AN INVESTIGATION ON THE POTENTIAL RECOVERY OF COPPER FROM A LOW-GRADE COPPER RESOURCE CONTAINING HIGH SILICA CONTENT

2.1. Materials

The studied sample (about 150 kg) was collected from the Kuh Khairy mine, Iran. The received sample was then subjected to preparation operations. The sample was firstly crushed using a jaw crusher (Fritsch 01.703, Fritsch, Idar-Oberstein, Germany), and then by a cone crusher to attain particles smaller than 4 mm in diameter. After mixing and dividing the sample into four parts, half of the sample was coded, packaged, and archived, and then the other half was homogenized and ground in a ball mill so that above 90% of particles were smaller than 2 mm in diameter. After that, the sample was remixed and homogenized and then split by means of cones and riffles to obtain a representative sample. Finally, the chemical composition of the representative sample was identified using x-ray fluorescence spectrometer (Philips PW1480 x-ray spectrometer, Netherlands) as shown in Table 1. As can be seen in Table 1, the studied sample contains high levels of silicate and clay components, and the copper content in the sample is about 0.57%. The content of Si (0.467×52.88= 24.69) and Fe (0.7×14.81= 10.367) is also found to be 26.69, and 10.367%, respectively. In addition, the representative sample (less than 2 mm in diameter) was analyzed by an atomic absorption spectrometer (Thermo Elemental’s SOLAAR S Series, Waltham, MA, USA), and the copper content was found to be ~0.54%.

Table 1. Chemical compositions of the studied representative sample

2.2. Leaching experiments

Leaching tests were conducted on the prepared representative sample in 1 L beakers using a mechanical stirrer with adjustable stirring speeds. For each leaching test, initially, a sulfuric acid solution was prepared at a predetermined concentration. Then, 200 g of the representative sample along with a specified volume of the sulfuric acid solution based on the desired solid percentage was transferred into the beaker. After adjusting the stirring speed, the solution was stirred for a specific duration. After finishing leaching time, the resulting pulp was filtered, and the leach liquor was analyzed by atomic absorption spectrometer to specify the copper concentration. Ultimately, the leaching efficiency of copper (R) for each experiment was calculated utilizing the following formula:

[TeX:] R = \frac{C_{f} \times V}{C_{i} \times m} \times 100$R=\frac{C_f\times V}{C_i\times m}\times 100$ (1)

in which C_f describes the concentration of copper in the leach liquor or PLS (Pregnant Leach Solution) (g/L), V denotes the volume of the leach solution (L), C_i represents the copper content in the solid sample (%, i.e. here C_i was 0.54/100 based on atomic absorption spectrometer (AAS)), and m is the weight of the sample (g).

2.3. Experimental design based on RSM-CCD

Modelling and process optimization were accomplished utilizing response surface methodology (RSM) combined with central composite design (CCD) to obtain more accurate information about the influence of effective parameters on the dissolution process, especially interactive effects between the parameters. RSM-CCD is one of the most important and efficient DOE (design of experiments) techniques used to develop, improve, model, and optimize the processes and assess the relative importance of the operating parameters (Montgomery, 2001; Myers and Montgomery, 2002; Bezerra et al., 2008; Motamedizadeh et al., 2021). Thus, according to the RSM-CCD model and Equation 2, a set of 27 experiments (N=2^(5-1) +2×4+3=27) consisting of 16 partial factorial runs, 8 axial runs (α = 2) and 3 repeated experiments at the middle point was designed and conducted. Table 2 shows the opted factors and the operating conditions for performing leaching tests.

[TeX:] N = 2^{(n - p)} + 2n + n_{c}$N=2^{(n-p)}+2n+n_c$ (2)

where N is the total number of experiments, n depicts the number of parameters, nc exhibits the number of central experiments, and P is a fraction of the number of parameters (here P= 1) (Motamedizadeh et al., 2021).

Table 2. The experimental conditions for performing the leaching tests using RSM-CCD strategy and the determined values of recovery and the content of copper leached

2.4. Kinetic study

The reaction kinetics was investigated to identify and understand the mechanism of copper leaching process. The reactions occurring between the ore particles and leach liquor during the leaching process are typically heterogeneous and, in such a system, the expression of the overall reaction rate is complicated by the interactions between physical and chemical processes (Habbache et al., 2009; Ghasemi and Azizi, 2018). Based on previous studies, the reaction rate between mineral particles and acidic leach solution mainly follows the shrinking core models (SCMs). SCM models are preferred for the kinetic modelling due to their accurate representation for the actual leaching process (especially closer approximation of real particles in the different situations) compared to other models (Levenspiel, 1999; Nozari and Azizi, 2020; Apua and Madiba, 2021; Houshmand et al., 2024). In heterogeneous SCM models, the reactant particles are assumed to be nonporous, spherical shape, and their size does not change during reaction. Also, the reaction between the mineral particles and the leaching Lixiviant occurs on the outer surface of the particles (Ekmekyapar et al., 2015; Bai et al., 2018). However, these assumptions may not exactly match with the reality. Therefore, SCM models were utilized for kinetic modelling of copper leaching process. In this regard, the diverse SCM models were applied to the experimental data as shown in Equations 7-9.

[TeX:] 1 - (1 - X)^{\frac{1}{3}} = k_{\text{sc}} \times t$1-(1-X{)}^{\frac{1}{3}}=k_{\mathrm{\text{sc}}}\times t$ Surface chemical reaction (3)

[TeX:] 1 - (1 - X)^{\frac{2}{3}} = k_{\text{lf}} \times t$1-(1-X{)}^{\frac{2}{3}}=k_{\mathrm{\text{lf}}}\times t$ Liquid film diffusion (4)

[TeX:] 1 - 3(1 - X)^{\frac{2}{3}} + 2(1 - X) = k_{\text{pl}} \times t$1-3(1-X{)}^{\frac{2}{3}}+2(1-X)=k_{\mathrm{\text{pl}}}\times t$ Diffusion through a product layer (5)

In the aforementioned equations, k_sc, k_lf, and k_pl are the kinetic rate constants, individually, for surface chemical reaction, diffusion through liquid film, and product layer diffusion model (min⁻¹), X is the conversion fraction of copper recovered and t implies the dissolution time (min). Meanwhile, the process kinetics is commonly controlled by the slowest stage.

3.1. Impact of major parameters on copper leaching

Figures 1, and 4-7 display the impact of operating parameters including particle size, sulfuric acid dose, solid percentage (pulp density), stirring rate, and leaching time on the leaching efficiency and the content of copper in PLS solution. These factors were assessed on the basis of OFAT method when one parameter was changed and the other parameters were fixed at a constant value. The following observations can be made on the impact of the parameters on the leaching yield of copper.

The optimal and desirable particle size has a great impact on the dissolution amount of copper. As can be seen in Figure 1, the 0-500 µm particle size has a higher copper content and extraction efficiency. Interestingly, contrary to the expectation that copper dissolution should reduce with increasing particle size, it is observed that for particle sizes of 0–2 mm the dissolution rate increases. It is generally accepted that the leaching rate increases with decreasing particle size due to an improved liberation degree of the mineral, increased contact surface area, and enhanced permeability of the leaching agent (Janyasuthiwong et al. 2016; Hao et al. 2022; Tanaydin et al. 2022). However, this behaviour may vary among the size fractions of a given ore, depending on the mineralogy, mineral associations, and the dissemination of various species (Hansen et al., 2005).

Figure 1. The impact of particle size on the leaching efficiency under conditions: 25°C temperature, 70 kg/t acid concentration, 30% solid percentage, 150 rpm stirring speed, and 90 min leaching time

According to XRF and XRD analyses (see Figure 2), the studied sample significantly contains silicate and clay minerals, which affect the copper dissolution performance due to their different and complex structures. The presence of clay and silicate compounds in the sample can influence the consumption of acid (leaching agent), and this is more severe in the fine particles. For instance, some silicate minerals (like mica, and clay) can consume acid generated by the oxidation phenomenon, and/or can quickly adsorb acid (such as kaolinite, and montmorillonite) (Toro et al., 2021). In addition, clays and silicates have negative effects on most mineral processing units owing to their adverse impacts on the pulp rheology and cause many problems in the filtration stage (Luo et al., 2024). Thus, regarding the presence of clay and silicate compounds in the sample, as well as the high fines (which creates filtration problems) produced in the smaller size fraction, the fraction of 0-2000 µm was selected for further study. The particle size analysis for the fraction of 0-2000 µm is given in Figure 3.

Figure 2. X-ray diffraction (XRD) spectra of the studied sample

Figure 3. Particle size distribution for the investigated 0-2 mm sample

The nature of the leaching operations is based on the penetration of the lixiviant into the very small pores in the ore. Theoretically, an increasing concentration of lixiviant will increase the reaction rate. However, in leaching operations, especially acid leaching, an excessive increase in the concentration of the leaching agent, in addition to the non-optimal consumption of acid, leads to the dissolution of gangue compounds into the PLS solution. Figure 4 indicates the impact of acid dosage ranging from 50 to 130 kg/t on the leaching performance. It can be seen that the recovery rate and the value of leached copper increase with an increasing acid concentration from 50 to 80 kg/t, but a further increase in acid concentration has little effect on copper extraction efficiency and remains almost constant. This trend can be attributed to the change in leaching mechanism due to the change in acid concentration, and more importantly, due to the mineralogical composition of the studied sample (presence of silicate and clay compounds in the sample) which increases acid consumption. Thus, considering the copper content of about 1650 ppm in PLS liquor and also from an economic point of view, a concentration of 80 kg/t of acid was selected to continue the investigation.

Figure 4. The impact of acid consumption on the leaching efficiency under conditions: 25°C temperature, 0-2000 µm particle size, 30% solid percentage, 150 rpm stirring speed, and 90 min leaching time

The pulp density (solid percentage) depends on the leaching system and its role in the tank leaching is very important. Therefore, six leaching tests were conducted to determine the effect of pulp solids percentage at the values of 20, 25, 30, 35, 40, and 45% under constant operating conditions including ambient temperature, 0-2 mm particle size, pulp mixing rate of 150 rpm, sulfuric acid concentration of 80 kg/t, and leaching time of 90 min. The behaviour of solid percentage on the extraction rate from the studied sample is shown in Figure 5. It is observed that as the solid percentage increases from 20 to 40%, the amount of leached copper increases sharply, and with further increase, the leaching efficiency decreases due to lack of proper mixing. The reason for the increase in recovery with increasing solid percentage can be attributed to the better performance of the agitation process due to more and more effective collision of particles in the solution. Therefore, low solid percentages reduce the dissolution efficiency and require a large amount of acid for a small amount of the sample and, on the other hand an excessive increase in the solid percent creates problems for slurry mixing, pumping, and filtration operations. Hence, a solid percentage of 35 (given the recovery rate of about 70% and the copper content of about 2.15 g/L in PLS) was selected for further studies.

Figure 5. The impact of solid content on the leaching efficiency under conditions: 25°C temperature, 80 kg/t acid concentration, 0-2000 µm particle size, 150 rpm stirring speed, and 90 min leaching time

Figure 6 shows the leaching recovery as a function of pulp agitation rate. As can be considered, the highest amount of copper is leached in a stirring rate of 150 rpm, and high mixing speeds have a negative impact on the dissolution efficiency. In general, the pulp agitation increases the dissolution rate at the beginning of the operation, but has little effect as the operation continues. In many systems, it is sufficient to have an agitation speed that suspends the solid particles and prevents them from settling. Therefore, based on the results received and observations, a stirring speed of 150 rpm was used to continue the study.

Figure 6. The impact of pulp stirring rate on the leaching efficiency under conditions: 25°C temperature, 80 kg/t acid concentration, 0-2000 mm particle size, 35% solid percentage, and 90 min leaching time

Each leaching project has an economic time depending on the concentration of the dissolving agent, particle size, and leaching method, and if the leaching time is longer than this desired time, the project tends to become uneconomical. Thus, the effect of contact time of the received sample with the leaching agent (sulfuric acid) was studied from 15 to 180 minutes as shown in Figure 7. As expected, the extraction efficiency (copper recovery and copper content in the PLS solution) increased with leaching time rising and reached its maximum value (about 71% with a concentration of 2192 mg/L) after 2 h. Thereafter, the copper dissolution rate increased slightly with increasing time, and the changes in the copper leaching rate were very small.

Figure 7. The impact of leaching time on the leaching efficiency under conditions: 25°C temperature, 80 kg/t acid concentration, 0-2000 µm particle size, 35% solid percentage, and 150 rpm pulp stirring rate

3.2. Modelling and process optimization

3.2.1. Modelling and interactive effects between the operating parameters

After designing and carrying out the tests based on RSM-CCD strategy (see Table 2), the experimental data was statistically analyzed in the Design Expert software environment. In this regards, a quadratic polynomial model based on Equation (6) (Bezerra et al., 2008) was fitted to the data.

[TeX:] Y = \theta_{0} + \sum_{i = 1}^{k}\theta_{i}x_{i} + \sum_{i = 1}^{k}\theta_{\text{ii}}x_{i}^{2} + \sum_{1 \leq i \leq j}^{k}{\theta_{\text{ij}}x_{i}x_{j}} + \varepsilon$Y={\theta }_0+{\sum }_{i=1}^k{\theta }_i{x}_i+{\sum }_{i=1}^k{\theta }_{\mathrm{\text{ii}}}{x}_i^2+{\sum }_{1\le i\le j}^k{\theta }_{\mathrm{\text{ij}}}{x}_i{x}_j+\epsilon$ (6)

where Y denotes the process responses (leaching recovery of copper (%) or copper concentration leached (mg/L), k is the number of parameters, θ₀ depicts a constant term, θ_i is the coefficients of the linear terms, x_i and x_j display the parameters, θ_ii, represents the coefficients of the quadratic terms, α_ij is the coefficients of the interactive terms among the parameters, and ε implies the residual error (the difference between the measured and the predicted values).

Ultimately, after removing the very negligible terms (p-value > 0.05), the final regression models developed to approximate the leaching rate (R_Cu) and the content of copper leached were obtained according to Equations 7 and 8, respectively.

[TeX:] R_{\text{Cu}} = + 66.69 + 1.91 \times A + 2.09 \times B + 2.19 \times C - 0.12 \times D - 0.76 \times A \times D - 0.9 \times B \times C - 0.58 \times B \times D - 0.88 \times A^{2} - 0.74 \times C^{2} - 0.61 \times D^{2}$R_{\mathrm{\text{Cu}}}=+66.69+1.91\times A+2.09\times B+2.19\times C-0.12\times D-0.76\times A\times D-0.9\times B\times C-0.58\times B\times D-0.88\times A^2-0.74\times C^2-0.61\times D^2$ (7)

[TeX:] C_{\text{Cu}} = + 1569.52 + 46.87 \times A + 436.07 \times B + 50.62 \times C - 5.37 \times D + 52.51 \times B^{2}$C_{\mathrm{\text{Cu}}}=+1569.52+46.87\times A+436.07\times B+50.62\times C-5.37\times D+52.51\times B^2$ (8)

In the above Equations, A, B, C, and D represent the sulfuric acid concentration, solid percentage, leaching time, and pulp agitation rate, respectively. Also, all parameters are based on the coded units. The coded amounts were used for simplifying the calculations and uniform comparison and determined according to the following formula (Motamedizadeh et al., 2021).

[TeX:] X_{i} = \frac{x_{i} - x_{0}}{\text{Δx}},i = 1,2,,3,...$X_i=\frac{x_i-x_0}{\mathrm{\text{Δx}}},i=1,2,,3,...$ (9)

in which X_i is the value of coded parameter, x_i is the value of real parameter, x₀ is the magnitude of real parameter at the centre point, and (x) is the step change in the actual parameter.

The final quadratic polynomial models developed were statistically evaluated employing analysis of variance (ANOVA) at a 95% confidence level (p-values less than 0.05) as presented in Tables 3 and 4.

Table 3. The results of ANOVA of the model proposed to approximate the leaching rates of copper and determine the relative importance of the influential operating parameters

Table 4. The results of ANOVA of the model proposed to approximate the copper concentration leached and determine the relative importance of the influential operating parameters

The high F-values and low probability levels (p < 0.05) in the ANOVA tables indicate the model proposed and the parameter is statistically significant. As can be seen, the developed models show an excellent relationship between the operational parameters and the process responses with p-values less than 0.0001, and so they can be used to predict the response variable. Meanwhile, the difference between (R²) and adjusted (R²) of 0.0379 (The leaching efficiency) and 0.0026 (the copper content leached) indicate a very good correlation between experimental data and the fitted model. Additionally, the larger F-values and smaller p-values indicate which the parameter has a greater influence than the other parameters. It is observed from Table 3 that the leaching time, solid percentage and acid concentration have the greatest impact on the copper leaching recovery, respectively, while pulp mixing rate does not have much effect on the efficiency of process. It is also seen that the leaching efficiency of copper highly depends on the interactive impacts among the parameters. The significance degree of the parameters on the copper concentration leached is found to be in the order of the quadratic effect of solid percent (B²) > the linear effect of solid percent (B) > the linear effect of leaching time (C) > the linear effect of acid concentration (A) > the linear effect of stirring rate (D). The perturbation graph (see Figure 8) showing the effect of all operating parameters on the process responses (the leaching rate and the content of copper leached) also confirms these results. This graph displays the impacts of all the parameters at a special point in the design space. The sharp slope or curve in graph implies the sensitivity of the leaching rate and the content of copper leached to that parameter. As observed in Figure 3, all parameters except pulp stirring rate significantly affect the process responses, confirming the results derived from ANOVA tables.

Figure 8. Perturbation plot showing the impact of all operating parameters on the leaching recovery and the concentration of copper leached

In addition to the above, the three-dimensional (3D) response surface plots were utilized to attain a better understanding of the main and synergistic effects of parameters on the leaching recovery and content of copper leached, as shown in Figures 7-10. These graphs are obtained based on the change in two variables when other variables are fixed at their middle level, and display the nature and magnitude of possible interactions between various parameters.

Figure 9. 3D graphs displaying the combined impact of acid concentration and solid percentage on the recovery and concentration of copper leached

Figure 9 shows the simultaneous effect of acid concentration and solid percentage on the recovery and concentration of copper in the PLS solution. As can be seen, copper recovery is a quadratic function of acid concentration and has a linear relationship with solid percentage. It is obvious that at low and high solids percentages, copper recovery enhances with an increase in the acid concentration, and this increase is more intense at lower solid percentages. The highest copper recovery is observed at a solid percentage of 35 and an acid concentration of 90 kg/t, which, under these conditions, reaches more than 71% with a copper concentration of ~2170 mg/L in PLS solution. It is noteworthy that under these conditions, the mixing rate was 150 rpm and the leaching time was 2.5 hours. Additionally, it was found that the effect of acid concentration on copper recovery is greater and the effect of solid percentage on the leached copper concentration is higher. At the lower solid percentage, the acid has more surface area and effective contact to react with copper-bearing minerals, leading to higher leaching rate, but only if the acid is not overly concentrated, as it may cause excessive dissolution of other unwanted elements. While, at high solid percentages, the leaching pulp becomes denser, reducing the effectiveness of acid due to less surface area exposure. In this case, the higher acid concentrations may be required to compensate for the reduced contact, though this can increase costs and side reactions. Wang et al. (2019) attributed the increase in copper dissolution with increasing acid concentration to the enhanced activity of hydrogen ions (H⁺). They also reported that the upward trend in copper dissolution rate decreases with a further increase.

The combined effect of acid concentration and solid percentage with pulp mixing rate are shown in Figures 10 and 11, individually. It can be seen that the dissolution rate of copper increases with the pulp mixing speed at the beginning of the operations, and generally the pulp stirring speed had a smaller effect on the dissolution performance compared to the other two parameters, showing that the copper dissolution process is affected by a certain value of pulp mixing rate. Additionally, according to Figures 10 and 11, the dissolution efficiency increases greatly by increasing the acid concentration and solid percentage at a lower mixing rate within 2.5 h. At lower mixing rates, high acid concentration compensates for slower mass transfer by offering strong chemical reactivity, leading to increased leaching efficiency. A high solids percentage provides more material for the acid, and increases overall dissolution at low stirring rates. However, an excessive solid percentage will cause the leaching pulp to be denser and reduce the copper recovery. Nevertheless, the combination of these factors can still significantly improve efficiency, even if stirring is not optimal. It is also observed that when the acid concentration and stirring speed are simultaneously low, the recovery decreases. Similar results were found in the work reported by Maleki et al. (2023).

Figure 10. 3D graphs displaying the combined impact of acid concentration and pulp mixing rate on the recovery and concentration of copper leached

Figure 11. 3D graphs displaying the combined impact of solid percentage and pulp mixing rate on the recovery and concentration of copper leached

Figure 12 exhibits the interaction between the solid percentage (pulp density) and the contact time of leaching agent with the sample. As can be seen, these two parameters have a synergistic effect in increasing copper leaching efficiency, and this synergistic effect is greater at lower solid percentages, although it reaches its maximum value at 35% solids and a contact time of 2.5 hours. In general, increasing the leaching rate with a decrease in solid/liquid ratio can be due to the greater amount of acid per unit amount of solid sample and the decrease in mass transfer resistance (Hosseinzadeh et al., 2021).

Figure 12. 3D graphs displaying the combined impact of solid percentage and leaching time on the recovery and concentration of copper leached

3.2.2. Process optimization

The numerical optimization of leaching process was conducted using Design Expert (DX) software and employing the desirability function procedure. The optimal operating conditions proposed by RSM modelling for achieving the highest leaching efficiency and the leached copper content are illustrated in ramp plots in Figure 13. As observed, the optimized values of leaching parameters for -2 mm size fraction as follow: 90 kg/t sulfuric acid concentration, 35% solid content, 150 rpm agitation rate, and leaching time of 2.5 hours. It is found that under these optimal conditions, a PLS solution with a copper concentration of ~2160 mg/L and an efficiency of more than 71% can be achieved. It is noteworthy that in the confirmation experiment conducted under the proposed optimum conditions similar to Run 21 (see Table 2), a recovery of about 70.5% with a copper content of 2165 mg/L in PLS was achieved.

Figure 13. The optimized values of operating parameters to achieve maximum copper dissolution efficiency (recovery and content of copper in PLS solution)

3.3. Leaching kinetic modelling

To model, the left side of Equations 3-5 was plotted as a function of leaching time for each operational variable, including the concentration of leaching agent, solid percentage (solid/liquid ratio), pulp agitation speed, and leaching temperature. For instance, the results obtained from the temperature parameter are shown in Figure 14, in which the slope of the straight lines represents the kinetic rate constant for copper leaching. The values of kinetic constants and correlation coefficients for each parameter are provided in Table 5. According to Figure 14 and Table 5, the diffusion equation through a product layer exhibits the best correlation relative to other kinetic equations, demonstrating that this model is the controlling step of the reaction rate.

Figure 14. The graph of the SCMs models versus leaching time at different temperatures (25, 30, 40, and 50)

Table 5. The kinetic analysis of the shrinking core models fitted to the experimental data

To better distinguish the reaction mechanism, the activation energy was determined by analyzing the data and applying the Arrhenius equation according to the following equation (Levenspiel, 1999; Adebayo and Olasehinde, 2015):

[TeX:] k = A \times e^{\frac{- E_{a}}{R \times T}} \Rightarrow Ln(k) = Ln(A) - \frac{E_{a}}{R}.\frac{1}{T}$k=A\times e^{\frac{-E_a}{R\times T}}\Rightarrow Ln(k)=Ln(A)-\frac{E_a}{R}.\frac{1}{T}$ (10)

where k implies the reaction rate constant (min⁻¹), A illustrates the frequency factor (min⁻¹), E_a denotes the activation energy (J/mol), R depicts the universal gas constant (8.314 J·K⁻¹·mol⁻¹), and T is for the leaching temperature (K).

To measure the activation energy, Ln(k) versus (1/T) for each temperature was plotted, as shown in Figure 15. The slope of the straight line in this graph represents [TeX:] - \frac{E_{a}}{R}$-\frac{E_a}{R}$. Thus, the activation energy of the diffusion model through a product layer was calculated 14.15 kJ/mol (-E_a/R=-1702.4→E_a=-8.314×-1702.4=14154J=14.15kJ). According to Espiari et al. (2006) and Ekmekyapar et al. (2012), the values of E_a less than 40 kJ/mol demonstrate that the leaching process is controlled by diffusion phenomena. Ekmekyapar et al. (2015) investigated the leaching kinetics of malachite ore in ammonium sulfate solutions and reprotted the diffusion model through the product layer was the rate-controlling step of the dissolution kinetics. Hosseinzadeh et al. (2021) performed a similar kinetic study on copper leaching from a low-grade copper oxide source, and reported that the predominant mechanism was a difusion process with the activation energy of 11.72 kJ/mol. Therefore, by combining Equations 5 and 10 and ascertaining the influence of parameters on the apparent rate constant given in Table 5, the mathematical equation showing the kinetics of copper leaching can be described as follows:

[TeX:] 1 - 3(1 - x)^{\frac{2}{3}} + 2(1 - x) = A \times (SA)^{s} \times (SP)^{p} \times (SR)^{r} \times e^{\frac{- \text{Ea}}{R \times T}} \times t$1-3(1-x{)}^{\frac{2}{3}}+2(1-x)=A\times (SA{)}^s\times (SP{)}^p\times (SR{)}^r\times e^{\frac{-\mathrm{\text{Ea}}}{R\times T}}\times t$ (11)

in which A, SA, SP, and SR represent frequency factor, sulfuric acid concentration, solid percentage, and pulp stirring rate, individually, and the constants s, p, and r depict the reaction order for each the corresponding parameter. The reaction orders (constants) were determined by plotting Ln(SA), Ln(SP), and Ln(SR) against Ln(kpd) and approximating the slope of straight line each curve. The constants s, p, and r were estimated to be 1.3021, 2.1337, and 0.8689, respectively. Ultimately, the ultimate model describing the leaching kinetics of copper was understood as Equation 12:

[TeX:] 1 - 3(1 - x)^{\frac{2}{3}} + 2(1 - x) = A \times (SA)^{1.3021} \times (SP)^{2.1337} \times (SR)^{0.8689} \times e^{\frac{- 14.15 \times 10^{3}}{8.314 \times T}} \times t$1-3(1-x{)}^{\frac{2}{3}}+2(1-x)=A\times (SA{)}^{1.3021}\times (SP{)}^{2.1337}\times (SR{)}^{0.8689}\times e^{\frac{-14.15\times {10}^3}{8.314\times T}}\times t$ (12)

Figure 15. Arrhenius graph for product layer diffusion model to approximate the activation energy