# Optimization of Electric Field Assisted Mining Process Applied to Rare Earths in Soils
## 3 Results and Discussion
### Electromining Process Optimization  
The EM process optimization was performed considering the standard problem presented in Equation (3). In this problem, the objective function \(-\text{m}_{\text{Y}}\left(\overline{\text{C}},\ \overline{\varepsilon}\right)\) and the inequality constraint function associated with the electric current density, \(\overline{\text{I}}\left(\overline{\text{C}},\ \overline{\varepsilon}\right)\), were obtained by a fit using the least squares method, considering the values of the variables in their coded form, \(\overline{\text{C}}\ \overline{\varepsilon}\). The objective function, presented in Figure 4, can be explicitly written as:  
\[-\ \text{m}_{\text{Y}}\left(\overline{\text{C}},\ \overline{\varepsilon} \right)=-\ \left(9.31593-\ 3.27450\ \overline{\text{C}}+1.94153\ \overline{\varepsilon}-\ 2.23245\ \overline{\text{C}}\ \overline{\varepsilon}\right) \tag{4}\]  
Defining the vector of design variables in the form \(\text{x}=\left\{\overline{\text{C}},\ \overline{\varepsilon}\right\}^{\text{T}}\), the objective function can be rewritten in matrix form as:  
\[-\ \text{m}_{\text{Y}}\left(\text{x}\right)=-\ 9.31593+\left[\ 3.27450\ \ \ -1.94153\ \ \ \right]\text{x}+\frac{1}{2}\text{x}^{\text{t}}\left[ \begin{array}{cc}0.0&2.23245\\ 2.23245&0.0\end{array} \right]\text{x} \tag{5}\]  
On the other hand, the current density function presented in Figure 5, can be explicitly defined as:  
\[\text{I}_{\text{m}}\left(\overline{\text{C}},\ \overline{\varepsilon}\right)=0.08271+0.01995\ \overline{\text{C}}+0.04968\ \overline{\varepsilon}+0.01100\ \overline{\text{C}}\ \overline{\varepsilon} \tag{6}\] \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Experiment** & **C\({}_{\text{Y}}\) (mol L\({}^{-1}\))** & **m\({}_{\text{Y}}\) (mg)** & \(\underline{\varepsilon}\) (\%)** & **I\({}_{\text{m}}\) (mA cm\({}^{-2}\))** \\ \hline EM-01 & 55.175 & 8.276 & 21.221 & 0.022 \\ EM-02 & 41.281 & 6.192 & 15.877 & 0.040 \\ EM-03 & 110.828 & 16.624 & 42.626 & 0.099 \\ EM-04 & 37.402 & 5.610 & 14.385 & 0.161 \\ EM-05 & 65.845 & 9.877 & 25.325 & 0.091 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: EMS analysis parameters. In matrix form, this function is written as  
\[\text{I}_{\text{m}}\left(\textbf{x}\right)=0.08271+\left[ \begin{array}{cc}0.01995 &0.04968\end{array} \right]\textbf{x}+\frac{1}{2}\textbf{x}^{\textbf{t}}\left[ \begin{array}{cc}0.0&0.01100\\ 0.01100&0.0\end{array} \right]\textbf{x} \tag{7}\]  
It is important to highlight that, due to the model chosen to represent the objective and constraint functions, their second-order Taylor-series approximations at any point result in their own Hessian functions. Thus, to analyze the convexity of these functions it is necessary to evaluate only the principal values of their respective Hessian matrices. Solving the eigenvalues/eigenvectors problems associated with the objective and constraint function's Hessian matrices, the principal values \(\pm\) 2.23245 and \(\pm\) 0.01100 were, respectively, obtained. Positive and negative eigenvalues characterize an indefinite matrix, in other words, a non-convex function. Therefore, it can be concluded that the P\({}_{\text{Y}}\) optimization problem is not convex. Aiming to analyze the EM process, six optimization problems were proposed (P\({}_{\text{Y1}}\), P\({}_{\text{Y2}}\)-P\({}_{\text{Y6}}\)) and characterized by different limit electric current density values (I\({}_{\text{lim}}\)). The optimal configuration and the extracted mass value for each problem are presented in Table 4. For any I\({}_{\text{lim}}\) value, it is observed that the best configuration always occurs with the lowest electrolyte concentration value. In all optimization problems P\({}_{\text{Y1}}\) (i = 1-6) the objective function is the same (Equation (5). Each problem P\({}_{\text{Y1}}\) is characterized by the limit value of the constraint function associated to the electric current density, which redefines the viable set.