## 2 Mathematical Modelling
### Model Equations
\begin{table}
\begin{tabular}{c c c c} \hline \hline  & & MOF-177 & \\ \hline \(q_{sat}\) & \(1.03\times 10^{1}\) & mol/kg & [PERSON] et al. [84] \\ \(b_{o}\) & \(2.51\times 10^{-12}\) & 1/Pa & [PERSON] et al. [84] \\ \(\Delta H_{\text{CO2}}\) & \(-1.4\times 10^{4}\) & J/mol & [PERSON] et al. [64] \\ \hline \multicolumn{4}{c}{MOF-5} \\ \hline \(q_{sat}\) & \(4.8\times 10\) & mol/kg & Eytrope [75] \\ \(b_{o}\) & \(8.06\times 10^{-10}\) & 1/Pa & Eytrope [75] \\ \(\Delta H_{\text{CO2}}\) & \(-3.4\times 10^{4}\) & J/mol & Eytrope [75] \\ \hline \multicolumn{4}{c}{MIL-101} \\ \hline \(q_{\text{sat}}\) (298 K) & \(3.32\times 10^{1}\) & mol/kg & [PERSON] et al. [65] \\ \(b\) (298 K) & \(3\times 10^{-2}\) & 1/bar & [PERSON] et al. [65] \\ \(q_{\text{sat}}\) (373 K) & \(3\times 10^{-5}\) & mol/kg & [PERSON] et al. [65] \\ \(b\) (373 K) & \(7.32\times 10^{2}\) & 1/bar & [PERSON] et al. [65] \\ \(\Delta H_{\text{CO2}}\) & \(-1.63\times 10^{4}\) & J/mol & [PERSON] et al. [85] \\ \hline \hline \end{tabular}
\end{table} 
Table 7: Langmuir parameters for MOF sorbents. For the SI-AEATPMS sorbent, the adsorption stage, considered at 298 K, is described by the Freundlich isotherm model, as in the following relation [74] (see Equation (9)):  
\[q^{*}=1.001\cdot Pco_{2}^{\frac{1}{8.788}} \tag{9}\]  
with \(P_{\text{CO2}}\) being the CO2 partial pressure (bar) and \(q^{*}\) the adsorbed amount in equilibrium (mol/kg). The desorption stage, taking place at 383 K, is, on the other hand, defined (within the literature used) by the Langmuir model as in Equations (7) and (8), with a value for \(q_{sat}\) of 0.295 mol/kg and a value for \(b\) parameter of 16.1 1/bar [74]. The transition to the Langmuir model from the Freundlich one occurs in the temperature range of 343-383 K, because adsorbate-adsorbate interactions are insignificant above this temperature [86]. The Toth model is used to describe the adsorption and desorption stages, respectively, at 298 K and 373 K, for the APDES-NFC-FD sorbent, as defined in the following correlations [65] (see Equations (10)-(13)):  
\[q^{*}=ns\cdot\frac{b\cdot P_{co_{2}}}{\left(1+\left(b\cdot P_{co_{2}}\right)^{ t}\right)^{\frac{1}{t}}} \tag{10}\]  
\[b=b_{o}\cdot e^{\frac{\Delta H_{co_{2}}\rho}{RT\cdot 2b}\cdot\left(\frac{T}{T}-1 \right)} \tag{11}\]  
\[ns=ns_{o}\cdot e^{\chi\cdot\left(1-\frac{T}{TS}\right)} \tag{12}\]  
\[t=t_{o}+\alpha\cdot\left(1-\frac{T}{T}\right) \tag{13}\]  
with \(P_{CO2}\) being the CO2 partial pressure (bar), \(q^{*}\) the adsorbed amount in equilibrium (mol/kg), \(T\) the temperature in K, and \(\Delta H_{CO2,\rho}\) the negative of the isosteric heat of adsorption at zero fractional loading (J/mol), To the reference temperature of 296 K, \(ns\) (mol/kg) and \(b\) (1/bar) the constants in the equation, which are a function of temperature, referring, respectively, to the maximum adsorption capacity and adsorption affinity, while \(t\) characterizes the system heterogeneity; \(t_{o}\), \(b_{o}\) (1/bar), and \(ns_{o}\) (mol/kg) are the Toth parameters at the reference temperature, while \(\alpha\) and \(\chi\) are dimensionless constants. Table 8 shows the value of each parameter [74]. To solve the partial differential equation, initial and boundary conditions are first required. For the ordinary differential equation describing the loading only one initial condition is fixed. These boundary conditions are different for the adsorption and desorption stages. For the adsorption stage, the following initial and boundary conditions are considered (see Equations (14)-(16)):  
\[c_{co_{2}}=q=0\quad t=0 \tag{14}\]  
\[c_{co_{2}}=c_{co_{2},\,intel}\quad\quad z=0,\;t>0 \tag{15}\]  
\[\frac{\partial c_{co_{2}}}{\partial z}=0\quad\quad z=L,\;t>0 \tag{16}\] \begin{table}
\begin{tabular}{c c c} \hline \hline \(b_{0}\) & \(2.25\times 10^{4}\) & 1/bar \\ \(\Delta H_{CO2}\) & \(6\times 10^{4}\) & J/mol \\ \(t_{o}\) & \(4.22\times 10^{-1}\) & \\ \(\alpha\) & \(9.49\times 10^{-1}\) & \\ \(ns_{o}\) & \(1.97\) & mol/kg \\ \(\chi\) & \(2.37\) & \\ \hline \hline \end{tabular}
\end{table}