The flux of granular material though an aperture in a silo configuration has been analysed for frictional and frictionless ellipses of varying aspect ratio.

There is considerable interest amongst bulk solids handling industries to better understand the controls on hopper/silo discharge rates and incidence of hang-ups or “jamming”. Mining companies have particular interest in granular flow through given aperture geometries to better understand isolated extraction zones, (IEZ) and isolated movement zones, (IMZ) above draw points in block caving operations. Understanding the flow and origins of naturally fragmented material arriving at a series of draw points is of great importance in draw point design in caving operations. To model the processes involved, study methods have included; large scale physical models, continuum models of granular flow and increasingly, researchers have turned to discrete particle DEM models.

A remarkable richness of mechanical behaviour for the silo discharge flow problem can be seen in the simulations by Guises (2008) using 2D FEMDEM models of elliptical particles with friction. Experiments were run with 900 particles discharging from a rectangular silo with an opening of ~5.7 times the effective diameter of elliptical particles, i.e. sufficiently narrow to observe intermittent jamming for the frictional case. Discharge rates are dependent on particle shape, with faster flows observed for elongated particles when there is zero friction, see Fig. 1. With friction, the periods of time over which significant jamming occurs is greater for increasingly elongated particles.

Fig.1 (a) Total cumulative mass discharged from the silo aperture for frictional particles, m = 0.5, see also Fig.2 and Fig 3. (b) frictionless particles with insets plotting the fluctuations in value of the discharge rate.

FEMDEM methods allow a more fundamental study of flow of granular materials with realistic shape and friction properties.  Aperture shapes, opening widths and spacings can be optimised for flow rates that retain the most uniform flow within the granular mass of the caved ore. There is also potential to parameterise continuum models, compare results with less costly FEMDEM models employing rigid interior properties, or less costly DEM models based on superquadrics or clustered spheres. Transient phenomena leading to silo wall collapse can be studied with this approach.

Fig. 2 Snapshots of the differential stress field during the flux of grains. It is observed that each system goes through periods of relatively jammed flows with long stress chain arches, which consequently reduces the flux of grains. a) α = 1.0, t = 0s, b) α = 1.2, t = 0s, c) α = 2.5, t = 0s, d) α = 1.0, t = 1s, e) α = 1.2, t = 1s, f) α = 2.5, t = 1s, g) α = 1.0, t = 2s, h) α = 1.2, t = 2s, i) α = 2.5, t = 2s, j) α = 1.0, t = 4s, k) α = 1.2, t = 4s, l) α = 2.5, t = 4s

Fig. 3 Snapshots of the velocity field during the flux of grains through the aperture of the silo at different time for 3 types of frictional (μ = 0.5) particles. The silo width is D = 5.656 times the equivalent grain diameter. a) α = 1.0, t = 0s, b) α = 1.2, t = 0s, c) α = 2.5, t = 0s, d) α = 1.0, t = 1s, e) α = 1.2, t = 1s, f) α = 2.5, t = 1s, g) α = 1.0, t = 2s, h) α = 1.2, t = 2s, i) α = 2.5, t = 2s, j) α = 1.0, t = 4s, k) α = 1.2, t = 4s, l) α = 2.5, t = 4s

Reference

Guises (2008) PhD Thesis

For granular geomaterials with very large particles such as quarried rock-fill or armourstone rock blocks and concrete units used for breakwater protection, both the complex angular shapes and transient stress levels (sometimes approaching material strengths, especially tensile strengths) have an important impact on the granular material behaviour. DEM simulation struggles to capture the shape details and the deformations associated with stress wave transients during multi-body collisions. The 3D FEMDEM code, (Xiang, Munjiza and Latham 2009) introduced and uploaded to VGW in 2008, has precisely these necessary modelling attributes.

The potential ability to capture an entire system of boulders during dumping/placement is illustrated below. An irregular-shaped rock block is obtained from a library of quarried aggregate shapes captured by laser scanner. The shape chosen here is not dissimilar to a possible armourstone shape and is represented at a suitable resolution in order to produce a simulation with ~300 blocks relatively quickly.

These figures show the FEMDEM capability to model such a system of monosized angular rock-like boulders (bodies of about 40 kg) during dumping in a 5.26m×5.3m×3.87m bin (front face not shown). Part of the motion history is captured and stress development in time and space has been contoured with a default colour scale indicating Von Mises stress. The simulation is set up for convenience as a space filling array of identical rocks.  The random disorder soon takes over, and a highly energetic compressive phase is followed by a net dilation phase of bouncing back. The movie shows a simulation where a realistic coefficient of friction acts on the contacting boulders resulting in rapid energy dissipation as the particles quickly come to rest. The simulation results point towards a future where FEMDEM analyses can look at the accurate and detailed dynamic stress transients within flowing granular systems as well as pseudo-static responses to loaded systems.

Reference

Xiang, J., Munjiza, A. and Latham, J.-P., 2009. Finite strain, finite rotation quadratic tetrahedral element for the combined finite-discrete element method. International Journal for Numerical Methods in Engineering. 79(8), 946-978. doi: 10.1002/nme.2599

The development of a fast clustered overlapping sphere algorithm to represent real particles in discrete element modelling (Garcia et al. 2009a) provided the means to study the influence of shape on 3D grain packing, using DEM. Both ellipsoids and natural rounded grain shapes selected from the VGeST shape library were studied. Care was taken to introduce realistic dynamics into the grain accumulation process, shown in the figure below while porosity was determined using a point sampling method that achieves accuracies in porosity values greater than 0.25%.

In this construction of the granular packs, a small number of randomly oriented grains are dropped in batches (light colours), one batch at a time, to simulate a low settling rate. Three stages during the construction are shown after settling 1000, 2000 and 3000 grains.

As a consequence new research using dynamic ballistic deposition simulations of the packing of prolate ellipsoids, including spheres, a great many contradictory results for pack porosities reported in the literature covering experimental and numerical simulation can now be explained, as shown in the graph below. The conditions favouring packs of lower porosity than 36.7%, the widely reported value for dense random sphere packs, are well described in the plot that shows how friction effects compete with aspect ratio or sphericity. For low friction (μ) our results show that the porosity of slightly elongated particles is smaller than in sphere packs. Numerical algorithms of a geometrical nature have also been devised that will rearrange different particles into packs denser than for spheres, however without mechanical realism in the simulation there has always remained some doubt whether grain packs in nature can find these algorithmically created space-filling configurations with any natural sedimenting or packing processes. For high friction the spheres pack more densely than the rest of the particles considered, in agreement with simulations employing algorithms that constrain the dynamics or non-dynamic methods that invoke “frozen” status for all previously settled grains in a continuous packing process.

The porosity of ellipsoid packs are shown as a function of aspect ratio and sphericity for various friction coefficients determined by DEM simulations and compared with non-dynamic simulations.

The porosity minimum is better developed here for low friction coefficients, whereas in the 2D FEMDEM simulations of ellipse packing, the better developed porosity minimum occurred when the friction coefficient was 0.5 compared to the zero friction case. These 3D DEM and 2D FEMDEM methods are very different and neither are creating perfect ellipses or ellipsoids. There therefore remain some contradictory trends that need further investigation.

Cluster representations of aspherical grains that were used in DEM simulations of realistic rounded sand grains are shown above, the left is an elongated grain with aspect ratio α = 2.09, and sphericity Ψ= 0.90, the right has low aspect ratio grain α = 1.30, Ψ = 0.92.

Non-symmetrical rounded natural grain shapes selected from the shape library were also investigated (using the above representations) to see if packs denser than for spheres could be obtained with particles with reduced rotational freedom compared with ellipsoids, but with the same packing conditions.

Above are two examples of packs of non-symmetrical grains identically constructed by DEM with friction coefficient μ = 0.2. Both these non-symmetrical particles packed at lower porosity than that achieved for sphere packs constructed under the same conditions with μ = 0.2. Left: Porosity = 35.4%, Elongated grain α = 2.09, Ψ= 0.9, Right: Porosity = 34.9%, low aspect ratio grain α = 1.3, Ψ = 0.92.

These simulations demonstrate the ability to isolate particle Form as distinct from roundness and angularity in a manner rare if not impossible for experiments on natural sediments. Discussion as to whether porosity can be captured by just one function of Form, such as sphericity or aspect ratio, and a measure of friction is included in Garcia’s (2009) PhD thesis.The details of the DEM packing simulation and clustering algorithm are given in the paper entitled numerical study of the effects of particle shape and polydispersity on permeability, (Garcia et al. 2009b) where the purpose of the simulations is to create a grain pack skeleton for fluid flow and permeability modelling.

References

Garcia, X., Latham, J.-P., Xiang, J., Harrison., J. 2009a. A clustered overlapping sphere algorithm to represent real particles in discrete element modelling, Geotechnique, 59, No. 9, 779-784 doi:10.1680/geot.8.T.037

Garcia, X., Lateef, A., Blunt, M., Matthai, S., Latham J.-P., 2009b. Numerical study of the effects of particle shape and polydispersity on permeability, Physics Review E, Vol 80, doi/10.1103/PhysRevE.80.021304

The packing of particles is recognised to be a critically important process in many industrial fields, especially in powder technology, powder metallurgy, ceramics, pharmaceutical and the mineral industries. It also remains a focus of intense research in physical, chemical and environmental science.

By coupling the DEM with a finite element formulation (FEMDEM), the realism of particulate modelling has been greatly improved by taking into account the internal deformations of the particle that are a physical consequence of the stress generated by the interactions.

To study the influence of shape, the particle form was examined using ellipses with a range of aspect ratios for friction and frictionless cases. The emergent properties of the granular packs considered in the study were packing density, coordination number, and force distribution. The spatial distribution of the maximum differential stress can also be examined. A set of sedimentation experiments was carried out as summarised below.

Packs of 900 particles are shown where on the left the packs are frictionless and on the right, they are frictional (μ = 0.5). The aspect ratios of the ellipses vary from 1.0 (circle) top, 1.4 middle and 5.0 bottom. The colour scheme varies between blue for (unstressed) to red colour (highly stressed) and the scale is identical for each figure.

The influence of aspect ratio and friction on porosity, left and on coordination number, right, is rigorously examined in the graphs below. The increase in the packing density above that for random packed frictional circular particles for aspect ratios of ~1.6 supports 3D experimental results reported in the literature for ellipsoids.

The influence of friction on coordination number is dramatically shown by the distribution of the number of contacts per particle for a packing of ellipses of aspect ratio 2.5 for frictionless (left) and frictional systems (right). It is shown that the eccentricity of the particles not only significantly influences the final density of the pack but also the distribution of the stress and the contact forces. The presence of surface friction increases the amount of disorder within the granular system.

Stress heterogeneities and force chain patterns propagate through the particles more efficiently than for the frictionless systems. The results shown below also suggest that for the monodisperse systems investigated, the coordination number is one of the factors that control the distribution of the stress within a granular medium. The figure shows distributions of the average value of the differential stress over the area of the grain (depth normalised) for two granular packs of 900 grains. Identical stress distributions are observed for two packs with identical z but different particle shapes, suggesting a strong control of z on stress distributions. However, the difference in packing density of 5% is significant. FEMDEM provides a powerful simulation technology to probe curious granular phenomena. Further details may be found in Guises PhD Thesis (2008) and Guises et al. (2009).

References

Guises PhD Thesis

Guises, R., Xiang , J., Latham, J.-P., Munjiza, A. 2009. Granular packing: Numerical Simulation and Characterization of the Effect of Particle Shape, Granular Matter, , Vol 11, 281-292  doi:10.1007/s10035-009-0148-0.