Convective diffusion velocity profile derivation

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. 

For a number of flow situations, the mass-transfer rate can be derived directly from the equation of convective diffusion (see Table VII, Part A). The velocity profile near the electrode is known, and the equation is reduced to a simpler form by appropriate similarity transformations (N6). These well-defined flows, therefore, are being exploited increasingly by electrochemists as tools for the kinetic characterization of electrode reactions. Current distributions at, or below, the limiting current, transient mass transfer, and other aspects of these flows are amenable to analysis. Especially noteworthy are the systematic investigations conducted by Newman (review until 1973 in N7 also N9b, N9c, H6b and references in Table VII), by Daguenet and other French workers (references in Table VII), and by Matsuda (M4a-d). Here we only want to comment on the nature of the velocity profile near the electrode, and on the agreement between theory and mass-transfer experiment. 

The first step in solving convective diffusion problems is the derivation of the velocity profile. In this case, the flow arriving with velocity v is modified by... 

The false-transient method can be applied to convective diffusion equations in a manner similar to that used for velocity profiles. Finite-difference approximations are written for the spatial derivatives. Second-order approximations can be used for first derivatives since they involve only the same five points needed for the second derivatives. The result is a set of simultaneous ODEs with (false) time as the independent variable. The computational template of Figure 16.3 is unchanged. The next two examples illustrate its application to problems where axial diffusion is negligible. Such problems are also readily solved by the method of lines as described in Chapter 8. Cases with significant axial diffusion are troublesome for the method of lines and require special boundary conditions for the method of false transients. They are treated in Section 16.2.4. 

The exact solution of the convection-diffusion equations is complicated, because the theoretical treatments involve solving a hydrodynamic problem, that is, the determination of the solution flow velocity profile by using Navier-Stokes equation. For the calculation of a velocity profile, the solution viscosity, densities, rotation rate, or stirring rate, as well as the shape of the electrode should be considered. Exact solution has been derived for the rotating disc electrode (RDE) ... 

A remarkable property of the solution that we have derived, clearly evident in Eq. (3-244) for the cross-sectionally averaged temperature profile, is that the position in the tube where the cross-sectionally averaged temperature is maximum moves as if there were convection downstream at the mean velocity U. The temperature pulse also spreads about this plane as though there were axial conduction with an effective thermal diffusivity of... 

---