reosato lecture 3 - applications

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Transcript reosato lecture 3 - applications

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 1

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, proto geneticist, half-cousin of Charles Darwin and best selling 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 Granular Science Lab - NJIT

Lecture 3: Rapid Granular Flow Applications

Galton’s Board: EXPERIMENTS

5  16 5  32

d p

 1 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

Granular Science Lab - NJIT 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.

Lecture 3: Rapid Granular Flow Applications

Experimental Parameters

Materials of the sphere

Aluminum Brass Stainless Steel

Release Height H

(max = 15.53”)

Board Tilt Angle

q ( 30 o to 70 o ) – measured from horizontal

Measurements Made

Residence Times Distribution of Exit Positions

Computed Quantities

Average downward velocity (cm/sec) Lateral dispersion (cm 2 /sec)

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Lecture 3: Rapid Granular Flow Applications

Sampling of Experimental Results

Average residence time

av stainless steel spheres. as a function of release height

for 1 2 1 1 1 0 9 8 7 6 5 4 1 0 3 2 5 1 0 1 5 2 0 2 5 Release Height (cm ) 3 0 3 5 4 0 Board Ang le Degree 70 Degree 60 Degree 50 Degree 40 Degree 30

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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

, 163-169 (1971) ] 

- concentration of particles at (

) for an infinitely wide board - delta-distribution centered at

= 0 and height



c



t



D



c



x

,

t

 0,

lim





N



 0,

t

 0

Solution …



2 N

πDt e



2 4

Dt Granular Science Lab - NJIT

Lecture 3: Rapid Granular Flow Applications ( )

N

(

x

,

t

)

 - number of particles in the interval [-

]

2 N

o



Dt





e



4 dz



N

o

erf



x

2 Dt



D = 1.85 cm 2 /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 = 70 o . The origin of the

-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 = 70 o ) 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 from simulation in which

= 0.6

= 70 o

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Lecture 3: Rapid Granular Flow Applications Lateral Dispersion Computed from Limiting Slope of Mean Square Displacement



lim



1 2





2.43 cm sec

(m 2 )

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Lecture 3: Rapid Granular Flow Applications Comparison of Simulated Results with Experiments

Quantity

Simulations ( q = 70 o ) Simulations ( q = 90 o ) Experiments ( q = 70 o )

av (s) 7.12

6.70

Stainless Steel Aluminum Brass 7.22

6.84

6.81

(cm/s) 5.6

5.92

(cm 2 /s) 2.43

1.48

5.49

5.77

5.79

1.85

1.96

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

Averaging layer

Instantaneous layer diagnostic

: Mass weighted average taken over all particles that occupy the layer at time

. A layer is ‘identified’ by its center y-coordinate.

m t





2 6

Mass hold-up: bulk mass supported by the floor of cross-sectional area

N = # of spheres Long term cumulative mean velocity of layer-y taken over the time interval (

o ,

1 ). Instantaneous fluctuating (or

deviatoric

velocity) of the

th particle in layer-y L 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

 1 3

(

)

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 Continuum Mechanical 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

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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, m t = 5.0

 sin

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

 2 /

= 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 m  m b = 0.8, Parameters

f = 7 Hz, a/d = 0.5,

Width = 20d G = 10 Superimposed Trajectory of Large Intruder 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

4 3.5

3 2.5

2 1.5

1 0.5

0 0 2 4 6

Gamma

8 10 12 0.15

0.25

0.275

0.3

0.4

0.5

Poly. (0.15) Poly. (0.25) Poly. (0.275) Poly. (0.3) Poly. (0.4) Poly. (0.5)

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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

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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).

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 (

) 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 side walls are smooth.

As the cell width (

) 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. Granular Science Lab - NJIT

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

16.0

0.8

12.0

0.6

0.4

8.0

4.0

0.0

-0.4

-0.2

0.0

V / U

0.2

0.4

0.2

0.0

0 30 60 90 120 150

Average Number of Collisions

180

Average # of collisions/sec ~ 30 for each particle

4 5 6 1-top bdry 2 3 7 8-middle

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Lecture 3: Rapid Granular Flow Applications

Granular temperature - kinetic energy of the velocity fluctuations

T t v

'  1  3

v u

' 



' 

' “Peculiar” Velocity

v v

Particle Velocity Mean Velocity Dimensionless   

T t



T t

( ) 2 Effective shear rate = 2U/H

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Lecture 3: Rapid Granular Flow Applications 1

Granular Temperature Profiles

0.9

0.8

0.7

0.6

0.5

0.4

0.3

0.2

0.1

0 0 0.5

1 1.5

T t

2 2.5

H=8d H=16d H=32d

Mean Velocity Profiles

0.9

1 0.8

0.7

0.6

0.5

0.4

0.3

0.2

0.1

0 -1 -0.5

n = 0.45

H=8d H=16d H=32d 0



u U

0.5

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Lecture 3: Rapid Granular Flow Applications

Solids Fraction Profiles

1 0.9

0.8

0.7

Y/H 0.6

0.5

0.4

0.3

0.2

0.1

0 0

n

bulk

 0 .

45

0.1

H=16d H=32d 0.2

n 0.3

0.4

0.5

0.6

Pressure - P

16 14 12 10 8 6 4 2

* yy 0 0.1

0.3

0.5

0.7

0.9



P k



 1

 

i N



 1 m i



 1 2





 

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2 y / d Lecture 3: Rapid Granular Flow Applications

Secondary Velocity Field

(

 (

), v (

) )

U U

= 8

= 8  2 /

s U

x/ d Slab used to compute v Averaging layer used for profiles H y x z D x

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L D x D y



( ) ( ) ( )

 41

Lecture 3: Rapid Granular Flow Applications Velocity field for

= 8 d/s (W/d=64) v

(

)

Figure 7

: Plot of v as a function of

(

= 64,  2 /

) showing development to the steady state velocity. This appears in the inset, where the horizontal axis is 2

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Lecture 3: Rapid Granular Flow Applications v

for U

= 8

d

/s (

W

d

=64)

Peak at

= =15 R

Auto-Correlation FFT spectrum analysis Peak at

= 7.5 d

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Lecture 3: Rapid Granular Flow Applications

Wavelength vs. Effective Shear Rate

40 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 = 8. The solid line is included to show the trend.

H / d

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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

- U

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Lecture 3: Rapid Granular Flow Applications

Size and Mass Ratios

Size Ratio

f =

D/d

1 1.5

2 3

Mass Ratio

f m ~ f 3

1 3.375

8 27

= Intruder diameter/Flow particle diameter

f m

= 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 2.5

2 1.5

U = 64 r/s U = 32 r/s U = 16 r/s U = 64 r/s U = 32 r/d U = 16 r/s 60 50 40 30 U=16 r/s U=32 r/s 20 1 U=64 r/s 10 0.5

0 0 0.0

1.0

2.0

Size Ratio f 3.0

4.0

0 1 f 2 3 4 (a) Crossing time T c (seconds) versus f at U = +/-16, 32, 64 r/s; (b) Average intruder velocity

V av



S T

c is the distance traveled by the mass center from its initial position near the wall to the mid-plane of the cell.

, where S

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Lecture 3: Rapid Granular Flow Applications 1 0.8

0.6

0.4

0.2

Tc =

Rise Time for the intruder to reach the middle layer from bottom.

= 1.0

= 2.0

0 0 20 Tc=48 s 40 60

Time (s)

D / d = 1.0

80 100 1 0.8

0.6

0.4

0.2

0 0 Tc=14 s 20 40 60

Time (s)

D / d = 2.0

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 (**

=1.0)





 2 /

s





 2 /

s Y

* 

(

Y

( f )



Y

m f

H



2*

Y

m Closest distance possible between the center of the intruder and boundary plane

(

) 

1  lg(

) lg(

)  



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Lecture 3: Rapid Granular Flow Applications

Power spectrum of Intruder

-Trajectories

f =

h  = -(2 h + 1) 1.0

0.7

-2.4

2.0

0.9

-2.8







-2.4



-2.8

3.0

1.06

-3.06



-3.06

(

) 

1  lg(

) lg(

)  



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Background: Noise Signals

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

 1

2 h  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 moment t+1 .

t is not likely to be followed by a similar trend at next  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,

=2.0

 h

=0.9

,

=3.0

 h

=1.06

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Lecture 3: Rapid Granular Flow Applications

Intruder Histograms of its y-location

1.0

0.8

0.6

0.4

0.2

0.0

0 0.01

= 1.0

0.02

0.03

Frequency

D / d =1.0

0.04

0.05

1.0

0.8

0.6

0.4

0.2

0.0

0 0.01

= 1.5

: 0.02

0.03

Frequency

D / d =1.5

0.04

0.05

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Lecture 3: Rapid Granular Flow Applications

Intruder Histograms - Continued

1.0

0.8

0.6

0.4

0.2

0.0

0 0.01

=2.0

0.02

0.03

Frequency

D / d =2.0

0.04

0.05

1.0

0.8

0.6

0.4

0.2

0.0

0 f

=3.0

0.01

“Trapping” in region of low granular temperature 0.02

0.03

Frequency

D / d =3.0

0.04

0.05

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S. Dahl, C. Hrenya, Physics of Fluids 16, 1-24 (2004).

ZONE 1 2 3 4 5 6 7 8 9 10 Lecture 3: Rapid Granular Flow Applications

Net resultant force averaged over time interval

D + U

F y

( )

net



N c

1 (

)

j N





F y j

F x F y F net X

D is three orders of magnitude smaller than time scale over which the dynamics evolve, but much larger than the integration step.

Normalized



F y

 

(

d F y

2 ( )

net U H

2 ) - U

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Lecture 3: Rapid Granular Flow Applications

Evolution of

F y

and

V y F y V y F y V y

Figure 13

: Evolution of

F y Granular Science Lab - NJIT

for (a) f = 1, (c) f = 3, and velocity

V y

for (b) f = 1 and (d) f = 3. 56

Lecture 3: Rapid Granular Flow Applications

Root-mean-square force and Velocity versus

f 0.30

0.25

y rms = 0.0772 f + 0.0114

2 = 0.9931

0.50

0.40

0.20

0.15

0.30

V y rms

= 0.2023

f -1.2731

2 = 0.9925

0.20

0.10

0.05

0.00

0 1 f 2 3 0 1 f 2 3

Figure 14

: Steady state graphs of

F y rms

(left) and

V y rms R

2 are shown for each fitted curve. (right) versus size f . Correlation coefficients

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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

= 1

= Intruder diameter

flow particle diameter

f m

= Mass Intruder

Mass of flow particles

n = 0.45



= constant 3.0

2.0

1.0

0.0

0.5

1.0

1.5

f

2.0

2.5

3.0

Original System: Vary size ratio

and maintain constant particle density



.

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Lecture 3: Rapid Granular Flow Applications 3.0

2.0

1.0

0.0

0 f = 1 5 f m 10 3.0

2.0

1.0

0.0

f

m = 1 1.0

f

2.0

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15 3.0

Vary mass ratio

f m

and maintain size ratio

.

Vary size ratio

and maintain mass ratio

f m.

Lecture 3: Rapid Granular Flow Applications

Progression of Velocity Distributions for

= 1

1500 f m =8.0

f m =27.0

1000 f m =1.0

f m =3.375

500 f m =0.5

0 -6 -4 -2 0

V yrms

( d / s )

2 4 6 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 “densification”.

“density relaxation” or 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

Cylinder Controller

Acrylic

spheres:

= 1/8″  = 1200 kg/m 3 Frequency: 25 – 100 Hz r Shake Amplifier Accelerometer Power Amplifier Amplitude

(

/

):

0.04 – 0.24

G

: 0.94 – 11.0 (relative acceleration) T = 10 minutes (vibration duration) Aspect Ratio:

~ 20

“Maximum” Solids Fraction:

= 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).

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Details →

Lecture 3: Rapid Granular Flow Applications

Details of the Experimental Procedure …

Compute bulk solids fraction and improvement in solids fraction.

= 0.125 inch

Pour Slice

Improvement in Solids Fraction

= Vibrate

relaxed

poured

 1   100

Solids Fraction

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Lecture 3: Rapid Granular Flow Applications 0.64

1 0.635

0.63

0.625

0.62

0.615

0.61

0.605

Results: Systems vibrated for 10 minutes

2 3 2 3 4 5 0.635

a/d=0.04

5.5

5 4.5

4 3.5

3 0.63

0.625

0.62

a/d=0.06

4 2.5

2 1.5

1 0.5

0 2 3 4 Relative Acceleration 5 0.615

0.61

0.605

0.6

2 3 5 4 5 Relative Acceleration 6 6 0.64

2 0.636

0.632

0.628

0.624

0.62

0.616

0.612

0.608

2 0.64

2 3 4 4 6 8 10 12

a/d=0.16

4 5.5

5 4.5

4 3.5

3 0.635

0.63

0.625

0.62

a/d=0.24

2.5

2 1.5

0.615

0.61

1 0.605

6 8 Relative Acceleration 10 12 2 Data points are averages of 4 trials 3 4 Relative Acceleration 5 5

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7 7 8 0 2 1 8 5 4 3 6 6 5 4 1 0 3 2 64

Lecture 3: Rapid Granular Flow Applications

Do Simulations Results Agree with Physical Experiments ?

Comparisons:

Poured Bulk Density

Vibrated Bulk Density

H d

z Averaging Layer Vibrating Floor

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

-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

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Select literature: Sphere packing

Lecture 3: Rapid Granular Flow Applications

Select Literature on Packing of Spheres Coord. No. Solids Fraction

0.601  0.001/ 0.637  0.001 0.625 6.1 6.0 6.4 6.01 6.0 6.0 6.0 6.0 0.6366  0.004 0.59 0.628 0.61 0.582 0.58 0.606  0.006 0.6099 / 0.6472 0.59  0.01 0.58  0.05 5.64 0.634 0.582 0.6366 0.610 – 0.658

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System

Steel spheres in cylinder

Reference

Scott (1960) Steel spheres in glass container McGeary (1961) Steel spheres in cylinder Computer Computer Computer Computer Computer Computer Statistical Model Computer Computer Finney Tory, et al. (1968) Adams & Matheson (1972) Bennett (1972) Visscher & Bolsterli (1972) Tory, Church, et al. (1973) Matheson (1974) Gotoh & Finney (1974) Powell (1980) Rodriguez, et al. (1986) Computer Computer Computer Computer Mason (1967) Gotoh, Jodrey & Tory (1978) Jodrey & Tory (1981) Zhang & Rosato (2004) 66

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

0.62

0.15

0.01

0.02

0.2

0.03

0.25

0.04

0.62

0.61

0.6

0.59

0.615

0.61

0.605

0.615

0.6

0.61

0.605

0.59

0.6

0 0.01

0.02

0.03

0.04

0.6

0.58

0.57

0.56

n 

= 0.6102

0.56

0.55

0.05

0.1

0.15

Aspect Ratio d/L 0.2

0.25

0.55

Solids fraction as a function of the inverse aspect ratio (

) for a system of particles with friction coefficient m = 0.1. The inset shows the extrapolated value 0.6102 as

→ 0.

depends on

→

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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

. 0.6

0 0.2

0.4

0.6

0.8

0.6

0.59

0.58

0.57

0.56

0.59

0.58

0.57

0.56

0.55

0 0.2

0.4

Friction Coefficient 0.6

0.8

0.55

Variation of the solids fraction with friction coefficient m (

= 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

0.64

30 40 50 60 70 80 0.635

0.63

0.625

90 100 110 120 0.64

a=0.01' ' a=0.02' ' a=0.03' ' a=0.04' ' a=0.06' ' a=0.005' ' 0.635

0.63

0.625

0.62

0.615

0.62

0.615

0.61

30 40 50 100 110 120 0.61

60 70 80 Frequency (Hz) 90 n

= 0.6582

is the solids fraction of random dense packing, in good agreement with the experimental result of Nowak et al. (0.656)

Simulated Trends

 0.66

0 0.655

0.65

0.645

0.64

0.635

0.63

0.625

0.62

0.615

0.61

0 5 10 15 5 10 Vibration Time(s) 15 0.67

0 0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

0.09

0.1

0.11

0.67

0.668

0.666

0.664

0.666

0.664

0.662

0.66

0.658

0.656

0.662

0.66

0.658

0.656

0.654

0 0.01

0.02

0.03

0.04

0.05

0.06

1/Tv 0.07

0.08

0.09

0.1

0.11

0.654

Granular Science Lab - NJIT

0.66

0.655

0.65

0.645

0.64

0.635

0.63

0.625

0.62

0.615

0.61

Lecture 3: Rapid Granular Flow Applications

Simulated Trends versus Frequency ….

0.61

25 50 75 0.605

0.6

0.595

0.59

0.585

0.58

=0.02

40 50 60 Frequency(Hz) 70 80 5.5

5 4.5

4 3.5

3 2.5

2 1.5

1 90 0.5

0.612

0.61

0.608

0.606

0.604

0.602

0.6

0.598

0.596

0.594

0.592

30 30 40 50 60 40

=0.08

50 60 Frequency(Hz) 70 80 70 80 90 3.5

3 90 2.5

5.5

5 4.5

4 0.615

0 0.61

0.605

0.6

0.595

0.59

0.585

0.58

0.575

40 =0.24

60 0.57

0 20 40 Frequency(Hz) 60

Granular Science Lab - NJIT

80 80 0 -1 2 1 4 3 6 5 0.588

30 0.586

0.584

0.582

0.58

0.578

0.576

0.574

0.572

0.57

30 40 40 50 50

60 70 =0.48

60 Frequency(Hz) 70 80 80 90 1.5

1 0.5

0 -0.5

-1 90 72

Lecture 3: Rapid Granular Flow Applications

Simulated Densification Phase Map: L/d = 25, N = 8000

0.12

0.11

0.1

0.09

0.08

0.07

0.06

0.05

0.04

0.03

0.02

0.01

Granular Science Lab - NJIT

0.459431

1.39462

2.32982

3.26501

4.20021

40 Frequency (Hz) 60 80 73

Lecture 3: Rapid Granular Flow Applications

Experimental Evidence

L. Vanel, A. Rosato, R. Dave,

Phys. Rev. Lett

, 1255 (1997).

Cylinder Controller Shaker Amplifier Accelerometer Power Amplifier Small amplitudes

< 0.25, and high frequencies (40 – 80 Hz)

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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.”

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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 Hilbert

– Found analytical expressions for the porosity in beds of uniform spheres - Found a structure for which n

=0.123

Heesch and Laves

: Created a stable arrangement of spheres with n

= 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.

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Apollonian Gasket 76

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 Periodic BC in lateral dimensions

=0.02

Particle material density  = 1.2 g/cm 3 Restitution coefficient

= 0.9

Particle-particle friction m p =0.1

Integration Time step ~ 10 -5 s

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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

and frequency

. Allow system to relax until quiescent.

Kinetic Energy ~ 0 System tapped Particles bounce back

= 0.4;

= 7.5 Hz

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Lecture 3: Rapid Granular Flow Applications

Animation of Pouring from DEM Simulation

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Lecture 3: Rapid Granular Flow Applications

Tapping Sequence: Motion of the Plane Floor

t C y t

0 ( )

 

 (  0, otherwise

t b

t p



(

relaxation time, 1 )

t C t b

, )

t i t C

 tapping time

Granular Science Lab - NJIT p



Dimensionless acceleration G 

 2

a g

 4  2

 1.35

Lecture 3: Rapid Granular Flow Applications

System Response to Taps

The vertical positions of particles in a given layer (y i /

) after tap are monitored

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Lecture 3: Rapid Granular Flow Applications

Simulated Effect of Particle Friction

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

 0 .

6110 n

 0 .

7018 n

 0 .

6118  0 .

6884 n

 0 .

6116  0 .

Average of 20 realizations

Granular Science Lab - NJIT

 0 .

6118 n

 0 .

7077 83

Lecture 3: Rapid Granular Flow Applications

Distribution of Particle Centers

Number of particle centers inside a layer of thickness 

total number of system particles normalized by the Poured System

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Ordering adjacent to plane floor 84

Lecture 3: Rapid Granular Flow Applications

Center Distribution

averaged over 10 consecutive taps

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Lecture 3: Rapid Granular Flow Applications

Ensemble-Averaged Center Distribution

Granular Science Lab - NJIT f

 7 .

Hz a

G /

  0 .

5 0 .

44 86

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 O.Dybenko, A. Rosato, V. Ratnaswamy, D. Horntrop, L. Kondic, “Density Relaxation by Tapping”, in preparation.

Granular Science Lab - NJIT

g Although not presented here, similar findings were observed in DEM simulations.

g 87

Lecture 3: Rapid Granular Flow Applications

Evolution of Structure: Effect of Displacement Amplitude

f

 7 .

5 Hz a

/



d



0 .

5 0 .

11

Granular Science Lab - NJIT

Lecture 3: Rapid Granular Flow Applications

Evolution of Structure: Effect of Displacement Amplitude



7 .

Hz a

G / 

 0 .

5 0 .

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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 ( D t ~ 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]:

1665

 Hooke [2]: Interested in the geometry of the snowflake 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]

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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. 1 st analytical expressions.

attempt to find

“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 18   cos  1

( ) ( )

   0 .

7797 

Alternative: What is the minimal solids fraction of rigid assembly of uniform spheres?

Rigidity  E ach 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 to large N. 1

n loose  0.59

n dense  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.

n

dense

 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

)

, 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

0.58

0.56

0.54

0.52

0.5

0 0.005

0.01

1/N

0.015

0.02

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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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Lecture 3: Rapid Granular Flow Applications

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. 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).

A. E. R. Westman and H. R. Hugill, “The packing of particles,” J. Am. Ceram. Soc. 13 [10], 767-779 (1930).

233-242 (1978). 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).

G.D. Scott, “Packing of spheres,” Nature 188, 908-909 (1960).

C. H. Bennett, “Serially deposited amorphous aggregates of hard spheres,” J. Appl. Phys. 6, 2727-2733 (1972).

, 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).

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, “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).

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, W. S. Jodrey & E. M. Tory, “Computer simulation of isotropic, homogeneous, dense random packing of equal spheres,” Powder Technol. 30, 111-118 (1981).

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).

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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Lecture 3: Rapid Granular Flow Applications

Appendix L3-B: Basic Terminology on Packing

V b : = Bulk volume V p : = Volume of particles r p = Particle density

= Fractional voidage (void fraction)  A = Voids ratio (

) (

)

A =

Fractional Free Area

area of the plane = Ratio of the free area in a plane parallel to the layers in regular packing to the total n =

Solids fraction

= V p /V b  b =

Bulk density

= Weight of the particles/V b =  p V p /V b Substitute V p = V b (1-

) obtained from (

) into the above … V s  The void ratio

 b =  p ( 1 -

) = Apparent specific volume  1  

( 1 1 

can be expressed in terms of

e e

) as follows: (

) (

) Substitute V b /V p = 1/(1 -

) obtained from (

) into (

)    1 1 

 1  1



(

)

Granular Science Lab - NJIT