## 4 Conclusions  
This work has identified, assessed, and solved the root cause of the research community's lack of confidence in two-phase mixture models for predicting the performance of a nanofluid cooled minichannel heat sink, compared to the more widely adopted and simpler single-phase approach. It has done so by implementing both single-phase and two-phase mixture models for the forced convection in a minichannel heat sink of rectangular cross-section, cooled by water and Al\({}_{2}\)O\({}_{3}\) nanoparticles, at a nanoparticle volume fraction of \(\phi\) =1%, in the Reynolds number range 92 to 455. The key to a successful two-phase mixture model implementation is using water as one phase and, for the second phase, a Newtonian fluid of equivalent thermal and physical properties of a suspension of nanoparticles in water. The second phase nanoparticle volume fraction must be less than 100% to provide a physical fluid, rather than a solid, for the second phase. The same conclusion is arrived at by modelling the flow and heat transport through a microscale backwards-facing step, with a uniformly heated bottom wall. The flow separation and recirculation near the step shows that, in the presence of greater flow complexity, the two-phase mixture model with a 100% phase two volume fraction of nanoparticles is still at variance with the other models. Specifically, it over-predicts the local Nusselt number peak by 35%. Various combinations of second phase volume fraction \(\alpha_{nf}\) and second phase nanoparticle volume fraction \(\Phi\) were used in the two-phase mixture model to represent the same 1% Al\({}_{2}\)O\({}_{3}\) nanoparticle water mixture. For all thermal and flow performance quantities considered, increasing the second phase volume fraction above 2% in the minichannel heat sink provided a monotonic and incremental approach to the predictions from the single-phase model. This clearly shows that the two-phase mixture model converges towards the single-phase predictions for a homogeneous nanofluid. The minichannel mean heat transfer coefficient predictions and, to a good extent, the rates of exergy expended and destroyed, as well as the second law efficiency, were found to be relatively insensitive to the second phase volume fraction, within the range \(\alpha_{nf}\) for which the model remains physically sound. The agreement with experiment in the mean heat  
Figure 16: Local Nusselt number along the heated bottom wall of a microscale backward-facing step at \(Re=280\). transfer coefficient was within 2.5% and 1.76% at \(Re=455\) with the single-phase approach and with the two-phase mixture model \(\left(\alpha_{nf}\geq 2\%,\Phi\leq 50\%\right)\), respectively. On the other hand, the minichannel friction factor and pressure drop display variations with appreciable \(\alpha_{nf}\) dependence. The agreement with experiment varies from \(-2\%\) at \(\alpha_{nf}=4\%\) to 14.7% at \(\alpha_{nf}=50\%\). The latter is similar to the 15.4% agreement with experiment that is obtained using the single-phase approach. These results, albeit limited in applicability by the minichannel simple geometry used in this study, have important practical implications for the progress of heat sink technology. It is shown that in the rectangular minichannel at the tested flow conditions, the lack of significant flow curvature to drive phase segregation develops a condition close to that of a homogeneous mixture. This is evidenced by the average heat transfer coefficient predicted by the two-phase mixture model being substantially invariant over a range of \(\alpha_{nf}\) and \(\Phi\) combinations. This provides an important performance feature to rectangular minichannel heat sinks required to work within tight temperature tolerances. As the current model resolves neither sedimentation nor agglomeration nor other direct particle-particle interaction, the lack of uniformity in the nanofluid due to these processes was not modelled. Therefore, heat sinks featuring such phenomena may well not have this performance feature.