## 4 Results and Discussion
### Forced Convective Heat Transfer Study
Using the obtained experimental data, the following empirical correlation was derived to predict the internal \(N_{Nu}\) of the TiO\({}_{2}\)-W NFs flowing in the CFI geometry using least-squares regression analysis. \[N_{Nu}=0.002524N_{Re}.^{1.1622}N_{Pr}.^{0.4}\phi^{0.1514} \tag{12}\]  
Equation (12) is valid over the \(0.002\leq\phi\leq 0.01\) range of NP concentrations, the \(1400\leq N_{Re}\leq 9500\) range of Reynolds numbers, the \(4.5\leq N_{Pr}\leq 5.2\) range of Prandtl numbers, and a curvature ratio of \(\lambda=12.8\). Note that Equation (12) excludes the effect of \(\lambda\) on \(N_{Nu}\), as \(\lambda\) remained unchanged throughout the present experiments. In order to extend Equation (12) for the experimental data of the NF containing 1.5 _v/v_% TiO\({}_{2}\) NPs (\(\phi=0.015\)) and also to take into consideration the CFI curvature ratio (\(\lambda\)), the approach reported by [PERSON] et al. [55] was adapted from the relationship proposed by [PERSON] et al. [22] and the following correlation was obtained:  
\[N_{Nu}=0.03N_{Re}.^{0.7}N_{Pr}.^{0.4}\lambda^{-0.1}\left\{1+\alpha_{0}N_{Pr} \hat{\phi}_{0}\left[\phi(\phi_{L}-\phi)^{0.25}\right]^{Y}\right\} \tag{13}\]  
Figure 10: Experimental \(N_{Nu}\) versus \(N_{Re}\) in water and TiO\({}_{2}\)–W NFs. where \(N_{Pe}\) is the Peclet number, \(\phi_{L}\) is the NP concentration enhancement limit for the present study (\(\phi_{L}\) = 0.015), and the constants \(\alpha_{0}\), \(\beta_{0}\), and \(\gamma\) are determined by nonlinear regression with tolerance and convergence of 5% and 0.0001, respectively. The analysis results are listed in Table 2.  
Equation (13) is then valid over the 0.002 \(\leq\phi\leq\) 0.015 range of NP concentrations, the \(\lambda\geq\) 10 values of the curvature ratio, the 1400 \(\leq N_{Re}\leq\) 9500 range of Reynolds numbers, and the 4.5 \(\leq N_{Pr}\leq\) 5.2 range of Prandtl numbers. Figure 11a,b shows the deviation of the \(N_{Nu}\) predicted by the two proposed correlations, Equations (12) and (13), from the experimental data, respectively. As observed by the lines projected from the central line representing the perfect fit, the maximum deviation was approximately \(\pm\)10% for both correlations. The predicted \(N_{Nu}\) values of the NF with 1.5 \(\
icefrac{{v}}{{\%}}\) NPs shown in Figure 11a are displayed for reference only, as Equation (12) is exclusively valid for NP concentrations from 0.2 to 1.0 \(\
icefrac{{v}}{{\%}}\). The fitting is of sufficient quality for obtaining the correlations from the cumulative experimental data. \begin{table}
\begin{tabular}{c c c c} \hline \hline
**NP Volume Concentration in TiO\({}_{2}\)–W NFs** & \multicolumn{2}{c}{**Constants**} \\ \hline \(0.002\leq\phi\leq\) 0.015 & \(\alpha_{0}\) & \(\beta_{0}\) & \(\gamma\) \\
0.000783 & 0.8933 & 0.24 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Constant values of Equation (13). Figure 11: Deviation between the experimental and predicted \(N_{Nu}\) values of TiO\({}_{2}\)–W flows. The predicted values were obtained by the proposed empirical correlations: Equation (12) in (**a**) and Equation (13) in (**b**). Figure 12 compares the \(N_{Nu}\) of the present experimental data and that obtained by the proposed correlations, Equation (12) (Figure 12a-a.iv), and Equation (13) (Figure 12b-b.iv). The plots are consistent for the TiO\({}_{2}\) NP concentrations at which the correlations are valid: 0.2-1.0 \(v\)/\(\%\) for Equation (12) (see Figure 12a.ii-a.iv) and 0.2-1.5 \(v\)/\(v\)\(\%\) for Equation (13) (see Figure 12b.i-b.iv). The \(N_{Nu}\) data of the NF with 1.5 \(v\)/\(\%\) NPs shown in Figure 12a.i are included to confirm that Equation (12) is exclusively valid for NP concentrations of up to 1.0 \(v\)/\(v\)\(\%\), as previously established. At any given \(N_{Re}\) within the range of this study, increasing the NP concentration (up to 1.0 \(v\)/\(v\)\(\%\)) increased the predicted \(N_{Nu}\).