## 2 Materials and Methods  
### Problem Formulation  
By suitable arrangement of the heating zone and the cooling zone, the formed thermal buoyancy drives the working fluid to achieve heat transfer in the thermosiphon. The self-adjusting mechanism and stability enable extensive thermosyphon applications, such as solar heating and cooling systems, coolers for reactors in nuclear power plants, geothermal energy systems, waste heat recovery systems,and electronic cooling systems. The physical modeling of this study was inspired by the cooling section of the thermosipon. The physical configuration of the conjugate heat transfer problem considered, as illustrated schematically in Figure 1a, is a circular pipe of inner radius \(r_{i}^{+}\) and wall thickness \(t_{w}^{+}\) (\(=r_{o}^{+}-r_{i}^{+}\)). A tube is divided into three sections: (1) an upstream adiabatic section of length \(L_{u}^{+}\), (2) a heat exchange area with a convection boundary (\(T_{\infty}\), \(h_{\infty}\)) and a length of \(L_{h}^{+}\), and (3) a downstream adiabatic section of length \(L_{d}^{+}\). The PCM suspension enters the upstream adiabatic section at a fully developed velocity \(u_{m}^{+}\), and the fluid has a uniform temperature \(T_{in}\). The following basic assumptions are made in the model. 1. The working fluid is a suspension of PCM particles. The PCM particles can be completely solid phase, completely liquid phase, or a mixture of solid and liquid phases. 2. A uniform diameter is assumed for the PCM particles suspended in the working fluid. 3. All of the physical properties of the PCM suspension are assumed to be constant. 4. The PCM suspension is regarded as a thin mixture. 5. The PCM suspension is regarded as an incompressible Newtonian fluid. 6. Viscous dissipation is neglected. 7. The PCM particles are neutral and evenly distributed in the working fluid. 8. The PCM suspension enters the tube in fully developed laminar flow. 9. The flow and temperature distribution of the PCM suspension in the tube are radially axisymmetric. 10. The change in the density of the PCM particles during solid-liquid phase changes is neglected. Figure 1: (**a**) Physical model and (**b**) energy dissipation processes. Based on the foregoing assumptions, the following dimensionless governing equations and boundary conditions are used in this study. (1) Governing equation  
The fluid energy equation and the tube wall heat conduction equation are as follows:  
\[\mathrm{a}\frac{\partial\theta}{\partial F_{o}}+\frac{u}{2}\frac{\partial\theta}{ \partial x}=\frac{1}{r}\frac{\partial}{\partial r}\bigg{(}r\frac{\partial \theta}{\partial r}\bigg{)}+\bigg{(}\frac{k_{bf}^{*}}{\rho_{bf}^{*}c_{p,bf}^{* }}\bigg{)}^{2}\frac{1}{Pe^{2}}\frac{\partial^{2}\theta}{\partial x^{2}}-b \frac{c_{p}}{Ste}\bigg{(}\frac{\partial\xi_{b}}{\partial F_{o}}+\frac{u}{2} \frac{\partial\xi_{b}}{\partial x}\bigg{)} \tag{1}\]  
where for the fluid region (0 \(<\) r \(<\) 1), \(\mathrm{u}=\frac{u^{+}}{u_{m}^{+}}=2\big{(}1-r^{2}\big{)}\), \(\mathrm{a}=1\), and \(\mathrm{b}=\rho_{bf}^{*}/\rho_{bf}^{*}\); and for the wall region (1 \(\leq\) r \(\leq\) 1 + \(t_{w}\)), \(\mathrm{u}=0\), \(\mathrm{a}=(k_{bf}^{*}/k_{bf}^{*})/(\rho_{w}^{*}/\rho_{bf}^{*})\) (\(c_{p,wf}^{*}/c_{p,bf}^{*}\)), and \(\mathrm{b}=0\). The governing equation for the volumetric liquid fraction of a PCM particle \(\xi_{b}\) is as follows:  
\[\frac{\partial\xi_{b}}{\partial F_{o}}+\frac{u}{2}\frac{\partial\xi_{b}}{ \partial x}=\frac{12}{d_{p}^{2}}\bigg{(}\frac{k_{bf}^{*}}{k_{bf}^{*}}\bigg{)} \bigg{(}\frac{\rho_{bf}^{*}}{\rho_{bf}^{*}}\bigg{)}SteB_{ip}\theta \tag{2}\]  
A detailed treatment of the solid-liquid phase change of particles can be found in [PERSON] et al. (2010). (2) Boundary conditions  
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