Application of projection methods to simulating mass transport in reverse osmosis systems

Reverse Osmosis has important applications to seawater desalination and advanced water treatment. Its efficiency depends, however, on unsteady fluid flow and solute transport that are challenging to simulate. The challenges arise due to interactions between solute boundary layers and unsteady vortical flow structures generated by complicated geometries. These flow structures also interact with semi-permeable membranes through which the permeate flow depends on the local pressure. We show that this additional pressure coupling causes the temporal accuracy of traditional projection methods to drop to first-order. We track the source of this accuracy drop to the treatment of viscous terms in the derivation of the Poisson equation used to update the velocity and pressure fields. This allows us to propose a modified projection method that recovers second order temporal accuracy. Finally, we show that the modified projection method can be coupled to convective outlet conditions and immersed boundary conditions to simulate reverse osmosis in steady and unsteady flow regimes.