Frictional-kinetic models for steady cohesionless granular flow through a wedge-shaped hopper

Abstract

The present work attempts to combine a newly developed kinetic constitutive theory with an older frictional theory to model.the two-dimensional gravity flow of a cohesionless granular material such as sand through a wedge-shaped hopper. Frictional theories work best for slow flows and high bulk densities, when the constituent grains are in sustained contact with each other as they flow. These theories use the idea of plastic yield to describe the flow behaviour, and are based on the theories of plasticity in soil mechanics. On the other hand, kinetic theories have been developed for fast flows at low bulk densities, when the particles no longer maintain sustained contact with each other. Here, stress build-up is because of collisions between the grains as they flow. A characteristic quantity of the kinetic theories is 'grain temperature’ which is defined in a manner analogous to true temperature of a dense gas. It is proportional to the mean kinetic energy of velocity fluctuation of the grains, called the pseudo-thermal energy. Since interparticle collisions are inelastic, pseudo-thermal energy is not conserved. The particle velocity fluctuations can be sustained only by the presence of a mean flow of the ensemble, or with the help of vibrating boundaries. The present work is motivated by the fact that earlier frictional flow models used to describe hopper flow, fail to give a physically reasonable solution near the hopper exit. The solution is not continuous across the exit, and discharge rates predicted are invariably higher than the experimental values. The idea of modifying the constitutive equations to include kinetic effects is suggested by density measurements reported in the literature. They show that the material dilates as it flows down the hopper, assuming values of density near the exit below the loosest random packing of spheres. Ac ruch densities, the particles are so far apart that it is no longer reasonable to assume tha-. friction is the mechanism of stress build-up. The kinetic theories appear to fit in v/ell with the conditions prevailing near the hopper exit. Therefore, a combined frictionalkinetic theory is used here to model the flow in the upper regions o f the hopper, as well as near the exit. T h e equations are first tested on a hypothetical one-dimensional approxim ation called the sm ooth-wall, radial gravity problem. Here the assum ptions used are: (i) the m aterial is incompressible, (ii) gravity is directed radially towards the virtual vertex o f the hopper, (iii) the hopper walls are smooth, and (iv) particle-wall collisions are elastic and specular. A s a result of these assumptions, the governing partial differential equations are reduced to ordinary differential equations. W hen they are suitably nondimensionalised, the pseudo-thermal energy balance is found to be decoupled from the m om entum balance. Further, a sm all parameter e appears, which m ultiplies the kinetic stress term in the momentum balance, and the conduction term in the energy balance. T h e energy balance is first solved using the method o f matched asym ptotic expansions. T he tem perature profile so obtained is then used to solve the m om entum balance by a regular perturbation method. The results appear to justify the inclusion o f kinetic stresses in the model. T h is is because, particle fluctuation energy is found to increase as the material flows down the hopper, so that kinetic stresses dom inate in the region o f the exit. Further, the discharge rate predicted is lower than the purely frictional value. However, since the kinetic term s are m ultiplied by e, their effects are not felt quantitatively. Nevertheless, the approach is extended to the realistic hopper problem, in the hope that with the inclusion of compressibility and a two-dim ensional model, the kinetic effects w ill be large enough to overcome the problems with the purely frictional equations. For the two-dimensional model, the assumptions o f incom pressibility and radial gravity are dropped, but that o f smooth walls is retained. T he particle je t below the hopper is included in the analysis, and downstream conditions are specified far below the hopper exit. Here too, the kinetic stress term s in the m om entum balance are m ultiplied by a small parameter e '. T he governing equations and boundary conditions are highly non-linear, and resist solution. Therefore, they are sim plified further by quasi one-dinnensionalisation. Sinnple transverse profiles are assumed for the variables, and the equations are integrated along the hopper cross-section. For sm all hopper wall angle, it is convenient to assume no transverse variation for all variables except the horizontal com ponent o f velocity. T h e quasi one-dim ensional equations are solved in two regions. In the upper region o f the hopper, the frictional lim it o f the equations is solved as an initial-value problem. These equations break down when the density decreases to the value corresponding to the loosest packing o f spheres. T he frictionalkinetic equations are then solved below as a boundary-value problem. T h is non-linear problem is also very difficult to solve, and is therefore sim plified by linearisation. The solution finally obtained reveals some interesting features, but is not entirely satisfactory. T h e m ost encouraging result is that it is possible to obtain a continuous solution across the hopper exit that goes sm oothly from the frictional solution in the hopper to free-fall in the jet. The material dilates for a little distance below the hopper, but compacts later. T h is is not a serious drawback o f the solution, since com paction could have been prevented if the transverse kinetic stress gradients had been large enough to counter the converging inertial flow, and the particle trajectories had moved to vertical paths. Thus it is concluded that the inclusion o f kinetic effects in a frictional model helps to obtain a continuous solution across the hopper exit. However, in order to predict the experim entally observed dilation in the jet. it appears th a t either the model must be modified to include transverse gradients, or a superior kinetic theory must be employed, since the present one was developed for small mean field gradients.