Potential distribution in a parallel-plate

Figure 3. Potential distribution in a parallel plate plasma etcher with grounded surface area larger than powered electrode area, (Reproduced with permission from Ref 12 J...

Diaphragm cells used in chlor-alkali production are also effectively a parallel-plate flow reactor but they are constructed in a very different way they will be discussed in the next chapter. While the potential distribution in a parallel-plate cell is good and the mixing conditions can be made to meet most requirements, the space time yield leaves much to be desired and it is often difficult to reduce the inter-electrode gap sufficiently to give the required space time yield and energy efficiency. These problems have led to the development of many novel cell designs at the present time they remain laboratory or pilot-plant concepts but it is to be expected that some will eventually have an impact on the industrial scene. 

Equation 16 can be easily solved using numerical techniques with the boundary conditions specified in Eqs. 17a, b. A typical potential distribution in a parallel plate microchannel obtained through numerical methods is shown in Fig. 3. 

Equation (48) is identical in form to Equation (13) for a parallel plate capacitor, with pQ replacing A p and k 1 replacing 8. This result shows that a diffuse double layer at low potentials behaves like a parallel plate capacitor in which the separation between the plates is given by k . This explains why k 1 is called the double-layer thickness. It is important to remember, however, that the actual distribution of counterions in the vicinity of a charged wall is diffuse and approaches the unperturbed bulk value only at large distances from the surface. 

FIGURE 18 11 Transient potential and current distributions within a parallel plate reactor in the presence of a double layer (a) Distribution of the potential near the surface of a spherical catalyst pellet at short times (b) Distribution of the transient current density near the surface of a spherical catalyst pellet. 

Fig. 6. Potential distribution in a DC discharge sustained between two closely-spaced parallel plates. After [10].

Fig. 8. Spatiotemporal profiles of the potential distribution in a 13.56 MHz argon discharge sustained between two parallel plates (diode). The left electrode is driven by a sinusoidal voltage of 100 V amplitude. The right electrode is grounded. Note that the plasma potential (potential in the bulk away from walls) is more positive than either electrode potential. After [26].

Numerical Method Considering a symmetric electrolyte solution such as KCl solution in a parallel plate microchannel with a height of 2H, the electrical potential distribution is described by the one-dimensional Poisson-Boltzmann equation ... 

Consider two parallel identical plates coated with a charged polymer brush layer of intact thickness at separation h immersed in a symmetrical electrolyte solution of valence z and bulk concentration n as shown in Fig. 17. la [2]. We assume that dissociated groups of valence Z are uniformly distributed in the intact brush layer at a number density of No- We first obtain the potential distribution in the system when the two brushes are not in contact h > 2d. We take an x-axis perpendicular to the brushes with its origin 0 at the core surface of the left plate so that the region... 

As was shown in Fig. 2.5, the primary current distribution is only uniform when all points on the electrode surface are strictly equivalent and the current density is low. This is possible only with two reactor designs, a parallel-plate reactor having electrodes of equal area and occupying opposite walls and the concentric-cylinder reactor. There will be a variation of potential and current density over the surface for all other electrode arrangements an example is shown in Fig. 2.16(a) where the broken lines join equipotential points and the current densities are inversely proportional to the lengths of the arrowed lines. The highest current density is between the points closest together on the two electrodes and almost no current flows on the reverse side of the anode. 

The simplest model of an electrical double layer is one in which a charge 4-Q is uniformly distributed over a plane of surface area A and is separated by a distance d from a similar plane with a charge -Q. Writing cr = Q/A for the surface charge density, we obtain the potential difference across the layer from the formula for a parallel-plate capacitor... 

We start with the simplest problem of the plate-plate interaction. Consider two parallel plates 1 and 2 in an electrolyte solution, having constant surface potentials i/ oi and J/o2, separated at a distance H between their surfaces (Fig. 14.1). We take an x-axis perpendicular to the plates with its origin 0 at the surface of one plate so that the region 0solution phase. We derive the potential distribution for the region between the plates (0linearized Poisson-Boltzmann equation in the one-dimensional case is... 

---