Response Surface Methodology for Copper Flotation Optimization in Saline Systems

Abstract

Response surface methodology (RSM) is one of the most effective tools for optimizing processes, and it has been used in conjunction with the Analysis of Variance (ANOVA) test to establish the effect of input factors on output factors. However, when this methodology is used in mineral flotation, its polynomial model usually performs poorly. An alternative is to use artificial neural networks (ANNs) in such situations. Within this context, the ANOVA test is not the best option for these model types; moreover, it requires statistical assumptions that are difficult to satisfy in flotation. This work proposes replacing the polynomial model of the RSM with ANNs and the Sobol methods to determine the influential input factors instead of the ANOVA test. This proposal is applied to two porphyry copper ores with a high content of pyrite, clay, and dilution media. In addition, this study shows how other computational intelligence techniques, such as swarm intelligence, can be incorporated into this type of problem to improve the learning process of ANNs. The results gave an adjustment of over 0.98 for R² using ANNs, in comparison to values of around 0.5 when the polynomial model of RSM was utilized. On the other hand, the application of Global Sensitivity Analysis (GSA) identified the aeration rate and P₈₀ size as the most influential variables in copper recovery under the conditions studied. Additionally, we identified significant interactions that affect the recovery of copper, with the interactions between the aeration rate, frother concentration, and P₈₀ size being the most important.

1. Introduction

The copper mining industry faces challenges associated with deposits, scarcity of resources, and pressure to reduce environmental impacts. One of the problems associated with the ores corresponds to the low grades of the deposits and their mineralogical associations, such as pyrite and clay, which reduce the value of the concentrate. For example, clays generate problems throughout the copper value chain, including the sedimentation stage and water recovery [1,2]. By recirculating this water to the flotation stages, non-sedimented clays are dragged, increasing their concentration. Once in the flotation stage, they are trapped in the foam zone because they are not heavy enough to settle [3]. The worldwide scarcity of water has made the need for the efficient use of water in mining critical. The mining industry uses water primarily for mineral processing and other activities such as dust suppression and slurry transport. Groundwater, rivers, and lakes are used in most mining operations and by commercial water service providers. However, mining sites are often located where water is already scarce, and other water sources must be used. For example, in northern Chile, one of the largest copper-producing areas, the use of seawater, with and without desalinization, has increased significantly despite the high costs of transporting water from the coast to mining operations because of the high elevation of mining operations [4]. Seawater contains ions that physically and chemically interact with mineral species, making it difficult for the flotation and thickening stages to operate [5,6,7]. Climate change has been increasing the extent of water scarcity in Chile, and, along with this, there are an increasing number of conflicts with other water users such as farmers [8]. The use of this critical resource adds to other environmental problems, such as the generation of massive waste, e.g., overburden, slags, leaked ores, and tailings [9]. To face the challenges indicated above, it is necessary to use tools that allow the optimization of operations to improve their efficiency.

The application of optimization requires models that represent the relationships between the variables of interest. However, the relationships between the variables of interest in mineral processing are not fully understood, making it difficult to predict their behavior. For this reason, it is common to use empirical or semi-empirical models based on experimental tests at the laboratory level or plant data. One technique used to model these systems empirically is the response surface methodology (RSM) based on some experimental designs [10].

Since RSM was first proposed in 1951 by Box and Wilson [11], it has been widely applied successfully in different disciplines, including the separation of radioactive material [12], the extraction of essential oils [13], biological wastewater treatment processes [14], and energy applications [15], to name a few. The RSM consists of three steps: the design of experiments (DoE) to define the number of experiments required and the value of the independent and dependent variables, the determination of the factors that affect the variable of interest using ANOVA and RSM-based modeling, and the optimization of the variable of interest [16]. Despite its wide use, the RSM sometimes presents a poor fit to the data because the relationship between the variables does not follow a second-order polynomial behavior. The immediate consequence of poor fitting is incorrect optimization.

For example,

Table 1. Examples of papers on RSM in mineral flotation are available from Web of Science journals.

DoE	Mineral/Ore type	Input	Output	R-Squared (R2)	Adjusted R-Squared (Adj. R2)	Ref.
CCD	Not defined	Particle size, particle density, bubble size, bubble velocity, and turbulence dissipation rate	Flotation rate constant, particle–bubble encounter efficiency	0.947	0.933	[28]
BBD	REE	Frother dosage, pulp density, and collector dosage	REE recoveries and concentrations	0.930	0.900	[29]
D-optimal design	Iron	Collector dosage, frother dosage, depressant dosage, dispersant dosage, and pH	Grade and recoveries of iron	0.970	0.958	[30]
CCD	Coal	Collector dosage, frother dosage, solid ratio, and airflow rate	Ash contents and combustible recoveries of clean coal	0.898	Information not available	[31]
CCD	Lepidolite	Collector dosage, flotation time, and pH	Recoveries and grade of lepidolite	0.96	Information not available	[32]
CCD	Sphalerite	Collector dosage, activator dosage, and pH	Recoveries and grade of sphalerite and lead	0.879	0.794	[33]
BBD	Graphite	Scrubbing medium’s particle size, stirring speed, and solid concentration	Enrichment efficiency of scrubbing flotation (α)	0.883	0.787	[34]
CCD	Chalcopyrite	Pulp pH, depressor dosage, and collector dosage	Copper and sulfur recoveries	0.990	0.981	[35]
BBD	Chalcopyrite	Primary collector dosage, secondary collector dosages, and frother dosages	Grade, recovery, and separation efficiency of copper	0.955	0.900	[36]
CCD	Copper ore	pH and nanoparticle dosage	Grade, recovery, and flotation rate constant	0.982	0.974	[37]
BBD	Lead	Pulp pH, depressor dosage, and collector dosage	Grade and recovery of lead	0.910	Information not available	[38]
CCD	Copper ore	Collector dosage, depressant dosage, frother dosage, pulp pH, and agitation rate	Copper recoveries	0.810	Information not available	[39]
CCD	Copper ore	Three collectors and two frother dosages	Copper grade and recovery	0.940	0.910	[40]
Factorial design	Complex sulfide ores	Collector concentration, pH value, and depressant concentration	Grade and recovery of copper and zinc	0.640	0.590	[41]
Factorial design	Lead and zinc	Collector type and the particle size	Grade and recovery of lead and zinc	0.950	Information not available	[42]
CCD	Fluorite	Aeration flow rate, time of flotation, agitator speed, and pH	Grade and recovery of fluorite	0.834	Information not available	[43]
Custom design of experiment	Talc ore	Collector, frother, and depressant	Flotation kinetics	0.9	Information not available	[44]
Factorial design	Covellite	pH and collector concentration	Covellite recovery	0.9363	0.87	[45]
Central composite design	Hematite	Field intensity, drum speed, separating gate position, and particle size	Fe recovery	0.9447	0.9192	[46]
Central composite design	Magnetite	Bowl speed, fluidizing water rate, and solid feed rate	Concentrate grade	0.93	0.87	[47]
Central composite design	LiO2	Na2CO3, NaOH, CaCl2, and NaOL	Grade of LiO2 using hard-hardness water (HHW)	0.98	0.96	[48]
Box–Behnken design	Coal	Solid concentration, oil dosage, type of oil, and agglomeration time	Ash rejection	0.9906	0.9891	[49]
Central composite design	Oxide zinc	Temperature, time, solid–liquid ratio, and NTA concentration	Zinc extraction	0.97	0.95	[50]
Central composite design	Chalcopyrite	Py/Cp ratio, Ag concentration, potential, and acid concentration	Cu recovery	0.9744	Information not available	[51]

shows examples of RSM applied in ore flotation from the available literature. The first column shows the types of DoE applied, with Central Composite Design (CCD) and Box-Behnken Design (BBD) being the most used. It can be seen in Table 1 that this type of methodology has been applied to several minerals and for the study of different phenomena in flotation. Additionally, Table 1 highlights that several studies present low R-squared (R²) and adjusted R-squared (Adj. R²) values, varying between 0.64 and 0.99 for the R² and between 0.59 and 0.91 for the adjusted R². In some fields, such as social science or finance, an R² above 0.7 can be considered a good correlation, but in mineral processing, the standards for a good R² must be 0.90–0.95 or above, depending on the problem [17,18,19]. For example, if there is a small variation in the output variable in the DoE values of the R², a value of ~0.90 can be acceptable, but if there is a large variation in the output variable values of the R², a value of ~0.98 is necessary to have a good representation of the data.

To eliminate low values of the R², several authors have proposed the use of different approaches for the RSM as alternatives to the polynomial models. The most popular approach to replacing the polynomial model has been the use of artificial neural networks (ANNs) [20].

Table 2. Examples of ANNs used in modeling DoE data.

Sample	Input	Output	DoE	Regression Model (R2)	ANN Type	Hidden Neuron	Regression Model (R2 Adjusted)	Training Algorithm	Ref.
Galegine	Temperature, pressure, flow rate, and extraction time	Experimental yield	CCD	0.934	MLP	6	0.9668	BP	[52]
L-asparaginase	L-asparaginase unit, sodium nitrate, L-aspargine, and glucose	L-asparaginase activity	CCD	0.973	MLP	- *	0.997	Genetic algorithm	[53]
Crystal violet	Concentration, pH, adsorbent dosage, and sonication time	Crystal violet removal	CCD	0.99	MLP	20	0.999	BP	[54]
Sodium sulfide	Initial NaOH concentration, scrubbing solution temperature, and liquid-to-gas volumetric ratio	Na2S production	CCD	0.9824	MLP	3	0.9953	BP	[55]
Wastewater	Initial pH, [H2O2]/[Fe2+] mole ratio, [Fe2+] dosage, and initial COD concentration	Chemical oxygen demand (COD) removal	CCD	0.9171 and 0.9617	RBF, MLP	10	0.9810 and 0.9980	BP and genetic algorithm	[56]
Biodiesel	Reaction time, catalyst concentration, and methanol concentration	Biodiesel yield	BBD	0.9657	MLP	9	0.999651	BP	[57]
Quercetin	Dynamic time, pressure, temperature, and flow rate of SC-CO2	Extraction of quercetin	CCD	0.9317	MLP	6	0.995	BP	[58]
Green tea	Pressure, temperature, flow rate, and dynamic time	EGCG extraction	CCD	0.984	MLP	6	0.997	BP	[59]
Olive leaves	pH, extraction time, temperature, and solid/solvent ratio	Olive leaf extraction	BBD	0.9315	MLP	60	0.9746	BP	[60]
Membrane distillation	NaCl, CaSO4, organics, and SDS	Time of wetting occurrence and recovery	CCD	0.983 and 0.985	MLP	6	0.999 and 0.999	BP	[61]
Flue gas	Flow rate of the additional oxygen, water temperature, pH, and aqueous concentration of H2O2	Nox removal in acidic and basic media	CCD	0.868 and 0.874	MLP	3	0.9747 and 0.992	BP	[62]
Mercury bioremediation	pH, temperature, and Hg concentration	Mercury removal	CCD	0.999	MLP-Fuzzy	20	0.997 and 0.996	(---)	[63]
Aqueous environment	pH, AC dose, contact time, initial concentration, and temperature	Chromium (VI) adsorption	CCD	0.9886	MLP	20	0.9911	BP	[64]
Artemisia absinthium leaves	Air temperature, air velocity, and drying time	Air drying	CCD	0.9423	MLP	15	0.9997	BP	[65]
Osmosis process	Osmotic pressure difference, feed solution velocity, draw solution velocity, temperature, and DS temperature	Membrane flux	BBD	0.9408	MLP	10	0.98036	BP	[66]
Oil yield	Methanol-to-oil ratio, reaction temperature, reaction time, and catalyst amount	Biodiesel production	BBD	0.9867	MLP	10	0.9976	BP	[67]
Wastewater	NFC amount, contact time, pH, and phenol concentration	Phenol removal	CCD	0.961	MLP	6	0.9934	BP	[68]
Wastewater	pH, contact time, temperature, CR concentration, and adsorbent dosage	Congo red removal	CCD	0.9954	MLP	9	0.9983	BP	[69]
Biodiesel	Reaction temperature, ethanol, and catalyst amount	Coconut oil ethyl ester (CNOEE) yield	CCD	0.9564	MLP	10	0.998	BP	[70]
Neem (azadirachta indica) oil	Methanol-to-oil molar ratio, catalyst concentration, reaction temperature, and reaction time	FAME conversion	CCD	0.919	MLP	9	0.999	BP	[71]
Wastewater	Dosage, pH, and stirring time	CTSP removal	BBD	0.9871	MLP	10	0.999	BP	[72]
Carbon dots	Temperature, dosage, time, and W/Ace/NaOH ratio	Fluorescent CDs synthesis	CCD	0.9563	MLP	20	0.944	BP	[73]
Biodiesel	Reaction temperature, ethanol-to-oil molar ratio, catalyst loading, and reaction time	FAME yield	34 full factorial	0.803	MLP	20	0.947	BP	[74]
Wastewater	pH, temperature, and b dose	Bioadsorption of arsenic	CCD	0.9843	MLP	-	0.9952	BP	[75]
Wastewater	Initial concentration of Cr(VI), varying dosages of photocatalyst, different irradiation times, and pHs	Cr(VI) photocatalytic reduction	CCD	0.9812	MLP	4	0.9858	BP	[76]
Cocoa shell	Temperature, time, acidity, and ratio	Extraction of phenolic compounds (TPA)	BBD	0.9256	MLP	10	0.98	BFGS, quasi-Newton backpropagation, gradient descent	[77]
Wastewater	pH, temperature, initial concentration, and adsorbent dosage	Cu2+ adsorption optimization	CCF	0.941	MLP	4	0.962	BP	[78]
Coal	Solid concentration, oil dosage, and agglomeration time	% ash rejection	BBD	0.9956	MLP	3	0.9965	Levenberg–Marquardt, Bayesian regularization, gradient descent with momentum, gradient descent	[79]
Rubia cordifolia	Ultrasonic frequency, solid-to-liquid ratio, and extraction time	Natural dye extraction	CCD	0.9805	MLP	10	0.9853	BP	[80]
Wastewater	pH, turbidity, BOD, and COD	COD removal (%)	BBD	0.9754	MLP	10	0.9924	BP

* (---) not reported value.

summarizes several applications of ANNs in the DoE for RSM modeling in different areas, demonstrating that low values of the R² occur in other areas. The values of the R² using ANNs are usually over 0.95, which indicates that ANNs are a superior technique for modeling the RSM. Based on Table 2, it is clear that many types of ANNs that differ in structure and operation have been utilized. The multi-layered perception (MLP), which is a fully connected class of feedforward ANNs, is commonly applied in predicting the performance of many processes [21]. Only a few applications use radial basis function (RBF) networks. Additionally, Table 2 shows that the traditional BP method is the one that is most commonly used as a training algorithm, and, in a few cases, the genetic algorithm and particle swarm optimization has been utilized. RBF is an ANN that uses radial basis functions as activation functions [18,19,20]. In this type of ANN, the output of the network is a linear combination of the radial basis functions of the inputs and neuron parameters. However, Lucay et al. [22] studied several examples of the application of ANNs for modeling the RSM using the DoE. They found that the RBF ANN gave better results than the MLP ANN.

Usually, ANNs are trained using the descent gradient-based error back propagation (BP) algorithm [19], such as the Broyden-Fletcher-Goldfarb-Shanno (BFGS) and Levenberg–Marquardt algorithms. These algorithms exhibit effectiveness in the local search but ineffectiveness in global search; perhaps this remark explains the complex arrangements reported in the literature. On the other hand, metaheuristic algorithms such as swarm intelligence and genetic algorithms exhibit effectiveness in the global search but ineffectiveness in local search. Therefore, the use of hybrid algorithms that combine the two previously mentioned algorithms can provide good effectiveness in global and local searches.

In addition, ANNs work best when a massive dataset is available to avoid overtraining, which is not the case with the DoE. To avoid this problem, Lucay et al. [22] used a training dataset generated from the experimental data using classical kriging as an interpolation method.

Although ANNs provide excellent R-squared (R²) values when applied to the DoE, their nonlinear nature requires the use of robust methods to determine the most influential variables on the output variable. Such methods are included in the global sensitivity analysis (GSA), which studies how the uncertainty in the output of a mathematical model can be divided and allocated to different sources of uncertainty in its inputs [23]. The interested reader can compare these in [22], where it is highlighted [24] that the Sobol method is more adequate than ANOVA-DoE for nonlinear models and that the choice of the DoE and the levels are critical in ANOVA-DoE. Thus, ANOVA requires that each sample is drawn from a normally distributed population, which is rare in mineral processing, especially in the presence of operational and geology uncertainty. It is worth mentioning that the Sobol method does not exhibit the abovementioned disadvantages. Therefore, in this work, a method included within the Sobol family was used to determine the influential factors on the model’s outputs, namely, the Jansen–Sobol method, which has been applied successfully to evaluate metallurgical processes such as flotation [25], heap leaching [26], and ore grinding [27].

This work aims to propose a methodology for the application of the RSM in mineral flotation systems where the adjustment results are not satisfactory. For this, we proposed the replacement of the polynomial model of the RSM with ANNs and the use of GSA to determine the influential input variables instead of the ANOVA test. To solve the problems of the application of ANNs to the DoE, a training dataset was generated using kriging from the experimental values, and a hybrid methodology (a gradient-metaheuristic algorithm) was used to generate the network. The proposed method was applied to two porphyry copper minerals with a high content of pyrite and clay that uses seawater as a dilution medium.

2. Methodology

The methodology consisted of four stages, as described below (see Figure 1). The first stage consisted of defining the input and output variables to be studied, including the value range of the variables to be studied. These were determined based on the requirements of the engineer or researcher and the problem to be solved. The lower and upper limits of the input variables were defined based on prior knowledge of the ore flotation and plant operating experience. In flotation, one can expect linear behavior if the range of the variables is very narrow or under conditions where the response converges to a given value. On the other hand, nonlinear behaviors different from those of the second-degree polynomial can be expected in a wide range of variables of interest. For example, the recovery of a mineral behaves exponentially with the flotation time and the collector dose [27]. For this reason, it is commonly found that the traditional application of the RSM delivers unsatisfactory results.

The second stage consisted of designing and carrying out the experiments. There are several techniques to design these experiments, with Full Factorial Design FFD, CCD, and BBD being the most commonly used techniques. For the selection of the best DoE, it is important to consider the number of variables, the type of the problem and the known information, cost and time restrictions, and the ease of implementation and understanding of the design. FFD and replicates are better because more data are available to generate the model, but variables such as cost and time may limit their use. There is extensive literature on these methods, and a review of them is outside the scope of this article. For more information, readers can are referred to works such as those in [81,82] and Supplementary Materials.

Once the experimental data were generated in stage three, the ANN was defined and trained. We proposed the use of the RBF ANN, as was explained in the Introduction. The first thing needed was a dataset to perform the training and a dataset to test whether the model worked properly. Classical kriging was used to generate the training data from the experimental data. This allowed us to generate a dataset large enough to ensure a good fit. Then, the experimental data were utilized to test and validate the obtained network. A hybrid algorithm was applied to optimize the weight values of the ANNs. Levanberg–Marquardt and swarm intelligence were utilized together for efficient local and global searches. There is extensive information available about kriging (see, for example, [81] and [83] hybrid ANN training (see, for example, [84]), and, therefore, the details are not discussed here. For more information on RBF ANN, see Table S1 in Supplementary Materials, and for information on hybrid algorithms, see Supplementary Materials.

Finally, in stage four, the diagnosis of the model was carried out to deepen our understanding of the behavior of the system under study. First, GSA was used to identify which input variables were more influential on the output variables and to identify if there was an interaction between the input variables. Then, uncertainty analysis (UA) was used to determine the effect of the input variables on the behavior of the output variables. Response surface was also performed at this stage to visualize the behavior of the system better. For information on GSA, see Table S2 in Supplementary Materials.

3. Application of the Methodology

3.1. Description of the Cases and Experimental Setup

The proposed methodology was applied to two porphyry copper minerals from Chile with a high content of pyrite and clay and using seawater. The water samples corresponded to the flow obtained from Instituto de Ciencias Naturales A. von Humboldt of Universidad de Antofagasta and were pretreated with a UV filter to eliminate organic compounds and successive physical filter (0.5 µm) for solid suspension removal. However, the literature about the salinity concentrate indicated a historical variability in San Jorge Bay between 34 and 36 g/L [85,86,87]. The methodology was applied to two mineralogical samples named Case 1 and Case 2. These samples had a high content of pyrite and clay and were floated with seawater without desalination. These samples came from the north of Chile, the Atacama Desert, which is the driest continental desert in the world. In this area, seawater is essential, given the limited water resources available. Some companies desalinate seawater by reverse osmosis, producing environmental impacts due to energy consumption and the generation of brine that is returned to the sea. Other companies, as in this case, use seawater without desalination, thereby reducing the environmental impacts but making the behavior of the system more complex.

The mineral samples were analyzed mineralogically using a Hitachi SU 5000 scanning electron microscope (Tokyo, Japan) with a Bruker Advance D8 X-ray Diffractor (Billerica, MA, USA) for chemical analysis of the sample X-ray. The characterization of the samples shown in

Table 3. Chemical composition of mineral ore samples (wt%).

Sample	Cu	Fe	Mo
Case 1	0.28	4.81	0.0085
Case 2	0.27	3.95	0.0087

indicates that Case 1 had a composition of 0.28% Cu, 4.81% Fe, and 0.0085% Mo. In contrast, Case 2 has a composition of 0.27% Cu, 3.95% Fe, and 0.0087% Mo. The mineral identification and quantification of the ores were conducted using a quantitative evaluation of the minerals by scanning electron microscopy (QEMSCAN^®) coupled with X-ray energy dispersive spectroscopy (ZEISS EVO series, Bruker detectors, AXS XFlash series 6, IDiscover 5.3.2.501 software, FEI Company, Brisbane, Australia). For both samples, the previously exposed chemical results were confirmed, and these are shown in

Table 4. QEMSCAN analysis for Case 1 and Case 2 samples in wt%.

Minerals	Case 1	Case 2	Minerals	Case 1	Case 2
Chalcocite/digenite	0.03	0.05	Siderite	0.01	0.01
Covellite	0	0.01	Calcite/dolomite	0.42	0.59
Chalcopyrite	0.63	0.46	Apatite	0.56	0.48
Bornite	0.01	0.02	Quartz	21.94	30
Tetrahedrite group	0	0.01	Orthoclase (K-feldspars)	2.68	4.12
Other Cu minerals	0.01	0.01	Plagioclase (Ca, Na-feldspars)	31.46	17.66
Cu-bearing phyllosilicates	1.3	1.27	Kaolinite group	1.92	1.07
Cu-bearing Fe Oxy/hydroxides	0	0.01	Muscovite/sericite	1.42	3.12
Cu-bearing wad	0.07	0.17	Illite	0.35	0.87
Pyrite	2.58	4.48	Smectite group (montmorillonite, nontronite)	0.34	0.35
Molybdenite	0.04	0.02	Pyrophyllite	0.21	0.25
Sphalerite	0.01	0.02	Chlorite group	15.39	21.82
Magnetite	0.88	0.44	Biotite/phlogopite	11.21	4.8.0
Hematite	0.74	0.47	Hornblende	0.11	0.15
Goethite	0.03	0.03	Tourmaline group	0.48	0.26
Other Fe Oxy/hydroxides	0.1	0.08	Titanite	0.14	0.09
Gypsum/anhydrite	3.97	5.6	Rutile	0.4	0.51
Jarosite	0.06	0.14	Ilmenite	0.24	0.19
Alunite	0.08	0.21	Other gangue	0.17	0.18
			Total	100	100

. This analysis indicates that the main copper minerals in both samples were chalcopyrite and chalcocite; the main sulfide gangue was pyrite. Case 2 had more pyrite than Case 1. Molybdenum was present as molybdenite. Clay associations were presented as chlorite, biotite, muscovite, and kaolinite. Case 1 had more clays (muscovite and chlorite) than Case 2. The F₈₀ values for Case 1 and Case 2 were 883 and 874 µm, respectively.

Seawater samples were obtained from the coast of the city of Antofagasta (San Jorge Bay), Chile. Initially, the seawater was filtered to remove coarse solids. This was then processed using a UV filter to remove organic material. The composition of seawater is shown in

Table 5. Seawater composition at San Jorge Bay.

Parameter	Value	Parameter	Value
Mg2+ (mg L−1)	1310 ± 38	NO3− (mg L−1)	3.62 ± 0.38
Na+ (mg L−1)	11,138 ± 12	HCO3− (mg L−1)	143 ± 5
K+ (mg L−1)	401 ± 4	SO42− (mg L−1)	2791 ± 18
Ca2+ (mg L−1)	415 ± 26	Conductivity (mS cm−1)	53.4 ± 1.2
Cl− (mg L−1)	19,867 ± 24	pH	7.98 ± 0.08

.

3.2. Stage 1: Definition of Input and Output Factors

The information provided by the Centinela operation of Antofagasta Minerals company was used to identify the input variables, as well as other experimental conditions. The definition of factors to be considered for the DoE are those typically used in mining companies. These include aeration rate ($x_a$), P₈₀ size ($x_p$), collector concentration ($x_c$), and frother concentration ($x_f$). The identified output variables were copper ($R_{Cu}$) and iron ($R_{Fe}$) recoveries, where it was desired to maximize copper recovery by keeping the iron recoveries as low as possible.

3.3. Stage 2: Design and Experimentation

BBD was selected as the DoE methodology.

Table 6. Factors and their levels for BBD experiments used in Case 1 and Case 2.

Factors	Code Factor Level
	Low	Center	High
	−1	0	1
$x_a$: Air, cm/s	0.34	0.51	0.68
$x_p$: P80, µm	150	210	250
$x_c$: Collector, gpt	8	12	16
$x_f$: Frother, gpt	7	13	19

presents the minimum, mean, and maximum values of the DoE generated by BBD. In the case of BBD, the number of experiments was determined according to the following equation: $N=2 k \left(k-1\right)+C_0$, where $N$ is the number of experimental trials, $k$ the number of input factors, and $C_0$ is the number of central points. In this case, there were four factors and three central points, for which 27 combinations were generated. The values of the parameters of the central point correspond to those used by the mining company that provided the samples in its daily plant control.

The flotation tests were carried out in a Metso model D-12 cell (Helsinki, Finland). Each test was carried out in batches and under ambient conditions (with average values of 1 atm and 20 °C). The volume of pulp was 3460 mL, and the pH for all the tests was that which was recorded naturally (pH $\cong$ 8.0). The floating time was 12 min, with paddling every 5 s. The collecting and frother reagents were of ethyl isopropyl thionocarbamate (TC-123) type and a blend of polyglycol alkyl alcohols (MatFroth 355), respectively, provided by Mathiesen Company (Huechuraba, Chile). The pulp densities were 2.75 g/cm³ and 2.78 g/cm³ for Case 1 and Case 2, respectively.

Once the flotations were completed, the collected concentrate and the generated tailings were vacuum-filtered and dried at 100 °C for 24 h. The dry masses were used to determine the cumulative recovery as shown in Equation (1):

$$R_i=\frac{c_i \left(f_i-t_i\right)}{f_i \left(c_i-t_i\right)}100$$

(1)

where $R_i$ is the accumulated recovery (%) $i$ based on the grades of feed ($f_i$), concentrate ($c_i$), and tailings ($t_i$), respectively.

The experimental copper and iron recovery results for the combinations delivered by BBD are shown in

Table 7. Summary of BBD experimental results for Case 1 and Case 2 flotations.

Run	Input Factors				Output Factors
	${\mathit{x}}_{\mathit{a}}$	${\mathit{x}}_{\mathit{p}}$	${\mathit{x}}_{\mathit{c}}$	${\mathit{x}}_{\mathit{f}}$	Case 1		Case 2
					${\mathit{R}}_{\mathit{C}\mathit{u}}$	${\mathit{R}}_{\mathit{F}\mathit{e}}$	${\mathit{R}}_{\mathit{C}\mathit{u}}$	${\mathit{R}}_{\mathit{F}\mathit{e}}$
15	0	0	−1	−1	87.8	42.1	57.1	37.0
11	−1	−1	0	0	84.8	43.6	69.7	43.7
4	0	0	0	0	84.8	38.3	66.6	43.3
9	0	1	0	−1	84.7	29.4	66.6	43.6
8	−1	0	0	1	84.9	39.1	60.9	41.9
7	1	0	0	1	84.8	41.6	71.5	40.8
23	1	0	0	−1	84.7	25.0	78.1	46.4
21	0	1	1	0	84.7	29.8	77.6	43.8
6	1	0	−1	0	84.7	39.2	77.6	44.4
2	1	0	1	0	87.9	42.1	80.9	45.3
3	−1	0	−1	0	81.9	35.0	79.2	34.3
16	−1	0	1	0	84.9	25.3	76.3	37.5
17	0	−1	−1	0	87.9	42.8	86.8	47.9
5	0	0	1	−1	87.9	33.1	82.6	37.3
18	0	1	−1	0	81.6	40.1	80.6	39.4
12	0	1	0	1	84.8	37.9	83.2	43.0
24	−1	1	0	0	84.9	37.3	76.3	37.9
1	0	−1	0	−1	84.9	41.0	86.5	46.4
19	1	1	0	0	87.8	27.4	83.5	38.9
26	0	−1	1	0	88.0	27.1	86.6	43.6
27	0	0	0	0	84.8	40.7	80.2	41.5
14	1	−1	0	0	88.0	37.2	89.8	49.4
13	−1	0	0	−1	84.8	41.7	79.6	37.8
10	0	0	0	0	87.8	39.6	83.3	44.7
25	0	−1	0	1	88.0	37.3	86.4	45.0
20	0	0	1	1	87.9	25.4	86.1	41.6
22	0	0	−1	1	87.9	32.1	82.8	37.8

. The maximum copper recoveries for Case 1 and Case 2 were 88.0 ± 1.7 and 89.8% ± 2.1, respectively. On the other hand, the minimum iron recoveries were 25.0% ± 1.2 and 37.0% ± 1.6 for Case 1 and Case 2, respectively. The recoveries applying the mining protocol for Case 1 were 85.8 and 39.5% for copper and iron, respectively. For Case 2, the recoveries were 76.7 and 43.2% for copper and iron, respectively.

The copper recoveries in both samples showed a small difference of 1.8%. The case of iron recovery showed significant differences of 12%. This can be attributed not only to the fact that the samples were different but also that, for Case 1, a more diluted pulp density was used, with values of 2.75 and 2.78 g/cm³ for Case 1 and Case 2, respectively. Lower pulp densities increase the dilution of the system, thereby minimizing the mechanical trapping of the clay generated by hydrodynamic drag in the froth zone [2,88]. Note that some results obtained in Case 1 provided good copper recoveries but low iron recoveries (e.g., test 20); however, this behavior was not observed in Case 2.

Before applying the methodology proposed here, the traditional form of the RSM must be applied, that is, the use of the second-degree polynomial in the adjustment of the experimental data. If the results are good, then the application of the traditional methodology may be an alternative worth considering. For this, traditional statistical analysis was applied using Minitab 18. The results were poor. The adjustments for copper recovery for Case 1 and Case 2 were 0.48 and 0.50, respectively (R²). These results should not be surprising; in a previous study by the authors [89], similar results were obtained for the recovery of pure chalcopyrite in the presence of kaolinite and an aqueous solution of NaCl and KCl. In this case, it was not possible to represent copper recovery with a quadratic polynomial. However, in a study with pure pyrite [89], it was possible to use a quadratic polynomial to adjust the recovery. In our case, for the real ore, the Fe recovery with the traditional method was not acceptable, with R² values of 0.54 and 0.79 for Case 1 and Case 2, respectively. This difference observed in the case of pure pyrite could be related to a large number of minerals in the flotation system and the use of seawater. The elements in the ore have interactions that affect flotation, giving a more complex behavior. Additionally, these interactions depend on the type of reagents, aeration, P₈₀, and water quality, among others. In the case of seawater, it is known that it contains a large number of ions [7,89,90].

3.4. Stage 3: ANN Fit

Given the poor results when applying the traditional techniques, the modeling of the experimental data was performed using neural networks. However, the experimental dataset is very small (25 rows), favoring the overfitting of neural networks. Under this context, we use the following procedure [24] to avoid this drawback: (a) we generate synthetic datasets using theoretical variogram and classical-kriging. The theoretical variogram describes how the data (flotation datasets) are correlated with distance, and the classical kriging supposes that these correlations can be used to interpolate new responses to the studied process (copper and iron recoveries). We compare the gaussian, spherical, linear, circular, power, and exponential theoretical variograms; in all instances, the gaussian variogram better fitted the experimental variogram constructed using the experimental dataset. Posteriorly, the gaussian variogram and ordinary kriging (one type of classical-variogram) were used to generate synthetic datasets of 1000 rows; (b) synthetic datasets were sampled using the Monte Carlo method to generate training datasets; depending on the instance analyzed, we did not observe improvements during testing of neural models (RBFNN) when the training dataset was larger than 190, 200, or 195 rows; then, we decided to round the value to 200. Note that testing of neural models was conducted using the experimental datasets. We did not analyze the heteroscedasticity and autocorrelation in the experimental datasets because neural models were separately constructed for each output. We obtained the following results using the procedure described; for Case 1 and Case 2, R²-testing of 0.9844 and 0.9989 were achieved, respectively, see Figure 2. The neural models considered 15 neurons in the hidden layer and a Gaussian activation function. The training algorithm implemented was a hybrid algorithm (differential evolution-backpropagation) confected in the R programming language.

3.5. Stage 4: Diagnose and Confirm Model

Although the neuronal models provide good estimates for copper and iron recoveries, these must be subjected to quantification of uncertainty to guarantee their robustness. The quantification of uncertainty includes uncertainty and sensitivity analyses. The results related to the former can be seen in Figure 3.

To study the effect of the input factors’ uncertainties on the copper and iron recovery, the input factors were represented by intervals. Their ranges were normalized and divided into ten subintervals, on which we conducted a box plot. Specifically, each subinterval was sampled via the Monte Carlo method, generating 1000 samples for each input factor, which were evaluated in the neuronal models and analyzed using a box plot. As shown in Figure 3A, the uncertainty had dissimilar effects on copper and iron. The uncertainty caused oscillatory and decreasing recoveries for copper and iron, respectively, as we went through the intervals. For copper, oscillations were accompanied by recoveries having normal distributions without outliers. While for iron, decreasing recoveries were accompanied by an increase in the box’s length without outliers, revealing that combinations of high values of the input factors generated dispersion in the recoveries with a normal distribution. Figure 3B shows behavior similar to that of Case 1 for the copper and iron recoveries, but the box plots exhibit outliers, which are residuals considering the sample size. In general, in both instances, the results show a stable behavior process despite uncertainty, allowing us also to demonstrate that neuronal models provide stable predictions against uncertainties, which is a key aspect for validating a metamodel [21]. Note that these figures could be used to establish operating ranges for the froth flotation process. Finally, we subjected the neuronal models to GSA for full validation, the results of which are shown in Figure 4.

Figure 4A shows the first-order and total sensitivity indices provided by the Sobol–Jansen method for copper recovery. The first-order index indicates the direct effect of an input factor on recovery. The total sensitivity index shows the average impact of a given input factor on recovery, considering all the possible interactions with all the other input factors. Within this context, the interpretation of the indices is simple: the higher the index, the greater the effect of the variable it represents. For instance, for Case 1 copper, as shown in Figure 4A, the decreasing influence of the input factors on recovery based on the total index is given by P₈₀ ($x_p$) $>$ aeration rate ($x_a$) $>$ frother concentration ($x_f$) $>$ collector concentration ($x_c$), while for the Case 2 copper, it is aeration rate ($x_a$) $>$ frother concentration ($x_f$) $>$ P₈₀ ($x_p$) $>$ collector concentration ($x_c$). The first outcome was that the range of the collector concentration analyzed in this study had little effect on copper recovery. The second outcome was that the aeration rate was the more influential input factor on copper recovery in both cases, and this can be explained by its direct impact on the flotation kinetics [91]. Note that each input factor’s first-order and total sensitivity indices differed, indicating that the model is nonlinear, which is consistent with previous results. In other words, the input factors interact; for estimating the importance of these interactions in recovery, second-order Sobol indices were determined and are shown in Figure 4B. Here, it is clear that $x_a$-$x_p$ and $x_a$-$x_f$ were the most important interactions for both cases, which is consistent with Figure 4A. In both cases, the $x_c$-$x_f$ interaction was the least important. To understand the flotation systems’ behavior, contour plots were confected and are shown in Figure 5. Here, it can be seen that the behaviors are complex and significantly different from second-order polynomial behavior.

3.6. Model Application

The application of the model depends on the objectives sought. A usual application is the optimization of the output variables, in our case, to maximize copper recovery and minimize iron recovery. To solve this multi-objective problem, the desirability function was optimized (see Table S2 in Supplementary Material).

Table 8. Optimal values of input variables for Case 1 and Case 2.

Case	Input Factors
	${\mathit{x}}_{\mathit{a}}$	${\mathit{x}}_{\mathit{p}}$	${\mathit{x}}_{\mathit{c}}$	${\mathit{x}}_{\mathit{f}}$
	cm/s	µm	gpt	gpt
Case 1	0.54	190.7	10.5	18.9
Case 2	0.52	150.1	11.0	14.2

shows the values obtained for both ore samples. Although the value of $x_a$ varied slightly between both samples, according to Figure 4, the recovery of copper was sensitive to changes in this variable. In any case, these results correspond to different samples, and therefore different values of the input variables are expected in both cases.

Under these optimal conditions, flotation kinetic tests were performed to analyze whether the conditions used in the experiments (12 min) were adequate. The mathematical model of García-Zúñiga [92] was applied to model the kinetics:

$$R=R_{\infty }\left(1-e^{-k t}\right)$$

(2)

where $R$ and $R_{\infty }$ are the accumulated and infinite recoveries as percentages, respectively; $k$ corresponds to the kinetic constant (1/min), and $t$ is the time (min).

Table 9. Kinetics coefficients in optimized conditions for Case 1 and Case 2.

Case	Kinetics Parameters
	Cu			Fe
	${\mathit{R}}_{\infty }$ %	$\mathit{k}$ 1/min	Standard Error	${\mathit{R}}_{\infty }$ %	$\mathit{k}$ 1/min	Standard Error
Case 1	87.5	1.4	100.0	35.3	0.5	63.8
Case 2	89.8	1.2	119.1	54.9	0.8	114.4

gives the results, including standard error (SE), and Figure 6 shows the kinetics. The maximum recovery was obtained between 6 and 7 min; therefore, the flotation test of 12 min was correct. For both samples, the copper kinetics reached the maximum at 6 and 7 min for Case 1 and Case 2, respectively. The kinetic curves for both minerals are shown in Figure 6. It is worth noting that models other than the García-Zúñiga model can be used if the data are not adequately represented or according to the future application of the kinetic model.

The copper recoveries were 87.5% and 89.8% for Case 1 and Case 2, respectively. On the other hand, the Fe recoveries were 35.3% and 54.9% for Case 1 and Case 2, respectively. The QEMSCAN analysis showed a higher iron grade in these samples (Case 2), and it knows the high floatability of pyrite with respect to other sulfide ores. This can explain the higher recovery of iron in Case 2 than in Case 1.

The mineralogical report showed a more significant amount of pyrite, with 4.48%, for Case 2 versus 2.58% for Case 1. Another difference lies in the amount of clay, with 21.82% versus 15.39% for Case 1 and Case 2, respectively, and corresponding to the group of chlorites. These clays can cause flotation due to mechanical trapping of the ore in the foam, fouling the concentrate [93,94].

Regarding pyrite, it should be noted that it has a cathodic behavior on its surface. However, when in the presence of chalcopyrite, the latter also behaves as a cathode because it has a lower resting potential than pyrite. These galvanic interactions between both minerals affect the flotation between the two as pyrite is a noble mineral and chalcopyrite is active. This makes chalcopyrite promote pyrite flotation, and pyrite inhibits chalcopyrite flotation [95]. In the case of Case 2, in the sensitivity analysis, we made discrete changes to the values of aeration, foaming, and collector dosages. In addition, we significantly reduced particle sizes. These changes slightly increased chalcopyrite recoveries and iron increases, as mentioned above. The difference between both cases studied lies in the effects that generate flotation at low percentages of solids, such as Case 1 with 32% and 36% for Case 2. The latter is the sample that presented the highest clay content, which, considering its higher pulp density, could generate drag flotation, as stated by [95,96,97].

4. Conclusions

The proposed methodology allows the application of DoE, kriging, ANN, GSA, and SA to analyze and develop a model to represent complex systems, such as the flotation of minerals in the presence of clays and using seawater. The results gave an adjustment of over 0.98 for R² using ANN, in comparison to values of around 0.5 when the polynomial model of RSM was utilized. On the other hand, applying GSA identified the aeration rate and P₈₀ size as the most influential variables in copper recovery under the conditions studied. Additionally, we identified significant interactions that affect the recovery of copper, with the interactions between the aeration rate, frother concentration, and P₈₀ size being the most important. The proposed methodology allows for representing systems that have complex behavior that does not fit the second-order polynomial model and solves the difficulties of applying ANNs when there are few data. Additionally, it introduces the SA and GSA for the analysis of the effect of the independent variables on the dependent variables in RSM. Also, it proposes the use of more efficient algorithms for ANN training and optimization.