## 3 Results and Discussion
### Validation of the Model  
To verify the correctness of the assumptions explained in Section 3.1, it is useful to simulate a simple breakthrough curve of a CO\({}_{2}\)/CH\({}_{4}\) mixture with 45% vol/55% vol. proportions, in order to reproduce the data obtained from an experiment by [PERSON] et al. involving a fixed bed track [8]. An inlet pressure of 3.2 bar and a temperature of 303 K have been set, with a total flow rate equal to 1 standard liter per minute (SLPM). The same dimensions with [PERSON]'s study were maintained for the adsorbent bed, i.e., a height of 0.83 m and a diameter of 0.021 m.  
In order to function, Aspen Adsorption must initialize the gas content inside the adsorbent bed. Since it was not possible to set the column to be empty at the initial conditions, it was decided to assume that the bed was completely filled with N\({}_{2}\), requiring that it could not be adsorbed (isotherm parameters = 0) and did not offer any resistance to diffusion (K\({}_{\text{N2}}\) = 0). In this way, a curve quite similar to the experimental data was obtained in Figure 5, but with an excessive slope regarding the CO\({}_{2}\) breakthrough curve. This is related to the fact that, while the kinetic parameters of methane and carbon dioxide have regressed to 308 K, the experimental temperatures were obtained at 303 K. This is because of the fact that the slope of the curve is not only changed due to the adsorption limit of adsorbent but also the temperature increase during the adsorption step, resulting in the non-symmetric S-shape curve due to a faster diffusion. However, the isothermal operation is presented in this study, so the thermal effects are lumped into the mass transfer coefficient, which is helpful in obtaining a faster solution. For this reason, manually decreasing the kinetic parameter of CO\({}_{2}\) allowed us to obtain K  
Figure 4: CH\({}_{4}\) and CO\({}_{2}\) concentrations as a function of time.  
= 0.01 s\({}^{-1}\), which was a better fitting of the curve. The results of the two simulations, first keeping the K\({}_{\rm CO2}\) regressed to 308 K and then correcting it to 303 K, are shown in Figure 5 for (K0.01) and (K0.027).  
As can be seen from the graphs, the starting point of the experimental data of CO\({}_{2}\) is linked to the adsorption isotherm and is therefore not changed. Until 1250 s, the slope of the curve is changed due to the adsorption limit of the adsorbent and is related to the influence of the mass transfer coefficient. Thus, once the value of the kinetic parameter of CO\({}_{2}\) was corrected at a temperature of 303 K, we proceeded with a fitting with MATLAB(r) to derive the temperature coefficient of K\({}_{\rm CO2}\) in an exponential form of the Arrhenius equation:  
\[K_{\rm CO2}=\mathrm{k}_{0}e^{(-\frac{E}{RT})} \tag{8}\]  
where R, k\({}_{0}\), \(E\) are the ideal gas constant, pre-exponential factor, and the activation energy, respectively, and T is temperature. In this way, we obtained k\({}_{0}\) = 1.89\(\cdot\)10\({}^{27}\) s\({}^{-1}\), \(E\) = 1.70 \(\times\)10\({}^{5}\) J/mol. This dependence of the kinetic coefficient of CO\({}_{2}\) on temperature has been included in Aspen Adsorption and used for subsequent simulations.