## 2 Materials and Methods  
### Theory  
In this section, the exergy analysis of the TES systems is presented. During the discharge of the TES system, the HTF is heated up by the storage material. Thus, the temperature of the HTF increased during the discharge tests; however, the circulation of the HTF through the TES system was subjected to a pressure drop. To account for both the temperature increase and the pressure drop, the exergy balance of the HTF between the inlet and the outlet of the TES system was calculated in order to determine the exergy increase \(\Delta E\) of the HTF circulating through the TES system, as follows:  
\[\Delta E=\dot{m}_{f}(\Delta h-T_{0}\Delta s)-\bigg{(}1-\frac{T_{0}}{T_{w}}\bigg{)} q_{loss}, \tag{1}\]  
where \(\dot{m}_{f}\) is the mass flow rate of the HTF, \(h\) and \(s\) are the specific enthalpy and entropy increase of the HTF in the TES system, respectively, \(T_{0}\) is the temperature of the dead state, \(T_{w}\) is the temperature of the wall of the TES system, and \(q_{loss}\) is the rate of heat loss through the TES system wall. Considering the HTF as an ideal gas with constant specific heat, the variations in specific enthalpy \(\Delta h\) and entropy \(\Delta s\) can be determined as a function of the temperature and pressure of the HTF at the inlet and outlet of the TES system, the specific heat at constant pressure \(c_{pf}\), and the ratio of specific heat \(\gamma\).  
\[\Delta h=c_{pf}(T_{out}-T_{in}), \tag{2}\]  
\[\Delta s=c_{pf}\bigg{(}\text{ln}\frac{T_{out}}{T_{in}}-\frac{\gamma-1}{\gamma }\text{ln}\frac{p_{out}}{p_{in}}\bigg{)}. \tag{3}\]  
Assuming negligible heat losses through the walls of the TES system, which were properly insulated, and considering both the temperature of the dead state \(T_{0}\) and the temperature of the HTF at the inlet of the TES system \(T_{in}\) equal to the ambient temperature \(T_{a}\), the exergy increase in the HTF circulating through the TES system during the discharge process can be computed as a function of the ambient temperature \(T_{a}\) and pressure \(p_{a}\), the temperature of the HTF at the outlet of the TES system \(T_{out}\), and the pressure drop of the TES system \(\Delta p\), as follows:  
\[\Delta E=\dot{m}_{f}c_{pf}\bigg{[}(T_{out}-T_{a})-T_{a}\bigg{(}\text{ln}\frac{ T_{out}}{T_{a}}+\frac{\gamma-1}{\gamma}\text{ln}\frac{p_{a}+\Delta p}{p_{a}} \bigg{)}\bigg{]}. \tag{4}\]  
In view of the exergy increase in the HTF during the discharge of the TES system as determined by Equation (4), the maximum possible exergy increase \(\Delta E_{max}\) is attained when the outlet temperature of the HTF is at its maximum, i.e., \(T_{out}=T_{max}\), which is the initial temperature for the discharge process, and the pressure drop of the TES system is null, i.e., \(\Delta p=0\):  
\[\Delta E_{max}=\dot{m}_{f}c_{pf}\bigg{[}(T_{max}-T_{a})-T_{a}\text{ln}\frac{T_ {max}}{T_{a}}\bigg{]}. \tag{5}\]  
Then, the exergy efficiency of the TES system during the discharge process can be determined as the ratio of the actual energy increase of the HTF \(\Delta E\) in the TES system and the maximum energy increase \(\Delta E_{max}\):  
\[\eta_{ex}=\frac{(T_{out}-T_{a})-T_{a}\bigg{(}\text{ln}\frac{T_{out}}{T_{a}}+ \frac{\gamma-1}{\gamma}\text{ln}\frac{p_{a}+\Delta p}{p_{a}}\bigg{)}}{(T_{ max}-T_{a})-T_{a}\text{ln}\frac{T_{max}}{T_{a}}}. \tag{6}\]