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

Accounting for concentration-polarization (CP is c ical for modeling solute transport in membrane separation processes. In a mixed-electrolyte solution, ion CP is af d not only by diffusion and advection but also by electromigration. Yet, the classic film model, la ing n electromigration term, is frequently used for modeling ion CP. Often, ion CP is altogethe cted to r uce 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 lectromigration film heory to obtain an analytical solution for the CP and membrane-transport of trace- ns in a dominan alt solution. Secondly, we consider limiting conditions relevant to reverse-osmosis and nanofi lation, from wh ch we derive two compact equations that emerge as a seamless extension to the classic m theor These eq ations can be used to account for the effect of electromigration on CP with minimal effort. T rdly, we our theory to quantify the effect of electromigration on ion CP in different dominant salt solutio Finally, by analyzing two environmental membrane processes, we demonstrate how our thory deviates from the onventional one and quantify the implications on membrane scaling potential and the tran ort of ionic contaminants.