Powder Technology 386 (2021) 154–165
Contents lists available at ScienceDirect
Powder Technology
journal homepage: www.elsevier.com/locate/powtec
DEM investigation of SAG mill with spherical grinding media
and non-spherical ore based on polyhedron-sphere contact model
Changhua Xie a, Huaqing Ma a, Tao Song b,c, Yongzhi Zhao a,⁎
a
b
c
Institute of Process Equipment, Zhejiang University, Hangzhou, 310027, China
State Key Laboratory of Process Automation in Mining & Metallurgy, Beijing, 100160, China
Beijing Key Laboratory of Process Automation in Mining & Metallurgy, Beijing, 100160, China
a r t i c l e
i n f o
Article history:
Received 6 January 2021
Received in revised form 15 March 2021
Accepted 16 March 2021
Available online 18 March 2021
Keywords:
Polyhedron-sphere contact model
Non-spherical
DEM
SAG mill
Collision energy
a b s t r a c t
In the ore milling process, the ore particles are usually irregular polyhedrons, and the grinding media are often
spherical. The need for accurate simulation of this process calls for a contact algorithm between the polyhedron
and sphere, achieving a balance of accuracy, stability, and efficiency. For this purpose, we propose a polyhedronsphere contact model based on the deepest point method. Experiments and simulation studies by discrete element method (DEM) are conducted to test this model's accuracy in a hexahedral box and a lab-scale horizontal
drum with lifters. Then, systematic research is carried out for an industrial semi-automatic grinding (SAG) mill to
explore the effect of particle shape. To reveal the advantage over the traditional spherical DEM model, the
polyhedron-sphere (PH-SP) grinding system is compared with a pure spherical grinding system. Results are
discussed from the motion behavior particles, power consumption, collision energy on ore and liner, and liner
wear. The results show that the charging of particles in the PH-SP grinding system is blocked to a certain extent
by polyhedral particles, both the power consumption and energy utilization efficiency increase somewhat, the
large energy collision between ore and liner has increased significantly. The liner wear of the PH-SP grinding system is more severe than the pure spherical grinding system, which indicates that the new model is necessary for
more accurate prediction of the grinding process.
© 2021 Published by Elsevier B.V.
1. Introduction
Ball mill is widely used in ore particles' comminution process for its
large processing capacity, strong adaptability, and simple operation
[1–4]. The comminution process can be divided into coarse and fine
grinding according to ore particle products' size. In the coarse grinding
process, the ore particle input is irregularly polyhedral, and they are
ground into fine particles by the impact and abrasion of steel balls. In addition to experimental research [5,6], numerical simulation has been
widely applied in the study of the milling process with the development
of computational technology. Because it can not only monitor the charging behavior of particles inside the mill but also be used to predict the
wear of liner and grinding media accurately [7]. Among those numerical
simulation methods, the discrete element method (DEM) has been
widely employed in the simulation investigation of the milling process
[5,8–15] since it was proposed by Cundall and Stack [16]. Because it
can record detailed information of particles, and it is easy to be coupled
with other numerical calculation methods. In the early research on ball
mills, due to the imperfection of the non-spherical model and the
⁎ Corresponding author.
E-mail address: yzzhao@zju.edu.cn (Y. Zhao).
https://doi.org/10.1016/j.powtec.2021.03.042
0032-5910/© 2021 Published by Elsevier B.V.
limitation of calculation, most researchers focused on spherical particles
[3,4,8,17–19], which was of significant difference from the real nonspherical ore particles. Later, with the development of non-spherical
models such as multi-sphere [20–23], super-ellipsoids (or superquadrics) [24–28], and polyhedron [29–34], the influence of ore particle
shapes on the milling process gradually received attention.
In the study of Cleary [3], particles were modeled by the superellipsoids in 2D, and the influence of ball and rock shape on the charging
behavior and power draw was initially explored. Kiangi et al. [26] combined experiments and simulation to conduct similar research in a 3D
mill. Furthermore, Xu et al. [35] used a 3D super-ellipsoid model in a
SAG mill to systematically study the impact of particle shape on collision
energy and liner wear. Cleary et al. [35] explored the relationship between collision energy and abrasion of particles based on the superellipsoids. The shape of particles modeled by super-ellipsoid was
rounding during the milling process. Although the super-ellipsoid
method can model lots of different non-spherical particles, the ore
(coal, rock and so on) in the grinding process are often irregular
polyhedrons, and the super-ellipsoid cannot accurately describe these
particles. Peng et al. [36] used a sphere-clump method (also named
multi-sphere method) to model irregular iron ore, and compared the
sensitivity of particle charge behavior to the speed and lifter height. In
C. Xie, H. Ma, T. Song et al.
Powder Technology 386 (2021) 154–165
a PH and a SE terminates in one of six cases for each contact events.
Xie et al. [47] proposed a composite contact algorithm to solve the contact detection between the polyhedron and the super-ellipsoid, in
which the super-ellipsoid is converted into polyhedron when it contacts
with polyhedral particle. In this composite algorithm, the contact between super-ellipsoid is dealt with the deepest point method, and the
contact between polyhedrons or between polyhedron and superellipsoid are directly or indirectly processed by the overlapping volume
method.
Since the spherical particles can also be modeled by the superellipsoid method, the above composite contact algorithm is also applicable to the polyhedral-spherical particle system. However, the number of
faces of the polyhedron converted from a sphere is large. The combined
contact algorithm would not be efficient enough to deal with the contact detection between polyhedron and sphere. On the other hand, the
sphere is more special than other non-spherical particles modeled by
super-ellipsoid. The sphere's normal contact force always points to the
sphere's center, making it feasible to use the deepest point method to
calculate the contact between polyhedral and spherical particles. The
deepest point method can also be used to resolve the contact between
polyhedrons, and it has been proven to be accurate and efficient [31].
That means all the contact in the polyhedron-sphere particle system
(including PH-PH, PH-SP, and SP-SP) can be handled under a unified
contact theory framework rather than a composite contact algorithm
contains two or more contact models.
We proposed a contact algorithm suitable for all contact types in the
polyhedral-spherical particle system in this paper. Experiments and
simulations were performed for particle packing in a box and particle
motion in a lab-scale ball mill with lifters to verify the algorithm's accuracy. After that, a systematical comparison was carried out to investigate
the influence of spherical ore and actual irregular polyhedral ore on the
mill's operating performance in an industrial SAG mill.
their research, irregular polyhedral particles can be modeled by the
sphere-clump method, and the number of clumps can be adjusted to
achieve different shape representation accuracy. Although nonspherical particles can be simplified as composite spheres because this
method is efficient and can describe many phenomena of
non-spherical particles [37]. However, in some cases that need accurate
simulation, such as scientific research about granular matter or some industrial applications need high prediction accuracy, a more precise
model is required. Liu et al. [34] proposed a dilated polyhedron model
with the characteristics of both sphere and polyhedron, thereby converted the contact search problem between polyhedrons into an optimization problem between the envelope functions of the dilated
polyhedron. Based on the principle of energy conservation principle
for elastic contact, Feng et al. [29–31] proposed a theoretical framework
suitable for polygons (2D) and polyhedrons (3D) of arbitrary shapes, in
which a potential function restricts the normal contact. In the past, most
researchers used this method to calculate the contact forces and torques
between polyhedrons, which has been proven to be a stable and effective method. For example, Govender et al. compared the difference of
polyhedral and spherical particles on the processes, including the
discharging in a hopper [38] and a blast furnace [39], the mixing of particles [40] in a rotating drum and a four-blade mixer [41].
When we discuss the simulation of a granular system via DEM, the
particle's shape and the contact detection and the calculation of forces
and torques related to it are two of the critical issues that need to be
considered. According to the characteristics of particle shape [42], particles can generally be divided into different categories, which includes
smooth continuous surface particles (like sphere, ellipsoid, superellipsoid, etc.), composite particles (such as multi-spheres, bonded
spheres, multi-super-ellipsoids [43], etc.), combined surface particles
(polyhedron, sphero-polyhedron, etc.), and so on. In the past decades,
most researchers focused on particle systems with the same shape
type to simplify the problem, such as spherical particle system, ellipsoidal particle system, or polyhedral particle system, in which particles of
different shapes (varies in aspect ratio or diameter) can still be modeled
in the same way. However, in the ball mill which we need to investigate
here, the ore particles are irregular polyhedrons with vertexes and
sharp edges, while the grinding media are spherical steel balls. Considering that different shape representation methods have unique advantages in modeling particles of specific shapes, it is more feasible to
develop a contact model suitable for different types of particles rather
than a shape model that can describe all particles. For example, the
polyhedron method may describe irregular ore particles with minimal
elements and significant computational efficiency. At the same time,
the traditional spherical model can efficiently and accurately model
spherical particles. Therefore, it is of great practical significance to develop a model to handle the contact between polyhedron and sphere
to veritably simulate the milling process of ore particles inside the
ball mill.
The calculation will be more complicated when dealing with contact
between polyhedrons and other shapes. For example, when a polyhedron is in touch with a sphere or a particle modeled by superellipsoid, the overlapping body may be a complex geometry with both
continuous and discrete surfaces rather than a polyhedron, and the effort to calculating the overlap volume may significantly increase.
Wachs et al. [44] used the Gilbert–Johnson–Keerthi (GJK) algorithm
[45] in the contact detection between polyhedrons and tried the possibility of applying this algorithm to the contact detection of particles of
arbitrary shape. In the lasted study of Feng et al. [32], both the GJK
and expanding polytope algorithm (EPA) were used to solve the particle
system containing multiple particle types. The results showed that although they can handle the contact between polyhedrons well, both
the GJK and EPA is less effective for smooth-surfaced shape. Peng et al.
[46] presented a contact detection algorithm for a convex polyhedron
(PH) and a super-ellipsoid (SE), where the contact detection between
2. Mathematical model
2.1. Particle motion equations
In DEM simulation, the motion of particles follows Newton's law of
motion, and the corresponding equations can be written as:
dv
¼ mg þ ∑F c ,
dt
ð1Þ
dω
¼ ∑T c ,
dt
ð2Þ
m
I
where m and I are the mass and inertia tensor of the particle, respectively; dv/dt and dω/dt are the translational acceleration and angular acceleration of the particle, respectively; g is the gravitational
acceleration. Fc and Tc correspond to the contact force and contact
torque, respectively. The Spring-dashpot contact model proposed by
Cundall and Strack [16] is adopted in this paper. It includes normal contact force Fc,n, normal contact torque Tc,n produced by Fc,n, tangential
contact force Fc,t, and tangential contact torque Tc,t produced by Fc,t.
Among them, the contact forces can be calculated by equations as
follows:
F c ¼ F c,n þ F c,t ,
ð3Þ
F c,n ¼ −kn δn −ηn vn ,
ð4Þ
F c,t ¼ −kt δt −ηt vt ,
ð5Þ
where kn and kt are the normal and tangential spring stiffness, respectively; δn and δt correspond to the displacement between two particles
(or a particle and a wall) in the normal and tangential direction, respectively. ηn and ηt refer to the normal and tangential damping coefficient,
155
C. Xie, H. Ma, T. Song et al.
Powder Technology 386 (2021) 154–165
respectively, which can be calculated from the restitution coefficient. vn
and vt are the relative velocities of two particles (or a particle and a wall)
in the normal and tangential direction, respectively. When the following
relationship is satisfied,
jF c,t j> f s jF c,n j,
the spherical particles. The polyhedral method is used to model the
polyhedral particle, which can be obtained by importing an STL file
modeled by any 3D modeling software.
Contact detection between particles is carried by the deepest point
method, which includes the contact detection between polyhedrons
(PH-PH), between spheres (SP-SP), and between polyhedron and
sphere (PH-SP). Since SP-SP contact has been described in detail in
many works of literature [2,16], the contact model in this section only
concerns the contact of PH-PH and PH-SP. The contact is figured out in
a 2D schematic diagram to simplify the description, where polyhedrons
are simplified as polygons and spheres are simplified as circles. In the
case of 2D, PH-PH contact can be divided into vertex-vertex contact
and vertex-edge contact. The contact of PH-SP can be divided into
vertex-edge contact and edge-edge contact, as shown in Fig. 1. In the
deepest point method, the contact between particles is obtained by
the following steps:
ð6Þ
the Coulomb friction model is adopted to calculate the tangential contact force as follows:
F c,t ¼ − f s jF c,n j
δt
,
jδt j
ð7Þ
where fs is the sliding friction coefficient.
The normal contact force cannot produce torque for the spherical
particle, and only the tangential contact force can produce torque. However, when the particle is non-spherical, the normal and tangential contact force may produce torques as follows:
T c ¼ L ðF c,n þ F c,t Þ,
1) finding the deepest points;
2) determining the direction of normal force;
3) determining the contact point and calculating the overlap.
ð8Þ
where L is the distance vector from the particle center to the contact
point. Rolling friction is neglected because its effect is small for the polyhedral particles.
Unlike the spherical particles, the calculation of the inertia moment I
and the contact torque Tc are more complicated for the non-spherical
particles because of the asymmetry of geometry. The inertia moment
is time-varying due to the rotation of non-spherical particle in the global
coordinate system, and it has off-diagonal non-zero entries, viz.:
0
1
Ixx
−Ixy −Ixz
B
C
Iyy
−I yz A
I ¼ @ −Iyx
ð9Þ
−Izx −Izy
Izz
2.2.1. Contact of PH-PH
As shown in Fig. 1a, when two polygons (PH-1 and PH-2) are in
vertex-vertex contact, a new polygon composed of intersection (g and
h) and vertices (p and q) will appear. Here, p is the deepest point inside
PH-1, and q is the deepest point inside PH-2. The unit vector (n) of the
normal direction force at point p is the sum of the normal vector of
the adjacent edges, which can be written as follows:
n¼
n 1 l1 þ n 2 l2
,
‖n1 l1 þ n2 l2 ‖
ð12Þ
where n1 and n2 are the normal unit vector of edge pg and ph, respectively; l1 and l2 correspond to the length of edge pg and ph, respectively. While the normal unit vector (n') at the deepest point q can
also be calculated in the same way, and it satisfies n' = − n. Once
the deepest points are determined, the lines (dotted line in Fig. 1a)
perpendicular to the unit normal vector's direction can be obtained.
The overlap between PH-1 and PH-2 (δ) is converted into the distance
0 0 I0z
between the dotted line, and the centroid of polygon phqg (pc) is the
0Z 1
y02 þ z02 dV
0
0
contact point.
B
C
Z
B
C
When PH-1 and PH-2 are in vertex-edge contact, the overlapping
02
B
C
02
¼ ρB
0
0
x þ z dV
ð10Þ
C
part contains a vertex (point p) and two intersections (point g and h),
C
B
Z
@
02
A
as shown in Fig. 1b. The unit vector (n) of the normal direction force
0
0
x þ y02 dV
at point p is calculated by Eq. 12, and the intersection (point q) of the reverse extension of n and the edge gh is the deepest point of PH-2 (point
After calculating the contact torques in the global coordinate system,
q). Since the normal unit vector (n') at point q is perpendicular to edge
a rotation matrix will be used to convert them from the global coordigh, the distance between point p and q is the overlap, and the centroid of
nate system to the local coordinate system, which can be written as:
triangle phg is the contact point.
0
1
cosα cosφ− sinα cosβ sinφ
sinα cosφ þ cosα cosβ sinφ
sinβ sinφ
@
A ¼ − cosα sinφ− sinα cosβ cosφ − sinα sinφ þ cosα cosβ cosφ sinβ cosφ A;
sinα sinβ
− cosα sinβ
cosβ
To calculate the inertia moment of a polyhedral particle, a local coordinate system that the inertia moment contains only non-zero entries in
its diagonal is introduced here, and the inertia moment in the local coordinate system can be written as:
0 0
1
Ix 0 0
0
0
I ¼ @ 0 Iy 0 A
ð11Þ
where α, β, and φ are Euler angles, which correspond to nutation angle,
precession angle, and spin angle. Then, the angular acceleration can be
determined in the local coordinate system. It is transformed back to
the global coordinate system in the last step by an inverse rotation matrix A−1. More details about non-spherical particles' modeling using
DEM can be seen in the literature [27,48–51].
2.2.2. Contact of PH-SP
As shown in Fig. 1c, A circle SP-1 is in contact with a polygon PH-2 in
the form of edge-vertex contact. The circle SP-1 can be described as F(x,
y) = 0, and points satisfied F(x, y) < 0 are inside SP-1. Among these
points, the vertex of PH-2 (point p) meets the smallest F(x, y) value,
so it is the deepest point inside SP-1. The direction of the normal force
at point p always points to the center of SP-1, and the intersection of
the reverse extension line of n and the SP-1 is the deepest point inside
PH-2 (point q). The distance between point p and q is the overlap, and
the middle point of them is the contact point pc. When the circle SP-1
contacts with the polygon PH-2 in the form of edge-edge contact, the
2.2. Contact model
There are spherical and polyhedral particles in the particle system
studied in this paper. The traditional spherical model is used to model
156
C. Xie, H. Ma, T. Song et al.
Powder Technology 386 (2021) 154–165
Fig. 1. Schematic diagram of the contact in 2D: (a) vertex-vertex contact between polygons, (b) vertex-edge contact between polygons, (c) edge-vertex contact between a circle and a
polygon, and (d) edge-edge contact between a circle and a polygon.
contact detection will be simpler, as shown in Fig. 1d. The way to find
the deepest point inside SP-1 is similar to the edge-vertex contact of
PH-SP, and the only difference is that the deepest point is on the edge
of the PH-2.
Once the deepest points and the normal direction of contact force
are determined, the contact forces and torques can be calculated similarly to traditional DEM.
the start and end time of the contact, respectively. Fc,t,ij is the tangential
contact force between particle i and particle j, vt,ij is the relative tangential velocity between them. vi is the velocity of particle i. According to
the research of Finnie et al. [52], the plastic flow pressure is set to 1.5
times the Vickers hardness in this paper. More details about the SIEM
can be found in our previous works [48,49,53].
2.3. Erosion model
3. Validation
Shear Impact Energy Method (SIEM) is used to obtain the liner wear,
which connects the shear impact energy with the volume removed
from the surface. The following equations are used to describe this
model:
The contact algorithm's accuracy between polyhedral and spherical
particles is verified by combining the experimental and simulation research of particle packing in a hexahedral box and particle motion in a
lab-scale ball mill with lifters. In the experiments, the polyhedral particles are made by 3D printing with resin by SLA (Stereo Lithography Apparatus), and the printing precision is ±0.1 mm. The spherical particles
are produced by PMMA. The density of both resin and PMMA is about
1190 kg·m−3. Here, the polyhedral particles are of the same shape
and size, an irregular polyhedron with 12 faces (40 triangle faces), as
shown in Fig. 2. The equivalent diameter of polyhedral particles is
13.75 mm, and the diameter of the spherical particle is 18 mm. In the
simulation, the elastic modulus of resin and PMMA are 2.5 GPa and
3.15 GPa, respectively; the Poisson's ratio of resin and PMMA are 0.41
and 0.35, respectively. The restitution coefficient between resin is
0.65, the restitution coefficient between PMMA is 0.75, and that between resin and PMMA is 0.70. The sliding friction coefficient and the
rolling friction are 0.3 and 0.001, respectively. The time step is
Vp ¼
Eshaer
4:0p
ð13Þ
Z t1
Eshear ¼ −
F c,t,ij ∙vt,ij dt
t0
ð14Þ
The shear impact energy in Eq. 13 will be accumulated to calculate
wear only when both Fc, t, ij ∙ vt, ij < 0 and Fc, t, ij ∙ vi < 0 are satisfied. In
these equations, Vp is the volume removed from the surface in the plastic deformation caused by shear impact energy, and Eshear is the shear
impact energy. p is the plastic flow pressure of the target surface, and
it is generally 1–5 times the Vickers hardness. t0 and t1 correspond to
157
C. Xie, H. Ma, T. Song et al.
Powder Technology 386 (2021) 154–165
Fig. 2. Polyhedral particle (a) without mesh and (b) with triangle mesh.
5 × 10−5 s, which is smaller than the critical time step calculated by the
natural oscillation periods [54].
Firstly, we compared the packing height of the polyhedron-sphere
system in a hexahedral box of 80 × 80 × 200 mm. As shown in Fig. 3,
the number of spherical and polyhedral particles is set to 0–300,
50–250, 100–200, and 140–160 to ensure the total number of particles
is consistent. Due to the randomness of particle orientation in packing,
this process has been repeated three times both in the experiment
and numerical simulation. Fig. 4 shows the average packing height,
and it can be seen that the packing height in the experiment and simulation is quite close.
Besides, we compared the motion behavior of polyhedral and spherical particles in a lab-scale rotation mill. The mill is also made by 3D printing with resin by SLA, and it is 300 mm × 40 mm in diameter and length.
There are 18 lifters inside the mill, the angle of the lifter face is 14°, the
height of the lifter is 10 mm, and the width of the top plate is 10 mm.
The mill's rotation speeds are set to 10 rpm, 20 rpm, 30 rpm, 40 rpm,
50 rpm, and 60 rpm to test this contact algorithm's stability. In the experiment, after 300 polyhedral particles and 80 spherical particles are put inside the mill, it rotates for 20 s to reach a quasi-steady state. Then, it runs
for 10 s to record the charging behavior, and a high-speed camera captures the mill motions. In the simulations, particles are randomly generated in the mill and settled by gravity. The simulation data are recorded
with a time interval of 1 s. As shown in Fig. 5, the motion behavior of
particles in the mill at different speeds is of significant difference. When
the mill rotates at 10 rpm, the particle bed's slumping occurs; as the
speed increases to 30 rpm, the transverse motion turns into cascading;
while the transverse motion will be converted into cataracting when
the rotation speed increases to as large as 60 rpm. As we can see, the particles' motion behavior in the simulation is in good agreement with that
in the experiment, which shows the accuracy of the contact algorithm between sphere and polyhedron proposed in this paper.
Furthermore, the shoulder and toe positions at different rotation
speeds in the experiment and simulation are compared. The definition
of the specific positions in a ball mill proposed by Cleary et al. [55] is
used, as shown in Fig. 6. The head is the highest point where the liner is
still in contact with the particles, and the shoulder is the top position of
the kidney-shaped charge. The impact toe is the top position of the region
where particles impact the liner directly, and the bulk toe is the end of the
kidney-shaped contiguous main body of the charge. In the experiment,
shoulder and toe positions are marked in the snapshots taken by a
high-speed camera. Same with the simulation, ten snapshots with a
physical time interval of 1 s were used. The position data are then converted into the angles on the mill's inner surface, and the average angle
of each rotation speed is figured in Fig. 7. Same as the experiment, the
shoulder and toe position in simulations is also obtained by the snapshots
of motion behavior. It can be seen that the height of both the shoulder and
toe increases with the rotation speed, which is consistent with the result
Fig. 3. Packing of particles in a box of 80 × 80 × 200 mm in experiment and simulation, and the number of spherical and polyhedral particles are set to (a) 0–300, (b) 50–250, (c) 100–200,
and (d) 140–160.
158
C. Xie, H. Ma, T. Song et al.
Powder Technology 386 (2021) 154–165
Fig. 4. Comparison of average packing height in experiment and simulation.
in Fig. 5. Moreover, the positions of the shoulder and toe in the simulation
are close to the experimental results. That is, the simulation can predict
the positions of the shoulder and toe quite accurately.
A comparison in simulation time between pure polyhedron DEM
and sphere-polyhedron DEM was also carried out in the mixing process
of 300 polyhedral and 80 spherical particles insides the lab-scale mill to
show the algorithm's efficiency. In the pure polyhedron case, balls were
Fig. 6. Schematic diagram of the definition of specific locations in the mill.
modeled by the polyhedron method with 20 × 10 slices (360 triangular
facets in total) in longitude and latitude. Under this condition, the pure
polyhedron particle system's simulation time is about 1.82 times that of
Fig. 5. Snapshots of 300 polyhedral particles and 80 spherical particles in a lab-scale ball mill rotated at 10 rpm, 30 rpm, and 60 rpm in the experiment and DEM simulation.
159
C. Xie, H. Ma, T. Song et al.
Powder Technology 386 (2021) 154–165
Fig. 7. Positions of (a) the shoulder and (b) the toe in the experiment and DEM simulation at different rotation speeds.
and spherical grinding media (SP grinding system). The grinding media
are steel balls with a constant diameter of 125 mm. The size distribution
of ore particles is referred to as the setting in the research of Cleary et al.
[3], and a cutoff of 15 mm is used to ignore the effect of fine ore powder.
Within each diameter range, the diameter of the ore particle is randomly
distributed. More detailed information about the grinding media and ore
can be found in Table 1. Only ore particles with equivalent diameters
larger than 75 mm are polyhedral to reduce the computational effort.
Such a setting is reasonable because the shape of the small size ore particles in the real mill gradually tends to be spherical under the action of
abrasion. The size of the mill is 9.75 × 4.88 m in diameter and length, as
shown in Fig. 8. A slice of 0.5 m in length is used here to reduce the number of particles, and periodic boundaries are set at the two ends. All the
simulations are carried out via our self-developed code (DEMSLab 4.0)
on a workstation. Except for the ore particles' shape, all the simulation
parameters are same in the two cases. More detailed parameters about
the device and simulation can be found in Table 2. After all the particles
are randomly generated and settled by gravity, the mill rotates for four
revolutions counterclockwise with a speed of 10.5 rpm to reach the
quasi-steady state. During this process, it's worth noting that the PH-SP
particle system's simulation time is only about 1.85 times that of the SP
Table 1
Size of ore particles and grinding media.
Type
Diameter (mm)
Amount
Volume fraction (%)
Mass fraction (%)
Ore
15–25
25–35
35–45
45–75
75–105
105–150
125
298,410
66,227
23,616
6683
2051
513
2500
25
18
15
15
15
12
–
54.49
Media
45.51
the polyhedron-sphere particle system, which means the proposed
algorithm is quite efficient.
4. Solution and simulation condition
The simulations are carried out in an industrial-scale SAG mill, which
includes a case with polyhedral ore particles and spherical grinding
media (PH-SP grinding system), and a case with spherical ore particles
Table 2
Simulation parameters in an industrial SAG mill.
Fig. 8. Schematic diagram of the SAG mill.
160
Parameters
Value
Parameters of the liners
Vickers hardness
Length of the liners/m
Height of the lifters/m
The angle of the lifters face
NO. of lifters
HV370
0.5
152
14°
60
Parameters of the simulation
Shape of particles
Density of particles/kg·m−3
Vickers hardness
Coefficient of restitution between ore particles
Coefficient of restitution between ore and steels
Coefficient of restitution between balls and liners
Coefficient of sliding friction
Normal spring constant kn/N·m−1
Tangential spring constant kt/N·m−1
Timestep/s
Polyhedron, Sphere
4500 (ore), 7800 (media)
HV160 (ore), HV370 (media)
0.3
0.5
0.8
0.5
2 × 107
8 × 106
10−5
Parameters of the mill
Rotation speed Ω/rpm
Ore filling level
Media filling level
10.5
20%
15%
C. Xie, H. Ma, T. Song et al.
Powder Technology 386 (2021) 154–165
Fig. 9. Snapshots of charge behavior in SAG mill of (a) SP particle system, (b) PH-SP particle system.
lifted to a higher position and obtain larger gravitational potential energy. The flow of particles in the mill containing polyhedral ore is
more sluggish than the mill containing only spherical particles. This
phenomenon is most apparent in the region around the bulk toe,
where spherical particles continue to fill the bulk toe under the action
of gravity. In contrast, the polyhedral particles cannot load the bulk
toe in time due to blockage between particles, which results in a pit in
the bulk toe.
Also, considering that the main difference in the motion behavior of
irregular polyhedral particles and spherical particles is rotation, the average angular velocities of ore particles in different sizes during the milling process are compared here, as shown in Fig. 10. In general, the ore
particles' angular velocity in the PH-SP grinding system is smaller than
that of the SP grinding system, which means the spherical particles
are more easily to rotate. In particular, this difference gradually decreases as the diameter of the ore particles increases. It is also worth
noting that although the particles smaller than 75 mm in the PH-SP
grinding system are still spherical, their angular velocity is also significantly reduced by the influence of irregular polyhedral particles. During
the milling process, spherical particles are more likely to slide and be
projected from the shoulder than non-spherical particles. The decrease
in angular velocity is an essential feature of the PH-SP grinding system
because it affects the shoulder's height and is closely related to ore particles' abrasion during the milling process.
particle system. After that, it runs at 10.5 rpm for four more revolutions to
obtain the simulation results.
5. Results and discussion
5.1. Charge behavior
As the SAG mill enters a stable operation state, the grinding system's
charging behavior in the PH-SP and SP grinding systems is different. As
shown in Fig. 9, the shoulder and the head in the SAG mill containing
polyhedral ore particles are slightly higher than that of the SAG mill containing only spherical particles. It means polyhedron particles can be
5.2. Energy consumption
Besides, the power consumption of the PH-SP and SP grinding systems is discussed. As shown in Table 3, when part of the ore particles
is irregular polyhedrons, the total power increases about 6.7% compared
to that of spherical ore particles, from ~834 kW to ~890 kW. When the
collision type is considered, it is easy to see that all collision types' specific power consumption slightly increases. Among them, the increment
in the power consumption of collision between ore and media is the
largest, which is sufficient for the ore particles' breakage. The increased
power consumption in media-media and media-liner collisions also
means more severe wear the steel balls and the liner.
Fig. 10. The average angular velocity of ore particles in different particle size ranges in the
milling process.
161
C. Xie, H. Ma, T. Song et al.
Powder Technology 386 (2021) 154–165
Table 3
Power consumption of different types of collision.
Type of collision
PH-SP
SP
Power/kW
Percent/%
Power/kW
Percent/%
Ore-Ore
Ore-Media
Ore-Liner
Media-Media
Media-Liner
Total
641.55
72.02
634.05
76.01
216.96
24.36
177.24
21.25
12.04
1.35
9.62
1.15
18.30
2.06
12.35
1.48
1.86
0.21
0.90
0.11
890.71
100
834.16
100
Fig. 11. Energy spectra of the collision on ore in PH-SP and SP particle system: (a) frequency distribution of different collision energies, and (b) energy dissipation rate at each collision
energy.
influence of ore shape. Among the collisions happened in the mill, collisions on ore directly affect the breakage and abrasion of ore, including
the ore-ore, ore-media, and ore-liner collision. The collision energy on
ore is discussed from the relationship between collision energy and collision frequency and energy dissipation rate, as shown in Fig. 11a and
Fig. 11b. It can be seen from Fig. 11a, the collision frequency first increases with the collision energy and then decreases after reaching the
maximum frequency, and the collision with the energy of about
0.001 J has the maximal frequency. The collision frequency is quite different in the PH-SP and SP grinding systems, and there are two dividing
points in the frequency curve. When the collision energy is smaller than
~3 × 10−5 J or larger than ~0.6 J, the collision frequency on the ore in the
PH-SP grinding system is higher than that of the SP grinding system;
when the collision energy is between ~3 × 10−5 J and ~ 0.6 J, the collision frequency in the PH-SP grinding system is smaller than that of
the SP grinding system. As we look through the relationship between
energy dissipation rate and collision energy, as shown in Fig. 11b, it
Furthermore, there are also apparent differences in the energy proportions of different collisions in the PH-SP and SP grinding systems.
Table 3 shows that the collision energy ratio between ore particles in
the PH-SP grinding system is about 72%. It is a significant decrease compared to 76% in the SP grinding system. Correspondingly, the percentage
of collision energy between ore and grinding media increases from
21.25% to 24.36%. Since the steel ball's kinetic energy is much larger
than that of the ore particles, it is generally believed that steel balls' impact mainly causes ore particles' breakage. Therefore, replacing polyhedral ore particles with spheres will lead to a conservative prediction of
power consumption and particle breakage.
5.3. Collision energy on ore
Since we are studying a complex grinding system containing many
ore particles with different diameters and velocities in a SAG mill, the
energy spectrum analysis is an important way to understand the
Fig. 12. Energy spectra of different types of the collision on ore in PH-SP and SP particle system within the collision energy of (a) 10−12 J – 3 × 10−5 J, (b) 3 × 10−5 J – 0.6 J, and
(c) 0.6 J – 1000 J.
162
C. Xie, H. Ma, T. Song et al.
Powder Technology 386 (2021) 154–165
Fig. 13. Energy spectra of the collision on the liner in PH-SP and SP particle system: (a) frequency distribution of different collision energies, and (b) energy dissipation rate at each collision
energy.
Fig. 12c, the collision frequency between ore-ore and ore-media in the
SP grinding system is similar in Range C (> 0.6 J). However, in the PHSP grinding system, ore-media collisions' frequency is significantly
higher than that of ore-ore collisions. Besides, since the particles in the
PH-SP grinding system are lifted to a higher position, the number of
large energy collisions in the toe area is significantly larger than that
in the SP grinding system.
would be seen that the two dividing points are the same as the collision
frequency. The collision energy corresponding to the maximal energy
dissipation rate is about 0.01 J.
Also, collisions within different energy ranges are figured out according to their collision types to understand the collision on ore in
the PH-SP and SP grinding system, including the ore-ore, ore-media,
and ore-liner collision. It can be seen from Fig. 12a that the collision frequency of ore-media and ore-liner is equivalent in Range A (10−12 J –
3 × 10−5 J), and they are much smaller than that of ore-ore. Within
this energy range, the collision frequency of ore-ore and ore-media in
the PH-SP grinding system is larger than that of the SP grinding system,
and the difference between them narrows as the collision energy increases. As shown in Fig. 12b, in Range B (3 × 10−5 J – 0.6 J), ore-ore collisions' decisive role weakens as collision energy increases. The
frequency of the three types of ore-related collisions in the PH-SP grinding system is smaller than that of the SP grinding system. Among them,
the difference in collision frequency of ore-ore and ore-media collisions
first increase and then decrease, and that of ore-liner collisions stabilizes
at a relatively large level. In the SAG mill, the collisions with large energy
acting on the ore mainly occur in the toe area, where the dropped steel
balls and ore with high kinetic energy hit the ore and liner. As shown in
5.4. Collision energy on the liner
Fig. 13 figures out the collision frequency and the liner's energy dissipation rate, including the collision of ore-liner and collision of medialiner. As shown in Fig. 13a, the main difference of collision frequency on
the liner in the PH-SP and SP grinding systems mainly exists within the
energy range of 10−7 J and 1 J. Within this range, the collision frequency
on the liner in the SP grinding system is significantly larger than that in
the PH-SP grinding system. Collision energy with a maximum frequency
in the SP grinding system is about 2 × 10−6 J, and the maximal collision
frequency is about 2 × 104 Hz. However, collision energy with a maximum frequency in the PH-SP grinding system is about 6 × 10−7 J, and
the maximal collision frequency is about 1.5 × 104 Hz. It means that
once irregular polyhedrons model the ore particles, both the collision
frequency and collision energy between particles (both ore and
media) and liners reduce. The decrease in collision frequency and collision energy is caused by the blockage effect of polyhedral particles in
the space between two adjacent lifters, which protects the liner from
particles' impact and reduces the relative velocity between the particles
and the liner. As shown in Fig. 13b, the main difference in the liner's energy dissipation rate also exists in the range of 10−7 J and 1 J. The energy
dissipation rate within this range of the PH-SP grinding system is lower
than that of the SP grinding system.
5.5. Liner wear
In this section, the liner's wear calculated by SIEM is compared and
analyzed in the PH-SP and SP grinding systems, and it is an average
value of 60 groups wear data. As shown in Fig. 14, the maximal wear
rate occurs in the lifter's upper-right edge because the SAG mill rotates
counterclockwise. The difference in wear rate distribution between PHSP and SP grinding systems mainly exists in the lifter's top plate, where
the liner's wear rate in the PH-SP grinding system is larger than that of
the SP grinding system. It first increases with the position, reaches the
maximum at the middle of the lifter's top surface, and then gradually
decreases to zero at the lifter's right inclined plane. When we model
Fig. 14. Distribution of the wear rate on the liner of SAG mill in PH-SP and SP particle
system.
163
C. Xie, H. Ma, T. Song et al.
Powder Technology 386 (2021) 154–165
Fig. 15. During the milling process, (a) the average number of different types of particles, and (b) the average kinetic energy of each particle in the toe area near the liner.
almost the same in simulation and experiment. The simulation could accurately predict the position of the shoulder and toe at different rotation
speeds. The computational efficiency of the contact detection algorithm
is also tested, and the calculation time of the polyhedron-sphere particle
system is obviously shorter than that of the pure polyhedron particle
system.
Modeling the ore particles by the spherical model instead of the
polyhedral model will lead to a noticeable deviation in the prediction
of power consumption and liner wear. The angular velocity of the ore
particles in the PH-SP grinding system is smaller. Both the power consumption and energy utilization rate in the PH-SP grinding system are
larger than those in the SP grinding system. The ore particles and the
grinding media are lifted to a higher height, which leads to more large
energy collisions at the toe area, and finally results in more severe
liner wear.
the ore particles with a spherical method, the prediction value of liner
wear would be smaller than the actual ones.
Although the collision energy on the liner in Section 5.4 can reflect
the liner's wear to some extent, this phenomenon has not been effectively explained because of the low frequency of high energy collision.
As we know, most wear behavior occurs in the toe area, where the particles thrown from the shoulder hit the liner violently. For this reason,
the average number and average kinetic energy of steel balls and ore
particles with large diameters (d ≥ 75 mm) within this area are analyzed. Since the SAG mill's inner diameter is 4.8 m and the lifter's height
is 0.152 m, we concern about the area near the wall, where the distance
to the center is larger than 4.4 m. The toe area is the space where the
tangent angle in the x-y plane is between 205° and 245°. As shown in
Fig. 15a, the average number of both Ore 75–105 (ore particles with a
diameter between 75 mm and 105 mm) and steel balls appearing in
the toe area in the SP grinding system are larger than that in the PHSP grinding system, while the average number of Ore 105–150 is almost
the same. When we look through the average kinetic energy of each
particle appearing in the toe area, as shown in Fig. 15b, it can be seen
that the average kinetic energy of ore particle increases with the diameter. The average kinetic energy in the PH-SP grinding system of both
Ore 105–150 and steel ball is significantly larger than that in the SP
grinding system. The results show that although there are fewer steel
balls and large-diameter ore in the toe area of the PH-SP grinding system, each particle's kinetic energy is larger. In particular, the steel ball's
kinetic energy in the PH-SP grinding system is significantly higher than
that of the SP grinding system, and the steel ball in the toe area is the
main reason for liner wear and ore breakage. As for that, the liner
wear in the PH-SP grinding system is more severe than that in the SP
grinding system.
CRediT authorship contribution statement
Changhua Xie: Conceptualization, Writing - review & editing, Data
curation, Visualization, Writing - original draft. Huaqing Ma: Writing review & editing, Resources. Tao Song: Validation, Writing - review &
editing. Yongzhi Zhao: Supervision, Methodology, Software, Writing review & editing.
Declaration of Competing Interest
The authors declared that they have no conflicts of interest to this
work. We declare that we do not have any commercial or associative interest that represents a conflict of interest in connection with the work
submitted.
6. Conclusion
Acknowledgment
A contact detection algorithm based on the deepest point method is
proposed in this paper to deal with the contact in particle system containing both polyhedral and spherical particles. Both validation and application researches are carried out to study this algorithm, and some
conclusions have been obtained.
A comparison between experiments and simulations has verified the
polyhedron-sphere contact algorithm's accuracy in calculating the contact and collision between polyhedral and spherical particles. For the
particle packing in a box, the packing height in simulation is basically
same as that of the experiment. In a lab-scale horizontal drum containing polyhedral and spherical particles, particles' motion behavior is
The authors would like to acknowledge the support provided by
the National Key Research and Development Program of China
(2019YFC1805600), and the support provided by the State Key Laboratory of Process Automation in Mining & Metallurgy and Beijing Key Laboratory of Process Automation in Mining (KY20192069000002).
References
[1] C. Suryanarayana, Mechanical alloying and milling, Prog. Mater. Sci. 46 (2001)
1–184.
[2] B.K. Mishra, R.K. Rajamani, The discrete element method for the simulation of ball
mills, Appl. Math. Model. 16 (1992) 598–604.
164
C. Xie, H. Ma, T. Song et al.
Powder Technology 386 (2021) 154–165
[30] Y.T. Feng, K. Han, D.R.J. Owen, An energy-based polyhedron-to-polyhedron contact
model, Conference on Computational Fluid and Solid Mechanics, 3rd M.I.T. 2005,
pp. 210–214.
[31] Y.T. Feng, K. Han, D.R.J. Owen, Energy-conserving contact interaction models for arbitrarily shaped discrete elements, Comput. Method Appl. M. 205 (2012) 169–177.
[32] Y.T. Feng, Y. Tan, On Minkowski difference-based contact detection in discrete/discontinuous modelling of convex polygons/polyhedra algorithms and implementation, Eng. Comput. 37 (2020) 54–72.
[33] G. Ma, W. Zhou, X. Chang, W. Yuan, Combined FEM/DEM modeling of triaxial compression tests for rockfills with polyhedral particles, Int. J. Geomech. 14 (2014)
4014014–4014025.
[34] L. Liu, S. Ji, A fast detection algorithm based on the envelope function of dilated polyhedron, Sci. Sin.-Phys. Mech. Astron. 49 (2019) 1–15.
[35] L. Xu, K. Luo, Y. Zhao, J. Fan, K. Cen, Influence of particle shape on liner wear in tumbling mills: a DEM study, Powder Technol. 350 (2019) 26–35.
[36] Y. Peng, X. Ni, Z. Zhu, Z. Yu, Z. Yin, T. Li, S. Liu, L. Zhao, J. Xu, Friction and wear of liner
and grinding ball in iron ore ball mill, Tribol. Int. 115 (2017) 506–517.
[37] Y. Guo, C. Wassgren, W. Ketterhagen, B. Hancock, B. James, J. Curtis, A numerical
study of granular shear flows of rod-like particles using the discrete element
method, J. Fluid Mech. 713 (2012) 1–26.
[38] N. Govender, D.N. Wilke, P. Pizette, N.E. Abriak, A study of shape non-uniformity and
poly-dispersity in hopper discharge of spherical and polyhedral particle systems
using the blaze-DEM GPU code, Appl. Math. Comput. 319 (2018) 318–336.
[39] N. Govender, D.N. Wilke, C. Wu, U. Tuzun, H. Kureck, A numerical investigation into
the effect of angular particle shape on blast furnace burden topography and percolation using a GPU solved discrete element model, Chem. Eng. Sci. 204 (2019) 9–26.
[40] N. Govender, R. Rajamani, D.N. Wilke, C. Wu, J. Khinast, B.J. Glasser, Effect of particle
shape in grinding mills using a GPU based DEM code, Miner. Eng. 129 (2018) 71–84.
[41] N. Govender, D.N. Wilke, C. Wu, R. Rajamani, J. Khinast, B.J. Glasser, Large-scale GPU
based DEM modeling of mixing using irregularly shaped particles, Adv. Powder
Technol. 29 (2018) 2476–2490.
[42] K. Dong, C. Wang, A. Yu, A novel method based on orientation discretization for discrete element modeling of non-spherical particles, Chem. Eng. Sci. 126 (2015)
500–516.
[43] Z. Liu, Y. Zhao, Multi-super-ellipsoid model for non-spherical particles in DEM simulation, Powder Technol. 361 (2019) 190–202.
[44] A. Wachs, L. Girolami, G. Vinay, G. Ferrer, Grains3D, a flexible DEM approach for particles of arbitrary convex shape - part I: numerical model and validations, Powder
Technol. 224 (2012) 374–389.
[45] E.G. Gilbert, D.W. Johnson, S.S. Keerthi, A fast procedure for computing the distance
between complex objects in three space, IEEE J. Robot. Auto. 4 (1988) 1883–1889.
[46] D. Peng, K.J. Hanley, Contact detection between convex polyhedra and
superquadrics in discrete element codes, Powder Technol. 356 (2019) 11–20.
[47] C. Xie, T. Song, Y. Zhao, Discrete element modeling and simulation of non-spherical
particles using polyhedrons and super-ellipsoids, Powder Technol. 368 (2020)
253–267.
[48] Y. Zhao, L. Xu, J. Zheng, CFD-DEM simulation of tube erosion in a fluidized bed,
AICHE J. 63 (2017) 418–437.
[49] L. Xu, Q. Zhang, J. Zheng, Y. Zhao, Numerical prediction of erosion in elbow based on
CFD-DEM simulation, Powder Technol. 302 (2016) 236–246.
[50] Y. You, M. Liu, H. Ma, L. Xu, B. Liu, Y. Shao, Y. Tang, Y. Zhao, Investigation of the vibration sorting of non-spherical particles based on DEM simulation, Powder
Technol. 325 (2018) 316–332.
[51] H. Ma, Y. Zhao, Investigating the flow of rod-like particles in a horizontal rotating
drum using DEM simulation, Granul. Matter 20 (2018) 15.
[52] I. Finnie, Erosion of surfaces by solid particles, Wear 3 (1960) 87–103.
[53] Y. Zhao, H. Ma, L. Xu, J. Zheng, An erosion model for the discrete element method,
Particuology 34 (2017) 81–88.
[54] J.M. Ting, B.T. Corkum, Computational laboratory for discrete element
geomechanics, J. Comput. Civ. Eng. 6 (1992) 129–146.
[55] P.W. Cleary, P. Owen, Effect of particle shape on structure of the charge and nature
of energy utilisation in a SAG mill, Miner. Eng. 132 (2019) 48–68.
[3] P.W. Cleary, Predicting charge motion, power draw, segregation and wear in ball
mills using discrete element methods, Miner. Eng. 11 (1998) 1061–1080.
[4] P.W. Cleary, Modelling comminution devices using DEM, Int. J. Numer. Anal.
Methods 25 (2001) 83–105.
[5] B.K. Mishra, R.K. Rajamani, Simulation of charge motion in ball mills. Part 1: experimental verifications, Int. J. Miner. Process. 40 (1994) 171–186.
[6] S. Rosenkranz, S. Breitung-Faes, A. Kwade, Experimental investigations and modelling of the ball motion in planetary ball mills, Powder Technol. 212 (2011) 224–230.
[7] D. Boemer, P. Ponthot, A generic wear prediction procedure based on the discrete
element method for ball mill liners in the cement industry, Miner. Eng. 109
(2017) 55–79.
[8] B.K. Mishra, R.K. Rajamai, Simulation of charge motion in ball mills. Part 2: numerical simulations, Int. J. Miner. Process. 40 (1994) 187–197.
[9] B.K. Mishra, C. Thornton, An improved contact model for ball mill simulation by the
discrete element method, Adv. Powder Technol. 13 (2002) 25–41.
[10] P.W. Cleary, R. Morrisson, S. Morrell, Comparison of DEM and experiment for a scale
model SAG mill, Int. J. Miner. Process. 68 (2003) 129–165.
[11] A.B. Makokha, M.H. Moys, M.M. Bwalya, K. Kimera, A new approach to optimising
the life and performance of worn liners in ball mills: experimental study and DEM
simulation, Int. J. Miner. Process. 84 (2007) 221–227.
[12] Y. Luo, H. Liu, L. Jia, W. Cai, Modeling and simulation of ball mill coal-pulverizing system, 6th IEEE Conference on Industrial Electronics and Applications, IEEE 2011,
pp. 1348–1353.
[13] P.R. Santhanam, E.L. Dreizin, Predicting conditions for scaled-up manufacturing of
materials prepared by ball milling, Powder Technol. 221 (2012) 403–411.
[14] S. Beinert, C. Schilde, G. Gronau, A. Kwade, CFD- discrete element method simulations combined with compression experiments to characterize stirred-media
mills, Chem. Eng. Technol. 37 (2014) 770–778.
[15] Y. Bai, F. He, Modeling on the effect of coal loads on kinetic energy of balls for ball
mills, Energies 8 (2015) 6859–6880.
[16] P.A. Cundall, O.D. Strack, A discrete numerical model for granular assemblies,
Géotechnique 29 (1979) 47–65.
[17] L. Xu, K. Luo, Y. Zhao, Numerical prediction of wear in SAG mills based on DEM simulations, Powder Technol. 329 (2018) 353–363.
[18] B.K. Mishra, C.V.R. Murty, On the determination of contact parameters for realistic
DEM simulations of ball mills, Powder Technol. 115 (2001) 290–297.
[19] R.M. de Carvalho, L.M. Tavares, Predicting the effect of operating and design variables on breakage rates using the mechanistic ball mill model, Miner. Eng. 43-44
(2013) 91–101.
[20] S. Naik, R. Malla, M. Shaw, B. Chaudhuri, Investigation of comminution in a Wiley
mill: experiments and DEM simulations, Powder Technol. 237 (2013) 338–354.
[21] P. Boehling, G. Toschkoff, S. Just, K. Knop, P. Kleinebudde, A. Funke, H. Rehbaum, P.
Rajniak, J. Khinast, Simulation of a tablet coating process at different scales using
DEM, Eur. J. Pharm. Sci. 93 (2016) 74–83.
[22] Y.X. Peng, T.Q. Li, Z.C. Zhu, S.Y. Zou, Z.X. Yin, Discrete element method simulations of
load behavior with mono-sized iron ore particles in a ball mill, Adv. Mech. Eng. 9
(2017).
[23] Y. You, Y. Zhao, Discrete element modelling of ellipsoidal particles using superellipsoids and multi-spheres: a comparative study, Powder Technol. 331 (2018)
179–191.
[24] P.W. Cleary, Charge behaviour and power consumption in ball mills: sensitivity to
mill operating conditions, liner geometry and charge composition, Int. J. Miner. Process. 63 (2001) 79–114.
[25] P.W. Cleary, DEM prediction of industrial and geophysical particle flows,
Particuology 8 (2010) 106–118.
[26] K. Kiangi, A. Potapov, M. Moys, DEM validation of media shape effects on the load
behaviour and power in a dry pilot mill, Miner. Eng. 46-47 (2013) 52–59.
[27] G. Lu, J. Third, C. Müller, Discrete element models for non-spherical particle systems:
from theoretical developments to applications, Chem. Eng. Sci. 127 (2015) 425–465.
[28] Y. Zhao, L. Xu, P.B. Umbanhowar, R.M. Lueptow, Discrete element simulation of cylindrical particles using super-ellipsoids, Particuology 46 (2019) 55–66.
[29] Y.T. Feng, D.R.J. Owen, A 2D polygon/polygon contact model: algorithmic aspects,
Eng. Comput. 21 (2004) 265–277.
165