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Generalized Ion Exchange

The generalized ion exchange model is based on the steric mass action model [9, 10]. In addition to the first component \(c_{p,0}\), which represents salt, the second component \(c_{p,1}\) represents another non-binding modifier (e.g., pH). In comparison to the SMA model, the characteristic charge \(\nu\) and the adsorption and desorption rate constants are modified:

\[\begin{split}\begin{aligned} q_0 &= \Lambda - \sum_{j=2}^{N_{\text{comp}} - 1} \nu_j(c_{p,1}) q_j \\ \frac{\partial q_i}{\partial t} &= k_{a,i}(c_{p,0},c_{p,1}) \: c_{p,i} \: \left( \frac{\bar{q}_0 }{q_{\text{ref}}} \right)^{\nu_i(c_{p,1})} - k_{d,i}(c_{p,0},c_{p,1}) \: q_i \: \left( \frac{c_{p,0}}{c_{\text{ref}}} \right)^{\nu_i(c_{p,1})} & &i = 2, \dots, N_{\text{comp}} - 1, \end{aligned}\end{split}\]

where

\[\begin{aligned} \bar{q}_0 &= \Lambda - \sum_{j=2}^{N_{\text{comp}} - 1} \left( \nu_j(c_{p,1}) + \sigma_j \right) q_j = q_0 - \sum_{j=2}^{N_{\text{comp}} - 1} \sigma_j q_j \end{aligned}\]

The dependence of the parameters on \(c_{p,0}\) and \(c_{p,1}\) is given for \(i = 2, \dots, N_{\text{comp}} - 1\) by

\[\begin{split}\begin{aligned} \nu_i(c_{p,1}) &= \nu_{i,\mathrm{base}} + c_{p,1} \nu_{i,\mathrm{lin}} + c_{p,1}^2 \nu_{i,\mathrm{quad}} \\ k_{a,i}\left(c_{p,0}, c_{p,1}\right) &= k_{a,i,\mathrm{base}} \exp\left(k_{a,i,\mathrm{lin}} c_{p,1} + k_{a,i,\mathrm{quad}} c_{p,1}^2 + k_{a,i,\mathrm{salt}} \frac{c_{p,0}}{c_{\text{ref}}} + k_{a,i,\mathrm{prot}} c_{p,i}\right) \\ k_{d,i}\left(c_{p,0}, c_{p,1}\right) &= k_{d,i,\mathrm{base}} \exp\left(k_{d,i,\mathrm{lin}} c_{p,1} + k_{d,i,\mathrm{quad}} c_{p,1}^2 + k_{d,i,\mathrm{salt}} \frac{c_{p,0}}{c_{\text{ref}}} + k_{d,i,\mathrm{prot}} c_{p,i}\right) \end{aligned}\end{split}\]

The concept of reference concentrations (\(c_{\text{ref}}\) and \(q_{\text{ref}}\)) is explained in the respective paragraph in Section Reference concentrations.

For more information on model parameters required to define in CADET file format, see Generalized Ion Exchange.