Comparative study by PEPT and DEM for flow and mixing
in a ploughshare mixer
1
1
B. F. C. Laurent and 2 P. W. Cleary
Late of Institute for Manufacturing, Department of Engineering,
Cambridge University, Mill lane, Cambridge, CB2 1RX, UK.
2
CSIRO Mathematical and Information Sciences,
Private Bag 33, Clayton South, Clayton, Vic, 3169, Australia.
Email: Paul.Cleary@csiro.au
Abstract
Investigation of the granular flow induced by a single plough mixer was performed using
positron tomography (PEPT) and Discrete Element Modelling (DEM). Both approaches
showed the development of two loops of circulation, one on either side of the plough. This
involves particles being pushed forward through the bed by the plough and then picked up
and thrown through the space above the bed, falling to the surface and the slowly flowing
back into the trench left behind the plough blade. The flow patterns obtained
experimentally were compared to those obtained using DEM for a speed of 2.25 Hz and
showed reasonably good qualitative agreement. The angle of repose of the trench down
which the avalanching back flow occurs was found to be too low in the DEM predictions.
This resulted from the modelling of the rice grains as being spherical particles. Mixing rates
for different blade speeds were characterised for both PEPT and DEM. Good quantitative
correlation was found and a consistent picture of increasing mixing efficiency with
increasing plough speed was observed. Finally, observations on the averaging of PEPT and
DEM data and on the comparison results from these two methods are made.
Keywords
Granular flow; particle; mixing; ploughshare; DEM; PEPT
1
1
Introduction
Mixing is a critically important industrial process, ranging from blending on stockpiles in
mineral processing to powder and grain mixing in pharmaceuticals and food processing.
One common device used for smaller scale industrial mixing is the ploughshare mixer.
These have one or more blades attached via radial arms to a central shaft that rotates within
the cylindrical mixing vessel. Such mixers have been studied using positron emission
particle tracking (PEPT) by Broadbent et al., [1] and more recently by Jones and
Bridgwater [2]. This technique permits visualisation of the granular flow in the mixer and
allows quantitative characterisation of the mixing by following the long term trajectories of
an irradiated tracer particle.
Discrete Element Modelling (DEM) is a computational method that allows particles flows
in such equipment to be simulated and the flow patterns and mixing rates to be predicted.
DEM, in principle, provides the ability to optimise such equipment by evaluating design
parameters such as blade speed, the shape of the blades, the number of blades, their angular
offsets from neighbouring blades and the axial blade spacing.
To date, there has been limited comparison of such experimental results with DEM
predictions for mixing. Kaneko et al., [3] studied granular flow in a vertical cylindrical
mixer stirred by a ribbon agitator. The comparative results showed good agreement
between the experimental and the numerical techniques. Work from Stewart et al., [4]
presented a comparative study of experimental and numerical results of investigations for a
single horizontal blade high shear powder mixer. Early findings again showed good
qualitative agreement between the two methods. Other recent findings are reported by Kuo
et al., [5] who investigated particle motion in a V-blender and compared results obtained
with positron tomography with simulation work using DEM. They showed some
similarities between experimental and numerical results such as the flow fields and particle
occupancy within the mixer.
The present work describes qualitatively and quantitatively the powder flow and the
particle mixing generated by a single plough in a horizontal drum using positron
tomography and DEM. The investigation also provides a comparative study of these
experimental and numerical methods. Both methods have different strengths and
weaknesses and there are challenges in comparing the quite different types of mixing
characterisation commonly used for these two methods.
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2
2.1
Apparatus, experimental and simulation techniques
Mixing apparatus
The mixer shell used in this study was a horizontal cylinder with a diameter of 250 mm and
a length 450 mm (see Figure 1). The plough (shown in Figure 2) is mounted on a shaft of
diameter 30 mm about which it rotates. The blade somewhat resembles a chevron bent
down the middle with the middle bend being the leading edge of the blade. The mixer is
partially filled by a granular material forming a bed which is then mixed by the interaction
with the plough blade. For these experiments rice grains of approximately 2 mm by 4 mm
with a density of 1400 kg/m3 were used. The fill level was 25% of the volume of the mixer.
The tracer used in the PEPT experiments was a glass sphere of diameter 2 mm and a
density of 2500 kg/m3. Although there is a slight difference in density and size of the tracer
compared to the bed particles, earlier work in a horizontal mixer stirred by a flat blade
showed that the discrepancy in size had a minimal effect in relation to the general flow
patterns [6]. Indeed, these investigations performed using three tracers, one smaller, one
larger than the bulk particles and one of the same size as the bulk particles showed that the
flow patterns in the trans-axial plane were not dependent upon the size of tracer. It was also
observed in previous studies [7, 8] that tracer size does not have a noticeable effect in
rapidly stirred systems where momentum effects are dominant. This means that information
extracted from tracking the motion of this single tracer inside the bulk should fairly well
represent the behaviour of the bulk particles.
2.2
The PEPT technique
Positron emission particle tracking (PEPT) is a non-invasive method of investigation for
opaque systems, using a positron-emitting tracer introduced into the system. Each positron
emitted annihilates with a surrounding electron, emitting two back to back -rays. Their
impacts on two detectors situated on either side of the system permit the construction of the
line on which the annihilation event has occurred. In theory, two events give the location of
the positron emitter. The spatial position of the tracer is reconstructed from the events by an
algorithm, fully described by Parker et al. [9]. The PEPT data provided are the spatial coordinates (x,y,z) of the tracer as a function of time t. An average of 2000 events per second
allows around 20 location points per second to be determined. These numbers vary with the
3
activity of the tracer. The uncertainty of the data is governed by the spatial resolution which
for these experiments was approximately 2 mm for a speed of 0.2 m/s. This increases with
the speed of the positron emitter to about 5 mm at a speed of 1 m/s.
The data is collected in a Lagrangian form and is averaged into an Eulerian grid structure.
The quality of this Eulerian representation of the steady average flow field depends on the
amount of averaging data which in turn depends on the duration of the experiment and the
nature of the flow. Typically this was around an hour, so that the tracer has a reasonable
possibility of accessing all permitted moving regions of the system. Note that the tracer is
unable to move into or out of dead or stationary regions, so sampling these regions requires
a specific experiment where the tracer is directly introduced into these.
2.3
DEM method
The DEM method was first applied to geotechnical problems by Cundall and Strack [10].
Over the next two decades it has grown in popularity and has been used to study many
particle flow problems. DEM simulation involves following the motion of every particle
and modelling each collision between the particles and between the particles and their
environment. Early review articles [11, 12] give more information on rapid granular flows
and the use of DEM for modelling these. The DEM implementation used in this paper uses
a linear spring-dashpot collision model and is described in Cleary [13, 14, 15]. DEM has
been used extensively for analysing mixing in many types of mixers [16]. For broader
examples of the use of this DEM code, see [14, 15, 17].
The plough share blade and containment vessel were constructed to exactly match the
specifications of the laboratory mixer (Figures 1 and 2). The particles in the DEM model
are represented as spheres with sizes distributed between 2.5 and 3.0 mm, approximating
the rice grains used in the PEPT experiments above on a matching volume basis. The
simulations consisted of around 103,000 particles for the 25% fill level used in the
experiment. The coefficient of restitution used was 0.3 and the coefficient of friction used
was 0.75. Simulation predictions have previously been shown to be insensitive to the
precise choice of these material constants in dynamically similar rotating cylinders [18].
4
3
Comparison of transient PEPT and DEM flow patterns
Figure 3 shows the PEPT measured flow field induced by the motion of the plough as it
moves though the particle bed for a rotation rate of 2.25 Hz. The dotted lines in the front
view (rectangular shaped container) shows the region covered by the plough as it moves
though the bed. Initially, the plough is out of the bed and no significant motion is visible.
As it penetrates the bed, the arrows representing the velocity vectors indicate that the bed is
divided into two distinct parts, one on each side of the axis of the plough. This creates two
waves of material which are moved first upwards and then flow downwards as the plough
leaves the bed. Material is subsequently lifted by the plough out of the bed and thrown
through the space above. The bed then relaxes to rest after both waves of material have
collided in the empty trench behind the plough. This avalanching motion is thought to
induce exchange of particles from one half to the other, as reported in [2] for a multi-plough
powder mixer. The velocity arrows show the motion of particles in the agitated part of the
bed. There is evidence of the non-planar flow structure generated by the plough as it stirs
the particle bed. In particular, on the axial (side) view in the (x,y) plane, the flow field
shows two loops of circulation which are reflectively symmetric about the plane in which
the plough moves. Both cells ascend and move away from the plane of the plough and then
move down towards the bottom of the slope.
Figure 4 shows the DEM predictions of the flow for the same case as shown in the PEPT
experiment in Figure 3. Figure 4a shows the initial setup of the simulated mixer with the
particles coloured according to the quadrant of the bed that they were originally located in,
so that the extent and nature of the mixing can be observed. Figure 4b, shows the mixer
after 0.5 blade pass at a 2.25 Hz blade speed. The following frames are then at 0.25 blade
pass increments thereafter. After 0.5 revs, the blade has risen above the initial surface level
of the particle bed and is lifting reasonably large numbers of particles upwards. These are
predominantly from the rear red (dark grey in greyscale print) and yellow (pale grey)
quadrants. The first passage of the blade has pushed significant volumes of blue (black) and
green (mid-grey) particles from the front quadrants into the rear ones, this having started
the mixing. The rising blade continues to lift and throw the particles sideways. A coherent
front of particles is clearly visible at 0.75 revs travelling to either side away from the blade.
By 1 rev these ballistic particles are approaching the surface of the bed. At 1.25 revs, the
blade is forcing its way through the bed leaving an open trench behind and pushing
particles in front. The previously thrown particles have now all landed and have piled up
5
nearer to the ends of the mixer forming a bed surface that slopes back down towards the
trench through which the blade passes.
As the blade moves from vertical to horizontal, it again lifts and throws masses of particles
to either side. The thrown particles now consist of all four colours indicating a degree of
mixing has already occurred. Meanwhile the trench opened behind the blade has been filled
by surface material avalanching from surrounding areas. This two part flow continues with
each blade pass. First, the surface back flow from around the sides fills the newly opened
trench behind the blade and then the blade throws a symmetric cascade of particles up and
to either side. After several passes the bed surface becomes inclined back towards the entry
point of the blade into the bed. With each passage of the blade progressively more mixing
is produced.
The flow patterns found in these DEM simulations are in close accord with the butterfly
shaped pattern and the bed relaxation observed in the PEPT experiments. The low
occupancy rates found behind the blade and in the surface regions of the bed are in accord
with the surface back flow after the passage of the blade through the bed, whilst the high
ballistic trajectories shown in the PEPT velocity field are consistent with the high arcing of
material thrown by the blade in the upward part of its motion.
4
Comparison of average flow fields for PEPT and DEM
Both the PEPT trajectory data and the DEM particle flow can be averaged onto a stationary
cubic Eulerian grid. This allows significant reductions in the noise (variability) of the flow
characterisation that results from the removal of the transient details of the specific particles
involved at specific times. This provides a more robust basis for quantitative comparison.
This is done by averaging the PEPT and DEM data for all plough positions across multiple
blade cycles. The averaged cells are 5 mm x 5 mm for the cross-section and for all positions
in the orthogonal direction.
Figure 5 shows the PEPT time-occupancy diagram (shaded) superimposed with the velocity
field (arrows) in an axial (side) view and the front view on the left. The scale gives the
ratio of time the tracer spends in a given bin to the total experimental time, which here
about an hour. It also shows the matching averaged DEM modified solid fraction
distributions on the right. The solid fraction is the fraction of each volume occupied, on
average, occupied by particles. This has then been scale by a Heaviside function with based
6
on the normalised velocity of the bed so that the dead regions are removed. This is done to
try to mimic the PEPT behaviour where the tracer cannot enter the dead regions and so
these particles cannot be included in the PEPT occupancy measure.
The PEPT time-occupancy diagrams (left side of Figure 5) show the general shape of the
mobile part of the bed where the tracer was found during the experiment, developing a
butterfly structure in the side view. Assuming that the scale is proportional to the local bulk
density, as was shown to be reasonable in [17], it appears that two specific dynamic
regions, a layer about 5 mm deep at the free surface and the region situated near the radial
plane of the plough, have a local bulk density about ten times lower than that in the core of
the particle bed. This indicates that the free surface is actively flowing and is dilated as is
the region disrupted by the passage of the blade. To either side of the agitated region of the
bed, seen best in the side view, are large dead zones into which the marker is unable to
travel.
The DEM solid fraction distributions (right side of Figure 5) show that the bed occupies a
region in the bottom of the mixer with an inclined nearly flat free surface. In the axial view,
the surface becomes more flat towards the left side of the mixer. This occurs because the
surface is more steeply inclined in the active blade passing region and but remains
undisturbed with a horizontal free surface further away from the blade. The averaging of
these two different regions along the axial direction gives the composite surface shown. In
the front view, the trench formed by the blade is visible in the center of the mixer, with
dilated bed (due to flow induced by the blade in a V shaped region. To either side the bed
density increases (becoming) red as the dead regions remain more densely packed.
Comparing the cross-sectional and the side views of the mixer using PEPT and DEM
results, we observe reasonable qualitative similarities in the structures, but some key
differences in the detail are identifiable. Specifically, the angle of surface to the horizontal
(in the axial view) is much lower for the DEM. In the front view, the angle of the trench is
also different with the PEPT showing a much steeper angle (defined by the bottom of the
data shown) and the DEM showing a much shallower angle (given by the upper green
contour). This is attributed to the use of spherical shaped particles used in the DEM
simulation which leads to a lower angle of repose than for the experimental case where the
particles are essentially ellipsoids. This comparison highlights the importance of including
particle shape in the DEM modelling and the quantitative penalty that is associated with the
conventional use of spherical approximations.
7
There are also superficial differences present that relate to the inherently different methods
required to define and calculate the occupancy for the experiments and solid fractions for
the simulations. The volume fraction in DEM is an instantaneous average over all particles
in space, which has been filtered by the normalized velocity field to remove contributions
from stationary particles (trying to mimic the PEPT occupancy measure), whereas the
PEPT time occupancy is an explicit average of the trajectory of one particle over time.
There is an implicit assumption that if the PEPT is performed for long enough, that all parts
of the phase space will be sampled equally. Since there are substantial dead zones this is
clearly not true. It is quite possible that other regions such as the very active region around
the blade and the more quiescent region of surface flow back into the trench will be
sampled with different probabilities leading to non-uniformly weighted averages. In
contrast, the DEM solid fraction is an average over all particles and so gives equal
weighting to each region. These quantities are related, but are not the same.
Despite this, there is good agreement in the distribution of the volume fraction and the time
occupancy, particularly in the axial view. Both clearly show a circulatory motion with an
inclined free surface and higher occupancy at the core of the bed than on the surface or
around the shell corresponding to lower velocities in the middle of the bed. The front view
shows a qualitatively similar bed shape for both methods with an inclined free surface
starting at a low point at the location of the trench produced by the plough blade and
leading up to the peak bed height at the ends of the mixer. Again the DEM surface
inclination is lower due to the use of spherical particles. Overall, both sets of results show
the same outwardly directed flow generated by the blade upward passage and the return
inward flow due to the avalanching along the free surface into the trench left by the blade.
The axial flow patterns are shown in Figure 6 for both PEPT and DEM. The immediately
obvious difference between these is that the DEM data fills almost the entire volume of the
mixer while the PEPT data is absent from the upper half of the mixer and from the dead
zones in the bed to either side of the plough trench. The reasons for the absence of the dead
regions is as discussed earlier with the tracer not being able to enter these and therefore no
signal is recorded here for the PEPT. The very visible blue and red/yellow regions in the
DEM flow field are from the coherent ballistic trajectories of particles picked up and
thrown by the blade. These form two spatially large but very dilute clouds of particles. The
PEPT analysis requires a minimum threshold occupancy in order to measure a reliable
speed. In the upper half of the mixer, this condition is not met and so no reliable PEPT data
8
was available here for comparison. This is a direct result of the high speeds of the particles
on high ballistic trajectories. PEPT data is only available in the lowest parts of the ballistic
cloud. These are shown as the blue and yellow/red regions of the PEPT plot. These closely
match the DEM flow field both in location and in maximum speed which is around 0.5 m/s
for both. At the bottom of the V shaped region in the PEPT are some yellow/red on the left
and blue on the right. These represent the return avalanching flow at the bottom of the
plough trench. In the DEM flow field, there is a corresponding yellow region on the left and
dark green on the right which represent the return flow. It is again clear that the angle of the
trench is much lower with the reduced angle of repose due to the use of the spherical
particles. Setting aside this difference, a comparison of the PEPT and DEM in this region
also indicates that the return flow in the DEM occurs over a much larger area and is much
slower. This is also expected to be a result of using spherical particles which produces a
broad slumping deformation of the bed rather than a narrow avalanching flow (as seen in
the PEPT). This is consistent with the observations of mixing in a rotating drum [19] where
circular DEM particles were used to model salt cubes and produced a similar slumping flow
rather than avalanching.
Figure 7 shows the axial profiles of speed for both PEPT and DEM. The PEPT shows a
region of strong motion in the trench formed by the passage of the plough with speeds
higher than 0.1 m/s. As was the case for the axial motion, not data is available above the
surface of the bed. In contrast, the DEM data is dominated by the large high speed coherent
cloud formed by the ballistic particles thrown by the plough blade. In comparison, the
particle speeds in the bed are low and are consistent with the speeds observed in the PEPT.
Again, it is clear that the angle of the free surface of the bed is far too low in the DEM flow.
So overall, there is quite good agreement between PEPT and DEM in the region around the
plough. But PEPT has issues in data quality outside this region and cannot measure the
spatially large high speed ballistic cloud of particles. Also, the use of spherical particles in
the DEM leads to both too low an angle of repose and a return flow into the trench that is a
broad slumping deformation rather than a more localised surface avalanche.
5
Particle scale mixing kinematics and tracer dispersion
PEPT data analysis permits the reconstruction of the dispersion of a labelled slice of
material as the blade moves through the system, as was reported in [20]. The region
9
investigated here is an annular volume close to the cylindrical shell in the radial direction
and in an axial slice between the centreline of the plough blade and its right hand edge. The
region is 15 mm deep in the radial direction and 50 mm long in the axial direction. The
points considered are shown in Figure 8a. The locations of particles initially in this region
are tracked and their dispersion after 1, 5 and 10 blade passes is shown in the other parts of
Figure 8.
After one blade rotation (Figure 8b), the material has started to disperse in the right hand
side of the mixer. Some material has been lifted out of the bed and has been transported to
the left hand side half of the mixer and is now located near the top of the free surface. After
5 and 10 blade rotations (Figures 8c and d), the cross-sectional view is similar to the crosssectional view averaged time occupancy diagram shown in Figure 5a. Little material has
been transferred to the other half of the mixer. Results for the dispersion of a labelled slice
of material situated initially in the left hand side of the mixer are found to be similar, due to
symmetry of the system. These observations suggest that mixing in each loop of circulation
on either side of the plough is rapid. However, the plough forms a separatrix between the
two loops and inhibits the exchange of material from one half of the mixer to the other.
This is consistent with the DEM predictions of the flow pattern in Figs 4-6.
Detailed features of the internal convection of material that leads to mixing can be assessed
by looking at the azimuthal, radial and axial displacement of the tracer, which is shown in
Figure 9. The horizontal dashed line in the axial displacement plot marks the radial plane of
the mixer where the plough operates. Over the 1400 s (622 blade revs) of experiment
duration shown here, the tracer has crossed the symmetry plane of the mixing chamber only
five times. This demonstrates that particles passing across the separatrix is a very infrequent
event and that the blade acts to separate the bed in the mixer into two parts with
independent axial flow. One significant event can be observed in the time interval 600 - 850
s. Here, the tracer is thrown onto the free surface far from the region where the flow is
directly affected by the pass of the blade and slowly slides back down into the trench
created by the plough and rejoins the main flow. This is entirely consistent with the slow
surface flow backflow into the trench observed in the DEM simulation (Figure 4).
We use the term shunt to describe an event where a particle is picked up by the blade and
rapidly ballistically above the bed so that it rapidly changes its azimuthal position with only
modest changes in the radial and axial locations. For the majority of the time particles are in
the bed and moves predominantly relatively slowly either being bulldozed forward by the
10
blade or flowing slowing back into the trench. So most particles that remain close to the
blade are pushed around relatively gently and experience only slow changes in their radial
and axial positions. When the tracer enters the large ballistic butterfly loops then its
position changes rapidly and appears as a shunt in the position time series (the large vertical
step changes in the azimuthal position shown in Figure 9). This is again consistent with
observed DEM motion. The shunts have little enduring impact on any of the tracer
coordinates. This shows that although the big butterfly loops are highly visible, their total
contribution to mixing is modest because of the low occupancy of these parts of the phase
space. They are large amplitude but low probability events (from each particles
perspective). Most mixing appears to result from the much more frequent but much less
dramatic (energetic) pushing through the bed by through the blade.
Figure 9b shows a more detailed view of the azimuthal, radial and axial displacement of the
tracer for the time interval 140 – 220 s. All three displacement plots show distinct shunts.
Consider as an example the time interval of 156 – 164 s, 18 shunts can be identified in the
axial displacement, corresponding to a frequency of 2.25Hz which is exactly the agitator
frequency. The radial and axial displacement plots also exhibit another periodic event.
During the time interval 156 – 212s, around 12 events can be identified, corresponding to a
period of 4.7 s. This period corresponds to a cycle of 10 blade passes. Visualisation of the
trajectory of the tracer reveals that each cycle corresponds to the bulldozed passage of the
tracer from the toe of the bed to the shoulder (top) of the bed before it falls back into the
trench created by the plough ready for another passage through the mixer. This means that,
on average, it takes 10 blade passes to push a particle within the part of the bed it agitates
from the bottom of the bed to the top and back, so this is the period of the in-bed
recirculation generated by the blade.
6
6.1
Long term mixing of materials at different ploughshare mixer speeds
Mixing characterisation from DEM
In the DEM simulations, the distance of the centres of mass or centroids of each of the four
colours (shown initially in Figure 4a) from the overall centre of mass can be used to
characterise the degree of mixing. This was first used in experiment by McCarthy et al.,
[21] and for DEM mixing quantification by Cleary et al., [19]. For more details and
examples on use to quantify mixing in DEM simulation, see [16].
11
Figure 10 shows the influence of agitation time on the relative axial and radial centroid
locations for the front left quadrant colour. This is normalised by its initial maximum value
to give a non-dimensional measure between 0 and 1, with 0 corresponding to the case
where both centroids coincide (perfectly mixed) and 1 being the maximum possible
distance (perfectly segregated). As the particles mix, some cross the blade path in the axial
direction or change sides of the mixer in the blade motion direction leading to reductions in
the centroid measures as the bed mixes.
Radial mixing (Figure 10a) is initially rapid and then declines gradually as the material near
the blade is progressively better mixed. The rate of decrease of the radial centroid is the
similar for 1 Hz and for 2.25 Hz up to 10 blade passes with the radial measure having
declined to 0.65. After this, the rate of mixing for the 1 Hz speed is very slow. For 2.25 Hz,
the early rapid mixing phase continues for longer (until 15 revs) at which point the radial
centroid measure has declined to 0.5 (meaning that the bed is half mixed). For 4 Hz, there is
a sharp initial drop as the first ballistic material is thrown across the surface of the bed, but
the rate of mixing is then slower than for the lower speeds. It is not until 15 revs that the
mixing in the 4 Hz speed exceeds that of the 1 Hz speed and not until about 45 revs that it
exceeds the 2.25 Hz case. After this time, the amount of mixing in the 4 Hz is better than
for other speeds even though the initial rate of mixing was slower. The asymptotic state of
mixing is improves with increasing speed as shown by the state after 300 blade passes.
Axial mixing (Figure 10b) is significantly slower than the radial mixing for all speeds. For
1 Hz, the axial measure declines monotically and steadily. For 2.25 Hz, there is an early
rise in the measure as the ballistic throwing of material from the central region out onto the
bed moves the axial centroid initially further from the blade than at the start. The mixing
rate though is higher and by XX revs, the axial mixing for 2.25 Hz is better than for 1 Hz.
For a 4 Hz speed, the amount of material thrown ballistically in the butterfly loops above
the bed is much larger so there is a much larger early net migration of material away from
the plough giving a larger increase in the axial centroid measure. The axial mixing rate for
this speed is higher than for the lower speeds with this speed having the highest mixing
after 120 revs. The long term radial mixing appears to be limited by the rate of axial
transport of particles from the dead zones. The radial mixing is relatively effective at
mixing particles that are in the active mixing zone near the plough and these become
quickly well mixed. The unmixed material near or in the dead regions cannot be mixed
radially until it is transported axially into this mixing zone. This transport is a slow process
12
controlled by the slow steady deformation of the bed as ballistic material settles on top
causing it to slowly creep back towards the plough. Eventually, particles that started in
locations between the dead regions and the mixing zone move into the active zone where
they are rapidly radially mixed.
6.2
Mixing characterisation from PEPT data
Axial and radial mixing in granular flow can be characterised for PEPT using the mean
square of axial dispersion coefficient and root mean square of radial displacement
respectively, as shown in [22]. We define D to be the dispersion coefficient using the
Einstein law:
D = lim
Δx 2
(1)
Δt→0 2Δt
where < x2> is the mean square axial displacement considered during the time interval t.
Figure 11 shows the mean square of axial displacement which increases linearly with the
number of blade passes. D is then the gradient of the line fitting the mean square of axial
displacements. It increases with blade speed from 30 mm2 per blade pass for 1.0 Hz to 90
mm2 per blade pass for 4.0 Hz. Earlier studies (see [22]) for Froude numbers lower than 1
showed a linear dependence of mixing with agitator speed. Here, the Froude number varies
from 0.5 for a speed of 1.0 Hz to 8.0 for a speed of 4.0 Hz and a more complex behaviour is
observed.
Figure 12 shows No.P(N), the scaled power function of the Fourier Transform P(N) of the
axial displacement of the tracer, versus the non-dimensional frequency N/ No, where No is
the agitator speed which varies between 1 Hz and 4 Hz. All three curves exhibit a peak at
the frequency of rotation of the agitator corresponding to an axial impulse at every blade
pass. The integral of the curve under this peak decreases with increase of speed which is
especially noticeable for the 4 Hz case suggesting that the particle motion is now more
chaotic and that the type of flow regime is different from that at 1.0 and 2.25 Hz. The
asymptotic behaviour of the spectra shows a characteristic 1/N dependence for dispersed
systems in accordance with molecular dynamic simulations by Savage [23], mathematical
calculations by Bak et al., [24] and by Shlesinger and West [25] and experimental work by
Miller et al. [26].
13
Figure 13a shows the influence of the agitator speed on RMSr (the root mean square of the
radial displacement). It first increases with number of blade passes but then decreases
modestly after about 10 blade passes for 1 Hz. For 2.25 Hz, it first increases strongly and
then pauses for several revolutions also after about 10 blade passes. Such reversal or
pausing features, already observed in a powder mixer of similar geometry (see [22]) are
thought to be characteristic of radial agitation in systems stirred by a single mixing element.
The same behaviour is observed in the DEM radial centroid motion with a reversal for 1 Hz
at 10 revolutions and a sharp reduction in the radial mixing rate for 2.25 Hz. The radial
dispersion decreases with increase of speed. The limit reached after 200 blade rotations is
27 mm, 15 mm and 9 mm for an agitator speed of 1.0 Hz, 2.25 Hz and 4.0 Hz respectively
These values can be compared to two limiting cases. The first case is the theoretical limit
for RMSr based on the mixer geometry and fill level, which is 40 mm. The other extreme
case is where all the material is centrifuged and fills an annular volume against the inner
cylindrical wall of the mixing chamber. With a fill of 25%, this corresponds to a thickness
of 20 mm and yields a minimum value for RMSr of 7 mm. As agitator speed increases, the
centrifugal forces generated by the plough increase and the RMSr tends towards this value.
Figure 13b shows the temporal variation of the RMS of axial displacement for the three
agitation speed investigated. After 300 blade passes, RMSx reaches asymptotic values of 50
mm, 60 mm and 120 mm for the 1 Hz, 2.25 Hz and 4 Hz blade speeds. The axial
dispersion increases with agitation speed. This is consistent with the trend for ratios of axial
centroid motion predicted by the DEM simulation (see Figure 10b). The asymptotic values
may be compared to the theoretical limit of RMSx corresponding to the ideal case where a
tracer starting from any point in the axial direction has an equal probability of reaching any
point in the cylinder length of length L. This limit is L/ 6 , which gives a theoretical limit
of 180 mm for this mixer. The asymptotic PEPT values are significantly lower than this
reflecting the impact of the large dead regions at either end of the mixer. The measured
limit for 4 Hz is 120 mm which is equivalent to an axial length of 300 mm. This indicates
that the dead regions occupy about 1/3 the length of the bed for this high blade speed. This
rises to 2/3 of the length of the mixer for the lowest speed of 1 Hz.
6.3
Relating PEPT and DEM mixing measures
Table 1 shows the variation of the PEPT axial dispersion and the DEM relative change in
centroid location for the three blade rotation speeds. The PEPT dispersion coefficient
14
increases by 17% when the speed increases from 1.0 to 2.25 Hz. This compares to a 22%
increase in the centroid migration rate. The increase in the dispersion coefficient for 4 Hz
compared to 1 Hz is a factor of 3.0 which is quite similar to the 2.56 factor change observed
in the centroid location. There appears to be a very good correlation between the two quite
differently calculated measures, which indicates that they are similarly characterising the
mixing produced by the plough.
7
Asymptotic states
Figure 14 shows the DEM predictions of the near asymptotic state of the bed for the three
mixer speeds when viewed from above the bed. For the 1 Hz speed, the central mixed
region can be seen extending from the either side of the plough path. The mixed region is
broadest at the far side of the shell to which the blade is pushing the particles and narrowest
on the closest side where the plough first enters the bed. On this leading side the mixed
region is only about 1/3 of the length of the mixer and on the far side it is about ¾ of the
length. For the other two speeds, the surface of the bed appears to be very well mixed.
Figure 15 shows the same bed states but now from a lower viewpoint and with the front
part of the bed removed from the visualisation so that the mixing state below the surface
can be seen. For the 1 Hz speed, the mixed region occupies just the V shaped trench close
by on either side of the blade path. For 2.25 Hz, the mixed region is now quite wide, but
there are very clear large dead regions of undisturbed material to either side. For the 4 Hz,
speed, the central mixed region is again somewhat broader, but again there remain large
dead zones at both ends. These were not visible in Figure 14 because the ballistic
trajectories in the upper butterfly circulation results in a relatively thin layer of well mixed
material being deposited over the surface of the bed, making it appear to be fully mixed.
The presence of the large dead regions is consistent with the asymptotic limits of the radial
centroid measures of 0.55, 0.47 and 0.24 for the three speeds, as shown in Figure 10. They
are also consistent with the PEPT axial RMSx values which have asymptotic values of 50
mm, 60 mm and 120 mm for the three speeds. So the DEM flow visualisation, the DEM
centroid mixing measures and the PEPT tracer dispersion measures all give a consistent
picture of large dead regions on either side of the plough blade with the size of the dead
regions decreasing moderately with increasing blade speed.
15
8
Issues concerning validity and comparison of PEPT and DEM data
The comparison between experimental and numerical results provided by PEPT and DEM
respectively show reasonable qualitative similarities for flow and mixing in a single blade
ploughshare mixer. DEM modelling of laboratory scale mixing equipment with 100,000’s
to millions of mm size particles is now feasible for hundreds of revolutions and quantitative
mixing rates can be predicted. The comparison of the DEM and PEPT reveals reasonable
qualitative and quantitative similarities. Indeed, the topology of the flow structure was
observed to be practically identical in both sets of results. However, this work also
highlights some current technological limits of both the experimental and the simulation
methods, issues that need to be resolved in making such comparative work more rigorous in
the future.
The comparisons demonstrate the critical importance of the particle shape and highlight the
strong penalties for modelling particles with spheres in DEM when the particles are not
spherical. Specifically, the use of spherical particles in the DEM leads to sharply lower
angle of repose and shorter dead regions at the ends of the mixer. In the PEPT experiments,
it was also difficult to irradiate the rice grains and so a tracer with a different size and a near
spherical shape had to be used. This introduces the probability that the tracer motion has
been subject to some degree of segregation, which means that the tracer is not equally
likely to sample all parts of the phase space. This leads to different weighting of different
parts of the flow in the PEPT predictions of the flow pattern, which in turn distort the
results to some degree. Here the differences in tracer and bed materials was not severe and
was within the ranges previously investigated and found to not have strong effects.
Nonetheless, the PEPT results would be more reliable if tracers could correspond exactly to
material used for the beds in the experiments. Some important limits on the current PEPT
experiments were also revealed. Low data counts in the dilute high speed areas makes
PEPT averaging less reliable in the space above the bed where particles move on high
ballistic trajectories. This is ultimately an issue of the maximum speeds that can be
accurately resolved by a given PEPT camera. In this case, it lead to more than half the
physical space not giving usable results. Similarly, the PEPT tracer was unable to enter the
dead regions which are therefore invisible in the experimental results.
An important, but subtle, issue is that both methods use different ways of averaging the
data. In DEM, averaging is typically performed over all the particles and perhaps some
16
short term time averaging. In general, this is done because the DEM method is
computationally expensive and so long simulations are typically not desirable. In this work,
the simulations were performed over 300 revolutions so individual particle tracking,
mimicking the PEPT could have been performed, but typically this is not possible. The
averaging over all particles means that all physically accessible parts of the phase space are
included in the averaged properties and they have equal mass weight. For PEPT, the
averaging is over time for one particle. If the motion of this tracer is ergodic, that is it
samples all parts of the phase space with equal probability, then this trajectory average
should give the same outcome as the average over all the particles in the system. However,
the system is not ergodic. The phase space typically has complex structure, with potentially
multiple non-interconnected flow regions and where some trajectories have higher
probabilities than others. Size and shape induced segregation, both near the moving plough
blade and in the recirculating bed, influence the trajectories of the tracer. For example,
since the tracer is smaller, its will migrate towards the plough blade in the shear flow of this
granular boundary layers. This means that the tracer will be predisposed to be closer to the
blade. Particles closer to the blade are more often lifted out of the bed and more often
thrown on high trajectories. Similarly, segregation in the avalanching flow back into the
trench means that the tracer will not equally sample all the possible trajectories in this part
of the phase space. Since the PEPT data is averaged over these trajectories, it means that
the tracer will be more predisposed to some spatial areas which will then have higher
average weightings in the averages. The PEPT analysis also assumes that the flow of the
bulk particles in the bed is the same as that of the tracer, which is then not quite true. These
issues mean that there will always be distortions in the PEPT averaged data. It is currently
assumed that these are small. These averaging issues therefore deserve attention in future
development of the PEPT method.
9
Conclusions
The investigation of flow and mixing in a single plough powder mixer showed the
existence of a re-circulation zone on either side of the blade with dead regions further from
the blade. Material is either pushed on both sides of the plough at low agitation speed or
lifted and thrown in the open space above the free surface of the bed at high speeds. This
motion induced by the blade leaves a trench behind that fills by a slower surface avalanche
17
flow. The plough itself acts as a separatrix between the two loops of circulation and
prevents axial convection between the two halves of the mixer.
Mixing rates were found to be speed dependent in terms of number of blade passes to mix
to a certain state. Radial mixing is initially rapid and then slows substantially with increase
of speed. Axial mixing is very slow, limiting the long term radially mixed state. The rate of
axial mixing was found to be nearly speed independent at low speeds and to increase at
high speed. Radial mixing was found to be improving with increase of speed.
Reasonable qualitative agreement was found between the PEPT and DEM averaged flow
data (specifically the occupancy/solid fraction and the bed speed). However, the angle of
repose of the plough trench was found to be far too low in the DEM simulations, which is
the result of approximating the particles as spherical. In future work, more realistic particles
shapes need to be used to improve the quantitative agreement. Good quantitative agreement
was found between PEPT and DEM for the mixing behaviour. Mixing was characterised
for the PEPT data using dispersion coefficients and RMS displacements of the trajectory of
the tracer over time. In the DEM, mixing was characterised using the movement of the
normalised relative centroids of different initial parts of the bed. The mixing measures of
the PEPT and DEM correlated well and gave consistent variations in mixing with
increasing blade speed. Finally, a range of subtle issues relating to the quality and
averaging of PEPT data were identified and discussed.
Notation
D
[mm2/s]
Dispersion coefficient
L
[mm]
Characteristic axial length of the mixer
N
[Hz]
agitator speed
RMSr
[mm]
Root mean square of radial displacement
RMSx
[mm]
Root mean square of axial displacement
< x2>
[mm2]
Mean square of axial displacement
10 Acknowledgements
This paper is dedicated to his memory of the first author Bruno Laurent who sadly passed
away before it was completed. The final revisions of the paper were completed by Paul
18
Cleary on behalf of his co-author. Thanks are extended to D.J. Parker of the Positron
Imaging Centre, The University of Birmingham, for his technical support to the first author.
11 References
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Sensitivity to mill operating conditions, liner geometry and charge composition'', Int.
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20
Tables
Blade Speed
(Hz)
PEPT axial
dispersion
coefficient
DEM change in
relative axial
centroid
PEPT
normalised by 1
Hz value
DEM centroid
normalised by 1
Hz value
1.0
30
0.18
1.0
1.0
2.25
35
0.22
1.17
1.22
4.0
90
0.46
3.0
2.56
Table 1: Comparison of the PEPT axial dispersion coefficient and the change in DEM
relative centroid.
21
Figures
tracer
trajectory
Plan View
y
Vax
D = 250 mm
Vrad
x
Vtg
z
Plough
Cross-sectional
View
L = 450 mm
Side View
Figure 1: Configuration of the mixer showing the external shell and the plough blade.
22
30
90
10
65
125
5
107
50
50
65
107
40
Figure 2: (top) Sketches of the top view and side view of the plough (dimensions in mm)
and (bottom) two views of the CAD model used in the DEM.
23
40mm/s
Figure 3: Side and cross-sectional views of the PEPT velocity fields for the particle bed
induced by the motion of the plough at different phases of the plough blade for a 25% fill
and 2.25 Hz rotational speed. The dotted lines mark the loci of the edges of the plough.
24
a) Initial set-up
b) After half a blade pass
c) After ¾ blade pass
d) After one blade pass
e) After one and a quarter blade pass
f) After one and a half blade pass
Figure 4: Particle distributions predicted by DEM in the mixer with plough blade revolving
at 2.25 Hz at a) the start, b) after 0.5 rev, c) 0.75 rev, d) 1.0 rev, e) 1.25 rev and f) 1.5 rev.
25
Direction of rotation
a)
50 mm
0.470
> 0.2%
0.18-0.20%
0.16-0.18%
0.14-0.16%
0.12-0.14%
0.10-0.12%
0.08-0.10%
0.06-0.08%
0.04-0.06%
0.02-0.04%
0.00-0.02%
0
0.402
0
b)
Figure 5: Time occupancy (PEPT) and modified solid fraction of the bed (DEM) for a blade
speed of 2.25 Hz; a) cross-sectional view of the mixer, and b) side view of the mixer.
26
>. 480.
384.- 480.
288.- 384.
192.- 288.
96.- 192.
0.- 96.
-96.- 0.
-192.- -96.
-288.--192.
-384.--288.
< -384.
a)
b)
Figure 6: Average axial velocity distribution when viewed from the front of the mixer for,
a) PEPT with maximum axial speeds of 0.48 m/s (pink), and b) DEM with blue being
negative, green around zero and red being positive with peak axial speed of 0.5 m/s. The
DEM data fills the most of the volume of the mixer. Valid PEPT data was only available in
the lower half of the mixer near the plough blade, with the horizontal lines showing the top
and bottom of the mixer shell.
27
direction of rotation
a)
> 100.
90.- 100.
80.- 90.
70.- 80.
60.- 70.
50.- 60.
40.- 50.
30.- 40.
20.- 30.
10.- 20.
0.- 10.
b)
Figure 7: Axial averaged speed distribution for; a) PEPT, and b) DEM.
28
direction of rotation
Side view
a)
Cross-sectional view
> 0.5%
0.45- 0.50%
0.40- 0.45%
0.35- 0.40%
0.30- 0.35%
0.25- 0.30%
0.20- 0.25%
0.15- 0.20%
0.10- 0.15%
0.05- 0.10%
0.00- 0.05%
Plan view
50 mm
direction of rotation
Side view
Cross-sectional view
> 0.5%
0.45- 0.50%
0.40- 0.45%
0.35- 0.40%
0.30- 0.35%
0.25- 0.30%
0.20- 0.25%
0.15- 0.20%
0.10- 0.15%
0.05- 0.10%
0.00- 0.05%
b) Plan view
Figure 8: Reconstruction of the dispersion of a labelled slice of material near the wall of the
mixer for a blade speed of 2.25 Hz; a) initial configuration of the labelled slice, and b)
dispersion of the labelled slice after 1 blade rotation.
29
direction of rotation
Side view
c)
Cross-sectional view
> 0.5%
0.45- 0.50%
0.40- 0.45%
0.35- 0.40%
0.30- 0.35%
0.25- 0.30%
0.20- 0.25%
0.15- 0.20%
0.10- 0.15%
0.05- 0.10%
0.00- 0.05%
Plan view
direction of rotation
Side view
Cross-sectional view
> 0.5%
0.45- 0.50%
0.40- 0.45%
0.35- 0.40%
0.30- 0.35%
0.25- 0.30%
0.20- 0.25%
0.15- 0.20%
0.10- 0.15%
0.05- 0.10%
0.00- 0.05%
d) Plan view
Figure 8 (ctd): Reconstruction of the dispersion of a labelled slice of material near the wall
of the mixer for a blade speed of 2.25 Hz; c) dispersion of the labelled slice after 5 blade
rotations, d) dispersion of the labelled slice after 10 blade rotations.
30
Figure 9: Azimuthal, radial and axial displacement of the tracer from the PEPT experiment
showing, a) the full time range, and b) a narrow time range so that the shunts in bed
recirculation can be seen.
31
1
0.9
1Hz
2.25Hz
Normalised centroid position
0.8
4Hz
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
0
20
40
60
80
100
120
140
Number of blade passes
1.2
Normalised centroid position
1
0.8
0.6
0.4
1Hz
2.25Hz
0.2
4Hz
0
0
50
100
150
200
250
300
350
Number of blade passes
Figure 10: Influence of the agitator speed on the displacement of the centroid in the DEM
simulation using the front quadrant of the mixer, a) radial, and b) axial.
32
2000
1800
1600
1400
1200
1000
800
600
1Hz
400
2.25Hz
4Hz
200
0
0
2
4
6
8
10
Number of blade passes
Figure 11: Influence of the blade speed on the mean square of axial displacement.
33
0.01
0.01
0.1
1
10
1Hz
2.25Hz
4Hz
No.P(N/No) (mm2.s)
0.001
0.0001
0.00001
N/No
Figure 12: Scaled power spectrum NoP(N/No) of the axial displacement vs. scaled
frequency N/No for blade speeds of 1.0, 2.25 and 4.0 Hz.
34
30
25
RMSr (mm)
20
15
10
1Hz
5
2.25Hz
4Hz
0
0
50
100
150
200
Number of blade passes
a)
120
RMSx (mm)
100
80
60
40
1Hz
2.25Hz
20
4Hz
0
0
b)
100
200
300
400
Number of blade passes
500
600
700
Figure 13: Influence of the agitator speed on mixing in the PEPT experiments, a) RMSr, the
root mean square of the radial displacement, and b) RMSx, the root mean square of the axial
displacement.
35
a)
b)
c)
Figure 14: Near asymptotic bed states as predicted by DEM and viewed from above for; a)
1 Hz, b) 2.25 Hz and c) 4 Hz.
36
a)
b)
c)
Figure 15: Near asymptotic bed states as predicted by DEM and viewed from the front
showing significant dead regions at both ends of the mixer for: a) 1 Hz, b) 2.25 Hz and c) 4
Hz.
37