Equivalence between Nernst-Planck and ''corrected'' Fick's law in modeling fixed-bed ion exchange processes

An equivalence between the Nernst-Planck approach, which takes into account the diffusive and electrical contributions for intraparticle mass transfer, and Fick's law, which considers only the diffusive contribution, is developed in this paper. This equivalence leads to the determination of a ''corrected'' diffusivity that enables the simulation of ion-exchange processes with the Fick approach. The development is based on the comparison of the behavior of the processes in an infinite bath, measured by the filling times for the particles; the ''corrected'' Fick diffusivity obtained is tested against the solution with the Nernst-Planck model for column ion-exchange processes. It is able to simulate the processes in all but the cases when favorable equilibrium and kinetics coexist. To cope with this situation, another equivalence is developed based on the variance of the derivative of the breakthrough curves. The same test shows that this last approach is able to simulate all the situations.