Unsteady Convective Diffusion

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

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... 

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. 

Fo being the Fourier number and d the diameter of the disk. The mass transfer coefficient k can be considered as interpolating between the steady-state convective diffusion at large times (t - oo) and unsteady-state diffusion at short times (t — 0 and v = 0). The constants A and B of Eq. (147) follow from the solutions for these two limiting cases. For these two limiting cases... 

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

Measurements of the rate of deposition of particles, suspended in a moving phase, onto a surface also change dramatically with ionic strength (Marshall and Kitchener, 1966 Hull and Kitchener, 1969 Fitzpatrick and Spiel-man, 1973 Clint et al., 1973). This indicates that repulsive double-layer forces are also of importance to the transport rates of particulate solutes. When the interactions act over distances that are small compared to the diffusion boundary-layer thickness, the rate of transport can be computed (Ruckenstein and Prieve, 1973 Spiel-man and Friedlander, 1974) by lumping the interactions into a boundary condition on the usual convective-diffusion equation. This takes die form of an irreversible, first-order reaction on tlie surface. A similar analysis has also been performed for the case of unsteady deposition from stagnant suspensions (Ruckenstein and Prieve, 1975). 

We can extend the hyperbolic model to cases in which the solute diffuses in more than one phase. A common case is that of a monolith channel in which the flow is laminar and the walls are coated with a washcoat layer into which the solute can diffuse (Fig. 4). The complete model for a non-reacting solute here is described by the convection-diffusion equation for the fluid phase coupled with the unsteady-state diffusion equation in the solid phase with continuity of concentration and flux at the fluid-solid interface. Transverse averaging of such a model gives the following hyperbolic model for the cup-mixing concentration in the fluid phase ... 

A large number of explicit numerical advection algorithms were described and evaluated for the use in atmospheric transport and chemistry models by Rood [162], and Dabdub and Seinfeld [32]. A requirement in air pollution simulations is to calculate the transport of pollutants in a strictly conservative manner. For this purpose, the flux integral method has been a popular procedure for constructing an explicit single step forward in time conservative control volume update of the unsteady multidimensional convection-diffusion equation. The second order moments (SOM) [164, 148], Bott [14, 15], and UTOPIA (Uniformly Third-Order Polynomial Interpolation Algorithm) [112] schemes are all derived based on the flux integral concept. 

Because of the symmetry of the problem, we employ the binary, unsteady, axially symmetric convective diffusion equation with a constant diffusion coefficient ... 

The dispersion model discussed next is that treated by Taylor, where the times are such that axial convection is important, but where radial diffusion can be assumed large in comparison with axial diffusion. For the unsteady concentration term Sc/dt in the axially symmetric convective diffusion equation to be of the same order as D(d ddr ), we have with r a that... 

The functions M (t) were determined from the complete unsteady axially symmetric convective diffusion equation (Eq. 4.6.7), and M (f) were obtained from the Taylor dispersion equation, which was used as the model equation. The phenomenological coefficients U and in the equation were determined by matching the first three moments of the infinite sequence M (t) to M (t) for asymptotically large times [t>a lD). Applying his scheme to the circular capillary problem, Aris showed that D fj, where axial molecular diffusion is not neglected, is given by Eq. (4.6.35). Fried Combarnous (1971) later showed that the satisfaction of the first three moments for t—implies that c x, t), obtained as a solution of the Taylor dispersion equation with = D + Pe /48), is asymptotically the solution of the complete, unsteady, axially symmetric convective diffusion equation averaged over the cross section. 

DIMENSIONLESS FORM OF THE GENERALIZED MASS TRANSFER EQUATION WITH UNSTEADY-STATE CONVECTION, DIFFUSION, AND CHEMICAL REACTION... 

A quantitative strategy is discussed herein to design isothermal packed catalytic tubular reactors. The dimensionless mass transfer equation with unsteady-state convection, diffusion, and multiple chemical reactions represents the fundamental starting point to accomplish this task. Previous analysis of mass transfer rate processes indicates that the dimensionless molar density of component i in the mixture I must satisfy (i.e., see equation 10-11) ... 

The fluxes of mass and energy required in expression 8.7 can be obtained from the equation of continuity per volume unit and energy balance, respectively, for an open unsteady state multicomponent system. Let (a) be a given phase, then the fundamental equation of continuity is given by convective, diffusive and chemical reaction contributions. 

Steady-state heat transfer Unsteady-state heat transfer Convective heat transfer (heat transfer coefficient) Convective heat transfer (heat transfer coefficient) Radiative heat transfer (not analogous with other transfer processes) Steady-state molecular diffusion Unsteady-state molecular diffusion Convective mass transfer (mass transfer coefficients) Equilibrium staged operations (convective mass transfer using departure from equilibrium as a driving force) Mechanical separations (not analogous with other transfer processes) ... 

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