## 3 Results and Discussion  
### Experimental Results  
Figure 4 shows the thermal conductivity of the nanofluid containing 4 and 8 wt% of nanoparticles for several temperatures and three sonication times (_ST_): 1.5, 4, and 16.5 h. From Figure 4, it can be observed that a higher nanofluid temperature is associated with a higher thermal conductivity and the temperature dependence can be approximated as linear. A good agreement with the linear fit is obtained for both nanoparticle concentrations. Moreover, both straight lines are nearly parallel, indicating that the temperature influence on the thermal conductivity is the same; so, it is independent of the nanoparticle concentration. In these experiments, the improvement obtained in the thermal conductivity is about a 0.001 W/m\(\cdot\)K per \({}^{\circ}\)C increase. In the experimental tests performed in this research, the increment in the sonication time had no appreciable effect on the thermal conductivity. Similar results were obtained by [28; 30] using other types of nanoparticles. Table 4 contains the average values for the temperature and thermal conductivity obtained at different sonication times and the corresponding standard deviations. The statistical analysis of data modifying the sonication  
Figure 3: (**a**) Experimental thermal conductivity of the base fluid compared with data from NIST. (**b**) Deviation of experimental with respect to NIST data. times shows that the changes are not statistically significant. Standard deviations are negligible in comparison with the thermal conductivity values. The impact of the nanoparticle concentration on the nanofluid thermal conductivity, for two different temperatures (around 24.1 and 49.4 \({}^{\circ}\)C), is presented in Figure 5. Data shown correspond to a sonication time of 1.5 h. As reported in previous studies, thermal conductivity enhances as the concentration of nanoparticles rises. In this case, the improvement obtained in the thermal conductivity is about 0.0027 W/m\(\cdot\)K per unit of wt% increase. Thermal conductivity results obtained in the current research for the aqueous theta-phase aluminum oxide nanofluids are compared with the results available in the open literature for other water-based aluminum oxide nanofluids. The comparison is shown in Figure 6, representing the effect of the nanoparticle volume concentration. For the conversion from wt% to vol%, the density of the nanoparticles considered was 3970 kg/m\({}^{3}\), according to [39], and the following formula was applied:  
\[\frac{1}{X_{v}}=1+\frac{\rho_{np}}{\rho_{b}}\left(\frac{1-X_{wt}}{X_{wt}}\right) \tag{2}\]  
where \(X_{wt}\) and \(X_{v}\) are the mass and volume fraction, respectively\(\rho_{b}\) and \(\rho_{np}\) are the base fluid and nanoparticles density. The base fluid density, at different temperatures, was computed using EES [40]. \begin{table}
\begin{tabular}{c c c c c} \hline \hline \multirow{2}{*}{**Nanoparticle Mass Fraction (\%)**} & \multicolumn{2}{c}{**Temperature (\({}^{\circ}\)C)**} & \multicolumn{2}{c}{**Thermal Conductivity (W/m K)**} \\ \cline{2-5}  & **Average** & **SD** & **Average** & **SD** \\ \hline
2 & 23.50 & 0.40 & 0.615 & 0.003 \\
2 & 49.38 & 0.08 & 0.643 & 0.007 \\
4 & 23.7 & 0.62 & 0.622 & 0.004 \\
4 & 49.39 & 0.08 & 0.645 & 0.005 \\
6 & 49.41 & 0.08 & 0.651 & 0.007 \\
8 & 23.27 & 1.14 & 0.629 & 0.003 \\Figure 5: Thermal conductivity vs. nanoparticles wt% for two temperatures, \(ST=1.5\) h.  
Figure 6: Thermal conductivity vs. nanoparticle volume fraction at different temperatures [14; 18; 21; 24; 25; 26; 29; 32]. **(a)** Temperatures between 23 and 32 \({}^{\circ}\)C. **(b)** Temperatures between 47 and 52 \({}^{\circ}\)C.  
Figure 6 presents the thermal conductivity variation of the studied nanofluid with respect to the nanoparticle volume fraction for two temperature ranges.