## 3 Mathematical Model
### Numerical Procedure  
In the present study, the numerical analysis is performed using a commercially available computational fluid dynamics (CFD) program, Fluent 14.5 [23]. A two-dimensional and transient model is built to simulate the melting process of paraffin in the container. The Second-Order Implicit Method for Pressure-Linked Equations (SIMPLE) is employed for the pressure-velocity coupling field. The discretization method for pressure uses the Pressure Staggering Option (PRESTO) scheme, which is advisable for calculating the natural convection. The grids are uniformly structured with quadrilateral cells. Then, grids consisting of 3000, 6000 and 10,000 grids are all tested for the liquid fraction, and relative errors are calculated in Equation (11) [24]. Figure 6 shows the numerical results and relative errors of vertical heating. \[e=\left|\frac{f_{n}-f_{std}}{f_{std}}\right| \tag{12}\]  
in which \(f_{n}\) is the liquid fraction of measurement value, which is the data at 3000 and 6000 grids. \(f_{std}\) is the standard value, which is the data at 10,000 grids. It is concluded that the differences of liquid fraction at 6000 and 10,000 grids are less than 2%, while the result at 3000 grids has larger relative error. Besides, the effect of time step size is examined by considering time step sizes of 1 s, 2 s and 4 s with the grid number of 6000. Figure 7 shows the numerical results and relative errors with these time steps. It is depicted that the differences of liquid fraction between 2 s and 1 s are less than 0.1%, while relative error of 4 s is less than 0.5%. Therefore, a grid of 6000 and a time step of 2 s are appropriate for the numerical calculation to balance the accuracy of results and the computational time. \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Definite Conditions** & **Locations** & **Velocity** & **Conditions** \\ \hline Initial condition & \(0\leq x\leq 100\), \(0\leq y\leq 30\) & \(U=v=0\) & \(T=T_{init}=25\)\({}^{\circ}\)C \\ \hline Boundary conditions & \(0\leq x\leq 100\), \(y=0\) & \(U=v=0\) & UDF from 25 to 80\({}^{\circ}\)C \\ \(x=0\), \(0\leq y\leq 30\) & \(U=v=0\) & \(\dfrac{\partial T}{\partial x}=0\) \\ \(x=0\), \(0\leq y\leq 30\) & \(U=v=0\) & \(\dfrac{\partial T}{\partial x}=0\) \\ \(x=100\), \(0\leq y\leq 30\) & \(U=v=0\) & \(\dfrac{\partial T}{\partial x}=0\) \\ \hline \hline \end{tabular}
\end{table} 
Table 2: The initial and boundary conditions. UDF—user defined function. Figure 6: Results of grid-independent analysis. In Table 3, it can be found that the solid liquid interfaces begin to curve at 30 min, because the natural convection starts to play a dominant role from this time. However, the bending degree is very small at each inclination angle to compare with the experimental result. At 90 min, the paraffin is almost completely melted at 0\({}^{\circ}\) and 30\({}^{\circ}\), which cannot be well compared. In addition, at 60 min, natural convection plays the dominant role of heat transfer. At this time, the characteristics of solid liquid interface with each inclination angle can be clearly observed, so that the 60 min after the start of heating is selected to compare shapes of solid liquid interfaces between the numerical results and the experimental result. Table 4 shows both numerical results of liquid fraction and pictures of solid liquid interface location taken by the camera at 60 min at different inclination angles. It can be found that the shapes of the solid liquid interfaces are different when the inclination angle varies. At 0\({}^{\circ}\) and 30\({}^{\circ}\), the solid liquid interfaces are wavy, because there are several Benard convection cells inside the specimen. At 60 to 150\({}^{\circ}\), the shapes of solid liquid interfaces are similar, because there is only one natural heat flow cycle inside the specimen. However, because of the different inclination angles, the natural convection has different effects on the melting process.