Fermi age model

Better to demonstrate some of the suggestions on the use of the Fermi age model, let us consider the problem of determining the critical size and minimum fuel mass of a uniform bare reactor. For simplicity, assume that the basic constituents fuel, moderator, structure, and coolant have all been selected and an estimate of the relative proportions of these materials in the reactor is available. Our problem is to find the dimensions of the reactor with these basic characteristics which has the smallest fuel mass. The procedure is straightforward and involves the calculation of a number of systems each of different size. 

As before, the term is obtained from the relation (6.163). The nh term we compute from Eq. (6.47). This implies that the Fermi age model is applicable to the system in question and therefore all the requirements for the validity of this model are satisfied. Of immediate concern to the present calculation is the condition that 2a(u) <3C S,(w). Thus, if the integral expression for the age... 

S A point source in an infinite medium is emitting neutrons of zero lethargy isotropically at a constant rate in time. Assume that the slowing-down process may be described by means of the Fermi age model once the first collision has occurred. Thus at points of first collision, the neutrons enter a continuous slowing-down process. 

The analytical method employed in both of these presentations involves the use of expansions in spherical harmonics. In each case the original integrodifferential equation, which states the neutron-balance condition, is reduced to an infinite set of coupled equations in the various harmonics of the function . By truncating the series expansion for the flux, this set can be further reduced to finite size (depending on the accuracy desired), and it is shown that in first approximation the resulting equations yield the diffusion theory and Fermi age models. 

The purpose of the analysis which follows is to establish a suitable neutron-balance relation which takes into account the dependence of the relevant functions upon the neutron energy, direction of motion, time and spatial coordinates. The balance condition, which is an integro-differential equation, is then reduced to an infinite, but simpler, set of coupled integro-dilTerential equations by expanding the flux, the neutron sources, and the scattering function in spherical harmonics. The derivation of the Fermi age model from these results is then demonstrated. 

A set of equations which has proved to be very useful in reactor calculations, wherein the Fermi age model is applicable, is... 

We conclude the present section by summarizing the basic assumptions that have been introduced in arriving at the Fermi age model as represented by (6.5) and (6.6). (1) The medium is isotropic and homo-... 

The purpose of this section is to display three computational methods for the neutron age that are especially useful for hand calculation. These methods are based on various approximate models which involve some rather gross idealizations of the physical system. It is convenient to discuss these at this point since they serve well to demonstrate the application of the Fermi age model and the transport-theory methods developed in the preceding sections. 

We know from our earlier study of the Fermi age model that the age t u) is proportional to the quantity r (w), the mean radial distance squared at which fast neutrons from a specified source reach lethargy u the constant of proportionality in this relation is dependent upon the nature of the source [see, for example, Eq. (6.52)]. Our ultimate objective, then, is to compute r (w). For this purpose we introduce the following model ... 

The first term at the right is immediately recognized as the r u) computed for a point source on the basis of the Fermi age model [see Eqs. (6.49) through (6.52)], since ... 

The removal cross sections may be identified from a similar analysis for the bare multiplying system. It follows from our study of the Fermi age model that the spatial variation in the neutron flux is of the same functional form at all energies. Thus we may take... 

It is of interest to note that the general method outlined here is entirely analogous to the one used for the bare-reactor calculations, using either the one-velocity or Fermi age models. For example, in the case of the Fermi age model, the problem was to solve the equation... 

These results are entirely consistent with those of our previous analysis of the bare reactor using the Fermi age model (refer to Sec. 6.3). In this formulation, Eq. (8.281a) describes the neutron-energy spectrum and is easily recognized as the integral equation for the collision density in an infinite homogeneous system. If we select, for example [cf. Eq. (4.36)],... 

Apart from this limitation, the principal restriction in the use of the Fermi age model is due to the large number of conditions which must be satisfied by the physical system if the model is to be valid (see Sec. 7.3i). However, even with this shortcoming, the model is still very useful for preliminary studies and for gaining insight into the essential features of many reactor problems. The Fermi age or continuous-slowing-down diffusion theory, then, offers some generality without introducing excessive computational difficulties. 

The authors have attempted to present a discussion of all the principal topics of reactor analysis, with an entire chapter devoted to each. In many instances several analytical methods are presented in order to provide as wide a treatment as possible. These include Chap. 2 on probability concepts Chap. 3 on the neutron flux Chap. 4 on slowing down Chap. 5 on diffusion theory Chap. 6 on the Fermi age model Chap. 7 on transport theory Chap. 8 on reflected reactors Chap. 9 on reactor kinetics and Chap. 10 on heterogeneity. It is important to mention that the remaining chapters represent in main part extensions and applications of these general topics. 

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