### Nomenclature  
\begin{tabular}{l l} \(l_{z}\) & dimensionless mean elevation of cooling section relative to heating section, \(l_{z}^{+}/l_{c}^{+}\) \\ \(q_{n}^{\prime\prime}\) & local heat flux \\ \(q_{i}^{\prime}\)\({}_{1}^{*}\) & local dimensionless heat flux, \(q_{i}^{\prime*}=q_{i}^{\prime}/q_{h}^{\prime\prime}\) \\ \(q_{h}^{\prime}\) & total convection heat-transfer \\ \(R_{L}\) & dimensionless pipe radius, \(r_{i}^{+}/l_{L}^{+}\) \\ \(Ra^{*}\) & modified Rayleigh number, \((g\beta b_{i}^{\prime\prime}r_{i}^{+}l_{L}^{+3})/(a_{b}\
u_{b}k_{b})\) \\ \(r_{i}^{+}\) & radial coordinate \\ \(r_{i}^{+}\) & inner radius of loop pipe \\ \(r_{0}^{+}\) & outer radius of loop pipe \\ \(r_{p}^{+}\) & PCM particle radius \\ r & dimensionless radial coordinate, \(r^{+}/r_{i}^{+}\) \\ \(r_{p}\) & dimensionless particle radius, \(r_{p}^{+}/r_{i}^{+}\) \\ \(s^{+}\) & axial coordinate \\ \(s\) & dimensionless axial coordinate, \(s^{+}/l_{c}^{+}\) \\ \(s^{*}\) & modified dimensionless axial coordinate, \(s^{+}/l_{L}^{+}\) \\ \(Sb^{*}\) & modified subcooling factor, \((T_{M}-T_{c})/(q_{h}^{*}r_{i}^{+}/k_{b})\) \\ \(Ste^{*}\) & modified Stefan number, \(c_{p}(q_{h}^{\prime\prime}r_{i}^{+}/k_{b})/h_{l}\) \\ \(t_{w}^{+}\) & pipe-wall thickness \((r_{o}^{+}-r_{i}^{+})\) \\ \(t_{w}\) & dimensionless pipe-wall thickness (=\(l_{w}^{+}/r_{i}^{+}\)) \\ \(T\) & temperature \\ \(u^{+}\) & axial velocity \\ \(u\) & dimensionless axial velocity, \(u^{+}/(Ra^{*1/4}a_{b}/r_{i}^{+})\) \\ \(G\) & _break symbols_ \\ \(\alpha\) & thermal diffusivity \\ \(\beta\) & thermal-expansion coefficient \\ \(\varepsilon_{T_{\text{max}}}\) & effectiveness of maximal wall temperature control, \((T_{w,b,\text{max},f}-T_{M})/(T_{w,b,\text{max},b}-T_{M})\) or \\  & \((\rho_{w,b,\text{max},f}/\theta_{w,b,\text{max},b})(k_{b}^{*})\) \\ \(\gamma_{L}\) & correction factor for loop length, \(l_{L,eff}^{+}/l_{L}^{+}\) \\ \(\
u\) & kinematic viscosity \\ \(\theta\) & dimensionless temperature, \((T-T_{M})/(q_{h}^{\prime\prime}r_{i}^{+}/k_{b})\) \\ \(\rho\) & density \\ \(\rho_{bf}^{*}\) & base fluid-density ratio, \(\rho_{p}/\rho_{f}\) \\ \(p_{pf}^{\prime}\) & particle-density ratio, \(\rho_{p}/\rho_{f}\) \\ \(\xi_{b}\) & mean liquid fraction of PCM suspension \\ \(\xi_{\ell}\) & liquid-phase volume fraction of PCM particles \\ \(\Phi\) & latent-heat contribution \\ _Subscripts_ \\ \(b\) & bulk fluid \\ \(bf\) & bulk-to-base-fluid ratio \\ \(c\) & cooling section \\ \(f\) & base fluid (water) \\ \(h\) & heating section \\ \(i\) & inner \\ \(\ell\) & PCM liquid phase \\ \(lat\) & latent heat \\ \(M\) & melting point \\ \(max\) & maximum \\ \(p\) & particle \\ \(pf\) & particle-to-base-fluid ratio \\ \(sen\) & sensible heat \\ \(w\) & pipe wall \\ \(w\) & pipe-wall-to-base-fluid ratio \\ _Superscripts_ \\ \({}^{*}\) & ratio of quantities \\ \({}^{+}\) & dimensional quantity \\ \(T\) & temperature \\ \(u^{+}\) & axial velocity \\ \(u\) & dimensionless axial velocity, \(u^{+}/(Ra^{*1/4}a_{b}/r_{i}^{+}/k_{b})\) \\ \(u^{+}\) & dimensionless axial velocity, \(u^{+}/(Ra^{*1/4}a_{b}/r_{i}^{+}/k_{b})\) \\ \(G\) & thermal diffusivity \\ \(\beta\) & thermal-expansion coefficient \\ \(\varepsilon_{T_{\text{max}}}\) & effectiveness of maximal wall temperature control, \((T_{w,b,\text{max},f}-T_{M})/(T_{w,b,\text{max},b}-T_{M})\) or \\  & \((\rho_{w,b,\text{max},f}/\theta_{w,b,\text{max},b})(k_{b}^{*})\) \\ \(\gamma_{L}\) & correction factor for loop length, \(l_{L,eff}^{+}/l_{L}^{+}\) \\ \(\
u\) & kinematic viscosity \\ \(\theta\) & dimensionless temperature, \((T-T_{M})/(q_{h}^{\prime\prime}r_{i}^{+}/k_{b})\) \\ \(\rho\) & density \\ \(\rho_{bf}^{*}\) & base fluid-density ratio, \(\rho_{p}/\rho_{f}\) \\ \(p_{pf}^{\prime}\) & particle-density ratio, \(\rho_{p}/\rho_{f}\) \\ \(\xi_{b}\) & mean liquid fraction of PCM suspension \\ \(\xi_{\ell}\) & liquid-phase volume fraction of PCM particles \\ \(\Phi\) & latent-heat contribution \\ _Subscripts_ \\ \(b\) & bulk fluid \\ \(bf\) & bulk-to-base-fluid ratio \\ \(c\) & cooling section \\ \(f\) & base fluid (water) \\ \(h\) & heating section \\ \(i\) & inner \\ \(\ell\) & PCM liquid phase \\ \(lat\) & latent heat \\ \(M\) & melting point \\ \(max\) & maximum \\ \(p\) & particle \\ \(pf\) & particle-to-base-fluid ratio \\ \(sen\) & sensible heat \\ \(w\) & pipe wall \\ \(w\) & pipe-wall-to-base-fluid ratio \\ _Superscripts_ \\ \({}^{*}\) & ratio of quantities \\ \({}^{+}\) & dimensional quantity \\ \(T\) & temperature \\ \(u^{+}\) & axial velocity \\ \(u