Theoretical analysis of ultrasonic vibration spectra from multiple particle-plate impacts

Many industrial processes involve particles in a carrier fluid, and it is often of interest to monitor the size of these particles noninvasively. The aim of this paper is to develop a theoretical model of multiple particle-wall impact vibrations that can be used to recover the particle size from experimental data. These vibrations have been measured by an ultrasonic transducer attached to the exterior of a vessel containing a stirred-particle-laden fluid. A linear systems model is derived for the response of the piezoelectric ultrasonic transducer, which has a single matching layer. The acceleration power spectrum of these vibrations has been shown experimentally to contain information on the size of the impacting particle. In particular, the frequency of the main spectral lobe is inversely proportional to the particle size. We present a theoretical model that agrees with this empirically observed phenomenon. The theoretical model is then used to simulate multiple particle-wall impacts, with each particle impacting at a randomly chosen location. A set of theoretical vibration spectra arising from multiple particle-wall impacts are integrated and compared with the experimental data. The ability of this approach to distinguish between different particle sizes is clearly shown.