## 3 Characterization of Nanofluid Sprays: An Example
Figure 8 shows examples for the distilled water spray in two radial locations, where the fitting functions that best described the probability of presence of droplets in the spray at \(r=0\) mm the Log-Normal--Equation (6)--and at \(r=6\) mm Gamma- Equation (7), probability distribution functions. Using a Kolmogorov-Smirnov test to evaluate the best fitting, including combinations of probability distribution functions (pdf), Figure 9 maps the results for \(Z=10\) and \(20\) mm. On the right, it shows an example for distilled water of the pdf characteristics allowing the reconstruction of spray data at the measured locations, where \(C_{1}\) and \(C_{2}\) are the scale and shape parameters for each kind, also included in the legend. It is noteworthy that the Nukiyama-Tanasawa empirical distribution did not produce the best fitting in any measurement point, spray condition, or contribution to a mixed cumulative distribution function. The spray analyzed is a hollow-cone, which means the formation of a liquid sheet at the nozzle exit further disintegrates into ligaments, and these fragment into droplets. In the central region, the momentum exchanges between smaller droplets and air leads to a dragging effect of these droplets into the central region of the spray, as expressed by the lower mean diameter shown in Figure 7. According to the interpretation delineated in Section 2.2, the Log-Normal predominance in the central region would be a expression of these transport phenomena, while in the regions of higher velocity, likely the result of spray formation through the fragmentation of ligaments, the gamma produces the best fitting. In the outer regions of the spray, the best fittings of the Weibull, in the case of  
Figure 7: Mean velocity, \(\langle u_{d}\rangle\) [m/s] of the spray droplets for \(Z=10\) mm (**top**) and \(Z=20\) mm (**bottom**). Distilled water, and combination of pdfs indicate transition regions and, despite being unclear what is the physical meaning of these transitions, the identification of their location within the spray structure is a valuable insight. The final observation is the effect of the nanoparticles shape on the spray structure. While in the mean sizes, these differences were negligible, in dynamic terms, they were not. The results in Figure 9 indicated as WATER + CTAB + wt% Au correspond to the spray of the base fluid and the base fluid plus different percentages of gold nanospheres (0.1, 0.5, 1 [wt%]), but the best pdf fitting map was the same, which means that gold nanospheres are not likely to alter the spray structure of the base fluid. However, gold cylindrical nanoparticles and silver triangular led to changes in the spray structure, and these results are coherent with the mean velocity values, but adds a substantial interpretative value. Figure 8: Example of the results of the cumulative curve fitting for Distilled water, \(r=0\) mm where the best fitting occurred with a Log-Normal, Equation (6), and \(r=6\) mm with a best fitting through a Gamma function, Equation (7). Figure 9: Radial profiles of the mathematical probability functions that best fitted the experimental results for all experimental conditions and measurement planes considered. The table on the right exemplifies the data that allows the reconstruction of each cumulative distribution. Some included more than one function in different drop size ranges. Finally, in this work, we introduced the concept of drop size diversity to better understand the many different sizes relevant in a spray, as well as how different the relevant sizes in a spray are. The polydispersion degree given by the normalized Shannon entropy, \(H_{n}\), and spray heterogeneity degree given by the volume-weighted standard deviation, \(SD_{v}\) [\(\blacktimes\)], can describe this diversity, as depicted in Figure 10.  
In the case of distilled water, the results show a higher polydispersion degree around \(r=8\) mm, where the spray heterogeneity degree also becomes higher.