Prediction of the concentration polarization in the nanofiltration/reverse osmosis of dilute multi-ionic solutions

The aim of this work was to develop a simple and accurate model for predicting the concentration polarization index in the nanofiltration (NF)/reverse osmosis (RO) of dilute multi-ionic solutions. On the grounds of this model, the total flux of the ion i at the feed-solution/membrane interface consists of the sum of the diffusion, convection and migration fluxes, the former of which is determined by conventional mass-transfer correlations duly corrected to take into account the permeation through the membrane (suction effect). The coupling of the ionic fluxes is enforced by the electroneutrality requirement at the feed-solution/membrane interface. The model developed dispenses with the arbitrary assumption of the thickness of a film layer in the vicinity of the membrane surface.

Assessing the accuracy/validity of this model with multi-ionic solutions would be rather harsh, thus the model accuracy and ranges of validity were ascertained for a benchmark case: NF/RO of single salt solutions. The model predicts approximate concentration polarization indexes of the salts A⁺B⁻, A⁺₂B²⁻ and A⁺₃B³⁻ (or A²B⁻₂ and A³⁺B⁻₃) with positive deviations lower than 10% with respect to the benchmark concentration polarization index, for ions diffusivities ratios, D₁/D₂ (or D₂/D₁), in the range 0.16–5.5 and ≡J_v/k_c<3, where J_v is the permeation flux and k_c is the mass-transfer coefficient of the salt for vanishing mass-transfer rates at impermeable walls. The main assumption of the model – the individual mass-transfer coefficients of the ions are independent of each other – appears to hold in a broad range of conditions, for single salt solutions.

The model developed was expeditely applied to predict the concentration polarization in the nanofiltration of solutions containing Na⁺, Cl⁻ and a dye³⁻ (experimental data of Bowen and Mohammad [AIChE J. 44 (8) (1998) 1799–1812]), and its predictions are in fair agreement with the predictions of the extended Nernst–Planck equations in the film layer of the “slowest” ion.