## 1 Introducing the Statistical Organization of Spray Data
For example, in laser diffraction measurement systems, like [PERSON]'s Spraytec, the ratio \((d_{upperBoundary}-d_{lowerBoundary})/d_{k}\) is constant, leading to broader classes for larger diameters representing each class. However, a systematic method for using irregular bin width to improve the description of drop size and velocity distributions is still open for further research. These details seldom appear reported in the literature when authors present the results of spray characterization and compare sprays obtained in different operating conditions. In this sense, the approach less prone to error would be to present the statistical data of the spray characteristics in terms of cumulative probability distributions, as explored later in this introduction. A final remark on the presentation and analysis of spray data in the form of statistical distributions is to consider the weight given to each class. In the case of the velocity of spray droplets, a number-weighted probability distribution is the most adequate. However, for histograms of drop size, other weighting factors, such as the  
* area-weighted \(p_{a,k}=\frac{s_{k}}{5}\) with \(s_{k}=\pi n_{k}d_{k}^{2}\) and \(S=\sum s_{k}\);
* and volume-weighted \(p_{v,k}=\frac{v_{k}}{V}\) with \(v_{k}=(\pi/6)n_{k}d_{k}^{3}\) and \(V=\sum v_{k}\);  
applied to probability distributions improve the interpretation of its moments, as reported in the work of [PERSON] [8]. The reason for organizing spray data with other weight values besides the number-weighted case relates to the physics associated with the research question. Namely, if the investigation involves phenomena occurring at the droplet surface area, such as heat and mass transfer events, an area-weighted drop size distribution is more adequate. However, if the spray liquid volume is more important, such as spray cooling applications, the most adequate is the volume-weighted drop size distribution. The characterization of a spray often involves moments of the measured discrete probability distribution using single-point diagnostic techniques, like the Phase-Doppler Interferometry, or field diagnostic techniques using imaging. In the case of drop size, the calculation of each moment from drop size raw data corresponds to  
\[d_{ab}=\left(\frac{\sum_{i=1}^{N}d_{i}^{a}}{\sum_{i=1}^{N}d_{i}^{b}}\right)^{ \frac{1}{a-b}},\forall_{a>b,\{a,b\}\in\mathbb{Z}^{+}} \tag{2}\]  
where \(d_{i}\) is a measurement of drop size in the sample acquired. If these moments use, instead, the number-weighted probability distribution values, the expression is  
\[d_{ab}=\left(\frac{\sum_{k}n_{k}(d_{k})^{a}}{\sum_{k}n_{k}(d_{k})^{b}}\right)^ {\frac{1}{a-b}},\forall_{a>b,\{a,b\}\in\mathbb{Z}^{+}} \tag{3}\]  
with \(n_{k}\) as the number of measurements counted within the size class \(k\) where \(d_{k}\) represents the mid-point in the interval between a lower and an upper bound. The characterization of the spray droplets using the average size obtained from a number-weighted probability distribution--\(d_{10}\)--means considering all droplets have, on average, the same size of \(d_{10}\). For example, when comparing \(d_{10}\) for different locations in the spray, or the same location for different sprays, any increase implies more droplets of larger sizes. When analyzing the physics of droplet transport while interacting with the surrounding environment, the arithmetic mean diameter might be a valuable characteristic measure to consider.