## 2 PSA Plant Model and Simulation  
The PSA system comprises two columns interconnected by ten proportional valves (see Figure 1). The size and nominal conditions for running this plant under nominal conditions are described in Table 1.  
The PSA plant consists of two parallel columns packed with 5A zeolite. Ten proportional valves are utilized to facilitate gas transfer between columns. Figure 1 illustrates the nominal initial parameters. The PSA process involves a feed of the synthesized gas containing the following composition: H\({}_{2}\) (69%), CH\({}_{4}\) (28%) and CO (11%). The process consists of four steps: adsorption, depressurization, repressurization, and purge. Each stage has its own time, which makes this process cyclical until reaching the production objective. The first step consists of feeding Bed 1 with the mixture of gases, where an adsorption step takes place for 70 s to initiate the hydrogen production. Simultaneously, Bed 2 passes through the purge step to realese the absorbent in the column. Figure 1: Flow diagram of the PSA plant. Following that, both beds undergo the depressurization and repressurization steps for 3 s. In this step, the pressure drops or rises as follows; depressurization from 980 kPa to 150 kPa, and repressurization from 150 kPa to 980 kPa. Then, the purge step occurred in the Bed 1 for 70 s, and the Bed 2 undergoes the adsorption process for 70 s, absorbing and retaining the compounds to purify hydrogen. The purge step carried out during the cycle performs the operation of depressurizing the beds. Finally, repressurization and depressurization were carried out again in both beds. In conclusion, the four steps are repeated until the operation cycle is complete. This process achieves a biohydrogen purity of 0.9485 in molar fraction. Table 2 summarizes the times necessary for each step and pressure. The type 5A zeolite exhibits exceptional resistance to high temperatures and pressure fluctuations, experiencing minimal degradation compared to other adsorbents. Its adsorption capacities per gram of adsorbent are as follows: CH\({}_{4}\): 0.030, CO: 0.049, and N\({}_{2}\): 0.238, allowing the production of high-purity hydrogen. The model process used in this study is based on insights from previous work [39; 40; 41]. The plant model incorporates partial differential equations (PDEs) governing mass and energy balance in the gas phase, differences in pressure across the system, and kinetic models for the solid adsorbent. These PDEs are coupled with the Langmuir isotherm equations to represent the adsorption dynamics accurately. The following concerns were taken into account:  
1. The gas within the beds adheres to the ideal gas law. 2. Variations in pressure, concentration, and temperature along the radial axis are neglected. 3. The pressure drop along the axial direction is considered negligible. 4. The mass transfer rate is calculated using the Linear Driving Force (LDF) model. \begin{table}
\begin{tabular}{l l l} \hline \hline  & \multicolumn{2}{c}{**Main Flows**} \\ \hline  & Feed & Product \\ \hline Flow (kmol/h) & 0.162 & 0.162 \\ \hline Temperature (\({}^{\circ}\)K) & 298.15 & 298.15 \\ \hline Pressure (kPa) & 100 & 980 \\ \hline H\({}_{2}\) molar fraction & 0.69 & 0.9485 \\ \hline  & \multicolumn{2}{c}{**Sizing**} \\ \hline Bed diameter & 0.037 m & \\ \hline Bed height & 1.000 m & \\ \hline Zeolite type & 5 A & \\ \hline Zeolite radius & 0.0015 m & \\ \hline Packed density & 850 kg/m\({}^{3}\) & \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Data required for the biohydrogen production. \begin{table}
\begin{tabular}{l l l l l l} \hline \hline  & **Step 1** & **Step 2** & **Step 3** & **Step 4** \\ \hline Bed 1 & Adsorption (70 s) & 980 kPa & Despressurization (3 s) & Purge (70 s) & 150 kPa & Repressurization (3 s) \\ \hline Bed 2 & Purge (70 s) & 150 kPa & Repressurization (3 s) & Adsorption (70 s) & 980 kPa & Despressurization (3 s) \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Process steps and times.