Granular Convection and Size Separation

Convection and size separation occur when granular material is shaken in the vertical direction. Convective flow takes the form of rolls that continuously transport grains up and then down again. In cylindrical or rectangular vessels the flow will be upwards in the center and downwards in a thin stream along the side walls, leading to the formation of a central heap. With different boundary conditions, such as side walls that are slanted outward, it is possible to reverse the sense of the convection roll and to induce downward flow in the center, resulting in two heaps flanking a central depression.

Exponential depth-dependence of the convection velocity

Exponential depth-dependence of the convection velocity

This data set, obtained from MRI data shows the vertical component, vz(z), of the convection velocity (displacement per tap) of granular material along the central axis in a vibrated cylinder for several values of the applied, peak acceleration (in units of the Earth's acceleration, g = 9.81 m/s2). Positive values for v(z) indicate upward movement of the particles. Note that in this plot a straight line indicates an exponential increase of the velocity as the depth, z, below the top surface is decreased. Near the very top surface, i.e. near z = 0, v(z) is reduced since the material no longer is moving only vertically upward, but also moves out toward the cylinder walls (this gives rise to a large horizontal component which is not shown here). This data set, as well as the corresponding MR images are described further and analyzed in a recent paper. More detailed data on the depth dependence of the axial flow (along the cylinder axis) can be obtained by direct visual particle tracking (see Knight et al, Experimental Study of Granular Convection). Our experiments show the existance of a typical lengthscale over which the convection velocity decays below the top surface, and thus also the depth of the convection rolls as well as a typical timescale that characterizes the convection speed. Both paramters scale with vibration parameters, such as shaking amplitude and frequency.

Radial velocity profile

MRI can also provide detailed information about the radial convection velocity profile, vz(r), as shown below:

radial velocity profiles for various size container

In this data set, the three curves show data obtained from MRI data for three different size containers. The radial coordinate, r, is measured from the central axis of the cylindrical container (in units of the bead diameter, d). Also shown are fits to a parabolic flow profile (as one would expect for simple laminar flow of a liquid in a pipe; clearly this does not work for granular flow!), and to a functional form (cosh or certain Bessel functions) that exhibit an exponential decay of the velocity away from the walls. Such exponential, radial decay appears to fit extremely well and naturally leads to the blunt velocity profiles observed in the wider containers. In all cases the stippled lines give the location of the control lines before the shake was applied, and their widths indicate the widths of the containers. Note how the flow is upward (positive v) in the middle of the container, and downward (negative v) in a narrow region near the walls. This and many more detailed data sets, a scaling analysis, and a simple model for both the vertical and radial velocities are contained in our recent paper(Knight et al, Experimental Study of Granular Convection).

Reversal of convection rolls

One way to reverse the sense of the convection rolls is to change the boundary conditions. Specifically, if instead of a straight cylindrical vessel (as in the examples above) a vessel with flared-out walls is used, the flow can be reversed (depending on the applied acceleration and other parameters such as the wall angle) and now beads move down in the center and up along the walls. Below we show this for a quasi two-dimensional geometry (about ten particle diameters deep, a few hundred wide) with circular walls:

evolution of conv. rolls for circular boundaries

Top left (a) shows the system before vertical shaking began (the layering, using different color sand in the actual experiment, is for easier visualization). In image (b) shaking has commenced and clearly the upward deformation of the stripes along the walls is visable. After a few minutes of shaking the sand is all mixed (c), and particles flow up along the sides and avalanche down towards the central depression. (d)At higher applied accelerations the heap angle steepens. A color image of this situation shows that there is radial segregation. (Note the white particles are smaller and/or rougher than the colored ones). These pictures are adapted from an overview article that appeared in the MRS Bulletin in 1994. A detailed investigation of roll reversal can be found in our paper (Knight, External Boundaries and Internal Shear Bands in Granular Convection).

Size Separation: The Convection Connection

If there are larger grains in a mixture of granular media confined to a vessel with rough, vertical walls then they will invariably rise to the top if the mix is subjected to vertical vibrations. This is often referred to as the Brazil Nut Effect, and it can be a nuisance or a blessing for industry, depending on the application. The main question is: if shaking fluidizes a granular material, why do larger objects move up and why don't they sink down as they would in any ordinary fluid? What happened to entropy if shaking induces de-mixing? The answer is that we are dealing with a system which is far from equilibrium, and that dynamic effects outweigh any thermodynamical considerations.
Instead of a nut mix with lots of large Brazil nuts, consider instead a bunch of coffee beans embedded in smaller coffee grounds. This system is easily imaged via MRI as shown here in a cross-section :

MRI of coffee beans and grounds (image by J. Knight and D. Kovar)

Our research shows an intimate connection of the Brazil nut phenomenon with granular convection: the larger particles are carried along with the convective motion of the smaller, background particles. Once at the top surface, the larger ones become trapped when they cannot fit into the narrow region along the walls reserved for the downward flow. The set of images below demonstrates this for five large coffee beans that were placed inside a background of poppy seeds (the poppy seeds provide the stronger MRI signal, so the beans appear as dark holes):

MRI of rising coffee beans

Time in this picture runs from left to right; the initial placement of the beans is visible in the left column of images, the next columns show snapshots at later times. The four rows show four vertical cuts through the cylindrical container: one close to the front, two near the center (that's why they are slightly wider!), and one closer to the back. You can clearly see the beans rise as time progresses, and you can also see an indication of the radial velocity profiles (the beans near the central container axis rise faster). The above image is taken from unpublished work by Jim Knight.

Publications on granular convection and size separation:

  • J. B. Knight, H. M. Jaeger, and S. Nagel, "Vibration-Induced Size Separation in Granular Media: The Convection Connection", Physical Review Letters, 70, 3728 (1993).
  • E. E. Ehrichs, H. M. Jaeger, G. S. Karczmar, J. B. Knight, V. Y. Kuperman, and S. R. Nagel, "Granular Convection Observed by Magnetic Resonance Imaging," Science, 267, 1632 (1995).
  • V. Y. Kuperman, E. E. Ehrichs, H. M. Jaeger, and G. S. Karczmar, "A New Technique for Differentiating Between Diffusion and Flow in Granular Media using Magnetic Resonance Imaging," Review of Scientific Instruments, 66, 4355 (1995).
  • J. B. Knight, E. E. Ehrichs, V. Y. Kuperman, J. K. Flint, H. M. Jaeger, and S. R. Nagel, "Experimental Study of Granular Convection," Phys.Rev. E, 54, 5726 (1996). This paper has an extensive set of data on granular convection in vertically vibrated cylinders with vertical walls. Both data taken with tracer beads and MRI are included, as are (heuristical) approaches to model the data.
  • J. B. Knight, "External Boundaries and internal shear bands in granular convection", Phys. Rev. E, 55, 6016 (1997).
  • J. B. Knight, H. M. Jaeger, and S. R. Nagel, "Magnetic Resonance Imaging of Granular Convection", AIChE Symposium Series on Fluidization and Fluid-Particle Systems, to appear.
  • E. L. Grossman, "The Effects of Container Geometry on Granular Convection",Phys. Rev. E 56, 3290(1997).