One-velocity Diffusion Equation

The basis of the following analysis, then, is the one-velocity approximation. This model was introduced in Chap. 3, but our attention there w as confined to the infinite medium. In the present treatment we will 

1/ total track length of neutrons of speed v per unit 

Note that these functions refer to neutrons of a single speed v according to our model. In all the analyses of this chapter the speed v will be implied in our definitions of n and and therefore the number v will not appear as an argument of these functions. Also note that the argument r denotes a general position vector in some appropriate coordinate system set up in 

If we introduce the absorption cross section 2a for neutrons of speed v and the source function S(r,0, where 

The function S(r,0 dr denotes the net losses from the volume element dr due to transport of neutrons through the boundaries. 

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The general one-velocity diffusion equation for the multiplying medium is then... 

Assume that the neutron flux satisfies the one-velocity diffusion equation in the material of the shell and that appropriate boundary conditions are to be applied. 

One-velocity diffusion equation gives an adequate representation of the neutron physics. 

We went from the one-dimensional velocity distribution to the three-dimensional speed distribution by adding in an extra v2 factor to account for the added degeneracy in velocity space. The one-dimensional diffusion equation 7.36 has a form which is mathematically very similar to the one-dimensional velocity distribution... 

In Eq. (5.5.3) the radial velocity has been replaced by the sedimentation coefficient s, from the definition of Eq. (5.5.2). The fluid dynamicist should be aware that this one-dimensional diffusion equation is known in the ultracentrifuge literature as the Lamm equation (Fujita 1975). In the limit of infinitely dilute solutions D and s are independent of concentration and may be taken out of the derivative to give... 

As an illustrative example consider a spherical reactor of core radius a and an infinite reflector. In the case of the one-velocity diffusion-theory model we found that the appropriate characteristic equation was the transcendental relation (8.11). For the reactor with an infinite reflector, this relation reduces to... 

This section derives a simple version of the convective diffusion equation, applicable to tubular reactors with a one-dimensional velocity profile V (r). The starting point is Equation (1.4) applied to the differential volume element shown in Figure 8.9. The volume element is located at point (r, z) and is in the shape of a ring. Note that 0-dependence is ignored so that the results will not be applicable to systems with significant natural convection. Also, convection due to is neglected. Component A is transported by radial and axial diffusion and by axial convection. The diffusive flux is governed by Pick s law. 

In comparison with the large amount of literature that is available on the deposition of particles from laminar fluid flows, literature on turbulent deposition is virtually non-existant [114]. It was mentioned that the trajectory and convective diffusion equations also apply when the fluid inertial effects are considered, including the case of turbulent flow conditions, provided one is able to express the fluid velocities explicitly as a function of position and time. 

Solving the purely advective equation or even introducing an advection term into the diffusion equation is a source of numerical difficulties. The simplest advection equation of a medium moving at velocity v in one dimension can be written... 

According to measurements made in the atmosphere, the Lagrangian time scale is of the order of 100 sec (Csanady, 1973). Using a characteristic particle velocity of 5 m sec", the above conditions are 100 sec and L > 500 m. Since one primary concern is to examine diffusion from point sources such as industrial stacks, which are generally characterized by small T and L, it is apparent that either one (but particularly the second one) or both of the above constraints cannot be satisfied, at least locally, in the vicinity of the point-like source. Therefore, in these situations, it is important to assess the error incurred by the use of the atmospheric diffusion equation. 

For one-dimensional diffusion and laminar flow with constant velocity along the direction, the above diffusion-flow equation can be written as... 

In the case of one-dimensional crystal dissolution with u = Uq, if the reference frame is fixed at the faraway melt (x = oo), the melt does not flow even though the melt is generated at the interface at velocity u. (The interface moves as a rate of u.) Hence, the diffusion equation is Equation 3-9 without a velocity term ... 

Still in the case of one-dimensional dissolution, if the reference frame is fixed at the non dissolving part of the crystal x = - oo), the interface moves at a velocity of However, any point in the melt is moving at a velocity of w > That is, relative to the reference frame fixed to the nondissolving part of the crystal, the melt flows at a velocity of (w — w ). Hence, the equation to describe diffusion in the melt is the flow-diffusion equation (Equation 3-19b),... 

Problems in forced convection are solved in two steps first one solves the equation of motion to obtain the velocity distribution, and then one puts the expression for v back into the diffusion equation [usually as given in Eq. (50)]. An illustration of this type of problem is that of the absorption of a gas by a liquid film flowing down a... 

Because of the success encountered by finite elements in the solution of elliptic problems, it was extended (in the 80s) to the advection or transport equation which is a hyperbolic equation with only one real characteristic. This equation can be solved naturally for an analytical velocity field by solving a time differential equation. It appeared important, when the velocity field was numerically obtained, to be able to solve simultaneously propagation and diffusion equations at low cost. By introducing upwinding in test functions or in the discretization scheme, the particular nature of the transport equation was considered. In this case, a particular direction is given at each point (the direction of the convecting flow) and boundary conditions are only considered on the part of the boundary where the flow is entrant. 

So far, the concept of mass conservation has been applied to large, easily measurable control volumes such as lakes. Mass conservation also can be usefully expressed in an infinitesimal control volume, mathematically considered to be a point. Conservation of mass is expressed in such a volume with the advection—dispersion-reaction equation. This equation states that the rate of change of chemical storage at any point in space, dC/dt, equals the sum of both the rates of chemical input and output by physical means and the rate of net internal production (sources minus sinks). The inputs and outputs that occur by physical means (advection and Fickian transport) are expressed in terms of the fluid velocity (V), the diffusion/dispersion coefficient (D), and the chemical concentration gradient in the fluid (dC/dx). The input or output associated with internal sources or sinks of the chemical is represented by r. In one dimension, the equation for a fixed point is... 

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