Theory for salt transport in charged reverse osmosis membranes: Novel analytical equations for desalination performance and experimental validation

Reverse osmosis (RO) is one of the most successful membrane technologies for desalination and contaminant removal from water. RO is applied globally, and can be used for both small- and large-scale applications. To characterize membrane performance, standard testing uses membrane coupons and a NaCl solution in a labscale setup under controlled conditions. Ideally, experiments are done for a range of applied hydrostatic pressures and salt concentrations, with water flux and salt rejection measured in each experiment. This full dataset can then be checked for internal consistency, and all these data must then be described by a comprehensive theoretical framework, i.e., we need an appropriate set of equations to parametrize these data. Parameters derived from this procedure, such as water and salt permeability, can then be compared to those obtained in other studies, for other membranes, salts, or temperatures. If this theory indeed correctly describes data for water flux and salt flux, it can also be applied in larger scale models for RO modules and combinations of modules, which are the basis of engineering design and economic optimization. Herein, we present a novel equation for salt flux that we derive from the full solution-friction (SF) theory. This equation interpolates between an equation for neutral membranes on the one hand, and an equation for highly-charged membranes on the other hand, and thus it is more generally applicable. We apply this new equation to several datasets of seawater RO membranes, and we propose an accurate method to compare the salt permeability of different membranes.