*2.7. Estimation of Linear and Interaction Coe*ffi*cients for Factorial Designs of Experiments of 23*

Two independent variables were chosen for the factorial design of 36 experiments, where: time and ratio MnO2/reducing agent represent the independent variables that explain the extraction of Mn for a certain type of reducing agent. The analysis through a factorial design allowed us to study the effect of the factors and their levels in a response variable, helping to understand which factors are the most relevant [30,31]. Four factorial designs were carried out that involved two factors with three levels each, with a total of 36 experimental tests (Table 4). The Minitab 18 software (version 18, Pennsylvania State University, State College, PA, USA) was used for modeling, experimental design and adjustment of a multiple regression [32].

**Table 4.** Experimental conditions.


The expression of the response variable according to the linear effect of the variables of interest and considering the effects of interaction and curvature, is shown in Equation (19).

$$\text{Cu }\text{Recovery}(\%) = \alpha + \sum\_{i=1}^{n} \beta\_i \times \mathbf{x}\_i + \sum\_{i=1}^{n} \beta\_i^2 \times \mathbf{x}\_i^2 + \beta\_{1,2} \times \mathbf{x}\_1 \times \mathbf{x}\_2. \tag{19}$$

where α is the overall constant, *xi* is the value of the level "*i*" of the factor, β*<sup>i</sup>* is the coefficient of the linear factor *xi*, β<sup>2</sup> *<sup>i</sup>* is the coefficient of the quadratic factors, β1,2 is the coefficient of the interaction, *n* are the levels of the factors and Mn recovery is the dependent variable.

Table 4 shows the values of the levels for each factor, while Table 5 shows the recovery obtained for each configuration.


**Table 5.** Experimental configuration and Mn extraction data.

#### **3. Results**

#### *3.1. Statistical Analysis*

From the analysis of the main components, the time and ratio factors MnO2/Reducing agent showed a main effect, since the variation between the different levels affected the response in a different way, as shown in Figure 3.

**Figure 3.** Main effect plots of Mn extraction in function of Time (min) and MnO2/Reductant agent ratio for (**a**) FeS2, (**b**) FeC, (**c**) Fe2O3 and (**d**) Fe2<sup>+</sup> agents.

By developing the ANOVA test and the multiple linear regression adjustment for each of the configurations, it is necessary to recover the Mn as a function of the time predictor variables, and MnO2/reducing agent, which is given by:

$$\text{Mn Extraction (\%)} \,\text{[FeS}\_2\text{]} = 13.867 + 6.330 \times \text{Time} + 4.648 \text{MnO}\_2 / \text{FeS}\_2 \times \text{ratio.} \tag{20}$$

$$\text{Mn Extraction} \left( \% \right) \left[ \text{Fe}^{2+} \right] = 58.49 + 10.39 \times \text{Time} + 29.22 \text{ MnO}\_2 / \text{Fe}^{2+} \times \text{ratio.} \tag{21}$$

$$\text{Mn Extraction (\%)} \,\text{[FeC]} = 62.42 + 10.04 \times \text{Time} + 31.16 \,\text{MnO}\_2 / \text{FeC} \times \text{ratio.} \tag{22}$$

$$\text{Mn Extraction (\%)} \text{ [Fe}\_2\text{O}\_3] = 57.22 + 6.01 \times \text{Time} + 17.37 \text{ MrO}\_2\text{/Fe}\_2\text{O}\_3 \times \text{ratio.} \tag{23}$$

The time and ratio MnO2/reducing agents were coded according to low and medium high levels. From the adjustment of multiple regression models, the interactions of the factors together with the curvature of the time factor and MnO2/reducing agent did not contribute to explain the variability in any of the adjusted models.

From Equations (20)–(23) and from the main effect graphs in Figure 3, the factor that had showed a higher marginal contribution in Mn recovery was the MnO2/Reducing agent ratio for the experimental design whose reducing agent was Fe2<sup>+</sup>, FeC and Fe2O3, while in case of using FeS2 as a reducing agent, the factor that has a greater impact on recovery is time.

The ANOVA test indicates that the models adequately represent Mn extraction for the set of sampled values. The model does not require additional adjustments and is validated by the goodness-of-fit statistics shown in Table 6. The *p* value (*p* < 0.05) and the significance tests F (FRegression >> (F Table= F2,6(5 .1432)) for a level of significance of α = 0.05 (95% confidence level) indicate that all models generated for the representation of the experimental tests were statistically significant. The normality tests indicate that the assumption of normality of the residuals was met. The low values of the S statistic indicate that there were no large deviations between the experimental data and the values of the adjusted model.


**Table 6.** Goodness of fit statistics.

The value of the *R*<sup>2</sup> statistics was greater than 90%, which indicates that a large part of the total variability was explained by the models, while the similarity between the R<sup>2</sup> and R<sup>2</sup> predictive statistics indicates that the model could adequately predict the response to new observations.
