Model Description
SovovaMulti implements eight kinetic models for supercritical fluid extraction (SFE). All models relate cumulative extracted mass $m_e(t)$ to time $t$. The total extractable mass is $m_T = x_0 \cdot m_s$, where $x_0$ is the initial solute loading (kg/kg) and $m_s$ is the solid mass (kg).
Sovová (1994) — Broken and Intact Cells
The only mechanistic model in the set. It distinguishes two fractions of the solid: easily accessible solute (outside broken cells) and solute trapped inside intact cells. Extraction proceeds through three consecutive phases: constant extraction rate (CER), falling extraction rate (FER), and diffusion-controlled (DC).
Mass balance equations
Coupled PDEs along the bed height $h$:
Fluid phase:
\[\varepsilon \, v \, \frac{\partial Y}{\partial h} = J(X, Y)\]
Solid phase:
\[(1 - \varepsilon)\,\rho_s\,\frac{\partial X}{\partial t} = -\rho_f\,J(X, Y)\]
Mass transfer rate
\[J = \begin{cases} k_Y a \,(Y^* - Y) & X > x_k \quad \text{(CER)} \\[4pt] k_X a \, X \!\left(1 - \dfrac{Y}{Y^*}\right) & X \le x_k \quad \text{(FER)} \end{cases}\]
Fitted parameters (per curve, except xk/x0 which is shared across curves):
| Symbol | Description |
|---|---|
kya ($k_Y a$) | Fluid-phase volumetric mass-transfer coefficient (1/s) |
kxa ($k_X a$) | Solid-phase volumetric mass-transfer coefficient (1/s) |
xk/x0 ($x_k/x_0$) | Fraction of easily accessible solute |
The PDE is solved numerically by the method of lines (upwind finite differences + explicit Euler).
Reference: Sovová, H. (1994). Chem. Eng. Sci., 49(3), 409–414. doi:10.1016/0009-2509(94)87012-8
Esquível (1999)
Single-exponential empirical model derived from a simplified mass balance:
\[m_e(t) = m_T \left(1 - e^{-k_1 t}\right)\]
| Symbol | Description |
|---|---|
| $k_1$ | Rate constant (1/s); physically related to solubility and flow conditions |
Reference: Esquível, M.M.; Bernardo-Gil, M.G.; King, M.B. (1999). J. Supercrit. Fluids, 16(1), 43–58. doi:10.1016/S0896-8446(99)00014-5
Zekovic (2003)
Two-parameter model separating the accessible yield fraction from the extraction rate:
\[m_e(t) = m_T \, k_1 \left(1 - e^{-k_2 t}\right)\]
| Symbol | Description |
|---|---|
| $k_1$ | Accessible yield fraction (dimensionless, 0–1) |
| $k_2$ | Rate constant (1/s) |
Reference: Zeković, Z.P. et al. (2003). Acta Period. Technol., 34, 125–133. doi
PKM — Parallel Reaction Kinetics (Maksimovic, 2012)
Interprets extraction as parallel first-order "reactions" from two solid fractions:
\[m_e(t) = m_T \left[ k_1 \left(1 - e^{-k_2 t}\right) + (1 - k_1)\left(1 - e^{-k_3 t}\right) \right]\]
| Symbol | Description |
|---|---|
| $k_1$ | Easily accessible solute fraction (dimensionless, 0–1) |
| $k_2$ | Fluid-phase rate constant (1/s) |
| $k_3$ | Solid-phase rate constant (1/s), $k_3 < k_2$ |
Reference: Maksimović, S.; Ivanović, J.; Skala, D. (2012). Procedia Eng., 42, 1767–1777. doi:10.1016/j.proeng.2012.07.571
Spline — Piecewise-linear CER/FER/DC (Rodrigues, 2003)
Fits the extraction curve with three straight-line segments, one per extraction phase:
\[m_e(t) = \begin{cases} m_T\,k_1\,t & t \le k_2 \quad \text{(CER)} \\[4pt] m_T\,k_1\,k_2 + m_T\,k_3\,(t - k_2) & k_2 < t \le k_4 \quad \text{(FER)} \\[4pt] m_T\,k_1\,k_2 + m_T\,k_3\,(k_4 - k_2) & t > k_4 \quad \text{(DC, flat)} \end{cases}\]
| Symbol | Description |
|---|---|
| $k_1$ | CER extraction rate (1/s) |
| $k_2$ | End time of CER phase (s) |
| $k_3$ | FER extraction rate (1/s), $k_3 < k_1$ |
| $k_4$ | End time of FER phase (s), $k_4 > k_2$ |
Reference: Rodrigues, V.M. et al. (2003). J. Agric. Food Chem., 51(6), 1518–1523. doi:10.1021/jf0257493
Parameter estimation
All models minimize the sum of squared residuals (SSR):
\[\text{SSR} = \sum_{i=1}^{N_{\text{curves}}} \sum_{j=1}^{m_i} \left( m_{e,\text{cal},j}^{(i)} - m_{e,\text{exp},j}^{(i)} \right)^2\]
For empirical models, all parameters are shared across curves. For the Sovová model, kya and kxa are per-curve while xk/x0 is shared.
Optimization uses BlackBoxOptim.jl, a derivative-free global optimizer that handles non-convex, bound-constrained problems without manual multi-start.