## 3 Validation
### Validation of Breakthrough Curves on Adsorption Beds  
By solving the mathematical equations presented in Section 2, the breakthrough curves on the adsorption bed can be obtained. Table 2 provides the physical property parameters used in the simulation. It is important to note that the structural parameters of the adsorption bed were selected to match typical laboratory-scale PSA setups. Figure 2 illustrates the breakthrough curve results for a lean hydrogen mixture (N\({}_{2}\)/CO\({}_{2}\)/H\({}_{2}\)/CO = 44.6/35.4/19.9/0.1 mol%) on the AC single bed, the zeolite 13X single bed, and an AC/zeolite 13X = 6:4 layered bed, respectively. The simulation conditions include a temperature of 298.15 K, a feed flow rate of 3 LSTP/min, and a feed pressure of 10 bar. It should be noted that, before the simulation begins, pure hydrogen gas is fed into the adsorption bed to saturate the adsorbent with hydrogen. Therefore, the adsorption bed is completely filled with pure hydrogen gas under the initial simulation conditions. The breakthrough curve reflects the distinct adsorption characteristics of various adsorbents and the adsorption kinetics between different adsorbates. The breakthrough times of N\({}_{2}\), CO, and CO\({}_{2}\) are as follows: 260 s, 360 s, and 1000 s for the AC single bed; 310 s, 750 s, and 1485 s for the zeolite 13X single bed; and 265 s, 500 s, and 1225 s for the AC/zeolite 13X = 6:4 layered bed. Whether on a single-layer or layered bed, the breakthrough sequence of impurity gases is N\({}_{2}\), CO, and CO\({}_{2}\). The breakthrough sequence of impurity gases is determined by the adsorption kinetics equation (Equation (11)), which reflects the combined influence of adsorption capacity and the mass transfer coefficient \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Adsorbate** & **CO\({}_{2}\)** & **N\({}_{2}\)** & **H\({}_{2}\)** & **CO** \\ \hline
**AC** & & & & \\ \(w_{l}\) [s\({}^{-1}\)] & 0.050 & 0.220 & 0.900 & 0.350 \\ \(k_{1}\) [mol/kg] & 6.611 & 8.755 & 248.5 & 4.65 \\ \(k_{2}\) [kPa\({}^{-1}\)] & \(1.04\times 10^{-7}\) & \(8.75\times 10^{-7}\) & \(1.38\times 10^{-7}\) & \(3.27\times 10^{-6}\) \\ \(k_{3}\) [K] & 3306 & 1777 & 765 & 1720 \\ \(k_{4}\) [mol/kg] & 0.011 & 0.121 & 0.017 & 0.099 \\ \(k_{5}\) [kPa\({}^{-1}\)] & \(7.76\times 10^{-7}\) & \(2.74\times 10^{-6}\) & \(1.06\times 10^{-11}\) & \(2.24\times 10^{-7}\) \\ \(k_{6}\) [K] & 3300 & 2805 & 7010 & 3390 \\ \(\Delta H_{\rm ads}\) [K/mol] & \(-\)22.5 & \(-\)14.0 & \(-\)12.0 & \(-\)15.0 \\ \hline
**Zeolite 13X** & & & & \\ \(w_{l}\) [s\({}^{-1}\)] & 0.120 & 0.650 & 0.900 & 0.750 \\ \(k_{1}\) [mol/kg] & 3.395 & 2.761 & 3.101 & 2.708 \\ \(k_{2}\) [kPa\({}^{-1}\)] & \(1.80\times 10^{-5}\) & \(1.18\times 10^{-6}\) & \(6.64\times 10^{-7}\) & \(2.66\times 10^{-7}\) \\ \(k_{3}\) [K] & 2964 & 1955 & 1294 & 2502 \\ \(k_{4}\) [mol/kg] & 2.305 & 0.547 & 1.118 & 1.165 \\ \(k_{5}\) [kPa\({}^{-1}\)] & 6.30\(\times 10^{-9}\) & \(7.33\times 10^{-8}\) & \(3.82\times 10^{-5}\) & \(3.92\times 10^{-7}\) \\ \(k_{6}\) [K] & 4291 & 3064 & 52.41 & 3006 \\ \(\Delta H_{\rm ads}\) [K/mol] & \(-\)32.2 & \(-\)18.0 & \(-\)8.00 & \(-\)21.0 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: LDF parameters, TD DSL parameters, and enthalpy of adsorption [28]. (\(w_{i}\)). As indicated by the LDF equation, both a higher \(w_{i}\) and a higher adsorption capacity can enhance the adsorption rate, delaying the breakthrough. This effect is particularly evident when comparing the breakthrough times of CO\({}_{2}\) on AC and zeolite 13X. For the two adsorbents studied, the order of \(w_{i}\) is CO > N\({}_{2}\) > CO\({}_{2}\), while the adsorption capacities follow CO\({}_{2}\) > CO > N\({}_{2}\). Ultimately, the observed breakthrough sequence corresponds to the order of adsorption capacities, suggesting that adsorption capacity has a greater impact on breakthrough behavior than kinetic limits here.