## 4 Sensitivity Analysis for Factors and Temperature Relationship Derivation  
Following the analysis of the five variables critical to the charging characteristics of vertically installed hydrogen tanks, a design of experiments approach was employed to derive equations predicting the maximum temperature within the internal hydrogen, shell, and outer surface of the tank. Initially, key factors influencing the tank's maximum temperature were identified through sensitivity analysis using a factorial design. Subsequently, equations for predicting these maximum temperatures were formulated via response surface methodology. Given the varied materials and permissible temperature ranges across different sections of the tube skid, the tank was segmented into hydrogen, shell, and outer surface regions for analysis. The validity of the charging variables for each region's maximum temperature was assessed through factorial design. The charging time and temperature ranges were maintained at 60 to 240 min and \(-\)20 to 40 \({}^{\circ}\)C, respectively, consistent with the previous sections. The experimental combinations derived from factorial design are listed in Tables 8 and 9, with analysis results presented in Tables 10-12. Adj SS represents the temperature variance analyzed statistically, while Adj MS is derived by dividing Adj SS by the degrees of freedom (DF). The F-value correlates with the ratio of each variable's Adj MS to the error's Adj MS, and the \(p\)-value inversely relates to the F-value. A \(p\)-value < 0.05 generally indicates a factor's significance. Applying this criterion, charging time, ambient temperature, and initial temperature were identified as significant factors, whereas inlet temperature and initial pressure were considered nonsignificant, as evidenced in the tables. The analysis results are detailed in Figure 15.  
Figure 14: Effect of initial pressure on maximum temperature for each region of tube skid; (**a**) hydrogen, (**b**) tank shell, and (**c**) outer surface. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Source** & **DF** & **Adj SS** & **Adj MS** & **F-Value** & _p-Value_ \\ \hline
**Model** & 5 & 10,368 & 2074 & 11.96 & 0.001 \\
**Linear** & 5 & 10,368 & 2074 & 11.96 & 0.001 \\ Charging Time & 1 & 3978 & 3978 & 22.94 & 0.001 \\ Inlet Temperature & 1 & 113 & 113 & 0.65 & 0.438 \\ Ambient Temperature & 1 & 2382 & 2382 & 13.74 & 0.004 \\ Initial Temperature & 1 & 3888 & 3888 & 22.43 & 0.001 \\ Initial Pressure & 1 & 7 & 7 & 0.04 & 0.846 \\
**Error** & 10 & 1734 & 173 & & \\
**Total** & 15 & 12,102 & & & \\ \hline \hline \end{tabular}
\end{table} 
Table 10: Factorial analysis for maximum temperature of inner gas. \begin{table}
\begin{tabular}{c c c} \hline \hline
**Parameter** & **Low Limit Value** & **Upper Limit Value** \\ \hline Charging Time & 60 & 240 \\ Inlet Temperature & \(-20\) & 40 \\ Ambient Temperature & \(-20\) & 40 \\ Initial Temperature & \(-20\) & 40 \\ Initial Pressure & 5 & 20 \\ \hline \hline \end{tabular}
\end{table} 
Table 8: Range of parameters for factorial analysis. \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Case** & **Charging Time** & **Inlet Temperature** & **Ambient Temperature** & **Initial Temperature** & **Initial Pressure** \\ \hline
**1** & 60 & \(-20\) & \(-20\) & \(-20\) & 20 \\
**2** & 240 & \(-20\) & \(-20\) & \(-20\) & 5 \\
**3** & 60 & 40 & \(-20\) & \(-20\) & 5 \\
**4** & 240 & 40 & \(-20\) & \(-20\) & 20 \\
**5** & 60 & \(-20\) & 40 & \(-20\) & 5 \\
**6** & 240 & \(-20\) & 40 & \(-20\) & 20 \\
**7** & 60 & 40 & 40 & \(-20\) & 20 \\
**8** & 240 & 40 & 40 & \(-20\) & 5 \\
**9** & 60 & \(-20\) & \(-20\) & 40 & 5 \\
**10** & 240 & \(-20\) & \(-20\) & 40 & 20 \\
**11** & 60 & 40 & \(-20\) & 40 & 20 \\
**12** & 240 & 40 & \(-20\) & 40 & 5 \\
**13** & 60 & \(-20\) & 40 & 40 & 20 \\
**14** & 240 & \(-20\) & 40 & 40 & 5 \\
**15** & 60 & 40 & 40 & 40 & 5 \\
**16** & 240 & 40 & 40 & 40 & 20 \\ \hline \hline \end{tabular}
\end{table} 
Table 9: Factorial analysis for hydrogen tank filling conditions.