Introduction & Context

In process engineering, the solid‑liquid extraction of oil from oilseeds is a mass‑transfer‑limited operation. The efficiency of this process is heavily dependent on the physical preparation of the raw material, known as comminution (grinding or flaking). By reducing the particle size, the diffusion path length for the solvent is shortened, significantly accelerating the extraction kinetics, which must be considered when defining the carrousel extractor operation parameters.

This calculation is critical for optimizing industrial extractors, where residence time is limited. It allows engineers to predict the fractional yield based on the effective diffusivity of the oil within the porous matrix and the characteristic half-thickness of the solid particles. Proper sizing ensures high recovery rates while avoiding excessive pressure drops or fines generation that can impede solvent percolation.

Methodology & Formulas

The system is modeled as unsteady-state diffusion in a slab geometry. The extraction progress is governed by the Fourier number (Fo), which relates the rate of diffusion to the square of the characteristic length and is essential for extraction time determination.

The effective diffusivity (D_eff) is calculated by adjusting the molecular diffusivity (D_AB) for the internal structure of the solid:

\[ D_{eff} = D_{AB} \cdot \frac{\varepsilon}{\tau} \]

The Fourier number (Fo) is defined as:

\[ Fo = \frac{D_{eff} \cdot t}{L^2} \]

The fractional yield (Y) is determined based on the regime of the extraction process:

Regime	Condition	Formula
Early Stage (Coarse)	\( Fo < 0.1 \)	\( Y = 2 \sqrt{ \frac{Fo}{\pi} } \)
Later Stage (Fine)	\( Fo \geq 0.1 \)	\( Y = 1 - \frac{8}{\pi^2} \exp\left( -\frac{\pi^2 \cdot Fo}{4} \right) \)

The improvement factor, which quantifies the particle size effect on extraction rate, is calculated as the ratio of the yield of the fine particles (Y₂) to the yield of the coarse particles (Y₁).

\[ \text{Improvement Factor} = \frac{Y_{2}}{Y_{1}} \]

Empirical Validity Constraints

Parameter	Symbol	Valid Range
Temperature	T	40.0 - 70.0 °C
Particle Porosity	ε	0.2 - 0.4
Tortuosity	τ	1.5 - 5.0
Effective Diffusivity	D_eff	1 × 10^-12 - 1 × 10^-10 m²/s

Reducing particle size increases the surface area to volume ratio, which significantly enhances mass transfer rates. For process engineers, this involves:

Shortening the diffusion path for the solvent into the solid matrix.
Increasing the contact area between the solvent and the target compounds.
Managing potential pressure drops in packed bed reactors caused by excessive fines.

High moisture content can act as a physical barrier to non-polar solvents, effectively blocking pores and reducing yield. To optimize the process, consider the following:

Pre-drying the raw material to remove interstitial water.
Evaluating if the moisture acts as a co-solvent for polar target analytes.
Monitoring the water activity to prevent microbial degradation during storage.

Thermal pre-treatment, such as blanching or steam treatment, denatures enzymes and softens the plant cell wall matrix. This process provides several technical advantages:

Inactivation of endogenous enzymes that may degrade the target compounds.
Increased permeability of the cell membrane to facilitate solvent ingress.
Reduction of the viscosity of internal fluids, allowing for faster extraction kinetics.

Worked Example: Effect of Flake Thickness on Extraction Yield

Scenario: A batch extraction of oil from soybeans using hexane at 50 °C for 1800 s. The effective diffusivity of oil in the porous matrix is \(D_{\text{eff}} = 1 \times 10^{-10}\,\text{m}^2/\text{s}\). The initial flake half-thickness is \(L_1 = 0.001\,\text{m}\) (1 mm), which is reduced by grinding to \(L_2 = 0.00025\,\text{m}\) (0.25 mm). The goal is to compute the improvement in fractional extraction yield \(Y(t) = M_t / M_\infty\) after the pre-treatment.

Knowns:

Extraction time, \(t = 1800\) s
Effective diffusivity, \(D_{\text{eff}} = 1 \times 10^{-10}\) m²/s
Initial half-thickness, \(L_1 = 0.001\) m
Reduced half-thickness, \(L_2 = 0.00025\) m
Fourier number for coarse flakes, \(F_{o1} = 0.18\)
Fourier number for fine flakes, \(F_{o2} = 2.88\)

Step-by-step calculation:

Select the appropriate yield formula.
The governing equation for a slab geometry uses the Fourier number \(F_o = D_{\text{eff}} t / L^2\). Since \(F_{o1} = 0.18 > 0.1\) and \(F_{o2} = 2.88 > 0.1\), the long-time expression applies for both cases: \[ Y(t) = 1 - \frac{8}{\pi^2} \exp\left( -\frac{\pi^2 D_{\text{eff}} t}{4 L^2} \right) \]
Calculate the fractional yields.
For the coarse flakes (\(F_{o1} = 0.18\)): \[ Y_1 = 1 - \frac{8}{\pi^2} \exp\left( -\frac{\pi^2 \cdot 0.18}{4} \right) = 0.48 \] For the fine flakes (\(F_{o2} = 2.88\)): \[ Y_2 = 1 - \frac{8}{\pi^2} \exp\left( -\frac{\pi^2 \cdot 2.88}{4} \right) = 0.999
Determine the improvement factor.
The improvement in yield due to grinding is the ratio: \[ I = \frac{Y_2}{Y_1} = \frac{0.999}{0.48} = 2.081 \]

Final Answer:
Reducing the flake half-thickness from 1 mm to 0.25 mm increases the 30-minute extraction yield from 48 % to 99.9 %, an improvement factor of 2.081 (a 108 % increase).

"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