Steady Compressible Plane Flow Of Cohesionless Granular Materials in Hoppers and Bunkers

Abstract

A continuum model based on the critical state theory of soil mechanics is used to generate stress, density, and velocity profiles, and to compute discharge rates for the flow of granular materials in wedge?shaped hoppers and bunkers. The importance of compressibility effects in such flows is also examined. The one?dimensional case of a hopper with smooth walls and gravity directed radially toward its apex is considered first. The results are found to be relatively insensitive to the shape of the yield locus, the location of the upper traction?free surface, and the density specified at this surface. This insensitivity arises from the existence of asymptotic stress and density fields toward which the solution converges as the material flows downward through the hopper. Approximate expressions for these asymptotic fields are derived, and discharge rates estimated using these expressions are within 13?% of the exact (numerical) values. It is also shown that assuming incompressibility leads to discharge rates that are significantly higher than those obtained when density variations are incorporated. The two?dimensional case of mass flow in a bunker is considered next. The bin–hopper transition region is idealized as a shock across which all variables change discontinuously. Comparison with the work of Michalowski (1987) shows that the experimentally determined rupture layer lies between the predictions of the present theory and those of Michalowski, though it more closely resembles the latter. The conventional condition of a traction?free surface at the hopper exit is replaced by an exit shock, below which the material falls vertically with zero frictional stress. The governing equations are not classifiable under standard types and require excessive computational effort. This difficulty is reduced by introducing the Mohr–Coulomb Approximation (MCA). Stress, density, and velocity profiles obtained by integrating the MCA converge to asymptotic fields as the material moves downward. These fields are derived using a perturbation method. Computational difficulties arise for bunkers with wall angles greater than 15°, which are overcome by adjusting the initial conditions. Predicted discharge rates are significantly lower than the measured values reported by Nguyen et?al. (1980), ranging from 38?% at a wall angle of 15° to 59?% at 32°. This discrepancy is largely attributed to the exit condition used in the model. Interestingly, incompressible discharge rates are closer to the measured values. An approximate semi?analytical expression for the discharge rate is derived, predicting values within 9?% of the exact numerical results for the compressible case and within 11?% for the incompressible case. Both the approximate and exact analyses confirm that incorporating density variation reduces the discharge rate.