Determination of concentration-dependent transport coefficients in nanofiltration: Defining an optimal set of coefficients

The currently used equation for solute transport in nanofiltration contains two parameters (ω and 1 - σ) that may be concentration-dependent. A force balance equation allows interpretation of these parameters (as well as L_p) in terms of distribution and friction coefficients, as was demonstrated for a neutral solute and a single 1:1 salt. It is generally assumed in model calculations that it is the distribution coefficient that determines the concentration dependence of ω and 1 - σ. This suggests that a more practically convenient form of the equation may be proposed, in which only one concentration-dependent parameter, ω, appears, while the other is replaced with the ratio of the two, A, which has the meaning of the membrane Peclét number divided by the volume flux and may be assumed to be constant. This may facilitate the analysis of flux-rejection curves and parameter evaluation including concentration dependence, which is a crucial and unavoidable step towards predictive NF modeling. The direct connection between transport parameters and distribution coefficients also suggests that experimentally measured concentration dependence may help to discriminate between different exclusion mechanisms. An approximate analysis based on the connection between A and solute-water friction shows that for presently used NF membranes and realistic fluxes the expected contribution of convection to solute flow cannot become dominant so that the limiting value for salt rejection, R = σ, cannot be reached.