THE UNIVERSITY OF MICHIGAN INDUSTRY PROGRAM OF THE COLLEGE OF ENGINEERING DIFFUSION LENGTHS IN BIETEROGENEOUS MEDIA:Joel H. Ferziger: G e~ eo rge. C'' Wang -i Paul F. Zweifel' Department of Nuclear Engineerring The University of Michigan Ann Arbor, Michigan IP -468

IJ.i/ I A

ACKNOWLEDGMENTS We wish to thank Professor R. K. Osborn for several illuminating discussions and the National Science Foundation and the Babcock and Wilcox Company, Atomic Energy Division, for their support. ii

TABLE OF CONTENTS Page ACKNOWLEDGMENTS............................................ ii LIST OF FIGURES.............................................. iv DIFFUSION LENGTHS IN HETEOGENEOUS MEDIA I. INTRODUCTION...................................... 1 II. CALCULATION OF THE DIFFUSION LENGTH. 4 A. Isotropic Scattering, Flat Flux.................. 5 B. Anisotropic Scattering........................... 12 C. Small Moderator to Absorber Ratio................. 14 D. Anisotropy Factors................................ 19 E. Flux Depression................................... 20 III. RESULTS............................................ 22 IV. CONCLUSIONS........................................ 24 BIBLIOGRAPHY................................................ 40 iii

LIST OF FIGURES Figure Page 1 Geometry for Calculation of io and PAo 7......... 2 Geometry for Calculation of k1 and PAl............ 17 3 Natural U02(10g/cc) in Pure H20.................. 27 4 Natural U02(10g/cc) in Pure O20................... 28 5 Natural U02(10g/cc) in Pure D20................... 29 6 Natural UO2(10g/cc) in Pure D20............ 30 7 l1.00% Enriched Uranium in Pure H20................ 31 8 l1.00% Enriched Uranium in Pure H20................ 32 9 1.15% Enriched Uranium in Pure H20................ 33 10 1.15% Enriched Uranium in Pure H20................. 34 11 1.30% Enriched Uranium in Pure H20................. 35 12 1.3 % Enriched Uranium in Pure H20................. 36 13 1.00% Enriched Uranium in Pure D20................. 37 14 1.00% Enriched Uranium in Pure D20................. 38 15 1.00% Enriched Uranium in Pure H20................. 39 iv

ABSTRACT An adaptation of Behrens' Method(l) to the calculation of diffusion lengths in heterogeneous media is given. In all cases, the diffusion length in a medium containing absorbing lumps can be related to the self-shielding factor of the lumps. Calculations are presented only for the simplest case given but the results display considerable disagreement with a frequently used formula. On grounds which are mainly intuitive, it is believed that this method is more accurate, particularly for large moderator to absorber ratio. Final conrlcusions cannot be drawn, however, until more experimental data becomes available. Calculations using some of the corrections and evaluation of some of the integrals shown here will be given in a future paper.

DIFFUSION LENGTHS IIN HETEOGENEOUS.MEDIA I Introduction A knowledge of thermal diffusion lengths for nonmhomogeheous media is of particular importance in survey type criticality and shielding studies for which an accurate estimate of the thermal leakage may be required~ In cases in, which inhomogeneiLties are not very large compared to a neutron near-free path the use of the homogeneaous reactor thqory with appropriate self-shielded eross sections probably gives reasonable results- Hownever, in a typical low enrichment reactor, the inhomogeneities may be large, and some other approach musat be sought~ Weinberg and WignerO) and Russell2( have defined the diffue sion length, L, as that value which makes the well known formula l+L2B 2 () yield the correct thermal leakage, Bg being the geometric buckling of the system. As the leakage is the quantity in which one is directly interested, this is a very convenient means of defining the diffusion length. By applying diffusion. theory to alternating slabs of moderator and fuel, Weinberg and Wigner arrive at the result~ 2 2 L2 = L~ f + L;M (l:f (2) where 2 and I are the diffusion areas for the pure fuel and moderator respectively and f is the thermal utilization, defined conventionally. 1

In slightly enriched uranium-water systems, the two terms on the right hand side of Equation(2) are of comparable magnitude, whereas is a similar homogeneous medium, the second term alone represents the cora rect diffusion area, Thus, this method gives L2 incorrectly in the homogeneous limit. The difficutly apparently lies in the application of diffusion theory to thin. regionso By using a variational principle, Russell arrives at the result L~2'= D/ a (3) where D and 1a are the flux-volume averaged diffusion coefficient and macroscopic absorption cross section, respectively. This method does not appear to be completely satisfactory either. First, the details of the enexngy-space distribution. of the flux are required for computing the required averages and second, directional effects are neglected. In rodded media,, one expects a large diffusion length parallel to the rods than perpendicular *to them. This effect has been found to be significant and is taken into account by the French( in the design of the their graphite reactorso By retaining diffusion theory, with. a tensor rather than. a scalar diffusion coefficient, one finds the leakage to be 1+ B 4 2 2

-5Here L~ and B2 are the diffusion area and the buckling in the i direction, the i being the principal axes of the diffusion tensor~ Spinrad() and Shevelev(5) are able to predict different L2 in various directions when anisotropies exist0 They define L in terms of the asymptotic decay of the neutron flux from a source and apply diffusion theory to thin region.s. There is some question as to the validity of this procedure~ In. fact) Spinrade s method has the wrong homogeneous limit. (6) Triflaj - applied a variational principle to the transport equation and arrived at an. expression for the diffus3on length, which, while apparently accurate is rather cumbersome0 A result amenable to hand calculation would appear to be more desirableo In searching for a new method for calculating diffusion lengths) one is struck by the remarkable agreement of Beherense theory(7) with experimental resul~ts(8) for the case of a homogeneous -medium containing empty holes. The metlhod employed here is in essence an adaptation of Behrens' method to the more general case of a heterogeneous medi.um.

II CALCULATION OF THE DIFFUSION LENGTH As a starting point we take the definition of the diffusion length as one-sixth of the mean square crow flight distance that a neutron travels from the time it is thermalized to the time it is absorbed. For convenience, we shall, deal only with infinite media in which the structure is periodic, so that a unit cell may be studied. In such. a medium the thermal flux shape is also periodic and it is assumed that this shape can be calculated. First, we seek ~2, the me-an "squ are distance travelled by neutrons in the stationary distribution. (It is assumed here that the neutrons have the same spatial distribution after each fight, and that the effect of sources and sinks balance so as to maintain this distribution.) Then, PA' the probability that a neutron is absorbed in a.single flight~ is to be calculated. UnTder the assumptions made above ~ and PA are identical for each flight made by a neutron, and one can write ~2 L2 = (5) provided that scattering isisotropic, ioeo, provided that the direction of a neutron's travel does not depend on. its past history. Equation (5) results from the fact that under the assumption of a stationary distribution PA is constant for each flight, and thus 1/PA is the average number of flights made by thermal neutron. before being absorbed. The assumption of a stationary distribution is very nearly fulfilled when a neutron makes many eol.li.sions before being absorbed.

In calculating B and PA, it will be further assumed that the lumps are widely separated and thus a neutron has a negligibly small probability of entering more than one lump of a given flight. (This restriction may be relaxed in a subsequent paper.) Then contributions to Y2 and PA come from five sources: 1) Neutrons suffering their ith. collisions in the moderator and their (i+l)st collisions in. a lump; 2) Neutrons suffering their ith collisions in the moderator which pass through a lump and suffer their (i+l)st collisions in the moderator; 3) Neutrons which do not enter a lump at all. or a given flight; 4) Neutrons which suffer their ith collisions in a lump, leave the lump and suffer their (i+l)st collisions in the moderator; and 5) Neutrons which do not leave the lump at all on a given flight. The probabilities of each type of event will be calculated, and the mean square flight distance and absorption probability will be averaged over the events to yield the required value of 2 and PAo A. Isotropic Scattering, Flat Flux As a first example, the following highly idealized case will be considered (some of the previous assumptions repeated for clarity). a) The medium is composed of only two materials, one of whic'hs highly absorbing; b) The absorber is in the form of widely separated lumps, distributed periodically in the moderator;

c) Scattering is isotropic (Equation (5) applies); d) The neutron flux is uni.form in the moderator; e) Neutrons scattered in the absorber have the same 22 and PA as those scattered in the moderator, (Contributions (4) and (5) above can be neglected, it can be assumed that all. neutrons start in the moderator.):Some of these restrictions will be relaxed later. Macroscopic cross sections will be denoted by Z for the heavily absorbing material and a for the moderating material, both with the conventional subscripts a, s, and tr for absorption, scattering, and transport respectively. Lack of a subscript will denote total cross section~ We define the directional chord length distribution?f(R,)~, such that r(R,_ )dR is the number of chords in the lump with lengths between R and R+dR in direction.n, (see Figure 1)o It is related to the conventional (normalized) chord length distribution qj(R) defined by Case, Placzek and DeHoffmann(9) by~ p(.R) f (Rn Q)d. (6) where S is the surface area of a lump; cp(R)dR is the relative n:umber of chords with lengthsbetween R and R+dR in a lumpo A volume v of moderator is associated with each. lumpo From Figure 1, one sees that the neutrons which travel a distance between z and z+dz in direction 2 before entering a lump, and then enter the lump such that they travel along a chord of length between.R and R+dR, must have started their

(, I' " ~'"'/' ~dy F ~ ~~~igure~~~~~d 1u Figure 1. Geometry for Calculation of io and PA

flights in the element of volume dav r(R),Q)dRdz The fraction. ofneutrons in volume elements of this type is dv/v, under the assumption of a flat flux distribution~ But these neutrons are attenuated by a factor e- z in travelling to th.e lumpo Therefore, -the probability that a neutron travels a distance between z.and z+dz, in direction 2, and then enters the lump along a chord with length between R and R+dR is, q(z,R, 2)dRdz e t(,n ) dRdz (7) v Since the probability of travelling a distance between y and y+dy(y<R) in the lump and then colliding is Z e' ZYdy, the square distance travelled by those neutrons which make their next collisions within a lump is~ fdRf nfdz dy(y+z )2q(z,R, 2)L e-Y (8) o 4II o 0 The z integral is allowed to go to infinity as a result of the assumed wide separation of the lumps; the factor 41T is for normalization purposes. On the other hand, the neutron has probability e- R of going through the lump and it may then travel a distance between x and x+dx,, with probability ae-oxdx, in the moderator after leaving the lump. These neutrons contribute 00 00 00 jdfodRfdzfdx (z+R+x )q ( z, R, 2) e (9) 4ITo o o to the square distance travelled (12).

-9Lastly, some of the neutrons may not enter the lump at all on a given flight. For these neutrons, it does not matter what is in the lump. Thus, it may thus be assumed to be a vacuum, and we may follow Behrens' argument to say that the probability of enteri.ng the hole for a neutron which travels total distance between w and w+dw in the moderator (w = z+x as defined above) in direction 2 on a given. flight is given by: w P(w Q)d= dw f dR f 4z' dx q(zRJ2 )e- Z0e -x (z+x-w) o o 0 (.10) 0 =,we"~W dw /,(R,[)dR a result derived by Behrens'. Then the conrtribution to i2 of neutrons which do not enter the lump is~ 00 f d f w2 [ -P(w, ) ] ae Wdw (11) 11 0 ( It may be noted that the probabilities of the three processes described above add to unity~ The integrals may'be carried out and expressed in. terms of the escape probability (self-shielding factor) of the lumps~ The result, obtained by adding expressions (8), (9), and (1l), is: 2 = s( lg) [~ + ]2-aY~ -'] + -( ) 1. ) 1R 1 1 ~+(~) + 2 (12) v [2 V dZ2C 2-'2'2

where the function g is defined by: g = f df fe *R(R, 2)dR = 1 - aR>Po (13) Here, Po is the usual escape probability and is tabulated for spheres, slabs and cylinders in Reference 0 The average chord length <R> is simply the surface area of a lump, S, divided by four times the volume, V, of the lump. To obtain the diffusion length, one needs only to calculate the probability that a neutron is absorbed on a given flight, PAo From previous arguments, the probability that a neutron in the assumed flat distribution outside the lump will suffer its next collision in a lump is given by Expression (8) with the (y+z)2 term deleted from the integrand: PCa=-f m U dR f dz f dy q(z,R Q) e- - S (l-g) (1.4) It would be unreasonable to assume that all of these neutrons are absorbed, especially since the absorption and scattering cross sections are approximately equal for low enrichment uranium. A better approximation is to assume that only Za/. of them are absorbed. This explains why assumption (e) above was invoked. Neutrons do scatter from the lumps, but if they have the same 22 and PA as neutrons scattered from the moderator, thei:r effect on. L2 can. be neglected. When the ratio of moderator to absorber is high, relatively few neutrons are scattered from the moderator and these neutrons canbe safely ignored,, This is clearly a poor approximation for -small moderator to absorber ratios but

but is an excellant approximation for very heavily absorbing materials, eg. highly enriched uranium. Neutrons which collide in the moderator have probability aa/a of being absorbed, so the absorption probability becomes~ PAo =(2/)Pc + (aa/a) (lPc) (1) 4vS (1 g) + aa 1 4v (zg)] Z 41-va 4va By inserting Equations (12) and (15 into Equation 5)) one obtains a formula for the diffusion length. In addition to the neglect of scattering from the lump, there is another major flaw in the argument All o~f the integrals within the moderator have been extended to inrfinity as a consequence of the assumed wide separation. of the lumps. Again, this'breaks down if the moderator'to absorber ratio (M/A) is smailo Means of avoiding these difficulties will be given in section. Co In the limit of small lump dimensions, for constant moderator to fuel ratio, the result should go to -the homogeneous diffusion length. One can show that this is so, provided that (M/A) is sufficiently large. For small lumps, the simplified case Es = 0, aa 0 yields~ LZa/ + A (16) 0 3Zacr which, for large (M/A), goes to the result (3Z6Z)-1, which is correct for isotropic scattering in a weakly absorbing homogeneous medium, The validity, of course, depends on the cross sections to some extent through the ratio (Za/o)D

-12Bo Anisotropic Scattering If anisotropic scattering is present Equation (5) is invalid since the direction of the (n+l.)st flight depends on the direction of the nth flighto The method employed in the previous section is essentially a random walk technique. Generalization of this problem to anisotropic scattering if very difficult if a solution is to be obtained in closed form. Instead we have done the random walk problem for a homogeneous medium and obtained the not unexpected result~ L2 = 1 (17) where - is the average cosine of the laboratory scattering angle.For the same problem with isotropic scattering using the notation of section A, we have: 22 2/= PA = Za/Zt From which one obtains the well known result L2 = (13Zat ) (18) Comparing (17) and (18), one sees that (17) is derived from (18) by replacing the total cross section by the transport cross section,'-r = St - ZsT. From this it was assumed that for anisotropic scattering one need only replace the total cross section in the moderator by the transport cross section.

135 The derivation of Equation Cl( is of some interest~ Let P(ri.) be the probability that the ith neutron flight be represented by the vector r. Then the mean square distance travelled by a neutron in exactly n flights is <r2>n f 4 f (r1 r + orn)2(r P(rn)rloo (19) rl r2 rn nd nn Now, the P(ri) can, be decomposed into the product of the probability P(ri) that ri have length ri and the probability P(_Qi) that it have direction _io For integration purposes, it is convenient to use Q as the polar axis of the coordinates. Terms of the following type arise from Equation 19. 00 00 fdrl. o fdrnfdS21. fdan(ri rj)P(rl)P(_ l).,P(ri)P(-i )~~ o o (20) P(rj)P(aj)..P(rn)P(%) Now r r. depends only on the lengths of r and r. and the scattering angles for collisions i + 1, i + 2.. j. When the various probabilities P(rk) and P(Sk) are properly normalized, all of the integrals over rk for kii, j and all of integrals over Qk for k<i and k>j may be carried out to yield unityo Then Equation(20)reduces tofriP(ri)adrf rjP(rj)dr jdnfi+lo ~ofdj(nio )P(Qi+ml). P(j) (21) ~~~~o 0~ ~i Hone ck(10) has shown that fdai+lo o.. fd (_ni u j )P(.i+l)oo(j) ( J (22() -- j (32)

a result that has been verified'by using the properties of rotation groups~ Finally, since P(ri) expression (21) becomes (including the case i j)'+b6ji <ii (23) Now, collecting terms and noting that there all nr terms of the type rir ri n-l of the type iril r1, etc, we have~ 2>[2+ + n i(24) <r2>n = [n+(n-l)Y + (n,-2)2 + +.nl] 2 (24) The probability tthat a neutron will make exactly n collisions is~ = s () (>) (25) Pn n 2 and, finally, L is given by 2<r2 =6L2 = Pn <r >n) ml (26) n=l n Z n=l C m=l The double sum may be carried out in closed form by rearranging the series and yields (17)o C, Small Moderator to Absorber Ratio It was shown that the method of section A is best for large (M/A) and it was pointed out that themethod has at least two difficulties~ 1) That integrals in the moderator extend to infinity and

-l52) that scattering from the lump is treated in a very approximate manner. The first of these difficulties is overcome by cutting the integrals off at some limit point Q. The determination of Q is rather difficult and it would be best to obtain it, or at least'verify any guess at it, by comparison with experiment. As experimental data were lacking at the writing of this paper, no authoritiative value can be giveno A reasonable just guess might be o0/4-v where So and v are the surface area and volume of a cello By allowing the iTtegrals in the moderator to range only from zero to Q one obtains the correction: 2 S _-(lg)e [+3 + 2 Q +[4 2 ] 2 2 ] 2 11v 22CF S dg Q 2 2Q c2 2 5 2 + ae [2Q + —+ [Q +-+ -Q e- Q' 4v r (Y 4aZ 4v d2'27) eQt [Q2 + 2Q, + 2 ] which is to be added to Equation (12) and give a corrected, The probability of collision in the lump now becomes p S (-g) (!-e Q) (28) Pc 4vI which should be used in place of (14) in computing P AO The second difficulty mentioned, that of treating scattering in the lumps, may be handled as follows: on. each flight a fraction PC of the neutrons which started outside the lump suffer their next

collisions within the lump. Of these, (s/t)PC = tl are scattered and we assume that they have probability (lPo) of having their next collision within the lump. This is equivalent to assuming that the entering neus trons have a uniform collision density- in the lumpo Then (Cs/Ct)(lPo)tl = t2tl of the neutrons in the moderator are twice scattered in the lumpo Continuing in this manner, always assuming a flat distribution, we find2 _ t1 tl(l+t+ + t2 t 22 + oo) 1 28) as the ratio of the number of neutrons scattering in the lump to those scattering outside. The mean square distance travelled by a neutron starting in the lump may then be calculated by arguments similar to those used in.section A, Consider the neutrons originating in the volume element *pfR,O) dR, in the lump (see Figure 2), travelling.in dn about A. They- are.(R,2)dR5s _. of the total neutrons scattering in the lump, V being the _4H volume of the lump, again assuming uniform collision density and i.so tropic scattering. Those which do not leave the lump on a single flight contribute fdf dR f dRs f dx x2 & (R) (29) oii o ~' o: - V

-17dR _1 L R K Q Ad MID R, Figure 2. Geometry for Calculation of.1 and PA

to the mean square length travelled. Those that do leave contributeo f o f dR f dRs f dy (Rs + y) 2e-Y t(R,) eZRs (0) o0 0 v Adding these'up we haveo 2 2 (1-g) -4 1 2Q I, 2} + 42 g _ 1+-+e +4 e+ [ C + d. e- 2 -22 As the square distance travelled in one flight by neutrons originating in the lump. Lastly, PA for neutrons originating in the lump is needed. Since the collision density in the' lump is assumed uniform, the probability of the next collision being in the lump is just (1lPo) and the probability of absorptiont.bor nar3onsscattered from the lump is~o =a (1 o ( ) + a Po(52) PAn L for all neutrons, it seems more reasonable to.To obtain an L2 for all neutrons, it seems more reasonable to average R2 and PA individually rather than averaging L2 This results from an argument that a neutron generally spends part of its life in the moderator and part of its life in the:full- and is subject to mean square flights of 22 and i2 and absorption probabilities P and P and flights of and 3 1 p A0 A1

Il9 not 21 - on ea.chl -fi;ght,,. The fraction of neutrons scattering in the fuel is: tl/l-t2 t2.T/l t - t21 (+t)t2 while the fraction of neutrons scattering in the moderator is~ tj _ lt2 g(34) l+tl~t2 1 + tltt2 Thus, the proper values of ~2 and PA should be: 2 2 1-t~ 2 L1 0o l+tl-t2 1 l+tl-t2 i-t2 t _ P P +P (36) A Ao l+tl-t2 + PA1 1+tlt2 where the value of i2 is to be obtained from (12) and (27), PA is given 0 o by (28) and (15), iC2 may be taken from (31) and PA from (32). Do Anisotropy Factors;Earlier in the paper it was stated that neutrons generally travel further in some directions than others when the dffusing medium is anisotropic. A tensor'- diffusion coefficient is then required and Equation(4) is used to describe the leakage from such a system. The method proposed here may be used to obtain the diagonal elements of this tensor, which, if the coordinate system is aligned with the principal axes, are the only necessary elementso The L2of Equation 4) are

-20the projections of L2 onto the three principal axes of the diffusion tensor. Then, all one need do is to calculate the projections of L2o This is readily accomplished by multiplying the integrands used to obo tain m 2 by (a.i), where i is a unit vector in the direction in which the diffusion length is required. Thus as an examples (9) would read~0 f 4dQ dR dZ adx i)2(Z+R+x)2q(Z,R,)e (37) 2 as one contribution to Y2 projected on the i direction.> Of' course, PA will also vary with direction, but one may follow a suggestion of Beeler(lL) and determine a projected PA by multiplying the integrands of the integrals for PA by 2Qi. The integrals obtained in this manner are related to g but have not been tabulated. In a subsequent paper these integrals will be developed and a comparison of the results with those of Beeler will be given. E. Flux Depression In all of the previous discussion, the effect of flux shape has been ignoredo This is particularly important in the case of widely separated black lumps; the neutrons tend to distribute themselves away from black lumps, resulting in a decrease in the p'robability of entering the lumps, and hence decreasing PAo A smaller effect on 22 is also introduced.

-21In general, the previous calculations become very difficult for a neutron density which is not uniform. As a first attempt, one might use a diffusion theory flux and obtain values of Y2 and PAo Even this appears to be a rather formidable task. It is possible to carry out the integr~als in:Iabgeometryo This has been done, assuming alternate layers of absorber and moderator, using a flux calcualted from an albedo boundary condition: _+ = 03 (38) J-.at the moderator-fuel interfaceso The method -:Ifoll6Ws;. the method previously used except that the probability that a lettron is found in the volume element r(R,SQ)dRdZ is: r )(Rn) dRd.(39) where the integral extends over the moderator associated with one lump. It is difficult to apply these calculations to more complicated geometry. It does not appear that a reasonable attempt could be made until more data are available

III RESULTS The results obtained"from the method of section II-A are plotted as L2 vs M/A, and L2 vs rod radius for UO2 and uranium metal containing natural and enriched uranium in both H2~0 and D20 moderator (see Figures 3-14)o Extension of the L2 vs rod radius plots to zero rod radius should give the L2 values for a homogeneous mixture. The comparison of the extrapolated homogeneous diffus iorn. length (L2 ex) against the calculated homogeneous diffusion length (L2 ) provides a check on the validity of the results. For example. cal. for the case of 1l0 percent enriched uranium in pure D20, Lex i 2 on the average, about 355 percent lower than Li Cal' but for light water the difference is somewhat larger o At low M/A, as discussed in section. II.C the L= obtained by the methods of section II-A are incorrect, and are, in fact, below the homogeneous values. Physically, one expects self-shielding effects to increase L2 in. a heterogeneous media. Corrections could be made using the methods of section INC and would be in the ri.ght d.irection These calculations have not been made as yet but will be reported on. in a subsequent paper. 2 For purposes of comparison, L was calculated using Equation (2). To obtain. f, the thermal utilization, we used. the following methodo The disadvantage factor G, the ratio of flux at the surface of the element in the mean flux in the element, was calculated using (12) the method by P. Lehamann, et al.- and Amouyal., Benoist, and -22

-235 Horowitz -()o ma ~Zs ()2 (4o) G =+E A [ I+ i + 4 ( ]0 The coefficients A, a, and f are functions of the parameter aZt (where a is the rod radius) In turn, the thermal utilization is given byl)5 T = 1 + (A) G o (41,) Use of these formulas in Equation(2) yields the values of L2 shown in Figure 15o As M/A increases, L drops below the value for a homogeneous mixtureo Sher and Kouts (14) used Equation2) with measured values of f and obtained the values shown in Figure 1.95 These results appear more reasonable but the above mentioned d~iscrepancy is observed if the graphs are extrapolated. The difference between our results and those of Kouts and Sher may be due: to a different choice of nuclear constants

IV CONC LUSIONS The results given by the method proposed here appears to be more satisfactory than those obtained by the older method, Equation (2) (see Figure 15). The major effect of the heterogeneity of the system on L2 is due to a decrease in PA which is, in turn, due to self-shielding and flux depression. Thus, one expects that for a heterogeneous medium, the diffusion length will be greater than for a homogeneous medium with the same (M/A). As can be seen in Figure 15, Equation (2) predicts L which are sometimes lower than the homogeneous L2 For this reason it is believed that the present calculation represents a better approximation of L2 No calculations have been made using the corrections derived in sections II-B through II-E. As experimental data are not as yet available, it-was impossible to determine which of these corrections is necessary and we have therefore decided to delay further calculations until such time as it will be possible to compare results with experiment'-... From Figure 15, it is seen that the results of section II-A are too small for low M/A. This is as. anticipated but detailed calculations have not been carried out for this case. Sample calculations indicate that the corrections are in the right direction and yield reasonable results.o

V ACKNOWLEDGEMENTS We wish to thank Professor Ro K. Osborn for several, illuminating discussions and the National Science Foundation and the Babcock and Wilcox Company, Atomic Energy Division~ for their supporto -25

-26NUCLEAR CONSTANTS USED Enrichment NU02 S8 Natural (l.Og/cc) 2 23x 1022 1 0 % 4 971xlO 4 732x022 1o 15% 5. 540x1020 4 731x1022 1o 3 % 6 288x1020 4. 719x1022'Element Microscopic Cross Sections (barns) Total Absorption Scattering Uj235 475 465 10 u238 10o74 2o44 8o3 U (Natural) 14o 05 6~76 8~3 0 T0 4~2 D20 1. 355 10 13o55 D 491. 0 4091 ~H20 90~5 22008 6066 15o42 H 31 o292 30071 log/cc, Natural Enriched Uranium H20 DO U02 l % 115% 1 ~ 3% o - So 38L3 1o04 5-40 10300 lo 65 1n020

16 14 __ _ ______l_ / 12 0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 M/A Figure 3. Natural U02(lOg/cc) in Pure HF20 rI = Rod Radius

14 0 0 0.25 0.50 0.75 1.0 1.25 1.5 1.75 2.0 ROD RADIUS rm, Cm L2 vs Rod Radius Figure 4. Natural U02 (lOg/cc) in Pure H20

-29260' 240 220 200 180 160 140 120 100 60 5 10 15 20 25 30 M/A L2 vs M/A Figure 5. Natural UO2 (10g/cc) in Pure D20 rl = Rod Radius

-30260 240 220 200 180 NJ 160 140 120 100 80 60 0 0.25 0.5 0.75 1.0 1.25 1.50 1.75 ROD RADIUS r, Cm L2 vs Rod Radius Figure 6. Natural U02 (10g/cc) in Pure D20

3 0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 M/A T2 vs M/A Figure 7. 1.00% Enriched Uranium in Pure H20 rl = Rod Radius

0 0.5 0.75 1. 1.25 1.5 1.75 2.0 2.25 "~j 5 0 0 0.25 0.5 0.75 1.0 1.25 1.5 1.75 2.0 2.25 ROD RADIUS r!,Cm Lf vs Rod Railus Figure 8. 1.00% Enriched Uranium in Pure H20 M/A = Water to Metal Ratio

10 9,N.J 5 \0 3 25 0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0.5 M/A L vs M/A Figure 9. 1.15% Enriched Uranium in Pure H20 r1 = Rod Radius

10 7o - - o - 6 _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ __O_ _ _ _ _ 0 0. 25 0.5 0.75 1.0 1.25 1.50 1.75 2.0 2.25 ROD RADIUS rl, Cm L2 vs Rod Radilis Figure 10. 1.15% Enriched Uranium in Pure H20O M/A = Water to Metal Ratio

10 ~. Tni, 8 0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 M/A L vs M/A Figure 11. 1.50% Enriched Uranium in Pure H20 rl = Rod Radius

10 9 8 N~j 5 3v _ _I i. __-.2 _. o 0 0.25 0.5 0.75 1.0 1. 75 1.5 1.75 2.0 2.25 ROD RADIUS r,, Cm L2 vs Rod Radius Figure 12. 1.3% Enriched Uranium in Pure H20 W/M = Water to Metel Ratio

-37140 M/A 120 I00 80 60 40 20 5 10 15 20 25 30 M/A L2 vs M/A Figure 13. 1.00% Enriched Uranium in Pure D20 rl = Rod Radius

-38140 120 l l l I / I / I 100 80 N~J 40 20 0 0.25 0.5 0.75 1.0 1.25 1.5 1.75 ROD RADIUS r,,Cm 2 L vs Rod Radius Figure 14. 1.00% Enriched Uranium in Pure D20 M/A = Water to Metal Ratio

M/A vs M/A FigRESULTSFROM PROPOSED CALCULm TION (SEC I-A) RESULTS FROM Eq. (2) BNL 486 0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0 M /A T? vs M/^ Comparison of Results rI = Rod Radius Figure 15. 1.00% Enriched Uranium in Pure H20

BIBLIOGRAPHY 1. Weinberg, A. M. and E. PO Wigner, "The Physical. Theory of Neutron. Chain Reactors," University of Chicago Press (1958). 2. Russell, J. Lo, Trans. Amer. Nuc. Soc, 2, (Nov. 1959), Paper 17.-8. 3. Schmitt, A. P., et al. Proc. Int 1. Conf., Geneva, (1958) P. 1191. 4. Spinrad, B. I., J. Appl. Phys.,O 26, 548, (1.955). 5. Shevelev, Y. V., Jo N ucoEnergy, 6, 132, (1957). 6. Triflaj, Y., J. Nuco- Energy, 6, 126, (1957). 7. Behrens, D. J., Proco Phys. Soc., 52, 607, (1949). 8. Seren, L., Trans. Amer:. Nuc.o Soc., 1, 2, Paper 3-6, (Nov. 1959), also Trans. Amer. Nuc. Soc, TJune 1960). 9. Case, K. Mo, F. DeHoffmann, G. Placzek, "Information to the Theory of Neutron Diffusion," U, S. Government Printing Office, (1953) 10. Honeck, H, "Monte Carlo Calculation of Fermi Age," ORNL Report CF 55-8-193, (1955). 11. Beeler, J. R. "Anisotropy of Neutron Diffusion in Slab Systems with Simple Discontinuities", Report XDC-60o6-176, (1960)o 12. Lehmann, P., R. Wo Meier and J. Po Schneeberger, "The Physical Properties of Some Composite Fuel. -Elements in U-D20 Type Reactors," Proco Into Con:f~, Geneva, P. 245 (1958). 13. Amouyal, Benoist, and Horowitz, J. Nuc. o Energy, 6, 79 (1957). 14. Kouts, H. and Ro Sher, "Experimental Studies of Slightly Enriched Uranium, Water Moderated Lattices", BNL 486. a, o

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