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A consistent approach of heterogeneous theory leading to a dipolar formulation—I. Formulation
Journal of Nuclear Encrsy, Vol. 26, pp. 303 to 317. Permtmn Press 1972. Priatcd in Northern Ireland
A CONSISTENT APPROACH OF HETEROGENEOUS THEORY LEADING TO A DIPOLAR FORMULATION-I. FORMULATION* S.E.R.M.A.,
PH. BERNAt B.P. No 2, 91 Gif-sur Yvette, France (Received 14 October 1971)
Abstract-Starting from a theorem established in transport theory, we developed a new dipolar heterogeneous formulation where channels are characterized by parameters depending on the environThe formulae for these parameters can be calculatedin less than ment only through cross-sections. 0.3 set (by channel type) on an IBM 360/91. The FODITH code based on this formulation gives an acceptable value for the experimental reactivity of a reactor having 3 narrow zones of different scattering properties (deviation lower than 200 pcm). The code therefore gives results in better agreement with the experiment than other code-s based on diffusion or source-sink theory. The new formulation appears as the extension of the ABH method initially developed for cell calculations to heterogeneous reactor calculations. 1. INTRODUCTION
crossing the moderator of a nuclear reactor, each channel: and its contents introduce two discontinuities : one for absorption and fission and the other for scattering. These discontinuities can be dealt with in three different ways
FOR A NEUTRON
(a) Their effects in the reactor are made uniform: this is the homogeneous theory. (b) Only the effects of scattering discontinuities are made uniform and the geometric locations of absorption and fission discontinuities are correctly taken into account: this is the source-sink theory of FEINBERG(1955) and GALANIN(1955). (c) The geometric locations of both discontinuities are taken into account: this is the heterogeneous theory. Codes using the first theory are convenient only for the calculation of reactors with large regular lattices. With the codes using the second theory, reactors presenting many irregularities in channel distribution can be calculated provided that the scattering properties are uniform over the whole reactor. The third theory, on the other hand, aims at the development of methods applying to the calculation of any type of reactors,5 in particular of reactors presenting many irregularities in channel distribution, even if scattering, absorption and fission properties are not the same over the whole reactor. Existing heterogeneous formulations had the disadvantage, either of introducing parameters depending on the flux structure or of needing an auxiliary time-consuming code to calculate the parameter values (AUERBACH,1967a,b; ALPIAR, 1969). * Paper for presentationto the E.A.C.R.P. WorkingGroup Meetingon HeterogeneousMethods for Reactor Calculations-Paris,March 1971. 7 Present address: SECPR, C.E.N. Cadasache,B.P. No. 1, Saint Paul les Durance 13, France. $ By channel we mean the cavity and the rod. 0 Weconsideronly thermal reactors wherethe channel spacingexceedsseveralmean free paths in the moderator. 303
304
PH. BERNA
This paper presents a new heterogeneous formulation where channels are characterized by parameters strictly independent of the environment, which can be calculated in less than O-3 set (by channel type) on an IBM 360/91. We start from a theorem proved in transport theory: the separation theorem, which shows that each channel contribution may be obtained independently of those of the other channels. Each channel contribution may be calculated using the ‘open cell technique. By open cell (term introduced by ALPIAR, 1969), we mean the geometry obtained by surrounding the channel by moderator up to infinity. The effect of the other channels is taken into account by the introduction of a compensating source in the above channel. The compensating source and the channel contribution are expanded in Fourier series around the channel axis. These expansions are limited to the first order (dipolar approximation). Asymptotic theory is used in the moderator while transport theory is used within the channels. The problem is treated in multigroup theory, with one thermal group. Heterogeneous parameters are introduced to calculate the coefficients of the Bessel functions in the asymptotic expression of channel contributions in the moderator. The formulae for these parameters are obtained by comparing the results of a cell calculation to those given by the new heterogeneous formulation for the corresponding infinite periodic lattice. They do not involve the lattice pitch since the parameters have been defined in an open cell. It can be shown that, by the use of the et al., new formulation, the ABH formula of the thermal utilization factor (AMOUYAL 1957; AMOUYALand BENOIST,1957), and the formula of the extrapolation length of the channels are preserved (that was not possible with the source-sink theory). The new formulation appears then as the extension of the ABH method initially developed for cell calculations to heterogeneous reactor calculations. The code FODITH (for IBM 360/91) is based on this new formulation. Comparison of results given by FODITH, HECTOR (source-sink theory) and RIFIFI (homogeneous theory) with experimental results showed that FODITH was quite satisfactory for reactivity calculations of reactors, such as MARIUS. As a matter of fact, when the reactor has many irregularities in channel distribution and when scattering, absorption and fission properties are not the same over the whole reactor, FODITH gives an acceptable value of the reactivity, while the values given by HECTOR and RIFIFI differ from the real value by more than 1500 x 10e6 (in opposite directions). FODITH was applied to the computation of the average flux gradient across the clusters of an experimental AGR type reactor with an empty channel at its centre. FODITH is able to describe conveniently the flux fine structure in an irregular lattice. 2. SEPARATION THEOREM The establishment of the separation theorem aims essentially at proving the rightness of the idea contained in the Feinberg-Galanin formalism, that is: the effects of the channels can be separated in the flux calculation of a heterogeneous reactor. It is proven by the transport theory that each channel contribution to the flux can be calculated in the open cell, built by surrounding the channel with moderator up to infinity, provided that a source proportional to the sum of the contributions of the other
A consistent approach of heterogeneous
theory leading to a dipolar formulation-I
305
channels is introduced within the above-mentioned channel.* This source compensates for the fact that the channel is replaced by moderator when the other channel contributions are calculated. Let Y be the angular flux in the reactor, vk the contribution of the kth channel, and S, the compensating source to be introduced in the kth channel. We have V=xVj
(24
(the sum is extended to the number n of cdannels of the reactor) if, and only if, S, has the intensity S, = (8X, - %3%k
(2.2)
in the medium i contained by the kth channel with vOk =
(2.3)
z: VI. 5#k
The operator e,, combines symbolically the total absorptiont, I&, scattering, 2 .s(,and fission$, Pit, operators for the medium i; m is the subscript for the moderator. We have
e,,= X,i-2e,i-LP
47r ji
(2.4)
Proof
In the stable state, the flux v satisfies the equation div Qv + &,,v = 0 in the reactor moderator,
(2.5)
and the equation div Qv + &v = 0
(2.6) The flux V~ satisfies the equation
in the medium i belonging to the kth channel.
div Szv, + &,,,vk: = 0
in the moderator
(2.7) of the open cell centred on the kth channel and the equation div
QRyk
+
&Vk
=
sik
(2.8)
in the medium i belonging to the kth channel. The open cells are made to overlap in such a way as to preserve the channel distribution as it is in the reactor. We add term to term at each point the equations satisfied separately by all the contributions of the reactor channels. We find (2.9) * It is assumed than the medium external to the reactor can be simulated by to an annulus of black rods. t This operator is scalar. $ The fission operator is assumed to be isotropic.
306
PH. BERNA
on the area of the reactor moderator and div Q 2 ~5 + f;,dvk + &, 2 ~5 = Si, i
(2.10)
i#k
the area of the medium i belonging to the kth channel. The comparison of equations (2.6) and (2.10) shows that the necessary and sufficient condition in order that 2, vj be equal to v is to define Si, by equation (2.2), which demonstrates the separation theorem. on
3. APPLICATION CALCULATION
OF THE SEPARATION THEOREM TO THE OF A REACTOR OF INFINITE HEIGHT, IN MONOKINETIC THEORY
To present in a simple way the new heterogeneous formulation we can restrict ourselves to the monokinetic theory in applying the separation theorem to the calculation of a reactor of infinite height. In the following section, we shall see that the multigroup formalism reduces to monokinetic theory within each group. 3.1. Flux equations in the open cell centred on the kth channel We shall now assume the collision to be isotropic* and we shall put
(a) equation in the moderator div Pv, + &,,vk =‘&
(3.1)
4lr
(b) equation in the medium i belonging to the kth channel (3.2) with Ln sik
=
(%I
-
&bOk
-
If we group the fission terms, equation (3.2) becomes &#k
div SLY,+ Z,,v, = - 4r
+
‘ik
(3.3)
with z,k
=
&a
-
&ihk
-
(3.4)
3.2. Description of v,,~on the area of the kth channel The area of the kth channel is in the asymptotic region of the moderator of the other open cells.? Hence, we can replace vok in the definition (3.4) by the sum of the * The finite height of the reactor and the collision anisotropy in the moderator will be taken into account separately by a correction. t See footnote (9) p. 303.
A consistent approach of heterogeneous theory leading to a dipolar formulation-I
asymptotic contributions of the other channels [see equation written (CASEet al. 1953)
(2.3)];
307
this can be
(3.5) where rj is the centre of the jth channel in the reactor section which contains r; u is an unit vector contained in the same section; fi a function depending only on u; Km is the inverse of the diffusion length in the moderator and is given by z e
&I tanh-’ F = 1. (3.6) ( tm1 111 The functionsf, are eliminated when the equation (3.5) is expanded in a Fourier series as a function of the angle that brings I - r, on the gradient V+,,(rJ o f& at the point r,. To the first harmonic, we have vOk
-
:
$Ok(rk)
-
+
[%
s
r
m
. v+Ok(rk)
+
h.2
s
. Rv+Ok(rk)]
9
(3.7)
where R represents - (7r/2) rotation around the channel axis
s 2n
e-Kmp
cxmaO
0 c,, - K, cos (go + q~)sin 8 2r
z
e-K,,,p
(3.8)
duo
cos q,
dor,
(3.9)
sin go dcr, 4?T2s 0 X,, - K, cos (a0 + 9) sin 13
(3.10)
ql(p~
v,
0)
=
-$
h,,(f,
y,
0)
=
5E
s 0 &?n - K, cos (CQ,+ q~)sin 0 2a
e-Kmp
cos
u.
cosn,,
p is the modulus of r - r, ; q~is the angle that brings the projection of S2 in the reactor section on r - r,; 13is the angle that brings Sz on the channel axis. It is easy to see that
s s
(4 ,aodQ = r
(4a)
~1.1 da
a,,,
2ZoKnd
= --2Z1Knp)
da = 0.
s (4a)
Hence, by integration of equation (3.7) over the solid angle 47~ +Ok(r)
=
zO(Km
b
-
rki)+Ok(rk)
+
j$
ZdK,
ir
-
bl)
E,
. v+ok(r&
(3.11)
m
Justification of the dipolar approximation : If the expansion (3.11) had been extended to further orders, terms proportional to
PH. BERNA
308
Z,(K,p) cos 2cr, Z,(K,p) cos 3a . . . would have been obtained; &,Ok is interesting only for values of p less than the channel radius c. As the functions Z, tend rapidly to zero when n tends to infinity, the expansion (3.7) will be sufficient. 3.3. Expansion of the fission sources In a similar way, we shall assume it valid to calculate the fission sources in the linear flux approximation within the medium i (if necessary, this medium will be divided in thin annuli). This gives (3.12) where
(3.13) and where the x component of the vector Vo,, is defined by:
vail, =
s
(x - QGW ({’ s
$
dV
- G)~ dl’
(3.14) -
3.4 Formula for the compensating source within the medium i belonging to the kth channel From equations (3.7) and (3.12), we have
h@tm- &9 La - u%k(r,) + 4Ln - &Yw%k(r,)l
- f m
x s
+ ‘2.
(r - r),
(3.15)
with z,(t, s) = t ;
-.;
zdt, s> = tal.1 +
k
~~(0 = ral,,.
ZiKnp)
(3.16)
Z1(K,p)
(3.17) (3.18)
Equations (3.16), (3.17) and (3.18) are invariant under a rotation of the vector couple r - r,, S2 around the channel axis. 3.5. AsymptoticJlux in the moderator of the open cell centred on the kth channel. Introduction of heterogeneous parameters The open cell is symmetric with respect to the channel axis. Since the sources of equation (3.3) have a term of revolution and an antisymmetrical term with respect to
309
A consistent approach of heterogeneous theory leading to a dipolar formulation-I
this axis, it is the same for the flux &. The asymptotic part of this flux in the moderator of the open cell centred on the kth channel can be written
(3.19) Here 7, yc, 8 and 8*are constants; jj and yi are the polar heterogeneous parameters of the kth channel and ,8 and 19~the dipolar heterogeneous parameters. It can be seen that they are defined in the following way. In the moderator of the open cell centred on the kth channel, $7and yi are respectively the coefficients which must be applied to K,,(K, Ir - rkl) to obtain the asymptotic part of the fluxes respectively created : by the source s,, located in the channel and having the intensity GL?a - &P %I - U in the medium i, and by the isotropic and uniform source located in i and having the intensity 1/4rr. Similarly, ,!I and Bi are respectively the coefficients which must be applied to K,,$,(K, Ir - r, I) cos ccto obtain the asymptotic part of the flux respectively created: by the source v1 located in the channel and having the intensity* +
M%,
- &AL
112
- %) cos cc + z@:,, - &) sin
al
in the medium i, and by the isotropic source located in i and having the intensity p cos u/477. Calculation of these parameters will be described in part II of this paper. It must be mentioned that they are, by definition, independent of the environment of the kth channel. 3.6. Relation between ai, and theflux $,,k(rk) The calculation of oiO reduces to an absorption rate calculation in the medium i. Taking into account the equality d =
we have
&i
r
$0,
+
r
r
+dv =&
J(i)
J
(3.20)
$kk,
&kd~+&
(i)
J&dV. (i)
(3.21)
From equation (3.1 I), the following equality is easily obtained:
where pt and pi’ are respectively the internal and external radii of the medium i. * As the source with an intensity of l/K,,, zg sin a is odd with respect to Q [see equation (3.10)], it induces a flux having a cos a shape. 3
310
PH. BERNA
The integral Z& jfij & dV is no other than the capture in i due to the compensating source in the open cell centred on the kth channel. From equation (3.15), we see that (3.23) where Poi Vi is the capture in i due to the source s,, and Pz, the probability for a neutron born according to an uniform and isotropic distribution in I to be captured in i after zero or any number of collisions in the open cell. The calculation of Poi will be found in BERNA(1971). Substituting equation (3.22) and equation (3.23) in equation (3.21), we find + Pll*Wl&)
(3.24)
with
Equation (3.13) leads to; (3.26) 3.7. Relation between Vail and the gradient V+,,(r& In a similar way, it can be shown that
where PtE*is the probability for a neutron born according to the ‘x mode’t in the medium I to be captured according to the ‘X mode’ in the medium i; and where -PIrVi represents the capture in i, according to the x mode, due to the source sl for which a is the angle that brings r - r, on Ox. The formula of Plz* will be found in part II. The calculation of PIi is described in BERNA(1971). 3.8. Characteristic relations for the kth channel Let us assume, for the sake of simplicity, that the channel contains only one fissionable medium u and let us put, as in the multigroup formalism (3.29)
% = r’+oOk(rk) + yuoUo V, = - JG(~ Wo_&J t See part II of this paper.
- 6, Vo,J.
(3.30)
A consistent approach of heterogeneous theory leading to a dipolar formulation-I
311
Eliminating CI,,,using equation (3.26) and Vrsul using equation (3.27), we find ak”$oAJ akl W,,(rJ
+ CkONk+ ruPkO$o&) + c,lV, + A&l W&r,)
(3.31)
+ &ON,1 = 0 +
(3.32)
43,l = 0,
with akl = 4KW2
LzKO = y CkO-
-
1 Ck =
1
-1
f&j* =$',,K,z 1 LPI,+ 1 bk' - 1
b,O = TY~ pou - r P:u:u, [ * Yu
(3.33)
U
dk’ = yP,,*
d,’ = Y%,
We have extracted from r,~the parameter ,u by which the fission sources must be
multiplied in order to make the reactor critical (,u = l/k&. 3.9. Geometrical relations Substituting equations (3.29) and (3.30) in equation (3.19) we obtain &&r) = N&,(K,
Ir - rA>+ Vk. El
(3.34)
KdK, Ir - d.
To calculate #ok on the area of the kth channel, $j can be replaced in the equality #Jam=jg4j by its asymptotic part (see Section 3.2)
[N,K,(K, Irk- rjl> + Vj . ‘si
400k(d =,z
N&Kl(L
1%- rjl) + V, - WV,h - rj>l ??
K1(K,
(3.35)
Irk - rjl)]
3
GW, 1%- r,l> lb - rjl
1
LKdL Irk- rjl) (rk - rJ . (3.36) Ir, _ rj12
3.10. Critical equation for the reactor
The system given by equations (3.31) and (3.32) relative to the n reactor channels can be written concisely in the matrix form A#0 + CN + PW,
(3.37)
+ DNl = 0,
where A, B, C and D are 3n diagonal matrices and $. and N the vectors defined by
Wo&> -,*..,G N = [N,, . . . , N,, v,,, . . . , v,,, K:y, . . . 9 v,,].
‘+“$)I m
(3.38) (3.39)
Similarly, the system of geometrical relations (3.35) and (3.36) expressing the interactions between all reactor channels can be written do = GN, where G is a 3n symmetric matrix.
(3 40)
A consistent approach of heterogeneous theory leading to a dipolar formulation-I
313
and where $iSasis the asymptotic part of the flux induced in the kth open cell moderator by the compensating source, in the absence of source in the moderator. The formulae for the constant coefficients ,~(g’+ g) and ~(g’ -+ g) are given in BERNA(1971). Obviously the process leading from equation (4.1) to equation (4.2) can be further pursued. As a result, in the kth open cell moderator the flux at the level of the g group has an asymptotic part, which can be written
+
Vkk’,g’>. ,: I
z,KICK;’Ir -
rkl) , 1
(4.4)
with m(g’, g) solution of the triangular system
m(g’g) y = k;;,~~
1
(4.5)
and
mk’, d
Nkk,d =15z_Pk’--+d v&L g) =lSJ<#&
k/
Nkk’,8’)+ Mk”
mw, s> Ug’, -+ g) kpO*
8’) + %‘,
(4.6)
(4.7)
where Mng and Tkgare respectively the polar and dipolar components of the asymptotic flux induced at the level of the g group in the kth open cell moderator by the compensating source, in the absence of source in the moderator
4.3. Expansion to$rst order in Fourier series of the compensating source around the axis of the kth channel
Incorporating in the compensating source the fission and slowing down terms, _ the equation which governs V~‘in J the medium i is div S&0 + &:,iovkg= ‘2 with
The number of groups is denoted NG.
&” + Z,;,
314
PH. BERNA
From the result obtained in the above paragraph, the reasoning followed in Sections 2.2,2.3 and 2.4 to expand the compensating source to first harmonic gives
Y&%-J = &
[
N,(g, g)&(L
Ir, - ri I) + V,(g, g) . fk &(K,‘Ir, Irk - rA
+ v,k?, d K”~~g~r~
3
ri’) - (Vj(g,
- rA)
1
(4.13)
g) . (r* - r,))
x LgKdLg Irk - r,l) (rh - r,)] I5 - rr12
linear flux approximation of the fission and slowing-down sources within i.
(4.14)
(4.15)
The formulae for ~$0, z$g and z$‘J are respectively deduced from those of z,, z, and z2 defined in monokinetic theory by replacing K, x s,,, by
K,,,O’ 1 - &+.g Ln“ + k;’ ’
by
where 8’~~is the Kronecker symbol, and c tm by Lg. 4.4. Introduction of heterogeneous parameters in multigroup theory From (4.12), it follows that l,
T; = --K,,,g ls~sgm(g’,
(4.16) (4.17)
and eig are generalizations for the parameters p, yi, fl and 8,. where j@lg, yig, ,8”‘*”
A consistent approach of heterogeneous theory leading to a dipolar formulation-I
315
4.5. Critical equation for the reactor in multigroup theory
By an approach similar to that used in monokinetic theory for proving the relations (3.31) and (3.32) it can be shown that the kth channel is characterized by the polar relation
+ PXq<~~$,O(g’
+ .!Z)Yodk’W+ d,O(g’-+ g)N,(g’, s’)l = 0, (4.18)
and by the dipolar relation
in group g, where the constants a”, b”, co, do, al, bl, cl and d1 depend only on the physical and geometrical properties of the channel and on the physical properties of the moderator surrounding it. These relations grouped with the geometrical relations (4.13) and (4.14) make a system of order 3n x NG, which can be put in the matrix form (4.20)
(H+/.K)N=O
with block diagonalization for the Green’s functions matrix. This equation represents the reactor critical equation. It is solved on an IBM 360/91 by the FODITH code, with a number of groups not exceeding six, by an iterative method with convergence acceleration, the basis of which is described in LIVOLANT(1968). 5. NUMERICAL RESULTS-COMPARISON OF THE FODITH HETEROGENEOUS CODE TO THE RIFIFI DIFFUSION CODE AND TO THE HECTOR SOURCE SINK CODE-COMPARISON WITH EXPERIMENT 5.1. Reactivity comparisons
We used the following codes:
RIFIFI: diffusion, one dimension, two groups (AMOUYAL et al. 1965); HECTOR: source-sink, age-diffusion (BOIVINEAUand CHABRILLAC,1965), infinite reflector effect corrected by RIFIFI; FODITH : two groups, infinite reflector effect corrected by RIFIFI, except in the case of the three-region reactor where the reflector boundary is simulated by an annulus of black rods. Four natural uranium-graphite characteristics : (A) square (symmetry (B) square (symmetry
reactors were calculated.
They have the following
pitch: 192 mm, channel dia.: 70 mm, rod dia.: 28 mm, 332 channels order: 8). pitch : 317 mm, channel dia.: 140 mm, rod dia.: 50 mm, 130 channels order: 4.)
316
PH. BERNA
(C) square pitch: 317 mm, rod dia.: 50 mm central region : channel dia. : 170 mm, 86 channels peripheral region : channel dia. : 140 mm, 76 channels (symmetry order: 4). (D) central region: square pitch : 384 mm, channel dia. : 140 mm rod dia.: 50 mm, 45 channels. intermediate region: pitch: 271.5 mm, channel dia. : 70 mm rod dia. : 31 mm, 16 channels. peripheral region : square pitch : 271.5 mm, channel dia. : 70 mm rod dia. : 50 mm, 72 channels (symmetry order: 8). TABLEl.-REACTIVITIES ( x 1O-6) RIFIFI A
+ 388
B C D
+ 148 + 114 + 1730
HECTOR
FODITH
+ 364 +2 - 385 - 2237
Experiment + 232 -8 - 38 + 159
+366 +159 + 66 - 86
It can be observed that even in the case of the most irregular reactor (D), FODITH gives a satisfactory result, while RIFIFI and HECTOR do not seem always suitable. 5.2. Calculation of the averagefrux gradient through the cluster of an A.G.R. type reactor having an empty channel at its centre
I
I p = 40.386
cm
FIG. l.-Reactor map. @I Empty channel-radius : 8.89 cm. o Fuelled channel-cluster radius: S-189 cm; outer radius: 8.89 cm. -Q+ Fueied channelfor which the ave;rag;p2x gradient through the cluster is given in
The following codes were compared with experiment: SQUIFID:
dipolar heterogeneous code built from a two dimensional diffusion code
(BLACKBURN, 1967); HET: supercell code (BLACKBURN,
1967) using the dipolar parameters calculated from the results given in (BRIDGE and HOWARTH, 1970); FODITH: used with 3 groups, reflector boundary simulated by an annulus of black rods.
A consistent approach of heterogeneous theory leading to a dipolar formulation-I
317
TABLE2.-RATIO (MAX - MIN)/MOY OF THEFLUXMEASURED BYA MANGANESE DETEC’rOR AROUND THECLUSTER Channel
EXP
FODITH ~ EXP
I II III IV V
0.092 0.076 0.078 0.090 0.170
0.730 1.030 0.896 1.179 1.023
SOUIFID .m
HET
EXP
EXP
1.130 1.408 1.038 1.756 1.600
051 O-81 0.70 l-00 0.85
The experimental uncertainty on this ratio can be estimated to he between 10 and 20 per cent. Let us remark that FODITH gives a result outside this range only for the cluster nearest to the empty channel. Acknowledgements-The results from SQUIFID and HET have been communicated by Mr. M. J. BRIDGE,whom I should like to thank.
REFERENCES ALPIARR. (1969) Heterogeneous reactor lattice programmes using the ‘open cell’ technique, EIR Bericht no 160. AMOUYAL A. and BENOI~TP. (1957) Interpretation des r&hats obtenus par S. A. Kushneriuk et C. MC. KAY dans le rapport AECL 137, Internal Report SPM. AMOWAL A., BENOI~TP. and HOROWITZJ. (1957) J. nucl. Energy 6,79. AMOUYAL A., BADOSM., LENATH. and MENGINF. L. (1965) RIFIFI II: Programme de resolution sur IBM 7090 des equations de la diffusion en theorie a dew groupes d’energie et une variable d’espace, Note CEA-N-518. AUERBACH T. (1967a) Two-dimensional multigroup theory with one thermal group and flux-independent cross-sections, EIR Bericht no 100; (1967b) A consistent heterogeneous theory of reflected lattices N.S.E. Vol. 29, pp. 317-324. BERNAPH. (1971) Formulation coherente de la theorie heterogene en approximation dipolaire, Thesis. BLACKBURN D. (1967) Heterogeneous methods for advanced gas cooled reactors, Ad hoc specialist meeting on the application of heterogeneous methods to bum-up calculations, Saclay 16th-18th January. BOMNEAUA. and CHABRILLAC M. (1965) Le code HECTOR (Methode heterogene de calcul de piles), Note CEA-N-523. BRIDGEM. J. and HOWARTHN. (1970) Corrections to the Benoist diffusion coefficients for CAGRTYPE lattices, Report of Central Electricity Generating Board-RD/B/N 1645. CASEK. M., DE HOFFMANN F. and PLACZEKG. (1953) Introduction to the theory of Neutron Diffusion, Vol. 1, LASL. FErNBERGS. M. (1955) Heterogeneous methods for calculating reactors: survey of results and comparison with experiment, P/669 (U.S.S.R.) 1st ZnttirnationalConference on the Peaceful Uses of Atomic Energy, Geneva, Proc. Vol. 5. GALANINA. D. (1955) The thermal coefficient in a heterogeneous reactor P/666 (U.S.S.R.) 1st International Conference on the Peaceful Uses of Atomic Energy, Geneva, Proc. Vol. 5. LIVOLANT M. (1968) Mtthodes iteratives simples pour la resolution de problemes lineaires, Internal Note SPM.
A CONSISTENT APPROACH OF HETEROGENEOUS THEORY LEADING TO A DIPOLAR FORMULATION-I. FORMULATION* S.E.R.M.A.,
PH. BERNAt B.P. No 2, 91 Gif-sur Yvette, France (Received 14 October 1971)
Abstract-Starting from a theorem established in transport theory, we developed a new dipolar heterogeneous formulation where channels are characterized by parameters depending on the environThe formulae for these parameters can be calculatedin less than ment only through cross-sections. 0.3 set (by channel type) on an IBM 360/91. The FODITH code based on this formulation gives an acceptable value for the experimental reactivity of a reactor having 3 narrow zones of different scattering properties (deviation lower than 200 pcm). The code therefore gives results in better agreement with the experiment than other code-s based on diffusion or source-sink theory. The new formulation appears as the extension of the ABH method initially developed for cell calculations to heterogeneous reactor calculations. 1. INTRODUCTION
crossing the moderator of a nuclear reactor, each channel: and its contents introduce two discontinuities : one for absorption and fission and the other for scattering. These discontinuities can be dealt with in three different ways
FOR A NEUTRON
(a) Their effects in the reactor are made uniform: this is the homogeneous theory. (b) Only the effects of scattering discontinuities are made uniform and the geometric locations of absorption and fission discontinuities are correctly taken into account: this is the source-sink theory of FEINBERG(1955) and GALANIN(1955). (c) The geometric locations of both discontinuities are taken into account: this is the heterogeneous theory. Codes using the first theory are convenient only for the calculation of reactors with large regular lattices. With the codes using the second theory, reactors presenting many irregularities in channel distribution can be calculated provided that the scattering properties are uniform over the whole reactor. The third theory, on the other hand, aims at the development of methods applying to the calculation of any type of reactors,5 in particular of reactors presenting many irregularities in channel distribution, even if scattering, absorption and fission properties are not the same over the whole reactor. Existing heterogeneous formulations had the disadvantage, either of introducing parameters depending on the flux structure or of needing an auxiliary time-consuming code to calculate the parameter values (AUERBACH,1967a,b; ALPIAR, 1969). * Paper for presentationto the E.A.C.R.P. WorkingGroup Meetingon HeterogeneousMethods for Reactor Calculations-Paris,March 1971. 7 Present address: SECPR, C.E.N. Cadasache,B.P. No. 1, Saint Paul les Durance 13, France. $ By channel we mean the cavity and the rod. 0 Weconsideronly thermal reactors wherethe channel spacingexceedsseveralmean free paths in the moderator. 303
304
PH. BERNA
This paper presents a new heterogeneous formulation where channels are characterized by parameters strictly independent of the environment, which can be calculated in less than O-3 set (by channel type) on an IBM 360/91. We start from a theorem proved in transport theory: the separation theorem, which shows that each channel contribution may be obtained independently of those of the other channels. Each channel contribution may be calculated using the ‘open cell technique. By open cell (term introduced by ALPIAR, 1969), we mean the geometry obtained by surrounding the channel by moderator up to infinity. The effect of the other channels is taken into account by the introduction of a compensating source in the above channel. The compensating source and the channel contribution are expanded in Fourier series around the channel axis. These expansions are limited to the first order (dipolar approximation). Asymptotic theory is used in the moderator while transport theory is used within the channels. The problem is treated in multigroup theory, with one thermal group. Heterogeneous parameters are introduced to calculate the coefficients of the Bessel functions in the asymptotic expression of channel contributions in the moderator. The formulae for these parameters are obtained by comparing the results of a cell calculation to those given by the new heterogeneous formulation for the corresponding infinite periodic lattice. They do not involve the lattice pitch since the parameters have been defined in an open cell. It can be shown that, by the use of the et al., new formulation, the ABH formula of the thermal utilization factor (AMOUYAL 1957; AMOUYALand BENOIST,1957), and the formula of the extrapolation length of the channels are preserved (that was not possible with the source-sink theory). The new formulation appears then as the extension of the ABH method initially developed for cell calculations to heterogeneous reactor calculations. The code FODITH (for IBM 360/91) is based on this new formulation. Comparison of results given by FODITH, HECTOR (source-sink theory) and RIFIFI (homogeneous theory) with experimental results showed that FODITH was quite satisfactory for reactivity calculations of reactors, such as MARIUS. As a matter of fact, when the reactor has many irregularities in channel distribution and when scattering, absorption and fission properties are not the same over the whole reactor, FODITH gives an acceptable value of the reactivity, while the values given by HECTOR and RIFIFI differ from the real value by more than 1500 x 10e6 (in opposite directions). FODITH was applied to the computation of the average flux gradient across the clusters of an experimental AGR type reactor with an empty channel at its centre. FODITH is able to describe conveniently the flux fine structure in an irregular lattice. 2. SEPARATION THEOREM The establishment of the separation theorem aims essentially at proving the rightness of the idea contained in the Feinberg-Galanin formalism, that is: the effects of the channels can be separated in the flux calculation of a heterogeneous reactor. It is proven by the transport theory that each channel contribution to the flux can be calculated in the open cell, built by surrounding the channel with moderator up to infinity, provided that a source proportional to the sum of the contributions of the other
A consistent approach of heterogeneous
theory leading to a dipolar formulation-I
305
channels is introduced within the above-mentioned channel.* This source compensates for the fact that the channel is replaced by moderator when the other channel contributions are calculated. Let Y be the angular flux in the reactor, vk the contribution of the kth channel, and S, the compensating source to be introduced in the kth channel. We have V=xVj
(24
(the sum is extended to the number n of cdannels of the reactor) if, and only if, S, has the intensity S, = (8X, - %3%k
(2.2)
in the medium i contained by the kth channel with vOk =
(2.3)
z: VI. 5#k
The operator e,, combines symbolically the total absorptiont, I&, scattering, 2 .s(,and fission$, Pit, operators for the medium i; m is the subscript for the moderator. We have
e,,= X,i-2e,i-LP
47r ji
(2.4)
Proof
In the stable state, the flux v satisfies the equation div Qv + &,,v = 0 in the reactor moderator,
(2.5)
and the equation div Qv + &v = 0
(2.6) The flux V~ satisfies the equation
in the medium i belonging to the kth channel.
div Szv, + &,,,vk: = 0
in the moderator
(2.7) of the open cell centred on the kth channel and the equation div
QRyk
+
&Vk
=
sik
(2.8)
in the medium i belonging to the kth channel. The open cells are made to overlap in such a way as to preserve the channel distribution as it is in the reactor. We add term to term at each point the equations satisfied separately by all the contributions of the reactor channels. We find (2.9) * It is assumed than the medium external to the reactor can be simulated by to an annulus of black rods. t This operator is scalar. $ The fission operator is assumed to be isotropic.
306
PH. BERNA
on the area of the reactor moderator and div Q 2 ~5 + f;,dvk + &, 2 ~5 = Si, i
(2.10)
i#k
the area of the medium i belonging to the kth channel. The comparison of equations (2.6) and (2.10) shows that the necessary and sufficient condition in order that 2, vj be equal to v is to define Si, by equation (2.2), which demonstrates the separation theorem. on
3. APPLICATION CALCULATION
OF THE SEPARATION THEOREM TO THE OF A REACTOR OF INFINITE HEIGHT, IN MONOKINETIC THEORY
To present in a simple way the new heterogeneous formulation we can restrict ourselves to the monokinetic theory in applying the separation theorem to the calculation of a reactor of infinite height. In the following section, we shall see that the multigroup formalism reduces to monokinetic theory within each group. 3.1. Flux equations in the open cell centred on the kth channel We shall now assume the collision to be isotropic* and we shall put
(a) equation in the moderator div Pv, + &,,vk =‘&
(3.1)
4lr
(b) equation in the medium i belonging to the kth channel (3.2) with Ln sik
=
(%I
-
&bOk
-
If we group the fission terms, equation (3.2) becomes &#k
div SLY,+ Z,,v, = - 4r
+
‘ik
(3.3)
with z,k
=
&a
-
&ihk
-
(3.4)
3.2. Description of v,,~on the area of the kth channel The area of the kth channel is in the asymptotic region of the moderator of the other open cells.? Hence, we can replace vok in the definition (3.4) by the sum of the * The finite height of the reactor and the collision anisotropy in the moderator will be taken into account separately by a correction. t See footnote (9) p. 303.
A consistent approach of heterogeneous theory leading to a dipolar formulation-I
asymptotic contributions of the other channels [see equation written (CASEet al. 1953)
(2.3)];
307
this can be
(3.5) where rj is the centre of the jth channel in the reactor section which contains r; u is an unit vector contained in the same section; fi a function depending only on u; Km is the inverse of the diffusion length in the moderator and is given by z e
&I tanh-’ F = 1. (3.6) ( tm1 111 The functionsf, are eliminated when the equation (3.5) is expanded in a Fourier series as a function of the angle that brings I - r, on the gradient V+,,(rJ o f& at the point r,. To the first harmonic, we have vOk
-
:
$Ok(rk)
-
+
[%
s
r
m
. v+Ok(rk)
+
h.2
s
. Rv+Ok(rk)]
9
(3.7)
where R represents - (7r/2) rotation around the channel axis
s 2n
e-Kmp
cxmaO
0 c,, - K, cos (go + q~)sin 8 2r
z
e-K,,,p
(3.8)
duo
cos q,
dor,
(3.9)
sin go dcr, 4?T2s 0 X,, - K, cos (a0 + 9) sin 13
(3.10)
ql(p~
v,
0)
=
-$
h,,(f,
y,
0)
=
5E
s 0 &?n - K, cos (CQ,+ q~)sin 0 2a
e-Kmp
cos
u.
cosn,,
p is the modulus of r - r, ; q~is the angle that brings the projection of S2 in the reactor section on r - r,; 13is the angle that brings Sz on the channel axis. It is easy to see that
s s
(4 ,aodQ = r
(4a)
~1.1 da
a,,,
2ZoKnd
= --2Z1Knp)
da = 0.
s (4a)
Hence, by integration of equation (3.7) over the solid angle 47~ +Ok(r)
=
zO(Km
b
-
rki)+Ok(rk)
+
j$
ZdK,
ir
-
bl)
E,
. v+ok(r&
(3.11)
m
Justification of the dipolar approximation : If the expansion (3.11) had been extended to further orders, terms proportional to
PH. BERNA
308
Z,(K,p) cos 2cr, Z,(K,p) cos 3a . . . would have been obtained; &,Ok is interesting only for values of p less than the channel radius c. As the functions Z, tend rapidly to zero when n tends to infinity, the expansion (3.7) will be sufficient. 3.3. Expansion of the fission sources In a similar way, we shall assume it valid to calculate the fission sources in the linear flux approximation within the medium i (if necessary, this medium will be divided in thin annuli). This gives (3.12) where
(3.13) and where the x component of the vector Vo,, is defined by:
vail, =
s
(x - QGW ({’ s
$
dV
- G)~ dl’
(3.14) -
3.4 Formula for the compensating source within the medium i belonging to the kth channel From equations (3.7) and (3.12), we have
h@tm- &9 La - u%k(r,) + 4Ln - &Yw%k(r,)l
- f m
x s
+ ‘2.
(r - r),
(3.15)
with z,(t, s) = t ;
-.;
zdt, s> = tal.1 +
k
~~(0 = ral,,.
ZiKnp)
(3.16)
Z1(K,p)
(3.17) (3.18)
Equations (3.16), (3.17) and (3.18) are invariant under a rotation of the vector couple r - r,, S2 around the channel axis. 3.5. AsymptoticJlux in the moderator of the open cell centred on the kth channel. Introduction of heterogeneous parameters The open cell is symmetric with respect to the channel axis. Since the sources of equation (3.3) have a term of revolution and an antisymmetrical term with respect to
309
A consistent approach of heterogeneous theory leading to a dipolar formulation-I
this axis, it is the same for the flux &. The asymptotic part of this flux in the moderator of the open cell centred on the kth channel can be written
(3.19) Here 7, yc, 8 and 8*are constants; jj and yi are the polar heterogeneous parameters of the kth channel and ,8 and 19~the dipolar heterogeneous parameters. It can be seen that they are defined in the following way. In the moderator of the open cell centred on the kth channel, $7and yi are respectively the coefficients which must be applied to K,,(K, Ir - rkl) to obtain the asymptotic part of the fluxes respectively created : by the source s,, located in the channel and having the intensity GL?a - &P %I - U in the medium i, and by the isotropic and uniform source located in i and having the intensity 1/4rr. Similarly, ,!I and Bi are respectively the coefficients which must be applied to K,,$,(K, Ir - r, I) cos ccto obtain the asymptotic part of the flux respectively created: by the source v1 located in the channel and having the intensity* +
M%,
- &AL
112
- %) cos cc + z@:,, - &) sin
al
in the medium i, and by the isotropic source located in i and having the intensity p cos u/477. Calculation of these parameters will be described in part II of this paper. It must be mentioned that they are, by definition, independent of the environment of the kth channel. 3.6. Relation between ai, and theflux $,,k(rk) The calculation of oiO reduces to an absorption rate calculation in the medium i. Taking into account the equality d =
we have
&i
r
$0,
+
r
r
+dv =&
J(i)
J
(3.20)
$kk,
&kd~+&
(i)
J&dV. (i)
(3.21)
From equation (3.1 I), the following equality is easily obtained:
where pt and pi’ are respectively the internal and external radii of the medium i. * As the source with an intensity of l/K,,, zg sin a is odd with respect to Q [see equation (3.10)], it induces a flux having a cos a shape. 3
310
PH. BERNA
The integral Z& jfij & dV is no other than the capture in i due to the compensating source in the open cell centred on the kth channel. From equation (3.15), we see that (3.23) where Poi Vi is the capture in i due to the source s,, and Pz, the probability for a neutron born according to an uniform and isotropic distribution in I to be captured in i after zero or any number of collisions in the open cell. The calculation of Poi will be found in BERNA(1971). Substituting equation (3.22) and equation (3.23) in equation (3.21), we find + Pll*Wl&)
(3.24)
with
Equation (3.13) leads to; (3.26) 3.7. Relation between Vail and the gradient V+,,(r& In a similar way, it can be shown that
where PtE*is the probability for a neutron born according to the ‘x mode’t in the medium I to be captured according to the ‘X mode’ in the medium i; and where -PIrVi represents the capture in i, according to the x mode, due to the source sl for which a is the angle that brings r - r, on Ox. The formula of Plz* will be found in part II. The calculation of PIi is described in BERNA(1971). 3.8. Characteristic relations for the kth channel Let us assume, for the sake of simplicity, that the channel contains only one fissionable medium u and let us put, as in the multigroup formalism (3.29)
% = r’+oOk(rk) + yuoUo V, = - JG(~ Wo_&J t See part II of this paper.
- 6, Vo,J.
(3.30)
A consistent approach of heterogeneous theory leading to a dipolar formulation-I
311
Eliminating CI,,,using equation (3.26) and Vrsul using equation (3.27), we find ak”$oAJ akl W,,(rJ
+ CkONk+ ruPkO$o&) + c,lV, + A&l W&r,)
(3.31)
+ &ON,1 = 0 +
(3.32)
43,l = 0,
with akl = 4KW2
LzKO = y CkO-
-
1 Ck =
1
-1
f&j* =$',,K,z 1 LPI,+ 1 bk' - 1
b,O = TY~ pou - r P:u:u, [ * Yu
(3.33)
U
dk’ = yP,,*
d,’ = Y%,
We have extracted from r,~the parameter ,u by which the fission sources must be
multiplied in order to make the reactor critical (,u = l/k&. 3.9. Geometrical relations Substituting equations (3.29) and (3.30) in equation (3.19) we obtain &&r) = N&,(K,
Ir - rA>+ Vk. El
(3.34)
KdK, Ir - d.
To calculate #ok on the area of the kth channel, $j can be replaced in the equality #Jam=jg4j by its asymptotic part (see Section 3.2)
[N,K,(K, Irk- rjl> + Vj . ‘si
400k(d =,z
N&Kl(L
1%- rjl) + V, - WV,h - rj>l ??
K1(K,
(3.35)
Irk - rjl)]
3
GW, 1%- r,l> lb - rjl
1
LKdL Irk- rjl) (rk - rJ . (3.36) Ir, _ rj12
3.10. Critical equation for the reactor
The system given by equations (3.31) and (3.32) relative to the n reactor channels can be written concisely in the matrix form A#0 + CN + PW,
(3.37)
+ DNl = 0,
where A, B, C and D are 3n diagonal matrices and $. and N the vectors defined by
Wo&> -,*..,G N = [N,, . . . , N,, v,,, . . . , v,,, K:y, . . . 9 v,,].
‘+“$)I m
(3.38) (3.39)
Similarly, the system of geometrical relations (3.35) and (3.36) expressing the interactions between all reactor channels can be written do = GN, where G is a 3n symmetric matrix.
(3 40)
A consistent approach of heterogeneous theory leading to a dipolar formulation-I
313
and where $iSasis the asymptotic part of the flux induced in the kth open cell moderator by the compensating source, in the absence of source in the moderator. The formulae for the constant coefficients ,~(g’+ g) and ~(g’ -+ g) are given in BERNA(1971). Obviously the process leading from equation (4.1) to equation (4.2) can be further pursued. As a result, in the kth open cell moderator the flux at the level of the g group has an asymptotic part, which can be written
+
Vkk’,g’>. ,: I
z,KICK;’Ir -
rkl) , 1
(4.4)
with m(g’, g) solution of the triangular system
m(g’g) y = k;;,~~
1
(4.5)
and
mk’, d
Nkk,d =15z_Pk’--+d v&L g) =lSJ<#&
k/
Nkk’,8’)+ Mk”
mw, s> Ug’, -+ g) kpO*
8’) + %‘,
(4.6)
(4.7)
where Mng and Tkgare respectively the polar and dipolar components of the asymptotic flux induced at the level of the g group in the kth open cell moderator by the compensating source, in the absence of source in the moderator
4.3. Expansion to$rst order in Fourier series of the compensating source around the axis of the kth channel
Incorporating in the compensating source the fission and slowing down terms, _ the equation which governs V~‘in J the medium i is div S&0 + &:,iovkg= ‘2 with
The number of groups is denoted NG.
&” + Z,;,
314
PH. BERNA
From the result obtained in the above paragraph, the reasoning followed in Sections 2.2,2.3 and 2.4 to expand the compensating source to first harmonic gives
Y&%-J = &
[
N,(g, g)&(L
Ir, - ri I) + V,(g, g) . fk &(K,‘Ir, Irk - rA
+ v,k?, d K”~~g~r~
3
ri’) - (Vj(g,
- rA)
1
(4.13)
g) . (r* - r,))
x LgKdLg Irk - r,l) (rh - r,)] I5 - rr12
linear flux approximation of the fission and slowing-down sources within i.
(4.14)
(4.15)
The formulae for ~$0, z$g and z$‘J are respectively deduced from those of z,, z, and z2 defined in monokinetic theory by replacing K, x s,,, by
K,,,O’ 1 - &+.g Ln“ + k;’ ’
by
where 8’~~is the Kronecker symbol, and c tm by Lg. 4.4. Introduction of heterogeneous parameters in multigroup theory From (4.12), it follows that l,
T; = --K,,,g ls~sgm(g’,
(4.16) (4.17)
and eig are generalizations for the parameters p, yi, fl and 8,. where j@lg, yig, ,8”‘*”
A consistent approach of heterogeneous theory leading to a dipolar formulation-I
315
4.5. Critical equation for the reactor in multigroup theory
By an approach similar to that used in monokinetic theory for proving the relations (3.31) and (3.32) it can be shown that the kth channel is characterized by the polar relation
+ PXq<~~$,O(g’
+ .!Z)Yodk’W+ d,O(g’-+ g)N,(g’, s’)l = 0, (4.18)
and by the dipolar relation
in group g, where the constants a”, b”, co, do, al, bl, cl and d1 depend only on the physical and geometrical properties of the channel and on the physical properties of the moderator surrounding it. These relations grouped with the geometrical relations (4.13) and (4.14) make a system of order 3n x NG, which can be put in the matrix form (4.20)
(H+/.K)N=O
with block diagonalization for the Green’s functions matrix. This equation represents the reactor critical equation. It is solved on an IBM 360/91 by the FODITH code, with a number of groups not exceeding six, by an iterative method with convergence acceleration, the basis of which is described in LIVOLANT(1968). 5. NUMERICAL RESULTS-COMPARISON OF THE FODITH HETEROGENEOUS CODE TO THE RIFIFI DIFFUSION CODE AND TO THE HECTOR SOURCE SINK CODE-COMPARISON WITH EXPERIMENT 5.1. Reactivity comparisons
We used the following codes:
RIFIFI: diffusion, one dimension, two groups (AMOUYAL et al. 1965); HECTOR: source-sink, age-diffusion (BOIVINEAUand CHABRILLAC,1965), infinite reflector effect corrected by RIFIFI; FODITH : two groups, infinite reflector effect corrected by RIFIFI, except in the case of the three-region reactor where the reflector boundary is simulated by an annulus of black rods. Four natural uranium-graphite characteristics : (A) square (symmetry (B) square (symmetry
reactors were calculated.
They have the following
pitch: 192 mm, channel dia.: 70 mm, rod dia.: 28 mm, 332 channels order: 8). pitch : 317 mm, channel dia.: 140 mm, rod dia.: 50 mm, 130 channels order: 4.)
316
PH. BERNA
(C) square pitch: 317 mm, rod dia.: 50 mm central region : channel dia. : 170 mm, 86 channels peripheral region : channel dia. : 140 mm, 76 channels (symmetry order: 4). (D) central region: square pitch : 384 mm, channel dia. : 140 mm rod dia.: 50 mm, 45 channels. intermediate region: pitch: 271.5 mm, channel dia. : 70 mm rod dia. : 31 mm, 16 channels. peripheral region : square pitch : 271.5 mm, channel dia. : 70 mm rod dia. : 50 mm, 72 channels (symmetry order: 8). TABLEl.-REACTIVITIES ( x 1O-6) RIFIFI A
+ 388
B C D
+ 148 + 114 + 1730
HECTOR
FODITH
+ 364 +2 - 385 - 2237
Experiment + 232 -8 - 38 + 159
+366 +159 + 66 - 86
It can be observed that even in the case of the most irregular reactor (D), FODITH gives a satisfactory result, while RIFIFI and HECTOR do not seem always suitable. 5.2. Calculation of the averagefrux gradient through the cluster of an A.G.R. type reactor having an empty channel at its centre
I
I p = 40.386
cm
FIG. l.-Reactor map. @I Empty channel-radius : 8.89 cm. o Fuelled channel-cluster radius: S-189 cm; outer radius: 8.89 cm. -Q+ Fueied channelfor which the ave;rag;p2x gradient through the cluster is given in
The following codes were compared with experiment: SQUIFID:
dipolar heterogeneous code built from a two dimensional diffusion code
(BLACKBURN, 1967); HET: supercell code (BLACKBURN,
1967) using the dipolar parameters calculated from the results given in (BRIDGE and HOWARTH, 1970); FODITH: used with 3 groups, reflector boundary simulated by an annulus of black rods.
A consistent approach of heterogeneous theory leading to a dipolar formulation-I
317
TABLE2.-RATIO (MAX - MIN)/MOY OF THEFLUXMEASURED BYA MANGANESE DETEC’rOR AROUND THECLUSTER Channel
EXP
FODITH ~ EXP
I II III IV V
0.092 0.076 0.078 0.090 0.170
0.730 1.030 0.896 1.179 1.023
SOUIFID .m
HET
EXP
EXP
1.130 1.408 1.038 1.756 1.600
051 O-81 0.70 l-00 0.85
The experimental uncertainty on this ratio can be estimated to he between 10 and 20 per cent. Let us remark that FODITH gives a result outside this range only for the cluster nearest to the empty channel. Acknowledgements-The results from SQUIFID and HET have been communicated by Mr. M. J. BRIDGE,whom I should like to thank.
REFERENCES ALPIARR. (1969) Heterogeneous reactor lattice programmes using the ‘open cell’ technique, EIR Bericht no 160. AMOUYAL A. and BENOI~TP. (1957) Interpretation des r&hats obtenus par S. A. Kushneriuk et C. MC. KAY dans le rapport AECL 137, Internal Report SPM. AMOWAL A., BENOI~TP. and HOROWITZJ. (1957) J. nucl. Energy 6,79. AMOUYAL A., BADOSM., LENATH. and MENGINF. L. (1965) RIFIFI II: Programme de resolution sur IBM 7090 des equations de la diffusion en theorie a dew groupes d’energie et une variable d’espace, Note CEA-N-518. AUERBACH T. (1967a) Two-dimensional multigroup theory with one thermal group and flux-independent cross-sections, EIR Bericht no 100; (1967b) A consistent heterogeneous theory of reflected lattices N.S.E. Vol. 29, pp. 317-324. BERNAPH. (1971) Formulation coherente de la theorie heterogene en approximation dipolaire, Thesis. BLACKBURN D. (1967) Heterogeneous methods for advanced gas cooled reactors, Ad hoc specialist meeting on the application of heterogeneous methods to bum-up calculations, Saclay 16th-18th January. BOMNEAUA. and CHABRILLAC M. (1965) Le code HECTOR (Methode heterogene de calcul de piles), Note CEA-N-523. BRIDGEM. J. and HOWARTHN. (1970) Corrections to the Benoist diffusion coefficients for CAGRTYPE lattices, Report of Central Electricity Generating Board-RD/B/N 1645. CASEK. M., DE HOFFMANN F. and PLACZEKG. (1953) Introduction to the theory of Neutron Diffusion, Vol. 1, LASL. FErNBERGS. M. (1955) Heterogeneous methods for calculating reactors: survey of results and comparison with experiment, P/669 (U.S.S.R.) 1st ZnttirnationalConference on the Peaceful Uses of Atomic Energy, Geneva, Proc. Vol. 5. GALANINA. D. (1955) The thermal coefficient in a heterogeneous reactor P/666 (U.S.S.R.) 1st International Conference on the Peaceful Uses of Atomic Energy, Geneva, Proc. Vol. 5. LIVOLANT M. (1968) Mtthodes iteratives simples pour la resolution de problemes lineaires, Internal Note SPM.