## 3 Results and Discussion
### Tube Wall and Fluid Temperature  
The wall conduction effect is illustrated in Figure 3 in terms of the axial distributions of the outer tube wall temperature (\(\theta_{w,o}\)), the inner tube wall temperature (\(\theta_{w,i}\)), the average fluid temperature (\(\theta_{b}\)), and \(\xi_{b}\) for different values of \(t_{w}\) and \(k_{wf}^{*}\). The dimensionless distance \(\left(x^{+}+L_{u}^{+}\right)/(D_{i}Pe)\) of 0 represents the starting point \(x^{+}=-L_{u}^{+}\) of the tube. \(\left(x^{+}+L_{u}^{+}\right)/(D_{i}Pe)\) values of 0-0.025 represent the adiabatic section just prior to the cooling section. Values of 0.025-0.175 represent the cooling section. Values of 0.175 represent the adiabatic section after the cooling section. The dimensionless temperatures (\(\theta_{w,o}\), \(\theta_{w,i}\), and \(\theta_{b}\)) at the tube entrance \(\left(\left(x^{+}+L_{u}^{+}\right)/(D_{i}Pe)=0\right)\) are set to 0. Wall conduction appears to have a strong influence on the post-cooling of the wall in the downstream regions adjacent to the cooled section. In the downstream indirectly cooled regions, the inner and outer wall temperatures increase with increases in various convex curvatures.  
Figure 3a shows that there is little change in the respective temperatures and \(\xi_{b}\) between \(t_{w}\) = 0.2 and 0.5. Thus, \(t_{w}\) does not directly affect the heat transfer performance of the heat exchange section. By contrast, the temperature distributions are quite sensitive to changes in \(k_{wf}^{*}\). The heat transfer enhancement by the suspension fluid at fixed \(Ste\) or \(c_{v}\) is markedly impeded by a lower \(k_{wf}^{*}\), resulting in considerably higher inner and outer wall temperatures.  
In Figure 3b, the post-cooling effect is increasingly discernible with increasing \(k_{wf}^{*}\). For \(k_{wf}^{*}\) = 10, the PCM completes heat exchange when the dimensionless parameter \(\left(x^{+}+L_{u}^{+}\right)/(D_{i}Pe)\) = 0.16. The length required for the entire PCM to dissipate heat is 91% of the length of the heat exchange section. The axial propagation of conduction heat transfer along the tube wall results in an over twofold decrease in the inner and outer wall temperatures at the exit of the cooled section. The PCM suspension microparticles still undergo an exothermic reaction in the downstream adiabatic section, as shown by \(\xi_{b}\) (the yellow line) in Figure 3. Thus, the temperature increases the fluid temperature and inner and outer walls, as reflected by the change in \(\theta_{w,i}\), as illustrated by the yellow region.  
Figure 2: GCI for \(Nu\) of the inner wall and sampling value of the extrapolated relative error for different numbers of grid points.  
For \(k_{wf}^{*}=1\), the reaction in the PCM suspension microparticles remains exothermic in the downstream adiabatic section, as shown by the red line for the change in \(\xi_{b}\) in Figure 3. Even at the terminal exit of the downstream adiabatic section, the PCM still contains unsolidified liquid, which increases the fluid temperature and thus the temperature of the inner and outer walls, as shown by the red region in Figure 3 for the change in \(\theta_{wj}\). Additionally, when the working fluid exits the adiabatic section, the temperatures are not yet steady (the steadiness corresponds to no further increase in the  
Figure 3: Effects of the (**a**) dimensionless wall thickness (\(t_{w}\)) and (**b**) thermal conductivity ratio (\(k_{wf}^{*}\)) on the outer tube wall temperature (top), fluid cross section temperature (middle), inner tube wall temperature (middle), and volumetric liquid fraction in a PCM particle (bottom).  
temperature). Thus, the temperature at the downstream outlet and its steadiness depend on whether the PCM latent heat is completely released in the heat exchange section.