## 3 Results and Discussion
With the already validated CFD results under the slip flow, as shown in Figure 2a, the effects of the slip flow on the value of (L/D\({}_{\text{h}}\)) were examined and compared with the no-slip condition results for the same Re and Pr values (\(\Delta\)(L/D\({}_{\text{h}}\)) = (L/D\({}_{\text{h}}\))\({}_{\text{no}}\) slip \(-\) (L/D\({}_{\text{h}}\))\({}_{\text{slip}}\)), as shown in Figure 10b. It can be seen that as the Kn value increased, the reduction in the (L/D\({}_{\text{h}}\)) value increased due to the decreased value of U/U\({}_{\text{m}}\) for all investigated Re values. Therefore, the distance required for the nanofluid to travel toward the fully developed value of U/U\({}_{\text{m}}\) was decreased. Consequently, the hydrodynamic entrance length became shorter with the increase in Kn values when compared with the no-slip conditions, starting from a reduction of about 0.8 diameters at Kn = 0.1 for Re = 250 to a maximum value of around five diameters at Kn = 0.1 for Re = 1750. Therefore, the maximum reduction due to the increase in the Kn value was 5 diameters at Re = 1750 and due to the increase in Re value was 5 \(-\) 0.8 = 4.2 diameters at Kn = 0.1, as shown in Figure 10b. Therefore, the presence of a slip wall reduced the hydraulic entrance length with the increase in the Kn and Re values. The values of C\({}_{\text{f}}\) obtained from CFD simulations were estimated based on Equation (19) and shown in Figure 11. The highest C\({}_{\text{f}}\) value was at the inlet of the pipe due to the highest velocity gradient at the wall, which resulted in the highest wall shear stress. These values decreased as the nanofluid traveled downstream the pipe, causing the velocity gradient to decrease until it reached the fully developed value of 16/Re. As the slip flow became active, with Kn values greater than 0.01 and less than 0.1, the C\({}_{\text{f}}\) value started to drop below the no-slip condition. The lowest ReC\({}_{\text{f}}\) value obtained at Kn = 0.1 was around 8.79, and the highest value under slip flow conditions was equal to 14.78 at Kn = 0.01, as shown in Figure 11a. Similar to the U/U\({}_{\text{m}}\) results, it has been found that ReC\({}_{\text{f}}\) was insensitive to the value of \(\times\)/(D\({}_{\text{h}}\) Re Pr), resulting from the addition of nanoparticles to the base fluid with different percentages, as shown in Figure 11b,c. As explained earlier, the change in \(\lambda\) value has to be significant to cause a significant change in Kn that would increase the slip velocity value. The increase in slip velocity would decrease the wall shear stress, resulting in a decrease in the value of C\({}_{\text{f}}\). Figure 5:0 Effects of nanoparticles volume fraction represented by (Pr) on (**a**) hydrodynamic entrance length (L/D\({}_{\text{h}}\)) at Kn = 0, compared with Shah and London [40]; and (**b**) variations in hydrodynamic entrance length (\(\Delta\)(L/D\({}_{\text{h}}\)) = (L/D\({}_{\text{h}}\))\({}_{\text{no}}\) slip \(-\) (L/D\({}_{\text{h}}\))\({}_{\text{slip}}\)) at 0.01 \(\leq\) Kn \(\leq\) 0.1. The addition of nanoparticles to air would change the thermophysical properties of the resulting nanofluid, the value of Pr, and the flow inlet velocity. These changes would result in changing the magnitude of nanofluid bulk temperature (T\({}_{\rm m}\)) in accordance with Equation (15) and shown in Figure 12a. Increasing the nanoparticle volume fraction would result in a decrease in Pr and an increase in the specific heat of the nanofluid (C\({}_{\rm p}\)), which increases the ability of the nanofluid to absorb more heat. Therefore, T\({}_{\rm m}\) decreased with the decrease in Pr and increased in the flow direction due to the increase in the traveled axial distance. Consequently, the wall temperature (T\({}_{\rm w}\)) would decrease to maintain the enforced constant q\({}^{\prime\prime}\) value at the wall boundary condition, as shown in Figure 12b.