Diffusion-convection constants

Equation (8.12) is a form of the convective dijfusion equation. More general forms can be found in any good textbook on transport phenomena, but Equation (8.12) is sufficient for many practical situations. It assumes constant diffusivity and constant density. It is written in cylindrical coordinates since we are primarily concerned with reactors that have circular cross sections, but Section 8.4 gives a rectangular-coordinate version applicable to flow between flat plates. 

Crystal growth rate may be constant, which could happen if temperature is decreasing or if there is convection. Smith et al. (1956) treated the problem of diffusion for constant crystal growth rate. In the interface-fixed reference frame, the diffusion equation in the melt is... 

The Cottrell equation states that the product it,/2 should be a constant K for a diffusion-controlled reaction at a planar electrode. Deviation from this constancy can be caused by a number of situations, including nonplanar diffusion, convection in the cell, slow charging of the electrode during the potential step, and coupled chemical reactions. For each of these cases, the variation of it1/2 when plotted against t is somewhat characteristic. 

TDFRS allows for experiments on a micro- to mesoscopic length scale with short subsecond diffusion time constants, which eliminate almost all convection problems. There is no permanent bleaching of the dye as in related forced Rayleigh scattering experiments with photochromic markers [29, 30] and no chemical modification of the polymer. Furthermore, the perturbations are extremely weak, and the solution stays close to thermal equilibrium. 

Table 1.4 Mass transport coefficients m,, for different experimental conditions. The values of m, correspond to the application of a constant potential. The expressions corresponding to the Rotating Disc Electrode (convective mass transport) under stationary conditions and to Dropping Mercury Electrode with the expanding plane model (diffusive-convective mass transport) have also been included...

The modeling of the electrochemical response corresponding to the application of a constant potential to an RDE is similar to that discussed in the case of a DME since in this electrode it is imperative to consider the convection caused by the rotation of the electrode. This problem was solved by Levich under stationary conditions [76]. To do this, the starting point is the diffusive-convective differential... 

Steady-state solutions for diffusion/convection When the air is moving, it becomes more difficult to calculate the interception rate. The mass transfer under these circumstances is generally expressed in dimensionless terms. Adam and Delbriick (1968) were able to generate a formula by making the simplifying assumption that the velocity of the air as it passed around the hair was everywhere constant (U) and very similar to the ambient air flow farther away (U0) ... 

Diffusion time (diffusion time constant) — This parameter appears in numerous problems of - diffusion, diffusion-migration, or convective diffusion (- diffusion, subentry -> convective diffusion) of an electroactive species inside solution or a solid phase and means a characteristic time interval for the process to approach an equilibrium or a steady state after a perturbation, e.g., a stepwise change of the electrode potential. For onedimensional transport across a uniform layer of thickness L the diffusion time constant, iq, is of the order of L2/D (D, -> diffusion coefficient of the rate-determining species). For spherical diffusion (inside a spherical volume or in the solution to the surface of a spherical electrode) r spherical diffusion). The same expression is valid for hemispherical diffusion in a half-space (occupied by a solution or another conducting medium) to the surface of a disk electrode, R being the disk radius (-> diffusion, subentry -> hemispherical diffusion). For the relaxation of the concentration profile after an electrical perturbation (e.g., a potential step) Tj = L /D LD being - diffusion layer thickness in steady-state conditions. All these expressions can be derived from the qualitative estimate of the thickness of the nonstationary layer... 

The third and fourth condition are fulfilled by Tarhan [25]. Axial dispersion is fundamentally local backmixing of reactants and products in the axial, or longitudinal direction in the small interstices of the packed bed, which is due to molecular diffusion, convection, and turbulence. Axial dispersion has been shown to be negligible in fixed-bed gas reactors. The fourth condition (no radial dispersion) can be met if the flow pattern through the bed already meets the second condition. If the flow velocity in the axial direction is constant through the entire cross section and if the reactor is well insulated (first condition), there can be no radial dispersion to speak of in gas reactors. Thus, the one-dimensional adiabatic reactor model may be actualized without great difficulties. ... 

In Chap. 26, concentration polarization in reverse osmosis was treated using a simple mass-transfer equation, Eq. (26.48), which is satisfactory where the surface concentration is only moderately higher than the bulk concentration. For UF, the large change in concentration near the surface requires integration to get the concentration profile. The basic equation states that the flux of solute due to convection plus diffusion is constant in the boundary layer and equal to the flux of solute in the permeate ... 

Internal one-dimensional transient conduction within infinite plates, infinite circular cylinders, and spheres is the subject of this section. The dimensionless temperature < ) = 0/0/ is a function of three dimensionless parameters (1) dimensionless position C, = xlZF, (2) dimensionless time Fo = otr/i 2, and (3) the Biot number Bi = hiElk, which depends on the convective boundary condition. The characteristic length IF, is the half-thickness L of the plate and the radius a of the cylinder or the sphere. The thermophysical properties k, a, the thermal conductivity and the thermal diffusivity, are constant. 

The rate of convective mass transfer relative to the rate of mass transfer via interpellet axial dispersion is eqnivalent to the ratio of the diffusion time constant relative to the residence time for convective mass transfer. The interpellet Damkohler nnmber for reactant A is... 

Diffusion layer of finite thickness (diffusion + convection). We now use the Nemst hypothesis, which assumes that the concentration of the reacting species that diffuse changes linearly in a layer of thickness 5n and is constant thereafter. 

The ideal models presented here allow us to study the processes at constant values of temperature and concentration in a reactor, i.e., under steady state conditions. Under the real conditions of chemical production, the physical processes of heat and mass transfer (heat conduction, diffusion, convection, turbulence, and so on) play an essential role. 

TwizeU EH (1985) The extrapolation of implicit methods for the constant coefficient diffusion-convection equation. Commun Appl Numer Methods 1 129-135... 

Semi-infinite linear diffusion conditions The rate of an electrochemical process depends not only on electrode kinetics but also on the transport of species to/from the bulk solution. Mass transport can occur by diffusion, convection or migration. Generally, in a spectroeiectrochemicai experiment, conditions are chosen in which migration and convection effects are negligible. The solution of diffusion equations, that is the discovery of an equation for the calculation of oxidized form [O] and reduced form [R] concentrations as functions of distance from electrode and time, requires boundary conditions to be assumed. Usually the electrochemical cell is so large relative to the length of the diffusion path that effects at walls of the cell are not felt at the electrode. For semiinfinite linear diffusion boundary conditions, one can assume that at large distances from the electrode the concentration reaches a constant value. 

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