## 2 Drop Size Distribution Functions and Spray Science
### Underlying Physics of Probability Distribution Functions Applied to Sprays
However, if we consider theinteraction between droplets and the continuous gaseous phase, in time, the dragging of droplets leads to secondary flows which eventually produce a vortical effect on the transport of subsequent droplets. Namely, smaller droplets, with lower response times, dragged by secondary flows, may return to upward locations, redistributing the counts of certain drop size classes in locations further downstream of the spray trajectory. Therefore, even if it is not the result of a multiplicative breakup process, the presence of these smaller drops affects the probability distributions describing the spray, having an effect similar to that of a cascade of multiple breakup stages. One could even speculate whether or not the reason for a Log-Normal distribution best fitting experimental results expresses the way the multiphase flow organizes the transport of droplets according to their size in a cascade pattern. Besides several breakup stages and transport phenomena, some atomization processes result from the disintegration of liquid sheets or jets into ligaments, and those ligaments further fragmenting into droplets. In this case, [PERSON] et al. [16] argues each ligament constituted of several blobs, and when it fragments into several droplets, the size distribution that reasonably fits is a Gamma probability distribution function, which cumulative form is expressed by  
\[F_{\Gamma}(d,a,b)=\frac{1}{b^{a}\Gamma(a)}\int_{0}^{d}x^{a-1}\exp(-x/b)dx \tag{7}\]  
with \(a\) and \(b\) as the shape and scale parameter, respectively. In this case, a characteristic size corresponds the product of both: \(d_{c}=a\cdot b\). Considering empirical distribution functions, this work focuses on the Weibull distribution, first applied to describe the distribution of drop sizes by [PERSON] and [PERSON],  
\[F_{WB}(d,d_{c},q)=1-\exp\left(-\left(\frac{d}{d_{c}}\right)^{q}\right) \tag{8}\]  
where \(d_{c}\) is a scale parameter related to a characteristic drop size and \(q\) is the shape parameter and considered a measure of the spreading in drop sizes. The accuracy of this empirical probability distribution is best related to drop size distributions with fewer smaller droplets or narrower size distributions [15]. A final note considers the Nukiyama-Tanasawa empirical distribution that is often used to fit experimental data. Here, particular attention is given to the work of [PERSON] and [PERSON] [18] that derived the expression using an information-theory approach, and for the spray liquid volume, its cumulative form results in  
\[F_{NT}(d,d_{c},q)=1-(1+qd^{3})\exp\left(-qd^{3}\right) \tag{9}\]  
where \(q\) is also the shape parameter. The average quantities referred to so far are related to moments in probability distributions, but considering the cumulative distribution, any quantity corresponds to a _representative diameter_, generally expressed as \(D_{xw}\), where \(x\) is the type of distribution, and \(w\) is the percent cumulative value related to the representative diameter. Therefore, if the cumulative distribution is  
* number-based, \(D_{mw}\) represents the size containing w% of the droplets in the spray;
* area-based, \(D_{aw}\) represents the size containing w% of the spray surface area;
* volume-based, \(D_{vw}\) represents the size containing w% of the liquid spray volume. One of the most relevant representative diameters corresponds to 50% (\(D_{n0.5},D_{a0.5}\), or \(D_{v0.5}\)) because dividing the classes by that number allows a better comparison between cumulative distributions, useful for analyzing the effect of parametric variations in the spray.