Modeling and Optimization of Galena Dissolution in Hydrochloric Acid: Comparison of Central Composite Design and Artificial Neural Network

Response surface methodology (RSM) and Artificial neural network (ANN) were used for the simulation and optimization of galena dissolution in hydrochloric acid. The galena ore was characterized for structure elucidation using FTIR, SEM and X-ray diffraction spectroscopic techniques and the results indicate that the galena ore exists mainly as lead sulphide (PbS). A feed-forward neural network model with Leverberg-Marquardt back propagating training algorithm was used to predict the response (lead yield). The leaching temperature, acid concentration, solid/liquid ratio, stirring rate and leaching time were defined as input variables, while the percentage yield of lead was labelled as output variable. The multilayer perceptron with architecture of 5-9-1 provided the best performance. All the process variables were found to have significant impact on the response with p-values of <0.0001. The performance of the RSM and ANN model showed adequate prediction of the response, with AAD of 0.750% and 0.295%, and R of 0.991 and 1.00, respectively. 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.

It focused on optimizing chemical reactions in order to obtain, e.g., high yield and purity at low costs. This was accomplished through the use of sequential experimentation, involving factors such as temperature, pressure, duration of reaction, and proportion of reactants. The same methodology can be applied to model or optimize any response that is affected by the levels of one or more quantitative factors.
The most popular RSM is the central composite design (CCD). A CCD has three groups of design points: 1) two-level factorial or fractional factorial design points, 2) axial points (sometimes called star points), and 3) centre points. CCDs are designed to estimate the coefficient of a quadratic model. All point description is in terms of coded values of the factors.
Artificial Neural Network (ANN) is an empirical tool, which is analogous to the behavior of biological neural structures [3]. Neural networks are powerful tools that have the abilities to identify underlying highly complex relationships from input-output data only [4]. For the past two decades, artificial neural networks (ANNs), and, in particular, feed-forward artificial neural networks (FANNs), have been extensively studied to present process models, and their use in industry has been rapidly growing [5].
Although response surface methodology (RSM) and artificial neural network (ANN) have been applied in several areas, there hasn't been any reported work, to the best of our knowledge, of their application on lead dissolution from Nigerian galena ore. The present work therefore intends to identify the most signifi-

Material Sample Mining and Preparation
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.

Spectrophotometric Analysis
The X-ray fluorometer (XRF), X-supreme 600 oxford instruments was used for the elemental analysis of the ore. The mineralogical analysis of the ore was done using ARL X'TRA X-ray Diffractometer, Thermoscientific with the serial number 197492086 with CuKα (1.54 Å) radiation generated and 40 mA and 45 kV.
This unit comprises of a single compact cabinet. The cabinet houses a high speed, high precision Goniometer; high efficiency generator (X-ray) and an automatic sample loading facility.
The petrographic slides of galena ore were prepared using Epoxy and Lakeside 70 media according to the method of Hutchison [6].

FTIR and SEM Analysis
FTIR analysis was carried out using Buck Scientific M530 Infrared Spectrophotometer. SEM analysis was carried out using Q250 by FEI model from the Netherlands.

Experimental Procedure
Leaching experiments were performed in a 500 ml glass reactor fitted with a condenser to prevent losses through evaporation. The two major variables (heat and stirring rate) necessary for accelerating the rate of chemical reaction was provided by the aid of a magnetically-stirred hot plate (Model 78HW-1). For every leaching experiment, the solution mixture was freshly prepared by dissolving the required mass of the ore sample in the acid solution at the required temperature. At the end of each reaction time, the undissolved materials in the suspension was allowed to settle and separated by filtration. The resulting solutions were diluted and analyzed for lead using atomic absorption spectrophotometer (AAS).
The mole fraction of lead passing into the solution from galena was calculated by the formula given in Equation (1), where x designates quantity dissolution.

Amount of Pb passing into the solution
Amount of Pb in original sample x = (1)

Design of Experiment
The process variables affecting the dissolution of galena in hydrochloric acid  Table 1.
A total of 32 runs were carried out to optimize the process variables and experiments were performed according to the actual experimental design matrix shown in Table 2. The experiments were performed randomly to avoid systemic error. The results were analyzed using the analysis of variance (ANOVA), contour, and response surface plots. In RSM, the most widely used second-order polynomial equation (Equation (2)) developed to fit the experimental data and identify the relevant model terms may be written as: where Y is the predicted response variable which is the% yield of lead in this the different interaction coefficients between the input variables i x and j x and ε is the error of the model.

Elemental Composition by XRF
The results of the elemental composition of galena by X-ray fluorescence technique showed that the galena mineral exist mainly as PbS with metals such as Na, Mg, Al, Ca, Fe and Zn occurring as minor elements, and K, Cr, and Sr as traces. The elemental analysis gave Pb (60.01%), S (14.66%), Fe (4.32%), Na

Phase Studies by XRD
The analysis of galena by X-ray diffraction gives a better description in terms of  Table 2 present the results of the X-ray diffractogram of the ore and shows that the ore exist mainly as lead sulphide (PbS).
The galena ore gave three major peaks at 2.96, 3.42, and 2.09 Å, respectively as shown in Figure 1. All these supported the results of the elemental analysis by XRF.

FTIR Analysis of Galena
The FTIR spectra of galena ore is shown in Figure 2.  The FTIR result is in agreement with XRF and XRD results which confirmed the presence of the minerals detected.

SEM Analysis of Galena
The scanning electron micrograph (SEM) of galena ore was obtained with magnifications of 240×, 520×, 1000×, and 1500× respectively as shown in Figure 3.
The average cell diameter of the ore ranges from 8 to 62 µm while the average cell density ranges from 0.0042 to 1.13 cells/mm. The results indicate that the ore particles are very cohesive, forming an aggregate mass that appeared to have been formed by several flaky particles stacked together in form of agglomerates [7]. The particles have irregular shapes with rough edges, and are highly crystalline due to the high level of purity of the ore. The central composite design (CCD) for the leaching of lead from galena using hydrochloric acid is shown in Table 3 with the experimental values.

RSM Modelling
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 I. A. Nnanwube et al.

Statistical Analysis
To determine the adequacy of the models depicting the removal of lead by the 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  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) The results were analyzed by using the analysis of variance (ANOVA) suitable for experimental design used and shown in X , 2 5 X (where X 1 = leaching temperature, X 2 = acid concentration, X 3 = solid/liquid ratio, X 4 = stirring rate, and X 5 = 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).
In terms of the actual factors the model equation (Equation (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   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.

Combined Effects of Operating Parameters on the Response
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.     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 Pb 2+ 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.

ANN Modeling
Artificial neural networks (ANNs) are machine-based computational techniques J. Minerals and Materials Characterization and Engineering

Comparison of RSM and ANN A comparison between Artificial Neural Network (ANN) and Response Surface
Methodology was carried out using the Absolute average deviation (AAD) and Coefficient of determination observed for both models. The AAD observed for both models gives an indication of how accurate the model predictions can be [16].
where n is the number of sample points, artpred R is the predicted value of lead dissolution and artexp R is the experimentally determined value for lead dissolution [16]. Since the value of COD for both RSM and ANN are approximately equal to 1, it is evident that both models could efficiently predict the recovery of lead from galena [13]. RSM was then adopted for the optimization studies.

Process Optimization Using CCD
The optimization exercise for the dissolution process was conducted separately using the flexibility of the design expert tool function. Equation (4)  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

Conclusion
The