# Modern Techniques for Flavonoid Extraction--To Optimize or Not to Optimize?
## 5 Statistical and Mathematical Modeling Techniques for Optimization Flavonoid Extraction Parameters
The literature states that a kinetic model describing solid-liquid extraction can be developed using an empirical and mechanistic method [216]. The mechanistic method involves a hypothetical description of the occurrences, whereas the empirical approach is based on facts from trials. According to [PERSON] and [PERSON] [216], solid-liquid extraction includes two stages; the first stage is the washing stage, during which the solute and solvent are first mixed and the second step uses a diffusion technique for a significantly slower solute transport. As described by [PERSON] et al. [217], understanding the kinetics of the \begin{table}
\begin{tabular}{p{113.8 pt} p{113.8 pt} p{113.8 pt}} \hline \hline
**Kinetic Model** & **Equation** & **Parameters** \\ \hline Equilibrium-dependent model [206] & \(\frac{d(t)}{dt}=k\cdot(c_{e}-c(t))\) & \(c\) is the concentration of the dissolved substance in the liquid phase, \(k\) is the transport coefficient and \(c_{e}\) is the dissolved substance concentration at the equilibrium \\ \hline Second-order model [207] & \(\frac{d(t)}{dt}=k\cdot(c_{e}-c(t))^{2}\) & \(c\) is the concentration of the dissolved substance in the liquid phase, \(k\) is the transport coefficient and \(c_{e}\) is the dissolved substance concentration at the equilibrium \\ \hline [PERSON]’s model [208] & \(c(t)=c_{0}+\frac{t}{K_{1}+K_{2}t}\) & \(c(t)\) presents the concentration of the dissolved substance at the time \(t\), \(K_{1}\) is [PERSON]’s rate constant (relates to extraction rate at the very beginning of the extraction process), \(K_{2}\) is [PERSON]’s capacity constant (relates to maximum dissolved substance concentration and \(c_{0}\) is the concentration of dissolved substance at time \(t=0\). \\ \hline [PERSON]’s model [209] & \(c(t)=e^{-k\cdot t^{n}}\) & \(k\) and \(n\) are [PERSON]’s model constants and \(c(t)\) represents the concentration of dissolved substance at time \(t\) \\ \hline Logarithmic model [209] & \(c(t)=a\cdot\log(t)+b\) & \(a\) and \(b\) are Logarithmic model constants and \(c(t)\) represents the concentration of dissolved substance at time \(t\) \\ \hline [PERSON]’s model [210] & \(1-\frac{c(t)}{c_{e}}=b+k\cdot t\) & \(b\) and \(k\) are [PERSON]’s model constants, \(c(t)\) represents the concentration at time \(t\) and \(c_{e}\) is the equilibrium concentration \\ \hline [PERSON] empirical model [211] & \(c(t)=e^{-a\cdot t}\) & \(a\) is [PERSON]’s model constants, and \(c(t)\) represents the concentration at time \(t\) \\ \hline [PERSON] and [PERSON] empirical model [212] & \(c(t)=a\cdot e^{-b\cdot t}\) & \(a\) and \(b\) are [PERSON] and [PERSON] model constants and \(c(t)\) represent the concentration at time \(t\) \\ \hline Power-law model [213] & \(y=B\cdot t^{n}\) & \(y\) represents extraction yield or final concentration, \(B\) extraction solvent constant, \(n\) diffusion exponent and \(t\) extraction time \\ \hline Two-site kinetic model [214] & \(\frac{c(t)}{c_{w}}=1-F\cdot e^{-k_{1}\cdot t}-(1-F)\cdot e^{-k_{2}\cdot t}\) & \(c_{o}\) represents the maximum extracted concentration, \(F\) denotes the number of biocompounds discharged quickly, \((1-F)\) denoted the amount of biocompound components released gradually, \(k_{1}\) and \(k_{2}\) are first-order rate constants of rapid and slow phases \\ \hline Mass transfer model [215] & \(\frac{\partial c}{dt}=D\frac{\partial^{2}c}{\partial x^{2}}\) & \(c\) is concentration, \(t\) is time, \(D\) is the diffusion coefficients and \(x\) is the diffusion distance \\ \hline \hline \end{tabular}
\end{table} 
Table 5: Kinetic models mostly used for the description of the solid–liquid extraction process of bioactive molecules. extraction depends heavily on the concentration of the substances (_Cs_), the extraction order (n), the activation energy (Ea), and the rate constant (_k_).