## 5 Results and Analysis
#### Single-Phase and Solid-Liquid Phase Change with High Prandtl Number  
To investigate the effect of phase change on the entropy generation, we conduct the simulations on octadecane, whose parameters are summarized in Table 4. Octadecane is known for having a relatively high convective coefficient. This increases the heat transfer and the fluid velocities. The generation of latent heat during the melting process increases the energy stored in the system. However, we will investigate whether this is provoked by an increase or decrease in the total entropy generation. \begin{table}
\begin{tabular}{c c c c c} \hline
**Parameter** & **Pr** & **Ra** & \(\varphi\) & **Ste (Figure 1b)** \\ \hline Value & 50 & \(10^{4}-10^{5}\) & \(10^{-5}\) & 1.0 \\ \hline \end{tabular}
\end{table} 
Table 4: The simulation parameters of Octadecane. Figure 8: Bejan number _vs._ Prandtl number. Figures 9 and 10 show the temporal evolution of the entropy generation during heat transfer with no phase change (the problem described in Figure 1a). Here, these quantities further exhibit oscillations due to the high value of Ra  before reaching their asymptotic values. Such behavior is due to boundary conditions [3]. At first, the isotherms are almost vertical and parallel to the walls since the heat transfer is dictated by the conduction. As time passes, the advection mode becomes more dominant and distorts isotherms, inducing vertical temperature gradients. This will result in a decrease in horizontal temperature gradients, which eventually become locally negative. Consequently, the central streamline elongates to give rise to a second eddy. Thus, for smaller Rayleigh numbers, the steady state is sufficiently close to the equilibrium state, allowing the system to return towards the steady state. Thereby, [PERSON]'s theorem of minimum entropy production is satisfied. However, for larger Ra's, the equilibrium state being relatively far, the system oscillates increasingly. Note that this finding has already been reported by [PERSON] [6]. We also observe that the steady state is reached sooner at larger Ra (Figure 10), whereas at lower Ra (Figure 9), oscillations of transient entropy generation are both apparent and smooth. This is due to the effect of the heat transfer coefficient. For more information on this, we plot in Figures 11 and 12 the Nusselt number (Nu) _versus_ the dimensionless time for the cases of Ra = 10\({}^{4}\) and 10\({}^{5}\). The rate of  
Figure 10: Entropy generations with no phase change (problem in Figure 1a) for Pr = 50 and Ra = 10\({}^{5}\). Figure 9: Entropy generations with no phase change (problem in Figure 1a) for Pr = 50 and Ra = 10\({}^{5}\). variation of Nu affects the losses due to heat transfer. However, the increase of Nu values at steady state has a significant effect on the fluid velocities and, hence, fluid friction losses. Figure 13 shows the Nusselt number (Nu) _versus_ the different values of (Ra). We deduce that somewhere after \(\mathrm{Ra}=10^{4}\), the heat transfer coefficient at steady state increases dramatically. This leads to higher fluid velocities and, therefore, higher fluid friction losses, as revealed in Figure 10.  
Figure 14 shows the temporal variation of the entropy generation in the case of melting with convection. Note that the entropy generation oscillates at the beginning with about a 25% decrease compared to the case of heat transfer with no phase change. Nevertheless, at the steady state, the total entropy generation decreases by more than 50% with the presence of phase change and latent heat dissipation, as illustrated in Figure 14. This is also justified by tracing the Nu _versus_ time in Figure 15. The values of Nu decrease in comparison with the case of no phase change. The reason for that is that the fluid volume increases with time as the melting front recedes.