# Theory of ion and water transport in reverse osmosis membranes

## TL;DR

This paper develops a comprehensive theoretical model for ion and water transport in reverse osmosis membranes using a Maxwell-Stefan framework, accounting for various forces, membrane charge, and chemical equilibria, with results aligning well with experimental data.

## Contribution

It introduces a detailed, physically grounded model that incorporates ion-fluid and ion-wall frictions, chemical dissociation, and pH effects to predict transport and rejection in RO membranes.

## Key findings

- Model agrees well with experimental data on fluid flow and rejection.
- Incorporates chemical equilibria and membrane charge effects.
- Predicts effluent composition as a function of flux and pressure.

## Abstract

We present theory for ion and water transport through reverse osmosis membranes based on a Maxwell-Stefan framework combined with hydrodynamic theory for the reduced motion of particles in thin pores. We include all driving forces and frictions both on the fluid (water), and on the ions, including ion-fluid friction as well as ion-wall friction. By including the acid-base character of the carbonic acid system, the boric acid system, H$_3$O$^+$/OH$^-$, and the membrane charge, we locally determine pH and thus the effective charge of the membrane as well as the dissociation degree of boric acid. We present calculation results for a dead end experiment with fixed feed concentration, where effluent composition is a self-consistent function of fluxes through the membrane. Comparison with experimental results from literature for fluid flow vs. pressure, and for salt and boron rejection, shows that theory agrees well with data. Our model is based on realistic assumptions for the effective sizes of the ions and for the diameter of the RO membrane pore in the polyamide toplayer ($\sim$0.75 nm).