## 3 Results
### Prediction by BBD Method  
The BBD method was employed within the Design Expert software (version 10) to establish the relationship between the performance and operating parameters of a PSA system, with the objective of identifying the optimal conditions for the system. In the PSA process, the adsorption duration determines the overall cycle time, which is closely associated with the breakthrough point, where impurities reach the end of column. The ideal adsorption time is influenced by factors such as the adsorber's bed length and operating conditions [34; 35]. [PERSON] and colleagues emphasized the importance of incorporating a pressure-equalization step to enhance the purity both of H\({}_{2}\) and CO\({}_{2}\)[36]. Consequently, in this work, the adsorption time, feed flow rate, and pressure equalization time were selected as independent factors for the BBD design. The system responses of interest were hydrogen purity and productivity. For the BBD method, a one-factor design was used first to define the range of independent factors. Each factor was coded as \"-1\", \"0\", or \"+1\", corresponding to low, center point, and high levels, respectively, as detailed in  
\begin{table}
\begin{tabular}{c c c c c c c} \hline \hline
**Step** & **I** & **II** & **III** & **IV** & **V** & **VI** \\ \hline Schematic diagram & & & & & & \\ of adsorption bed & & & & & & \\ \hline Bed 1 & AD & DPE & DP & PG & PPE & FP \\ Bed 2 & PG & PPE & FP & AD & DPE & DP \\ Time (s) & 180 & 20 & 8 & 180 & 20 & 8 \\ \hline \hline \end{tabular}
\end{table}
Table 1: Six-step cyclic sequences of the PSA process.  
Figure 6: Molar fraction of H\({}_{2}\) and pressure in the layered bed of one cycle.  
Through the analysis, quadratic regression models were derived to describe the relationships between hydrogen purity and productivity:  
\[purity=85.26+0.15A-0.09B+0.70C-7.11\times 10^{-5}AB-3.47\times 10^{-3}AC\]  
\[+0.01 BC-3.59\times 10^{-4}A^{2}+7.97\times 10^{-4}B^{2}-0.03C^{2}\]  
\[productivity=9.86-0.21A+0.23B+2.43C+3.28\times 10^{-4}AB+\]  
\[6.98\times 10^{-4}AC-0.03 BC+5.73\times 10^{-4}A^{2}-2.53\times 10^{-3}B^{2}-0. 05C^{2}\]  
where \(A\) is adsorption time, \(B\) is pressure equalization time, and \(C\) is feed flow rate.  
Figure 7 presents 3D response surface plots illustrating the effects of adsorption time, feed flow rate, and pressure equalization time on hydrogen purity (a) and productivity (b). The results indicated that, at a feed flow rate of 8 L/min, hydrogen purity increases with decreasing adsorption time and increasing pressure equalization time. Conversely, productivity increases as pressure equalization time decreases and adsorption time increases. When the adsorption time is 180 s, hydrogen purity increases as the feed flow rate decreases and the pressure equalization time increases. Meanwhile, productivity increases with both an increase in pressure equalization time and feed flow rate. At a pressure equalization time of 20 s, hydrogen purity increases as adsorption time decreases and feed flow rate increases. In this case, productivity also increases when both adsorption time and feed flow rate are increased.  
\begin{table}
\begin{tabular}{c c c c} \hline \hline \multirow{2}{*}{**Factors**} & \multicolumn{3}{c}{**Variables and Ranges**} \\ \cline{2-4}  & **Adsorption Time (s)** & **Pressure Equalization Time (s)** & **Feed Flow Rate (L/min)** \\ \hline \(-1\) & 160 & 10 & 6 \\ \(0\) & 180 & 20 & 8 \\ \(+1\) & 200 & 30 & 10 \\ \hline \hline \end{tabular}
\end{table}
Table 2: The ranges of the independent factors and variables in BBD method.  
Figure 7: _Cont._