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Composite reactor lattices
J. Nucl. Energy, Part A: Reactor Science. 1959, Vol. 10, pp 44-52.
Pergamon Press Ltd. Printed in Northern Ireland
,..,
COMPOSITE
REACTOR
Ll4TTICES
A. N. BUCKLERand C. CARTER Atomic Energy Research Establishment,Harwell, Didcot, Berks (Received 5 November 1958)
Abstract-A procedure is presentedfor the homogenizationof a large heterogeneousreactor consistingof a composite lattice of several different types of fuel rod immersed in a moderator. The properties of the composite lattice are expressedas suitable averagesof the properties of singlelattices containing only one type of fuel rod.
1. INTRODUCTION FOR small heterogeneous reactors, it is possible to calculate critical sizes and neutron flux distributions directly by considering the cumulative effect of the individual fuel rods. This approach has been considered by, amongst others, FEINBERG(1955), GALANIN (1955) and HASSITT (1958). However, for large reactors in which all fuel rods are identical, it has been usual to reduce the heterogeneous core to an equivalent homogeneous model. Thus the thermal utilization factor, i.e. the fraction of thermal neutrons absorbed in the fuel, is calculated for a single cell either taking a cylindrical cell (GLASSTONE and EDLUND, 1952) or considering the correct lattice geometry as has been done by COHEN(1956). When the reactor contains several types of fuel rod, these cellular methods are not directly applicable, but if each type of fuel rod forms a regular periodic lattice, it is still possible to obtain an equivalent homogeneous model for the reactor, as we show in this paper. FEINBERG(1955) and GALANIN(1955) have justified the homogenization procedure for a single lattice, but they neglected the fuel rod size wherever possible and took an age theory slowing down model. We employ similar techniques, but without their restrictions, to derive a method for the homogenization of a composite lattice. For the purposes of this paper a composite lattice is defined as follows : A composite lattice is a two dimensional regular and periodic array of several different types of rod, such that all the rods of each type also form a regular and periodic array, called a sub-lattice. All the rods in a sub-lattice must have geometrically equivalent positions relative to any other sub-lattice. The region around a rod enclosed by the perpendicular bisectors of the lines joining the rod to each of its nearest neighbours in the composite lattice is called the 44
composite lattice cell. The sub-lattice cell is defined similarly. The properties of the homogeneous model of the composite lattice are first expressed in term‘s of the lattice geometry and basic nuclear properties of the rods, but we then go on to express them in terms of the properties of the single lattices when all the rods in the composite lattice are equivalent to each type of rod in turn. This latter form makes it possible to apply any empirical corrections considered desirable in single lattice calculations. THEORY We consider a composite lattice consisting of N sub-lattices, the kth sub-lattice having cell area V,, which must be an integral multiple of the basic cell area V, of the composite lattice. All the fuel channels are taken to be infinite in length and of radius a, so that the moderator area in each basic cell is V, = V, - 7Ta2. Each fuel rod acts as a sink of thermal neutrons and a source of fast neutrons. We shall assume that the thermal sink at the nth rod gives a contribution to the thermal flux at the point r proportional to -I&+ - r&L). In a finite reactor there should also be terms in I& - r&5), but we have omitted them, since they would vanish later when we transform the finite medium problem into an infinite medium problem. The thermal flux due to the fission neutrons produced in a rod will depend on the particular slowing down model assumed. We shall write the contribution to the thermal flux at r due to a unit fast neutron source at the nth rod as &(r, rJ. In general the explicit expression for F,(r, r,) will be very complicated, but we shall indicate later how, by making plausible assumptions, we can obtain sufficient knowledge of its properties for our purpose. For the 2.
Composite reactor lattices time being we will write F,(r, r,) = U,F(r, r,J, where 0, is some operator which acts on F(r, r,J, the thermal flux due to a unit line fast neutron source at r,. We denote a fuel rod by two suffices, the first one
45
where 13is the angular co-ordinate of the vector B. In equation (l), the summation over u is over all rods in the reactor and the function F(r, rn) is derived from a finite medium slowing down kernel, but an infinite medium slowing down kernel may be used provided the summation over u is extended over an infinite lattice (see Appendix A). Then the function F(r, rku) is replaced by the function F,(lr - rk2$, and using (3), equation (1) becomes
(i, j or k) referring to the sub-lattice to which it belongs and the second one (s, t or u) giving its position in the sub-lattice. qk is defined as the ratio of fast neutrons leaving to thermal neutrons entering a rod in the kth sub-lattice, and iku as the thermal sink-strength of the uth rod in the kth sub-lattice, i.e. ik, is the number of thermal neutrons per unit length 449= ; vkrkiko, u=-* per unit time absorbed in the rod. To allow for e resonance capture during slowing down, we introduce _t x F,( Ir - rkul ) exp (iB . rktL) - f$ a quantity 7~~, defined as the fraction of the fast u m neutrons produced in a rod of the kth sub-lattice X K,,( Ir - rkuj/L) exp (iB . rku) (4) which escape resonance capture. Unlike qk, 7~~is not just a property of the kth sub-lattice but depends on all We now take the axis of the jth rod as origin of a the other sub-lattices. Methods of calculating 7rk primed co-ordinate system and define &t(r’) to be the are discussed in Appendix C, where it is shown that it average flux over the surface of a cylinder, radius r’. is usually satisfactory to take all
[=dekgl ;[
1.
“k
=
p
=
f
j=l
Then, 4it(r’) = &
v&/I/,,
where pi is the resonance escape probability for a lattice consisting entirely of jth type rods. The thermal flux at r is now written as the sum of the contributions from all the rods in the composite lattice:
s
02rexp (iB . rit) de : k=l
,,>_ b: Jpm(lr’ lr’
-
r’kul
r’kul
/L>
exp
) exp
(iB
(iB . r’,,,,j.
. r’kd
(5)
The evaluation of the summations over u is discussed in detail in Appendix A, where we show that, under the conditions L2, 7
The thermal sink strengths iku and the arbitrary constants Zk,,are as yet undetermined. In order to obtain an equivalent homogeneous model it is usual to introduce a macroscopic flux q5Jr) which satisfies the equation :
(V2 + B2>4,W= 0,
(2)
with +m = 0 at the boundary of the reactor, but we can equally well require i,, and Ik, to be discrete values of functions ik(r) and G(r) which satisfy
equation (2). In a cylindrical reactor, the appropriate axially symmetric solutions of (2) give iklL =
;
ikJO(B”k,) e
exp (iB . rku) d0,
’
and
(3) Iku
=
;
lk,J,@k,,) c
exp (iB . rku) de, J
.=t cop;,( lr’ -
r’ktcl
and ) exp
V,/~T~,
B-2> (iBe
r’ku)
1
Wco(4Tk)
_E,
(6)
V, 1 + B2L2
Jo@-‘) (7)
and
2&_Fm Rotlr’-
r’kul
IL)
hp
GB
. r’kJ
+ Gjk(rf)]Jo(Br’).
(8)
W,(B, TV) is the Fourier transform of the slowing down kernel w,(r, TV) from which F,(r) is obtained. rk is a quantity with dimensions of area which describes the slowing down process. For a Fermi Age slowing down kernel, 7k is the mean age to thermal for neutrons leaving a kth type rod, and for a single diffusion kernel it is the slowing down area. For a multi-group diffusion kernel, 7k represents a set of numbers which are the slowing down areas from each group to all lower groups. In all cases 7k depends principally on the moderator but also will depend slightly on the size
46
A. N. BUCKLERand C. CARTER
and constitution of the kth type rod since the spectrum of neutrons leaving the rod will be the original fission spectrum modified by inelastic collisions in the rod. Equation (7) shows that there is no fine structure in the thermal flux due to line fast neutron sources. There is no reason to expect that the substitution of rods for lines would introduce any fine structure, and we shall assume that this is the case. Since the difficulties in passing to a homogeneous model for a heterogeneous reactor arise entirely from the flux fine structure, we can now include the effect of the finite rod size simply by weighting all cross-sections by the relative volumes. Thus the effect of the operator 0, on (7) is to change C, to (V,/ V,)X,, L2 to (V,/ V,) L2 and TV to ( VC/V&lc. This result can in fact be proved if we use a multi-group diffusion model for the slowing down process and neglect certain small terms of order a2/L2. The thermal flux due to the thermal neutron sinks has a fine structure given by the term GJr’) in equation (8). The formulae for Gj, for the particular composite lattices of Figs. l-5 are given in Table 1. These were obtained by the method explained in Appendix B.
k
1
k
1
k
k
2,
k
2,
k
2,
1
k
1
k
1
2,
1
2,
1
2,
1
k
1
k
1
k
k
2,
k
21 k
21
1 k
1
k
1
2,
1
2,
1
1
1 k
1
k
k
21 k
21 k
21
2,
1
1
1
k
FIG. l.-Vk
= 2V,
3,
1
3,
k
3,
1
k
4,
3,
2,
3,
2,
3,
2,
34
49
1
3,
k
3,
1
3,
k
41
3,
22
3,
21
3,
2,
3,
k
3,
1
3,
k
3,
3,
21
34 2a
1
3,
k
3,
FIG. 3.-V,
F
E
= 2,4 and 8.
4,
44 k
44 k
4,
4,
49
42
4,
4,
k
41
44 k
4,
4,
4,
41
1
42
4,
44 k
41
42
33 21
3,
k
41
42
44 k
1
k
3,
4,
4,
FIG. 4.-V,
= 8 VE
(Figs. 1, 2 and 3).
2,
FIG. 2.- Vk = 4 V,
k
TABLEI.-VALUES OF Gk, FORSEVERAL SQUARE COMPOSITE LATTICES (a)
2,
2,
k
5,
44
5,
4,
1
4,
5,
4,
5,
k
5,
4,
5,
4,
1
4,
5,
4,
5,
k
5,
44
51 4,
1
4,
54
4,
5,
k
5,
4,
5,
4,
1
4,
5,
4,
53 k
4,
5,
4,
1
FIG. S.-V,
4,
= 5 V,
5,
54 4,
= IOV,
N.B. In all the figures, lattice points with the same number all have the same geometrical relationship to the kth sub-lattice and hence the same G&,, but points with the same number and different subscript may belong to different sub-lattices.
Substituting (7) and (8) into (5) and including the effect of the operator 0, gives
(b)
F
0
= 5 and 10.
(Figs. 4 and 5).
-
zlc
1
+
i2 L2
+
1
Gdr’ J&W. ))
(9)
_I
Now ijt =
$’ ijJt@rd =2mD [G~,x~J)]~,=~ . (10)
By hypothesis the first summation in (9) gives no contribution to ijt since it is just the thermal flux due
Composite reactor lattices
47
to fast sources, so that using the result given in Appendix A for the derivative of the second summation in (9) equation (10) gives iJ,
x
a
0 L,
=
Ij
2g(V,) +&2))/(l C
+ B2;L2) 112
-
and Ajk = 1 + 6, ;
E,Vjyj
+ 2
e
t
l+s
a2
n-a2 N ij+ykzlik
1
1
(
1 +
Gik + g2 c
+
On inverting (1 l), we obtain to order u2/L2
Ij=
2g(VJ g .
IgpL” +Gik) 3
For convenience, we will choose the normalization of i, so that
(12)
kz, P& = 1
(19)
and (16) becomes z Ajkik = 1,
(20)
k=l
= yjij,
(18)
e
where we have written Gj, for G,,(a). In equation (9) we now substitute for Ik from equation (12) and apply the boundary condition $&)
(17)
(13)
where yj is a rod property which can be determined from one velocity neutron transport theory (see Section 5). We obtain
for allj = 1 to N. We now operate on equation (20) by the reciprocal matrix of A, whose elements are Aij-l Aji is the co-factor
Aji
= -,
where
I4
of Aji in the determinant
[Ai.
Then i, = j$IAkj-l
= jzIAjk//Al.
(21)
Substituting equation (17) for pk into (19), we obtain -2
V
VC
1
+;2p+Gjk)(1 +G) + Gji
The term rf (14)
We now neglect terms of order G2 and use the result proved in Appendix B that
2g + is small, and is of the same li &) order as terms neglected in a multi-group diffusion model for the slowing down of the fast neutrons to give the effect of the operator 0, as used above. Hence, we shall neglect this term. Then, with the definitions : Mean thermal utilization factor, J;= :
ik,
k=l
Mean multiplication
constant,
15, = vpf=
k.?lik[pk
$ rjk7rkik, k=l
Equation (14) becomes
Mean ? defined by -
Ajkl
=
O?
(23)
(24)
A. N. BUCKLER and C. CARTER
48
Equation (25) for Lc2 is valid for a single lattice if we replace fby J As is to be expected, equation (26) reduces to the form obtained by FEINBERG,
Effective diffusion length in the core, L,2 = L2;
(1 -p>, m
equation (22) may be written in the standard form for the critical equation for a homogenized core : (27)
3. PARAMETERS
FOR
A SINGLE
LATTICE
We will first apply the theory of the previous Section to a lattice in which all the rods are the same. Then the matrix Aj, defined by equation (18) reduces to a single term A = 1 + V,J,y
+
Equations (21) and (23) give 1 1 ---= f i
A
Using the expression for g given in (15), we obtain
L,2 = L2(1 -f),
when rod radius a tends to zero. We are not aware of any other theoretical expression for Lc2 which can be compared with equation (26) although DAVISON (1958) has considered the effective diffusion length in a sandwich reactor. Unfortunately he was not able to reach any firm conclusions, but he does show that none of the semi-intuitive formulae which are in current use are of universal validity. All of these formulae are similar to equation (26) in that they differ from (31) only by a small correcting term, but this correcting term varies considerably. We believe that equation (26) has a better theoretical foundation than any other formula we know, but, as we shall show in the next Section, the effective diffusion length and other parameters for a composite lattice may be expressed as suitably weighted averages of single lattice parameters, which themselves can be evaluated by any method currently in favour. 4.
COMPOSITE EXPRESSED
vo
K
775y-1-2vc
5
+ 0.024 + 112
This expression agrees exactly with that given by COHEN (1956) for a square cell lattice. As he points out, it differs from the expression obtained in the cylindrical cell approximation only in the term 0.024, which is usually not significant. The expression given by FEINBERGis
;_ 1 =
v&J
+
In s2 - 1.484 + F
(30) c
GALANIN
states that V, should be replaced by V, in the first term on the right hand side of equation (30) but this does not give the complete correction for fuel rod size, although it will usually be adequate. We might mention that the discrepancy between the 1.476 of equation (29) and l-484 of equation (30) arises from the neglect of terms of order exp (-2~) in the work of FEINBERGand GALANIN.* * In his book GALANIN (1957) does consider the correction for He also falsely.‘corrects’ the constant in equation (30)
fuel rod size. ~ ._^ to 1’41.2.
LATTICE
PARAMETERS
IN TERMS
OF SINGLE
LATTICE
7ra2\
+ 0.524 v,\
(31)
PARAMETERS
We now wish to express Fj,.f J?, L:, ?, in terms of the parameters for single lattices. Let &, pk, Lczk be the parameters for the single lattice with cell area V,, formed when all the rods in the composite lattice are of the kth type. Using equations (23) and (24), we may write V=
2 r&
k=l
I
(32)
and
It is shown in Appendix C that it is usually adequate to take 7~~as a constant T Vcpj/Vj, independent of k. j=l Then equation (33) becomes
Thus p is just the mean of the single lattice p’s, weighted with the relative number of each type of rod in the composite lattice. To express f in terms of fk, we note that using equations (19) and (23) we may write (34)
49
Composite reactor lattices
When all rods are of the kth type, all pi are equal to a constant, which (34) shows to be l/’ Also ij is proportional to l/Vj and Akj is unchanged for all j, since Akj (for j # k) depends only on the relative geometry of the kth and jth sub-lattices and not on the nuclear properties of the jth type rod. Thus, equation (16) gives j
,i llVj =jgl h3
A?JVi
1
or N 1 X = C A,j VP,)
(35)
j=l
since :
l/Vi = l/Ve.
(36)
j=l
Now, it is shown in Appendix B that Akj = Ajk, so that equation (35) gives
kg1 && zj !$ (VJ vj)kzl A&k =,!1 using equation
(16).
P&
(37)
Hence (34) may be written
f’= $1iJ~1
(38)
&IL
For Lc2, equation (26) gives L,2 = P(Vc/Vm)(l -f) and L,7c2= L2(~Pm)(1
-_&).
Using equation (38) we obtain
;i is defined by equation (25) which does not in general permit an explicit expression for 7 in terms of TV. However, -rkwill not usually vary very much from 7, the value for the moderator alone. Then we may expand W, ( B, v‘lTk)andW,(B,2i)
inTaylor
series about T, retaining only the first two terms. Equation (25) then gives (? - T) = 5 (Tk -
T)
k=l
N kzl
rjk.rr,ik
5.
vknkik
REFERENCES
Tkri,rkik/k!l
With the approximation, in evaluating jj, we have f 4
vkrkik.
(40)
nfi = constant, used above
~k~Tkvkik/k,f&ikike
Wa)
E. R. (1956) Nuclear Sci. and Engng.
1, 268. DAVJSONB. (1958) J. Nucl. Energy 7, 51. FEINBERGS. M. (1955) Geneva Conference Paper P.669. GALANINA. D. (1955) Geneva Conference Paper P.666. GALANINA. D. (1957) The Theory of Thermal-neutron Nuclear Reactors. Consultants Bureau, New York. GLASSTONE S. and EDLUNDM. C. (1952) The Elements ofNuclear Reactor Theory. Macmillan. HAWTT A. (1958) Progress in Nuclear Energy, Series 1, 2, 271. &HEN
=,zl
we
Acknowledgements-The authors are indebted to Miss P. A. DEE, Dr. A. KEENEand Dr. G. ROWLANDSfor many stimulating discussions, and to Mr. J. CODD for his helpful advice.
I
or T
CONCLUSIONS
have not only developed a theory of composite reactor lattices, but have also shown in Section 3, how to obtain the equivalent homogeneous core model for a heterogeneous reactor containing only one type of fuel rod. In particular, we have included the effect of fuel rod size and reproduced the result of COHENfor the thermal utilization factor. We have given in equation (26), a formula for the effective diffusion length in a heterogeneous core. For a composite reactor lattice, we have given formulae for q, jj,x Lc2 and ? in terms of single lattice parameters in equations (32), (33), (38), (39) and (40) respectively. In all the formulae, ik, the relative number of thermal neutrons absorbed in the kth sub-lattice, enters as a weighting factor and its calculation is the key to the evaluation of the properties of the composite lattice. Given i,, all the mean value formulae of Section 4 could have been obtained by plausible physical arguments. The procedure for calculating i, is as follows : (i) Evaluate the matrix Ajk from equation (18), where g( V,) is given by equation (15) and the Gjk are given in Table 1 for the simplest composite lattices. yj in equation (18) can be related to a previously calculatedJ;: by equation (29) or it can be calculated from the nuclear properties of the rod using transport theory. The formulae of KUSHNERIUK and MCKAY (1954), obtained from an approximate variational solution of the transport equation, may often be used to calculate the extrapolation length, 1, at the fuel channel boundary. Then il is simply related to y by the equation 31 = 2nay, where 1 is expressed in units of the mean free path in the moderator as in the report of KUSHNERIUKand MCKAY. (ii) Evaluate the determinant IAl and the cofactors Ajk, or calculate the reciprocal matrix A&’ by a direct numerical method. Then calculate il, from equation (21). In this paper,
50
A. N. BUCKLER and C. CARTER
S. A. and MCKAY C. (1954) Chalk River Report. Unpublished. WEINBERGA. M. and SCHWEINLERH. C. (1948) Phys. Rev. 74, ~USHNERrUK
851.
For
the
u =,, APPENDIX
The thermal@
thermal
sources
w&(r() -+ 6( Irl) and
[K,( Ir - rkul /L)/2nD] exp (iB . rku)
A
in an infinite square lattice of line neutron sources
The thermal neutron fhrx #(r) due to a lattice of line fast sources is the solution of the equation
v’+(r) - +(rMa +
we take
W, (w) --f 1. Then
2 S,od/r - r&D u=--(x)
= 0 (Al)
=
exp (iB . r) B,V,
1 1
+ C(r)
where
G(r) =
G
1,W
vk 2 “,?I
#O
where S,, is the strength of the kuth source and w,(lrl) is the slowing down density. If we put S,, = exp (iB . rku) then 4(r) is equal to the sum I&( Ir - rkul)exp (iB . rku). The flux due to a lattice of thermal sources
I&(
Ir - r,,l /L) exp (iB . rku)/2nD
is obtained if we further let ;a( Irl) -+ 6( Irl). To solve equation (l), we take 2-dimensional forms : D(o) = JJ exp (if2 . r)+(r)&
With the further condition that Lz > V&b? then approximately Fourier trans-
(.42)
W&co) = JJ exp(iw . r) w& (rJ)&. I By Green’s theorem, JJ exp (io . r) V2+(r)dZr= -cu*+(co)
(A3)
Equation (1) gives therefore, taking Sk” = exp (iB . rku), (w” + l/LyTqw) =
2 exp (iB + 4 . rkuW&lo I)/0 (A4 uz--0D
1 cos $ Vk G(r) = 4gLS Y1’YZ #O
1.476 + F;
. ss
. r)@(co)d%
exp C--i0 r)W,(lwl) 1 + uPL2
1
=-
(27v&I
,=z,
exp(@ + @) . bu)d2m (As)
In rectangular Cartesian co-ordinates, Is = Vn, and we use the result that
s_, exp
u
(iwAul)
1
=F 1
rk,, = (lu,, ha), where
.,=$, (s(wl -7)
(A61
If we define
then qw _ w, + B), and substituting I
=
into equation (AS) we obtain
expGB . 4
znv,
2
v=-cc
exp(-
iw, . r) Wa C/B- ~1)
(A7)
1 + LZ(B - w,,)”
Under the conditions that 7 and B-2 > VJ479, only the term y1 = vz = 0 gives a significant contribution to the summation, and we have
2
u,=--co
Fm (Ir - rkul)exp (iB . rkd =
exp (is . r)
~,v,
W,(B) 1.
648)
(Al’3
VI= + v2
G&r’) as defined in equation (8) in the text is the value of G(r) averaged over the surface of a cylinder of radius r’ about the axis of a rod in thejth sub-lattice. Here we will consider only the diagonal terms Gklo where r’ = r, and defer the discussion of the non-diagonal terms to Appendix B. The evaluation of the double summation in (AlO) has been discussed by GALANIN. After allowing for the terms of order exp (-2x) which GALANIN neglected, we find to order r2/L2,
Now, take the inverse transform #(r) = (271)-2JJ exp (-io
@1x + v*y)
(All)
From equations (AS), (AY) and (AlO), we see that, for l/L and LIZ<<27r, the solution of equation (Al) with both fast and thermal sources may be written in the form +(r) = exp (iB . r)f(P, r), with S,, = exp (iB . rku) and f (IS*, r) is periodic in r with the same period as the lattice. More generally, if S,, = S(r&, where S(r) = 0 over a boundary C and satisfies (V2 + Bz) S(r) = 0, we may write #(r) = S(r)f(Ba, r) (A12) Thus +(r) = 0 over the same boundary C. If C is the boundary of an unreflected heterogeneous reactor, then 4(r) given by (A12) is the solution for the flux in this reactor, since it satisfies the correct boundary condition and is derived from the same sources inside the boundary. The sources S,, outside the boundary are image sources which give an approximate solution for +(r) valid if l/L and BI < 2~. They are not necessarily the correct image sources, which would depend on the detailed shape of the boundary. A fuller discussion of this result for a homogeneous reactor is given by GLASSTONE and EDLUND (1952) and by WEINBERG and SCHWEINLER (1948). A more detailed treatment for a heterogeneous reactor will be given in a later paper. To conclude this appendix, we prove the result used in equation (11) of the text for the derivative of the second summation in equation (9). Let yjk(r) =
2
u=-cc
&jr
- r&L)
exp (iB . r&/2x0
(A13)
51
Composite reactor lattices Use the result that, on averaging over all directions of r (A14)
&(jr - rlcvjIL) = Z(rL) &(r&). Then we find that 2xDly,,(r) = Z&/L)
; u=-Cc exp (iB . r,,)
x K&,/L)
for
j i k (A15)
x Ko(rku/L)exp (iB . cdl On differentiating,
J
and using the identity
K&&(x)
+ &(x) Z(x) =
l/x,
(A16)
Thus G is a symmetric matrix, and from the definition (18), A is also a symmetric matrix. By applying equation (B3) to various sub-lattices of the composite lattice, and using (B4), it is possible to determine G completely for the simpler composite lattices. In a square lattice we must’ have V,/V, = mz + n2, where m and n are integers or zero. Then the above method determines GLj for allj for V,/VC = 1, 2, 4, 5, 8 and 10 and the results are given in Table 1 of the text. To illustrate the above method, we derive Glcjfor Vk/Ve = 8. In Fig. 3, all the lattice positions marked with the same number have a similar geometrical relation with the kth sub-lattice and have the same matrix element Gkj, but points with the same number and different subscripts may belong to different sublattices. We first apply equation (B3) to the kth sub-lattice on its own to obtain G,, = g( V,) = -!-!
In 5 - 1,476 + y “1 i7a2 V,< j
we have
a r
ay
Next we consider the lattice with cell area +V, consisting of lattice points k and 1. Equation (B3) gives
lMr/L)
,Yj,.(r)= 1
=
-[s,, - ~~ZIWlx L
Jo@9 2rrD/Z,(r/L)
f G&) APPENDIX
G,, = 2g(V,./2) - Gas = 2@‘,/2)
- g(v,)
-Sk;++ (A17)
Considering the lattice with cell area $V, consisting of lattice points k, 1 and 2, we obtain
B
Evaluation of the matrix G All the elements Gjk may be obtained from equation (AlO) by taking suitable averages, but for all the simpler composite lattices a less laborious method may be used. In equation (8) of the text, if we let B = 0, we have
G,, = ; {4p(V,/4) - GM - GM<}= 2g(Va/4) - gO’&) =&(~ln~+$) Finally, applying lattice, we have
equation
(B3) to the complete
composite
Gkz = ;@gU’/@ - 2G, - G, - GA This equation gives the average flux over the surface of a rod in thejth sub-lattice due to unit thermal neutron sources at each rod in the kth sub-lattice. If we now sum equation (Bl) over all k, we obtain the average flux over the surface of a rod in the composite lattice due to unit thermal neutrons sources at each rod in the composite lattice. This is clearly equivalent to the diagonal term in (Bl) but with a lattice cell area V, instead of V,. Corresponding to equation (Al 1) for GIIK,we define
= 2gWA
- gO’d4
APPENDIX
C
Resonance escape probability in a composite lattice lnz - 1.476 + f 57a2 C
032)
Then, by the above argument we have $,
(1 + G,li>lV~= 11 + gWAl/K
or, using (36), $
GdV,
=g(K)lK.
(B3)
k=l
Now, the flux at rod 1 due to unit source at rod 2 is equal to the flux at rod 2 due to rod 1. Since the number of rods in the kth sub-lattice is proportional to l/V,, using equation (Bl) we have 1 + Gjk 1 + G,, VL. x,vK = v,.
Llv*
or
Grr
=
GM.
034)
As delined in the text, =k is the resonance escape probability for neutrons produced in the kth sub-lattice. Also, let pk(V) be the resonance escape probability for a single lattice of kth type rods with cell volume V, so thatp, as defined in the text is pr(VC). We assume that the lattice spacing is sufficiently large so that neutrons with energies in the resonance region which escape capture at one rod will have slowed down out of the region when they reach a neighbouring rod. Then the fraction of neutrons formed in the kth sub-lattice which are captured in the jth sub-lattice will depend on the relative geometry of the sub-lattice and on the capture properties of the jth type rods. We will write this fraction as Xkj = YkjWj, where Ykj depends only on the relative geometry and W, depends only on the neutron capture resonances of thejth type rod. In fact Ykj may be considered as a resonance neutron flux and W, as a macroscopic resonance capture cross-section. Consider a situation in which all the rods of the composite lattice are producing fast neutrons at the same rate and neglect
52
A. N. BUCKLER and C. CARTER
any difference in the fission spectrum from the different types of rod. Then the resonance flux into each rod is the same. Now the number of rods in the jth lattice is proportional to l/Vi, so that the resonance flux into the ith lattice from the jth lattice is proportional to Y,,/ Vj and the total resonance flux into a rod of the ith lattice from all the lattices is proportional to
Thus we have that ,1 independent
Vi Yji/ Vj = constant,
of i.
(Cl)
where Ak is independent of V. If the resonance capture is small, so that pk is close to 1, we can write approximately 1 -p(V)
= A&’
(C7)
Since l/V is proportional to the number of rods in the lattice, equation (C7) states that the resonance capture per rod is independent of the lattice spacing. This is only possible if the resonance neutron flux is constant in space, so that V,Yj, is independent of bothj and k. Then from equation (C5) we have that VkXik = V,(l - p*) (C8) and substituting
in equations (C2) and (C3),
Now, from the definitions of vk and Xkj, we have
W2)
1 - lr* = ? Xkj j=l
Example Consider the composite lattice of Fig. 6, in which there are
Using equation (33) for & we have
1 2 1 2 1 3 1 3 1 3 1 2 1 2 1 3 1 3 1 3 1 2 1 2 1 FIG. 6.
(C3) When all the rods are of the kth type, @+pk( V,), all 7, + qr Xji --f Yji W,, and i, is proportional to 1IV,. Then (C3) becomes 1 -pK(vC) = [
Wk $ $ yj,/vj j=li=l
$ l/C. II
j-1
(C4)
three sub-lattices with V, = 2V,, V, = V, = 4V,. apply (C5) to each of the sub-lattices in turn to give
We first
Using (Cl) this gives 1 -p1(2V,) 1 -pk(V,)
= WkVk 2 Yj,/Vj = 5 VkXj*/Vj. j=l
j=l
63
By applying equations (C5) to various sub-lattices of the composite lattice it is usually possible to express all the Xj, in terms of the pk for single lattices of different cell area. Then Q is given by (C2) and J by (33) or (C3). An example of the use of equations (C5) to determine the Xj, is given at the end of this appendix. However, if the resonance absorption is small, the expression for @reduces to a much simpler form as we shall now show. It is shown in GLADSTONE and EDLUND, Chapter 9 that we can write (UT) p&9 = exp (--Ad V
= X1*, 1 -p2(4V,)
Now consider the composite 2 and 3 only. We obtain
= X,, and 1 - ps(4V,) = X,,. lattice consisting
of sub-lattices
1 - pa(2Vc) = X,, + X,,, and 1 - p3(2 V,) = X,, + X,,. Finally consider the complete composite lattice. Equation (C.5) gives 1 -pdv,) = Xl1 + THX,, + X,1), 1 -p,(Vc)
= Xx + X32 + 2X1,,
1 -ppwc)
= X3, + X23 + 2X1,.
Together with the symmetry requirement that X,, = Xsl, these eight equations determine the nine components of Xkj.
Pergamon Press Ltd. Printed in Northern Ireland
,..,
COMPOSITE
REACTOR
Ll4TTICES
A. N. BUCKLERand C. CARTER Atomic Energy Research Establishment,Harwell, Didcot, Berks (Received 5 November 1958)
Abstract-A procedure is presentedfor the homogenizationof a large heterogeneousreactor consistingof a composite lattice of several different types of fuel rod immersed in a moderator. The properties of the composite lattice are expressedas suitable averagesof the properties of singlelattices containing only one type of fuel rod.
1. INTRODUCTION FOR small heterogeneous reactors, it is possible to calculate critical sizes and neutron flux distributions directly by considering the cumulative effect of the individual fuel rods. This approach has been considered by, amongst others, FEINBERG(1955), GALANIN (1955) and HASSITT (1958). However, for large reactors in which all fuel rods are identical, it has been usual to reduce the heterogeneous core to an equivalent homogeneous model. Thus the thermal utilization factor, i.e. the fraction of thermal neutrons absorbed in the fuel, is calculated for a single cell either taking a cylindrical cell (GLASSTONE and EDLUND, 1952) or considering the correct lattice geometry as has been done by COHEN(1956). When the reactor contains several types of fuel rod, these cellular methods are not directly applicable, but if each type of fuel rod forms a regular periodic lattice, it is still possible to obtain an equivalent homogeneous model for the reactor, as we show in this paper. FEINBERG(1955) and GALANIN(1955) have justified the homogenization procedure for a single lattice, but they neglected the fuel rod size wherever possible and took an age theory slowing down model. We employ similar techniques, but without their restrictions, to derive a method for the homogenization of a composite lattice. For the purposes of this paper a composite lattice is defined as follows : A composite lattice is a two dimensional regular and periodic array of several different types of rod, such that all the rods of each type also form a regular and periodic array, called a sub-lattice. All the rods in a sub-lattice must have geometrically equivalent positions relative to any other sub-lattice. The region around a rod enclosed by the perpendicular bisectors of the lines joining the rod to each of its nearest neighbours in the composite lattice is called the 44
composite lattice cell. The sub-lattice cell is defined similarly. The properties of the homogeneous model of the composite lattice are first expressed in term‘s of the lattice geometry and basic nuclear properties of the rods, but we then go on to express them in terms of the properties of the single lattices when all the rods in the composite lattice are equivalent to each type of rod in turn. This latter form makes it possible to apply any empirical corrections considered desirable in single lattice calculations. THEORY We consider a composite lattice consisting of N sub-lattices, the kth sub-lattice having cell area V,, which must be an integral multiple of the basic cell area V, of the composite lattice. All the fuel channels are taken to be infinite in length and of radius a, so that the moderator area in each basic cell is V, = V, - 7Ta2. Each fuel rod acts as a sink of thermal neutrons and a source of fast neutrons. We shall assume that the thermal sink at the nth rod gives a contribution to the thermal flux at the point r proportional to -I&+ - r&L). In a finite reactor there should also be terms in I& - r&5), but we have omitted them, since they would vanish later when we transform the finite medium problem into an infinite medium problem. The thermal flux due to the fission neutrons produced in a rod will depend on the particular slowing down model assumed. We shall write the contribution to the thermal flux at r due to a unit fast neutron source at the nth rod as &(r, rJ. In general the explicit expression for F,(r, r,) will be very complicated, but we shall indicate later how, by making plausible assumptions, we can obtain sufficient knowledge of its properties for our purpose. For the 2.
Composite reactor lattices time being we will write F,(r, r,) = U,F(r, r,J, where 0, is some operator which acts on F(r, r,J, the thermal flux due to a unit line fast neutron source at r,. We denote a fuel rod by two suffices, the first one
45
where 13is the angular co-ordinate of the vector B. In equation (l), the summation over u is over all rods in the reactor and the function F(r, rn) is derived from a finite medium slowing down kernel, but an infinite medium slowing down kernel may be used provided the summation over u is extended over an infinite lattice (see Appendix A). Then the function F(r, rku) is replaced by the function F,(lr - rk2$, and using (3), equation (1) becomes
(i, j or k) referring to the sub-lattice to which it belongs and the second one (s, t or u) giving its position in the sub-lattice. qk is defined as the ratio of fast neutrons leaving to thermal neutrons entering a rod in the kth sub-lattice, and iku as the thermal sink-strength of the uth rod in the kth sub-lattice, i.e. ik, is the number of thermal neutrons per unit length 449= ; vkrkiko, u=-* per unit time absorbed in the rod. To allow for e resonance capture during slowing down, we introduce _t x F,( Ir - rkul ) exp (iB . rktL) - f$ a quantity 7~~, defined as the fraction of the fast u m neutrons produced in a rod of the kth sub-lattice X K,,( Ir - rkuj/L) exp (iB . rku) (4) which escape resonance capture. Unlike qk, 7~~is not just a property of the kth sub-lattice but depends on all We now take the axis of the jth rod as origin of a the other sub-lattices. Methods of calculating 7rk primed co-ordinate system and define &t(r’) to be the are discussed in Appendix C, where it is shown that it average flux over the surface of a cylinder, radius r’. is usually satisfactory to take all
[=dekgl ;[
1.
“k
=
p
=
f
j=l
Then, 4it(r’) = &
v&/I/,,
where pi is the resonance escape probability for a lattice consisting entirely of jth type rods. The thermal flux at r is now written as the sum of the contributions from all the rods in the composite lattice:
s
02rexp (iB . rit) de : k=l
,,>_ b: Jpm(lr’ lr’
-
r’kul
r’kul
/L>
exp
) exp
(iB
(iB . r’,,,,j.
. r’kd
(5)
The evaluation of the summations over u is discussed in detail in Appendix A, where we show that, under the conditions L2, 7
The thermal sink strengths iku and the arbitrary constants Zk,,are as yet undetermined. In order to obtain an equivalent homogeneous model it is usual to introduce a macroscopic flux q5Jr) which satisfies the equation :
(V2 + B2>4,W= 0,
(2)
with +m = 0 at the boundary of the reactor, but we can equally well require i,, and Ik, to be discrete values of functions ik(r) and G(r) which satisfy
equation (2). In a cylindrical reactor, the appropriate axially symmetric solutions of (2) give iklL =
;
ikJO(B”k,) e
exp (iB . rku) d0,
’
and
(3) Iku
=
;
lk,J,@k,,) c
exp (iB . rku) de, J
.=t cop;,( lr’ -
r’ktcl
and ) exp
V,/~T~,
B-2> (iBe
r’ku)
1
Wco(4Tk)
_E,
(6)
V, 1 + B2L2
Jo@-‘) (7)
and
2&_Fm Rotlr’-
r’kul
IL)
hp
GB
. r’kJ
+ Gjk(rf)]Jo(Br’).
(8)
W,(B, TV) is the Fourier transform of the slowing down kernel w,(r, TV) from which F,(r) is obtained. rk is a quantity with dimensions of area which describes the slowing down process. For a Fermi Age slowing down kernel, 7k is the mean age to thermal for neutrons leaving a kth type rod, and for a single diffusion kernel it is the slowing down area. For a multi-group diffusion kernel, 7k represents a set of numbers which are the slowing down areas from each group to all lower groups. In all cases 7k depends principally on the moderator but also will depend slightly on the size
46
A. N. BUCKLERand C. CARTER
and constitution of the kth type rod since the spectrum of neutrons leaving the rod will be the original fission spectrum modified by inelastic collisions in the rod. Equation (7) shows that there is no fine structure in the thermal flux due to line fast neutron sources. There is no reason to expect that the substitution of rods for lines would introduce any fine structure, and we shall assume that this is the case. Since the difficulties in passing to a homogeneous model for a heterogeneous reactor arise entirely from the flux fine structure, we can now include the effect of the finite rod size simply by weighting all cross-sections by the relative volumes. Thus the effect of the operator 0, on (7) is to change C, to (V,/ V,)X,, L2 to (V,/ V,) L2 and TV to ( VC/V&lc. This result can in fact be proved if we use a multi-group diffusion model for the slowing down process and neglect certain small terms of order a2/L2. The thermal flux due to the thermal neutron sinks has a fine structure given by the term GJr’) in equation (8). The formulae for Gj, for the particular composite lattices of Figs. l-5 are given in Table 1. These were obtained by the method explained in Appendix B.
k
1
k
1
k
k
2,
k
2,
k
2,
1
k
1
k
1
2,
1
2,
1
2,
1
k
1
k
1
k
k
2,
k
21 k
21
1 k
1
k
1
2,
1
2,
1
1
1 k
1
k
k
21 k
21 k
21
2,
1
1
1
k
FIG. l.-Vk
= 2V,
3,
1
3,
k
3,
1
k
4,
3,
2,
3,
2,
3,
2,
34
49
1
3,
k
3,
1
3,
k
41
3,
22
3,
21
3,
2,
3,
k
3,
1
3,
k
3,
3,
21
34 2a
1
3,
k
3,
FIG. 3.-V,
F
E
= 2,4 and 8.
4,
44 k
44 k
4,
4,
49
42
4,
4,
k
41
44 k
4,
4,
4,
41
1
42
4,
44 k
41
42
33 21
3,
k
41
42
44 k
1
k
3,
4,
4,
FIG. 4.-V,
= 8 VE
(Figs. 1, 2 and 3).
2,
FIG. 2.- Vk = 4 V,
k
TABLEI.-VALUES OF Gk, FORSEVERAL SQUARE COMPOSITE LATTICES (a)
2,
2,
k
5,
44
5,
4,
1
4,
5,
4,
5,
k
5,
4,
5,
4,
1
4,
5,
4,
5,
k
5,
44
51 4,
1
4,
54
4,
5,
k
5,
4,
5,
4,
1
4,
5,
4,
53 k
4,
5,
4,
1
FIG. S.-V,
4,
= 5 V,
5,
54 4,
= IOV,
N.B. In all the figures, lattice points with the same number all have the same geometrical relationship to the kth sub-lattice and hence the same G&,, but points with the same number and different subscript may belong to different sub-lattices.
Substituting (7) and (8) into (5) and including the effect of the operator 0, gives
(b)
F
0
= 5 and 10.
(Figs. 4 and 5).
-
zlc
1
+
i2 L2
+
1
Gdr’ J&W. ))
(9)
_I
Now ijt =
$’ ijJt@rd =2mD [G~,x~J)]~,=~ . (10)
By hypothesis the first summation in (9) gives no contribution to ijt since it is just the thermal flux due
Composite reactor lattices
47
to fast sources, so that using the result given in Appendix A for the derivative of the second summation in (9) equation (10) gives iJ,
x
a
0 L,
=
Ij
2g(V,) +&2))/(l C
+ B2;L2) 112
-
and Ajk = 1 + 6, ;
E,Vjyj
+ 2
e
t
l+s
a2
n-a2 N ij+ykzlik
1
1
(
1 +
Gik + g2 c
+
On inverting (1 l), we obtain to order u2/L2
Ij=
2g(VJ g .
IgpL” +Gik) 3
For convenience, we will choose the normalization of i, so that
(12)
kz, P& = 1
(19)
and (16) becomes z Ajkik = 1,
(20)
k=l
= yjij,
(18)
e
where we have written Gj, for G,,(a). In equation (9) we now substitute for Ik from equation (12) and apply the boundary condition $&)
(17)
(13)
where yj is a rod property which can be determined from one velocity neutron transport theory (see Section 5). We obtain
for allj = 1 to N. We now operate on equation (20) by the reciprocal matrix of A, whose elements are Aij-l Aji is the co-factor
Aji
= -,
where
I4
of Aji in the determinant
[Ai.
Then i, = j$IAkj-l
= jzIAjk//Al.
(21)
Substituting equation (17) for pk into (19), we obtain -2
V
VC
1
+;2p+Gjk)(1 +G) + Gji
The term rf (14)
We now neglect terms of order G2 and use the result proved in Appendix B that
2g + is small, and is of the same li &) order as terms neglected in a multi-group diffusion model for the slowing down of the fast neutrons to give the effect of the operator 0, as used above. Hence, we shall neglect this term. Then, with the definitions : Mean thermal utilization factor, J;= :
ik,
k=l
Mean multiplication
constant,
15, = vpf=
k.?lik[pk
$ rjk7rkik, k=l
Equation (14) becomes
Mean ? defined by -
Ajkl
=
O?
(23)
(24)
A. N. BUCKLER and C. CARTER
48
Equation (25) for Lc2 is valid for a single lattice if we replace fby J As is to be expected, equation (26) reduces to the form obtained by FEINBERG,
Effective diffusion length in the core, L,2 = L2;
(1 -p>, m
equation (22) may be written in the standard form for the critical equation for a homogenized core : (27)
3. PARAMETERS
FOR
A SINGLE
LATTICE
We will first apply the theory of the previous Section to a lattice in which all the rods are the same. Then the matrix Aj, defined by equation (18) reduces to a single term A = 1 + V,J,y
+
Equations (21) and (23) give 1 1 ---= f i
A
Using the expression for g given in (15), we obtain
L,2 = L2(1 -f),
when rod radius a tends to zero. We are not aware of any other theoretical expression for Lc2 which can be compared with equation (26) although DAVISON (1958) has considered the effective diffusion length in a sandwich reactor. Unfortunately he was not able to reach any firm conclusions, but he does show that none of the semi-intuitive formulae which are in current use are of universal validity. All of these formulae are similar to equation (26) in that they differ from (31) only by a small correcting term, but this correcting term varies considerably. We believe that equation (26) has a better theoretical foundation than any other formula we know, but, as we shall show in the next Section, the effective diffusion length and other parameters for a composite lattice may be expressed as suitably weighted averages of single lattice parameters, which themselves can be evaluated by any method currently in favour. 4.
COMPOSITE EXPRESSED
vo
K
775y-1-2vc
5
+ 0.024 + 112
This expression agrees exactly with that given by COHEN (1956) for a square cell lattice. As he points out, it differs from the expression obtained in the cylindrical cell approximation only in the term 0.024, which is usually not significant. The expression given by FEINBERGis
;_ 1 =
v&J
+
In s2 - 1.484 + F
(30) c
GALANIN
states that V, should be replaced by V, in the first term on the right hand side of equation (30) but this does not give the complete correction for fuel rod size, although it will usually be adequate. We might mention that the discrepancy between the 1.476 of equation (29) and l-484 of equation (30) arises from the neglect of terms of order exp (-2~) in the work of FEINBERGand GALANIN.* * In his book GALANIN (1957) does consider the correction for He also falsely.‘corrects’ the constant in equation (30)
fuel rod size. ~ ._^ to 1’41.2.
LATTICE
PARAMETERS
IN TERMS
OF SINGLE
LATTICE
7ra2\
+ 0.524 v,\
(31)
PARAMETERS
We now wish to express Fj,.f J?, L:, ?, in terms of the parameters for single lattices. Let &, pk, Lczk be the parameters for the single lattice with cell area V,, formed when all the rods in the composite lattice are of the kth type. Using equations (23) and (24), we may write V=
2 r&
k=l
I
(32)
and
It is shown in Appendix C that it is usually adequate to take 7~~as a constant T Vcpj/Vj, independent of k. j=l Then equation (33) becomes
Thus p is just the mean of the single lattice p’s, weighted with the relative number of each type of rod in the composite lattice. To express f in terms of fk, we note that using equations (19) and (23) we may write (34)
49
Composite reactor lattices
When all rods are of the kth type, all pi are equal to a constant, which (34) shows to be l/’ Also ij is proportional to l/Vj and Akj is unchanged for all j, since Akj (for j # k) depends only on the relative geometry of the kth and jth sub-lattices and not on the nuclear properties of the jth type rod. Thus, equation (16) gives j
,i llVj =jgl h3
A?JVi
1
or N 1 X = C A,j VP,)
(35)
j=l
since :
l/Vi = l/Ve.
(36)
j=l
Now, it is shown in Appendix B that Akj = Ajk, so that equation (35) gives
kg1 && zj !$ (VJ vj)kzl A&k =,!1 using equation
(16).
P&
(37)
Hence (34) may be written
f’= $1iJ~1
(38)
&IL
For Lc2, equation (26) gives L,2 = P(Vc/Vm)(l -f) and L,7c2= L2(~Pm)(1
-_&).
Using equation (38) we obtain
;i is defined by equation (25) which does not in general permit an explicit expression for 7 in terms of TV. However, -rkwill not usually vary very much from 7, the value for the moderator alone. Then we may expand W, ( B, v‘lTk)andW,(B,2i)
inTaylor
series about T, retaining only the first two terms. Equation (25) then gives (? - T) = 5 (Tk -
T)
k=l
N kzl
rjk.rr,ik
5.
vknkik
REFERENCES
Tkri,rkik/k!l
With the approximation, in evaluating jj, we have f 4
vkrkik.
(40)
nfi = constant, used above
~k~Tkvkik/k,f&ikike
Wa)
E. R. (1956) Nuclear Sci. and Engng.
1, 268. DAVJSONB. (1958) J. Nucl. Energy 7, 51. FEINBERGS. M. (1955) Geneva Conference Paper P.669. GALANINA. D. (1955) Geneva Conference Paper P.666. GALANINA. D. (1957) The Theory of Thermal-neutron Nuclear Reactors. Consultants Bureau, New York. GLASSTONE S. and EDLUNDM. C. (1952) The Elements ofNuclear Reactor Theory. Macmillan. HAWTT A. (1958) Progress in Nuclear Energy, Series 1, 2, 271. &HEN
=,zl
we
Acknowledgements-The authors are indebted to Miss P. A. DEE, Dr. A. KEENEand Dr. G. ROWLANDSfor many stimulating discussions, and to Mr. J. CODD for his helpful advice.
I
or T
CONCLUSIONS
have not only developed a theory of composite reactor lattices, but have also shown in Section 3, how to obtain the equivalent homogeneous core model for a heterogeneous reactor containing only one type of fuel rod. In particular, we have included the effect of fuel rod size and reproduced the result of COHENfor the thermal utilization factor. We have given in equation (26), a formula for the effective diffusion length in a heterogeneous core. For a composite reactor lattice, we have given formulae for q, jj,x Lc2 and ? in terms of single lattice parameters in equations (32), (33), (38), (39) and (40) respectively. In all the formulae, ik, the relative number of thermal neutrons absorbed in the kth sub-lattice, enters as a weighting factor and its calculation is the key to the evaluation of the properties of the composite lattice. Given i,, all the mean value formulae of Section 4 could have been obtained by plausible physical arguments. The procedure for calculating i, is as follows : (i) Evaluate the matrix Ajk from equation (18), where g( V,) is given by equation (15) and the Gjk are given in Table 1 for the simplest composite lattices. yj in equation (18) can be related to a previously calculatedJ;: by equation (29) or it can be calculated from the nuclear properties of the rod using transport theory. The formulae of KUSHNERIUK and MCKAY (1954), obtained from an approximate variational solution of the transport equation, may often be used to calculate the extrapolation length, 1, at the fuel channel boundary. Then il is simply related to y by the equation 31 = 2nay, where 1 is expressed in units of the mean free path in the moderator as in the report of KUSHNERIUKand MCKAY. (ii) Evaluate the determinant IAl and the cofactors Ajk, or calculate the reciprocal matrix A&’ by a direct numerical method. Then calculate il, from equation (21). In this paper,
50
A. N. BUCKLER and C. CARTER
S. A. and MCKAY C. (1954) Chalk River Report. Unpublished. WEINBERGA. M. and SCHWEINLERH. C. (1948) Phys. Rev. 74, ~USHNERrUK
851.
For
the
u =,, APPENDIX
The thermal@
thermal
sources
w&(r() -+ 6( Irl) and
[K,( Ir - rkul /L)/2nD] exp (iB . rku)
A
in an infinite square lattice of line neutron sources
The thermal neutron fhrx #(r) due to a lattice of line fast sources is the solution of the equation
v’+(r) - +(rMa +
we take
W, (w) --f 1. Then
2 S,od/r - r&D u=--(x)
= 0 (Al)
=
exp (iB . r) B,V,
1 1
+ C(r)
where
G(r) =
G
1,W
vk 2 “,?I
#O
where S,, is the strength of the kuth source and w,(lrl) is the slowing down density. If we put S,, = exp (iB . rku) then 4(r) is equal to the sum I&( Ir - rkul)exp (iB . rku). The flux due to a lattice of thermal sources
I&(
Ir - r,,l /L) exp (iB . rku)/2nD
is obtained if we further let ;a( Irl) -+ 6( Irl). To solve equation (l), we take 2-dimensional forms : D(o) = JJ exp (if2 . r)+(r)&
With the further condition that Lz > V&b? then approximately Fourier trans-
(.42)
W&co) = JJ exp(iw . r) w& (rJ)&. I By Green’s theorem, JJ exp (io . r) V2+(r)dZr= -cu*+(co)
(A3)
Equation (1) gives therefore, taking Sk” = exp (iB . rku), (w” + l/LyTqw) =
2 exp (iB + 4 . rkuW&lo I)/0 (A4 uz--0D
1 cos $ Vk G(r) = 4gLS Y1’YZ #O
1.476 + F;
. ss
. r)@(co)d%
exp C--i0 r)W,(lwl) 1 + uPL2
1
=-
(27v&I
,=z,
exp(@ + @) . bu)d2m (As)
In rectangular Cartesian co-ordinates, Is = Vn, and we use the result that
s_, exp
u
(iwAul)
1
=F 1
rk,, = (lu,, ha), where
.,=$, (s(wl -7)
(A61
If we define
then qw _ w, + B), and substituting I
=
into equation (AS) we obtain
expGB . 4
znv,
2
v=-cc
exp(-
iw, . r) Wa C/B- ~1)
(A7)
1 + LZ(B - w,,)”
Under the conditions that 7 and B-2 > VJ479, only the term y1 = vz = 0 gives a significant contribution to the summation, and we have
2
u,=--co
Fm (Ir - rkul)exp (iB . rkd =
exp (is . r)
~,v,
W,(B) 1.
648)
(Al’3
VI= + v2
G&r’) as defined in equation (8) in the text is the value of G(r) averaged over the surface of a cylinder of radius r’ about the axis of a rod in thejth sub-lattice. Here we will consider only the diagonal terms Gklo where r’ = r, and defer the discussion of the non-diagonal terms to Appendix B. The evaluation of the double summation in (AlO) has been discussed by GALANIN. After allowing for the terms of order exp (-2x) which GALANIN neglected, we find to order r2/L2,
Now, take the inverse transform #(r) = (271)-2JJ exp (-io
@1x + v*y)
(All)
From equations (AS), (AY) and (AlO), we see that, for l/L and LIZ<<27r, the solution of equation (Al) with both fast and thermal sources may be written in the form +(r) = exp (iB . r)f(P, r), with S,, = exp (iB . rku) and f (IS*, r) is periodic in r with the same period as the lattice. More generally, if S,, = S(r&, where S(r) = 0 over a boundary C and satisfies (V2 + Bz) S(r) = 0, we may write #(r) = S(r)f(Ba, r) (A12) Thus +(r) = 0 over the same boundary C. If C is the boundary of an unreflected heterogeneous reactor, then 4(r) given by (A12) is the solution for the flux in this reactor, since it satisfies the correct boundary condition and is derived from the same sources inside the boundary. The sources S,, outside the boundary are image sources which give an approximate solution for +(r) valid if l/L and BI < 2~. They are not necessarily the correct image sources, which would depend on the detailed shape of the boundary. A fuller discussion of this result for a homogeneous reactor is given by GLASSTONE and EDLUND (1952) and by WEINBERG and SCHWEINLER (1948). A more detailed treatment for a heterogeneous reactor will be given in a later paper. To conclude this appendix, we prove the result used in equation (11) of the text for the derivative of the second summation in equation (9). Let yjk(r) =
2
u=-cc
&jr
- r&L)
exp (iB . r&/2x0
(A13)
51
Composite reactor lattices Use the result that, on averaging over all directions of r (A14)
&(jr - rlcvjIL) = Z(rL) &(r&). Then we find that 2xDly,,(r) = Z&/L)
; u=-Cc exp (iB . r,,)
x K&,/L)
for
j i k (A15)
x Ko(rku/L)exp (iB . cdl On differentiating,
J
and using the identity
K&&(x)
+ &(x) Z(x) =
l/x,
(A16)
Thus G is a symmetric matrix, and from the definition (18), A is also a symmetric matrix. By applying equation (B3) to various sub-lattices of the composite lattice, and using (B4), it is possible to determine G completely for the simpler composite lattices. In a square lattice we must’ have V,/V, = mz + n2, where m and n are integers or zero. Then the above method determines GLj for allj for V,/VC = 1, 2, 4, 5, 8 and 10 and the results are given in Table 1 of the text. To illustrate the above method, we derive Glcjfor Vk/Ve = 8. In Fig. 3, all the lattice positions marked with the same number have a similar geometrical relation with the kth sub-lattice and have the same matrix element Gkj, but points with the same number and different subscripts may belong to different sublattices. We first apply equation (B3) to the kth sub-lattice on its own to obtain G,, = g( V,) = -!-!
In 5 - 1,476 + y “1 i7a2 V,< j
we have
a r
ay
Next we consider the lattice with cell area +V, consisting of lattice points k and 1. Equation (B3) gives
lMr/L)
,Yj,.(r)= 1
=
-[s,, - ~~ZIWlx L
Jo@9 2rrD/Z,(r/L)
f G&) APPENDIX
G,, = 2g(V,./2) - Gas = 2@‘,/2)
- g(v,)
-Sk;++ (A17)
Considering the lattice with cell area $V, consisting of lattice points k, 1 and 2, we obtain
B
Evaluation of the matrix G All the elements Gjk may be obtained from equation (AlO) by taking suitable averages, but for all the simpler composite lattices a less laborious method may be used. In equation (8) of the text, if we let B = 0, we have
G,, = ; {4p(V,/4) - GM - GM<}= 2g(Va/4) - gO’&) =&(~ln~+$) Finally, applying lattice, we have
equation
(B3) to the complete
composite
Gkz = ;@gU’/@ - 2G, - G, - GA This equation gives the average flux over the surface of a rod in thejth sub-lattice due to unit thermal neutron sources at each rod in the kth sub-lattice. If we now sum equation (Bl) over all k, we obtain the average flux over the surface of a rod in the composite lattice due to unit thermal neutrons sources at each rod in the composite lattice. This is clearly equivalent to the diagonal term in (Bl) but with a lattice cell area V, instead of V,. Corresponding to equation (Al 1) for GIIK,we define
= 2gWA
- gO’d4
APPENDIX
C
Resonance escape probability in a composite lattice lnz - 1.476 + f 57a2 C
032)
Then, by the above argument we have $,
(1 + G,li>lV~= 11 + gWAl/K
or, using (36), $
GdV,
=g(K)lK.
(B3)
k=l
Now, the flux at rod 1 due to unit source at rod 2 is equal to the flux at rod 2 due to rod 1. Since the number of rods in the kth sub-lattice is proportional to l/V,, using equation (Bl) we have 1 + Gjk 1 + G,, VL. x,vK = v,.
Llv*
or
Grr
=
GM.
034)
As delined in the text, =k is the resonance escape probability for neutrons produced in the kth sub-lattice. Also, let pk(V) be the resonance escape probability for a single lattice of kth type rods with cell volume V, so thatp, as defined in the text is pr(VC). We assume that the lattice spacing is sufficiently large so that neutrons with energies in the resonance region which escape capture at one rod will have slowed down out of the region when they reach a neighbouring rod. Then the fraction of neutrons formed in the kth sub-lattice which are captured in the jth sub-lattice will depend on the relative geometry of the sub-lattice and on the capture properties of the jth type rods. We will write this fraction as Xkj = YkjWj, where Ykj depends only on the relative geometry and W, depends only on the neutron capture resonances of thejth type rod. In fact Ykj may be considered as a resonance neutron flux and W, as a macroscopic resonance capture cross-section. Consider a situation in which all the rods of the composite lattice are producing fast neutrons at the same rate and neglect
52
A. N. BUCKLER and C. CARTER
any difference in the fission spectrum from the different types of rod. Then the resonance flux into each rod is the same. Now the number of rods in the jth lattice is proportional to l/Vi, so that the resonance flux into the ith lattice from the jth lattice is proportional to Y,,/ Vj and the total resonance flux into a rod of the ith lattice from all the lattices is proportional to
Thus we have that ,1 independent
Vi Yji/ Vj = constant,
of i.
(Cl)
where Ak is independent of V. If the resonance capture is small, so that pk is close to 1, we can write approximately 1 -p(V)
= A&’
(C7)
Since l/V is proportional to the number of rods in the lattice, equation (C7) states that the resonance capture per rod is independent of the lattice spacing. This is only possible if the resonance neutron flux is constant in space, so that V,Yj, is independent of bothj and k. Then from equation (C5) we have that VkXik = V,(l - p*) (C8) and substituting
in equations (C2) and (C3),
Now, from the definitions of vk and Xkj, we have
W2)
1 - lr* = ? Xkj j=l
Example Consider the composite lattice of Fig. 6, in which there are
Using equation (33) for & we have
1 2 1 2 1 3 1 3 1 3 1 2 1 2 1 3 1 3 1 3 1 2 1 2 1 FIG. 6.
(C3) When all the rods are of the kth type, @+pk( V,), all 7, + qr Xji --f Yji W,, and i, is proportional to 1IV,. Then (C3) becomes 1 -pK(vC) = [
Wk $ $ yj,/vj j=li=l
$ l/C. II
j-1
(C4)
three sub-lattices with V, = 2V,, V, = V, = 4V,. apply (C5) to each of the sub-lattices in turn to give
We first
Using (Cl) this gives 1 -p1(2V,) 1 -pk(V,)
= WkVk 2 Yj,/Vj = 5 VkXj*/Vj. j=l
j=l
63
By applying equations (C5) to various sub-lattices of the composite lattice it is usually possible to express all the Xj, in terms of the pk for single lattices of different cell area. Then Q is given by (C2) and J by (33) or (C3). An example of the use of equations (C5) to determine the Xj, is given at the end of this appendix. However, if the resonance absorption is small, the expression for @reduces to a much simpler form as we shall now show. It is shown in GLADSTONE and EDLUND, Chapter 9 that we can write (UT) p&9 = exp (--Ad V
= X1*, 1 -p2(4V,)
Now consider the composite 2 and 3 only. We obtain
= X,, and 1 - ps(4V,) = X,,. lattice consisting
of sub-lattices
1 - pa(2Vc) = X,, + X,,, and 1 - p3(2 V,) = X,, + X,,. Finally consider the complete composite lattice. Equation (C.5) gives 1 -pdv,) = Xl1 + THX,, + X,1), 1 -p,(Vc)
= Xx + X32 + 2X1,,
1 -ppwc)
= X3, + X23 + 2X1,.
Together with the symmetry requirement that X,, = Xsl, these eight equations determine the nine components of Xkj.