## 4 Results and Discussion
After the fluid left the cooling section and entered the adiabatic section, the PCM could completely solidify, leading to \(\xi_{b}=0\) (as indicated by the blue circle). Compared with other cases (as shown by the solid red arrow), the lack of supercooling for \(k_{wf}^{*}=0.1,\ t_{w}=0.1\) facilitated early PCM melting in the heating section (as indicated by the dotted red arrow). The rapid completion of the melting process resulted in \(\xi_{b}=1\). For \(k_{wf}^{*}=0.1,\ t_{w}=0.2\)-0.5 (i.e., a slightly thicker pipe with poor thermal conductivity), \(\xi_{b}=1\) in all the parts of the loop, showing the utilization of latent heat in the loop (Please also refer to Figure 5a, \(\left(\Phi_{h}^{*}\right)_{lat}=0\): yellow area). Figure 5 shows latent heat contribution versus wall thickness, where \(\left(\Phi_{h}^{*}\right)_{lat}\) represents latent heat contribution to the overall heat transfer of the heating section (\(=\left(q_{h}^{*}\right)_{lat}/q_{h}^{*}\)). For \(Ra^{*}=10^{10}\) (Figure 5a), increasing the wall thickness from theoretical condition \(t_{w}=0\) (shown by \(\blacktriangleright\) in the figure) to \(t_{w}=0.1\) did not result in significant change in \(\left(\Phi_{h}^{*}\right)_{lat}\) (as indicated by the area enclosed by the dotted red line), because the final wall thickness was not large. However, increasing wall thickness affected \(\left(\Phi_{h}^{*}\right)_{lat}\) differently for different pipe-wall materials (corresponding to different thermal-conductivity levels). For \(k_{wf}^{*}=0.1\), \(t_{w}\geq 0.2\), loop wall temperature was too high because the wall had low thermal conductivity and was relatively thick. Thus, the PCM particles did not solidify in the cooling section (as indicated by the dotted red circle in the insert of Figure 5a), and latent heat did not play a role in the heating section. For \(k_{wf}^{*}=100\), increasing wall thickness (0.1-0.5) increased \(\left(\Phi_{h}^{*}\right)_{lat}\) in the heating section, effectively using latent heat. Figure 5b shows that, for \(k_{wf}^{*}=0.1\), \(t_{w}\geq 0.2\), the \(Ra^{*}=10^{11}\) and  
\(Ra^{*}=10^{10}\) cases were similar, and latent heat was not utilized. For \(k_{wf}^{*}\geq 1\), increasing wall thickness (0.1-0.5) increased \(\left(\Phi_{h}^{*}\right)_{lat}\) in the heating section, and latent PCM heat was effectively utilized. Figure 6 shows that, the thicker the pipe wall was, the higher the heat flux in the heating section was, as shown by the red arrow. At the entrance to the cooling section, the thicker the pipe wall was, the lower the maximal heat flux was, which reduced the peak heat flux (yellow area). However, the thicker wall also increased the heat flux in the cooling section outside the above-mentioned (dotted-red) area. For a relatively large \(k_{wf}^{*}\)s, the heat-flux curve was modified earlier because of preheating the heating section and precooling the cooling section, as shown by the dotted blue area. This precooling phenomenon was more marked for \(k_{wf}^{*}=100\) (Figure 6b). Figure 7 shows the \(\varepsilon_{\rm{Tmax}}\) of the heating section versus \(t_{w}\), where \(\varepsilon_{\rm{Tmax}}\), defined in Equation (16), represents the effectiveness of the maximal temperature control for the heating section. A high \(\varepsilon_{\rm{Tmax}}\) shows low maximal wall temperature in the heating section. In Figure 7a, for \(Ra^{*}=10^{10}\), \(\varepsilon_{\rm{Tmax}}\) increased with \(k_{wf}^{*}\). \(\varepsilon_{\rm{Tmax}}\) was higher than the theoretical value (\(t_{w}=0\); shown by ) for the following cases: \(k_{wf}^{*}=10\), \(t_{w}=0.1\) and \(k_{wf}^{*}=100\), \(t_{w}=0.1\)-0.5. The highest \(\varepsilon_{T\rm{max}}\) of 1.21 was obtained for \(k_{wf}^{*}=100\)  
\(t_{w}=0.4\). On the basis of the design parameters provided here (\(k^{*}_{wf}=100\), \(t_{w}=0.4\)), an optimized practical system could be designed and manufactured by readers.