Theoretical basis for ultrasonic instruments

A mathematical model, Allegra-Hawley [227,228], prediets the attenuation of ultrasonic waves as a funetion of frequency for each particle size distribution and concentration. Some mechanical, thermodynamic and transport properties of both phases are needed. The relationship between the size, concentration and frequency is obtained from the solution of the 

Heat losses occur at the surface of each particle when a suspension is radiated with ultrasonics due to velocity differences between the viscous liquid and suspended particles and this results [232] in absorption of energy. 

A viscous term predominates when the ratio of wavelength to particle size is so high that particles do not follow the liquid movement. The viscous losses term can be derived from Stokes equation for the effect of viscosity on a spherical pendulum swinging in a viscous liquid. 

Proassist Herbst and Alba [233], in developing the Proassist, measured the attenuation of the sound at ten different frequencies from 0.5 MHz to 6 MHz to retrofit five points on the size distribution instead of only one as with the original Autometrics instrument. They discretized the resulting equation for narrow size fractions and used a mathematical programming technique to find the set of fractions and concentrations which gave the best fit. 

Ultraspec Alba reported on an extension of this technology and termed it the Ultraspec particle size analyzer [234]. The prototype laboratory instrument covered the size range 0.01 pm to 1,000 pm at a volume concentration range from 0.1 to 70 employing a continuous frequency band of 1 MHz to 2000 MHz. In conjunction with the DuPont company, the technique was extended to the measure of undiluted emulsions and an online system was developed. 

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