1 lecture 3 rapid granular flow applications anthony d. rosato granular science laboratory me...

1

Lecture 3 Rapid Granular Flow Applications

Anthony D. RosatoGranular Science Laboratory

ME DepartmentNew Jersey Institute of Technology

Newark, NJ, USA

Presented at the University of Salerno: May, 2011

2

Presentation OutlinePresentation 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

Lecture 3: Rapid Granular Flow Applications

Granular Science Lab - NJIT

3

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



4

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



http://www.mugu.com/galton/start.html

5

Galton’s Board: EXPERIMENTS


16

5

32

5

1

16pd

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.


6Granular Science Lab - NJIT


7

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.



8

Experimental Parameters

Materials of the sphereAluminumBrassStainless Steel

Release Height H (max = 15.53”)Board Tilt Angle 30o to 70o– measured from horizontal

Measurements MadeResidence Times

Distribution of Exit Positions

Computed Quantities

Average downward velocity (cm/sec)

Lateral dispersion (cm2/sec)



9

Sampling of Experimental Results

Board Angle

Degree 70

Degree 60

Degree 50

Degree 40

Degree 30

0

1

2

3

4

5

6

Ave

rage

Res

iden

ce T

ime

(sec

)

7

8

9

10

11

12

5 10 15 20 25 30 35 40

Release Height (cm)

Average residence time Tav as a function of release height H for stainless steel spheres.



10

Distribution of Exit Positions for Stainless Steel Spheres



11

Lateral Dispersion or Diffusivity

Diffusion model [Bridgwater et al., Trans. Instn. Chem. Engrs. 49, 163-169 (1971) ]

- concentration of particles at (x, t) for an infinitely wide board ( , )c x t

- delta-distribution centered at x = 0 and height No o x2

2

o o

, 0,

( , ) 0, 0

( ,0) ( ),

lim x

c cD t x

t xc x t t

c x N x x

2 4o( , )2

x DtNc x t e

πDt

Solution …



12

DtxerfNdzeDt

NtxN

x

x

Dtz 2 2

),( o4o 2

- number of particles in the interval [-x, x] ,N x t

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 = 70o. The origin of the x-axis denotes the center of the board.

Summary of Dispersion Results



13

Sample Trajectories Generated by the Simulation

Figure 9: Three typical trajectories from the discrete element simulation ( = 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.



14

Simulated Exit Position Distribution

Exit distribution of the number of particles for 1/8” spheres at board angle = 70o from simulation in which e = 0.6



15

Lateral Dispersion Computed from Limiting Slope of Mean Square Displacement

(m2)

2 21lim 2.43 cm sec

2tD r

t



16

Quantity Tav (s) V (cm/s) D (cm2/s)

Simulations ( = 70o) 7.12 5.6 2.43

Simulations ( = 90o) 6.70 5.92 1.48

Experiments ( = 70o)

Stainless Steel 7.22 5.49 1.85

Aluminum 6.84 5.77 1.96

Brass 6.81 5.79 2.085

Comparison of Simulated Results with Experiments



17

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)



18

Simulation Parameters

Geometry of Periodic Computational Cell

Steady state computations performed.

Spheres are smooth (no friction) and inelastic, obeying the soft contact laws of Walton and Braun.



19

Steady-State Diagnostics

In computing depth profiles, the cell is partitioned into layers of thickness equal to approximately the particle diameter d.

Averaging layer

Instantaneous layer diagnostic: Mass-weighted average taken over all particles that occupy the layer at time t.

y

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

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

AdNmt 62 Mass hold-up: bulk mass supported by the floor of cross-sectional area A

L Designates the long-term average

N = # of spheres



20

Depth profile of the instantaneous RMS deviatoric velocity

Long-term cumulative, mass-weighted average deviatoric velocity depth profile

2),(3

1L

tyCT Granular Temperature depth profile

Measure of the kinetic energy per unit mass attributed to the particles’ fluctuating velocity components.

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.

dgyTyW /)()( Non-dimensional Granular Temperature depth profile



21

Comparisons with Kinetic Theory of Richman and Martin



22



23

Comparisons with Experiments of Hunt et al.

Simulation Parameters



Validation against Experiments

Paper lid

ta sin

M. Hunt et al., J. Fluids Eng. 116, pg. 785 (1994).

Relatively smooth spheres used in experiment136 grams of particles used, mt = 5.0

24



25



26

Summary of Findings

The behavior of the system depends on the magnitude of the floor acceleration = a2/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 < < 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.



27

Convection in a Vibrated Vessel of Granular Materials

Continuously shake the vessel up and down. Particles will flow upwards near the walls and downward in the center.

Rough, inelastic spheres obeying the Walton & Braun soft-particle models.



28

Velocity Field – Long-time Average Superimposed Trajectory of Large Intruder

Parameters b = 0.8, f = 7 Hz, a/d = 0.5, = 10

Width = 20d

SpheresVelocity

Y. Lan and A. D. Rosato, Phys. Fluids 9 (12), 3615-3624 (1997).



29



30



0

0.5

1

1.5

2

2.5

3

3.5

4

4.5

0 2 4 6 8 10 12

Gamma

Avg

. Cn

v. V

el. (

m/s

x .0

1)

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)

Average Convection Velocity as a Function of

31

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.



32

Instantaneous velocity fields and sphere center projections for the 100d cell (f = 7 Hz, a/d = 0.5, = 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).



33

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).



34

Summary of Findings

The onset of convection is controlled by (a/d) rather than by 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 (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.



35

Upper and lower bumpy walls move at constant velocity in opposite directions. Collisions with flow particles causes them to flow.

Application 3: Couette Flow



D:\Research\Presentations\Univ-of-Salerno(Italy)_June21-28(2009)\Videos\GranCouetteFlow.avi

36

General Features of the Flow- Steady-state Profiles -

VelocityGranular TemperatureSolids FractionPressure



37

Average # of collisions/sec ~ 30 for each particle

Steady State

0.0

4.0

8.0

12.0

16.0

-0.4 -0.2 0.0 0.2 0.4

V / U

Y /

d

0.0

0.2

0.4

0.6

0.8

0 30 60 90 120 150 180Average Number of Collisions

Vxm

ean

(d/s

)

1-topbdry2

3

4

5

6

7

8-middle



38

Granular temperature - kinetic energy of the velocity fluctuations

Mean Velocity

''''''

3

1wwvvuuTt

vvv '

v Particle Velocity

v

“Peculiar” VelocityDimensionless

2

d

TT t

t

Effective shear rate = 2U/H



39

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 0.5 1 1.5 2 2.5 3

Tt

Y /

H

H=8d

H=16d

H=32d

Granular Temperature Profiles

U

uu

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

-1 -0.5 0 0.5 1

Y /

H

H=8d

H=16d

H=32d

= 0.45

Mean Velocity Profiles



40

Solids Fraction Profiles

Y/H

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 0.1 0.2 0.3 0.4 0.5 0.6

H=16d

H=32d

45.0 bulk

N

i

N

jiijijii

k PPP1 2

11Fruuim

V

Pressure - Pyy

0

2

4

6

8

10

12

14

16

0.1 0.3 0.5 0.7 0.9

P*

yy

Y/d



41

Secondary Velocity Field

),( ),,( yxyxu v

2y/d

x/d

U

U

U = 8 d/s

H/d = 8

s/2

U

yuyxuyxu

,

,

Averaging layer used for profiles

y

x

z

H

Lx

x

y

Slab used to compute xv



42

Figure 7: Plot of xv as a function of x/d (L/d = 64, s/2 ) showing development to the

steady state velocity. This appears in the inset, where the horizontal axis is 2x/d.

xv

t (s) x/d

Velocity field for U = 8 d/s (W/d=64)



43

Auto-Correlation

FFT spectrum analysis

Peak at = 7.5 d

Peak at = =15 R

for U = 8 d/s (W/d=64) xv



44

Wavelength vs. Effective Shear Rate

0

5

10

15

20

25

30

35

40

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5

Effective shear rate

Wav

elen

gth

( / d

)

Figure 8: Wavelength/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.



45

Application 4: Intruder Dynamics in Couette Flows

- U

Intruder Properties• Different size, but

same density

• Different mass, but same size

• Different size, same mass



46

= Intruder diameter/Flow particle diameter

m = Intruder mass/Flow particle mass

Size Ratio = D/d Mass Ratio m ~

1 1

1.5 3.375

2 8

3 27

Size and Mass Ratios



47

0

0.5

1

1.5

2

2.5

3

3.5

0.0 1.0 2.0 3.0 4.0Size Ratio

Vav

U = 64 r/s

U = 32 r/s

U = 16 r/s

U = 64 r/s

U = 32 r/d

U = 16 r/s

0

10

20

30

40

50

60

70

0 1 2 3 4Tc

(s)

U=64 r/s

U=32 r/s

U=16 r/s

(a) Crossing time Tc (seconds) versus at U = +/-16, 32, 64 r/s; (b) Average intruder velocity , where S is the distance traveled by the mass center from its initial position near the wall to the mid-plane of the cell.

cTSVav

Intruder Velocity and Mid-Plane Crossing Time



48

0

0.2

0.4

0.6

0.8

1

0 20 40 60 80 100

Time (s)

Y / H

D / d = 1.0

Tc=48 s

0

0.2

0.4

0.6

0.8

1

0 20 40 60 80 100

Time (s)

Y /

H

D / d = 2.0Tc=14 s

= 1.0

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

= 2.0

As the relative size of the intruder increase, its rise time decreases.



49

*

m m( ( )) /( 2* ( ))Y Y Y H Y Ym() - Closest distance possible between the center of the intruder and boundary plane

ffP

1)(

)lg(

)lg(

f

P

s/2

Y* Trajectory + Power Spectrum ( =1.0)

s/2



50

Power spectrum of Intruder y-Trajectories

= D/d = -(2 + 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

Background: Noise Signals

ffP

1)(

)lg(

)lg(

f

P



51

= 0 White noise

= 2 Brownian noise

(random walk)

= 1 1/f noise

(often in processes found in nature)

General Information on Noise



52

= ½, Brownian Motion, is Hausdorff (or Hurst) exponent.

121 fPfffP

1)(

>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.

<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.

=1.0 =0.7, =2.0 =0.9, =3.0 =1.06



53

= 1.0

0.0

0.2

0.4

0.6

0.8

1.0

0 0.01 0.02 0.03 0.04 0.05

Frequency

Y /

H

D / d =1.0

= 1.5:

0.0

0.2

0.4

0.6

0.8

1.0

0 0.01 0.02 0.03 0.04 0.05

Frequency

Y /

H

D / d =1.5

Intruder Histograms of its y-location



54

Intruder Histograms - Continued

“Trapping” in region of low granular temperature

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

=2.0

0.0

0.2

0.4

0.6

0.8

1.0

0 0.01 0.02 0.03 0.04 0.05

Frequency

Y / H

D / d =2.0

=3.0

0.0

0.2

0.4

0.6

0.8

1.0

0 0.01 0.02 0.03 0.04 0.05

Frequency

Y /

H

D / d =3.0



55

Net resultant force averaged over time interval

+ UZONE

1

2

3

4

5

6

7

8

9

10

- U

X

Y

Fy Fnet

Fx

cN

j

jy

cnety F

tNtF

1

)(

1

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

22 HUd

tFtF nety

y

Normalized



56Figure 13: Evolution of tFy for (a) = 1, (c) = 3, and velocity tVy for (b) = 1 and (d) = 3.

yF

yF

yV

yV

Evolution of yF and yV



57

Figure 14: Steady state graphs of rmsyF (left) and rms

yV (right) versus size . Correlation coefficients

R2 are shown for each fitted curve.

F yrms

= 0.0772 + 0.0114

R 2 = 0.9931

0.00

0.05

0.10

0.15

0.20

0.25

0.30

0 1 2 3

(Fy )

rms V y

rms = 0.2023-1.2731

R 2 = 0.9925

0.00

0.10

0.20

0.30

0.40

0.50

0 1 2 3

Vyrm

s

Root-mean-square force and Velocity versus



58

Procedure: Vary particle density and maintain size ratio = 1

= Intruder diameter / flow particle diameter

m = Mass Intruder / Mass of flow particles = 0.45

= constant

0.0

1.0

2.0

3.0

0.5 1.0 1.5 2.0 2.5 3.0

Vyr

ms

( d /

s )

Original System: Vary size ratio and maintain constant particle density .

How does particle mass affect the fluctuation velocity in the direction perpendicular to the shear?



59

= 1

0.0

1.0

2.0

3.0

0 5 10 15m

Vyr

ms

( d

/ s

)

Vary mass ratio m and maintain size ratio .

m = 1

0.0

1.0

2.0

3.0

0.0 1.0 2.0 3.0

Vyr

ms

( d /

s )

Vary size ratio and maintain mass ratio m.



60

0

500

1000

1500

-6 -4 -2 0 2 4 6

V yrms ( d / s )

His

t(V

y)

m=27.0

m=1.0

m=8.0

m=3.375

m=0.5

Progression of Velocity Distributions for = 1

An increase in particle mass results in a narrower velocity distribution (qualitative agreement with the Maxwell-Boltzmann distribution).



61

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)



62

Cylinder

Power Amplifier

Controller

AmplifierShaker

Accelerometer

Acrylic spheres: d = 1/8″

= 1200 kg/m3

Frequency: 25 – 100 Hz

Amplitude (a/d): 0.04 – 0.24

: 0.94 – 11.0 (relative acceleration)

T = 10 minutes (vibration duration)Aspect Ratio: D/d ~ 20

“Maximum” Solids Fraction: = 0.6366 ± 0.0005 for uniform spheresG.D.Scott, D.M.Kilgour, British Journal of Applied Physics,1969.

Densification Experiments: Uniform Spheres

Details →

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).



63

Details of the Experimental Procedure …

Pour Slice Vibrate

Compute bulk solids fraction and improvement in solids fraction.

d = 0.125 inch

Solids Fraction

Improvement in Solids Fraction = 1001

poured

relaxed



64

Data points are averages of 4 trialsRelative Acceleration

Sol

ids

Fra

ctio

n

Impr

ovem

enti

nS

olid

sF

ract

ion

(%)

2 3 4 5 6

2 3 4 5 6

0.605

0.61

0.615

0.62

0.625

0.63

0.635

0.64

0

1

2

3

4

5a/d=0.24

Relative Acceleration

Sol

ids

Fra

ctio

n

Impr

ovem

enti

nS

olid

sF

ract

ion

(%)

2 4 6 8 10 12

2 4 6 8 10 12

0.608

0.612

0.616

0.62

0.624

0.628

0.632

0.636

0.64

1

1.5

2

2.5

3

3.5

4

4.5

5

5.5

a/d=0.16


Sol

ids

Fra

ctio

n

Impr

ovem

enti

nSol

ids

Fra

ctio

n(%

)

1 2 3 4 5

1 2 3 4 5

0.605

0.61

0.615

0.62

0.625

0.63

0.635

0.64

0

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5

5.5

a/d=0.04


Sol

ids

Fra

ctio

n

Impr

ovem

enti

nS

olid

sF

ract

ion

(%)

2 3 4 5 6 7 8

2 3 4 5 6 7 8

0.6

0.605

0.61

0.615

0.62

0.625

0.63

0.635

0

1

2

3

4

5a/d=0.06

Results: Systems vibrated for 10 minutes



65

Do Simulations Results Agree with Physical Experiments ?

Comparisons: Poured Bulk Density

Vibrated Bulk Density

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 vs. system size and extrapolated …

-loose = 0.59 -dense = 0.63

1969: G.D.Scott, D.M.Kilgour, British Journal of Applied Physics

= 0.6366 ± 0.0005 for uniform spheres1997: E. R. Nowak, M. Povinelli, et al., Powders & Grains 97, (Balkema, Durham, NC, 1997), pp. 377 - 380.

= 0.656 for vibrated column (d/D ~ 9) of uniform spheresSelect literature: Sphere packing

d

Z

Vibrating Floor

H

L

W

Y

x

z Averaging Layer



66

Select Literature on Packing of Spheres

Coord. No. Solids Fraction System Reference 0.601 0.001/ 0.637 0.001

Steel spheres in cylinder Scott (1960)

0.625 Steel spheres in glass container McGeary (1961)

0.6366 0.004 Steel spheres in cylinder Finney

6.1 0.59 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 0.606 0.006 Computer Matheson (1974)

6.0 0.6099 / 0.6472 Statistical Model Gotoh & Finney (1974)

6.0 0.59 0.01 Computer Powell (1980)

6.0 0.58 0.05 Computer Rodriguez, et al. (1986)

0.634 Computer Mason (1967)

0.582 Computer Gotoh, Jodrey & Tory (1978)

5.64 0.6366 Computer Jodrey & Tory (1981)

0.610 – 0.658 Computer Zhang & Rosato (2004)



67

Aspect Ratio d/L

Sol

ids

Fra

ctio

n

0.05 0.1 0.15 0.2 0.25

0.05 0.1 0.15 0.2 0.25

0.55

0.56

0.57

0.58

0.59

0.6

0.61

0.55

0.56

0.57

0.58

0.59

0.6

0.61

0 0.01 0.02 0.03 0.04

0 0.01 0.02 0.03 0.04

0.6

0.605

0.61

0.615

0.62

0.6

0.605

0.61

0.615

0.62

Solids fraction as a function of the inverse aspect ratio (d/L) for a system of particles with friction coefficient = 0.1. The inset shows the extrapolated value 0.6102 as d/L → 0.

Extrapolated solids fraction for infinitely wide container in good agreement with experiments reported in literature.

depends on →

= 0.6102



68

Friction Coefficient

Sol

ids

Fra

ctio

n

0 0.2 0.4 0.6 0.8

0 0.2 0.4 0.6 0.8

0.55

0.56

0.57

0.58

0.59

0.6

0.55

0.56

0.57

0.58

0.59

0.6

Variation of the solids fraction with friction coefficient (d/L = 0.1064, N = 600). Each point of the curve represents an average taken over 10 realizations, while the bars show the deviation.

Particles that are more frictional produce a less dense structure after pouring because of formation of bridges among particles.

As increases, there is approximately a 7% reduction in . At = 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.



69

Simulated Random Dense Packing

Frequency (Hz)

Sol

ids

Fra

ctio

n

30 40 50 60 70 80 90 100 110 120

30 40 50 60 70 80 90 100 110 120

0.61

0.615

0.62

0.625

0.63

0.635

0.64

0.61

0.615

0.62

0.625

0.63

0.635

0.64

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

Vibration Time(s)

Sol

ids

Fra

ctio

n

0 5 10 15

0 5 10 15

0.61

0.615

0.62

0.625

0.63

0.635

0.64

0.645

0.65

0.655

0.66

0.61

0.615

0.62

0.625

0.63

0.635

0.64

0.645

0.65

0.655

0.66

1/Tv

Sol

ids

Fra

ctio

n

0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1 0.11

0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1 0.11

0.654

0.656

0.658

0.66

0.662

0.664

0.666

0.668

0.67

0.654

0.656

0.658

0.66

0.662

0.664

0.666

0.668

0.67 = 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



72

Simulated Trends versus Frequency ….

Frequency(Hz)

Sol

ids

Fra

ctio

n

Impr

ovem

enti

nS

olid

sF

ract

ion(

%)

30 40 50 60 70 80 90

25 50 75

0.58

0.585

0.59

0.595

0.6

0.605

0.61

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5

5.5

Frequency(Hz)

Sol

ids

Fra

ctio

n

Impr

ovem

enti

nS

olid

sF

ract

ion(

%)

30 40 50 60 70 80 90

30 40 50 60 70 80 90

0.592

0.594

0.596

0.598

0.6

0.602

0.604

0.606

0.608

0.61

0.612

2.5

3

3.5

4

4.5

5

5.5

Frequency(Hz)

Sol

ids

Fra

ctio

n

Impr

ovem

enti

nS

olid

sF

ract

ion(

%)

0 20 40 60 80

0 20 40 60 80

0.57

0.575

0.58

0.585

0.59

0.595

0.6

0.605

0.61

0.615

-1

0

1

2

3

4

5

6

Frequency(Hz)

Sol

ids

Fra

ctio

n

Impr

ovem

enti

nS

olid

sF

ract

ion

(%)

30 40 50 60 70 80 90

30 40 50 60 70 80 90

0.57

0.572

0.574

0.576

0.578

0.58

0.582

0.584

0.586

0.588

-1

-0.5

0

0.5

1

1.5

a/d=0.24

a/d=0.48

a/d=0.08

a/d=0.02



73Frequency (Hz)

Am

plitu

de(i

nch)

20 40 60 80

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

0.09

0.1

0.11

0.12

0.459431 1.39462 2.32982 3.26501 4.20021

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



74



L. Vanel, A. Rosato, R. Dave, Phys. Rev. Lett. 78, 1255 (1997).

Cylinder

Power Amplifier

Controller

AmplifierShaker

Accelerometer

Experimental Evidence

Small amplitudes a/d < 0.25, and high frequencies (40 – 80 Hz)

75

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.”


76



The study of density relaxation has its foundations in the extensive literature on the packing of circles and spheres

1611: Kepler - Geometry of the snowflake

1665: Robert Hook - Circle and sphere packings

1694: Gregory, a Scottish astronomer, suggested that 13 rigid uniform spheres could be

packed around a sphere of the same size

1727: Hales - Packing of dry peas pressed into a container

1887: Thompson - How to fill Euclidean space using truncated octahedrons

1899: Slichter – Found analytical expressions for the porosity in beds of uniform spheres

1932: Hilbert - Found a structure for which m=0.123

1933: Heesch and Laves: Created a stable arrangement of spheres with m= 0.056

1944: Oman & Watson: ‘Loose’ and ‘dense’ random packing of spheres

1951: Stewart - Consolidated state of optimal bulk density

Much more …

Apollonian Gasket

Boyd, D. W. "The Residual Set Dimension of the

Apollonian Packing." Mathematika 20, 170-174, 1973.

77



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.

Number of particles: 3,456

Periodic BC in lateral dimensions

d=0.02m

Restitution coefficiente= 0.9

Particle-particle frictionp=0.1

Particle material density= 1.2 g/cm3

Integration Time step ~ 10-5 s

Parameters for discrete element simulation

78



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

79



Animation of Pouring from DEM Simulation

80



0

sin 2 1 , ;( )

0, otherwise

, relaxation time, tapping time

p C C p C

C p b p b

a f t t i t t i t t i ty t

t t t t t

22 24 1.35

a af

g g

Dimensionless acceleration

Tapping Sequence: Motion of the Plane Floor

81



The vertical positions of particles in a given layer (y i/d) after tap are monitored

System Response to Taps

82



Data points are averages over 20 realizations.

Red lines are error bars.

Particle that are more frictional produce a less dense structure after pouring because of formation of bridges among particles.

McGeary (1961): Steel spheres in glass cylinder

Simulated Effect of Particle Friction on Poured Bulk Solids Fraction

83



Sample Realizations

7018.0

6110.0

f

o6884.0

6118.0

f

o

7007.0

6116.0

f

o

7077.0

6118.0

f

oAverage of 20 realizations

84

Number of particle centers inside a layer of thickness H normalized by the total number of system particles

Distribution of Particle Centers

Ordering adjacent to plane floor

Poured System



85



Center Distribution averaged over 10 consecutive taps

86


Ensemble-Averaged Center Distribution


Hzf 5.7

5.044.0/ da

87



Evidence of a Critical Displacement Amplitude

Monte Carlo simulation results strongly suggest that there is a critical displacement amplitude that promotes an optimal evolution to a dense structure.

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

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

Before TapMC Simulated Tap Applied

88



Evolution of Structure: Effect of Displacement Amplitude

Hzf 5.7

5.011.0/ da

89



Evolution of Structure: Effect of Displacement Amplitude

Hzf 5.7

5.044.0/ da

90



Summary - Conclusions

Upward progression of “organized” layers induced by the plane floor as the taps evolve.

Evidence of a critical tap intensity that optimizes the evolution of packing density.

Discrete element model reveals the mechanism…

The configuration of the particles plays an important role in how the system evolves, rather than solely the value of the bulk solids fraction.

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 (t ~ 10-5 s), period of applied tap, single-tap system relaxation time, long-time relaxation scale

91



End of Lecture 3

92


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]

Rhombic Dodecahedron 12 faces


93




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

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 = 0.123

1933 Heesch & Laves [13] found looser packing with = 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).

7797.033/11cos18

94



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 vs. and extrapolated to large N.

loosedense

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.

dense

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).

31 N


95

0.5

0.52

0.54

0.56

0.58

0.6

0 0.005 0.01 0.015 0.02

1/N

Pa

ck

ing

Fra

cti

on

Random Loose Packing of SpheresExperiments: G. Onoda and Y. Liniger, PRL 64, 2727, 1990




96

References for Additional Reading1.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).



Appendix L3-A: Selected Literature on Packing Studies

97


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).

Appendix L3-A: Selected Literature on Packing Studies


98



Appendix L3-B: Basic Terminology on Packing

Vb : = Bulk volume

Vp : = Volume of particles

rp = Particle density

e = Fractional voidage (void fraction) (1)

= 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

= 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) (4)

Vs = Apparent specific volume (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)

(6)

ee

e

11

11

)1(11

eb

p

1 lecture 3 rapid granular flow applications anthony d. rosato granular science laboratory me...

Documents

galtons board application

steady application

couette flows application

vibrated systems application

board tilt angle q

stainless steel spheres

system of solenoids

optical sensor