Discrete element method is a computational method that is used for predicting the flow of particulates in circumstances where their collisions are the dominant physical process. The ball motion in planetary ball mill is examined by DEM modelling during high-energy ball milling of a titanium carbide powder. The effect of velocities of vial and plate on milling efficiency has been studied.

Figure 1 shows the high-energy planetary ball mill (Fritsch Pulverisette 7 classic line, Idar-Oberstein, Germany) equipped with two vials used in this work. It has an external structure made of steel, which isolates the rotating parts from the external environment for protecting the operator from the risk of accidents from detached parts.

Moreover, computer simulations are a useful method to explore the milling process, providing information about kinematic and dynamic quantities [21]. In

this work, the high-energy planetary ball mill process is modeled using the software Automated Dynamic Analysis of Mechanical System MSC Adams [22]. Figure 2 illustrates a schematic of a high-energy planetary ball mill equipped with two cylindrical vials. The mill was powered by a 0.37 kW electric motor. The main geometric and operation parameters include: R_p is the distance between the global rotation axis and the rotation axis of the vial; R_v is the radius of the vial cylinder; ω and Ω are the angular velocities of the vial and plate, respectively. The direction of the vials revolution was set counter to that of the plate revolution. The geometrical and physical properties of the vial and ball used for mill are given in Table 1.

The velocities used in the force-displacement law are obtained based on Newton’s second law of motion applied to disc x:

m ( x ) x i • • = m d ν ( x ) d t = ∑ F ( x ) i (1)

I ( x ) θ i • • = d ω ( x ) d t = ∑ M ( x ) (2)

where m ( x ) represents the mass of a particle, ν ( x ) is the translational velocity, ω ( x ) is the angular velocity, I ( x ) is the moment of inertia of disc x and ( • ) denotes a time derivative.

In this work, the Hertz-Mindlin force model was used to resolve particles interactions [23] [24] [25]. We used the commercially available software Experts in Discrete Element Modeling (EDEM 2.4) [26].

Therefore, the total force acting on particle i is obtained by taking the summation of the above forces:

F = F N + F T (3)

where F N is the normal force component and F T is the tangential force.

F N = 4 3 E * R * δ N 3 / 2 − 2 5 6 ln e ln 2 e + π 2 2 E * m * ( R * δ N ) 1 / 4 ν N r e l (4)

where δ N is the normal particle overlap, e is the restitution coefficient, ν N r e l is the normal component of the relative velocity between two particles, E * , R * and m * are the equivalent Young’s modulus, equivalent radius and equivalent mass of a single grinding ball, respectively, which are defined as following:

E * = ( 1 − ν a 2 E a + 1 − ν b 2 E b ) − 1 (5)

R * = ( 1 R a + 1 R b ) − 1 (6)

and

m * = ( 1 m a + 1 m b ) − 1 (7)

where ν , E, R, m are the Poisson’s ratio, Young’s modulus, radius, mass of a particle a in contact with particle b, respectively.

The tangential force is a function of the elastic force and damping force.

F T = − 8 G * R * δ N δ T − 2 5 6 ln e ln 2 e + π 2 8 G * m * ( R * δ N ) 1 / 4 ν T r e l (8)

where G * is the equivalent shear modulus calculated from the following Equations:

G * = ( 1 − ν a 2 G a + 1 − ν b 2 G b ) − 1 (9)

G = E 2 ( 1 + ν ) (10)

The torque applying on a particle due to contact with a surface is given by:

τ a = − μ R F N R a ω a (11)

where μ R is the coefficient of rolling and ω a is the angular velocity.

Table 2 summarizes the simulation parameters and their values used in this study.

The model proposed by Barrios et al. [27] assumes that the particles can be modeled as a monolayer bed. If the particles have spherical shapes and arranged according to a dense hexagonal packing, the number of particles trapped in each collision as a function of radius could be estimated by:

N c a p , i = 1 4 + 3 4 ( 2 r c d i ) 2 for r c ≥ d i 2 (12)

N c a p , i = 1 for r c < d i 2

where r_c is the radius of the bed and d_i is the mean size of the particles caught by the colliding balls.

They also proposed that the radius of capture can be obtained by adding the radius of contact due to the elastic deformation during the contact, and the radius of contact caused by the geometry of media (balls) and particle powders.

r c = ( 15 E 8 K e ) 1 / 5 K g 2 / 5 + [ K g 2 − ( K g − h Δ h ) 2 ] 1 / 2 (13)

where K_e is the elastic constant of the contact is given by the following equation [28]:

K e = K 1 K 2 K 1 + K 2 (14)

and K_g is the geometric constant of the contact given by the equation:

K g = r c 1 r c 2 r c 1 + r c 2 (15)

K_i is elastic stiffness of each of the bodies in contact, r_ci is the radius of curvature and Δ is the maximum deformation of the bed during collision.

Therefore, in the case of monolayer bed the initial bed height may be estimated equal to the mean size of the particles caught by the colliding balls, d_i.

Thus, the ratio Δ/h represents the maximum relative deformation of the bed [27], given by:

Δ h = 0.362 [ 1 − exp ( − 0.152 ( E E 50 b ) 0.413 ) ] (16)

This model includes grinding media (two balls) and particles (TiC powder) since the DEM software records only collisions and contact involving two bodies. In this approach, only ball-particle collisions are extracted (see Figure 3). Thus, the presence of the particles represents the powder captured between grinding media. When the powder particles (TiC) were milled by large balls (the

case of our study), only some particles are collided with the balls because of the difference in size between balls and powder particles.

The most important ingredient in the model of a ball mill is the contact law. In this context, Gilardi et al. [29] have reported the existence of both models (discrete and continuous).

The contact force is defined by the hard-coded impact function [22], based on the non-linear spring (F_k) and linear damper scheme (F_d) proposed by Dubowsky and Freudenstein [30] from following relation:

F c = F k + F d = k u n + c u ˙ (17)

where u and u ˙ the relative displacement and velocity of the colliding bodies, k and c are the spring generalized stiffness and the damping coefficient. With respect to the above formulation, to prevent discontinuities, the impact function, defined within MSC Adams, implements a damping coefficient which depends on the relative displacement of colliding bodies, given by:

c = − c m ( u d ) 2 ( 3 − 2 u d ) (18)

c varies from zero to cm, assuming the latter value when the relative displacement is greater than or equal to d. With the exponent n = 3/2, an estimate of k can be derived from the Hertzian theory of contact [31], as given in the following relation:

k = 4 3 π ( 1 η i + η j ) ( R i R j R i + R j ) 1 / 2 (19)

for the case of contact between sphere i and j, with radius R, and

k = 4 3 π ( 1 η i + η j ) ( R i ) 1 / 2 (20)

for the case of contact between sphere i and a plane.

In fact, the parameter, accounts for elastic properties of materials, is given by the following equation:

η i = 1 − ν i 2 π E i (21)

where n is the Poisson ratio and E represents the Young modulus. The solution of the classical problem of damped vibration of a mass spring system provides a guess of maximum the damping parameter [30],

c m = − 2 ln ( e ) [ 1 ln 2 ( e ) + π 2 ] 1 / 2 ( m e k ∗ ) 1 / 2 (22)

being e = (h/h₀)^1/2 the restitution coefficient, accounting for the difference from initial (h₀) and final (h) height in a drop test, 1/m_e = 1/m_i + 1/m_j and k^* [32]:

k * = ( 320 81 m e u ˙ n 0 E e 4 R e 2 ) 1 / 5 (23)

Geometrical and elastic properties are expressed, respectively, as 1/R_e = 1/R_i + 1/R_j and 1/E_e = 1/E_i + 1/E_j and u ˙ n 0 is the relative velocity during the collision.