2.10.1. Governing equations
Some assumptions are introduced to derive the governing equations of the IEC theory (Gu, 2015; Yamamoto et al., 1983a, 1983b). 1. Plug flow with axial dispersion in the column. 2. Total and species material balances for the liquid phase. 3. Isothermal condition throughout the column. 4. The ion components are in dilute concentration, and then the effect of adsorption on the total mass balance is negligible. Thus, superficial velocity and volumetric flow rate remain constant. 5. Ideal mixing happens in the aqueous phase. Further, the overall molar volume is constant according to the dilute concentration of ionic components. 6. Total exchange capacity of the resin inside the bed is constant.7. Kinetic model was used to specify the overall mass transfer of ionic components between the bulk liquid and adsorbed phase. Additionally, the mass transfer should be overcome in mass transfer resistance within the boundary layer surrounding the particle, as well as mass transfer resistance within the resin particle. The second resistance usually controls the overall mass transfer rate. Therefore, a lumped mass transfer rate was employed along with solid-film resistance.
Generally, the model describing the IEX column includes material and momentum transport in the bulk liquid (mobile phase), mass transfer from the bulk liquid to the adsorbent (stationary phase), and mass transfer resistances which influence this transport.
2.10.2. General mass balance equation for IEX
The following equation is introduced for the general mass balance equation for describing the behavior of the exchanged counter-ion in the bulk liquid phase, i.e. each ionic species in the liquid phase enters into the ion-exchange column (Mehay & Gu, 2014).