Terminal velocity formula for spheres in a viscous fluid

Various attempts have been made to develop a general expression for the terminal velocity of spheres in a viscous fluid (Stokes, Prandtl, Oseen, Rubey, etc.: see Bogardi, 1974 and Vanoni, 1975). All of these formulae show a lack of accuracy and/or are restrict ed to a relatively small range of Reynolds numbers. Since sieving is a time consuming method for determining particle size the interest in the use of settling tubes has increased as a mean of particle-size analysis. A measure of the size of an irregular shaped particle is the sedimentation diameter defined as the diameter of the sphere that has the same density and the same terminal settling velocity as the given particle in the same sedimentation fluid. It is therefore very convenient to have a general and accurate expression for the conversion from settling velocity to sedimentation diameter. In this note new formulae for the terminal velocity as well as the sedimentation diameter are derived having a high accuracy and a wide range of validity. A relatively simple analytical expression is derived for the terminal velocity of sphere that depends on the diameter, the density and the kinematic viscosity, due to an acting force. The accuracy of the derived formula is better than 2% for Reynolds numbers up to 2000