4.6.3. Reverchon (1996) Model

The model proposed by Reverchon [25] was developed from a mass balance for the solid and fluid phases, according to Equations (12) and (13). In the study, the extract is considered a pseudo component, which is not readily available on the surface of the particles after the milling process. As a result, the mass transfer is controlled by internal resistance. The axial dispersion is considered negligible. The density and solvent flow rate are said to be constant along the bed:

$$
\mu V \frac{\partial c}{\partial t} + \varepsilon V \frac{\partial c}{\partial t} + (1 - \varepsilon) V \frac{\partial q}{\partial t} = 0 \tag{12}
$$

$$(1 - \varepsilon)V\frac{\partial q}{\partial t} = -A\_p k\_{TM} (q - q^\*) \tag{13}$$

where *u* is the interstitial velocity of the fluid, *ε* is the bed porosity, and *ρ<sup>s</sup>* is the plant density. The previous differential equations satisfy the initial conditions described by *c*(*h*, 0) = 0 and *q*(*h*, 0) = *q*<sup>0</sup> for all *h*, and the following boundary conditions *C*(0, *t*) = 0 for all *t*. A linear relationship describes the equilibrium behavior between the phases during the supercritical fluid extraction process [25]:

$$q^\* = \text{K} \, \text{C} \tag{14}$$

where *K* is the volumetric partition coefficient of the extract between fluid and solid phases at the equilibrium condition. Reverchon (1996) sets the internal diffusion time as:

$$t\_i = \frac{(1 - \varepsilon)V}{A\_p k\_{TM}}.\tag{15}$$

and Equation (13) can be rewritten as:

$$\frac{\partial q}{\partial t} = -\frac{1}{t\_i} \left( q - q^\* \right);\tag{16}$$

and the internal diffusion time (Equation (17)) is related to the internal diffusion coefficient (*Di*):

$$t\_i = \frac{\mu \text{ l}^2}{D\_i} \tag{17}$$

where *μ* is a constant related to the particle geometry (equal to 3/5 for spherical particles) and *l* is the characteristic dimension given by the ratio between the particle volume and the particle superficial area.

## **5. Conclusions**

The Factorial 2<sup>2</sup> design indicated that the crude extract yield was higher when ethanol was used as the cosolvent, and the maximum tested pressure (240 bar) was applied in the extraction. The crude extract obtained showed antimicrobial action against *S. aureus*. The purification by silica gel column chromatography generated a fraction rich in a compound identified as *p*-anisic acid and this fraction improved the antimicrobial performance against *S. aureus*. The use of the selectivity criterion as an objective function in the response surface method demonstrated that the condition that includes 278.8 bar, 40 ◦C, and 42 mesh is the condition that produces the highest amount of *p*-anisic acid per unit of extract mass. This result is important, as this optimal condition is obtained with the use of eco-friendly solvents. The three mathematical models used to simulate the extraction kinetics were adequate for the supercritical CO2 extraction, with aqueous ethanol as the cosolvent, from *A. meanrsii* flowers. Thus, the supercritical extraction is an adequate and clean method to obtain *p*-anisic acid from *Acacia mearnsii* flowers and this work demonstrates that the potential usage of this plant material is abundant but not exploitative. As a new source of bioactive compounds, the use of this flower extract also contributes to reducing the volume of solid waste generated by the cultivation of this plant in forests.

**Supplementary Materials:** The following supporting information is available online, Figure S1: chromatogram of extract 1 (P = 120 bar, cosolvent: water), Figure S2: chromatogram of extract 2 (P = 120 bar, cosolvent: ethanol), Figure S3: chromatogram of extract 3 (P = 240 bar, cosolvent: water), Figure S4: chromatogram of extract 4 (P = 240 bar, cosolvent: ethanol), Figure S5: chromatogram of extract 5 (P = 180 bar, cosolvent: ethanol:water), Figure S6: chromatogram of extract 6 (P = 180 bar, cosolvent: ethanol:water), Figure S7: chromatogram of extract 7 (P = 180 bar, cosolvent: ethanol:water), Figure S8: chromatograms of (a) subfraction 4 and (b) subfraction 5 obtained by silica gel column chromatography separation, Figure S9: TLC for the ethyl acetate fraction and subfractions 1–10 after application of sulfuric vanillin as color reagent (the marks drawn represent the spots visualized under U.V. light).

**Author Contributions:** Conceptualization, G.F.d.S., E.C. and R.M.F.V.; data curation and formal analysis, G.F.d.S., E.T.d.S.J., R.N.A., A.L.B.F., A.T.d.E.S. and A.M.L.; investigation, G.F.d.S., E.T.d.S.J., A.L.B.F. and A.M.L.; writing of the first draft, G.F.d.S.; writing—review and editing, G.F.d.S., R.N.A., E.C. and R.M.F.V.; supervision, A.M.L., E.C. and R.M.F.V. All authors have read and agreed to the published version of the manuscript.

**Funding:** This work was supported by the National Council for Scientific and Technological Development (CNPq-425933/2018-0) and the National Council for the Improvement of Higher Education (CAPES-Finance Code 001).

**Institutional Review Board Statement:** Not applicable.

**Informed Consent Statement:** Not applicable.

**Data Availability Statement:** Not applicable.

**Acknowledgments:** The authors are grateful to the CNPq and CAPES for financial support and to TANAC S/A for flower supply.

**Conflicts of Interest:** The authors declare no conflict of interest.

**Sample Availability:** Samples of the extracts are not available from the authors.
