UFDC Home  Search all Groups  World Studies  Federal Depository Libraries of Florida & the Caribbean   Help 
Material Information
Subjects
Notes
Record Information

Full Text 
31A7
7. /' 7/ MDDC 1071 UNITED STATES ATOMIC ENERGY COMMISSION CALCULATION OF NEUTRON DISTRIBUTIONS IN HETEROGENEOUS PILES by A. M. Weinberg d. Clinton Laboratories Date of Manuscript: Date Declassified: May 27, 1947 May 29, 1947 Issuance of this document does not constitute authority for declassification of classified copies of the same or similar content and tiee I, 'N:'. FL l p and by the same author. Styled, typed, and reproduced from copy as submitted to this office. Technical Information Division, ORE, Oak Ridge, Tennessee AEC, Oak Ridge, Tenn., 31750900A20187 PRINTED IN USA PRICE 10 CENTS 174rfthiA Now r i r, CALCULATION OF NEUTRON DISTRIBUTIONS IN HETEROGENEOUS PILES By A. M. Weinberg The calculation of the slow neutron density in a pile which consists of lumps of uranium disposed in a lattice is a problem of fundamental importance in elementary pile theory. The method used for such a calculation is rather similar to the method used for calculating the distribution around control rods; it is therefore doubly worthwhile to examine the calculation in detail. In a thermal neutron heterogeneous pile, fast neutrons produced as the result of fission in the uranium lumps are moderated by, for example, the graphite matrix in which the rods are embedded until they become slow. The neutrons then diffuse in accordance with laws of diffusion theory until they are finally captured in the moderator, or in the uranium, or until they escape from the boundaries of the pile. We consider an infinite prototype of an actual pile. Our problem is to calculate the fraction of the thermal neutrons absorbed inside this infinite pile which are absorbed in the uranium. We call the ratio: neutrons absorbed in uranium neutrons absorbed in uranium and moderator the thermal utilization; in accordance with usual practice we denote this quantity of f. or simply f. The ratio of slow neutrons absorbed in uranium to neutrons produced in a finite pile, called the ef fective thermal utilization, and denoted by feff, is evidently less than the ratio of slow neutrons ab sorbed in uranium to slow neutrons produced as a result of slowing downthe reason being that not all slow neutrons produced are absorbed inside the pile. However, in pile theory, it is customary to first examine the infinite prototype of a pile, then to deal with the corrections introduced by the finite ness of the pile, i.e., it is customary to separate microscopic pile theory from macroscopic pile theory. It will be our purpose here to consider only the calculation of the thermal utilization in an infinite lattice. The thermal utilization in the finite lattice depends on the fraction of neutrons which escape from the pile, and therefore upon the size of the pile in which the lattice is placed. Its calculation is very much more difficult than the calculation of f. The calculation of f according to Plementary diffusion theory is very straightforward. The method and the assumptions are quite analogous to those used for the calculation of wave functions in crystal lattices. We list the assumptions as follows: 1) Elementary diffusion theory is applicable. 2) The production of thermal neutrons is uniform throughout the moderator, and is zero in the uranium. 3) The units cell of a lattice can be replaced by a sphere in case the lattice is a point lattice, e.g., simple cubic, or a cylinder in case the lattice is a rod lattice. We now discuss the validity of these assumptions: MDDC 1071 MDDC 1071 1) The assumption of elementary diffusion theory as is well known, is accurate first, if the dimensions of the system are large compared to a mean free path; second, if the system does not absorb neutrons very heavily and third, if the system contains no sources. None of these assumptions are strictly correct in a uranium lattice. The errors which are involved because elementary diffusion theory is used, can be estimated. It turns out that they are sizeable in lattices of interest but the labor involved in extending calculations to include the more accurate theory is prohibitive if a large number of calculations, such as are needed in calculating optimum lattice dimensions, are to be performed. 2) The assumption that thermal neutrons are produced uniformly throughout a cell is accurate if the cell size, i.e., the separation between lumps, is not large compared to a slowing down distance. It is easy to compute the slowing down density in an infinite lattice at each of whose points neutrons, which slow down according to a Gaussian picture, are produced. It is rather remarkable how uniform the slow neutron production rate is in such a lattice. 3) This procedure of replacing e.g., a cubic cell by a spherical one is called sphericizing of the unit cell; it is entirely analogous to the WignerSeitz method for calculating the first approximation to the wave functions in a crystal lattice. The sphericizing assumption is made essential in order to simplify the calculational procedure. Thus if the lattice is a simple cubic lattice the unit cell is a cube and the neutron distribution involves all spherical harmonics of order 4n. The sphericizing approximation implies that only the spherical harmonic of zero order is considered. Evidently the nearer the actual cell is to the "sphericized" cell, the closer is the approximate neutron density to the actual one. In case the uranium is disposed in parallel plates, the "sphericized" cel and the actual cell are identical, both being defined by the planes of symmetry between the uranium plates. The details of the calculation are now easy to write down. We must solve a simple diffusion equa  tion for the thermal neutron density in the moderator and a similar diffusion equation in the uranium and must apply the appropriate boundary conditions at the boundary of the sphericized unit cell and at the interface between the uranium and the metal. Thus in the moderator we have the equation NI cal S A #1N1,al 1l+q=0 (1) K2 1 where O1 is the sum of the speeds of all the thermal neutrons in a cubic centimeter, NI oai is the macroscopic absorption cross section of the moderator, KI is the reciprocal of .he diffusion length in the moderator, and q is the number of thermal neutrons produced per cubic centimeter per second. The quantity N1lal is recognized as the diffusion coefficient according to the elementary diffusion 2 1 theory. In the uranium a similar equation is written down, but since practically no neutrons are pro duced there, the uranium atomic weight being so large, it is appropriate to set q = 0. Thus in the uran ium we have No2ao A 0oNo rao o =0 (2) Ko where the subscript zero refers to the uranium region. The cell we take to be either a composite sphere, a composite cylinder, or a composite slab (see Figure 1); in all cases we denote the onehalf diameter of the uranium by ro, the onehalf thickness of the ceUll by r1. The boundary conditions at the interface between the uranium and the moderator, i.e., at r = ro, must be continuity of flux and continuity of net current. (This corresponds to ensuring continuity of the first two harmonics in a spherical harmonic expansion of the slow neutron angular distribution.) The boundary conditions can therefore be written: Nj=ai or l at r = ro (3) 2 2 2 C1 V 1 K0 at r = ro (4) *1 = o0 MDDC 1071 Figure 1. where v represents the outward normal. Finally, at the outer boundary of the cell there can be no net flow of neutrons, i.e., 6 i 0 at r = rI. (5) To solve these equations is straightforward. We set 00o = A Zol(Kor) (6) and 01 a 1 B K1r1 Zl2(1 r) Z (ll r) Z 2( rlI] (7)1 where A and B are arbitrary constants, Zij( KIr) is a function which solves the equation SZj KZi = 0 (8) in the appropriate geometry and denotes derivative. Since (8) is a second order differential equation it has two solutions which we distinguish by the subscript j = 1 or 2. One of these solutions will be singular at the origin (in cylindrical or spherical geometry), or will be nonsymmetric about the origin (plane geometry). This function we denote by j = 2. The other function will be symmetric and nonsingular at the origin; this we denote by j = 1. We tabulate the functions Zij and their derivatives for the three geometries in the following table: The Functions Zij Geometry Zil(Kir) Z 1(l ir Zi2(Kir) Zi2(Kir) Plane Cosh Kir sinh Kir sinh Kir cosh Kir Cylinder lo(Kir) I1 (Ocir) Ko( Klr) Kl( ir) Sphere sinh Kir Kir cosh Kir sinh Kir cosh Kir *r cosh Kir coshwir Kir ( Kir)2 Kir (Kgr)2 The linear combination of Z11 and Z12 chosen in (7) automatically satisfies boundary condition (5) i.e., no net current at the edge of the unit cell. There are still the two boundary conditions (3) and (4) MDDC 1071 to be satisfied, and these two conditions are just enough to determine the arbitrary constants A and B. Substituting (6) and (7) into (3) and (4) we get the following pair of equations for A and B: B N a [Z11( Kirl) Zi2( xlro) + Z'll Ilro) Z'12( Krl)] + A No ao Zo1(~o,ro) = 0 (8) KO B [Zil( Klrl) Z12( Kro) Zl1( lro) Z'12( Kirl)] (92 + A Z01(Koro)= N (9) from which we obtain I/A: No ao Z0'1(Koro) C(r r 1, ro) + a Zo(Koro) C'( K rl, K ro) A 0 1(10) C'( Kirl, Kiro) q/ K1 where C(x,y) = Z'l(x) Z12() Z' I(y) Z12(x) (11) C'(x,y) = Z11(x) Z'2(y) Z'I(y) Z'12(x) (12) From the expression for A and the corresponding expression for B, the nuetron fluxes (6) and (7) can be calculated. The general shape of the flux distributions are given in Figure 2. It is seen there that the density is highest in the moderator where the slow neutrons are produced, and dips in the uranium where the neutrons are absorbed but are not produced. The thermal utilization is defined as the ratio of slow neutrons absorbed in the uranium to slow neutrons produced (by moderation) in the moderator. Since the number of neutrons absorbed in the uranium must equal the number which flow into the uranium 1 qlV1 = qlV 1 N So 4ao 01 r No a SZ'01( oro) A where So is the surface of the uranium lump and Vi is the volume of the moderator. We substitute (10) into (13), and use Gauss's theorem: f 1 f So Z0((or)o ZoVo = Z01( or)dVo = AZo01 or)dV So z f I f AC(lr)dV SoCi (Klro) CIVI V1 CI( cKr)dVI =  AC( 1r)dVI where bar denotes average. Thus = N alVIF + E =+N al F+E1 (15) f No vaoVo No oaoVo MDDC 1071 Figure 2. where F oVo Z01( Koro) Z01(Koro) (16) So Z01( or ) Z01( Koro) and E = i Cl( Kro, cirl) = Cl( Kr, Kr) (17) So C l( lro, Kgrl) ( lro, 1rl() The actual formulas for F and C depend upon the geometry of the cell. We list these formulas in the following table: Table of F and E Plane Koro coth Koro K1(rl ro) coth al(r1 ro) Coro lo( oro) 2(riro) II( Kirl)Ko( Kro) Io( lro)Ki(xiri) 2 Il( Koro) 2K iro Ii(Klrl)Kl(Kiro) + Ii( Kro)Kl(clrl) 3 2 3 Sphere Koro tanh Koro Kl(r r3) 0lro coth Ki(rl ro) 3 Koro tanh 10ro 3 K2(r1 r2) K1ro + coth Ki(rl ro) The structure of the thermal utilization formulas, as pointed out by J. A. Wheeler, is easily under stood. If the diffusion coefficient in the moderator were infinite (Figure 3) then there could be no gradient of neutrons in the moderator, i.e., the neutron density there would be uniform. In this case K1 = 0, and as may be easily verified, E = 1. Thus the quantity I + N Ia iV F represents the reciprocal of the thermal utilization if the neutron No aaoVo density throughout the moderator were just the same as its value at the interface between moderator and metal. This can also be seen if we observe that the quantity F according to (16) is the ratio of the N1ualV1 neutron density at the interface to the average inside the lump. The quantity NoaV F is therefore No aa0Vo the ratio of the absorptions in moderator and uranium if the neutron density in the moderator is uni form and equal to its value at the uranium surface. This quantity is often called the "relative absorption". Actually the diffusion coefficient is finite and therefore the neutron density in the moderator is higher than it is in the uranium. This causes an additional absorption of neutrons in the moderator and Digilized by the Iniernei Archive in 2011 with funding Irom University ol Florida, George A. Smathers Libi aies wilh support from LYRASIS and the Sloan Foundalion Ilp: www.archiive.org details c alculaiionoIlreuOOciin MDDC 1071 Figure 3. the magnitude of this socalled excess absorption is measured by the quantity (E 1). For K1(r1 ro) =0,E 1 = 0. For finite but small Kl,E 1 can be expanded in a series which involves powers of K 1(rl ro). The reciprocal thermal utilization can therefore be written  = 1 t relative absorption plus excess absorption. It is worthy to note, as a matter of computation that E involves only the constants of the moderator and F only the constants of the moderator region. It is feasible therefore to prepare tables of F and E which can be used for many lattice calculations. Such tables were prepared under the supervision of J. A. Wheeler. From the behavior of the functions E and F, it is easy to make some general statements about the behavior of the therm..l utilization. Evidently, as the volume ratio of moderator to uranium increase, the thermal utilization becomes smaller. Furthermore, for a given volume ratio the thermal utilization is smaller in a system containing large uranium lumps than in a system containing small uranium lumps. This is so because the function F, being the ratio of surface density to average density, is larger in a large lump than in a small one. One might prepare a series of curves in which the quantity 1/f is plotted as a function of V1/Vo for various ro. The general trend of these curves is given in the accompanying figure. (See Figure 4). Since the resonance absorption goes in just the reverse way, i.e., the resonance absorption is large when the volume ratio is small, the multiplication constant, having the product of p and f as factors, must have a highest value at some point which represents a compromise between the value of p and the value of f. The choice of this compromise in any actual case is of course complicated by questions of heat transfer, fabricability of lattice units, and other matters of an engineering nature; the solution of these conflicting demands between engineering and nuclear physics is a perpetual problem which always arises in the design of a lattice. 'o, r r02 r3 r02 \ / // 01> 02> 0>o Figure 4. END OF DOCUMENT UNIVERSITY OF FLORIDA 3 1262 08909 7710 
Full Text 
xml version 1.0 encoding UTF8
REPORT xmlns http:www.fcla.edudlsmddaitss xmlns:xsi http:www.w3.org2001XMLSchemainstance xsi:schemaLocation http:www.fcla.edudlsmddaitssdaitssReport.xsd INGEST IEID ECI9LHN92_YU62Q5 INGEST_TIME 20120302T23:09:38Z PACKAGE AA00009342_00001 AGREEMENT_INFO ACCOUNT UF PROJECT UFDC FILES 