## 4 Results and Discussion  
All the results presented in the following have been obtained using the mixture model with temperature-dependent properties and Equations (7) and (10) to model the nanofluid properties. The influence of the Reynolds number \(Re\), the volume fraction \(\varphi\), the diameter \(d_{np}\) of the nanoparticles and the type of nanoparticles on the hydrodynamic and the thermal fields are successively discussed in details in the following sections. Influence of the Volume Fraction of Nanoparticles and Reynolds Number for \(Al_{2}O_{3}\)/Water-Based Nanofluids  
The combined effects of the Reynolds number and the volume fraction of nanoparticles on the average heat transfer coefficient are plotted in Figure 7. It can be clearly observed that an average enhancement of the convective average heat transfer coefficient of about 40% is achieved when the Reynolds number increases from \(Re=600\) to 1600 for all nanoparticle concentrations \(\varphi\). A linear dependency of \(h_{av}\) is obtained against to \(\varphi\) for the three \(Re\) numbers. Figure 8 shows the axial distribution of the heat transfer coefficient ratio \(h_{nf}/h_{bf}\) along the pipe for \(d_{np}=42\) nm, \(Re=1600\) and three volume fractions of \(Al_{2}O_{3}\) nanoparticles. It clearly indicates an average thermal enhancement of 28%, 48% and 75.6% for \(\varphi=0.006\), 0.01 and 0.016, respectively. This ratio \(h_{nf}/h_{bf}\) is rather constant in the axial direction \(z/D\) with a local maximum around \(z/D\simeq 85\) whatever \(\varphi\) and a second local maximum at \(z/D\simeq 175\) for \(\varphi=0.01\). Figure 7: Effects of the \(Al_{2}O_{3}\) nanoparticle concentration \(\varphi\) and Reynolds number \(Re\) on the average heat transfer coefficient \(h_{av}\) for \(d_{np}=42\) nm. Figure 9 illustrates the temperature contours at four axial positions \(z=0.2\) m (\(z/D=44.4\)), 0.4 m (\(z/D=88.9\)), 0.6 m (\(z/D=133.3\)) and 0.8 m (\(z/D=177.8\)) for \(d_{np}=42\) nm, \(Re=1600\) and four concentrations \(\varphi\). The temperature contours change from a circular form at \(z=0.2\) m, to an elliptical one at \(z=0.4\) m then to a kidney shape from \(z=0.6\) m to the tube outlet for all concentrations of nanoparticles. The circumferential wall temperature appears to be non-uniformly distributed in the tangential direction, especially at \(z=0.8\) m, with maximum values at the top of the tube. It can be simply explained by density variations, since the warm nanofluid has a lowest density and can rise due to the buoyancy force to the upper half of the tube inducing a stratification of the fluid temperature. This suggests the necessity to consider temperature-dependent properties for the nanofluid in order to predict this effect. The hot temperature region located at the top of tube for \(\varphi=0\) shown in Figure 9d progressively disappears when increasing the nanoparticle volume fraction. For example, the wall temperature \(T_{w}\) decreases noticeably when \(\varphi\) increases: at \(z=0.8\) m, the maximum wall temperature decreases from 332 K to 293 K for \(\varphi=0\) (Figure 9d) and \(\varphi=0.016\) (Figure 9p), respectively. The maximum temperature difference is inversely proportional to the nanoparticle volume fraction. It is reduced almost by a factor 2 between \(\varphi=0.016\) and \(\varphi=0\). More generally, the introduction of even higher volume fractions (for this range of parameters) tends to homogeneize the temperature distribution at a given cross-section. It may be explained by considering nano-convection effect, which is linearly related to \(\varphi\)[43]. This effect is induced by the Brownian motion of the nanoparticles. Brownian motion caused by the thermal interaction between the nanoparticles and the base fluid is stronger within regions of higher fluid temperature that is why the upper half of the tube is more affected.