Mathematical Modeling of Particle Terminal Velocity for Improved Design of Clarifiers, Thickeners and Flotation Devices for Wastewater Treatment

Abstract

The prediction of the terminal velocity of a single spherical particle is essential to realize mathematical modeling useful for the design and adjustment of separators used in wastewater treatment. For non-spherical and non-single particles, terminal velocity can be traced back to that of single spheres using coefficients and Kynch’s theory, respectively. Because separation processes can involve small or large particles and can be carried out using gravity, as with clarifiers/thickeners, or by centrifugation in centrifuges where the acceleration can exceed 10,000× g, the Reynolds number of the particle can be highly variable, ranging from 0.1 to 200,000. The terminal velocity depends on the drag coefficient, which depends, in turn, on the Reynolds number containing the terminal velocity. Because of this, to find the terminal velocity formula, it is preferable to look first for a relationship between the drag coefficient and the Archimedes number which does not contain the terminal velocity. Formulas already exist expressing the relationship between the drag coefficient and the Archimedes number, from which the relationship between the terminal velocity and the Archimedes number may be derived. To improve the accuracy obtained by these formulas, a new relationship was developed in this study, using dimensional analysis, which is valid for Reynolds number values between 0.1 and 200,000. The resulting mean relative difference, compared to the experimental standard drag curve, was only 1.44%. This formula was developed using the logarithms of dimensionless numbers, and the unprecedented accuracy obtained with this method suggested that an equally accurate formula for the drag coefficient could also be obtained with respect to the Reynolds number. Again, the resulting level of accuracy was unprecedentedly high, with a mean relative difference of 1.77% for Reynolds number values between 0.1 and 200,000.