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To characterize solute transport in nanofiltration (NF) the Spiegler–Kedem equation requires that two coefficients be determined for two-component solutions (a solute in water), solute permeability ω and reflection coefficient σ. For salts both coefficients strongly and in a complex way depend on concentration, which greatly complicates their evaluation from experiments. For this reason, the parameters are usually assumed constant for a given feed and the concentration dependence is assessed from flux–rejection curves for several feeds. This procedure however ignores the fact that the solute concentration and hence the coefficients significantly vary across the membrane. One way to overcome this inconsistency and address concentration dependence is to use physical models explicitly introducing exclusion mechanism(s) and fitting relevant membrane-specific parameters, such as fixed charge or dielectric properties. This procedure often fails to produce unique values of parameters for a given membrane and different salts. In the present study a new phenomenological approach is proposed and critically analyzed, based on the assumption of a similar concentration dependence of ω and 1 − σ, previously shown to be valid under fairly general conditions, thereby the Peclet coefficient A = (1 − σ)/ω may be assumed to be independent of concentration. The coefficients and their concentration dependence for a given solute may be directly and consistently evaluated by fitting flux–rejection data for several feeds and fluxes to numeric solution of the modified transport equations without the need to invoke specific physical models. The values of transport parameters deduced in this way for representative membranes and salts allow important conclusions regarding the transport mechanism. In particular, the roles of different mechanisms in overall salt exclusion could be addressed directly from the variation of ω or 1 − σ with concentration. On the other hand, the value of the Peclet coefficient, free of the effect of salt partitioning, may be analyzed in terms of hindered transport. Using the proposed method, this value was found to be very small for studied thin-film composite membranes, which may significantly simplify the transport equations. |
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