## 4 Sensitivity Analysis for Factors and Temperature Relationship Derivation
The R\({}^{2}\) values for these equations exceeded 0.99, confirming their accuracy in capturing the analytical trends. \[T_{gas} = 96.69-0.3956\cdot t_{chr}+0.073\cdot T_{amb}+0.9605\cdot T_{ini}+0.00 0722\cdot t_{chr}^{2} \tag{8}\] \[+0.002501\cdot t_{chr}\cdot T_{amb}-0.002692\cdot t_{chr}\cdot T_{ini}\]  
\[T_{shell} = 90.63-0.3586\cdot t_{chr}+0.0934\cdot T_{amb}+0.9330\cdot T_{ini}+0.0 00647\cdot t_{chr}^{2} \tag{9}\] \[+0.002413\cdot t_{chr}\cdot T_{amb}-0.002578\cdot t_{chr}\cdot T_{ini}\]  
\[T_{surface} = 71.047-0.2381\cdot t_{chr}+0.2023\cdot T_{amb}+0.791\cdot T_{ini}+0.0 00391\cdot t_{chr}^{2} \tag{10}\] \[+0.001269\cdot T_{ini}^{2}+0.00198\cdot t_{chr}\cdot T_{amb}-0.0 02138\cdot t_{chr}\cdot T_{ini}\] \begin{table}
\begin{tabular}{l c c c c c} \hline \hline
**Source** & **DF** & **Adj SS** & **Adj MS** & **F-Value** & \(p\)**-Value** \\ \hline
**Model** & 6 & 8205 & 1368 & 1370 & 0.000 \\
**Linear** & 3 & 7246 & 2415 & 2421 & 0.000 \\ Charging Time (t\({}_{chr}\)) & 1 & 2649 & 2649 & 2655 & 0.000 \\ Ambient Temperature (T\({}_{amb}\)) & 1 & 1808 & 1808 & 1812 & 0.000 \\ Initial Temperature (T\({}_{ini}\)) & 1 & 2789 & 2789 & 2795 & 0.000 \\
**Square** & 1 & 171 & 171 & 171 & 0.000 \\
**t\({}_{chr}\times\) t\({}_{chr}\)** & 1 & 171 & 171 & 171 & 0.000 \\
**Interaction** & 2 & 788 & 394 & 395 & 0.000 \\
**t\({}_{chr}\times\) T\({}_{amb}\)** & 1 & 365 & 365 & 366 & 0.000 \\
**t\({}_{chr}\times\) T\({}_{ini}\)** & 1 & 423 & 423 & 424 & 0.000 \\
**Error** & 13 & 13 & 1 & & \\
**Total** & 19 & 8218 & & & \\ \hline \hline \end{tabular}
\end{table} 
Table 14: Response surface analysis for maximum temperature of inner gas. \begin{table}
\begin{tabular}{l c c c c c} \hline \hline
**Source** & **DF** & **Adj SS** & **Adj MS** & **F-Value** & \(p\)**-Value** \\ \hline
**Model** & 6 & 7650 & 1275 & 1549 & 0.000 \\
**Linear** & 3 & 6785 & 2262 & 2748 & 0.000 \\ Charging Time (t\({}_{chr}\)) & 1 & 2233 & 2233 & 2713 & 0.000 \\ Ambient Temperature (T\({}_{amb}\)) & 1 & 1866 & 1866 & 2267 & 0.000 \\ Initial Temperature (T\({}_{ini}\)) & 1 & 2686 & 2686 & 3263 & 0.000 \\
**Square** & 1 & 137 & 137 & 167 & 0.000 \\
**t\({}_{chr}\times\) t\({}_{chr}\)** & 1 & 137 & 137 & 167 & 0.000 \\
**Interaction** & 2 & 727 & 364 & 442 & 0.000 \\
**t\({}_{chr}\times\) T\({}_{amb}\)** & 1 & 340 & 340 & 412 & 0.000 \\
**t\({}_{chr}\times\) T\({}_{ini}\)** & 1 & 388 & 388 & 471 & 0.000 \\
**Error** & 13 & 11 & 1 & & \\
**Total** & 19 & 7661 & & & \\ \hline \hline \end{tabular}
\end{table} 
Table 15: Response surface analysis for maximum temperature of tank shell. Figure 16: _Cont._ \begin{table}
\begin{tabular}{l c c c c c} \hline \hline
**Source** & **DF** & **Adj SS** & **Adj MS** & **F-Value** & _p-Value_ \\ \hline
**Model** & 7 & 6242 & 892 & 2367 & 0.000 \\
**Linear** & 3 & 5668 & 1889 & 5016 & 0.000 \\ Charging Time (t\({}_{\rm chr}\)) & 1 & 1213 & 1213 & 3220 & 0.000 \\ Ambient Temperature (T\({}_{\rm amb}\)) & 1 & 2244 & 2244 & 5957 & 0.000 \\ Initial Temperature (T\({}_{\rm ini}\)) & 1 & 2212 & 2212 & 5872 & 0.000 \\
**Square** & 2 & 78 & 39 & 104 & 0.000 \\  & t\({}_{\rm chr}\)\(\times\) t\({}_{\rm chr}\) & 1 & 32 & 32 & 85 & 0.000 \\ T\({}_{\rm ini}\)\(\times\) T\({}_{\rm ini}\) & 1 & 4 & 4 & 11 & 0.000 \\
**Interaction** & 2 & 495 & 248 & 657 & 0.000 \\  & t\({}_{\rm chr}\)\(\times\) T\({}_{\rm amb}\) & 1 & 229 & 229 & 607 & 0.000 \\
**Error** & 1 & 167 & 267 & 708 & 0.000 \\
**Total** & 12 & 5 & 0 & & \\ \hline \hline \end{tabular}
\end{table} 
Table 16: Response surface analysis for maximum temperature of outer surface. Figure 16: Main effects plot for maximum temperature of hydrogen tube skid.