## 2 Materials and Methods  
### Mathematical Model for PSA Process  
To accurately depict the adsorption process, it is necessary to formulate equations for mass balance, energy balance, momentum balance, and adsorption isotherms. The mathematical models refer to reference [26]. The mass balance for the bulk phase within the column consists of two components: for individual components and for the overall mass balance. \[-D_{L}\frac{\partial^{2}y_{i}}{\partial x^{2}}+\frac{\partial y_{i}}{\partial t }+u_{z}\frac{\partial y_{i}}{\partial x}+\frac{RT}{p}\frac{1-\epsilon_{b}}{ \epsilon_{b}}\rho_{p}\left(\frac{\partial q_{i}}{\partial t}-y_{i}\sum_{j=1}^{ N}\frac{\partial q_{j}}{\partial t}\right)=0,\ i=1,\cdots,N, \tag{1}\]  
\[-D_{L}\frac{\partial^{2}p}{\partial x^{2}}+\frac{\partial p}{\partial t}+p \frac{\partial u}{\partial z}+u_{z}\frac{\partial p}{\partial z}\frac{p}{T} \left(-D_{L}\frac{\partial^{2}T}{\partial x^{2}}+\frac{\partial T}{\partial t }+u_{z}\frac{\partial T}{\partial z}\right)+\frac{1-\epsilon_{b}}{\epsilon_{b} }\rho_{p}RT\sum_{j=1}^{N}\frac{\partial q_{j}}{\partial t}=0, \tag{2}\]  
where \(D_{L}\) represents the axial dispersion coefficient; \(u_{z}\) represents the axial physical velocity; \(y_{i}\) denotes the molar fraction of species \(i\); \(q_{i}\) denotes the concentration of adsorbed phase of species \(i\); \(\epsilon_{b}\) represents the interparticle void fraction; \(\rho_{p}\) represents the adsorbent pellet density; R is the gas constant; \(T\) represents the temperature within the adsorption bed; \(P\) represents pressure in the adsorption bed; \(t\) represent the time; \(z\) is the axial position in the adsorption bed. The energy balance consists of two components in a PSA system: the energy balance between the gas and solid phases during adsorption process, and the energy balance at the wall of the adsorption bed. \[-K_{L}\frac{\partial^{2}T}{\partial z^{2}}+\left(\epsilon_{i}\rho_{g}C_{pg}+ \rho_{b}C_{ps}\right)\frac{\partial T}{\partial t}+\rho_{g}C_{pg}\epsilon_{b}h _{z}\frac{\partial T}{\partial z}-\rho_{b}\sum_{i}^{N}Q_{i}\frac{\partial q_{ i}}{\partial t}+\frac{2h_{i}}{R_{bi}}(T-T_{w})=0 \tag{3}\]  
\[\rho_{w}C_{pw}A_{w}\frac{\partial T_{w}}{\partial t}=2\pi R_{bi}h_{i}(T-T_{w} )-2\pi R_{bo}h_{o}(T_{w}-T_{atm}), \tag{4}\]  
\[A_{w}=\pi\left(R_{bo}^{2}-R_{bi}^{2}\right) \tag{5}\]  
The axial dispersion coefficient of thermal is denoted by \(K_{L}\); \(C_{pg}\) is the heat capacity of gas phase; \(C_{ps}\) is the specific heat capacity of adsorbent; \(\epsilon_{t}\) and \(\epsilon_{b}\) are the total and the interparticle void fraction, respectively; \(Q_{i}\) is the heat adsorption for species \(i\); \(h_{i}\) and \(h_{o}\) are, respectively, the coefficient of heat transfer of inner wall and outer wall; \(T_{w}\) and \(T_{atm}\) are, respectively, the wall and the ambient temperature; \(R_{bi}\) and \(R_{bo}\) are, respectively, the inside radius and outside radius of adsorption column. Drawing from the work in [27], the momentum balance is calculated using [PERSON]'s equation:  
\[\frac{dp}{dz}=a\mu v_{z}+b\rho v_{z}\left|\vec{v}\right|, \tag{6}\]  
where the coefficients \(a\) and \(b\) are determined by the following equations:  
\[a=\frac{150}{4R_{p}^{2}}\frac{(1-\epsilon_{b})^{2}}{\epsilon_{b}^{3}} \tag{7}\]  
\[b=1.75\frac{(1-\epsilon_{b})}{2R_{p}\epsilon_{b}^{3}}. \tag{8}\]  
where \(\mu\) represents the dynamic viscosity; \(v_{z}\) is the Darcy's velocity; \(R_{p}\) stands for the radius of the particle. In this study, the extended Langmuir-Freundlich (L-F) model is employed to describe the adsorption isotherms:  
\[q_{i}^{*}=\frac{q_{mi}B_{i}p_{i}^{n_{i}}}{1+\sum_{j=1}^{N}B_{j}p_{j}^{n_{j}}}, \ i=1,\cdots,N, \tag{9}\]  
where \(q_{i}^{*}\) represents the equilibrium concentration of absorbed phase; \(q_{mi}\) and \(B_{i}\) are the isotherm parameters of extended L-F model; \(p_{i}\) is the partial pressure of species \(i\).