Engineering & Science
Thiele Modulus (phi) & Catalyst Pore Diffusion Effectiveness Factor (eta)
Calculate the isothermal first-order catalyst Thiele modulus phi = L * SQRT(k / D_eff) and internal pore-diffusion effectiveness factor eta = tanh(phi) / phi (%) using exact exponential hyperbolic tangent.
Stop wasting precious platinum inside diffusion-starved catalyst pellets!
Pack expensive platinum or palladium nanoparticles deep inside porous alumina catalyst pellets: if intrinsic reaction rate k (s^-1) is much faster than reactant pore diffusivity D_eff (m^2/s) across characteristic half-thickness L (m, where L = Volume / Surface_Area = R / 3 for a sphere or half-thickness for a washcoat slab), reactant molecules get consumed near the outer rim—leaving the expensive interior core starved of reactant!
Ernest Thiele (1939) captured that reaction-vs-diffusion race in the dimensionless Thiele Modulus phi = L * SQRT(k / D_eff), which determines the Internal Effectiveness Factor eta = (Actual Pellet Rate) / (Surface-Concentration Rate) = tanh(phi) / phi! Look at a 2.0 mm catalyst layer (L = 0.002 m, k = 2.5 s^-1, D_eff = 1.0e-6 m^2/s): phi = 3.162, so eta = 31.51% (nearly 70% of internal catalyst sits underutilized!); crushing or thinning the washcoat 4-fold to L = 0.50 mm (0.0005 m) drops phi to 0.791 and restores catalyst utilization to 83.32%!
| Catalyst Characteristic Thickness L (k = 2.5 s^-1, Deff = 10^-6 m^2/s) | Thiele Modulus phi | Effectiveness Factor eta (%) | Mass-Transfer Regime |
|---|---|---|---|
| Thick Pellet / Washcoat (L = 2.0 mm) | 3.162 (> 3) | 31.51% (~1 / phi) | Strong Internal Pore Diffusion Limit! |
| Thin Monolith Washcoat (L = 0.5 mm) | 0.791 (< 1) | 83.32% | Near-kinetic utilization |
| Invalid Diffusivity D_eff <= 0 | Division by Zero Guard | Guarded Input Error | "Invalid Catalyst Inputs" |
Dimensionless First-Order Catalyst Thiele Modulus phi = L * SQRT(k / D_eff)
Compares surface reaction speed against internal Knudsen/bulk pore diffusion.
=IF(OR(B2 <= 0, B3 <= 0, B4 <= 0), "Invalid Catalyst Inputs", ROUND(B2 * SQRT(B3 / B4), 3))Internal Catalyst Effectiveness Factor eta (%) = 100 * tanh(phi) / phi
Evaluates exact exponential identity tanh(phi) = (EXP(2*phi) - 1) / (EXP(2*phi) + 1) divided by phi.
=IF(OR(B2 <= 0, B3 <= 0, B4 <= 0), "Invalid Catalyst Inputs", ROUND(100 * (((EXP(2 * B2 * SQRT(B3 / B4)) - 1) / (EXP(2 * B2 * SQRT(B3 / B4)) + 1)) / (B2 * SQRT(B3 / B4))), 2))Try the example
Use the original values or change them to test your understanding.