## 3 Results and Discussion
### Effect of the Volume Fraction of the Second Phase \(\alpha_{nf}\) on the Thermal Characteristics
\(\left(\alpha_{nf}\geq 6\%,\Phi\leq 16.67\%\right)\) underpredict the pressure drop compared to the experiments, and so does the \(\left(\alpha_{nf}=1\%,\Phi=100\%\right)\) two-phase mixture model, as previously reported by [PERSON], et al. (2011) and [PERSON] and [PERSON] (2014). In the two-phase mixture model with \(\left(\alpha_{nf}=1\%,\Phi=100\%\right)\), the second phase is made up entirely of nanoparticles, yet the phase is modelled as a Newtonian fluid, with a finite molecular viscosity. This is an unphysical implementation of the two-phase mixture model, as a volume of nanoparticles packed at 100% volume fraction would respond to stress by a strain rather than by a strain rate. As such, Equation (19) appears to be used beyond its range of applicability. Figure 9 shows that the \(\left(\alpha_{nf}=1\%,\Phi=100\%\right)\) prediction is an outlier. Specifically, Figure 9 shows a monotonic increment of the pressure drop predicted by the two-phase mixture model with decreasing \(\alpha_{nf}\), down to \(\alpha_{nf}=2\%\), followed by a sharp drop in pressure loss, as \(\alpha_{nf}\) is reduced from 2% to 1%, across the whole Reynolds number range 92 to 455. This behaviour typically suggests an inconsistency in the use of the physical model, which this work has identified in the definition of the second phase viscosity. Concerning the two-phase mixture model with \(\alpha_{nf}\geq 2\%\), Figure 9 shows that increasing \(\alpha_{nf}\) produces pressure drop predictions that monotonically reduce at increasing \(\alpha_{nf}\) and that incrementally approach the pressure drop predicted by the single-phase simulation. The closest two-phase mixture model prediction to that from the single-phase approach is that with \(\left(\alpha_{nf}=50\%,\Phi=2\%\right)\), which most closely represents the homogeneous mixing of the single-phase approach among the two-phase mixture simulations. This convergence in predictions provides further confidence in using the two-phase mixture model to perform minichannel flow predictions, provided its modelling assumptions are upheld. Figure 9 shows that, unlike the average heat transfer coefficient, the pressure drop depends on the combination of the phase two volume fraction \(\alpha_{nf}\) and of the phase two Al\({}_{2}\)O\({}_{3}\) nanoparticle volume fraction \(\Phi\). As the phase two volume fraction decreases below 50%, the pressure drop approaches the experimental results. For \(\alpha_{nf}=4\%\), the mixture corresponds to the case with the highest \(\Phi\) for which the homogenous model used within  
Figure 9: Minichannel inlet to outlet pressure drop \(\left(\Delta p\right)\) from the single-phase approach and the two-phase mixture model at \(92\leq\mathrm{Re}\leq 455\). a single bubble is valid. For \(\alpha_{nf}=2\%\), the volume fraction of nanoparticles within a bubble is above 30%, beyond the range of applicability of the homogenous model [4], and this is reflected by a steep increase in pressure drop compared to the \(\alpha_{nf}=4\%\) case. These results confirm the challenge of modelling nanofluids. As the homogeneous mixture relation between viscosity and nanoparticle volume fraction is not linear, Equation (19), the overall pressure drop is sensitive to the second phase nanoparticle volume fraction \(\Phi\) in the two-phase mixture model. This means, in principle, that it is possible to cluster nanoparticles to obtain a desired pressure drop independently from the average heat transfer coefficient. This may be obtained by adding a coagulant to the nanofluid to promote flocculation or by adding a surfactant to promote nanofluid particle dispersion. Surfactants appear to be of greater potential benefit to a minichannel heat sink than coagulants, since, by promoting the nanoparticle dispersion, the condition of a homogeneous mixture of the single-phase model could be induced, which Figure 9 shows being that associated with the smallest pressure drop, among the physically valid models.