# Ultrasonic Deicing Efficiency Prediction and Validation for a Flat Deicing System
## 4 Results and Discussion  
### Deicing Energy Efficiency  
After harmonic analysis, several nodes were selected as reference points to confirm optimal frequencies for each configuration. Due to the amount of data, only configuration 1 (\(r\) = 4 mm, \(t\) = 6.25 mm) was shown as an example to interpret the whole process. Here, only the two reference points A and B shown in Figure 4, which are located at the boundary and center of the interface between the host plate and ice layer, respectively, were presented as an example to output shear stress amplitude-frequency curves. The shear stress amplitude-frequency curves are shown in Figure 7. The optimal frequency results for the rest nodes were the same, so they were not presented here. From the results of Figure 7, it can be found that the optimal frequency for deicing is 221 kHz. Then, the same processes were performed for the remaining six configuratio  
Figure 7: Shear stress amplitude-frequency curves of node: (**a**) node A (edge point); (**b**) node B (middle point). frequency of each model. For the purpose of estimating the deicing effect of each model, the average shear stresses for delamination \(\tau_{13}\) and \(\tau_{23}\) at the interface of the ice layer were calculated. The results are shown in Table 3.  
From the results of Table 3, the total shear stress \(\tau_{\text{total}}\) for delamination can be calculated from Formula (10):  
\[\tau_{\text{total}}=\sqrt{{\tau_{13}}^{2}+{\tau_{23}}^{2}}. \tag{10}\]  
The total shear stresses for seven different configurations are shown in Figure 8. These shear stresses were enough to overcome ice adhesion strength measured in lab condition, which was 0.36 \(\pm\) 0.06 MPa at temperature \(-\)12 \({}^{\circ}\)C (detailed information is in part 3). Besides deicing shear stresses analysis, energy consumed for the deicing system is another important aspect needed to take into consideration. For simplicity, the energy dissipation and heat loss were ignored, only the kinetic energy and strain energy for the whole system were considered to estimate the energy consumed for each model at optimal frequency. Therefore, the energy density for deicing can be written as Formula (11):  
\[w=\frac{E_{k}+U}{V}, \tag{11}\]  
where \(w\), \(E_{k}\), \(U\), and \(V\) represent energy density, kinetic energy, strain energy, and volume of system, respectively, for deicing. From these data, the energy density needed to detach ice from the host plate for each model can be estimated. From computation results, the conclusion can be drawn that the energy density needed for deicing varies from the radius of the piezoelectric wafer with the system  
Figure 8: Shear stress for delamination of seven different configuration. \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Configuration Radius (mm)** & **Optimal Frequency (kHz)** & \(\tau_{13}\) (MPa) & \(\tau_{23}\) (MPa)** \\ \hline
4 & 221 & 4.33 & 3.01 \\
5 & 213 & 0.64 & 0.85 \\
6 & 236 & 2.99 & 2.75 \\
7 & 222 & 1.92 & 2.02 \\
8 & 232 & 7.22 & 6.94 \\
9 & 212 & 1.75 & 1.77 \\
10 & 190 & 2.27 & 2.42 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Optimal frequency and the average shear stresses at the interface for seven configurations. at optimal frequency for each configuration. Meanwhile, the deicing effect also differs from one to another. Therefore, it is difficult to distinguish which model is better. The ideal goal is to achieve deicing with minimum energy consumption. Therefore, a coefficient called deicing efficiency factor (DEF) was proposed to describe the relationship between deicing effect and deicing energy density. DEF was defined as Equation (12):  
\[\text{DEF}=\frac{\tau_{\text{total}}}{w}=\frac{\sqrt{{\tau_{13}}^{2}+{\tau_{23} }^{2}}}{\frac{{E_{\text{k}}}+U}{V}}=\frac{\sqrt{{\tau_{13}}^{2}+{\tau_{23}}^{ 2}}\cdot V}{{E_{\text{k}}}+U}, \tag{12}\]  
where \(\tau_{\text{total}}\), \(\tau_{13}\), \(\tau_{23}\), \(w\), \(E_{k}\), \(U\), and \(V\) have the same meaning as mentioned prior. This coefficient can describe deicing efficiency for each model.