Diffusion layer limitation

Figure 11. Position of the diffusion layer limit (dashed line computed with equation (45) ) for a species diffusing towards a spherical surface (continuous line) when creeping flow coming from the right-hand side is considered. Notice that the ensuing flux will be maximum at positions facing the flow. Parameters r0 — 10 4m, Dt = 10 9m2 s-1, and v — 10-3 ms-1...

Therefore, in tire limiting case—tire surface concentration of tire reacting species is zero as all tire arriving ions immediately react—tire current density becomes voltage independent and depends only on diffusion, specifically, on tire widtli of tire Nerstian diffusion layer S, and of course tire diffusion coefficient and tire bulk concentration of anions (c). The limiting current density (/ ) is tlien given by... 

Fig. 3. Steady state concentration profiles of catalyst and substrate species in the film and diffusion layer for for various cases of redox catalysis at polymer-modified electrodes. Explanation of layers see bottom case (S + E) f film d diffusion layer b bulk solution i, limiting current at the rotating disk electrode other symbols have the same meaning as in Fig. 2 (from ref.

The transient response of DMFC is inherently slower and consequently the performance is worse than that of the hydrogen fuel cell, since the electrochemical oxidation kinetics of methanol are inherently slower due to intermediates formed during methanol oxidation [3]. Since the methanol solution should penetrate a diffusion layer toward the anode catalyst layer for oxidation, it is inevitable for the DMFC to experience the hi mass transport resistance. The carbon dioxide produced as the result of the oxidation reaction of methanol could also partly block the narrow flow path to be more difScult for the methanol to diflhise toward the catalyst. All these resistances and limitations can alter the cell characteristics and the power output when the cell is operated under variable load conditions. Especially when the DMFC stack is considered, the fluid dynamics inside the fuel cell stack is more complicated and so the transient stack performance could be more dependent of the variable load conditions. 

In electrochemical systems with flat electrodes, all fluxes within the diffusion layers are always linear (one-dimensional) and the concentration gradient grad Cj can be written as dCfldx. For electrodes of different shape (e.g., cylindrical), linearity will be retained when thickness 5 is markedly smaller than the radius of surface curvature. When the flux is linear, the flux density under steady-state conditions must be constant along the entire path (throughout the layer of thickness 8). In this the concentration gradient is also constant within the limits of the layer diffusion layer 5 and can be described in terms of finite differences as dcjidx = Ac /8, where for reactants, Acj = Cyj - c j (diffusion from the bulk of the solution toward the electrode s surface), and for reaction products, Acj = Cg j— Cyj (diffusion in the opposite direction). Thus, the equation for the diffusion flux becomes... 

It follows from Eqs. (4.37) and (4.38) that the diffusion-layer thickness will increase without limits and the diffusion flux will decrease to zero when the electrolyte is not stirred (v = 0) or the electrode not rotated (co = 0). This implies that a steady electric cnrrent cannot flow in such cells. But this conclusion is at variance with the experimental data. 

Most successful is a rotating Pt wire microelectrode as illustrated in Fig. 3.75 as a consequence of the rotation, which should be of a constant speed, the steady state is quickly attained and the diffusion layer thickness appreciably reduced, thus raising the limiting current (proportional to the rotation speed to the 1/3 power above 200 rpm140 and 15-20-fold in comparison with a dme) and as a result considerably improving the sensitivity of the amperometric- titration. 

It is assumed that the quantity Cc is not a function of the electrolyte concentration c, and changes only with the charge cr, while Cd depends both on o and on c, according to the diffuse layer theory (see below). The validity of this relationship is a necessary condition for the case where the adsorption of ions in the double layer is purely electrostatic in nature. Experiments have demonstrated that the concept of the electrical double layer without specific adsorption is applicable to a very limited number of systems. Specific adsorption apparently does not occur in LiF, NaF and KF solutions (except at high concentrations, where anomalous phenomena occur). At potentials that are appropriately more negative than Epzc, where adsorption of anions is absent, no specific adsorption occurs for the salts of... 

A rigorous solution of this problem was attempted, for example, in the hard sphere approximation by D. Henderson, L. Blum, and others. Here the discussion will be limited to the classical Gouy-Chapman theory, describing conditions between the bulk of the solution and the outer Helmholtz plane and considering the ions as point charges and the solvent as a structureless dielectric of permittivity e. The inner electrical potential 0(1) of the bulk of the solution will be taken as zero and the potential in the outer Helmholtz plane will be denoted as 02. The space charge in the diffuse layer is given by the Poisson equation... 

The dependence of the limiting current density on the rate of stirring was first established in 1904 by Nernst (N2) and Brunner (Blla). They interpreted this dependence using the stagnant layer concept first proposed by Noyes and Whitney. The thickness of this layer ( Nernst diffusion layer thickness ) was correlated simply with the speed of the stirring impeller or rotated electrode tip. 

Only a few reviews have appeared in which application of the limiting-current method is discussed from a chemical engineering viewpoint. In the review of Tobias et al (T3) mentioned earlier, the authors examined the knowledge available on electrochemical mass transport during the early stages of its application in 1952. Ibl (II) reviewed early work on free convection, to which he and his co-workers contributed notably by development of optical methods for study of the diffusion layer. A discussion of the application of optical techniques for the study of phase boundaries has been given by Muller (M14). 

The diffusivities thus obtained are necessarily effective diffusivities since (1) they reflect a migration contribution that is not always negligible and (2) they contain the effect of variable properties in the diffusion layer that are neglected in the well-known solutions to constant-property equations. It has been shown, however, that the limiting current at a rotating disk in the laminar range is still proportional to the square root of the rotation rate if the variation of physical properties in the diffusion layer is accounted for (D3e, H8). Similar invariant relationships hold for the laminar diffusion layer at a flat plate in forced convection (D4), in which case the mass-transfer rate is proportional to the square root of velocity, and in free convection at a vertical plate (Dl), where it is proportional to the three-fourths power of plate height. 

Because of the likely high ionic concentration and the small dielectric constant of the oxide, the diffuse layer thickness is expected to be small, and hence this space charge is limited to a few nanometers. 

If q is negligibly small a stagnant diffusion layer in aqueous phase is a rate-limiting step. This case is often the most useful from an analytical point of view. The well-known equation for a reversible polarographic wave can be obtained as ... 

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