## 4 Results and Discussion
### Influence of the Geometrical Parameters
#### 4.3.3 Effect of Coil Height (HC)  
In this section, the effect of coil height is investigated. Accordingly, two cases including case E and case F are considered for numerical simulation. The geometrical parameters of these cases  
Figure 15: Contours of the liquid fraction and temperature for cases C and D at four different times in the plane \(x=0\). (coils and shell) are listed in Tables 3 and 4. Two different coil heights including 0.24 and 0.42 m and two calculated helix diameters as a free parameter (in order to keep constant the heat transfer area) including 0.43 and 0.25 m are selected for cases E and F, respectively. The coil diameter and coil pitch are considered 0.016 and 0.04 m, respectively. The dimensions of the shell are kept constant as listed in Table 4. The configurations of the two investigated cases (E and F) are illustrated in Figure 11. The liquid fraction (LF) of two cases of E and F are depicted in Figure 16. The LF is drawn for 180 min. These cases are compared with together in four point including 40, 80, 120, and 150 min. According to Figure 16, there is no significant difference between cases E and F. As illustrated, at time = 40 min, case E has the best performance with LF improvement of 2.9%. At time = 80 min, case E has a better thermal performance in comparison with the case F with LF improvement of 3.1%. At time = 120 min, cases E and F have the same value of LF. After that, the melting rate changes and case F has a better thermal performance in comparison with case E. At time = 150 min, case F has a better melting rate with LF improvement of 1.3%. Generally, the difference between the cases E and F is low, but the melting rate of case F after 150 min is to some extent better than case E with 1.3% difference. The liquid fraction and temperature contours for these two cases at different time are illustrated in Figure 17. The contours are shown at a slice of x = 0. As it can be seen, because of the buoyancy effect and natural convection, the melting rate of shell top region is faster than bottom region. By increasing the coil height, the helix diameter decreases which can be attributed to the fact that at the start point of the melting process, the melting rate of the case E with low coil height (high helix diameter) is more than case F with high coil height (low helix diameter). Before about the time = 40 min, the LF trend is the same because the melting rate is under effect of the conduction and the melting region is a small region near the coil walls, which is the same in both cases. After time = 40 min, the small difference appears and case E has a better melting rate because of the geometry of the case E that covers the top and bottom regions of the shell so that the natural convection shows its influence on the melting rate. At time = 40 min, the LF is the same for both cases (E and F). Figure 16: Temporal evolution of the liquid fraction (LF) for cases E and F.  
It can be deduced from the results that the melting process depends on the conduction mechanism near the high temperature (in comparison with the melting temperature) walls of the coil tube (Figure 17). At time = 80 min, the melted region of the shell top region for case E is more than case F. The case E covers the bottom region of the shell more than case F so that the LF of the case E is more than case F. At time = 120 min, the LF of both cases (E and F) are the same while the melted regions of both cases are different. In case E, the top and bottom regions of the shell are almost completely melted. However, in case F, although the center region of the shell is significantly melted, a small region of top and a huge region of the bottom is in solid phase yet. After time = 120 min, the melting rate changes so that at time = 150 min, the LF of the case F is more than case E with small difference (about 1.3%). According to Figure 17, the temperature distribution difference between the cases (E and F) has the same trend with LF. However, in the case E, the top region of the shell melts faster.