## 3 Results and Discussion
The nanofluid temperature at the centerline of the pipe (T\({}_{\rm min}\)) was extracted from the CFD simulations directly without any further calculations and followed the same trend of T\({}_{\rm m}\) and T\({}_{\rm w}\), as shown in Figure 12c. Figure 5:1 Values of (_ReC\({}_{f}\)_) at different values of (**a**) Kn; (**b**) Re; and (**c**) Pr. The dual variation in \(\rm T_{m}\) and \(\rm T_{w}\) as a function of \(\rm Pr\) resulted in making the dependence of \(\rm Nu_{x}\) on \(\rm Pr\) at a constant \(\rm Kn\) negligible, as shown in Figure 13a. A similar trend was found for different \(\rm Re\) values, as shown in Figure 13b. At low \(\rm Re\) values, the nanofluid velocity was low, which resulted in a higher \(\rm T_{m}\) slope, based on Equation (15), at a relatively short entrance distance. As the \(\rm Re\) value increased, the nanofluid velocity increased and resulted in a low \(\rm T_{m}\) slope at a relatively long entrance distance, as shown in Figure 13c. The fully developed value of \(\rm Nu_{x}\) was reached at different entrance distances with similar trends. This explains why all simulated \(\rm Re\) values followed the same curve of \(\rm Nu_{x}\) vs. (x/(D\({}_{\rm h}\) Re Pr)) at a constant \(\rm Kn\), as shown in Figure 13b. Moreover, the dependence of \(\rm Nu_{x}\) on \(\rm Kn\) was more significant than the dependence on \(\rm Pr\) or \(\rm Re\), as shown in Figure 13d. It was confirmed that as \(\rm Kn\) increased, the value of \(\rm Nu_{x}\) decreased along the axial distance traveled by the nanofluid. The slip flow and temperature jump effects at the wall were caused by different values of \(\rm Kn\). As a result, they changed both the shear stress values and the \(\rm T_{w}\) values, which resulted in a decrease in the value of \(\rm Nu_{x}\). At \(\rm Kn=0\), the fully developed value of \(\rm Nu_{x}\) was equal to 4.36, and the minimum \(\rm Nu_{x}\) value obtained was equal to 2.5 at \(\rm Kn=0.1\) for all investigated \(\rm Pr\) and \(\rm Re\) values. Therefore, \(\rm Kn\) has negatively affected both the value of \(\rm Nu_{x}\) and the value of the entrance length (\(\rm L/D_{\rm h}\)). Figure 5:2 Temperature profiles due to different nanoparticle volume fractions: (**a**) nanofluid bulk temperature (\(\rm T_{m}\)); (**b**) wall temperature (\(\rm T_{w}\)); (**c**) pipe centerline temperature (\(\rm T_{min}\)). The thermal entrance length (L/D\({}_{\rm h}\))\({}_{\rm th}\) was defined as the hydrodynamic entrance length multiplied by Pr, and it was related to the development of the thermal boundary layer inside the pipe. For the current study, the flow was considered thermally fully developed when the temperature ratio (\(\theta\)), defined in Equation (20), was equal to 0.8. The inlet temperature (T\({}_{\rm in}\)) was constant at 300 K, while T\({}_{\rm w}\) and T\({}_{\rm min}\) were extracted from the CFD simulations. The current approach was validated against the published work of [PERSON] and [PERSON] [40] for the no-slip flow cases and shown in Figure 14a. As can be seen, the CFD results fitted well with the published correlation for all the investigated nanofluid volume fractions under the no-slip condition (Kn = 0). Building upon this conclusion, the variation in (L/D\({}_{\rm h}\))\({}_{\rm th}\) was examined and presented in Figure 14b as (\(\Delta\)(L/D\({}_{\rm h}\))\({}_{\rm th}\) = (L/D\({}_{\rm h}\))\({}_{\rm th-slip}\) - (L/D\({}_{\rm h}\))\({}_{\rm th-no\,slip}\)). The increase in \(\Delta\)(L/D\({}_{\rm h}\))\({}_{\rm th}\) with Re, when compared with same conditions at no-slip conditions, was clear, with a maximum value of 14.3 diameters. This is translated to an increase in the thermal entrance length by 14.3 diameters for the slip flow, with Kn = 0.1 at Re = 1750. At Re = 250 and Kn = 0.1, however, the maximum increase was around 2.0 diameters. For the remaining Kn values, the maximum increase in the thermal entrance length was between 2 diameters and 14.3 diameters.