Development of a grinding model based on flotation performance
Graphical abstract
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.
Section snippets
Methodology
The proposed methodology is based on the use of a flotation kinetics modeling strategy to represent ore grinding indirectly (see Fig. 1). The idea is to consider the change in flotation kinetics that occurs between different species of particles before and after grinding. The species are represented by fine–slow, medium–fast, and coarse–slow particles. The change in the faction of fine–slow, medium–fast, and coarse–slow particles in flotation kinetics is considered to be a direct effect of the
Case study
Sulfurized copper ore from the copper mining industry in the Antofagasta region of Chile was used as a case study. The mineral, which features 100% granulometry down to a 10-mesh sieve (2 mm), has an average density of 2.60 g/mL, which was determined by employing a pycnometer and an electronic scale (model LT5001E). The mineralogical composition of the sulfide copper ore is presented in Table 1, where it can be seen that the mineral contains chalcocite and covelite, as well as pyrite as a
Results
The particle size distribution is shown in Fig. 3 after 5, 10, 15, 22, 35, and 40 min of grinding; each grinding time was performed in duplicate. For each grinding time, the particle size distribution was adjusted with Eq. (1), using the minimum of the sum of the squared deviations as the objective function. The value featured no significant changes; it remained constant and equal to 1.0166 throughout. The values obtained are shown in Fig. 4. The values were adjusted using Equation (2),
Discussion
The results obtained represent the experimental copper and iron recovery values. In general, the model predicts that the recovery of both copper and iron increases as grinding and flotation times increase (Fig. 7). Moreover, the kinetic constant values are in agreement with their definitions—that is, medium–fast species have higher values than the other species, and the coarse–slow species are the slowest. Fig. 8 shows the contributions of each of these flotation kinetics for grinding times of
Conclusions
In this paper, a strategy based on using flotation kinetics to model grinding units was developed with an integrated approach. The methodology allows one to determine the effects of grinding on flotation. The methodology was validated based on its application to a copper sulfide mineral with acceptable results. Given its nature, the application of this model can help preliminary studies on the design, improvement, and simulation of flotation circuits. This model’s primary advantage is its
CRediT authorship contribution statement
Enoque Mathe: Validation, Formal analysis, Investigation, Writing - original draft, Writing - review & editing. Constanza Cruz: Writing - original draft, Writing - review & editing, Supervision. Freddy A. Lucay: Software, Validation. Edelmira D. Gálvez: Methodology, Resources, Supervision. Luis A. Cisternas: Conceptualization, Methodology, Validation, Formal analysis, Resources, Writing - original draft, Writing - review & editing, Visualization, Supervision, Project administration, Funding
Declaration of Competing Interest
The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.
Acknowledgements
This work was supported by financial assistance from ANID (Fondecyt 1180826). E.M. thanks the Agencia Chilena de Cooperación Internacional para el Desarrollo (AGCID) for the Scholarship for the master's degree. C.C. thanks the Universidad de Antofagasta, grant ANT 1956- MINEDUC. The authors thank Lorena Cortés for her assistance in the experimental tests.
References (62)
- et al.
Effect of pyrite content of feed and configuration of locked particles on rougher flotation of copper in low and high pyritic ore types
Int. J. Min. Sci. Technol.
(2018) - et al.
Particle size distribution models, their characteristics and fitting capability
J. Hydrol.
(2015) - et al.
A strategy for the identification of optimal flotation circuits
Miner. Eng.
(2016) - et al.
Verification, validation, qualification and certification of enterprise models: Statements and opportunities
Comput. Ind.
(2008) - et al.
Approximate recovery values for each stage are sufficient to select the concentration circuit structures
Miner. Eng.
(2015) - et al.
Assessment of global sensitivity analysis methods for project scheduling
Comput. Ind. Eng.
(2016) - et al.
Study of the process of stirred ball milling of poorly water soluble organic products using factorial design
Powder Technol.
(2010) - et al.
Identification of the grinding mechanisms and their origin in a stirred ball mill using population balances
Chem. Eng. Sci.
(2009) - et al.
Froth flotation of scheelite – A review
Int. J. Min. Sci. Technol.
(2018) - et al.
Verification, validation, and uncertainty quantification of a sub-grid model for heat transfer in gas-particle flows with immersed horizontal cylinders
Chem. Eng. Sci.
(2018)
Global sensitivity analysis for identifying critical process design decisions
Chem. Eng. Res. Des.
Stochastic analysis of heap leaching process via analytical models
Miner. Eng.
Optimizing for grade or profit in mineral processing circuits - Circuit analysis
Int. J. Miner. Process.
State of the art in the conceptual design of flotation circuits
Int. J. Miner. Process.
Arsenic-rejection flotation circuit design and selection based on a multiple-objective evaluation
Miner. Eng.
The effects of stage recovery uncertainty in the performance of concentration circuits
Int. J. Miner. Process.
Grinding of calcite suspensions in a stirred media mill: Effect of operational parameters on the product quality and the specific energy
Powder Technol.
First-order flotation kinetics models and methods for estimation of the true distribution of flotation rate constants
Int. J. Miner. Process.
Flotation technique: Its mechanisms and design parameters
Chem. Eng. Process. – Process Intensif.
Variance based sensitivity analysis of model output. Design and estimator for the total sensitivity index
Comput. Phys. Commun.
A review of froth flotation control
Int. J. Miner. Process.
Integrated simulation of grinding and flotation application to a lead-silver ore
Miner. Eng.
Nanomilling in stirred media mills
Chem. Eng. Sci.
Batch flotation behaviour of composite particles
Miner. Eng.
An appreciation of galena concentration using a steady-state flotation model
Int. J. Miner. Process.
Identification of the fragmentation mechanisms in wet-phase fine grinding in a stirred bead mill
Chem. Eng. Sci.
Product size distribution in stirred media mills
Miner. Eng.
Modelling and simulation of rougher flotation circuits
Int. J. Miner. Process.
Short-cut method for flotation rates modelling of industrial flotation banks
Miner. Eng.
Two phases optimization methodology for the design of mineral flotation plants including multi-species, bank or cell models
Miner. Met. Process. J.
The effect of regrinding on the design of flotation circuits
Miner. Eng.
Cited by (7)
Copper recovery from copper slags through flotation enhanced by sodium carbonate synergistic mechanical activation
2022, Journal of Environmental Chemical EngineeringStatistical distributions for modeling mineral liberation
2023, E3S Web of Conferences