Introduction

Aluminium is the most common metal in the earth’s crust and the second most popular metal for making things. Due to its many unusual combinations of properties such as low density, corrosion resistance, electrical conductivity, ductility, and strength in alloys, aluminium and its compounds have found application in many industrial and commercial sectors such as food and drink packaging, aviation, automobile, building and construction, clothing, power cables, cooking utensils, to mention a few. On the other hand, alumina has been extensively deployed in progressive technological sectors, which include its application for the manufacture of microelectronics, good semiconductors, ceramics, biofuel and cell-fuel, insulators, high-strength materials, fireproof plastics, refractories, and high-grade polishes1. The metallic aluminium is not found in nature but occurs in the form of hydrated oxide or silicate (clay). The increasing demand for aluminium globally and the unavailability of bauxite in many countries increases the need to develop alternative technologies for the production of aluminium from low-grade ores2.

Researchers in many countries have put in rigorous effort into alumina extraction from natural sources other than bauxite, particularly clays3. The use of clay as the natural resource for alumina recovery is preferred due to the merits of the process such as its environment friendliness, ready availability, and cost-effectiveness. This is in contrast to the Bayer process which is characterized by the generation of toxic residues and their attendant negative environmental consequences as well as the incurred cost of residue disposal. The choice of this process (clay process) would also help create job opportunities in communities where it is found as well as improve the gross domestic product (GDP) of Nigeria and attract foreign investment to Africa in line with African Union Agenda 2063, as well as promote responsible consumption and production, innovation, economic growth and industrial revolution, in line with the United Nations (UN) sustainable development goals4.

Clays are hydrous aluminium silicates that usually contain small amounts of impurities such as calcium, magnesium, potassium, sodium, or iron. Many clay contain as much as 30–40% alumina5. One of the most common processes of clay formation is the chemical decomposition of feldspar. Among the various clay minerals, microcline proves to be a feasible alternative for alumina production owing to its substantial aluminium content. Microcline (KAlSi3O8) is a key rock-forming mineral in many rocks, especially pegmatite, granite, syenite, and metamorphic gneisses. Microcline is usually found jointly with feldspar; as well as in syenite, granite, and pegmatites, critically characterizing acidic and neutral core igneous rocks6.

Leaching involves materials extraction from a solid by dissolving them in a liquid, either through natural means, or through an industrial process. Leaching has various commercial uses in the chemical processing industry, including recovery of metal from its ore with the aid of acid, and sugar from sugar beets with the aid of hot water2. In extractive metallurgy, leaching is normally applied on account of its cost-effectiveness, environment-friendliness, low-energy requirement, and the capability of processing ores of low-grade. The three procedures generally applied for the recovery of alumina from clays include: (1) leaching using acids such as H2SO4, HNO3 or HCl to remove alumina from clay after calcination, (2) sulphatization by sintering clay with ammonium sulphate and the subsequent removal of alumina by leaching using hot water, (3) alkali roasting by sintering of the clay with ammonium sulphate and the removal of alumina and silica by leaching using hot water1.

Most acid-leaching procedures for alumina recovery have been conducted with the conventional one factor per time1. This procedure takes a long time and wastes a lot of reagents. To overcome this challenge, statistical and optimization techniques such as RSM which is made up of advanced statistical and mathematical procedures applied for process enhancement and optimization are adopted7,8,9. Using this method, optimum points are determined for given independent variables with the aid of a second-order polynomial prediction equation. Widely applied in the chemical industry, RSM helps in the improvement of manufacturing processes as well as optimizing chemical reactions to obtain products of high yield and purity at reduced cost10,11.

On the other hand, PSO is a stochastic computation method12. Motivated by the social behaviour and intelligence of the insect swarm, PSO has continuously gained a rising reputation due to its ease and efficiency in performing hard optimization problems. It has been effectively applied in solving various optimization and engineering problems13,14,15,16. As obtainable in comparable evolutionary algorithms, PSO is an optimization technique that relies on a population often referred to as a swarm. Such a population comprises individual particles having specific positions and velocities produced at random during initialization. Particles look for a solution space and recall the overall best position it has established. Contrary to genetic algorithms, PSO does not need the process of encoding and special operators like crossover or mutation. The distinguishing feature of PSO is derived from particles’ individual experiences, collaboration among particles as well as their aptitude to share knowledge12.

In general, many investigators have tried to recover alumina from other alternative sources, such as kaolinitic clays using both alkaline and acid routes17,18,19. It was a common consensus that acids were more effective than bases in aluminium extraction1. Our previous publication on alumina recovery from microcline focused on the kinetics of the leaching process in nitric acid and hydrogen peroxide solution6. The process parameters were not optimized. Hence, in the present investigation, aluminium leaching from microcline through the nitric acid route was explored via optimization of process parameters. Process parameters that affect the leaching process were optimized and optimum points were determined using RSM and PSO.

Materials and methods

Materials

Sourcing and Preparation of samples

Microcline ore from Amagunze in south-eastern Nigeria was deployed for this work. The sample was ground, dried in the sun for two days, and calcined at 700 °C for 1 h to enhance its reactivity. The calcined microcline was further pulverized and sieved with < 75 μm ASTM standard test sieve and kept for further use. The solutions used in the experiment were prepared using analytical-grade nitric acid and deionized water.

Leaching experiments

Experiments were carried out according to the method described by Nnanwube and Onukwuli20 and Nnanwube et al.6. The stoichiometry of the process is illustrated in Eq. (1).

$$\rm A{l_2}{O_{3(s)}} + 6HN{O_{3(aq)}}\; \longrightarrow \;\;2Al{\left( {N{O_3}} \right)_{3(aq)}} + 3{H_2}{O_{(l)}}$$
(1)

Design of experiment

The experimental design for alumina leaching from microcline via nitric acid leaching was executed with the central composite design of RSM. RSM is a set of mathematical and statistical tools which are helpful for the development of empirical models, enhancing and optimizing process parameters, as well as identifying the interaction between the factors that affect the response of interest, with a minimal number of experiments21,22. It utilizes numeric data from an associated experiment to establish a regression model as well as optimize a response which is affected by several of input factors.

The central composite design (CCD) of RSM is usually made up of a 2n factorial runs with 2n axial runs. The error in the experiment is estimated by centre runs (nc). The variables are examined at two levels. However, as the variable number (n) rises, the number of runs necessary for a whole duplicate of the design increases speedily. Since the individual effect of second-order cannot be evaluated independently by 2n factorial design, CCD was deployed to investigate the quadratic effects of the model for alumina recovery from Amagunze microcline21. In the statistical analysis, the response alongside the relevant parameters utilized in the process was modelled to optimize the process parameters for the response of interest. Statistical factors were evaluated using the analysis of variance (ANOVA)23.

RSM involves three main steps which include the design of experiments, evaluation of coefficients in a developed mathematical model, and lastly, prediction of the response(s) and assessing the sufficiency of the model in the design space24. In this work, the five variables selected for the design include the reaction temperature (oC), S/L ratio (g/mL), nitric acid concentration (M), stirring speed (rpm), and reaction time (min). The range and levels of the factors deployed in the design are depicted in Table 125.

Table 1 Factors and level of various parameters of CCD design for alumina recovery from Amagunze microcline.
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In carrying out the design, the variables in the experiments are presumed to be uninterrupted and measured by experiments with minimal errors. The design of the experiment was targeted at optimizing the response factor21. Hence, there is a need to find a proper estimate for the relation connecting the independent factors and the response. To minimize the error and the influence of unrestrained factors, the experimental run was arranged arbitrarily26. An empirical model generated using the response which relates it to the experimental variable is depicted by a second-degree polynomial equation presented as Eq. 227.

$${\rm{Y}}=\mathop \beta \nolimits_0 + \sum\limits_{i = 1}^n {\mathop \beta \nolimits_i } \mathop x\nolimits_i + \sum\limits_{i = 1}^{n - 1} {\sum\limits_{j = 2}^n {\mathop \beta \nolimits_{ij} \mathop x\nolimits_i \mathop x\nolimits_j } } + \sum\limits_{i = 1}^n {\mathop \beta \nolimits_{ii} } \mathop x\nolimits_i^2 + \varepsilon$$
(2)

where Y denote the response predicted (alumina yield), \(\:{\beta\:}_{0}\),\(\:\:{\beta\:}_{i}\), \(\:{\beta\:}_{ij}\), \(\:{\beta\:}_{ii}\), n denotes the constant coefficient, linear coefficient, interaction coefficient, quadratic coefficient, and number of factors in the experiments, respectively; \(\:{x}_{i}\) and \(\:{x}_{j}\) denote the coded values of the variable parameters which affect the response, while ε denotes the models’ random error28,29. The codes are estimated as a function of the range of interest of each factor as depicted in Table 1. Equation (3) represents the coding of the test variables in developing the regression model30,31.

$$\:{x}_{i}$$= $$\:\frac{{X}_{i}-{X}_{i}^{*}}{{\varDelta\:X}_{i}}$$
(3)

where \(\:{x}_{i}\) denotes the \(\:{i}^{th}\) independent factor of the dimensionless coded value; \(\:{X}_{i}\) denotes the uncoded value of the \(\:{i}^{th}\) independent factor; \(\:{X}_{i}^{*}\) denotes the uncoded value of the \(\:{i}^{th}\) independent variable at the centre point, while \(\:{\varDelta\:X}_{i}\) denote the step change32.

The final equation for the five factors and the error of the model is depicted as Eq. (4).

$$\begin{array}{l} {\rm{Y}}={\beta\:}_{o}+{\beta\:}_{1}A +{\beta\:}_{2}B+{\beta\:}_{3}C+{\beta\:}_{4}D+{\beta\:}_{5}E+{\beta\:}_{12}AB+{\beta\:}_{13}AC+{\beta\:}_{14}AD\\ \quad\,\,\,+{\beta\:}_{15}AE+{\beta\:}_{23}BC+{\beta\:}_{24}BD+{\beta\:}_{25}BE+{\beta\:}_{34}CD+{\beta\:}_{35}CE+{\beta\:}_{45}DE\\ \quad\,\,\,+{\beta\:}_{11}{A}^{2}+{\beta\:}_{22}{B}^{2}+{\beta\:}_{33}{C}^{2}+{\beta\:}_{44}{D}^{2}+{\beta\:}_{55}{E}^{2}+{\epsilon\:}_{i} \end{array}$$
(4)

The standard particle swarm model

PSO was motivated by the swarming activities of animals and the social conduct of humans. A particle swarm consists of a populace of particles in which every particle is an object in motion33. It flies across the search space and is drawn to former locations it visited with a high fitness. Contrary to the entities in evolutionary estimation, particles do not reproduce nor be replaced by others34,35. In this model, the particles are prepared with a populace of arbitrary entrant solutions. The particles move iteratively across the d-dimension problem space to seek new solutions. Every particle possesses a position denoted by a position vector \(\:\overrightarrow{{p}_{i}}\) (i denotes the particle index), as well as a velocity denoted by a velocity-vector \(\:\overrightarrow{{v}_{i}}\). Every single particle recalls its own finest position so far in a vector \(\:\overrightarrow{{p}_{i}^{\#}}\) and its jth dimensional value is \(\:{p}_{j}^{\#}\). The overall best position vector in the swarm until now is then deposited in a vector \(\:\overrightarrow{{p}^{*}}\) and its jth dimensional value is \(\:{p}_{j}^{*}\). In the course of the iteration time t, the updated velocity from the former velocity to the new velocity is estimated by Eq. (5). The new position is then estimated by summing the former position and the new velocity using Eq. (6)36.

$$\:{v}_{ij}(t)=w{v}_{ij}\text{(t-1)}+{\epsilon}_{1}{r}_{1}({p}_{ij}^{\#}\left(t-1\right)-\:{p}_{ij}(t-1))+{\epsilon}_{2}{r}_{2}({p}_{j}^{\#}\left(t-1\right)-\:{p}_{ij}(t-1))$$
(5)
$$\:{p}_{ij}(t)={p}_{ij}\text{(t-1)}+{v}_{ij}(t)$$
(6)

where \(\:{r}_{1}\) and \(\:{r}_{2}\) are arbitrary numbers, equally spread within the interval [0, 1] for the jth dimension of the ith particle. \(\:{\epsilon}_{1}\) and \(\:{\epsilon}_{2}\) are positive constants representing the coefficient of the self-recognition component and the coefficient of the social components, respectively36. w denotes the inertia factor which controls the extent the particles tend to follow their present direction in comparison to the learnt positions34,37. It varies linearly from 1 to near zero for the period of the iterated processing. According to Eq. (5), a particle chooses where next to move, bearing in mind its peculiar experience, which depicts the reminiscence of its best former position and the experience of its most effective particle in the swarm36.

Results and discussion

Characterization

XRF analysis

The XRF results of both the raw and calcined samples had previously been reported6. The result shows the main oxides in the clay as Al2O3, SiO3 and Fe2O3, while the minor oxides are MgO, K2O and TiO2. Oxides found in traces include P2O5, SO3, CaO, Cr2O3, ZnO, Mn2O3 and SrO, as shown in Table 2.

Table 2 XRF result of Raw and calcined Amagunze microcline6.
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Mineralogical analysis

The XRD result of the clay confirms the presence of microcline (KAlSi3O8) with major peaks at 21.07, 26.44, 26.64, 27.11, and 27.45o 2θ, as earlier reported6. The XRD data is shown in Table 3 while the XRD pattern is shown in Fig. 1.

Fig. 1
figure 1

XRD pattern of microcline.

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Table 3 The X-ray diffraction data of Microcline showing the angle 2θ and d-values of the compounds identified, with their relative intensity (%).
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Statistical analysis

Statistical analysis of experimental data on alumina dissolution from microcline was performed with a rented version of Design Expert (DE) software 10.0.0. The CCD of RSM found in DE software was deployed to perform the regression analysis of the data generated from the experiment and plot the response surface and contour plots at the optimum conditions21,38. Several statistical indicators were used to fit the model and test the significance. F-test was used to examine the statistical importance of the models while the precision of the fitted polynomial model was established using the R2 values39. The probability value (p-value) was deployed to assess the significant model terms at a 95% confidence interval. The experimental design, the coded, predicted and experimental values for this study, are depicted in Table 428.

Table 4 Experimental design for alumina leaching from microcline using nitric acid.
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For good quality model fitting, a test of significance for the regression model and separate model coefficients and lack-of-fit test is normally carried out. The sequential model sum of squares (Table 5) indicates that the quadratic vs. 2FI model is suggested with p-value of < 0.0001 and F-value of 76.71.

Table 5 Sequential model sum of squares.
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From the model summary statistics, it is shown that the R2, adjusted R2 and the predicted R2 values of the quadratic model (0.9936, 0.9819, and 0.9198) gave a better fitting in comparison with the 2FI (0.7696, 0.5535 and − 2.0753) and the linear model (0.7493, 0.7010 and 0.6776) as depicted in Table 67. The results above show that the quadratic model gave a good description of the correlation between the independent factors and the response (dependent variable).

Table 6 Model summary statistics.
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For the model not to be over-fitted, the variance between the adjusted R2 and predicted R2 should not be more than 0.2. From the results obtained in this work, the variance between the adjusted and predicted R2 for the quadratic model is 0.0117, indicating that the model is not over-fitted40,41. The adjusted R2 measures the extent of difference around the mean described by the model, adjusted for the model number of terms; while the predicted R2 measures the extent of difference in new data described by the model42.

The quadratic model also gave a minimum standard deviation of 0.69, indicating a close connection between the predicted and experimental values of the responses43.

The degree the model fits every point in the design is designated by the predicted residual error sum of squares (PRESS). It is estimated by first predicting where every single point should be from a model that comprises every other point except the one being considered.

To investigate if the process variables are statistically important or not, statistical analysis of variance (ANOVA) was performed. The ANOVA results are shown in Table 7.

Table 7 ANOVA for response surface quadratic model.
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The F-value for each variable indicates which variable had a considerable effect on the response which is the percentage of alumina leached44,45. The residual row shows the extent of disparity in the response that is yet to be explained while the Lack of Fit is the degree of variance between the model predictions and the observations10,46,47.

The F-value of 85.06 from the model implies the model is significant. There is just a 0.01% probability that an F-value as large as this could occur because of noise. The change associated with a term and its residual variance is indicated by the F-value of that term. The F-value of the independent factors A, B, C, D, and E were obtained as 259.00, 260.27, 300.11, 240.19 and 223.28, respectively, signifying that the effect of all the independent variables on the response was significantly high. “Prob > F” values less than 0.050 implies that the model terms are important. The “Lack of Fit F-value” of 0.66 indicates that the Lack of Fit is not significant when compared to the pure error. There is a 68.91% chance that a “Lack of Fit F-value” this large could happen because of noise48,49. The lack of fit being insignificant is good since it indicates that the model fits. The significant and non-significant terms were determined from the ANOVA table (Table 5) in line with their p-values. From the ANOVA results, reaction temperature (A), nitric acid concentration (B), S/L ratio (C), stirring rate (D), reaction time (E), reaction temperature and stirring rate (AD), reaction temperature and reaction time (AE), reaction temperature squared (A2), nitric acid concentration squared (B2), S/L ratio squared (C2), stirring rate squared (D2), reaction time squared (E2) are all significant10. The final polynomial predictive equation of the second order, after eliminating the insignificant terms is given in Eq. (7).

$$\begin{array}{l}Y{\rm{ }}\left( \% \right)= \rm 80.36 + 3.38*A+ 3.39*B-3.64*C+ 3.25*D+ 3.14*E- 0.59*AD-0.68*AE\\\quad\quad\quad\quad\rm- 1.47*{A^2}-2.25*{B^2}-1.93*{C^2} - 2.27*{D^2}-1.43*{E^2}\end{array}$$
(7)

In terms of the actual factor values, the percentage yield of alumina is obtained and shown in Eq. (8).

$$\begin{array}{l}{\rm{Y }}\left( \% \right)= \rm- 51.90+ 1.35*Leaching\,temp. + 9.62*Acid\,conc.+ 162.72*S/L\,ratio\\\quad\quad\quad\quad\rm+ 0.12*Stirring\,rate+ 0.53*Leaching\,time-2.55E - 004*Leaching\,temp.*Stirring\,rate\\\quad\quad\quad\quad\rm-1.51E - 003*Leaching\,temp.*Leaching\,time-6.54E - 003*Leaching\,temp{.^2}\\\quad\quad\quad\quad\rm-1.00*Acid\,conc{.^2}-{\rm{ }}19329.55*S/L\,rati{o^2}-{\rm{ }}9.45E - 005*Stirring\,rat{e^2} - 1.59E - \\\quad\quad\quad\quad\rm003*Leaching\,tim{e^2}\end{array}$$
(8)

The coefficient of variation is estimated by dividing the standard deviation by the mean and multiplying by 10010,48. Values less than 15% indicate a practically reproducible model. The coefficient of variation (CV) value of 1.40% obtained from this study illustrates that the model is practically reproducible. The signal-to-noise ratio depicted as the adequate precision is 37.846 as presented in Table 8. Ratios greater than 4 indicate that a good connection of signal-to-noise ratio exists. The mean is the overall average of all the response data. A value of 73.34 was recorded from this study7,30.

Table 8 Regression values.
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The experimental results were also analyzed to verify the connection between the experimental and predicted alumina recovery. Figure 2 shows an adequate relationship between the experimental and predicted alumina yield. The result indicates also that the chosen model was sufficient for carrying out the prediction and optimization exercise50,51,52.

Fig. 2
figure 2

Plot of the predicted values versus the actual experimental values.

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Effect of process parameters

The effect of the variables that influence the leaching of alumina from microcline in HNO3 solution was studied by conducting batch experiments at the pre-determined conditions obtained according to the experimental design. Experimental design was performed using the CCD of RSM, facilitated by the DE software by Stat Ease Inc., Minneapolis, USA. The interactive effect of the process variables was represented with contour and surface plots made with the software, according to the experimental results50,53,54,55. From the ANOVA results depicted in Table 7, the squared and individual effects of the factors were all significant with p-values < 0.0001, with the S/L ratio, acid concentration and leaching temperature, having the most significant effects, as specified by their F-values10. The interactive effects of stirring rate and leaching temperature (AD) as well as leaching temperature and leaching time (AE) were also found to be significant. Higher temperatures provide more energy for chemical reactions to occur faster, resulting in quicker dissolution of clay minerals. As temperature rises, the diffusion of ions through the clay particle’s surface layer is facilitated, allowing for better access of the acid to the reactive sites. Increased temperature can increase mass transfer between the liquid phase (acid solution) and the solid clay particles, further improving the leaching process. While higher temperatures generally lead to better leaching efficiency, there is usually an optimum temperature range beyond which other factors like excessive energy consumption or potential side reactions might become significant. As acid concentration rises, more hydrogen ions are available to react with the clay minerals, breaking down the chemical bonds and releasing elements like aluminium and silica into the solution. Excessively high acid concentrations can result in significant damage to the clay’s crystalline structure, potentially hindering further leaching by creating a less accessible surface area. For efficient clay leaching, finding the optimal acid concentration is crucial to maximizing the extraction of desired elements while minimizing unnecessary damage to the clay structure. When the solid/liquid ratio is high, fewer acid molecules are available to interact with each clay particle, limiting the reaction sites and slowing down the dissolution process. As more clay particles are present, diffusion of the dissolved ions away from the clay surface becomes more difficult, creating a concentration gradient that hinders further leaching. A lower solid/liquid ratio is generally preferred for efficient acid leaching of clays to maximize the contact between acid and clay particles. However, an optimum solid/liquid ratio is preferred for more efficient leaching. As stirring speed increases, the boundary layer surrounding clay particles thins, facilitating faster diffusion of the acid solution to the clay surface, thereby accelerating the leaching process. Better contact between the acid and clay particles due to increased stirring leads to a higher reaction rate, resulting in more minerals being dissolved within a given time frame. For more efficient leaching, optimum stirring speed should be used. As leaching time increases, more targeted elements are dissolved and removed from the clay structure, resulting in higher extraction efficiency. The initial phase of leaching often shows a rapid increase in extraction as the acid readily attacks the clay surface, with the rate gradually slowing down as the reaction progresses. Eventually, an equilibrium is reached where the rate of further extraction becomes negligible as most accessible minerals are already dissolved, indicating an optimal leaching time56.

The contour and response surface plots are represented in Fig. 3 (a-d). Figure 3 (a and b) depict the joint effect of temperature and stirring speed on alumina leaching from microcline10. The results reveal that the highest alumina yield was achieved within a reaction temperature range of (72-78oC) and a stirring rate range of (354–416) rpm. Other variables were kept constant at a nitric acid concentration of 3.5 M, S/L ratio of 0.03 g/mL, and leaching time of 90 min. The percentage of alumina leached increased as the stirring rate was increased and attained a maximum at a stirring rate of 385 rpm. This shows that the stirring rate had a mild influence on the leaching rate of alumina57. On the other hand, the percentage alumina recovery rate increased with an increase in temperature up to a temperature of 75 o. This is ascribed to the destruction of the clay’s structure at a higher temperature. The plot displaying the collaborative effect of temperature and time is depicted in Fig. 3 (c and d). The plot reveals that maximum alumina recovery was achieved within a temperature range of (78-84oC) and a reaction time value of (90–100 min) while other factors were held constant at a HNO3 concentration of 3.5 M, S/L ratio of 0.03 g/mL and stirring speed of 385 rpm. The amount of alumina recovered increased with an increase in leaching temperature up to a leaching temperature of 75oC and then decreased. The decrease in alumina recovery upon attainment of a leaching temperature of 75oC can be ascribed to the obliteration of the clay’s structure at a higher temperature value10. The quantity of alumina leached was also found to increase as the leaching time was increased, up to 98 min and then decreased. The decrease in alumina recovery after 98 min may be due to the attainment of steady-state at a leaching time of 98 min.

Fig. 3
figure 3

3D response surface (a, c) and contour (b, d) plots on the effects of process variables on alumina yield.

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Optimization using RSM and PSO

The optimization exercise was performed with the optimization apparatuses of the CCD of RSM in the design expert software and the particle swarm optimization in Matlab. In performing the RSM optimization exercise, the optimum points were selected based on cost implications which were aimed at reducing the cost of reagents, energy, and leaching time. Based on the above-named conditions, RSM predicted optimum conditions of 82.11oC reaction temperature, 3.53 M nitric acid concentration, 0.023 g/mL S/L ratio, 356.14 rpm stirring speed, and 99.92 min reaction time, at which about 83.21% alumina leaching rate was recorded10. The result obtained above was authenticated by performing three separate experiments at which an average alumina leaching rate of about 82.12% was recorded. PSO optimization was performed using a PSO algorithm developed in Matlab. Equation 8 was used to develop the objective function which was carried out in a Matlab file. The five factors used for RSM optimization were selected as assessment variables for particle swarm optimization. Other factors necessary for the simulation work were properly set in the PSO algorithm10. The initial weight damping ratio, the weight, and the swarm size were set to be 0.99, 1.0, and 30, respectively. The personal learning coefficient was set to be 1.2 while the global learning coefficient was set to be 2.058. At the conditions stated above, PSO predicted optimum conditions of 90oC reaction temperature, 4 M nitric acid concentration, 0.021 g/mL S/L ratio, 420 rpm stirring speed, and 118 min reaction time, at which about 82.56% alumina leaching rate was predicted. The optimum conditions above were authenticated by performing three separate experiments which contributed an estimated average value of 81. 37%. The PSO plot of the percentage alumina yield versus the number of iterations is depicted in Fig. 4. In the RSM procedure, the optimum conditions were chosen from an array of options based on economic considerations. However, in PSO optimization, the minimum likely obtainable values of the experimental parameters (factors) are predicted, since PSO is a minimization tool. From the results attained in both optimization methods, a difference 7.89oC (8.77%), 0.47 M (11.75%), 0.002 g/mL (8.70%), 63.86 rpm (15.20%), and 18.08 min (15.32%) were recorded for the reaction temperature, HNO3 concentration, S/L ratio, stirring speed, and reaction time, respectively. Hence, it can be inferred that the reaction time and stirring rate had a mild effect on the leaching process while the reaction temperature, HNO3 concentration, and solid/liquid ratio had a more significant influence on the leaching process. When compared to the results obtained from the kinetic study, the two optimization techniques were found to be viable for alumina recovery from microcline.

Fig. 4
figure 4

PSO plot of alumina yield versus the number of iterations.

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Conclusions

The optimum conditions for alumina dissolution from Amagunze microcline were determined in this study. The mineral phases found in the clay as determined by XRD showed the presence of microcline as the major mineral present. XRF analysis result showed that the main oxides in the clay include Al2O3, SiO3 and Fe2O3. Other oxides present in the clay sample include MgO, K2O, CaO, Mn2O3, TiO2, Cr2O3, ZnO, and SrO. Two optimization techniques which include the optimization tool found in the CCD of RSM of design expert software and the particle swarm optimization tool found in Matlab were used for the study. The optimum predicted conditions by RSM include reaction temperature of 82.11oC, HNO3 concentration of 3.53 M, S/L ratio of 0.023 g/mL, stirring speed of 356.14 rpm and reaction time of 99.92 min, at which about 83.21% alumina leaching rate was predicted, while about 82.56% alumina leaching prediction was recorded by PSO at a reaction temperature of 90oC, nitric acid concentration of 4 M, S/L ratio of 0.021 g/mL, stirring speed of 420 rpm, and reaction time of 118 min, respectively. Hence, the two optimization methods proved to be feasible for alumina leaching.