Effect of particle shape on particle breakage inside rotating cylinders

. We study the inﬂuence of particle shape on the evolution of particle breakage process taking place inside rotating cylinders. Extensive particle dynamics simulations taking into account the dynamics of the granular ﬂow, particle breakage, and polygonal particle shapes were carried out. We ﬁnd that the rate of particle breakage is faster in samples composed of initially rounder particles. The analysis of the active ﬂowing layer thickness suggests that for samples composed of rounder particles a relatively lower dilatancy and higher connectivity lead to a less curved free surface proﬁle. As a result, rounder particles rolling down the free surface have a higher mobility and thus higher velocities. In consequence, the faster breakage observed for rounder initial particles is due to the larger particles kinetic energy at the toe of the ﬂow.


Introduction
In numerous applications e.g. food engineering, pharmaceutical, mining, bio-fuels manufacture, powder metallurgy, tumbling devices are used to perform several industrial processes such as milling, grinding, drying, granulation [1][2][3]. Drum rotation implies particle flow and it can also lead to desired or undesired particle fragmentation. Therefore, the continuous evolution of the particle size and shape together with the randomness of the initial particle shape that one can find in nature, render particle breakage inside tumbling devices a complex subject that needs a multiscale detailed analysis.
The rotating drum flow configuration presents a continuous, dense free surface flow. Such conditions make this configuration suitable for studying the rheology of granular flows [4]. Indeed, several research works have focused on the flow characterization [5][6][7][8] and some have studied the e↵ect of the particle shape [9][10][11][12], but there is few work on the e↵ect of particle breakage during the process [13].
This work focuses on the influence of the initial particle shape on the grinding process taking place inside rotating cylinders. Discrete-element simulations applying the Bonded-cells method were performed in 2D to model the dynamics of particle breakage. To capture the e↵ect of particle shape, several samples composed of regular polygons but increasing number of sides were tested under identical system configurations.
The roundness increases with the number of sides and the particles approach a circular shape. The e↵ect of roundness on particle breakage is not intuitive. Indeed, particle breakage is initiated mainly at vertex-side contacts where high stress concentration is expected, and this ⇤ e-mail: franck.radjai@umontpellier.fr should lead to lower probability of particle breakage as the roundness increases. On the other hand, when grinding occurs as a result of dense granular flow, rounder particles exhibit larger mobility and hence more energetic collisions that may enhance particle breakage. The grinding evolution was tracked through specific surface measurements (i.e. sum of the particles perimeters normalized by the total mass).
In the following, the simulation methodology and sample characteristics are introduced. Then, the grinding evolution will be presented and analyzed.

Simulation Procedures
Numerical 2D simulations were performed by means of the contact dynamics method together with the Bondedcell method (BCM) that deals with particle breakage. Contact Dynamics is a discrete-element method in which perfectly rigid particles interact through frictional contacts, and the particle motions are calculated by a step-wise implicit scheme [14][15][16][17][18]. A particularity of Contact Dynamics is that the velocities and contact forces are simultaneously calculated through an iterative process accounting for the contacts as unilateral constraints. From the obtained solution, the contact network and the particles positions are updated.
In the framework of BCM, a Voronoï tessellation is applied to each particle in order to subdivide them into smaller independent elements called cells. The average cell size d cell is fixed such that each primary particle has around 20 cells, but the cell shapes are random. As a consequence, the particle volume is exactly equal to the sum of cell volumes [19][20][21][22][23]. In the generated sample, interfaces between adjacent cells belonging to a same particle are side-side, and each contact is represented by two cohesive bonds. However, other types of contacts such as vertex-side and vertex-vertex are represented by a single cohesionless contact point. The breakage of a cohesive bond takes into account two criteria: 1) A tensile stress threshold C n in the normal contact direction and a shear stress threshold C t in the tangential direction. The relative movement between two cells in contact is forbidden as long as the stress remains below the corresponding threshold. 2) A fracture energy W that must be consumed by the relative displacements at the stress threshold as in the classical fracture mechanics. Further details about the breakage model can be found in [13].
For the sake of geometric consistency, the external shapes of the particles are polygonal, as the cells. We considered di↵erent samples composed of initial particles with a given number of sides n sides ; See fig. 1. The sample particle size follows a uniform distribution in particle volumes in a range between d min 0 and d max 0 . The generated sample is deposited inside a hollow drum (ring in 2D) of internal radius R under the action of gravity g (See Fig. 1). A constant angular velocity ! = 3.14 rad/s is applied to the drum until a full breakage state is achieved (i.e. the fragments size reach d cell ). The Froude number Fr= ! 2 R/g, common for all tested systems, has a value of 0.21 and a cascading flow regime is obtained. The values of all parameters considered in this work are presented in Table 1.

Particle breakage
Since no grinding media (as in ball mills) was considered, particle breakage is a mere consequence of collisions between particles and with the drum walls. The grinding can be characterized through the evolution of specific surface area S , defined as particle surface per unit mass, presented in Fig. 2. The linear-like monotonic evolution reflects the surface generation during particle breakage. Indeed, once a particle is broken the cohesion is not restored, so that nor re-agglomeration neither healing processes are considered  in the model. Once a fragment size reach d cell , no further fragments can be generated so that S gradually tends to a plateau.
In the linear regime, the specific surface increases all the more faster that the number of sides is larger. This is what we observe more clearly in Fig. 3, in which the grinding rateṠ /S 0 is presented as a function of n sides . The grinding rate is the slope of the linear trend presented as dashed lines in Fig. 2.

Active layer
The active layer or flowing layer is the portion of the granular bed that flows in the opposite direction to the imposed wall velocity (i.e. red portion in Fig. 4). The thickness of the the active layer h act is measured in the velocity profile calculated at the mid-chord length from the free surface in the direction perpendicular to the flow. In Fig. 5 we observe a monotonic increase on the average active layer thickness regarding n sides . This means that for samples  composed of rounder particles an increasing amount of particles flow down the slope. This is also a signature of the higher mobility that these samples present as compared to samples composed of more angular particles. Note that in Figs. 3 and 5 the values for the configurations with n sides = 6 increase drastically, so that they are out of the trend followed by the rest of the data. This behavior is due to the long-range correlations enhanced by the regular shape. Such irregularities in the flow are more present in samples with n sides = 5 and n sides = 6 and they vanish for rounder particles. On the other hand, the samples composed of more angular particles present steeper free surface profiles (e.g. a more pronounced S-shape, see Fig. 6). The microstructure may also play a key role, that we investigate in more detail below.

Connectivity
The connectivity of particles is linked to several rheological properties such as packing fraction and macro- y/D n sides =5 n sides =6 n sides =7 n sides =9 n sides =12 n sides =25 Figure 6. Free surface profiles for each sample of unbreakable polygons with di↵erent number of sides (n sides ).
scopic friction coe cient [24]. The coordination number, a typical measurement of the connectivity, is defined as Z = 2N c /N p , where N c is the number of interactions and N p the number of particles. Each interaction involves only two particles, so that Z represents the number of touching neighbors per particle irrespective of the nature (side-side, side-vertex, vertex-vertex) of the contacts. The partial coordination number or connectivity number Z c is defined in a similar way but by counting each side-side contact as two simple contacts. This is because a line contact is equivalent to two constraints concentrated at two points belonging to that line whereas a vertex-side contact is a point constraint. Fig. 7 displays the time evolution of Z and Z c (inset) for samples composed of unbreakable polygons with different values of n sides . We see that samples composed of rounder particles exhibit smaller values of Z, and Z c shows larger changes as the initial particle shape gets more angular. This means that angular particles are more tightly connected and these samples present lower volume change. This is consistent with the free surface shape for di↵erent   Figure 7. Evolution of the particle connectivity Z and Z c of nonbreakable samples of regular polygons with di↵erent shapes.
values of n sides (See Fig. 6), where rounder particles samples present a flatter profile. The e↵ect of roundness as analyzed here can be compared to the cushioning e↵ect observed in polydisperse flows [25]. Smaller particles in the pores between larger particles lead to a higher coordination number of large particles, and hence a more isotropic distribution of stresses, which reduces their breakage rate. In the case of a rotating drum and during flow, a lower connectivity together with higher mobility leads to higher particle breakage.

Discussion and conclusion
In this paper, we studied the e↵ect of the initial roundness of particles on the particle breakage evolution during grinding inside rotating drums. Simulations of breakable samples composed of polygons of di↵erent shapes were performed in the framework of DEM coupled with the Bonded-Cell Method. The results show that the initial particle shape plays a role on the grinding rate inside rotating drums. In particular, we found that the grinding occurs faster for samples of rounder initial particles.
The monotonic increment of the active layer thickness with the initial particle roundness means that there is a better mobility for rounder particles down the slope. Due to their higher mobility, rounder particles flow easier leading to higher velocities and higher collision forces. Also, rounder particles are better connected and therefore, a lower sample dilatancy and less steep free surface profiles are exhibited during the flow. These two features (i.e. higher mobility and lower dilatancy) seem thus to be at the origin of larger particle fragmentation inside rotating cylinders.
To sum up, the increasing grinding rate found for samples composed of rounder initial particles can be explained by the fact that rounder particles flow easier down the slope, show lower dilatancy and are better connected. These flow features lead to higher velocities and collision forces at the toe of the flow and thus higher likeliness of particle breakage. These features hold also in 3D and we therefore expect also higher grinding rate in 3D, which needs to be checked either with experiments or 3D simulations.