## 3 Results and Discussion
### Design by \(\varepsilon\)-NTU Method (Validation Using Experimental Data)  
This section employs the published data of several LHTES units to validate the \(\varepsilon\)-NTU method.  
According to the applicable requirements of the \(\varepsilon\)-NTU method:  
* HTF on the tube side and PCM on the shell side,
* phase change takes place at or near the melting temperature range with no obvious supercooling,
* smooth tubes evenly and parallelly distributed,  
The lab-scale LHTES units developed by [PERSON] et al. at the University of Lleida [38], [PERSON] and [PERSON] at Loughborough University [39], as well as [PERSON] et al. [35] at the University of South Australia based on which the \(\varepsilon\)-NTU method was proposed, are selected and used to validate the accuracy and applicable range of the \(\varepsilon\)-NTU method.  
These units represent a wide range of shell-and-tube LHTES designs and operation conditions by covering different tank geometries, tube arrangements (number of tubes and passes, tube diameter and length), HTFs, PCMs, temperature ranges, and flow schemes (laminar or turbulent). The configurations and key parameters of these shell-and-tube LHTES units are presented in Table 5.  
\begin{table}
\begin{tabular}{c c c c} \hline \hline
**[PERSON]** & **[PERSON] et al.**[38] & **[PERSON] \& [PERSON]**[39] & **[PERSON] et al.**[35] \\ \hline \hline Configuration & & & \\ \hline Number of tubes & 49 & 1 & 1, 2, 4 \\ Number of passes & 2 & 55 & 14, 32, 64 \\ Tube length & 2.485 m & 27 m & 5.46, 11.62, 23.83 m \\ Tube outer diameter & 17.2 mm & 15 mm & 10 mm 1 \\ Tube inner diameter & 13.2 mm & 13 mm & 8 mm \\ Shell dimension & 1273 (length) \(\times\) 273 (depth) \(\times\) & 520 (length) \(\times\) 560 (height) \(\times\) & 290 (diameter) \(\times\) 330 mm \\  & 527.5 mm (width) & 160 mm (width) & (height) \\ PCM & RT58 & RT62 HC & PCM0 and PCM27 \\ Melting temperature & 53–59 \({}^{\circ}\)C & 62–63 \({}^{\circ}\)C & 0 and 27 \({}^{\circ}\)C \\ HTF & Syltherm 800 & Water & Aqueous based fluid \\ Temperature difference & & & \\ between the HTF and the & Approximately 10 \({}^{\circ}\)C & Approximately 13 \({}^{\circ}\)C & Approximately 18 and 40 \({}^{\circ}\)C \\ PCM & & & \\ HTF flow rate & 500 kg.h\({}^{-1}\) & 2–6 L.min\({}^{-1}\) & Around 0–0.07 kg.s\({}^{-1}\) \\ Flow scheme & Laminar & Turbulent & Laminar and turbulent \\ \hline \hline Average error & 2.2\% & 8.5\% & 13\% 2 \\ \hline \hline \end{tabular}  
\({}^{1}\) Calculated based on Table 1 in [PERSON] et al. [40]; \({}^{2}\) Calculated by [PERSON] et al. [35].  
\end{table}
Table 5: Average absolute error and key parameters of experimental cases used to validate the \(\varepsilon\)-NTU.  
The calculated results generated by the \(\varepsilon\)-NTU method and the experimental results are compared and presented in Table 5. The average absolute error of the three cases ranges from 2.2% to 13%. The [PERSON] et al. (2018) and [PERSON] (2019) cases have lower discrepancies probably because of lower temperature differences between the HTF and the PCM, hence smaller heat transfer enhancement due to natural convection. Despite the highest temperature differences, the average error of the [PERSON] et al. (2019) case is still limited. Therefore, the conclusion can be drawn that the \(\varepsilon\)-NTU method provides modest errors on a broad range of shell-and-tube LHTES designs and operating conditions. It can be used to calculate the performance of a shell-and-tube LHTES unit as long as the applicable requirements are met.  
Future research can try to include the influence of the natural convection during the phase change into the design methodology, as well as develop similar methods for finned units and other types of TES designs, for example, packed-bed and plate type.