Polarization effects in modelling unsteady-state reverse osmosis

Desalination and ultrafiltration processes have gained a growing interest in recent years and a very rich literature has been developed to model both large scale plants and local convectiondiffusion phenomena governing the single-step units [1-2]. In the latter context, membrane separation technologies by reverse osmosis are a very promising tool to realize separation of ions from aqueous solutions and in recovery of valuable or toxic cations from wastewaters. The most important modelling techniques in describing reverse osmosis rely upon different schemes that, as a first approximation, can be summarized in semi-emipirical mass-transfer models, film theory and more accurate differential-algebraic models accounting for hydrodynamics and interface thermodynamics [3]. Semi-empirical models represent a first and maybe rough attempt where the concentration profile is studied by analogies with heat transfer correlations. Film theory can be considered a satisfactory trade-off between computational simplicity and reliability of results. In fact, this theory allows to treat a complex mass transfer process as a simple one-dimensional problem provided the axial convection close to the membrane surface is assumed negligible. However, the most interesting approach to better foresee the concentration trend near the membrane surface is based on an unsteady-state modelling of the diffusion-convection equation in one or two space variables [4-5]. According to this strategy, a progressive deterioration in separation yield owing to both polarization effects and membrane fouling can be predicted with good accuracy. Unfortunately, this technique is often affected by numerical instabilities related, for instance, to a bad matching between initial and boundary conditions that may severely endanger the applicability of the method. In this paper, we propose a numerical simulation of a single-step separation in a spiral wound module in unsteadystate regime with axial and radial diffusion. The scheme is organized as follows. In section 2, we outline the essentials of the model and we describe the numerical algorithm. In section 3, we present the results of the simulations obtained with different values of the parameters conditioning the yield of the global process. In particular, we estimate the role of the diffusivity in tuning the onset of the polarization regime. In section 3 we draw the conclusion and trace the direction for future works.