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Rate Constant Distribution Theory

The rate constant distribution (RCD) theory describes a heterogeneous system by assuming a continuum of different adsorption sites, in contrast to the single site presumed in the Thomas Model. The traditional RCD model assumes that each binding site is governed by a Langmuir isotherm model, while in theory any single-component isotherm models can be used.

The RCD model reads:

\[\begin{aligned} \frac{\mathrm{d} q}{\mathrm{d} t} = \int_{0}^{\infty} \int_{0}^{\infty} k_a c(t) (\mathbf{q_{\text{max}}}(k_a, k_d) - \mathbf{q}(k_a, k_d, c, t) ) \mathrm{d} k_a \mathrm{d} k_d - \int_{0}^{\infty} \int_{0}^{\infty} k_d \mathbf{q}(k_a, k_d, c, t) \mathrm{d} k_a \mathrm{d} k_d. \end{aligned}\]

where \(q\) is the total solid phase concentration, \(\mathbf{q_{\text{max}}}(k_a, k_d)\) and \(\mathbf{q}(k_a, k_d, c, t)\) are the binding capacity distribution and the solid phase concentration distribution in the \([k_a^{\text{min}}, k_a^{\text{max}}]\) and \([k_d^{\text{min}}, k_d^{\text{max}}]\) domain, respectively, where \(k_a^{\text{max}}\), \(k_a^{\text{min}}\) and \(k_d^{\text{max}}\), \(k_d^{\text{min}}\) are the maximum and minimum adsorption and desorption rate constants considered.

To solve the above integro-differential equation, the \(k_a\) and \(k_d\) domains can be separately discretized on an equidistant logarithmic grid: \(\ln k_a^{\text{min}} = \ln k_a^{1} < \ln k_a^{2} < ...< \ln k_a^{N_{ka}-1} < \ln k_a^{N_{ka}} = \ln k_a^{\text{max}}\) and \(\ln k_d^{\text{min}} = \ln k_d^{1} < \ln k_d^{2} < ...< \ln k_d^{N_{kd}-1} < \ln k_d^{N_{kd}} = \ln k_d^{\text{max}}\). \(N_{ka}\) and \(N_{kd}\) are the number of nodes considered for the \(k_a\) and \(k_d\) domains, respectively.

The discretized RCD model reads:

\[\begin{aligned} \frac{\mathrm{d} q}{\mathrm{d} t} = \sum_{i=1}^{N_{ka}} \sum_{j=1}^{N_{kd}} k_{a}^i c (q_{\text{max}}^{i, j} - q^{i,j}) - \sum_{i=1}^{N_{ka}} \sum_{j=1}^{N_{kd}} k_{d}^{j} q^{i,j}, \end{aligned}\]

where \(q^{i,j}\) and \(q_{\text{max}}^{i, j}\) are the solid phase concentration and binding capacity for binding site \((i,j)\).

By definition, the total solid phase concentration \(q\) is a collective value of the amount absorbed by each site, rendering

\[\begin{aligned} q = \sum_{i=1}^{N_{ka}} \sum_{j=1}^{N_{kd}} q^{i, j}. \end{aligned}\]

Therefore, we obtain a governing equation for each discretized binding site:

\[\begin{aligned} \frac{\mathrm{d} q^{i,j}}{\mathrm{d} t} = k_{a}^i c (q_{max}^{i, j} - q^{i,j}) - k_{d}^j q^{i,j}. \end{aligned}\]

This equation is essentially a single-component Langmuir isotherm model. Hence, to configure the entire RCD model, one can utilize the Multi Component Bi-Langmuir implementation and the RCD model is thus not implemented as a standalone model.

The associated mass balance equation is given by:

\[\begin{aligned} \frac{\mathrm{d} c}{\mathrm{d} t} = -\frac{1}{\beta} \sum_{i=1}^{N_{ka}} \sum_{j=1}^{N_{kd}} \frac{\mathrm{d} q^{i,j}}{\mathrm{d} t}. \end{aligned}\]

Similar to the Thomas Model, the RCD model can also be configured in different unit operation models like Continuous stirred tank reactor model (CSTR) and Lumped rate model without pores (LRM).

We note that a significant issue with the RCD model is the difficulty in uniquely determining its parameters. Therefore, careful handling of the model is essential.