A Direct Numerical Method of Lines Approach for Predicting Primary and Secondary Current Density Distributions: Linear and Nonlinear Boundary Conditions

A direct numerical method of lines approach is presented for solving Laplace’s equation to obtain primary and secondary potential and current density distributions in electrochemical cells.  

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.

Acknowledgements

The work presented herein was funded in part by the Clean energy institute at the University of Washington, Seattle.

References

1. Carnahan, B., Luther, H.A., Wilkes, J.O., 1969. Applied Numerical Methods. Wiley, New York.
2. Schiesser, W.E., 1991. The Numerical Method of Lines. Academic Press, New York.
3. Schiesser, W.E., 1994. Computational mathematics in Engineering and Applied Science: ODEs, DAEs and PDEs. CRC Press, Boca Raton.
4. Schiesser, W.E., Silebi, C.A., 1997. Computational Transport Phenomena. Cambridge UniversityPress, New York.
5. Davis, M.E., 1984. Numerical Methods and Modeling for Chemical Engineers. Wiley, New York.
6. Rice, R.G., Do, D.D., 1995. Applied mathematics and modelling for chemical Engineers. John Wiley& Sons Inc., New York.
7. Constantinides, A., Mostoufi, N., 1999. Numerical Methods for Chemical Engineers with Matlab Applications. Prentice-Hall, New Jersey.
8. Cutlip, M.B., Shacham, M., 1998. The numerical method of lines for partial differential equations. CACHE News 47, 18–21.
9. Taylor, R., 1999. Engineering computing with maple: solution of PDEs via method of lines. CACHE News 49, 5–8.
10. Taylor, R., Krishna, R., 1993. Multi Component Mass Transfer. Wiley, New York.
11. Subramanian, V.R., White, R.E., 2000a. Solving differential equations with maple. Chemical Engineering Education 34, 328–336.
12. Subramanian, V.R., White, R.E., 2000b. A semianalytical method for predicting primary and secondary current density distributions: linear and non-linear boundary conditions. Journal of Electrochemical Society 147 (5), 1636–1644.
13. Subramanian, V.R., White, R.E., 2004. Semianalytical method of lines for solving elliptic partial differential equations. Chemical Engineering Science 59, 781–788.