## 2 Numerical Analysis Model
### Analysis of Effect of Initial Pressure  
The residual hydrogen quantity and pressure within a tube skid following unloading may differ based on when the release is halted, implying that the initial pressure might vary for each subsequent charging cycle. Alterations in the initial pressure can influence the enthalpy and temperature of the internal hydrogen at the end of charging, as they affect the compression ratio needed to reach the maximum pressure. Therefore, an analysis was conducted to determine the impact of initial pressure variations, with the assumption that the tank was nearly depleted to set a benchmark for maximum discharge. The investigation focused on observing any significant temperature disparities resulting from initial pressure adjustments within the 5 to 20 bars range.  
\begin{table}
\begin{tabular}{c c c c} \hline
**Case** & **Initial Temperature** & \begin{tabular}{c} **Maximum Temperature** \\ **(Hydrogen, \({}^{\circ}\)C)** \\ \end{tabular} & \
\begin{tabular}{c} **Maximum Temperature** \\ **(Tank Shell, \({}^{\circ}\)C)** \\ \end{tabular} &
\begin{tabular}{c} **Maximum Temperature** \\ **(Outer Surface, \({}^{\circ}\)C)** \\ \end{tabular} \\ \hline
**10** & \(-\)**20 \({}^{\circ}\)C** & 55.6 & 53.4 & 47.4 \\
**11** & \(0\) \({}^{\circ}\)C & 66.9 & 64.4 & 57.1 \\
**Ref** & \(20\) \({}^{\circ}\)C & 79.1 & 76.4 & 67.7 \\
**12** & \(40\) \({}^{\circ}\)C & 91.8 & 88.9 & 79.1 \\ \hline \end{tabular}
\end{table}
Table 6: Maximum temperature for each region of tube skid by initial temperature.  
Figure 12: Effect of initial temperature on maximum temperature for each region of tube skid; (**a**) hydrogen, (**b**) tank shell, and (**c**) outer surface.  
According to Figure 13, which illustrates the central section's temperature distribution upon charging completion, variations in the initial pressure from 5 to 20 bars did not significantly affect the final temperature distribution, with discrepancies under 4 \({}^{\circ}\)C. This phenomenon likely occurred because a lower initial pressure necessitated more compression as the tank filled, yet the relevance of compressing the initially present hydrogen diminished as its initial mass decreased. As shown in the case 13 graph in Figure 14a, the temperature initially surged when starting at a lower pressure, but then the rate of increase moderated, aligning with other cases as more hydrogen was introduced. Consequently, Figure 14 confirms a marginal uptrend in maximum temperature with decreasing initial pressure, although with relatively low sensitivity to the pressure changes. Detailed results are provided in Table 7.  
\begin{table}
\begin{tabular}{c c c c c} \hline \hline \multirow{2}{*}{**Case**} & \multirow{2}{*}{**Initial Pressure**} & **Maximum Temperature** & **Maximum Temperature** & **Maximum Temperature** \\  & & **(Hydrogen, \({}^{\circ}\)C)** & **(Tank Shell, \({}^{\circ}\)C)** & **(Outer Surface, \({}^{\circ}\)C)** \\ \hline
**13** & **5 bar** & 81.5 & 78.6 & 69.5 \\
**14** & **10 bar** & 80.1 & 77.2 & 68.4 \\
**Ref** & **15 bar** & 79.1 & 76.4 & 67.7 \\
**15** & **20 bar** & 78.1 & 75.3 & 66.7 \\ \hline \hline \end{tabular}
\end{table}
Table 7: Maximum temperature for each region of tube skid by initial pressure.  
Figure 13: Temperature distribution of central section at completion of charging by change in initial pressure.  
Figure 14: Cont.