## 2 Drop Size Distribution Functions and Spray Science
### Introducing Drop Size Diversity in a Spray
Therefore, if \(w=0\), all droplets have the size of 20 \(\upmu\)m, and if \(w=1\), all droplets have 10 \(\upmu\)m. Figure 3 shows the result for the Relative Span (\(\Delta_{v}\)), normalized Shannon entropy based on the volume-weight drop size distribution (\(H_{n,v}\)),and the volume-weighted standard deviation (\(SD_{v}\)) with the variation of the weight \(w\) attributed to the presence of droplets from each monosize source. In the extreme cases where all droplets have the same size, the absence of polydispersed sizes results in null values for all indicators. However, while a small portion of droplets of a different size leads to a discontinuity in \(\Delta_{v}\), the normalized Shannon entropy and the volume-weighted standard deviation show a continuous behavior. In Figure 3, points A and D in \(\Delta_{v}\) mark the limits of 10% and 90% of the spray volume, meaning until A, 10 \(\upmu\)m droplets do not make 10% of the spray volume, and, at point D, these small droplets reached the mark of 90% of the spray volume. The maximum value of \(\Delta_{v}\) at point C has no evident meaning. The evolution of \(H_{n,v}\) is simple to interpret. The maximum normalized Shannon, \(H_{n,v}=1\), occurs when all droplets have the same probability of being present in the spray (\(w=0.5\)), corresponding to the uniform distribution. It is noteworthy that \(H_{n,v}\) has no discontinuities as \(\Delta_{v}\), and the variation with \(w\) shows a transition between the monosize spray droplets with 20 \(\upmu\)m to the monosize spray of 10 \(\upmu\)m droplets. It means that the size of droplets has no influence on \(H_{n,v}\), only their relevancy in the spray. On the contrary, the volume-weighted standard deviation depends on the maximum distance between relevant drop size in the spray. In this case, with an evolution similar to \(H_{n,v}\), the maximum occurs at 5 \(\upmu\)m corresponding to the maximum distance between a mean quantity (15 \(\upmu\)m) and the sizes of the monodispersed sprays (10 and 20 \(\upmu\)m). The Relative Span raises several questions, eventually pointing to its limitation if used to quantify the droplets size diversity. Namely, the values depend on drop sizes. Why would a monosize spray of smaller droplets, with a small portion of a monosize spray of larger droplets have a higher Relative Span? In fact, Figure 4 helps to understand why this outcome is a result of the evolution of the diameter representing 50% of the liquid volume of the spray (\(D_{v0.5}\)). The first observation from Figure 4 is the similarity between the other indicators (\(H_{n,v},SD_{v}\)) and the evolution of the difference \(D_{v0.9}\)-\(D_{v0.1}\), in the numerator of \(\Delta_{v}\), as a function of \(w\) with a maximum value at \(w=0.5\). However, because the relative span normalizes this difference by \(D_{v0.5}\), which continues to decrease as the number of 10 \(\upmu\)m droplets in the mixed spray increases, \(\Delta_{v}\) grows and indicates, for no reason, an increase of the drop size diversity until point C, when it is not the case. This theoretical example shows the limitations of using the relative span as a reliable tool to quantify the drop size diversity in a spray. Figure 3: Variation of three Drop Size Diversity Indicators, the Relative Span (\(\Delta_{v}\)), the normalized Shannon Entropy (\(H_{n,v}\)), and the volume-weighted standard deviation (\(SD_{v}\)) for the mixing of two monosize droplet clusters. Although for the previous example, the \(H_{n,v}\) and \(SD_{v}\) led to the same value of \(w\) = 0.5 corresponding to the maximum polydispersity and heterogeneity degrees, the parameters express different meanings relative to the drop size diversity. Therefore, a second theoretical simulation uses three monosize stream of droplets with 10, 20, and 40 \(\upmu\)m, with weights \(w_{1},w_{2}\), and \(w_{3}\), respectively, varying from 0.01 to 0.98 to ensure that a percentage of all three sizes, no matter how small, is present in the spray.