# Numerical Evaluation of Ultrasonic Detection of Solid Occlusions in CSP Molten Salt Piping
## 4 Results and Discussion  
Simulations were conducted to evaluate the capability of ultrasound technology to detect solid inclusions within the piping section under investigation. The pulse frequency \(f_{0}\) was fixed to 2.5 MHz. In line with existing ultrasonic monitoring technologies for molten salt systems, as demonstrated by Ionix Advanced Technologies [29], the frequency of ultrasonic waves in this study was chosen based on the optimal balance between penetration depth and resolution. Frequencies in the range of 1-5 MHz have been shown to effectively penetrate molten salts, while providing the necessary resolution to detect solid inclusions. The chosen frequency range ensured that the waves could traverse the molten salt effectively while maintaining sensitivity to variations in the internal structure, such as inclusions and porosity. Previous studies on ultrasonic inspection of liquid metals and molten salts support this choice, as these frequencies allow for precise monitoring of material characteristics without excessive attenuation or distortion of the signals. The temperature of the system was fixed at 240 \({}^{\circ}\)C for all the simulations. This choice was based on the observation that the thermodynamic properties of the medium, including both the fluid and the solid phases, exhibit negligible sensitivity to temperature variations, particularly within the phase transition range. As such, the analysis focused on evaluating the dimensionless attenuation, which depends on the input and output signals, allowing for a simplified and consistent approach to assessing the behavior of the system. The analyses presented below were based on 7437 simulations, where the position of the solid inclusion was systematically varied within the reference system, as illustrated in Figure 1a. The position is described using the dimensionless coordinate  
\[x_{p}=\frac{x_{s}}{L_{d}}, \tag{17}\]  
where \(x_{s}\) is the leftmost coordinate of the solid inclusion and \(L_{d}\) is the axial distance between the ultrasonic source and receiver. Another parameter that was varied was the percentage of the solid inclusion within the pipe cross-section, referred to here as the section porosity:  
\[\varphi=1-\frac{S_{s}}{S_{t}}, \tag{18}\]  
where \(S_{s}\) is the area of the solid inclusion and \(S_{t}\) is the total cross-sectional area of the pipe. Note that the section porosity \(\varphi\) was complementary to the degree of solid inclusion, meaning that \(\varphi=0\) indicated a section completely obstructed by the solid. The signals analyzed to evaluate the dimensionless attenuation were the input signal, applied as a pressure boundary condition, Equation (15), at the surface corresponding to the emitter (marked in purple, Figure 1b), and the output signal recorded at the receiver surface (marked in light blue, Figure 1b). To better understand the temporal evolution of the ultrasonic wave propagation, we present a series of snapshots of the acoustic wave at different time steps: \(T_{0}\), \(3T_{0}\), \(9T_{0}\), \(15T_{0}\), and \(20T_{0}\). These time steps were selected to highlight the different stages of the wave interaction, from the initial propagation to the scattering and reflection phenomena that occurred as the wave interacted with solid inclusions within the pipe. Figure 4 illustrates this temporal evolution. At \(T_{0}\), the impulse began its propagation from the emitter and started to travel through the medium. At this stage, the wavefront was still relatively coherent, and there  
not generate reflections, as expected. This was because the PML was specifically designed to recreate free-field conditions, preventing any artificial reflections that could distort the results. This sequence of images demonstrates the physical process at play as the ultrasonic waves interacted with the solid inclusions, providing insight into the dynamics of the signal attenuation and scattering that will be further analyzed in the following sections.