Flow and jamming of granular matter through an orifice

Flow and jamming of granular
matter through an orifice
Dpto. de Física y Matemática Aplicada,
Universidad de Navarra
31080 Pamplona, Spain
http://fisica.unav.es/granular/
R. Arévalo, D. Maza, A. Garcimartín,
C. Mankoc, A. Janda, I. Zuriguel & M. Pastor
Thanks to:
Eric Clément (PMMH, ESPCI)
Luis A. Pugnaloni (CONICET, Argentine)
Tom Mullin (Univ. Manchester, U.K.)
Andrés Santos (Univ. Extremadura, Spain)
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
Questions:
• 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
A. Garcimartín - Flow and jamming of granular matter through an orifice
Traffic and granular flow '07

The experiment
Silo
Oscilloscope
Valve
Microphone
Scales
Air jet
GPIB
BUS
The relevant parameter is the ratio
between the orifice diameter and the
particle diameter: R=D/φ
A. Garcimartín - Flow and jamming of granular matter through an orifice
Traffic and granular flow '07

The experiment
Silo
Oscilloscope
Valve
Microphone
Scales
Air jet
GPIB
BUS
The relevant parameter is the
ratio between the orifice diameter
and the particle diameter: R=D/φ
A. Garcimartín - Flow and jamming of granular matter through an orifice
Traffic and granular flow '07

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

Avalanche statistics: PDF of the number of beads
700
600
500
10 3
10 2
The exponential tail denotes that
the phenomenon is governed by a
characteristic magnitude of the
system.
n R
(s)
400
300
200
100
10 1
10 0
0 200 400 600 800
0
0 200 400 600 800
s
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,
) can be used to rescale the
histograms.
A. Garcimartín - Flow and jamming of granular matter through an orifice
Traffic and granular flow '07

Avalanche statistics: PDF of the number of beads
log n R
(s)
1
0.1
0.01
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
0 2 4 6 8
s/
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,
) can be used to rescale the
histograms.
A. Garcimartín - Flow and jamming of granular matter through an orifice
Traffic and granular flow '07

Jamming probability: definitions
A simple model
Independent events: nearby beads do not influence the probability that a bead passes through the orifice
n
R
1 − p =
no
n + n
o
n
p =
•
• no
+ n•
n R
( s)
=
( 1−
p) ⇒ log( n ( s))
= s log( p)
+ log(1 − )
number of avalanches of size s
s
( s)
= p
p ( p can be obtained from the histogram slope )
n
−
moments of the distribution: ( ) ( ) 1
Let us define
jamming
bead
falls
R
s
⎛ ∂ ⎞
= 1−
p ⎜ p ⎟ 1−
p
⎝ ∂p
⎠
* s
avalanche
of size s
678
n
n o
= p
⇒ p = 1+
1−
p
s
; 1 st −1
−
moment: ( ) 1
s * = s / s , and nR ( s
* ) = s nR(
) . Then if s >> 1, the rescaled PDF can be written as
s
n
*
R
( s
*
) = ⎜ ⎛ 1
⎝
+
s
−1
⎟⎞
⎠
−1
exp
⎡
− s
⎢⎣
*
s
ln⎜⎛ 1+
⎝
s
−1
⎟⎞
⎤
→ e
⎠⎥⎦
−s
*
Jamming probability
bility: probability that a jamming event occurs before N beads fall:
J ( R) = 1 n ( s)
, where n ( s) ( − p) p
s
N
J
N
− ∑ ∞
=
s
( R)
N
R
R
= 1 , and substituting n,
N
p
⎧
−
N s
⎫
⎨ ⎜⎛ 1
= 1−
= 1−
exp ln 1+
⎟⎞
⎬ → 1 − e
⎩ ⎝ ⎠ ⎭
−N
s
A. Garcimartín - Flow and jamming of granular matter through an orifice
Traffic and granular flow '07

Jamming probability: results
1
0.9
0.8
J(R,N)
0.7
0.6
0.5
0.4
0.3
N=100
Fits:
J ( R)
N
=1− e
−
N
s
0.2
0.1
0
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5
R
A. Garcimartín - Flow and jamming of granular matter through an orifice
Traffic and granular flow '07

Jamming probability: results
1
J(R,N)
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
N=20
N=50
N=100
N=200
N=500
N=1000
N=2000
N=5000
N=10000
N=20000
N=50000
N=100000
Fits:
J ( R)
N
=1− e
−
N
s
0
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5
R
A. Garcimartín - Flow and jamming of granular matter through an orifice
Traffic and granular flow '07

Jamming probability: results
1
0.8
Jamming
J(R,N)
0.6
0.4
N∞
0.2
0
0 1 2 3 4 5 6
R
No jamming
R c (critical size
of the orifice)
A. Garcimartín - Flow and jamming of granular matter through an orifice
Traffic and granular flow '07

Critical size of the outlet orifice
10 8
s
=
A
( R ) γ
c − R
10 5
10 2
For spheres, the critical radius is
R c = 4.94 in 3D, R c = 8.5 in 2D
The critical exponent is very high:
γ ≈ 6.9 in 3D, γ ≈ 12.7 in 2D
10 -1
0.5 1 2 3
1/(R c
-R)
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)
A. Garcimartín - Flow and jamming of granular matter through an orifice
Traffic and granular flow '07

Critical size of the outlet orifice
glass
pasta grains
rice
lentils
10
6
10 4
10 2
10 0
1/(R -R) c
0.5 1 2 3
s
=
A
( R ) γ
c − R
For spheres, the critical radius is
R c = 4.94 in 3D, R c = 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)
A. Garcimartín - Flow and jamming of granular matter through an orifice
Traffic and granular flow '07

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

The flow rate (number of beads per unit time)
Beverloo’s law
Let us assume that the mass flow rate W depends on g, R, µ and ρ.
5
A simple dimensional calculation leads to
2
W
= C( µ )
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
2 2
times the outlet area; then W = v ⋅ A ∝ R ⋅ R = R
The functional dependence of W on R has been checked for big orifices.
Beverloo’s law does not work well for small
R. In fact, for R1, W should go to 0. Even
substituting R-1 for R does not provide a good fit. Empirical mend:
W
= C( µ )
ρ
g
5
(R - k) 2
5
ρ
g
R
where k can take any value between 1 and 3
A. Garcimartín - Flow and jamming of granular matter through an orifice
Traffic and granular flow '07

Flow data for 3D and 2D silos
10 7 Glass d p
=0,5mm
10 6
10 5
W b
10 4
10 3
10 2
10 1
Glass d p
=1mm
Glass d p
=2 mm
Glass d p
=3mm
Lead d p
=3mm
Delrin d p
=3mm
1 10 100
R
A. Garcimartín - Flow and jamming of granular matter through an orifice
Traffic and granular flow '07

Flow data for 3D and 2D silos
10 7
10 6
10 5
W
=
C( µ )
ρ
g
5
(R - k) 2
W b
10 4
10 3
10 2
k=1.16
10 7
10 6
10 1
1 10
R
100
10 4
W b
10 5
10 3
k=1
10 2
10 1
1 10 100
A. Garcimartín - Flow and jamming of granular matter through an orifice
Traffic and granular flow '07
R

Flow data for 3D and 2D silos
600
500
400
W
=
C( µ )
ρ
g
5
(R - k) 2
W 2/5
300
200
100
0
0 20 40 60 80 100
R
70
60
2
W 5 ∝
(R - k)
W 2/5
50
40
30
20
10
0
0 2 4 6 8 10 12 14
R
A. Garcimartín - Flow and jamming of granular matter through an orifice
Traffic and granular flow '07

A new proposal for the flow rate
W/ W Bev
1.0
0.8
0.6
0.4
0 20 40 60 80 100
R
e 0
W
10 7
10 6
10 5
10 4
10 3
10 2
1 - W / W Bev
e -1
e -2
e -3
e -4
e -5
e -6
0 20 40 60 80 100
R-1
10 1
10 0 10 1 10 2
R
W
⎛ 1 *
= C( µ ) ρ 1 −
−l
R
⎜
⎝
e
2
⎟
⎠
⎞
g
R
*
5
2
C. Mankoc, A. Janda, R. Arévalo, J.M. Pastor, I. Zuriguel, A. Garcimartín and D. Maza, submitted to Granular Matter (2007)
A. Garcimartín - Flow and jamming of granular matter through an orifice
Traffic and granular flow '07

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

The velocity profile inside a 2D silo
Kinematic model for the velocity field inside a silo
∂ v
Let us assume that u = −B
(from Reynolds’ dilatancy principle)
∂ x
∂ u ∂ v
Q
plus the continuity equation: + = 0 , then v = -
∂ x ∂ y
4πB y
where Q is the volumetric flow rate, and B is a characteristic length
e
2
x
-
4 B y
u ∝ ∂ x v
v (mm/s)
0
-20
-40
-60
-80
24 measurements involving > 3000 beads each
-100
-100 -50 0 50 100
x (mm)
R.M. Nedderman, “Statics and Kinematics of
Granular Materials”, Cambridge (1992)
This model fits the data nicely:
For an orifice R=15.9, at y=155
the 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
A. Garcimartín - Flow and jamming of granular matter through an orifice
Traffic and granular flow '07

Diffusive models
J. Litwiniszyn, Bull Acad. Pol. Sci. 11, 593 (1963)
W. W. Mullins, J. Appl. Physics 43, 665 (1972)
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.
ρ
φ
D =
∆y
2
2 ∆t
;
if
there is a
well defined
v
y
=
∆y
∆t
then
D
v
y
=
∆y
2
2 ∆y
≡ B
The spot model M. Z. Bazant, Mechanics of Materials 38, 717 (2006)
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 B
obtained in experiments.
A. Garcimartín - Flow and jamming of granular matter through an orifice
Traffic and granular flow '07

Our results
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 > R c
R < R c
1
0.1
0.01
1E-3
1E-4
1E-5
0.01 0.1 1 10
v y
∆t/d
A. Garcimartín - Flow and jamming of granular matter through an orifice
Traffic and granular flow '07

Experimental results
10 0 v y
∆ t / φ
10 -2
10 -4
for big orifices:
−transition from a ballistic
regime to normal diffusion
always:
−non-gaussian fluctuations
with long tails
10 -2 10 -1 10 0 10 1
10 3 ∆ x / σ ∆ x
10 3 ∆ y / σ ∆ y
10 2
R=15.9
10 2
#
#
10 1
10 1
10 0
-2 0 2
10 0
-2 0 2
A. Garcimartín - Flow and jamming of granular matter through an orifice
Traffic and granular flow '07

Experimental results
10 0 v y
∆ t / φ
10 -2
10 -4
for big orifices:
−transition from a ballistic
regime to normal diffusion
always:
−non-gaussian fluctuations
with long tails
10 -2 10 -1 10 0 10 1
10 3 ∆ x / σ ∆ x
10 3 ∆ y / σ ∆ y
10 2
R=4.8
10 2
#
#
10 1
10 1
10 0
-3 0 3
10 0
-2 0 2
A. Garcimartín - Flow and jamming of granular matter through an orifice
Traffic and granular flow '07

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 R c .
The flow for small orifices does not follow the same law than for big orifices. The
behavior R 5/2 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
A. Garcimartín - Flow and jamming of granular matter through an orifice
Traffic and granular flow '07