Steady incompressible flow of cohesionless granular materials through a wedge-shaped hopper: Frictional-kinetic solution to the smooth wall, radial gravity problem

Jyotsna, R. ; Rao, K. Kesava. (1991) Steady incompressible flow of cohesionless granular materials through a wedge-shaped hopper: Frictional-kinetic solution to the smooth wall, radial gravity problem Chemical Engineering Science, 46 (8). pp. 1951-1967. ISSN 0009-2509

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Official URL: http://linkinghub.elsevier.com/retrieve/pii/000925...

Related URL: http://dx.doi.org/10.1016/0009-2509(91)80156-S

Abstract

Hybrid frictional-kinetic equations are used to predict the velocity, grain temperature, and stress fields in hoppers. A suitable choice of dimensionless variables permits the pseudo-thermal energy balance to be decoupled from the momentum balance. These balances contain a small parameter, which is analogous to a reciprocal Reynolds number. Hence an approximate semi-analytical solution is constructed using perturbation methods. The energy balance is solved using the method of matched asymptotic expansions. The effect of heat conduction is confined to a very thin boundary layer near the exit, where it causes a marginal change in the temperature. Outside this layer, the temperature T increases rapidly as the radial coordinate r decreases. In particular, the conduction-free energy balance yields an asymptotic solution, valid for small values of r, of the form T ∝ r-4. There is a corresponding increase in the kinetic stresses, which attain their maximum values at the hopper exit. The momentum balance is solved by a regular perturbation method. The contribution of the kinetic stresses is important only in a small region near the exit, where the frictional stresses tend to zero. Therefore, the discharge rate is only about 2.3% lower than the frictional value, for typical parameter values. As in the frictional case, the discharge rate for deep hoppers is found to be independent of the head of material.

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