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Multi Component Sips

The Sips binding model is a combination of the Freundlich and the Langmuir adsorption model.

\[\begin{aligned} \frac{\mathrm{d} q_i}{\mathrm{d} t} = k_{a,i}\: \left( \frac{c_{p,i}}{ c_{\text{ref}} }\right)^{1 / n_i}\: q_{\text{max},i} \left( 1 - \sum_{j=0}^{N_{\text{comp}} - 1} \frac{q_j}{q_{\text{max},j}} \right) - k_{d,i} \left( \frac{q_i}{q_{\text{ref}}} \right) && i = 0, \dots, N_{\text{comp}} - 1. \end{aligned}\]

Here, \(c_{\text{ref}}\) is a reference concentration, \(n_i\) is the Freundlich exponent, \(k_{a,i}, k_{d,i}\) are the adsorption and desorption rates, and \(q_{\text{max},j}\) is the adsorption capacity.

As for the Freundlich isotherm, the first order Jacobian \(\left(\frac{dq^*}{dc_p}\right)\) tends to infinity as \(c_{p} \rightarrow 0\) for \(n>1\). Additionally, the isotherm is undefined for \(c_{p} < 0\) if \(\frac{1}{n_i}\) can be expressed as \(\frac{p}{q}\) with \(p,q \in \mathbb{N}\) where \(q\) is an even number. Negative concentrations can arise during simulations due to numerical fluctuations. To address these issues an approximation of the isotherm is considered below a threshold concentration \(c_p < \varepsilon\). This approximation matches the isotherm in such a way that \(q=0\) at \(c_p=0\) and also matches the value and the first derivative of the istotherm at \(c_p = \varepsilon\), where \(\varepsilon\) is a very small number, for example \(1e-10\). The form of approximation and its derivative is given below for \(c_p < \varepsilon\):

\[\begin{split}\begin{aligned} c_{p,i,lin} &= \left(\frac{\varepsilon}{c_{p,\text{ref}}}\right)^{\frac{1}{n_i} - 2} \frac{c_{p,i}}{{c_{p,\text{ref}}}^2} \left( \left(2-\frac{1}{n_i}\right)\varepsilon + c_{p,i}\left(\frac{1}{n}-1\right) \right) \\ \frac{\mathrm{d}q_i}{\mathrm{d}t} &= k_{a,i} c_{p,i,lin} q_{\text{max},i} \left( 1 - \sum_j \frac{q_j}{q_{\text{max},j}} \right) - k_{d,i} \frac{q_i}{q_{i,\text{ref}}} \end{aligned}\end{split}\]

For more information on model parameters required to define in CADET file format, see Multi Component Sips. For more information on the model and its origin, please refer to [21].