## 2 Materials and Methods
### Dimensionless Thermal Quantities of Interest  
Several dimensionless thermal quantities of interest can be derived from the solution to the governing differential equations.  
(1) Local mean temperature of the suspension fluid at a given cross section of the tube  
\[\theta_{b}(x)=\frac{\int_{0}^{1}u\theta(r,x)rdr}{\int_{0}^{1}urdr} \tag{4}\](2) Mean liquid-phase or melted fraction of the PCM particles at a given cross section of the tube  
\[\xi_{b}(x)=\frac{\int_{0}^{1}u\xi(r,x)rdr}{\int_{0}^{1}urdr} \tag{5}\]  
(3) Heat transfer on the inner wall surface  
The energy dissipated outward from the tube is shown in Figure 1b.  
The dimensionless local heat flux at the inner tube wall can be obtained as follows:  
\[{q^{\prime\prime}}_{w,j}^{*}=\frac{{q^{\prime\prime}}_{w,j}}{{k_{b}(T_{m}-T_{ \infty})}/{r_{i}^{+}}}=-\left(\frac{\partial\theta}{\partial r}\right)_{r=1} \tag{6}\]  
Integrating the heat flux along the heat exchange section yields the heat transfer through the inner wall surface:  
\[q_{w,j}^{*}=\frac{1}{L_{h}^{+}}\frac{\int_{0}^{L_{h}^{+}}{q^{\prime\prime}}_{ w,j}dx^{+}}{{k_{b}(T_{m}-T_{\infty})}/{r_{i}^{+}}}=\frac{1}{L_{h}}\int_{0}^{L_{h}}{ q^{\prime\prime}}_{w,j}^{*}dx \tag{7}\]  
Neglecting the axial heat conduction, the following equation can be deduced from the equation for the fluid energy:  
\[\left(q_{w,i}^{*}\right)_{L_{h,conv}}=\frac{-1}{4L_{h}}\left\{\left[\theta_{b }(x=L_{h})-\theta_{b}(x=0)\right]+\frac{\rho_{pf}}{\rho_{bf}}\frac{c_{v}}{Ste }[\xi_{b}(x=L_{h})-\xi_{b}(x=0)]\right\} \tag{8}\]  
where \(\left(q_{w,j}^{*}\right)_{L_{h,conv}}\) is the dimensionless total heat transfer rate to the tube wall from fluid convection heat transfer that has two components:  
\[\left(q_{w,i}^{*}\right)_{L_{h,conv}}=\frac{1}{4L_{h}}[\theta_{b}(x=0)-\theta _{b}(x=L_{h})] \tag{9}\]  
\[\left(q_{w,i}^{*}\right)_{L_{h,lat}}=\frac{1}{4L_{h}}\frac{\rho_{pf}}{\rho_{bf }}\frac{c_{v}}{Ste}[\xi_{b}(x=0)-\xi_{b}(x=L_{h})] \tag{10}\]  
The axial dimensionless heat transfer rate in the fluid is as follows:  
\[\left(q_{w,i}^{*}\right)_{L_{h,conv}}=q_{w,i}^{*}-\left(q_{w,i}^{*}\right)_{L _{h,conv}} \tag{11}\]  
(4) Heat transfer for the outer wall  
The total heat transfer dissipated through the outer tube wall of the heat exchange section is as follows:  
\[q_{w,\rho}^{*}=\frac{\left(2\pi r_{o}^{+}\right)\int_{0}^{L_{h}^{+}}{h_{ \infty}(T_{w,\rho}-T_{\infty})}}{{k_{b}(T_{m}-T_{\infty})}/{r_{i}^{+}}\left(2 \pi r_{o}^{+}\right)}=\frac{Bi_{o}}{1+t_{w}}\frac{k_{wf}^{*}}{k_{bf}^{*}}\left( 1+\overline{\theta}_{w,\rho}\right) \tag{12}\]  
where  
\[\overline{\theta}_{w,\rho}=\frac{1}{L_{h}}\int_{0}^{L_{h}}{\theta_{w,\rho}}dx \tag{13}\]  
(5) Axial heat conduction in the tube wall  
\[\left(q_{w}^{*}\right)_{cond}=\frac{2}{p_{e}}\frac{k_{wf}^{*}}{k_{bf}^{*}} \frac{1}{t_{w}(2+t_{w})}\left[\int_{1+t_{w}}^{1}\left(\frac{\partial\theta}{ \partial x}\right)_{x=0}rdr-\int_{1+t_{w}}^{1}\left(\frac{\partial\theta}{ \partial x}\right)_{x=L_{h}}rdr\right] \tag{14}\](6) Nusselt number (\(Nu\)) of the inner wall surface  
\[Nu_{i}=\frac{h_{i}\Big{(}2r_{i}^{+}\Big{)}}{k_{b}}=\frac{-2\Big{(}\frac{\partial \theta}{\partial r}\Big{)}_{r=1}}{\theta_{b}-\theta_{w,i}} \tag{15}\]  
(7) Nusselt number of the outer wall surface  
\[Nu_{o}=\frac{U_{o}\Big{(}2r_{i}^{+}\Big{)}}{k_{b}}=\frac{2 Bi_{o}}{(1+t_{w})} \frac{k_{wf}^{*}}{k_{bf}^{*}}\frac{(\theta_{w,o}+1)}{\theta_{in}+1} \tag{16}\]  
where the overall heat transfer coefficient is  
\[U_{o}=\frac{h_{\infty}(T_{w,\rho}-T_{\infty})}{(T_{in}-T_{\infty})} \tag{17}\]