## 3 Numerical Model
### Initial and Boundary Conditions and Model Parameters  
The Skarstrom PSA cycle was employed comprising four basic steps named as pressurization with air (Pr), adsorption (Ad), countercurrent blowdown (Cb), and low pressure purge (Pu). The Skarstrom cycle in its basic form utilizes two packed adsorbent beds, which both undergo these four operations alternately. The initial conditions of the PSA oxygen production process of the CP-\(\pi\) RFA are shown in Table 4. The Skarstrom cyclic sequences and the boundary conditions are summarized in Table 5. The inlet of the bed is set to the mass flow rate boundary condition, and the pressure-flow curve formula of the compressor is:  
\[\text{Q}=7.23-5.633\times 10^{-6}\text{ P} \tag{16}\]  
In the following, we take Adsor 1 as an example to explain the setting of the entrance and vent boundary conditions. The PSA process starts from the Pr step of Adsorber 1. During the Pr process, the vent of Adsorber 1 is closed, and air flows into the CP-\(\pi\) RFA at a certain mass flow rate, and the pressure in Adsorber 1 is constantly rising. After the Pr process has completed, the valve at the vent is opened and product gas is obtained. The high operating pressure in Pr process is not an input operating pressure, but is determined by other process variables. During the Ad process, the pressure at the vent is set to the pressure value at the end of the Pr step. At the end of Ad process, the product vent is closed and the feed is turned to Adsorber 2. During the Cb process, the pressure at the feed end is set to 101,325 Pa. When the pressure in Adsorber 1 drops to a lower pressure value, the Cb process is considered to complete. During the Pu process, the product gas of Adsorber 2 is used to purge Adsorber 1, and the pressure at the feed end of Adsorber 1 is kept at 101,325 Pa. The purge mass flow rate is determined by the product recovery.  
\begin{table}
\begin{tabular}{l c} \hline \hline  & **Parameter** & **Values** \\ \hline Pressure (Pa) & 101,325 \\ Air-temperature (K) & 298 \\ Solid-temperature (K) & 298 \\ \hline Mass fraction of O\({}_{2}\) in the gas-phase & 0.233 \\ Amount of O\({}_{2}\) adsorbed per unit mass of sorbent (mol\(\cdot\)kg\({}^{-1}\)) & 0.0262832 \\ Amount of N\({}_{2}\) adsorbed per unit mass of sorbent (mol\(\cdot\)kg\({}^{-1}\)) & 0.6328067 \\ \hline \hline \end{tabular}
\end{table}
Table 4: Initial conditions.  
\begin{table}
\begin{tabular}{c c c c c c c} \hline \hline \multirow{2}{*}{**Step**} & **Schematic** & \multirow{2}{*}{**Duration(s)**} & \multicolumn{5}{c}{**Boundary Conditions for Adsorber 1**} \\ \cline{3-6}  & & & **Entrance** & **Vent** & **Adsorber Wall** & **Adsorber Axis** \\ \hline \multirow{2}{*}{Pr Step} & 1 & 2 & & mass flow & wall & wall & axis \\  & & & inlet & wall & wall & axis \\ \hline \multirow{2}{*}{Ad Step} & 1 & 2 & & mass flow & pressure & wall & axis \\  & & & inlet & outlet & wall & axis \\ \hline \multirow{2}{*}{Cb Step} & 1 & 2 & & pressure & wall & wall & axis \\  & & & outlet & wall & wall & axis \\ \hline \multirow{2}{*}{Pu Step} & 1 & 2 & & pressure & mass flow & wall & axis \\  & & & outlet & inlet & wall & axis \\ \hline \hline \end{tabular}
\end{table}
Table 5: Skarstrom cyclic sequence and boundary conditions for Adsorber 1.