## 2 Numerical Model
### Governing Equations  
The governing equations are consistent with previous research [27; 28], and the simulation results were verified experimentally by use of the product of the O\({}_{2}\) concentration on the axial flow adsorber [28]. (1) Mass balance  
The overall mass balance can be expressed as:  
\[\frac{\partial\varepsilon\rho_{f}}{\partial\tau}+\
abla\Big{(}\varepsilon\rho_ {f}\cdot u\Big{)}=-S_{m} \tag{1}\]  
\[S_{m}=\sum_{i=1}^{n}S_{i} \tag{2}\] \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Structure Parameters** & **Values** & **Structure Parameters** & **Values** \\ \hline Diameter of adsorber/m & 219 & Sorbent filling height/mm & 187 \\ Inner diameter of the adsorber/m & 211 & Diameter of outer distribution hole/mm & 8 \\ Outer diameter of sorbent layer/m & 153 & Opening ratio of outer distribution cylinder \% & 3.6 \\ Diameter of inner flow channel/m & 26 & Diameter of inner distribution hole/mm & 2 \\ Wall thickness of distribution tube/mm & 3 & Opening ratio of inner distribution cylinder \% & 24.5 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Structure parameters of the adsorbers. Figure 1: Models of radial-flow adsorbers: (**a**) CP-Z flow; (**b**) CF-Z flow; (**c**) CP-\(\pi\) flow; (**d**) CF-\(\pi\) flow; (**e**) the model of the 1/4 sector radial-flow adsorber; (**f**) schematic of radial-flow adsorbers. and the mass balance equation for component i can be expressed as:  
\[\frac{\partial}{\partial\tau}\Big{(}\varepsilon\rho_{f}Y_{i}\Big{)}+\
abla\cdot \Big{(}\varepsilon\rho_{f}uY_{i}\Big{)}-\
abla\cdot\Big{(}\varepsilon\rho_{f} D_{i}\
abla Y_{i}\Big{)}=-S_{i} \tag{3}\]  
\[S_{i}=(1-\varepsilon)\rho_{p}M_{i}\frac{\partial q_{i}}{\partial\tau} \tag{4}\]  
where, \(Sm\) is the total mass source term that results from bulk sorption; \(\varepsilon\) is the bed porosity; \(\rho_{f}\) is the density of the gas mixture, km m\({}^{-3}\); \(Y_{i}\) is the mass fraction of component \(I\); \(u\) is the velocity vector, m s\({}^{-1}\); \(D_{i}\) is the mass dispersion rate, m\({}^{2}\) s\({}^{-1}\); and \(M_{i}\) is the molar weight of component i, kg mol\({}^{-1}\). (2) 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:  
\[\frac{\partial}{\partial\tau}\Big{(}\rho_{f}u_{i}\Big{)}+\
abla\cdot\Big{(} \rho_{f}uu_{i}\Big{)}=\rho_{f}g-\
abla p+\mu\
abla^{2}u+\frac{\mu}{3}\
abla( \
abla\cdot u)+S_{v} \tag{5}\]  
\[S_{v}=-\Big{(}\mu\frac{u}{\alpha}+C_{2}\rho_{f}|u|+S_{m}u\Big{)} \tag{6}\]  
\[\alpha=\frac{\varepsilon^{3}d_{p}^{2}}{150(1-\varepsilon)^{2}},C_{2}=1.75 \frac{1}{d_{p}}\frac{1-\varepsilon}{\varepsilon^{3}} \tag{7}\]  
where \(\mu\) is the dynamic viscosity of the fluid; \(p\) is the gas pressure, and \(S_{v}\) is the momentum source term. The momentum source term \(Sv\) is written as Equation (6) with the consideration of gas adsorption. The coefficients \(\alpha\) and \(C_{2}\) are given by the relations developed by [PERSON] [29] for the flow in a packed bed. (3) Energy Balance  
For an adiabatic bed with no heat transfer with the surroundings, the overall heat balance can be written as  
\[\frac{\partial}{\partial t}\Big{(}\varepsilon\rho_{f}e_{f}+(1-\varepsilon) \rho_{p}e_{p}\Big{)}+\
abla\cdot\Big{(}\overrightarrow{v}\Big{(}\rho_{f}e_{ f}+p\Big{)}\Big{)}=\
abla\cdot\Big{[}k_{eff}\
abla T+\Big{(}\overrightarrow{\pi} \cdot\overrightarrow{u}\Big{)}\Big{]}+S_{f}^{h} \tag{8}\]  
\[S_{f}^{h}=(1-\varepsilon)\rho_{p}\underset{i}{\sum}\bigg{(}-\Delta H_{i} \frac{\partial q_{i}}{\partial t}\bigg{)} \tag{9}\]  
where \(e_{f}\) is the total fluid energy, J kg\({}^{-1}\); \(e_{p}\) is the total solid medium energy; \(\Delta H_{i}\) is the heat of adsorption, kJ mol\({}^{-1}\); \(k_{\text{eff}}\) is the effective thermal conductivity of the bed, W m\({}^{-2}\) K\({}^{-1}\).