Generalization of membrane reflection coefficients for nonideal, nonisothermal, multicomponent systems with external forces and viscous flow

Abstract

The notion of reflection coefficients is generalized from dilute ideal solutions to apply to virtually any kind of solution and any kind of membrane whose properties are not affected by the solution. The crucial points in the generalization are the selection of a suitable definition of partial osmotic pressure and the inclusion of separative viscous flow as a transport mechanism (necessary to obtain the correct semipermeability limit). The latter can lead to loss or concealment of Onsager reciprocity, so that the reflection coefficients for volume flow and for solute fluxes are not necessarily equal. Two choices of reference state are presented: the traditional choice of zero reflection coefficient for solvent volume flow, and a more symmetric choice of an average reflection coefficient equal to zero. Several examples are worked out for binary and ternary solutions and compared with results from experiments and from model calculations. Thermal gradients are included for completeness, but play no essential role in the reflection coefficients. The development is given entirely in terms of differential equations of transport, and problems and inconsistencies associated with the use of finite-difference equations are briefly discussed.