A mathematical model of multicomponent ion transport through a cation-exchange membrane is developed based on the Nernst-Planck equation. A correlation for the non-linear potential gradient is derived from current density relation with fluxes. The boundary conditions are determined with the Donnan equilibrium at the membranesolution interface, taking into account the convective flow. Effective diffusivities are used in the model based on the correlation of tortuosity and ionic diffusivities in free water. The model predicts the effect of an increase in current density on the ion concentrations inside the membrane. The model is fitted to the previously published experimental data. The effect of current density on the observed increase in voltage drop and the decrease in permselectivity has been analyzed using the available qualitative membrane swelling theories. The observed nonlinear behavior of the membrane voltage drop versus current density can be explained by an increase in membrane pore diameter and an increase in the number of active pores. We show how the membrane pore diameter increases and dead-end pores open up when the current density is increased.
A mathematical model based on a generalized Maxwell-Stefan equation has been developed to describe multicomponent ion and water transport inside a cation-exchange membrane. This model has been validated using experimental data and has been used to predict concentration profiles, membrane potential drop, and transport numbers of ions and water for the chlor-alkali process at increased current densities. Several improvements have been made to previously developed Maxwell-Stefan models. In our model, the generalized Maxwell-Stefan equation is written in terms of concentration instead of mole fraction and the fixed group (membrane) concentration is assumed to be constant. We have adapted the Augmented matrix method using the built-in partial differential equation parabolic elliptic (pdepe) solver in Matlab®, and both the concentration and the electrical potential gradients have been solved simultaneously. The boundary conditions are determined with the Donnan equilibrium at the membrane-solution interface. We have also employed semi-empirical correlations to define the Maxwell-Stefan diffusivities inside the membrane. For the bulk diffusivities, we applied the correlations for the concentrated solution instead of the values at infinite dilution. With the diffusivities presented in this work, the model shows a better fit to the experimental data than with previously reported fitted diffusivities. Prediction of the sodium transport number and water transport number is generally good, whereas the deviations with regard to membrane potential might also be related to issues with the experimental data. The model predicts an increase in both sodium and water transport numbers at increased current density operation of chlor-alkali production.
This work describes a model for bilayer cationexchange membranes used in the chlor-alkali process. The ion transport inside the membrane is modeled with the Nernst-Planck equation. A logistic function is used at the boundary between the two layers of the bilayer membrane to describe the change in the properties of each membrane layer. The local convective velocity is calculated inside the membrane using the Schlögl equation and the equation of continuity. The model calculates the ion concentration profiles inside the membrane layers. Modeling results of mono-and bilayer membranes are compared. The changes in membrane voltage drop and sodium selectivity are predicted. The concentration profile of sodium ions in the bilayer membrane is significantly different from the monolayer membrane. Without the applied current, a linear change in the sodium concentration is observed in the monolayer membrane and in each layer of the bilayer membrane. With an increase in current density, the stronger electromotive force in the carboxylate layer causes a decrease in the sodium concentration in the sulfonate layer, down to the fixed ionic group concentration. This significant decrease of sodium ion concentration in the sulfonate layer results in low concentrations of counter ions and as a consequence a higher permselectivity of the bilayer membrane is obtained when compared to the single-layer membrane. As a drawback, the resistance in the bilayer membrane increases.
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