- Email: [email protected]
Space and angle dependent collision probability in cell problems
Journal of Nuclear Energy, Vol. 27, pp. 15 to 24. Persamon Press 1973. Printed in Northern Ireland
SPACE AND ANGLE DEPENDENT COLLISION PROBABILITY IN CELL PROBLEMS T. TAKEDA and T. SEKIYA Department of Nuclear Engineering, Faculty of Engineering, Osaka University, Yamada-kami, Suita-shi, Osaka, Japan (Received 29 March
1972)
Abstract-The integral transport equation is formulated in terms of the generalized collision probability in a cylindrical lattice cell in one velocity. The anisotropy of the neutron angular distribution and the scattering is considered by representing them with spherical harmonics series respectively. The space dependence of the neutron distribution is taken into account by representing it with a Legendre series in each annular region. The isotropic return condition or white boundary condition is adopted for the generalized collision probability. One group disadvantage factors are calculated for cylindrical lattices in both cases of isotropic and linearly anisotropic scattering using the generalized first-flight collision probability. 1NTRODUCTION
THE EFFECT of the anisotropic scattering on the disadvantage factor has been treated by us (1971) and by ROYL and EMENDORFER(1970) in a cylindrical lattice cell. They
have used the generalized first-flight collision probability with the isotropic return boundary condition by dividing the cell into some regions. But they assumed that the flux and the current are flat in each region. Although this assumption is improved by the increment of the number of partition, the improvement is very slow for the calculation of the effect of the anisotropic scattering. This is due to the fact that, from Fick’s law, the assumption of flat flux in a given region is contradictory to the assumption of flat current in the region. Further, different from the case of calculating the average flux for isotropic scattering, it is difficult to determine the way how to divide a cell for the case where we calculate the effect of anisotropic scattering since the current shows appreciable change over the cell. BRUN and KAVENOKY (1971) also studied the anisotropic effect using the conservation law which relates the flux and current, but his method is limited up to the linear anisotropic scattering. In this paper, we will take account of the spatial dependence of the neutron angular flux in each annular region by expanding it into a Legendre series. This is an extension of the method for a slab cell by CORNCOLD (1956) and CARLVIK (1968). In a slab cell, the periodicity of the lattice can be treated exactly by adopting the perfect reflection condition for a neutron escaping from a cell. But in a cylindrical cell the condition is not satisfactory and the isotropic return condition is proved to be good for isotropic and anisotropic scatterings (TAKEDA et al. 1972) by numerical calculations. Expanding the neutron angular distribution into a spherical harmonics series, we will derive a generalized collision probability with use made of the isotropic return condition at the cell boundary. DERIVATION
OF GENERALIZED
COLLISION
PROBABILITY
One speed neutron distribution is considered in a circular cylindrical cell with inf?nite axial distance. We choose the z-axis in the axial direction and assume that 0 is the polar angle of the direction of a neutron 8 from the z-axis and TVis the azimuthal angle of the projection of S2 onto the x-y plane from the radial direction. In such a 15
16
T. TAKEDA and T. SEKIYA
system we expand the neutron angular flux ~$(r, a) and the source distribution S(r, SZ) into the spherical harmonics series about the neutron angle and the Legendre series about co-ordinate space respectively. We divide a cell into N concentric annular regions and take the inner and outer radii of i’th region as ri_r and ri respectively. Then the neutron distribution at radius r in the i’th region may be expanded by a Legendre series Pk(2r2 - ri2 - &Jri2 - rf_J (k = 0, 1,2, . . .). Namely the neutron angular distribution and the source distribution in the i’th region have the forms:
(1) S(r, $2) = 2
$
~(Hnm)2S~n,m~k’Psna(~os0) cos map,
““,T
n=Om=Ok=O
:2ri t
r’-l (2 - LJ9 1
(2) where (2n $ l)(n - m)!
H m= n
J
497(n + m)!
and 6,, is the Kronecker delta. Since we are dealing with an infinitely long cylindrical cell, there remain in equations (1) and (2) only the terms of which n + m is an even number. The scattering kernel z8(r, GY -+ 51) has no spatial dependence in a given region (for example in i’th region) and may be expressed in the form &(r, Q’ -Q)
= 2 2 (Hnm)21Z;sinP.m(cos0) cos m(a - y.‘)(2 - S,,). n=Om=o
(3)
The integral Boltzmann equation takes the form #r, a) =rdRe-‘(S(r’,
Sz) +SdP’Z,(r’,
a2 -+ Q)+(r’, p’))
(4)
where c”R is the optical distance from a point r’ to r. In terms of the orthogonal relation : daH,“P,“(cos s 4a
0) cos maH,tm’P,fm’(cos 0) cos m’o! =
(2 - 6,J1 (0
for II = n’ and 111= m’ (,_) otherwise
the scattering term in equation (4) reduces to dQ’&(r’, G?’+ !Z)+(r’, Sz’) s =jo
2,
,~$H,‘“)2+~“*“*k’Pk (““,
Tarp_ “-I) (2 - 6,0)X:d”(r’)P,m(cos 0) cos ma. z
El
(6)
17
Space and angle dependent collision probability in cell problems
We assume that the points r and r’ in equation (4) are located in j’th and i’th region respectively. The region number i changes with the distance R. Though the direction Q in the left-hand side of equation (4) is identical with that in the term of emission density in the right-hand side of the equation, the spherical harmonics expansions at both sides about azimuthal angles are different because of the difference of radial directions at r and r’. To express this difference we write the azimuthal angle at r’ by CC’.Substituting equations (I) through (3) and (6) into equation (4) we obtain
so2, =
~~H,m)2~~~-~k~~,m(cos 0) cos maP, (2r2 j
s
I
y ,- “-l) (2 - S,,) 3 I
OmdlZe-z $ i 5 (H,.m’)‘P,.““‘(cos f3) cos m’a’Pk, n’=Orn’%D 5’4l
>
(7) ?( (2 - fim,&$n’,m’.k’) + &in’C#Ijn’*m’*k’)}. Multiplying the above equation by P,“(cos 0) cos ma and integrating over dS2, we have
ss m
=
dR d8P,“(cos
6) cos nlxe
-E
0
i< 2 i 2 (H,,“‘)ZP,,,“n’(cos 0) cos m’u’Pk, 2r’2ry z2rTlr’-‘) C n’=_O rn’S07.‘;=0 2 E ><(2 _ ~m,O){s~n’.~‘.k’) _t &n’+;n’.m’.k’)).
(8)
Further we multiply the above equation by Pk(2r2 - rj2 - r:_,/r,” - ri_,) integrate with 2w dr over the volume ofj’th region, and we obtain
and
2r2 - rj2 - r:_,
(
(Hn?)2Pn.m’(COS e> cos m’a’ ) r.1? - rf_I 2rf2 - ri2 - r,“_, (2 _ ~m,O),@n’>m’~k’ + ) _=q’&‘.m’.k’)) ‘J Pkf r.2 r,‘ _, z
:-; P,
(9) where Vi is the volume ofj’th region. Here we define the following generalized first-flight collision probability: p!~‘.m’,k’)4n.m.k) 13
Gj
= v_
ss
z vi
vi
x Pk’ 2r”ry “4.: (
L
e-m
dr ~2
dr’
(H,?“)2P,fm’(~~~ 8) cos m’a’
“-I) (2 - Sm~O)P,m(cos0) cos ma 21 x Pk
2
2r2
(
-
rj2 - r:_-,
rj2 -
rtl
1
3 (10)
18
T. TAKEDA and T. SEKIYA
where Zti is the total cross section in j’th region. The explicit formulaforp~~‘,m’,k’)‘(n,~~~) is given in Appendix. In terms of the above equation, equation (9) reduces to
If the scattering and the source distribution are isotropic it is sufficient to take account of the contribution from the factor with (n, m) = (0,O). If we consider up to the linear anisotropic scattering it will suffice to treat the contribution from the factors with (n, m) = (0,O) and (1, 1). The quantity P$‘~“~o)+-(o,o,o) corresponds to the usual first-flight collision probability.
flight between neighboring cells.
Fm. I.-Neutron GENERALIZED
COLLISION
PROBABILITY
IN A LATTICE
CELL
We calculate the generalized first-flight collision probability P$(R’*m’,k’)+(n*m,k) that a neutron born in an i’th region in a cell undergoes its first collision in aj’th region in any cell so as to reduce the infinite sum over i to finite sum 1 I i I N. First we will calculate the generalized first-flight collision probability for a neutron born in a cell to make its first collision in the nearest neighboring cell. As shown in Fig. 1, we divide the distance R from the region i to the region j into two parts R, and R,; R, being the path length in an original cell 1 and R, being the path length in subsequent cell 2. Then we obtain ph’.m’.k’)+h.m.k) fj
2rt2 -
x P, (
x p
k
ri2 -
ri2 - $-I 29 -
rtl
)1
rj2 - r3_1
rj2 -
rtl
>
(2 - Bm.o)e-‘l. e-x’zP,m(cos 9
0) cos ma
(12)
where dR is a line element in the j’th region along the direction a. We describe the solid angle subtended by the neutron direction from the interface of cell by sZ* and
19
Space and angle dependent collision probability in cell problems
the normal vector at the interface by n. We make an approximation to calculate the above equation as follows: Expressing the surface area of the cell per unit height by S we insert an identical operator 1
S
dfi*(n s 277
(13) . Q*)
between the terms e-% and emcY2m . equation (12) and neglect the r’ and S2dependence of the latter term. This is based on the assumption that all neutrons impinging on the cell surface may return isotropically. Then we obtain p+‘.n’.h-‘Hn,m.l) =
p,!~‘m’,k”p,‘wn>kJ,
(14)
23
where P@‘*nL’,k’ is )the probability that a neutron born in the i’th region in a cell 1 in mode (9, m’, k’) escapes the cell without collision, and is given by p!n’ sm’ek’ =) i 18
s sdne-~~l(H,,n~‘)2p,.“‘(cos 0) cos m’u’
vdr’
t
’
2r’” ;(
Pk.
ri2 -
$_,
(2 - &%),
ri= - r;_,
and P$,llLsk)is the probability that a neutron entering the cell 2 isotropically in thej’th region in mode (n, m, k), and is expressed in the form p(?f.m.k) II
1
=
collides
dRe-Lr;;apnm(cos 0)
dS dQ*(n . sl*)
S dQ*(n . Q*)
(15)
s
ss
.
-
2r2
> cos muP,
rj2 -
r.2 , -
r5_1
?$_I
C,j*
(16)
Equation (15) can be transformed into the form ,;~‘.m’.k’)
=
1vi
l,clr’ J‘dB (1- rdRZ(R)e-‘)
X cos m’a’Pkf
2rr2 -
(
dr’ = dk&,,,,Bm,,, - r Vi s Vi f X cos m’a’Pkr
rf_,
ri2 -
ri2 -
(H,tm’)2Pli,m’(cos0)
rF_1
1
(2 - 4nxl)
drC(r)e- G (HaSm’)2Pntnz’(~~~ 0) cell
1
2rf2 - ri2 -
&
ri2 - rf_1
(2 -
LA
(17)
where the integration about r is made over the volume of the cell 1. In terms of the definition of P@‘ Y ~nb’~l’-,m,k),we obtain
P;p-) = Bn,Odm'OBK'O
(18)
20
T.
and
TAKEDA
T.
SEKIYA
In the following we calculate the probability Pijn,m,k).In a circular cylinder
s . . da*(n
S2*) = 7.
(19)
277
We denote the solid angle subtended by the neutron direction from the collision point r in the j’th region by 8’. Then using the relations (n Sit*) ___ dS = da’, Rz2
(20)
Rz2 dR dQ* = dr,
we have the relation (n * S*) da* dS = dS2’dr.
(21)
Adopting these relations we rewrite equation (16). At this point it should be noticed that, if we express the azimuthal angle by CIfor a neutron starting from point r toward the cell surface, we must replace u in equation (16) by rr - t( because the u in equation (16) is measured for a neutron directing from the cell boundary to the point r. Then equation (16) reduces to s
8)( - 1)” cos map,
4adS2’Vdret’lsP,“(cos s 3
4Ec,jvj = BnOBm,&,- 2 s
4CtjVj
2r2 -
rj2 -
rj2 -
rja-
$1
r;_,\ J
$I
r5-1
(2 - S&-l $ Pi;*m,r,-(“.o*o)].
BnOBm08k0 - 4;;;)m)a
1
S
rj2 -
~~~r~~pn.~~~hRz(R)e-~
X Pnnz(COs6) COsma(-l)“P,
= -
2r2 -
n
l=l
(22)
Considering P$m,k) in the case of II = m = k = 0, the probability for a neutron entering a cell isotropically to escape the cell without collision is given by N 4&jvi
P,,=l-z----
j=l
s
1 _ fj p~,0.0,40.0.0, . (
(23)
1
Z=l
From equations (18), (22) and (23), the generalized first-flight collision probability that a neutron born in an i’th region in a cell makes its first collision in a j’th region in any cell is given by p~(n’.m’.k’bh,m.k) $2
=
pbAm’.k’b+h.m.k)+ ii
X-
4&Vj S
BnOB,oc3ko (
(2 - &,)-I
4;;;);)2
12
2 P$m.x)-ro*o.o,). I’=1
(24) Using this probability $(n*m*k) is obtained by solving the equation
Space and angle dependent collision probability TABLE l.-LEGENDRE Upper limit of Legendre polynomials
in cell problems
EXPANSION COEFFICIENT OF FLUX FOR ISOTROPIC WAITBRING Case 1
Expansion coefficient
0
+
1
+ %O,Ol
21
Fuel
(0 O,O)
,#lO,O,l,
Moderator
Fuel
Case 2 Inner moderator
Outer moderator
4.5298
5.1556
96.0
161.66
177.95
4.5294 0.2546
5.1654 0.1316
96.0 24.339
170.68 21.657
192.10 3.4884
2
(0,0,1l 4 (O,O,?)
4.5292 0.2532 0.04008
5.1679 0.1323 -0.1120
96.0 23.954 3.8506
172.5’2 22.103 -9.9745
193.75 3.6622 -2.2599
(O,O,O,
4.5292
3
f i0,O.l) 10.0,2)
5.1681 0.1320 -0.1120 0.03056
96.0 23.879 3.7706 1.0457
172.82 21.874 -10.108 4.5623
193.99 3.6185 -2.2494 0.3406
to,o,o,
0.2531 0.03991 0.01323
lO.O,3J
TABLE 2.--MODERATOR-TO-FUEL
FLUX RATIO FOR ISOTROPIC SCATTERING
Upper limit of Legendre polynomials 0 1 2 3
TABLE 3.-LEGENDRE
Upper limit of Legendre polynomials 0
Case 2
1.1382 1.1404 1.1410 1.1412
1.7971 1.9267 1.9445 1.9472
EXPANSION COEFFICIENT OF ANGULAR ANISOTROPIC SCAlTERING @ = 1/3) Case 1
Expansion coefficient ,+0,0,0, 4 ,1,1,0)
p,o,o, p,o,1,
3
Case 1
#o,o,z, +to,o,w + 11,1.01 p,1.1, p.1.2,
4 U,l,W
Fuel
Moderator
Fuel
FLUX FOR LINEAR
Case 2 Inner moderator
Outer moderator
157.37 -8.2352
169.70 -1.8469
4.5305 -0.2067
5.1367 -0.1343
96.0 -12.43
4.5306 0.2545 -0.2111 -0.1422
5.1362 0.1121 -0.1390 0.1594
96.0 24.224 -13.856 - 10,696
161.95 15.976 -10.177 6.2690
177.30 2.1676 -2.1651 2.4177
4.5305 0.2533 0.04008 -0.2110 -0.1427 0.02606
5.1382 0.1135 -0.1027 -0.1392 0.1621 -0.02874
96,O 23.875 3.8034 -13.836 - 10.887 1.3677
162.97 16.541 -7.9287 - 10.262 7.0200 -2.9097
178.18 2.3404 -1.5783 -2.1682 2.5546 -0.4560
4.5305 0.2532 0.03991 0.01324 -0.2110 -0.1427 0.02594 -0.01364
5.1383 0.1132 -0.1029 0.02868 -0.1392 0.1621 -0.02926 OGO6809
96.0 23.806 3.7278 I.0453 -13.843 -10.876 1.3309 -0.8810
163.15 16.392 -8.1376 3.9630 - 10.262 7.0756 -3.1932 1 a4839
178.33 2.3138 -1.5738 0.1497 -2.1619 2.5542 - 0.4673 o-1034
22
T. TAKEDA and T. SEKIYA TABLE a.-LEGENDRE
Upper
EXPANSION COEFFICIENT OF ANGULAR FLUX FOR LINEAR ANISOTROPIC SCATTERING (iu = 2/3)
limit
of Legendre
polynomials 0
Case 1
Expansion coefficient
10,0,0) (1,1,01
_~_. -1
2
(O,O,Ol f Wl,O,l)
4.5313 -0.2068
Moderator 5.1176 -0.1352
4.5318 0.2545
5.1069 0.09248
Fuel 96.0 - 12.505 96.0 24.104
$ (1,1,11 I1,l.O)
-0.1422 -0.2113
-0.1392 0.1597
--13.939 10.608
$ l0,0,11 (O.O,O, : i0,0,21
4.5317 0.2533 0.04007 -0.2112 -0.1427 0.02611
0.09455 5.1083 -0.09342 -0.1394 0.1621 -0.02875
9 lO,O,3)
4.5317 0.2532 0.03991 0.01324
0.09430 5.1084 -0.09375 0.02680
f r1,1,w (l,l,ll (1,1,2) (l,l,S)
-0.2112 -0.1426 0.02599 -0.01365
(l,l,O)
9 (1.1,1, 9 (1.1,21
f lO,O,OL 10,O.l) ,#p,O,Z,
3
Fuel
Case 2
NUMERICAL
m--o.1394 0.1622 -0.02924 OX)06784
RESULTS
AND
Inner moderator
Outer moderator
152.38 - 8.9029
160.07 -2.0294
152.88 10.120
- 162.01 0.8065
- 10.321 6.5214
-2.1976 2.4902
96.0 23.794 3.7551 -13.920 - 10.793 1.3496
153.18 10.846 -5.8355 -10.388 7.0983 -2.9595 ____
162.24 0.9870 -0.8869 -2.1970 2.5676 -0.4732
96.0 23.732 3.6841 1.0449
153.27 10.771 -6.1224 3.3570
162.31 0.9750 -0.8900 -0.04470
--10.781 13.927 1.3125 -0.8748
--lo.387 7.1461 -3.2045 1.4716
--2.1906 2.5659 -0.4708 0~1100
CONCLUSIONS
We calculate the flux distributions in the cases of two different lattice cells. The first cell called case 1 is a closely packed lattice cell and the second called case 2 is that with a large moderator, in both cases a cell is composed of a fuel rod and a moderator. For case 1 the outer radii of fuel and moderator are 0.38 1 and 0.64487 cm respectively and the total and scattering cross sections of the fuel are 0.78 and O-387 cm-l and those of the moderator are l-0618 and 1.053 cm-l. For case 2 the outer radii of fuel and moderator are 2-O and IO.0 cm respectively and the total and scattering cross sections of the fuel are 0.6 and 0.35 cm-l and those of the moderator are 0.3 and O-3 cm-l. The case 2 corresponds to the cell adopted by LEWIS (1969). In the calculation we divide the cell of case 1 into 2-regions and that of case 2 into 3-regions (one fuel rod and two moderator regions divided by a circle at radius 6 cm). In calculating flux distribution we assume a uniform and isotropic source per unit volume only in moderator. In the case of isotropic scattering the expansion coefficients #O,OPk) of the flux are given in Table 1 for the cases where the upper limit of k is taken to be 0, 1, 2 or 3. For cases where the upper limit of the Legendre polynomials is taken to be 3, the coefficients c$(O$O,~) (k = 0, 1,2,3) can reproduce an almost continuous flux curve at is very fast for case 1 but interfaces. The convergency of the coefficients c#(O,O,~) not so rapid for case 2. This shows that the deviation of flux in each region from the flat flux is very small for case 1 but is large for case 2. The moderator-to-fuel flux ratio is given in Table 2. From this table it can easily be seen that the value of the flux ratio converges very rapidly with the increase of the number of terms
Space and angle dependent collision probability
in cell problems
23
TABLE S.-MODERATOR-TO-FUEL FLUX RATIO FOR LINEAR ANISOTROPIC SCATTERING(.lC= 1/3) Upper limit of Legendre polynomials
-
0 1 2 3
Case 1
Case 2
I.1338 I.1337 1.1341 1.1342
1.1249 I.7936 1.8032 1.8049
--
TABLE 6.-MODERATOR-TO-FUEL FLUX RATIO FOR LINEAR ANISOTROPIC SCATTERING(,u = 2/3) Upper limit of Legendre polynomials 0 1 2 3
Case 1
Case 2
1.1294 1.1269 I.1272 1.1273
16407 1.6559 1.6585 1.6593
the Legendre expansion. And, for the ratio, it is sufficient to take into account only the first two terms for k = 0 and 1. The Legendre expansion coefficients in the case of linear anisotropic scattering are presented in Tables 3 and 4 where the average cosine of the scattering angle ,u are taken to be l/3 and 2/3 respectively. From these tables it can be seen that the convergency of the expansion coefficients ~#(lJ,~r(k = 0, 1, . . .) is very slow for both cases compared with that of 4 (“,o,k). Especially the term #rtryl) is nearly comparable with the term qY1*l,o).Then the flat current approximation is disagreeable to express the curve of the current over a cell. In the case of anisotropic scattering also, the consideration up to the term k = 1 yields good results for the expansion coefficients, The moderator-to-fuel flux ratio for the case of anisotropic scattering is presented in Tables 5 and 6 and shows that the error resulted from the flat flux and flat current approximation decreases with the increase of ,u. And if we include the term k = 1, remarkable improvement is obtained.
in
REFERENCES BRUN A. M. and KAVENOKYA. (1971) Paper presented to ANS National Topical Meeting New Developments in Reactor Mathematics and Applications. CARLVIKI. (1968) Nucl. Sci. Engng 31, 295. CORNGGLDN. (1957) J. nucl. Energy 4,293. KAVENOKY A. (1969) CEA-N-1077. LEWIS E. E. (1969) J. nucl. Energy 23,87. MAKINO K. (1967) Nucleonik 9, 351. ROYL P. and EMENDORFER D. (1970) Atomkernenergie 18, 151. TAKEDAT. and SEKIYAT. (1971) J. Nucl. Sci. Technol. 8,663. TAKEDAT., NAKAMURAK. and SEKIYAT. (1972) J. Nucl. Sci. Technol. 9,53.
APPENDIX We evaluate the generalized first-flight collision probability [equation (IO)] for a cylindrical cell using the same technique as KAVENOKY(1969) adopted for the calculation of the usual first-flight collision probability. First we calculate 4, (n’.m’.L‘--c’n.m.L’ for i < j. The region number is enumerated from the inner site of a cylindrical cell. The variables h, y and y’ are taken as in Fig. 2. In the integration about y’ over the i’th region we concurrently sum up the values at two points A and A’ located symmetrically about the h-axis. And we express the optical distances along the paths AB
24
T.
FIG.
TAKEDA
2.-Notations
and T.
SEKIYA
in a concentric cylindrical cell.
N N and A’B on the x-y plane by Cp, and Cpz respectively.
Then we obtain
X (2 - 6,~~) Pnm(cos 0) cos map,
2r2
(
-
rj2 -
rj”-,
rj 2 - f2j-1
)’
where yc and y, are given by z/rr2 - h2 and l/rj 2 - h”. For the case of m’ = 0 the integration over dy’ and dy can be achieved since the quantities t2 and ra are expressed by he + yle and h2 + y2 respectively. The integration over the polar angle 6 is expressed by the Bickley function. The rational expression of the function is presented by MAKINO (1967). In the calculation of Pj;‘.m’,k’)-+‘n*m*k) for i > j, the following reciprocity relation can be used (H/n’)2 (2 - am,& &,V, p;Lm.kMn’.m’.li’) = (_1)“tm’(~,m)2 (2 - a,,) &Vi p;;‘.d.k’Mn.m.Ll. 6.2)
SPACE AND ANGLE DEPENDENT COLLISION PROBABILITY IN CELL PROBLEMS T. TAKEDA and T. SEKIYA Department of Nuclear Engineering, Faculty of Engineering, Osaka University, Yamada-kami, Suita-shi, Osaka, Japan (Received 29 March
1972)
Abstract-The integral transport equation is formulated in terms of the generalized collision probability in a cylindrical lattice cell in one velocity. The anisotropy of the neutron angular distribution and the scattering is considered by representing them with spherical harmonics series respectively. The space dependence of the neutron distribution is taken into account by representing it with a Legendre series in each annular region. The isotropic return condition or white boundary condition is adopted for the generalized collision probability. One group disadvantage factors are calculated for cylindrical lattices in both cases of isotropic and linearly anisotropic scattering using the generalized first-flight collision probability. 1NTRODUCTION
THE EFFECT of the anisotropic scattering on the disadvantage factor has been treated by us (1971) and by ROYL and EMENDORFER(1970) in a cylindrical lattice cell. They
have used the generalized first-flight collision probability with the isotropic return boundary condition by dividing the cell into some regions. But they assumed that the flux and the current are flat in each region. Although this assumption is improved by the increment of the number of partition, the improvement is very slow for the calculation of the effect of the anisotropic scattering. This is due to the fact that, from Fick’s law, the assumption of flat flux in a given region is contradictory to the assumption of flat current in the region. Further, different from the case of calculating the average flux for isotropic scattering, it is difficult to determine the way how to divide a cell for the case where we calculate the effect of anisotropic scattering since the current shows appreciable change over the cell. BRUN and KAVENOKY (1971) also studied the anisotropic effect using the conservation law which relates the flux and current, but his method is limited up to the linear anisotropic scattering. In this paper, we will take account of the spatial dependence of the neutron angular flux in each annular region by expanding it into a Legendre series. This is an extension of the method for a slab cell by CORNCOLD (1956) and CARLVIK (1968). In a slab cell, the periodicity of the lattice can be treated exactly by adopting the perfect reflection condition for a neutron escaping from a cell. But in a cylindrical cell the condition is not satisfactory and the isotropic return condition is proved to be good for isotropic and anisotropic scatterings (TAKEDA et al. 1972) by numerical calculations. Expanding the neutron angular distribution into a spherical harmonics series, we will derive a generalized collision probability with use made of the isotropic return condition at the cell boundary. DERIVATION
OF GENERALIZED
COLLISION
PROBABILITY
One speed neutron distribution is considered in a circular cylindrical cell with inf?nite axial distance. We choose the z-axis in the axial direction and assume that 0 is the polar angle of the direction of a neutron 8 from the z-axis and TVis the azimuthal angle of the projection of S2 onto the x-y plane from the radial direction. In such a 15
16
T. TAKEDA and T. SEKIYA
system we expand the neutron angular flux ~$(r, a) and the source distribution S(r, SZ) into the spherical harmonics series about the neutron angle and the Legendre series about co-ordinate space respectively. We divide a cell into N concentric annular regions and take the inner and outer radii of i’th region as ri_r and ri respectively. Then the neutron distribution at radius r in the i’th region may be expanded by a Legendre series Pk(2r2 - ri2 - &Jri2 - rf_J (k = 0, 1,2, . . .). Namely the neutron angular distribution and the source distribution in the i’th region have the forms:
(1) S(r, $2) = 2
$
~(Hnm)2S~n,m~k’Psna(~os0) cos map,
““,T
n=Om=Ok=O
:2ri t
r’-l (2 - LJ9 1
(2) where (2n $ l)(n - m)!
H m= n
J
497(n + m)!
and 6,, is the Kronecker delta. Since we are dealing with an infinitely long cylindrical cell, there remain in equations (1) and (2) only the terms of which n + m is an even number. The scattering kernel z8(r, GY -+ 51) has no spatial dependence in a given region (for example in i’th region) and may be expressed in the form &(r, Q’ -Q)
= 2 2 (Hnm)21Z;sinP.m(cos0) cos m(a - y.‘)(2 - S,,). n=Om=o
(3)
The integral Boltzmann equation takes the form #r, a) =rdRe-‘(S(r’,
Sz) +SdP’Z,(r’,
a2 -+ Q)+(r’, p’))
(4)
where c”R is the optical distance from a point r’ to r. In terms of the orthogonal relation : daH,“P,“(cos s 4a
0) cos maH,tm’P,fm’(cos 0) cos m’o! =
(2 - 6,J1 (0
for II = n’ and 111= m’ (,_) otherwise
the scattering term in equation (4) reduces to dQ’&(r’, G?’+ !Z)+(r’, Sz’) s =jo
2,
,~$H,‘“)2+~“*“*k’Pk (““,
Tarp_ “-I) (2 - 6,0)X:d”(r’)P,m(cos 0) cos ma. z
El
(6)
17
Space and angle dependent collision probability in cell problems
We assume that the points r and r’ in equation (4) are located in j’th and i’th region respectively. The region number i changes with the distance R. Though the direction Q in the left-hand side of equation (4) is identical with that in the term of emission density in the right-hand side of the equation, the spherical harmonics expansions at both sides about azimuthal angles are different because of the difference of radial directions at r and r’. To express this difference we write the azimuthal angle at r’ by CC’.Substituting equations (I) through (3) and (6) into equation (4) we obtain
so2, =
~~H,m)2~~~-~k~~,m(cos 0) cos maP, (2r2 j
s
I
y ,- “-l) (2 - S,,) 3 I
OmdlZe-z $ i 5 (H,.m’)‘P,.““‘(cos f3) cos m’a’Pk, n’=Orn’%D 5’4l
>
(7) ?( (2 - fim,&$n’,m’.k’) + &in’C#Ijn’*m’*k’)}. Multiplying the above equation by P,“(cos 0) cos ma and integrating over dS2, we have
ss m
=
dR d8P,“(cos
6) cos nlxe
-E
0
i< 2 i 2 (H,,“‘)ZP,,,“n’(cos 0) cos m’u’Pk, 2r’2ry z2rTlr’-‘) C n’=_O rn’S07.‘;=0 2 E ><(2 _ ~m,O){s~n’.~‘.k’) _t &n’+;n’.m’.k’)).
(8)
Further we multiply the above equation by Pk(2r2 - rj2 - r:_,/r,” - ri_,) integrate with 2w dr over the volume ofj’th region, and we obtain
and
2r2 - rj2 - r:_,
(
(Hn?)2Pn.m’(COS e> cos m’a’ ) r.1? - rf_I 2rf2 - ri2 - r,“_, (2 _ ~m,O),@n’>m’~k’ + ) _=q’&‘.m’.k’)) ‘J Pkf r.2 r,‘ _, z
:-; P,
(9) where Vi is the volume ofj’th region. Here we define the following generalized first-flight collision probability: p!~‘.m’,k’)4n.m.k) 13
Gj
= v_
ss
z vi
vi
x Pk’ 2r”ry “4.: (
L
e-m
dr ~2
dr’
(H,?“)2P,fm’(~~~ 8) cos m’a’
“-I) (2 - Sm~O)P,m(cos0) cos ma 21 x Pk
2
2r2
(
-
rj2 - r:_-,
rj2 -
rtl
1
3 (10)
18
T. TAKEDA and T. SEKIYA
where Zti is the total cross section in j’th region. The explicit formulaforp~~‘,m’,k’)‘(n,~~~) is given in Appendix. In terms of the above equation, equation (9) reduces to
If the scattering and the source distribution are isotropic it is sufficient to take account of the contribution from the factor with (n, m) = (0,O). If we consider up to the linear anisotropic scattering it will suffice to treat the contribution from the factors with (n, m) = (0,O) and (1, 1). The quantity P$‘~“~o)+-(o,o,o) corresponds to the usual first-flight collision probability.
flight between neighboring cells.
Fm. I.-Neutron GENERALIZED
COLLISION
PROBABILITY
IN A LATTICE
CELL
We calculate the generalized first-flight collision probability P$(R’*m’,k’)+(n*m,k) that a neutron born in an i’th region in a cell undergoes its first collision in aj’th region in any cell so as to reduce the infinite sum over i to finite sum 1 I i I N. First we will calculate the generalized first-flight collision probability for a neutron born in a cell to make its first collision in the nearest neighboring cell. As shown in Fig. 1, we divide the distance R from the region i to the region j into two parts R, and R,; R, being the path length in an original cell 1 and R, being the path length in subsequent cell 2. Then we obtain ph’.m’.k’)+h.m.k) fj
2rt2 -
x P, (
x p
k
ri2 -
ri2 - $-I 29 -
rtl
)1
rj2 - r3_1
rj2 -
rtl
>
(2 - Bm.o)e-‘l. e-x’zP,m(cos 9
0) cos ma
(12)
where dR is a line element in the j’th region along the direction a. We describe the solid angle subtended by the neutron direction from the interface of cell by sZ* and
19
Space and angle dependent collision probability in cell problems
the normal vector at the interface by n. We make an approximation to calculate the above equation as follows: Expressing the surface area of the cell per unit height by S we insert an identical operator 1
S
dfi*(n s 277
(13) . Q*)
between the terms e-% and emcY2m . equation (12) and neglect the r’ and S2dependence of the latter term. This is based on the assumption that all neutrons impinging on the cell surface may return isotropically. Then we obtain p+‘.n’.h-‘Hn,m.l) =
p,!~‘m’,k”p,‘wn>kJ,
(14)
23
where P@‘*nL’,k’ is )the probability that a neutron born in the i’th region in a cell 1 in mode (9, m’, k’) escapes the cell without collision, and is given by p!n’ sm’ek’ =) i 18
s sdne-~~l(H,,n~‘)2p,.“‘(cos 0) cos m’u’
vdr’
t
’
2r’” ;(
Pk.
ri2 -
$_,
(2 - &%),
ri= - r;_,
and P$,llLsk)is the probability that a neutron entering the cell 2 isotropically in thej’th region in mode (n, m, k), and is expressed in the form p(?f.m.k) II
1
=
collides
dRe-Lr;;apnm(cos 0)
dS dQ*(n . sl*)
S dQ*(n . Q*)
(15)
s
ss
.
-
2r2
> cos muP,
rj2 -
r.2 , -
r5_1
?$_I
C,j*
(16)
Equation (15) can be transformed into the form ,;~‘.m’.k’)
=
1vi
l,clr’ J‘dB (1- rdRZ(R)e-‘)
X cos m’a’Pkf
2rr2 -
(
dr’ = dk&,,,,Bm,,, - r Vi s Vi f X cos m’a’Pkr
rf_,
ri2 -
ri2 -
(H,tm’)2Pli,m’(cos0)
rF_1
1
(2 - 4nxl)
drC(r)e- G (HaSm’)2Pntnz’(~~~ 0) cell
1
2rf2 - ri2 -
&
ri2 - rf_1
(2 -
LA
(17)
where the integration about r is made over the volume of the cell 1. In terms of the definition of P@‘ Y ~nb’~l’-,m,k),we obtain
P;p-) = Bn,Odm'OBK'O
(18)
20
T.
and
TAKEDA
T.
SEKIYA
In the following we calculate the probability Pijn,m,k).In a circular cylinder
s . . da*(n
S2*) = 7.
(19)
277
We denote the solid angle subtended by the neutron direction from the collision point r in the j’th region by 8’. Then using the relations (n Sit*) ___ dS = da’, Rz2
(20)
Rz2 dR dQ* = dr,
we have the relation (n * S*) da* dS = dS2’dr.
(21)
Adopting these relations we rewrite equation (16). At this point it should be noticed that, if we express the azimuthal angle by CIfor a neutron starting from point r toward the cell surface, we must replace u in equation (16) by rr - t( because the u in equation (16) is measured for a neutron directing from the cell boundary to the point r. Then equation (16) reduces to s
8)( - 1)” cos map,
4adS2’Vdret’lsP,“(cos s 3
4Ec,jvj = BnOBm,&,- 2 s
4CtjVj
2r2 -
rj2 -
rj2 -
rja-
$1
r;_,\ J
$I
r5-1
(2 - S&-l $ Pi;*m,r,-(“.o*o)].
BnOBm08k0 - 4;;;)m)a
1
S
rj2 -
~~~r~~pn.~~~hRz(R)e-~
X Pnnz(COs6) COsma(-l)“P,
= -
2r2 -
n
l=l
(22)
Considering P$m,k) in the case of II = m = k = 0, the probability for a neutron entering a cell isotropically to escape the cell without collision is given by N 4&jvi
P,,=l-z----
j=l
s
1 _ fj p~,0.0,40.0.0, . (
(23)
1
Z=l
From equations (18), (22) and (23), the generalized first-flight collision probability that a neutron born in an i’th region in a cell makes its first collision in a j’th region in any cell is given by p~(n’.m’.k’bh,m.k) $2
=
pbAm’.k’b+h.m.k)+ ii
X-
4&Vj S
BnOB,oc3ko (
(2 - &,)-I
4;;;);)2
12
2 P$m.x)-ro*o.o,). I’=1
(24) Using this probability $(n*m*k) is obtained by solving the equation
Space and angle dependent collision probability TABLE l.-LEGENDRE Upper limit of Legendre polynomials
in cell problems
EXPANSION COEFFICIENT OF FLUX FOR ISOTROPIC WAITBRING Case 1
Expansion coefficient
0
+
1
+ %O,Ol
21
Fuel
(0 O,O)
,#lO,O,l,
Moderator
Fuel
Case 2 Inner moderator
Outer moderator
4.5298
5.1556
96.0
161.66
177.95
4.5294 0.2546
5.1654 0.1316
96.0 24.339
170.68 21.657
192.10 3.4884
2
(0,0,1l 4 (O,O,?)
4.5292 0.2532 0.04008
5.1679 0.1323 -0.1120
96.0 23.954 3.8506
172.5’2 22.103 -9.9745
193.75 3.6622 -2.2599
(O,O,O,
4.5292
3
f i0,O.l) 10.0,2)
5.1681 0.1320 -0.1120 0.03056
96.0 23.879 3.7706 1.0457
172.82 21.874 -10.108 4.5623
193.99 3.6185 -2.2494 0.3406
to,o,o,
0.2531 0.03991 0.01323
lO.O,3J
TABLE 2.--MODERATOR-TO-FUEL
FLUX RATIO FOR ISOTROPIC SCATTERING
Upper limit of Legendre polynomials 0 1 2 3
TABLE 3.-LEGENDRE
Upper limit of Legendre polynomials 0
Case 2
1.1382 1.1404 1.1410 1.1412
1.7971 1.9267 1.9445 1.9472
EXPANSION COEFFICIENT OF ANGULAR ANISOTROPIC SCAlTERING @ = 1/3) Case 1
Expansion coefficient ,+0,0,0, 4 ,1,1,0)
p,o,o, p,o,1,
3
Case 1
#o,o,z, +to,o,w + 11,1.01 p,1.1, p.1.2,
4 U,l,W
Fuel
Moderator
Fuel
FLUX FOR LINEAR
Case 2 Inner moderator
Outer moderator
157.37 -8.2352
169.70 -1.8469
4.5305 -0.2067
5.1367 -0.1343
96.0 -12.43
4.5306 0.2545 -0.2111 -0.1422
5.1362 0.1121 -0.1390 0.1594
96.0 24.224 -13.856 - 10,696
161.95 15.976 -10.177 6.2690
177.30 2.1676 -2.1651 2.4177
4.5305 0.2533 0.04008 -0.2110 -0.1427 0.02606
5.1382 0.1135 -0.1027 -0.1392 0.1621 -0.02874
96,O 23.875 3.8034 -13.836 - 10.887 1.3677
162.97 16.541 -7.9287 - 10.262 7.0200 -2.9097
178.18 2.3404 -1.5783 -2.1682 2.5546 -0.4560
4.5305 0.2532 0.03991 0.01324 -0.2110 -0.1427 0.02594 -0.01364
5.1383 0.1132 -0.1029 0.02868 -0.1392 0.1621 -0.02926 OGO6809
96.0 23.806 3.7278 I.0453 -13.843 -10.876 1.3309 -0.8810
163.15 16.392 -8.1376 3.9630 - 10.262 7.0756 -3.1932 1 a4839
178.33 2.3138 -1.5738 0.1497 -2.1619 2.5542 - 0.4673 o-1034
22
T. TAKEDA and T. SEKIYA TABLE a.-LEGENDRE
Upper
EXPANSION COEFFICIENT OF ANGULAR FLUX FOR LINEAR ANISOTROPIC SCATTERING (iu = 2/3)
limit
of Legendre
polynomials 0
Case 1
Expansion coefficient
10,0,0) (1,1,01
_~_. -1
2
(O,O,Ol f Wl,O,l)
4.5313 -0.2068
Moderator 5.1176 -0.1352
4.5318 0.2545
5.1069 0.09248
Fuel 96.0 - 12.505 96.0 24.104
$ (1,1,11 I1,l.O)
-0.1422 -0.2113
-0.1392 0.1597
--13.939 10.608
$ l0,0,11 (O.O,O, : i0,0,21
4.5317 0.2533 0.04007 -0.2112 -0.1427 0.02611
0.09455 5.1083 -0.09342 -0.1394 0.1621 -0.02875
9 lO,O,3)
4.5317 0.2532 0.03991 0.01324
0.09430 5.1084 -0.09375 0.02680
f r1,1,w (l,l,ll (1,1,2) (l,l,S)
-0.2112 -0.1426 0.02599 -0.01365
(l,l,O)
9 (1.1,1, 9 (1.1,21
f lO,O,OL 10,O.l) ,#p,O,Z,
3
Fuel
Case 2
NUMERICAL
m--o.1394 0.1622 -0.02924 OX)06784
RESULTS
AND
Inner moderator
Outer moderator
152.38 - 8.9029
160.07 -2.0294
152.88 10.120
- 162.01 0.8065
- 10.321 6.5214
-2.1976 2.4902
96.0 23.794 3.7551 -13.920 - 10.793 1.3496
153.18 10.846 -5.8355 -10.388 7.0983 -2.9595 ____
162.24 0.9870 -0.8869 -2.1970 2.5676 -0.4732
96.0 23.732 3.6841 1.0449
153.27 10.771 -6.1224 3.3570
162.31 0.9750 -0.8900 -0.04470
--10.781 13.927 1.3125 -0.8748
--lo.387 7.1461 -3.2045 1.4716
--2.1906 2.5659 -0.4708 0~1100
CONCLUSIONS
We calculate the flux distributions in the cases of two different lattice cells. The first cell called case 1 is a closely packed lattice cell and the second called case 2 is that with a large moderator, in both cases a cell is composed of a fuel rod and a moderator. For case 1 the outer radii of fuel and moderator are 0.38 1 and 0.64487 cm respectively and the total and scattering cross sections of the fuel are 0.78 and O-387 cm-l and those of the moderator are l-0618 and 1.053 cm-l. For case 2 the outer radii of fuel and moderator are 2-O and IO.0 cm respectively and the total and scattering cross sections of the fuel are 0.6 and 0.35 cm-l and those of the moderator are 0.3 and O-3 cm-l. The case 2 corresponds to the cell adopted by LEWIS (1969). In the calculation we divide the cell of case 1 into 2-regions and that of case 2 into 3-regions (one fuel rod and two moderator regions divided by a circle at radius 6 cm). In calculating flux distribution we assume a uniform and isotropic source per unit volume only in moderator. In the case of isotropic scattering the expansion coefficients #O,OPk) of the flux are given in Table 1 for the cases where the upper limit of k is taken to be 0, 1, 2 or 3. For cases where the upper limit of the Legendre polynomials is taken to be 3, the coefficients c$(O$O,~) (k = 0, 1,2,3) can reproduce an almost continuous flux curve at is very fast for case 1 but interfaces. The convergency of the coefficients c#(O,O,~) not so rapid for case 2. This shows that the deviation of flux in each region from the flat flux is very small for case 1 but is large for case 2. The moderator-to-fuel flux ratio is given in Table 2. From this table it can easily be seen that the value of the flux ratio converges very rapidly with the increase of the number of terms
Space and angle dependent collision probability
in cell problems
23
TABLE S.-MODERATOR-TO-FUEL FLUX RATIO FOR LINEAR ANISOTROPIC SCATTERING(.lC= 1/3) Upper limit of Legendre polynomials
-
0 1 2 3
Case 1
Case 2
I.1338 I.1337 1.1341 1.1342
1.1249 I.7936 1.8032 1.8049
--
TABLE 6.-MODERATOR-TO-FUEL FLUX RATIO FOR LINEAR ANISOTROPIC SCATTERING(,u = 2/3) Upper limit of Legendre polynomials 0 1 2 3
Case 1
Case 2
1.1294 1.1269 I.1272 1.1273
16407 1.6559 1.6585 1.6593
the Legendre expansion. And, for the ratio, it is sufficient to take into account only the first two terms for k = 0 and 1. The Legendre expansion coefficients in the case of linear anisotropic scattering are presented in Tables 3 and 4 where the average cosine of the scattering angle ,u are taken to be l/3 and 2/3 respectively. From these tables it can be seen that the convergency of the expansion coefficients ~#(lJ,~r(k = 0, 1, . . .) is very slow for both cases compared with that of 4 (“,o,k). Especially the term #rtryl) is nearly comparable with the term qY1*l,o).Then the flat current approximation is disagreeable to express the curve of the current over a cell. In the case of anisotropic scattering also, the consideration up to the term k = 1 yields good results for the expansion coefficients, The moderator-to-fuel flux ratio for the case of anisotropic scattering is presented in Tables 5 and 6 and shows that the error resulted from the flat flux and flat current approximation decreases with the increase of ,u. And if we include the term k = 1, remarkable improvement is obtained.
in
REFERENCES BRUN A. M. and KAVENOKYA. (1971) Paper presented to ANS National Topical Meeting New Developments in Reactor Mathematics and Applications. CARLVIKI. (1968) Nucl. Sci. Engng 31, 295. CORNGGLDN. (1957) J. nucl. Energy 4,293. KAVENOKY A. (1969) CEA-N-1077. LEWIS E. E. (1969) J. nucl. Energy 23,87. MAKINO K. (1967) Nucleonik 9, 351. ROYL P. and EMENDORFER D. (1970) Atomkernenergie 18, 151. TAKEDAT. and SEKIYAT. (1971) J. Nucl. Sci. Technol. 8,663. TAKEDAT., NAKAMURAK. and SEKIYAT. (1972) J. Nucl. Sci. Technol. 9,53.
APPENDIX We evaluate the generalized first-flight collision probability [equation (IO)] for a cylindrical cell using the same technique as KAVENOKY(1969) adopted for the calculation of the usual first-flight collision probability. First we calculate 4, (n’.m’.L‘--c’n.m.L’ for i < j. The region number is enumerated from the inner site of a cylindrical cell. The variables h, y and y’ are taken as in Fig. 2. In the integration about y’ over the i’th region we concurrently sum up the values at two points A and A’ located symmetrically about the h-axis. And we express the optical distances along the paths AB
24
T.
FIG.
TAKEDA
2.-Notations
and T.
SEKIYA
in a concentric cylindrical cell.
N N and A’B on the x-y plane by Cp, and Cpz respectively.
Then we obtain
X (2 - 6,~~) Pnm(cos 0) cos map,
2r2
(
-
rj2 -
rj”-,
rj 2 - f2j-1
)’
where yc and y, are given by z/rr2 - h2 and l/rj 2 - h”. For the case of m’ = 0 the integration over dy’ and dy can be achieved since the quantities t2 and ra are expressed by he + yle and h2 + y2 respectively. The integration over the polar angle 6 is expressed by the Bickley function. The rational expression of the function is presented by MAKINO (1967). In the calculation of Pj;‘.m’,k’)-+‘n*m*k) for i > j, the following reciprocity relation can be used (H/n’)2 (2 - am,& &,V, p;Lm.kMn’.m’.li’) = (_1)“tm’(~,m)2 (2 - a,,) &Vi p;;‘.d.k’Mn.m.Ll. 6.2)