*2.2. Step II—The Optimization of SFE of p-Anisic Acid and the Mathematical Modeling of Mass Transfer*

Once the *p*-anisic acid was identified in the extract, the next step was to maximize the compound selectivity due to its wide applicability. Until this point, there was no data available in the literature on the supercritical extraction of flowers from *A. mearnsii*, which justified the study.

From the Step I results, a new study regarding the extraction process conditions was evaluated. The use of ethanol as the cosolvent was maintained, considering the results obtained from the factorial design. As larger extract yields were observed for higher CO2 pressures set in the factorial design, the pressure range was increased from 200 to 300 bar. The solvability and diffusivity of the supercritical fluid are directly related to its density, which is a function of temperature and pressure. Thus, another factor selected for the optimization process was the temperature range, from 40 to 60 ◦C. The final parameter evaluated was the particle size. The flowers were ground and passed through a series of six sieves (24, 32, 42, 60, 150, and 325 mesh). Based on the amount retained in each sieve, the fractions retained in the 42, 60, and 150 mesh sieves (0.423, 0.303, and 0.125 mm, respectively) were used. From these variables, a Box–Behnken design was established and the results for the global extract yield and the *p*-anisic acid selectivity are presented in Table 3, where extractions have the solvent-to-feed ratio (S/F) equal to 61.8 gsolvent/gplant.


**Table 3.** Design matrix in the Box–Behnken model and observed responses.

**<sup>a</sup>** grams of *p*-anisic acid in 100 g of crude extract; <sup>b</sup> grams of crude extract from 100 g of dried flowers.

The *p*-anisic acid selectivity ranged from 0.57 to 2.48% *w*/*w* (g *p*-anisic acid/g extract) in the crude extracts and the crude extract yield ranged from 0.86 to 7.84% *w*/*w* (g extract/g plant). The global extract yield using the Box–Behnken design had a significant increase since the highest yield was 2.49% *w*/*w* in the factorial design.

To evaluate the response surface model, an analysis of variance (ANOVA) was performed with the statistical software Minitab® using the results of Table 3. The statistical significance and the influence of the extraction parameters were estimated by the analysis of variance regarding the *p*-anisic acid selectivity, which are presented in Table 4.


**Table 4.** Analysis of variance of the *p*-anisic acid selectivity from the crude extract.

DF: Degrees of freedom; Seq SS: sequential sum of squares; Adj SS: adjusted sum of squares; *F*: *F*-statistics; *p*: *p*-value. T, P, and G correspond to the variables: temperature, pressure, and medium particle size, respectively. ∗: the interaction between factors.

According to the ANOVA, only the interaction between pressure (P) and the average particle size (G) and the interaction between the temperature (T) and particle size (G) were significant (*p* < 0.05), considering a significance level of 95% (α = 0.05). The analysis also indicated that the regression was statistically significant and could be applied to describe the variation in the *p*-anisic acid amounts in the extract. However, *p*-values larger than 0.05 were obtained for the quadratic and linear regressions, indicating that part of the data behaved linearly and in a partly quadratic fashion. Thus, the regression coefficients from the response surface were estimated for the full quadratic Box–Behnken model [58], resulting in Equation (2) (for variables not coded):

$$\begin{array}{l}\text{s velocity} \left(\% \frac{w\_{p-\text{mid}}}{w\_{\text{etmax}}}\right) = -9.742 + 8.782 \times 10^{-2} \,\text{P} + 5.495 \times 10^{-2} \,\text{T} + 1.660 \times 10^{-2} \,\text{G} - 1429 \times 10^{-4} \,\text{P}^2 \\\ -1.631 \times 10^{-3} \,\text{T}^2 - 3.523 \times 10^{-5} \,\text{G}^2 - 1.003 \times 10^{-5} \,\text{P} \,\text{T} - 1.858 \times 10^{-4} \,\text{P} \,\text{G} - 7.097 \times 10^{-4} \,\text{T} \,\text{G} \end{array} (2)$$

The model given by Equation (2) fits the experimental data with a coefficient of determination equal to 0.9125. The validity of the model was further confirmed by the nonsignificant value (*p* = 0.199 > 0.05), which indicated the quadratic model as a statistically significant model for the response. Through this equation, the response surfaces shown in Figure 2 were generated. The higher selectivity of *p*-anisic acid was obtained at lower temperatures, smaller mean particle sizes, and higher-pressures, as can be observe in the Figure 2. The combination of high pressure with smaller particles led to bed compaction and preferential flow paths, so a better result was observed with high pressure and larger particle size (smaller mesh).

**Figure 2.** Response surfaces and contour plots for effects of two independent variables on the yield of *p*-anisic acid in the extract obtained by supercritical extraction: (**a**) CO2 pressure (P) and CO2 temperature (T); (**b**) CO2 temperature (T) and medium particle size of milled flowers (G); (**c**) CO2 pressure (P) and medium particle size of milled flowers (G).

From the response optimizer of the Minitab® software, the optimal parameters to maximize the *p*-anisic acid selectivity were defined as 278.8 bar, 40 ◦C, and 42 mesh and, thus, the yield would be 2.76% (Figure 3). Under these operating conditions, adjusting the pressure to 279 bar, the extraction was carried out in triplicate and the yield curves versus time were determined. The average selectivity of *p*-anisic acid was 2.51%, indicating an error of 9.06% to the value estimated by the model, confirming again the good fit. This result is also very close to the value found for extraction in the conditions of 300 bar, 40 ◦C, and 60 mesh (*p*-anisic acid selectivity of 2.48%) due to the slight variation in the extraction conditions. However, when working with lower pressure, there is an increase in the energy efficiency involved in the process, which is another important factor regarding process optimization.

**Figure 3.** Process parameters optimized to maximum yield of *p*-anisic acid.

The mass transfer mathematical modeling was performed using the three selected models and the global yield versus time curve. The experimental data and the fitted models are shown in Figure 4. The modeling was performed considering the plant particle as a sphere and its average diameter as the average particle size (mesh).

The three models presented a good fit for the experimental data. However, as shown in Figure 4, some differences between the models were noticed. The Sovová model notes that the grinding process breaks the cell walls, making the solute easily accessible, while the Reverchon model notes that there is little solute that is easily accessible. Comparing the models with the experimental data, the Sovová model fits better in the initial extraction stage, which suggests that the grinding of *A. mearnsii* flowers increases the solute availability. The linear behavior at the beginning of extraction is observed by several authors [40,59–63] and is associated with the saturation of the particle surface, caused by grinding and its subsequent exposure to the extract. Despite being a simplified model, the model proposed by Crank presented a better fit with the data. Considering the good fit of the models, it is possible to say that internal diffusion controls the supercritical fluid extraction process of *A. mearnsii* flowers.

The MATLAB® optimization tool was used for the Crank model fitting. The order of magnitude for the Crank model internal diffusion coefficient was the same as was found by Goto et al. [64] and Hornovar et al. [65]. The same software was also used to estimate the four parameters of the Sovová model: *Z*, *W*, *xk*, and *yr*, which were obtained by the least-squares method and were minimized by the Nelder–Mead simplex method [66]. Once these parameters were estimated, the mass transfer coefficients for solid and liquid phases were calculated. The solid phase mass transfer coefficient presented an order of magnitude of 10<sup>−</sup>9(m/s) which agrees with the values obtained by Scopel et al. [67] and Nagy et al. [68]. The mass transfer coefficient for the solvent was found to equal 9.6 × <sup>10</sup>−<sup>10</sup> m/s, whose order of magnitude is the same as found by Gallo et al. [69] when studying the supercritical

extraction of *pyrethrum* flowers. The Reverchon model was implemented in the simulation software EMSO [70] and the system of equations was solved by an integrator of multiple steps, optimized by a flexible polyhedron. Thus, the values of 5.8 × <sup>10</sup>−<sup>4</sup> (s−1) for the internal mass transfer coefficient and 5.3 × <sup>10</sup>−<sup>3</sup> for the equilibrium constant were estimated by the least-squares method. The order of magnitude found for the parameters coincides with the values determined by Silva et al. [32], Garcez et al. [71], Almeida et al. [60], Scopel et al. [67], and Campos et al. [72]. All the parameters and the coefficients of determination for each model are presented in Table 5.

**Figure 4.** The yield curve for supercritical fluid extraction at 40 ◦C, 279 bar, and milled flower (42 mesh): mathematical models and experimental data.


**Table 5.** Adjusted and calculated parameters for mathematical models of mass transfer.
