## 3 Results and Discussion
### Extraction Model for The Extraction Process of Glycyrrhizic Acid  
For ultrasonic power, the relationship of \(\ln(C_{e}-C_{t})\) against time was shown in Figure 4 which presented a good linear relationship. The values of kinetic parameters (\(C_{e}\), \(\lambda\)) for different ultrasonic powers from the logarithm of \(\ln(C_{e}-C_{t})\) were obtained and listed in Table 1. As can be seen, these kinetic parameters increased with the increase of ultrasonic power as expected in accordance with the experimental analysis results. Figure 4: The relationship of \(\ln(C_{e}-C_{t})\) against extraction time (\(t\)) under different ultrasonic powers  
Figure 3: Evolution of extraction yields of glycyrrhizic acid with different extraction temperatures at ultrasonic frequency of 20 kHz and ultrasonic power of 125 W.  
Because these kinetic parameters were extremely dependent on ultrasonic powers, the \(C_{e}\) and \(\lambda\) values for different \(P\) values were successfully fitted by linear and second-order polynomial functions due to the high values of correlation coefficient (\(R^{2}\)). The functions were plotted in Figure 5 and expressed as follows:  
\[C_{e,P}=0.0164P+2.143\ \ R^{2}=0.995 \tag{7}\]  
\[\lambda_{P}=0.0004P^{2}-0.0171P+0.667\ \ R^{2}=0.985 \tag{8}\]  
Substituting Equations (7) and (8) into Equation (4), the total bufadienolides concentration \(C_{t,p}\) as a function of \(P\) therefore could be obtained. Their relationship was expressed as:  
\[C_{t,P}=(0.0164P+2.143)[1-e^{-(0.0004P^{2}-0.0171P+0.667)t}] \tag{9}\]  
This power extraction model (i.e., Equation (9)) could be employed to predict the extraction of glycyrrhizic acid from _Glycyrrhizic varalensis_ at a given ultrasonic frequency of 55 kHz and extraction temperature of 25 \({}^{\circ}\)C with a solvent/solid ratio of 30 (mL/g). For ultrasonic frequency, the relationship of \(\ln(C_{e}-C_{t})\) against time was shown in Figure 6 which also presented a good linear relationship. The values of kinetic parameters (\(C_{e},\lambda\)) for different ultrasonic frequencies from the logarithm of \(\ln(C_{e}-C_{t})\sim t\) were obtained and listed in Table 2. \begin{table}
\begin{tabular}{c c c c c c} \hline
**Ultrasonic Power (W)** & **20** & **50** & **75** & **100** & **125** \\ \hline \(C_{e}\) (g/L) & 2.397 & 2.754 & 3.425 & 3.696 & 4.208 \\ \(\lambda\) (10\({}^{-1}\) s\({}^{-1}\)) & 0.482 & 0.876 & 1.440 & 2.759 & 4.516 \\ \(R^{2}\) & 0.998 & 0.999 & 0.997 & 0.999 & 0.996 \\ \hline \end{tabular}
\end{table} 
Table 1: Parameters of Kinetic model of glycyrrhizic acid from _Glycyrrhizic varalensis_ with different ultrasonic powers. Figure 5: The equilibrium concentration (\(C_{e}\)) of glycyrrhizic acid, and integrated influence coefficient (\(\lambda\)) with different ultrasonic powers. As can be seen, the extraction at a frequency of 70 kHz displayed the highest \(\lambda\) value compared to those at 20, 30, 40, and 55 kHz. For the \(C_{e}\), however, the highest \(C_{e}\) value was obtained at 55 kHz instead of 90 kHz. The result was in agreement with the results in Section 3.1.2. Based on the model assumption, these kinetic parameters \((C_{e},\lambda)\) were expressed by the variance of ultrasonic frequency (\(f\)). Therefore, the relationships among \(C_{e}\), \(\lambda\), and \(f\) were nonlinearly fitted with three-order polynomial, and second-order polynomial functions, respectively. The two functions were plotted in Figure 7 and expressed as follows: \begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Ultrasonic Frequency** & **20** & **30** & **40** & **55** & **90** \\ (kHz) & & & & & \\ \hline \(C_{e}\) (g/L) & 3.688 & 3.755 & 5.086 & 7.121 & 5.827 \\ \(\lambda\) (10\({}^{-1}\) s\({}^{-1}\)) & 1.088 & 1.026 & 1.275 & 1.922 & 2.645 \\ \(R^{2}\) & 0.999 & 0.999 & 0.997 & 0.998 & 0.998 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Parameters of kinetic model of glycyrrhizic acid from _Glycyrrhiza uralensis_ with different ultrasonic frequencies.