Separation of variables is a classic method of solving partial differential equations, such as the convective diffusion equations that are ubiquitous in electrochemistry. It leads to infinite series solutions, which are exact analytical solutions. However, various parameters involved in the series have to be numerically evaluated, and the series converge rather slowly, which has meant that this method has not been used for practical numerical calculations of concentration profiles for electrochemical problems. We show here that these difficulties can be overcome, and demonstrate the use of the symbolic algebra program Maple in a practical implementation of this for steady-state convective diffusion past electrodes in a 2-D channel [1].

The key to successful implementation has been the use of asymptotic formulas to reliably locate the eigenvalues, so that the numerical solver does not miss any.

The method has the following advantages:

1. It is a mesh-free method, so a separate step of validating the mesh or grid is not necessary.

2. The global error can be estimated by evaluating the series to more terms.

3. Each segment (above an electrode or insulating section) is solved at once, so changing the length of an electrode does not change the difficulty of the calculation.

4. Workup of the concentrations to derived quantities such as currents or collection efficiencies can be done without any degradation of accuracy. 

The present implementation assumes diffusivities of reactant and product are equal, which allows the generation of both concentration profiles in a single calculation by using a recently developed transformation method [2]. It also neglects axial diffusion. Future work will seek to remove these limitations, and extend the results to 3-D and time-dependent systems.

References: 

[1] T. Holm, S. Sunde, F. Seland and D.A. Harrington, A Semianalytical Method for Simulating Mass Transport at Channel Electrodes, J. Electroanal. Chem.,745 (2015) 72-79.

[2] D.A. Harrington, Rules to Transform Concentrations and Currents for Irreversible Reactions to those of Quasireversible Reactions, Electrochim. Acta., 152 (2015) 308-314.

Figure 1. Concentration Profile for a Collection Efficiency Calculation at Electrodes in a Channel. X = distance along channel, Y = distance across channel (dimensionless variables). Electrode between X = 0 and X = 1 producing species at the limiting current, Electrode between X = 2 and X = 3 consuming species at the limiting current.