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A numerical study on the influence of particle shape on hopper dischargewithin the polyhedral and multi-sphere discrete element method

D. Hohner, S. Wirtz, V. Scherer

PII: S0032-5910(12)00209-4DOI: doi: 10.1016/j.powtec.2012.03.041Reference: PTEC 8834

To appear in: Powder Technology

Received date: 14 December 2011Revised date: 13 March 2012Accepted date: 24 March 2012

Please cite this article as: D. Hohner, S. Wirtz, V. Scherer, A numerical study on theinfluence of particle shape on hopper discharge within the polyhedral and multi-spherediscrete element method, Powder Technology (2012), doi: 10.1016/j.powtec.2012.03.041

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A numerical study on the influence of particle shap e on hopper discharge with-

in the polyhedral and multi-sphere discrete element method

D. Höhner 1, S. Wirtz, V. Scherer

Department of Energy Plant Technology, Ruhr University Bochum, 44780 Bochum,

Germany

Abstract

In this study 3D DEM-simulations of hopper discharge using non-cohesive, monodisperse spherical

and polyhedral particles as well as particle shapes generated by the multi-sphere method are carried

out. For this purpose an overview of the Common Plane algorithm for contact detection between poly-

hedral particles is given and an important refinement of the contact point definition is presented. In the

hopper the effect of increasing particle angularity on the flow properties is investigated. Moreover,

three different hopper designs are chosen, to further examine the influence of hopper angle and hop-

per opening size on the flow properties in combination with varying particle shapes. It is demonstrated

that particles with an increasing angularity reduce the mass flow rate from the hopper and in case of

the flat bottom hopper (α = 0°) increase the residual quantity after dischar ge. In all simulations signifi-

cant differences between polyhedral and clustered particles were observed, which indicates that the

type of particle shape approximation is a parameter that has to be considered in DEM-simulations of

hopper discharge.

Keywords : Discrete Element Method, hopper discharge, multi-sphere & polyhedral particles

1 Introduction

An estimated 10% of the worldwide converted energy per year is utilized for the processing of different

kinds of granulates [1] which are usually stored in hoppers, silos or bunkers [2]. Since the flow charac-

teristics of the granular material is of vital importance for the operational stability and structural integri-

ty of these units [3], a deeper knowledge and understanding of the mechanical behavior within a hop-

per is necessary for best possible design and operation. Nevertheless a general approach for design-

ing a hopper is not available [4].

In recent years the discrete element method has proven to be a capable tool for predicting the

mechanical behavior of granular materials. Approximation of the particle shape is one of the key chal-

1 Corresponding author E-mail address: hoehner@leat.ruhr-uni-bochum.de

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lenges of DEM-simulations [5]. Still many discrete element codes represent particle shape by discs (in

2D) or spheres (in 3D). The advantage of using discs or spheres is their simplicity especially in terms

of interparticle contact detection. Moreover, the contact force models for contacting spheres and

spheres contacting flat walls are well known [6-9]. While spheres are easy to handle in a DEM-Code,

they show a different mechanical behavior on the single grain level as well as in larger assemblies,

compared to the actual particle behavior in most application areas [10-11]. For example interlocking of

particles cannot be simulated using spherical shape representation. Therefore the physical meaning of

results obtained from these simulations is questionable [12].

In the following an overview of existing experimental and numerical studies on granular mate-

rial in hoppers is given. Kohring et al. [13] conducted 2D simulations of convex polygons in a hopper.

They identified four generic flow regimes, depending on the inflow rate of the hopper. (A) Even with

fluctuating inflow rate the outflow rate remains steady; (B) while the outflow is steady, blocking can

occur, if the fluctuations of the inflow rate are sufficiently large; (C) even without fluctuations blocking

will occur in the hopper, since the outflow varies with time and the mean time to blockage is nonzero;

(D) the outflow is unsteady and the hopper is always blocked.

Cleary and Sawley [14] investigated the outflow of superquadrics from a hopper in 2D DEM-

simulations. They found that particles with a higher aspect ratio, compared to spherical particles, lead

to a reduction of the outflow of up to 30 %. Moreover, a difference in the flow profile was detected.

Particles with a higher angularity increased the flow resistance of the granular material as well and

resulted in reduced mass flows of up to 28 %, while no specific influence on the flow profile could be

detected.

Li et al. [15] investigated sphero-discs in a rectangular hopper numerically as well as experi-

mentally. A quantitative comparison of the flow behavior, arching and discharge within the hopper for

three different opening sizes was conducted and in general a good agreement between experiments

and DEM-simulations could be accomplished. By also testing spherical particles, the effect of particle

shape on the discharge rate of the hopper was shown. While a comparison of the experimental data

was difficult, since both particle shapes have slightly different volumes and material properties, the

DEM-simulations led to 20-30% higher discharge rates for sphero-discs. Li et al. attributed this to discs

having an easier flow path over each other, while spheres have a “bumpier ride”.

Fraige et al. [16] conducted numerical and experimental investigations with 200 spherical and

cubical particles in a rectangular flat bottom hopper. In particular the flow patterns, flow rates and stat-

ic packings were closely examined. Experimental and numerical results were generally in a good

agreement and demonstrated that cubes pack more efficiently than spheres, while flowing less readily.

Fraige et al. acknowledged that a larger system (i.e. a larger number of particles) needs to be exam-

ined for a more comprehensive comparison between both particle shapes.

Tao et al. [17] conducted DEM-simulations with spherical and corn-shaped particles in a rec-

tangular flat bottom hopper. The corn-shaped particles were represented by four overlapping spheres

applying the multi-sphere approach [18]. Simulation results indicate that the vertical velocity difference

between center and side wall and horizontal velocity of corn-shaped particles is smaller than for spher-

ical particles, while the mean voidage in the hopper is smaller. Moreover, Tao et al. demonstrated by

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varying the ratio of width and length as well as height and width of the hopper that the wall effect influ-

ences the voidage of spheres more than of corn-shaped particles.

González-Montellano et al. [19] investigated glass beads and maize grains in a rectangular

hopper. In the 3D DEM-simulation the glass beads were represented as spheres, while the maize

grains were approximated as multi-sphere particles consisting of six overlapping spheres. In DEM-

simulations as well as in corresponding experiments the mean bulk density at the end of the filling

phase, the discharge rate and the flow pattern were recorded. A comparison between experiments

and simulations revealed reasonable predictions for the glass beads, while the simulations of the

maize grains required calibration. By modifying the friction properties of the material, acceptable pre-

dictions were obtained.

Most recently Mack et al. [20] investigated 322 non-cohesive spheres as well as a mixture of

polyhedral dices with different numbers of faces in hoppers of three different angles α. The polyhedra

had rounded vertices in experiments and simulations in order to acknowledge the fact that particles in

technical material rarely exhibit perfect vertices. The results indicated that polyhedra flow generally

slightly faster than spheres and it was assumed that the flat surfaces enable them to slide past each

other more easily. Moreover, for α = 30° the polyhedra showed a greater tendency for bridging. For a

flat bottom hopper they additionally observed a higher tendency for polyhedral particles to pile up in

the corner of the hopper. A reasonable agreement between experiments and simulations in terms of

static packing, flow behavior and hopper discharge rates was accomplished.

The research mentioned is insufficient to fully understand the effect of particle shape on the

flow behavior in hoppers. Most numerical studies were conducted in 2D or quasi 2D systems and/or

considered only a small number of different particle shapes. Moreover, a thorough comparison be-

tween polyhedral and multi-sphere particles, as two of the most commonly employed approximations

for arbitrarily shaped particles, has not been carried out so far.

In this study 3D DEM-simulations of hopper discharge using non-cohesive, monodisperse

spherical and polyhedral particles as well as particle shapes generated by the multi-sphere method

are carried out. For this purpose an overview of the Common Plane algorithm for contact detection

between polyhedral particles is given and an important refinement of the contact point definition is

presented. In contrast to Mack et al. [20] the vertices of the polyhedra are sharply-edged, in order to

highlight the influence of particle geometry on the flow properties. Differences between the particle

shapes in terms of flow pattern and discharge rate are examined in the hopper-simulations and the

influence of the contact point definition is presented. Similar to Cleary and Sawley [14] the effect of

increasing particle angularity on the flow properties is investigated. Moreover, three different hopper

designs are chosen, to further examine the influence of hopper angle and hopper opening size on the

flow properties in combination with varying particle shapes. Thus, this paper is aimed at broadening

the understanding of the parameter ‘particle shape’, especially in the limit of sharply edged particles,

on the mechanical behavior of granulates during hopper discharge.

2 Discrete element simulation model

2.1 Force Laws applied

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The discrete element method monitors the movement and interaction of all particles in the investigated

system. By calculating the acting forces for all particle-particle and particle-wall contacts, the transla-

tional and rotational velocities as well as the positions of all particles can be determined. For this pur-

pose the discrete element method allows particles to overlap and computes the resulting contact forc-

es as a function of the particle overlap. The particle overlap leads to the compression of a spring in

normal and tangential direction. An overview of force schemes is given by Di Renzo and Di Maio [7,9]

as well as Kruggel-Emden et al. [6,8].

The most frequently employed normal force laws in DEM-simulations are viscoelastic spring

dashpot models [21-23]. The normal force Fn consists of two parts: the elastic component (Fn,el) mod-

els elastic repulsion, while the dissipative component (Fn,diss) accounts for the dissipation of energy

during the collision. According to its dependence on the overlap (ξn) between the particles, these mod-

els can be classified in either linear or non-linear force models. The linear viscoelastic spring dashpot

model has the advantage of having a simple analytic solution, which guarantees a simple relation be-

tween its model parameters and the macroscopic collision properties.

Figure 1 shows a schematic of the model:

Figure 1: Schematic of the normal viscoelastic spring dashpot model.

Linear force models are also frequently applied for the tangential direction [7,24]. The earliest ap-

proach was proposed by Cundall and Strack in 1979 [25]. In their model, the tangential contact force is

calculated based on a linear elastic spring unless the related Coulomb force is exceeded.

Note that in all DEM-simulations, which are presented in this study, the linear models were applied to

determine contact forces.

2.2 Multi-sphere approach

In the multi-sphere model a single particle is represented by a set of connected spheres, which are

inscribed into the shape of the particle such that at each contact point of sphere and real body a tan-

gential plane can be constructed. The component spheres forming a cluster-particle may vary in size

and overlap each other [12,26].

In contrast to spherical particles, the orientation of any more complexly shaped particle has to

be accounted for. Therefore, apart from the global frame of reference, an additional body-fixed frame

is required to describe translational and rotational body motion of the master particle (MP) (see figure

2). The location of the component spheres (CSP) can be determined as

jPMPCSPj rRxx��� ⋅+=, (1)

With =PR Rotational matrix converting vectors from the body-fixed frame to the

global coordinate system

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=jr�

Vector from the clustered particle’s center of gravity to the center of

gravity of component sphere j

Analogously the velocity of the component spheres can be determined as

( )jMPPMPCSPj rRxx��

ɺ�ɺ� ×+= ω, (2)

With =MPω� Rotational velocity of the clustered particle

Figure 2: Relevant position and velocity vectors of a clustered particle, approximated with a

multi-sphere approach.

Contact detection for multi-sphere particles is similar to spherical particles. Each component sphere of

particle A is checked for contact against each component sphere of particle B or in the case of a parti-

cle-wall contact each component sphere is checked for contact against the wall. In order to determine

contact forces for the component spheres, force laws and contact detection schemes can be applied

as for regular spherical particles (see section 2.1). In contrast to spheres, clustered particles can have

more than one contact point. It is a common approach to determine overall contact forces / moments

acting on the master particle by averaging the contact forces / moments at all existing contact points of

the component spheres [27].

∑=

=num

sCSPsMP F

numF

1,

1 �� (3)

∑=

=num

sCSPsMP M

numM

1,

1 �� (4)

With num = number of component spheres for which a contact was detected

2.3 Polyhedral particles

The basic representation of a polyhedral body is similar to a clustered-particle, as outlined in section

2.2. A global frame of reference and a body-fixed frame have to be employed, in order to keep track of

translational and rotational body movement. The location Vjx ,

� of each vertex on the surface can be

determined as

VjPPVj rRxx ,,

��� ⋅+= (5)

With =Px�

Position vector of the particle’s center of gravity

=Vjr ,

� Vector from the particle’s center of gravity to the vertex i

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For the velocity at each vertex of the particle follows

( )VjPPPVj rMxx ,,

��ɺ�ɺ� ×+= ω (6)

With =Pω� Rotational velocity of the particle

Similar to multi-sphere particles, the overall contact forces and moments acting on the polyhedron can

be determined as the mean value of the contact forces and moments acting at each vertex [27].

∑=

=num

ssF

numF

1

1 �� (7)

∑=

=num

ssM

numM

1

1 �� (8)

With =num number of vertices for which a contact was detected

Detecting contact between polyhedral bodies can be complicated and many approaches have been

described in literature [28-31]. One of the most effective methods is the Common Plane algorithm

which was utilized in the DEM-simulations presented in this study.

3 Common Plane algorithm

Since polyhedra may have various shapes and contact geometries, determining the contact properties

can be complicated. Two of the main problems regarding the contact of two polyhedral bodies are the

definition of the normal direction as well as the overlap. The idea behind the Common Plane algorithm

is to substitute the complicated contact detection between two polyhedra by two separate contact

checks between each polyhedron and a plane.

The Common Plane (CP) can be understood as a plane which bisects the space between two

bodies. By convention, one half-space is denoted with a positive and the other one with a negative

sign (see figure 3).

Figure 3: 2D-sketch of the CP between two polyhedral bodies.

The CP needs to satisfy the following conditions:

1. The centroids of the polyhedra are located on opposite sides of the CP.

2. d1 and d2 are the distances of the vertices of polyhedron 1 and 2, which are closest to the CP.

d1 and d2 have equal absolute values and opposite signs.

3. The total gap which is defined as d2-d1 is a maximum, i.e. every rotation of the CP would result

in a smaller total gap.

The distance d of a polyhedron’s vertex V�

to the CP can be determined as

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( )RVnd CP

��� −⋅= (9)

With =R�

Reference point of the Common Plane

Taking the sign convention of the CP into account, d1 is the maximum distance of all vertices on poly-

hedron 1 to the CP, while d2 is defined as the minimum distance of polyhedron 2 respectively.

{ }idd max1 = (10)

{ }jdd max2 = (11)

After the correct position of the CP has been determined, each polyhedron is checked separately for

contact with the CP. For this purpose the distances of all vertices to the CP are calculated according to

equation 9. A polyhedron’s vertex is considered a contact point, if located on a different side of the CP

than the center of its polyhedron.

3.1 Algorithms to find the Common Plane

Cundall [28] describes a two-stage algorithm for positioning the CP. In a first stage an initial CP is

placed mid-way between the centroids of the polyhedral bodies, perpendicular to the line connecting

them. The initial CP is translated along the connection line, i.e. its normal vector, in a way that condi-

tion 2 is satisfied.

The second stage of the algorithm is an iterative scheme for rotating the CP in the correct

position. For this purpose, two axes u�

and v�

, both orthogonal to the CP normal vector CPn�

, are

defined. The unit normal vector of the CP is now rotated in positive and negative direction to the or-

thogonal axes, accounting for four rotations and thus four candidate common planes. Their respective

unit normal vectors are defined as

uknn iCPiCP

��� ⋅+= −1,,

uknn iCPiCP

��� ⋅−= −1,,

vknn iCPiCP

��� ⋅+= −1,,

vknn iCPiCP

��� ⋅−= −1,, (12)

With =i iteration step of the algorithm

=k rotation parameter

For each candidate CP the total gap is determined by searching the vertex with the smallest

distance to the CP. If the total gap is larger than the one from the previous iteration step, iCPn ,

�

is reset

to the normal vector of the candidate common plane, which produced the largest gap in the previous

step.

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If none of the candidate planes increases the total gap the rotation parameter is reduced to

k=k/2 and the four resulting candidate common planes are tested once again. Cundall suggests set-

ting k initially to a value that corresponds to a rotation angle of 5° for contacts being created for the

first time and choosing the minimum value corresponding to 0.01° at which the iteration process halts.

For contacts which already exist, k can initially be set to kmin.

One big disadvantage of this algorithm to find the CP is the requirement of large numbers of

iteration steps which makes it quite time consuming. Moreover own implementations have shown that

the algorithm is not always stable and may return a CP which does not produce the largest gap.

3.2 Fast Common Plane algorithm

The Fast Common Plane (FCP) algorithm, as introduced by Nezami et al. [32], speeds up finding the

CP by identifying a limited amount of candidate planes which need to be tested. As a consequence,

the number of necessary iterations as well as overall operations decreases considerably. Nezami et al.

identified four types of candidate planes:

a. The CP is located mid-way between the two vertices, one on each polyhedron, which are

closest to each other and it is perpendicular to their connection line (PB = perpendicular bisec-

tor).

b. The CP is parallel to a face of at least one polyhedron.

c. The CP is parallel to one edge of each polyhedron.

d. The CP is parallel to one edge of only one polyhedron.

Assuming the particles are not in contact, finding the CP requires the following operations:

1. Initial CP: If a contact between particle 1 and 2 already exists, the CP from the previous time

step is used as initial common plane; otherwise it is the PB.

2. Shortest Link: Find the closest vertices of polyhedron 1 and 2 to the CP and select those two

with the shortest distance from each other (one on each polyhedron). In case that more than

one pair of closest vertices has the shortest distance, choose any of them.

3. Identify relevant faces and edges: Find all faces and edges on both polyhedra of which the

closest vertices are a part of. Only these faces and edges are relevant for the position of the

CP.

4. Check all four types of candidate planes (a - d) and select the plane which produces the larg-

est total gap.

Figure 4: FCP algorithm

This is an iterative procedure in which steps 2 – 4 are repeated until the CP found in the cur-

rent iteration step is equal to the CP form the previous step (see figure 4). Since the iteration mainly

serves for finding the closest vertices rather than finding the CP itself, the number of iteration steps is

in most cases very small [32].

In case the two particles are in contact, Nezami et al. recommend an additional step to tempo-

rarily separate them. In case of an already existing contact, the particles are translated in opposite

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directions along the unit normal vector of the CP from the previous time step or otherwise the PB. It is

crucial for the algorithm to translate the particles just enough to separate them, so that the contact

geometry of the separated particles resembles the actual contact geometry. According to Nezami et

al., for already existing contacts the total gap of the previous time step can serve as translation dis-

tance ‘tran’.

Implementing and testing the FCP algorithm has shown that the value for ‘tran’ as proposed

by Nezami et al. does not guarantee a separation of the particles. Thus the FCP algorithm requires

another iteration scheme within the first iteration loop cycle (i.e. i = 1), which checks whether the parti-

cles are actually separated and finds the optimal translation distance (see figure 4). Here κ is a num-

ber greater than 1, which increases the translation distance with every loop cycle in which the particles

have not been successfully separated. Although this adjustment of the CP algorithm implicates an

increase in computing time (which is a function of the selected value for κ), it is essential for a stable

contact detection.

Nezami et al. observed by means of 2D and 3D DEM-simulations that the FCP algorithm is

about 12 – 40 times faster than the conventional algorithm and hence a definite improvement of the

conventional Common Plane method. Based on this, the FCP algorithm was implemented in the au-

thors’ existing DEM-Code and utilized in the simulations, which will be presented later in this study.

3.3 Refinement of the contact point definition

After correct positioning of the CP, the contact points and contact forces must be determined. A simple

and computationally cheap approach (later labeled as approach #1) is to check each polyhedron’s

vertices for contact with the CP, using the sign convention outlined in the beginning of section 3. Then

contact forces at each contact point of polyhedron with the CP can be computed. As a result, a situa-

tion may occur where the detected points are quite far away from the actual contact area (see figure

5).

Figure 5: 2D-sketch of two cubic particles in contact (red arrows indicating contact forces at the

detected contact points for cube 2).

Cube 2 is placed on cube 1 such that its center of mass is located above the edge of cube 1 in x-

direction. Apart from that, both cubes have neither translational nor rotational initial velocities. Physi-

cally correct behavior implicates that cube 2 tilts over the edge over cube 1, i.e. experiences a rotation

about the y-axis. But this is not necessarily the case when using the CP algorithm without further ad-

justment of the contact force calculation. For the situation depicted in figure 5 the CP algorithm deliv-

ers a CP that is either parallel to the upper face of cube 1 or to the bottom face of cube 2. Four contact

points are detected for cube 1 and cube 2 with all having the same overlapping distance and thus

normal contact force. Due to the cube’s symmetry it does not experience a rotation about the y-axis,

because the calculated moments at all four contacting vertices compensate each other. Thus cube 2

does not tilt over the edge of cube 1 and physically correct behavior cannot be predicted.

Moreover relative velocities at the contact points cannot be calculated correctly since lever

arms, which are required to account for rotational velocities at the contact points, could lie outside of

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the polyhedron. Since relative velocities at the contact points are relevant for the force laws applied

(see eq. 1 and 5), this situation will result in less accurate contact force predictions.

A better approach (later labeled as approach #2) to predict realistic contact behavior of two con-

tacting polyhedra can be outlined as follows:

• Find all vertices of each polyhedron, which are in contact with the CP. These vertices are

considered as candidate contact points.

• Check whether candidate contact points of polyhedron 1 are inside polyhedron 2 and vice

versa. Only candidate contact points which satisfy this condition are considered as contact

points.

• Consider each edge of the polyhedron where two endpoints are candidate contact points as a

relevant edge.

• Project all relevant edges onto the CP and check for intersection points. Each intersection

point defines an additional contact point.

This approach of defining the contact points in the CP algorithm is more intensive with regards to

computing time, yet delivers a better approximation of the actual contact area. In the further study both

described contact point definitions are utilized in DEM simulations of hopper discharge, in order to

evaluate and quantify their influence on the investigated flow properties.

4 Simulation setup

The simulations were carried out for non-cohesive, monodisperse particles with a diameter of d = 25

mm. The selected material properties correspond to polyethylene (PE) particles as used in [4,33].

Table 1 summarizes the particle properties for the conducted DEM-simulations:

Table 1: Particle properties

Figure 6 depicts the investigated non-spherical particle shapes:

Figure 6: Investigated particle shapes. Clustered particles consisting of 30, 20 and 10 sub-

spheres with polyhedra comprising 112, 24 and 12 triangular surface elements respec-

tively (from left to right).

For all simulations the hopper was filled with approximately 15 kg of PE-particles. Along with spheres,

three clustered (CLS) and three polyhedral (PS) sphere-approximations at different accuracy levels

with increasing angularity were investigated, with the third and most angular polyhedron being a cube

(see figure 6). Thus, similar to Cleary and Sawley [14] the influence of increasing particle angularity on

the flow properties inside a hopper can be investigated. But taking it a step further, this influence is not

only examined for roundly-edged particles (CLS) but for particles showing sharp edges (PS) as well.

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According to Cleary [34], the stresses in a particle flow do not depend on the details of the par-

ticle shape, but on the size of the particles. To make spheres and non-spherical particles comparable,

they must have the same characteristic dimension, which Cleary identifies as the diameter. Using a

constant diameter implies varying particle masses for the investigated particle types, so that different

numbers of particles had to be used for filling the hopper with the same amount of mass in all simula-

tions. Table 2 lists the mass and quantity for all investigated particle types:

Table 2: Mass of single particle and quantity in the DEM-simulation for all investigated particle

types.

The investigated hoppers had a rectangular cross section with a height of H = 1 m, a width of W = 0.4

m and a depth of D = 0.2 m (see figure 7).

Figure 7: Hopper designs

In order to obtain results comparable to Mack et al. [20], a flat bottom hopper (α = 0°) as well

as hopper angles of α = 30° and α = 60° were investigated ( α being measured from the horizontal). By

variation of the hopper angle α as well as the orifice size, different hopper designs and their influence

on the flow rates and flow properties were examined in combination with the particle shape. Table 3

summarizes the investigated hopper designs; note that the square cross section of the hopper open-

ing is specified as a multiple of the particle diameter d:

Table 3: Investigated hopper designs

5 Results

5.1 Contact point definition

As described in section 3, there are two basic approaches for defining contact points between polyhe-

dral bodies within the CP algorithm. Approach 2 has been introduced by the authors for better approx-

imation of the actual contact area. With the example of hopper design 1, the influence of the contact

point definition on the outflow of polyhedral particles (PS 1, PS 2, PS 3) from a hopper is investigated.

Figure 8: Comparison of the residual mass in the hopper for DEM-simulation with polyhedral

particles using approach #1 and #2, as outlined in section 3 of this study.

The results depicted in figure 8 indicate increasing differences between approach #1 and #2 with de-

creasing sphere-approximation accuracy, i.e. decreasing number of vertices and triangular surface

segments of the polyhedral particles. Obviously a proper contact point definition within the Common

Plane method has a considerable influence on the mechanical behavior of the granular material, es-

pecially for simple particle geometries.

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In the following simulations only approach #2 is employed for simulating polyhedral sphere-

approximations, since it represents a better approximation of the actual contact geometry.

5.2 Flat bottom hopper (Design no. 1 and 2)

In all simulations the particles were left to rest under gravity before the hopper orifice was opened

(opening starts at t = 1 s and ends at t = 1.2 s). Figure 9 depicts the particle mass over time for the flat

bottom hopper (α = 0°) with an orifice cross section of 6 x 6 d. Fo r the sake of comparison the results

for spherical particles are added to each diagram:

Figure 9: Residual mass design no. 1 (α = 0°, orifice cross section = 6x6 d)

The results show that the hopper discharge is fastest and most complete using spherical particles.

Spheres approximated with the multi-sphere method show a similar discharge curve as real spheres,

while the remaining amount of residual material in the hopper after discharge increases with decreas-

ing approximation accuracy. Similar results are obtained for polyhedral sphere-approximations: with

decreasing approximation accuracy, i.e. with increasing particle angularity, the hopper discharge be-

comes slower and less complete. Cubes, as particles with the highest angularity, have the largest

deviations from the results obtained for spherical particles. This stands in contrast to the experimental

and numerical results of Mack et al. [20], who identified a faster flow for polyhedral particles than for

spheres. Since their experiments and simulations were conducted with polyhedra with rounded verti-

ces, a possible explanation for this discrepancy may be found in the sharp edges of the polyhedral

particles used in the simulations presented here.

Using the example of cubes, figure 10 demonstrates that more polyhedral particles than

spheres pile up in the corners of the flat bottom hopper. This is in compliance with the observations

made by Mack et al. [20].

Figure 10: Pile-up of cubes in the corners of the flat bottom hopper (at t = 20 s) in comparison to

spheres.

In comparison to polyhedra the discharge curves of the clustered particles appear smoother,

indicating a more continuous discharge from the hopper. It seems that the flow resistance of the inves-

tigated polyhedral particles is in general greater than for the clustered particles. This can be attributed

to multi-sphere particles having relatively smooth surfaces, while polyhedral surfaces have multiple

sharp vertices and edges. As a result, in addition to the particle shape parameter “angularity”, as in-

vestigated by Cleary and Sawley [14], the type of particle-shape approximation (clustered particles or

polyhedral; resolution of edges and corners) has a distinct influence on the discharge from the hopper.

Figure 11 illustrates the influence of particle shape on the flow resistance as well as the overall

flow pattern inside the two investigated flat bottom hoppers. Particles are colored depending on their

velocity in vertical direction. The snapshots were taken 0.5 s after full opening of the hopper orifice:

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Figure 11: Vertical velocity distribution of spheres (top), CLS 1 – 3 (middle row) and PS 1 – 3

(lower row) for hopper design 1.

As shown in figure 11, spherical particles develop an almost V-shaped flow across the hopper

with significant movement reaching to the side walls. This indicates mass flow behavior, which is not

expected for real granular material in a flat bottom hopper [14]. For this hopper design so called core

flow can be expected with movement mainly localized in its center. This behavior is much better pre-

dicted by the other investigated non-spherical particle shapes and especially by cubes, where the flow

is localized in the area above the orifice. This indicates that spheres are inaccurate for modeling gran-

ular material flow, especially when angular and sharply-edged particles are present.

Table 4 summarizes the residual mass in the flat bottom hopper as percentage of the initial fill-

ing quantity at t = 20 s for design no. 1 and 2:

Table 4: Residual mass in the hopper as percentage of the initial filling quantity at t = 20 s for

the two investigated flat bottom hoppers.

As expected, for the flat bottom hopper a reduction of the orifice size results in an increase of

the remaining material after discharge for all particle shapes. Nevertheless the increase is more pro-

nounced for polyhedral particles, which indicates a sharper increase of flow resistance and partial

bridging due to the formation of arch-like structures directly above the orifice (evident in figures 13 and

15).

5.3 30° hopper (Design no. 3 and 4)

Figure 12 shows the particle mass over time for hopper design no. 3 with a hopper angle of α =30°

and an orifice cross section of 6x6 d:

Figure 12: Results for hopper design no. 3 (α = 30°, opening cross section = 6x6 d)

As expected, the larger hopper angle leads to a faster and more complete discharge of the

granular material for all investigated particle shapes. Surprisingly, Clustered particles show only mar-

ginal deviations from the discharge of spheres from the hopper.

Table 5 shows the deviations of the mass flow rates for the two investigated hoppers with an

angle of α = 30° as percentage of the mass flow rates obtaine d in the respective simulations with

spherical particles. Note that the deviations are averaged over the first few seconds of discharge,

when there is still a representative filling quantity left in the hopper:

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Table 5: Deviation of the mass flow rates for non-spherical particles as percentage of the re-

sults for spheres. The mass flow rates are averaged over the first few seconds of dis-

charge.

Similar to the results for the flat bottom hopper, an increase of the particle angularity results in

a smaller discharge rate. Moreover, for hopper design no. 3 table 5 indicates smaller deviations, i.e.

higher discharge rates for the clustered particles when compared to polyhedra.

The results for hopper design no. 4 supports the hypothesis that a reduction of the orifice size

has a stronger influence on non-spherical than spherical particles. All clustered particles and

polyhedra experience a greater reduction of the discharge rate than spheres. Obviously the increased

flow resistance of non-spherical particles and especially polyhedra increases the aptitude for bridging

for small orifice sizes. Figure 13 supports this observation:

Figure 13: Vertical velocity distribution of spheres (top), CLS 1 – 3 (middle row) and PS 1 – 3

(lower row) for hopper design 4.

While no clear-cut differences between spheres and clustered sphere-approximations can be

observed, polyhedral particles show a significantly deviating flow behavior. As for the flat bottom hop-

pers core flow is very pronounced for angular polyhedra (PS 2, PS 3). Moreover, for angular polyhe-

dral particle shapes small void structures can be observed in the flow area, indicating non-continuous

particle flow caused by increased flow resistance. Similar to the observation of Cleary and Sawley [14]

in the case of superquadrics, the polyhedral shape of the particles gives the microstructure of the

granular material more strength and slows the rate of yielding and flow. In comparison to figure 11 this

influence is more pronounced as a result of the smaller orifice size.

5.4 60° hopper (Design no. 5 and 6)

Figure 14 shows the particle mass over time for hopper design no. 5 with a hopper angle of α = 60°

and an orifice cross section of 6x6 d:

Figure 14: Results for hopper design no. 5 (α = 60°, opening cross section = 6x6 d)

Table 6 summarizes the deviations of the mean mass flow rate for non-spherical particles as

percentage from the spherical solution. Due to the short duration of the discharge, the mass flow rates

for the spheres and clustered sphere-approximations of hopper design no. 5 are averaged over the

first second of the discharge while for all other particle shapes it is averaged over the first two se-

conds.

Table 6: Deviation of the mass flow rates for non-spherical particles as percentage of the re-

sults for spheres.

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A comparison of the mass flow rates for hopper design 5 and 6 basically supports the obser-

vations already made for the 30°-hoppers. An increa se of the hopper angle increases the mass flow

rate of all particle shapes. While a reduction of the orifice size leads to a greater reduction of the mass

flow rate for non-spherical than spherical particles, a clear tendency of the influence of an increased

hopper angle on different particle shapes cannot be observed. Summarizing the results, clustered

particles predict a greater mass flow rate than polyhedra, which is also less influenced by a reduction

of the orifice size. The greater tendency of polyhedral particle shapes for bridging is again supported

by a velocity plot as shown in figure 15.

Figure 15: Vertical velocity distribution of spheres (top), CLS 1 – 3 (middle row) and PS 1 – 3

(lower row) for hopper design 6.

The polyhedral tendency for bridging is especially evident for cubic particles (PS 3), which

move rather in clumps than as single discrete elements. Moreover, the movement of the cubes in the

hopper is limited to the area above the orifice, in contrast to spheres and clustered particles. Similar to

the flat bottom and 30° hoppers, spheres and cluste red particles exhibit distinctive mass flow across

the hopper cross section. For this angle even polyhedral particles with only little angularity (PS 1) pre-

dict slightly less pronounced mass flow behavior, which is not unrealistic for this specific hopper de-

sign.

5.5 Arching

Bridging or arching is a well-known phenomenon in hoppers. Due to the micro-structure of the packed

bed as well as adhesion, particles can form an arch-like formation directly above the orifice, which can

lead to an unsteady mass flow out of the hopper or even a complete blockage of the orifice. In order to

compare the mass flows of different particle shapes, the orifice sizes in the simulations presented in

this paper are large enough to avoid significant arching. Nevertheless results for polyhedral particles

and in particular cubes show a higher tendency for arching, which can be observed especially for hop-

per angles of 30° and 60°.

6 Discussion

The discrete element method was used to study the granular flow within a hopper during discharge.

For this purpose six different hopper geometries were chosen with commonly used hopper angles and

orifice sizes. Material parameters for the granular material (polyethylene) were chosen according to

Yang and Hsiau [4] as well as Kruggel-Emden et al. [33]. The main focus of the study was a closer

inspection of the parameter “particle shape” in a hopper and its influence on granular flow. In particu-

lar, polyhedral particles and particles generated with the multi-sphere method of different shapes were

investigated and the results were compared with each other as well as with the results obtained for

spheres. In order to make the different particle geometries comparable, a constant radius was chosen

as proposed by Cleary and Sawley [14]. In contrast to Mack et al. [20], the vertices of the polyhedra

were not rounded for the sake of examining the effect of sharp edges on the flow properties. Due to

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the complexity of interpolyhedral contact detection, a review and amendment of the Common Plane

method was presented.

The improved contact point definition was quantified in DEM-simulations of hopper discharge.

Especially for simple polyhedral geometries the selected contact point definition had a considerable

influence on the discharge behavior of the granular material.

It was shown that particles with an increasing angularity reduce the mass flow rate from the

hopper and in case of the flat bottom hopper (α = 0°) increase the residual quantity after dischar ge. An

increased flow resistance could be observed which resulted in arch-like void structures above the ori-

fice of the hopper indicating bridging. Moreover angular particles predicted core flow with particle

movement mainly localized in the center of the hopper, while spheres and clustered particles predicted

mass flow behavior with significant movement reaching far to the side walls of the hopper.

As expected, an increase of the hopper angle increased the mass flow rate for all particles

shapes. Different tendencies for the influence of the hopper angle on particles with increasing angular-

ity were not detected. On the other hand a reduction of the orifice size led to a greater reduction of the

mass flow rate for non-spherical and especially polyhedral particles. Moreover, smaller orifice sizes

favor bridging of the hopper when filled with polyhedra of high angularity.

In all simulations significant differences between polyhedral and clustered particles were ob-

served. Clustered particles did not reduce the mass flow rate and increase the flow resistance in the

same degree polyhedral particles did. This can be attributed to multi-sphere particles having relatively

smooth surfaces while polyhedral surfaces have multiple sharp vertices and edges. Obviously the type

of particle shape approximation is another parameter that has to be considered in DEM-simulations of

hopper discharge.

Further experimental studies will be conducted to review the conclusions drawn in this paper.

With regards to hopper discharge, it is of interest to investigate the effect of different material proper-

ties and wall friction coefficients along with different particle shapes on the flow properties. Also the

influence of particle shape on the disposition for arching needs to be more thoroughly addressed.

Moreover we will examine particles with higher aspect ratio which may represent pellets or wooden

chips.

Acknowledgement

The current study has been funded by the German Science Foundation (DFG) within the project

SCHE 322/6-1. The authors would like to acknowledge the generous support.

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Table 1.

Table 2

Table 3

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Table 4

Table 5

Table 6

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Fig. 1

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Graphical absract

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Highlights

• Simulation of hopper discharge with multi-sphere and polyhedral particles.

• Reduction of mass flow with increasing particle angularity.

• Significant differences between multi-sphere and polyhedral particles.

• Improved contact point definition within the Common Plane algorithm for polyhedra.