The General Reactor Problem

Generally speaking, the dependence of the neutron density upon all seven independent variables implied above is customarily indicated in the initial, very broad statement of the problem. Very seldom is such generality required in actual practice. In most situations, attention is confined to certain specific aspects of the neutron problem, and less precise descriptions involving only a few variables will often suffice. For example, the time-dependent behavior of the neutron population is of interest only in regard to the nuclear stability and control of the reactor, and in many cases an analysis of the steady-state problem is entirely adequate. Likewise, the directional dependence of the neutron density (i.e., the specification of the direction of motion of the neutrons being studied) is of little interest except in regions of the reactor system close to discontinuities in the material composition (boundaries, etc.). 

Only when a complete description including all seven variables is required is it necessary to solve the Boltzmann equation in all its generalities, Frequently, simplifying assumptions and limiting conditions can be imposed which reduce the integrodifTerential equation to more tractable form. Thus much of the subject of reactor analysis is devoted to the development and the application of simplified analytical models which define, within the limits of engineering needs, the nuclear characteristics of the reactor complex. 

Frequently coupled with this problem is the determination of the optimum nuclear configuration which yields a minimum fuel mass. Reasonable estimates for preliminary studies can be made with relatively little effort, and many crude analytical models are available for this purpose. Accurate estimates require more elegant methods or the use of critical experiments. Although precise mass figures per se are only infrequently required in modern practice, this information is usually available in every reactor study as the by-product of solutions to more essential problems involving neutron-density distributions. As a practical matter, relatively large discrepancies in mass estimates can be readily accommodated with the increased availability of high-enrichment fuel samples. 

Another class of time-dependent problems of concern to the reactor physicist are questions on fuel burnup, poison production and burnup, breeding ratio, and the like. These problems differ from those on reactor stability in that they involve time scales measured in hours (or years) in contrast to stability problems which are concerned with fractions of a second. Reactor-analysis problems, such as the determination of critical mass and neutron-density distributions, are based on the steady-state operating condition of the reactor. The day-to-day operation of the reactor at steady state involves, however, long-time changes in the fuel concentration. Except in the case of circulating-fuel reactors, the fuel is introduced into the reactor according to some predetermined cycle. As the fuel is consumed, some gradual adjustments can be made by means 

The general field of problems described above, except in some special areas, may be treated by the well-known methods and analytical models of mathematical physics. It has already been noted that the most general description of the neutron population usually starts with a neutron-balance relation of the Boltzmann type. The Boltzmann equation was developed in connection with the study of nonuniform gas mixtures, and the application to the neutron problem represents a considerable simplification of the general gas problem. (Whereas in gas problems all the particles are in motion, in reactor problems only the neutrons are in motion. ) The fundamental equation of reactor physics, then, is already a familiar one from the kinetic theory. Further, many of the most useful neutron models obtained from approximations to the Boltzmann equation reduce to familiar forms, such as the heat-conduction, Helmholtz, and telegraphist s equations. These simplifications result from the elimination of various independent variables in the 

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The purpose of the present chapter is to develop a more general model for describing the neutron population in a reactor, a model which combines the principal features of the two already presented. That this is indeed a prerequisite for treating the general reactor problem may be... 

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