# Ultrasonic Power and Data Transfer through Multiple Curved Layers Applied to Pipe Instrumentation
## 3 Acoustic Channel Analysis
### Experimental Setup  
An experimental setup was created in order to investigate power and data transfer across multiple curved layers (water and steel). The setup has two concentric pipes with water in between them with dimensions shown in Figure 4a,b, respectively. The geometric parameters such as diameters and thickness of the pipes are based on the actual dimensions used in wellbore systems. For a proper operation, the piezoelectric transducers must be aligned to reduce the insertion loss of the acoustic channel [17; 18]; this means that the energy transfer to the inside block is dramatically dependent on the alignment of both transducers. According to this, and knowing that the control of alignment between the transducers is difficult for a wellbore application, a higher area of the piezoelectric transducer was used to reduce the loss of power due to misalignment. The dimensions of the rectangular-shaped piezoelectric transducers were \(70\,\mathrm{mm}\times 25\,\mathrm{mm}\times 2\,\mathrm{mm}\), and are also shown in the inset plot of Figure 4a. Furthermore, generally the acoustic channel has a high selectivity in frequency [16; 17; 18], meaning that the system must operate in an optimum frequency that provides sufficient power to the inside block and an adequate voltage variation to allow a reliable communication. As it can be seen in Figure 4b, the experimental setup comprised a flat stainless steel case that covers the internal transducer, keeping its electrodes from being in contact with the water and also leaving the piezoelectric back face in contact with air. This case is \(7.5\,\mathrm{mm}\) thick and is attached on the internal pipe. The water layer and the external pipe section are \(20.5\,\mathrm{mm}\) and \(20\,\mathrm{mm}\) thick, respectively. The external and internal pipes are made of stainless steel. The physical properties of the different materials used in the experimental setup are shown in Table 1. It is worth noticing that, despite the whole structure being cylindrical in nature, the PZTs were bonded to flat surfaces in order to provide better coupling for generated and received vibrations, as shown in Figure 4. Consequently, in what concerns the ultrasonic propagating path, there is only one curved interface in the channel, namely the interface at the inner radius of the outer steel pipe and the water annulus, as can be seen by the outer dashed circle in Figure 4a. \begin{table}
\begin{tabular}{l c c c c} \hline \hline \multicolumn{2}{c}{**Piezoceramic: PZT5**} & \multicolumn{2}{c}{**Fluid: Water (20 \({}^{\circ}\)C)**} \\ \hline Density & \(\rho\) (kg/m\({}^{2}\)) & \(7733\) & Density & \(\rho\) (kg/m\({}^{3}\)) & \(1000\) \\ Sound speed & \(v\) (m/s) & \(4419\) & Sound speed & \(v\) (m/s) & \(1484\) \\ Quality factor & \(Q_{m}\) & \(48\) & Acoustic damping coefficient & \(\alpha_{v}\) (Np/m) & \(0\) \\ Electrical permittivity & \(\varepsilon_{33}\)\({}^{s}\)/\(\varepsilon_{0}\) & \(1101\) & Thickness & \(t_{w}\) (m) & \(20.5\times 10^{-3}\) \\ Piezoelectric coupling factor & \(\varepsilon_{33}\) & \(0.58\) & Area & A (m\({}^{2}\)) & \(175\times 10^{-3}\) \\ Piezoelectric coefficient & \(C_{33}\) (N/m\({}^{2}\)) & \(15.1\times 10^{10}\) & & \\ Area & A (m\({}^{2}\)) & \(175\times 10^{-3}\) & & \\ Thickness & \(t\) (m) & \(2\times 10^{-3}\) & & \\ \hline \multicolumn{2}{c}{**Metal: Stainless Steel**} & \multicolumn{2}{c}{**Adhesive: Epoxy**} \\ \hline Density & \(\rho\) (kg/m\({}^{3}\)) & \(7894\) & Density & \(\rho\) (kg/m\({}^{3}\)) & \(1400\) \\ Sound speed & \(v\) (m/s) & \(5619\) & Sound speed & \(v\) (m/s) & \(2344\) \\ Acoustic damping coefficient & \(\alpha_{v}\) (Np/m) & \(4\) & Acoustic damping coefficient & \(\alpha_{v}\) (Np/m) & \(172\) \\ Area & A (m\({}^{2}\)) & \(175\times 10^{-3}\) & Area & A (m\({}^{2}\)) & \(175\times 10^{-3}\) \\ Outer pipe thickness & \(t_{0}\) (m) & \(7.5\times 10^{-3}\) & Thickness & t (m) & \(150\times 10^{-6}\) \\ Steel cover thickness & \(t_{e}\) (m) & \(20\times 10^{-3}\) & & \\ \hline \hline \end{tabular}
\end{table}