## 1 Introduction  
Hydrogen energy represents a promising solution for improving energy efficiency and advancing the use of clean energy. Due to the limited availability and environmental drawbacks of fossil fuels, hydrogen fuel cells have gained increasing attention as a sustainable power source. The operation of fuel cells requires a continuous supply of high-purity hydrogen gas. Industrially, hydrogen is predominantly produced through large-scale processes such as petroleum cracking, methane reforming, and coke oven gas utilization [1, 2, 3]. However, the hydrogen generated by these methods often contains impurities that do not meet the stringent purity standards required for fuel cell applications [4]. The impurity gas CO can poison metal catalysts within fuel cells, significantly impairing their catalytic performance [5]. Therefore, additional separation and purification steps are essential to ensure hydrogen purity. Adsorption technology has become a widely adopted method for gas separation due to its advantages, including low cost, high efficiency, operational simplicity, and environmental friendliness. Based on this, pressure swing adsorption (PSA) is extensively employed to produce high-purity hydrogen. A substantial body of research has focused on utilizing PSA technology to achieve hydrogen with high purity [6; 7; 8; 9]. [PERSON] investigated three different vacuum pressure swing adsorption (VPSA) systems for extracting hydrogen from compressed natural gas (CNG) and achieved a hydrogen purity of 68% [10]. The VPSA process enables the efficient concentration of H\({}_{2}\) from an initial level of 20% to 68%, with a recovery of up to 92%. Tail gas is commonly used as a combustion fuel in hydrogen production processes. However, this gas mixture, which contains a small amount of hydrogen, can also be purified and recycled. Recent studies have demonstrated that the recovery and purification of such mixtures are economically viable [11]. Lean hydrogen mixtures are typically derived from various industrial off-gases and contain hydrogen at concentrations of 13% to 24%. These mixtures may also include impurity gases such as CO\({}_{2}\), N\({}_{2}\), CO, and CH\({}_{4}\). Such multi-component gases often exhibit complex competitive adsorption behavior on adsorption beds, which is described by breakthrough curves. The formation of breakthrough curves and their temporal changes are closely linked to adsorption isotherms, gas diffusion rates, and adsorption bed parameters. The breakthrough time of different gases reflects the adsorption selectivity of the adsorbents for specific components, as well as the efficiency of the adsorption bed. Researchers have also investigated the impact of various parameters on the breakthrough curve, which can further inform the optimization of PSA performance [12; 13; 14]. [PERSON]'s study of the breakthrough curves for H\({}_{2}\)/CH\({}_{4}\)/CO\({}_{2}\) on CuBTC revealed that a higher feed flow rate led to earlier breakthrough, and increased adsorption pressure resulted in a delayed breakthrough [15]. As the core part of PSA, the direct performance of adsorbents in removing impurity gases (purity and recovery) has been widely studied. Adsorbents are usually porous materials, including activated carbons, zeolites, and MOFs [16; 17; 18]. [PERSON] used CaX zeolite to purify hydrogen from nitrogen-rich off-gas, obtaining hydrogen with a purity higher than 99.7% and a recovery of over 65% [19]. Additionally, some researchers have also proposed the use of layered beds for the PSA process, achieving good results [4; 20; 21]. At the same time, researchers have continuously optimized the operating conditions and other parameters in the PSA process to explore more satisfactory results. [PERSON] investigated the PSA process for a gas mixture with a composition of H\({}_{2}\)/CH\({}_{4}\)/CO/N\({}_{2}\)/CO\({}_{2}\) = 56.4/26.6/8.4/5.5/3.1 mol% on a layered bed consisting of activated carbon and zeolite 5A and employed the Box-Behnken design method to optimize the PSA process [22].