A Semianalytical Method for Predicting Primary and Secondary Current Density Distributions: Linear and Nonlinear Boundary Conditions

Potential distributions and their associated current density distrib- utions (primary and secondary) are typically obtained by solving Laplace's equation. 1-3 The methods used to solve Laplace's equation include analytic and numerical methods. Analytic methods ( e.g., con- formal mapping4) provide the maximum insight into the problem and usually yield closed-form potential and current density distributions. Unfortunately, analytic techniques are system specific and are often difficult to obtain. Numerical techniques are very general but usually give a numerical value for the potential at a particular location. We present here a semianalytical method (or analytic method of lines), which is for a two-spatial-coordinate problem analytic in y and nume- rical in x; thereby the technique is more general than a particular ana- lytic solution technique and gives better insight than numerical tech- niques for a certain class of problems (Laplace's equation, which has constant coefficients in at least one of the independent variables). It is important to note that the method presented here for solving Laplace's equation in two spatial coordinates with nonlinear boundary conditions does not require iterations for interior node points as is usu- ally the case. 5 The reason for this is that our method does not require interior node points, but instead, only has node points in the bound- aries. The nonlinearities of the boundary conditions are removed by solving for the constants that appear in the solution of Laplace's equa- tion as explained in the following discussion. The technique is an extension of the method presented by De Vidts and White,6 who presented the semianalytical method for solving the one-dimensional, unsteady-state diffusion equation. The semianalyti- cal method presented by De Vidts and White consists of using the method of lines7 to solve the diffusion equation with finite differences used in the spatial direction. (This method was mentioned by Smith et al.,8 but they did not present any results.) The resulting system of linear ordinary differential equations is then solved analytically using the matrix-exponential method.9 This technique is extended here to solve Laplace's equation with two spatial coordinates. The second de- rivative of the potential in the x direction is cast into finite differences accurate to order h2 (h 5D x), and the second derivative in the other direction is replaced by two first-order-derivative equations. The resulting system of ordinary differential equations is solved analyti- cally using the matrix-exponential approach. The method requires de- termining constants of integration in a manner described previously by Subramanian et al.10 The method is illustrated by first solving La- place's equation for a rectangle in which a cathode faces an anode be- tween two insulators. The method employed for this simple case is then extended to other current-distribution problems. Semianalytical Method Consider the rectangle ABCD shown in Fig. 1 of dimensionless length 1 (AB 5 CD 5 1) and dimensionless height b (AC 5 BD 5 b). The dimensionless governing equation for the dimensionless potential inside the rectangle is given by Laplace's equation (1) A semianalytical solution technique is presented for solving Laplace's equation to obtain primary and secondary potential and cur- rent density distributions in electrochemical cells. The potential distribution inside a rectangle with the electrodes facing e ach other between two insulators is presented to illustrate the method. It is shown that the method yields analytic equations for the pot ential and the potential gradient along the lines. The unique attribute of the technique developed is that the solution once obtained is valid for nonlinear boundary conditions also. The procedure is applied to some realistic problems encountered in electrochemical engi - neering to illustrate the utility of the technique developed.