## 3 Results and Discussion
### Analysis of Flow and Heat Transfer Characteristics Under Sloshing Conditions
#### 3.4.2 Effect of Different Sloshing Period  
To comprehensively investigate the influence of sloshing excitation on the hydrodynamic characteristics and boil-off gas (BOG) generation in the CO\({}_{2}\) storage tank, additional sloshing conditions are examined. This section focuses on the study of different sloshing periods. As is well-known, the impact of sloshing reaches its maximum at resonance, where the sloshing intensity is most pronounced. The resonance period typically occurs near the natural period of the system. For the Type C CO\({}_{2}\) tank studied in this work, the first-mode natural period is 8.6 s under an 80% filling ratio. Consequently, in addition to the default sloshing period of 6 s analyzed in Section 3.4.1, the hydrodynamic characteristics and BOG generation behavior are also investigated at sloshing periods of 8 s, 10 s, and 15 s.  
Figures 19 and 20, respectively, present the phase distribution diagrams and free surface elevation profiles under sloshing periods of 6 s, 8 s, 8.6 s, 10 s, and 15 s. At a sloshing period of 6 s, the free surface exhibits higher oscillation amplitudes but remains relatively stable, with the liquid surface maintaining an almost consistent level. In contrast, at 15 s, the free surface displays more pronounced fluctuations, indicating the formation of  
Figure 18: Temperature in static and sloshing conditions. \begin{table}
\begin{tabular}{c c} \hline
**Filling Ratio (\%)** & **Average BOG Generation Rate (kg/s)** \\ \hline
50 & 0.0000865 \\
60 & 0.0000987 \\
70 & 0.0001236 \\
80 & 0.0002314 \\
90 & 0.0002612 \\
95 & 0.0003001 \\ \hline \end{tabular}
\end{table} 
Table 5: Average BOG generation rate at different filling ratios. multiple wave patterns within each oscillation cycle. This behavior suggests that lower-frequency sloshing (15 s) induces more complex and dynamic interfacial motion compared to higher-frequency sloshing (6 s). For sloshing periods close to the natural period, distinct differences are observed. At 10 s, the free surface exhibits higher oscillation amplitudes, while at the natural period of 8.6 s, the liquid surface height variations are the most intense. Specifically, at 8.6 s, the free surface undergoes continuous transformation into a highly fragmented state, accompanied by the formation of intense jets along both sides of the tank wall. This phenomenon is attributed to the resonance effect, which amplifies the sloshing energy and leads to pronounced interfacial disturbances, including wave breaking and jet formation. Figure 21 illustrates the temperature distribution under sloshing periods of 6 s, 8 s, 8.6 s, 10 s, and 15 s. At periods far from the natural period (6 s and 15 s), significant temperature gradients are observed. Specifically, in the gas phase region, a temperature difference of approximately 0.3 K is noted, while a smaller but noticeable temperature gradient also exists in the liquid phase region. Notably, at 6 s, the temperature gradient in the gas phase is more pronounced, whereas at 15 s, the temperature gradient in the liquid phase becomes more dominant. Figure 19: Phase contours at 80% filling rate under different sloshing periods. Figure 20: Free liquid level height distribution curve at 80% filling rate. To quantify these observations, we calculated the temperature difference (\(\Delta\)T) between the maximum and minimum temperatures within each phase at 120 s for each sloshing period. The results are summarized in Table 6.  
As shown in Table 6, the temperature difference in the gas phase is significantly larger than in the liquid phase across all periods. The gas phase \(\Delta\)T is largest at 6 s (0.216 K) and 15 s (0.151 K), indicating more pronounced thermal stratification away from the natural period. Near the natural period (8 s, 8.6 s, 10 s), the gas phase \(\Delta\)T is reduced to approximately 0.1 K or less, suggesting a more uniform temperature distribution. The liquid phase \(\Delta\)T remains small across all periods.