Thiele Modulus & Catalyst Effectiveness Factor Calculator

Understanding the Thiele modulus and effectiveness factor calculator — concept and principle

Inside a porous catalyst pellet, reactants must diffuse through the pores before they react. If the reaction is fast compared with diffusion, the centre of the pellet is starved and much of the catalyst does little work. The Thiele modulus measures this competition, and the effectiveness factor η gives the fraction of the ideal rate that the pellet actually delivers.

Reactant is used up before it can diffuse far into the pellet, so the core sits idle.

How it is calculated

Thiele modulus φ = R × √(k / D_e), with R converted from mm to m.
Sphere: η = (3/φ) × (1/tanh φ − 1/φ)
Slab: η = tanh φ ÷ φ, with R taken as the half-thickness.

k is the first-order rate constant per unit pellet volume (1/s) and D_e the effective diffusivity inside the pores (m²/s).

The principle

Thiele solved the steady diffusion–reaction balance inside a pellet. The concentration falls from the surface towards the centre, and φ² is the ratio of a characteristic reaction rate to a characteristic diffusion rate. When φ is small, diffusion keeps the whole pellet near surface concentration and η approaches 1. When φ is large, reactant is consumed in a thin shell near the surface and η falls towards 3/φ for a sphere (1/φ for a slab). The effectiveness factor is the observed rate divided by the rate the pellet would have if its whole interior were at surface concentration.

Worked example

Inputs: R = 2 mm, k = 5 s⁻¹, D_e = 1 × 10⁻⁸ m²/s, sphere.
√(k/D_e) = √(5 × 10⁸) = 22,360.7 m⁻¹.
φ = 0.002 × 22,360.7 = 44.72.
tanh 44.72 = 1, so η = (3/44.72) × (1 − 1/44.72) = 0.06708 × 0.97764 = 0.06558.

Each point is the calculator re-run with only “Pellet radius, or half-thickness for a slab (mm)” changed; the gold point is the default example. Change the inputs above and this chart follows your numbers.

Pellet radius, or half-thickness for a slab (mm)	Thiele modulus φ
0.5	11.18033989
1	22.36067977
1.5	33.54101966
2	44.72135955
2.5	55.90169944
3	67.08203932
4	89.4427191

How to read the result

Only about 6.6% of the pellet's potential is used: the reaction happens in a shell roughly R/φ ≈ 0.045 mm deep. As a rough guide, φ below about 0.5 means negligible diffusion limits (η > 0.9), while φ above about 5 means strong limits. Because η ≈ 3/φ in that region, halving the pellet radius roughly doubles η. Smaller pellets, egg-shell catalysts with the active metal near the surface, or larger pores are the usual remedies, traded against pressure drop and strength.

Common mistakes

Entering the diameter instead of the radius (or the full slab thickness instead of half).
Using the molecular diffusivity in the gas instead of the much smaller effective diffusivity in the pores.
Comparing φ values from texts that define the sphere modulus with R/3 instead of R; their thresholds differ.

Keep in mind: These results assume an isothermal pellet, first-order irreversible kinetics and uniform properties. Strong heat effects, other reaction orders and external film resistance change η; reactor design needs measured kinetics and expert review.

Frequently asked questions

Can I check for diffusion limits without knowing k?

Yes, the Weisz–Prater criterion uses the observed rate instead of k to judge whether internal diffusion matters.

Why does the observed activation energy look halved?

In the strongly limited region the observed rate scales with √(k D_e), so the apparent activation energy is about half the true value.

Sources: E. W. Thiele, Industrial and Engineering Chemistry 31 (1939) 916 · H. S. Fogler, Elements of Chemical Reaction Engineering, Pearson · O. Levenspiel, Chemical Reaction Engineering, 3rd edition, Wiley

Checked 8 Nov 2026. Standard textbook formulas in SI units, computed in your browser. For learning and estimates — real designs must follow your local codes and be checked by a qualified engineer.

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