A particle scale model based on a full two-way coupling of the Discrete Element Method (DEM) and Smoothed Particle Hydrodynamics (SPHs) methods is applied to SAG mills. Motion and collisions of resolved coarser particles within an SAG mill are performed by the DEM component. Fine particles in the feed combine with the water to form a slurry, which is represented by the SPH component of the model. Slurry rheology is controlled by solid loading and fine particle size distribution for each volume of slurry. Transport, dispersion, and grinding of the slurry phase particle size distribution are predicted by solving additional coupled advection–diffusion equations in the SPH component of the model. Grinding of the finer particles in the slurry due to collisions and shear of the coarser particles (rocks and grinding media) is achieved via the inclusion of population balance terms in these equations for each SPH particle. This allows prediction of the transport of both coarser and finer material within the grinding and pulp chambers of an SAG mill, including the discharge performance of the mill. This particle-scale model is used to investigate the relative performance (throughput, product size distribution, resident particle size distribution, net power draw, wear) for an SAG mill at a pilot scale and a 36 ft industrial scale. The 36′ SAG mill considered is a geometrically scaled-up version of the 1.8 m Hardinge pilot scale mill but with a longer belly length, reflecting current SAG mill design preferences. The belly lifters are scaled to a lesser degree with a larger number of lifters used (but still many fewer liners than would typically be used in a large SAG mill based on conventional liner selection rules). The model shows that despite reasonable qualitative similarities, many aspects of the charge structure, slurry transport, coarse particle and slurry discharge through the grates, and the collision energy spectra vary in important ways. This demonstrates that a near purely geometric scale-up of an SAG mill is not sufficient to produce a comparable performance at the two physical mill scales.

This paper presents a set of related methods for performing a computational geometry analysis of a cone crusher shape that provides geometric information for use in understanding cone crusher flows and breakage and their variation with key crusher geometric parameters. Algorithms that are able to calculate geometric measures such as the variation of the cross-sectional area available for flow at each height in the crusher, open and closed side areas, geometry based reduction ratios, eccentric throw as well as the open and closed side setting are described. These are demonstrated in a worked example of a specific short head cone crusher showing how systematic variation in the cone vertical displacement and eccentric angle affects geometric measures. Both the OSS and CSS are found to vary almost linearly with cone height and eccentric angle. The eccentric throw is found to be nearly invariant for changes in cone height but to vary strongly with eccentric angle. There is close to a linear relationship between variation in choke area (representing throughput) and the CSS (representing maximum product size) when the cone height is changed. In contrast, the choke area is almost independent of the CSS when the eccentric angle is varied meaning that throughput and breakage top size are independently controllable. Linear and area-based estimates of the reduction ratio can be used to bound the likely breakage performance. This analysis can be used in conjunction with DEM models to help to guide the evolution of liner configurations.

This second part paper explores rock breakage mechanisms, the life cycle of rocks in mills and the strong influence of end walls on charge motion within mills. We present recent advances in particle-based modelling of mills for comminution focused around wear and the effect of slurry and slurry phase grinding. Three mill scenarios are considered: 1. Media flow and the resulting wear evolution of the belly and end wall liners and the resulting change in mill performance for a full industrial scale dry ball mill (modelled using DEM) 2. Axial slurry transport and mixing in a wet overflow industrial scale ball mill (modelled using fully coupled DEM and SPH) 3. Effect of mill speed on slurry and solid charge motion and the resulting grinding of fine particles in a 1.8 m diameter wet Hardinge pilot mill (modelled using fully coupled DEM and SPH with advection-diffusion-population balance equations solved for the slurry size distribution for each SPH particle) These demonstrate the nature and level of fidelity that is now possible to include in particle-scale comminution models. They provide insights into the critical importance of curtain flows generated by the end walls of tumbling mills, on wear behaviour on liners, on the structure of slurry pools and mill discharge and on the adverse effect on grinding of increasing mill speed. Crown Copyright (C) 2020 Published by Elsevier Inc. All rights reserved.

A coupled DEM + SPH model can be used to predict the motion and breakage of resolved coarser particles within a SAG mill. The fine product from coarse particle fracture can then be included in the slurry phase modelled using SPH. This allows, in principle, the prediction of the breakage and transport of coarser material and the transport of the finer material within the grinding and pulp chambers of a SAG mill including discharge performance of the mill. It also allows the effect of the changing solids loading on the slurry theology to be included. In this paper we will explore the development of an extension of this model that also allows prediction of the grinding of the finer particles embedded in the slurry phase due to the collisions and shear of the coarser particles (rocks and grinding media). The size distribution of the slurry fines is discretised into a set of size fractions so that its change due to grinding can be tracked at each point in the slurry. This is formulated as a system of coupled advection-diffusion equations. An SPH discretisation of this system is then developed. The resulting coupled SPH ODE'S are solved using the SPH method in a way that is fully coupled to the DEM and SPH parts of the model. The proposed model includes a diffusive component that allows for the shear induced dispersion of the slurry size fractions and allows prediction of the spatial distribution of these fine size fractions within the slurry phase. The advection of the slurry is automatically accounted for by the motion of the SPH particles which is an important benefit of using the SPH method for such wet mill modelling. The local fine grinding behaviour arising from the coarse DEM resolved components of the charge are characterised at each location in terms of the local energy dissipation rate. This information is used in conjunction with a first order grinding law to predict the grinding of each slurry size fraction at each location in the mill due to the collisional action of the coarser particles. The ability of this new model to predict fine particle grinding and transport within the slurry phase is demonstrated for an industry standard 1.8 m diameter by 0.6 m long AG/SAG pilot mill.

Particle scale modelling of comminution processes can provide significant insight into the flow of particles, their breakage, the effect of slurry, wear and energy utilisation within these machines. The ability to use such models to assist in faster and lower cost design of new comminution devices and in the improvement of existing ones will be critical to the ability of industry to respond to the substantive challenges facing mineral processing in the next decade. These challenges are reviewed and drivers for change are discussed. Under-standing individual unit process performance needs to be in the context of the flowsheets in which they are used so this is also reviewed. Advances in particle based comminution modelling are presented with this work divided into two parts. This first part focuses on recent advances in particle based modelling of crushing. Three crusher types are used to demonstrate these capabilities: 1. Twin roll crusher 2. Cone crusher 3. Vertical Shaft Impactor (VSI) These show the nature and level of fidelity that is now possible to include in particle scale crusher models including breakage of non-spherical particles and prediction of the product size distribution and throughput. Crown Copyright (c) 2020 Published by Elsevier Inc. All rights reserved.

Coupled DEM + SPH models are able to predict the motion of the coarse particulates and the slurry phase in a SAG mill. An extension of this model is proposed in which breakage of coarser particles (which are resolved in the DEM sub-model) in accordance with the incremental damage theory is included. This allows direct prediction of the resident size distribution of the rock component of the charge. It also allows the finer unresolved progeny from the breakage to be added to the slurry whose rheological properties then become a dynamic prediction of the model via the evolving local solids density of the slurry. It also allows transport of the fine material within and from the mill to be predicted. This model is demonstrated for an industry standard 1.8 m diameter by 0.6 m long pilot SAG mill. It allows the nature of the damage accumulation by the rock particles and its linkage to the flow structure of the charge to be explored. A flux of particles rebounding from the liner in the impact zone of the mill colliding with cataracting material produces the strongest incremental damage. Damage is also generated by cascading material arriving in the toe region. The presence of a slurry pool restricts the opportunity for incremental damage from the rebounding particles in the impact zone by damping rebound in places where particles fall directly into slurry. Finally, the coupled DEM + breakage + SPH model enables a mechanistic linkage between the slurry properties and the rock breakage.

Applying DEM to prediction of tumbling mill performance is challenging because several different modes of breakage are active in the process. Here we use measured data from a well characterised ore in a well instrumented, 1.2 m diameter pilot scale mill to validate direct DEM prediction of particle size reduction. The key comminution mechanisms involved for a SAG mill are: (1) incremental breakage where parent particles break into progeny based on the cumulative energy absorption above the elastic damage threshold, (2) abrasion, and (3) chipping/rounding due to preferential contact and breakage of corners and edges of non-round particles. In this paper, a method for including incremental damage breakage in DEM is presented. The inclusion of all the size reduction mechanisms in the same DEM framework allows direct prediction of the evolution of the resident rock particle size and shape distributions and the product throughput rate. The surface mass loss mechanisms are shown to be critical for reducing the particle size to the point where the accumulation of incremental damage becomes significant leading to body breakage of these damaged particles. The energy split between ball and rock is also important for exceeding the elastic threshold and creating damage. Comparison of the predicted particle sizes at the completion of ten minutes of grinding operation with the measured experimental values from the pilot mill provides quantitative validation of the breakage predictions of this DEM breakage model.

Discharge of finer rock, pebbles, ball scats and slurry from mills and its flow through trommels and into other processing operations all affect the performance of overflow ball mills. Modelling of the coarser rock components and the grinding media is best done using the Discrete Element Method (DEM) while modelling of the slurry component is best done using a compatible particle method such as SPH (Smoothed Particle Hydrodynamics). This combination of methods allows both these critical components and their interactions to be included in flow models for the mill and discharge arrangements. Information from such models can be used to both understand the flows in these typically closed and data poor environments and to help optimise designs for improved performance and superior wear life. In this paper, a typical discharge/trommel arrangement for an overflow ball mill is analysed using this modelling approach with opportunities for use in process improvement discussed. In the mill grinding chamber the addition of slurry was found to lower the charge shoulder and toe positions due to drag forces on the media, as well as produce a large slurry pool above the toe. Slurry surging out of the grinding chamber carries mill product as well as a small amount of balls through the trunnion and into the trommel for classification. The trunnion spiral was found to be very effective at collecting and returning balls back into the mill grinding chamber. The rate of balls flowing into the trommel was found to be size-dependent favouring the discharge of smaller worn scats for removal as waste. Trammels with and without baffles were compared to establish linkages between transport and classification performance and trommel geometry. The inclusion of baffles inside the trommel was found to spread the slurry sideways and axially and provide improved screening characteristics. Crown Copyright (C) 2017 Published by Elsevier Ltd. All rights reserved.

Discrete Element Method (DEM) simulation with non-round particles and including breakage has been used to understand the breakage behaviour and operating performance of an industrial scale cone crusher using a representative ore. The breakage model uses a replacement strategy and impact energy specific progeny size data from a Drop Weight Test (DWT). There is a strong variation in the breakage behaviour with height in the compression region as the differing profiles of the concave and mantle create five different regions with monotonically decreasing width and differing degrees of convergence between the surfaces. These control the rate of motion and the ability to load and break the particles, and determine whether high forces are generated via multi-particle stress chains or as single particle loading directly from the liner surfaces. The larger feed particles jam in the compression zone prior to breakage and cause observable obstruction to the flow of finer material and strong non-uniformity in the flow of product down the lower part of the mantle. Trends in the coarseness of product and changes in steady state throughput are identified with changes in material properties (rock breakage energy and friction coefficient) and crusher operating parameters (Closed Side Setting and crusher rotation rate). Crown Copyright (C) 2016 Published by Elsevier Ltd. All rights reserved.

A computational model for use with DEM (Discrete Element Method) is proposed for the five mechanisms applying to particle comminution in AG, SAG and coarse feed ball mills. Chipping and rounding are mechanisms that lead to preferential mass loss from corners and edges of particles and which produce shape change for the particles as well as size reduction. These are controlled by the energy dissipation at a contact and the location of the contact on the surfaces of the particles. These mechanisms lead to rounding or conditioning of angular feed particles. Single impact body breakage is a very weak contributor to overall size reduction with most body breakage occurring via damage accumulation over many contacts. This incremental damage mechanism is inherently less inefficient but leads to size reduction from the many weak collisions experienced by particles within the mill. Finally, size reduction from abrasion is well represented by the shear energy absorption of the particles. Particle size and shape evolution due to chipping, rounding and attrition is demonstrated using a well characterised pilot mill for which detailed particle mass data is available. The relative contributions of the five mechanisms and a quantification of the wasted energy for both AG and SAG charges and for new and substantially comminuted material are reported. Changes in the energy spectra with the decreasing particle sizes over time are described with increases in the fraction of collisions above the elastic energy threshold leading to faster damage accumulation and reduced energy wastage. Finally, it is shown that the attribution of dissipated energy between particles in a collision is material dependent with significantly more energy absorbed by rock particles than balls. This needs to be accounted for in DEM models, particularly when attempting to explicitly predict particle size reduction. Crown Copyright (C) 2015 Published by Elsevier Ltd. All rights reserved.

Predictions of particle flow and compression breakage of non-round rock passing through an industrial scale cone crusher are presented. The DEM (Discrete Element Method) particle breakage model is generalised to allow non-round particles to be broken into non-round progeny. Particles are broken in this DEM model when the elastic energy of a contact is sufficiently high to initiate fracture. Progeny size distribution data from JKMRC Drop Weight Test (JKDWT) or JKMRC rotary breakage test (JKRBT) is used to generate the specific daughter fragments from each breakage event. This DEM model is able to predict the production of both coarser progeny which are resolved in the DEM model and finer progeny which are not. This allows the prediction of product down to very small sizes, limited only by the fineness of the fragments measured in the breakage characterisation. The predicted flow of material through the crusher, product size distribution and liner wear are discussed. The generalised breakage model demonstrated here is suitable for modelling all forms of crushers. Crown Copyright (C) 2015 Published by Elsevier Ltd. All rights reserved.

•The uniformity of heating for batch cavities is well described by S parameter.•S parameter was unable to fully describe heating uniformity for MW applicators.•New parameter (SP) was defined and compared to S parameter through FDTD simulations.•One possible application of SP parameter is to compare different MW applicators.