Artifact of “Breakthrough” osmosis: comment on the local Spiegler-Kedem-Katchalsky equations with constant coefficients

Abstract

Osmosis—solution (solvent) flow through non-perfectly (perfectly) semipermeable membranes—is a fundamental classical phenomenon of major practical importance. One of its potentially useful technological applications is the Pressure Retarded Osmosis (PRO) employed for energy harvesting from salinity variations1. In this process the flow resulting from the osmotic pressure drop between fresh and saline water is used to drive a turbine. Unfortunately, at the current stage, in spite of its extreme simplicity and conceptual beauty, this process does not appear to be practically viable due to insufficient power efficiency2. This assessment could be radically changed by the recently theoretically predicted “Breakthrough” operation mode of PRO3. In this mode, the solute concentration at the interface between the porous support and the dense selective barrier layer of a non-perfect (‘leaky’) asymmetric membrane employed in PRO decreases with the increase of draw concentration, and, thus, the impeding effect of internal concentration polarization is eliminated, Fig. 1. The existence of this mode was predicted by Yaroshchuk based on the accurate analysis of the system of classical local Spiegler-KedemKatchalsky (SKK) equations of membrane transport with three constant coefficients for the barrier layer: solute permeability (diffusivity), solute reflection coefficient and hydraulic permeability4. In a still more recent study Wu and Field5, contested the physical feasibility of “Breakthrough mode” and casted doubt upon the suitability of SKK equations with constant coefficients to PRO. In this note we re-derive the SKK equations based on a very simple capillary friction model of membrane transport in the dense barrier layer and identify the problem with the constant coefficients’ assumption resulting in the occurrence of “Breakthrough mode”. Our derivation results in recovering the SKK equations in the dilute solution limit, albeit with hydraulic permeability dependent on the local solute concentration in the barrier layer (modified SKK equations, MSKK). Taking into account this dependence, necessary for preserving the detailed force balance in the barrier layer, eliminates the existence of the “Breakthrough mode”.