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The preferential removal of sodium sulfate over sodium chloride (fractionation) by nanofiltration (NF) is studied. A fractionation metric, M, is defined, and a simple set of equations are derived to describe its variation due to operating parameters, such as temperature and pressure. The use of these equations to explain known behavior of NF systems and suggest improvements in system or membrane design is demonstrated. Furthermore, the derived framework is applicable to all membranes used in pressure-driven technologies, without detailed characterization of specific membrane properties, such as structural parameters (pore radius, active layer thickness, tortuosity, porosity) or charge-based parameters. Given that the principal applications of NF involve the selective removal of multivalent ions over monovalent ions, the introduced metric provides a direct measure of its efficacy in such applications, and is shown to be more accurate in some cases than comparing salt rejection ratios, as usually done. Finally, the concept of ‘breakthrough’, commonly observed as the point at which rejection ratio begins to decrease with increase in pressure, is mathematically described, and its implication on selectivity discussed.

The present work demonstrates that experimental salt passage data for reverse osmosis (RO) shows better agreement with models capturing solute-solvent transport coupling (convective coupling), especially as the applied pressure is increased. This conclusion is drawn based on five RO data sets using cellulose acetate and polyamide membranes that were modeled using the classical solution diffusion (SD) model, which does not include convection, as well as the convection-inclusive SD model introduced by Paul in 2004 and the pore-based model, which also includes convective coupling. The improved model-to-experimental data agreement with solute-solvent coupling is more easily noticed from salt passage variation with pressure than from the most commonly studied RO metric, salt rejection ratio. The importance of solute-solvent coupling in RO indicates that free volume in the membrane influences the description of membrane transport, and that these voids *open up as applied pressure is increased. The derivation of the pore-based and SD models from the Maxwell-Stefan equations is shown, and major differences in assumptions used in their derivation are discussed. A study of these differences should aid membrane researchers in selecting the most appropriate modeling approach for a given solute-solvent-membrane system.

Nanofiltration performance as a function of feed temperature is relevant to several industrial settings including pretreatment for scale control in thermal desalination. Understanding of solute transport as a function of temperature is critical for effective membrane and system design. In this study, nanofiltration is modeled at 22, 40 and 50 °C using the Donnan Steric Pore Model with dielectric exclusion (DSPM-DE). This modeling includes the temperature dependence of the three modes of solute transport, namely the convective, electromigrative, and diffusive modes, and the three mechanisms of solute exclusion, namely Donnan, steric, and dielectric exclusion. The effect of temperature is captured through the variation of membrane parameters and solvent and ionic mobilities with temperature. We compare the most abundant ionic compound in natural water, sodium-chloride with magnesium-chloride to portray how the salt passage and rejection change for a 1:1 salt compared to a 2:1 salt, and we analyze Arabian Gulf seawater to understand how rejection of scale-forming ions, such as Mg2 + and Ca2 +, is affected by feed temperature. In all cases, solute transport increases with temperature, attributed predominantly to the cumulative effect of membrane parameters and only to a small extent (up to 5%) to the solvent viscosity and ion diffusivity together.

The Donnan Steric Pore Model with dielectric exclusion (DSPM-DE) is implemented over flat-sheet and spiral-wound leaves to develop a comprehensive model for nanofiltration modules. This model allows the user to gain insight into the physics of the nanofiltration process by allowing one to adjust and investigate effects of membrane charge, pore radius, and other membrane characteristics. The study shows how operating conditions such as feed flow rate and pressure affect the recovery ratio and solute rejection across the membrane. A comparison is made between the results for the flat-sheet and spiral-wound configurations. The comparison showed that for the spiral-wound leaf, the maximum values of transmembrane pressure, flux and velocity occur at the feed entrance (near the permeate exit), and the lowest value of these quantities are at the diametrically opposite corner. This is in contrast to the flat-sheet leaf, where all the quantities vary only in the feed flow direction. However it is found that the extent of variation of these quantities along the permeate flow direction in the spiral-wound membrane is negligibly small in most cases. Also, for identical geometries and operating conditions, the flat-sheet and spiral-wound configurations give similar results. Thus the computationally expensive and complex spiral-wound model can be replaced by the flat-sheet model for a variety of purposes. In addition, the model was utilized to predict the performance of a seawater nanofiltration system which has been validated with the data obtained from a large-scale seawater desalination plant, thereby establishing a reliable model for desalination using nanofiltration., Center for Clean Water and Clean Energy at MIT and KFUPM