## 2 PSA Plant Model and Simulation
The adsorption and desorption behavior of the gas is characterized using the extended Langmuir model. The adsorbent utilized in this study is type 5A zeolite, and the mathematical model of Langmuir characterizes the adsorption isotherm. \[\zeta_{i}^{*}=\frac{(IP_{1}-IP_{2}T_{s})IP_{3}e^{ip_{4}/T_{s}}P_{i}}{1+\sum_{i}( IP_{3}e^{ip_{4}/T_{s}}P_{i})} \tag{1}\]  
where \(\zeta_{i}^{*}\) denotes the saturation capacity for compound \(i\), \(T\) is the temperature during adsorption, \(P_{i}\) is the partial pressure, and \(IP_{n}\) with \(n=1,\ldots,4\) are the Langmuir parameters. Material, energy, and momentum balances are fundamental in chemical and process engineering, enabling a comprehensive analysis of system behavior. The material balance represented by Equation (2), based on mass conservation, accounts for inputs, outputs, and consumption of compounds. The energy balance represented by Equation (4), grounded in the first law of thermodynamics, evaluates energy exchanges and accumulation. Additionally, the momentum balance represented by Equation (3), derived from [PERSON]'s laws, describes fluid motion and pressure distribution within a system. Mass transfer principles further quantify the transport of species due to diffusion and convection, playing a critical role in separation processes and reaction engineering, which is represented Equation (5). These balances collectively help quantify flows, identify inefficiencies, optimize equipment performance, and enhance overall process efficiency. The set of equations used to solve the corresponding PDEs are listed below:  
* **Material balance** \[-D_{L}\frac{\partial^{2}c_{i}}{\partial z^{2}}+\frac{\partial(U_{2}c_{i})}{ \partial z}+\frac{\partial c_{i}}{\partial t}+\frac{(1-\epsilon_{b})}{\epsilon _{b}}J=0\] (2)
* **Moment balance** \[-\frac{\partial P}{\partial z}=\left(\frac{150\mu V_{2}(1-\epsilon_{b})^{2}}{4 R_{p}^{2}\epsilon_{b}^{3}}+\frac{1.75\rho_{g}(1-\epsilon_{b})}{(2R_{p})\epsilon_{b}^{3}}v_{g }^{2}\right)\] (3)
* **Energy balance** \[K_{L}\frac{\partial^{2}T}{\partial z^{2}}+\left[\epsilon_{b}cC_{pg}+(1- \epsilon_{b})\rho_{p}C_{pg}\right]\frac{\partial T}{\partial t}-\epsilon_{b}cC _{pg}U_{z}\frac{\partial T}{\partial z}=(1-\epsilon_{b})\rho_{p}\sum_{j=1}^{N }\Delta H_{j}\frac{\partial n_{j}}{\partial t}+\frac{2h_{lin}}{R_{in}}(T_{w} -T)\] (4)
* **Mass transfer** \[\frac{\partial W_{i}}{\partial t}=MTC_{si}(W_{i}^{*}-W_{i}),MTC_{si}=\frac{ \Omega D_{ci}}{r_{p}^{2}}\] (5) Initial and boundary conditions must be taken into account; Table 3 summarizes them. The energy, recovery of H\({}_{2}\), and the productivity of H\({}_{2}\) of the plant were evaluated using the following Equations (6)-(8):  
\[productivity_{H_{2}}(\%)=\frac{\int_{0}^{t_{ads}}cH_{2}Pu_{s}Pdt}{\Sigma_{i =1}^{n}\int_{0}^{t_{ads}}ciPu_{s}dt} \tag{6}\]  
\[Recovery_{H_{2}}(\%)=\frac{\int_{0}^{t_{ads}}cH_{2}Pu_{s}p\ dt-\int_{0}^{t_{ purge}}cH_{2}Pu_{s}Purge\ dt}{\int_{0}^{t_{ads}}cH_{2}fu_{s}\ dt+\int_{0}^{tARP}cH_{2}fu_{s}f\ dt} \tag{7}\]\[Energy\ efficiency\ (\%)=\frac{n_{s}}{n_{s}-1}RT_{gf}\Bigg{[}\bigg{(}\frac{Patm}{ Pva}\bigg{)}^{\frac{n_{s}-1}{n_{b}}}-1\bigg{]}u_{s}r_{gf}\pi r^{2}bi \tag{8}\]  
The operating conditions for the process were established based on methodologies reported by [39; 40], utilizing activated carbon for hydrogen purification and production. The schematic diagram of the PSA process, illustrated in Figure 1, shows the cycles of the PSA process. The PSA process incorporates inputs such as pressure, temperature, flow, and composition (see Figure 1). Consequently, each of the formulated equations provides a solution for these variables. Material balance allows for the determination of each component's purity in terms of concentration or molar fraction. The momentum balance provides insights into pressure profiles within the beds, highlighting potential pressure drops that may occur during transitions between steps.