Analysis of Ion Transport in Nanofiltration Using Phenomenological Coefficients and Structural Characteristics

Abstract

The analysis of salt transport in nanofiltration using extended Nernst-Planck equations or similar models often suffers from the difficulties to establish and independently and transparently verify the consistency between the filtration results, assumed mechanism, and fitted values of parameters. As a general alternative, we propose here a procedure that reduces filtration data to two general phenomenological coefficients, concentration-dependent salt permeability omega(s) and Peclet coefficient A, which does not require that a specific exclusion mechanism be assumed and thus allows a transparent test on consistency with commonly used models. This approach was demonstrated using concentration polarization-corrected filtration data for NF-200 membrane and four monovalent salts, NaCl, NaBr, KBr, and KCl. The coefficient A was found to be very small, which points to the negligible contribution of convection to salt transport. The smallness of A was verified through estimates of the effective pore radius of the membrane, found to be between 0.2 and 0.3 nm, and comparing them with similar independent estimates from the hydraulic permeability L(p) using the data on the thickness and swelling of the selective polyamide layer obtained by AFM. The concentration dependence of omega(s) and its variation for different salts suggested that in the concentration range above 0.01 M the salt exclusion may be dominated by a combination of Donnan and dielectric mechanisms. The values of omega(s) obtained for single salts were also consistent with the selectivity observed for equimolar feed mixtures of NaCl and NaBr. However, the observed variation of omega(s) with concentrations of single salts below 0.01 M reveals a new regime that is inconsistent with all commonly used models of NF based on a Donnan mechanism modified with dielectric and steric effects. In particular, omega(s) appeared to approach a constant value at low salt concentrations, whereas the standard mechanisms predict a linear or even steeper decrease as concentration decreases. This puzzling discrepancy could have passed unnoticed in the standard multiparameter fitting extended Nernst-Planck equations and demonstrates the benefits of the present phenomenological analysis.

Phenomenological analysis of transport of mono- and divalent ions in nanofiltration

“…By generalizing arguments for monovalent salts presented earlier [33,34,36] it may be shown that for a single n 1 :n 2 salt (the subscripts "1" and "2" denote the counter-and co-ions, respectively) the coefficients A and ω s may be related to frictional, thermodynamic and membrane structure characteristics as follows:…”

“…(7) in Ref. [36] supposed to be analogous to Eq. (2) here, erroneously contained a tortuosity factor˛.…”

“…(7) in Ref. [36], which gives the subsequently used expression for A through the hindered transport theory. In the latter case the tortuosity factor explicitly accounts for the increased velocities and decreased thermodynamic potential gradients in the pores.…”

“…This analysis is based on the SK equation, i.e., essentially, the ENP equations for a single salt. Its utility has been already demonstrated by analyzing NF data for several monovalent salts [36]. Perhaps the most unexpected finding was that the variation of the coefficients with salt concentration showed features incompatible with the regular SDE picture at low salt concentrations pointing to the need for its modification.…”

Aplicaciones estratégicas de la Nanofiltración para el tratamiento de las aguas en la industria cerámica

“…También se ha utilizado el modelo (SC) ("Space-Charge") para predecir la retención de las sales en capilares cargados [100]. También ha sido empleado en la nanofiltración uno de los modelos más extensamente admitido en la ósmosis inversa, el modelo de disolución-difusión (SDM), [101][102] [103].…”