Introduction & Context

Extraction time determination is a critical calculation in Process Engineering, specifically within the oilseed processing and pharmaceutical industries. It defines the duration required for a solvent to leach a solute (such as oil) from a porous solid matrix, and it is closely linked to the pre‑treatment effects on extraction yield, which can significantly influence the required contact time. This calculation is essential for sizing industrial extractors, optimizing solvent‑to‑solid ratios, and ensuring that target recovery yields are met without excessive energy consumption or solvent usage.

Methodology & Formulas

The extraction process is modeled as unsteady-state diffusion within a slab geometry, assuming the solid particles are thin flakes. The rate-limiting step is the internal diffusion of the solute through the porous structure of the solid. The fractional oil recovery, R, is determined by the long-time approximation of Fick’s second law, which is valid when the dimensionless Fourier number (Fo) exceeds 0.1.

The relationship between fractional recovery and time is expressed as:

\[ R = 1 - \frac{8}{\pi^2} \exp\left(-\frac{\pi^2 \cdot D_{\text{eff}} \cdot t}{4 \cdot L^2}\right) \]

To determine the required extraction time t for a specific recovery target, the equation is rearranged as follows:

\[ t = -\frac{4 \cdot L^2}{\pi^2 \cdot D_{\text{eff}}} \cdot \ln\left[\frac{\pi^2}{8} \cdot (1 - R)\right] \]

Where the variables are defined as:

t: Extraction time (s)
L: Half-thickness of the flake (m)
D_eff: Effective diffusivity of the solute in the porous solid (m²/s)
R: Fractional oil recovery (dimensionless)

The validity of this model is governed by specific physical regimes and constraints, summarized in the table below:

Parameter	Condition / Threshold	Significance
Fourier Number (Fo)	Fo > 0.1	Ensures the long-time approximation is valid.
Effective Diffusivity (D_eff)	10^-12 ≤ D_eff ≤ 10^-9 m²/s	Empirical range for oil in organic solvents at 50–70°C.
Recovery Target (R)	R > 0.189	Required for the first-term series expansion to be accurate.

To determine the optimal extraction time, you must perform a kinetic study to identify the point of equilibrium. Follow these steps:

Conduct a series of batch extractions at varying time intervals.
Analyze the solute concentration in the extract phase for each interval.
Plot the concentration against time to identify the plateau where mass transfer rates stabilize.
Select the time point that achieves 95 percent of the maximum theoretical yield to balance efficiency and throughput.

Scaling up from laboratory to industrial equipment introduces variables that significantly alter mass transfer kinetics. Key factors include:

Particle size distribution and surface area of the solid matrix.
Agitation intensity and the resulting Reynolds number within the vessel.
Solvent viscosity and its impact on the diffusion coefficient.
Temperature fluctuations affecting the solubility limit of the target compound.

As the solvent approaches saturation, the concentration gradient between the solid and liquid phases decreases, which slows the mass transfer rate according to Fick's Law. To maintain an efficient extraction time, process engineers should:

Monitor the driving force, defined as the difference between the saturation concentration and the current bulk concentration.
Implement a multi-stage counter-current extraction process to maintain a high concentration gradient throughout the operation.
Adjust the solvent-to-feed ratio to prevent premature saturation of the solvent phase.

Worked Example: Diffusion-Controlled Extraction Time for Soybean Oil from Hexane-Flakes System

Scenario: A batch extraction process is used to recover soybean oil from thin flakes immersed in hexane at 60°C. The effective diffusivity of oil within the porous flakes is known, and the fraction recovery target is 95%. The bulk solvent is well-mixed, so external mass transfer is negligible. The rate-limiting step is internal diffusion, described by the transient, one-dimensional Fickian model for a slab of half-thickness L.

Known Parameters:

Flake half-thickness, L = 0.0002 m
Effective diffusivity, D_eff = 3×10^-11 m²/s
Target fractional oil recovery, R = 0.95
Process temperature, T = 60.0 °C (isothermal)

Step-by-Step Calculation

Compute the fraction of oil remaining: 1 - R = 0.050.
Determine the logarithmic term from the long-time approximation:
\[ \ln\left(\frac{\pi^2}{8}(1-R)\right) = \ln\left(\frac{9.869604401089358}{8} \times 0.050\right) = -2.785714043535026 \approx -2.786. \]
Apply the rearranged extraction time equation for a slab:
\[ t = -\frac{4 L^{2}}{\pi^{2} D_{\text{eff}}} \ln\left(\frac{\pi^{2}}{8}(1-R)\right). \]
Substitute the known values:
\[ t = -\frac{4 \times (4\times10^{-8}~\text{m}^{2})}{(9.869604401089358) \times (3\times10^{-11}~\text{m}^{2}/\text{s})} \times (-2.785714043535026). \]
The computed extraction time is t = 1505.8 s (rounded to one decimal place).
Convert to minutes:
\[ t_\text{min} = \frac{1505.8~\text{s}}{60~\text{s/min}} = 25.10~\text{min}. \]
Verify validity of the long-time approximation: Calculate the Fourier number (dimensionless time):
\[ \text{Fo} = \frac{D_{\text{eff}} t}{L^{2}} = \frac{(3\times10^{-11}~\text{m}^{2}/\text{s}) \times 1505.8~\text{s}}{4\times10^{-8}~\text{m}^{2}} = 1.129. \]
Since Fo = 1.129 > 0.1, the single-term approximation is valid. Additionally, the effective diffusivity lies within the empirical range 10^-12 to 10^-9 m²/s, and the target recovery R=0.95 exceeds the threshold of 0.189 for applicability of the long-time equation.

Final Answer: The required extraction time to achieve 95% oil recovery is approximately 25.10 minutes (equivalent to 1505.8 seconds).

"Un projet n'est jamais trop grand s'il est bien conçu." — André Citroën

"La difficulté attire l'homme de caractère, car c'est en l'étreignant qu'il se réalise." — Charles de Gaulle

Extraction Time Determination

Introduction & Context

Methodology & Formulas

Worked Example: Diffusion-Controlled Extraction Time for Soybean Oil from Hexane-Flakes System

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