## 4 Conclusions  
In this work, we simulated the pressure swing adsorption process for the separation of biogas mixed with 55% of CH\({}_{4}\) and 45% of CO\({}_{2}\). The parametric analysis was performed with inputs from a real PSA process using the Aspen Adsorption software. A number of key conclusions are drawn from this study:  
* After determining the model parameter of the adsorbent, the Langmuir isotherm was used to determine the CH\({}_{4}\) and CO\({}_{2}\) isotherm of CMS-3K by using MATLAB(r) software with an r-square of 0.98. The mass transfer coefficients of CH\({}_{4}\) and CO\({}_{2}\) were then determined to be 0.0008 and 0.027 for the porosity of 0.33. With a set of parameters, the simulated concentration of CO\({}_{2}\) and CH\({}_{4}\) is extremely close to the experimental data measured at a temperature of 308 K. However, we found that the mass transfer coefficient changed in relation to variation in temperature. * The kinetic parameters were regressed with MATLAB(r), and by using the Arrhenius equation, the pre-exponential factor and activation energy were determined to be 1.89\(\cdot\)10\({}^{27}\) s\({}^{-1}\) and 1.70\(\cdot\)10\({}^{5}\) J/mol, respectively. * The six-step PSA process was then set, in which the step time was fixed as the same as real process conditions. The adsorption and desorption pressures were 6.5 and 0.5 bar, respectively. The comparison between the simulation and real process gives a good agreement. The change \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Parameters** & **Case 1** & **Optimal Case** & **Case 2** & **Case 3** \\ \hline Pressure (bar) & 5 & 6 & 6.5 & 8 \\ PR (s) & 20 & 23 & 25 & 30 \\ AD (s) & 250 & 250 & 250 & 250 \\ DPE (s) & 5 & 5 & 5 & 5 \\ BL (s) & 20 & 23 & 25 & 30 \\ PU (s) & 250 & 250 & 250 & 250 \\ PPE (s) & 5 & 5 & 5 & 5 \\ Recovery of CH\({}_{4}\) (\%) & 85.32 & 84.11 & 83.45 & 81.17 \\ Purity of CH\({}_{4}\) (\%) & 91.46 & 92.01 & 92.1 & 92.45 \\ Number of cycles & 98 & 95 & 94 & 92 \\ \hline \hline \end{tabular}
\end{table} 
Table 7: Biomethane purity, recovery and number of cycles necessary to reach the CSS as a function of pressure. Figure 14: Effect of pressure on biomethane purity. in CH\({}_{4}\) and CO\({}_{2}\) mole fractions along the adsorption bed length in a steady-state cycle followed opposite directions. * Finally, the key operating parameters such as the flowrate, adsorption pressure and adsorption time were analyzed to find the optimal conditions. As the flowrate decreases, the purity can rise up to around 98%, while it slightly decreases when increasing the biogas flowrate. With the increase in the adsorption pressure, the amount of CO\({}_{2}\) adsorption is increased, as well as the purity of the biomethane. However, this leads to a decrease in the recovery. The purity of the biomethane increased when increasing the adsorption time until 300 s, at which point, the adsorbent performance dropped and the purity decreased. Therefore, the optimal conditions of biogas are a 5000 Nm\({}^{3}\)/h flowrate, an adsorption pressure of 6 bar and an adsorption time of 250 s, which results in the purity complying to the minimum standard of 92% CH\({}_{4}\). It is worth noting that in order to obtain a comprehensive optimization, the energy consumption should be taken into account. This will be a factor of consideration in future work. Conceptualization, B.K. and Q.T.P. ; data curation, B.K. and N.P. ; funding acquisition, T.S. ; investigation, B.K. ; methodology, B.K. ; resources, N.P. ; supervision, T.S. and Q.T.P. ; validation, T.S., N.P. and Q.T.P. ; writing--original draft, B.K. ; writing--review and editing, T.S., N.P. and Q.T.P. All authors have read and agreed to the published version of the manuscript. This research was funded by Center of Excellence on Petrochemical and Materials Technology, Thailand.