# Discretizing Multi-species Ion Exchange Equations by Finite Volume Method

I'm solving a system of multispecies ion exchange equations (diffusion+drift fluxes) in 1-d spherical domain using finite volume method to obtain the ion concentrations at the next time step. After polishing the generic equation with so many assumptions, I ended up with this equation for species A $\frac{{\partial {C_A}}}{{\partial t}} = D\frac{{{\partial ^2}{C_A}}}{{\partial {r^2}}} - {\mu _{_A}}{C_A}\frac{{{D_A}\frac{{{\partial ^2}{C_A}}}{{\partial {r^2}}} + {D_B}\frac{{{\partial ^2}{C_B}}}{{\partial {r^2}}} + 2{D_C}\frac{{{\partial ^2}{C_C}}}{{\partial {r^2}}}}}{{{\mu _A}{C_A} + {\mu _B}{C_B} + 4{\mu _D}{C_C}}} + \frac{2}{r}{D_A}\frac{{\partial {C_A}}}{{\partial r}} - \frac{2}{r}{\mu _A}{C_A}\frac{{{D_A}\frac{{\partial {C_A}}}{{\partial r}} + {D_B}\frac{{\partial {C_B}}}{{\partial r}} + 2{D_C}\frac{{\partial {C_C}}}{{\partial r}}}}{{{\mu _A}{C_A} + {\mu _B}{C_B} + 4{\mu _D}{C_C}}}$

D is the diffusivity and mu is the ion mobility and they are all constants. There are two more similar equations for B and C. My question is: how can I discretize the second term in the RHS? should I assume that CA, CB and CD are known at initial time and discretize second order derivatives? knowing that the transient term in LHS can be solved using forward Euler's methods for time first order derivative (n is time index):

$\frac{{\partial {C_A}}}{{\partial t}} = \frac{{{C_A}^{n + 1} - {C_A}^n}}{{\Delta t}}$
• I don't understand your question. There are several options. Do you want a time resolved discretization? Jun 25, 2021 at 20:17
• @covexHull No, I want a space discretization which is resolved by FVM. I've mentioned the time discretization for the sake of transient term.
– Matt
Jun 25, 2021 at 22:04
• I don't think the equation is correct, unless the unit of charge is 1. The prefactor and demoninator don't cancel $q$, while the nominator has the correct unit. Same is true for the 4th term.
– Bort
Jul 5, 2021 at 17:43
• Thanks @Bort for your comment. Yes I will check my derivation again.
– Matt
Jul 6, 2021 at 14:47
• @Bort So the assumption for my model is zi (unitless) instead of qi
– Matt
Jul 6, 2021 at 15:54