Reference ID: MET-357E | Process Engineering Reference Sheets Calculation Guide
Introduction & Context
Effective diffusivity quantifies how fast a species migrates through the void space of a porous solid when driven by a concentration gradient. In process engineering it is the key parameter linking intrinsic reaction rates to observed rates in catalyst pellets, adsorbent beads, membrane separators, and biomass supports. Underestimating this value leads to oversized reactors; overestimating it masks intraparticle diffusion limitations and yields optimistic conversion predictions.
Methodology & Formulas
Effective diffusivity
The void fraction is reduced by tortuosity to give the transport area actually available in the direction of the gradient:
\[ D_{\text{eff}} = \frac{\varepsilon \, D_{\text{bulk}}}{\tau} \]
Symbol
Meaning
Units
\( \varepsilon \)
porosity (void volume / total volume)
—
\( \tau \)
tortuosity (actual path / straight-line distance)
—
\( D_{\text{bulk}} \)
binary diffusion coefficient in the fluid phase
m2 s−1
Mass-transfer flux
For steady-state diffusion across a slab of thickness \( L \) with constant driving force \( \Delta C \):
\[ N = \frac{D_{\text{eff}} \, \Delta C}{L} \]
Symbol
Meaning
Units
\( N \)
molar flux
mol m−2 s−1
\( \Delta C \)
concentration difference across the pellet
mol m−3
\( L \)
characteristic diffusion length (pellet half-thickness for symmetric slab)
m
Effective diffusivity (De) is the transport rate of a species inside a porous medium relative to its concentration gradient, corrected for porosity (ε) and tortuosity (τ) of the solid. It matters because:
It sets the Thiele modulus and effectiveness factor, which decide how much of the catalyst pellet is actually working.
Low De shifts the regime from kinetic- to diffusion-limited, lowering observed reaction rates and selectivity.
Accurate De values let you size pellets, choose pore sizes, and avoid over-designing downstream units.
Use the Bosanquet or random-pore model: De = (ε/τ) · [1/(1/DAB + 1/DKn)]. Obtain ε from mercury porosimetry or He pycnometry, estimate τ from ε via τ ≈ 1/ε or independent diffusion measurements, and calculate DKn for each species using the mean pore radius. For bidisperse solids, sum the contributions from macro- and micro-pores weighted by their respective void fractions.
Steady-state Wicke–Kallenbach cells give direct De at reaction conditions, but require sealing and can be slow. Transient pulse or frequency-response techniques are faster and work for microporous solids, yet need chromatographic calibration. For quick plant support, match the method to the dominant diffusion regime:
Macropore-controlled: use counter-diffusion with binary gas mixtures.
Micropore-controlled: use zero-length column (ZLC) or gravimetric uptake.
"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
