The One-velocity Transport Equation

The direction (unit vector) is specified by means of the co-latitude 

It is important to recognize that the function defined by (7.3) is related 

Thus our present definition simply selects from the entire population of neutrons of speed v those which are moving in a particular direction. 

With the foregoing definitions, we may proceed to construct, by considering the neutron balance, an equation governing the neutron density. Consider, then, the set of neutrons of speed V in the volume element dr about r at time t which have velocity vectors that lie in dQ about Q. From conservation requirements it must be true that 

Number of neutrons moving in direction O which appear per unit time from sources in dr 

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L. S. Bohl and F. D. Judge, Variational treatment of the one-velocity transport equation for an arbitrary slab lattice. Transactions of the American Nuclear Society, vol. 1, no. 2, June, 1958. 

P. B. Daitch et al., CEPTR An IBM-704 Code to Solve the P, Approximation to the One-Velocity Transport Equation in Cylindrical Geometry, Combustion Engineering report MPC-20 (Sept. 1959). 

The one-velocity calculations require as Input a collision cross section ( t). end a neutron production per collision. Both were determine from soluUons of the one- velocity transport equation for two delayed critical experiments a 7-in.-diam (17.78 cm) by 4.971-in.-hi (12 63 cm) cylinder of -the- U(93) metal and a IS-in. (38.1)cm) diam by 3.03-ln.-h di (T.Bg cm) cylinder. The solutUms, Which were obtaii from Su transport calculations with the DDK cote , using a 16 X 16. grid of space points, yielded values of vZt/7 as a function oi Zt. The pair Of values common to both experiments was then used as input to the Monte Carlo calculations for all the geometries.. ... 

In the presentation which follows, we derive the formal solution to the one-velocity transport equation (7.15) for several media of practical importance. The general treatment follows that of Case, de Hoffmann, and Placzek. The reader is referred to the work of these authors for a much more complete development of the general methods and results. 

For the functions 0c(r) and ji(r) which appear in (8.95) we use first-order approximations of the solutions to the transport equation. The basis for this choice is derived from the following argument. It was shown previously in Sec. 7.4d that the elementary solution to the one-velocity transport equation for the source-free infinite multiplying... 

The solution to the general time-dependent system is easily developed from the above results. In the notation of the present method, the non-stationary one-velocity transport equation is... 

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. 

Only for an isothermal, first-order reaction where Sa = —k a will the chemical source term in (3.102) be closed, i.e., ++(<+ = h (u,(pa). Indeed, for more complex chemistry, closure of the chemical source term in the scalar-flux transport equation is a major challenge. However, note that, unlike the scalar-flux dissipation term, which involves the correlation between gradients (and hence two-point statistical information), the chemical source term is given in terms of u(x, t) and 0(x, t). Thus, given the one-point joint velocity, composition PDF /u,chemical source term is closed, and can be computed from... 

The effluent concentration can be predicted by solving the mass conservation equation. The conservation equations of particulate matter consider the change in concentration of particulate and change of porosity with time. The amount of fines retained in the porous medium is represented by a, while u signifies the superficial velocity of the incompressible transport fluid. For constant volumetric, incompressible flow, neglecting dispersion and gravitational effects, the one dimensional conservation equation follows. 

All symbols have their usual meaning and only more important ones are defined here. Cj is the concentration of component j in the aqueous phase (e.g. polymer, tracer, etc.). The viscosity of the aqueous phase, rj, may depend on polymer or ionic concentrations, temperature, etc. Dj is the dispersion of component j in the aqueous phase Rj and qj are the source/sink terms for component j through chemical reaction and injection/production respectively. Polymer adsorption, as described by the Vj term in Equation 8.34, may feed back onto the mobility term in Equation 8.37 through permeability reduction as discussed above. In addition to the polymer/tracer transport equation above, a pressure equation must be solved (Bondor etai, 1972 Vela etai, 1974 Naiki, 1979 Scott etal, 1987), in order to find the velocity fields for each of the phases present, i.e. aqueous, oleic and micellar (if there is a surfactant present). This pressure equation will be rather more complex than that described earlier in this chapter (Equation 8.12). However, the overall idea is very similar except that when compressibility is included the pressure equation becomes parabolic rather than elliptic (as it is in Equation 8.12). This is discussed in detail elsewhere (Aziz and Settari, 1979 Peaceman, 1977). Various forms of the pressure equation for polymer and more general chemical flood simulators are presented in a number of references (Zeito, 1968 Bondor etal, 1972 Vela etal, 1974 Todd and Chase, 1979 Scott etal, 1987). 

A more quantitative discussion requires the use of transport equations for the density matrix. To begin, we consider a beam of one-level atoms, each of mass m and incident velocity v, scattering from a collection of perturbers with volume density W, The transport equation, in the form of Eq. (3,14b) below, could be written down without derivation simply on the basis of physical principles, How -ever, we sketch a derivation here since it facilitates the derivation of the somewhat more complicated transport equations for two-level atoms. The atoms in the incident beam are assumed to have wave-packets of similar form but random impact parameters. Thus the normalized incoming wavepacket of a typical atom is... 

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 many respects, the solutions to equations 12.7.38 and 12.7.47 do not provide sufficient additional information to warrant their use in design calculations. It has been clearly demonstrated that for the fluid velocities used in industrial practice, the influence of axial dispersion of both heat and mass on the conversion achieved is negligible provided that the packing depth is in excess of 100 pellet diameters (109). Such shallow beds are only employed as the first stage of multibed adiabatic reactors. There is some question as to whether or not such short beds can be adequately described by an effective transport model. Thus for most preliminary design calculations, the simplified one-dimensional model discussed earlier is preferred. The discrepancies between model simulations and actual reactor behavior are not resolved by the inclusion of longitudinal dispersion terms. Their effects are small compared to the influence of radial gradients in temperature and composition. Consequently, for more accurate simulations, we employ a two-dimensional model (Section 12.7.2.2). 

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