## 3 Results and Discussion
### Optimization of Microalgal Harvesting Variables: Dosage, Settling Time, and Mixing Times  
The efficiency of microalgal harvesting (response variable) obtained for the 2\({}^{4}\) full factorial experimental design are in Table 11 (Appendix A). The results were analyzed by ANOVA (\(p\)-value \(<\) 0.05) to quantify statistical differences among the variables. The main effects and interaction analysis between the factors (dosage, settling time, RM time, and SM time) are in Table 12 (Appendix A). The effects describe the size and direction of the relationship between a factor and the response variable; that is, they represent the predicted change in the mean response when the factor changes from the low to the high level [31]. The signs \"+\" and \"\(-\)\" of the effects indicate if the factor positively or negatively influences the harvesting efficiency, respectively. For instance, increasing the aluminum sulfate dosage from 50 to 200 mg\(\cdot\)L\({}^{-1}\) positively affected efficiency. According to the \(p\)-values, settling time was statistically not significant (\(p\)-value \(>\) 0.05) for aluminum sulfate and chitosan. The effect of dosage was also statistically not significant (\(p\)-value \(>\) 0.05) for chitosan and ferric sulfate. The empirical models for each flocculant are in Table 6. These models describe how the different factors and their main interactions affect the harvesting efficiency. They were obtained by eliminating the not statistically significant parameters and the ones with coefficients close to zero. The high R\({}^{2}\) values of each model indicate that the models appropriately fitted the data. However, it does not necessarily mean that the models predict new observations well. That information is given by the predicted R\({}^{2}\) values, which are close to 100%. The predicted R\({}^{2}\) is derived by systematically excluding each observation from the dataset, estimating the regression equation, and assessing how effectively the model predicts the omitted observation [32]. To improve the flocculation process, the regression equations were used to predict the values that the factors should have to obtain the maximum harvesting efficiency. These optimal parameters were tested to harvest _C. vulgaris_, and the results are in Table 7. All flocculants showed excellent biomass recoveries. The experimental data were very close to the predicted data, showing once again the ability of the models to predict new observations. \begin{table}
\begin{tabular}{c c} \hline
**Floculant** & **Optimal Dosage Range (mg\(\cdot\)L\({}^{-1}\))** \\ \hline Aluminum sulfate & 50–200 \\ Ferric sulfate & 150–200 \\ Chitosan & 10–50 \\ \hline \end{tabular}
\end{table} 
Table 5: Optimal dosage range for flocculation with aluminum sulfate, ferric sulfate, and chitosan. [PERSON] and [PERSON] [26] reported harvesting efficiencies of 87.33% with 600 mg\(\cdot\)L\({}^{-1}\) ferric sulfate and 82.27% with 800 mg\(\cdot\)L\({}^{-1}\) of aluminum sulfate. The authors used higher dosages and only performed an RM stage. In the present study, lower flocculant dosages (150-250 mg\(\cdot\)L\({}^{-1}\)) combined with two mixing stages allowed the achievement of better efficiencies. [PERSON] et al. [30] also included two mixing stages and obtained biomass recoveries of 95% with only 100 mg\(\cdot\)L\({}^{-1}\) of aluminum sulfate. The mentioned articles studied different species of microalgae at different concentrations; therefore, the comparison is not straightforward. Nevertheless, it is still possible to perceive that the mixing stages significantly impact the flocculation process. Inorganic flocculants were most efficient when an RM and an SM phase were included. On the contrary, chitosan reached a maximum efficiency of 99.5% with just 15.5 min of SM. When a fast mixing of the culture was provided, higher dosages of chitosan (200 mg\(\cdot\)L\({}^{-1}\)) were needed to harvest _C. vulgaris_[22]. [PERSON] et al.