Equation of Convective Diffusion

It IS sometimes possible to predict rates of deposition by diffusion from flowing fluids by analysis of the equation of convective diffusion. This equation is derived by making a material balance on an elemental volume fixed in space with respect to laboratory coordinates (Fig. 2.1). Through this volume flows a gas carrying small particles in Brownian motion. 

The rate at which particles are carried by the flow into the volume element across the face A BCD is given by 

The net rate of particle accumulation for the flow in the. v direction is given by subtracting the rate leaving from the rate entering  

Analogous expressions are obtained for the other four faces summing up for all three pairs, the result for the net accumulation of particles in the volume element is given by 

Tbe rate of particle accumulation in the volume SxSySz taking into account the flow, diffusion, and external force fields (Chapter 2) is obtained by summing the three effects  

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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 general equation of convective diffusion in liquids, equation (15), is a second-order, partial differential equation with variable coefficients. Its solution yields the spatial distribution of c, as a function of time, namely its transient behaviour. On an analytical level, solution of equation (15) into the transient c(t) is possible only for a number of relatively simple systems with well-defined geometry and flow properties. The problem is greatly simplified if the concentration function Cj(x,y,z) is essentially independent of time t, i.e. in the steady-state. Then equation (15) reduces to ... 

The concentration distribution in the diffusion layer is governed by the one-dimensional unsteady equation of convective diffusion ... 

The mathematical model comprises a set of partial differential equations of convective diffusion and heat conduction as well as the Navier-Stokes equations written for each phase separately. For the description of reactive separation processes (e.g. reactive absorption, reactive distillation), the reaction terms are introduced either as source terms in the convective diffusion and heat conduction equations or in the boundary condition at the channel wall, depending on whether the reaction is homogeneous or heterogeneous. The solution yields local concentration and temperature fields, which are used for calculation of the concentration and temperature profiles along the column. 

Referring to the nondimensional equation of convective diffusion (3.3), it is of interest to examine the conditions under which the diffusion term, on the one hand, or convection, on the other, is the controlling mode of transport. The Peclet number, dUfD, for flow around a cylinder of diameter r/ is a measure of the relative importance of (he two term.s. For Pe 1, transport by llte flow can be neglected, and the deposition rate can be determined approximately by solving the equation of diffusion in a non flowing fluid with appropriate boundary conditions (Carslaw and Jaeger, 1959 Crank, 1975). 

Even though the Reynolds number is small, there are many practical situations in which Pe = Re Sc is large because the Schmidt number, Sc. for aerosols is very large. For Pe I, two important simplifications can be made in the equation of convective diffusion. First, diffusion in the tangential direction can be neglected in comparison with convective transport ... 

In addition, a concentration boundary layer develops over the surface of the cylinder with its thinnest portion near the forward stagnation point. When the thickness of the concentration boundary layer is much less than the radius of the cylinder, the equation of convective diffusion simplifies to the familiar form for rectangular coordinates (Schlichling, 1979, Chapter XII) ... 

The concentration boundary condition n = is set at y = oo, even though the boundary layer form of the equation of convective diffusion (3,10) is valid only near (he surface of the cylinder. This can be justified by noting that the concentration approaches Oao very near the surface for high Pe. 

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. 

The diffusion battery consists of banks of tubes, channels, or screens through which a submicron aerosol passes at a constant flow rale. Particles deposit on the surface of the battery elements, and the decay in total number concentration along the flow path i measured, usually with a condensation particle counter. The equations of convective diffusion (Chapter 3) can be solved for the rate of deposition as a function of the particle diffusion coefficient. Because the diffusion coefficient is a monotonic function of particle size (Chapter 2), the measured and theoretical deposition curves can be compared to detennine the size for a monodisperse aerosol. 

The change in the discrete distribution function with time and position is obtained by generalizing the equation of convective diffusion (Chapter 3) to include terms for particle growth and coagulation ... 

Equations and a condition on the surface. Nonsteady-state distribution of surfactants in the volume V and on the surface S is described by the equation of convective diffusion in the bulk and on the surface, respectively [250] ... 

The supplements contain tables with exact solutions of the heat equation. In addition, the equation of convective diffusion, the continuity equation, equations of motion in some curvilinear orthogonal coordinate systems, and some other reference materials are given. 

At first, the equations of convective diffusion for a system of three ions has to be considered. Under the condition (7.9), the diffuse layer is approximately electrically neutral ... 

In the diffusion boundary layer, we have d Cfdx d C/dy, so the equation of convective diffusion in the layer becomes ... 

Consider convection diffusion toward a spherical particle of radius R, which undergoes translational motion with constant velocity U in a binary infinite diluted solution [3], Assume the particle is small enough so that the Reynolds number is Re = UR/v 1. Then the flow in the vicinity of the particle will be Stoke-sean and there will be no viscous boundary layer at the particle surface. The Peclet diffusion number is equal to Peo = Re Sc. Since for infinite diluted solutions, Sc 10 and the flow can be described as Stokesian for the Re up to Re 0.5, it is perfectly safe to assume Pec 1. Thus, a thin diffusion boundary layer exists at the surface. Assume that a fast heterogeneous reaction happens at the particle surface, i.e. the particle is dissolving in the liquid. The equation of convective diffusion in the boundary diffusion layer, in a spherical system of coordinates r, 6, (p, subject to the condition that concentration does not depend on the azimuthal angle [Pg.128]

So consider the equation of convective diffusion in a developed flow,... 

It is convenient to look for the solution in a coordinate system moving with velocity U relative to the pipe wall. In this system of coordinates, the equation of convective diffusion (6.118) subject to the inequalities L/R 1 and Pe 1 takes the form... 

For a binary mixture, in the absence of chemical reactions in the volume, mass transport equation reduces to the equation of convective diffusion. Let C denote the concentration of dissolved substance. Then the mass transport equation is... 

We assume that deposition on the sphere is ideal, that is, each collision of a particle with the sphere results in the particle being captured. The factor of Brownian diffusion Dj,r = kTwhere Oj, is the particle s radius, is much smaller than the factor of molecular diffusion, therefore the Peclet diffusion number is Peo = Ua/Dhr 1- By virtue of this inequality (see Section 6.5), the diffusion flux of particles toward the sphere can be found by solving the stationary equation of convective diffusion with a condition corresponding to a thick or thin diffusion-boundary layer. Particles may then be considered as point-like, and the diffusion equation will become ... 

Consider a multicomponent solution. The distribution of concentrations of the dissolved substances is described by the equations of convective diffusion... 

Consider now the diffusion growth of an isolated motionless bubble in a binary solution. Denote by pn the mass concentration of the dissolved component in the liquid. Suppose the bubble consists only of component 1, the process is spherically symmetrical, and the distribution p i is described by the equation of convective diffusion (22.1), in which should be should take Ug = 0 and Ur = (R/r) R under the condition p Q p. ... 

According to the selected model of multicomponent chemical kinetics the distribution of concentration for each component was calculated from the system of interconnected equations of convective-diffusion transfer ... 

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