## 3 Results and Discussion  
### Validation of the Experimental Design  
Based on previous research, it has been identified that four factors, namely fermentation temperature, initial soluble solids, inoculum amount, and initial pH, have a significant impact on the fermentation of Huaniu apple cider. These factors have been carefully screened and found to play a crucial role in the overall fermentation process and the quality of the cider. Using Design Expert 12.0 software, a response surface analysis test was designed with 29 points, considering the four factors at three levels. The fermentation rate was chosen as the response value (Table 2). The fermentation rate ranged from 2.29 to 2.94.  
\begin{table}
\begin{tabular}{c c c c c} \hline \hline \multirow{2}{*}{**Symbol**} & \multirow{2}{*}{**Independents**} & \multicolumn{3}{c}{**Coded Level**} \\ \cline{3-5}  & & \(-\alpha(-1)\) & 0 & +\(\alpha(+1)\) \\ \hline X\({}_{1}\) & temperature/\({}^{\circ}\)C & 25 & 28 & 31 \\ X\({}_{2}\) & initial soluble solids/\({}^{\circ}\)Brix & 13 & 16 & 19 \\ X\({}_{3}\) & inoculum amount/\% & 8 & 10 & 12 \\ X\({}_{4}\) & initial pH & 3.1 & 3.6 & 4.1 \\ \hline \hline \end{tabular}
\end{table}
Table 1: Coded levels of independent variables for the Box–Behnken design (BBD).  
\begin{table}
\begin{tabular}{c c c c c c} \hline \hline \multicolumn{1}{c}{**Run**} & \(\mathbf{X_{1}}\) & \(\mathbf{X_{2}}\) & \(\mathbf{X_{3}}\) & \(\mathbf{X_{4}}\) & **R (Predicted): Fermentation Rate** & **R (Actual): Fermentation Rate** \\ \hline
1 & 25 & 13 & 10 & 3.6 & 2.53 & 2.62 \\
2 & 31 & 13 & 10 & 3.6 & 2.60 & 2.51 \\
3 & 25 & 19 & 10 & 3.6 & 2.89 & 2.93 \\
4 & 31 & 19 & 10 & 3.6 & 2.80 & 2.67 \\
5 & 28 & 16 & 8 & 3.1 & 2.36 & 2.29 \\
6 & 28 & 16 & 12 & 3.1 & 2.44 & 2.41 \\ \hline \hline \end{tabular}
\end{table}
Table 2: BBD with responses of the dependent variables to fermentation rate.  
The regression model for the fermentation process parameters of Huaniu apple cider was established. The fermentation rate was used as the response value, and the independent variables were fermentation temperature (X\({}_{1}\)), initial soluble solids (X\({}_{2}\)), inoculum amount (X\({}_{3}\)), and initial pH (X\({}_{4}\)).  
\[\text{Fermentation Rate R}=2.54+0.14X_{2}+0.081X_{4}+0.11X_{24}+0.087X_{11}+0.077X_{2 2}-0.11X_{44}\]  
The model was analyzed for variance, and the significance of the model coefficients was tested, taking the fermentation rate as the evaluation index (Table 3). The regression model was statistically significant (\(p<0.001\)), indicating its significance. The initial soluble solids content had a significant effect on the fermentation rate (\(p<0.001\)), and the initial pH had a significant effect as well (\(p<0.01\)). The lack-of-fit term (\(p=0.0692\)) was not significant, suggesting a good fit of the experimental model and reasonable model selection. The R\({}^{2}\) value of the model was 0.8693, indicating a good fit between the measured and predicted values of Huaniu apple cider's fermentation rate. The accuracy of the test (C.V.) was 3.13%, demonstrating the reliability of the experimental procedure.  
\begin{table}
\begin{tabular}{c c c c c c} \hline
**Run** & **X\({}_{1}\)** & **X\({}_{2}\)** & **X\({}_{3}\)** & **X\({}_{4}\)** & **R (Predicted): Fermentation Rate** & **R (Actual): Fermentation Rate** \\ \hline  
\hline \end{tabular}
\end{table}
Table 2: _Cont._  
\begin{table}
\begin{tabular}{c c c} \hline
**Source** & **_F Value_** & **_p-Value_** \\ \hline Model & 6.65 & 0.0005 *** \\ X\({}_{1}\) & 0.046 & 0.8334 \\ X\({}_{2}\) & 36 & 0.0001 *** \\ X\({}_{3}\) & 1.07 & 0.3179 \\ X\({}_{4}\) & 12 & 0.0038 ** \\ X\({}_{12}\) & 0.86 & 0.3692 \\ X\({}_{13}\) & 0.034 & 0.8554 \\ X\({}_{14}\) & 0.38 & 0.5461 \\ \hline \end{tabular}
\end{table}
Table 3: Model summary and ANOVA of fermentation rate.