K. Kesava Rao

An introduction to granular flow

Preface. 1. Introduction 2. Theory for slow plane flow 3. Flow through hoppers 4. Flow through wedge-shaped bunkers 5. Theory for slow three-dimensional flow 6. Flow through axisymmetric hoppers and bunkers 7. Theory for rapid flow of smooth, inelastic particles 8. Analysis of rapid flow in simple geometries 9. Theory for rapid flow of rough, inelastic particles 10. Hybrid theories A. Operations with vectors and tensors B. The stress tensor C. Hyperbolic partial differential equations of first order D. Jump balances E. Discontinuous solutions of hyperbolic equations F. Proof of the coaxiality condition G. Material frame-indifference H. The evaluation of some integrals I. Linear stability J. Pseudoscalars, vectors, and tensors K. Answers to selected problems References.

A frictional Cosserat model for the slow shearing of granular materials

A rigid-plastic Cosserat model for slow frictional flow of granular materials, proposed by us in an earlier paper, has been used to analyse plane and cylindrical Couette flow. In this model, the hydrodynamic fields of a classical continuum are supplemented by the couple stress and the intrinsic angular velocity fields. The balance of angular momentum, which is satisfied implicitly in a classical continuum, must be enforced in a Cosserat continuum. As a result, the stress tensor could be asymmetric, and the angular velocity of a material point may differ from half the local vorticity. An important consequence of treating the granular medium as a Cosserat continuum is that it incorporates a material length scale in the model, which is absent in frictional models based on a classical continuum. Further, the Cosserat model allows determination of the velocity fields uniquely in viscometric flows, in contrast to classical frictional models. Experiments on viscometric flows of dense, slowly deforming granular materials indicate that shear is confined to a narrow region, usually a few grain diameters thick, while the remaining material is largely undeformed. This feature is captured by the present model, and the velocity profile predicted for cylindrical Couette flow is in good agreement with reported data. When the walls of the Couette cell are smoother than the granular material, the model predicts that the shear layer thickness is independent of the Couette gap H when the latter is large compared to the grain diameter d p . When the walls are of the same roughness as the granular material, the model predicts that the shear layer thickness varies as (H/d p ) 1/3 (in the limit H/d p [dbl greater-than sign] 1) for plane shear under gravity and cylindrical Couette flow.

Fully developed flow of coarse granular materials through a vertical channel

Some recent studies have proposed a simple manner of combining the frictional contributions to the stress with the kinetic theory based models for granular flow. In this paper, kinetic and frictional-kinetic models are used to study the gravity flow of granular materials through a vertical channel. The results show that as the mean density increases, the velocity profile predicted by the kinetic model shows a region of plug how near the centre, and a shear layer adjacent to the wall. At high densities ( approximate to 0.64), the thickness of the shear layer is a few particle diameters. The centerline velocity and the mass flow rate exhibit maxima as increases. The maxima are due to the opposing effects of the weight of the material and the shear viscosity. For the parameter values used here, the grain temperature profile exhibits a maximum near the wall and, correspondingly, the density profile exhibits a minimum. Predictions of the frictional-kinetic model are qualitatively similar, except that (i) the profiles are discontinuous at the interface between the plug and the shear layer, and (ii) at high densities, the thickness of the shear layer is much less than in the kinetic case. When mean densities are matched with measured values, velocity profiles (scaled by the centerline velocities) predicted by both models are found to be in fair agreement with the experimental data. In contrast, the absolute centerline velocities are higher than the reported values by a factor of 10-50. This discrepancy cannot be attributed to air-drag, as calculations show that the latter has a negligible effect on the velocity profiles. When either the mass flow rates or the centerline velocities are matched with the experimental data, the kinetic profiles fit the data fairly well. This is a puzzling result, as at high densities, many of the assumptions of the kinetic model are unlikely to hold. Though the frictional-kinetic model is expected to be more realistic, the shear layer is much thinner than that observed. Both models underestimate the shear stresses at the channel walls by a factor of 1.5-3, and overestimate the normal stresses by a similar factor.

A frictional Cosserat model for the flow of granular materials through a vertical channel

SummaryA rigid-plastic Cosserat model has been used to study dense, fully developed flow of granular materials through a vertical channel. Frictional models based on the classical continuum do not predict the occurrence of shear layers, in contrast to experimental observations. This feature has been attributed to the absence of a material length scale in their constitutive equations. The present model incorporates such a material length scale by treating the granular material as a Cosserat continuum. Thus, localized couple stresses exist, and the stress tensor is asymmetric. The velocity profiles predicted by the model are in close agreement with available experimental data. The predicted dependence of the shear layer thickness on the width of the channel is in reasonable agreement with data. In the limit of small ε (ratio of the particle diameter to the half-width of the channel), the model predicts that the shear layer thickness scaled by the particle diameter grows as ɛ-1/3.

Steady compressible flow of granular materials through a wedge-shaped hopper: The smooth wall, radial gravity problem

A continuum model based on the critical state theory of soil mechanics is used to generate stress and density profiles, and to compute discharge velocities for the plane flow of cohesionless materials. Two types of yield loci are employed, namely, a yield locus with a corner, and a smooth yield locus. The yield locus with a corner leads to computational difficulties. For the smooth yield locus, 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 on this surface. This insensitivity arises from the existence of asymptotic stress and density fields, to which the solution tends to converge on moving down the hopper. Numerical and approximate analytical solutions are obtained for these fields and the latter is used to derive an expression for the discharge velocity. This relation predicts discharge velocities to within 13% of the exact (numerical) values. While the assumption of incompressibility has been frequently used in the literature, it is shown here that in some cases, this leads to discharge velocities which are significantly higher than those obtained by the incorporation of density variation.

A frictional-kinetic model for the flow of granular materials through a wedge-shaped hopper

A quasi-one-dimensional model is used to examine the steady flow of granular materials through a wedge-shaped hopper with smooth, steep walls. Hybrid frictional-kinetic equations are used in an attempt to overcome some of the difficulties faced by earlier works, which were based on frictional equations. Owing to computational difficulties, two different solution procedures are used: (i) in the upper region, where frictional effects dominate, and (ii) the lower region which includes the exit slot and a part of the particle jet below the hopper, where kinetic and frictional effects are expected to be comparable. The equations are integrated numerically in (i). In (ii), they are linearized, and a semi-analytical solution is constructed. In contrast to the works of Kaza & Jackson (1982a) and Prakash & Rao (1991), the density varies smoothly across the exit slot. The density profile is qualitatively similar to the data of Fickie, Mehrabi & Jackson (1989). However, the range of density variation is much smaller than that observed. Owing to the approximations used, and perhaps also to the form of the kinetic constitutive equations, kinetic effects are dominated by frictional effects, except close to the downstream boundary.

STEADY COMPRESSIBLE FLOW OF COHESIONLESS GRANULAR MATERIALS THROUGH A WEDGE-SHAPED BUNKER

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 material in a mass flow bunker. The bin–hopper transition region is idealized as a shock across which all the variables change discontinuously. Comparison with the work of Michalowski (1987) shows that his experimentally determined rupture layer lies between his prediction and that of the present theory. However, it resembles the former more closely. The conventional condition involving a traction-free surface at the hopper exit is abandoned in favour of an exit shock below which the material falls vertically with zero frictional stress. The basic equations, which are not classifiable under any of the standard types, require excessive computational time. This problem is alleviated by the introduction of the Mohr–Coulomb approximation (MCA). The stress, density, and velocity profiles obtained by integration of the MCA converge to asymptotic fields on moving down the hopper. Expressions for these fields are derived by a perturbation method. Computational difficulties are encountered for bunkers with wall angles θw [gt-or-equal, slanted] 15° these are overcome by altering the initial conditions. Predicted discharge rates lie significantly below the measured values of Nguyen et al. (1980), ranging from 38% at θw = 15° to 59% at θw = 32°. The poor prediction appears to be largely due to the exit condition used here. Paradoxically, incompressible discharge rates lie closer to the measured values. An approximate semi-analytical expression for the discharge rate is obtained, which predicts values within 9% of the exact (numerical) ones in the compressible case, and 11% in the incompressible case. The approximate analysis also suggests that inclusion of density variation decreases the discharge rate. This is borne out by the exact (numerical) results – for the parameter values investigated, the compressible discharge rate is about 10% lower than the incompressible value. A preliminary comparison of the predicted density profiles with the measurements of Fickie et al. (1989) shows that the material within the hopper dilates more strongly than predicted. Surprisingly, just below the exit slot, there is good agreement between theory and experiment.

A low-cost device for the estimation of fluoride in drinking water

Fluorosis is endemic in many regions of the world due to high (>1.5 mg I-1) levels of fluoride in groundwater. This problem is aggravated by the lack of a simple and inexpensive test for fluoride in drinking water. We discuss the design and testing of a low-cost LED-based colorimeter for estimating the fluoride concentration in water. The device was calibrated with the use of the cerium-alizarin complexone (Ce-AC) and the sodium 2-(parasulfophenylazo)-1,8-dihydroxy-3,6-naphthalene disulfonate (SPADNS) methods, which show color changes in the visible region on addition of water containing fluoride. Raw water samples collected from different parts of India were analyzed for fluoride. The fluoride concentration, as estimated by using a fluoride electrode, was in the range 0.7-1.20 mg I-1. The results obtained by the colorimeter showed good agreement with the electrode method for most of the samples. The cost of chemicals per test was

A Model For Heat Transfer In A Honey Bee Swarm

0.14 for the Ce-AC method and

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

0.018 for the SPADNS method. The cost of the colorimeter was about