## 2 Numerical Analysis Model
### Analysis of Effect of Inlet Temperature  
The inlet temperature of hydrogen introduced into the tube skid is primarily influenced by external environmental conditions. However, this temperature may increase as the hydrogen is compressed and fed into the tube skid. In scenarios involving rapid filling, precooling measures are often implemented prior to hydrogen injection to mitigate excessive temperature rise within the hydrogen tank [15]. Thus, an investigation was conducted to examine the effect of the inlet temperature on the temperature distribution within the tube skid.  
Figure 7 illustrates the temperature distribution across the central section of the tube skid following the charging process according to the inlet temperature. Contrary to the observations in Figure 6, where temperature varied throughout the tank depending on the case, alterations in the inlet temperature predominantly affected the temperature distribution in the tank's lower section, with no significant impact on the higher-temperature region at the top of the tank. This phenomenon is attributed to the increased influence of buoyancy, a consequence of the low flow velocity and diminished momentum of hydrogen  
\begin{table}
\begin{tabular}{c c c c c} \hline \hline \multirow{2}{*}{**Case**} & \multirow{2}{*}{**Charging Time**} & \multicolumn{2}{c}{**Maximum Temperature**} & \multicolumn{2}{c}{**Maximum Temperature**} & \multicolumn{2}{c}{**Maximum Temperature**} \\  & & **(Hydrogen, \({}^{\circ}\)C)** & **(Tank Shell, \({}^{\circ}\)C)** & **(Outer Surface, \({}^{\circ}\)C)** \\ \hline
**1** & **1 h (3600 s)** & 96.1 & 91.7 & 79.0 \\
**Ref** & **2 h (7200 s)** & 79.1 & 76.4 & 67.7 \\
**2** & **3 h (10,800 s)** & 69.6 & 67.6 & 61.2 \\
**3** & **4 h (14,400 s)** & 62.8 & 61.3 & 56.3 \\ \hline \hline \end{tabular}
\end{table}
Table 3: Maximum temperature for each region of tube skid by charging time.  
Figure 6: Effect of charging time on maximum temperature for each region of tube skid; (**a**) hydrogen, (**b**) tank shell, and (**c**) outer surface.  
over extended charging periods. As the inlet temperature decreases, the density difference between the incoming hydrogen and that within the tank's upper portion increases, suggesting that changes in inlet temperature primarily influence the temperature difference between the tank's upper and lower sections without markedly affecting the upper region's temperature. The regional maximum temperature analysis, as depicted in Figure 8 and detailed in Table 4, further corroborated that the tank's maximum temperature varied by only 0.5 \({}^{\circ}\)C in response to different inlet temperatures.  
Figure 8 also reveals a distinctive pattern in case 6 during the initial phase of charging on the tank's outer surface, diverging from other cases. This anomaly arose because the peak surface temperature initially appeared in the aluminum section of the inlet, which was in direct contact with hydrogen at 40 \({}^{\circ}\)C. However, as charging advanced, the location of the highest surface temperature shifted to the upper part of the tube skid, aligning with the trends observed in other cases.  
\begin{table}
\begin{tabular}{c c c c c} \hline \hline \multirow{2}{*}{**Case**} & \multirow{2}{*}{**Inlet Temperature**} & \multicolumn{1}{c}{**Maximum Temperature**} & \multicolumn{1}{c}{**Maximum Temperature**} & \multicolumn{1}{c}{**Maximum Temperature**} \\  & & **(Hydrogen, \({}^{\circ}\)C)** & **(Tank Shell, \({}^{\circ}\)C)** & **(Outer Surface, \({}^{\circ}\)C)** \\ \hline
**4** & \(-\)**20 \({}^{\circ}\)C** & 79.1 & 76.3 & 67.7 \\
**5** & **0 \({}^{\circ}\)C** & 79.2 & 76.5 & 67.9 \\
**Ref** & **20 \({}^{\circ}\)C** & 79.1 & 76.4 & 67.7 \\
**6** & **40 \({}^{\circ}\)C** & 79.6 & 76.8 & 68.1 \\ \hline \hline \end{tabular}
\end{table}
Table 4: Maximum temperature for each region of tube skid by inlet temperature.  
Figure 7: Temperature distribution of central section at completion of charging by change in inlet temperature.  
Figure 8: _Cont._