## 3 Results and Discussion
### Effect of the Volume Fraction of the Second Phase \(\alpha_{nf}\) on the Thermal Characteristics  
Figure 8 shows the average heat transfer coefficient obtained with single-phase approach and with the two-phase mixture model. Different combinations of phase two volume fraction \(\alpha_{nf}\) and of phase two Al\({}_{2}\)O\({}_{3}\) nanoparticle volume fraction \(\Phi\), as defined in Table 2, provide the same 1% Al\({}_{2}\)O\({}_{3}\) nanoparticle volume fraction in the mixture. The results are compared with the experiments by [PERSON] and [PERSON] [25], over the Reynolds number range 92 to 455. Overall, the average heat transfer coefficient in the minichannel heat sink increases with the Reynolds number, due to the higher velocity providing a stronger heat convection. This improvement in \(h\) is similar to that reported in the literature [57; 58]. The mixture model with \(\left(\alpha_{nf}=1\%,\Phi=100\%\right)\) is shown to overpredict the average heat transfer coefficient when compared to the experiments and to any of the other models, due to the overestimation of the phase interaction between nanoparticles and the base fluid, consistent with the results by [PERSON], et al. [11], [PERSON] and [PERSON] [26], and [PERSON], et al. [31]. On the contrary, the average heat transfer coefficient from the single-phase approach and from the two-phase mixture model with \(\left(\alpha_{nf}\geq 2\%,\Phi\leq 50\%\right)\) is underestimated, particularly at lower Reynolds numbers. This is in line with the results  
Figure 7: Validation of numerical results (**a**) the average heat transfer coefficient and (**b**) friction factor. obtained for water reported in Section 2.9, which show a similar underestimation at low Reynolds numbers. It is observed that the predictions from the single-phase approach and from the two-phase mixture with \(\left(\alpha_{nf}\geq 2\%,\Phi\leq 50\%\right)\) essentially overlap. The close agreement between the two-phase mixture model at \(\left(\alpha_{nf}=50\%,\Phi=2\%\right)\) and the single-phase approach at \(\phi=1\%\) shows that the two-phase mixture model predicts the same heat transport as the nanoparticles approach a uniform distribution in the flow, which is what the single-phase approach simulates. The convergence between the two models in their heat transport prediction is an important outcome of this investigation. It shows that the reservations in the application of the two-phase mixture model to minichannel heat transport problems can be overcome by a judicious model use, as further elaborated later in this paper. Figure 8 shows that the average heat transfer coefficient predicted by the two-phase mixture model is substantially invariant over a range of \(\alpha_{nf}\) and \(\Phi\) combinations. This is likely to result from the simple geometry of the channel, which promotes the nanoparticle dispersion towards a uniform distribution. The absence of significant flow curvature to drive phase segregation indicates that the mass transport mechanism in this minichannel is unlikely to form areas of high nanoparticle concentration, which the two-phase mixture model is designed to resolve. Effect of the Volume Fraction of the Second Phase \(\alpha_{nf}\) on the Hydraulic Characteristics  
Figure 9 shows the pressure drop along the minichannel predicted by the single-phase approach and by the two-phase mixture model at the conditions defined in Table 2. The results are compared to the experiments by [PERSON] and [PERSON] [25] in the Reynolds number range 92 to 455 at the same Al\({}_{2}\)O\({}_{3}\) nanofluid volume fraction \(\phi=1\%\). The pressure drop in the minichannel heat sink is shown to increase with the increasing Reynolds number. The single-phase approach and the two-phase mixture model with  
Figure 8: Average heat transfer coefficient of the minchannel heat sink from using the single-phase approach and the two-phase mixture model at \(92\leq Re\leq 455\).