## 2 Drop Size Distribution Functions and Spray Science
### Introducing Drop Size Diversity in a Spray  
Drop size distributions are a way to organize the data acquired to characterize a spray. The characterization of the diversity of drop sizes answers two distinct questions: (1) _how many different sizes_ are relevant in a spray; (2) and _how different_ are the relevant sizes in a spray. The word _relevant_ links to the probability of presence of certain drop size classes relative to others. In the known textbook on _Atomization and Sprays_, [PERSON] and [PERSON] [15] refer to this diversity as _drop spray dispersion_ associated with the size range of droplets. On the other hand, several research articles address the different sizes of the spray droplets as a _polydispersed spray_. In the authors' opinion, these are two different things, which is why we introduce the concept of _Drop Size Diversity_ (DSD) measured by two different degrees:  
* **the polydispersion degree** to quantify the multitude of different sizes that are relevant in a spray. Thus, the maximum for the case where all different sizes have the same probability of being present in the spray, and
* **the heterogeneity degree** to quantify how different are the relevant sizes in the spray. Therefore, it is related with the size range or size dispersion. The challenge is to devise the right indicators to measure both degrees. Among the several indicators available and synthesized in [PERSON] and McDonell [15], the most known and used indicator is the _Relative Span_ obtained from the representative diameters of a volume-based cumulative size distribution (\(D_{vX}\) with \(0<X<1\) as the fraction of the spray liquid volume) as  
\[\Delta_{v}=\frac{D_{v0.9}-D_{v0.1}}{D_{v0.5}} \tag{10}\]  
Considering the normalization of a representative diameter by the value representing half of the liquid volume-\(D_{v0.5}\)-as \(D_{vX}^{*}=D_{vX}/D_{v0.5}\), the interpretation of Equation (10) relative to the range of drop sizes corresponds to a difference-\(\Delta_{v}=D_{v0.9}^{*}-D_{v0.1}^{*}\) which is equal to zero when all droplets in the spray have the same size, and maximum when all droplets have the same probability of occurrence (a limit unrealistic case). [PERSON] [19] proposed a different approach based on information theory, through the concept of the normalized Shannon entropy, already defined in Section 2.1 as  
\[H_{n}=\frac{H}{\ln(N_{bins})} \tag{11}\]  
In the information theory terminology applied to spray characterization, a spray where all droplets have the same size, \(p=1\), resulting in a null normalized Shannon entropy, \(H_{n}=0\), while an unrealistic spray with all classes having the same probability of presence (uniform distribution), \(H_{n}=1\), because the Shannon entropy-numerator in Equation (11)-is maximum. Finally, [PERSON] et al. [20] proposed a third approach based on the standard deviation of the volume-weighted drop size distribution expressed as  
\[SD_{v}=\sqrt{d_{53}^{2}-d_{43}^{2}} \tag{12}\]  
where \(d_{53}\) and \(d_{43}\) are the second- and first-order moments of the volume-weighted drop size distribution, respectively. The authors compared this standard deviation with Shannon entropy and found inconsistencies. They state that the Shannon entropy has a main drawback, since the information of the drop sizes representing each class is not explicitly included; thus, if these probability values would be randomly rearranged, the H value would be the same. This is an important insight because it allows to understand the difference between polydispersion and size dispersion in a spray. To compare the three approaches, consider the simulation of a spray mixing two monosize droplet streams of 10 \(\upmu\)m and 20 \(\upmu\)m. A weight parameter \(w\), varying between 0 and 1, sets the percentage of drops present in the mixed spray from each of the monosize sources.