# Numerical Evaluation of Ultrasonic Detection of Solid Occlusions in CSP Molten Salt Piping
## 2 Theoretical Background
### Ultrasonic Attenuation and Absorption  
When ultrasonic waves propagate through a material, their intensity decreases due to attenuation and absorption. This energy reduction is expressed mathematically as [8]  
\[E_{x}=E_{0}e^{-ax} \tag{5}\]  
where \(E_{x}\) is the energy of the wave after traversing a material of thickness \(x\), \(E_{0}\) is the energy of the incident wave, and \(\alpha\) is the _linear absorption coefficient_, commonly referred to as the _attenuation coefficient_ expressed in dB/m. Attenuation refers to the reduction in wave intensity as ultrasound propagates through a medium. This phenomenon is due to the conversion of acoustic energy into heat through viscous and relaxation processes, known as **absorption**, and the deflection of waves by inhomogeneities such as grain boundaries, porosity, and material heterogeneities, referred to as **scattering**. The total attenuation coefficient \(\alpha\) combines contributions from absorption (\(\alpha_{a}\)) and scattering (\(\alpha_{s}\)):  
\[\alpha=\alpha_{a}+\alpha_{s} \tag{6}\]  
In homogeneous materials, absorption dominates, due to viscoelastic damping. In heterogeneous materials, scattering contributes significantly. Frequency-dependent attenuation, typically proportional to \(f^{n}\) (where \(n\) ranges from 1 to 2), is a critical parameter in material characterization [7; 8]. From Equation (5), the attenuation coefficient can be explicitly expressed, leading to an operational formula for determining it experimentally (or with the aid of numerical simulations). In particular, for a global assessment, the energies of the signals emitted, \(E_{1}\), and received, \(E_{2}\), by the two transducers are considered. The attenuation coefficient can be calculated as [8]  
\[\alpha=\frac{10}{d}\log_{10}\left(\frac{E_{1}}{E_{2}}\right), \tag{7}\]  
where \(d\) is the distance between the transducers. The energy of a signal is evaluated from the waveform of the detected acoustic pressure:  
\[E=\int_{\Delta T}|p|^{2}dt. \tag{8}\]  
In this expression, \(p\) represents the acoustic pressure, and the integral is taken over the time interval \(\Delta T\), corresponding to the duration during which the signal is detected. Thismethod allows for the calculation of energy lost due to attenuation as the ultrasonic wave propagates through the medium. The formula can be applied to both experimental measurements and numerical simulations to evaluate the attenuation properties of the medium. The concept of signal energy and attenuation played a crucial role in this analysis, as the study focused on the energy characteristics of the ultrasonic signals generated and received by the emitter and receiver, respectively. By evaluating the energy loss due to attenuation, it was possible to assess how the presence of solid inclusions influenced the propagation of the ultrasonic wave through the medium. As will be demonstrated in the subsequent sections, this energy-based approach allowed for the identification of solid inclusions within the molten salt by comparing the attenuation patterns of waves traveling through a homogeneous medium versus those interacting with heterogeneous regions. This distinction in signal attenuation served as a key indicator of the presence, location, and characteristics of solid occlusions, providing valuable insights into the internal structure of the molten salt system.