The theory of convective diffusion

The success of the analysis in correlating experimental data for clean filters offers convincing support for the theory of convective diffusion of particles of finite diameter to surfaces. As particles accumulate in the filter, both the efficiency of removal and the pressure drop increase, and the analysis no longer holds. Some data on this effect are available in the literature. Care must be taken in the practical application of these results because of pinhole leaks in the filters or leaks iU ound the frames. 

In connection with the development of the theory of convective diffusion in liquids the foundation of the theory of diffusion boundary layers and dynamic adsorption layers are given by Levich (1962) in his works on physico-chemical hydrodynamics. A variety of problems of convective diffusion in liquids was solved which are of essential interest for the description of different heterogeneous processes in liquids the rate of which is limited by diffusion kinetics. In connection with the objectives of the present chapter, only a general approach to problems of diffusion boundary layers and their concrete results (Levich 1962) are reported. These are of direct interest for the theory of dynamic adsorption layers of bubble. 

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 development of convective-diffusion theories is due principally to Prandtl9 and Schlichting10, and their application in electrochemistry to Levich11. Levich was the first to solve the equations for the rotating disc electrode. 

In what follows, the equation of diffusion derived in Chapter 2 is generalized to take into account the effect of flow. For point particles (dp = 0), rates of convective diffusion can often be predicted from theory or from experiment with aqueous solutions because the Schmidt numbers are of the same order of magnitude. There is an extensive literature on this subject to which the reader is directed. For particle diffusion, there is a difference from the usual theory of convective diffusion because of the special boundary condition The concentration vanishes at a distance of one particle radius from the surface. This has a very large effect on particle deposition rates and causes considerable difficulty in the mathematical theory. As discussed in this chapter, the theory can be simplified by incorporating the particle radius in the diffusion boundary condition. 

For large Reynolds numbers, the function K depends on the particular shape of the obstacle and can be calculated by standard methods of boundary layer theory (Schlichting, 1979, Chapter IX). Then, the equation of convective diffusion is, in the boundary layer approximation. 

It should be pointed out that in the formulation of the problem of convective diffusion one should know the velocity distribution v(z) since it appears on the lefk-hand side of Eq. (8.8). A solution of the hydrodynamic problem is possible in the limiting cases of small and large Reynolds numbers. Therefore, the theory of the boundary diffusion layer for the two limiting cases Re l, Pe l and Re l, Pe l has to be developed. 

The main distinction of the theory of a dynamic adsorption layer formed under weak and strong retardation arises when formulating the convective diffusion equation. At weak retardation the Hadamard-Rybczynski hydrodynamic velocity field is used while at strong retardation the Stokes velocity field. Different formulas for the dependence of the diffusion layer thickness on Peclet numbers are obtained. The problem of convective diffusion in the neighbourhood of a spherical particle with an immobile surface at small Reynolds numbers and condition (8.74) is solved, so that the well-known expression for the density distribution of the diffusion flow along the surface can be used. As a result, Eq. (8.10) takes the form (Dukhin, 1982),... 

Membrane transport represents a major application of mass transport theory in the pharmaceutical sciences [4], Since convection is not generally involved, we will use Fick s first and second laws to find flux and concentration across membranes in this section. We begin with the discussion of steady diffusion across a thin film and a membrane with or without aqueous diffusion resistance, followed by steady diffusion across the skin, and conclude this section with unsteady membrane diffusion and membrane diffusion with reaction. 

Very much more is known about the theory of concentration gradients at electrodes than has been mentioned in this brief account. Experimental methods for observing them have also been devised, based on the dependence of refractive index on concentration (the Schlieren method) by means of interferometry (O Brien, 1986). Nevertheless, the basic concept of an effective diffusion-layer thickness, treated here as varying in thickness with fi until the onset of natural convection and as constant with time after convection sets in (though decreasing in value with the degree of disturbance, Table 7.10), is a useful aid to the simple and approximate analysis of many transport-controlled electrodic situations. A few of the uses of the concept of 8 will now be outlined. 

If diffusive combustion occurs and the flame does not move with respect to the gas, then convective flows will carry it upward such a flame can not propagate downward. Therefore we relate the dependence of the limits on the direction of propagation to the possibility of a diffusive mechanism, and we compare the theory of limits of normal propagation ( 1.4 and 1.5) with data relating to downward propagation. 

In the operator L, the first term represents convection and the second diffusion. Equation (44) therefore describes a balance of convective, diffusive, and reactive effects. Such balances are very common in combustion and often are employed as points of departure in theories that do not begin with derivations of conservation equations. If the steady-flow approximation is relaxed, then an additional term, d(p(x)/dt, appears in L this term represents accumulation of thermal energy or chemical species. For species conservation, equations (48) and (49) may be derived with this generalized definition of L, in the absence of the assumptions of low-speed flow and of a Lewis... 

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