Reaction layer approximation

At the channel electrode, this problem has been treated by Matsuda, initially employing the reaction-layer approximation [101], but subsequently the full coupled convective-diffusion equations were solved [102] (at the level of the Leveque approximation). It was shown that, to within 1%, the equation... 

The equations governing the steady-state current as a function of tq can be obtained by solving the ordinary differential equations for spherical diffusion governing the appropriate kinetic scheme or by using the reaction layer approximation (7, 91-94). The relevant behavior at microspherical electrodes in any time regime can also be obtained through digital simulation (17). 

Finally we emphasise the approximate nature of Eq. (8.15) and stress that in modern work, numerical simulation is used to determine the behaviour of ECE and other reaction mechanisms at rotating disc electrodes without recourse to the reaction layer approximation. 

If the effect of the electrical double layer is neglected (e.g. at higher indifferent electrolyte concentrations), the rate constant of the cathodic reaction is approximately given by the equation... 

The second term represents a correction for spherical diffusion. This result is approximate and assumes that there is a steady state in the reaction layer. Numerical solution using the expanding-plane model leads to the approximate equation... 

A similar discussion can be made, although only in an approximate way, for disc microelectrodes by assuming as valid the analogy between the disc and sphere radius, i.e., by making the change rs = nr /A in the expressions of the diffusion and reaction layers (see Sect. 3.4.7). 

The GCK approximation includes the reaction layer adjacent to the contact in the reaction sphere and thus magnifies its external radius to the size of R considered as a fitting parameter. Using it instead of a, we obtain 4%RD instead of kn and transform Eq. (3.22) into the following expression ... 

The R obtained from this condition is approximately equal to Rq and follows the same logarithmic dependence on diffusion (3.62). All excitations crossing the reaction layer of width 1/2 adjacent to the black sphere are quenched during their residence time there, l2/AD. However, most of those that were initially inside this sphere are quenched where they were at the moment of excitation. These excitations disappear first, during static quenching, which is often considered as instantaneous compared to subsequent stages limited by diffusion, which delivers the excitations from outside into the black sphere. 

In the contact approximation (CA) the electron transfer proceeds in the thin reaction layer adjacent to the nontransparent sphere of radius a [Fig. 3.22(a)], However, the ions do not necessarily start from there as in EM. When their initial separation exceeds a, the ions do not recombine until they are delivered to the contact by encounter diffusion. Their distribution obeys the diffusional equation similar to (3.19) but for ions in the Coulomb well ... 

Figure 3.25. The dependence of the quantum yield cp(ro) on the contact quantum yield

To proceed further with the analytic investigation, we have to confine ourself to the contact recombination, which is a reasonable approximation for the remote start and narrow reaction layer adjacent to the contact. In this approximation the reaction term in Eqs. (3.568) can be omitted... 

A general mathematical formulation and a detailed analysis of the dynamic behavior of this mass-transport induced N-NDR oscillations were given by Koper and Sluyters [8, 65]. The concentration of the electroactive species at the electrode decreases owing to the electron-transfer reaction and increases due to diffusion. For the mathematical description of diffusion, Koper and Sluyters [65] invoke a linear diffusion layer approximation, that is, it is assumed that there is a diffusion layer of constant thickness, and the concentration profile across the diffusion layer adjusts instantaneously to a linear profile. Thus, they arrive at the following dimensionless set of equations for the double layer potential, [Pg.117]

THE BOUNDARY-LAYER APPROXIMATION FOR LAMINAR FLOWS WITH CHEMICAL REACTIONS... 

Near the point where the two streams first meet the chemical reaction rate is small and a self-similar frozen-flow solution for Yp applies. This frozen solution has been used as the first term in a series expansion [62] or as the first approximation in an iterative approach [64]. An integral method also has been developed [62], in which ordinary differential equations are solved for the streamwise evolution of parameters that characterize profile shapes. The problem also is well suited for application of activation-energy asymptotics, as may be seen by analogy with [65]. The boundary-layer approximation fails in the downstream region of flame spreading unless the burning velocity is small compared with u it may also fail near the point where the temperature bulge develops because of the rapid onset of heat release there,... 

FORTRAN computer program that predicts the species, temperature, and velocity profiles in two-dimensional (planar or axisymmetric) channels. The model uses the boundary layer approximations for the fluid flow equations, coupled to gas-phase and surface species continuity equations. The program runs in conjunction with CHEMKIN preprocessors (CHEMKIN, SURFACE CHEMKIN, and TRAN-FIT) for the gas-phase and surface chemical reaction mechanisms and transport properties. The finite difference representation of the defining equations forms a set of differential algebraic equations which are solved using the computer program DASSL (dassal.f, L. R. Petzold, Sandia National Laboratories Report, SAND 82-8637, 1982). 

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