Finite planar geometry

This section deals with transport between two parallel electrodes. The reaction at one electrode is the converse of that at the other, so that no overall change occurs to the contents of the cell. We consider only the simplest instance of this behaviour in which the anode is a metal, M, dissolved by the reaction 

there is only one electroactive species its concentration prior to the electrolysis is uniform and equal to c. The cell geometry is shown in Fig. 20. Note that, for the reasons discussed in Sect. 2.1, the positioning of the cathode above the anode will act to prevent natural convection. We therefore ignore convective transport but we will, in most of Sect. 4, be concerned with both migration and diffusion. 

Because we admit migration, we must be concerned with all ionic species present. We consider a total of N ionic species, and utilize subscripts to distinguish them. Thus, we let C2,C3.Cjv be the initial 

This electrode geometry has also been used as the basis of detection in high performance liquid chromatography [70, 71]. 

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Transport by combined migration—diffusion in a finite planar geometry can achieve a true steady state when only two ions are present, as we saw in Sect. 4.2. The same holds true when there are three or more ions present. Under simplifying conditions [see eqn. (89) below], it is possible to predict the steady-state behaviour with arbitrary concentrations of many ions. However, the corresponding transient problem is much more difficult and we shall not attempt to derive the general transient relationship, as we were able to do in deriving eqn. (82) in the two-ion case. 

We are now in a position to examine, in quantitative detail, the effect of the supporting electrolyte, at least for steady-state homovalent transport in finite planar geometry. The amount of supporting electrolyte present, relative to electroactive electrolyte, is characterized by the support ratio that was defined as... 

OTHER EXPERIMENTS WITH FINITE PLANAR GEOMETRY... 

Constant-current experiments, such as that described in the preceding subsection, are not as popular as experiments in which the potential is the controlled variable. In this section, we shall address controlled-potential experiments in cells with finite planar geometry, even though such experiments do not lend themselves readily to kinetic studies. We shall assume, as in previous sections, that the anode reaction is the converse of the cathode reaction. 

The above equations are valid for any experiment in a cell with finite planar geometry. For example, they apply to the experiment described in Sect. 4.7 in fact, eqns. (116) can be derived from eqns. (123) and (124) by setting i(t) equal to the constant i and performing a Laplace inversion. The Laplace inversion is difficult in this derivation and the interested reader is referred to ref. 79 for guidance. 

For reasons already mentioned in Sect. 3.1, cells with finite planar geometry are usually thin cells (i.e. L is small, usually a fraction of a millimetre) and there are only two electrodes. A controlled-potential experiment thus usually involves fixing the potential between the two electrodes, though this does not necessarily mean fixing the potential across either electrode. That is, the way in which the applied potential divides itself between the anode and the cathode will, in general, change with time, even if the total applied potential remains constant. For this reason, the simplification that normally attends experiments carried out at constant applied potential is not achieved with finite planar cells. 

Fig. 13.17 Comparison of advancing front shapes for Newtonian and Power Law fluids (planar geometry). Power Law parameters m — 10,000 Pa s , n — 0.5, yw = 500s-1. [Reprinted by permission from H. Mavridis, A. N. Hrymak, and J. Vlachopoulos, A Finite Element Simulation of Fountain Flow, Polym. Eng. Sci., 26, 449 (1986).]...

The Hele-Shaw equation for the determination of pressure has been derived for a two-dimensional geometry. To solve the pressure problem for a thin eavity of general planar geometry in three-dimensional space, we use a finite-element (FE) representation on the midplane of the cavity (Fig. 8.1). Each element is assigned a thickness. The Hele-Shaw equation is diseretized on eaeh element using the local coordinate system associated with that element. The unknown node pressure and the volumetric flow rate are all scalar quantities and they are not linked to the coordinate system. In addition, we use a finite dififerenee (FD) method to discretize the time- and gap-wise coordinates to solve the energy equation for the temperature field. In the following derivation of the FE/FD equations, only the cavity planar flow is considered. Derivation of the axisymmetrie form of the equations for the runner flow can be done in the same manner. This approach deals with a 2-D pressure field, eoupled to a 3-D temperature field, and therefore it is called a 2.5D simulation. 

Because of the symmetry of the H-PDLC formation process, liquid crystal domains phase separated from the polymer host are axially symmetric ellipsoids, which are usually compressed in the direction of the symmetry axis. It is reasonable to expect that for such oblate ellipsoids the spectrum of director fluctuations is intermediate between the spherical and the planar geometry and to facilitate the analysis, we examine the situation of finite cylindrical cavities of the radius R and length d < R. 

The DigiSim program enables the user to simulate cyclic voltanunetric responses for most of the common electrode geometries (planar, full and hemispherical, and full and hemicylindrical) and modes of diffusion (semiinfinite, finite and hydrodynamic diffusion), with or without inclusion of IR drop and double-layer charging. 

These mathematical representations are complex and it is necessary to use numerical techniques for the solution of the initial-boundary value problems associated with the descriptions of fluidized bed gasification. The numerical model is based on finite difference techniques. A detailed description of this model is presented in (11-14). With this model there is a degree of flexibility in the representation of geometric surfaces and hence the code can be used to model rather arbitrary reactor geometries appropriate to the systems of interest. [The model includes both two-dimensional planar and... 

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