Particle sizing using passive ultrasonic measurement of particle–wall impact vibrations

In continuously stirred reactor vessels the non-invasive recovery of the particle size could be used to monitor the reaction process. Experimental and numerical investigations have shown empirically that the frequency of the peak vibration response arising from the particle–wall impact is inversely proportional to the particle size. The passive monitoring of these impact vibrations using an ultrasonic transducer has the potential therefore of non-invasively recovering the particle size. However, the vessel geometry, fluid loading, variable impact position and velocity, stirrer and transducer effects, and noise levels make this problem very complex. There are a large number of system parameters and this makes empirical derivations of cause and effects extremely difficult. The first objective of this paper is to derive an analytical expression for the vibrations arising from a spherical particle impacting a circular plate. Using a series expansion in terms of the plate loss parameter, an expression for the frequency of the peak pressure in terms of the system parameters is derived. In particular, its explicit dependency on the impacting particle size and the impact velocity is found. The inverse problem of recovering the particle size from the experimental data is then investigated. A set of experiments are described where the impact vibrations are recorded using an ultrasonic transducer attached to the rear of a thin plate. The results show that it is possible to recover the particle size using this approach. Data from a second set of experiments, involving multi-particle impacts with a vessel wall in a continuously stirred reactor, are then used. The inverse problem of recovering the particle size from the vibration spectrum was then investigated with encouraging results.