Diffusion

Diffusion is the movement of molecules from high to low concentration. Learn Fick's law, diffusivities in gases, liquids and solids, steady diffusion through films, mass-transfer coefficients and dimensionless numbers (Sc, Sh), and how diffusion limits separation and reaction rates.

Drop ink into still water and it slowly spreads — that's diffusion. In process equipment, molecules must diffuse across thin fluid films between phases, so diffusion rates control how big absorbers, extractors and catalyst pellets must be.

Diffusion in gases is about 10 000 times faster than in liquids, and solids are slower still.

An analogy

Diffusion is like people spreading out from a crowded room into an empty hall — nobody pushes them, they just randomly wander toward the emptier space.

Fick's law

Flux is proportional to the concentration gradient. Typical diffusivities: gases 10⁻⁵ m²/s, liquids 10⁻⁹ m²/s, solids 10⁻¹²–10⁻¹⁴ m²/s. Diffusivity rises with temperature and (for gases) falls with pressure.

Mass-transfer coefficients

In flowing fluids, most resistance is in thin boundary layers near interfaces. Engineers lump this into k_c (film theory k = D/δ; penetration theory k ∝ √(D/t)). Correlations relate Sh to Re and Sc, just as Nu relates to Re and Pr in heat transfer.

Two-film theory

Transfer between gas and liquid passes through a gas film and a liquid film in series. Overall coefficients combine them (1/K_G = 1/k_G + m/k_L). Soluble gases are gas-film controlled; sparingly soluble ones (O₂ in water) are liquid-film controlled.

Diffusion with reaction and in porous media

In catalyst pellets, reactants diffuse through pores while reacting; the Thiele modulus and effectiveness factor tell whether diffusion limits the rate. Knudsen diffusion dominates in very small pores; Maxwell–Stefan equations handle multicomponent diffusion.

Formulas

• Fick's first law: J_A = −D_AB dC_A/dz
• Film model: N_A = k_c (C_A,s − C_A,bulk), k_c = D/δ
• Diffusion time scale: t ≈ L² / D
• Gas concentration: C = p / (R T)
• Correlation form: Sh = k_c L/D = a Re^b Sc^(1/3)

Worked example: Flux across a gas film

Given: Partial-pressure difference 10 kPa at 300 K; Film 5 mm, D = 2 × 10⁻⁵ m²/s

Find: Diffusive flux.

• ΔC = 10 000/(8.314 × 300) = 4.01 mol/m³
• J = 2 × 10⁻⁵ × 4.01/0.005 = 0.016 mol/m²·s

Answer: ≈ 0.016 mol/m²·s

Worked example: Diffusion time

Given: Distance 1 mm

Find: Time in a gas and a liquid.

• Gas: (10⁻³)²/2 × 10⁻⁵ = 0.05 s
• Liquid: (10⁻³)²/10⁻⁹ = 1000 s ≈ 17 min

Answer: 0.05 s vs 17 min — stirring and mixing matter in liquids

Common mistakes

• Wrong: Using gas diffusivities for liquids. — Right: Liquid D is ~10⁴× smaller — check the phase.
• Wrong: Ignoring the controlling film. — Right: Identify whether gas or liquid film dominates before improving equipment.

Applications

• Gas absorption & stripping: Film coefficients size packed columns.
• Catalysis: Pore diffusion sets catalyst pellet size and effectiveness.
• Membranes & drug delivery: Diffusion through polymers controls permeation and release.

Key takeaways

• J = −D dC/dz; D_gas ≫ D_liquid ≫ D_solid.
• t ≈ L²/D.
• k = D/δ; correlations Sh = f(Re, Sc).
• Two-film resistances in series.

Going further

• Chilton–Colburn analogy: Heat, mass and momentum transfer correlate: j_D = j_H = f/2.
• Thiele modulus: Ratio of reaction to diffusion rate in catalysts; large values mean diffusion limits.
• Maxwell–Stefan: Rigorous multicomponent diffusion with interaction between species.