Flotation tests were conducted on a copper sulfide ore sample from the low-grade ore storage area at the Kakula mine in the southeastern part of the Democratic Republic of Congo (DRC). During the flotation tests, Sodium Isobutyl Xanthate (SIBX) was used as the collector (product of Axis House), Hydrofroth (HDF) was employed as a frother, and AERO (promoter) served as the secondary collector. The water used in our experiments originated from the Kakula wastewater treatment plant. The chemical characterization of the sample was carried out for various metals using the X-ray Fluorescence Spectroscopy (Niton™ XL5 Plus Handheld, Japan).

The study began with developing the Taguchi experimental design, followed by sample collection at the Kakula mine. The experiments were conducted in the Kakula concentrator laboratory. The research was concluded by statistical analysis of results using Minitab 21.4.0 software.

The tests were conducted using an Altso flotation machine with a parallelepiped cell capacity of 2.8 liters. The unit features a variable digital display of shaft rpm (revolutions per minute), allowing for precise agitator speed control, which is crucial for mastering aeration and agitation intensity. It also includes air pressure, flow control, and a digital timer for accurate test duration. The cell agitator’s rotation speed was set at 600 rpm. Various materials were employed to facilitate our experiments on the flotation machine, starting from sample collection to chemical analysis of the samples. A scoop and a bag were used for sample collection from the low-grade ore storage area at the mine. A laboratory jaw crusher and a RockLab were utilized for particle size reduction from + 8 mm to 2 mm. A rotary sample divider ensured sample representativeness. A series of sieves (25, 38, 53, 75, 106, 150 μm) was also used for particle size analysis and grindability studies. A laboratory rod mill was used for mineral grinding. A vacuum pump, filter paper, pans, an oven, and a plate, as well as a balance, pipettes (2 μl - 20 μl; 50 μl - 200 μl), and syringes (3, 5, 10 ml), were used respectively for filtering the rough and waste materials, containing the samples, drying the samples, weighing the samples, cleaning the mill, adding water to the cell, extracting the quantity of reagents, preparing the reagents, taking water to put in the mill, and holding the pulp.

To successfully conduct the tests, a sample of 100 kg of low-grade ore was collected from the low-grade ore storage area on the Kakula underground mine. Fifteen sample points were randomly targeted on the low-grade ore storage area to ensure representativeness of the sample.

Based on the previous study on the mineralogy of the Kakula deposit conducted by XPS Expert Process Solutions, the Kakula ore is rich in chalcocite. Other major mineral phases are bornite and chalcopyrite. Gangue minerals are quartz, feldspar, micas, and chlorite. Fe oxides and carbonates are minor phases [13] . This work assumes that the ore sample contains the same mineral phase.

The sample, as received, consisted of coarse particles larger than 8 mm. Initially, it underwent primary crushing using a laboratory jaw crusher and secondary crushing in the rock lab. Subsequently, the ore was subjected to wet grinding at various intervals to determine the correlation between particle size and grinding time. The mill was wet fed with 0.835 kg, 1.035 kg, and 1.295 kg, respectively, corresponding to 25%, 30%, and 35% solid percentages. Grinding time was determined for the 15%, 20%, and 25% rejection range on the 53 µm sieve, following variations in solid percentage.

Based on previous studies and literature, the air flow rate is not considered as a parameter. This study chose the percentage solids, air flow rate, oversize at 53µm, collector dosage, frother dosage, and secondary collector dosage as the six main parameters to be investigated. According to the Taguchi design methodology, three levels were chosen for each parameter as illustrated in Table 1 . Considering the six parameters and the relevant three levels, a Taguchi orthogonal array (L27) was established to execute the experiment. The Kakula Concentrator's operational Conditions inspired the selection of three levels for each factor. Twenty-seven experiments were therefore conducted as shown in Table 2 . Numbers one (1) to three (3) in Table 2 are the relative levels of the parameters indicated in Table 1 . In this study, interactions between parameters were not taken into consideration.

Note: ^aLitter per min, ^bgram per ton.

In experiments based on the Taguchi design, signal-to-noise (SN) ratios are recommended for optimizing process parameters [20] [26] . The term Signal-to-Noise Ratio (SN) establishes a relationship between the average of the response (signal) and the dispersion of the response concerning the noise (background noise) [15] . The SN ratios can then be calculated based on the required response characteristics, i.e., “larger is better”, “smaller is better”, and “nominal is better”. In this study, the aim is to increase recovery yield and copper content. Therefore, SN ratios for “larger is better” and smaller is better were selected and calculated using response values. The SN ratio for this scenario was determined using Equations (1) and (2) [22] :

$V_i=\log \left(\frac{1}{n}{\displaystyle \underset{j=1}{\overset{n}{\sum }}{Y}_{ij}^2}\right)$ (1)

$V_i=\log \left(\frac{1}{n}{\displaystyle \underset{j=1}{\overset{n}{\sum }}\frac{1}{Y_{ij}^2}}\right)$ (2)

Where n is the number of repetitions of experiments, and $Y_i$ is the response value (recovery) of the i-th experiment, and j successively takes the values $1,2,\cdot \cdot \cdot ,n$.

The flotation test was conducted according to the Kamoa standard. The flotation time was 40 minutes, with an additional 14 minutes of conditioning (5 minutes for the first adding reagent, 3 minutes for the second, and 2 minutes for the third until five adding reagent). Before beginning the flotation test, the pH of the pulp was fixed at 8.66. For each trial, five (5) concentrate fractions were removed, along with the tailing fraction. The flotation test process carried out during the study can be summarized in Figure 1 . The method of obtaining pulp with a well-defined solid percentage was first carried out through wet grinding. Next, a 5-minute conditioning step was performed by adding 60% SIBX, 57% Hydrofroth, and 100% AERO. The reagents, such as SIBX, Hydrofroth, and AERO, were prepared at 1%, 97%, and 50%, respectively.

The amount of metal copper recovered was calculated using Equation (3), which is based on assays alone of the feed (f), tailings (t), and concentrate (c) [9] :

$\text{Recovery}\text{}=100\left(\frac{c}{f}\right)\left(\frac{f-t}{c-t}\right)$ (3)

The theory of analysis of variance (ANOVA) aims to analyze a situation where the effect of one or more factors on a variable is investigated. In other words, it is a method for studying the variability of a product based on a set of production factors that can be systematically controlled [22] It uses statistical tests to evaluate the variation between groups (intergroup variance) compared to the variation within groups (intragroup variance). More formulas used in ANOVA can be shown in Table 3 .

In Equations (4) and (5), N is the total number of observations or experiments, Y_i is the metal recovery for the i^tch experiment. In Equations (6) and (7), p is one of the experiment’s parameters, and j is the level number of this p parameter. NK is the number of levels for each parameter (3 in this case), and $N_{k-1}=f_p$ is the degrees of freedom for the p parameter. In Equation (8), (f_T) is the total degrees of freedom of the result and is equal to the total number of experiments (27 in this case) minus one. In Equation (9), V_E is the error’s mean square (variance). This is used to evaluate the significance of the parameters on the response. Minitab breaks down the SS Regression or SS treatments component into the amount of variation explained by each term using both the sequential sum of squares (Seq SS) and adjusted sum of squares (Adj SS) after Minitab defines the mean square Adjusted (MS Adj) to determine the F-Value as presented in Equations (10) and (11).

The Taguchi methodology operates on the principle that the trial corresponding to the optimal conditions found may or may not be conducted during the experimentation phase, but the prediction of the performance value of the experiment can be made based on the prediction function represented by Equation (12) [8] .

$Y_{opt}=\frac{T}{n}+\left(A_j-\frac{T}{n}\right)+\left(B_j-\frac{T}{n}\right)+\cdot \cdot \cdot$ (12)

With T representing the sum of all trial responses, n the total number of trials, and $A_j,B_j,\cdot \cdot \cdot$ the average responses at level i, j, etc.

To optimize all outcomes simultaneously, a compromise is necessary. Based on [22] , the desirability function is the most effective way to achieve this. Minitab Software was used to calculate composite desirability as the weighted geometric mean of the individual desirability for each response. This mathematical approach ensures that if any individual’s desirability is zero (implying that one response is unacceptable), the composite desirability will also be zero, thereby highlighting a critical failure point. The formula for composite desirability is given by (Equation (13)):

$D={{\displaystyle \underset{i=1}{\overset{n}{\prod }}\left(d_i^{w_i}\right)}}^{\frac{1}{{\displaystyle \underset{i=1}{\overset{n}{\sum }}{w}_i}}}$ (13)

$d_i$ represents the individual desirability score for the i-th response. $w_i$ denotes the importance value assigned explicitly to the i-th response, reflecting its relative priority. n is the total number of responses included in the optimization. We fixed $w_i$ at 10, 10, 5, 5 for the recovery, silica yield, mass pull, and copper content responses.

A quadratic model was established for each response as a function of the control variables. This was done to determine the relationship between the technological performance factors (copper recovery, mass pull, copper content, and silica yield) and the control parameters. Several models were tested before considering the models presented in Table 17 .

$y={\beta }_0+{\displaystyle \underset{i=1}{\overset{k}{\sum }}{\beta }_i{x}_i}+{\displaystyle \underset{i=1}{\overset{k}{\sum }}{\beta }_{ii}{x}_i^2}+{\displaystyle \underset{i=1}{\overset{k}{\sum }}{\displaystyle \underset{j=1}{\overset{k}{\sum }}{\beta }_{ij}{x}_i{x}_j}}+\epsilon$ (14)

Where $x_1,x_2,\cdot \cdot \cdot ,x_k$ are input factors that influence the responses y; ${\beta }_o,{\beta }_{ii}\left(i=1,2,\cdot \cdot \cdot ,k\right),{\beta }_{ij}$ $\left(i=1,2,\cdot \cdot \cdot ,k\right)$ are constant parameters of the model, and $\epsilon$ is the random error. Furthermore, cross-factors and variables were applied to the models to ensure fit. Statistical parameters such as R² and S (a parameter accounting for residuals) were a guide for good model fit.

Table 4 details the chemical analysis of the sample examined in this study. The analysis revealed a copper concentration of 4.29%, alongside a significant silica presence of 55.81%, which poses a considerable challenge for the process. Figure 2 visualizes the results of the particle size chemical analysis conducted on the ore sample. This figure shows that a significant proportion of copper is below 53 µm. These findings corroborate the metallurgical test results reported recently during feasibility studies related to the Kakula deposit [13] .

Based on the grindability study results presented with the equations of the curves below ( Figure 3 ), the times required were determined (see Table 5 ) to achieve 15%, 20%, and 25% oversize at 53 µm sieve.

The average recovery of copper by flotation and the copper content for the various combinations of process parameters and levels are shown in Table 6 . The highest recovery of copper of 95,45% into the concentrate occurred in the 24th experiment run (% solids = 35; oversize at 53 µm = 20%; air flow rate = 6l/min; SIBX dosage = 306 g/t; HDF dosage = 163g/t; AERO dosage = 30 g/t) and the lowest (86.88%) was recorded in the 27th experiment run ((% solids = 35; oversize at 53 µm = 25%; air flow rate = 8l/min; SIBX dosage = 206g/t; HDF dosage = 163g/t; AERO dosage = 30g/t). The highest copper content of 12.30% of the concentrate, occurred in the 18th experiment run.

To determine the effect that each parameter has on the responses (copper recovery, copper content, mass pull, and Silica yield), the signal-to-noise ratio (SN) was calculated for each experiment. The SN ratio is based on “larger is better” (Equation (1)) or “smaller is better” (Equation (2)). The average SN ratios of each level of the six parameters for the flotation of copper were calculated and listed in Tables 7-10 . Delta values, which show the relative significance of the factors, were also calculated as the difference between the maximum and minimum average values. Based on this, the collector SIBX dosage was the most influential factor affecting the copper recovery response. The Hydrofroth dosage had the least influence on copper recovery response. Furthermore, Solid mass percentage is the most influential factor affecting the copper content, Mass pull, and Silica yield responses.

The level of a parameter with the highest average SN ratio corresponds to a better performance and gives the best combination level. Therefore, the optimal levels of flotation parameters have the most significant average SN ratio. The optimum levels of parameters for recovery of copper are solid mass percentage, 25% (Level 1); oversize at 53 µm, 20% (Level 2); air flow rate, 10 l/min (Level 3); SIBX dosage, 256 g/t (Level 2); Hydrofroth dosage, 133 g/t (Level 2); and AERO dosage, 50 g/t (Level 3). As far as, the optimum levels of parameters for copper content response are solid mass percentage, 30% (Level 2); oversize at 53 µm, 15% (Level 1); air flow rate, 6 l/min (Level 1); SIBX dosage, 206g/t (Level 1); Hydrofroth dosage,133 g/t (Level 2); and AERO dosage, 10 g/t (Level 1). The optimum levels of parameters for mass pull response are solid mass percentage, 35% (Level 3); oversize at 53 µm, 20% (Level 2); air flow rate, 10 l/min (Level 3); SIBX dosage,256 g/t (Level 2); Hydrofroth dosage,163 g/t (Level 3); and AERO dosage, 30 g/t (Level 2). The optimum levels of parameters for silica yield response are solid mass percentage, 30% (Level 2); oversize at 53 µm, 20% (Level 2); air flow rate, 10 l/min (Level 3); SIBX dosage,206g/t (Level 1); Hydrofroth dosage, 163 g/t (Level 3); and AERO dosage, 10 g/t (Level 1). A plot of the graph of the parameter against its response on each level is referred to as the main effect, and it indicates the general trend of influence of each parameter ( Figure 4 ).

The parameters that significantly affected responses were investigated using the analysis of variance (ANOVA). The analysis of variance for copper recovery shows that at a confidence level of α = 0.95, the calculated F-values for parameters A (F = 0.62), B (F = 1.12), C (F = 1.54), D (F = 2.05), E (F = 0.26), and F (F = 0.42) not exceed the critical F-value read from Fisher tables (F (2,14) = 3.739) ( Table 11 ). None of these factors significantly affects the recovery of copper at the tested confidence level of 95%. Regarding importance, the critical probabilities or p-values for A, B, C, D, E, and F parameters are 0.553, 0.354, 0.249, 0.165, 0.776, and 0.667, respectively. This allows us to judge the D parameter as the most significant, as it has the least critical probability.

Analysis of variance for copper content in concentrate shows that at a confidence level of α = 0.95, the calculated F-values for parameters A (F = 12.7) exceed the critical F-value read from Fisher tables (F (2,14) = 3.739) ( Table 12 ). This parameter is therefore statistically significant for copper content. Regarding importance, A’s critical probabilities or p-value is 0.001 (<5%). This allows us to judge the A parameter as the most significant. Furthermore, the A parameter is statistically significant for mass pull and silica yield ( Table 13 and Table 14 ).

The established optimal conditions, defined by specific levels of factors A to F, enabled the maximization of the signal-to-noise ratio (SN) for copper recovery and content, as well as for mass pull (M_Pull), while minimizing silica yield (S_Y) ( Table 15 ). This optimization suggests improved process stability in the face of operational parameter fluctuations. Although copper recovery reaches a high level (96.4%) under these ideal conditions, a trade-off is evident with the silica yield (35.4%). To overcome this imbalance, a multi-objective approach would prove essential to simultaneously improve copper recovery and minimize silica entrainment in the concentrate.

The Taguchi method identified the best values for each outcome. However, a multi-objective approach was necessary for this study to prove essential to obtain optimal parameters that can improve copper recovery and minimize silica yield. Table 16 provides statistical parameters such as R² and S (a parameter accounting for residuals). A quadratic model was established for each response as a function of the control variables ( Table 17 ).

Note: ^aCopper Recovery, ^bMass pull, ^cSilica yield.

The obtained models were applied to the multi-objective method’s algorithm using Minitab 21.4.0 software. Thus, the optimized results are visualized in Figure 5 . It can be seen that to achieve the best overall results (95.4% copper recovery, 48.0% mass pull, 12.2% copper content, and 24.8% silica yield), the following control variables should be used: 28.53% mass solid percentage, 21.1% oversize at 53 µm sieve, 10 l/min air flow rate, 249.4 g/t of SIBX, 163.0 g/t of Hydrofroth, and 10 g/t of AERO.

Comparing the results obtained by the Taguchi and multi-objective methods, the multi-objective approach contributes more to minimizing silica, 24% versus 35%. This means a small percentage of silica would be carried into the concentrate.