## 3 Results and Discussion
### Comparison Model with Real Operating Data  
Once the validity of the model was verified, it was applied for real PSA process simulations. The base case was developed with a supply of 5000 Nm\({}^{3}\)/h of biogas of agricultural origin (45% vol. CO\({}_{2}\), 55% vol. CH\({}_{4}\)) that reproduced the specifications made by the manufacturer as much as possible. In this case, the purity of the biomethane was acceptable and equals 92%. The reason for this decrease in purity is linked to the feed flowrate, adsorption time and adsorption pressure of the real process. All of these parameters influence the purity and recovery of CH\({}_{4}\). Moreover, the decrease in CH\({}_{4}\) recovery is particularly important considering its high greenhouse gas emission. It is useful at this point to clarify the definitions of purity and recovery of methane adopted in the following equations [10]:  
\[Purity = \frac{\int_{0}^{t_{abs}}c_{CH4}v_{g}\big{|}_{z=H}dt}{\int_{0}^{t_ {abs}}c_{CH4}v_{g}\big{|}_{z=H}dt+\int_{0}^{t_{abs}}c_{CO2}v_{g}\big{|}_{z=H} dt} \tag{9}\] \[Recovery = \frac{\int_{0}^{t_{abs}}c_{CH4}v_{g}\big{|}_{z=H}dt-\int_{0}^{t_{ phys}}c_{CH4}v_{g}\big{|}_{z=H}dt}{\int_{0}^{t_{abs}}c_{CH4}v_{g}\big{|}_{z=H}dt} \tag{10}\]  
Figure 5: Breakthrough curves obtained with Aspen Adsorption compared with the experimental data of [PERSON] et al. [8], (- -) curve of CH\({}_{4}\) obtained with K\({}_{\rm CH4}\) = 0.0008 s\({}^{-1}\) (.....) curve of CH\({}_{4}\) obtained with K\({}_{\rm CO2}\) = 0.027 s\({}^{-1}\), (—) curve obtained with K\({}_{\rm CO2}\) = 0.01 s\({}^{-1}\). In this case, \(t_{ads}\) and \(t_{purgo}\) represent, respectively, the times of the adsorption and purging phases, \(c_{CH4}\) and \(c_{CO2}\) are the concentrations in (mol/m\({}^{3}\)) of methane and CO\({}_{2}\), respectively, \(v_{g}\) is the speed of the gas phase moving in the column (m/s) and \(H\) (m) represents the height of the bed. These are, in fact, integral means of the parameters over time since the flow rates and purities in the various currents are not constant. From the data provided by REBiofuel, it has been observed that, in general, six columns are used to treat 5000 Nm\({}^{3}\)/h of biogas. It was therefore decided to divide the total supply flow rate by six, assuming that the columns work in pairs. The results presented below are therefore intended for a single pair of columns, which is fed a flow rate of 37.17 kmol/h, corresponding to 1/6 of the total. The chosen adsorption pressure of 6.5 bar followed the suggestion from REBiofuel, while the one set for desorption is equal to 0.5 absolute bar due to the steep slope of the carbon dioxide adsorption isotherm, which would not allow a substantial release at higher operating pressures. A temperature of 306 K was set, intermediate between 301 and 311 K, among which the behavior of the mass transfer coefficient of CO\({}_{2}\) was regressed, and it equaled 0.018 s\({}^{-1}\). The dimensions of the column were initially hypothesized to correspond to a scale-up with a ratio of fed biogas flow rate/volume of the column constant with respect to the conditions of the study by [PERSON] et al. [8]. The height to diameter ratio was set to five, in accordance with the work of [PERSON] and [PERSON] [10]. In those studies, they used the condition of purity of at least 98% and, based on this, the estimated bed design from Aspen Adsorption should be have a diameter and height of 0.698 and 3.49 m, respectively, resulting in a very long adsorption time of around 700 s. In the real process, it is required only 92% purity, and REBiofuel indicated that the maximum duration of the cycle is ten minutes, thereby indicating a probable oversizing of the column. For this reason, there was a slight decrease in the volume used for adsorption, while the same height/diameter ratio was maintained.