## 3 Mathematical Model
### Influence of Inclination Angle on Melting Process
Then, the liquid paraffin flows down along the solid liquid interface to the bottom. Therefore, a natural heat flow cycle is formed. In Figure 12b, the temperature variations of all motoring points are similar to that in Figure 11b. The only difference is that the temperatures of \(T_{2}\), \(T_{3}\), and \(T_{4}\) are approximately the same because they have the same ordinate. Figure 11: The melting process of paraffin at \(60^{\circ}\). (**a**) Liquid fraction; (**b**) temperature variation. Figure 10: The melting process of paraffin at \(30^{\circ}\). (**a**) Liquid fraction; (**b**) temperature variation. Figures 13 and 14 visualize the liquid fraction of the melting process and the temperature variation of monitoring points when the inclination angles are 120\({}^{\circ}\) and 150\({}^{\circ}\), respectively. For these inclination angles, the bending degrees of the solid liquid interface are smaller than that in the vertical container, which are shown in Figures 13a and 14a. There is also only one natural heat flow cycle inside the liquid paraffin. Furthermore, the melting rate of paraffin at 150\({}^{\circ}\) is much slower. This is because the heat flow of gravity direction component is inhibited by buoyancy. In Figure 13b, the temperature variation of \(T_{1}\) is the same as that in Figure 12b, because the heat conduction plays the dominate role at this monitoring point. The temperature variations of \(T_{2}\), \(T_{3}\), \(T_{4}\) and \(T_{5}\) stay longer in the phase change region in Figure 13b, and the slopes of these temperature curves are smaller than those at 90\({}^{\circ}\). In Figure 14b, the range of temperature constant area is further expanded. The effect of natural convection is gradually decreasing. It can be seen that the paraffin at monitoring points 4 and 5 do not melt in 10,000 s.  
Figure 12: The melting process of paraffin at 90\({}^{\circ}\). (**a**) Liquid fraction; (**b**) temperature variation. Figure 13: The melting process of paraffin at 120\({}^{\circ}\). (**a**) Liquid fraction; (**b**) temperature variation. Figure 15 visualizes the melting process of paraffin with top horizontal heating (\(\alpha\) = 180\({}^{\circ}\)). In Figure 15a, the part of fully melted paraffin is growing very slowly. Because buoyancy drives the hot liquid paraffin moving upward, the paraffin melting downward needs to overcome the effect of buoyancy in the opposite direction. In Figure 15b, it can be depicted that temperature on the top of the liquid paraffin is higher than that on the bottom. Besides, the temperatures of monitoring points (2, 3 and 4) are almost the same. Therefore, in the design of steady-state method, the heat source is always at the top of the specimen to overcome the effect of buoyancy. Figure 16 depicts the curves of liquid fraction varying with time at different angles. In the initial heating stage (about 2000 s), the liquid fractions of all inclination angles are approximately the same. This is because the heat conduction is the domain heat transfer mode during this period. After that, the convective heat transfer begins to dominate the melting process. Compared with the melting process of other inclination angles, the melting rate at 0\({}^{\circ}\) is the fastest and the speed of melting at 180\({}^{\circ}\) is the lowest. Figure 14: The melting process of paraffin at 150\({}^{\circ}\). (**a**) Liquid fraction; (**b**) temperature variation. Figure 15: The melting process of paraffin with top horizontal heating (180\({}^{\circ}\)). (**a**) Liquid fraction; (**b**) temperature variation. Figure 17 shows the relationship between the complete melting time and the inclination angles. By means of observing the liquid fraction curves at different inclination angles, it can be concluded that the melting rates of paraffin are non-linearly decreased with the increase of angle, which is expressed in Equation (13). Based on this relationship, the inclination angle can be selected according to the actual technical requirements of temperature control system.