## 4 Results and Discussion  
The parametric study performed in this paper looked at varying the thermal conductivity of dodecanoic acid by multiplying both the solid and liquid thermal conductivities by factors of 1, 1.05, 1.1, 1.2, 1.5, 2, 5, 10, 20, and 50, and determining the resulting system equivalent power over time. Simulations were performed for 12-, 8-, and 4-finned-tube geometries, with a system HTF flow rate of 9 LPM and over two sets of initial and final system temperatures: \(T_{i}\) = 65 \({}^{\circ}\)C to \(T_{f}\) = 21 \({}^{\circ}\)C and \(T_{i}\) = 55 \({}^{\circ}\)C to \(T_{f}\) = 31 \({}^{\circ}\)C.  
Firstly, Figure 9 shows the temperature distribution, as well as the solidification interface, at six specific times (\(t\) = 1200, 1800, 2400, 3000, 4500, and 6000 s) for the original dodecanoic acid, and PCMs with thermal conductivities of 2\(k\), 10\(k\), and 50\(k\) obtained in the eight-finned-tube geometry at a starting temperature of 65 \({}^{\circ}\)C and discharged with an HTF at 21 \({}^{\circ}\)C. In the figure, the solidification interface is indicated by a white line in each image at the phase change temperature of dodecanoic acid (\(T_{m}\) = 43.3 \({}^{\circ}\)C = 316.45 K). From the temperature legend, purple and dark-pink colors correspond to temperatures below the phase change temperature and correspond to the PCM being solid; orange and yellow colors correspond to the PCM being liquid. Figure 9: Temperature distribution snapshots at timesteps of 1200 s, 1800 s, 2400 s, 3000 s, 4500 s, and 6000 s for parametric study of 8 finned tubes at an HTF flowrate of 9 LPM, being discharged from 65 to 21 \({}^{\circ}\)C, showing thermal conductivities of \(k\), 2\(k\), 10\(k\), and 50\(k\). As expected, solidification started on the surface of the finned tube and progressed away as time went on, with faster solidification around the tube section of the fin compared to solidification from the fins. Therefore, solidification ended at the top and bottom of the system around the symmetry line. The progression of the solidification interface for the PCM with a thermal conductivity of \(2k\) was very similar to the original dodecanoic acid, although slightly faster, and a solid bridge was created by the time the simulation reached 6000 s. Increasing the thermal conductivity to \(10k\), the PCM was fully melted by 6000 s. Further increasing the thermal conductivity to \(50k\), the PCM was fully melted by 4500 s, but it can also be observed that with this increased thermal conductivity, the fins play a lesser role in conducting heat away from the solidifying PCM; the solidification interface is much more centered on the tube. This points to the fact that an increase in the thermal conductivity of the PCM could allow a change in geometry (the removing of fins in this case), reducing the necessary heat exchange material and potentially reducing the initial cost of the TES infrastructure. Figure 10 presents the system equivalent power obtained for each finned-tube configuration and operating temperatures for the various PCM thermal conductivity values used in this parametric study. Figure 10a,c,e show the results when the systems were at 65 \({}^{\circ}\)C initially and were discharged with an HTF at 21 \({}^{\circ}\)C for the 12-, 8-, and 4-finned-tube configurations, respectively. The system with 12 fins always had a larger discharging power compared to the other two. For the original dodecanoic acid (\(k\)), the maximum system equivalent power was approximately 500 W for 12 fins, 400 W for 8 fins, and 200 W for 4 fins. Increasing the thermal conductivity to 2\(k\) led to a maximum power of approximately 600 W for 12 fins, 450 W for 8 fins, and 260 W for 4 fins. The increase in absolute power was only by approximately 12 to 30%, with a doubling of the PCM thermal conductivity. Finally, increasing the thermal conductivity by a factor of 50 (50\(k\)) led to a maximum power of approximately 1075 W for 12 fins, 1050 W for 8 fins, and nearly 800 W for 4 fins.