Model Description

SFEModeling implements 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

A mechanistic model that 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):

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

Shrinking Core Model (Moreno-Pulido et al., 2026)

A mechanistic model describing diffusion-limited leaching from a spherical solid particle whose extractable core shrinks as solute is removed. The pseudo-steady-state (PSS) analytical solution relates the core radius $s$ to non-dimensional time via:

\[\frac{s^3 - 1}{3} - \frac{s^2 - 1}{2} - \frac{s - 1}{T_m} = t\]

where $T_m = R k / D$ is the Thiele modulus, and $t$ is non-dimensional time scaled by the growth time-scale $\tau_g$: $t = t_{\mathrm{dim}} / \tau_g$.

The reacted (extracted) volume fraction is $X(t) = 1 - s^3$.

Reference: Moreno-Pulido, C.; Olwande, R.; Myers, T.; Font, F. (2026). Approximate solutions to the shrinking core model and their interpretation. Appl. Math. Model., 154, 116715. doi:10.1016/j.apm.2025.116715

Esquível (1999)

Single-exponential empirical model derived from a simplified mass balance:

\[m_e(t) = m_T \left(1 - e^{-k_1 t}\right)\]

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:

\[m_e(t) = m_T \, k_1 \left(1 - e^{-k_2 t}\right)\]

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:

\[m_e(t) = m_T \left[ k_1 \left(1 - e^{-k_2 t}\right) + (1 - k_1)\left(1 - e^{-k_3 t}\right) \right]\]

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:

\[m_e(t) = \begin{cases} m_T\,k_1\,t & t \le k_2 \quad \text{(CER)} \\[4pt] m_T\,k_1\,k_2 + m_T\,k_3\,(t - k_2) & k_2 < t \le k_4 \quad \text{(FER)} \\[4pt] m_T\,k_1\,k_2 + m_T\,k_3\,(k_4 - k_2) & t > k_4 \quad \text{(DC, flat)} \end{cases}\]

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.