Transport equation for a membrane based on a frictional model

Abstract We propose transport equations for a membrane bounded by binary solutions based on a frictional model. Several general transport equations in a differential form have been previously proposed, for example, the frictional model, Stefan-Maxwell and Nernst-Plank equations. However, further knowledge, e.g. the relationship between the chemical potential and the concentration of the solution in the membrane, is necessary in order to integrate these equations. Transport equations with an integrated form (finite-difference form) have been developed by Spiegler, Kedem, and Katchalsky on the basis of the frictional model, and these are widely used for the analysis of membrane transport. These equations are obtained on the assumption that the solution in a membrane is in equilibrium with an imaginary free solution. The reflection coefficient for the volume flux is equal to that for the solute flux in these equations. We have improved these equations by eliminating this assumption. These equations show that the reflection coefficient for the volume flux depends on the volume flux. Two reflection coefficients for the solute flux were derived. One is independent of the volume flux and the other is equal to that for the volume flux. The difference between the two reflection coefficients becomes larger with an increase in the volume flux.