## 3 Results and Discussion
### Analysis of the Statistical Design of Experiments
#### 3.6.2 Graphical Analysis
In the interaction plot, each pair of curves represents the variation of the dye removal percentage by changing one variable at the extremes of the design, i.e., with its value equal to +1.0 (maximum value, marked with the \"+\" sign) and equal to \(-1.0\) (minimum value, marked with the \"\(-\)\" sign), while the other variables are held constant at their central value. If the curves of a variable pair are parallel, modifying one variable does not affect the other. On the other hand, if two lines intersect, the interaction is evident. There can also be an intermediate situation where no clear behavior of the aforementioned types is observed. The data shown in Figure 8c reaffirm the hypotheses described earlier, demonstrating a significant interaction between the concentration of polyphenols and hydrogen peroxide. Additionally, a slight interaction between the iron (II) and iron (III) concentrations can also be observed, which, as previously mentioned, although not remarkable, does exist. Among all the graphs analyzed in this section, perhaps the most important one is the response surface plot. From a mathematical standpoint, it is a graphical representation of Equation (10) (regression equation and correlation coefficients). From a results analysis perspective, it provides data along two different lines: the specific estimation of the optimum and the differential influence of each study factor. It allows for a qualitative understanding of the behavior of the entire system under study, the shape of the function defining the mathematical model, which factors are more influential than others, and in which regions of the studied space this influence is increased or decreased. To obtain the response surface plot, two out of the four variables must be fixed while modifying the other two, and then the efficiency of dye removal against both operational variables is graphically represented in the form of a surface plot. Furthermore, this graph allows us to observe the region where optimal elimination (above 95%) is achieved. In other words, it visualizes the value of the optimum 1 mentioned earlier (see Table 2). By keeping the values of initial concentrations of hydrogen peroxide and polyphenols fixed and equal to those corresponding to the optimum, Figure 8d has been plotted. Figure S2 (Supplementary Material) shows the results of the graphical analysis corresponding to the removal of methylene blue. Most of the considerations made when dealing with the removal of chrysoidine are also applicable to the removal of methylene blue so these results will be discussed more succinctly. One of the most relevant observations made by the Pareto diagram, Figure S2a, is that while the concentrations of Fe(II), Fe(III), and hydrogen peroxide have a positive influence on the elimination of methylene blue (i.e., increasing these concentrations increases the efficiency of the process), an increase in the initial concentration of polyphenols leads to a decrease in the percentage of dye removed. One possible explanation for this is that the use of polyphenols as reducers in the synthesis of iron nanoparticles removes Fe(II) and Fe(III) ions from the solution through their reduction and subsequent formation of nanoparticles. This, in turn, implies a decrease in the efficiency of the Fenton process since part of the catalyst necessary for the decomposition of hydrogen peroxide and the generation of hydroxyl radicals is being removed. However, it is worth mentioning that experimenting in the complete absence of hydrogen peroxide (where all dye removal from the medium would be attributable to its adsorption on the iron nanoparticles) achieves an elimination efficiency of 53% (experiment 21 in Table 1), which is much higher than other cases previously reported in the literature [87; 88; 89]. Additionally, polyphenols play an important role in the elimination of methylene blue, not so much in terms of the efficiency of elimination but rather in terms of the kinetics of the process.