## 3 Results and Discussion
### Comparison between the Horizontal and Vertical Positions  
The influence of the ice storage tank position on the ice formation is investigated numerically for the three types of tubes. The temporal variations of the liquid fraction (LF) for three different types of tube including a simple U-shaped tube, a coil tube with an inner return line (model 1), and a coil tube with an outer return line (model 2) are illustrated in Figure 13a-c, respectively, for two different positions of the ice storage tank. Figure 13a shows that for the simple U-shaped tube's case, until 8 h, there is no noticeable difference between the two positions. However, as time passed, the difference between the models became significant. Finally, after 20 h, this type of ice storage system in its horizontal position produces more ice compared to the tank in the vertical position.  
Figure 13b shows that for the case with a coil tube with an inner return line (model 1), before time = 12.6 h, the horizontal position produces more ice than the vertical one. At time = 12.6 h, the amount of produced ice for the two positions are the same. After that, as time passes, the vertical position exhibits better performance in comparison with the horizontal position. It should be noted that the differences between the two positions (for this set of operating conditions and geometrical parameters) remain very small. Contrary to the results obtained in Figure 13b, for the case with a coil tube and an outer return line (model 2), the vertical position shows higher produced ice, as compared  
Figure 11: Temporal evolution of the liquid fraction for three types of tubes in the ice on coil ice storage system in the vertical position.  
Figure 12: Contours of liquid fraction and velocity magnitude (m/s) for three types of tube in the vertical position after 14 h and at midplane Y = 0.  
with the horizontal one (Figure 13c). The effect of position on the ice formation in the case with a coil tube and an outer return line is significantly more pronounced than the case with a coil tube and an inner return line.  
In order to better show the effect of natural convection on the solidification process, the 3D contours of the velocity magnitude for different cases are illustrated in Figure 14 with the system in the vertical position. Natural convection clearly affects the velocity field. However, because the solidification process is here investigated for only 14 h, the ice is formed just around the tube. The simulation time is too short for the liquid fraction to highlight a significant effect of natural convection.  
As shown in Figure 13, the position of the storage system (vertical and horizontal) does not affect significantly the ice formation. Natural convection has indeed no noticeable influence on the solidification process, as previously discussed.  
To go a little bit further, the three-dimensional streamlines and the contours of the velocity magnitude are illustrated in Figures 15 and 16 for the horizontal and vertical positions, respectively. In the horizontal position (Figure 15), whatever the geometry of the HTF tube, the streamline patterns remain the same. The main vortexes are produced around the Z-axis because of the direction of gravity along the X-axis. In the vertical position (Figure 16), the streamlines get completely different. Streamlines are then a better indicator compared to the average liquid fraction to highlight the influence  
Figure 13: Comparison between two positions of the ice-on-coil ice storage system: (**a**) single U-shaped, (**b**) coil tube-model 1, (**c**) coil tube-model 2.  
of natural convection on the solidification process. This is mainly due to the relatively short simulation time (16 h) for such a storage volume and to the external melt type.  
Figure 14: Three-dimensional contours of the velocity magnitude (m/s) for the three models.  
Figure 15: Three-dimensional streamlines with contours of the velocity magnitude for the three different cases in the horizontal position.