# Improvement of Mesh Atomizer Performance by Electrolytic Polishing
## 2 Theoretical Model of Atomization Amount and Electrolysis Time
### Mathematical Expressions of Atomization Amount and Polishing Time  
Given that the metal sheet is made of 304 stainless steel, the quality of the whole metal sheet is  
\[m_{1}=\rho\pi R^{2}\delta, \tag{1}\]  
where \(\rho\) is the density of the stainless steel, \(R\) is the radius of the metal sheet, and \(\delta\) is the thickness of the metal sheet. The micro-tapered hole of the metal sheet has a circular shape; given \(D_{1}=D^{2}+d^{2}+Dd\), \({D_{1}}^{\prime}={D^{\prime}}^{2}+{d^{\prime}}^{2}+D^{\prime}d^{\prime}\), the mass of the micro-tapered hole is  
\[m_{2}=\frac{\pi}{12}\rho\delta n_{1}D_{1}, \tag{2}\]  
where \(D\) is the diameter of the large end of a micro-tapered hole, \(d\) is the diameter of the small end of a micro-tapered hole, and \(n_{1}\) is the number of micro-tapered holes. The actual metal sheet mass is the mass of the entire metal sheet without laser processing minus the mass of micro-tapered holes:  
\[m_{3}=\frac{\pi}{12}\rho\delta\Big{(}12R^{2}-n_{1}D_{1}\Big{)}. \tag{3}\]  
According to [PERSON]'s law of electrolysis, the mass of a metal sheet with chemical reaction on the electrode is  
\[m_{4}=\frac{M1t}{n_{2}F}, \tag{4}\]  
where \(M\) is the molar mass of the reactive substance, \(F\) is the Faraday constant, \(n_{2}\) is the electron measurement coefficient in the electrode reaction measurement equation, \(t\) is the reaction time, and \(I\) is the current intensity. By the law of conservation of mass,  
\[\frac{\pi}{12}\rho\delta\Big{[}12\Big{(}R^{2}-{R^{\prime}}^{2}\Big{)}+n_{1} \big{(}{D_{1}}^{\prime}-D_{1}\big{)}\Big{]}=\frac{M1t}{n_{2}F}, \tag{5}\]where \(R^{\prime}\) is the radius of the metal sheet after electropolishing, \(D^{\prime}\) is the diameter of the large end of a micro-tapered hole after electropolishing, and \(d^{\prime}\) is the diameter of the small end of a micro-tapered hole after electropolishing. Figure 1 shows a structural drawing of the atomization sheet. Given that the radius \(R\) of the metal sheet minimally changes in electrolytic polishing, it can be ignored. Hence, Equation (5) can be simplified as  
\[\frac{\pi}{12}\rho\delta n_{1}\big{(}{D_{1}}^{\prime}-{D_{1}}\big{)}=\frac{Mlt }{n_{2}F} \tag{6}\]  
The volume of micro-tapered holes is the theoretical volume of the metal sheet minus the volume of burr. The burr height inside the hole cannot be measured. Only the burr height near the hole can be determined to represent the burr height inside the hole, and the burr is regarded as evenly distributed on the surface of the hole. Thus, this paper adds a correction coefficient \(\gamma\) when calculating the burr. Given \(D_{2}=D+d\), the following equation is obtained:  
\[V=\frac{\pi}{12}\delta{D_{1}}-\frac{\pi}{2}\gamma h{D_{2}}, \tag{7}\]  
where \(\gamma\) represents the correction coefficient, \(h\) is the burr height, and \(l\) is the busbar length. The busbar length is expressed as follows,  
\[l=\frac{D-d}{2\sin\frac{\theta}{2}} \tag{8}\]  
where \(\theta\) represents the conical angle of a micro-tapered hole. It is expressed as follows:  
\[\theta=2\text{arctan}\frac{D-d}{2\delta} \tag{9}\]  
Figure 1: Structural drawing of the atomization sheet: (**a**) Vertical view; (**b**) Side view; (**c**) Large end of a micro-tapered hole before polishing; (**d**) Large end of a micro-tapered hole after polishing. Given \(D_{3}=D^{2}-d^{2}\), the difference between the volume of the cavity where the atomization sheet reaches the maximum deformation when vibrating after electrolytic polishing and the maximum deformation when vibrating before electrolytic polishing is  
\[m_{2}=\frac{\pi}{12}\rho\delta n_{1}D_{1}, \tag{2}\]0  
For the micro-tapered hole, the fluid flows from the large hole side to the small hole side. The structure is a conical diffuser, and the flow from the small hole side to the large hole side is a conical nozzle.