Two-Dimensional Formulation on an Orthogonal Grid

A two-dimensional control volume is shown in Fig. 3. The control volume centered at the node point (Jx, jy) is rectangular. A material balance for species i at node point (jx, jy) can be vritten as 

Analogous equations are written for fluxes along the other faces. Equation (8) is assumed to be linear. Commonly, in electrochemical systems, the constitutive relationship may be nonlinear. For example, assuming dilute-solution theory, the flux of species j is given by 

Since the electrical potential is an unknown, the first term on the right-hand side is nonlinear. However, a numerical method to be employed is likely to linearize that term around a guess so that for a given iteration, the flux relationship (in deviation form) is given by Eq. (8). 

To obtain an algebraic approximation to Eq. (7), the for fluxes expressions on each face must be discretized. The optimal interpolation formula used to evaluate the variables and their derivatives depends on the local Peclet number. Nevertheless, the formulas for the east and west faces will have the following forms  

Similar expressions would be utilized along the north and south faces. For a low Peclet number, a linear interpolation is most appropriate. For this case, 

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