The Convective Diffusion Equation

FIGURE 8.9 Differential volume element in cylindrical coordinates. 

Applying Equation (1.2), dividing everything by [2jrr Ar Az], and rearranging 

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Hyperbolic Equations The most common situation yielding hyperbohc equations involves unsteady phenomena with convection. Two typical equations are the convective diffusive equation... 

The effect of using upstream derivatives is to add artificial or numerical diffusion to the model. This can be ascertained by rearranging the finite difference form of the convective diffusion equation... 

Another method often used for hyperbohc equations is the Mac-Cormack method. This method has two steps, and it is written here for the convective diffusion equation. 

Another approach to modeling the particle-collection process is based on the convective diffusion equation... 

Example 8.8 Explore conservation of mass, stability, and instability when the convective diffusion equation is solved using the method of lines combined with Euler s method. 

Radial motion of fluid can have a significant, cumulative effect on the convective diffusion equations even when Vr has a negligible effect on the equation of motion for V. Thus, Equation (8.68) can give an accurate approximation for even though Equations (8.12) and (8.52) need to be modified to account for radial convection. The extended versions of these equations are... 

The convective diffusion equations for mass and energy are given detailed treatments in most texts on transport phenomena. The classic reference is... 

The appropriateness of neglecting radial flow in the axial momentum equation yet of retaining it in the convective diffusion equation is discussed in... 

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. 

Include the radial velocity term in the convective diffusion equation and plot streamlines in the reactor. 

Solution The problem requires solution of the convective diffusion equation for mass but not for energy. Rewriting Equation (8.71) in dimensionless form gives... 

The unsteady version of the convective diffusion equation is obtained just by adding a time derivative to the steady version. Equation (8.32) for the convective diffusion of mass becomes... 

Axial Dispersion. Rigorous models for residence time distributions require use of the convective diffusion equation. Equation (14.19). Such solutions, either analytical or numerical, are rather difficult. Example 15.4 solved the simplest possible version of the convective diffusion equation to determine the residence time distribution of a piston flow reactor. The derivation of W t) for parabolic flow was actually equivalent to solving... 

A second approach is to assume that the drop surface approaching the electrode is a moving plane. This is appropriate, since the diffusion layer is almost always considerably smaller than the size of the drop, for most of its lifetime under practical conditions. To a good approximation, the convective effect close to the moving front is then calculated based on velocities which are twice those determined from Eq. (23), in order to account for the moving center of the drop. The convective-diffusion equation which describes this case is given by... 

At the RRE the derivation123 of the Levich equation requires reconsideration of the convection-diffusion equation, which results in... 

The convective diffusion equations presented above have been used to model tablet dissolution in flowing fluids and the penetration of targeted macro-molecular drugs into solid tumors [5], In comparison with the nonequilibrium thermodynamics approach described below, the convective diffusion equations have the advantage of theoretical rigor. However, their mathematical complexity dictates a numerical solution in all but the simplest cases. 

The convective diffusion equation is simply cast in terms of radius (r) instead of x. 

See Partial Differential Equations. ) If the diffusion coefficient is zero, the convective diffusion equation is hyperbolic. If D is small, the phenomenon may be essentially hyperbolic, even though the equations are parabolic. Thus the numerical methods for hyperbolic equations may be useful even for parabolic equations. 

Since turbulent fluctuations not only occur in the velocity (and pressure) field but also in species concentrations and temperature, the convection diffusion equations for heat and species transport under turbulent-flow conditions also comprise cross-correlation terms, obtained by properly averaging products of... 

Eggels and Somers (1995) used an LB scheme for simulating species transport in a cavity flow. Such an LB scheme, however, is more memory intensive than a FV formulation of the convective-diffusion equation, as in the LB discretization typically 18 single-precision concentrations (associated with the 18 velocity directions in the usual lattice) need to be stored, while in the FV just 2 or 3 (double-precision) variables are needed. Scalar species transport therefore can better be simulated with an FV solver. 

For problems involving gradients in chemical species, the convection-diffusion equations for the species are also solved, usually for N— 1 species with the Nth species obtained by forcing the mass fractions to sum to unity. Turbulence can be described by a turbulent diffusivity and a turbulent Schmidt number, Sct, analogous to the heat transfer case. 

Then an approximate analytical solution of the convective diffusion equation (43), which satisfies the boundary conditions, equation (44), is available under the assumption that the thickness of the diffusion layer <5, is small compared with the body radius r0 (p. 80 in [25]). As in the example of Section 4.1 (see equation (33)), the results of the derivation can be formally written in terms of the diffusion layer thickness, which now is ... 

The uniformflux of oxygen, S, into the fluid along fhe Pt surface and zero flux along the Au surface provide boundary conditions for the convection-diffusion equation. 

The starting point is the convective-diffusion equation suitably modified to account for wall effects and potential field effects (25). 

In what follows, the preceding evaluation procedure is employed in a somewhat different mode, the main objective now being to obtain expressions for the heat or mass transfer coefficient in complex situations on the basis of information available for some simpler asymptotic cases. The order-of-magnitude procedure replaces the convective diffusion equation by an algebraic equation whose coefficients are determined from exact solutions available in simpler limiting cases [13,14]. Various cases involving free convection, forced convection, mixed convection, diffusion with reaction, convective diffusion with reaction, turbulent mass transfer with chemical reaction, and unsteady heat transfer are examined to demonstrate the usefulness of this simple approach. There are, of course, cases, such as the one treated earlier, in which the constants cannot be obtained because exact solutions are not available even for simpler limiting cases. In such cases, the procedure is still useful to correlate experimental data if the constants are determined on the basis of those data. 

In this case, the convective diffusion equation can be written as... 

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