Granular Materials: Observing S-systems
through the use of Soda-Lime Beads
Tommy Boykin
Department of Physics, Berea College
April 30, 2013
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Contents
1 Introduction
3
2 Experimental Design and Methods
4
3 Results and Data Analysis
7
4 Conclusion
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2
1
Introduction
Granular materials are defined as the collection of discrete particles that can
behave like liquids or solids under certain conditions [1]. Within the field of
granular materials, one of the phenomena studied is the segregation among
bidisperse particle systems. Along with using the name bidisperse particle
systems, mixtures may also be referred to as binary systems. The reason for
using names such as bidisperse and binary is due to the mixtures containing
particles that possess properties that differ from one another such as the size
of the particles.
The study of segregation appeals to many industries such as pharmaceutical industries, papermaking industries, construction industries, and even
nuclear fuel industries. For example, in pharmaceutical industries, companies want to produce drugs with the correct amount of ingredients. In order
to produce drugs with the correct amount of ingredients, the products must
be mixed with equal amounts of active and inactive ingredients. The active
ingredients give off the drug’s desired effect. The inactive ingredients help
deliver the active ingredients to the desired parts of the body. If a product
contains too much of the active ingredient, the patient could possibly overdose. If a product contains too much of the inactive ingredient, the patient
would not experience the desired effect of the drug [1].
The process pharmaceutical companies follow to make a product contain
the correct proportions of active and inactive ingredients is mixing. However,
within the process of mixing there seems to be a problem. Pharmaceutical
companies have noticed during the process of mixing that the active and
inactive ingredients are not mixing; in fact the active and inactive ingredients are segregating. Due to the ingredients segregating, the companies are
receiving concentrated regions of active and inactive ingredients. As stated
above, having concentrated regions of the ingredients is not desirable due to
the patients health being at risk. It is due to patients health being at risk
that segregation and the conditions under which it occurs are of great interest. Through this research, one condition which will be observed is the filling
fraction of the container and whether the filling fraction has a important role
in observing segregation. The filling fraction of the container refers to the
volume of the container occupied by the mixture of particles.
One important distinction to make clear is the difference between mixing
and segregating. Mixing is a process in which there are no changes in the
concentration of a mixture even after the mixture has been rotated at a
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fixed angular speed. Segregating is a process in which changes occur in the
concentration of a mixture after being rotated at a fixed angular speed due
to the interaction between like particles. One way to define segregation for
a mixture is as the departure away from the mixture’s initial homogeneity.
To study bidisperse particle systems, researchers have defined two types
of systems. These systems are called S-systems and D-systems. S-systems
are systems in which the particles have the same density but differ in size.
D-systems are systems in which the particles have the same size but differ
in density. The system chosen to study through this experiment is the Ssystem. The particles chosen to fuel this study of S-systems were Soda-Lime
glass beads. While glass beads may not seem like a granular material, they
are often used for research purposes. Using glass beads also allows for certain
health hazards to be avoided that pharmaceutical powders bring if inhaled
[1].
One main goal for this project was to verify that segregation occurs for
different size glass beads with varying filling fractions.
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Experimental Design and Methods
The experimental setup used for this study of soda-lime glass beads is shown
in figure 1. For this experiment, we needed a way to study and observe
bidisperse systems. Therefore, we made the experimental setup shown in
figure 1 along with a plexiglass drum which was used to contain and observe
the glass beads under rotation. Prasad uses a rotating cylinder drum to
study glass beads, but also uses a coloring method to distinguish beads from
one another [2]. Through using a similar technique as Prasad, we were able
to construct our own plexiglass drum and devise a method for observing the
beads while rotating. The plexiglass drum we built has a length of 11.5cm
and an inner diameter of 10.1cm. To hold the glass beads inside of the drum
while rotating, two circular covers were made from plexiglass. One of the
glass covers is fixed to the drum with spray adhesive. The second glass cover
is attached to the drum with tape as this allows for beads to be easily added
and removed from the drum.
In terms of viewing the beads while the drum was rotating, we devised
a method for making the beads distinguishable and visible. This goal was
achieved through using water color paint. To be sure the water color paint
did not affect the motion of the glass beads, we placed the beads in the
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plexiglass drum and observed whether the beads motion was affected by the
paint. It was observed that the beads were not affected by being painted. In
order to record data, a Logitech HD 1080p webcam was used. The webcam
recorded data from the side of the plexiglass drum held together by the spray
adhesive. Figure 2 shows an example of the beads contained in the plexiglass
drum using the 4mm and 7mm beads.
Figure 1: A picture of the experimental setup. On the left side of the picture,
there is a rotary motor attached to a piece of teflon. There are two screws
holding the teflon to the motor and the steel rod on the right side of the
teflon. The steel rod connected to the motor is placed through two wheel
bearings which are fixed in the wooden blocks. The wheel bearings serve as
a way for the rods to move easily while the system is rotating. Between the
steel rod connected to the motor and the other rod are three black rubber
bands. These rubber bands allow the steels rods to move in a synchronized
manner. The rod not connected to the motor is also placed through wheel
bearings which are fixed in the wooden blocks. On the end of the rod not
connected to the motor, there are two cut pieces of black rubber. The black
rubber is placed here in order to make suffient contact with the plexiglass
drum while the system is rotating.
In all of the mixtures the smaller beads were painted with a bright red
color and the larger beads a dark green. The choosing of these colors did not
depend on another experiment. It was through performing our experiment
that red and green colors showed the best contrast between beads. The
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Figure 2: A picture of 4mm and 7mm beads contained in the plexiglass
drum at t=0 seconds. For all of the experiments, the red and green beads
were mixed throughout the container and data was recorded for t=0 seconds.
There is a black marker located on the edge of the plexiglass drum. This
maker served two purposes. One purpose was as a standard for the scale
when using Logger Pro to track the particles position. The second purpose
of the marker was to provide an indication when the drum has completed
one revolution. The length of the marker is 0.517cm.
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smaller beads used for this experiment were the 2mm, 3mm, 4mm, 5mm,
and 6mm size beads. The larger beads used for this experiment were the
7mm size beads.
The first set of experiments we performed were with a filling fraction of
0.5. One of the mixtures we observed with a filling fraction of 0.5 was a
mixture containing the 2mm and 7mm size beads. In order to carry out this
experiment, all of the red 2mm size beads and the green 7mm size beads
needed to fill the bottom of the plexiglass drum first. Once the bottom of
the drum was covered with a sufficient amount of painted 2mm and 7mm size
beads, the remaining filling fraction of the drum was filled with unpainted
2mm and 7mm size beads. This same process was repeated with 3mm, 4mm,
5mm, and 6mm size beads also painted red.
We kept the number of 7mm size beads within the drum fixed for each of
the repeated mixtures. When all of the data was collected for 2mm, 3mm,
4mm, 5mm, and 6mm size beads with a filling fraction of 0.5, we decided to
lower the filling fraction of the drum to 0.375. The same process as stated
in the previous paragraph was repeated for all five mixtures.
For recording data, the plexiglass drum was allowed to rotate one revolution at 0.022 revolutions per second. Once one revolution was completed,
Logger Pro analysis software was used to track the x-coordinate and ycoordinate of a number of particles initially at t=0 seconds and t=43.89
seconds. It is important to mention t=0 seconds is not the beginning of the
video for any of the mixtures. The frame for t=0 seconds is the frame within
a video where the plexiglass drum has not rotated, but in the very next frame
the drum has begun rotating. An example of how the frames look at t=0
seconds is shown in figure 3. An example of how the frames look at t=43.89
seconds is shown in figure 4. The mixture in both figures 3 and 4 is with 4mm
and 7mm size beads. Once the x-coordinate and y-coordinate of a particle
were found we could observe whether the smaller size beads moved closer or
farther away from the axis of rotation.
3
Results and Data Analysis
From the report by Tapia-McClung, ”[i]t is known that particles of higher
density or smaller size segregate to the core of the granular bed while particles
of lower density or larger size segregate to the outer edges and to the flowing
layer” [4]. From this statement, we devised a systematic method to collect
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Figure 3: A picture representing a
homogeneous mixture at t=0 seconds.
Figure 4: A picture representing a segregated mixture at t=43.89 seconds.
data which would show whether particles of smaller size segregate to the core
and particles of larger size segregate to the outer edges of a granular bed.
The method used to investigate this theory was one we developed ourselves.
The method involved finding the average distance a population of particles
has travelled around the axis of rotation. For example, if a mixture contains
2mm and 7mm size beads we are interested in knowing the average distance
the 2mm size beads are away from the axis of rotation after one revolution.
There is not a single theory which describes the behavior of granular
materials. However, based on empirical evidence that smaller size beads segregate to the core of a granular bed, we expected to see that as the difference
in the size of bead increases the average distance away from the axis of rotation decreases. When we report the difference in bead size, the 7mm size
beads are compared to all of the other beads. For example, the size difference
of 0.1cm on the x-axis of figure 5 corresponds to a mixture of the 6mm and
7mm size beads. Figure 5 shows the trend for the average distance from the
axis of rotation versus the difference in the size of the beads with a filling
fraction of 0.5.
From figure 5, it can be seen from the data that as the difference in
the size of the beads increases the average distance away from the axis of
rotation decreases. However, with the data taken for the 2mm size bead, it
would seem as though there is a point at which segregation is not observed.
At this point, we decided to record data for all of the mixtures with a filling
fraction of 0.375. Figure 6 shows the trend for the average distance from the
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axis of rotation versus the difference in the size of the beads with a filling
fraction of 0.375. In figure 6, the trend for the average distance away from
the axis of rotation decreasing as the difference in the size of the beads is
increasing begins with the 5mm size beads. Therefore, in figure 6 the range
for the size of the beads which will segregate seems to have become smaller.
In figures 7 and 8, we took the data from figures 5 and 6 and made plots
for the difference between the average distance specific beads are away from
the axis of rotation. For example, we have data for the average distance the
2mm beads are away from the axis of rotation at t=0 seconds and t=43.89
seconds. If we take the difference between these two values, we have the
graphs pictured in figures 7 and 8. The
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Conclusion
The mixing of granular materials is usually thought to have no significance.
This is due to granular materials always being seen as homogeneous and
never as inhomogeneous. In this study, we made a systematic way to study
S-systems and show the distance smaller beads are from the axis of rotation
becomes smaller as the difference in bead size increases.
References
[1] Jesse Debes, Brittany Friedman, Grace Kim, and Kristina Vishnevetskaya, Segregation of Particulate Mixtures, Line Building Products.
2010.
[2] D. V. N. Prasad and D. V. Khakhar, Mixing of granular materials in
rotating cylinders with noncircular cross sections, Physics of Fluids. 22,
103302 (2010).
[3] G. Seiden, P.J. Thomas, Complexity, segregation, and pattern formation
in rotating-drum flows, Rev. of Modern Physics. 83, (2011).
[4] H. Tapia-McClung, Numerical Simulation of Granular Materials in a
Rotating Tumbler, Journal of Mechanics of Materials and Structures. 2,
8 (2007).
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Figure 5: This graph
displays the trend
between the average
distance smaller size
beads move relative
to axis of rotation
versus the difference
in size. The graph
was obtained using
2mm, 3mm, 4mm,
5mm, and 6mm size
beads mixed with
7mm size beads in a
filling fraction of 0.5.
The y-axis displays
the average distance
the smaller beads
are away from the
axis of rotation as a
percentage of the radius of the plexiglass
drum.
The marks
on the y-axis are in
incremental steps of
5 percent of the total
radius. The radius of
the plexiglass drum is
5.07cm.
Figure 6: The graph was
obtained using 2mm, 3mm,
4mm, 5mm, and 6mm size
beads mixed with 7mm size
beads in a filling fraction of
0.375. The same display as
used on the y-axis in figure
5 is used here.
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Figure 7: This
graph represents
the
difference
between the averages distances
each bead is
away from the
axis of rotation
for
a
filling
fraction of 0.5.
In other words,
to calculate the
values on the
y-axis the average distance for
each size bead
at t=0 seconds
and the average distance at
t=43.89 seconds
were used. The
same
display
as used on the
y-axis in figure 5
is used here.
Figure 8: This graph represents the difference between the averages distances each bead is away
from the axis of rotation for a filling fraction of
0.375. As in figure 7, to calculate the values on the
y-axis the average distance for each size bead was
at t=0 seconds and t=43.89 seconds. The same
display as used on the y-axis in figure 5 is used
here.
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