# Detection of Pneumatic Conveying by Acoustic Emissions
## 2 Theoretical Model  
Solid particle collisions with the probe generate transient elastic stress waves that propagate away from the point of impact. These waves can be detected by a piezoelectric transducer located a certain distance from the collision site.  
According to the Hertz contact theory, the impact frequency relates to the duration of the collision. [PERSON] [19] deduced the frequency of the resulting acoustic signal, assuming that the particle is spherical and that the impact is a completely elastic collision. Accordingly, the frequency can be approximated by:  
\[f=\frac{1}{\tau}=\frac{2}{5.7}\frac{v_{0}^{\frac{1}{5}}}{D^{\frac{3}{5}}d_{p} \rho_{s}^{\frac{2}{5}}} \tag{1}\]  
where \(\mathrm{D}=\frac{3}{4}\left(\frac{1-\sigma_{1}^{2}}{E_{1}}+\frac{1-\sigma_{2 }^{2}}{E_{2}}\right)\) is the elastic modulus parameter, and \(E\) and \(\sigma\) are the Young's modulus and the Poisson ratio, respectively; subscripts 1 and 2 refer to the material of the particle's bulk and its surface, respectively; \(\rho_{s}\), \(d_{p}\), and \(v_{0}\) represent the particle's density, radius, and approach velocity, respectively. With a certain particle velocity and known material properties, the particle size can be related to the signal frequency, and the particle size distribution can be described by the Rosin-Rammler function. From the equation, it is obvious that the larger the particle size, the smaller the corresponding frequency. Therefore, when the particles in pneumatic conveying collide with the probe or the tube wall, there will be different resulting frequency distributions due to different particle sizes.  
Based on the research of [PERSON] [20], [PERSON] [21] deduced the acoustic energy driven by contact between the particle and wall. The energy can be approximately by:  
\[E=\int_{0}^{t}\int_{0}^{S_{p}}2\eta mC_{p}\sum_{j=1}^{n}v_{0}^{3}x_{j}dAdt=2 \eta tmC_{p}S_{p}\sum_{j=1}^{n}v_{0}^{3}x_{j} \tag{2}\]  
where \(\eta\) is the transformation efficiency from collision pressure to acoustic pressure; \(t\) is the time interval; \(m\) is the mass of the particle; and \(C_{p}\) is the particle concentration dispersed within the air volume. \(S_{p}\) and \(x_{j}\) represent the considered interaction area where dispersed particles collide with the wall and the percentage of particles impacting it at an angle of \(\alpha_{j}\), respectively.  
In a real system, the AE signal traveling from sensor to the computer is dependent not only upon this theoretical analysis, but also upon wave propagation effects and the system's response to surface vibrations. The complete AE signal can be expressed as [22]:  
\[V(t)=S(t)*G(t)*R(t) \tag{3}\]  
where \(V(t)\) represents the measured AE voltage signal, and \(S(t)\), \(G(t)\), and \(R(t)\) are the original acoustic signal, wave propagation function, and the system response, respectively, where the symbol * denotes convolution. The wave propagation medium and the system response function are non-linear and time variant; therefore, it is difficult to develop a direct relationship between the AE signal and pneumatic conveying without signal processing.