Flow and jamming of granular matter through an orifice

A. Garcimartín - Flow and jamming of granular matter through an orifice Traffic and granular flow '07

Flow and jamming of granular

matter through an orifice

R. Arévalo, D. Maza, A. Garcimartín,

C. Mankoc, A. Janda, I. Zuriguel & M. Pastor

Dpto. de Física y Matemática Aplicada,

Universidad de Navarra

31080 Pamplona, Spain

http://fisica.unav.es/granular/

Thanks to:

Eric Clément (PMMH, ESPCI)

Luis A. Pugnaloni (CONICET, Argentine)

Tom Mullin (Univ. Manchester, U.K.)

Andrés Santos (Univ. Extremadura, Spain)


Flow and jamming of granular

matter through an orifice

•Granular matter can behave as a solid, a liquid or a gas.

•Jamming can be likened to a phase transition:A. Liu and S. Nagel, “Jamming is not just cool any

more”, Nature 396, 21-22 (1998).

– Are there distinct, different regimes (jamming / non jamming)?

– If so, does the law for the flow rate change or not?

– Do the grains behave the same way when they are going to get jammed? Does

this knowledge offer any hint to avoid jamming?

Questions:


The experiment

The relevant parameter is the ratio between the orifice diameter and the particle diameter: R=D/φ

Scales

Air jet

Microphone

Silo

Oscilloscope

Valve

GPIBBUS


The experiment

The relevant parameter is the ratio between the orifice diameter

and the particle diameter: R=D/φ

Scales

Air jet

Microphone

Silo

Oscilloscope

Valve

GPIBBUS


Are there distinct, different regimes

(jamming / non jamming)?

– Statistics of avalanches.

– Measurement of the jamming probability.

– Asymptotic behavior of the jamming probability for a large number of grains.


Avalanche statistics: PDF of the number of beads

The exponential tail denotes that the phenomenon is governed by a characteristic magnitude of the system.

All the histograms have the same shape except for very small avalanches – and this can be ignored for large orifices.

A single parameter (for example, the mean size of the avalanche, <s>) can be used to rescale the histograms.

0 200 400 600 8000

100

200

300

400

500

600

700

s

nR

(s)

0 200 400 600 80010

0

101

102

103


Avalanche statistics: PDF of the number of beads

The exponential tail denotes that the phenomenon is governed by a characteristic magnitude of the system.

All the histograms have the same shape except for very small avalanches – and this can be ignored for large orifices.

A single parameter (for example, the mean size of the avalanche, <s>) can be used to rescale the histograms.

0 2 4 6 8

0.01

0.1

1

log

nR(s

)

s/<s>

R=2.23 R=2.43 R=2.84 R=2.96 R=3.01 R=3.11 R=3.2 R=3.3 R=3.4 R=3.5 R=3.54 R=3.64 R=3.74 R=3.83


Jamming probability: definitions

A simple model Independent events: nearby beads do not influence the probability that a bead passes through the orifice

•

•

• +=

+=−

nn

np

nn

np

oo

o1 on

ssnR

size ofavalanches ofnumber)( =

( ) )1log()log())(log(1)( ppssnppsn Rs

R −+=⇒−= ( p can be obtained from the histogram slope )

moments of the distribution: ( ) ( ) 111 −−

∂∂−= pp

ppsn

n ; 1st moment: ( ) 111

1

−−+=⇒−

= spp

ps

Let us define sss /* = , and )()( ** snssn RR = . Then if 1>>s , the rescaled PDF can be written as

*1*

11** 1lnexp1)( sR esssssn −−−− →

+−

+=

Jamming probaJamming probaJamming probaJamming probabilitybilitybilitybility: probability that a jamming event occurs before N beads fall:

( ) ( )∑∞

=

−=Ns

RN snRJ 1 , where ( ) ( ) sR ppsn −= 1 , and substituting n,

( ) sN

NN esNpRJ

−− −→

+−=−= 11lnexp11 1

jammingbead

falls 678

avalancheof size s


Jamming probability: results

Fits:

s

N

eRN

J

−−=1)(

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

R

J(R

,N) N=100



Fits:

s

N

eRN

J

−−=1)(

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

R

J(R

,N)

N=20N=50N=100N=200N=500N=1000N=2000N=5000N=10000N=20000N=50000N=100000



0 1 2 3 4 5 6

0

0.2

0.4

0.6

0.8

1

R

J(R

,N)

Jamming

No jamming

N∞

Rc (critical size of the orifice)


Critical size of the outlet orifice

( )γc RR

A

−=s

0.5 1 2 310-1

102

105

108

<s>

1/(Rc-R)

For spheres, the critical radius is Rc = 4.94 in 3D, Rc = 8.5 in 2D

The critical exponent is very high:

γ ≈ 6.9 in 3D, γ ≈ 12.7 in 2D

Qualitatively unnafected by specific gravity, rugosity, friction coefficient, shape (moderate changes).

I. Zuriguel, L. A. Pugnaloni, A. Garcimartín and D. Maza, Phys. Rev. E – RC 68, 030301 (2003)

I. Zuriguel, A. Garcimartín, D. Maza, L. A. Pugnaloni and J. M. Pastor, Phys. Rev. E 71, 051303 (2005)


Critical size of the outlet orifice

( )γc RR

A

−=s

For spheres, the critical radius is Rc = 4.94 in 3D, Rc = 8.5 in 2D

The critical exponent is very high:

γ ≈ 6.9 in 3D, γ ≈ 12.7 in 2D

Qualitatively unnafected by specific gravity, rugosity, friction coefficient, shape (moderate changes).

I. Zuriguel, L. A. Pugnaloni, A. Garcimartín and D. Maza, Phys. Rev. E – RC 68, 030301 (2003)

I. Zuriguel, A. Garcimartín, D. Maza, L. A. Pugnaloni and J. M. Pastor, Phys. Rev. E 71, 051303 (2005)

0.5 1 2 3100

102

104

106

1/(Rc-R)

<s>

glass pasta grains rice lentils


Is the law for the flow rate the same for small

and big orifices?

– The Beverloo law for the mass flow rate.

– The flow for small orifices.

– A new proposal.


The flow rate (number of beads per unit time)

Beverloo’s lawBeverloo’s lawBeverloo’s lawBeverloo’s law

Let us assume that the mass flow rate W depends on g, R, µ and ρ.

A simple dimensional calculation leads to

25

Rg )C(W ρµ=

which is known as the Beverloo law. (For two dimensions, the exponent is 3/2)

Another way of reasoning: W must be proportional to the velocity of the beads falling freely from a vault of radius R

times the outlet area; then 252 RRRAvW =⋅∝⋅=

The functional dependence of W on R has been checked for big orifices.

Beverloo’s law does not work well for sdoes not work well for sdoes not work well for sdoes not work well for small mall mall mall R. In fact, for R1, W should go to 0. Even substituting R-1 for R does not provide a good fit. Empirical mend:

2

5k)-R(g )C(W ρµ= where k can take any value between 1 and 3


Flow data for 3D and 2D silos

1 10 100101

102

103

104

105

106

107

Glass dp=0,5mm

Glass dp=1mm

Glass dp=2 mm

Glass dp=3mm

Lead dp=3mm

Delrin dp=3mm

Wb

R



1 10 100101

102

103

104

105

106

107

Wb

R

k=1.16

25

k)-R(g )C(W ρµ=

1 10 100101

102

103

104

105

106

107

Wb

R

k=1



0 20 40 60 80 1000

100

200

300

400

500

600

W

2/5

R

0 2 4 6 8 10 12 140

10

20

30

40

50

60

70

W2/

5

R

25

k)-R(g )C(W ρµ=

k)-R(W 52

∝


A new proposal for the flow rate

0 20 40 60 80 100

0.4

0.6

0.8

1.0

W/ W

Be

v

R

0 20 40 60 80 100

e-6

e-5

e-4

e-3

e-2

e-1

e0

1 -

W /

WB

ev

R-1

100 101 102101

102

103

104

105

106

107

WR

25

**Rge

21

1 )C(W

−ρµ= − Rl

C. Mankoc, A. Janda, R. Arévalo, J.M. Pastor, I. Zuriguel, A. Garcimartín and D. Maza, submitted to Granular Matter (2007)


Do the grains behave the same way when they are going

to get jammed? Does this knowledge offer any hint to

avoid jamming?

– The velocity profile inside the silo.

– Models.

– Statistical properties of the fluctuations.


The velocity profile inside a 2D siloKinematic model for the velocity field inside a siloKinematic model for the velocity field inside a siloKinematic model for the velocity field inside a siloKinematic model for the velocity field inside a silo

Let us assume that x

vBu

∂∂−= (from Reynolds’ dilatancy principle?)

plus the continuity equation: 0y

v

x

u =∂∂+

∂∂

, then yB4

x-

2

eyB4

Q-v

π=

where Q is the volumetric flow rate, and B is a characteristic length

This model fits the data nicely:

For an orifice R=15.9, at y=155the fit gives

Q=4100 beads / sec. (measured: 4600)B=2.46 (Nedderman measured B=2.3)(B changes a little with y)

but no prediction for B is offered

24 measurements involving > 3000 beads each

-100-100-100-100 -50-50-50-50 0000 50505050 100100100100-100-100-100-100

-80-80-80-80

-60-60-60-60

-40-40-40-40

-20-20-20-20

0000

x (mm)x (mm)x (mm)x (mm)

v (m

m/s

)v

(mm

/s)

v (m

m/s

)v

(mm

/s)

R.M. Nedderman, “Statics and Kinematics of

Granular Materials”, Cambridge (1992)

u ∝ ∂xv


Diffusive models

A characteristic time scale (the time it takes for a particle to travel its own diameter) and a characteristic length (the particle diameter) define a mean velocity v.

Coupled with some assumptions (e.g. biased random walk) these models imply a normal diffusion and gaussian fluctuations for the particle displacements.

They provide a value for a diffusive length which is a fraction of the particle diameter, instead of 2-3 φ as found in experiments.

ρφ

J. Litwiniszyn, Bull Acad. Pol. Sci. 11, 593 (1963)

W. W. Mullins, J. Appl. Physics 43, 665 (1972)

The spot model M. Z. Bazant, Mechanics of Materials 38, 717 (2006)

B∆y2

∆y

v

D then

∆t

∆yv defined wella is thereif ;

∆t2

∆yD

2

yy

2

≡===

The above model reformulated for “spots” , where a void is spreaded through a cluster of grains that share the void collectively. Adjusting the size of the spot gives the value for Bobtained in experiments.


Our results

0.01 0.1 1 101E-5

1E-4

1E-3

0.01

0.1

1

vy∆t/d

<(∆

y(t)

-∆y(

0))2 >

<(∆

x(t)

-∆x(

0))2 >

D. Maza, A. Garcimartín, R. Arévalo, EPJ – Special Topics 143, 191 (2007)

R. Arévalo, A. Garcimartín, D. Maza, EPJ E (accepted for publication)

Numerical simulations (DEM) of the discharge of a 2D silo

For big orifices, developed flow:

•Non gaussian statistics for the particle displacements

•Ballistic regime for small displacements, normal diffusion for large displacements

R > Rc R < Rc


Experimental results

10-2

10-1

100

101

10-4

10-2

100

vy ∆ t / φ

<(∆

y(t

) -

∆ y(

0))2 > for big orifices:

−transition from a ballistic regime to normal diffusionalways:−non-gaussian fluctuations with long tails

-2 0 210

0

101

102

103

∆ x / σ∆ x

#

-2 0 210

0

101

102

103

∆ y / σ∆ y

#

R=15.9


Experimental results

10-2

10-1

100

101

10-4

10-2

100

vy ∆ t / φ

<(∆

y(t

) -

∆ y(

0))2 >

for big orifices:−transition from a ballistic regime to normal diffusionalways:−non-gaussian fluctuations with long tails

-3 0 3 10

0

101

102

103

∆ x / σ∆ x

#

-2 0 210

0

101

102

103

∆ y / σ∆ y

#R=4.8


Conclusions

� There are two distinct regimes for the granular flow through an orifice, one in which the flow will eventually get arrested, and another which will never jam.

� The boundary between both situations is well defined, at a “critical” size of the orifice Rc.

� The flow for small orifices does not follow the same law than for big orifices. The behavior R5/2 is an asymptotic law that must be corrected for small R.

� The fluctuations of the grains display some features that depend on the size of the orifice.

Outlook

� Find an explanation for the exponential term included in the law for the flow.

� Look more closely to the statistics for the fluctuations of the grains.

� Can we derive the velocity profile from the PDF of the fluctuations?

� If small fluctuations are introduced, does the granular flow elude jamming?

Documents

Flow and jamming of granular matter through an orifice