Mixing of granular material in rotating cylinders with noncircular cross-sections

Abstract

We study, experimentally and theoretically, the mixing of monodisperse colored beads in rotating cylinders with different cross-sectional shapes (square, star, and a circle with two and four triangular wedges), operating in the continuous flow regime, to understand the role of cross-sectional shape on the mixing process. The evolution of the mixed state is quantified using two measures: the intensity of segregation and the centroid distance. The mixing rate index, defined as the specific rate of change of the intensity of segregation with time, is initially lower in all the noncircular cross-sections compared to a circle. It increases monotonically in noncircular cross-sections, whereas it is nearly constant in a circle. This corresponds to a faster than exponential decay of the intensity of segregation for noncircular cross-sections as compared to an exponential decay for a circular cross-section. The decay of centroid distance is oscillatory in a circle, whereas it is monotonic in noncircular cross-sections. A significant improvement in the mixing rate is obtained for noncircular cross-sections relative to circular cross-sections. The mixing rate is highest for the circle with four wedges; the mixing rates for square and star cross-sections are slightly lower. The circle with two wedges has the lowest mixing rate among the noncircular cross-sections. Experiments with smaller glass beads, in which the particle diffusivity is significantly smaller, yield a mixing rate very close to that for the larger particles. Mixing patterns obtained experimentally at short times are well predicted by a convective diffusion model that includes a time-periodic velocity field for the noncircular cross-sections. A quantitative comparison of model predictions with experiments in terms of the intensity of segregation and the centroid distance shows a reasonably good agreement for the different shapes and for the two particle sizes. A scaling analysis is presented to explain the insensitivity of the mixing rate to particle size despite a significant variation in diffusivity. Mixing patterns for a tracer blob and the mixing rates obtained for the different mixers are found to be related to the regions of the chaotic advection in computed Poincaré maps. For the systems studied, the mixing rate increases with an increase in the number of corners in the geometry, but is relatively unaffected by the ratio of the maximum to minimum diameter of the cross-section.