# Innovative Acoustic-Hydraulic Method for High-Performance Fine Liquid Atomization
## 3 Mathematical Model of the Acoustic-Hydraulic Nozzle
### Droplet Size upon Expiration of Cavitated Liquid  
What happens to the bubble after its nucleation? As shown in [18], under conditions of continuous flow, cavitation bubbles do not collapse, but only grow. It can be assumed that when the cavitated liquid flows into the atmosphere, each bubble will carry with it a layer of liquid, inversely proportional to the number of bubbles:\(W_{w}/N\). This volume will determine the maximum volume of the drop. Then, taking into account Equations (9) and (11), its diameter is:  
\[D_{0}=\frac{2}{\omega}\sqrt[3]{\frac{p_{a}}{\mathrm{K}p}}\sqrt{\frac{I}{pc}}= \frac{D_{c}}{\sqrt[3]{\mathrm{K}}}. \tag{12}\]  
Thus, the maximum droplet diameter is 1.5-1.7 times larger than the diameter of the cavitation bubble. This model does not take into account the surface energy of the liquid carried by the cavitation bubble. On the other hand, a gas bubble surrounded by liquid, when exposed to atmospheric pressure, will behave like a soap bubble. It will expand to a certain maximum size, determined by surface tension forces, and only then will it collapse into fragments with sizes on the order of the thickness of the liquid layer at that moment [19]. Below we will describe this model in more detail. Since the outflow occurs quickly, the cavitation expansion can be considered adiabatic:  
\[\frac{D_{\mathrm{max}}}{D_{c}}=\left(\frac{p}{p_{\mathrm{min}}}\right)^{\frac {1}{3\gamma}}, \tag{13}\]  
where \(\gamma\) is the adiabatic index of liquid vapor, \(p_{\mathrm{min}}\) is the minimum pressure in the bubble at the moment of destruction, and \(D_{\mathrm{max}}\) is the maximum diameter of the bubble before destruction. The moment of bubble destruction can be explained using the principle of energy correlation of bubble surface, as in [20] where the potential collapse mechanism of foam was explained. The bubble expands as long as the surface energy of the liquid layer \(\sigma\)-\(S_{c}\) is greater than the internal energy of the vapor inside it \(p_{\mathrm{min}}W_{c}\), where \(W_{c}\) is the volume of the cavitation bubble, \(S_{c}\) is its surface, and \(\sigma\) is the surface tension. When these energies become equal, it will collapse. From here, we obtain the minimum pressure in the bubble at the moment of destruction:  
\[p_{\mathrm{min}}=\frac{6\sigma}{D_{\mathrm{max}}}. \tag{14}\]  
Solving Equations (9), (13) and (14) together, we obtain the value of the bubble diameter before destruction:  
\[D_{\mathrm{max}}=\left(\frac{p}{6\sigma}\right)^{\frac{1}{3\gamma-1}}D_{c}^{ \frac{3\gamma}{3\gamma-1}}. \tag{15}\]  
The volume of liquid that the bubble carried with it is \(\frac{\pi D_{0}^{3}}{6}\), where the diameter \(D_{0}\) is determined by expression (12). We denote the outer diameter of the bubble before destruction by \(D_{end}\). Then from the law of conservation of mass it follows: \(D_{0}^{3}-D_{c}^{3}=D_{end}^{3}-D_{\mathrm{max}}^{3}\). From here,  
\[D_{end}=\sqrt[3]{D_{\mathrm{max}}^{3}+D_{c}^{3}(1/\sqrt[3]{\mathrm{K}}-1)}. \tag{16}\]  
It can be assumed that the minimum droplet size will be equal to the thickness of the liquid layer upon destruction of the cavitation bubble. The thickness of the water layer at the moment of destruction, and therefore the size of the droplets, will be \(D_{\mathrm{drop}}=(D_{end}-D_{\mathrm{max}})/2\). Or, taking into account Equation (13):  
\[D_{drop}=k\bigg{(}\sqrt[3]{D_{\mathrm{max}}^{3}+D_{c}^{3}(1/\sqrt[3]{\mathrm{ K}}-1)}-D_{\mathrm{max}}\bigg{)}/2. \tag{17}\]  
A joint solution of Equations (9), (15) and (17) will give an estimate of the minimum size of droplets formed during spraying, depending on the characteristics of ultrasonic influence, pressure, and properties of the liquid. Analysis of these equations shows that the process of disintegration of cavitation bubbles in the air and the formation of droplets, as well as the size of the formed droplets, significantly depends on the intensity and frequency of ultrasound (9) and the surface tension of the liquid (15).