## 2 Drop Size Distribution Functions and Spray Science
### Underlying Physics of Probability Distribution Functions Applied to Sprays  
An important consideration when characterizing sprays is the effect of the interaction between droplets and the continuous phase on the local (or even overall) size and velocity distributions. This interaction involves different transport phenomena, with momentum and energy exchanges between the dispersed phase (spray) and the carrier phase (surrounding environment), generating eventual secondary breakup of the spray droplets leading to changes in the shape and scale of drop size distributions, or acceleration (positive or negative) captured by changes in local velocity probability distributions. However, the physical reason why a certain distribution function might fit better than another is still open for further research. This work advances an argument in favor of distinguishing between modeling and characterizing drop size distributions. The purpose of modeling drop size distributions is to _predict_ them from the information of the atomizer geometry and operating conditions. The simulation of sprays using a numerical approach [12], or a statistical or stochastic approach [13] can produce data on the droplets characteristics, but it is different from statistically processing such data. Therefore, according to [PERSON] et al. [14], there are four methods for modeling drop size distributions:  
* the empirical;
* the Maximum Entropy Formalism (MEF);
* the Discrete Probability Function (DPF);
* and the Stochastic. However, the purpose of characterizing a spray is to _describe_, as accurately as possible, the polydispersion of sizes and velocities of its droplets. This description aims at obtaining mean quantities for analyzing heat transfer and flow processes--or its aim is to improve our understanding of the nature underlying the atomization mechanisms. There are two categories of probability distribution functions used to describe droplets' characteristics: mathematical and empirical. [PERSON] and [PERSON] [15] provide a synthesis of the main probability distributions in each category. Except for the Rosin-Rammler or Weibull, the Nukiyama-Tanasawa and Upper-Limit empirical distribution functions are complex and problems arise when determining the best-fit values for their parameters. As to the mathematical distribution functions, the simplest is the Log-Normal, while the Log-Hyperbolic is also complex and problems arise with finding the best fitting parameters. One distribution absent from [PERSON] and [PERSON] [15] and other review works is the Gamma distribution function which [PERSON] et al. [16] associated with the distribution of droplets resulting from the fragmentation of ligaments, generating a spray, reviewed later in this section. While most research on spray characterization focuses on the fitting process, few works such as [PERSON] [17] and [PERSON] et al. [16] address the meaning of the mathematical distribution function used and the physical background for such fitting. As mentioned in the Introduction, instead of focusing our attention on probability or probability density functions, we propose a greater focus on cumulative distribution functions (\(F(d)\)). Therefore, any comparison between different \(F(d)\) becomes universal and if such distribution properly describes the local or global experimental results, its digitization to simulate a spray is relatively accessible. In the case of the Log-Normal distribution, its cumulative form given by  
\[F_{LN}(d,d_{g},\gamma)=\frac{1}{2}\left(1+\text{erf}\left(\frac{\ln(d/d_{g})}{ \sqrt{2}\gamma}\right)\right) \tag{6}\]  
includes \(d_{g}\) as the scale parameter, the geometric mean diameter, and \(\gamma\) as the shape parameter of the distribution. Applied to spray characterization, the reason for using a Log-Normal distribution function is related to the multiplicative effect of subsequent stages of droplets breaking up during atomization, as a cascade process, where one drop breaks into two or more and so on.