A Direct Numerical Method of Lines Approach for Predicting Primary and Secondary Current Density Distributions: Linear and Nonlinear Boundary Conditions
Steady-state heat transfer in solids is governed by Laplace’s equation and has been solved directly by various numerical methods such as the successive over-relaxation method and the implicit alternative direction method (see e.g., Carnahan et al., 1969). Another method that has been used to solve Laplace’s equation is the method of false transients (Schiesser, 1991, 1994). In this method, one adds a time derivative of the dependent variable to Laplace’s equation, uses finite differences to approximate the spatial derivatives, and then solves the resulting system of equations by the method of lines (Davis, 1984; Schiesser, 1991, 1994; Rice and Do, 1995; Schiesser and Silebi, 1997; Cutlip and Shacham, 1998; Constantinides and Mostoufi, 1999;Taylor, 1999; Subramanian and White, 2000a).
In our earlier paper, we presented a method for solving Laplace’s equation using a semi analytical method of lines. This method consists of using a central difference approximation for the second-order derivative in one of the spatial directions followed by solving analytically the resulting system of second-order differential equations by an analytical method. That is, the system of second-order, two-point boundary value problems are solved analytically by casting them in first-order form.
As of today, solvers have developed to a point where the resulting two point boundary value problems in y (when discretization is used in x) can be directly and efficiently solved numerically. We present and analyze the results of this approach, called Numerical two dimensional method of lines for analyzing current distributions in Hull cell and other geometries of interest for electrochemical engineers. Another key point is that instead of finite difference methods, staggered grid and finite volume methods with variable node spacing in x gives much superior results even compared to state of the art finite element software such as COMSOL.
The work presented herein was funded in part by the Clean energy institute at the University of Washington, Seattle.
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