## 3 Numerical Model
### Mathematical Model  
The PSA separation process includes four stages: co-current pressurization (Pr) with feed gas, selective adsorption (Ad) of N\({}_{2}\) from a compressed air stream to produce an oxygen enriched gas, countercurrent blowdown (Cb) of column to desorption pressure and purge (Pu). In order to establish a mathematical model of the molecular sieve adsorption process for nitrogen-oxygen mixed gas, the actual PSA process [30] must be simplified as the process followed. Mass conservation equation:  
\[\frac{\partial\varepsilon\rho_{f}}{\partial\tau}+\
abla\Big{(}\varepsilon\rho _{f}\cdot u\Big{)}=-S_{m} \tag{1}\]  
\[S_{m}=\sum_{i=1}^{n}S_{i} \tag{2}\]  
The mass balance equation for component \(i\)[25]:  
\[\frac{\partial}{\partial\tau}\Big{(}\varepsilon\rho_{f}Y_{i}\Big{)}+\
abla \cdot\Big{(}\varepsilon\rho_{f}uY_{i}\Big{)}-\
abla\cdot\Big{(}\varepsilon\rho _{f}D_{i}\
abla Y_{i}\Big{)}=-S_{i} \tag{3}\]  
\[S_{i}=(1-\varepsilon)\rho_{p}M_{i}\frac{\partial q_{i}}{\partial\tau} \tag{4}\]  
According to the momentum balance, the porous media are modeled by the addition of a momentum source term to the standard fluid Navier-Stokes equations (laminar flow). The modified N-S equation is written as:  
\[\frac{\partial}{\partial\tau}\Big{(}\rho_{f}u_{i}\Big{)}+\
abla\cdot\Big{(} \rho_{f}uu_{i}\Big{)}=\rho_{f}g-\
abla p+\mu\
abla^{2}u+\frac{\mu}{3}\
abla( \
abla\cdot u)+S_{v} \tag{5}\]  
In the porous-media zone, apart from the usual momentum sources, there would be momentum changes in the bed due to gas sorption, which was not considered in the previous relevant studies. Hence, an additional source term is added in the momentum equation and the momentum source term is as follows:  
\[S_{v}=-\Big{(}\mu\frac{u}{\alpha}+C_{2}\rho_{f}|u|u+S_{m}u\Big{)} \tag{6}\]  
In this study, coefficients \(\alpha\) and C\({}_{2}\), which are given by the relations developed by Ergun for flow in a packed bed, are as follows:  
\[\alpha=\frac{\varepsilon^{3}d_{p}^{2}}{150{(1-\varepsilon)}^{2}}C_{2}=1.75 \frac{1}{d_{p}}\frac{1-\varepsilon}{\varepsilon^{3}} \tag{7}\]  
According to the energy balance, for an adiabatic bed with no heat transfer with the surroundings, the overall heat balance may be written as follows. Gas energy Equation (8):  
\[\frac{\partial}{\partial t}\Big{(}\varepsilon\rho_{f}e_{f}+(1-\varepsilon) \rho_{p}e_{p}\Big{)}+\
abla\cdot\Big{(}\overset{\rightarrow}{v}\Big{(}\rho_ {f}e_{f}+p\Big{)}\Big{)}=\
abla\cdot\Big{[}k_{eff}\
abla T+\Big{(}\overset{ \rightarrow}{\tau}\cdot\overset{\rightarrow}{u}\Big{)}\Big{]}+S_{f}^{h} \tag{8}\]  
\[S_{f}^{h}=(1-\varepsilon)\rho_{p}\underset{i}{\sum}\left(-\Delta H_{i}\frac{ \partial q_{i}}{\partial t}\right) \tag{9}\]  
The effective thermal conductivity is taken as:  
\[k_{eff}=\varepsilon k_{f}+(1-\varepsilon)k_{p} \tag{10}\]Solid energy equation:  
\[S_{m}=\sum_{i=1}^{n}S_{i} \tag{2}\]0  
The equation of state:  
\[S_{m}=\sum_{i=1}^{n}S_{i} \tag{2}\]1  
Mass transfer rate [31, 32]:  
\[S_{m}=\sum_{i=1}^{n}S_{i} \tag{2}\]2  
\[S_{m}=\sum_{i=1}^{n}S_{i} \tag{2}\]3  
Adsorption equilibrium:  
\[S_{m}=\sum_{i=1}^{n}S_{i} \tag{2}\]4  
The isotherm parameters for LiX zeolite are listed in \[S_{m}=\sum_{i=1}^{n}S_{i} \tag{2}\]5  
The adsorption isotherm of CP-\(\pi\) RFA was confirmed to follow the Langmuir relationship in \[S_{m}=\sum_{i=1}^{n}S_{i} \tag{2}\]6 These isotherms were measured in our laboratory apparatus by using a high-pressure volumetric apparatus up to 350 kPa, where the adsorption isotherms of pure O\({}_{2}\) and N\({}_{2}\) on LiX zeolite were measured at two different temperatures (278 K and 293 K). \[S_{m}=\sum_{i=1}^{n}S_{i} \tag{2}\]7 Adsorption isotherms for pure O\({}_{2}\) and N\({}_{2}\) on LiX zeolite.