## 4 Sensitivity Analysis for Factors and Temperature Relationship Derivation
\begin{table}
\begin{tabular}{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 11: Factorial analysis for maximum temperature of tank shell. Figure 15: _Cont._ \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 & 9222 & 1844 & 15.61 & 0.000 \\
**Linear** & 5 & 9222 & 1844 & 15.61 & 0.000 \\ Charging Time & 1 & 2077 & 2077 & 17.58 & 0.002 \\ Inlet Temperature & 1 & 118 & 118 & 1.000 & 0.341 \\ Ambient Temperature & 1 & 3539 & 3539 & 29.96 & 0.000 \\ Initial Temperature & 1 & 3488 & 3488 & 29.52 & 0.000 \\ Initial Pressure & 1 & 0 & 0 & 0.000 & 0.985 \\
**Error** & 10 & 1181 & 118 & & \\
**Total** & 15 & 10,403 & & & \\ \hline \hline \end{tabular}
\end{table} 
Table 12: Factorial analysis for maximum temperature of outer surface. Further, a response surface methodology analysis was conducted to establish a regression equation for predicting the tank's maximum temperature using the three significant factors identified. A face-centered design was utilized, with data from an additional 15 cases analyzed, as shown in Table 13.  
This design facilitated the derivation of the regression equation (Equation (7)) from the analysis results. In the regression equation, the validity of the terms can be identified by applying the reference value of the \(p\)-value (0.05) to each term, and the terms with no influence can be excluded from the equation. Here, \(X_{i}\) represents the design factors, \(Z\) denotes the result, and \(\beta_{i}\) and \(\beta_{\bar{y}}\) are the coefficients for each term. \[Z=\beta_{0}+\Sigma_{i=1}^{k}\beta_{i}X_{i}+\Sigma_{i=1,j\leq i}^{k}\beta_{i}X_ {i}X_{j} \tag{7}\]  
The response surface methodology's analysis results for the tube skid are summarized in Tables 14-16, with Figure 16 illustrating the correlations between each variable and the \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Case** & **Charging Time (min)** & **Ambient Temperature (\({}^{\circ}\)C)** & **Initial Temperature (\({}^{\circ}\)C)** \\ \hline
**1** & 60 & \(-\)20 & \(-\)20 \\
**2** & 240 & \(-\)20 & \(-\)20 \\
**3** & 60 & 40 & \(-\)20 \\
**4** & 240 & 40 & \(-\)20 \\
**5** & 60 & \(-\)20 & 40 \\
**6** & 240 & \(-\)20 & 40 \\
**7** & 60 & 40 & 40 \\
**8** & 240 & 40 & 40 \\
**9** & 60 & 10 & 10 \\
**10** & 240 & 10 & 10 \\
**11** & 150 & \(-\)20 & 10 \\
**12** & 150 & 40 & 10 \\
**13** & 150 & 10 & \(-\)20 \\
**14** & 150 & 10 & 40 \\
**15** & 150 & 10 & 10 \\ \hline \hline \end{tabular}
\end{table} 
Table 13: Response surface analysis for hydrogen tank filling conditions. Figure 15: Responsivities of performance factors using fractional factorial design. tank's maximum temperature. The charging time is represented by \(t_{chr}\), the ambient temperature by \(T_{amb}\), and the initial temperature by \(T_{ini}\). By isolating only the impactful terms, the regression equation for the internal hydrogen's maximum temperature is expressed as Equation (8), with a coefficient of determination (R\({}^{2}\)) of approximately 0.9977--surpassing the often-referenced benchmark of 0.8 [25]. This high R\({}^{2}\) value suggests that the regression equation reliably predicted the maximum temperature, facilitating the establishment of charging standards for tube skids without necessitating further analysis. Similarly, regression equations for the shell and outer surface temperatures were derived, resulting in Equations (9) and (10).