Scaling surface velocities in rotating cylinders as a function of vessel radius, rotation rate, and particle size

In industrial practice, scale-up of granular flows in tumbling devices has been largely attempted using one of two parameters, either the vessel tangential speed (ω) or the Froude (Fr) number. In this communication, we measure surface velocities of 1.6-mm particles in half-filled rotating cylinders and find that neither ω nor Fr accurately scales changes in particle velocity with changes in vessel rotation rate, diameter, or particle size. New non-dimensional scaling criteria using a simplified model produce agreement in both the magnitude and shape of the velocity profiles. A strong dependence on both rotation rate and vessel radius is found and a small but measurable effect of particle size is also demonstrated. We find that there are two different scaling regimes that depend on whether or not the cascading layer reaches a symmetric equilibrium state. At lower rotation rates, Ω, cascading particles can reach equilibrium, and granular surface speeds scale as Ω2/3; at higher rotation rates, particle velocities scale as Ω1/2. New effects of relative particle size with respect to the cylinder diameter are also reported.