Reference ID: MET-51EF | Process Engineering Reference Sheets Calculation Guide
Introduction & Context
The calculation of effective diffusivity in porous solids is a fundamental requirement in Process Engineering, particularly for operations involving mass transfer within porous solid matrices. In systems such as solid‑liquid extraction, leaching, or catalyst bed design, solutes must migrate through a complex network of fluid‑filled pores rather than through an open, unobstructed medium.
Because the internal structure of a solid matrix restricts the path of diffusing molecules—due to both the reduction in available cross-sectional area and the increased path length caused by pore curvature—the observed rate of diffusion is significantly lower than in a bulk fluid. This calculation is essential for determining the time required for extraction processes, sizing equipment, and predicting the performance of packed-bed reactors.
Methodology & Formulas
The effective diffusivity is determined by adjusting the bulk diffusion coefficient for the geometric constraints of the porous medium. The physics relies on the ratio of the void fraction to the tortuosity of the matrix.
The primary governing equation is defined as:
\[ D_{eff} = \frac{\epsilon \cdot D}{\tau} \]
Where the variables are defined as follows:
\( D_{eff} \): Effective diffusivity in the porous matrix (m²/s).
\( \epsilon \): Porosity, representing the volume fraction of the solid occupied by pores (dimensionless).
\( D \): Bulk diffusion coefficient of the solute in the solvent (m²/s).
\( \tau \): Tortuosity factor, representing the ratio of the actual path length of a diffusing molecule to the straight-line distance (dimensionless).
To ensure the physical validity of the calculation, the following empirical constraints must be satisfied:
Parameter
Symbol
Valid Range
Engineering Significance
Porosity
\( \epsilon \)
0.1 ≤ \( \epsilon \) ≤ 0.9
Defines the available cross-sectional area for mass transfer.
Tortuosity
\( \tau \)
1.5 ≤ \( \tau \) ≤ 10.0
Accounts for the increased path length due to pore complexity.
Bulk Diffusivity
\( D \)
10-10 ≤ \( D \) ≤ 10-8
Standard range for liquid-phase molecular diffusion.
To accurately determine the diffusion coefficient within a solid matrix, process engineers should follow these standard analytical steps:
Perform a series of controlled sorption or desorption experiments to measure mass uptake over time.
Apply Fick's Second Law of Diffusion to the experimental data, ensuring the boundary conditions match your specific geometry.
Utilize non-linear regression analysis to fit the experimental curve to the theoretical model.
Validate the calculated coefficient by comparing results across different sample thicknesses to ensure the value is independent of geometry.
The diffusion coefficient is highly sensitive to the physical and chemical properties of the matrix. Key variables include:
Porosity and tortuosity of the solid structure.
Temperature, which typically follows an Arrhenius relationship.
Molecular size of the penetrant relative to the pore size distribution.
Chemical affinity or interactions between the penetrant and the matrix surface.
If your experimental data shows that the diffusion coefficient changes as concentration increases, you are likely dealing with non-Fickian diffusion. This often occurs due to:
Plasticization of the polymer matrix by the penetrant.
Concentration-dependent mobility of the penetrant molecules.
Structural swelling of the matrix that alters the free volume available for transport.
Worked Example: Effective Diffusivity for Oil Extraction
A process engineer is designing a solid-liquid extraction unit to recover oil from crushed sunflower seeds using hexane solvent at 25°C. The diffusion of oil through the solvent-filled pores of the seed matrix is assumed to be the rate-controlling step. The effective diffusivity must be calculated to model the extraction kinetics.
Knowns (Input Parameters):
Porosity of the seed matrix, \( \epsilon = 0.45 \) (dimensionless)
