# Numerical Evaluation of Ultrasonic Detection of Solid Occlusions in CSP Molten Salt Piping
## 3 Materials and Methods
### Numerical Simulations  
The numerical study aimed to evaluate the degree of solidification of the molten salts within the storage sections of the CSP system, where the fluid remains stagnant. When the temperature drops below the liquefaction point (approximately 240 \({}^{\circ}\)C at ambient pressure), solidification begins in the NaNO\({}_{3}\)-KNO\({}_{3}\) mixture, leading to the formation of solid inclusions within the piping sections. The pipes are typically insulated with either rock wool or Pyrogel, to slow the thermal transient. However, the inner surfaces of the pipes are the first to reach temperatures near the solidification point of the salts, initiating the formation of solid inclusions along the walls. The numerical setup consisted of a segment of pipe equipped with a through-transmission ultrasonic device (emitter-receiver); see Figure 1a. At the ends of the fluid domain, two perfectly matched layer (PML) domains were placed, to simulate free-field propagation within the pipe, preventing reflections at the numerical domain boundaries. The spatial and temporal distribution of the acoustic field generated by the ultrasonic emitter was obtained by solving the wave equation in the time domain:  
\[\frac{1}{\rho c^{2}}\frac{\partial p}{\partial t}+\
abla\cdot \mathbf{u} =Q_{m}, \tag{13}\] \[\rho\frac{\partial\mathbf{u}}{\partial t}+\
abla\cdot(p\mathbf{I }) =\mathbf{q}_{d}, \tag{14}\]  
where \(p\) is the acoustic pressure, \(\mathbf{u}\) is the acoustic velocity vector, and \(\rho\) and \(c\) are the fluid density and the speed of sound, respectively. \(Q_{m}\) and \(\mathbf{q}_{d}\) are the monopole domain source and the dipole domain source terms, respectively. An impulse boundary condition was applied at the interface between the pulse emitter and the fluid domain (the purple surface in Figure 1b) and is mathematically expressed as  
\[p(t)=\sin(2\pi f_{0}t)\;\mathrm{rect}(t). \tag{15}\]  
To reduce the computational domain size and the system's degrees of freedom, only half of the domain was considered, by applying a symmetry condition on the light orange surface in Figure 1b. Furthermore, to facilitate the convergence of the free-field condition at the extreme boundaries of the PML domain, an impedance condition equal to the fluid medium impedance (\(Z=\rho c\)) was imposed. It is worth noting that without the use of PML, the signal would be reflected by the rigid walls of the numerical domain, distorting the received signal. These unwanted reflections would scatter the signal, leading to inaccurate attenuation results that do not correspond to the physical scenario being studied. The PML boundary condition effectively absorbed the outgoing waves, preventing reflections and ensuring that the simulation results reflected a realistic physical system. The interface relied on the discontinuous Galerkin method (dG-FEM) with a fourth-order Runge-Kutta explicit time solver. Fourth-order elements were employed for discretizing both the acoustic pressure and velocity fields. Two sensitivity analyses were performed, to ensure the numerical results' convergence: a spatial resolution analysis (mesh refinement) and a temporal resolution analysis (time-step adjustment).