Transport property measurements in sheared granular flows

Experiments were performed in a shear cell device under four different solid fractions. The glass spheres with a mean diameter of 3 mm were used as granular materials. The motions of the granular materials were recorded by a high-speed camera. By using image processing technology and a particle tracking method, the average and fluctuation velocities in the streamwise and the transverse directions could be successfully measured and analyzed. Three bi-directional stress gages were used to measure the normal and shear stresses along the upper boundary. The effective viscosity of the granular material flow can be calculated. By tracking the movements of particles continually, the curves of the mean-square diffusive displacements versus time were plotted and were used to determine the self-diffusion coefficients from the slopes of the curves. The fluctuations and the self-diffusion coefficients in the streamwise direction were much higher than those in the transverse direction. The fluctuations were found to increase with the solid fraction, but the diffusion coefficients were greater in a more dilute flow system.     

A mathematical model for dilating, non-cohesive granular flows in steep-walled hoppers

A new variable-density plastic flow model is developed, in which the Drucker Prager yield condition holds identically, but the corresponding flow condition contains the time derivative of density (or the divergence of mass flux), in order to satisfy mass conservation. This “softening” model is applied to the steady radial flow of a cohesionless granular material from steep-walled wedge and conical hopper. Density is assumed to vary with pressure. The variation of density within the hopper is shown to decrease the mass discharge rate, relative to the incompressible model, by a similar amount to the fractional reduction in voidage about the orifice. The predicted mass discharge decreased with increasing internal friction angle. This paper assumed that the inclination of the stagnant region in hopper flow is described by regression curves fitted to data from Brown and Richards. Approximate agreement between the theory of this paper and voidage measurements by Fickie et al. was obtained. Approximate agreement was also obtained with the published mass discharge rates of Nedderman and Beverloo for wedge and conical hoppers, respectively, and our results were insensitive to variations in internal angles of friction between about 25° and 35°. The steady equations considered here can only be satisfied approximately, supporting observations that granular flows are intrinsically transient.     

Simulation of dense granular flows: Comparison with experiments

A comparison of the predictions of a rheological model that we recently developed with experimental results of stress and flow profiles in a pilot scale silo is presented in this work. Experiments were performed to collect information on the flow field by means of a tracer method and on wall normal stresses at several different positions along the vessel. The silo (2.5 m high, 0.5 m wide) had the possibility of inserting internal devices; the model was first validated on data without internals and then used to predict the profiles for the case with them. Both stress and flow profiles with and without internals agree with the experimental results within the experimental error that locally could be rather significant due to the difficulty of large scale experiments with granular materials.     

Effects of particle size distribution in a three-dimensional micropolar continuum model of granular media

A particle size distribution is incorporated into a three-dimensional homogenisation scheme, devised on the scale of a particle and its immediate (or first ring) of neighbours. Based on this scheme, micropolar continuum models for polydisperse, dry, and densely packed granular assemblies of spherical particles undergoing quasi-static deformation are developed for various particle size distributions. Three different cases are considered: (1) a monodisperse assembly, (2) a defect particle in an otherwise monodisperse assembly, and (3) an assembly of a given particle size distribution. In Case 1, an additional dependence on particle radius is found in 3D systems, compared with previous 2D constitutive laws. In Case 2, it is found that a small (large) particle in an otherwise monodisperse system increases (decreases) the stress compared to a purely monodisperse assembly, but the couple stress may increase or decrease depending on the relative size of the rolling resistance compared with the tangential stiffness coefficients. On the other hand, if the defect particle is substantially smaller or larger than the monodisperse particle size, the stress and couple stress are always increased. In Case 3, three different distributions are examined, i.e. square, normal and a lognormal distribution. For Cases 2 and 3, both the stress and the couple stress increased with the degree of dispersity, from the lower bound value corresponding to the monodisperse system considered in Case 1. Finally, the paper highlights areas that will need to be addressed to enable the future advancement of micromechanical continuum models.     

Wet granular materials in sheared flows

The transport properties of wet granular materials in a shear cell apparatus have been studied. If the particles are wet, the flow becomes more viscous forming liquid bridges between particles. The dynamic liquid bridge forces are considered as the cohesive forces between particles to restrict their movements. The cohesive forces make the particles stick tighter with each other and hamper the movement of particles. The mixing and transport properties are influenced seriously by the amount of moisture added in the flow. This paper discusses a series of experiments performed in a shear cell device with five different moisture contents using 3-mm glass spheres as the granular materials. The motion of granular materials was recorded by a high-speed camera. Using the image processing technology and particle tracking method, the average and fluctuation velocities in the streamwise and transverse directions could be measured. The self-diffusion coefficient could be found from the history of the particle displacements. The self-diffusion coefficients and fluctuations in the streamwise direction were much larger than those in the transverse direction. Three bi-directional stress gages were installed to the upper wall to measure the normal and shear stresses of the granular materials along the upper wall. For wetter granular material flows, the fluctuation velocities and the self-diffusion coefficients were smaller.     

Self-fluidization of subaerial rapid granular flows

The fast motion of gravity currents of granular solids is studied with a focus on the dynamical structure of the frontal zone. The front of the current is “immobilized” and observed in a fixed frame of reference by letting the current flow inside a rotary drum, big enough to make curvature effects negligible. The study addresses the motion of beds made of particles of different size and density, corresponding to different values of the incipient fluidization velocity. The establishment of a variety of flow regimes, including intermittent avalanching, periodic “plunging breaking” and permanent fluidization of the granular solids in the frontal zone has been recorded. Flow regimes have been related to flow conditions and to the nature of the granular solids with an attempt to define a criterion for the self-fluidization of the current. Results suggest that such a criterion should include the canonical Froude number, determining the onset of front instabilities, and the ratio of the incipient fluidization velocity of the bed solids to the velocity of the current. The relative importance of the establishment of a purely “granular liquid” state versus fluidization due to gas entrainment is addressed and discussed with a focus on the effects on solids flowability.     

Application of incomplete similarity theory for estimating maximum shear layer thickness of granular flows in rotating drums

Granular flow in a rotating drum provides a convenient system for investigating mixing, segregation and general properties of granular materials. This study is concerned with the maximum thickness of the flowing surface layer observed at the rolling regime. A scaling relation is first derived with the consideration of incomplete similarity associated with the drum-particle size ratio and the Froude number. Calibration is then carried out with published laboratory data, which were collected for the case of rotating drums half-filled with glass beads. The scaling relation is also compared with other kinds of datasets, showing good agreement and possibilities of the proposed approach to be further extended to more complex cases.     

Polydisperse granular flows in a bladed mixer: Experiments and simulations of cohesionless spheres

The flow and segregation of polydisperse, spherical particle mixtures in a bladed mixer was investigated using experimental and computational techniques. Discrete element simulations were able to reproduce the qualitative segregation profiles and surface velocities observed experimentally. For a binary system with a 2:1 size ratio, segregation by size occurs due to a sieving mechanism. Segregation in the binary system is fast, with a fully segregated system observed after just 5 revolutions. However, the numerical simulations showed that the extent of segregation in the bladed mixer can be reduced by introducing intermediate particle sizes in between the smallest and the largest particles. Addition of intermediate particle sizes increases convective and diffusive particle motion promoting a mixing mechanism that reduces segregation via the sieving mechanism. Void fraction within the bladed mixer increases as the degree of polydispersity is increased allowing the particles to move more freely throughout the particle bed. Higher void fractions also increase the ability of large particles to penetrate deeper into the particle bed. Normal and shear stresses are also affected by particle size distributions, with lower average values obtained for the system with the largest number of particle species. Differences in the amount of stress generated by each particle species were observed. However, the difference in stresses is reduced as the number of particle species in the system is increased.     

Shock waves in granular materials: Discrete and continuum comparisons

Small amplitude compression and shock waves in granular materials were examined from the point of view of an analytical model and discrete element simulations. The barotropic behaviour of granular materials was discussed in terms of the mechanisms behind the formation of shock fronts. This discussion leads to the development of a one dimensional continuum model which was used together with the method of characteristics to describe the nature of shock waves. The model provides a relationship between the states of the material on either side of a shock and an equation that defines the velocity of shock waves.Discrete Element Modelling was used to demonstrate the shock forming process in granular materials and to confirm the predictions of the analytical model. During this process it is demonstrated that an assembly based on a linear contact model does not show any barotropy and consequently cannot describe dynamics of granular materials in terms of shock forming mechanisms. This observation may have important consequences in the application of the linear contact model to dynamic systems. The Hertz-Mindlin contact model did not suffer from this issue and was able to demonstrate both barotropy and shock formation.The DEM assemblies were characterised in terms of the material properties of the analytical model which allowed direct comparisons to be made. In all respects the analytical model performed well, predicting the change in state of the material and the shock wave speed with a good deal of accuracy.     

Contour-variable model of constitutive equations for polymer melts

Based on a modified expression of the rate of the convective constraint release, we present a new contour-variable model of constitutive equations in which the non-uniform segmental stretch and the non-Gaussian chain statistical treatment of the single chain are considered to describe the polymer chain dynamics and the rheological behavior of an entangled system composed of linear polymer chains. The constitutive equations are solved numerically in the cases of steady shear and transient start-up of steady shear. The results indicate that the orientation and stretch, as well as the tube survival probability, have strong dependence on the chain contour variable, especially in the high-shear-rate region. However, the inclusion of the non-uniform features in the constitutive models has little modification comparing with the uniform models in determining the rheological properties both qualitatively and quantitatively.     

Vane rheology of cohesionless glass beads

The rheology of a single coarse granular powder has been studied with shear vane rotational viscometry. The torque required to maintain constant rotation of a vane tool in a loose bed of glass beads (with a mean particle size of 203 μm) is measured as a function of vane immersion depth and rotational speed. The resulting torque profiles exhibit both Coulombic behavior at low rotational rates and fluid-like, collisional behavior at high rotational rates. Analyzing vane shaft and end effects allows the flow dynamics at the cylindrical and top and bottom disk surfaces of vane rotation to be determined. Disk surfaces show a uniform torque profile consistent with Coulombic friction over most of the rotational rates studied. In contrast, cylindrical surfaces show both frictional and collisional torque contributions, with significant dynamic torque increases at deep immersion depths and fast vane rotation.     

Stresses of cohesionless granular flows in liquids

We construct constitutive relations for friction stress, slide stress and collision stress of cohesionless granular flows. We propose that the friction stress and slide stress could be neglected when τ_c/τ_e < 0.02, where τ_c represents the interval time experienced by a given particle between two consecutive binary collisions and τ_e the particle relaxation time. We applied these constitutive relations to a simulation of solid-liquid flow in an inclined chute and found the simulation results agree with experimental data.     

The maximum thickness of upper shear layers of granular materials in rotating cylinders

Published data from Henein et al. (Metallurgical Transactions 14B (1983a) 191-206), Woodle and Munro (Powder Technology 76 (1993) 241-245), Boateng and Barr (Journal of Fluid Mechanics 330 (1997) 233-249), Van Puyvelde et al. (Transactions of the Institute of Chemical Engineers 78A (2000) 643-650), and Felix et al. (Powder Technology 128 (2002) 314-319) on the maximum thickness of shear layers at the upper surface of a freely flowing granular material being rotated in a cylindrical drum are modeled and analyzed. A theory is developed which suggests that all distances should be scaled by x_m, or half the length of the shear layer, which should remove most of the explicit dependence on the filled fraction. The rolling regime is predicted to contain two extreme regimes. The dispersive (inertial) regime occurs when the parameter Fr_s(x_m/σ)^4/3 is small (large), and is characterized by the dominance of dispersive (inertial) effects. Here Fr_s is a particle-dependent Froude number scaled by x_m, and σ is the particle diameter. An author-dependent and particle-dependent parameter λ is introduced to account for the different definitions and particle types used. Regression analysis shows that most of the data sets above (for the rolling regime) are approximately described by the dispersive regime. Our theory predicts that shear layers in the rolling regime should be almost identical for fill fractions symmetric about the half-filled level. All of the data analyzed satisfy essentially the same scale-up relationship, and does not support the idea that the maximum shear layer thickness should be about ten particle diameters.     

Conduction and dissipation in the shearing flow of granular materials modeled as non-Newtonian fluids

After providing a brief review of the constitutive modeling of the stress tensor for granular materials using non-Newtonian fluid models, we study the flow between two horizontal flat plates. It is assumed that the granular media behaves as a non-Newtonian fluid (of the Reiner-Rivlin type); we use the constitutive relation derived by Rajagopal and Massoudi [Rajagopal, K. R. and M. Massoudi, “A Method for measuring material moduli of granular materials: flow in an orthogonal rheometer,” Topical Report, DOE/PETC/TR-90/3, 1990] which can predict the normal stress differences. The lower plate is fixed and heated, and the upper plate (which is at a lower temperature than the lower plate) is set into motion with a constant velocity. The steady fully developed flow and the heat transfer equations are made dimensionless and are solved numerically; the effects of different dimensionless numbers and viscous dissipation are discussed.     

A rate-dependent constitutive model for brittle granular materials based on breakage mechanics

Modeling the rate-dependent mechanical behavior of brittle granular materials is of interest to defense applications, civil and mining engineering, geology, and geophysics. In particular, granulated ceramics in armor systems play a significant role in the overall dynamic material response of ceramics, particularly in their penetration resistance. This paper presents a rate-dependent constitutive model for brittle granular materials based on a recent reformulation of breakage mechanics theory. The rate-dependency is introduced via the overstress theory of viscoplasticity. The proposed formulation incorporates the effects of relative density and particle grading on strength and porous compaction/dilation, and is capable of tracking their evolution. The model is devised with internal variables linked to underlying dissipative micromechanisms including configurational reorganization, particle breakage and frictional dissipation. A strategy for calibrating model parameters and required experiments are described. The impact of loading rate on shear strength and grading evolution are explored through a sensitivity analysis. The presented model is capable of capturing several key features of the experimentally observed behavior of brittle granular materials including stress-, rate- and density-dependent stress-strain and volume change responses, the competition between dilation and breakage-induced compaction, the evolving particle grading due to particle breakage, and the evolution toward a critical (steady) state under shearing. A possible application of this micromechanics-inspired modeling framework involves integrating it into rate-dependent models for ceramics to assist in improving the impact performance of next-generation ceramics.     

On the relative importance of the kinetic and frictional contributions to granular motion in an annular Couette flow

A quantitative assessment is made on the relative importance of the kinetic and frictional contributions to the motion of dry granular materials under shear in an annular Couette flow configuration. The assessment is based on comparing the modelling results using the kinetic-frictional model with the experiments. It is shown that the kinetic-theory-based model with equal weight of the collisional and frictional contributions, commonly used in the literature, gives a great deviation from the experimental results in the point of view of the dominant solids motion, while an increase in the weight of the frictional contribution improves the modelling towards the experimental results. An increase in the weight of the frictional contribution by 25-50% leads to the best match, suggesting the current constitutive relationship with equal weight of the kinetic and frictional contributions need to be refined in order to reflect the real dense granular flows.     

Digital particle image velocimetry (DPIV) technique in measurements of granular material flows, Part 2 of 3-converging hoppers

The flow evolution of an amaranth seed is being investigated in a wedge-shaped model made of Plexiglas. The objective of this paper is to recognise flow patterns in the flowing material, and also to depict evolution of velocity fields, flow field discontinuities, velocity profiles for cross-sections of the model, shear zones and flow streamlines using the digital particle image velocimetry (DPIV) optical technique. It is demonstrated that the DPIV technique used in the experiments enables quantitative analysis of the flow zones geometry. The technique also allows to reveal boundaries between flowing and stagnant zones and to extract velocity profiles at any selected sections of the model.     

Practical validation of the two-fluid model applied to dense gas-solid flows in fluidized beds

This paper discusses the simulation of bubbling gas-solid flows by using the Eulerian two-fluid approach. Predictions of particle motion, bed expansion, bubble size and bubble velocity in bubbling beds containing Geldart B particles are compared with experimental results and correlations found in the literature. In addition, gas mixing in a bed of Geldart A particles is investigated.An in-house code has been developed based on the finite-volume method and the time-splitting approach using a staggered grid arrangement. The velocities in both phases are obtained by solving the 2D Reynolds-averaged Navier/Stokes equations using a partial elimination algorithm (PEA) and a coupled solver. The k-ε turbulence model is used to describe the turbulent quantities in the continuous phase.In general, the model predictions are in good agreement with experimental data found in the literature. Most important observations are: the level of the restitution coefficient was found to be crucial in order to obtain successful results from 2D axisymmetric simulations of a system containing Geldart B particles. Bubble size and bubble rise velocities are not as sensitive to the restitution coefficient. The turbulence model is of outmost importance concerning gas mixing in a fluidized bed of Geldart A particles.From these numerical analyzes an optimized granular flow two-fluid model can be designed for the purpose of simulating reactive systems in fluidized bed reactors.     

Measurements of granular flows in two-dimensional hoppers by particle image velocimetry. Part I: experimental method and results

The objective of the present investigation was to test the applicability of particle image velocimetry (PIV), which is normally used for measuring velocities in liquids or gases, to measurement of velocities in granular flow. A second objective was to use PIV to provide experimental data for comparison with mathematical models. The flow of zinc particles, of size 0.4, 0.61 and 0.76 mm size, in a flat-bottomed two-dimensional hopper was measured by PIV. The particles were characterized using ASTM procedures for angle of repose, packing density and flow rate through a funnel. Through PIV, velocities and mass flow rates were determined for exit apertures 5 and 7.5 mm in width and 10, 30 and 50 mm long. The bed of particles in the hopper showed the expected stagnant zones on either side of the aperture. There was a continual avalanche of particles at the “V’’ which forms at the surface of the bed and some images of this avalanche, obtained with a boroscope, are included.