Diffusion, generally constant

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. 

Figure 18.1—Electrochemical measurement string using an ion selective (or specific) electrode (ISEJ. The membrane potential varies with the concentration of the specific ion in solution. Other potentials are fixed by the construction of the electrode. The junction (diffusion) potential Ej has a low value and is generally constant. Measurement is typically conducted with an ionometer. Manufacturers also supply combined electrodes that include both electrodes (external and ion selective) in the same device. Commonly employed pH electrodes are of this type. The schematics of combined electrodes are much less clear due to the proximity of the membrane to other electrode components.

When Fick s law applies, the concentration profile generally contains information about the concentration dependence of the diffusivity. For constant D, step-function initial conditions have the error function (Eq. 4.31) as a solution to dc/dt = Dd2c/dx2. When the diffusivity is a function of concentration,... 

When the catalyst is a porous solid, most of the surface area of the catalyst is the surface area of the inner surface of the pores. Therefore, most of the reaction proceeds in the pore. Gas molecules are transferred to the outer surface of the catalyst by diffusion. Generally speaking, the diffusion is faster than the diffusion inside the pores. Gas molecules collide with the inner wall of the pore before they collide with another molecule for the porous catalyst having an average pore radius rp of a few nm. Such diffusion is called Knudsen diffusion and its diffusion constant D is given by ... 

Two experimental runs were performed. The H2S- and CO2 mole fluxes were obtained from the measured concentration curves by numerical differentiation and are plotted in figure 8a,b together with penetration and film model calculations. It is evident that forced desorption can be realized under practical conditions and can be predicted by the model. In general, measured H2S mole fluxes are between the values predicted by the models, whereas the CO2 forced desorption flux is larger than calculated by the models. The CO2 absorption flux, on the other hand, can correctly be calculated by the models. This probably implies that the rate of the reverse reaction, incorporated in equation (5), is underestimated. Moreover, it should be kept in mind that especially the results of the calculations in the forced desorption range are very sensitive to indirectly obtained parameters (diffusion, equilibrium constants and mass transfer coefficients) and the numerical differentiation technique applied. 

Diffusion-model constants were recalculated using the results from in vitro flow-cell studies with suture punctured, Dacron mesh enclosed, valve-rim inserts. In general, all the model results fall short of predicting the actual serum-gentamicin concentrations determined experimentally over the observed valve implantation period. 

Fickian diffusion with constant effective diffusivities, which is the simplest and most widely used approach. This is quite useful in studying the general characteristics of the problem, but is not valid for realistic simulation of industrial systems except for cases which can be approximated by the assumption of components diffusion in a stationary bulk phase. 

While the Gaussian equations have been widely used for atmospheric diffusion calculations, the lack of ability to include changes in windspeed with height and nonlinear chemical reactions limits the situations in which they may be used. The atmospheric diffusion equation provides a more general approach to atmospheric diffusion calculations than do the Gaussian models, since the Gaussian models have been shown to be special cases of that equation when the windspeed is uniform and the eddy diffusivities are constant. The atmospheric diffusion equation in the absence of chemical reaction is... 

The general equation of one-directional diffusion with constant diffusivity in each layer is ... 

A rapid increase in diffusivity in the saturation region is therefore to be expected, as illustrated in Figure 7 (17). Although the corrected diffusivity (Dq) is, in principle, concentration dependent, the concentration dependence of this quantity is generally much weaker than that of the thermodynamic correction factor d ap d a q). The assumption of a constant corrected diffusivity is therefore an acceptable approximation for many systems. More detailed analysis shows that the corrected diffusivity is closely related to the self-diffusivity or tracer diffusivity, and at low sorbate concentrations these quantities become identical. 

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