Reaction-diffusion equation with electric field

It was observed in the previous section that a certain limit case of non-reactive binary ion-exchange is described by the porous medium equation with m = 2 in other words, a weak shock is to be expected at the boundary of the support. Recall that this shock results from a specific interplay of ion migration in a self-consistent electric field with diffusion. Another source of shocks (weak or even strong in the sense to be elaborated upon below) may be fast reactions of ion binding by the ion-exchanger. 

In this paper we combine the approach of [6], which consists in solving the equations for the electric fields in the anode, cathode and the electrolyte under steady state conditions, with our own approximation of the electrochemical reaction and the transport of reactants. We solve a 2D problem for the Laplace equation coupled with a system of the convection-diffusion equations through use of the boundary conditions. Therefore om problem becomes non-stationary. We study the time period of about one horn and observe the formation of the C02 boundary layer and the variation of the Galvani potential caused by it. 

To increase the electrolyte conductivity, an additional ionic component that does not participate in the electrochemical reactions is often added to a solution. This nom-eactive component is called a supporting electrolyte or indifferent electrolyte. In the presence of a supporting electrolyte, there is a lowering of the electric field in solution, due to the electrolyte s high conductivity. Transport of the minor ionic species in solution is due primarily to diffusion and convection, in accordance with Equation (26.54) with VO = 0. Also, in the presence of a supporting electrolyte, the convective diffusion equation for a minor component in solution is written as... 

Reactions between species, where the interaction energy is large compared with thermal energies, is markedly different from those reactions where no such interaction occurs. The energetics of reaction of encounter pairs, the timescale for approach of reactants, and the relative importance of other factors are all changed. In principle, these modifications to reaction processes enable more information to be obtained about the whole range of factors complicating any analysis of diffusion-limited reaction rates. However, in practice, the more important factors (such as initial distribution of pair separations, hydrodynamic repulsion, and electric field-dependent mobility) are of themselves unable to explain all the differences between experimental results and theoretical predictions. There is a clear need for further work. Finally, it can be remarked that when interactions between reactants are specifically included in an analysis of these rates of reaction in solution, the chosen theoretical techniques has been almost exclusively the Debye—Smoluchowski equation... 

The steady-state flux of hydrated protons in the agglomerate is due to diffusion and migration in the internal electric field. It is dictated by the Nernst-Planck equation, with a sink term, ip, due to electrochemical reactions at the dispersed Pt water interfaces. 

Once the local concentration overpotential is known, the activation overpotential, ria, is obtained by subtracting Tjc from total Tj. The local activation overpotential is the actual driving force of the electrochemical reaction. It is related to the local current density at any point of the reaction zone by an electrochemical rate equation such as the Butler-Volmer equation (Eq. (10a)). Therefore, the rate equation, the Nernst equation (Eq. (37)), and the potential balance in combination couple the electric field with the species diffusion field. In addition, the energy balance applies also at the electrode level. Although this introduces another complication, a model including a temperature profile in the electrode is very useful because heat generation occurs mainly by electrochemical reaction and is localised at the reaction zone, while the... 

This is the simplest model of an electrocatalyst system where the single energy dissipation is caused by the ohmic drop of the electrolyte, with no influence of the charge transfer in the electrochemical reaction. Thus, fast electrochemical reactions occur at current densities that are far from the limiting current density. The partial differential equation governing the potential distribution in the solution can be derived from the Laplace Equation 13.5. This equation also governs the conduction of heat in solids, steady-state diffusion, and electrostatic fields. The electric potential immediately adjacent to the electrocatalyst is modeled as a constant potential surface, and the current density is proportional to its gradient ... 

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