# Vibration Analysis of a Piezoelectric Ultrasonic Atomizer to Control Atomization Rate
## 4 Model of Atomization Volume Flow  
In mesh atomization, the atomizer releases energy into the liquid to break the surface tension and to allow the liquid droplets to escape from the surface. To control the droplet size distribution and to make the atomization process more controllable, micro-apertures are added to form a meshed thin plate driven by a piezoelectric ring. In [PERSON] et al. [14], the atomizer performance as a function of the number of micro-apertures was studied. A rough approximation of the deformation for the lowest symmetric modes can be found following the thin plate theory [24]. The coordinate system assumes that the neutral surface is in the \(xy\) plane, perpendicular to the \(z\)-axis. The flow in the micro-tapered aperture is closely connected with the chamber volume change given by the deformation of the thin plate [25]. In [PERSON] et al. [12], the atomization rate was studied, focusing on the effect of the volume change at the liquid chamber (container) due to the deformation of the thin plate during the vibrations at the lowest vibration mode. Additionally, the contribution by the small deformation of the micro-apertures of the mesh during the vibration to the mass flow was studied for that mode. This idea is illustrated in Figure 10, where the conical micro-aperture deforms by the global deformation of the thin plate. During the periodic vibration, any point on the non-neutral surface moves in two possible conditions. First, the point is bent in the upward direction and then is released back to equilibrium, followed by a compressed state to its lower limit and back to equilibrium. This mechanism acts as a micro-pump, promoting the generation of atomized drops [26]. Figure 10: Dynamic deformation of a micro-aperture with a conical shape, (**a**) prior to deformation, (**b**) with deformation in upward direction, and (**c**) downward direction. Figure 9: Distribution of the displacements along the cross-section of the thin plate for the 110 kHz (axisymmetric mode) resonance frequency as a function of the voltage. Thus, the total flow could have two contributions--namely, one from the global volume change of the thin plate \((q_{wa})\) and another possible from the micro-aperture volume change \((q_{vc})\), which could then be expressed as:  
\[Q=q_{wa}+q_{vc}\approx(n\Delta V_{D}+\Delta V_{S})\;f\xi \tag{7}\]  
where \(\Delta V_{D}\) is the change in the volume in the micro-aperture during vibration that can be obtained by considering the deformation of the neutral surface and the estimation of a triple integral to calculate the change of volume of the micro-cone aperture. \(\Delta V_{S}\) is the corresponding change in volume in the chamber created by the thin plate vibration, \(f\) is the oscillation frequency of the plate, and \(\xi\) is a pressure loss coefficient described below. The volume of the atomized liquid displaced by the mechanical oscillation of the plate for the lowest vibration mode is given by Reference [12]:  
\[\Delta V_{S}=\frac{\pi hr_{out}^{2}}{2} \tag{8}\]  
where \(r_{out}\) is the radius of the plate, and \(h\) is the thickness of the thin plate. However, since the vibrational modes studied in this work are of a higher order, there is a nonhomogeneous displacement distribution across the radius. In this case, numerical integration techniques were used to approximate the displaced volume, as will be described later. The effective pressure loss coefficient, \(\xi\), is expressed as:  
\[\xi=\frac{\xi_{Dd}-\xi_{Dn}}{2+\xi_{Dn}+\xi_{Dd}} \tag{9}\]  
where \(\xi_{Dd}\) is the pressure loss coefficient related to the diffuser effect, and \(\xi_{Dn}\) is related to the nozzle effect. To relate the volume changes to the mass flow, it is necessary to consider the resistance of the flow through the conical aperture. In a dynamic cycle, the cone aperture acts as nozzle or as diffuser, and the difference between the two flow resistances determines the flow rate. In [PERSON] et al.