The Discrete Element Method

The micromechanics of breakage and attrition of particulate systems can be simulated with a numerical method called Discrete Element Method (DEM), initially developed by Cundall and Strack (1979) for two-dimensional granular media. According to the Guo and Curtis (2015) review on DEM for complex granular flows, the principal DEM models to simulate the particle breakage and attrition are the Bonded Particle Model (BPM), the Fragment Spawing (FS) and the Attrition Prediction (AP). These three approaches are illustrated in Figure 2.12. In the Bonded Particle Model, a particle or grain (sphere, square, polygon) is composed of children rigid particles connected by bonds, through which the forces and moments are transmitted. The bonds, which can represent the real cement between grains in sedimentary rocks, break when the bond stress exceeds the material strength (Potyondy and Cundall, 2004). BPM was implemented in DEM to investigate the breakage of elongated particles (needle-shaped) in uniaxial compression tests (Grof et al., 2007), and the attrition of

42

squared particles in shear cell simulations (Potapov and Campbell, 1997a). Although the internal stress distribution and the fracture propagation between the sub-particles can be solved in BPM, this method is limited by the finite number of sub-particles forming each grain, implying a limited fragment size distribution. The Fragment Spawing method was therefore proposed to avoid a limited fragment size distribution (as in BPM) (Guo and Curtis, 2015). In FS, a mother particle is replaced by smaller siblings particles when the particle breakage occurs, and the number and size of the daughter fragments is decided by a breakage function. Despite the successful application of FS to pneumatic conveying and jet mills, this method strongly depends on empirical breakage functions and can be applied only to spherical mother particles (not yet to non-spherical particles) (Brosh et al., 2011). The third computational model of particle breakage, called Attrition Prediction, was presented by Hare et al. (2011) to predict the particle attrition (a small quantity of material worn down from the particle surface), in agitated particles beds. From DEM the stress and strains distributions of the bed could be estimated and then combined to an empirical relationship between extent of attrition and the prevailing stresses an strains. This empirical correlation was derived experimentally in shear cell tests (Ghadiri et al., 2000; Neil and Bridgwater, 1994). The AP approach is restricted to narrow particle size distributions of the bed, and should be applied only to systems with a relatively small extent of attrition, since the particle change during the process is not considered. In their DEM review, Guo and Curtis (2015), do not consider the cohesive particle models as computational modelling of particle breakage, but they cover that topic separately. The cohesive forces depend on the separation distance between the particles and become relevant for particle size less than 100 𝜇𝑚. The principal sources of particle cohesion are liquid bridges (capillary forces and viscous forces), electrostatic (Coulomb’s forces) and Van der Waals forces (expressed through interfacial or surface energy) and can be added in DEM as contact forces between particles. In this thesis, the Van der Waals forces are used to produce virtual agglomerates in DEM and their effect on the particle breakage is investigated.

43

Figure 2.12 Three major DEM models for particle breakage and attrition (Guo and Curtis, 2015).

The DEM code used in this study derives from the Cundall and Strack (1979) model of two- dimensional granular media, which was extended to three-dimensional spherical assemblies and named TRUBAL (Cundall, 1988). Later, the TRUBAL program was further extended at the Aston University to simulate particle agglomerates and it was re-named GRANULE. This software was able to model elastic, frictional, adhesive or non-adhesive spherical particles, with or without plastic deformation. Particles were treated as indestructible discrete entities and were capable to interact among themselves and with other elements, such as walls. The particle interactions were based on theories of contact mechanics. The frictional elastic particle interactions were described by the contact laws of Hertz (1882a) (Johnson, 1985) to model the normal force-dispalcement relationship. The Mindlin and Deresiewicz (1953a) theory was employed to model the non-linear tangential force-displacement relationship (Thornton, 1999). In presence of adhesive forces , the JKR theory (Johnson et al., 1971) used as extension of the Hertz (1882a) theory for frictional adhesive contacts. The tangential interactions in presence of adhesion were modelled combining the models of Savkoor and Briggs (1977) and Mindlin and Deresiewicz (1953a). A detailed description of the implementation of these models into DEM can be found in Thornton and Barnes (1986); Thornton and Ning (1998); Thornton and Randall (1988); Thornton and Yin (1991). A brief summary of the Hertzian model and the JKR model are presented here.

44

Considering two interacting particles, 1 and 2, with elastic moduli 𝐸₁ and 𝐸₂, with Poisson ratio 𝜈₁ and 𝜈₂, radii 𝑅₁ and 𝑅₂, the normal force-displacement relationship, 𝑃 − 𝛼 is given by: 𝑃 =4 3𝐸∗(𝑅∗)1 2⁄ 𝛼3 2⁄ (2) where _𝐸1_∗= 1−𝜈1 2 𝐸1 + 1−𝜈22 𝐸2 and 1 𝑅∗= 1 𝑅1+ 1

𝑅2 and 𝛼 is called the relative approach and it is related to the contact radius 𝑎 through the equation:

𝑎 = (𝛼𝑅∗₎1 2⁄ ₍₃₎

In case of autoadhesive particles, the JKR model provides a relationship between the contact force and the relative approach. The contact radius can be written as:

𝑎 = (3𝑅

∗_𝑃′

4𝐸∗ ) 1 3⁄

(4)

where 𝑃′ is called effective Hertzian force. This quantity is related to the adhesive force 𝑃_𝑐 and to the applied force 𝑃:

𝑃′ _{= 𝑃 + 2𝑃}

𝑐 ± √4𝑃𝑃𝑐+ 4𝑃𝑐2 (5)

The adhesive force is also called pull-off force and can be written as follows: 𝑃𝑐 =

3

2𝜋Γ𝑅∗ (6)

where Γ is the work of adhesion, often expressed with the Dupré equation, Γ = 𝛾₁+ 𝛾₂− 𝛾12, where 𝛾1 and 𝛾2 are the surface energies of the two solids and 𝛾12 is the interface energy

(Israelachvili, 2011). For the same material, 𝛾₁ = 𝛾₂ = 𝛾 and Γ = 2𝛾. From the contact area 𝑎 it is possible to derive the relative approach of the two spherical particles 𝛼, with the following relationship:

𝛼 =𝑎2 𝑅∗− √

2𝜋Γ𝑎

45

An increment in the relative approach Δ𝛼 corresponds to an increment in the normal force ΔP:

ΔP = 2𝐸∗_{𝑎 [}3√𝑃 − 3√𝑃𝑐

3√𝑃 − √𝑃𝑐

] Δ𝛼 (8)