## 2 Drop Size Distribution Functions and Spray Science
### Introducing Drop Size Diversity in a Spray
Figure 5 shows the contour plots for each Drop Size Diversity indicator as a function of \(w_{1}\) and \(w_{2}\), and plots on the right are the distributions corresponding to the maximum value obtained by each indicator. This simulation clarifies the difference between the polydispersion and heterogeneity degrees. Namely, the maximum normalized Shannon entropy obtained for the maximum number of droplets relevant in the spray corresponds to the uniform distribution where all relevant drop sizes have the same probability of being present in the spray--thus an adequate parameter to quantify the polydispersion degree. The results for the volume-weighted standard deviation prove a maximum for the maximum difference between relevant drop sizes. In this case, between 10 \(\upmu\)m and 40 \(\upmu\)m droplets, with the standard deviation value of 15 \(\upmu\)m relative to the mid-point around 25 \(\upmu\)m--thus an adequate parameter to quantify the heterogeneity degree. The results for the relative span show its meaning is closer to the heterogeneity degree, expressed by previous authors as size dispersion. However, it is a limited parameter to evaluate Drop Size Diversity, as shown by the distribution representing its maximum value closer to 5, which is difficult to interpret. The next step would be to simulate a spray resulting from the mixture of two polydispersed sprays, for example, using the Log-Normal distribution with different geometric mean diameters (\(d_{g}\)) and equal shape parameters \(\gamma\) (see Equation (1)), and change the mixing through the weight parameter \(w\) to simulate the transition between distributions. Considering a total sample of \(N=10^{5}\) droplets, the simulated spray fixes the shape of both distributions mixed with \(\gamma=0.5/\sqrt{6}\), and uses distinct geometric mean diameters of \(d_{g}=40\)\(\upmu\)m and 70 \(\upmu\)m to mix sprays around larger and smaller drop sizes. The weight \(w=0\) corresponds to the Log-Normal distribution of \(d_{g}=70\)\(\upmu\)m and \(w=1\) to the distribution around smaller sizes. Figure 6 shows on the top left the volume-weighted drop size distributions simulated with \(w\) = 0, 1 and the distributions corresponding the the maximum values obtained for the normalized Shannon entropy (\(H_{n,v}\)), volume-weighted standard deviation (\(SD_{v}\)), and relative span (\(\Delta_{v}\)). The number of classes used the Rice Rule (see Section 1). On the top right are the corresponding volume-based cumulative drop size distributions. Below is the comparison of all indicators for the Drop Size Diversity with the variation of the weight \(w\) attributed to each distribution. Figure 4: Variation of the numerator difference of the relative span (\(D_{v0,g}\)–\(D_{v0,1}\)), and the 50% spray liquid volume representative diameter (\(D_{v0,5}\)), with the weight parameter \(w\). Figure 5: Variation of three Drop Size Diversity Indicators, the Relative Span (\(\Delta_{v}\)), the normalized Shannon Entropy (\(H_{n,p}\)), and the volume-weighted standard deviation (\(SD_{v}\)) for the mixing of two monosize droplet clusters, and the corresponding distributions for the maximum value obtained by each indicator. Figure 6: Results for the volume-weighted Log-Normal mixed spray, corresponding cumulative drop size distributions for \(w=0\), \(1\), and the values corresponding the maximum \(H_{n,p}\) (\(w=0.735\)), \(SD_{v}\) (\(w=0.790\)) and \(\Delta_{v}\) (\(w=0.915\)). Below is the variation of \(H_{n,p}\), \(SD_{v}\), and \(\Delta_{v}\) with the weight parameter \(w\). The first observation concerns the results obtained for each distribution with \(w=0\) and \(w=1\). Since the Log-Normal shape parameter \(\gamma\) is the same for both distributions, the cumulative distributions are equal when classes are divided by \(D_{v0.5}\), which is why one obtains equal values for the relative span \(\Delta_{v}\). The second observation are the implications of probability values for each isolated distribution for the normalized Shannon entropy and the volume-weighted standard deviation.