## 3 Results and Discussion
### Comparison Model with Real Operating Data
A height of 3.00 m and a diameter of 0.60 m were set for the simulated process, and these values are actually the same as those in the real process, as show in Table 4. The velocity of gas can be calculated from the data of the surface (A, in m\({}^{2}\)) and volumetric input flow rate. It is possible to obtain the initial speed of the gas phase along the column, as show in the following equation:  
\[v_{g,in}=\frac{V}{A\cdot\varepsilon_{i}} \tag{11}\]  
where \(A\cdot\varepsilon_{i}\) represents the free section of the column (m\({}^{2}\)) and \(V\) is the volumetric flowrate (m\({}^{3}\)/s). In this way, a gas flow velocity of 0.18 m/s was obtained. Note that values of feed surface speed lower than or close to 0.2 m/s are able to allow a reasonable solid-gas contact and to reduce the effects of axial dispersion [12]. We then need to determine the step time of the pressurization step (PR), the adsorption step (AD), the depressurizing equalization step (DPE), the purging step (PU), the blowdown step (BL) and the pressure equalization step (PPE). For the pressurization step, we can calculate it from the flowrate to increase the pressure until 6.5 bar and, in this case, the pressurization step time is 25 s. Then, in order to decide the time of the adsorption phase, 15 cycles at different (long) adsorption times were processed in advance to observe that the biological methane concentration could be reduced at any point in time. The result is shown in Figure 6. \begin{table}
\begin{tabular}{c c c} \hline \hline
**Parameter Name** & **Value** & **Unit** \\ \hline Bed length & 3.00 & m \\ Bed inner diameter & 0.60 & m \\ Bed outer diameter & 0.608 & m \\ Bed porosity & 0.33 & - \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Bed design used in simulation. It can be observed that the curves obtained at the initial adsorption period have a 92% purity until 300 s, after which the molar fraction of CH4 begins to drop. The effect of the adsorption time on purity and recovery has therefore been investigated with greater precision by varying the duration of the adsorption phase (and consequently, of the regeneration phase) shorter than 300 s. The program was iterated until the cyclic steady state (CSS) was reached, at which time the process performance was stable over time. For this purpose, a relative tolerance has been set for the closure of the material and the energy balances equal 0.0005. It has been noted that CSS is achieved after 94 iteration cycles, as shown in Table 5. This is due to the slowness of the CH4 adsorption process. In fact, it takes a long time for the amount of methane adsorbed inside the CMS to stabilize. Having obtained a good agreement of the simulation and real process, it was decided to adopt an adsorption time of 250 s for a total cycle duration of 560 s (9.3 min). This allows us to obtain a biomethane with a purity slightly higher than 92% molar. Moreover, in this way, the cycle time is similar to the operating conditions of REBiofuel. The simulation of the pressure cycle was compared with the data from the real process, as shown in Figure 7. \begin{table}
\begin{tabular}{c c c} \hline \hline
**Case** & **Simulation** & **Real Conditions** \\ \hline Pressure (bar) & 6.5 & 6.5 \\ PR (s) & 25 & 25 \\ AD (s) & 250 & 250 \\ DPE (s) & 5 & 5 \\ BL (s) & 25 & 25 \\ PU (s) & 250 & 250 \\ PPE (s) & 5 & 5 \\ Recovery of CH4 (\%) & 81.63 & - \\ Purity of CH4 (\%) & 92.08 & 92.10 \\ Number of cycles & 94 & - \\ \hline \hline \end{tabular}
\end{table} 
Table 5: Biomethane purity, recovery and number of cycles necessary to reach the cyclic steady state (CSS) as a function of the adsorption time. Figure 6: Effect of time on the purity of the CH4 output. Result after 15 cycles and by varying the time of the adsorption phase to verify the correctness of the result.