## 3 Results and Discussion
### Analysis of Flow and Heat Transfer Characteristics Under Sloshing Conditions
#### 3.4.3 Sloshing Characteristics at Different Filling Rates  
In this section, the sloshing dynamics of liquid in a tank under different filling rates are systematically analyzed, with natural periods identified as 16 s for 30% and 11.89 s for 50% filling rates. At a 30% filling rate (Figure 22a), when the sloshing periods (6 s, 10 s) are significantly shorter than the natural period (16 s), the free surface remains stable and continuous, with only a few wave peaks observed. As the period approaches the natural period (15 s, 16 s), resonance effects intensify, leading to severe surface fragmentation accompanied by wall-adherent jets and splashing (Figure 22a, T15-T16). Notably, at T = 15 s (close to the natural \begin{table}
\begin{tabular}{c c c} \hline \hline
**Periods (s)** & **Liquid \(\Delta\)T (K)** & **Vapor \(\Delta\)T (K)** \\ \hline
6 & 0.012 & 0.216 \\
8 & 0.009 & 0.106 \\
8.6 & 0.009 & 0.101 \\
10 & 0.003 & 0.068 \\
15 & 0.011 & 0.151 \\ \hline \hline \end{tabular}
\end{table} 
Table 6: Temperature difference (\(\Delta\)T) at 120 s for different sloshing periods. Figure 21: Temperature distribution at 80% filling rate under different sloshing periods. period of 16 s), significant surface breakup already occurs, indicating higher sensitivity to resonance at lower filling rates. In contrast, at a 50% filling rate (Figure 1(b)), when the period (15 s) exceeds the natural period (11.89 s), inertial forces drive wall-adherent jets while maintaining surface continuity (Figure 1(b), T15), contrasting sharply with the intense splashing observed at the natural period (11.89 s). This behavior aligns with observations at an 80% filling rate, suggesting that higher filling rates suppress surface rupture by enhancing liquid inertia. Near the natural period (Figure 1(a), T15-T16; Figure 1(b), T10-T11.89), both filling rates exhibit interfacial instabilities, but with distinct manifestations. At 30% filling, gas-liquid interactions dominate energy dissipation, leading to surface breakup and splashing (Figure 1(a)). At 50% filling, liquid inertia prevails, resulting in more persistent jets during resonance, while reduced gas cushioning exacerbates energy accumulation, ultimately causing more violent splashing (Figure 1(b), T11.89). Additionally, in non-resonant regimes (e.g., T = 15 s at 50% filling), higher liquid mass delays surface rupture, allowing jets to propagate stably along the walls. Figure 23 illustrates distinct temperature evolution patterns at 30% (Figure 1(a), natural period: 16 s) and 50% (Figure 1(b), natural period: 11.89 s) filling rates. To quantify these observations, the temperature difference (\(\Delta\)T) between the maximum and minimum temperatures within the gas phase at a representative time point was calculated for different filling rates and sloshing periods (Table 7). At periods far from resonance (e.g., 6 s), the high sloshing frequency and large wave amplitudes result in a significantly increased gas-liquid interfacial area and enhanced convective heat transfer between the phases. This intensified heat exchange can influence the vapor temperature distribution by potentially limiting the rise of the maximum vapor while possibly increasing the minimum vapor temperature or maintaining it near the initial value, leading to a smaller temper  
Figure 22: Phase contours at 30% (**a**) and 50% (**b**) filling rate under different sloshing periods. in Table 7, the vapor temperature difference (\(\Delta\)T) is 0.310 K at 6 s (30% fill) and 0.329 K at 6 s (50% fill), suggesting that while interfacial heat transfer is strong, other factors also contribute to the overall temperature stratification.