Figure: 1
Movement of a wedge of material. When a granular material
exceeds its maximum angle of stability (left), an avalanche occurs and
the surface relaxes to the material's angle of repose (right). When
this avalanche occurs, the dark gray material moves from the top of the
surface to the bottom as shown.
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Figure: 2
Schematic of the motion of wedges in a circle and square shaped
tumbling mixer. Shown are three fill levels, f = 0.25,
f = 0.5, f = 0.75. During an avalanche, the dark wedge
falls to the position of the light wedge, and random mixing occurs.
Note the wedge intersections for f = 0.25 (e.g. material in one
wedge will enter a new wedge upon further rotation, enhancing
inter-wedge mixing) , there are no wedge intersections for f =
0.5, and the formation of a core for f = 0.75 (e.g. centrally
located material which never participates in an avalanche, therefore
never mixes).
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Figure: 3
Mixing patterns from simulation and experiment after two revolutions
at the indicated fill levels, f. Simulations (a,c) and
experiments (b,d) for a circular and triangular cross section,
respectively. To initialize an experiment (simulation), different
colored material (red, blue; red, yellow) is loaded side by side with
red on the right and blue (yellow) on the left.
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Figure: 4
Mixing efficiency in a 2-D mixer with a circular cross-section.
For each fill level, the growth of the interface was fit to an
exponential curve. The rate constant for the growth is multiplied by
the amount of material being mixed to obtain the overall mixing
efficiency. From this curve the optimum fill level is found to occur at
approximately f = 0.25, and core onset occurs at f=0.5.
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Figure: 5
Mixing efficiency in a 2-D mixer with a triangular
cross-section. The overall mixing efficiency was calculated as in
figure 4. This geometry exhibits an overall optimum at approximately
f = 0.15, a minimum at core onset (f=0.3), and a second
minimum at f = 0.5.
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Figure: 6
Mixing in a three dimensional wedge of material. For
mechanically identical particles, when the upper wedge (gray) falls to
the lower position (white), mixing along the axis of rotation is
diffusive, and mixing in the radial direction is random.
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Figure: 7
Axial mixing in a 3-D drum. Axial mixing of granular materials
can be modeled as a diffusive process. Shown is a comparison plot of
the normalized position of the centroid of one of the species versus
dimensionless time for an experiment (Hogg et al. 1966), a
simulation using the 3-D model, and the analytical solution obtained
using the diffusion equation.
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Figure: 8
Schematic of core erosion. Geometries with an odd number of
uniformly spaced, outwardly protruding baffles erode the core. As the
mixer rotates (a-d), an avalanche cannot completely fill the baffle and
a void forms. Arching of the material will preserve this void for some
time. When the arch collapses, the core shifts and a portion of the
core enters the mixing region.
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Figure: 9
Schematic of core erosion in an asymmetric container.
Prediction: core erosion in chiral containers can occur for one
direction of rotation and not another. Notice that for clockwise
rotation, avalanches can freely fill 'arms' of star, but not for
counter-clockwise rotation.
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Figure: 10
Core erosion in an asymmetric container. Results of two
core-erosion experiments with a bent star. In the first case the star
was rotated counter-clockwise and the core eroded by approximately 50%,
in the second case the core eroded essentially completely.
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Figure: 11
Schematic of wobbling motion. Wobbling enhances axially mixing.
When a drum is wobbled, avalanches occur along the mixer axis as well
as perpendicular to the axis (as in a purely rotated drum).
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Figure: 12
Axial mixing enhancement as a function of the ratio of the number of
avalanches. Ratio of the effective diffusion coefficient for a
wobbling simulation to that of pure diffusion -- no wobbling, are
plotted versus the ratio of the number of avalanches -- wobbling to
rotational. Note that a maximum occurs at one wobbling avalanche per
two rotational avalanches.
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Figure: 13
The geometrical origin of the core. The materials being mixer
are small, dense, cubic particles and large, light, spherical
particles. Mixing within wedges is dramatically different than in
figure 3; however, the core is evident despite this difference.
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Figure: 14
Including a boundary layer in a mixing simulation. A comparison
of an experiment and a simulation of a 2-D mixer with a square
cross-section. Here, the simulation is modified in order to include a
thin surface boundary layer.
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Figure: 15
Core precession. A comparison between two experimental photos of
a 2-D mixer with a circular cross-section filled with large cubic
particles. The first photo -- a -- is after 3 revolutions and the
second -- b -- is after 40 revolutions. Note that the interface within
the core has precessed by 42 degrees.
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Figure: 16
Precession rates by material. Different materials precess at
different rates. Shown is a comparison of the precession rates of large
spherical particles, large cubic particles, and small cubic particles.
The slope of the lines yield precession rates. Note that there is more
than an order of magnitude difference between the precession rate of
the large spherical particles and the large cubic particles.
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Figure: 17
Schematic of particle dynamic force model. Typical normal and
tangential forces in a soft particle model, composed of a spring,
dashpot and slider.
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Figure: 18
Comparison of experiment and simulation using the hybrid method.
Shown is a comparison of experiments (a) and simulations (b) of
segregating materials at f = 0.4 and f = 0.75. The
simulations easily capture the large scale, geometrical features, such
as the core, as well as the smaller scale, dynamical characteristics
like radial segregation.
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Table 1
Efficiency data for various 2-D shapes.
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