Convective dispersion illustration

We can illustrate the salient features of convective dispersal by choosing a simple velocity distribution in a rectangular convection cell (0associated with the onset of Benard instability in the conditions of Boussinesq approximations (e.g., Turcotte and Schubert, 1982). Let us make the calculation for the so-called free-slip conditions, which permit free movement along the boundaries, both vertical and horizontal, such as a convection cell which would be limited by no rigid boundary. From Turcotte and Schubert (1982), we take the velocity field to be... 

In the present problem (illustrated in Figure 6.2.2(a)), we will also find that the solute pulse introduced at z = 0 win show up (on a radially averaged basis) as a concentration peak with a broadened base as z becomes large. However, this broadening of the solute profile in the z-direction is not due to the molecular diffusion coefficient Djs of species i in the solvent. Rather, it arises primarily due to the radially nonuniform axial velocity profile (6.1.1b, c) of flow in a tube. It is identified as an axial dispersion or convective dispersion. This phenomenon was first studied by Taylor (1953, 1954), and is often described also as Taylor dispersion. 

That notorious pair, the Danckwerts boundary conditions for the tubular reactor, provides a good illustration of boundary conditions arising from nature. Much ink has been spilt over these, particularly the exit condition that Danckwerts based on his (perfectly correct, but intuitive) engineering insight. If we take the steady-state case of the simplest distributed example given previously but make the flux depend on dispersion as well as on convection, then, because there is only one-space dimension,/= vAc — DA dddz), where D is a dispersion coefficient. Then, as the assumption of steady state eliminates... 

Both the data in Table 1 and the surface Q(u,D) in Fig. 3 illustrate well that the sensitivity of the objective function with respect to the convective flow velocity is significant. At the same time, variation of Q as a function of the axial dispersion coefficient seems to be smaller, which means that the mean residence time has stronger influence on the properties of the particulate product than the axial mixing of particles. 

It is worth a short detour into fluid mechanics to explore some details of this approach and how it fits into the reactor conservation equations. For moderate flow velocities the dispersion of a tracer in laminar flow will occur by axial and radial diffusion from the flow front of the tracer and, in the absence of eddy motion, this will be via a molecular diffusion mechanism. However, the net contribution of diffusion in the axial direction can be taken as small in comparison to the contribution of the flow velocity profile. This leaves us with a two-dimensional problem with diffusion in the radial direction and convection in the longitudinal direction the situation is considered in illustrated in Figure 5.7. 

In Fig. 6 a velocity fields are shown for a system of four Rushton turbines. In addition to the velocity vector field, large arrows are used to illustrate the flow behavior. Each impeller creates a more or less independent symmetrical flow field. The multiple impeller system therefore shows very poor axial convection. The transport between the individual cells is performed mainly with the aid of axial turbulent dispersion. 

This chapter discusses finite-difference techniques for the solution of partial differential equations. Techniques are presented for pure convection problems, pure diffusion or dispersion problems, and mixed convection-diffusion problems. Each case is illustrated with common physical examples. Special techniques are introduced for one- and two-dimensional flow through porous media. The method of weighted residuals is also introduced with special emphasis given to orthogonal collocation. 

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