## 4 Results and Discussion
### Forced Convective Heat Transfer Study
At 1.5 \(v\)/\(v\)\(\%\) this trend was reversed by a drastic drop in the experimental results (see Figure 12a,b), indicating the development of complex flow and heat transfer behavior (as already discussed). This particular behavior is accurately captured by Equation (13). For \(N_{Re}\) values below 6000 the \(N_{Nu}\) was relatively insensitive to NP concentration. In fact, the results of pure water (see Figure 10) indicate that within this \(N_{Re}\) range, the heat transfer was not obviously enhanced by adding NPs to the base fluid, as discussed before. The proposed correlations also capture this behavior over the \(1400\leq N_{Re}\leq 6000\) range of Reynolds numbers for ranges of NP concentrations of 0.2-1.0 \(v\)/\(v\)\(\%\) (see Figure 12a.ii-a.iv) and 0.2-1.5 \(v\)/\(v\)\(\%\) (see Figure 12b.i-b.iv) as predicted by Equations (12) and (13), respectively. Figure 12: Plots of \(N_{Nu}\) versus \(N_{Re}\). The symbols are the experimental data and the curves were calculated by Equation (12) in (**a–a.iv**) and Equation (13) in (**b–b.iv**). Cumulative results (**a,b**), and the results of NFs with a NP concentration of 1.5 \(v\)/\(v\)\(\%\) (**a,i,b.i**), 1.0 \(v\)/\(v\)\(\%\) (**a,ii,b.ii**), 0.5 \(v\)/\(v\)\(\%\) (**a,iii,b.iii**), and 0.2 \(v\)/\(v\)\(\%\) (**a,iv,b.iv**). Legends as shown in (**a,b**). Figure 13 plots the ratio \(N_{Nu}/N_{Pr}^{0.4}\) as a function of \(N_{Re}\), which, in contrast to Figure 12, does not require average values of \(N_{Pr}\) for each NP concentration. Moreover, as both \(N_{Nu}\) and \(N_{Pr}\), were readily obtained in the experiments, the proposed correlations can be better compared with the experimental results in Figure 13 than in Figure 12. Consistent with the earlier comparison, the agreement is high at NP concentrations of 0.2, 0.5, and 1.0 \(v\)/\(v\)% for Equation (12) (see Figure 13a.ii-a.iv) and at 0.2, 0.5, 1.0, and 1.5 \(v\)/\(v\)% for Equation (13) (see Figure 13b.i-b.iv). As the correlation expressed in Equation (13) extends to the NP concentration of 1.5 \(v\)/\(v\)%, the quality of agreement of the experimental and predicted values at this concentration is confirmed again (see Figure 13b.i). In contrast, Figure 13a.i validates the exclusion of this concentration from Equation (12). Figure 13: Plots of \(N_{Nu}/N_{Pr}^{0.4}\) versus \(N_{Re}\). The symbols are the experimental data and the curves were calculated by Equation (12) in (**a**–**a**:**iv) and Equation (13) in (**b**–**b**:**iv). Cumulative results (**a**,**b**), and the results of NFs with an NP concentration of 1.5 \(v\)/\(v\)% (**a.i,b.i**), 1.0 \(v\)/\(v\)% (**a.ii,b.ii**), 0.5 \(v\)/\(v\)% (**a.iii,b.iii**), and 0.2 \(v\)/\(v\)% (**a.iv,b.iv**). Legends as shown in (**a**,**b**). The potential role that the CFI geometry may play in heat transfer enhancement when using TiO\({}_{2}\)-W NFs as working fluids was explored. To this end, the experimental \(N_{Nu}\) values of the NF containing 1.0 \(v\)/\(v\)% NPs from the present study were contrasted with the convective heat transfer experimental results of the TiO\({}_{2}\)-W NFs in a straight tube and to water flowing in a helical coiled tube. For the horizontal straight tube case, the correlation of Duangthongsuk and Wongwises [54] was utilized, which is given below:  
\[N_{Nu}=0.074N_{Re}{}^{0.070}N_{Pr}{}^{0.385}\phi^{0.074} \tag{14}\]  
Equation (14) is valid for TiO\({}_{2}\)-W NFs with NP volume concentrations in the range of 0 and 1.0 \(v\)/\(v\)% and \(N_{Re}\) between 3000 and 18,000. For the helical coil, the correlation derived by [PERSON] [56] was used:  
\[N_{Nu}=0.0575N_{Re}{}^{0.33}N_{Pr}{}^{0.43}N_{De}{}^{0.42} \tag{15}\]  
which is valid for water as the working fluid over the ranges of \(26<N_{De}<7000\) and \(6.2<\lambda<62.5\). Since the previous correlations (i.e., Equations (14) and (15)) can predict values for \(N_{Nu}\) that might not necessarily coincide with those of actual experiments, the following analysis highlights only potential benefits of using a CFI.