Diffusion pseudo-steady state approximation

A transformation of the dependent variables Cjt, and Cs allowed DelBorghi, Dunn, and Bischoff [9] and E>udukovic [25] to reduce the coupled set of partial differential equations for reactions first-order in the fluid concentration and with constant porosity and diffusivity, into a single partial differential equation. With the pseudo-steady-state approximation, this latter equation is further reduced to an ordinary differential equation of the form considered in Chapter 3 on diffusion and reaction (sk Problem 4.2). An extensive collection of solutions of such equations has been presented by Aris [7]. 

Note that these equations are again based on a pseudo-steady-state approximation such that the deactivation rate must be much slower than the diffusion or chemical reaction rates. These equations can be easily solved, as in Chapter 3, and the result substituted into the definition of the effectiveness factor, with the following results ... 

The reasoning can be used for radial diffusions through concentric or cyhndrical spherical coaxial layers. We ultimately observe that pseudo-steady state approximation is equivalent, in terms of flux, to the steady state condition through a constant thickness layer. 

Higuchi s approximation of the pseudo-steady state is no longer valid when A < Cs. Instead of assuming a linear concentration gradient within the drug-depleted layer, Fick s second law of diffusion is used to calculate the concentration profiles of the dissolved drug as ... 

In the majority of the practical cases and taking into accoimt the precision and the reproducibility of measurements, we can be satisfied, at least after a certain time, of a category of solutiorrs described as pseudo-steady state modes. In the case of diffusions of particles charged under electric field, the approximatiorrs of a null total cmrent, local electric neutrality and of an electric mobility (in general those of the interstitial iorts or vacancies) small compared to the other (in general the one of the electrons or electron holes), i.e. approximations that we used with section 5.5.3, are sufficient. 

Unless otherwise stated, and in particular section 15.2.5 devoted to mixed modes, we will be using the approximation of the pseudo-steady state modes with a rate determining step. Except in certain cases of abnormal diffusions, the rate will be separable and we will be able to devote ourselves to the study of the reactivity, the space functions, of very simple structures in the case of plates, having been determined in Chapter 10. 

For the case of a microdisc equation the following very approximate equation has been derived for the effective number of electrons transferred, eff > as a function of the electrode radius, fe, under steady-state conditions. Ox and Red are assumed to have the same diffusion coefficient D and the homogeneous kinetic step is assumed to show pseudo-first-order kinetics with a rate constant k s = k[Z]. 

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