## 3 Results and Discussion
### Analysis of Flow and Heat Transfer Characteristics Under Sloshing Conditions
#### 3.4.3 Sloshing Characteristics at Different Filling Rates
\begin{table}
\begin{tabular}{c c c c} \hline
**Filling Rate (\%)** & **Periods (s)** & **Liquid \(\Delta\)T (K)** & **Vapor \(\Delta\)T (K)** \\ \hline \multirow{4}{*}{30} & 6 & 0.006 & 0.310 \\  & 8 & 0.004 & 0.410 \\  & 10 & 0.007 & 0.430 \\  & 15 & 0.002 & 0.084 \\  & 16 & 0.003 & 0.105 \\ \hline \multirow{4}{*}{50} & 6 & 0.008 & 0.329 \\  & 8 & 0.005 & 0.219 \\  & 10 & 0.003 & 0.084 \\  & 11.89 & 0.002 & 0.054 \\  & 15 & 0.004 & 0.075 \\ \hline \multirow{4}{*}{80 (from Section 3.4.2)} & 6 & 0.012 & 0.216 \\  & 8 & 0.009 & 0.106 \\ \cline{1-1}  & 8.6 & 0.009 & 0.101 \\ \cline{1-1}  & 10 & 0.003 & 0.068 \\ \cline{1-1}  & 15 & 0.011 & 0.151 \\ \hline \end{tabular}
\end{table} 
Table 7: Temperature difference (\(\Delta\)T) for different filling rates and sloshing periods (K). Figure 23: The temperature distribution at a 30% (**a**) and 50% (**b**) filling rate. Conversely, as the period lengthens away from resonance (e.g., 10 s), the reduced wave amplitude and potentially smaller number of wave peaks lead to a decreased gas-liquid interfacial area and less efficient heat transfer. This allows the maximum vapor temperature to rise more significantly due to weaker cooling from the liquid phase, resulting in larger temperature gradients (Vapor \(\Delta\)T reaches 0.430 K at 10 s period for 30% fill). As the period approaches the natural frequency (15 s, 16 s for 30%; 10 s, 11.89 s for 50%), resonance-induced liquid surging and jet formation enhance gas-liquid mixing, effectively homogenizing the temperature distribution and significantly reducing the vapor temperature gradient. For the 50% filling rate, similar trends are observed, with inertial forces at super-resonance periods (e.g., 15 s) also playing a role in temperature distribution. The slightly lower overall temperatures in the 50% case are attributed to improved convective mixing from greater liquid inertia. As shown in Table 7, the temperature difference within the liquid phase (Liquid \(\Delta\)T) remains minimal (typically less than 0.015 K) across all tested sloshing periods and filling rates. This indicates that the liquid phase maintains a relatively uniform temperature distribution, with minimal thermal stratification, even under dynamic sloshing conditions. The interplay between filling rate and sloshing period governs thermal stratification. Lower filling rates (30%) can exhibit larger gas-phase temperature gradients at off-resonance conditions due to pronounced interfacial interactions, while higher fills (50%, 80%) prioritize inertial-driven mixing, reducing stratification, especially near resonance. In summary, the sloshing-induced thermal behavior is strongly dependent on both the filling rate, which influences the liquid inertia and interfacial area, and the sloshing period, which determines the intensity of interfacial motion and mixing efficiency. Figure 24 illustrates the liquid height level curves at different periods. At 30% filling (Figure 24a), the liquid level curves for 6 s, 8 s, and 10 s exhibit smooth and continuous profiles, with amplitudes slightly increasing as the oscillation period lengthens. At 15 s and 16 s (near the natural period of 16 s), the free surface undergoes significant fragmentation, characterized by elevated peaks corresponding to topping phenomena and wall-adherent jets. These resonance-driven instabilities result in larger amplitude fluctuations compared to non-resonant periods. Notably, the amplitude at 15 s approaches the maximum observed value, reflecting intensified fluid motion near the natural period. For 50% filling (Figure 24b), the liquid level curves at 6 s, 8 s, and 15 s maintain smooth profiles with moderate amplitude variations. However, near the natural period (10 s and  
Figure 24: Free liquid level height curves at 30% (**a**) and 50% (**b**) filling rate. 11.89 s), the free surface exhibits pronounced distortions.