## 4 Results and Discussion
### Modeling and Validation of PSA Cycle  
As shown in Figure 6, a dual-bed, six-step PSA cycle model using the Aspen Adsorption platform was developed. The PSA flow diagram is presented in Table 3. In Figure 6, F1, W1, and P1 represent the inlet, waste outlet, and product outlet, respectively. VF1, VW1, VP1, and VD1 are valves that regulate gas flow. Bed1 refers to the AC/zeolite 13X layered bed. D1 is an interactor, representing a virtual bed. Actually, D1 functions as Bed2. Figure 5: Effect of AC/zeolite 13X ratio on the breakthrough curve of AC/zeolite 13X layered bed. Figure 6: Schematic diagram of the PSA cycle model in Aspen Adsorption. The control of each step in the PSA cycle is implemented through a series of valve operations, described as the following: (1) Adsorption (AD) step: Valves VF1, VP1, and VD1 are opened to let high-pressure feed gas enter Bed1. Impurities are adsorbed, while hydrogen, which is less adsorbed, flows out. Some H\({}_{2}\) flows towards P1, and the rest enters Bed2 for purge. (2) Depressurizing pressure equalization (DPE) step: Valves VF1 and VP1 are closed to stop the feed gas. The pressure in Bed1 drops, while the pressure in Bed2 rises until both reach the same level. (3) Blowdown (BD) step: Valve VD1 is closed, and VW1 is opened. The pressure in Bed1 quickly drops to atmospheric level, releasing the adsorbed impurities through W1. (4) Purge (PG) step: Valves VF1, VP1, and VD1 remain closed, and VW1 stays open. Hydrogen from Bed2 flows into Bed1 to remove the remaining impurities. (5) Pressurizing pressure equalization (PPE) step: Valve VW1 is closed to stop the purge. The pressure in Bed2 decreases, while that in Bed1 increases until they equalize. (6) Feed pressurization (FP) step: Valve VF1 is reopened to feed gas into Bed1, raising its pressure to the required level. This marks the beginning of the next PSA cycle. The boundary conditions of each step in the simulation are detailed below:  
(1) Boundary conditions in the AD and FP step:  
\[-\epsilon_{b}D_{L}\frac{\partial c_{i}(0)}{\partial z}=v_{g}(0)(c_{in,i}(0)-c_ {i}(0)) \tag{13}\]  
\[\frac{\partial c_{i}(L)}{\partial z}=0 \tag{14}\]  
\[-\epsilon_{b}K_{g}\frac{\partial T(0)}{\partial z}=v_{g}(0)\rho_{g}(0)C_{ pg}(T_{in}-T(0)) \tag{15}\]  
\[\frac{\partial T(L)}{\partial z}=0 \tag{16}\]  
(2) Boundary conditions in the PG and PPE step:  
\[-\epsilon_{b}D_{L}\frac{\partial c_{i}(L)}{\partial z}=v_{g}(L)(c_{in,i}(L)- c_{i}(L)) \tag{17}\]  
\[\frac{\partial c_{i}(0)}{\partial z}=0 \tag{18}\]  
\[-\epsilon_{b}K_{g}\frac{\partial T(L)}{\partial z}=v_{g}(L)\rho_{g}(L)C_{pg} (T_{in}-T(L)) \tag{19}\]  
Figure 5:0  
(3) Boundary conditions in the DPE and BD step:  
Figure 5:1  
Figure 5:2  
The initial operating conditions for the cyclic process are set as follows: feed pressure of 10 bar, feed flow rate of 3 LSTP/min, temperature of 298.15 K, P/F ratio of 0.3, and a layer ratio of AC/zeolite 13X = 6:4. The process runs for 20 cycles. The P/F ratio \begin{table}
\begin{tabular}{l l l l l l l} \hline
**Step** & **1** & **2** & **3** & **4** & **5** & **6** \\ \hline Bed1 & AD & DPE & BD & PG & PPE & FP \\ Bed2 & PG & PPE & FP & AD & DPE & BD \\ Time [s] & 40 & 10 & 25 & 40 & 10 & 25 \\ \hline \end{tabular}
\end{table} 
Table 3: Cyclic configuration of the dual-bed, six-step PSA process. is a critical parameter in the cyclic process, yet it has not been adequately addressed in the study of breakthrough curves. The P/F ratio refers to the ratio of the hydrogen purge during the PG step to the hydrogen feed during the AD step in the PSA cycle, as detailed in Equation (23). The value of the P/F ratio significantly influences the purification performance of the PSA process.