## 3 Mathematical Model
### Influence of Inclination Angle on Melting Process  
Figure 9a visualizes the liquid fraction of the melting process when the paraffin is heated from the bottom (\(\alpha=0^{\circ}\)). The solid liquid interface is almost flat during the early stage of melting because the dominant mode of heat transfer is heat conduction. As time goes on, the shape of the solid liquid interface becomes wavy because of Benard convection cells formed by the combination of buoyancy and gravity inside the liquid paraffin [25]. During the melting process, hot liquid paraffin rises away from the heating wall vertically along the central area of the cell under the action of buoyancy. After reaching the solid liquid interface, the solid paraffin is melted and fell along the cell edge under the action of gravity. It is noted that the mean diameter of the cells increases so that the number of cells per unit area decreases as the melting time increased. Figure 9b shows the temperature variations of monitoring points. Owing to the effect of the Benard convection cells, the temperature of each monitoring point is almost the same after the paraffin is completely melted. Figures 10 and 11 visualize the liquid fraction of the melting process and the temperature variation of monitoring points when the inclination angles are \(30^{\circ}\) and \(60^{\circ}\), respectively. In the initial period of melting, the temperature variations are almost the same as the inclination angle was \(0^{\circ}\) at both angles. This is because the heat conduction is the dominant mode of heat transfer in this period of time. Furthermore, in Figure 10a, the paraffin with a higher position melts faster because of the effect of natural convection. Further, there are also Benard convection cells inside the paraffin during the melting process. Besides, Figure 10b refers to the temperature histories of monitoring points inside the specimen. During the melting procedure, \(T_{1}\) increases sharply after 2000 s, and the temperatures of \(T_{2}\), \(T_{3}\) and \(T_{4}\) are almost the same until the paraffin is completely melted in these monitoring points. This is because of the influence of the Benard convection cells. The temperature of \(T_{5}\) increases slowly in the phase change region (approximately between 4000 s and 6000 s), because \(T_{5}\) is far away from the heating wall and is heated much slower. In Figure 11a, the solid liquid interface becomes smoother. The melting rate on the top is much faster than that on the bottom, and Benard convection cells no longer exist in the specimen. In Figure 11b, it is found that the temperature variations of all thermocouples are hierarchical. In the phase change region, \(T_{1}\) has an increase because of the influence of the nature convection. Compared with \(T_{1}\) at 30 \({}^{\circ}\)C, \(T_{1}\) at 60 \({}^{\circ}\)C increases much slower. Besides, the shapes of \(T_{2}\), \(T_{3}\) and \(T_{4}\) curves are very similar, and \(T_{2}\), \(T_{3}\) and \(T_{4}\) increase to 75.0 \({}^{\circ}\)C, 74.5 \({}^{\circ}\)C and 73.5 \({}^{\circ}\)C after melting, respectively. At 7000 s, \(T_{5}\) jumps to 76.7 \({}^{\circ}\)C, which is only lower than \(T_{2}\) because of its location. Figure 9: The melting process of paraffin at \(0^{\circ}\). (**a**) Liquid fraction; (**b**) temperature variation. Figure 12 visualizes the melting process at \(90^{\circ}\). In Figure 12a, the trends of temperature variation and liquid fraction are very similar to the melting process of paraffin at \(60^{\circ}\). The hot liquid paraffin near the heating wall rises up vertically because of the force of the buoyancy with no velocity component in the \(x\)-direction. It leads to a higher temperature on the top than the temperature on the bottom. However, as the buoyancy hinders the liquid paraffin from flowing downward, the liquid paraffin flows transversely. When the liquid paraffin arrives at the solid liquid interface, it melts the solid paraffin near the solid liquid interface.