Discharge of elongated grains in silos under rotational shear Tivadar Pong o12Tam as B orzs onyi2and Ra ul Cruz Hidalgo1 1F sica y Matem atica Aplicada Facultad de Ciencias Universidad de Navarra Pamplona Spain

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Discharge of elongated grains in silos under rotational shear
Tivadar Pong´o,1,2Tam´as B¨orzs¨onyi2and Ra´ul Cruz Hidalgo,1
1F´ısica y Matem´atica Aplicada, Facultad de Ciencias, Universidad de Navarra, Pamplona, Spain
2Institute for Solid State Physics and Optics, Wigner Research Centre for Physics, P.O. Box 49, H-1525 Budapest,
(Dated: October 26, 2022)
The discharge of elongated particles from a silo with rotating bottom is investigated numerically.
The introduction of a slight transverse shear reduces the flow rate Qby up to 70% compared to
stationary bottom, but the flow rate shows a modest increase by further increasing the external
shear. Focusing on the dependency of flow rate Qon orifice diameter D, the spheres and rods show
two distinct trends. For rods, in the small aperture limit Qseems to follow an exponential trend,
deviating from the classical power-law dependence. These macroscopic observations are in good
agreement with our earlier experimental findings [Phys. Rev. E 103, 062905 (2021)]. With the help
of the coarse-graining methodology we obtain the spatial distribution of the macroscopic density,
velocity, kinetic pressure, and orientation fields. This allows us detecting a transition from funnel to
mass flow pattern, caused by the external shear. Additionally, averaging these fields in the region
of the orifice reveals that the strong initial decrease in Qis mostly attributed to changes in the flow
velocity, while the weakly increasing trend at higher rotation rates is related to increasing packing
fraction. Similar analysis of the grain orientation at the orifice suggests a correlation of the flow
rate magnitude with the vertical orientation and the packing fraction at the orifice with the order of
the grains. Lastly, the vertical profile of mean acceleration at the center of the silo denotes that the
region where the acceleration is not negligible shrinks significantly due to the strong perturbation
induced by the moving wall.
I. INTRODUCTION
Granular materials are everywhere in nature and they
are often used in industrial processes. Since long time,
humans have employed containers like silos and bins to
store them, so it is technologically important to un-
derstand their mechanical response under these specific
boundary conditions. Thus, notable research efforts have
been undertaken in this direction, where the ultimate aim
is to derive the macroscopic response of a granular sam-
ple from the contact interactions of the whole particle
ensemble [1, 2].
Developing technological solutions, several types of silo
flow patterns have been detected. For instance, a funnel
flow pattern is characterized by the initial particle flow in
the central region of the silo. Consequently, funnel flow
silos often have stagnant grains near the walls, leading to
undesired in-service issues. In contrast, mass flow pat-
tern provides a uniform outflow without a central flow
channel, and the material flows down as a continuum
column. Achieving mass flow condition is ideal, in par-
ticular, for mixtures of particles that are susceptible to
segregation.
The dependence of the particle flow rate Qon the ori-
fice diameter Dalso has a significant technological inter-
est. Due to its simplicity, the most used expression is
the well-known Beverloo’s correlation: Q∝(D−kd)5/2
[3]. In the formulation dis the particle diameter and the
parameter kenables a good fit of the experimental data
for small orifices. In the large-orifice limit (Dkd),
however, a simple power-law Q∝D5/2is obtained. Re-
cently, Janda et al. presented a different approach to pre-
dict the particle flow rate [4]. Examining the kinetic
spatial profiles of density and velocity at the orifice of
a two-dimensional (2D) silo, they obtained self-similar
functions, when using the orifice size Das a character-
istic length. This analysis led them to the formulation
of the expression Q∝(1 −α1e−D/α2)D5/2in which the
term in the parentheses accounts for the scaling of the
packing fraction and thus the density. They found the
fitting parameters α1,α2to be around 1/2, and 6 particle
diameters, respectively. In all the described approaches,
the value of the exponent 5/2 can be justified by arguing
that once a particle reaches a distance Dto the orifice,
it starts accelerating. The region of accelerating flow is
historically known as the free fall arch region.
To control the outflow in silos, several approaches have
been used. Typically, the silos and hoppers have been
perturbed, for instance, using electric fields to control
the outflow of metallic beads [5], magnetic fields to in-
troduce vibrations in the orifice region [6], or inducing a
repulsive force between the grains [7, 8]. The impact of
external vibration on the macroscopic flow rate of a silo
is far from being understood. More than thirty years ago,
Hunt et al. experimentally observed a flow rate enhance-
ment when increasing the intensity of a horizontal vibra-
tion [9]. Vertical vibrations, however, produce a more
complex response, showing a flow rate decrease when ris-
ing the dimensionless acceleration amplitude Γ. On the
contrary, when using a higher oscillation frequency, f,
a slight increase of Qagainst the same parameter was
encountered [10]. Pascot et al. have recently obtained
experimentally and numerically the existence of two dif-
ferent regimes when varying the oscillation amplitude A,
and fixing the frequency [11]. In particular, when ana-
lyzing small orifices, it is accepted that introducing vi-
brations alters the arches’ stability [12, 13], and the dis-
tributions of the unclogging times [14, 15].
Imposing an external shear is also a promising alter-
arXiv:2210.14115v1 [cond-mat.soft] 25 Oct 2022
2
native to avoid the formation of stable arches in silos.
In the past, the discharge of a cylindrical silo with ro-
tating bottom was explored [16]. However, the authors
only focused their attention on the dynamics of the sys-
tem’s surface [16]. Later on, Hilton and Cleary [17] ex-
amined the impact of the external shear on the flow rate
Q, finding that it is unaffected when a low shear rate
is applied. However, after reaching a critical value, Q
increases monotonically with the rotational frequency.
Very recently, the discharge of wooden rods from a
cylindrical silo perturbed by a rotating bottom wall was
investigated experimentally [18]. It was mainly found
that, for small orifices, the flow rate deviates from the
classical power-law correlation Q∼D5/2, and an expo-
nential dependence Q∼eκD is detected. More inter-
estingly, in the continuous flow regime, the introduction
of transversal shear induced by the bottom wall’s move-
ment decreases the flow rate significantly [18] compared
to spheres with similar effective dimensions [19]. Fur-
ther increasing the rotation rate results in an increase in
the flow rate, which is more significant for smaller ori-
fice sizes, where the flow is intermittent. Since elongated
particles in a shear flow develop orientational ordering
[20, 21], it is expected that changes in particle orienta-
tions due to the external shear will alter the discharge
rate. The complex behavior observed in the experiments
was not fully explained since the experimental approach
has the limitation of not accessing the velocity field, the
density distribution and the particle orientations inside
the silo.
In this work, we introduce a numerical approach,
which replicates the experimental scenario examined in
Ref. [18], reproducing their main macroscopic observa-
tions. Our aim is to extract relevant features of the pro-
cess, which were not accessible experimentally. Thus,
our approach shed light on understanding, how silo dis-
charge of nonspherical grains is influenced by an external
transversal shear.
II. NUMERICAL MODEL
The numerical simulation consists in, using a Discrete
Element Method (DEM) implementation [22], modeling
the mechanical behavior of elongated particles. The mod-
eled system is a cylindrical flat silo with a diameter of
Dc= 2Rc= 19 cm, and bottom wall with a circular
orifice in the center, with diameter D. As a novelty, an
external transversal shear is imposed by the rotation of
the silo bottom wall (see Fig. 1).
We examine the silo discharge, varying systematically
the particle elongation. For simplicity, we employ sphe-
rocylinders to model the experimental wooden rods (see
Ref. [18]). The model defines pairwise forces between
contacting particles. The particle-particle interaction is
computed using an algorithm for interacting spheropoly-
hedra [23, 24]; in particular, an implementation on GPU
architecture [22]. A spherocylinder is defined by the loca-
FIG. 1. Visualization of the rotating bottom silo setup dur-
ing the discharge of elongated particles of L/d = 3.3 (left).
The system dimension replicates the experimental setup ex-
amined in Ref. [18]. On the right, we display the spherocylin-
ders used within this work, together with their corresponding
dimensions.
tion of its two vertices and its sphero-radius r=d/2. The
spherocylinder’s surface is depicted by the set of points
at a distance rfrom the segment which connects its two
vertices. The contact detection between two spherocylin-
ders involves finding the closest point between the two
edges [25]. The overlap distance δnbetween two sphe-
rocylinders equals the overlap distance of two spheres of
radius r, located at each edge. Then, the contact force
is determined as the sum of the normal and tangential
components Fij =Fnˆ
n+Ftˆ
t. The normal unit vector ˆ
n
points from the center of one sphere to another, while the
tangential ˆ
tis parallel to the tangential velocity. The nor-
mal force is calculated by a simple linear spring-dashpot
model Fn=−knδn−γnvrel
nwhere knis a spring coef-
ficient, γna damping coefficient, δnis the overlap be-
tween two particles, and vrel
nis the relative velocity in
the normal direction. The tangential force represents the
friction between two particles, it is also modeled with a
spring-dashpot but taking into account the constraint of
Coulomb friction: Ft= min {−ktδt−γtvrel
t, µFn}. Here
the first term represents a spring with coefficient ktand
tangential deformation δt=|δt|which is integrated from
the following dδt/dt =vrel
twhere vrel
tis the relative tan-
gential velocity. The vector δtis kept parallel to the
contact plane by truncation [26]. The second term is
proportional to the tangential velocity with a coefficient
γt, while the particle friction µsets the constraint for the
force.
3
The DEM algorithm integrates the equations of mo-
tion for both translational and rotational degrees of free-
dom, accounting for gravity gand the force Fij acting
between contacting particles. A velocity Verlet method
[27] is used to integrate the translational, and a modified
leapfrog [28] resolves the rotational ones.
As initial conditions, packing of monodisperse rods
were created by letting a dilute granular gas settle in-
side the closed silo by gravity g= 9.8 m/s. The sys-
tem is composed of N= 10000 particles, and only in
the case of L/d = 4.0Nis set to 22500 to reach a bit
higher initial packing (note that in this case the parti-
cle’s volume is smaller). The discharge process starts by
opening the orifice and simultaneously setting the rota-
tion frequency of the bottom wall to one of the following
values: f= 0.0,0.16,0.32,0.64,0.96,and 1.28 Hz. Sim-
ulation time for the discharge was in the range of 1–7
hours depending on the flow rate, thus overall discharge
time, particle shape and the type of GPU used (NVIDIA
GeForce RTX 2070 and 3080).
The simulation mimics the behavior of wooden parti-
cles, thus, the particle density ρpis set to 620kg/m3, a
spring stiffness kn= 2 ·105mpg/d is used [29]. Other
model parameters were en= 0.9, kt= 2/7kn, γt=γn
and µ= 0.5. The normal damping constant is dependent
on the coefficient of restitution enin the following way:
γn=q2knmp
(π/ ln (en))2+ 1 [30]. The time step is ∆t= 10−6
s, which is smaller than 2% of the contact time between
two colliding particles [31].
The DEM algorithm provides the evolution of the tra-
jectories of all the particles, and their contact network,
with the desired time resolution. Post-processing this
data employing a coarse-graining methodology enables a
well-defined continuous description of the granular flows,
via the packing fraction
φ(r, t) = 1
ρp
N
X
i
miϕ(r−ri(t)),(1)
linear momentum P(r, t), and velocity
V(r, t) =
N
X
i
viϕ(r−ri(t)) (2)
fields [32, 33]. Here mi,viare the mass and velocity
of particle i, and ϕ(r) is non-negative integrable func-
tion that serves to coarse-grain our particles. In par-
ticular, we use a truncated Gaussian function ϕ(r) =
A−1
ωexp[r2/2ω2], with A−1
ωchosen so that the integral of
ϕ(r) over all the space results in 1, the cutoff distance
is rc= 4ωand ω=d∗/2, where d∗is the sphere equiva-
lent diameter d∗= ( 3Vp
4π)1
3, with Vpcorresponding to the
volume of the particle. Additionally, the contact σc(r, t)
and kinetic σk(r, t) stress tensor fields are calculated in
the following way:
σc
αβ(r, t) = −1
2
N
X
i,j
fijαrijβ Z1
0
ϕ(r−ri+srij ) ds, (3)
σk
αβ(r, t) = −
N
X
i
miv0
iαv0
iβϕ(r−ri).(4)
While the former is computed by a line integral over
rij =ri−rjconsidering the force acting between parti-
cles iand j, the kinetic stress represents a granular tem-
perature due to the multiplication of the velocity fluc-
tuation terms v0
i(r, t) = vi(t)−V(r, t). In the case of
elongated particles, the components of the mean orien-
tational tensor Oalso provide very useful information
during the analysis. We have used the following formula
for its coarse-graining:
Oαβ(r, t) =
N
X
i
liαliβ ϕ(r−ri),(5)
where liis the unit vector representing the particle’s di-
rection. By definition, the diagonal elements of the ori-
entation tensor are non-negative and fall between 0 and
1, representing the degree of alignment in the specific di-
rection. In this way, the quantity expressing the nematic
order Sof the ensemble of particles is the largest eigen-
value of the Otensor. With our convention Stakes values
from 1/3 (completely disordered) up to 1 (fully ordered).
In this work, we discuss the orientation of the particles in
the cylindrical coordinate system due to the specific sym-
metry of the silo. Furthermore, for all the coarse-grained
quantities we applied an averaging in the azimuthal direc-
tion in the following way: X(r, z, t) = 1
2πR2π
0X(r, t) dθ.
More details about the data post-processing can be found
in Ref. [26, 32–35].
In our analysis, we are interested in steady-state con-
ditions, namely, the part of the process in which the flow
is stationary. In the case of the static bottom, the dis-
charge slows down near the end of the process due to the
lack of particles outside the stagnant zone. While in the
case of the quickly rotating bottom, when the height of
the material in the silo goes below about Rc, the whole
ensemble starts spinning faster. To exclude these initial
and final effects, we focus on a time interval, where the
flow is stationary in all cases. To have a better statistics,
we ran simulations for each set of parameters, starting
from three slightly different initial configurations, and
apply time and ensemble averaging. Different seeds have
been used for the random generator in order to construct
different dilute granular gases at the beginning, which
yielded distinct initial packings. In the case of small ori-
fices with rotating bottom, we observed intermittent flow.
To handle this, during the calculation of the flow rate we
excluded the intervals larger than 1 s where there was no
flow.
摘要：

DischargeofelongatedgrainsinsilosunderrotationalshearTivadarPongo,1;2TamasBorzsonyi2andRaulCruzHidalgo,11FsicayMatematicaAplicada,FacultaddeCiencias,UniversidaddeNavarra,Pamplona,Spain2InstituteforSolidStatePhysicsandOptics,WignerResearchCentreforPhysics,P.O.Box49,H-1525Budapest,(Dated:Octob...

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