## 4 Results and Discussion
### Summary  
For engineering applications, empirical correlations are of primary importance to predict the average heat transfer coefficient as a function of the flow and geometrical parameters for an effective design of thermal systems such as heat exchangers. This is particularly challenging for convective nanofluid flows as opposed to single-phase flows, because of the influence of various parameters due to the presence of the solid nanoparticles. An attempt has been done in the following. First, the well-known correlation proposed by [PERSON] (1980) for laminar flows under a constant heat flux boundary condition and used by [PERSON] (1981) is considered:  
\[Nu_{[PERSON]}=1.953\left(RePr\frac{D}{x}\right)^{1/3}\qquad 33.3\leq RePr\frac{D}{x} \tag{29}\]  
A second correlation was proposed by [PERSON] (1980) for \(RePr\frac{D}{x}\leq 33.3\) but it was carefully checked here that it led to very similar results. For simplicity, only Equation (29) will be used in the following. All the results obtained in this paper using direct numerical simulations may be expressed in terms of Nusselt number using the following correlation:  
\[Nu_{DNS}=Nu_{[PERSON]}\left(1+1.7\left(\frac{Re}{Pr^{4}}\right)^{1/3}\varphi \right)\qquad 6.6\leq RePr\frac{D}{x}\leq 46.5\qquad Pr\geq 1 \tag{30}\]  
It is noteworthy that the validity range of Equation (30) has been extended to \(6.6\leq RePr\frac{D}{x}\leq 46.5\) compared to Equation (29). The present correlation is valid for all values tested in the present work except for the simulations involving copper nanoparticles for which \(Pr\leq 1\). Both geometrical (through \(x\) and \(D\)) and flow (\(Re\) and \(Pr\)) parameters are considered in Equation (30). The influence of the solid nanoparticles is taken into account through \(\varphi\) but also through their thermophysical properties used to define \(Re\) and \(Pr\). Figure 15 confirms that Equation (30) fits particularly well with all simulations for \(Pr\geq 1\). Figure 15: Verification of Equation (30). The second term \(1.7\left(Re/Pr^{4}\right)^{1/3}\varphi\) in Equation (30) is small compared to 1 in all cases. The results are then well fitted with Equation (29) and so with the correlation provided by [PERSON] et al. [35] for \(Al_{2}O_{3}/\)water-based nanofluids, \(431\leq Re\leq 2000\) and \(\varphi\leq 0.06\). It corresponds to Equation (29) with a prefactor equal to \(2.0398\) instead of \(1.953\). The second interesting quantity for engineering applications is the variations of the average friction factor \(f_{av}\) as a function of the Reynolds number \(Re\). The friction factor is calculated as proposed by [PERSON] and [PERSON] [51]:  
\[f=\frac{8\tau_{av}}{\rho u_{m}^{2}} \tag{31}\]  
where \(u_{m}\) is the mean fluid velocity and \(\tau_{av}\) is the wall shear stress. It may be convenient to find correlations under the form: \(f=ARe^{\alpha}\). Figure 16 summarizes the results obtained for \(Al_{2}O_{3}/\)water-based nanofluids with alumina nanoparticles of diameter \(d_{np}=42\) nm and for four volume fractions \(\varphi\). As for pure water flows, the average friction factor \(f_{av}\) decreases for increased values of \(Re\). Complementary calculations have been performed for turbulent flows up to \(Re=15,000\). In the laminar regime, \(f_{av}\) varies according to the relation \(f_{av}=11.381 Re^{-0.756}\), whereas, in the turbulent regime, \(f_{av}\) follows: \(f_{av}=1.05 Re^{-0.358}\). A discontinuity is also observed in the transitional regime. The presence of nanoparticles in the base fluid affects the variations of the average friction factor compared to pure water flows. However, the friction factor \(f_{av}\) appears to be insensitive to the nanoparticle concentration \(\varphi\). The results are to be compared to the classical Darcy relations for single-phase flows in pipes, where: \(A=64\) and \(\alpha=-1\) for laminar flows and \(A=0.3164\) and \(\alpha=-0.25\) for turbulent flows in smooth pipes. In the laminar regime, the results agree particularly well with the correlation of [PERSON] et al.