## 3 Validation of the Numerical Model
### Comparative Analysis of Single-Phase and Mixture Models  
Many authors [18; 19; 30; 41] have already shown that the mixture model performs better than single-phase or other two-phase models like the Eulerian or volume of fluid models. Nevertheless, it is crucial to use appropriate correlations for the effective nanofluid properties to obtain accurate results with single-phase models. First, the single-phase (using Equations (7) and (10) for \(\mu_{nf}\) and \(k_{nf}\) respectively) and the mixture models with temperature-dependent are compared in Figure 5 for \(Re=1600\) and four concentrations of nanoparticles. The single-phase model fails to predict the right axial distributions of the local heat transfer coefficient \(h(z)\). An exponential decrease of \(h(z)\) with the axial distance \(z\) is observed however with a noticeable underestimation for all cases with \(\varphi\
eq 0\). As expected, for \(\varphi=0\), both models predict the same profile. Though the nanofluid properties take into account the volume fraction of nanoparticles \(\varphi\), the single-phase model appears insensitive to \(\varphi\) as the same heat transfer coefficient distribution is obtained whatever the value of \(\varphi\leq 0.016\). This confirms the previous results of [PERSON] et al. [19] and [PERSON] et al. [17]. On the other hand, the heat transfer coefficient predicted by the  
Figure 4: Influence of the concentration of \(Al_{2}O_{3}\) nanoparticles \(\varphi\) on the average heat transfer coefficient for \(Re=1600\) and \(d_{ny}=42\) nm. Comparisons with the experimental data of [PERSON] and [PERSON] [11] and the numerical simulations of [PERSON] et al. [19]. mixture model clearly increases with increased nanoparticle concentration in agreement with previous observations [18; 19; 30; 41]. The superiority of the mixture model may be easily explained. In fact, the latter ensures a more accurate treatment of the two-phase mixture, compared to the single-phase model, which does not take into account either the spatial variations of the distribution in nanoparticles in the base fluid, nor the relative velocity of each phase. The mixture model seems a better model to describe the nanofluid flow. In fact, the slip velocity between the fluid and the nanoparticles is not zero due to several factors such as the Brownian motion or gravity, which induces for example the sedimentation of the solid particles. One could argue that the single-phase model does not perform well due to inappropriate correlations for the thermal conductivity and the dynamic viscosity of the nanofluid. Thus, several correlations have been tested and the results are summarized in Table 2 for \(Re=1600\), \(d_{np}=42\) nm and \(Al_{2}O_{3}\)-water based nanofluids. Two volume fractions \(\varphi=0.006\) and \(0.016\) have been considered. Simulation 1 combines Equations (7) and (10), whereas simulation 2 uses Equations (8) and (9). It is clear that the two sets of correlations provide rather the same results in terms of the average value of the heat transfer coefficient with differences of about 17% and 33% for \(\varphi=0.006\) and \(0.016\), respectively, compared to the experiments. It shows in particular that for these sets of parameters, the correlations do not considerably influence the accuracy of the single-phase model. As shown previously, the results are not influenced by the particle volume fraction. Though the effect of Brownian motion is accounted for in Equation (10), it has no noticeable influence in laminar flows. This confirms the previous work of [PERSON] et al. [42], who suggested that the motion of nanoparticles due to the Brownian motion is too slow to transport a significant amount of heat through a nanofluid. They ignored the effect of Brownian motion in the enhancement of thermal conductivity of nanofluids. Figure 5: Local heat transfer coefficient obtained for \(Re=1600\), \(d_{np}=42\) nm and \(Al_{2}O_{3}\)-water based nanofluids: comparison between the single-phase and mixture models with temperature-dependent properties.