Optimization Model for Counter-Flow Membrane-Based Processes

Optimization models provide a useful framework to identify low cost design and operation of membrane processes. In this work, we present an optimization model for pressure driven membrane-based processes, including, reverse osmosis, pressure retarded osmosis, forward osmosis, and osmotically assisted reverse osmosis. Our model uses a finite difference modeling approach to represent the non-linear and variable boundary conditions associated with solution density, osmotic pressure, concentration polarization, salt permeation, and counter-flow operation. We verify that our finite difference approach has less than 1% error compared to numerical solutions of reverse osmosis. We find that common simplifications in current literature optimization models, such as constant density, ideal solution, no salt flux, and single node approximations, result in average water flux errors of up to 30% for reverse osmosis and up to 80% for higher salinity processes like forward osmosis and osmotically assisted reverse osmosis. Our model is formulated without these simplifications, but remains tractable for complex optimization problems with many decision variables.