## 2 Materials and Methods
### Methodology
#### 2.2.5 Design of Experiment  
In this study, to design the experiments from central composite rotatable design (CCRD) using four independent variables viz. acetone (%), time of extraction (t), liquid-solid ratio (R\({}_{\text{lsr}}\)), and number of extraction cycles (N\({}_{\text{x}}\)) was employed. The independent variables and their levels are shown in Table 1. The CCRD consisted 30 experimental runs on the basis of \(2^{x}+2x+n\), where \(x\) = numbers of variables and \(n\) = number of replicate center points. In this study, six center points in each face of factorial space have been considered. The experimental design matrix for 30 experimental runs is shown in Table 2. The output responses were measured as free and total gossypol reduction in percentage. The error of measured values for free and total gossypol reduction was determined using statistical analysis because measured values are complete only if it is accompanied by details of uncertainty in the measurements. The standard errors of each measured response are shown in Table 3.  
Figure 2: Schematic layout of process of gossypol extraction from cottonseed used in this study.  
RSM was used to develop the regression models between output responses and independent variables. The functional relationship between output responses and independent variables can be shown as:  
\[Y=\mu\ (X_{1}\,X_{2},\ X_{3},X_{4}\ldots X_{k})+e_{r} \tag{1}\]  
In Equation (1), \(Y\) is a single dependent response variable, and \(X_{1}\), \(X_{2}\), \(X_{3}\), \(X_{4}\) are independent variables, and \(\mu\) is the response function. Residual \(e_{r}\) measures the experimental  
\begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Run** & **Acetone (A)\%** & **Time of Extraction** & **Liquid-Solid** & **No. of Extraction** & **Free Gossypol** & **Total Gossypol** \\  & **(t) (min)** & **Ratio(R\({}_{\text{lsr}}\))** & **Cycles (N\({}_{\text{x}}\))** & **Reduction, \% (F_{\text{g}}\)** & **Reduction, \% (T_{\text{g}}\))** \\ \hline
1 & 70 & 30 & 10 & 2 & 86.7 & 62.0 \\ \hline
2 & 80 & 45 & 7.5 & 5 & 98.2  
error [33]. The second-order regression equation has developed, as shown in Equation (2), which gives a response model or regression model.  
\[Y=\beta_{o}+\sum_{i=1}^{3}\beta_{i}X_{i}+\sum_{i=1}^{3}\beta_{ii}X_{i}{}^{2}+ \sum_{i=1}^{3}\sum_{j=1}^{3}\beta_{ij}X_{i}X_{j} \tag{2}\]  
where \(Y\) represents output responses viz. free gossypol and total gossypol percentage, \(\beta_{o}\) is constant, \(\beta_{i}\) is the linear coefficient, \(\beta_{ii}\) is the quadratic coefficient, \(\beta_{ij}\) is the cross-product coefficient, and \(X_{i}\) is the input variable [34]. The regression equations (Equations (3) and (4)) for the prediction of FG and TG were developed based on the experimental data.  
\[\begin{split}\text{FG}=-352.19+11.405\text{ A}-0.1519\text{ t}-0.8733\text{ R}_{\text{lsr}}-7.3958\text{ N}_{\text{x}}-0.00025\text{ At}-0.029\text{ A R}_{\text{lsr}}+0.05375\text{ A}\text{ N}_{\text{x}}+\\ 0.00166667\text{ t}_{\text{lsr}}-0.045\text{ t}\text{ N}_{\text{x}}+0.985\text{ R}_{\text{lsr}}\text{ N}_{\text{x}}-0.071708\text{ A}2+0.00385185\text{ t}2+0.10267\text{ R}_{\text{lsr}}2+0.22917\text{ N}_{\text{x}}\text{ 2}\end{split} \tag{3}\]  
\[\begin{split}\text{TG}=-474.78229+13.10979\text{ A}+0.082083\text{ t}-2.7675\text{ R}_{\text{lsr}}-9.01458\text{ N}_{\text{x}}+0.00395833\text{ At}+0.02975\text{ A}\text{ R}_{\text{lsr}}+\\ 0.083125\text{ A}\text{ N}_{\text{x}}-0.019167\text{ t}\text{ R}_{\text{lsr}}-0.00375\text{ t}\text{ N}_{\text{x}}+0.6575\text{ R}_{\text{lsr}}\text{ N}_{\text{x}}-0.080323\text{ A}2-0.00381019\text{ t}2+\\ 0.090833\text{ R}_{\text{lsr}}2+0.11771\text{ N}_{\text{x}}\text{ 2}\end{split} \tag{4}\]