Modeling of fracture surface of the quasi solid-body zone of motion of the granular fill in a rotating chamber

Abstract

Large-tonnage processing of granular materials is carried out in drum-type machines. Utmost simplicity of the design solutions of such equipment is paradoxically combined with the behavior of machined medium that is extremely complex to describe. The efficiency of working processes in drum machines is determined by the mode of motion of the fill of a rotating chamber. The character of this mode is predetermined by the position of a transition surface of quasi solid-body motion zone to the zone of fall and subsequent shift, which determines dynamic activity of the filler’s movable part. A traditional hypothesis on the implementation of processing granular materials in the drum machines is based on the concept of a separate element of the fill in a rotating chamber isolated from the surrounding medium. According to this hypothesis, the fracture surface of a solid-body zone is of cylindrical shape with a diameter that depends only on the rotation speed. Therefore, performing numerical calculations is associated with insurmountable computational difficulties. Obtaining experimental data is hampered due to the complexity of hardware control. That is why results, obtained recently, approach real motion modes of the examined medium only by qualitative characteristics. We constructed an analytical model of behavior of a granular fill in the transition from the circular, during ascent, to the quasi-parabolic, while non-free falling, trajectory of motion in the cross-section of a cylindrical chamber that rotates around a horizontal axis. Equation of a slip line coordinates in parametric form is received. They make it possible to approximately determine the shape and position of the transition surface depending on the kinematic, geometric and rheological parameters of the system. A granular fill is considered as a whole medium with parameters that are averaged by volume. We employed the plastic rheological model. Based on the modeling performed, we formalized a stress field in the mass of fill in the cross-section of a rotating chamber by using a system of differential equations of flat boundary equilibrium of a granular medium. A condition for the boundary equilibrium is obtained. It is demonstrated that disequilibrium is accompanied by sliding layers of the fill. We defined position of the slip lines in stress field. It was found that the motion zone transition is accompanied by the destruction of a quasi solid-body zone at the border that is a slip line in the motion picture of plastic medium in the cross-section of a chamber. It is established that the position of transition limit depends not only on the angular velocity. It is a function of radius and degree of filling the chamber, specific weight, angle of friction and ascent angle of the fill.