Primary Current Distribution and Resistance of a Slotted-Electrode Cell

The primary current distribution and the resistance of a cell containing a slotted electrode were calculated using numerical methods coupled with the Schwarz-Christoffel transformation. Results are presented and compared to asymptotic solutions. An approximate analytic expression for the cell resistance is presented. Primary current and potential distributions apply when the surface overpotential can be neglected and the solution adjacent to the electrode can be taken to be an equipotential surface. Calculation of a primary current distribution and resistance represents a first step toward analyzing an d optimizing an electrochemical system. The cell resistance calculated can be coupled with calculations including mass-transfer and kinetic effects to optimize approximately a given cell configuration. The objective of this work is to calculate the primary current distribution and resistance of a cell containing a slotted electrode. Calculation of the primary current and potential distributions involves solution of Laplace's equation, V2q ~ = 0, which is not trivial, even for relatively simple geometries. The method of images (1), separation of variables (2), and superposition (3, 4) have been used to solve Laplace's equation for a number of systems. A review of analytic solutions has been presented by Fleck (5). The Schwarz-Christoffel transformation (6-8) is a powerful tool for the solution of Laplace's equation in systems with planar boundaries. This method was used by Moulton (9) to derive the current distribution for two electrodes placed arbitrarily on the boundary of a rectangle. Hine et al. (10) used this method to describe the primary current distribution for two plane electrodes of infinite length and finite width confined between two infinite insulating planes, perpendicular to but not touching the electrodes. Wagner (11) presented the primary and secondary current distribution for a two-dimensional slot in a planar electrode. Newman (12) has presented the primary current distribution for two plane electrodes opposite each other in the walls of a flow channel. These solutions made use of the Schwarz-Christoffel transformation. Application of the Schwarz-Christoffel transformation is generally limited by the ability to generate solutions to the resulting integrals. Analytic solutions allow calculation of the primary current and potential distribution throughout the cell but are possible for a limited number of system geometries. Numerical evaluation of these integrals allows calculation of both the primary current distribution along the electrodes and the cell resistance. Cell Geometry A cell geometry is presented in Fig. 1 which may be well suited for photoelectrochemical applications. This cell contains a slotted semiconductor with the semiconductor-electrolyte interface open to illumination. A glass cover plate protects the cell. Sunlight passes through the cover plate and the electrolyte to illuminate the semiconductor surface. Electrical current flows from the semiconductor surface to the counterelectrode through the slots of the semiconductor. This configuration has the advantages that no shadows are cast upon the semiconductor, reaction products can be separated, absorption of light by the electrolyte can be minimized, and *Electrochemical Society Student Member. **Electrochemical Society Active Member. 1Present address: Department of Chemical Engineering, University of Virginia, Charlottesville, Virginia 22901. an enhanced-surface-area counterelectrode (perhaps a porous electrode) can be used. The slotted electrode cell can be sectioned and, under the assumption that the cell width W (in the direction perpendicular to the plane of the paper in Fig. lb) is large as compared to the spacing between slots, has the electrochemical characteristics of the two-dimensional cell presented in Fig. 2a. Here, the cell has been turned from Fig. lb so that illumination now comes from the right in Fig. 2a; the semiconductor electrode is represented by AB, the counterelectrode by EF, and all other boundaries of the cell are considered to be insulators. The lines DE and BC and FG represent two planes of symmetry in the cell in Fig. lb; one plane bisects the semiconductor electrode, and the other bisects the slot. The coordinate system of Fig. 2a is transformed through an intermediate half-plane t (see Fig. 2b) to a coordinate system (Fig. 2c) in which Laplace's equation can be solved easily. Theoretical Development The primary current distribution along the electrodes and the cell resistance can be calculated through application of the Schwarz-Christoffel transformation. Complex coordinate systems are used; thus