## 4 Conclusions  
The application of nanofluid is directed toward enhancing heat transfer within engineering systems in an effort to save energy. Improving heat transfer efficiency can make energy more affordable and accessible, particularly in developing regions where energy costs are high relative to income. These efforts could contribute to enhancing the quality of life and compliance with sustainable development goals (SDGs). With this motivation in mind, the effects of Al2O3 nanoparticles and rarefied flow at the entrance of a circular pipe on heat transfer under constant wall flux conditions were estimated using the CFD approach. The steady-state, two-dimensional, and single-phase simulations were validated against published research, and the results were satisfying. Based on the comprehensive study, the following conclusions were made:  
* The adopted approach of the low-pressure slip flow method for the laminar flow of Ansys-Fluent was proven to be sensitive to the value of q\({}^{\prime\prime}\) at the wall that causes a change in the wall temperature value. The change in \(\lambda\) resulting from the change in the wall temperature changes the value of Kn along the line of the nanofluid flow and becomes variable rather than constant. Therefore, changes in the resulting fluid bulk temperature have to be insignificant for the approach to be valid, and extra care should be taken when adopting this approach in order to avoid producing inaccurate results on the basis of constant Kn values. Consequently, the highest error obtained from current simulations was 4% when the wall heat flux increased from 5 W/m\({}^{2}\) to 50 W/m\({}^{2}\);
* The effects of different Pr values on U/U\({}_{\text{m}}\) and ReC\({}_{\text{f}}\) values were negligible. However, the effects of Kn values on the aforementioned parameters were significant;
* The hydrodynamic entrance length (L/D\({}_{\text{h}}\)) became shorter with the increase in Kn and Re when compared to the no-slip flow case. However, the thermal entrance length (L/D\({}_{\text{h}}\))\({}_{\text{th}}\) became longer with the increase in Kn and Re when compared to the no-slip flow case;
* Increasing Re and/or nanoparticle volume fraction values resulted in increased Nu\({}_{\text{x}}\) values due to rises in nanofluid thermal conductivity values. However, increasing x/D\({}_{\text{h}}\) values caused the Nu\({}_{\text{x}}\) values to decrease due to the dissipation effects at the entrance region. Consequently, the effects of slip flow on heat transfer, represented by Nu, in a pipe was adverse and reduced heat transfer from the pipe wall to moving nanofluids by 40% at Kn = 0.1. The maximum heat transfer enhancement due to the addition of nanoparticles was found to be minimal, with a 2.5% increase at \(\phi\) = 0.05, Kn = 0.1, and Re = 250;
* Correlations between U/U\({}_{\text{m}}\), ReC\({}_{\text{f}}\) and Nu\({}_{\text{x}}\) and the key investigated parameters (x/D\({}_{\text{h}}\), Re, Pr, Kn) with a high coefficient of determination (R\({}^{2}\)) were developed to \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**y** & **Nu\({}_{\text{avg}}\)** & **Nu\({}_{\text{x}}\)** & **U/U\({}_{\text{m}}\)** & **Re C\({}_{\text{f}}\)** \\ \hline a & 51.852499 & 69.008891 & 0.982444 & 34.438304 \\ b & \(-\)231.989750 & \(-\)333.523624 & 57.947560 & \(-\)182.496147 \\ c & 1485.741079 & 9873.998042 & 212.973373 & 990.447757 \\ d & 89.668426 & 106.415922 & 53.569393 & 7.232419 \\ e & 445.780079 & 2729.351897 & 101.564399 & 96.175369 \\ Coefficient of Determination (R\({}^{2}\)) & 0.93 & 0.92 & 0.99 & 0.87 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Constants utilized in Equation (23). produce a reliable correlation and to be used in the field of the laminar slip flow of nanofluid inside the entrance region of a circular pipe. Conceptualization, R.A.-W. and M.S.N. ; data curation, D.B.M. and J.M. ; formal analysis, M.A.-R. and R.A.-W.; methodology, M.A.-R., R.A.-W. and J.M. ; project administration, R.A.-W. and M.S.N. ; resources, R.A.-W. and J.M.