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# Generalized Ion Exchange¶

The generalized ion exchange model is based on the steric mass action model [11, 12]. 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.