## 3 Results and Discussion
Hence, the presence of a slip wall resulted in a proportional increase in the thermal entrance length as both the Kn and Re values increased. Figure 5:3  
Figure 5:4 Local Nusselt number (Nu\({}_{x}\)) at different values of (a) Pr; (b) Re; (c) Re function of x/D\({}_{\rm h}\); and (d) Kn. The increase in \(\mathrm{Kn}\) caused the wall-adjacent cell temperature to decrease due to the temperature jump effect, as defined in Equation (13) and shown in Figure 5:5 At the inlet of the pipe, \(\theta\) was equal to one and decreased as the nanofluid traveled downstream along the axial direction. Both effects required the nanofluid to travel longer downstream distances to achieve the \(\mathrm{T}_{\mathrm{min}}\) value that satisfied the condition of (\(\theta\) = 0.8) for a thermally fully developed flow. With the increase in \(\mathrm{Re}\), the amount of nanofluid flowing inside the pipe increased, which caused a decrease in \(\mathrm{T}_{\mathrm{min}}\) and resulted in an increase in the value of \((\mathrm{L}/\mathrm{D}_{\mathrm{h}})_{\mathrm{th}}\). The enhancement in heat transfer due to the inclusion of nanoparticles (\(\mathrm{Al}_{2}\mathrm{O}_{3}\)) into air was represented by values of the local Nusselt number, as defined in Equation (21). The equation presents how much more heat was transferred from the pipe walls to the nanofluid due to the addition of nanoparticles under slip flow conditions when compared with no-slip conditions. Based on the current CFD simulations, the maximum enhancement due to the addition of nanoparticles was found to be 2.5% at \(\mathrm{Kn}\) = 0.1, \(\mathrm{Re}\) = 250, and \(\mathrm{\phi}\) = 10% or \(\mathrm{Pr}\) = 0.729, as shown in Figure 5:6 The variation in the simulated \(\mathrm{Pr}\) was around \(-\)4.7%, between the maximum volume fraction of nanoparticles within the nanofluid and air. Also, increases in the nanofluid conductivity and viscosity of around 33% and 30%, respectively, were reported. Therefore, the increase in thermal conductivity would enhance  
Figure 5:7 Thermal entrance length (\(\mathrm{L}/\mathrm{D}\mathrm{h}\)) due to different nanoparticle volume fractions as functions of (**a**) \(\mathrm{Re}\) under no-slip conditions at \(\mathrm{Kn}\) = 0.0, compared with Shah and London [40]; (**b**) slip condition values (\(\Delta(\mathrm{L}/\mathrm{D}\mathrm{h})\) = (\(\mathrm{L}/\mathrm{D}\mathrm{h}\))\({}_{\mathrm{th-slip}}\) – \((\mathrm{L}/\mathrm{D}\mathrm{h})_{\mathrm{th-o slip}}\)) at 0.01 \(\leq\mathrm{Kn}\)\(\leq\) 0.1. Figure 5:8 Temperature profiles at different sections, ranging from 10D to the fully developed flow at \(\mathrm{Re}\) = 1750.  
the heat transfer rate, while the increase in viscosity would reduce the rate of heat transfer. Furthermore, with regard to the reduction in nanofluid-specific heat due to the addition of nanoparticles, the convective heat transfer tends to increase and cause an overall heat transfer enhancement. Therefore, there was a compromise in the resulted heat transfer enhancement for the simulated values in favor of a slight enhancement. The effects of slip flow conditions on the local Nusselt number were defined in Equation (22). The equation presents how much less heat was transferred from the pipe walls to the nanofluid due to the increase in the slip wall conditions (increase in Kn value) when compared with no-slip conditions. As the Kn value increased, the resulted slip velocity increased, which increased the convective effect of heat transfer. However, higher Kn values caused higher temperature jump values, which resulted in a significant reduction in heat transfer from the pipe walls to the flowing nanofluid. The results showed a relatively linear degradation in the amount of heat transfer as a function of the Kn value in a proportional manner. As a result, the highest reduction due to slip flow conditions was found to be 40% at Kn = 0.1. This conclusion highlights the importance of Kn over the volume fraction of nanoparticles for slip flow conditions inside a pipe for air and Al\({}_{2}\)O\({}_{3}\) nanoparticles.