## 4 Exploring the TES Performances Improvement by a Numerical Parametric Study
### Materials Conductivity Sensitivity Analysis  
As previously mentioned, the reference case was a charge up to 70 \({}^{\circ}\)C at 0.05 kg/s (180 L/h), storing 4.4 kWh, and the targeted parameter for minimization was the charge duration \(\tau_{95\%}\).  
A low global thermal resistance causes the heat transfer rate to increase and \(\tau_{95\%}\) to decrease. Three thermal resistances affect the storage process: convective between the HTF and the nodule, conductive in the capsule wall, and conductive in the PCM (convection in PCM is neglected). In the reference case, the two first resistances were respectively equal to 0.15 K/W and 0.06 K/W. The last one developed as the PCM melted, increasing up to 80 K/W; the limited transfer was due to the low thermal conductivity of the PCM, which is thus the main property to optimize. The PCM's conductivity may be enhanced by various methods, a widely studied approach being loading with expanded graphite. As summarized in Figure 11, the conductivity of stearic acid-based PCM increased with the mass fraction of expanded graphite (left), while the latent heat decreased (right). Thus, a larger storage tank will be needed to store the same amount of energy, decreasing the storage energy density.  
Figure 10: Charging time \(\tau_{95\%}\) (left axis) and valuable discharged energy above 45 \({}^{\circ}\)C (right axis) as function of the mass flow rate between 43 \({}^{\circ}\)C and 70 \({}^{\circ}\)C.  
Figure 9: Charging time \(\tau_{95\%}\) and energy stored (left axis) and storage efficiency (right axis), as functions of the inlet temperature during a charge from 43 \({}^{\circ}\)C at 0.05 kg/s.  
The industrial-grade SA-PA mixture used to fill the prototype has similar thermophysical properties to the one studied by [28], whose results showed intermediate values compared to the other works, and were partially repeated by [49]. Their experimental results on conductivity enhancement were then used in the following numerical results. An extrapolation was conducted, in accordance with other studies, with 28% expanded graphite to explore the effect of a conductivity value up to 50 times the standard. The heat capacity varied with the proportion of expanded graphite proportionally to its mass percentage, but the density was kept constant, as opposite influences were found in the literature [27; 49].  
Figure 12 scrutinizes the effects on the charging time and the system energy density (left), as well as the valuable energy available at the discharge (right). As expected, the charging time was minimized by a higher conductivity, but this improvement reached a _plateau_ after a multiplication by \(\sim\)10 of the PCM conductivity. Meanwhile, the energy density continued to decrease when expanded graphite was added: an optimum loading may be around 4%, leading to a conductivity of near 1.8 W/m\(\cdot\)K. Besides, after this conductivity threshold, the valuable energy was no longer improved, and even started to slightly decrease after 8 W/m\(\cdot\)K. That threshold is dependent on the system's features, such as the HTF. For instance, in a system with air as the HTF, the convective heat transfer between the HTF and the capsule wall starts to impose limitations earlier, and it is no longer relevant to improve the conductivity of the PCM when it exceeds 1 W/m\(\cdot\)K [50].  
Figure 11: Influence of the expanded graphite (EG) mass fraction on the PCM’s conductivity (**left**) and latent heat of melting (**right**) in the works of [PERSON] et al. (2014) [48], [PERSON] et al. (2016) [27], [PERSON] et al. (2019) [28] and [PERSON] et al. (2019) [49].  
Figure 12: Charging time \(\tau_{QS\%}\), energy density and valuable discharge energy above 45 \({}^{\circ}\)C for different values of the PCM thermal conductivity.