Development of a grinding model based on flotation performance

Highlights

• A grinding model is developed for preliminary studies in flotation circuits.

• The model uses fine-slow, medium-fast, and coarse-slow species.

• The grinding model is easy to integrate into flotation simulations.

• The model is applied to a copper sulfide mineral with acceptable results.

• The significant advantage of the modeling method is its simplicity and low cost.

Abstract

Usually, the concentration of minerals is carried out with a set of flotation and grinding units. Most of the modeling strategies for the flotation and grinding stages have followed separate developmental paths. This paper presents a strategy based on using flotation studies to model flotation and grinding via an integrated approach. The methodology, which is an approximate method that allows one to study of the effects of grinding on flotation circuits, is applied to a copper sulfide mineral with appropriate results. Given its nature, the application of this method will help preliminary studies on the design, improvement, and simulation of flotation circuits. The major advantages of this method are its simplicity and low cost. Thus, the main contribution of this work is a new strategy to model grinding for integration into the modeling of flotation circuits. This new strategy can be extended to other concentration technologies that include grinding.

Introduction

Flotation is the most widely used technology in mineral processing, and thousands of tons of ore are produced worldwide every day. This technology’s application affects several minerals, including sulfide minerals of lead, zinc, molybdenum, and copper, which are either polymetallic or not (Agheli et al., 2018, Prakash et al., 2018, Shean and Cilliers, 2011). Given the large-scale production of ore concentrates, a small improvement in recovery or product quality can produce a large increase in revenue. Therefore, permanent improvements are sought to increase recovery and reduce gangue in concentrates (Asghari et al., 2019, Kupka and Rudolph, 2018). Flotation processes include several flotation stages, with a specific configuration called a flotation circuit. The flotation circuit plays an important role in the observed recovery of valuable species and in the grade of concentrates. It has been shown that certain flotation circuits may be superior to others in terms of their economic gains, despite large variations in the recovery at each flotation stage (Cisternas et al., 2015). In other words, efforts to achieve better operating conditions—for example, with better collectors or foam handling—are marginal compared to the selection importance of the circuit structure. Whereas each flotation stage can include several flotation cells, the set of circuit alternatives for a given problem can total thousands, millions, or even billions of possibilities (Hu et al., 2013, Sepúlveda et al., 2017). For this reason, it is not possible to examine all possibilities through experimentation. The simulation of flotation circuits could be a good alternative, as demonstrated in several studies (Montenegro et al., 2013, Yianatos et al., 2012). Flotation circuits usually consider regrinding stages to liberate valuable species from the gangue. However, most flotation circuit simulation strategies cannot be applied to grinding operations, thus limiting their application.

The modeling of flotation stages and grinding stages has followed different directions, which makes the simulation and design of flotation circuits difficult. There are several models for flotation kinetics, and several reviews have analyzed these models (Gharai and Venugopal, 2015, Jovanović et al., 2015, Jovanović and Miljanović, 2015, Mendez et al., 2009, Polat and Chander, 2000, Yianatos and Henríquez, 2006). First-order kinetic models are the most commonly used model, both in the literature and in practice. In these models, the flotation system is represented by species, where a species refers to a group of particles that have the same floatability or can be represented by the same kinetics. Mostly, these species are minerals classified as slow or fast (Kelsall, 1961, Sutherland, 1989, Sutherland, 1977). In the fast fraction, the particles are of medium size, whereas in the slow fraction, the particles are fine and coarse (Bu et al., 2017). Notably, the first-order kinetic model has been extended to consider multiple components, namely, multiple fractions, with each one having its own kinetics (Nguyen, 2003). This type of model has the advantage of better adjusting to flotation kinetics that integrate with a distribution of flotation constants (Polat and Chander, 2000). The central purpose of grinding models is to obtain mathematical relationships between the size of the feed and the size of the product (Monov et al., 2012). Size reduction is a result of three fragmentation mechanisms: abrasion, cleavage, and fracturing (Hennart et al., 2009). Different size classes are used because the grinding system includes particles that differ significantly in size. In the theory of the breakage of solids, the fragmentation process is decomposed into the selection of a fraction of the material to be broken and the breakage of the selected material, producing a given distribution of fragment sizes (Varinot et al., 1997). These two operations are characterized by selection and breakage functions. Three types of models are commonly accepted in the literature: matrix, kinetic, and energy models (Toneva and Peukert, 2007). A general principle in the development of each model is to establish mass balance or energy balance equations relating to the mass components or the energy involved in the process. Since the main objective of the grinding process is to obtain a desired particle size distribution in the final product, most models focus on particle size, an important variable for flotation, but it is insufficient in terms of relating grinding to flotation. Other models have been developed to include mineral liberation. However, in these models, the classes are defined based on particle size and liberation (Pérez-García et al., 2018), which can yield a large number of classes. Furthermore, the identification of the liberation can require expensive characterization analysis, e.g., QEMSCAN. Sosa-Blanco et al. (Sosa-Blanco et al., 1999) presented a procedure to simulate a grinding–flotation system. They applied a population balance model to describe the grinding, classification, and flotation processes. In the grinding model, particles were characterized by their size, while in the flotation models, particles were characterized by their size and mineral composition. The link between grinding and flotation circuits was made using an empirical model that generates the mineral size particle population from the size distribution of the grinding circuit product. Once again, a large amount of experimentation is needed because of the representation by particle size and mineral composition.

Thus, because most flotation circuit simulation strategies cannot be applied to grinding operations, and vice versa, most studies on flotation circuits do not include grinding operations (Cisternas et al., 2018). For example, Méndez et al. (Mendez et al., 2015) modeled the flotation kinetics of an ore using a three-component kinetic model, redesigning a zinc flotation circuit using mathematical modeling techniques. Similarly, Calisaya et al. (Calisaya et al., 2016) applied the slow/fast model to design flotation circuits. However, both works did not consider grinding. Another method, linear circuit analysis (Meloy, 1983, Noble et al., 2019, Williams et al., 1986), also does not consider grinding. Unfortunately, predictive models are not available; therefore, the models used in practice are empirical and depend on the designed circuit. This is like the paradox of the chicken or the egg, where the circuit structure is required to determine the conditions to develop the flotation and grinding models, but the circuit structure cannot be developed without the flotation and grinding models. Because this is an iterative process, models that include particle size and liberation are too expensive. Significantly, it was demonstrated that accurate flotation and grinding models are not needed to identify the most optimal structures (Acosta-Flores et al., 2020, Acosta-Flores et al., 2018, Calisaya et al., 2016, Cisternas et al., 2015). Therefore, only approximate values for recovery in each flotation stage and particle size conversion in grinding are required to obtain a set of optimal solutions. Consequently, approximate and straightforward models can be useful for circuit structure identification and preliminary evaluation when processing a new ore or an ore with new mineralogy.

This work aims to develop a procedure for the modeling of grinding units based on a representation of the system by species with low and fast flotation kinetics. The procedure is intended for approximate simulation and could be applied to preliminary simulation, operation, and design problems for grinding units that operate together with flotation units.