## 3 Results and Discussion
### Empirical Correlation  
Up to now, it has been shown that the thermal conductivity of the aqueous theta-phase aluminum oxide nanofluids depends on the temperature and nanoparticle concentration, while the effect of sonication time is negligible. In order to use these results for possible industrial or research applications, a predictive correlation is considered a useful tool. With this objective, the thermal conductivity ratio defined in Equation (3) is used as follows:  
\[k_{r}=\frac{k_{nf}}{k_{b}} \tag{3}\]  
In previous studies [17; 23; 27; 41], correlations including the nanoparticle volume fraction, \(X_{v}\), have been presented. In addition to these, some correlations also include the temperature, \(T\)[19; 25]. These correlations are shown in Table 5, which also includes the size and volume fraction of the nanoparticles and the temperature used in the experiments to derive each correlation. It can be observed that many of the empirical correlations use a single value for the temperature, while the maximum temperature is limited to 50 \({}^{\circ}\)C [25]. In addition to this, only two correlations use small nanoparticle sizes (10 and 13 nm, in [25; 41], respectively). Thermal conductivity experimental data obtained in this work for the alumina-water nanofluid have been compared with those predicted by the empirical correlations shown in Table 5. The relative errors between predicted and experimental data are shown in Table 6. Very large deviations are obtained between predicted and experimental values, with the best agreement obtained using the correlation of [27], even though this correlation was developed for a different nanoparticle size and a very small temperature range. In this work, by means of statistical analysis of multiple linear correlation, using the specialized Engineering Equation Solver software (Professional version V10.612-3D), the correlation presented in Equation (4) for the thermal conductivity as a function of the nanoparticle mass fraction, \(X_{wt}\), and the temperature, \(T\), has been obtained as follows:  
\[k_{nf}=a+bX_{wt}T+\sum_{i=1}^{2}\left(c_{i}X_{wt}^{i}+d_{i}T^{i}\right) \tag{4}\]  
The validity ranges of the variables used in this equation are as follows: 1% \(<X_{wt}<\) 10% and 221 \({}^{\circ}\)C \(<T<\) 59.3 \({}^{\circ}\)C. The numerical values of the constants that appear in Equation (4) are shown in Table 7. This correlation is characterized by a goodness of fit parameter R\({}^{2}\) = 94%. The comparison presented in Figure 7 shows experimental values of the ratio between thermal conductivities, \(k_{r}\), and values predicted using Equation (4). An excellent agreement between predicted and experimental data can be seen, with a mean absolute percentage error of 0.55%. The new correlation significantly improves agreement with respect to previous empirical correlations, as shown in Table 6.  
Taking into account the experimental conditions used in [14; 21; 32], where the alumina phase is not specified, the thermal conductivity of the alumina-water nanofluid is predicted using the empirical correlation obtained in the present work, Equation (4). Figure 8 shows a comparison between the values predicted and the experimental data obtained by the authors in [14; 21; 32]. Results show a very good agreement between predictions and experimental values, with a maximum relative error of 5.3% and a mean absolute \begin{table}
\begin{tabular}{c c c} \hline \hline
**Reference** & \multicolumn{2}{c}{**Relative Error (\%)**} \\ \cline{2-3}  & **Average of Absolute Values** & **Range** \\ \hline
[17] & 21.7 & 1.8–45 \\
[19] & 31.4 & \(-\)3.5–65.8 \\
[23] & 23.6 & 4.3–42.5 \\
[25] & 20.2 & 7–32.2 \\
[27] & 7.0 & 0.4–14 \\
[41] & 38.7 & 6.1–69.9 \\ \hline \hline \end{tabular}
\end{table} 
Table 6: Relative error between empirical correlations in Table 5 and current experimental data.