## 2 Materials and Methods
The authors of, for example, [20,31], and even Minitab 18's support website [32], which is the software used for the statistical analysis in this paper, generally agree on the fact that the sample size of an experiment depends on (1) the size of the effects, (2) the choice of type I error, and (3) the power of the statistical test to be achieved. Even though the first and second criteria may always depend on the specificities of a given experiment, in the case of the third criterion, all of these works further state that the sample size should usually be able to yield a power (level of confidence) of more than 0.8 or 80%, while at the same time, these authors, and those of this paper, also agree on the fact that the bigger the number of replicates and size of the experiment, the better the chance the experimenter has to recognize smaller effects or simply increase the level of confidence of the whole design. In this sense, based on the authors' experience and several published works with similar experiments, for example, [20,26], where one can find some examples, an experiment of the \(2^{3}\) type is usually sufficient, with simply two or three replicates. However, in the case of this paper, the number of replicates was defined as four for two of the variables and three for the last one, and this was due to the fact that the authors wanted to guarantee a better analysis of the factors, and they mostly took into account the necessary monetary resources and time for the realization of all the measurements. As it has been mentioned before, the fact that the authors of this research opted to implement a factorial design-based DOE as the best course of action reflects their interest in and need for studying the joint effect of three specific factors and their interactions on given response variables. Analyzing more than a single response variable often leads to solution alternatives, which, in this case, translates to the decision of the authors to elaborate three separate \(2^{k}\) designs which are described and analyzed in detail later in this document. On the other hand, despite the existence of several variations or types of factorial designs, the authors specifically decided to implement one of the \(2^{3}\) type, and this was mainly for two reasons: (1) it fits the needs of the study, and (2) this type of design has been shown to be superior to other alternatives such as the \(3^{k}\) type, as discussed, for example, in [20]. The author of [20] also presented a very detailed and generally well-accepted sequence of steps for the realization of this type of design, and thus the experiment that was carried out in this paper was in accordance with his proposed \"Analysis Procedure for a \(2^{k}\) design\", which was slightly modified by the authors, as follows:  
1. Estimate the effects of the factors and their interactions;
2. Form the initial model: 1. If the design is replicated, fit the full model; 2. If there is no replication, form the model using a normal probability plot of the effects;
3. Perform statistical testing using the selected software (generally ANOVA testing);Refine the model (non-significant variables may be excluded from the model);
* Analyze residuals and verify the model adequacy and fulfilment of the assumptions;
* Interpret the results using proper plots (main effect and interaction plots, response surface and contour plots). The statistical model for a \(2^{\rm k}\) design includes k main effects, \(\left( \begin{array}{c}k\\ 2\end{array} \right)\) two-factor interactions, \(\left( \begin{array}{c}k\\ 3\end{array} \right)\) three-factor interactions, and one k factor interaction. In the case of our \(2^{3}\) design, if applying the commonly used letter or label notation, it can be interpreted as follows: a, b, and c are the main effects; ab, ac, and bc are the two-factor interactions; abc represents the three-factor interactions; and, finally, (1) is the k factor interaction. The following Table 2 shows the general experiment table for a \(2^{3}\) design.