The galena ore used for this study was collected from Abakaliki, Enyigba mining site in Ebonyi State of Nigeria. The galena ore was finely pulverized and sieved with a 75 µm size sieve. All experiments were performed with the 75 µm fraction. HCl solutions were prepared from analytical grade reagents with deionized water.

The responses obtained from different experimental runs carried out by combination of five variables are tabulated in the response column of Table 3. The five

experimental variables gave a total of 32 experimental runs comprising of 26 individual runs and 6 similar runs. The responses obtained from various experimental runs were quite unique, indicating that each of the factors have considerable effect on the response.

To determine the adequacy of the models depicting the removal of lead by the dissolution of galena in hydrochloric acid, two different tests, i.e. the lack of fit test and the model summary statistics, were conducted. From the lack of fit test, the highest order polynomial with non-significant p-value and is not aliased would normally be chosen. This test is used in the numerator in an F-test of the null hypothesis and indicates whether a proposed model fits well or not. It compares the variation around the model with pure variation within replicated observations and measures the adequacy of the model based on response surface analysis [8] . The results from the lack of fit test indicated that the 2FI, linear and the cubic model (aliased) did not provide a good description of the experimental data. The results are tabulated in Table 4. From the model summary statistics, it can be seen that the R-squared, adjusted R-squared and the predicted R-squared values for the quadratic model (0.9916, 0.9762, and 0.8565) showed a better correlation when compared with the cubic model (0.9948, 0.9732 and −1.4789) and the linear model (0.7281, 0.6758 and 0.6545) as shown in Table 5. The measure of how efficient the variability in the actual response values can be explained by the experimental variables and their interactions is given by the R-squared value. If the R-squared predicted and adjusted are too far from each other, there may be a problem with either the data or the model [8] . The results are tabulated in Table 5. The afore-mentioned results indicate that the quadratic model provided an excellent explanation for the relationship between the independent variables and the corresponding response. With respect to these results, the effect of each

df = degree of freedom.

PRESS = Predicted residual sum of squares.

parameter was evaluated using the quadratic model as shown in Table 6.

The second-order model tested at the 95% confidence level obtained for extraction of lead from galena is shown in Equation (3):

Yield = 83.52 + 3.44 X 1 + 3.39 X 2 − 3.57 X 3 + 3.71 X 4 + 3.41 X 5 − 0.31 X 1 X 2     + 0.24 X 1 X 3 − 0.28 X 1 X 4 − 0.35 X 1 X 5 + 0.37 X 2 X 3 − 0.14 X 2 X 4     − 0.24 X 2 X 5 + 0.36 X 3 X 4 − 0.26 X 3 X 5 − 0.77 X 4 X 5 − 2.23 X 1 2     − 2.25 X 2 2 − 2.07 X 3 2 − 2.26 X 4 2 − 1.96 X 5 2 (3)

The results were analyzed by using the analysis of variance (ANOVA) suitable for experimental design used and shown in Table 6. The ANOVA of the quadratic regression model indicates that the model is significant. The model F-value of 64.67 implied the model to be significant and there is only 0.01% chance that an F-value this large could occur due to noise. The F-value of the independent variables X₁, X₂, X₃, X₄, and X₅ were estimated as 182.94, 177.66, 197.39, 212.39 and 179.41 respectively, showing that the effects of most independent variables on the dependent variable were significantly high. Model F-value was calculated as ratio of Adj. mean square of the regression and Adj. mean square of the residual. The model P-value (Prob. > F) is very low which shows that the model is significant. The P-values were used as a tool to check the significance of each of the model coefficients. The smaller the P-value the more significant is the corresponding coefficient. Values of P < 0.05 indicate the model terms to be significant. The values of P for the coefficients estimated indicate that among the

tested variables used in the design, X₁, X₂, X₃, X₄, X₅, X₄X₅, X 1 2 , X 2 2 , X 3 2 , X 4 2 , X 5 2 (where X₁ = leaching temperature, X₂ = acid concentration, X₃ = solid/liquid ratio, X₄ = stirring rate, and X₅ = leaching time) are significant model terms. The “Lack of Fit F-value” of 1.59 implies that the Lack of Fit is not significant relative to the pure error. There is a 31.32% chance that a “Lack of Fit F-value” this large could occur due to noise. Non-significant lack of fit is good because it indicates that the model is well fitted. Since many insignificant model terms have been eliminated, the improved model can be used to predict effectively, the responses of the percentage recovery of lead from galena. The model equation with the significant coefficient is shown in Equation (4).

Yield = 83 . 52 + 3 . 44 X 1 + 3 . 39 X 2 − 3.57 X 3 + 3.71 X 4 + 3.41 X 5 − 0.77 X 4 X 5     − 2.23 X 1 2 − 2.25 X 2 2 − 2.07 X 3 2 − 2.26 X 4 2 − 1.96 X 5 2 (4)

In terms of the actual factors the model equation (Equation (5)) is as follows:

Yield = − 1244 .0 9 + 7 .0 5 ∗ Leaching temperature + 16 . 19 ∗ Acid concentration     + 98 . 32 ∗ Solid / Liquid ratio + 0. 15 ∗ Stirring rate     + 0. 82 ∗ Leaching time − 1 . 67E − 00 4 ∗ Stirring rate ∗ Leaching time     − 9 . 92E − 00 3 ∗ Leaching temperature 2 − 1 . 44 ∗ Acid concentration 2     – 2 0 7 0 4 . 55 ∗ Solid / Liquid ratio 2 − 9 . 4 0 E − 00 5 ∗ Stirring rate 2     – 2 . 18E − 00 3 ∗ Leaching time 2 (5)

The CV called coefficient of variation which is defined as the ratio of the standard deviation of estimate to the mean value of the observed response is independent of the unit. It is also a measure of reproducibility and repeatability of the models [9] [10] . The results indicated the CV value of 1.65% which illustrated that the model can be considered reasonably reproducible [9] . The signal to noise ratio which is given as the value of the adequate precision is 31.307 as shown in Table 7. This indicates that an adequate relationship of signal to noise ratio exists. The result shows that the model can be used to navigate the design space.

The data were also analyzed to check the correlation between the experimental and predicted dissolution yield (Y%), as shown in Figure 4. The experimental values were the measured response data for the runs designed by the CCD model, while the predicted values were obtained by calculation from the quadratic equation. It can be seen from Figure 4 that the data points on the plot were reasonably distributed near to the straight line (R² = 0.9916), indicating a good relationship between the experimental and predicted values of the response. This

shows that the model chosen is appropriate and that the central composite design (CCD) can be used to perform the optimization operation of the process.

The dissolution process for the extraction of lead from galena was analyzed based on the various solutions obtained at possible reaction conditions from the model predictive Equation (2). RSM was considered appropriate owing to its flexibility in navigating the design space. The model equations were solved for the various interaction effects on lead yield considering at any instance the interaction between two factors only, assuming the other variables are set at their mean coded value of zero (0). The combined effects of adjusting the process variables within the design space were monitored using the 3D surface plots and contour plots.

As the leaching temperature is increased from 326 K to 342 K, the percentage recovery of lead increased from 75% to 85% as seen in Figure 5. This linear relationship is as a result of increase in kinetic energy with higher temperature which allows the solvent molecules to more effectively break apart the solute molecules that are held together by intermolecular attractions, hence increasing the diffusion rate from the solid bulk phase into the solvent region. The same trend was observed in Figures 6-8. The results obtained here are in agreement with the results of statistical analyses shown in Table 6 which reveals that the quadratic effect of temperature (X₁ squared) on the response is significant within the factor range of the experiment with a p-value of <0.0001. This is in line with

the report of several researchers [11] [12] [13] who agree that increase in temperature substantially increases the dissolution rate of solutes.

As the acid concentration is increased from 1.8 M to 3.18 M, the percentage recovery of lead increased from 75% to 85% as seen in Figure 5. This is attributed to the increase in the diffusion rates of Pb²⁺ from the solid to the solution as the concentration and diffusion of hydronium ion rises. The same trend was observed in Figures 9-11.

A plot for the combined interactive effects of reaction temperature and solid/liquid ratio on the recovery of lead is shown in Figure 6. As the solid/liquid ratio is decreased from 0.039 to 0.028 g/ml, the percentage recovery of lead increased from 75% to 85%. This could be attributed to the decrease in the fluid reactant per unit weight of the solid [11] . The same trend was observed in Figure 9, Figure 12 and Figure 13.

The plot for the combined effects of leaching temperature and stirring rate on the recovery of lead is shown in Figure 7. As the stirring rate is increased from 270.4 rpm to 429.5 rpm, the percentage recovery of lead increased from 75% to 85%. This could be attributed to the decrease in the thickness of the film layer as the stirring rate is increased [11] . The same trend was observed in Figure 10, Figure 12 and Figure 14.

As the leaching time is increased from 65 min to 93.2 min, the percentage recovery of lead increased from 74% to 84% as seen in Figure 8. The same trend was observed in Figure 11, Figure 13 and Figure 14. The results obtained above also indicate that the quadratic effects of acid concentration, solid/liquid ratio, stirring rate and leaching time on response are also very significant within the factor range of the experiment with p-values of <0.0001.

Artificial neural networks (ANNs) are machine-based computational techniques

which attempt to simulate some of the neurological processing abilities of the human brain [14] . They can be considered as a nonlinear regression tool for making a relationship between input and output variables. Multilayer perceptron, radial basis function networks, linear networks, Bayesian networks (generalized regression), and Kohonen networks (probabilistic regression) are the most well-known ANN types.

Multilayer perceptrons (MLP) is perhaps the most popular network architecture in use today [15] . Neurons perform a weighed sum of their inputs and pass it through a transfer function f to produce their output. The MLP model used in this work was developed in MATLAB (The math works Inc. 2007b). For galena dissolution in HCl, the selected MLP model had three layers, where the first had five input units representing the process independent variables (temperature, acid concentration, solid/liquid ratio, stirring rate and leaching time), the second layer had nine hidden units, and the third layer had one output unit representing the percentage yield of lead. The MLP architecture is shown in Figure 15.

The optimization exercise for the dissolution process was conducted separately using the flexibility of the design expert tool function. Equation (4) was solved for the best solutions such that the responses (Y_n) are maximized within the design space. No unique solutions were possible. A usual approach, which involves selecting the best solution based on economic considerations, was adopted and

the chosen optimal solutions are presented in Table 8. The selection of desired optimum solution of Table 8 was mainly influenced by the cost of reagents and energy. By using the numerical optimization technique which is a feature of CCD in the design expert software, a combination of factors that concurrently satisfy the requirements placed on each of the responses and factors could be determined by the software [13] . In choosing the goal for each of the factors for the numerical optimisation, a number of considerations were made. The significance of each of the factors on the final response was the most important consideration. The response was set at maximum goal, all other factors were kept in range apart from the reaction time which was set to a minimum goal target. Based on these, the software predicted optimum reaction conditions with a desirability of 1.00 tabulated in Table 8.

In order to confirm the accuracy of this model, an experimental run was conducted under these optimal conditions. The experimentally obtained value for the% yield of lead was 84.54% and this was in reasonable agreement with that of CCD model of design expert (85.25%) as shown in Table 9. A non-dominated optimal response of 85.25% yield of lead at 343.96 K leaching temperature, 3.11 M hydrochloric acid concentration, 0.021 g/ml solid/liquid ratio, 362.27 rpm stirring speed and 87.37 min leaching time was established as a viable route for reduced material and operating cost using RSM.