# Study on Molybdenum-Rhenium Alloy Ultrasonic Resonance Temperature Sensor
## 2 Materials and Methods
### Material Selection and Parameter Design
\begin{table}
\begin{tabular}{c c c c c c c} \hline \hline
**Materials** & **Density/(g/cm\({}^{3}\))** & **Melting Point/\({}^{\circ}\)C** & **Young’s Modulus/GPa** & **Poisson’ Ratio** & **Thermal Conductivity W/(m-K)** & **Thermal Expansion Coefficient/K** \\ \hline Molybdenum-Rhenium & 10.4 & 2620 & 320 & 0.28 & 130 & \(5.3\times 10^{-6}\) \\ \hline Sapphire & 3.98 & 2053 & 400 & 0.28 & 45 & \(5.8\times 10^{-6}\) \\ \hline Spinel & 3.58 & 2135 & 280 & 0.26 & 15 & \(8.5\times 10^{-6}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Physical properties of waveguide materials. dispersion [24]. The relationship between phase velocity and group velocity can be expressed by the following function:  
\[C_{g}=\frac{C_{p}^{2}}{C_{p}-(2\pi fd_{2})\frac{dC_{p}}{d(2\pi fd_{2})}} \tag{6}\]  
where \(d_{2}\) represents the diameter of the waveguide rod and \(f\) denotes the guided wave frequency. In the ultrasonic resonance temperature-measurement study, in order to deeply investigate the acoustic-wave-propagation characteristics of the molybdenum-rhenium alloy waveguide rods, the Pcdisp module in the MATLAB 2014 program for guided wave analysis written by [PERSON] and [PERSON] [25] was used to simulate the signal transmission, based on the physical properties listed in Table 1 and the input of the molybdenum-rhenium alloy material properties, including those such as Young's modulus, Poisson's ratio, density and so on, and also to set different diameters, frequency ranges, and modal selection. At the same time, the different diameters of the waveguide rods, frequency range, and mode selection were set. In view of the propagation of acoustic waves in the metal and its alloy, the degree of attenuation is mainly related to the frequency. In the low-frequency band below 1000 kHz, the acoustic-wave transmission distance is relatively far and the signal attenuation is small, which is conducive to obtaining stable test signals. The lower frequency of the waveguide size of the change in the sensitivity of the waveguide size is relatively low, effectively reducing the error due to the size of the waveguide;  
(3) Resonator size  
In the design of resonant ultrasonic temperature sensors, the waveguide rod and the sensitive area section are integrated structures, which makes the relationship between the resonant frequency and the temperature different when the acoustic wave is transmitted to the sensitive elements with different structural parameters. In order to achieve optimized design and efficient processing, a half-wavelength resonator consisting of a combination of cones and cylinders was chosen in this study, and its structure is shown in Figure 3.  
The conical structure concentrates the ultrasonic signal, thereby effectively transferring the sound wave's energy to the cylinder. According to Formula (1), when the frequency f is set to 140 kHz, the wavelength is 38 mm, the cone length \(L1=\lambda/4\), and the resonator length \(L2=\lambda/2\) to ensure that the resonator is effectively coupled to the waveguide rod and transmits sound waves properly. In resonant ultrasonic temperature measurement, the quality factor \(Q\)[26] of the resonator is the key parameter, which is expressed as follows:  
\[Q=2\pi\frac{\omega}{\Delta\omega} \tag{7}\]  
\[Q_{c}=(2m-1)\times\pi/2r^{2} \tag{8}\]  
\[r=\frac{\rho_{1}C_{1}A_{1}}{\rho_{2}C_{2}A_{2}} \tag{9}\]  
Figure 3: Structure of the sensitive area. where \(\omega\) is the energy stored in the echo, and \(\Delta\omega\) is the energy lost per cycle of the echo. \(Q_{c}\) is the energy-loss coefficient, which indicates that in the \(m\)-th order mode, the energy loss of the resonator is determined solely by the \(Q\) factor.