flux mode is nowhere negative. If none of the columns of R is zero, then G is irreducible. W e then conclude there is a smallest real ν for which the reactor is critical, and the corresponding flux mode is positive every
where. Analogous results may be derived for higher dimension problems.
T h e adjoint equations may also be differenced and a similar eigenvalue relation obtained. It is important to notice that the formulation of the adjoint equations in Section 4.4 will produce a matrix eigenvalue equation of the form
*1 = ΰ * ψ * , (4.6.27) but G * will not be the transpose of matrix G as defined in Eq. (4.6.24).
T h i s circumstance occurs because we computed the adjoint of the age-diffusion equations and then formed the difference equations in energy and space, instead of vice versa. T h i s phenomenon is another illustration of the noncommutability of the operation of computing the adjoint with other operations. However, if, in the formation of the multigroup adjoint equations, we have assumed q* to be approximately constant from one group to another, instead of then the G and G * would be transposed.
4.7 Numerical Solution of the Multigroup Equations
T h e basic equations given in the last section may be solved by a variety of methods. T h e usual procedure is iterative, rather than direct inversion. As an example of the procedure, we outline the steps for one possible method.
Based upon the reactor materials and the relevant cross sections, the lethargy intervals are selected and the coefficients are computed. T h e coefficients may be computed by hand, graphically, or by numerical integration, depending upon the information and equipment available.
A n initial estimate of the flux in each group, say ψ° , is made.3 W i t h the initial estimates and coefficients, the vectors w|J are computed using
3 One of the many commonly used ways of estimating the flux in each group is to calculate it for an infinite assembly first. This calculation is performed explicitly very easily by starting from the group of highest energy and working down. The size of the thermal source of fission neutrons can be selected arbitrarily.
Only the fission spectrum is important at this stage of the calculation, and this spectrum is known. By knowing the flux in all groups of higher energy than the one in question, the flux in the group of interest can be computed. The thermal flux so calculated will not in general agree with that assumed, because the com
position of the infinite system assumed is not critical.
Eq. (4.6.6). Recall that the vectors w£ are the sources for the gth group.
T h e flux in group 1, ψ ? , is now corrected, i.e., we solve the diffusion equation
A ^ , = w ? (4.7.1) for the flux at all points of the reactor. For a one-dimensional problem,
Eq. (4.7.1) might be solved by matrix factorization since A1 is then tridiagonal. ( T h e equations could also be solved by inversion or iteration).
L e t the solution be denoted ψ } . In like manner, solve the diffusion equations
Α , ψ , = w» (4.7.2) in the remaining groups. T h e vector ψ1
ψΐ = Ψ1* (4.7.3)
| ψισ +ι . onve ( ζ ι Ψ1, ί ι Ψ1) = ( Ψ ° , Ψ ° ) ,
represents the first iteration. It is convenient to rescale the vector ψ1 such that
(4.7.4) where ζ1 is the scale factor. T h e scaling, sometimes called renormaliza-tion, prevents the solution from diverging. W i t h the new scaled vectors we recompute the source vectors in each group. T h u s we find . W i t h the corrected sources, we again solve the group diffusion equations at all points of the reactor to obtain ψ2, etc. After a sufficient number of iterations, say p> then has converged, that is
|ψρ+ι _ ψ Ρ I < e , (4.7.5)
where e is a convergence criterion, chosen by the user. In words, ine
quality (4.7.5) requires the flux on the p + 1st iteration to differ from that on the pth iteration by less than some arbitrarily chosen constant at all points of the reactor and in all speed groups. In testing for convergence one must be very careful in the application of the above inequality. T h e inequality merely states that the difference between two successive itera
tions be less than some quantity. It does not state that the error in any particular iteration is less than some criterion. Convergence may be slow, or the difference may have a maximum or minimum. T h e iteration pro
cess just described is sometimes called the "inner iteration."
4.7 N U M E R I C A L S O L U T I O N O F T H E M U L T I G R O U P E Q U A T I O N S 177
F I G . 4.7.1. Flow diagram for computing the critical size or mass of a reactor.
T h e scale factors ζμ will approach an asymptotic value determined by the state of the assembly. I f ζμ < 1, then the flux is growing with each iteration; the assembly is supercritical. Similarly, for ζμ > 1, the assembly is subcritical. If the critical size of a reactor is sought, then the adjustable parameters are altered according to 1 — ζμ . T h e adjustment of the gross properties is called an "outer iteration." After completing the outer iteration, the inner iterations are begun again. T h e process continues until the growth factors ζρ equal 1, which implies the system is critical. T h e scale factors ζμ are in fact the reciprocal of the multi
plication factor.
It is convenient to outline the logic of the above process in a flow chart.
Figure 4.7.1 is a pictorial representation of the steps.
It is important to recall that the method just outlined is one of many possibilities. Frequently programs are written for which there is only one inner iteration per outer iteration. Other variants exist, and we refer to some operating codes in the references for more detail (see References
10 through 14).
References
The derivations and physical interpretations for the age-diffusion equations are exhaustively considered in / and 2. A very general discussion of adjoint operators is found in 4. T h e use of the importance function in reactor physics is well discussed in I and 2. Further interpretations of the importance function are found in 3. Additional discussion is found in 5. T h e formation of the multi-group equations is considered in many references. For particularly interesting discussions, see 2, 5, 6, and 7. T h e multigroup difference equations are considered in 5 through 8. T h e development in the text is predominantly from 5. References 8 and 9 contain very rigorous discussions of the numerical solution of the multi-group difference equations. A brief list of particular multimulti-group codes is given in 10 through 13. Reference 14 is an abstract of many different nuclear codes.
1. Weinberg, A . M . , and Wigner, E. P., " T h e Physical Theory of Neutron Chain Reactors." Univ. of Chicago Press, Chicago, 1956.
2. Meghreblian, R. V., and Holmes, D . K., "Reactor Analysis." M c G r a w -Hill, N e w York, 1960.
3. Robkin, Μ . Α., and Clark, M . , Integral reactor theory; orthogonality and importance. Nuclear Set. Eng. 8, 437 (1960).
4. Morse, P. M . , and Feshbach, H., "Methods of Theoretical Physics." M c G r a w -Hill, N e w York, 1953.
5. Marchuk, G . I., "Numerical Methods for Reactor Calculations" (translation).
Consultants Bureau, N e w York, 1959.
6. Ehrlich, R., and Hurwitz, H., Multigroup methods for neutron diffusion problems. Nucleonics 12, N o . 2, 23 (1954).
7. BirkhofT, G., and Wigner, E. P., eds., "Proceedings of the Eleventh Symposium in Applied Mathematics." A m . Math. S o c , Providence, Rhode Island, 1961.
8. Birkhoff, G., and Wigner, E. P., eds., "Proceedings of the Eleventh Symposium
P R O B L E M S 179 in Applied Mathematics," pp. 164-189. A m . Math. S o c , Providence, Rhode Island, 1961.
9. BirkhofF, G . , and Varga, R. S., Reactor criticality and non-negative matrices.
J. Soc. Ind. Appl. Math. 6, 354 (1958).
JO. Wachspress, E. L., C U R E : a generalized two-space dimension multigroup coding of the 704. KAPL-1724 ( M a y , 1957).
II. Cadwell, W . R., Dorsey, J. P., Henderson, H . B., Liska, J. M . , Mandell, J. P., and Suggs, M . C , P D Q - 3 : a program for the solution of the neutron diffusion equation in two dimensions on the I B M 704. W A P D - T M - 1 7 9 . J2. Brinkley, F. W . , and Mills, C. B., A one dimensional intermediate reactor
computing program. LA-2161 ( M a y , 1959).
J3. Stuart, R. N., Canfield, Ε. H., Dougherty, Ε. E., and Stone, S. P., Zoom, a one dimensional, multigroup neutron diffusion theory reactor code for the I B M 204. U C R L - 5 2 9 3 (November 1958).
14. A m . Nuclear S o c , Math. Comp. Div., "Abstract of Nuclear Codes," ( M . Butler, ed.), N o . 1-80 (1962).
1. Derive the critical equation from the age-diffusion equation for an infinite reactor with thermal fissions only. Interpret the results.
2. Generalize the results of problem 1 to an infinite reactor with fissions occurring at all lethargies between 0 ^ u ^ utu .
3. Write the slowing down equation for an infinite medium in terms of the slowing down density, and derive an integral equation for the solution. Find the adjoint of the integral equation and show that solutions of the integral equations and its adjoint are biorthogonal.
4. For problem 3 above, derive the adjoint to the differential slowing down equation and then integrate the adjoint equation. Is the result the same as the adjoint equation in problem 3 ? Explain.
5. Derive the multigroup equations for a two-group approximation and express the criticality condition as a critical determinant.
6. Show that the multigroup equations obtained by using Eq. (4.3.19) are of the form of Eq. (4.3.26) with the following definitions:
Problems
Λ = - ( ^ s ) g.
and
7. Consider the basic group diffusion equation (4.3.26 in text) in a two-region slab. Derive a difference approximation which is accurate to order (Δχ)2 in each region and across the interface.
8. Derive a five-point coefficient matrix for a two-dimensional diffusion problem in r, ζ coordinates. Show that the coefficient matrix is symmetric.
9. For elastic scattering, calculate Sitj in terms of the energies et , e i +, , 1
and €,·+ 1, assuming that the slowing down density is constant within a group, for the following cases:
such that the G * is the transpose of the corresponding eigenvalue problem for the flux.
The following sequence of problems provides an elementary discussion of the numerical solution of the reactor kinetics equations.
12. T h e infinite medium kinetics equations for G delayed groups may be written dn(t)
_
p —dt ~~ Λ
Έ£± = h η ( ί) _ λ , Ο ( ί ) , « = 1 , 2 G ,
where η is the neutron density, Cl the zth delayed group precursor, p the reactivity, Λ the generation time, At the zth group decay factor, j3, the ith group fission yield fraction, and β = Σ , j8, the total delayed yield.
(a) Define the vector
η C1
Ψ
I ^CG
P R O B L E M S 181