# Evaluation of the Acoustic Noise Inside the Main Steam Line of a BWR/5 Nuclear Reactor
## 3 Materials and Methods
### Computational Method  
The acoustic analysis was conducted with the ANSYS 2024 R1 academic research program. The acoustic loads and frequencies were obtained with CFD in conjunction with the FSI one-way model and the equations of the quarter wave resonator tube, as seen in Equation (11). They were implemented using MATLAB R2023a code. The methodology followed in this research is shown in the flowchart in Figure 5.  
Table 1 shows the dimensions of the geometric model used in this paper. The scale that was used was 1:8 of the original dimensions. The advantage of the proposed numerical model is its low computational cost due to the use of a steady-state numerical model. The fluctuations of the pressure, which will be transformed into acoustic noise independently, can be obtained using the strains in Equations (5) and (6) [19; 26]. Figure 4: Instrumentation on the main steam line. Figure 5: Flowchart of the methodology. In the development of the 3D model, the boundary conditions for a steady-state analysis considered the velocity, turbulence, and continuity equations. Furthermore, the fluid was considered incomprehensible and under isothermal conditions. The velocity at the inlet was constant. The inlet and outlet gage pressure were set to zero. The turbulence intensity was calculated based on the hydraulic's diameter. The strains were obtained with a model (FSI one-way) using ANSYS 2024 R1 Mechanical. These results were compared with those obtained with the theory of thick-walled cylinders. Finally, the amplitudes and resonance frequencies were obtained with MATLAB R2023a code, using the model of the quarter-wave resonator and the numerical results of the finite element analysis. Figure 5 summarizes the methodology used in this paper. The mathematical formulation of the conditional Navier-Stokes equations of the finite volume method is shown in Equation (12), where \(\rho\) is the density, \(\phi\) is a specific numerical property, \(\Gamma\) is the diffusion coefficient, \(S\) is the gross generation of the variable \(\phi\) per unit volume, and, finally, \(\
abla\) which is the nabla operator. The Eulerian method was used because the fluid was analyzed from a fixed perspective in space. This allows for the use of simple stationary meshes that do not need to adapt to the structure deformations as it occurs in the unidirectional fluid-structure interaction. \[ \begin{array}{c}\frac{\partial}{\partial t}\int_{V}\rho\phi dV+\oint_{A}\rho \phi dV\cdot dA=\oint_{A}\Gamma_{\phi}\
abla\phi dV\cdot dA+\int_{V}S_{\phi}dV \\ \mbox{Transient Convective Diffusive Source}\end{array}  \tag{12}\]  
Figure 6 shows the flowchart of the MATLAB R2023a code. There were two methods used for the evaluation of the acoustic noise. The first one used the strains obtained in the FSI numerical simulation or the strains obtained experimentally in Equations (6), (11) and (13). The second method considered the pressures obtained by the CFD and Equations (11) and (13). The power acoustic was obtained in the lateral branch, where \(P_{a}\) is the acoustic power (W), \(I\) is the sound intensity (W/m\({}^{2}\)), \(A\) is the sound propagation area in the lateral branch (m\({}^{2}\)), \(P_{\rm rms}^{2}\) is the effective sound pressure (Pa), \(\rho\) is the density of the fluid (kg/m\({}^{3}\)), \(c\) is the velocity of the depropagation of sound within the fluid (m/s\({}^{2}\)) and \(P_{\rm ref}\) is the reference acoustic power (watts). \[P_{a}=I\cdot A\ \ I=\frac{P_{\rm rms}^{2}}{\rho c}\ L_{W}=10\log_{10}\left( \frac{P_{a}}{P_{\rm ref}}\right) \tag{13}\]  
Regarding the boundary conditions of the CFD analysis, the model (k-e) was used with a standard wall function. Table 2 shows the velocities considered in this paper. The Strouhal number was considered based on Equation (14) [1]. \(S_{t}\) is defined in terms of the inner diameter branch of the stub pipe (d), the flow velocity of the main pipe (\(U\)), and the resonance frequency of the system (\(f_{n}\)).