## 4 Conclusions  
This study simulates the freezing process in ballast tanks of a polar LNG carrier. Utilizing Fluent fluid dynamics software, we integrate interphase mass, momentum, and heat transfer models into the Volume of Fluid (VOF) multiphase flow model to establish a numerical calculation framework for ballast water freezing. The accuracy of this numerical model is validated by comparing it with the ice phase volume fraction reported in the literature. We investigate the effects of the cold air temperature, seawater temperature, and ballast water level height on the temperature distribution and freezing process within the ballast tank. The main conclusions are as follows:  
1. The decrease in cold air temperature significantly impacts the temperature of ballast water. As the cold air temperature drops, the rate of temperature reduction at the ballast water level accelerates markedly. When the cold air temperature reaches \(-40\)\({}^{\circ}\)C, the temperature of the ballast water reaches its lowest point at \(-1.34\)\({}^{\circ}\)C. Additionally, changes in the seawater temperature greatly affect the temperature near the bulkhead. As the seawater temperature decreases, the rate of temperature reduction in the ballast water at the right bulkhead also accelerates significantly. When the seawater temperature is \(-4\)\({}^{\circ}\)C, the ballast water temperature at the right bulkhead reaches its lowest point at \(-1.518\)\({}^{\circ}\)C. The change in the height of the ballast water level has a substantial influence on the heat transfer rate. As the height of the liquid level decreases, the time required for heat transfer increases, causing the heat transfer rate to slow down. When the height of the liquid level is \(5\) m, the heat transfer rate is at its fastest, and the temperature of the ballast water at that level is the lowest, measuring \(-1.14\)\({}^{\circ}\)C.
2. The decrease in the cold air temperature significantly impacts the evolution of ice formation at the ballast water level. As the cold air temperature drops, the thickness of the ice layer formed by freezing at the surface of the ballast water increases. When the ambient temperature reaches \(-40\)\({}^{\circ}\)C, the volume fraction of the ice phase reaches its maximum, approximately \(84.33\%\). Additionally, changes in the seawater temperature primarily influence the evolution of ice formation at the right bulkhead. As the seawater temperature decreases, the thickness of the ice layer formed by freezing at the right bulkhead also increases significantly. When the seawater temperature is \(-4\)\({}^{\circ}\)C, the freezing degree of the ballast tank is at its highest, with the volume fraction of the ice phase in the ballast tank reaching approximately \(91.53\%\). The change in the height of the ballast water level primarily affects the volume of ballast water. As the liquid level decreases, the volume of ballast water diminishes, resulting in a shorter time required for the ballast water to freeze and an increase in the degree of freezing. When the liquid level height is \(2\) m, the volume fraction of the ice phase is approximately \(92.35\%\). 3. The variation in the temperature field has the most significant impact on the evolution of ice shape. As the rate of change in the ballast water temperature increases, the rate of ice shape evolution also shows an upward trend. Notably, in areas where the temperature field experiences substantial fluctuations, a faster freezing rate in ballast water leads to a thicker ice layer and a greater degree of freezing. Due to limitations in technical resources and research areas, this paper presents certain deficiencies and flaws in the numerical study of ballast water freezing. The proposed numerical model lacks experimental validation, which undermines its reliability. Furthermore, the effects of salinity and flow on the freezing process of ballast water were overlooked during the calculations, resulting in a significant deviation from the actual distribution of ice shapes.