Mathematical modeling of nanofiltration membranes with mixed electrolyte solutions

A one-dimensional mathematical model is developed and used to describe and predict the performance of commercial nanofiltration (NF) membranes (DDS HC-50, Hydranautics HN-4040-TFV-7450, FilmTec NF-45) for mixed electrolyte solutions. The model is based on three fundamental ideas: the extended Nernst–Planck equation, Donnan equilibrium model, and Gouy–Chapman theory. The membrane is characterized using only three fitting parameters (LP, rP, φ). Two solutions (flat and cylindrical surface) of the Gouy–Chapman equation (GCE) are used and compared. The pure water permeability (LP) is obtained experimentally (independently). The pore radius (rP) and surface electrical potential (φ) are fitted simultaneously to experimental data from mixed electrolyte solutions, and the ratio of membrane thickness/water content (λ/E) is calculated from Darcy’s law. The parameters are independent of operating conditions (solute, pressure, and concentration). Model calculations are in agreement with the experimental data, especially when a cylindrical surface (GCE) is assumed. At very low concentrations the flat surface assumption is unsatisfactory. This mechanistic model has good predicting capabilities. The fitted parameters are compared to independently measured parameters from the literature.