## 2 Materials and Methods
### Experimental Design Methods
#### 2.1.3 Response Surface Method (RSM)  
The RSM covers a collection of mathematical and statistical techniques that rely on fitting a polynomial equation to experimental data and need to describe the behavior of a data set in order to make statistical predictions. This approach can be well applied when a response or a set of responses of interest is affected by several variables. The aim is to optimize the levels of these variables simultaneously to achieve the best system performance [19]. The experimental work using the RSM is greatly reduced compared to the number of runs determined using the full factorial design. Besides the reduction in experimental studies, the results from the RSM are claimed to be statistically acceptable [20]. The response surface method (RSM) was used in the experimental design. The RSM is an engineering tool used in the design phase of a new product or process to improve and optimize its performance [21]. The RSM was first introduced by [PERSON] (1951) and later developed for an experimental design and data analysis [22]. The method includes a sequential process in which statistical and mathematical techniques are used. In this process, the researcher investigates the type of appropriate approximation function, the selection of the appropriate experimental design layout, the location and shape of the optimum region, and the necessity for transformation for the response or design variables [22]. The RSM design process can be summarized in three steps: (i) designing a set of experiments for the measurement of responses (elimination trials); (ii) determining the mathematical model with the best fit between the input variables and the response and obtaining the optimal set of experimental settings that yields the response's maximum or minimum values (region research); (iii) expressing the effects of the parameters affecting the process with 2D or 3D graphics [23]. The RSM is mostly about approximating an unknown complex function using a first-order or second-order model of a low-order polynomial [15]. If the response variable is expressed as a linear function of the independent variables, the use of the first-order model is appropriate [24]. If there is a curvilinear relationship in the system, the main effect models called first-order models are insufficient. In this case, there is the establishment of a higher order polynomial, such as a second-order polynomial. This model is also called the second-order model. The following first-order model (Equation (1)) is generated if the system response in the RSM fits the independent variable well as a linear function [25]:  
\[y=\beta_{0}+\beta_{1}x_{1}+\beta_{2}x_{2}+\ldots+\beta_{k}x_{k}+\varepsilon, \tag{1}\]  
A quadratic model (Equation (2)) could be more suitable if the system's response surface is curved:  
\[y=\beta_{0}+\sum\
olimits_{j=1}^{k}\beta_{j}x_{j}+\sum\
olimits_{j=1}^{k}\beta _{jj}x_{j}^{2}+\sum\
olimits_{i}^{k-1}\beta_{ij}x_{i}x_{j}+\varepsilon \tag{2}\]  
where \(y\) is the response variable, \(\beta_{k}\) is the unknown regression parameter, \(x_{i}\) and \(x_{j}\) are process variables, and \(\varepsilon\) is the error term. The RSM method uses two different designs, the Box-Behnken design and central composite design [25]. _Central Composite Designs (CCD):_ The design developed by [PERSON] and [PERSON] in 1951 can work with both linear and quadric models. The CCD can be considered as a good alternative to the three-level full factorial design, providing a smaller number of experiments [26]. _Box-Behnken design:_ This is a very important experimental design that allows for calculating the response function and estimating the system performance at any experimental point in the examined range by performing a small number of studies [16]. The Box-Behnken method, which is used to develop the models created on the basis of the designs and to determine the optimum experimental conditions, is a three-level design that yields a second-order multivariate polynomial with factor and response surfaces, the number of which can be increased.