Modeling breakage rates of coarse particles in ball mills Luís Marcelo Tavares * , Rodrigo M. de Carvalho Department of Metallurgical and Materials Engineering, Universidade Federal do Rio de Janeiro – COPPE/UFRJ, Cx. Postal 68505, CEP 21941-972, Rio de Janeiro, RJ, Brazil article info Article history: Received 13 August 2008 Accepted 23 March 2009 Available online 28 April 2009 Keywords: Ball mill Modeling Particle breakage abstract Breakage rates of coarse particles in ball mills generally follow non-first-order kinetics and the distribu- tion products from batch milling are often characterized by significant contributions of abrasion besides breakage by impact, which are not well described using traditional size–mass balance formulations. Under such conditions, particles are often subject to impacts of insufficient magnitude to produce break- age in each stressing event, so that they are broken by a combination of abrasion and impact and also particles undergo weakening due to unsuccessful stressing events. The paper presents a mathematical model of batch grinding which takes into account the distribution of stressing energies in the mill, the distribution of fracture energies of particles contained in the charge, describing breakage by impacts from grinding media producing catastrophic breakage, abrasion and weakening from repeated impacts. The model has been applied to describe the rate of disappearance of two materials in batch grinding with good results. Ó 2009 Elsevier Ltd. All rights reserved. 1. Introduction Most of the research in comminution is dedicated to tracking down the variation of the property of interest, namely particle size, along a comminution circuit or individual size reduction machine. The assumption that material within a given size range has identi- cal breakage behavior in a milling circuit is advantageously used in techniques such as the size–mass balance model. In some situa- tions, however, this assumption proves to be unrealistic. For in- stance, it is well known in plant practice that the mill circulating load is often more difficult to grind than the fresh feed. This has been demonstrated by Tavares and King (1998), who showed that fracture energies of particles coarser than about 0.5 mm were higher in the discharge when compared to the fresh feed of a ball mill grinding taconite. This is an indirect evidence of the lower breakage rates of tougher particles inside the mill, which survive the stressing events more readily and, therefore, build up in the cir- culating load. In fact, several researchers (Kapur et al., 1997; Aus- tin, 2004) have recognized that the widely invoked assumptions of time-invariant breakage rates and energy-independent breakage distribution functions used in traditional size–mass balance model formulations cannot be reconciled with evidence from single-par- ticle breakage data, except for resorting to some kind of gross sta- tistical averages. Although it is possible to account for non-first-order breakage rates by using empirical descriptions of the breakage process (Aus- tin et al., 1982), these are highly artificial and result in even more parameters that must be estimated from the scarce experimental data normally available, and do not appropriately represent the physical phenomena responsible for its appearance. The paper presents a mathematical model of grinding which takes into account the distribution of collision energies in the ball mill, the distribution of fracture energies of particles in the charge, describing breakage by impacts from media elements producing catastrophic breakage, breakage by abrasion and weakening due to repeated stressing. 2. Breakage of particles in batch grinding Grinding of carefully-prepared narrow size fractions in a batch mill is an excellent tool to investigate breakage kinetics in ball milling, since the rate of disappearance of material contained in the original size range is directly associated to the breakage rates. As such, it has been widely used to investigate the influence of material characteristics and grinding conditions on breakage rates, also being the basis of a widely used ball mill scale-up procedure (Austin et al., 1984). Batch grinding experiments have been conducted for selected materials and sizes under identical conditions in a test mill (Table 1). Size ranges tested were kept within a ffiffiffi 2 p sequence in order to limit the effect of interval size on the appearance of non-first-order breakage rates (Austin and Luckie, 1970–1971). The mill has sharp corners and is equipped with four lifters, with 27.5 mm width and 7 mm height. Typical experimental results are presented in Fig. 1. For several of the sizes tested, the fraction of material remaining in the original size range could be well described assuming first-order breakage: 0892-6875/$ - see front matter Ó 2009 Elsevier Ltd. All rights reserved. doi:10.1016/j.mineng.2009.03.015 * Corresponding author. Tel.: +55 2125628538. E-mail address: tavares@ufrj.br (L.M. Tavares). Minerals Engineering 22 (2009) 650–659 Contents lists available at ScienceDirect Minerals Engineering journal homepage: www.elsevier.com/locate/mineng