One-velocity model

Consider an infinite reactor composed of a uniform mixture of beryllium and natural uranium. The atomic ratio of isotopes in natural uranium is U U 993 7. Assume that the one-velocity model applies, and use the microscopic cross sections and densities of Table 2.2. 

A detailed description of the boundary conditions to be used in the one-velocity model involves the methods and results of transport theory (see Chap. 7). 

Equation (5.156) gives the flux distribution at steady state in an infinite-slab reactor (with fission sources only) based on the one-velocity model. This result may also be obtained by starting with the steady-state diffusion equation directly however, it is not possible, then, to prove rigorously that only the member of the set,... 

Fto. 5.21 Critical fuel concentration and total fuel mass for spherical reactor in the one- velocity model. 

Figure 5.21 shows a plot of Ny/Njr as a function of the radius of the sphere for a system of pure and beryllium. The nuclear density of the beryllium was taken as 0.124 X 10 nuclei/cm, and the microscopic cross sections are from Table 2.2. It is of interest to note the asymptotic behavior of the function Ny R)/Nm The asymptote corresponds to the minimum fuel concentration with which a system of and beryllium can go critical (on the basis of the one-velocity model) this is the con-... 

T v"o physical parameters required in every application of the one-velocity model are the diffusion coefficient D and the diffusion length L. So far, our interest in the one-velocity model has been confined to descriptions of the spatial distribution of thermal neutrons, and these parameters have therefore been defined in terms of the thermal cross sections. We will show later that similar quantities can be defined in the treatment of multivelocity systems by the energy-group method, or for that matter, in any system wherein it is convenient to describe the diffusion properties of a particular group of neutrons in terms of a single speed. 

Sample Reactor Calculation with One-velocity Model... 

As an example of the application of the one-velocity model to the analysis of nuclear reactors, we will determine the critical fuel mass and maximum flux level in a light-water-moderated reactor which is designed to produce 20 megawatts of heat. Let the core of this reactor be a 60- by 32.3- by 55.4-cm parallelepiped, and for the present analysis assume... 

The critical fuel mass is computed from the criticality condition for the one-velocity model. The appropriate relation is (5.181) with fc = 1. If we solve this equation for the macroscopic absorption cross section of the we obtain [cf. (5.185)]... 

A spherical shell of material has inner radius Ri and outer radius Rt. (There is a vacuum both inside and outside the spherical shell.) The material of the shell is described by neutron cross sections 2/, and 2u (one-velocity model). 

In this case fission neutrons are assumed to appear directly at thermal energy. Thus losses by absorption or leakage during slowing down are not allowed, and therefore we take th = pth = 1. Clearly, in the one-velocity model all fissions are due to thermal neutrons so < = 1. 

It is important to recognize that this result satisfies the requirements of the one-velocity model developed in Sec. 5.4g, namely, that the neutrons from fission appear in the same space form as the thermal flux. Accord-... 

We will lead up to the calculation of the neutron flux defined by (7.2) by introducing first a somewhat more limited description of the neutron population, namely, the one-velocity model. The appropriate function is < (r,O,0. By omitting, for a time, the problem of describing the distribution of neutrons in energy space, we can focus attention on the directional properties of the neutron motion. Thus our first objective is to construct a picture of the neutron population which gives its dis-... 

A second general technique for treating the angular distribution of the neutron flux is presented in Sec. 7.4. This is the method of integral equations. Solutions for the directed flux 0(r,Q) are derived on the basis of the one-velocity model for various media of infinite extent. The application of these solutions for the infinite medium to systems of finite size is demonstrated in the case of the homogeneous slab and sphere. 

Section 7.5 deals with the application of the spherical-harmonics method developed for the one-velocity model of Sec. 7.2 to two situations of interest to reactor analysis. The emphasis is placed on various aspects of the computational procedure involved in such calculations and on the selection of suitable boundary conditions in terms of the directed flux. 

In this section we illustrate the application of the spherical-harmonics method by considering two problems of some practical interest. Both problems deal with steady-state one-dimensional systems, and the calculation is carried out on the basis of the one-velocity model developed in Sec. 7.2f. In the present applications the general time-dependent relations given in (7.84) reduce to the following set of differential equations ... 

An accurate calculation for the reflector savings based on the solution of the criticality equation for the one-velocity model will yield a curve of the shape shown in Fig. 8.3. The approximations (8.20) and (8.22) apply for the extreme values of reflector thickness. 

The method based on the Serber-Wilson condition is especially suited to the treatment of reactor systems which are spherically symmetric. We will develop the basic ideas involved in this method by analyzing, on the basis of the one-velocity model, the simple case of a spherical core surrounded by an infinite reflector. However, the method is generally applicable to finite systems, to multiregion configurations, and to multigroup (energy) calculations. These problems will not be discussed here. In order to demonstrate the effect on the criticality requirements of the... 

Now, the form of the 03 equation is seen to be the same as the reflector relation used in the one-velocity model [cf. Eq. (8.56)], and it was found that the solution to that equation could be given in terms of a single function. It follows, then, that... 

As a first example, consider the analysis of a completely reflected cylindrical reactor on the basis of the one-velocity model. A typical... 

In general the removal densities A (r) are complicated functions, but in the present calculation we represent these by rough approximations based on the fundamental mode from the one-velocity model. In the... 

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