## 2 Methodology
### Shell-and-Tube LHTES Design Method  
With the top PCMs produced by the material assessment, the \(\varepsilon\)-NTU method is adopted to produce the design configuration that will serve the LHTES unit's needs. The configuration is later optimized to minimize the production cost of the LHTES unit as a whole. The typology of shell-and-tube is used in this study, but the same methodology can be extended to cover a wider range of configurations, including packed-bed and plate type LHTES. Several assumptions are considered in the \(\varepsilon\)-NTU method of this methodology:  
* The HTF is on the tube side and PCM on the shell side. * The tubes are aligned parallelly and distributed evenly in the shell space, regardless of the number of the tubes and passes. * The phase change process happens when the temperature of the PCM is fixed at the phase change temperature without supercooling, and the natural convection of the HTF during the phase change is neglected. The \(\varepsilon\)-NTU method is designed to be temperature independent, while the effect of natural convection increases with the temperature difference between the HTF and the PCM. Neglecting the natural convection may lead to underestimating the heat transfer coefficient, especially during the melting process. However, this effect will not significantly affect the accuracy of the \(\varepsilon\)-NTU method according to [PERSON] et al. [35]. Therefore, this method approximates the \"worst-case scenario\" of an LHTES unit since the PCM sensible heat is not utilized to ensure a constant HTF outlet temperature during cyclic operations, and the phase change process is conduction-dominated. * The designs are optimized for cost reduction purposes. Since the operational cost depends primarily on the application, only the initial cost of the entire LHTES unit is considered here, including the material cost of both PCMs and containment, as well as the production cost of the designed configuration. * Finned tube designs or rough heat exchanging surfaces are not discussed within this study. However, future arrangements can be made to accommodate certain types of finned or surface designs. The mathematical models are introduced based on the basic shell-and-tube LHTES configuration shown in Figure 4. Figure 4a illustrates the 3D model, in which the PCMs resides on the shell side, and HTF flows inside the pipe. Figure 4b shows the single pipe configuration in 2D, while Figure 4c presents the storage unit configuration with multiple  
Figure 3: Illustration of an objective map used for the MODM examination [27] in which the \(f_{1}\) values of ten PCMs (marked as No. 1 to 10 in the map) are plotted against their \(f_{2}\) values. In this case, the preferred material alternatives are PCMs No. 1, 2, 3, and 4. If some of them are also ranked highly by the MADM, they will be recommended as the final decision of the material assessment step. tubes or passes. \(Dt\) is the tubes' outer diameter, \(Do\) stands for the shell's inner diameter, and \(P\) is the pitch distance between the tubes. The effectiveness of a heat exchanger used for the \(\varepsilon\)-NTU method is defined as:  
\[\varepsilon=\frac{DT}{T_{\text{mp}}-T_{\text{in}}} \tag{10}\]  
where \(DT\) [K] is the temperature difference between the outlet and inlet of HTF, \(T_{\text{mp}}\) [K] is the melting temperature of PCM, and \(T_{\text{in}}\) [K] is the inlet temperature of the HTF. The definition of the number of transfer units (\(NTU\)) is given below, as well as the correlation between effectiveness and \(NTU\):  
\[NTU=\frac{UA}{\left(\text{{{inc}}}_{\text{p,HTF}}\right)_{\text{per tube}}} \tag{11}\]  
\[\varepsilon=1-e^{-NTU} \tag{12}\]  
where \(UA\) [W\(\cdot\)K\({}^{-1}\)] stands for the heat transfer capacitance, which is the invert of the total thermal resistance. \[UA=\frac{1}{R_{\text{total}}}=\frac{1}{R_{\text{HTF}}+R_{\text{WALL}}+R_{ \text{PCM}}} \tag{13}\]  
The thermal resistances of the HTF, wall, and PCM are given by [PERSON] et al.