## 3 Results and Discussion
### The Second Law Efficiency of the Minichannel Heat Sink  
Figure 12 displays the exergy destruction rate \(\dot{E}_{xd}\) across the minichannel heat sink over the Reynolds number range 92 to 455 at a Al\({}_{2}\)O\({}_{3}\) nanoparticle volume fraction \(\phi=1\%\). All models show a reduction in \(\dot{E}_{xd}\) with increasing Reynolds number, which is due to the reduction in the minichannel bottom surface temperature at increasing Reynolds numbers [7, 8]. In fact, reducing \(T\) decreases \(\left(1-\frac{T_{r}}{T}\right)\) in the dominant first term of Equation (28) and hence it lowers \(\dot{E}_{xd}\). The exergy inflow transported by the nanofluid is zero since the dead state temperature \(T_{r}\) is equal to the inlet temperature \(T_{in}\). The minimum exergy destruction rate \(\dot{E}_{xd}\) is achieved at the highest Reynolds number, at which the minichannel average bottom surface temperature is lowest, owing to the average heat transfer coefficient being highest, as shown in Figure 8. The \(\dot{E}_{xd}\) predictions from the single-phase approach and from the  
Figure 11: Variation of the minichannel (**a**) average bottom wall temperature and (**b**) inlet to outlet pressure drop with the phase two bubble diameter \(d_{nf}\), over the range 85 nm \(\leq\ d_{nf}\ \leq\ 150\) nm. Two-phase mixture model with second phase volume fractions \(\alpha_{nf}\) 20% and 4%. two-phase mixture model with \(\left(\alpha_{nf}\geq 2\%,\Phi\leq 50\%\right)\) essentially overlap, confirming the good agreement among the models in predicting the minichannel thermal performance reported in Figure 8. The two-phase mixture model with \(\left(\alpha_{nf}=1\%,\Phi=100\%\right)\) produces predictions discordant with this data collapse. This indicates that the two-phase mixture model is improperly applied to a second phase made up solely of nanoparticles, as discussed in Section 3.2. This agrees with the finding in Figure 8, where the heat transfer coefficient of the mixture model with \(\left(\alpha_{nf}=1\%,\Phi=100\%\right)\) is reported as being well above that from all other models, as well as above the experimental measurements. Figure 13 displays the rate of exergy expended \(\dot{E}_{x}\) in the minichannel heat sink, predicted by the single-phase approach and by the two-phase mixture model, over the Reynolds number range 92 to 455, at a Al\({}_{2}\)O\({}_{3}\) nanoparticle volume fraction \(\phi=1\%\). The only contribution of the rate of exergy expended \(\dot{E}_{x}\) is the heat transfer through the heated base, since the dead state reference temperature \(T_{r}\) is equal to the inlet temperature T so that \(\psi=0\) across the inlet. Figure 13 shows the same decreasing trend in \(\dot{E}_{x}\) with increasing Reynolds number as for \(\dot{E}_{xd}\) in Figure 12. There is good agreement among the predictions from the single-phase approach and from the two-phase mixture model with \(\left(\alpha_{nf}\geq 2\%,\Phi\leq 50\%\right)\), which indicate that the lowest rate of exergy expended occurs at the highest Reynolds number. The two-phase mixture model with \(\left(\alpha_{nf}=1\%,\Phi=100\%\right)\) gives lower exergy destruction rate predictions that are lower than the ones shown in the data collapse with \(\left(\alpha_{nf}\geq 2\%,\Phi\leq 50\%\right)\). This supports the \(\left(\alpha_{nf}=1\%,\Phi=100\%\right)\) combination not being an appropriate two-phase mixture model setting, as discussed in the context of Figure 8.  
Figure 12: Variation of the exergy destruction rate \(\dot{E}_{xd}\) in the minichannel with Reynolds number, over the range \(92\leq Re\leq 455\). Predictions from the single-phase approach and from the two-phase mixture model. Figure 14 shows the second law efficiency \(\eta_{II}\). of the minichannel heat sink, estimated using the single-phase approach and the two-phase mixture model, at Reynolds numbers between 92 and 455, at a Al\({}_{2}\)O\({}_{3}\) nanoparticle volume fraction \(\phi=1\%\).