The maximum thickness of upper shear layers of granular materials in rotating cylinders

Published data from Henein et al. (Metallurgical Transactions 14B (1983a) 191-206), Woodle and Munro (Powder Technology 76 (1993) 241-245), Boateng and Barr (Journal of Fluid Mechanics 330 (1997) 233-249), Van Puyvelde et al. (Transactions of the Institute of Chemical Engineers 78A (2000) 643-650), and Felix et al. (Powder Technology 128 (2002) 314-319) on the maximum thickness of shear layers at the upper surface of a freely flowing granular material being rotated in a cylindrical drum are modeled and analyzed. A theory is developed which suggests that all distances should be scaled by x_m, or half the length of the shear layer, which should remove most of the explicit dependence on the filled fraction. The rolling regime is predicted to contain two extreme regimes. The dispersive (inertial) regime occurs when the parameter Fr_s(x_m/σ)^4/3 is small (large), and is characterized by the dominance of dispersive (inertial) effects. Here Fr_s is a particle-dependent Froude number scaled by x_m, and σ is the particle diameter. An author-dependent and particle-dependent parameter λ is introduced to account for the different definitions and particle types used. Regression analysis shows that most of the data sets above (for the rolling regime) are approximately described by the dispersive regime. Our theory predicts that shear layers in the rolling regime should be almost identical for fill fractions symmetric about the half-filled level. All of the data analyzed satisfy essentially the same scale-up relationship, and does not support the idea that the maximum shear layer thickness should be about ten particle diameters.