Flow and jamming of granular matter through an orifice
Flow and jamming of granular matter through an orifice
Flow and jamming of granular matter through an orifice
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<strong>Flow</strong> <strong><strong>an</strong>d</strong> <strong>jamming</strong> <strong>of</strong> <strong>gr<strong>an</strong>ular</strong><br />
<strong>matter</strong> <strong>through</strong> <strong>an</strong> <strong>orifice</strong><br />
Dpto. de Física y Matemática Aplicada,<br />
Universidad de Navarra<br />
31080 Pamplona, Spain<br />
http://fisica.unav.es/<strong>gr<strong>an</strong>ular</strong>/<br />
R. Arévalo, D. Maza, A. Garcimartín,<br />
C. M<strong>an</strong>koc, A. J<strong><strong>an</strong>d</strong>a, I. Zuriguel & M. Pastor<br />
Th<strong>an</strong>ks to:<br />
Eric Clément (PMMH, ESPCI)<br />
Luis A. Pugnaloni (CONICET, Argentine)<br />
Tom Mullin (Univ. M<strong>an</strong>chester, U.K.)<br />
Andrés S<strong>an</strong>tos (Univ. Extremadura, Spain)<br />
A. Garcimartín - <strong>Flow</strong> <strong><strong>an</strong>d</strong> <strong>jamming</strong> <strong>of</strong> <strong>gr<strong>an</strong>ular</strong> <strong>matter</strong> <strong>through</strong> <strong>an</strong> <strong>orifice</strong><br />
Traffic <strong><strong>an</strong>d</strong> <strong>gr<strong>an</strong>ular</strong> flow '07
<strong>Flow</strong> <strong><strong>an</strong>d</strong> <strong>jamming</strong> <strong>of</strong> <strong>gr<strong>an</strong>ular</strong><br />
<strong>matter</strong> <strong>through</strong> <strong>an</strong> <strong>orifice</strong><br />
Questions:<br />
• Gr<strong>an</strong>ular <strong>matter</strong> c<strong>an</strong> behave as a solid, a<br />
liquid or a gas.<br />
• Jamming c<strong>an</strong> be likened to a phase<br />
tr<strong>an</strong>sition:<br />
A. Liu <strong><strong>an</strong>d</strong> S. Nagel, “Jamming is not just cool <strong>an</strong>y<br />
more”, Nature 396, 21-22 (1998).<br />
– Are there distinct, different regimes (<strong>jamming</strong> / non <strong>jamming</strong>)<br />
– If so, does the law for the flow rate ch<strong>an</strong>ge or not<br />
– Do the grains behave the same way when they are going to get jammed Does<br />
this knowledge <strong>of</strong>fer <strong>an</strong>y hint to avoid <strong>jamming</strong><br />
A. Garcimartín - <strong>Flow</strong> <strong><strong>an</strong>d</strong> <strong>jamming</strong> <strong>of</strong> <strong>gr<strong>an</strong>ular</strong> <strong>matter</strong> <strong>through</strong> <strong>an</strong> <strong>orifice</strong><br />
Traffic <strong><strong>an</strong>d</strong> <strong>gr<strong>an</strong>ular</strong> flow '07
The experiment<br />
Silo<br />
Oscilloscope<br />
Valve<br />
Microphone<br />
Scales<br />
Air jet<br />
GPIB<br />
BUS<br />
The relev<strong>an</strong>t parameter is the ratio<br />
between the <strong>orifice</strong> diameter <strong><strong>an</strong>d</strong> the<br />
particle diameter: R=D/φ<br />
A. Garcimartín - <strong>Flow</strong> <strong><strong>an</strong>d</strong> <strong>jamming</strong> <strong>of</strong> <strong>gr<strong>an</strong>ular</strong> <strong>matter</strong> <strong>through</strong> <strong>an</strong> <strong>orifice</strong><br />
Traffic <strong><strong>an</strong>d</strong> <strong>gr<strong>an</strong>ular</strong> flow '07
The experiment<br />
Silo<br />
Oscilloscope<br />
Valve<br />
Microphone<br />
Scales<br />
Air jet<br />
GPIB<br />
BUS<br />
The relev<strong>an</strong>t parameter is the<br />
ratio between the <strong>orifice</strong> diameter<br />
<strong><strong>an</strong>d</strong> the particle diameter: R=D/φ<br />
A. Garcimartín - <strong>Flow</strong> <strong><strong>an</strong>d</strong> <strong>jamming</strong> <strong>of</strong> <strong>gr<strong>an</strong>ular</strong> <strong>matter</strong> <strong>through</strong> <strong>an</strong> <strong>orifice</strong><br />
Traffic <strong><strong>an</strong>d</strong> <strong>gr<strong>an</strong>ular</strong> flow '07
Are there distinct, different regimes<br />
(<strong>jamming</strong> / non <strong>jamming</strong>)<br />
– Statistics <strong>of</strong> aval<strong>an</strong>ches.<br />
– Measurement <strong>of</strong> the <strong>jamming</strong> probability.<br />
– Asymptotic behavior <strong>of</strong> the <strong>jamming</strong> probability for a large number <strong>of</strong> grains.<br />
A. Garcimartín - <strong>Flow</strong> <strong><strong>an</strong>d</strong> <strong>jamming</strong> <strong>of</strong> <strong>gr<strong>an</strong>ular</strong> <strong>matter</strong> <strong>through</strong> <strong>an</strong> <strong>orifice</strong><br />
Traffic <strong><strong>an</strong>d</strong> <strong>gr<strong>an</strong>ular</strong> flow '07
Aval<strong>an</strong>che statistics: PDF <strong>of</strong> the number <strong>of</strong> beads<br />
700<br />
600<br />
500<br />
10 3<br />
10 2<br />
The exponential tail denotes that<br />
the phenomenon is governed by a<br />
characteristic magnitude <strong>of</strong> the<br />
system.<br />
n R<br />
(s)<br />
400<br />
300<br />
200<br />
100<br />
10 1<br />
10 0<br />
0 200 400 600 800<br />
0<br />
0 200 400 600 800<br />
s<br />
All the histograms have the same<br />
shape except for very small<br />
aval<strong>an</strong>ches – <strong><strong>an</strong>d</strong> this c<strong>an</strong> be<br />
ignored for large <strong>orifice</strong>s.<br />
A single parameter (for example,<br />
the me<strong>an</strong> size <strong>of</strong> the aval<strong>an</strong>che,<br />
) c<strong>an</strong> be used to rescale the<br />
histograms.<br />
A. Garcimartín - <strong>Flow</strong> <strong><strong>an</strong>d</strong> <strong>jamming</strong> <strong>of</strong> <strong>gr<strong>an</strong>ular</strong> <strong>matter</strong> <strong>through</strong> <strong>an</strong> <strong>orifice</strong><br />
Traffic <strong><strong>an</strong>d</strong> <strong>gr<strong>an</strong>ular</strong> flow '07
Aval<strong>an</strong>che statistics: PDF <strong>of</strong> the number <strong>of</strong> beads<br />
log n R<br />
(s)<br />
1<br />
0.1<br />
0.01<br />
R=2.23<br />
R=2.43<br />
R=2.84<br />
R=2.96<br />
R=3.01<br />
R=3.11<br />
R=3.2<br />
R=3.3<br />
R=3.4<br />
R=3.5<br />
R=3.54<br />
R=3.64<br />
R=3.74<br />
R=3.83<br />
0 2 4 6 8<br />
s/<br />
The exponential tail denotes that<br />
the phenomenon is governed by a<br />
characteristic magnitude <strong>of</strong> the<br />
system.<br />
All the histograms have the same<br />
shape except for very small<br />
aval<strong>an</strong>ches – <strong><strong>an</strong>d</strong> this c<strong>an</strong> be<br />
ignored for large <strong>orifice</strong>s.<br />
A single parameter (for example,<br />
the me<strong>an</strong> size <strong>of</strong> the aval<strong>an</strong>che,<br />
) c<strong>an</strong> be used to rescale the<br />
histograms.<br />
A. Garcimartín - <strong>Flow</strong> <strong><strong>an</strong>d</strong> <strong>jamming</strong> <strong>of</strong> <strong>gr<strong>an</strong>ular</strong> <strong>matter</strong> <strong>through</strong> <strong>an</strong> <strong>orifice</strong><br />
Traffic <strong><strong>an</strong>d</strong> <strong>gr<strong>an</strong>ular</strong> flow '07
Jamming probability: definitions<br />
A simple model<br />
Independent events: nearby beads do not influence the probability that a bead passes <strong>through</strong> the <strong>orifice</strong><br />
n<br />
R<br />
1 − p =<br />
no<br />
n + n<br />
o<br />
n<br />
p =<br />
•<br />
• no<br />
+ n•<br />
n R<br />
( s)<br />
=<br />
( 1−<br />
p) ⇒ log( n ( s))<br />
= s log( p)<br />
+ log(1 − )<br />
number <strong>of</strong> aval<strong>an</strong>ches <strong>of</strong> size s<br />
s<br />
( s)<br />
= p<br />
p ( p c<strong>an</strong> be obtained from the histogram slope )<br />
n<br />
−<br />
moments <strong>of</strong> the distribution: ( ) ( ) 1<br />
Let us define<br />
<strong>jamming</strong><br />
bead<br />
falls<br />
R<br />
s<br />
⎛ ∂ ⎞<br />
= 1−<br />
p ⎜ p ⎟ 1−<br />
p<br />
⎝ ∂p<br />
⎠<br />
* s<br />
aval<strong>an</strong>che<br />
<strong>of</strong> size s<br />
678<br />
n<br />
n o<br />
= p<br />
⇒ p = 1+<br />
1−<br />
p<br />
s<br />
; 1 st −1<br />
−<br />
moment: ( ) 1<br />
s * = s / s , <strong><strong>an</strong>d</strong> nR ( s<br />
* ) = s nR(<br />
) . Then if s >> 1, the rescaled PDF c<strong>an</strong> be written as<br />
s<br />
n<br />
*<br />
R<br />
( s<br />
*<br />
) = ⎜ ⎛ 1<br />
⎝<br />
+<br />
s<br />
−1<br />
⎟⎞<br />
⎠<br />
−1<br />
exp<br />
⎡<br />
− s<br />
⎢⎣<br />
*<br />
s<br />
ln⎜⎛ 1+<br />
⎝<br />
s<br />
−1<br />
⎟⎞<br />
⎤<br />
→ e<br />
⎠⎥⎦<br />
−s<br />
*<br />
Jamming probability<br />
bility: probability that a <strong>jamming</strong> event occurs before N beads fall:<br />
J ( R) = 1 n ( s)<br />
, where n ( s) ( − p) p<br />
s<br />
N<br />
J<br />
N<br />
− ∑ ∞<br />
=<br />
s<br />
( R)<br />
N<br />
R<br />
R<br />
= 1 , <strong><strong>an</strong>d</strong> substituting n,<br />
N<br />
p<br />
⎧<br />
−<br />
N s<br />
⎫<br />
⎨ ⎜⎛ 1<br />
= 1−<br />
= 1−<br />
exp ln 1+<br />
⎟⎞<br />
⎬ → 1 − e<br />
⎩ ⎝ ⎠ ⎭<br />
−N<br />
s<br />
A. Garcimartín - <strong>Flow</strong> <strong><strong>an</strong>d</strong> <strong>jamming</strong> <strong>of</strong> <strong>gr<strong>an</strong>ular</strong> <strong>matter</strong> <strong>through</strong> <strong>an</strong> <strong>orifice</strong><br />
Traffic <strong><strong>an</strong>d</strong> <strong>gr<strong>an</strong>ular</strong> flow '07
Jamming probability: results<br />
1<br />
0.9<br />
0.8<br />
J(R,N)<br />
0.7<br />
0.6<br />
0.5<br />
0.4<br />
0.3<br />
N=100<br />
Fits:<br />
J ( R)<br />
N<br />
=1− e<br />
−<br />
N<br />
s<br />
0.2<br />
0.1<br />
0<br />
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5<br />
R<br />
A. Garcimartín - <strong>Flow</strong> <strong><strong>an</strong>d</strong> <strong>jamming</strong> <strong>of</strong> <strong>gr<strong>an</strong>ular</strong> <strong>matter</strong> <strong>through</strong> <strong>an</strong> <strong>orifice</strong><br />
Traffic <strong><strong>an</strong>d</strong> <strong>gr<strong>an</strong>ular</strong> flow '07
Jamming probability: results<br />
1<br />
J(R,N)<br />
0.9<br />
0.8<br />
0.7<br />
0.6<br />
0.5<br />
0.4<br />
0.3<br />
0.2<br />
0.1<br />
N=20<br />
N=50<br />
N=100<br />
N=200<br />
N=500<br />
N=1000<br />
N=2000<br />
N=5000<br />
N=10000<br />
N=20000<br />
N=50000<br />
N=100000<br />
Fits:<br />
J ( R)<br />
N<br />
=1− e<br />
−<br />
N<br />
s<br />
0<br />
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5<br />
R<br />
A. Garcimartín - <strong>Flow</strong> <strong><strong>an</strong>d</strong> <strong>jamming</strong> <strong>of</strong> <strong>gr<strong>an</strong>ular</strong> <strong>matter</strong> <strong>through</strong> <strong>an</strong> <strong>orifice</strong><br />
Traffic <strong><strong>an</strong>d</strong> <strong>gr<strong>an</strong>ular</strong> flow '07
Jamming probability: results<br />
1<br />
0.8<br />
Jamming<br />
J(R,N)<br />
0.6<br />
0.4<br />
N∞<br />
0.2<br />
0<br />
0 1 2 3 4 5 6<br />
R<br />
No <strong>jamming</strong><br />
R c (critical size<br />
<strong>of</strong> the <strong>orifice</strong>)<br />
A. Garcimartín - <strong>Flow</strong> <strong><strong>an</strong>d</strong> <strong>jamming</strong> <strong>of</strong> <strong>gr<strong>an</strong>ular</strong> <strong>matter</strong> <strong>through</strong> <strong>an</strong> <strong>orifice</strong><br />
Traffic <strong><strong>an</strong>d</strong> <strong>gr<strong>an</strong>ular</strong> flow '07
Critical size <strong>of</strong> the outlet <strong>orifice</strong><br />
10 8<br />
s<br />
=<br />
A<br />
( R ) γ<br />
c − R<br />
<br />
10 5<br />
10 2<br />
For spheres, the critical radius is<br />
R c = 4.94 in 3D, R c = 8.5 in 2D<br />
The critical exponent is very high:<br />
γ ≈ 6.9 in 3D, γ ≈ 12.7 in 2D<br />
10 -1<br />
0.5 1 2 3<br />
1/(R c<br />
-R)<br />
Qualitatively unnafected by specific<br />
gravity, rugosity, friction coefficient,<br />
shape (moderate ch<strong>an</strong>ges).<br />
I. Zuriguel, L. A. Pugnaloni, A. Garcimartín <strong><strong>an</strong>d</strong> D. Maza, Phys. Rev. E – RC 68, 030301 (2003)<br />
I. Zuriguel, A. Garcimartín, D. Maza, L. A. Pugnaloni <strong><strong>an</strong>d</strong> J. M. Pastor, Phys. Rev. E 71, 051303 (2005)<br />
A. Garcimartín - <strong>Flow</strong> <strong><strong>an</strong>d</strong> <strong>jamming</strong> <strong>of</strong> <strong>gr<strong>an</strong>ular</strong> <strong>matter</strong> <strong>through</strong> <strong>an</strong> <strong>orifice</strong><br />
Traffic <strong><strong>an</strong>d</strong> <strong>gr<strong>an</strong>ular</strong> flow '07
Critical size <strong>of</strong> the outlet <strong>orifice</strong><br />
<br />
glass<br />
pasta grains<br />
rice<br />
lentils<br />
10<br />
6<br />
10 4<br />
10 2<br />
10 0<br />
1/(R -R) c<br />
0.5 1 2 3<br />
s<br />
=<br />
A<br />
( R ) γ<br />
c − R<br />
For spheres, the critical radius is<br />
R c = 4.94 in 3D, R c = 8.5 in 2D<br />
The critical exponent is very high:<br />
γ ≈ 6.9 in 3D, γ ≈ 12.7 in 2D<br />
Qualitatively unnafected by specific<br />
gravity, rugosity, friction coefficient,<br />
shape (moderate ch<strong>an</strong>ges).<br />
I. Zuriguel, L. A. Pugnaloni, A. Garcimartín <strong><strong>an</strong>d</strong> D. Maza, Phys. Rev. E – RC 68, 030301 (2003)<br />
I. Zuriguel, A. Garcimartín, D. Maza, L. A. Pugnaloni <strong><strong>an</strong>d</strong> J. M. Pastor, Phys. Rev. E 71, 051303 (2005)<br />
A. Garcimartín - <strong>Flow</strong> <strong><strong>an</strong>d</strong> <strong>jamming</strong> <strong>of</strong> <strong>gr<strong>an</strong>ular</strong> <strong>matter</strong> <strong>through</strong> <strong>an</strong> <strong>orifice</strong><br />
Traffic <strong><strong>an</strong>d</strong> <strong>gr<strong>an</strong>ular</strong> flow '07
Is the law for the flow rate the same for small<br />
<strong><strong>an</strong>d</strong> big <strong>orifice</strong>s<br />
– The Beverloo law for the mass flow rate.<br />
– The flow for small <strong>orifice</strong>s.<br />
– A new proposal.<br />
A. Garcimartín - <strong>Flow</strong> <strong><strong>an</strong>d</strong> <strong>jamming</strong> <strong>of</strong> <strong>gr<strong>an</strong>ular</strong> <strong>matter</strong> <strong>through</strong> <strong>an</strong> <strong>orifice</strong><br />
Traffic <strong><strong>an</strong>d</strong> <strong>gr<strong>an</strong>ular</strong> flow '07
The flow rate (number <strong>of</strong> beads per unit time)<br />
Beverloo’s law<br />
Let us assume that the mass flow rate W depends on g, R, µ <strong><strong>an</strong>d</strong> ρ.<br />
5<br />
A simple dimensional calculation leads to<br />
2<br />
W<br />
= C( µ )<br />
which is known as the Beverloo law. (For two dimensions, the exponent is 3/2)<br />
Another way <strong>of</strong> reasoning:<br />
W must be proportional to the velocity <strong>of</strong> the beads falling freely from a vault <strong>of</strong> radius R<br />
2 2<br />
times the outlet area; then W = v ⋅ A ∝ R ⋅ R = R<br />
The functional dependence <strong>of</strong> W on R has been checked for big <strong>orifice</strong>s.<br />
Beverloo’s law does not work well for small<br />
R. In fact, for R1, W should go to 0. Even<br />
substituting R-1 for R does not provide a good fit. Empirical mend:<br />
W<br />
= C( µ )<br />
ρ<br />
g<br />
5<br />
(R - k) 2<br />
5<br />
ρ<br />
g<br />
R<br />
where k c<strong>an</strong> take <strong>an</strong>y value between 1 <strong><strong>an</strong>d</strong> 3<br />
A. Garcimartín - <strong>Flow</strong> <strong><strong>an</strong>d</strong> <strong>jamming</strong> <strong>of</strong> <strong>gr<strong>an</strong>ular</strong> <strong>matter</strong> <strong>through</strong> <strong>an</strong> <strong>orifice</strong><br />
Traffic <strong><strong>an</strong>d</strong> <strong>gr<strong>an</strong>ular</strong> flow '07
<strong>Flow</strong> data for 3D <strong><strong>an</strong>d</strong> 2D silos<br />
10 7 Glass d p<br />
=0,5mm<br />
10 6<br />
10 5<br />
W b<br />
10 4<br />
10 3<br />
10 2<br />
10 1<br />
Glass d p<br />
=1mm<br />
Glass d p<br />
=2 mm<br />
Glass d p<br />
=3mm<br />
Lead d p<br />
=3mm<br />
Delrin d p<br />
=3mm<br />
1 10 100<br />
R<br />
A. Garcimartín - <strong>Flow</strong> <strong><strong>an</strong>d</strong> <strong>jamming</strong> <strong>of</strong> <strong>gr<strong>an</strong>ular</strong> <strong>matter</strong> <strong>through</strong> <strong>an</strong> <strong>orifice</strong><br />
Traffic <strong><strong>an</strong>d</strong> <strong>gr<strong>an</strong>ular</strong> flow '07
<strong>Flow</strong> data for 3D <strong><strong>an</strong>d</strong> 2D silos<br />
10 7<br />
10 6<br />
10 5<br />
W<br />
=<br />
C( µ )<br />
ρ<br />
g<br />
5<br />
(R - k) 2<br />
W b<br />
10 4<br />
10 3<br />
10 2<br />
k=1.16<br />
10 7<br />
10 6<br />
10 1<br />
1 10<br />
R<br />
100<br />
10 4<br />
W b<br />
10 5<br />
10 3<br />
k=1<br />
10 2<br />
10 1<br />
1 10 100<br />
A. Garcimartín - <strong>Flow</strong> <strong><strong>an</strong>d</strong> <strong>jamming</strong> <strong>of</strong> <strong>gr<strong>an</strong>ular</strong> <strong>matter</strong> <strong>through</strong> <strong>an</strong> <strong>orifice</strong><br />
Traffic <strong><strong>an</strong>d</strong> <strong>gr<strong>an</strong>ular</strong> flow '07<br />
R
<strong>Flow</strong> data for 3D <strong><strong>an</strong>d</strong> 2D silos<br />
600<br />
500<br />
400<br />
W<br />
=<br />
C( µ )<br />
ρ<br />
g<br />
5<br />
(R - k) 2<br />
W 2/5<br />
300<br />
200<br />
100<br />
0<br />
0 20 40 60 80 100<br />
R<br />
70<br />
60<br />
2<br />
W 5 ∝<br />
(R - k)<br />
W 2/5<br />
50<br />
40<br />
30<br />
20<br />
10<br />
0<br />
0 2 4 6 8 10 12 14<br />
R<br />
A. Garcimartín - <strong>Flow</strong> <strong><strong>an</strong>d</strong> <strong>jamming</strong> <strong>of</strong> <strong>gr<strong>an</strong>ular</strong> <strong>matter</strong> <strong>through</strong> <strong>an</strong> <strong>orifice</strong><br />
Traffic <strong><strong>an</strong>d</strong> <strong>gr<strong>an</strong>ular</strong> flow '07
A new proposal for the flow rate<br />
W/ W Bev<br />
1.0<br />
0.8<br />
0.6<br />
0.4<br />
0 20 40 60 80 100<br />
R<br />
e 0<br />
W<br />
10 7<br />
10 6<br />
10 5<br />
10 4<br />
10 3<br />
10 2<br />
1 - W / W Bev<br />
e -1<br />
e -2<br />
e -3<br />
e -4<br />
e -5<br />
e -6<br />
0 20 40 60 80 100<br />
R-1<br />
10 1<br />
10 0 10 1 10 2<br />
R<br />
W<br />
⎛ 1 *<br />
= C( µ ) ρ 1 −<br />
−l<br />
R<br />
⎜<br />
⎝<br />
e<br />
2<br />
⎟<br />
⎠<br />
⎞<br />
g<br />
R<br />
*<br />
5<br />
2<br />
C. M<strong>an</strong>koc, A. J<strong><strong>an</strong>d</strong>a, R. Arévalo, J.M. Pastor, I. Zuriguel, A. Garcimartín <strong><strong>an</strong>d</strong> D. Maza, submitted to Gr<strong>an</strong>ular Matter (2007)<br />
A. Garcimartín - <strong>Flow</strong> <strong><strong>an</strong>d</strong> <strong>jamming</strong> <strong>of</strong> <strong>gr<strong>an</strong>ular</strong> <strong>matter</strong> <strong>through</strong> <strong>an</strong> <strong>orifice</strong><br />
Traffic <strong><strong>an</strong>d</strong> <strong>gr<strong>an</strong>ular</strong> flow '07
Do the grains behave the same way when they are going<br />
to get jammed Does this knowledge <strong>of</strong>fer <strong>an</strong>y hint to<br />
avoid <strong>jamming</strong><br />
– The velocity pr<strong>of</strong>ile inside the silo.<br />
– Models.<br />
– Statistical properties <strong>of</strong> the fluctuations.<br />
A. Garcimartín - <strong>Flow</strong> <strong><strong>an</strong>d</strong> <strong>jamming</strong> <strong>of</strong> <strong>gr<strong>an</strong>ular</strong> <strong>matter</strong> <strong>through</strong> <strong>an</strong> <strong>orifice</strong><br />
Traffic <strong><strong>an</strong>d</strong> <strong>gr<strong>an</strong>ular</strong> flow '07
The velocity pr<strong>of</strong>ile inside a 2D silo<br />
Kinematic model for the velocity field inside a silo<br />
∂ v<br />
Let us assume that u = −B<br />
(from Reynolds’ dilat<strong>an</strong>cy principle)<br />
∂ x<br />
∂ u ∂ v<br />
Q<br />
plus the continuity equation: + = 0 , then v = -<br />
∂ x ∂ y<br />
4πB y<br />
where Q is the volumetric flow rate, <strong><strong>an</strong>d</strong> B is a characteristic length<br />
e<br />
2<br />
x<br />
-<br />
4 B y<br />
u ∝ ∂ x v<br />
v (mm/s)<br />
0<br />
-20<br />
-40<br />
-60<br />
-80<br />
24 measurements involving > 3000 beads each<br />
-100<br />
-100 -50 0 50 100<br />
x (mm)<br />
R.M. Nedderm<strong>an</strong>, “Statics <strong><strong>an</strong>d</strong> Kinematics <strong>of</strong><br />
Gr<strong>an</strong>ular Materials”, Cambridge (1992)<br />
This model fits the data nicely:<br />
For <strong>an</strong> <strong>orifice</strong> R=15.9, at y=155<br />
the fit gives<br />
Q=4100 beads / sec. (measured: 4600)<br />
B=2.46 (Nedderm<strong>an</strong> measured B=2.3)<br />
(B ch<strong>an</strong>ges a little with y)<br />
but no prediction for B is <strong>of</strong>fered<br />
A. Garcimartín - <strong>Flow</strong> <strong><strong>an</strong>d</strong> <strong>jamming</strong> <strong>of</strong> <strong>gr<strong>an</strong>ular</strong> <strong>matter</strong> <strong>through</strong> <strong>an</strong> <strong>orifice</strong><br />
Traffic <strong><strong>an</strong>d</strong> <strong>gr<strong>an</strong>ular</strong> flow '07
Diffusive models<br />
J. Litwiniszyn, Bull Acad. Pol. Sci. 11, 593 (1963)<br />
W. W. Mullins, J. Appl. Physics 43, 665 (1972)<br />
A characteristic time scale (the time it takes for a particle to travel its own diameter) <strong><strong>an</strong>d</strong><br />
a characteristic length (the particle diameter) define a me<strong>an</strong> velocity v.<br />
Coupled with some assumptions (e.g. biased r<strong><strong>an</strong>d</strong>om walk) these models imply a normal<br />
diffusion <strong><strong>an</strong>d</strong> gaussi<strong>an</strong> fluctuations for the particle displacements.<br />
They provide a value for a diffusive length which is a fraction <strong>of</strong> the particle diameter,<br />
instead <strong>of</strong> 2-3 φ as found in experiments.<br />
ρ<br />
φ<br />
D =<br />
∆y<br />
2<br />
2 ∆t<br />
;<br />
if<br />
there is a<br />
well defined<br />
v<br />
y<br />
=<br />
∆y<br />
∆t<br />
then<br />
D<br />
v<br />
y<br />
=<br />
∆y<br />
2<br />
2 ∆y<br />
≡ B<br />
The spot model M. Z. Baz<strong>an</strong>t, Mech<strong>an</strong>ics <strong>of</strong> Materials 38, 717 (2006)<br />
The above model reformulated for “spots” , where a void is spreaded <strong>through</strong> a cluster <strong>of</strong><br />
grains that share the void collectively. Adjusting the size <strong>of</strong> the spot gives the value for B<br />
obtained in experiments.<br />
A. Garcimartín - <strong>Flow</strong> <strong><strong>an</strong>d</strong> <strong>jamming</strong> <strong>of</strong> <strong>gr<strong>an</strong>ular</strong> <strong>matter</strong> <strong>through</strong> <strong>an</strong> <strong>orifice</strong><br />
Traffic <strong><strong>an</strong>d</strong> <strong>gr<strong>an</strong>ular</strong> flow '07
Our results<br />
D. Maza, A. Garcimartín, R. Arévalo, EPJ – Special Topics 143, 191 (2007)<br />
R. Arévalo, A. Garcimartín, D. Maza, EPJ E (accepted for publication)<br />
Numerical simulations (DEM) <strong>of</strong> the discharge <strong>of</strong> a 2D silo<br />
For big <strong>orifice</strong>s, developed flow:<br />
•Non gaussi<strong>an</strong> statistics for the particle displacements<br />
•Ballistic regime for small displacements, normal diffusion for large displacements<br />
R > R c<br />
R < R c<br />
1<br />
<br />
0.1<br />
0.01<br />
1E-3<br />
1E-4<br />
<br />
1E-5<br />
0.01 0.1 1 10<br />
v y<br />
∆t/d<br />
A. Garcimartín - <strong>Flow</strong> <strong><strong>an</strong>d</strong> <strong>jamming</strong> <strong>of</strong> <strong>gr<strong>an</strong>ular</strong> <strong>matter</strong> <strong>through</strong> <strong>an</strong> <strong>orifice</strong><br />
Traffic <strong><strong>an</strong>d</strong> <strong>gr<strong>an</strong>ular</strong> flow '07
Experimental results<br />
10 0 v y<br />
∆ t / φ<br />
<br />
10 -2<br />
10 -4<br />
for big <strong>orifice</strong>s:<br />
−tr<strong>an</strong>sition from a ballistic<br />
regime to normal diffusion<br />
always:<br />
−non-gaussi<strong>an</strong> fluctuations<br />
with long tails<br />
10 -2 10 -1 10 0 10 1<br />
10 3 ∆ x / σ ∆ x<br />
10 3 ∆ y / σ ∆ y<br />
10 2<br />
R=15.9<br />
10 2<br />
#<br />
#<br />
10 1<br />
10 1<br />
10 0<br />
-2 0 2<br />
10 0<br />
-2 0 2<br />
A. Garcimartín - <strong>Flow</strong> <strong><strong>an</strong>d</strong> <strong>jamming</strong> <strong>of</strong> <strong>gr<strong>an</strong>ular</strong> <strong>matter</strong> <strong>through</strong> <strong>an</strong> <strong>orifice</strong><br />
Traffic <strong><strong>an</strong>d</strong> <strong>gr<strong>an</strong>ular</strong> flow '07
Experimental results<br />
10 0 v y<br />
∆ t / φ<br />
<br />
10 -2<br />
10 -4<br />
for big <strong>orifice</strong>s:<br />
−tr<strong>an</strong>sition from a ballistic<br />
regime to normal diffusion<br />
always:<br />
−non-gaussi<strong>an</strong> fluctuations<br />
with long tails<br />
10 -2 10 -1 10 0 10 1<br />
10 3 ∆ x / σ ∆ x<br />
10 3 ∆ y / σ ∆ y<br />
10 2<br />
R=4.8<br />
10 2<br />
#<br />
#<br />
10 1<br />
10 1<br />
10 0<br />
-3 0 3<br />
10 0<br />
-2 0 2<br />
A. Garcimartín - <strong>Flow</strong> <strong><strong>an</strong>d</strong> <strong>jamming</strong> <strong>of</strong> <strong>gr<strong>an</strong>ular</strong> <strong>matter</strong> <strong>through</strong> <strong>an</strong> <strong>orifice</strong><br />
Traffic <strong><strong>an</strong>d</strong> <strong>gr<strong>an</strong>ular</strong> flow '07
Conclusions<br />
<br />
<br />
<br />
<br />
There are two distinct regimes for the <strong>gr<strong>an</strong>ular</strong> flow <strong>through</strong> <strong>an</strong> <strong>orifice</strong>, one in<br />
which the flow will eventually get arrested, <strong><strong>an</strong>d</strong> <strong>an</strong>other which will never jam.<br />
The boundary between both situations is well defined, at a “critical” size <strong>of</strong> the<br />
<strong>orifice</strong> R c .<br />
The flow for small <strong>orifice</strong>s does not follow the same law th<strong>an</strong> for big <strong>orifice</strong>s. The<br />
behavior R 5/2 is <strong>an</strong> asymptotic law that must be corrected for small R.<br />
The fluctuations <strong>of</strong> the grains display some features that depend on the size <strong>of</strong> the<br />
<strong>orifice</strong>.<br />
Outlook<br />
<br />
<br />
<br />
<br />
Find <strong>an</strong> expl<strong>an</strong>ation for the exponential term included in the law for the flow.<br />
Look more closely to the statistics for the fluctuations <strong>of</strong> the grains.<br />
C<strong>an</strong> we derive the velocity pr<strong>of</strong>ile from the PDF <strong>of</strong> the fluctuations<br />
If small fluctuations are introduced, does the <strong>gr<strong>an</strong>ular</strong> flow elude <strong>jamming</strong><br />
A. Garcimartín - <strong>Flow</strong> <strong><strong>an</strong>d</strong> <strong>jamming</strong> <strong>of</strong> <strong>gr<strong>an</strong>ular</strong> <strong>matter</strong> <strong>through</strong> <strong>an</strong> <strong>orifice</strong><br />
Traffic <strong><strong>an</strong>d</strong> <strong>gr<strong>an</strong>ular</strong> flow '07