Particle Segregation of Inclined Plane Flows
Brian Lawney
Clarkson University, Potsdam, NY 13699-6261
Advisor: Dr. Hayley H. Shen
Department of Civil and Environmental Engineering, Clarkson University, Potsdam, NY
Abstract
Particle segregation is an important feature of non-uniform sheared granular flows, in which large particles
rise to the top of the flow regime. By changing the flow’s spatial composition, segregation has significant
consequences on the behavior of mixed particle systems. In an attempt to better understand the effects of
this mechanism of particle transport, experimental and computational studies have been developed to
model the dynamics of various geometrical arrangements. One particular problem of interest is the inclined
plane flow. Thus far, research on this topic has linked segregation rate to such factors as relative particle
size ratio, angle of incline, and shear rate. Other works on mixed flow have cited the importance of
boundary conditions on the rate of segregation, but none have studied this in particular. By utilizing an
existing numerical model modified for this specific incline geometry, this problem can be examined in
detail. Thus, future work will focus on studying the relationship between boundary roughness and
segregation rate. If time permits, additional studies will be performed to determine if there exists a
relationship between relative large particle concentration and segregation rate.
I. Introduction:
The study of segregation phenomena is of
particular importance in applications where the
features of non-uniform granular flow are of
interest.
Many processes involving granular
material are conducted with particles of varying
sizes, and an accurate model of the mechanics of
the system may be advantageous. The change in
the even dispersion of different size particles may
have some important consequences. In certain
applications, the result of this flow property may be
undesirable, as it affects the homogeneity of the
mixture. In yet other processes, the separation of
different sized particles could be an integral part of
system operations. It is for these reasons, among
others encountered in many disciplines, that the
mechanics of non-uniform granular flows must be
studied.
Particle segregation on an incline is characteristic
of a non-uniform, gravity driven granular flow. In
systems with varying particle sizes, it has been
shown that larger particles will generally segregate
to the top of the flow regime, while smaller
particles will remain at the bottom. This topic has
been studied in various ways, typically with the
system being excited with periodic vibration [1, 2],
sheared in a cylindrical geometry [3], or gravity
driven on an inclined slope [4]. Both experimental
[2, 3] and numerical computational methods [1, 4]
have been developed to model the behavior of the
particular systems. In the study of segregation, two
possible explanations for the process have been
proposed.
The first theory, supported by computer
simulations performed by Jullien and Meakin, [1]
contends that during flow movement, conical
shaped voids develop underneath large particles as
they are displaced vertically. It was found that
grains of smaller size would translate or
“avalanche” into the space, and subsequently build
up underneath the larger particles. Thus, in each
successive excitation, larger particles would
displace upwards, against gravity.
In their
simulations, vertical periodic shaking provided the
mechanism for particle transport.
The second theory, studied in a report by Knight et
al. [2], proposes that the mechanism of segregation
is a convection process, highly dependent upon the
boundary conditions of the flow.
In their
experiments, large particles translated to the top of
the flow in a fashion similar to a convection cycle.
Depending upon the relative size of the grains, it
was found that the large particles would eventually
return downward (with gravity) a short distance
before being forced to the top again. When
convective processes were controlled in
experimentation,
no
segregation
occurred,
suggesting the proposed mechanism. The authors
of the report also assert that friction and/or inelastic
collision of the flow particles with the boundaries
is essential to observing convective segregation
behavior.
They cite the periodic boundary
conditions of Jullien and Meakin’s [1] model as a
possible reason for the simulation’s inability to
observe convection in the flow.
In the experiments of Khosropour and Zirinsky [3],
glass beads of various sizes were sheared between
two glass cylinders to study the convective theory
of segregation. It was found that the larger beads
did indeed rise to the top of the flow regime, and
the smaller, more uniform assortment of particles
at the bottom displayed typical convection cycles
of rising and falling. Their findings supported the
assertion that boundary conditions were critical in
the development of segregation, but it was
concluded that the role of frictional boundaries
warranted further consideration.
In perhaps a more relevant (in terms of geometry
and method) study, Hirshfeld and Rapaport [4]
performed a two-dimensional numerical simulation
of non-uniform inclined plane flow. In their report,
the relationship between slope angle and
segregation rate, and the temporal and spatial
dependence on segregation was investigated. It
was reported that larger differences in relative
particle sizes and higher shear rates contributed to
faster segregation.
For the proposed study, a numerical simulation
code developed by Babić [5] will be utilized. In
the time since its initial formulation, the code has
been modified to investigate specific behavior of
shear granular systems, including the case of
inclined plane flows, which will be the focus of
future work. Currently, the model has been altered
to study non-uniform flows in other geometric
configurations. The specifics of this program and
the physical model will be discussed in the
methodology section.
A common theme expressed in much of the
literature was the need for further study into the
role of boundary conditions on segregation. Thus,
it may be of interest to study the segregation rate
under different roughness conditions imposed by
the incline, and possibly side walls. By varying the
size of the static particles composing the incline,
and including additional frictional losses due to
“walls”, it will be possible to investigate this.
Additionally, both [4] and [1] determined a general
relationship between relative particle size and
segregation rate. Both reports found that a critical
size ratio of greater than 2 is required to observe
appreciable segregation. It may also be interest to
determine a relationship between segregation rate
and the concentration of large particles. By
controlling the amount of large particles (in
relation to smaller ones), it may be possible to
change the segregation effects of the flow. If a
relationship exists, it could have significant
implications.
II. Methodology
The numerical code formulated by Babić [5] was
developed as a calculation tool for simulating the
behavior of various two-dimensional discrete
particle systems. By the inclusion of additional
code through further development, the program has
become quite modular, and has been used to study
several different granular flow cases. Thus far, it
has been used in the analysis of couette, channel,
and chute (inclined) flows [6].
This numerical
simulation will be used in the segregation study
proposed in this report.
The numerical code consists of three main parts. A
simple flow chart, appearing in Babić’s original
report [6] is given in figure 1.
From figure 1, it can be seen that the program
framework is adaptive and can be suited to
particular needs of study. To describe the program
in more detail, discussion of specific algorithms is
necessary.
Model Initialization and preparation:
In the Input/Initialization unit, the model is
prepared. Material properties, physical constants,
initial conditions, and boundary conditions are read
from an input file, and a computational grid is
defined. Walls and other features of the model are
implemented. The model space is then subdivided
into small cells into which only one particle can fit,
generally determined by the size of the particles.
This is important in defining where each particle is,
and where its closest neighbors are. Improper
sizing of the cells will cause problems within the
searching subroutine, and particles will fail to be
detected, resulting in severe overlaps. Thus, for
non-uniform flow, the cell dimensions must be
defined by the smallest particles present.
Start
By some simple trigonometry, it can be shown that
the length and width of the cell (2W), is 2 Rs ,
where Rs is the radius of the smallest particle.
Recent work by Bastien and Ji [7] suggests that
problems may be encountered with overlapping in
the initial packing arrangement, so some changes
may need to be implemented to avoid this
behavior.
Model calculation cycle:
Input and Initialization Unit
-input physical properties
-input or define problem specification
-input of prepare initial conditions
-input or prepare boundary conditions
Main Unit
Repeat:
{-detect all current contacts;
-calculate all current forces;
-sum all forces and moments acting on each
particle;
-integrate equations of motion for all particles to
obtain new positions and velocities;
-update wall positions;
-update statistical summations}
Until:
{Specified end condition is met}
The main unit of the code controls the particle
neighbor searches, force calculations, and the
subsequent particle movement. In the search
process, the program defines a neighborhood of
cells around each particle (based upon the grid that
was created), and constructs a list of possible
particle and/or boundary contacts. With this list in
mind, the distances between neighboring particles
are found, and force calculations are made if the
particles are in contact. With the known forces,
appropriate equations of motion are applied to find
the movement of each grain. This cycle of
searching, calculating forces, and moving the
particles repeats for each time step until the
program is complete. The specific details of the
kinematic equations are provided in the model
development section.
Model Development (formulated by Babić [6]):
Output and Statistics Unit
-print full printout at specified times
-perform time-averaging of
interesting variables and print results
Stop
Figure 1. Flow Chart-from reference [6]
Rs
W
W
Figure 2. Cell size determination
In the analysis of the system, a soft-particle
approach is used to model the collisions that occur.
Unlike the hard-particle model, which assumes
instantaneous collisions, the soft particle technique
features collisions that have finite durations.
Because of this relation, appropriate time steps
need to be chosen so that short contacts are not
missed. This is addressed in the input and
initialization of the program.
Collisions are modeled as viscoelastic, with elastic
and damping forces in both the normal and
tangential directions. The method of analysis is the
same for contact between moving particles, and
contact with walls. A schematic is given in figure
3.
Figure 3. Binary contact model [6]
Because of the nature of frictional forces acting as
moments, the particles in the model can rotate as
well as translate. It should be noted that when
particles are sliding past each other, the tangential
damping contribution (Cs) is ignored.
Each particle shown in the simple binary collision
model has its own spring constant which
contributes to the overall constants shown. To find
this overall stiffness, we recall that elastic elements
in series add as follows:
K eq
1
1
  
 k1 k 2 
1
(1)
It is in this manner that the normal and tangential
(shear) spring coefficients are calculated.
The expression for the damping terms can obtained
in a similar fashion.
C eq
1
1 
 

 C1 C 2 
1
(2)
To derive an appropriate equation of motion to
describe the system, a simple binary collision is
given in figure 4.
Figure 4. Collision model [6,7]
Position vectors, defining the centers of particles A
and B, are defined in the given coordinate system.
rA= (xA, yA)
rB= (xB, yB)
By inspection, the unit vector k lies along a line
connecting the centers of the two particles, and is
given as,
k
rB  rA
 (cos  , sin  )
rB  rA
(3)
The unit vector t, perpendicular to k as shown, is
defined as
t  ( sin  , cos  )
(4)
Locations PA and PB are points of intersection
between the line in the direction of vector k and the
circumference of the respective disks. The relative
velocity of PA with respect to PB can be written as,
VAB  (r A  rB )  ( R AA  RBB )t
(5)
By integrating this result, the relative positions of
the two particles can be obtained. For the purposes
of calculating the spring and damping forces in the
normal and tangential directions, VAB can be
broken into components. To do so, a dot product is
taken between VAB and k to find the normal
relative velocity, and VAB and t for the tangential
velocity.
n  V AB  k 
( x A  x B ) cos   ( y A  y B ) sin  (6)
q  V AB  t 
 ( x A  x B ) sin   ( y A  y B ) cos  
( R A A  R BB )
(7 )
(15)
S s  Fs  Ds
(16)
 S n sin   S s cos  (17)
Fx A  S n cos  S s sin  
(8)
M A  S s R A
(19)
Fx B  Fx A
(20)
F 
(9)
y B
The total spring force is the sum of these
incremental spring forces over the time of the
entire contact. In equation form,
( Fn ) T  ( Fn ) T 1  Fn
(10)
( Fs ) T  ( Fs ) T 1  Fs
(11)
 Fy A
M B  S s RB
(18)
(21)
(22)
It should be noted that the signs are opposite due to
the force convention of Newton’s third law.
Equations of motion:
By applying Newton’s second law for linear and
rotational systems, the equations of motion can be
found, and subsequently the dynamics of each
particle.
Viscous damping calculation:
The damping force associated with particle contact
is proportional to the relative velocities of particles
A and B. Therefore,
For translation,
mr   F  mg
(12)
(13)
Frictional effects:
As noted earlier, the tangential damping is
removed when the particles are sliding. To account
for this, a maximum tangential spring force is
defined as:
( Fs ) max  Fn
S n  Fn  Dn
Resolving these forces into x and y components,
y A
For each time step Δt, the normal and tangential
spring forces are,
Dn  C n n
Ds  C s q
The total force exerted on each particle is the sum
of the elastic and viscous force contributions.
F 
Elastic (spring) force calculation:
Fn  K n nt
Fs  K s qt
Total force:
(14)
where μ is the friction coefficient of the particles.
During a contact, if Fn develops a magnitude
greater than (Fs)max, Fn is equated to (Fs)max and the
tangential damping is ignored.
(23)
where mg is the weight force of the particle.
Naturally, for an accurate model, this weight force
should be specific for the size of particle in
question.
For rotation,
I   M
(24)
The forces and moments acting on the particles are
assumed to be acting for a time step of Δt from t NFor constant linear and angular
1/2 to tN+1/2 [8].
acceleration over this time period, equations (23)
and (24) can be integrated to obtain,
  F

r N 1 2  r N 1 2    N  g t
 m

 M N
 N 1 2   N 1 2 
t
I


(25)
(26)
Performing a second integration, equations for
particle position are given to be,
r N 1  r N  r N 1 2 t
 N 1   N  N 1 2 t
(27)
(28)
III. Future Work
The primary focus of study will be the examination
of the relationship between boundary roughness
and segregation rate. To accomplish this, the
current computer model will be updated to include
new code which will introduce non-uniform
particles to the system. Currently, this task has
been done for other flow cases at Clarkson
University, but not for the incline geometry. Study
of current research on this task will aid in efficient
and accurate implementation of changes.
To study the problem, trials will be performed with
varying wall roughness, and the output data will be
collected. Thus, it may become necessary to
develop additional code which will yield desired
results. The author has already added code which
will aid in the output of velocity profile
information. With the proper data, it may be
possible to develop a quantitative relation between
boundary roughness and segregation rate.
Additionally, frictional losses associated with side
walls will be added, to determine if it has any
bearing on the two dimensional simulation.
A secondary problem of interest, if time permits,
would concern the relationship of large particle
concentration on segregation rate. Based on the
amount of large particles contained in a mixed
flow, it may be possible to optimize or diminish the
rate of segregation.
By utilizing the same
computational model, the relative concentration of
large particles will be varied to investigate its
effects.
IV. Schedule
For the proposed study, a tentative timeline:
April-May 2004: Alter existing code to model
non-uniform flows and output desired information
June-July 2004: Using modified simulation, run
trials relating to the study of segregation rate vs.
boundary roughness. If time permits, run trials for
secondary study. Collect and organize data.
August-September:
Analyze and report data.
Begin initial draft of thesis
October-November:
final thesis
Complete rough draft and
V. References
[1] R. Jullien, P. Meakin, and A. Pavlovitch
(1992). “Three Dimensional Model for ParticleSize Segregation by Shaking”, Physical Review
Letters. Vol. 69, 640-643.
[2] Knight, H.M. Jaeger, and S.R. Nagel (1993).
“Vibration Induced Size Separation in Granular
Media: The Convection Connection”, Physical
Review Letters. Vol. 70, 3728-3731.
[3] R. Khosropour, J. Zirinsky, H.K. Pak, and R.P.
Behringer (1997). “Convection and Size
Segregation in a Couetter Flow of Granular
Material”, Physical Review E. Vol. 56, 44674473.
[4] D. Hirshfeld, D.C. Rapaport (1997).
“Molecular Dynamics studies of Grain Segregation
in Sheared Flow”, Physical Review E. Vol. 56,
2012-2018.
[5] Babić, M., H.H. Shen, H.T. Shen (1990) “The
Stress Tensor in Granular Shear Flows of
Uniform, Deformable Disks at High Solids
Concentrations”, Journal of Fluid Mechanics.
Vol. 219, 81-118.
[6] Babić, M. (1988) “Discrete Particle Numerical
Simulation of Granular Material Behavior”,
Report No. 88-11. Department of Civil and
Environmental Engineering, Clarkson
University, Potsdam, NY.
[7] Bastien, C (2003). “Simulating the Behavior
of Granular Materials in 2-D Shear Flow”.