A-4)]:

3 (r) ~_ t (r 3) P a s ( G , D ) = I 2 D ' 2 " D - - 5 - (4)

We see from formulas (A-6) and (A-8) of the Appendix that, for an act ive zone in the form of a cyl inder of a para l le lep iped , the sign in front of the main term <r> remains unchanged, and i t is thus reasonable to suppose that the foregoing conclusions are not dependent on the specific form of the act ive zone,

4. It follows from the results given that the theory of one group of decelera t ing neutrons describes the leakage of neutrons out of the reactor with a ccep t - able accuracy right up to volumes of water of the order of 10 liters in the act ive zone. Any at tempt to improve on the energy spectrum, stil l remaining with- in the framework of the Pl approximation will give worse agreement between theory and experiment . In par t icular , the split t ing up of the decelerat ing neu- trons into two groups, as proposed in [7], will cease to be valid for act ive zones containing less than 50 liters of water (see Figs. 1 and 2). The large error arising in calculat ions of uranium-water systems made in the Pl approximation by the multigroup method, using 21 energy groups of neutrons, is noted in the monograph of G. I. Marchuk [8].

On increasing the number of points in the e n - ergy scale, the theory must be of the nondiffusion type. The physical reason for this is that the neutron picks up the main part of the migrat ion area in water as a result of pr imary collisions. Inside the upper energy group there remain only those neutrons which were scattered at small angles, i . e . , the neutron dis- tribution is extended substantially forward, and full diffusion act ion fails to take place .

As seen from Figs. 1 and 2, nondiffusion cor rec- tions are especia l ly important for act ive zones with volumes considerably below 10 liters, since the free path of the fast neutrons becomes comparable with the dimensions of the act ive zone. Furthermore, in this case the reactor operates mainly with in te rme- diate neutrons, and the energy spectrum must be known more accurate ly .

It should be noted that the one group theory does not agree with exper imental values in respect of the ratio of the thermal neutron flux to the power of the point source [9]. The reason for this is the divergence of the one group slowing-down function Gi(r,r0) at the point r = r0. In order to remove the divergence, the following model may be proposed: The neutrons t ravel without energy loss, but after the first coll ision pass into the lower energy range, where, for example , the diffusion-age approximation may be used. The two parameters of the model (the effect ive free path of the fast neutrons, X, and the effect ive age, r l ) may be obtained from exper iment , if we pos- tulate the cor~servation of the total migrat ion area during the slowing-down process, 2X~+ 6rl = <r2> and the value of the slowing-down function G(0, 0). For neutrons dece lera t ing to the indium resonance, X = 7.88 cm and ~-1=6.56 cm 2.

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We see from the table that this f l ight-age model brings the theoret ica l and exper imenta l moments into satisfactory agreement. Of course this is not the only model , and may well be far from the best.

We note in concIusion that we have considered ei ther reactors with no reflector or those with a reflector as bad as water. The use of better reflectors lightens the requirements regarding the nondiffusion character of the theory.

The authors thank L. N. Yurova and A. A. Polyakov for permit t ing detai led study of their own results, and also G. A. Bat', A. N. Erykalov, and V. V. Orlov for discussing the results and for valuable comment .

A P P E N D I X

The probabil i ty of avoiding neutron leakage from the reactor during the slowing-down process, when the re- tarding properties of the active zone and infinite reflector are ident ica l , and When there are the same number of fissions at each point of the act ive zone, may be written in the following form:

p = I dr ~ droG(I r--ro I) ~ (rs--r) # (rso--ro)

i" dr i droG (I r - - r o [) O (rso--ro) (A-:)

where G(r) is the derivative of the slowing-down function, r s the radius vector of a point on the surface, ~(r s - r ) = 0 for r >rs a three-dimensional l~-function cutting out the corresponding integration range. i for r-<t s

Introducing new variables p = 1/2 ( r - r0), P0 = 1/2 (r + r0), and noting that drdr0 = 2 3 @ dp 0' we obtain

p = i d~ G ( I 2Q I ) I dOot~ (rs--Oo--O) e (rso--Qo@0)

V ~dQG(12QI ) (A-2)

where V is the volume of the reactor active zone.

Let us ca lcula te P for several part icular cases. For a spherical active zone of radius R, the integral with respect to @0 in (A-2) is taken over a range determined (Fig. 3) by the inequali t ies:

- - t < cos ~ < @qD (~0), - - ~ (eo) < cos ~ < + t,

where

29Qo

Carrying out the integration, we obtain (D = 2R)

D

i G(9) QZd~( t 23 DQ ~-2- 'D:XJ

p = o (A-3)

i a (9) Q2 d e 0

At large distances G(p) falls off at least exponent ial ly . Thus for large D the asymptotic expression

P a s = t 3 <r) t (ra) 2 " D @ - 2 " Da ' (A-4)

is val id to exponential precision, <rn> being the moments of the slowing-down function [see formula (1)].

For an active zone in the form of a rectangular para l le lepiped, the three-dimensional integral in @0 decom- poses into the product of three one-dimensional integrals over a region determined by the following inequali t ies

(Fig. 4): a i+ gi -> gi0 -> gi; a i - g i -> gi0 -> - -g i , where a i is the length of the i - t h edge of the para l le lepiped , and gi0 and gi are current Cartesian coordinates.

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;I \ , , ' vVej*cP, J

Fig. 3. Region of integration for circular section of the active zone.

Fig. 4. Region of integration for a rec- tangular section of the active zone.

Integrating over dp0, we obtain

i=3

i=I

P= f deG (e) (A -5)

If the mean path of a neutron in the zone /" = 4V/s is much larger than the slowing-down length, then the follow- ing asymptotic expression holds to exponential exactness:

P a s = t_a la2@a2a3@a3al , 2 a l @ a 2 @ a a ( r 2 ) 2ala2a 3 < r) ~.. 3~ ala2a 3

< r% 47~aia2a3 " (A-6)

For a cylindrical active zone of radius R and height H, the three-dimensional integral with respect to dp0 splits into the product of a one-dimensional integral over the range indicated in Fig. 4 and a two-dimensional integral over the region shown in Fig. 3. By integrating over alp0, we obtain

P =

2 I dQG(~)(I---~)t~(H--Z)~,(Z)[I_~ / t _ _ ( b ) 2 - - 2 a r c s i n ~ l

f dQC (e) (A-7)

If the dimensions of the active zone are large, then with an accuracy up to terms ,., <rt>/D 4 we obtain (D = 2R):

I ( 1 2 ) 4 D (r2) I <rS) &~=t- -~- i D ( ~ § D2 ~-~-" D~ " (A-8)

LITERATURE CITED

1. I. Hill, L. Roberts, and T. Fitch, Appl. Phys., 26, 1013 (1955). 2. D. Lombard and C. Blanchard, Nucl. Sci. and Engng, _7,448 (1960). 3. L .N. Yurova, A. A. Polyakov, and A. A. Ignator, Atomnaya gnergiya, 1__2,151 (1962). 4. A. Weinberg and E. Wigner, Physical Theory of Nuclear Reactors [Russian translation], Moscow, IL (1961). 5. K. In6nti, Nucl. Sci. and Engng, 5 , 2 4 8 (1959). 6. L. Dresner, Nucl. Sci. and Engng, 7_, 419 (1960) 7. R. Dentsch, Nucleonics, 15, 47 (1957). 8. G . I . Marchuk, Methods of Calculating Nuclear Reactors [in Russian], Moscow, Gosatomizdat (1961). 9. E .A. Garusov and Yu. V. Petrov, Atomnaya ~nergiya, 15; 71 (1963).

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