a relation of the coupling coefficient to the eigenvalue separation in the coupled reactors theory

Pergamon Ann. Nucl. Energy, Vol. 25, No. 4-5, 189-201, 1998 pp.

0 1998 Hsevier Science Ltd. All rights reserved PII: SO306-4549(97)00061-3 Printed in Great Britain

0306-4549/98 $19.00+0.00

A RELATION OF THE COUPLING COEFFICIENT TO THE EIGENVALUE SEPARATION IN THE COUPLED

REACTORS THEORY

KEISUKE KOBAYASHI

Department of Nuclear Engineering, Kyoto University, Yoshida, Sakyoku, Kyoto, Japan

(Received 12 June 1997)

Abstract-A relation of the coupling coefficient with eigenvalues for coupled reactors is given by making use of the rigorous expression for the coupling coefficient derived by using the region-wise importance function to produce fission neutrons. From this relation, an approximate relation of the coupling coefficient to the eigenvalue separation of the fundamental and first higher mode is obtained, in which the coupling coefficient is expressed simply in terms of the difference of the two eigenvalues. It is shown numerically for simple coupled reactors of one dimensional slab geometry by comparing with exact values that the accuracy of this relation becomes better as the strength of the coupling between cores becomes weaker. 0 1998 Elsevier Science Ltd.

1. INTRODUCTION

The relation of the instability of the flux distribution of large core reactors with the eigenvalue separation of the fundamental and the first higher mode for the flux distribution has been studied by many authors, for example, Mochizuki and Takeda (1960), Stacey (1969), Palmiotti (1983), and Palmiotti and Salvatores (1984). On the other hand, it has been shown that the flux distribution, or the flux tilt due to the perturbation, is sensitive to the coupling coefficients between reactors or between core regions, and the coupled reactors theory by Avery (1958) is used to analyze the instability of large cores of fast reactors by McFarlane er al. (1984), and Hashimoto (1995).

These facts suggest the existence of the relationship between the coupling coefficient and the eigenvalue separation. Kawai (1965) showed that the coupling coefficient defined by Baldwin (1959) for weakly coupled reactors had the relation to the eigenvalue separation of the fundamental mode and the first higher mode.

Recently, Nishina and Tokashiki (1996) investigated that this relation holds not only for weakly coupled reactors but also for strongly coupled reactors. According to their work, the relation of the coupling coefficient from core 2 to 1, Ai2 with the separation of the fundamental mode eigenvalue k. and the first higher mode kr expressed by

189

190 K. Kobayashi

holds with good accuracy independent of the strength of the coupling. Hashimoto discussed that the above coupling coefficient At2 defined by Nishina et al.

has the following relation to the coupling coefficient kf2 by Avery as

which suggests the relation of Avery’s coupling coefficient to the eigenvalue separation

He studied about this relation using the numerical and experimental data for large fast reactors.

Although many coupled reactor theories have been developed since Avery’s theory, all those theories are approximate, and hence the definition of the coupling coefficients are different depending on the theory.

Recently, rigorous multi-point equations for coupled reactors have been derived by Kobayashi (1991a,b, 1992) without making any approximations, where the region-wise importance functions which are describing the number of neutrons produced in each region due to a primary neutron born at a position in a particular region. Although the expression to calculate the coupling coefficients are different from those given by Avery, the physical meaning of the coupling coefficients is nearly the same as that given by Avery.

In the present work, using this rigorous theory for coupled reactors, the relation between the coupling coefficient and eigenvalue separation is studied, and for simple problems of slab geometry, the accuracy of this relation is examined numerically by comparing them with the exact coupling coefficients. A simple relation between the strength of the coupling and the flux tilt due to the perturbation is also given.

2. RELATION OF THE COUPLING COEFFICIENT TO THE EIGENVALUES

2.1. Exact relation

We will consider this problem using the group diffusion equation, solution can be obtained easily for simple problems.

We write the group diffusion equation of the steady state in a form,

since an analytical

where &g(i) is the neutron flux of the gth group at position 7, and the destruction operator A and the production operator B are defined as

A = -DgV2 + Xc, - c C,(g + g’), B = xg c i~E,-f. &?#g g’

(5)

The Coupled Reactors Theory 191

Here, Dgp Erg, xg, vxg, Wz t g’), and ks are the diffusion coefficient, removal cross- section, fission spectrum, fission cross section multiplied by the number of fission neutrons of the gth group, scattering cross section from group g’ to g, and criticality factor. We assume that the whole region of a reactor V is subdivided into appropriate subregions, V,, m = 1, 2 ,....

The importance function G,&, g), which expresses the number of neutrons produced in the region V, due to a neutron born in group g and at position r, is given by the equation

A+Gn(f, g) = v~fg4n(r?, (6)

where At is the adjoint operator of the destruction operator A, and the function S,(r) takes 1 when i is in the region V, and 0 when r is not in the region V,. We use the same boundary condition for both equations (4) and (6) that the flux and the importance function vanish at the outermost boundary.

Multiplying equation (4) by the importance function G,,,(t, g), integrating it over the whole region and summing up over all groups, we obtain the multi-point equations as

where S,,, is the number of fission neutrons produced in the region V, defined by

and k,,,,, is the coupling coefficient defined by

(7)

(9)

which expresses the number of neutrons produced in the region V, due to a fission neutron produced in a region V,. The physical meaning of the coupling coefficients kmn is nearly the same as that given by Avery, and the form of the coupled equations of equation (7) is also the same.

For simplicity, we use the vector p to express the variables (r, g) and write the integral over r and summation over g as J’dp.

We define eigen functions of the direct and adjoint equations by

ah(p)=kB+j@), j=O,1,2,... (10) /

A+@(p) = -t_B+,t, I ki I , i=o,1,2 ).... (11)

Comparing equation (10) with equation (4), we can see that #1~(3 = $&). Multiplying equation (10) by @i(p), equation (11) by @j(p), taking a difference of the

resulting two equations and integrating over the whole space, we obtain

192 K. Kobayashi

where (4) denotes J dp# = J, dr C, rPg(r). Then, we normalize the eigenfunctions as

(&.P)B$$P)) = $9 (13)

where 8~ is Kronecker’s delta. Using the adjoint eigenfunctions @j@), we expand the importance function G,(p) as

(14)

where gmj is the expansion coefficient. Substituting equation (14) into equation (6), using equations (11) and (13), multiplying

it by the eigenfunction $&) and integrating it over whole space, we obtain the expansion coefficient as

Using equations (14) in (9), the coupling coefficient k,, is expressed in terms of the expansion coefficients and eigenfunctions as

(16)

which gives the exact relation between the coupling coefficient and the eigenvalues kj which are included in the expansion coefficients gmj of equation (15).

2.2. Approximate relation to the eigenvalue separation

In the case of weakly coupled reactors, we can easily derive an approximate relation of the coupling coefficient to the eigenvalue separation as done by Kawai. For simplicity, let us consider coupled reactors of a slab geometry as shown in Fig. 1, which is symmetrical. In the figure, there is fuel in regions Vr and V3 and the region VZ is the moderator region with no fuel. We assume that the reactor is described by the one group diffusion equation and is critical with the eigenvalue ko.

The coupling coefficient from the core Vs to the core Vi is obtained by putting m = 1 and n = 3 in equation (16) as

(17)

We approximate the importance function of equation (14) by the first two terms of eigenfunctions as


Since the diffusion equation of one group is self-adjoin& the adjoint eigenfunctions &(x) are equal to the direct eigenfunctions I&). Neglecting higher order modes for the importance function, the coupling coefficient k13 of equation (17) becomes

k13~ J-2 WaoljroW + a1 lcIlWbWoo(4

J; dxuxfllro(x) ’ (19)

As seen in Fig. 1, the eigenfunction +1(x) of the first higher mode for x3 5 x I x4 can be approximated by the eigenfunction of the fundamental mode +0(x) as

$1 (x)+ - *a(x) for x3 SxSx4. (20)

Using this approximation, equation (19) becomes

(21)

Using equation (15) and the similarity of the eigenfunctions, qt (x)++a(x) for XI Ixsx2, the expansion coefficients become

f

m g10 = ko ~~q-$O(X)~

Xl

J -9

g11 = h dxuYZ,-+l (x)+/t, XI J

x2

dxuEf7Co(x). XI

Since the eigenfunction is normalized as in equation (13), we obtain

3r

x(m) 10 20 30X 40 50 po 70

x, x2 \ \ x3 f x,

l- \ / - Fundamental Mode , /

First Mode \ 0 --- I cd

(22)

Fig. 1. First two eigenfunctions in weakly coupled reactors of a slab geometry.

194 K. Kobayashi

(23)

Using equations (22) and (23) in equation (21), we obtain the coupling coefficient as

k,+ $ko - k,). (24)

As seen in equation (16), the coupling coefficient is given in terms of the higher modes, and the convergence of the series might be slower if the criticality factors were contained in the inverse form like l/5 as in equation (l), since kj becomes smaller and l/kj becomes larger for the higher modes. The form of equations (16) or (24) is advantageous, since the effect of the higher modes become smaller than with the form of l/kj.

The accuracy of equation (24) can be checked by comparing it with the exact coupling coefficient, which is considered in the sections 4 and 5.

3. THEORY OF COUPLED REACTORS BY AVERY

According to the theory of coupled reactors by Avery, the coupling coefficient is defined as “the expectation value that a fission neutron in reactor V,, gives rise to a next gen- eration fission neutron in reactor Vm”, or the fraction of the importance in reactor V, due to a neutron born in reactor V, to the total importance in reactor V,, which is given by

(25)

Here the adjoint function c##) is the solution of i=O of the adjoint equation, equation (1 l), namely

The function &(i, g) is the flux distribution obtained for a configuration where the fission source of the right hand side of equation (4) exists only in V,, namely, the solution of the following equation:

(27)

Summing up equation (27) with respect to all n and equating it with equation (4), we find the relation


Multiplying equation (25) by S,, and summing them up with respect to all m, we obtain equation (7). From this, we can deduce that the criticality factor ko calculated from the condition that equation (7) has a non-zero solution, is equal to that obtained from the original diffusion equation of equation (4).

In this argument, the fact that the function d@) satisfies equation (26) is not used. This means that any weighting function instead of the adjoint function #@ can be used to calculate coupling coefficients in equation (25), which give the same criticality factor ko of the original diffusion equation, namely, we can define many kinds of coupling coefficients which give the same criticality factor ko.

The advantage of the use of the adjoint function as the weighting function may commit in the fact that the first order error due to the use of the unperturbed flux for &(F, g) for the perturbed system can be eliminated as in the first order perturbation theory.

4. SIMPLE EXAMPLE FOR SLAB GEOMETRY

For the coupled reactors of the slab geometry described in section 2, we can calculate the flux and importance function analytically. We use the boundary condition that the flux and importance function vanishes at the outermost boundary x1 and x4, namely, #(x1) = #(x4) = 0.

The diffusion equations for the region V1 and V3 with index m for region V, are

i=O or 1, m= 1,3, (29)

where &, is the absorption cross-section. The diffusion equation for the region V2 is that the right hand side of equation (29) is 0. The flux in region Vi, V2 and V3 has a form

41 (x) = 42 -----sinBr(x - xl),

sin Bi 11 @3(x) =

$3 ____ sin B3(x4 - x), sin B3 13

(30) &X(x) =

sinh ~2(~3 - X) sinh KZ(X - x2) sinh ~212 #2+ sinh ~212 43,

B,,, = K,,,,/w~fm/(kj&) - 1, 1, = x,,,+~ - x,,,, Kz, = s m

where & = #(xm). The current must be continuous at the region boundaries as

_D &&> m ---&- Lx,+, = -&+1 d4;y(x) Ix=&+, , for m= 1,2.

(31)

(32)

Substituting the fluxes of equation (30) into equation (32), we obtain a linear algebraic equation for the boundary flux as

all&2 + a12h = 0, a2142 + 9243 = 0,

(33)

196 K. Kobayashi

where the coefficients ac are given by

all = DlBl cotBl1, +D2~2cothK212, D2K2 a12 = --,

sinh ~212

a22 = D2~2 coth ~212 + D3B3 cot B&, Dm (34)

a21 = --. sinh ~212

From the condition that equation (33) has a non-trivial solution, the eigenvalue kc, or kt is obtained, and then the fluxes at the region boundaries #Q and $3. Using these boundary fluxes, the flux inside the region can be obtained from equation (30).

For the importance function G,(X) of equation (6), using region-wise Green’s function for each region, we can obtain exact 3-point finite difference like equations whose unknowns are the fluxes at the region boundary:

b2Gt2 - c2Gl3 = d2r -a3G12 + b3G13 = 0,

(35)

where G, = Gm(X,,) and

b2 = Dl~l coth ~111 + D2~2 coth K212,

b3 = D2~2 coth K212 + Dj~j coth K313,

D2~2 c2 = - = as,

sinh ~212 x2

d2 ~1

sinh ~111 sinh KI(~ - Xt)uI$de

= uEB(cosh Kt1t - 1) K1 sinhKt11 *

The importance functions inside each region are given as

G](X) = p 1_ al ( cash Kl(T - x)

cash; Klfl

+ sinh K1 (x - xl) sinh ~111

G12r

G12(X) = sinh K2(X3 - X) G12 + sinh K~(X - x2)

sinh ~212 sinh ~212 G13,

(36)

(37)

G13(X) = sinh K3(X4 - x)

sinh ~313 G13.

Using these fluxes and importance functions in equation (9), we can obtain the exact coupling coefficient.

5. NUMERICAL EXAMPLES

The accuracy of the relation of the coupling coefficients to the eigenvalue separation of equation (24) is checked numerically by calculating the exact coupling coefficient for a one


dimensional slab geometry shown in Fig. 1, where the thickness of the regions Vr and V3 is 20 cm.

Transport cross-section Err = 0.15 cm-’ for all three regions and absorption cross-section I=,1 = Co3 = 0.08 cm-‘, &2 = 0.04 cm-’ are used, and k, = v &/ C, = 1.3 is used in regions Vr and Vs. The diffusion coefficient is derived from the transport cross-section as D = l/(3&).

The flux of the fundamental and the first higher mode and the importance function Cl(x) calculated by equations (35) and (37) are shown in Figs 1 and 2 , respectively, for the thickness of the region Vz, 12 = 20 cm.

Exact and approximate coupling coefficients calculated by equations (9) and (24) respectively are shown in Table 1 and Fig. 3, and the relative difference of the approximate relation of equation (24) to the exact coupling coefficients is shown in Fig. 4. As seen in Table 1 and Figs 3 and 4, the difference of the approximate relation of equation (24) increases as the separation of two reactors becomes smaller and then the coupling of the two reactors becomes stronger.

I

1 I

1 x (cm)

0 10 20 30 40 50 60 70

z-Coordinates (cm)

Fig. 2. Importance function G1 (x) for weakly coupled reactors of a slab geometry for the thickness of the region VZ, 12 = 20 cm.

, 0.15. ' \

y" z4 -0 9 0.1.

2 0.05.

0 5 10 15 20 25 30

Thickness ZS of Region I$ (cm)

Fig. 3. Exact and approximate coupling coefficients from equations (9) and (24), respectively.

198 K. Kobayashi

Table 1 Eigenvalues and coupling coefficients

Thickness Mode Calculation Coupled Reactors Theory ~2@d (Difference’ %) Present Theory Avery’s Theory

30

25

20

15

10

5

0

ko:

(ks-k,$i

:: (ks-k,),2i

ko:

(k,,-k,$!

iif (ks-k*),2i

ko:

(ks-k,;;;

ko:

(k,,-k,)$

0.92966 k& 0.92966 0.92966 0.92315 kll: 0.92645 0.92637 0.00326 (1.9) h 0.00320 0.00328

0.93278 k,,: 0.93278 0.93278 0.92005 hl: 0.92649 0.92631 0.00636 (1.1) kn: 0.00629 0.00647

0.93890 k,,: 0.93890 0.93890 0.91402 k,,: z-- 0.92649 0.92605 0.01244 (0.24) kn: 0.01241 0.01285

0.95093 ko: 0.95093 0.95093 0.90232 hi: 0.92628 0.92508 0.02431 (-1.4) km: 0.02466 0.02586

0.97463 k,,: 0.97463 0.97463 0.87993 kll: 0.92495 0.92143 0.04735 (-4.7) km: 0.04968 0.05320

1.02110 k,,: 1.02110 1.02110 0.83864 hl: 0.91863 0.90804 0.99123 (-11.0) kn: 0.10247 0.11306

1.10983 ko: 1.10983 1.10983 0.77134 hi: 0.89024 0.85990 0.16925 (-22.9) ha: 0.21960 0.24993

‘Relative difference from the k13 of the present theory. *k,,=k,, +k,3.

Fig. 4. Relative difference of the approximate coupling coefficients of equation (24).


From the condition that equation (7) for the present problem has a non-zero solution, we obtain the relation ks = kti + kis, which can be used to check whether the coupling coefficients ki 1 and ki3 are cakulated appropriately.

In Table 1 are also shown the coupling coefficients by Avery calculated using equation (25). They are close to those of the present method, when the coupling between cores is weak. This means that the relation of the coupling coefficients by Avery to the eigenvalue separation is also given by equation (24) but not equation (3) for this simple problem. As the coupling becomes stronger, the difference becomes larger. This difference may have some influence on the analysis of time dependent problems, since the kinetics equations may give a different time dependence.

The present result seems not to be consistent with the result given by Nishina and Tokashiki. This may be due to the fact that their definition of the coupling coefficient is different from the present one or Avery’s, and the coupling coefficient was calculated under the assumption that there was only the fund.mental mode and the first higher mode. In the present method, such an assumption is not used.

6. RELATION OF THE COUPLING COEFFICIENT TO THE FLUX TILT

Coupling coefficients or the eigenvalue separations are used to estimate the flux tilt in large power reactors. We will derive the relation of the coupling coefficients to the flux tilt.

If a small perturbation of SA and SB of the operators A and B is added to a system, equation (4) can be written in the form

(A + SA)&(r? = ; (B + SB)$,(fl. (38)

Multiplying equation (38) by the importance function of equation (6) for the unperturbed system and integrating it over the whole space, we obtain multi-point equations for the perturbed system

S,,,

where the coupling coefficients k,,,,, have the same form as equation (9) and

(39)

Here, 6vZfr is a perturbation to the fission cross-section vZ,-r. If we divide a reactor appropriately into two regions, and put a perturbation in the

region m = 1, equation (39) becomes

200 K. Kobayashi

SI = (;h + Ah+ +;k,,s2,

s2 = ($2, + Ak+, +;k22s2. (41)

The criticality factor k in equation (41) can be determined such that the determinant of the coefficients matrix vanishes. The ratio of the fission rate, namely the flux tilt due to the perturbation in the region Vi, is given by, for example, the first equation of equations (41) as

S2 k-k11 -kAkll -_= Sl kn *

(42)

This equation gives an exact relation of the coupling coefficients to the fiux tilt. For an illustration, if the system is symmetric and there is no perturbation, & should be

equal to $5, and then from equation (42) we obtain

S2 k--h1 -_=-= Sl ku

1, (43)

from which we have a relation

k=kll +kn. (44)

if a perturbation is added, the coupling coefficients kll and kl2 change due to the change of the flux distribution as seen in equation (40). However, if this change is small and equation (44) can be used approximately, equation (42) becomes

S2 h2 - kAh -& A.kll

Sl . h2 = 1 -k-.

h2 (45)

We can see that if the coupling coefficient k12 is small, the ratio &/Si will change con- siderably even if the perturbation Akii is small, and the flux tilt becomes large.

7. CONCLUSION

We can summarize the preceding results as follows.

The coupling coefficient used in the rigorous coupled reactors theory, which has the similar physical meaning as that of Avery’s, is approximately expressed simply by the difference of eigenvalues of the fundamental and first higher modes, but not the difference of the inverse of the eigenvalues. In order for this expression to be accurate, it is necessary that the region-wise importance function to produce fission neutrons is well approximated by the eigenfunctions of the fundamental and f&t higher modes. This condition may be well satisfied for weakly coupled reactors. However, the accuracy becomes worse as the strength of the coupling becomes stronger.

Coupling coefficients can be easily and exactly calculated by making use of the con- ventional multi-group diffusion or transport code by calculating importance functions to


produce fission neutrons, which is much simpler than to calculate the higher mode. From the magnitude of the coupling coefficient, we can deduce the sesitivity of the flux distribution to the perturbation.

Acknowledgement-The author wishes to express his sincere thanks to Dr. Edgar Kiefhaber of Forschungszentrum Karlsruhe for his useful comments on this paper.

REFERENCES

Avery, R. (1958) Theory of coupled reactors. In Proc. 2nd U.N. Znt. Conf Peaceful Uses of Atomic Energy, Vol. 12, p. 186. United Nations, Geneva.

Baldwin, G. C. (1959) Kinetics of a Reactor Composed of Two Loosely Coupled Cores. Nuclear Science and Engineering, 6, 320.

Hashimoto, K. (1995) Decoupling-phenomena, analyses of large nuclear reactor cores by the method of spatial higher harmonics. Doctoral thesis, Department of Nuclear Engineering, Nagoya University.

Kawai, T. (1965) Coupled Reactor Kinetics (II), Journal of Nuclear Science and Tech- nology 2, 285.

Kobayashi, K. (1991a) Rigorous Derivation of Nodal Equations for Coupled Reactors, Annals of Nuclear Energy 18, 13.

Kobayashi, K. (1991b) Rigorous Derivation of Static and Kinetic Nodal Equations for Coupled Reactors Using Transport Equation, Journal of Nuclear Science and Tech- nology 28, 389.

Kobayashi, K. (1992) Rigorous Derivation of Multi-Point Reactor Kinetics Equations with Explicit Dependence on Perturbation, Journal of Nuclear Science and Tech- nology 29, 110.

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Mochizuki, K. and Takeda, A. (1960) An Analysis of Neutron Flux Spatial Oscillation Due to Xenon Build-Up in a Large Power Reactor Core, Nuclear Science and Engineering 7, 336.

Nishina, K. and Tokashiki, M. (1996) Verification of More General Correspondence Between Eigenvalue Separation and Coupling Coefficient, Progress of Nuclear Energy 30, 277.

Palmiotti, G. (1983) Use of the Explicit High-Order Perturbation Formulation, Nuclear Science and Engineering 83, 28 1.

Palmiotti, G. and Salvatores, M. (1984) Use of Integral Experiments in the Assessment of Large Liquid-Metal Fast Breeder Basic Design Parameters, Nuclear Science and Engineering 87, 333.

Stacey, Jr., W. M. (1969) Space-Time Nuclear Reactor Kinetics. Academic Press, New York.

a relation of the coupling coefficient to the eigenvalue separation in the coupled reactors theory

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