## 5 Results Discussion
### Effect of Rayleigh Number
Therefore, the improvement of fluid motion is also more significant in the lower ranges. Figure 7 demonstrates the variation of Nuavg with Ra for different values of the volume fraction of the nano-solid particles (\(\phi\)). For Ra values below 100,000, values of Nuavg are almost equal with increasing values of Ra, besides increasing \(\phi\) does not significantly increase Nu. The almost equal behavior is due to the equilibrium between conduction and convection. The highest heat transfer rate occurs with the upper values of Ra, and we notice the disappearance of any difference of Cloud for different \(\phi\). To further examine the effect of permeability and its relation to the cross-sectional area in the problem, the Nuavg change with Ra at different values of Da is shown in Figure 8. The plots and conclusions are identical to those in Figure 6, where more fluid freedom of motion represents better heat transfer, which is attributed to better convection. Variations of total entropy generation with various values of Ra for Ha are shown in Figure 9. The entropy generation increases with Ra for higher Ra values regardless of Ha. However, and projecting it to Figure 5, it can be seen that for high values of Ra, the vortices and heat transfer decrease with increased Ha, and the Lorentz force is also accompanied by a decrease in the total entropy. This reflects the superior role of the thermal and fluid friction entropy as compared with the entropy, due to the magnetic field for that Ra range. For the small Ra range, the thermal and fluid friction entropy generation is small, reducing the total entropy generation, and therefore the Ha has no impact. Figure 10 represents variations of total entropy generated with various values of Ra for different \(\varepsilon_{p}\), and the results are similar to those in Figure 6.That means, the entropy generation (as mentioned upward) is governed mainly by thermal and fluid friction entropy generation (large heat transfer from more freedom of fluid motion) at high values of Ra, whereas at low values of Ra the entropy generation is indifferent to various \(\varepsilon_{p}\) values. Figure 3:0 Total entropy as a function of Rayleigh number (Ra) at various values of porosity (\(\varepsilon_{p}\)). Ha = 0, Da = 0.01 and \(\phi\) = 0.02. The total entropy generation, plotted against Ra for different \(\phi\) values, is plotted in Figure 3:1 Clearly, the variation of \(\phi\) does not impact total entropy generation significantly, which, once again, confirms that the total entropy generation is mainly governed by heat transfer, and heat transfer has shown small variations with different \(\phi\) values given in Figure 3:2  
Figure 12, which plots the total entropy generation against Ra for different Da numbers, can be compared again with Figure 8 to realize the dominance of the thermal and fluid motion (friction) contribution to the total entropy generation. Therefore, the main conclusion from Figures 9-12 is that whenever a parameter enhances heat transfer, total entropy generation is impacted in a similar manner (increase), as the thermal entropy generation contribution clearly is dominant against other entropy generation components, especially for high values of Ra.