Trans-membrane pressure in nanofiltration

Most studies to date of Donnan exclusion in membrane separation of mixed solutions of permeating and completely retained salts with shared mobile counter-ions have focused on its effects on negative salt rejection. A theoretical examination is presented here of the effects of Donnan exclusion on flux in general and the threshold trans-membrane pressure for nonzero flux in particular. These effects are expressed through the osmotic pressure of the different solutions in equilibrium with the membrane, which is directly affected by the value of the activity coefficient of the polyelectrolyte counter-ion, ϕp. The osmotic pressure of polyelectrolyte solutions is determined by measuring the threshold trans-membrane pressure for non-zero flux, and from this the activity coefficient is evaluated. The activity of the permeating salt in a mixed solution can then be predicted and used to estimate the partition coefficient, β, of the permeating salt between the mixed and pure salt solution separated by a semipermeable membrane. This value is found to be in reasonable agreement with partition coefficients determined directly in dialysis experiments. Finally, the depression of the threshold trans-membrane pressure for nonzero flux upon adding salt to a polyelectrolyte system is reasonably estimated from values of the permeating salt partition coefficient, β, previously determined in dialysis experiments.