## 3 Results and Discussion
### Comparison Model with Real Operating Data
The molar flowrates of the product and purge valve, respectively, are shown in Figure 8. Regarding the adsorption and desorption steps, when the product valve is opened, there is an almost instantaneous exit of a peak flow, mainly consisting of methane accumulated in the column during the pressurization phase. Then, the output flow remains almost constant until the end of the adsorption phase (PS CH\({}_{4}\) and CO\({}_{2}\)). Following the opening of the purge valve during the blowdown, there is a further peak in the flow rate of CH\({}_{4}\) (W CH\({}_{4}\)). As a result of the adsorption phase, there is a free part of CH\({}_{4}\) in the bed because CO\({}_{2}\) is still in the adsorbent. However, there is also a peak in the CO\({}_{2}\) flow rate at the outlet (W CO\({}_{2}\)). Although the molar flowrate of CO\({}_{2}\) is lower than CH\({}_{4}\) at an earlier time, it begins to gradually come out with a lower flowrate in the desorption phase. A zero flowrate is not reached at the end, indicating that the regeneration of the material is not complete. Although it is difficult to notice from the graph, in the purge phase, the flow of CH\({}_{4}\) is certainly not zero. In fact, the small portion of biomethane produced is used to facilitate desorption in the vacuum. Figure 8: Trend of molar flow rates of CO\({}_{2}\) and CH\({}_{4}\) during the cycle time in the product stream (PS CH\({}_{4}\) and CO\({}_{2}\)) and waste stream (W CH\({}_{4}\) and CO\({}_{2}\)) at the last cycle. Figure 7: The pressure in the adsorbent bed as a function of time, dotted symbols represent the real operating condition, while the continuous curve shows the simulated results from Aspen Absorption. The CH4 and CO2 loads adsorbed in the CMS at the end of each step are presented in Figures 9 and 10. These results refer to cycle 94, in which a CSS situation was reached. Figure 9 shows that the amount of CH4 adsorbed along the adsorption bed remains substantially constant through the cycle and slightly high on the top of adsorption bed, as CO2 is first absorbed. Therefore, the proportion of CH4 rises on the top of the adsorption bed and the adsorption amount of CH4 is also increased proportionally. On the other hand, the CO2 load varies significantly during the various phases, as shown in Figure 10. The adsorbed amount of CO2 at the end of the adsorption step is high, but at the blowdown and purge steps, CO2 comes out from the adsorbent due to the regeneration step. This is precisely linked to the kinetic nature of the process. Figure 10: Quantity of CO2 adsorbed along the column at the end of the various phases (cycle 94). Figure 9: Quantity of CH4 adsorbed along the column at the end of the various phases (cycle 94). Figure 11 shows the mole fraction of CH\({}_{4}\) along the three-meter length inside the adsorbent bed in the cyclic steady state. In the PR step, CH\({}_{4}\) has a low adsorption kinetics and is not adsorbed to the adsorbent at the inlet yet. Therefore, the mole fraction equals 0.55, the same as the biogas feed. The curve shows that the mole fraction of CH\({}_{4}\) inside the adsorption bed becomes slightly higher than the biogas along the adsorption bed length. At the AD step, the mole fraction also starts at 0.55, which has the same mole fraction of biogas as CH\({}_{4}\), as the adsorbents reaches the equilibrium at the inlet, subsequently causing CO\({}_{2}\) to be no longer adsorbed. However, CO\({}_{2}\) is adsorbed when moving up to the upper part. Consequently, the CH\({}_{4}\) mole faction rises up to more than 0.92. In the DPE step, only the valve between the two beds is opened, which makes the pressure of Bed1 decrease because the product gas moves from the high pressure bed to the low pressure bed without inlet flows. Then, the mole fraction of the DPE step is shifted up and becomes higher than the mole fraction in the AD step, because the inlet valve is closed and the remaining CO\({}_{2}\) in the adsorption bed is adsorbed without input.