Selective Salt Recovery from Reverse Osmosis Brine - University of ...

1.212 Selectivity Modeling
The thermodynamic approach for predicting ion exchange selectivity is based on
the fact that the change in Gibbs energy at equilibrium is equal to zero. Equation 6 shows
the pertinent Gibbs energy calculation for a standard ion exchange reaction [8]. The
terms in the equation are the chemical potentials of the ions in the solution and resin
phases. These can be calculated for the solution phase using Equation 7 and for the resin
phase using Equation 8 [8]. Resin phase partial molar excess free energies in Equation 8
can be calculated using the concentrated electrolyte solution model. Equation 9 describes
the calculation of the excess Gibbs free energy of the whole system which is then used to
calculate the partial molar free energies in a two-component system by Equation 10 and
Equation 11. The thermodynamic model of selectivity described previously and other
similar methods work well as approximations of selectivity of a resin, but they do not
take into account the specific qualities of the resin such as the degree of cross-linking
[12]. Depending on the method used, they may not account well for the low activity of
ions in the solution and resin phases and the effects of different ionic ratios in solution.
Other methods have been developed to deal with these parameters by including the
swelling pressure due to high degrees of cross-linking and the low activity of water in
highly concentrated solutions in the calculations [12, 13]. Marina et al, developed a
model to predict selectivity based on the degree of cross-linking of the resin, the water
activity of the solution, and the concentration ratio of the two counter-ions. He related
these parameters to selectivity using Equation 12 where
with regards to ions B and A,
11
B
K A is the selectivity coefficient
OB
K A is the thermodynamic constant of ion exchange, y and
y are the activity coefficients of the species in the resin and solution phases, respectively,
π is the resin phase swelling pressure, and V is the partial molar volume. The activity
coefficients are functions of water activity as shown by Equation 13 where a w is the
water activity and σ is the total concentration of dissolved species in solution. The water
activity is a function of swelling pressure, as shown by Equation 14 where Vw is the
partial molar volume of water and a w and aw are the water activities in the solution and