4.6.2. Sovová (1994) Model

The model proposed by Sovová [63] considers that the extraction of the solute by supercritical CO2 can be divided into three periods. The first period of extraction considers that only the easily accessible solute can be extracted, which has direct contact with the solvent; the second period considers that the easily accessible solute is gradually depleted from the inlet to the outlet of the bed, and the third period includes solutes that are difficult to access, contained within the particles. Therefore, the extract mass initially present in the solid phase (*O*) is the sum of the easily accessible mass of solute (*P*) and the inaccessible mass of solute contained within the solid particles (*K*). The solid phase, free from the solute (*N*), is the constant during the extraction and relates to the initial concentrations of the solute:

$$\mathbf{x}(t=0) = \mathbf{x}\_0 = \frac{O}{N} = \mathbf{x}\_p + \mathbf{x}\_k = \frac{P}{N} + \frac{K}{N} \tag{4}$$

The mass balance in the fluid phase and solid phase for a bed element is described by two differential equations that were analytically solved by Sovová [63], who simplified a few hypotheses. The final expression is given by Equation (5) in terms of the extract mass relative to the extract-free solid mass:

$$e = \left\{ \begin{array}{c} qy\_r[1 - \exp(-Z)] \\ y\_r[q - q\_m \exp(z\_w - Z)] \end{array} \right.$$

$$e = \left\{ \begin{array}{c} y\_r[q - q\_m \exp(z\_w - Z)] \\ \times\_0 - \frac{y\_r}{W} \ln\left\{ 1 + \left[ \exp\left(W \frac{x\_0}{y\_r} - 1\right) \exp\left[\mathcal{W}(q\_m - q)\right] \frac{x\_k}{x\_0} \right] \right\} \end{array} \right. \tag{5}$$

where:

$$q = \frac{Q \text{ t}}{N} \tag{6}$$

$$q\_m = \frac{(\mathbf{x}\_0 - \mathbf{x}\_k)}{y\_r \, Z} \tag{7}$$

$$q\_n = q\_m + \frac{1}{W} \ln \frac{\mathbf{x}\_k + (\mathbf{x}\_0 - \mathbf{x}\_k) \exp(\mathcal{W} \mathbf{x}\_0 / y\_r)}{\mathbf{x}\_0} \tag{8}$$

$$\frac{z\_w}{Z} = \frac{y\_r}{w \,\, \mathbf{x}\_0} \ln \frac{\mathbf{x}\_0 \, \exp[\mathcal{W}(q - q\_m)] - \mathbf{x}\_k}{\mathbf{x}\_0 - \mathbf{x}\_k} \tag{9}$$

$$Z = \frac{k\_f a\_0 \rho}{\dot{q}(1 - \varepsilon)\rho\_s} \tag{10}$$

$$\mathcal{W} = \frac{k\_s a\_0}{\dot{q}(1-\varepsilon)}\tag{11}$$

In the above equations, *yr* is the solubility; *Z* and *W* are the adjustable parameters for fast and slow periods, respectively, and are directly proportional to the mass transfer coefficients of each phase; the term *zw* corresponds to the boundary coordinate between fast and slow extraction; and *k <sup>f</sup>* and *ks* are the mass transfer coefficients of fluid and solid phases, respectively. The unknown quantities *xk*, *yr*, *ks*, and *k <sup>f</sup>* were estimated by the least-squares method.
