Modeling reverse osmosis element design using superposition and an analogy to convective heat transfer

Abstract

Accurate models for concentration polarization (CP), the buildup of solutes at the membrane–solution interface in reverse osmosis (RO) channels, are critical for predicting system performance. Despite its empirical success, many modeling approximations employed in the derivation of the often-used stagnant film model seem to limit the model's applicability to real systems. In addition, many existing models for CP use an average mass transfer coefficient with a local mass transfer driving force, which leads to incorrect predictions for the osmotic pressure at the membrane–channel interface. In this work, we reduce the Zydney-transformed governing equations for solute mass transfer to an analogous convective heat transfer problem. We then apply the principle of superposition to fit solutions from the heat transfer problem to the RO channel boundary conditions, yielding a solution that correctly and consistently combines a local transport coefficient with a local mass transfer driving force. The resulting expression for RO element sizing and rating shows good agreement with experimental data and provides a theoretical basis for CP modeling that captures the characteristic growth of the mass transfer boundary layer not accounted for by many existing, more empirical models. The model has important consequences for the design of RO systems with high permeability membranes, as the decrease in membrane resistance in these systems leads to a relative increase in the importance of CP in system performance.