# Physical and Mathematical Modeling of the Interaction of Water Droplets and High-Speed Gas Flow
## 2 Research Methodology
### Description of Data Processing Methods
For droplets with \(We\geq\text{We}_{\text{crit}}\) the change in their proportion was calculated using the formula proposed by the authors:  
\[\frac{dP_{i}}{dt}\ =\ -\widetilde{K}_{1}\cdot P_{i}\cdot We^{\widetilde{K}_{2}} \cdot Re^{0.025}, \tag{12}\]  
Here \(P_{i}\) is the relative number of particles of the considered fraction \(i\); \(\widetilde{K}_{1}\) and \(\widetilde{K}_{2}\)-empirical coefficients. In the proposed model a set of parameters \(\widetilde{K}_{1}\cdot P\cdot We^{\widetilde{K}_{2}}\cdot Re^{0.025}\) determines the number of particles separated from the droplet per unit time. In this case, it is assumed that the fragments detached from the droplets of the considered fraction move as independent droplets into a fraction with a diameter \(\text{d}_{1}\). The influence of the Re number for the breakup process is important since this parameter characterizes the effect of turbulent flow features. In addition, as shown in [36; 37] the Re number affects the droplet shape--the surface area of the liquid film correlates with the Re number. A similar effect was noted in a numerical study [5]. The work [8] describes the classification of droplet destruction modes depending on the Re number. In particular, a change in Re leads to a shift in the boundary denoting the bag and shear stripping types of drop destruction. The droplet velocity was calculated based on the obtained value of \(C_{x}\), the direction of the velocity vector, as well as the density and velocity of the gas phase in the corresponding longitudinal coordinate. It is assumed that droplet fragments obtained as a result of gas-dynamic fragmentation at the moment of their formation have a velocity equal to the velocity of the initial droplet. The distribution of diameters and droplet velocities in region 2 was determined based on the simulation results. Depending on the coefficients \(\widetilde{K}_{1}\) and \(\widetilde{K}_{2}\), an array of relative counting distributions of droplet diameter and velocity was formed. An automated search of solution, which is in the most extent similar to the experimental data, was realized in this array. The criterion of maximum coincidence is the minimum value of the standard deviation \(Er\) between the calculated and experimental distributions in the relative number of particles (\(P\)) for different fractions in the measurement region 2:  
\[Er\ =\ \sqrt{\frac{\sum\limits_{i=1}^{M}\left[\frac{p(\widetilde{K}_{1}, \widetilde{K}_{2})_{i}}{Norm}-\left(P_{\text{exp}}\right)_{i}\right]^{2}}{M}} \cdot 100\%, \tag{13}\]  
where \(M\) is the number of fractions, \(P(\widetilde{K}_{1},\widetilde{K}_{2})_{i}\)--simulated relative number of droplets of the fraction \(i\), (\(P_{\text{exp}}\))\(i\)--the experimentally recorded relative number of drops of the fraction \(i\), \(Norm\) is the normalizing coefficient. The normalizing factor is introduced due to the fact that droplets with a diameter less than \(\text{d}_{\text{min}}=5\)\(\upmu\)m were not taken into account. Therefore, in order to properly estimate the relative number of droplets with a diameter of more than \(5\)\(\upmu\)m it is necessary to normalize it to the \(Norm\) factor (usually \(\leq\)0.8) which depends on the proportion of unaccounted for droplets. The graphs of the change in the relative probability of finding droplets of three fractions from the longitudinal coordinate are shown in Figure 5.