## 2 Experimental Setup and Procedures
### Data Reduction  
The solid-liquid boundary standpoint with time is forecasted throughout the charging and discharging processes by tracking the temperature at numerous positions displayed in Figure 2. Consequently, with the average thickness \(t_{avg}\), the solidified or melted mass around the inner and outer coils can be predicted. As per Figure 2, \(t_{avg}=\frac{t_{1}+t_{2}}{2}\) for the inner coil of TCHC and \(t_{avg}=\frac{t_{3}+t_{4}}{2}\) for the exterior coil of TCHC. The calculations of the solidified and melting volume, depending on the average thickness of the solidified or melted mass, are categorized into two domains. The first domain is when the average thickness is within the half pitch around the coil turn, while the second domain is when it is higher than the half pitch. Accordingly, the solidified and melting volumes are calculated as follows:  
In the first domain, \([0<t_{avg}\leq(p/2-r_{t,\rho})]\):  
\[V_{\rm s}=(\rho_{w}/\rho_{i})\Big{(}\pi\Big{[}\big{(}r_{t,\rho}+t_{avg}\big{)} ^{2}-r_{t,\rho}^{2}\Big{]}*2\pi R_{coil,c}\Big{)} \tag{1}\]  
\[V_{\rm m}=N\pi\Big{[}\big{(}r_{t\rm{o}}+t_{avg}\big{)}^{2}-r_{t,\rm{o}}^{2} \Big{]}*2\pi R_{coil,c} \tag{2}\]  
In the second domain, \([t_{avg}>(p/2-r_{t,\rho})]\):  
\[V_{\rm s}=N(\rho_{w}/\rho_{i})\Big{(}\Big{[}\pi\Big{(}r_{2}^{2}-r_{1}^{2} \Big{)}\times p\Big{]}-\Big{(}\pi r_{t,\rho}^{2}\times 2\pi R_{coil,c}\Big{)} \Big{)} \tag{3}\]  
\[V_{\rm m}=N\Big{(}\Big{[}\pi\Big{(}r_{2}^{2}-r_{1}^{2}\Big{)}\times p\Big{]}- \Big{(}\pi r_{t,\rm{o}}^{2}\times 2\pi R_{coil,c}\Big{)}\Big{)} \tag{4}\]  
where \(r_{3}\) and \(r_{4}\) are used in place of \(r_{1}\) and \(r_{2}\) when computations are taken for the outer coil of TCHC. The solidified and melted mass fractions are calculated from the following forms, respectively:  
\[\frac{m_{\rm s}}{m_{\rm o}}=\frac{\rho_{\rm i}V_{\rm s}}{\rho_{\rm w}V_{\rm o}} \tag{5}\]  
\[\frac{m_{\rm m}}{m_{\rm o}}=\frac{\rho_{\rm i}V_{m}}{\rho_{\rm w}V_{\rm o}} \tag{6}\]  
The following formulas are used to calculate the cumulative and maximum thermal energy stored and regained:  
\[Q_{\rm st}=\rho_{\rm w}V_{\rm o}\bigg{\{}c_{\rm w}(T_{\rm o}-T_{\rm I})+ \bigg{(}\frac{m_{\rm s}}{m_{\rm o}}\bigg{)}\bigtriangleup H_{\it fus}+\bigg{(} \frac{m_{\rm s}}{m_{\rm o}}\bigg{)}c_{\rm i}\big{(}T_{\rm pc}-T_{\rm s}\big{)} \bigg{\}} \tag{7}\]  
\[Q_{\rm st,max}=\rho_{\rm w}V_{\rm o}\Big{\{}c_{\rm w}\big{(}T_{\rm o}-T_{\rm pc }\big{)}+\bigtriangleup H_{\it fus}+c_{\rm i}\big{(}T_{\rm pc}-T_{\rm HTF}\big{)} \Big{\}} \tag{8}\]  
\[Q_{\rm reg}=\rho_{\rm w}V_{\rm o}\bigg{\{}c_{\rm i}(T_{\rm s}-T_{\rm o})+ \bigg{(}\frac{m_{\rm m}}{m_{\rm o}}\bigg{)}\bigtriangleup H_{\it fus}+\bigg{(} \frac{m_{\rm m}}{m_{\rm o}}\bigg{)}c_{\rm w}\big{(}T_{\rm I}-T_{\rm pc}\big{)} \bigg{\}} \tag{9}\]  
\[V_{\rm s}=(\rho_{w}/\rho_{i})\Big{(}\pi\Big{[}\big{(}r_{t,\rho}+t_{avg}\big{)} ^{2}-r_{t,\rho}^{2}\Big{]}*2\pi R_{coil,c}\Big{)} \tag{1}\]0Additionally, the following forms are used to determine the average charging and discharging rates. \[V_{\rm s}=(\rho_{w}/\rho_{i})\Big{(}\pi\Big{[}\big{(}r_{t,\rho}+t_{avg}\big{)} ^{2}-r_{t,\rho}^{2}\Big{]}*2\pi R_{coil,c}\Big{)} \tag{1}\]1  
\[V_{\rm s}=(\rho_{w}/\rho_{i})\Big{(}\pi\Big{[}\big{(}r_{t,\rho}+t_{avg}\big{)} ^{2}-r_{t,\rho}^{2}\Big{]}*2\pi R_{coil,c}\Big{)} \tag{1}\]2  
The PCM (distilled water) properties and constant values applied in the current study in the two liquid and solid phases are scheduled in \[V_{\rm s}=(\rho_{w}/\rho_{i})\Big{(}\pi\Big{[}\big{(}r_{t,\rho}+t_{avg}\big{)} ^{2}-r_{t,\rho}^{2}\Big{]}*2\pi R_{coil,c}\Big{)} \tag{1}\]3  
The uncertainties of the measured solidified/melted mass fraction, percentage energy stored/regained, and average charging/discharging rate are calculated as per the procedure reported by [PERSON] [24], as follows, and for the charging processes, where similar equations are used for the discharging processes. The main parameter uncertainties are stated in Table 5 for all experimental runs.