## 3 Results and Discussion
### Extraction Model for The Extraction Process of Glycyrrhizic Acid
Figure 6: The relationship of \(\ln(C_{e}-C_{t})\) against extraction time (\(t\)) under different ultrasonic frequencies. Figure 7: The equilibrium concentration (\(C_{e}\)) of glycyrrhizic acid, and integrated influence coefficient (\(\lambda\)) with different ultrasonic frequencies. \[C_{e,f}=-0.0001f^{3}+0.0109f^{2}-0.26f+5.443\ R^{2}\ =\ 0.950 \tag{10}\]  
\[\lambda_{f}=0.0005f^{2}-0.0129f+0.1141\ R^{2}\ =\ 0.965 \tag{11}\]  
Substituting the \(C_{ef}\) and \(\lambda_{f}\) into Equation (4), the relationship was described as:  
\[C_{t,f}=(-0.0001f^{3}+0.0109f^{2}-0.26f+5.443)[1-e^{-(0.0005f^{2}-0.0129f+0.11 41)t}] \tag{12}\]  
This ultrasonic frequency extraction model (i.e., Equation (12)) could be employed to predict the glycyrrhizic acid extraction process at a given ultrasonic power of 125 W and extraction temperature of 25 \({}^{\circ}\)C with a solvent/solid ratio of 30 (mL/g). For extraction temperature, the relationship of \(\ln(C_{e}-C_{t})\) against time was shown in Figure 8. It can be seen that \(\ln(C_{e}-C_{t})\sim t\) presented a good linear relationship. The values of kinetic parameters (\(C_{e},\lambda\)) for different extraction temperatures from the logarithm of Figure 8 were obtained and listed in Table 3.  
As can been seen, both values of \(C_{e}\) and \(\lambda\) increased as the extraction temperature increased. The highest \(C_{e}\) and \(\lambda\) values were all achieved at a temperature of 70 \({}^{\circ}\)C, followed by those at 60, 35, 25, and 20 \({}^{\circ}\)C. This result was similar to the previous report [38], which was consistent with the fact that a significant increase of extraction temperature could cause the increase of diffusivity [39; 40]. The relationships among \(C_{e}\), \(\lambda\), and \(T\) were fitted well by two second-order polynomial functions because of the high values of \(R^{2}\). The functions were plotted in Figure 9 and written as:  
\[C_{e,T}=0.0013T^{2}-0.0439T+3.103\ R^{2}\ =\ 0.995 \tag{13}\]  
\[\lambda_{T}=0.003T^{2}-0.1707T+3.342\ R^{2}\ =\ 0.970 \tag{14}\] \begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Extraction Temperature (\({}^{\circ}\)C)** & **20** & **25** & **35** & **60** & **70** \\ \hline \(C_{e}\) (g/L) & 2.623 & 2.853 & 3.226 & 4.786 & 6.289 \\ \(\lambda\) (10\({}^{-1}\) s\({}^{-1}\)) & 0.971 & 1.022 & 1.258 & 3.556 & 6.267 \\ \(R^{2}\) & 0.996 & 0.998 & 0.997 & 0.999 & 0.998 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Parameters of kinetic model of glycyrrhizic acid from _Glycyrrhizic uralensis_ with different extraction temperatures. Figure 8: The relationship of \(\ln(C_{e}-C_{t})\) against extraction time (\(t\)) under different extraction temperatures. Substituting the \(C_{e,T}\) and \(\lambda_{T}\) into Equation (4), the relationship was obtained as:  
\[C_{t,T}=(0.0013T^{2}-0.0439T+3.103)[1-e^{-(0.003T^{2}-0.1707T+3.342)t}] \tag{15}\]  
This temperature extraction model (i.e., Equation (15)) could be employed to predict the glycvrrhizic acid extraction process at a given ultrasonic power of 125 W and frequency of 55 kHz with a solvent/solid ratio of 30 (mL/g).