# Vibration Analysis of a Piezoelectric Ultrasonic Atomizer to Control Atomization Rate
## 6 Results and Discussion
Thus, the effective volume used for the numerical results was given as \(\Delta V_{S}^{eff}=\Delta V_{s}-\Delta V_{s}^{th}\). The changes in volume due to deformation of the apertures (\(\Delta V_{D}\)) was neglected for the calculations of the atomization rate, since it was very small compared to the total volume change \(\Delta V_{S}^{eff}\). On the other hand, the pressure loss coefficient (\(\xi\)) is difficult to accurately assess its value, since the actual micro-apertures are manufactured by micro-abrasion electroforming and drilling techniques, which, in practice, give irregular conical shapes and edges [7]. Thus, in order to obtain an estimate of the approximate value of the loss coefficient (\(\xi_{a}\)), correlated to the experimental atomization rate (413 mg/min) obtained at 80 V and 110 kHz, the following procedure was implemented. Figure 16 shows the linear relation (solid line) between the atomization rate and the pressure loss  
Figure 14: FEM estimation of the maximum micro-aperture variations as a function of the cycling changes in the applied voltage. Figure 15: Approximation of the volume caused by the out-of-plane displacement distribution. On the left, the shell method for integration of the solid of the revolution about the z-axis is described. The function \(f(r)\) is integrated along \(r\), with its value subtracted by the threshold function \(g(r)\). coefficient \(\xi\), as given in Equation (7). By using this linear estimate, an approximate value of \(\xi_{a}=6.11\times 10^{-3}\) corresponding to the experimentally determined atomization rate (\(Q=413\) mg/min) was estimated. The value was found to be smaller than the theoretical one (\(\xi=4.18\times 10^{-2}\)) for an ideal conical shape micro-aperture, but this can be a consequence of the actual conditions found in the micro-apertures of the experimental ultrasonic atomizer. Using the found loss coefficient (\(\xi_{a}\)), an approximation of the atomization rate as a function of the voltage can be obtained using Equation (7). The variations of the volume \(\left(\Delta V_{D}^{eff}\), \(\Delta V_{S}\right)\) were calculated for every displacement distribution of the thin plate as given by the FEM at the resonance frequency of 110 kHz. For these calculations, the loss coefficient was kept constant. Using these estimations, the results of the atomization rate measurements at the resonant frequency of 110 kHz were compared with the numerical calculations. As it was shown, at this frequency, the displacement amplitude distribution of the vibrational mode was rather complex, with a harmonic decreasing spatial distribution as a function of the thin plate radius. Thus, it is reasonable to assume that the total deformation between the compression and tension cycles was expected to contribute to the atomization rate. In Figure 17, the obtained results showed a good correlation between the experimental (scatter points) and numerical predictions (solid line) by FEM. The predicted atomization rate was nearly linear, with a minimum at 20 V, which was the preset threshold. A small rate was observed below 40 V, which we attributed to the set constant threshold, thus indicating that the shape of the mode was not affected by the voltage, only its amplitude. The proposed approach assumes that the coefficient \(\xi_{a}\) is constant for all voltage values and that the distribution of aperture openings during the dynamic cycles is homogenous over all the area, whilst, in reality, it depends on the vibrational mode behavior. Despite these considerations, it is interesting to notice that the predictions predicted well the experimental results. The predicted behaviors appeared to follow the experimental results with a roughly linear behavior, indicating a directly proportional relationship between the voltage and the out-of-plane displacements through the center of the thin plate, which controls the atomization rate and can be used to optimize it. Figure 16: Approximation of an effective pressure loss coefficient using the experimental results.