New compact expressions for concentration-polarization of trace-ions in pressure-driven membrane processes

Accounting for concentration-polarization (CP) is critical for modeling solute transport in membrane separation processes. In a mixed-electrolyte solution, ions' CP is affected not only by diffusion and advection but also by electromigration. Yet, the classic film model, lacking an electromigration term, is frequently used for modeling ion CP. Often, ion CP is altogether neglected to reduce the computational load. Here, we study the CP of trace ions in a dominant salt solution, a case relevant for many reverse-osmosis and nanofiltration processes. First, we revisit the solution-diffusion-electromigration-film theory to obtain an analytical solution for the CP and membrane-transport of trace-ions in a dominant salt solution. Secondly, we consider limiting conditions relevant to reverse-osmosis and nanofiltration, from which we derive two compact equations that emerge as a seamless extension to the classic film theory. These equations can be used to account for the effect of electromigration on CP with minimal effort. Thirdly, we use our theory to quantify the effect of electromigration on ion CP in different dominant salt solutions. Finally, by analyzing two environmental membrane processes, we demonstrate how our theory deviates from the conventional one and quantify the implications on membrane scaling potential and the transport of ionic contaminants.