# Parameter Calibration of Discrete Element Model of Wine Lees Particles
## 3 Results and Discussion
### Response Surface Analysis Test
#### 3.3.2 Regression Model Analysis  
Design-Expert was utilized to establish the second-order regression equation between the angle of repose and the three parameters, which is expressed as:  
\[Y=31.57+17.48X_{1}-0.3X_{4}-183.72X_{0}+10.1X_{1}\times X_{4}-110.74X_{1}\times X _{0}-200.92X_{4}\times X_{0}-23.5X_{1}^{2}+5.17X_{4}^{2}+2618.92X_{0}^{2} \tag{10}\]  
The analysis of variance (ANOVA) of the quadratic regression model was performed as shown in Appendix A, Table A1, and based on the F-value and \(p\)-value, it can be obtained that the effects of the three factors on the angle of repose Y are in the order of \(X_{1}>X_{4}>X_{0}>X_{0}^{2}>X_{4}\times X_{0}>X_{1}^{2}>X_{1}\times X_{0}>X_{1 }\times X_{4}>X_{4}^{2}\), where the particle-particle recovery coefficient is \(p<0.05\), indicating that this parameter has a significant effect on the angle of repose. This linear regression model with \(p<0.05\) shows the validity of the regression model, indicating that the relationship between the angle of repose and the resulting regression equation is significant. The coefficient of variation of the test, CV = 1.41%, indicates that the reliability of this test is good; coefficient of determination \(R_{2}\) = 0.9595, Calibration coefficient of determination \(R_{2}\)adj = 0.8875, it shows that the regression equation is highly reliable; Adeq Precision = 11.9308, it shows that the accuracy of the regression model is good. \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Symbolic** & **Simulation Parameters** & **Low Level (\(-1\))** & **Mid-Level (\(0\))** & **High Level (\(1\))** \\ \hline \(X_{0}\) & Particle-particle surface & \multirow{2}{*}{0.0625} & \multirow{2}{*}{0.075} & \multirow{2}{*}{0.0875} \\  & energy & & & \\ \(X_{1}\) & Particle-particle recovery & & & \\ \(X_{4}\) & coefficient & & & \\ \hline \hline \end{tabular}
\end{table} 
Table 7: Factors and levels for Box-Behnken test. \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**No.** & \(X_{0}\) & \(X_{1}\) & \(X_{4}\) & **Resting Angle\({}^{c}\)** & **Relative Error** \\ \hline \(1\) & \(-1\) & \(-1\) & \(0\) & \(26.60\) & \(3.13\%\) \\ \(2\) & \(0\) & \(1\) & \(-1\) & \(24.46\) & \(5.18\%\) \\ \(3\) & \(0\) & \(-1\) & \(1\) & \(26.66\) & \(3.35\%\) \\ \(4\) & \(0\) & \(0\) & \(0\) & \(25.84\) & \(0.18\%\) \\ \(5\) & \(1\) & \(0\) & \(1\) & \(26.11\) & \(1.23\%\) \\ \(6\) & \(0\) & \(0\) & \(0\) & \(25.84\) & \(0.18\%\) \\ \(7\) & \(1\) & \(1\) & \(0\) & \(25.15\) & \(2.49\%\) \\ \(8\) & \(-1\) & \(0\) & \(-1\) & \(25.99\) & \(0.76\%\) \\ \(9\) & \(-1\) & \(1\) & \(0\) & \(24.79\) & \(3.89\%\) \\ \(10\) & \(0\) & \(1\) & \(1\) & \(23.76\) & \(7.87\%\) \\ \(11\) & \(1\) & \(-1\) & \(0\) & \(27.52\) & \(6.69\%\) \\ \(12\) & \(-1\) & \(0\) & \(1\) & \(26.15\) & \(1.37\%\) \\ \(13\) & \(0\) & \(0\) & \(0\) & \(25.84\) & \(0.18\%\) \\ \(14\) & \(1\) & \(0\) & \(-1\) & \(26.96\) & \(4.52\%\) \\ \(15\) & \(0\) & \(-1\) & \(-1\) & \(27.75\) & \(7.60\%\) \\ \hline \hline \end{tabular}
\end{table} 
Table 8: Design and results of Box-Behnken simulation test. The quadratic regression model is optimized by eliminating terms that do not significantly affect the results. The optimized regression equation is as follows:  
\[Y=30.80+15.71X_{1}+11.97X_{4}-246.34X_{0}-200.92X_{4}\times X_{0}-23.5X_{1}^{2}+25 93.45X_{0}^{2} \tag{11}\]  
The ANOVA results of the optimized regression model are presented in Appendix A, Table A2. At this stage, the coefficient of variation (CV) is 1.22%, the coefficient of determination (\(R_{2}\)) is 0.9522, the corrected coefficient of determination (\(R_{2\text{adj}}\)) is 0.9164, and the precision Adeq Precision is 16.5885. Compared to the pre-optimization period, there has been a significant improvement in terms of goodness of fit, reliability, and precision in this regression equation. The quadratic regression model was further optimized by again removing the terms with insignificant effects.