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.