# Innovative Acoustic-Hydraulic Method for High-Performance Fine Liquid Atomization
## 4 Results and Discussion
### Calculation Results: Comparison of Theoretical and Experimental Results
#### 4.2.3 Droplet Size  
The main achievement of mathematical modeling was the prediction of the critical (threshold) nature of droplet formation. The droplet size calculated in accordance with Equation (17) (free parameter \(k=300\)) is shown in Figure 9.  
Figure 9 demonstrates the critical nature of the occurring phenomena. Upon transition to the cavitation outflow mode, the droplet size becomes minimal (and with a further increase in pressure or intensity, it only slightly increases). A couple of process control parameters are fluid pressure and ultrasound intensity. At certain values of these parameters, the spraying transitions to cavitation mode. In this case, the droplet size becomes minimal and practically does not change with a further increase in pressure or intensity. Thus, to achieve a minimum droplet size, it is sufficient to increase the intensity at a given pressure or the pressure at a given intensity to threshold values, and not increase further. In accordance with Equation (10), the critical intensity is proportional to the hydrostatic pressure to the power of \(2/3\). In this case, the critical conditions in terms of pressure and intensity correspond to the observed ones: the higher the pressure, the higher the intensity required for the transition to the cavitation outflow regime, and vice versa. Therefore, the  
Figure 8: Critical ultrasound intensity \(l_{cr}\) as a function of pressure \(p/p_{a}\) for different values of the critical bubble size \(D_{cr}\). The model calculation at \(D_{cr}=30\)\(\upmu\)m most accurately describes the experimental points. Figure 9: Characteristic size of droplets \(D_{drop}\) depending on pressure \(p/p_{a}\) (**a**) and ultrasound intensity \(l\) (**b**). critical point in Figure 9a shifts to the right with increasing intensity, and in Figure 9b it shifts to the right with increasing pressure. The free model parameter \(k=300\) (Equation (17)) was chosen based on comparison with the experiment. The physical meaning of this coefficient is the excess of the characteristic size of an ensemble of droplets over the minimum possible size calculated in accordance with the \"soap bubble\" destruction model. Figure 10 shows experimental points (Table 1) and calculated curves for droplet size. The figure demonstrates the transition of spraying to cavitation mode, which is confirmed both theoretically and experimentally. This occurs when critical values for pressure and intensity are reached in accordance with Equation (10). It should be noted that there is no exact coincidence of the calculation results with the experiment (and such a goal was not set), since the experimental value of the droplet size is a statistical characteristic of the droplet size distribution (Sauter diameter), and the calculation uses a certain theoretical characteristic droplet size, and not the distribution by size. The size of droplets during ultrasonic spraying using the capillary wave mechanism is estimated using [PERSON]'s equation [21]:  
\[d_{us}=0.34\sqrt[3]{\frac{8\pi\sigma}{\rho\omega^{2}}}. \tag{18}\]  
Thus, the higher the frequency, the smaller the droplet. With the classic method of ultrasonic spraying, the droplet size at a frequency of 22.5 kHz, the Sauter mean diameter of the droplets is \(D_{32}=100\ldots 120\)\(\mu\)m. Moreover, the 22.5 kHz frequency used in our design is energetically more favorable than higher frequencies. When spraying with hydraulic nozzles, the size of the droplets depends on the liquid pressure in the nozzle: the higher the pressure, the smaller the droplets. On the other hand, the flow rate (or volumetric flow rate) of the liquid is important: the lower the flow rate, the higher the dispersion [22]. The corresponding empirical formulas depend on the fundamental design of the nozzle, but at a pressure of \(8\ldots 9\) atm and a volume flow rate of \(10\ldots 12\) mL/s, the droplet diameter does not decrease less than \(100\ldots 150\)\(\mu\)m (at lower pressure values, the droplet size is significantly larger).