New Relation for the Computation of Settling Velocities and Diameters of Spheres

Abstract

Single relations that can be used to calculate both the terminal settling velocities of spheres and the equivalent diameter of particles from their settling velocities are developed. The literature going back to Newton is reviewed and the relations developed tabulated. It is shown how the standard drag curve has developed into the dimensionless velocity versus dimensionless diameter curve. No relations that cover the full range that can conveniently be used for both velocity and diameter calculation were found, however a relation by Concha and Almendra covers most of the range.

The standard drag curve data are constructed by utilizing 535 data points available in the literature in a Reynolds number range of 2.4 × 10⁻⁵ to 2 × 10⁵. The settling velocities are corrected for experiments in finite width columns that do not satisfy the infinite medium dimensions. The data are converted to the dimensionless diameter and dimensionless velocity terms, which is more convenient for calculation purposes.

The data are analyzed using piecewise cubic functions. Data from sources with excessive scatter and a few outliers are removed leaving 443 data points. The resulting piecewise cubic can be used to obtain velocity from diameter or diameter from velocity. To give an algebraic expression a hyperbola is fitted to the data giving an expression that can be solved to give explicit relations for both dimensionless velocity and dimensionless diameter. This provides an accuracy that compares well with expressions given in the literature.

Keywords:

• fluid mechanics
• hydrodynamics
• multiphase flow
• sedimentation
• settling velocity

Acknowledgments

The authors deeply appreciate the invaluable efforts of librarians of The University of Queensland in supplying historical documents used in this study. The authors would like to acknowledge Prof. Alban J. Lynch for his continuous support and encouragement.

Notes

¹“The same things being supposed, I say, that the greater parts of the systems are resisted in a ratio compounded of the duplicate ratio of their velocities, and the duplicate ratio of their diameters, and the simple ratio of the density of the parts of the systems” (Newton 1726).

^a Equation re-arranged to fit template.

^b Quoted in Schiller and Naumann (1933).

^a Converted by Brown and Lawler (2003).

^b Converted by Turton and Clark (1987).

^a D < 2400 and R _e  < 2 × 10⁵.

^b 0.6 < D < 2400 and 10⁻² < R _e  < 2 × 10⁵.

^c D < 300 and R _e  < 2 × 10⁴.