## 2 Mathematical Modelling
### Model Equations
The mass transfer rate is evaluated by the film, microporous and microporous resistances of gases through the porous material as expressed by (see Equation (4)) [27]:  
\[\frac{1}{k_{l}}=\frac{R_{p}\cdot q_{o}^{*}}{3\cdot k_{f}\cdot C_{o}}+\frac{R_{ p}^{2}\cdot q_{o}^{*}}{15\cdot\varepsilon_{p}\cdot D_{e}\cdot C_{o}}+\frac{r_{c}^ {2}}{15\cdot D_{c}} \tag{4}\]  
where, \(R_{p}\) is the particle radius (m), \(D_{c}\) is the diffusivity in the micropore (m\({}^{2}\)/s), \(D_{e}\) is the effective diffusivity (m\({}^{2}\)/s), \(k_{f}\) is the film mass-transfer coefficient (m/s), \(r_{c}\) is the microparticle radius (m), \(\varepsilon_{p}\) is the particle porosity, and \(q_{o}\)\({}^{*}\) (mol/m\({}^{3}\)) is the equilibrium adsorbed phase concentration at the inlet concentration \(C_{o}\) (mol/m\({}^{3}\)). The film mass transfer coefficient is evaluated from Sherwood number as follows [81] (see Equations (5) and (6)):  
Figure 1: Operating principle of TSA process. \[Sh=2+1.1\cdot Re^{0.6}\cdot Sc^{0.33} \tag{5}\]  
\[k_{f}=\frac{Sh\cdot D_{m}}{d_{p}} \tag{6}\]  
where _Sh_ is the [PERSON] number, _Re_ is the [PERSON] number, _Sc_ is the [PERSON] number, _Dm_ is the molecular diffusivity (m\({}^{2}\)/s), and _d_\({}_{p}\) is the particle diameter (m). The correlation is verified for a range of Reynold number between 3 and 10\({}^{4}\), with values of dimensionless group in Table S1 of Supporting Information. Using these correlations, the values of _k\({}_{l}\)_ for MOF-177 at 298 K and 373 K are, respectively, 1.49 \(\times\) 10\({}^{-1}\) s\({}^{-1}\) and 1.69 \(\times\) 10\({}^{-1}\) s\({}^{-1}\). For MOF-5 this parameter is 0.019 s\({}^{-1}\) at 298 K and 3.3 \(\times\) 10\({}^{-2}\) s\({}^{-1}\) at 373 K. For MIL-101, the adsorption time constant is of 3.3 \(\times\) 10\({}^{-2}\) s\({}^{-1}\) at 298 K and 7.5 \(\times\) 10\({}^{-2}\) s\({}^{-1}\) at 373 K. The SI-AEATPMS sorbent has a value of _k\({}_{l}\)_ of 4 \(\times\) 10\({}^{-4}\) s\({}^{-1}\) at 298 K and 5 \(\times\) 10\({}^{-4}\) s\({}^{-1}\) at 383 K, while for the APDES-NFC-FD sorbent the same parameter has a value of 3.8 \(\times\) 10\({}^{-3}\) s\({}^{-1}\) at 298 K and 5.7 \(\times\) 10\({}^{-3}\) at 373 K.  
Different adsorption equilibrium isotherms have here been fitted according to the experimental data from the literature at different temperatures. For MOFs sorbents, equilibrium isotherms are described by the single site Langmuir model [65; 75; 84] utilizing the following relations (see Equations (7) and (8)):  
\[q^{*}=\frac{q_{sat}\cdot b\cdot\text{{P}co}_{2}}{1+b\cdot\text{{P}co}_{2}} \tag{7}\]  
\[b=b_{o}\cdot e^{-\frac{\Delta Hco}{kT}} \tag{8}\]  
with \(q_{sat}\) being the maximum adsorbed amount or monolayer capacity (mol/kg), _q\({}^{*}\)_ the adsorbed amount in equilibrium (mol/kg), _P\({}_{CO2}\)_ the CO\({}_{2}\) partial pressure (bar), \(b\) the adsorption constant as a function of temperature (1/bar), \(R\) the universal gas constant (J/molK), _A\(H_{CO2}\)_ the CO\({}_{2}\) adsorption enthalpy (J/mol), \(T\) the temperature (K), and \(b\) the adsorption constant at infinite dilution independent of temperature (1/bar). These above data are reported for each MOF in \[\frac{\partial c_{co_{2}}}{\partial t}=-v_{gas}.\frac{\partial c_{co_{2}}}{ \partial z}+D_{z}.\frac{\partial^{2}c_{co_{2}}}{\partial z^{2}}-\frac{1- \varepsilon}{\varepsilon}\cdot\rho_{p}.\frac{dq}{dt} \tag{1}\]0  
Equations (7) and (8) are used to describe the adsorption and desorption phases, evolving at two different temperatures: the adsorption is carried out at 298 K, while the desorption takes place at 373 K.