## 2 Numerical Analysis Model
### Verification of Analysis Model  
The numerical model used in this study incorporates the properties of high-pressure hydrogen and various governing equations, utilizing a 2D geometric representation of the hydrogen tank. To verify the model's reliability, we used a study by [PERSON] et al. [13], which similarly employed a 2D axisymmetric model alongside NIST property data. The tank geometry in [PERSON] et al.'s study was approximated using the digitizer function within Origin software. Given that [PERSON] et al.'s tank was oriented horizontally, our analysis model was adjusted by omitting the gravity component from the numerical model utilized in our research.  
In [PERSON] et al.'s experiment, hydrogen pressure exhibited an approximately linear increase; accordingly, we set the inlet pressure's rise from approximately 5 to 64 MPa over 180 s, based on the graph from the verification experiment presented in their paper. [PERSON] et al. did not specify values for the convective heat transfer coefficient of the tank's outer surface; consequently, we assumed a convective heat transfer coefficient of 6 W/(m\({}^{2}\) K). The temperature rise of the tank predicted by our model was then compared with the  
\begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Case** & **Charging Time** & \begin{tabular}{c} **Inlet** \\ **Temperature (\({}^{\circ}\)C)** \\ \end{tabular} & \begin{tabular}{c} **Ambient** \\ **Temperature (\({}^{\circ}\)C)** \\ \end{tabular} & \
\begin{tabular}{c} **Initial** \\ **Temperature (\({}^{\circ}\)C)** \\ \end{tabular} &
\begin{tabular}{c} **Initial Pressure** \\ **(bar)** \\ \end{tabular} \\ \hline
**Ref** & **2 h (120 min)** & **20** & **20** & **20** & **15** \\
**1** & **1 h (60 min)** & 20 & 20 & 20 & 15 \\
**2** & **3 h (180 min)** & 20 & 20 & 20 & 15 \\
**3** & **4 h (240 min)** & 20 & 20 & 20 & 15 \\ \hline
**4** & 2 h (120 min) & **\(-\)20** & 20 & 20 & 15 \\
**5** & 2 h (120 min) & **0** & 20 & 20 & 15 \\
**6** & 2 h (120 min) & **40** & 20 & 20 & 15 \\ \hline
**7** & 2 h (120 min) & 20 & **\(-\)20** & 20 & 15 \\
**8** & 2 h (120 min) & 20 & **0** & 20 & 15 \\
**9** & 2 h (120 min) & 20 & **40** & 20 & 15 \\ \hline
**10** & 2 h (120 min) & 20 & 20 & **\(-\)20** & 15 \\
**11** & 2 h (120 min) & 20 & 20 & **0** & 15 \\
**12** & 2 h (120 min) & 20 & 20 & **40** & 15 \\ \hline
**13** & 2 h (120 min) & 20 & 20 & 20 & **5** \\
**14** & 2 h (120 min) & 20 & 20 & 20 & **10** \\
**15** & 2 h (120 min) & 20 & 20 & 20 & **20** \\ \hline \hline \end{tabular}
\end{table}
Table 2: Numerical analysis list for operating conditions and environmental conditions.  
temperature increase derived from the verification experiment graph in [PERSON] et al.'s paper, with results illustrated in Figure 4. Although an error margin of up to 6.6 \({}^{\circ}\)C was observed in the tank's temperature rise--attributable to the reliance on estimated values from the graph--the overall trends between the two datasets were consistent, with an error of less than 1 \({}^{\circ}\)C at both the initial and final points. Because our study's focus was on the tank's maximum temperature at the end of charging, the error at the final charging point was considered a more critical evaluative criterion than the peak error observed throughout the charging process.