Velocity distribution infinite medium

The solution of (9.23) and (9.25) to obtain the velocity and pressure distribution around a sphere was first obtained by Stokes. The assumptions invoked to obtain the solution are (1) an infinite medium, (2) a rigid sphere, and (3) no slip at the surface of the sphere. For the solution details, we refer the reader to Bird et al. (1960, p. 132). 

II. Neutron velocity distribution in an infinite homogeneous medium. 

A. Velocity distribution following a fast neutron pulse. The physical processes occurring in neutron thermalization are best illustrated by considering the velocity distribution of neutrons as a function of time following a burst of fast neutrons at = 0. We consider an infinite homogeneous medium at a uniform temperature T with an absorption cross section varying inversely as the neutron velocity. The linearized Boltzmann equation describing the neutron distribution in velocity and time is... 

III. Determination of the energy-transfer cross sections and calculations of infinite medium velocity distributions. 

IV. Spatially dependent velocity distributions. When the spectrum is independent of position, the central problem is the determination of the energy-transfer cross sections. The calculation of the spectrum once these cross sections are known is a straightforward procedure. The cross-section aspect of the problem is both more difficult from a physics point of view, and more time consuming from the point of view of machine computation. This situation is reversed when we come to consider the spatial dependence of the slow neutron spectrum. The cross sections needed are the same ones that already have been computed for the infinite medium spectrum problem. The transport equation must now be solved in at least two variables, and in a form for which the existing approximate techniques are not very well adapted. The focus of the problem therefore shifts to the development of appropriate techniques for solution of the transport equation when the energy and position variables are coupled in such a way that neutrons can both gain and lose energy in a collision. 

In this treatment we examine the nuclear reactor with the aid of an infinite- medium model which describes the neutron-density distributions in terms of the kinetic energy of the neutrons. This model is the immediate extension of the one-velocity approximation introduced in Chap. 3. In the present analysis we discard the one-velocity restriction and attempt to describe the trajectory of the neutron in energy space as it is slowed down by elastic-scattering collisions with the nuclei of the medium. In... 

Let us examine now the analogous problem of the distribution of one-velocity neutrons in the space-time system. Consider, then, an isotropic plane source of neutrons in an infinite medium. The initial condition is that a burst of neutrons per unit area is released from the source (placed for convenience at the origin) at time, say, t = 0. These neutrons have speed i>, and they retain this speed for all subsequent time ... 

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. 

When there is a constant source of a reacting chemical species in the water column or at its boundaries (e.g., water-air and/or water-sediment interface) then, by a rule of thumb, a steady-state may be attained within a period of time equal to a few half-lives of the species. In detail, a steady-state concentration is attained after infinitely long time. The time required for the concentration to come close to the steady-state value at any point in the water column depends on its distance from the source, transport properties of the medium (i.e., its diffu-sivity and distribution of advective velocities), and the rates of the reactions removing the species from the water. A concentration of 95% of a steady-state value may be arbitrarily taken as sufficiently close to a steady-state and indicating that the transient state has effectively come to an end. The time required to attain this concentration level (i.e., when C = 0.95C ) at some point of a concentration-depth profile will be referred to as the time to steady-state. By way of generalization, a chemical species with a constant half-life would attain a steady-state concentration at any point in the water column sooner when the distance... 

---