# Numerical Evaluation of Ultrasonic Detection of Solid Occlusions in CSP Molten Salt Piping
## 4 Results and Discussion
Figure 5 shows the temporal acoustic pressure profile of the input signal (identical across all the simulations) and the three output signals at the receiver corresponding to the cases where \(x_{p}=0.5\) and \(\varphi=1\) (no occlusion present), \(\varphi=0.64\), and \(\varphi=0.25\). It is evident that in the case of \(\varphi=1\), the receiver signal was free from scattering phenomena, resulting in the energy being concentrated within a rectangular window consistent with the input signal (blue curve). Conversely, for cases with solid inclusions of increasing size (\(\varphi=0.25\) and \(\varphi=0.64\)), the output signal became scattered, meaning it was more spread out over time, due to multiple successive reflections extending its temporal duration at the receiver. Additionally, the amplitudes of the signals in the presence of occlusions were not only scattered but also significantly attenuated, due to the energy reflected by the solid inclusions. This observation supports the conclusion of greater attenuation in these cases. Figure 4: Temporal evolution of the acoustic wave at time steps \(T_{0}\), \(3T_{0}\), \(9T_{0}\), \(15T_{0}\), and \(20T_{0}\). The images show the beginning of propagation from the emitter at \(T_{0}\) and \(3T_{0}\), the initial scattering effects at \(9T_{0}\), the wave reaching the receiver around \(15T_{0}\), and the fully developed scattering and reflection at \(20T_{0}\). Figure 5: The blue curve represents the input impulse generated by the emitter. The yellow, red, and orange curves correspond to output signals for the cases \(x_{p}=0.5\) with \(\varphi=1\) (no occlusion present), \(\varphi=0.25\), and \(\varphi=0.64\), respectively. Figure 6a shows the attenuation of the signal as a function of the position of the occlusion formation (for varying degrees of porosity). Notably, the trend exhibits two peaks corresponding to the positions of the emitter and the receiver. When the solid occlusion formed at these sections, the detected attenuation was amplified compared to the cases where solidification occurred in the intermediate regions between the source and the receiver. A baseline attenuation level can be attributed to thermal relaxation and viscous stress effects in the liquid solution, as observed for \(x_{p}<0\) and \(x_{p}>1\). Similarly, Figure 6b illustrates the attenuation as a function of porosity (for different positions of the solid inclusion). The identified trend (particularly pronounced when the inclusion was near the emitter and receiver sections) can be approximated by an exponential profile. Conversely, the trend flattened when the inclusion formed in regions outside the propagation path between the emitter and receiver. By combining these two trends, a surface plot was obtained, as shown in Figure 7. This graph provides a comprehensive perspective on the attenuation behavior of the ultrasonic signal as a function of both the position and the degree of solid inclusion. From the results obtained, it is essential to derive statistical considerations to render this analysis operational. Once the signal attenuation is known, it will allow for the determination of the degree of solid occlusion and trigger the appropriate intervention strategies. Figure 6: (a) Dimensionless attenuation \(\bar{\alpha}\) as a function of the dimensionless position of the solid inclusion \(x_{p}\) for different values of section porosity \(\varphi\). Peaks in attenuation are observable near the emitter and receiver positions. (b) Dimensionless attenuation \(\bar{\alpha}\) as a function of section porosity \(\varphi\) for different values of \(x_{p}\). An exponential decay trend is noticeable where the inclusions are near the emitter or receiver, while attenuation levels off for inclusions outside the propagation path.