## 2 Materials and Methods
### Optimization of Sorption of Cu(II), Zn(II) and Pb(II) by Biosorbent Using Uniform Design  
With the purpose of the optimization of the removal of elements from water, the experimental uniform design was used, incorporating 3 factors, each with 3 levels. The factors considered for this study included the initial concentration of PTEs, the contact time (between the bean peel and solution), and the initial pH of the solutions. Each factor comprised the following 3 levels: 100 \(\upmu\)g/L, 250 \(\upmu\)g/L, and 500 \(\upmu\)g/L for the initial metal concentration; 2 h, 4 h, and 6 h for the contact time; and 4.0, 5.6, and 7.0, for the initial pHof the solutions. The response variable was the removal percentage of the elements (\(R_{t}\)), calculated as follows:  
\[R_{t}\ (\%)=\frac{(C_{0}-C_{t})}{C_{0}}\times 100 \tag{1}\]  
where \(C_{0}\) is the initial concentration of the element and \(C_{t}\) is the concentration of the element after the contact time (\(t\)). A U\({}_{12}\)(3\({}^{3}\)) design was applied, as shown in Table 2, and three replicates of the design were performed on different days. The uniform design was used due to its capacity to efficiently explore the interrelationships between the factors and the corresponding response variable while conducting a reasonable number of experiments [33]. This type of design has the advantage of simultaneously achieving efficiency and robustness in experimentation [34]. Furthermore, it facilitates the identification of factors that do not influence the response obtained [35]. Being considered a robust design indicates that it can be applied and is able to deliver good performance, even in cases where the exact regression form of the model is unknown [35]. In addition, it stands out for its ability to use values belonging to the experimental factors systematically, that is, uniformly distributed throughout the experimental domain, managing to obtain a high response area using a reduced number of experimental runs without loss of important information [34;35]. The experimental data were analyzed using the best subsets regression analysis to derive the most suitable models for the different predictors used in this study. The predictors are considered the independent variables, which in this work correspond to the initial concentration, contact time, and pH, and they were used to determine the removal percentage, that is, the response or the dependent variable. For each model obtained, the regression analysis was performed to determine the significance level of each predictor included in that model, thus allowing the selection of the most appropriate model for the acquired data. Subsequently, through the regression analysis of the chosen model, a second-order polynomial equation was obtained to describe the interaction between the various variables under investigation. The generalized form of this equation is described as follows:  
\[Y=\beta_{0}+\sum_{i=1}^{K}\beta_{i}x_{1}+\sum_{i=1}^{K}\beta_{ii}x_{ii}^{2}+\sum _{i\leq j}^{K}\sum_{j}^{K}\beta_{ij}x_{i}x_{j}+\epsilon \tag{2}\]  
\begin{table}
\begin{tabular}{c c c c c c c c c c c c} \hline \hline  & \multicolumn{3}{c}{**Factors**} & \multicolumn{3}{c}{**Zn(II) Removal (\%)**} & \multicolumn{3}{c}{**Cu(II) Removal (\%)**} & \multicolumn{3}{c}{**Pb(II) Removal (\%)**} \\ \cline{2-13}
**Run** & \( \begin{array}{c}\textbf{C}_{t}\\ \textbf{(\mu g/L)}\end{array} \) & \begin{tabular}{c} **Time** \\ **(h)** \\ \end{tabular}  & \begin{tabular}{c} **pH** \\ **pH** \\ \end{tabular}  & \begin{tabular}{c} **First** \\ **Replica** \\ \end{tabular}  & \begin{tabular}{c} **Second** \\ **Replica** \\ \end{tabular}  & \begin{tabular}{c} **Third** \\ **Replica** \\ \end{tabular}  & \begin{tabular}{c} **First** \\ **Replica** \\ \end{tabular}  & \begin{tabular}{c} **Second** \\ **Replica** \\ \end{tabular}  & \begin{tabular}{c} **Third** \\ **Replica** \\ \end{tabular}  & \begin{tabular}{c} **First** \\ **Replica** \\ \end{tabular}  & \ \begin{tabular}{c} **Second** \\ **Replica** \\ \end{tabular}  &