## 3 Validation of the Numerical Model
### Performances of the Mixture Model  
In order to first validate the selected numerical model, the local heat transfer coefficient \(h(z)\) is evaluated along the pipe length. The mixture model is used together with temperature-dependent properties for water and Equations (7) and (10) for the nanoparticle properties. Due to the lack of precise information on the temperature measurement procedure in the experiments of [PERSON] and [PERSON] [11], four different averaging methods are used to evaluate the wall temperature of the simulated cases. Comparisons with the measurements of [PERSON] and [PERSON] [11] and the numerical simulations of [PERSON] et al. [19], for \(Re=1600\) and \(\varphi=0.6\%\), are performed using the mixture model. Figure 3a illustrates that using an average over an upper arc of \(\pm 45^{\circ}\) leads to closer results to the experiments with an average error between 0.37% and 13.84% along the pipe. The temperature was probably measured experimentally near the top of the tube where the fluid is warmer due to buoyancy forces. This averaging method will be used adopted for the remainder of the study.  
Figure 3b compares the values of the convective heat transfer coefficient \(h(z)\) of the present simulations with the numerical results of [PERSON] et al. [19] and the experimental data of [PERSON] and [PERSON] [11] for \(Re=1600\) and three volume fractions of \(Al_{2}O_{3}\) nanoparticles \(\varphi=0.6\%,1\%\) and \(1.6\%\), respectively. The present numerical results agree fairly well with the experiments [11] with an exponential decrease of the heat transfer coefficient along the tube as expected from energy balance equation. The main discrepancy is observed close to the inlet due to the choice of the boundary conditions in the present calculations. It is noteworthy that the present  
Figure 3: Axial variations of the local heat transfer coefficient \(h(z)\) predicted by the mixture model for \(Al_{2}O_{3}\) water-based nanofluids: (**a**) different methods for the evaluation of the wall temperature \(T_{w}\) for \(\varphi=0.006\); (**b**) three nanoparticle concentrations (\(\varphi=0.006\), 0.01 and 0.016). Comparisons for \(Re=1600\) and \(d_{np}=42\) nm with the experimental data of [PERSON] and [PERSON] [11] and the numerical simulations of [PERSON] et al. [19].  
simulations improve the predictions of [PERSON] et al. [19] using exactly the same solver and methods. It points out in particular the necessity to use an appropriate mesh grid in the near-wall regions and globally a more dense grid. Compared to [19], the number of mesh points is multiplied by a factor 86. The mesh grid sensitivity studies are generally performed using small increments of values such that no noticeable effect is observed.  
Figure 4 provides further comparisons in terms of the average heat transfer coefficient \(h_{av}\) for the same three cases. The present results obtained using the mixture model show an acceptable agreement with the experimental ones. The numerical data exhibit an average enhancement of about 36% with increasing \(\varphi\) from 0.6% to 1.6%, which is to be compared to the value 24.3% in the experiments. Once again, the present simulations improve the previous ones of [PERSON] et al. [19] pointing out the influence of the mesh grid. This improvement may be attributed to the increase of the thermal conductivity and some authors [11; 12; 40] proposed that it is also associated to the decrease of the thermal boundary layer thickness.