Presented at the University of Salerno: May, 2011
Lecture 3
Rapid Granular Flow Applications
Anthony D. Rosato
Granular Science Laboratory
ME Department
New Jersey Institute of Technology
Newark, NJ, USA
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Lecture 3: Rapid Granular Flow Applications
Presentation Outline
 Application 1: Galton’s Board
 Application 2: Vibrated Systems
 Application 3: Couette Flows
 Application 4: Intruder Dynamics in Couette Flows
 Application 5: Density Relaxation -Continuous
Vibrations
 Application 6: Tapped Density Relaxation
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Lecture 3: Rapid Granular Flow Applications
Application 1:
Galton’s Board
Investigate the behavior of a single particle migrating
under gravity through an ordered, planar array of rigid
obstacles – a system known as a Galton’s board.
Examine subtle connections between the deterministic
particle simulations, physical experiments, and discrete
dynamical models
First step in a larger picture to extract generic dynamical
features of granular flows through the analyses of “simple”
models
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Lecture 3: Rapid Granular Flow Applications
Historical Background
Sir Francis Galton (1822 – 1911): British
scientist, Fellow of the Royal Society; Geographer,
meteorologist, tropical explorer, founder of
differential psychology, inventor of fingerprint
identification, pioneer of statistical correlation and
regression, convinced hereditarian, eugenicist, protogeneticist, half-cousin of Charles Darwin and bestselling author. http://www.mugu.com/galton/start.html
Developed “board” to describe biological processes statistically
“I have no patience with the hypothesis occasionally expressed, and often implied, especially in tales
written to teach children to be good, that babies are born pretty much alike, and that the sole agencies
in creating differences between boy and boy, and man and man, are steady application and moral
effort. It is in the most unqualified manner that I object to pretensions of natural equality. The
experiences of the nursery, the school, the University, and of professional careers, are a chain of
proofs to the contrary.”
-- Francis Galton, Hereditary Genius
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Lecture 3: Rapid Granular Flow Applications
Galton’s Board: EXPERIMENTS
5
16
5
32


1
dp 
16

Rendering of the board
depicting the pins, collection
slots, traverse, location of the
optical timer beams, and
detail of the triangular lattice
configuration of pins.
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Lecture 3: Rapid Granular Flow Applications
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Lecture 3: Rapid Granular Flow Applications
Experimental Apparatus
Schematic of the automatic Galton
Board data acquisition system
(AGB). Balls fed from the supply
hopper through a flexible tube are
dropped one at a time using a
system of solenoids. The
residence time is recorded via an
optical sensor (“stop eye”). The
exit position is also recorded with
an array of 49 custom-built optical
cell detectors.
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Lecture 3: Rapid Granular Flow Applications
Experimental Parameters
Materials of the sphere
Aluminum
Brass
Stainless Steel
Release Height
Board Tilt Angle
H (max = 15.53”)
q (30o to 70o) – measured from horizontal
Measurements Made
Residence Times
Distribution of Exit Positions
Computed Quantities
Average downward velocity (cm/sec)
Lateral dispersion (cm2/sec)
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Lecture 3: Rapid Granular Flow Applications
Sampling of Experimental Results
Average residence time Tav as a function of release height H for
stainless steel spheres.
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Average Residence Time (sec)
11
10
9
Board Angle
8
Deg ree 7 0
7
Deg ree 6 0
6
Deg ree 5 0
5
Deg ree 4 0
4
Deg ree 3 0
3
2
1
0
5
10
15
20
25
30
35
40
Release Height (cm)
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Lecture 3: Rapid Granular Flow Applications
Distribution of Exit Positions for Stainless Steel Spheres
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Lecture 3: Rapid Granular Flow Applications
Lateral Dispersion or Diffusivity
Diffusion model [Bridgwater et al., Trans. Instn. Chem. Engrs. 49, 163-169 (1971) ]
c ( x, t ) - concentration of particles at (x, t) for an infinitely wide board
o ( x ) - delta-distribution centered at x = 0 and height No
c
 2c
 D 2 , t  0,    x  
t
x
lim c( x, t )  0, t  0
x 
c( x,0)  N o o ( x),
 x
Solution …
c ( x, t ) 
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No
2 πDt
e
 x 2 4 Dt
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Lecture 3: Rapid Granular Flow Applications
N ( x, t )
- number of particles in the interval [-x, x]
x

2 4 Dt

z
N ( x, t ) 
dz  Noerf x 2 Dt
e
2 Dt  x
No
D = 1.85 cm2/sec

Least squares fit of the
stainless steel data (solid
circles) to the model. Spheres
were released from the top of
the board set at q = 70o. The
origin of the x-axis denotes the
center of the board.
Summary of Dispersion Results
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Lecture 3: Rapid Granular Flow Applications
Sample Trajectories Generated by the Simulation
Figure 9: Three typical trajectories from the discrete element simulation (q = 70o) obtained by slightly
varying the initial positions. Residence times are indicated for each trajectory. The center of the board is
located at X = 0.2032 meters.
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Lecture 3: Rapid Granular Flow Applications
Simulated Exit Position Distribution
Exit distribution of the number of particles for 1/8” spheres at board angle q = 70o
from simulation in which e = 0.6
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Lecture 3: Rapid Granular Flow Applications
Lateral Dispersion Computed from Limiting Slope of Mean Square
Displacement
1
D  lim
 r 2  2.43 cm2 sec
t  2t
(m2)
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Lecture 3: Rapid Granular Flow Applications
Comparison of Simulated Results with Experiments
Quantity
Tav (s)
V (cm/s)
D (cm2/s)
Simulations (q = 70o)
7.12
5.6
2.43
Simulations (q = 90o)
6.70
5.92
1.48
Experiments (q = 70o)
Stainless Steel
7.22
5.49
1.85
Aluminum
6.84
5.77
1.96
Brass
6.81
5.79
2.085
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Lecture 3: Rapid Granular Flow Applications
Application 2: Vibrated Systems
Investigate macroscopic behavior of granular materials subjected to
vibrations
Gravitationally loaded into a rectangular, periodic cell having an open top
and plan floor
Vibrations imposed through sinusoidally oscillated floor
Compare with kinetic theory predictions
Compare with physical experiments
Y. Lan, A. Rosato, “Macroscopic behavior of vibrating beds of smooth inelastic spheres, Phys. Fluids 7 [8], 1818 (1995)
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Lecture 3: Rapid Granular Flow Applications
Geometry of Periodic
Computational Cell
Spheres are smooth (no friction) and inelastic,
obeying the soft contact laws of Walton and Braun.
Steady state computations performed.
Simulation Parameters
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Lecture 3: Rapid Granular Flow Applications
Steady-State Diagnostics
In computing depth profiles, the cell is partitioned into layers of thickness equal to approximately
the particle diameter d.
Instantaneous layer diagnostic: Massweighted average taken over all particles that
Averaging layer occupy the layer at time t.
A layer is ‘identified’ by its center y-coordinate.
y
mt  Nd 2 6 A
Mass hold-up: bulk mass supported by the floor of
cross-sectional area A
N = # of spheres
Long term cumulative mean velocity of
layer-y taken over the time interval (to, t1).
Instantaneous fluctuating (or deviatoric velocity) of the
ith particle in layer-y
L
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Designates the long-term average
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Lecture 3: Rapid Granular Flow Applications
Depth profile of the instantaneous RMS
deviatoric velocity
Long-term cumulative, mass-weighted average deviatoric velocity depth profile
T
1
C ( y, t ) L 2
3
W ( y)  T ( y) / dg
Granular Temperature depth profile
Measure of the kinetic energy per unit mass attributed to the
particles’ fluctuating velocity components.
Non-dimensional Granular Temperature depth profile
S. Ogawa, “Multi-temperature theory of granular materials”, Proceedings of the US-Japan Seminar on ContinuumMechanical and Statistical Approaches in the Mechanics of Granular Materials, Tokyo, 1978, pp. 208-217.
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Lecture 3: Rapid Granular Flow Applications
Comparisons with Kinetic Theory of Richman and Martin
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Lecture 3: Rapid Granular Flow Applications
Comparisons with Experiments of Hunt et al.
Validation against Experiments
Paper lid
M. Hunt et al., J. Fluids Eng. 116, pg. 785 (1994).
Relatively smooth spheres used in experiment
136 grams of particles used, mt = 5.0
asin(t )
Simulation Parameters
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Lecture 3: Rapid Granular Flow Applications
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Lecture 3: Rapid Granular Flow Applications
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Lecture 3: Rapid Granular Flow Applications
Summary of Findings
The behavior of the system depends on the magnitude of the floor acceleration G =
a2/g
High accelerations: Dense upper region supported on a ‘fluidized’ lower-density
region near the floor
Granular temperature is maximum near the floor and attenuates (upwards)
towards the surface, and the solids fraction depth profiles peaks within the center
of the system.
Lower Accelerations: Granular temperature does not decrease monotonically
from the floor, and the solids fraction depth profile bulges near the floor. Upper
region of the system is highly agitated.
For accelerations less than (approx.) 1.2, the steady-state height of the system
remains constant.
For 1.2 < G < 2.0: System undergoes a large vertical expansion.
Computed steady-state granular temperature and solids fraction profiles in good
agreement with kinetic theory predictions when the system is sufficiently agitated, and
with physical experiments.
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Lecture 3: Rapid Granular Flow Applications
Convection in a Vibrated Vessel of Granular Materials
Rough, inelastic spheres obeying the Walton & Braun soft-particle models.
Continuously shake the vessel up
and down. Particles will flow
upwards near the walls and
downward in the center.
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Lecture 3: Rapid Granular Flow Applications
Velocity Field – Long-time Average
Superimposed Trajectory of Large Intruder
Parameters
m  mb = 0.8, f = 7 Hz, a/d = 0.5, G = 10
Width = 20d
Velocity
Spheres
Y. Lan and A. D. Rosato, Phys. Fluids 9 (12), 3615-3624 (1997).
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Lecture 3: Rapid Granular Flow Applications
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Lecture 3: Rapid Granular Flow Applications
Average Convection Velocity as a Function of G
4.5
0.15
0.25
4
0.275
Avg. Cnv. Vel. (m/s x .01)
3.5
0.3
0.4
3
0.5
Poly. (0.15)
2.5
Poly. (0.25)
Poly. (0.275)
2
Poly. (0.3)
1.5
Poly. (0.4)
Poly. (0.5)
1
0.5
0
0
2
4
6
8
10
12
Gamma
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Lecture 3: Rapid Granular Flow Applications
Long-term velocity field in a computational cell whose
lateral walls are smooth (no friction). Notice the downward
flow in the center and upward motion adjacent to the walls.
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Lecture 3: Rapid Granular Flow Applications
Instantaneous velocity fields and sphere
center projections for the 100d cell (f = 7
Hz, a/d = 0.5, G = 10) reveals the
formation of arches during the downwards
motion of the floor. The dashed line
represents the equilibrium position of the
floor. Although the arches are not very
distinct in (b), the corresponding
instantaneous velocity field reveals a
pattern where groups of particles are
moving collectively towards or away from
the floor. This has been marked by the
arrows in (c) whose directions indicate the
general sense of the flow at a time
subsequent to that shown in (b).
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Lecture 3: Rapid Granular Flow Applications
Comparison of trajectory of large intruder in a narrow and wide cell.
Notice the re-entrainment in (b), while the intruder is trapped at the
surface in (a).
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Lecture 3: Rapid Granular Flow Applications
Summary of Findings
The onset of convection is controlled by (a/d)G rather than by G alone.
When the lateral walls are frictional, a long-term convective flow develops that is
upward in the center of the cell and downward adjacent to the walls.
Reversal in the direction of the long-term convective flow occurs when the sidewalls are smooth.
As the cell width (w/d) is increased, a visible pattern in the long-term velocity field
is reduced and eventually it ceases to be evident.
Over the time scale of the period of vibration, adjacent internal convection fields with
opposed circulations were visible. Averaging over long time scales caused these flow
structures not to appear.
However, near the side walls, persistent vortex-like structures were attached, having a
length scale that appeared to be of the same order as the height of the static system.
A single, large intruder sphere placed on the floor in the center of the system was carried up
to the surface at nearly the same velocity as the mean convection. Upon reaching the surface,
it migrated toward the side-walls. There it was either trapped, or re-entrained into the bed,
depending on the width of the downward flow field near the wall relative to the particle
diameter.
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Lecture 3: Rapid Granular Flow Applications
Application 3:
Couette Flow
Upper and lower bumpy walls
move at constant velocity in
opposite directions. Collisions with
flow particles causes them to flow.
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Lecture 3: Rapid Granular Flow Applications
General Features of the Flow
- Steady-state Profiles -
Velocity
Granular Temperature
Solids Fraction
Pressure
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Lecture 3: Rapid Granular Flow Applications
Steady State
0.8
Vxmean (d/s)
16.0
Y/d
12.0
8.0
4.0
1-top
bdry
2
0.6
3
0.4
4
5
0.2
6
7
0.0
0.0
-0.4
-0.2
0.0
0.2
0.4
0
30
60
90
120
150
180
8-middle
Average Number of Collisions
V/U
Average # of collisions/sec ~ 30 for each particle
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Lecture 3: Rapid Granular Flow Applications
Granular temperature - kinetic energy of the velocity fluctuations
1 ' '
' '
' '
Tt  u u  v v  w w
3
v  vv
v
v
'
Dimensionless 
“Peculiar” Velocity
Particle Velocity

Tt
Tt 
(d )2
Effective shear rate = 2U/H
Mean Velocity
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Lecture 3: Rapid Granular Flow Applications
Granular Temperature
Profiles
Mean Velocity Profiles
1
0.9
1
0.8
0.9
0.7
0.8
0.5
H=16d
0.4
H=32d
0.3
Y/H
Y/H
0.7
H=8d
0.6
0.6
n = 0.45
0.5
0.4
H=8d
H=16d
H=32d
0.3
0.2
0.2
0.1
0.1
0
-1
0
0
0.5
1
1.5
Tt
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2
2.5
3
-0.5
0
u
0.5
1
u
U
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Lecture 3: Rapid Granular Flow Applications
Pressure - Pyy
Solids Fraction Profiles
1
0.9
16
0.8
14
n bulk  0.45
0.7
12
10
Y/H
Y /d
0.6
8
0.5
6
H=16d
0.4
4
H=32d
0.3
2
0.2
P
*
yy
0
0.1
0.3
0.5
0.7
0.9
0.1

1 N
1 N
P  P  P  mi ui ui   rij Fij 
V i 1
2ij

k
0
0
0.1
0.2
0.3
n
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0.4
0.5
0.6
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Lecture 3: Rapid Granular Flow Applications
Secondary Velocity Field (u( x, y), v ( x, y) )
U
U = 8 d/s
H/d = 8
2y/d
  2 / s
U
x/d
Slab used to compute
v (x )
Averaging layer used for profiles
H
u(x, y ) 
y
x
Dy
z
Dx
L
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u(x, y )  u ( y )
U
Dx
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Lecture 3: Rapid Granular Flow Applications
Velocity field for U = 8 d/s (W/d=64)
v (x )
t (s)
x/d
Figure 7: Plot of v (x ) as a function of x/d (L/d = 64,   2 / s ) showing development to the
steady state velocity. This appears in the inset, where the horizontal axis is 2x/d.
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Lecture 3: Rapid Granular Flow Applications
v (x ) for U = 8 d/s (W/d=64)
Auto-Correlation
Peak at l = =15 R
FFT spectrum analysis
Peak at l = 7.5 d
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Lecture 3: Rapid Granular Flow Applications
Wavelength vs. Effective Shear Rate
40
Wavelength (l/ d )
35
30
25
20
15
10
5
0
0
0.5
1
1.5
2
2.5
Effective shear rate
3
3.5
4
4.5


Figure 8: Wavelength l/d of the convection cells as a function of effective shear rate for a fixed shear gap H/d
= 8. The solid line is included to show the trend.
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Lecture 3: Rapid Granular Flow Applications
Application 4: Intruder Dynamics in
Couette Flows
Intruder Properties
•
Different size, but
same density
•
Different mass, but
same size
•
Different size, same
mass
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-U
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Lecture 3: Rapid Granular Flow Applications
Size and Mass Ratios
Size Ratio f = D/d
Mass Ratio fm ~ f3
1
1.5
1
3.375
2
3
8
27
f
= Intruder diameter/Flow particle diameter
fm = Intruder mass/Flow particle mass
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Lecture 3: Rapid Granular Flow Applications
Intruder Velocity and Mid-Plane Crossing Time
3.5
70
3
U = 32 r/s
2.5
Tc (s)
U = 64 r/s
U = 32 r/d
1.5
U=16 r/s
50
U = 16 r/s
2
Vav
60
U = 64 r/s
U = 16 r/s
40
U=32 r/s
30
1
20
0.5
10
U=64 r/s
0
0
0.0
1.0
2.0
Size Ratio f
3.0
4.0
0
1
f
2
3
4
(a) Crossing time Tc (seconds) versus f at U = +/-16, 32, 64 r/s; (b) Average intruder velocity Vav  S Tc , where S
is the distance traveled by the mass center from its initial position near the wall to the mid-plane of the cell.
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Lecture 3: Rapid Granular Flow Applications
Tc = Rise Time for the intruder to reach the middle layer from bottom.
f= 1.0
f= 2.0
1
1
Tc=14 s
0.8
0.8
0.6
0.6
Y/H
Y/H
Tc=48 s
0.4
D / d = 2.0
0.4
0.2
0.2
D / d = 1.0
0
0
20
40
60
Time (s)
80
100
0
0
20
40
60
Time (s)
80
100
As the relative size of the intruder increase, its rise time decreases.
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Lecture 3: Rapid Granular Flow Applications
Y* Trajectory + Power Spectrum (f =1.0)
  2 / s
  2 / s
Y *  (Y  Ym (f)) /( H  2* Ym (f))
Ym(f) - Closest distance possible between the center of the intruder and boundary plane
P( f ) 
1
f 
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lg( P )
 
lg( f )
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Lecture 3: Rapid Granular Flow Applications
Power spectrum of Intruder y-Trajectories
f = D/d
h
 = -(2h + 1)
P  f-
1.0
0.7
-2.4
P  f -2.4
2.0
0.9
-2.8
P  f -2.8
3.0
1.06
-3.06
P  f -3.06
P( f ) 
1
f 
lg( P )
 
lg( f )
Background: Noise Signals
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Lecture 3: Rapid Granular Flow Applications
General Information on Noise
= 0
White noise
= 1
1/f noise
(often in processes found in
nature)
= 2
Brownian noise
(random walk)
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Lecture 3: Rapid Granular Flow Applications
1
P( f )  
f
Pf  1 f
2h1
 h = ½, Brownian Motion, h is Hausdorff (or Hurst) exponent.
 h <1/2, anti-persistence fBm (fractional Brownian motion), trend of
motion at any time t is not likely to be followed by a similar trend at next
moment t+1.
 h >1/2, persistence fBm (fractional Brownian motion), trend of motion at
any time t is likely to be followed by a similar trend at next moment t+1.
 f=1.0 h=0.7,
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f=2.0  h=0.9,
f=3.0  h=1.06
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Lecture 3: Rapid Granular Flow Applications
Intruder Histograms of its y-location
1.0
1.0
0.8
0.8
D / d =1.5
0.6
Y/H
Y/H
D / d =1.0
0.4
0.6
0.4
0.2
0.2
0.0
0.0
0
0.01
f = 1.0
0.02
0.03
Frequency
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0.04
0.05
0
0.01
f = 1.5:
0.02
0.03
Frequency
0.04
0.05
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Lecture 3: Rapid Granular Flow Applications
Intruder Histograms - Continued
1.0
1.0
0.8
0.8
0.6
D / d =3.0
Y/H
Y/H
D / d =2.0
0.4
0.6
0.4
0.2
0.2
0.0
0.0
0
0.01
f =2.0
0.02
0.03
Frequency
0.04
0.05
0
0.01
f =3.0
0.02
0.03
Frequency
0.04
0.05
“Trapping” in region of low granular temperature
S. Dahl, C. Hrenya, Physics of Fluids 16, 1-24 (2004).
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Lecture 3: Rapid Granular Flow Applications
Net resultant force averaged over time interval D
ZONE
+U
Fy (t )net
1
2
Y
1 Nc j

 Fy
N c (t ) j 1
3
4
5
Fx
6
X
D is three orders of magnitude smaller than time scale
over which the dynamics evolve, but much larger than the
integration step.
7
Fy
Fne t
Normalized  Fy (t ) 
8
9
10
(
Fy (t )net
 d2U H
)
2
-U
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Lecture 3: Rapid Granular Flow Applications
Evolution of Fy and Vy
Fy
Vy
Fy
Vy
Figure 13: Evolution of Fy (t ) for (a) f = 1, (c) f = 3, and velocity V y (t ) for (b) f = 1 and (d) f = 3.
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Lecture 3: Rapid Granular Flow Applications
Root-mean-square force and Velocity versus f
0.50
0.25
F y rms = 0.0772 f + 0.0114
0.40
0.20
R 2 = 0.9931
V yrms
(F y)
rms
0.30
0.15
0.10
0.30
V y rms = 0.2023f-1.2731
0.20
R 2 = 0.9925
0.10
0.05
0.00
0.00
0
1
f
0
2
3
1
f
2
3
Figure 14: Steady state graphs of Fyrms (left) and V yrms (right) versus size f. Correlation coefficients
R2 are shown for each fitted curve.
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Lecture 3: Rapid Granular Flow Applications
How does particle mass affect the fluctuation velocity in the
direction perpendicular to the shear?
Procedure: Vary particle density and maintain size ratio f = 1
f = Intruder diameter / flow particle diameter
fm = Mass Intruder / Mass of flow particles
Vyrms ( d / s )
n = 0.45
 = constant
3.0
Original System: Vary
size ratio f and
maintain constant
particle density .
2.0
1.0
0.0
0.5
1.0
1.5
Granular Science Lab - NJIT
f
2.0
2.5
3.0
58
Vyrms ( d / s )
Lecture 3: Rapid Granular Flow Applications
3.0
f=1
2.0
Vary mass ratio fm and
maintain size ratio f.
1.0
0.0
Vyrms ( d / s )
0
5
fm
10
15
3.0
Vary size ratio f and
maintain mass ratio fm.
2.0
fm = 1
1.0
0.0
0.0
1.0
Granular Science Lab - NJIT
f
2.0
3.0
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Lecture 3: Rapid Granular Flow Applications
Progression of Velocity Distributions for f = 1
f m=8.0
Hist(Vy)
1500
1000
f m=27.0
f m=3.375
f m=1.0
500
f m=0.5
0
-6
-4
-2
0
2
4
6
V yrms ( d / s )
An increase in particle mass results in a narrower velocity distribution (qualitative
agreement with the Maxwell-Boltzmann distribution).
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Lecture 3: Rapid Granular Flow Applications
Application 5: Density Relaxation Under
Continuous Vibrations
•
After exposure to vibrations or taps, a bulk solid can attain an increase in
density. This phenomenon is often referred to as “density relaxation” or
“densification”.
•
Its occurrence depends on the behavior induced in the material, which in
turn is influenced by particle properties, vessel geometry and wall
conditions, strength of the vibrations, and the initial or “poured” state of the
material.
•
The importance of understanding densification is pertinent to solids handling
industries in which vibrations are often used to enhance the processing of
large quantities of bulk materials.
•
Density relaxation’s historical background can be traced in the literature on
the packing of spheres and disks (Appendix L3-A)
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Lecture 3: Rapid Granular Flow Applications
Densification Experiments: Uniform Spheres
Acrylic spheres: d = 1/8″
Cylinder
= 1200 kg/m3
Controller
Frequency: 25 – 100 Hz
Shake
r
Amplitude (a/d): 0.04 – 0.24
Amplifier
Accelerometer
Aspect Ratio: D/d ~ 20
“Maximum” Solids Fraction:
Power
Amplifier
G: 0.94 – 11.0 (relative acceleration)
T = 10 minutes (vibration duration)
n = 0.6366 ± 0.0005 for uniform spheres
G.D.Scott, D.M.Kilgour, British Journal of Applied Physics,1969.
D. J. D’Appolonia, and E. D’Apolonia, Proc.3rd Asian Reg. Conf. on Soil Mechanics, 1266~1268, Jerusalem Academic Press
(1967).
R. Dobry and R. V. Whitman, “Compaction of Sand …”, ASTM STP 523, 156~170, ASTM, Philadelphia (1973).
Granular Science Lab - NJIT
Details →
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Lecture 3: Rapid Granular Flow Applications
Details of the Experimental Procedure …
Compute bulk solids fraction and improvement in solids fraction.
d = 0.125 inch
Pour
Slice
Improvement in Solids Fraction =
Granular Science Lab - NJIT
Vibrate
Solids Fraction
 n relaxed 

 1  100
 n poured



63
Lecture 3: Rapid Granular Flow Applications
Results: Systems vibrated for 10 minutes
2
2
3
4
0.635
a/d=0.04
4.5
Solids Fraction
4
3.5
0.625
3
0.62
2.5
2
0.615
1.5
1
0.61
2
3
2
4
6
4
8
5
8
a/d=0.06
5
0.625
3
0.62
2
0.615
1
0
0
0.6
2
3
4
5
6
7
8
Relative Acceleration
10
0.64
12
4.5
4
0.628
3.5
0.624
3
0.62
2.5
2
0.616
1.5
0.612
3
4
5
6
a/d=0.24
5
0.63
4
Solids Fraction
5
2
0.635
Improvement in Solids Fraction (%)
a/d=0.16
0.632
Solids Fraction
7
0.605
5.5
0.636
6
0.61
Relative Acceleration
0.64
5
4
0.5
0.605
1
4
0.63
Solids Fraction
5
Improvement in Solids Fraction (%)
5.5
0.635
0.63
3
5
Improvement in Solids Fraction (%)
1
0.625
3
0.62
2
0.615
1
0.61
Improvement in Solids Fraction (%)
0.64
1
0.608
0.605
2
4
6
8
Relative Acceleration
10
12
2
3
4
5
6
0
Relative Acceleration
Data points are averages of 4 trials
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Lecture 3: Rapid Granular Flow Applications
Do Simulations Results Agree with Physical Experiments ?
Y
L
Comparisons:
Poured Bulk Density
W
Vibrated Bulk Density
H
d
x
Z
z
Vibrating Floor
Averaging Layer
Some Background ….
1944:
Oman & Watson [Natl. Patrol. News 36, R795-R802 (1944)] coined the terms random “random dense” and
“random loose” to describe the two limiting cases of random, uniform sphere packings.
1960: Scott carried out a number of different experiments with 3mm steel ball bearings to study dense & loose random
packings … Spherical containers and N steel ball bearing: He plotted packing fraction n vs. system size and extrapolated …
n-loose = 0.59
n-dense = 0.63
1969:
G.D.Scott, D.M.Kilgour, British Journal of Applied Physics
n = 0.6366 ± 0.0005 for uniform spheres
1997:
E. R. Nowak, M. Povinelli, et al., Powders & Grains 97, (Balkema, Durham, NC, 1997), pp. 377 - 380.
n = 0.656 for vibrated column (d/D ~ 9) of uniform spheres
Select literature: Sphere packing
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Lecture 3: Rapid Granular Flow Applications
Select Literature on Packing of Spheres
Coord. No.
System
Solids Fraction
Reference
0.601  0.001/ 0.637  0.001
Steel spheres in cylinder
0.625
Steel spheres in glass container McGeary (1961)
0.6366  0.004
0.59
Steel spheres in cylinder
Finney
Computer
Tory, et al. (1968)
0.628
Computer
Adams & Matheson (1972)
6.0
0.61
Computer
Bennett (1972)
6.4
0.582
Computer
Visscher & Bolsterli (1972)
6.01
0.58
Computer
Tory, Church, et al. (1973)
6.0
Computer
Matheson (1974)
6.0
0.606  0.006
0.6099 / 0.6472
Statistical Model
Gotoh & Finney (1974)
6.0
6.0
0.59  0.01
0.58  0.05
Computer
Powell (1980)
Computer
Rodriguez, et al. (1986)
0.634
Computer
Mason (1967)
0.582
Computer
0.6366
Computer
Gotoh, Jodrey & Tory
(1978)
Jodrey & Tory (1981)
0.610 – 0.658
Computer
Zhang & Rosato (2004)
6.1
5.64
Granular Science Lab - NJIT
Scott (1960)
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Lecture 3: Rapid Granular Flow Applications
Extrapolated solids fraction for infinitely wide container in good
agreement with experiments reported in literature.
0.61
0.05
0.1
0.15
0
0.62
Solids Fraction
0.6
0.59
0.01
0.2
0.02
0.03
0.25
0.04
0.62
0.615
0.615
0.61
0.61
0.605
0.605
0.6
0
0.01
0.02
0.03
0.04
0.61
0.6
0.59
0.6
0.58
0.58
0.57
0.57
n = 0.6102
0.56
0.55
0.05
0.1
0.15
0.56
0.2
0.25
0.55
Aspect Ratio d/L
Solids fraction as a function of the inverse aspect ratio (d/L) for a system of
particles with friction coefficient m = 0.1. The inset shows the extrapolated value
0.6102 as d/L → 0.
n depends on m →
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Lecture 3: Rapid Granular Flow Applications
Particles that are more frictional produce a less dense structure after pouring
because of formation of bridges among particles.
As m increases, there is approximately a 7% reduction in n. At m = 0, the solids fraction is within
the range of values normally ascribed for a loose or poured random packing of smooth spheres,
i.e., approximately between 0.59 to 0.608.
Solids Fraction
0.6
0
0.2
0.4
0.6
0.8
0.6
0.59
0.59
0.58
0.58
0.57
0.57
0.56
0.56
0.55
0
0.2
0.4
0.6
0.8
0.55
Friction Coefficient
Variation of the solids fraction with friction coefficient m (d/L = 0.1064, N = 600). Each point of
the curve represents an average taken over 10 realizations, while the bars show the deviation.
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Lecture 3: Rapid Granular Flow Applications
Simulated Random Dense Packing
40
50
60
70
80
90
110
a=0.01' '
a=0.02' '
a=0.03' '
a=0.04' '
a=0.06' '
a=0.005' '
0.635
0.63
Solids Fraction
100
0.635
0.63
0.625
0.625
0.62
0.62
0.615
0.615
0.61
30
40
50
60
70
80
90
100
110
0.66
120
0.64
0.61
120
Solids Fraction
30
0.64
Simulated Trends 
5
10
15
0.66
0.655
0.655
0.65
0.65
0.645
0.645
0.64
0.64
0.635
0.635
0.63
0.63
0.625
0.625
0.62
0.62
0.615
0.615
0.61
0.61
0
Frequency (Hz)
5
10
15
Vibration Time(s)
0
0.01
0.02
0.03
0.04
0.05
0.06
0.07
0.08
0.09
0.1
0.67
Solids Fraction
n = 0.6582 is the solids fraction of
random dense packing, in good
agreement with the experimental
result of Nowak et al. (0.656)
0
0.11
0.67
0.668
0.668
0.666
0.666
0.664
0.664
0.662
0.662
0.66
0.66
0.658
0.658
0.656
0.656
0.654
0
0.01
0.02
0.03
0.04
0.05
0.06
0.07
0.08
0.09
0.1
0.654
0.11
1/Tv
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Lecture 3: Rapid Granular Flow Applications
Simulated Trends versus Frequency ….
50
75
0.612
5.5
Solids Fraction
4
3.5
0.595
3
2.5
0.59
2
1.5
0.585
50
70
80
90
5.5
5
0.606
0.604
4.5
0.602
4
0.6
0.598
a/d=0.08
3.5
0.596
3
0.58
30
40
50
60
70
80
1
0.594
0.5
90
0.592
30
40
50
0
20
40
60
70
80
2.5
90
Frequency(Hz)
Frequency(Hz)
0.615
60
0.608
Solids Fraction
a/d=0.02
0.6
Improvement in Solids Fraction(%)
4.5
40
0.61
5
0.605
30
Improvement in Solids Fraction(%)
25
0.61
60
80
30
0.588
40
50
60
70
80
90
6
0.595
3
0.59
2
a/d=0.24
0.585
1
0.58
0
a/d=0.48
1
0.582
0.5
0.58
0.578
0
0.576
-0.5
0.574
0.572
-1
0.575
-1
0.57
0
20
40
Frequency(Hz)
Granular Science Lab - NJIT
60
80
Improvement in Solids Fraction (%)
4
0.584
Solids Fraction
0.6
Improvement in Solids Fraction(%)
5
0.605
Solids Fraction
1.5
0.586
0.61
0.57
30
40
50
60
70
80
90
Frequency(Hz)
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Lecture 3: Rapid Granular Flow Applications
Simulated Densification Phase Map: L/d = 25, N = 8000
0.459431
1.39462
2.32982
3.26501
4.20021
0.12
0.11
0.1
Amplitude (inch)
0.09
0.08
0.07
0.06
0.05
0.04
0.03
0.02
0.01
20
Granular Science Lab - NJIT
40
60
Frequency (Hz)
80
73
Lecture 3: Rapid Granular Flow Applications
Experimental Evidence
L. Vanel, A. Rosato, R. Dave, Phys. Rev. Lett. 78, 1255 (1997).
Cylinder
Controller
Shaker
Amplifier
Accelerometer
Power
Amplifier
Small amplitudes a/d < 0.25, and high frequencies (40 – 80 Hz)
Granular Science Lab - NJIT
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Lecture 3: Rapid Granular Flow Applications
Application 6: Density Relaxation under
Tapping
Taps applied
Rearrangement of particle positions so that
the bulk density of the material increases.
System is compacted
“Package sold by weight, not volume.
Contents may settle during shipment.”
Granular Science Lab - NJIT
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Lecture 3: Rapid Granular Flow Applications
The study of density relaxation has its foundations in the
extensive literature on the packing of circles and spheres
1611:
1665:
1694:
1727:
1887:
1899:
1932:
1933:
1944:
1951:
Kepler - Geometry of the snowflake
Robert Hook - Circle and sphere packings
Gregory, a Scottish astronomer, suggested that 13 rigid uniform spheres could be
packed around a sphere of the same size
Hales - Packing of dry peas pressed into a container
Thompson - How to fill Euclidean space using truncated octahedrons
Slichter – Found analytical expressions for the porosity in beds of uniform spheres
Hilbert - Found a structure for which nm=0.123
Heesch and Laves: Created a stable arrangement of spheres with nm= 0.056
Oman & Watson: ‘Loose’ and ‘dense’ random packing of spheres
Stewart - Consolidated state of optimal bulk density
Much more …
Boyd, D. W. "The Residual Set Dimension of the
Apollonian Packing." Mathematika 20, 170-174, 1973.
Granular Science Lab - NJIT
Apollonian Gasket
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Lecture 3: Rapid Granular Flow Applications
OBJECTIVE: Model the behavior of tapped a system of particles to
understand its evolution from a loose, disordered configuration to a
dense structure exhibiting order.
Parameters for discrete element
simulation
Number of particles: 3,456
d=0.02m
Periodic BC in lateral dimensions
Particle material density  = 1.2 g/cm3
Restitution coefficient e = 0.9
Particle-particle friction mp=0.1
Integration Time step ~ 10-5 s
Granular Science Lab - NJIT
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Lecture 3: Rapid Granular Flow Applications
Simulation Procedure
Randomly place spheres (diameter d) into the periodic volume
Turn on gravity – spheres collapse to a loose, random structure (pour)
Apply discrete tap of amplitude a/d and frequency f.
Allow system to relax until quiescent.
Kinetic Energy ~ 0
System tapped
Particles bounce back
a/d = 0.4; f = 7.5 Hz
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Lecture 3: Rapid Granular Flow Applications
Animation of Pouring from DEM Simulation
Granular Science Lab - NJIT
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Lecture 3: Rapid Granular Flow Applications
Tapping Sequence: Motion of the Plane Floor
(
)
a  sin 2 f t  t p  ( i  1) tC  , t  i  tC  t p ; i  tC 




y0 (t )  
0, otherwise
tC  t p  tb , t p  relaxation time, tb  tapping time
Granular Science Lab - NJIT
Dimensionless acceleration
a 2
a
G
 4 2 f 2  1.35
g
g
80
Lecture 3: Rapid Granular Flow Applications
System Response to Taps
The vertical positions of particles in a given layer (yi/d) after tap are monitored
Granular Science Lab - NJIT
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Lecture 3: Rapid Granular Flow Applications
Simulated Effect of Particle Friction m on Poured Bulk Solids Fraction
Particle that are more frictional produce a less dense structure after pouring because of
formation of bridges among particles.
Data points are averages over 20 realizations.
Red lines are error bars.
McGeary (1961): Steel
spheres in glass cylinder
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Lecture 3: Rapid Granular Flow Applications
Sample Realizations
n o  0.6118
n o  0.6110
n f  0.6884
n f  0.7018
n o  0.6116
Average
n f  0.7007
Granular Science Lab - NJIT
of 20 realizations
n o  0.6118
n f  0.7077
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Lecture 3: Rapid Granular Flow Applications
Distribution of Particle Centers
Number of particle centers inside a layer of thickness H normalized by the
total number of system particles
Poured System
Ordering adjacent to plane floor
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Lecture 3: Rapid Granular Flow Applications
Center Distribution averaged over 10 consecutive taps
Granular Science Lab - NJIT
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Lecture 3: Rapid Granular Flow Applications
Ensemble-Averaged Center Distribution
f  7.5 Hz
a / d  0.44
G  0.5
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Lecture 3: Rapid Granular Flow Applications
Evidence of a Critical Displacement Amplitude
Monte Carlo simulation results strongly suggest that there is a critical
displacement amplitude g that promotes an optimal evolution to a dense
structure.
Before Tap
MC Simulated
Tap Applied
g
g
O.Dybenko, A. Rosato, V. Ratnaswamy, D. Horntrop,
L. Kondic, “Density Relaxation by Tapping”,
in
preparation.
Granular Science Lab - NJIT
Although not presented here,
similar findings were observed in
DEM simulations.
87
Lecture 3: Rapid Granular Flow Applications
Evolution of Structure: Effect of Displacement Amplitude
f  7.5 Hz
a / d  0.11
G  0.5
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Lecture 3: Rapid Granular Flow Applications
Evolution of Structure: Effect of Displacement Amplitude
f  7.5 Hz
a / d  0.44
G  0.5
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Lecture 3: Rapid Granular Flow Applications
Summary - Conclusions
The parameter space of factors affecting the process is large: tap amplitude,
frequency, acceleration, particle properties, mass overburden, container aspect
ratio
Four time scales: particle collision duration (Dt ~ 10-5 s), period of applied tap,
single-tap system relaxation time, long-time relaxation scale
Discrete element model reveals the mechanism…
Upward progression of “organized” layers induced by the plane floor as the taps
evolve.
The configuration of the particles plays an important role in how the system
evolves, rather than solely the value of the bulk solids fraction.
Evidence of a critical tap intensity that optimizes the evolution of packing density.
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Lecture 3: Rapid Granular Flow Applications
End of Lecture 3
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Lecture 3: Rapid Granular Flow Applications
Appendix L3-A: Selected Highlights on Packing Studies
1611 Kepler [1]:
Interested in the geometry of the snowflake
1665 Hooke [2]:
Studied the packing of circles and spheres
1727 Hales [3]:
A botanist who carried out an experimental investigation of the packing of dry peas pressed in a
container – forming fairly regular polyhedra, which he erroneously assumed were regular dodecahedra. The experiment is
known as the “peas of Buffon” (based on similar experiments done by Comte de Buffon in 1753).
1694 Gregory [1]: Hypothesizes that 13 rigid uniform spheres can be packed around another sphere of the same size.
(Newton’s conjecture was 12)
1963 Proved that adequate space for 13.397 spheres exists around a single sphere, BUT this arrangement is
impossible (1956, Leech [7]).
1939 Marvin [2]: Repeated Hales’ experiment by applying pressure on uniform lead shot
Close-packed initial configuration  particles formed into regular dodecahedron (12 faces, each a rhombus)
Randomly poured initial configuration  predominant structure was irregular 14-faced polyhedra, and no rhombic
dodecahedra.
1883 Barlow [8]: Found hexagonal close packing where each sphere touches 12 others.
[1] Scottish astronomer (1661-1708)
[2] Also by Matzke [5]
Granular Science Lab - NJIT
Rhombic Dodecahedron
12 faces
92
Lecture 3: Rapid Granular Flow Applications
Appendix L3-A: Selected Highlights on Packing Studies
1887  Thompson: How to fill Euclidean space without voids can be done using truncated
octahedral (14 faces = 6 squares + 8 hexagons)
1899  Slichter: Studied porosity and channels in bed of uniform spheres. 1st attempt to find
analytical expressions.
“Practical Issue” - How dense can uniform spheres be packed?
1958  Rogers [11]: If there was a regular arrangement of uniform spheres more dense than that of a hexagonal
close packing (), it’s packing fraction could be no larger than
 1

18 cos

(1/ 3)  ( 3)  0.7797

Alternative: What is the minimal solids fraction of rigid assembly of uniform spheres?
Rigidity  Each sphere must touch at least 4 others, and the points of contact must not lie all
in one plane.
1932  Hilbert [12] found “loosest” packing with n = 0.123
1933  Heesch & Laves [13] found looser packing with n = 0.056.
1944: Oman & Watson * coined the terms random “random dense” and “random loose” to describe the two limiting
cases of random, uniform sphere packings.
* A. O. Oman & K. M. Watson, “Pressure drops in granular beds,” Natl. Patrol. News 36, R795-R802 (1944).
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Lecture 3: Rapid Granular Flow Applications
Appendix L3-A: Selected Highlights on Packing Studies
1960: Scott carried out a number of different experiments with 3mm steel ball bearings to study dense &
loose random packings …
Spherical containers and N steel ball bearing: He plotted packing fraction n vs. 1 3 N and extrapolated
to large N.
nloose  0.59
ndense  0.63
Pouring into cylindrical containers followed by 2 minutes of shaking to obtain dense random packing
Cylinder rotated about horizontal axis to obtain loose random packing.
Studies were also carried out in cylinders of various heights.
1969: Scott* carried out improved experiments for the solids fraction of a dense random packing.
ndense  0.6366  0.0005
* G.
D. Scott and D. M. Kilgour, “The density of random close packing of spheres,” Brit. J. Appl. Phys. (J. Phys. D) 2,
863-866 (1969).
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Lecture 3: Rapid Granular Flow Applications
Appendix L3-A: Selected Highlights on Packing Studies
Random Loose Packing of Spheres
Experiments: G. Onoda and Y. Liniger, PRL 64, 2727, 1990
0.6
Packing Fraction
0.58
0.56
0.54
0.52
0.5
0
0.005
0.01
0.015
0.02
1/N
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Lecture 3: Rapid Granular Flow Applications
Appendix L3-A: Selected Literature on Packing Studies
References for Additional Reading
1.J. A. Dodds, “Simplest statistical geometric model of the simplest version of the multicomponent random packing problem,” Nature, Vol. 267, 187-189 (1975).
2.W. A. Gray, The packing of solid particles, Chapman & Hall, London (1968).
3.D. N. Sutherland, “Random packing of circles in a plane,” Journal of Colloidal and Interface Science 60 [1], 96-102 (1977).
4.T. G. Owe Berg, R. I. McDonald, R. J. Trainor, Jr., “The packing of spheres,” Powder Technol. 3, 183-188 (1969/70).
5.C. Poirier, M. Ammi, D. Bideau, J.P. Troadec, “Experimental study of the geometrical effects in the localization of deformation,” Phys. Rev. Lett. 68 [2], 216-219 (1992).
6.D. R. Nelson, M. Rubinstein, F. Spaepen, “Order in two-dimensional binary random array,” Philosophical Magazine A 46 [1], 105-126 (1982).
7.A. Gervois, D. Bideau, “Some geometrical properties of hard disk packings,” in Disorder and Granular Media (ed. D. Bideau), Elsevier/North Holland (1992).
8.F. Deylon, Y.E. Lévy, “Instability in 2D random gravitational packings of identical hard discs,” J. Phys. A: Math Gen. 23, 4471-4480 (1990).
9.G. C. Barker, M. J. Grimson, “Sequential random close packing of binary disc mixtures,” J. Phys. Condens. Matter 1, 2279-2789 (1989).
10.D. Bideau, J. P. Troadec, “Compacity and mean coordination number of dense packings of hard discs,” J. Phys. C: Solid State Phys. 17, L731-L735 (1984).
11.M. Ammi, T. Travers, D. Bideau, Y. Delugeard, J. C. Messager, J. P. Troadec, A. Gervois, “Role of angular correlations on the mechanical properties of 2D packings
of cylinders,” J. Phys.: Condens. Matter 2, 9523-9530 (1990).
12.T. I. Quickenden and G. K. Tan, “Random packing in two dimensions and the structure of monolayers,” Journal of Colloidal and Interface Science 48 [3], 382-393
(1974).
13.G. Mason, “Computer simulation of hard disc packings of varying packing density,” Journal of Colloidal and Interface Science 56 [3], 483-491( 1976).
14.J. Lemaitre, J. P. Troadec, A. Gevois, D. Bideau, “Experimental study of densification of disc assemblies,” Europhys. Lett 14 [1], 77-83 (1991).
15.H. H. Kausch, D. G. Fesko, N. W. Tshoegl, “The random packing of circles in a plane,” Journal of Colloidal and Interface Science 37 [3], 603-611 (1971).
16.Y. Ueharra, T. Ree, F. H. Ree, “Radial distribution function for hard disks from the BGY2 theory,” J. Chem. Phys. 70, 1877-1883 (1979).
17.H. Stillinger, E. A. DiMarzio, R. L. Kornegay, ,”Systematic approach to explanation of the rigid disk phase transition,” J. Chem. Phys. 40[6], 1564-1576 (1964).
18.J. V. Sanders, “Close-packed structure of spheres of two different sizes I. Observations on natural opal,” Philosophical Magazine A 42 [6], 705-720 (1980).
19.E. Guyon, S. Roux, A. Hansen, D. Bideau, J-P. Troadec, H. Crapo, “Non-local and non-linear problems in the mechanics of disordered systems: application to
granular media and rigidity problem,” Rep. Prog. Phys. 53, 373-419 (1996).
20.W. M. Visscher, M. Bolsterli, “Random packing of equal and unequal spheres in two and three dimensions,” Nature 239, pg. 504 (1972).
21.J. G. Berryman, “Random close packing of hard spheres and disks,” Phys. Rev. A 27 [2], 1053-1061 (1983).
22.M. Shahinpoor, “Statistical mechanical considerations on the random packing of granular materials,” Powder Technol. 25, 163-176 (1980).
23.M. J. Powell, “Computer-simulated random packing of spheres,” Powder Technol. 25, 45-42 (1980).
24.L. Oger, J. P. Troadec, D. Bideau, J. A. Dodds, M.J. Powell, “Properties of disordered sphere packings, I. geometric structure: statistical model, numerical simulations
and experimental results,” Powder Technol. 45, 121-131 (1986).
A. P. Shapiro, R. F. Probstein, “Random packings of spheres and fluidity limits of monodisperse and bidisperse suspensions,” Phys. Rev. Lett 68 [9], 1422-1425 (1992).
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Appendix L3-A: Selected Literature on Packing Studies
G. T. Nolan, P. E. Kavanagh, “Computer simulation of random packings of spheres with log-normal distributions,” Powder Technol. 76, 309-316 (1993).
N. Standish, D. E. Borger, “The porosity of particulate mixtures,” Powder Technol. 22, 121-125 (1979).
T. Stovall, F. De Larrard, M. Buil, “Linear packing density model of grain mixtures,” Powder Technol. 48, 1-12 (1986).
A. B. Yu, N. Standish, “An analytical-parametric theory of the random packing of particles,” Powder Technol. 55, 171-186 (1988).
A. B. Yu, N. Standish, “A study of the packing of particles with a mixture size distribution,” Powder Technol. 76, 112-124 (1993).
M. Gardner, “Circles and spheres, and how they kiss and pack,” Scientific American 218 [5], 130-134 (1968).
H. J. Frost, “Cavities in dense random packings,” Acta Metall. 30, 889-904 (1982).
S. K. Chan, K. M. Ng, “Geometrical characteristics of the pore space in a random packing of equal spheres,” Powder Technology 54, 147-155 (1988).
K. Gotoh, S. Jodrey, E. M. Tory, “Average nearest-neighbor spacing in a random dispersion of equal spheres,” Powder Technol. 21, 285-287 (1978).
M. J. Powell, “Distribution of near neighbours in randomly packed hard spheres,” Powder Technol. 26, 221-223 (1980).
A. Marmur, “ A thermodynamic approach to the packing of particle mixtures,” Powder Technol. 44, 249-253 (1985).
A. E. R. Westman and H. R. Hugill, “The packing of particles,” J. Am. Ceram. Soc. 13 [10], 767-779 (1930).
A.E.R. Westman, “The packing of particles: empirical equations for intermediate diameter ratios,” J. Am. Ceram. Soc. 19, 127-129 (1936).
S. Yerazunis, J. W. Bartlett, A. H. Nissa, “Packing of binary mixtures of spheres and irregular particles,” Nature 195, 33-35 (1962).
S. Yerazunis, S. W. Cornell, B. Wintner, “Dense random packing of binary mixtures of spheres, Nature 207, 835-837 (1965).
D. J. Lee, “Packing of spheres and its effects on the viscosity of suspensions,” J. Paint Technol. 42 [550], 579-587 (1970).
T. C. Powers, “Geometric properties of particles and aggregates,” Journal of the Portland Cement Association 6 [1], 2-15 (1964).
D. J. Adams & A. J. Matheson, “Computation of dense random packing of hard spheres,” J. Chem. Phys. 56, 1989-1994 (1972).
C. H. Bennett, “Serially deposited amorphous aggregates of hard spheres,” J. Appl. Phys. 6, 2727-2733 (1972).
J. L. Finney, “Random packing and the structure of simple liquids. 1. The geometry of random close packing,” Proc. Roy. Soc. Lond. A. 319, 479-493 (1970).
K. Gotoh & J. L. Finney, “Statistical geometrical approach to random packing density of equal spheres,” Nature 252, 202-205 (1974).
K. Gotoh, W. S. Jodrey & E. M. Tory, “A random packing structure of equal spheres – statistical geometrical analysis of tetrahedral configurations,” Powder Technol. 20,
233-242 (1978).
W. S. Jodrey & E. M. Tory, “Computer simulation of isotropic, homogeneous, dense random packing of equal spheres,” Powder Technol. 30, 111-118 (1981).
R. K. McGeary, “Mechanical packing of spherical particles,” J. Am. Ceram. Soc. 44, 513-522 (1961).
G. Mason, “General discussion,” Discus. Faraday Soc. 43, 75-88 (1967).
A. J. Matheson, “Computation of a random packing of hard spheres,” J. Phys. 7, 2569-2576 (1974).
M. J. Powell, “Computer-simulated random packing of spheres,” Powder Technol. 25, 45-52 (1980).
J. Rodriquez, C. H. Allibert & J. M. Chaix, “A computer method for random packing of spheres of unequal size,” Powder Technol. 47, 25-33 (1986).
G.D. Scott, “Packing of spheres,” Nature 188, 908-909 (1960).
E.M. Tory, B. H. Church, M.K. Tam & M. Ratner, “Simulation random packing of equal spheres,” Can. J. Chem. Eng. 51, 484-493 (1973).
E.M. Tory, N. A. Cochrane & S.R. Waddell, “Anisotropy in simulated random packing of equal spheres,” Nature 220, 1023-1024 (1968).
W.M. Visscher & M. Bolsterli, “Random packing of equal and unequal spheres in two and three dimensions,” Nature 239, 504-507 (1972).
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Appendix L3-B: Basic Terminology on Packing
Vb : = Bulk volume
Vp : = Volume of particles
rp = Particle density
e = Fractional voidage (void fraction)
(1)
A = Voids ratio
(2)
eA = Fractional Free Area = Ratio of the free area in a plane parallel to the layers in regular packing to the total
area of the plane
n = Solids fraction = Vp/Vb
(3)
b = Bulk density = Weight of the particles/Vb = pVp/Vb
Substitute Vp = Vb(1-e) obtained from (1) into the above …
 b = p (1 - e)
Vs = Apparent specific volume
1  1
  p (1 e)
b
(4)
(5)
The void ratio e can be expressed in terms of e as follows:
Substitute Vb/Vp = 1/(1 - e) obtained from (1) into (2)


1
e
1 
1 e
1 e
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(6)
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