3.1. Statistical Analysis and Model Fitting
According to the Box-Behnken experimental design, all 29 experiments were performed under corresponding conditions, and the experimental results are listed in
Table 3. According to the data, two quadratic models with interdependent terms were obtained, as given by Equations (4) and (5).
The statistical results of ANOVA for the response surface quadratic models of V and Ni are listed in
Table 1. The results showed that the F-values of the respective model for V and Ni were 11.78 and 8.60, respectively. The data indicated the significance of the model. The prob > F values for the models were <0.0001, which indicate the significance of the model terms. Moreover, the model terms of V, A, B, C, AB, BC, A
2 and B
2 and the model terms of Ni, A, B, C, AC, BC, A
2 and B
2 were found to be significant. The values of R
2 were determined to be 0.959 and 0.971, as shown by the results presented in
Table 4. In the present study, R
2 and R
2 adj. coefficients guaranteed the satisfied adjustment of the quadratic model to the experimental data. The percentage recovery of coefficient of variance (CV) for V and Ni were confirmed to be (in %) 3.30 and 2.74, respectively. The ratio of the standard deviation to the mean value of the observed response, indicated by CV%, represents the degree of reproducibility of the models. The models produced a satisfactory degree of fitting and an adequate precision to measure the ‘signal-to-noise ratio’. They indicated an adequate signal.
3.2. Process Optimization
Comprehensive factor analysis of variance illustrated the main effects and interactions of the evaluated variables. In order to achieve a graphical interpretation of the interactions, it is strongly recommended to use the regression model with 3D surface plots and the 2D contour plots.
Interactions among the parameters of V leaching rate are shown in
Figure 3 and
Figure 4.
Figure 3A–C and
Figure 4A–C show the leaching rate of V for different H
2O
2 solution concentrations. The results demonstrate that the leaching efficiency of V improved significantly from 62.82% to 91.56%, with the increase in the concentrations of H
2O
2 solution from 0 to 1.5 mol/L, which suggests that the addition of H
2O
2 in the reaction system had a significant effect in increasing the extraction of V. This result is mainly attributed to the low valence of V (in the form of VO
2) that was oxidized to high valence (in the form of V
2O
5) in the SFCC catalysts [
27]. In this condition, the high valence species could easily react with oxalic acid. As seen in
Figure 3A,D,E and
Figure 4A,D,E, with different reaction times, the V recovery rate increased almost linearly with the increase in the concentration of oxalic acid from 1.5 mol/L to 2.0 mol/L. However, the slope curve lowered when the concentration of oxalic acid exceeded 2.0 mol/L. This is because a higher acid concentration provides enough H
+ to accelerate the leaching process, and therefore, more V could be recovered. However, after the reaction stabilized, simply increasing the H
+ could not further promote the reaction. As seen in
Figure 3B,D,F and
Figure 4B,D,F, the presented data explain that the factor of time has a more significant influence on leaching rate for vanadium compounds. When the leaching time extended from 0.25 to 1.5 h, a sharp increase in the productivity was observed. Any further extension in the leaching time could not increase the leaching rate significantly and it became practically unchanged. As shown in
Figure 3C,E,F and
Figure 4C,E,F, the leaching rate of V enhanced with the increasing microwave-assisted power (up to 500 W), and thereafter, the metal leaching stabilized.
Interactions among various parameters of Ni leaching rate are shown in
Figure 5 and
Figure 6. As seen in
Figure 5A–C and
Figure 6A–C, the data suggests that the addition of H
2O
2 in the reaction system had an insignificant effect on the extraction of Ni. This is due to the reason that H
2O
2 could not further oxidize Ni compounds in SFCC catalysts. Based on
Figure 5A,D,E and
Figure 6A,D,E, the recovery of Ni first increased, and then, decreased, which was due to the reason that the value was higher than the stoichiometric value due to the increase in acid concentration. This results in over-saturation due to the existence of large amounts of H
+ ions, which decreased the nickel recovery [
28]. As seen in
Figure 5B,D,F and
Figure 6B,D,F, extending the leaching time increased the Ni recovery. When the critical time of 1.5 h was extended and the leaching time was continuously extended, the Ni recovery decreased slightly. In this case, the reason could be attributed to the existence of impurities with some increase in the concentration of other ions. It is possible that some of the Ni
2+ species precipitated out. As shown in
Figure 5C,E,F and
Figure 6C,E,F, the leaching rate of Ni enhanced with the increasing microwave-assisted power (up to 500 W), and thereafter, the metal leaching stabilized. The phenomenon is similar to the leaching characteristics of V.
Based on the statistical modeling, the optimal conditions for vanadium and nickel recovery were found to be as follows: oxalic acid concentration of 1.8 mol/L; leaching time of 91 min; microwave-assisted power of 500 W; H
2O
2 concentration of 1.1 mol/L. As shown in
Figure 7, the actual values were evenly distributed around the predicted values. The desired values of Y
1 and Y
2 were 91.64%, 46.89%, respectively. The experimental results for Y
1 and Y
2 under the same conditions were 91.56% and 46.78%, respectively, which agreed well with the predicted values. In order to test the optimal experimental parameters estimated by the response surface model, three verification experiments were conducted in the optimal process conditions. At the same time, other experimental conditions were confirmed with a liquid-to-solid ratio of 50 g/L and a stirring speed of 500 r/min. The leaching rates of V were fixed as 91.62%, 91.41%, and 91.48% respectively, and the average value was 91.50% with the relative error of 0.07%. In addition, the leaching rates of Ni were 46.46%, 46.49%, and 46.83%, for which the average value of Ni leaching rate was 46.59%, and the relative error is 0.40%. The results above further explain that this model accurately predicts the leaching rates of V and Ni and they also prove the reliable optimal process conditions. The key influencing factors of the leaching rate and their corresponding optimum range can be identified by the single-factor experiment. The response surface method is very effective for studying the importance of each factor and the interaction among different factors and for figuring out the optimal process conditions.
3.3. Leaching Kinetics
The experimental data of leaching processes of V and Ni in oxalic acid are shown in
Figures S1 and S2. To determine the optimum kinetic model, the plots of the left-hand sides of Equations (1) and (2) against time for oxalic acid are shown in
Figure 8. The apparent ratio constant (k) and multiple regression coefficients (R
2) of V and Ni at various leaching temperatures using different kinetic models are listed in
Table 5.
As shown in
Figure 8 and
Table 5, a better fit is obtained using Equation (1) with a high R
2 value (>0.93) and there is a good agreement between the proposed kinetic model and the experimental data. This indicates that the control step in the reactions between oxalic acid and metals (V and Ni) under microwave-assisted is the surface chemical reaction.
To calculate the activation energy (Ea), lnk is plotted against 1/T according to the Arrhenius equation [
20]. The linear fitting of (lnk) versus (1/T) with a slope of (–Ea/R) is depicted in
Figure 9. The activation energies calculated from the Arrhenius plots are 3.28 kJ/mol for V and 34.41 kJ/mol for Ni during the leaching process under microwave-assisted. The apparent activation energy of V in the acid leaching process is lower than that of Ni, this indicates that V tends to be more soluble than Ni, because Ni tends to form both oxalate compounds and oxalate complexes, whereas V forms only oxalate compounds [
29].