## 4 Conclusions  
In this paper, numerical results have been obtained to investigate the conjugate heat transfer of a PCM suspension in a circular tube under an external cooling convection boundary. Of particular emphasis, in this study are the wall conduction effects on the heat transfer behavior. The parameters pertinent to the present problem are considered in the following ranges: the dimensionless tube wallthickness, \(t_{w}\) = 0-0.5, and the wall-to-fluid thermal conductivity ratio, \(k^{*}_{wf}\) = 0.1-10. Other parameters are kept constant (\(c_{v}\) (0.1), \(Bi_{o}\) (1), \(Ste\) (0.1), and \(Pe\) (1000)). The following conclusions are drawn from the results of the study. 1. The most significant of the considered parameters appears to be \(k^{*}_{wf}\), which affects the temperature of the outer tube wall, the temperature of the inner tube wall, the average temperature of the fluid, and the volumetric liquid fraction in a PCM particle. The various contributions to the heat transfer and \(t_{w}\) have a comparatively minor effect on the aforementioned variables. 2. For high \(k^{*}_{wf}\), the tube wall and fluid are significantly affected by postcooling in the downstream adiabatic section. Therefore, a substantial increase in \(k^{*}_{wf}\) creates an indirectly cooled region downstream of the heat exchange section where the PCM particles have completed solidification. Thus, there is a significant increase in the latent heat released in the solidification process and thus the efficacy of wall temperature control over the cooled section or even the heated section. 3. The results of the study show that wall conduction effects must be suitably accounted for to describe heat transfer in PCM suspensions in a circular tube under external cooling convection. **Author Contributions:** [PERSON] and [PERSON] conceived of and designed the model; [PERSON] and [PERSON] performed the simulation works; [PERSON], [PERSON], and [PERSON] analyzed the data; and [PERSON] and [PERSON] wrote the paper. All authors have read and agreed to the published version of the manuscript. **Funding:** This research received no external funding  
**Conflicts of Interest:** The authors declare that there are no conflicts of interest. **Nomenclature** \begin{tabular}{l l} \(a\) & parameter \\ \(b\) & parameter \\ \(Bi_{o}\) & Biot number at the outer surface of the tube (=\(h_{o}r_{o}^{+}/k_{w}\)) \\ \(Bi_{p}\) & Biot number of a PCM particle (=U\(r_{p}^{+}/k_{p}\)) \\ \(C_{p}\) & specific heat (kJ/kg K) \\ \(c^{*}_{p,bf}\) & specific heat ratio (=\(c_{p,b}/c_{p,f}\)) \\ \(c^{*}_{p,bf}\) & specific heat ratio (=\(c_{p,p}/c_{p,f}\)) \\ \(c_{o}\) & volumetric fraction of PCM particles \\ \(d_{p}\) & dimensionless diameter of a PCM particle (=\(d_{p}^{+}/r_{i}^{+}\)) \\ \(d_{p}^{+}\) & diameter of a PCM particle (m) \\ \(F_{o}\) & Fourier number (=\(\alpha_{b}t/\left(r_{i}^{+}\right)^{2}\)) \\ \(h\) & heat transfer coefficient (W/m\({}^{2}\) K); enthalpy of suspension (kJ/kg) \\ \(h_{h}\) & latent heat of fusion (J/kg) \\ \(k\) & thermal conductivity (W/m K) \\ \(k^{*}_{bf}\) & thermal conductivity ratio (=\(k_{b}/k_{f}\)) \\ \(k^{*}_{bf}\) & thermal conductivity ratio (=\(k_{p}/k_{f}\)) \\ \(k^{*}_{wf}\) & thermal conductivity ratio (=\(k_{w}/k_{f}\)) \\ \(L^{+}_{d}\) & length of downstream section (m) \\ \(L_{d}\) & dimensionless length of the downstream section (=\(L^{+}_{d}/\left(r_{i}^{+}Pe\right)\)) \\ \(L^{+}_{h}\) & length of heated section (m) \\ \(L_{h}\) & dimensionless length of the heated section (=\(L^{+}_{h}/\left(r_{i}^{+}Pe\right)\)) \\ \(L^{+}_{n}\) & length of upstream section (m) \\ \(L_{u}\) & dimensionless length of the upstream section (=\(L^{+}_{u}/\left(r_{i}^{+}Pe\right)\)) \\ \(Pe\) & Peclet number (=2\(r_{i}^{+}u_{m}^{+}/\alpha_{b}\)) \\ \(Nu\) & Nusselt number (=h\(2r_{i}^{+}/k_{b}\)) \\ \end{tabular}