## 2 Mathematical Modelling
### Model Equations  
The simulation model is based on a gas phase and adsorbed phase material balance, requiring one partial differential equation and one ordinal differential equation to be taken into account. The Maple tool was used for the solution of this equation system. Overall, the following assumptions are made for the analyzed system: the gas phase follows the idea of gas laws due to the low partial pressure, radial gradients are neglected, the bed is isothermal (due to the low partial pressures and concentrations of the adsorbable component, neglecting heat of adsorption and velocity [80]), constant bed porosity and physical properties, negligible N\({}_{2}\) and H\({}_{2}\)O adsorption compared to CO\({}_{2}\) (due to the low partial pressure and ideal conditions, interactions between molecules are neglected, then the adsorption can be considered as a single component, while we are assuming that water is removed before the adsorption as defined above), and the linear driving force (LDF) model is used for adsorption kinetics. Regeneration of the sorbent is conducted by changing the bed temperature, i.e., a Temperature Swing Adsorption (TSA) process, as the scheme in Figure 1. \begin{table}
\begin{tabular}{c c c} Area footprint excl. options & 2 \(\times\) 10\({}^{1}\) & m\({}^{2}\) \\ Height & 3.2 & m \\ CO\({}_{2}\) flow rate, inlet & 1.8 \(\times\) 10\({}^{2}\) & kg/day \\ YCO\({}_{2}\), inlet & 4 \(\times\) 10\({}^{-4}\) & \\ Air flow rate, inlet & 2.86 & m\({}^{3}\)/s \\ \end{tabular}
\end{table} 
Table 6: Data of one adsorbent bed unit. \begin{table}
\begin{tabular}{c c c c} \hline \hline \multicolumn{4}{c}{SI-AEATPMS} \\ \hline Particle diameter & 2 \(\times\) 10\({}^{-3}\) & m & [PERSON] et al. [77] \\ Particle density & 1.31 \(\times\) 10\({}^{3}\) & kg/m\({}^{3}\) & [PERSON] et al. [77] \\ Bed porosity & 6 \(\times\) 10\({}^{-1}\) & & \\ \hline \multicolumn{4}{c}{APDES-NFC-FD} \\ \hline Particle diameter & 5 \(\times\) 10\({}^{-3}\) & m & [PERSON] [73] \\ Particle density & 5.54 \(\times\) 10\({}^{1}\) & kg/m\({}^{3}\) & [PERSON] [73] \\ Bed porosity & 5 \(\times\) 10\({}^{-1}\) & & \\ \hline \hline \end{tabular}
\end{table} 
Table 5: Main data for amine-functionalized sorbents. The gas phase material balance for CO\({}_{2}\) is described by the 1D axial dispersed plug flow model, as in the following equation [81] (see Equation (1)):  
\[\frac{\partial c_{co_{2}}}{\partial t}=-v_{gas}.\frac{\partial c_{co_{2}}}{ \partial z}+D_{z}.\frac{\partial^{2}c_{co_{2}}}{\partial z^{2}}-\frac{1- \varepsilon}{\varepsilon}\cdot\rho_{p}.\frac{dq}{dt} \tag{1}\]  
where \(D_{z}\) is the axial dispersion coefficient (m\({}^{2}\)/s), \(z\) is the axial coordinate (m), \(v_{gas}\) is the interstitial gas velocity (m/s), \(t\) is time (s), \(\varepsilon\) is the bed void fraction, \(q\) is the loading (mol/kg), \(C_{CO2}\) is the CO\({}_{2}\) gas-phase concentration (mol/m\({}^{3}\)), and \(\rho_{p}\) is the particle density (kg/m\({}^{3}\)). The axial dispersion coefficient is evaluated by the following correlation [82] (Equation (2)):  
\[D_{z}=(0.45+0.55\cdot\varepsilon)\cdot D_{m}+0.35\cdot R_{p}\cdot v_{gas} \tag{2}\]  
with \(D_{m}\) being the molecular diffusivity of CO\({}_{2}\) in air (m\({}^{2}\)/s), \(\varepsilon\) the bed void fraction, \(v_{gas}\) the gas velocity (m/s), and \(R_{p}\) the particle radius (m). The behavior of the adsorbed phase is described by a linear driving force approximation, as in the following equation [83] (Equation (3)):  
\[\frac{dq}{dt}=k_{l}\cdot(q^{*}-q) \tag{3}\]  
where \(k_{l}\) is the LDF mass transfer coefficient or adsorption time constant (s\({}^{-1}\)), \(q^{*}\) is the equilibrium adsorbed amount (mol/kg) while \(q\) and \(t\) are, respectively, the loading (mol/kg) and the time (s) as defined before. According to this approximation, the adsorption rate is proportional to a linear gradient between the adsorbed amount on the sorbent and the amount adsorbed in equilibrium with the bulk concentration [52].