## 2 Materials and Methods
### Experimental Design for RSM  
In order to optimize the MUAE process, RSM as an optimization method was employed to determine the optimum conditions for the maximum recovery yield of quercetin and total flavonoids from red onion skin waste using the Box-Behnken design (BBD). The main factors for RSM design were selected based on the most effective factors on the results obtained from the single-factor experiments. The five effective variables on the extraction, namely ultrasound time (min, X1), ultrasound temperature (\({}^{\circ}\)C, X2), solvent-to-solid ratio (mL/g, X3) microwave time (s, X4), and ethanol concentration (% (\(v/v\), water), X5) are summarized in Table 1 at three levels (\(-\)1, 0, +1), with X1 (15, 30, and 45 min), X2 (60, 70, and 80 \({}^{\circ}\)C), X3 (20, 30, and 40 mL/g), and X4 (30, 60, and 90 s), and X5 (50, 60, and 70%). In general, 46 experimental runs (in triplicates) were carried out. The dependent variables (responses) were quercetin (Y1) and total flavonoid (Y2) yields.  
The design conditions and results of the 46 experimental runs to optimize quercetin and total flavonoid extraction conditions are summarized in Table 2. According to the variance analysis of the regression model, a \(p\)-value less than 0.05 indicates the significance of terms and a \(p\)-value less than 0.0001 indicates the high significance of the regression model terms. Moreover, \(p\)-values greater than 0.10 indicate that the model terms are insignificant. The regression analysis of the extraction responses was fitted using a second-order polynomial equation as expressed in Equation (2):  
\[Y_{n}=\beta_{0}+\sum_{i=1}^{k}\beta_{i}x_{i}\ +\ \sum_{i=1}^{k}\beta_{ii}\ x_{i}^{2 }\ +\sum_{i}^{k-1}\sum_{j}^{k}\beta_{ij}\ x_{i}\ x_{j} \tag{2}\]  
where \(Y_{n}\) represents the extraction responses, \(\beta_{0}\) is the intercept, \(\beta_{i}\) denotes the coefficient of the linear, \(\beta_{ii}\) is the quadratic term, \(\beta_{ij}\) represents the cross-product term, \(x_{i}\) and \(x_{j}\) denote the independent variables, and \(k\) is the number of variables (\(k=5\)).  
\begin{table}
\begin{tabular}{c c c c c c} \hline \hline \multirow{2}{*}{**Variables**} & \multirow{2}{*}{**Codes**} & \multirow{2}{*}{**Units**} & \multicolumn{3}{c}{**Levels**} \\  & & & \(-\)**1** & **0** & **+1** \\ \hline Ultrasound time & X1 & Min & 15 & 30 & 45 \\ Ultrasound temperature & X2 & \({}^{\circ}\)C & 60 & 70 & 80 \\ Solvent-to-solid ratio & X3 & mL/g & 20 & 30 & 40 \\ Microwave time & X4 & S & 30 & 60 & 90 \\ Ethanol concentration & X5 & \% (\(v/v\), water) & 50 & 60 & 70 \\ \hline \hline \end{tabular} \(\overline{\text{X}_{1}}\): ultrasound time; X2: ultrasound temperature; X3: solvent-to-solid ratio; X4: microwave time; X5: ethanol concentration.  
\end{table}
Table 1: Specifications of the independent variables.  
\begin{table}
\begin{tabular}{c c c c c c c} \hline \hline \multirow{2}{*}{**Run**} & \multicolumn{4}{c}{**Independent Variables**} & \multicolumn{4}{c}{**Responses (Dependent Variables)**} \\  & **X1: US** & **X2: US** & **X3: S/S Ratio** & **X4: MW** & **Y1:** & **Y2:** \\  & **Time (min)** & **Temp (\({}^{\circ}\)C)** & **(mL/g)** & **Time (min)** & **Coc** & **Quercetin Yield** & **Flavonoid** \\  & & & & & **(\(v/v\), Water)** & **(\%)** & **Yield (\%)** \\ \hline
1 & \(-\)1 & \(-\)1 & 0 & 0 & 0 & 9.63 \(\pm\) 0.16 & 12.52 \(\pm\) 0.21 \\
2 & 0 & \(-\)1 & 0 & 0 & 1 & 9.42 \(\pm\) 0.28 & 11.52 \(\pm\) 0.06 \\
3 & 0 & \(-\)1 & \(-\)1 & 0 & 0 & 8.41 \(\pm\) 0.09 & 10.41 \(\pm\) 0.7 \\
4 & 1 & 0 & 0 & \(-\)1 & 0 & 7.64 \(\pm\) 0.74 & 9.52 \(\pm\) 0.95 \\
5 & 0 & 0 & 0 & 0 & 0 & 8.51 \(\pm\) 0.40 & 10.49 \(\pm\) 0.8 \\
6 & 0 & 0 & 1 & \(-\)1 & 0 & 8.79 \(\pm\) 0.47 & 10.74 \(\pm\) 0.41 \\
7 & 0 & 0 & 0 & 0 & 0 & 8.35 \(\pm\) 0.27 & 10.53 \(\pm\) 0.16 \\
8 & \(-\)1 & 0 & 0 & 0 & \(-\)1 & 9.21 \(\pm\) 0.81 & 11.32 \(\pm\) 0.43 \\ \hline \hline \end{tabular}
\end{table}
Table 2: Box–Behnken design with coded independent variables and responses.