Modelling discrete fragmentation of brittle particles

Abstract

A novel discrete fragmentation method (DFM) for spherical brittle particles using the discrete element method (DEM) has been developed, implemented and validated. Trajectories of individual fragments can be studied from the moment of breakage where the progeny might originate from incremental, simultaneous and/or repetitive fragmentation events. A particle breaks depending on the applied dynamic impact forces from collisions, the damage history, the particle size and material properties. This 3D model requires setting parameters solely dependent on the particle material and is consequently independent of any empirical value. Mass, momentum and energy is conserved during breakage. A theoretically consistent description from the onset of fragmentation to the cloud formation after breakage is provided and model outcomes have been compared to experimental results and other model predictions where very little deviation has been encountered. All material parameters have been varied independently to study the sensitivity of the model under dynamic fragmentation of numerous particles in a semi-autogenous mill.

Introduction

Particle size reduction occurs in many engineering applications and requires a good understanding when it comes to process design optimisation. The discrete element method (DEM) is particularly adaptable for a variety of fragmentation applications and has been applied to particle comminution [1], blasting [2] and attrition [3]. Fragmentation can also be reasonably well categorised based on the degree of breakage which increases with the amount of strain energy [4], [5] and strain rate [5], [6] supplied. DEM-models for static (slow) stresses are widely applied in biaxial (2D) or triaxial (3D) tests for single global particles [7], [8], [9] and for a few crushable agglomerates [10], [11]. DEM-modelling of dynamic impact-induced fragmentation has been introduced by Potapov et al. [4] and has been brought forward to model particle breakage in tumbling mills [12]. In the latter model, particles are glued together by tetrahedra elements which detach during potential fragmentation. Inherently, this approach is computational expensive and seems to be unfeasible when it comes to scale-up or modelling of repetitive fragmentation.

Almost all discrete element models describing fragmentation, adopt an agglomeration framework [1], [2], [3], [4], [5], [6], [7], [8], [9], [10], [11], [12], where each parent particle consists of a number of smaller child particles. During the course of a simulation, the child particles can be separated from the parent particle to represent a fragment. The advantage of this approach is its simplicity in concept and implementation. The drawback of the approach are the limitation in the number of particles and the size distribution of fragments, as all fragments must consist out of the initial particles.

An alternative to the agglomeration framework is the discrete fragmentation method (DFM). In DFM models the number and size of the progeny is derived at the moment of breakage and replaces the parent particle. The advantage of this approach is the flexibility to produce any number and size distribution of fragments. The drawback is the increased complexity in describing the parent and children particles. The discrete fragmentation method combines accuracy and efficiency to cope with high particle numbers being favoured to model most applications containing granular materials. As far as the authors know, Cleary [13] is the only one who introduced a discrete fragmentation model. He stated that the actual rules used in his code are still crude (e.g. mass is not necessarily conserved) and progress beyond fragmentation involving high speed balls in cataracting streams is desired. No further detailed description of his model has been published lately.

This paper presents a full description of a DFM model for implementation into other soft-sphere DEM models. The onset of fragmentation is modelled by using a breakage probability which considers incremental impact breakage by summation of accumulated damage. In principle, this model can be provided with any particle size distribution (PSD) – however, a breakage index t₁₀ – a single value to determine the entire PSD has been used. This approach offers several advantages over others as it depends on material parameters only. Furthermore, it has been proven to be valid for multiple impact breakage and for many brittle materials. Discrete fragments are created depending on the given PSD and packed randomly into their parent particle volume with a minimum child–child particle overlap. Each fragment is assigned with a kinetic (velocity) component derived from the momentum conservation and an elastic (spring force) component derived from the energy equation. Material parameters for the discrete fragmentation model have been varied for sensitivity studies in a semi-autogenous mill and results are obtained according to our expectations. Furthermore, the model has been tested and validated with experimental data and other model predictions. The CFD code MultiFlow (www.multiflow.org) is used which has been successfully applied in previous CFD—soft-sphere DEM models [14], [15], [16]. A deterministic soft-sphere collision model is required for implementing the DFM code introduced in this paper.