# Flow rate of granular media: Breakdown of the Beverloo’s scaling

Marcos A. Madrid1, J. R. Darias2, Luis. A. Pugnaloni1 1Departamento de
Ingeniería Mecánica, Facultad Regional La Plata, Universidad Tecnológica
Nacional, CONICET, Av. 60 Esq. 124, 1900 La Plata, Argentina.
2Laboratorio de Óptica y Fluidos, Universidad Simón Bolívar, Apartado Postal
89000, Caracas 1080-A, Venezuela.

###### Abstract

The Beverloo’s scaling for the gravity flow of granular materials through
orifices has two distinct universal features. On the one hand, much
highlighted in the literature, the flow rate is independent of the height of
the granular column. On the other hand, less celebrated yet more striking, the
flow rate is fairly insensitive to the material properties of the grains
(density, Young’s modulus, friction coefficient, etc.). We show that both
universal features are lost if work is done on the system at a high rate. In
contrast with viscous fluids, the flow rate increases during discharge if a
constant pressure is applied to the free surface of a granular column.
Moreover, the flow rate becomes sensitive to the material properties.
Nevertheless, a new universal feature emerges: the dissipated power scaled by
the mean pressure and the flow rate follows a master curve for all conditions
of forcing and material properties studied. We show that this feature can be
explained if the granular flow in the silo is assumed to be a quasistatic
shear flow.

###### pacs:

45.70.-n, 45.70.Mg

The flow of granular matter (bulk solids such as sand, seeds, pellets, etc.)
presents rather peculiar features when compared with viscous fluid flows.
Prompted by the numerous industrial applications and by academic curiosity, an
entire area of research has been developed around granular matter. Despite
numerous advances, we are still far from producing a synthesis to provide a
theoretical framework that can be applied to all phenomena observed in
granular matter. Instead, we are still gaining knowledge from focusing on a
particular experimental observation and pushing to the limit our yet
incomplete models. One archetypal phenomenon in this sense is the discharge of
granular materials through an orifice at the bottom of a silo. This has been
considered in a number of studies since the 19th century (see for example
various ; beverloo ; NeddermanBook  and references therein). The most salient
feature, usually highlighted in the literature, is the fact that the flow rate
does not depend on the height $h$ of the column of grains in the container, in
clear contrast with the behavior of viscous fluids. A much more striking
peculiarity, yet little mentioned, is that the flow rate is not affected by
the material the grains are made of beverloo . This latter “universality” is
still poorly understood.

If the discharge orifice is circular and large enough to avoid clogging
Zuriguel2003 , the mass flow rate $W$ is described by the so called Beverloo’s
rule beverloo ; BrownBook

$W=C\rho_{\mathrm{b}}\sqrt{g}(D-k\,d)^{5/2},$ (1)

where $D$ is the diameter of the opening, $\rho_{\mathrm{b}}$ the bulk density
of the granular sample, $g$ the acceleration of gravity and $d$ the diameter
of the grains. Here, $k$ and $C$ are two fitting dimensionless constants.
Interestingly, while $k$ may vary up to a factor of 2, depending on the shape
of the grains used, $C\approx 0.58$ for virtually any material tested.
Initially, Beverloo et al. beverloo , and more recently Kondic kondic  and
Mankoc et al. Mankoc2007  have stressed this point. Even though this material
independence is not a fundamental law of nature for the flow of grains, it is
in fact a remarkable emerging feature still little investigated. Notice that
Eq. (1) does include $\rho_{\mathrm{b}}$, which depends on the material
density of the grains. However, this is only to calculate the mass flow rate
$W$. The particle flow rate $Q=W/m$, with $m$ the mass of one grain, does not
depend on the density of the material.

A simple explanation for the 5/2 power in the Beverloo’s rule is based on an
heuristic assumption so called _free fall arch_ NeddermanBook . It states that
most grains lose contact with the rest of the granular packing when they reach
a position that is about one orifice radius, $D/2$, above the outlet and that
particles at this point fall freely from an initial zero vertical velocity.
Then, the free fall of these grains over a distance $D/2$ leads to a flow rate
proportional to $\sqrt{g}D^{5/2}$, irrespective of their material properties.

In this letter, we show that adding a piston on top of the granular column the
universal feature described above breaks down. We observed that, when forced,
the flow rate increases during the discharge (in contrast with viscous fluids)
and the change in $Q$ depends on the material properties of the grains. We
will show that, for different materials and forcing conditions, the dissipated
power scaled by the mean internal pressure in the silo and by the flow rate,
falls onto a single master curve as a function of the number of grains
remaining in the silo. This scaling is consistent with the assumptions made
for a quasistatic shear flow in the so called $\mu(I)$-rheology, based on the
introduction of the inertial number $I$ dacruz . The analysis of these extreme
conditions of discharge allows us to put forward some ideas that help in
understanding the limits of the _free fall arch_ model NeddermanBook  and the
universal features of the unforced discharge.

Experiments— The experimental setup is shown in Fig. 1(a). The silo is a
cylindrical glass tube (300 $\pm$ 1) mm tall and (40.0 $\pm$ 0.5) mm in
internal diameter, which is attached to a metal stand that holds it fixed. The
bottom of the tube is bonded to an aluminum base that has a central orifice
(15.0 $\pm$ 0.5) mm in diameter. The silo is filled to a height (190 $\pm$ 1)
mm by pouring uniformly glass beads with diameter (1.00 $\pm$ 0.05) mm and
bulk density (1600 $\pm$ 20) kg/m3. We introduce a solid cylinder (piston) of
Plexiglas (37 $\pm$ 1) mm tall, (39.50 $\pm$ 0.05) mm in diameter and mass
(130 $\pm$ 1) g, which serves as a support for an extra overweight that can be
placed on top. The downward motion of the piston was recorded using a digital
camera (Pixelink PL-B741F) at 25 fps. The images were analyzed using a
commercial software to obtain the height of the column of grains as a function
of time. This allow us to calculate $W(t)$ under the approximation that the
packing fraction is roughly constant during the discharge. The internal
friction coefficient of the glass beads is $\mu=0.40\pm 0.03$.

Simulations— Most of the analysis is done on Discrete Element Method (DEM)
simulations. We use the implementation of YADE yade  with a particle–particle
interaction comprising a linear spring–dashpot model for the normal,
$F_{\mathrm{n}}=k_{\mathrm{n}}\xi-\gamma_{\mathrm{n}}\dot{\xi}$, and
tangential, $F_{\mathrm{t}}=\mathrm{min}(\mu
F_{\mathrm{n}},k_{\mathrm{t}}\zeta-\gamma_{\mathrm{t}}v_{\mathrm{t}})$,
contact forces. Here, $\xi$ is the particle–particle overlap, $\zeta$ the
total shear displacement of the contact and $v_{\mathrm{t}}$ the relative
tangential velocity. Unless otherwise specified, $k_{\mathrm{n}}=1000$ Nm-1,
$k_{\mathrm{t}}=2k_{\mathrm{n}}/7$, $\gamma_{\mathrm{n}}=0.03$ Nsm-1,
$\gamma_{\mathrm{t}}=0.02$ Nsm-1 and $\mu=0.55$. The same interaction applies
for the particle–walls contacts. $N=5\times 10^{4}$ spherical grains (diameter
$d=1$ mm) are poured in a silo. The sphere material density, $\rho$, was set
to different values ($500$, $1000$ and $2000$ kg/m3). The silo has a diameter
$D_{\mathrm{s}}=2R_{\mathrm{s}}=30d$ with an orifice of diameter $D=15d$ at
the center of the base. The orifice is initially blocked by a plug. After the
grains come to rest in the silo (we wait until the kinetic energy per particle
falls below $10^{-6}$ J), we remove the plug and count the number of grains
that pass through the aperture per unit time. The acceleration of gravity
$g=9.81$ ms-2 acts in the negative vertical direction. For the forced flow
simulations, we introduce a cylindrical piston ($78.54$ g) made of small
spheres ($0.33d$ in diameter) whose relative positions are fixed during the
simulation.

Figure 1: (Color online) (a) Sketch of the experimental setup. (b) Mass flow
rate $W$ as a function of the number of grains $N$ left in the silo measured
in the experiments during the forced discharge of glass beads for different
overweights (see labels in the figure). The full lines are only to guide the
eye. Each curve corresponds to the average over six realizations of the
experiment.

Results— Figure 1(b) shows the experimental mass flow rate for glass beads for
forced discharges using different overweights. As we can see, for light
overweights (with exception of the lighter overweight tested), $W$ is constant
during the initial part of the discharge and then grows in the final stages of
the discharge. As the overweight becomes heavier, the acceleration of the mass
flow rate is more significant and starts at an earlier stage. In contrast with
viscous fluids, which decrease in flow rate as the column empties (even with
an external constant forcing), granular samples forced with a constant applied
force increase their mass flow rate. This striking behavior is consistent with
enhanced flows observed in similar experiments carried out by Peng et al.
peng2009 . Wilson et al. showed a similar growth in $Q$ for a submerged silo
(interestingly, without the need of extra overweight) wilson2014 ; however,
this was latter proven to be due to an hydrodynamic effect Koivisto2016 . One
may speculate that the overweight induces a higher pressure at the bottom of
the silo (as shown by other authors Ovarlez2005 ) and this is responsible for
the increased flow rate. We will discuss this below. Importantly, the
increasing mass flow rate suggests that forced discharges cannot longer be
explained by a simple _free fall arch_ assumption and that properties related
to the specific material the grains are made of may become relevant in the
problem. Since we find difficult varying material properties in the
experiments in a controlled way, we turn in the rest of the discussion into
results obtained via the DEM simulations.

Figure 2(a) shows the particle flow rate during the discharge of grains with
different material properties, without external forcing, obtained via DEM
simulations. We have removed from the analysis the transients at the beginning
and end of the discharge by ignoring the initial and final $5000$ grains. As
we can see, $Q$ is constant throughout the discharge and the actual values do
not depend on the material properties. These observations are consistent with
results from many other authors that studied free discharges. Note, however,
that the sample with $\mu=0.31$ do display an slightly higher $Q$ which is
still independent of $h$ as seen recently by others in the case of low
friction kondic . The pressure at the bottom of the silo during the discharges
is shown in Fig. 2(c). It is clear that the bottom pressure falls
monotonically throughout the discharge while the flow rate remains constant.
This has been highlighted in previous studies aguirre2010 ; aguirre2011 .

Figure 2: (Color online) Particle flow rate $Q$ (panels (a) and (b)) and
bottom pressure $P$ (panels (c) and (d)) as a function of the number of
particles $N$ in the simulated silo discharge. Open symbols correspond to
unforced discharges (panels (a) and (c)) and solid symbols to forced
discharges with an overweight of $78.54$ g (panels (b) and (d)). Each curve
corresponds to a different material: $\rho=2000$ kg/m3, $\mu=0.31$ (red
circles); $\mu=0.55$ (blue squares); $\mu=0.85$ (green triangles); $\mu=1.0$
(orange diamonds); $\rho=1000$ kg/m3, $\mu=0.55$ (blue squares with pluses);
$\rho=500$ kg/m3, $\mu=0.55$ (blue squares with crosses). Most discharges are
for an orifice diameter $D=15.0$ mm. The pink pentagons correspond to a
smaller orifice ($12.0$ mm) with $\rho=2000$ kg/m3 and $\mu=0.55$.

In Fig. 2(b), we show results for the flow rate during the discharge of a silo
with an overweight (roughly 1.5 times that of the material inside the
container). The results are consistent with the experiments (see open circles
in Fig. 1), showing an increase of $Q$ towards the end of the discharge. Fig.
2(d) shows that the bottom pressure increases during the discharge since the
portion of granular column screening the overweight decreases. Results for
different overweights can be found in the Supplemental Material footnote2 .

An interesting result is that in the forced discharges, apart from the
accelerating $Q$, the actual acceleration becomes dependent on the material
properties of the grains. For example, Fig. 2(b) shows that denser materials
present a lower $Q$ during the acceleration phase. This seems counterintuitive
since the heavier grains induce in fact a higher bottom pressure (see Fig.
2(d)). We will put forward an explanation for this effect below. The friction
coefficient also affects, although to a lesser extent, the values of $Q$
achieved during the speedup. The larger the friction coefficient, the lower is
the speedup of $Q$ (compare filled orange diamonds and filled blue squares in
Fig. 2(b)). We conclude that the universal character of the Beverloo’s rule,
in which the particle flow rate is insensitive to the material properties,
breaks down in forced flows.

One may speculate that the increased pressure in the bottom of the silo due to
the use of a piston is the ultimate responsible for the increase in $Q$.
However, there are a few observations that indicate that this is not the case.
For the unforced discharges, materials with different density induce very
different pressures on the base, yet the flow rate is insensitive to this
footnote3 . For forced discharges, the bottom pressure increases during the
discharge. However, even higher bottom pressures can be also achieved without
an overweight (either by using a wider silo or by discharging materials with
very high densities). In these free flow case, $Q$ always coincides with the
one obtained for any unforced discharge for the given $D$, which is lower than
the enhanced $Q$ seen in the forced discharges. Although pressure is a
relevant parameter of the problem, it is clear that it does not control the
flow rate on its own.

Dissipated power— The flow in a discharging silo could be connected with
simpler geometries, such as a shear cell, where valuable scalings have been
found. It has been shown that for a shear cell of thickness $L$ the tangential
stress $\tau$ necessary to develop a flow at velocity $v$ of a granular sample
can be put in terms of the inertial number $I$ as dacruz2005

$\tau=\mu(I)P,$ (2)

where $P$ is the confining pressure, $\mu(I)$ is the effective friction
coefficient, $I=\frac{v}{L}\frac{d}{\sqrt{P/\rho}}$ is the inertial number
that characterizes the flow (if the gains are stiff and $L\gg d$) and $\rho$
is the density of the material of the grains. We assume here that the flow in
the silo can be roughly represented as a shear flow, where $v$ is the velocity
of the free surface, $L$ is the silo radius and $P$ is the mean stress in the
column footnote1 . Then, we can write the power, $W_{\mathrm{D}}$, dissipated
by friction at any given time during the discharge as

$\displaystyle W_{\mathrm{D}}(t)=\tau(t)A(t)v(t)$
$\displaystyle=\mu(I)P(t)A(t)v(t)$
$\displaystyle=\mu(I)P(t)A(t)\frac{Q(t)m}{\rho_{\mathrm{b}}A_{\mathrm{s}}},$
(3)

where $A(t)$ is the area of frictional contact between the grains and the
silo, $A_{\mathrm{s}}$ is the cross section of the silo, and we have used the
continuity relation $W(t)=mQ(t)=\rho_{\mathrm{b}}A_{\mathrm{s}}v(t)$.

In Eq. (3), $A(t)$ includes the contact with the lateral walls, which
decreases during the discharge, plus a fixed area due to the effect of the
base of the silo. Hence, we can express $A(t)$ as

$A(t)=2\pi R_{\mathrm{s}}z(t)+\alpha\pi R_{\mathrm{s}}^{2},$ (4)

where $R_{\mathrm{s}}$ is the silo radius, $z(t)$ is the height of the
granular column at time $t$, and $\alpha$ is a constant to correct the
frictional area of the base due to the cone generated by the stagnant zone.
Further, we can put $z(t)$ in terms of the number of grains in the silo as
$z(t)=mN(t)/(\rho_{\mathrm{b}}\pi R_{\mathrm{s}}^{2})$.

We can show that $\mu(I)$ is insensitive to the granular material properties.
Consider $A_{\mathrm{s}}\rightarrow\infty$, then $L\rightarrow\infty$ and
$I\rightarrow 0$. Da Cruz et al. have shown that in the quasistatic limit
(i.e., $I\rightarrow 0$) $\mu(I\rightarrow 0)\approx 0.26$ for all material
properties if the grain–grain friction coefficient is above 0.4 dacruz2005 .
In our simulations with a finite diameter silo the inertial number varies
during the discharge in the range $9.8\times 10^{-3}<I<1.6\times 10^{-2}$,
which is indeed in the quasistatic limit ($I<10^{-2}$) dacruz2005 . Therefore,
from Eqs. (3) and (4),

$\frac{W_{\mathrm{D}}}{PQ}(t)=0.26\frac{m}{\rho_{\mathrm{b}}}\frac{A(t)}{A_{\mathrm{s}}}=0.26\frac{m}{\rho_{\mathrm{b}}}\left[\frac{m}{\rho_{\mathrm{b}}}\frac{2N(t)}{\pi
R_{\mathrm{s}}^{3}}+\alpha\right],$ (5)

which is a quantity that should not depend on the material properties, nor
forcing conditions, nor orifice size $D$. Note that $\rho_{\mathrm{b}}/m$ is
the number density, which does not depend on the density of the material of
the grains.

Figure 3 shows $W_{\mathrm{D}}/(PQ)$ as a function of the number of grains in
the silo for all the discharges, forced and unforced, for all the values of
$\rho$ and $\mu$ considered. We have also included results for a discharge
with a smaller orifice (see pink pentagons in Fig. 3). With exception of the
low friction sample ($\mu=0.31$, red filled circles), all curves fall into a
single straight line (although with some scatter) that can be fitted
reasonably well with Eq. (5), where the only free parameter is $\alpha$ (see
full line in Fig. 3). For $\mu=0.31$, the observed deviation from Eq. (5) is
expected since $\mu(I\rightarrow 0)$ does depend on $\mu$ for $\mu<0.4$.
Equation (5) indicates that the slope of the straight line is inverse to
$R_{\mathrm{s}}^{3}$. Results for different silo radii, confirming this
prediction, can be found in the Supplemental Material footnote2 . The scaling
shown by Fig. 3 demonstrates that the flow in a silo discharge could be
modeled as a quasistatic flow, for unforced and forced conditions, without the
need of heuristic postulates such as the _free fall arch_. A first modeling
attempt in this direction, considering only free discharges, can be found in
Ref. madrid2016 .

Figure 3: (Color online) $W_{\mathrm{D}}/(PQ)$ during a discharge as function
of the number $N$ of particles in the silo for different material properties,
orifice size and forcing conditions (symbols as in Fig. 2). The inset shows
$W_{\mathrm{D}}$ as a function of $N$. The solid line corresponds to the best
fit to Eq. (5) yielding $\alpha=3.76$.

Discussion— From a “microscopic” perspective, we can understand the lack of
dependency on the material properties for free discharges based on an
_effective inelastic collapse_ mcnamara . When the density is high, the number
of collisions per unit time grows exponentially with the number of particles.
As a consequence, even if individual collisions dissipate little energy, the
kinetic energy of the entire system will be damped in a very short time. If
the system is in a state of _effective inelastic collapse_ , any work done on
the grains inside the silo will be dissipated in a very short time (in
particular the work done by the force of gravity). Therefore, with the
exception of the grains that set free from the pack as they reach the region
of the aperture, the system inside the silo moves, as a whole, at constant
velocity. The flow rate consistent with such velocity is imposed solely by the
size of the orifice, without the influence of the local or global pressure.

The increase in $Q$ for forced discharges can be attributed to the loss of
_effective inelastic collapse_. Since the piston is doing work on the system
at a high rate and the total number of contacts (collisions) decreases during
the discharge, these fewer contacts are eventually unable to fully dissipate
the energy input. As a result, the flow rate increases due to the not
dissipated kinetic energy. We suggest that the same phenomenon occurs even in
unforced discharges. However, the work done by the gravity force is low and
decreases as the silo discharges, so the accelerated $Q$ would be observed
only when a very small number of particles remain in the silo. Indeed, this is
consistent with previous preliminary observation wilson2014 ; geminard-private
; Koivisto2016  although no systematic study of the phenomenon has been
carried out to our knowledge.

The previous picture is consistent with the results in Fig. 2 for forced
flows. On the one hand, more dissipative materials (higher $\mu$) display a
slower acceleration in $Q$ as should be expected. On the other hand, denser
grains display a slower increase in $Q$, which is unexpected. Consider however
that heavier grains induce a higher average pressure across the packing. This
is consistent with stronger contact forces which lead to higher dissipation by
friction at each contact for a given $\mu$, since the tangential force is
proportional to the normal. As a consequence, denser grains are able to
dissipate more of the injected power and the increase in $Q$ is delayed until
fewer particles remain in the column to dissipate the energy.

Conclusions— Despite the little attention received, the universal free flow
rate of granular materials with respect to the different material properties
is a remarkable phenomenon. By injecting energy into the system at a high
rate, we have shown that not only the independence of $Q$ with the system
height is broken, also the universality with respect to material properties is
lost. In contrast with viscous fluids, the flow rate is accelerated during the
discharge due to the reduction of dissipation as the number of grains
decreases in the container.

We have shown that for both, forced and unforced discharges, the flow can be
described as a shear flow consistent with the $\mu(I)$-rheology in the
quasistatic limit; where the flow does not depend on the material properties
of the grains if the grain-grains friction coefficient is above 0.4. This
opens the opportunity for modeling silo discharges in a wide range of
conditions without the need of heuristic approximations such as the _free fall
arch_ and the _empty annulus_ , which have been recently challenged janda2012
; rubio2015 .

###### Acknowledgements.

We are grateful for valuable discussions with D. Maza, L. Kondic, I. Zuriguel,
J-C. Géminard, M. A. Aguirre, D. Durian and E. Clément. This work has been
supported by ANPCyT (Argentina) through grant PICT-2012-2155, Universidad
Tecnológica Nacional (Argentina) through grant PID-MA0FALP0002184, Centro
Argentino Francés de Ciencias de la Ingeniería (CAFCI, Argentina-Francia) and
FONACIT through grant 2015000072 (INVUNI2013-1563) Universidad Simón Bolívar
(Venezuela).

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## Supplemental Material

### Effect of the overweight

In the main text, we have shown simulation results for a single value of the
external force applied to the granular column. Our experimental data (see Fig.
1 in the main text), indicate that different weights applied to the piston
lead to different speedup of the flow rate. We have carried out additional
simulations with different overweights. The results are shown in Fig. 4. The
particle flow rate [Fig. 4(a)], as it was seen in the experiments, increases
more rapidly and at an earlier stage when heavier overweights are used.
However, $W_{\mathrm{D}}/(PQ)$ as a function of $N$ coincides for all values
of the overweight [see Fig. 4(b)] and is reasonably well fitted by Eq. (5) in
the main article. This confirms that the universal straight line holds for a
wide range of external forces and the ordinate $\alpha\approx 3.76$ is
consistent with the value obtained for different granular materials (see Fig.
3 in the main article).

Figure 4: (a) $Q$ as a function of $N$ for different pistons: $78$ g (blue
squares), $130$ g (green triangles) and $262$ g (orange diamonds) and $524$ g
(red circles). (b) Scaled dissipated power $W_{\mathrm{D}}/(PQ)$ as a function
of $N$. The solid line corresponds to a fit using Eq. (5) in the main text
with $\alpha=3.77$.

### Effect of the silo radius

As we mentioned in the main text, Eq. (5) predicts that the slope of the
universal straight line found for $W_{\mathrm{D}}/(PQ)$ as a function of $N$
must depend on the silo radius. To test this, we have carried out simulations
of unforced discharges with silos of different radii $R_{\mathrm{s}}$. Figure
5 displays the particle flow rate and scaled dissipated power. As we can see,
the flow rate does not depend on the silo radius. Note that in Fig 5(a), the
final stages of the discharge where the flow rate drops to zero is observed
earlier for the wider silos since the plot is against $N$ and not against the
column height.

In Fig. 5(b), we show $W_{\mathrm{D}}/(PQ)$ as a function of $N$. As predicted
by Eq. (5), the scaled dissipated power presents a different slope for each
value of $R_{\mathrm{s}}$. The solid lines in the figure correspond to the fit
of Eq. (5). We recall here that the slope of the straight line is not fitted
since Eq. (5) predicts its value. The fitted ordinate $\alpha$ is rather
insensitive to $R_{\mathrm{s}}$ in a first approximation. An increase of $66$
% in $R_{\mathrm{s}}$ induces a $16$ % reduction in $\alpha$. However, the
values do vary and are somewhat higher than for the forced discharges (compare
with Fig. 4(b)).

We speculate that the value of $\alpha$ may depend on geometrical details such
as the actual shape of the silo cross-section or the presence of a hopper-
shaped base. There is also an additional effect (neglected in our analysis)
due to the dissipation of the grains in contact with the piston. This makes
the forced discharges slightly “more dissipative” than the unforced flows.
These “second order” effects merit further study and may become crucial in
specific applications.

Figure 5: (a) $Q$ as a function of $N$ during unforced discharges for
different silo radius $R_{\mathrm{s}}$: $0.012$ mm (orange diamonds), $0.015$
mm (red circles) and $0.020$ mm (green triangles). (b) Scaled dissipated power
$W_{\mathrm{D}}/(PQ)$ as a function of $N$. The solid lines correspond to a
fit using Eq. (5) in the main text with $\alpha=5.0$ (orange), $4.59$ (red)
and $4.31$ (green).