## 3 Results and Discussion
### Analysis of the Statistical Design of Experiments
#### 3.6.1 Numerical Analysis
In other words, the order in which the experiments comprising the matrix were carried out does not influence the obtained results in any way. Regarding the removal of methylene blue, eight effects have a \(p\)-value below 0.05 and, hence, they have a statistically significant influence on the target variable. The R-squared statistic is as high as 93.21%, which suggests that the proposed model is suitable to fit the experimental data. The values of estimated standard error (6.16) and mean absolute error (3.66) corroborate this assertion. Again, according to the DW statistic and its \(p\)-value, no autocorrelation in the residuals is observed. Once it has been determined through the ANOVA test that the model is suitable for effectively representing the influence of the study variables, given their \(p\)-values, it is possible to obtain the coefficients of the polynomial that fits the experimental values, according to a general equation:  
\[\text{y}=\beta\circ+\sum\text{i}=1\text{n}\beta\text{i}\cdot\text{Xi}+\sum \text{i}=1\text{n}\beta\text{i}\cdot\text{X}\text{j}\cdot\text{Xi}+\sum\text {i}=1\text{n}\beta\text{ii}\cdot\text{X}\text{2i} \tag{9}\]  
In this particular case, the general equation is specialized into the following quadratic equations:  
\[\text{Removal \%(Chrysoidine)}=88.8425-0.572917\cdot[\text{Fe}^{2+}]+0.067083 3\cdot[\text{Fe}^{3+}]+6.41708\cdot[\text{H}_{2}\text{O}_{2}]+0.03625\cdot[ \text{Polyphenols}]+1.38927\cdot[\text{Fe}^{2+}]^{2}-0.118125\cdot[\text{Fe}^{2+ }]\cdot[\text{Fe}^{3+}]-0.231875\cdot[\text{Fe}^{2+}]\cdot[\text{H}_{2}\text{ O}_{2}]-0.530625\cdot[\text{Fe}^{2+}]\cdot[\text{Polyphenols}]+1.52177\cdot[\text{Fe}^{3+}]^{2}-0.154375 \cdot[\text{Fe}^{3+}]\cdot[\text{H}_{2}\text{O}_{2}]+0.661875\cdot[\text{Fe}^{3+} ]\cdot[\text{Polyphenols}]-2.14198\cdot[\text{H}_{2}\text{O}_{2}]^{2}-0.231875 \cdot[\text{H}_{2}\text{O}_{2}]\cdot[\text{Polyphenols}]-2.09948\cdot[\text{Polyphenols}]^{2}\]  
\[\text{for the removal of chrysoidine, and}\]  
\[\text{Removal \%(Methylene blue)}=85.1758+2.32172\cdot[\text{Fe}^{2+}]+5.57271\cdot[ \text{Fe}^{3+}]+9.99818\cdot[\text{H}_{2}\text{O}_{2}]-14.5478\cdot[\text{Polyphenols}]-3.62988 \cdot[\text{Fe}^{2+}]^{2}-0.985287\cdot[\text{Fe}^{2+}]\cdot[\text{Fe}^{3+}]+3.83 218\cdot[\text{Fe}^{2+}]\cdot[\text{H}_{2}\text{O}_{2}]-1.92516\cdot[\text{Fe}^{2+ }]\cdot[\text{Polyphenols}]-2.17238\cdot[\text{Fe}^{3+}]^{2}+2.55629\cdot[ \text{Fe}^{3+}]\cdot[\text{H}_{2}\text{O}_{2}]+2.71068\cdot[\text{Fe}^{3+} ]\cdot[\text{Polyphenols}]-2.95788\cdot[\text{H}_{2}\text{O}_{2}]^{2}+7.66436 \cdot[\text{H}_{2}\text{O}_{2}]\cdot[\text{Polyphenols}]-2.79897\cdot[\text{Polyphenols}]^{2}\]  
\[\text{for the removal of methylene blue.}\]  
According to the coefficients shown in Equation (10), the concentration of hydrogen peroxide has the greatest positive influence on the target variable, with a value of +6.41708. This means that an appropriate concentration of hydrogen peroxide is crucial for achieving maximum elimination. However, it is worth noting that an excess of this concentration (reflected by the quadratic term of hydrogen peroxide concentration) is undoubtedly detrimental to the proper removal of the dye, with a value of \(-2.14198\). This is in coherence with Equations (7) and (8). Another significant influence is worth highlighting. According to the obtained data, an excess of both iron solution concentrations (Fe\({}^{2+}\) and Fe\({}^{3+}\)) has a favorable effect on achieving maximum removal of chrysoidine. This can be deduced from the values accompanying the quadratic terms of iron (II) and iron (III) solutions, which are +1.38927 and +1.52177, respectively. To conclude with the variables of major influence, a coefficient of -2.09948 can be observed in the quadratic term of polyphenol concentration, indicating that an excess of these compounds greatly hinders dye removal. A plausible explanation for this fact is related to the ease of oxidation of polyphenols by -OH radicals.