## 4 Results and Discussion
As expected, the increase in thermal conductivity at this point made the number of fins less impactful. An increase in thermal conductivity by a factor of 50 only led to an increase in maximum power by factors of approximately 2.1 to 4. The increase in thermal conductivity was more impactful with fewer finned tubes in the system. The thermal conductivity increase led to a reduction in the total time needed to fully discharge the system. However, it took an increase in conductivity by a factor of 5 or 10 to bring the discharge time below 2 h, and by a factor of 5 in the case for the four-finned-tube geometry. Figure 10b,d,f show similar results when the systems were at 55 \({}^{\circ}\)C initially and discharged with an HTF at 31 \({}^{\circ}\)C. It has been demonstrated that the temperature differential between the HTF driving temperature and the PCM melting temperature plays a crucial role in power extraction [18]. In this case, the HTF temperature being higher led to smaller system equivalent power and a higher difference between power extracted with 12, 8, and 4 fins. For the original dodecanoic acid (\(k\)), the maximum power was approximately 375 W for 12 fins, 275 W for 8 fins, and 160 W for 4 fins. For 2\(k\), it was 450 W for 12 fins, 325 W for 8 fins, and 200 W for 4 fins; absolute power increased by approximately 20% to 25% when compared to a standard \(k\). When increased by 50\(k\), the power was 975 W for 12 fins, 825 W for 8 fins, and 600 W for 4 fins. In this case, the increase in thermal conductivity by a factor of 50 only led to an increase in system equivalent maximum power by factors of approximately 2.6 to 3.8, again with the largest impact of the increase in thermal conductivity being felt with the smaller number of finned tubes. Finally, the numerical model allows studies where just the solid or liquid thermal conductivity of the PCM is changed (with the thermal conductivity of the other phase staying at the original dodecanoic acid value of \(k\)). Figure 11a shows the system equivalent power when the liquid thermal conductivity remains at \(k\) and only the solid thermal conductivity is increased. Figure 10: System equivalent power over time as a function of the PCM solid and liquid thermal conductivities from \(k\) to 50\(k\) for simulations with an HTF flow rate of 9 LPM: 12 finned tubes from (**a**) 65 to 21 \({}^{\circ}\)C and (**b**) 55 to 31 \({}^{\circ}\)C; 8 finned tubes from (**c**) 65 to 21 \({}^{\circ}\)C and (**d**) 55 to 31 \({}^{\circ}\)C; 4 finned tubes from (**e**) 65 to 21 \({}^{\circ}\)C and (**f**) 55 to 31 \({}^{\circ}\)C.  
The power remains the same for each scenario until solidification starts (just before reaching the 0.4 h mark), which is consistent with the liquid thermal conductivity being the same for each scenario. In this scenario, the maximum power extraction reaches a value of 800 W when the solid PCM thermal conductivity is 50\(k_{s}\), compared to a maximum value of 1075 W when the conductivity of both phases is 50\(k\), as per Figure 10a. Figure 11b shows the system equivalent power when the solid thermal conductivity remains at k and only the liquid thermal conductivity is increased. This figure shows the impact of the liquid thermal conductivity when the PCM is entirely liquid prior to solidification (again just before reaching the 0.4 h mark). It was also observed that the liquid thermal conductivity increase on its own plays a very minor role in the overall power extracted from the system. From the information obtained through these studies and the results from Figure 11, it is clear that an increase in the solid PCM thermal conductivity plays the most important role overall, and any increase in the thermal conductivity of the liquid PCM then affirms this increase by further increasing the overall system equivalent power.