Uniformly accessible surface

The constancy of the diffusion layer over the entire surface and thus the uniform current-density distribution are important features of rotating-disk electrodes. Electrodes of this kind are called electrodes with uniformly accessible surface. It is seen from the quantitative solution of the hydrodynamic problem (Levich, 1944) that for RDE to a first approximation... 

When p = 1 (disc electrode), 5N is independent of r. This shows us immediately that the rotating disc electrode is a uniformly accessible surface, whereas the rotating ring electrode is not. Substitution of eqn. (24) into eqn. (22) and evaluation gives the well-known... 

Both Sq and S have the same value over the entire electrode surface, which has given rise to the description of the electrode as a uniformly accessible surface. The concepts of a hydrodynamic and a diffusion boundary layer have no theoretical significance as such but serve mainly to provide a suitable model for the hydrodynamic conditions related to the rotating electrode. 

TABLE 3 Bulk-Transfer Rate Constant = —/o/ 6 for Various Uniformly Accessible Surfaces... 

For uniformly accessible surfaces, Eq. (132) can be converted to the simple dimensionless form [7,18] suitable for numerical calculations ... 

FIG. 23 The dependence of the reduced hulk transfer rate k a/D (Sherwood number) on the Peclet number calculated numerically for a uniformly accessible surface in the case of no external force, < )q = —1(P curve 1, a/Le = 2 curve 2, a/Le = 5 curve 3, ajLe = 10 curve 4, ajLe = 100. The limiting values calculated from Eq. (156) are plotted by the dashed-dotted line and these calculated from Eq. (165) are depicted by the dashed line. (From Ref. 37.)... 

A different situation arises under the convection-controUed transport condition when the thickness of the diffusion boundary layer remains fixed after a short transition time. Then, for the uniformly accessible surfaces, one can integrate the bulk-transport equation with the nonlinear boundary conditions, Eq. (204). This results in the following expression [12,112,116] ... 

A solution to this problem is to use a rotating cone electrode (RConeE). It turns out that RCone electrodes are similar to RDE in that they act as uniformly accessible surfaces, producing a uniform rate of mass transport to all parts of the electrode surface. For a cone having an opening angle of 0, the current density is given by... 

Recently, many papers have been published on fiber catalysts and foam structures (Figure 9.2). Although, strictly speaking, fibers and foams might not be considered as structured systems, beds of such catalysts exhibit typical features of structured catalysts, namely, low pressure drop, uniform fiow, a good and uniform access to the catalytic surface, and they are definitely nonrandom. Therefore, we have included them in this chapter. 

In this equation the entire exterior surface of the catalyst is assumed to be uniformly accessible. Because equimolar counterdiffusion takes place for stoichiometry of the form of equation 12.4.18, there is no net molar transport normal to the surface. Hence there is no convective transport contribution to equation 12.4.21. Let us now consider two limiting conditions for steady-state operation. First, suppose that the intrinsic reaction as modified by intraparticle diffusion effects is extremely rapid. In this case PA ES will approach zero, and equation 12.4.21 indicates that the observed rate per unit mass of catalyst becomes... 

Unless the electrode is uniformly accessible, the mass transfer coefficient will vary over the electrode surface. This introduces certain complications in the quantitative analysis of experimental results. 

Like eqn. (121), this is only valid for the whole electrode when the surface is uniformly accessible. It is most commonly evaluated graphically by using a nomogram such as that shown in Fig. 12 [145]. 

An interesting way of evaluating rate constants and charge transfer coefficients is the technique of iso-surface concentration voltammetry (ISCVA) [164] where the surface concentration of reactant is held constant over the electrode surface. A uniformly accessible electrode such as the RDE is therefore a prerequisite. At the RDE, the value of Hto1/2 is kept constant and disc potential plotted against current for different ratios of i/co1/2. This yields the kinetic parameters as well as E and the number of electrons transferred. 

We see that, in principle, the overall reaction rate can be expressed in terms of coefficients such as the reaction rate constant and the mass transfer coefficient. To be of any use for design purposes, however, we must have knowledge of these parameters. By measuring the kinetic constant in the absence of mass transfer effects and using correlations to estimate the mass transfer coefficient we are really implying that these estimated parameters are independent of one another. This would only be true if each element of external surface behaved kinetically as all other surface elements. Such conditions are only fulfilled if the surface is uniformly accessible. It is fortuitous, however, that predictions of overall rates based on such assumptions are often within the accuracy of the kinetic information, and for this reason values of k and hD obtained independently are frequently employed for substitution into overall rate expressions. 

Note that at uniformly accessible electrodes, spheres and cylinders, the mass flux is identical at all the points of its surface, whereas at non-uniformly accessible ones, discs and bands, the mass flux varies through the radius and the width, respectively. Therefore, in these cases, the surface gradient should be calculated by integrating the flux over the electrode surface such that the current is given by (see Scheme 2.5) ... 

In this section, microdisc electrodes will be discussed since the disc is the most important geometry for microelectrodes (see Sect. 2.7). Note that discs are not uniformly accessible electrodes so the mass flux is not the same at different points of the electrode surface. For non-reversible processes, the applied potential controls the rate constant but not the surface concentrations, since these are defined by the local balance of electron transfer rates and mass transport rates at each point of the surface. This local balance is characteristic of a particular electrode geometry and will evolve along the voltammetric response. For this reason, it is difficult (if not impossible) to find analytical rigorous expressions for the current analogous to that presented above for spherical electrodes. To deal with this complex situation, different numerical or semi-analytical approaches have been followed [19-25]. The expression most employed for analyzing stationary responses at disc microelectrodes was derived by Oldham [20], and takes the following form when equal diffusion coefficients are assumed ... 

It is worth recalling that, as was indicated in Sect. 2.6, in the cases of non-uniformly accessible electrodes (discs and bands), the current is an average quantity resulting from an average flux over the electrode surface (see for example [17-19]). 

In general, the effects of mass-transport limitations are not as easy to characterize. The direction of fluid flow, the flow regime, and the local fluid velocity all influence the current distribution. Fluid flow to the rotating disk is unusual in that fluid velocity normal to the disk is dependent only on the normal distance from the disk surface, and not on radial distance. Because the disk surface is uniformly accessible to incoming reactants, mass-transport limitations tend to reduce the current density in regions of high... 

Normally, such as at stationary planar electrodes and at uniformly accessible hydrodynamic electrodes, for example the rotating disc, the flux over the electrode surface is constant in this case we have the simple relation... 

As described in Chapter 5, forced convection leads to a thin layer of solution next to the electrode, within which it is assumed that only diffusion occurs (i.e. it is assumed that all concentration gradients occur within this layer)—the diffusion layer of thickness <5. At a particular point on a hydrodynamic electrode and for constant convection, 6 is constant. If the value of <5 is constant over the whole electrode surface then the electrode is uniformly accessible to electroactive species that arrive from bulk solution. 

Fig. 16.3. Schematic streamlines at a wall-jet electrode illustrating the highly non-uniform accessibility of the electrode surface and that the electrode surface does not receive any solution from fluid recirculation (from Ref. [23]).

The aluminum industry consumes about 0.45 lb. of anode carbon for each pound of aluminum produced. The ideal carbon should have a moderately-isotropic structure with minimum oxidant-accessible surface of low, uniform reactivity, and a maximum ash content, excluding bath salts, of a few tenths of one percent. Industrial anode carbon is a baked composite usually made of calcined petroleum coke filler with a binder of coal-tar pitch coke. While there is no shortage of calcined petroleum coke, the quality is not very good and likely to become worse. Also, coke binders are subject to variability in both quality and supply. 

Maximum-performing anode carbon has minimum oxidant-accessible surface of low, uniform oxidation sensitivity to air (02) and CC ... 

Anode porosity is important because it affects the extent of oxidant-accessible surface. This surface is influenced both by coke microstructure and the fabrication process for converting the raw materials into baked carbon. The prime requirement for good anode carbon is minimum oxidant-accessible surface. It is also desirable that this surface have a low, uniform specific reactivity. Anode surface with pores having diameters in the 1-10 micron range are accessible to oxidation unless blocked in some manner. Submicron porosity, such as that produced by thermal desulfurization of coke, is oxidant diffusion-limited and will not affect carbon consumption significantly. Increasing anode carbon density will usually increase anode performance because the oxidant-accessible surface is reduced. 

The rotating disk electrode (RDE) is an important system in electrochemistry. Axial followed by radial flow across the disk brings fresh solution to all points across the disk (Fig. 6). The surface is therefore uniformly accessible to reacting species. The RDE operates under laminar flow for Re < 1.7 X 105. Flow is turbulent above 3.5 X 105 and is transitional in between (4). Thus the system is less practical for the study of corrosion under turbulent conditions but enjoys widespread use in research electrochemistry. For the rotating disk electrode, the laminar mass transport correlation obtained in the literature is given by Levich (10) ... 

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