Thiele Modulus & Catalyst Effectiveness Factor Simulator

Inside a porous catalyst pellet, reaction and diffusion compete with each other. Change the pellet shape, size, rate constant and effective diffusivity to see the Thiele modulus φ, effectiveness factor η, apparent reaction rate and catalyst utilisation update in real time, and judge whether the pellet has become diffusion-limited.

Parameters

Catalyst pellet shape

The characteristic length L_c formula depends on shape

Characteristic size (radius for sphere/cylinder, half-thickness for slab)

Rate constant k

1/s

First-order rate constant — how active the catalyst is

Effective diffusivity De

×10⁻⁷ m²/s

How fast the reactant diffuses through the pores

Surface concentration Cₛ

mol/m³

Reactant concentration at the pellet surface

Results

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Thiele modulus φ

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Effectiveness factor η

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Controlling regime

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Apparent rate (mol/m³·s)

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Characteristic length L_c (mm)

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Catalyst utilisation (%)

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Catalyst pellet cross-section — internal concentration profile

The colour of the pellet cross-section is the reactant concentration (bright = near Cₛ, dark = near zero). The larger φ is, the more starved the core becomes, with reaction confined to a thin surface shell. The particles show reactant diffusing inward.

Effectiveness factor η vs Thiele modulus φ

Internal concentration profile C(ξ)/Cₛ

Theory & Key Formulas

$$\phi=L_c\sqrt{\frac{k}{D_e}},\qquad \eta=\frac{\tanh\phi}{\phi}$$

Thiele modulus φ (dimensionless) and effectiveness factor η. L_c: characteristic length, k: rate constant, De: effective diffusivity. With the generalised modulus, η is a good approximation for spheres, cylinders and slabs.

$$r_{obs}=\eta\,k\,C_s,\qquad L_c=\frac{V_p}{S_p}\;\big(\text{sphere}:R/3,\ \text{cylinder}:R/2,\ \text{slab}:L\big)$$

Apparent reaction rate r_obs and the characteristic length L_c defined as pellet volume over surface area. Cₛ: surface concentration, Vₚ: pellet volume, Sₚ: external surface area.

$$\frac{C(\xi)}{C_s}=\frac{\cosh(\phi\,\xi)}{\cosh\phi}$$

Internal concentration profile for a slab (ξ = 0: centre, ξ = 1: surface). η → 1 for φ ≪ 1 (reaction-limited) and η ≈ 1/φ for φ ≫ 1 (diffusion-limited).

What is the Thiele Modulus and the Effectiveness Factor?

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A "catalyst" is the powder or pellet that speeds up a chemical reaction, right? Why do I have to worry about "diffusion" inside it?

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Good question. Industrial catalysts are made "porous" — sponge-like inside — to pack in a huge surface area. The reactant gas or liquid first reaches the outer surface of the pellet, then has to "diffuse" inward through the narrow pores. Only once it reaches the inside can the catalyst deep inside do any work. So reaction and inward diffusion are competing.

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Competing...? Isn't a faster reaction always better?

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That is exactly the trap. If the reaction is too fast, the reactant is consumed the instant it reaches the surface region. Then nothing reaches the core, and the inside of the pellet has "nothing to do". The ratio of "how fast it reacts" to "how fast it diffuses" packed into a single number is the Thiele modulus φ. It is φ = L_c·√(k/De). Raise the rate constant k on the left and you will see φ shoot up.

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So a large φ means the catalyst deep inside is loafing. Is that what the effectiveness factor η describes?

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Exactly. η is the fraction of the rate you actually get, compared with the rate you would get if the whole pellet could react at the surface concentration. It is η = tanh(φ)/φ. When φ is small, η is nearly 1 — the catalyst works at 100%. As φ grows, η keeps dropping. η = 0.3 means 70% of the expensive catalyst is idle. On the η-φ chart below you can see that beyond φ around 10, η falls off in a straight line.

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That's a problem... What do I change to raise η?

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Lower φ. The quickest move is to "make the pellet smaller" — shrink the characteristic length L_c and φ shrinks with it. Next is "enlarge the pores so diffusion is faster", i.e. raise De. The last is "slow the reaction" — lower the temperature or deliberately reduce activity. A common trick in practice: the more active the catalyst, the more it is made into small-diameter pellets, or into a "shell catalyst" with the active component painted only on the outer layer, so the reactant never has to diffuse far.

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I see! So I should think about catalyst design in terms of lowering φ and raising η.

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Right. But smaller is not unconditionally better — shrinking the pellet increases the pressure drop across the packed bed. You size the pellet by watching the trade-off between "catalyst utilisation" and "pressure drop". Use this tool to watch φ, η and the catalyst utilisation while you try different combinations.

Frequently Asked Questions

The Thiele modulus φ is a dimensionless number that expresses the ratio of the intrinsic reaction rate to the rate at which the reactant can diffuse into the catalyst pellet. It is calculated as φ = L_c·√(k/De), where L_c is the characteristic length, k is the rate constant and De is the effective diffusivity. When φ is small, diffusion is fast enough for the reactant to reach the whole pellet; when φ is large the reaction is so fast that the reactant is consumed near the surface and never reaches the core.

The effectiveness factor η is the actual reaction rate of the whole pellet divided by the rate that would occur if the entire interior were at the surface concentration. With the generalised Thiele modulus it can be approximated by η = tanh(φ)/φ for spheres, cylinders and slabs alike. η = 1 means the catalyst is working at 100% efficiency; η = 0.3 means only 30% of the pellet is effectively used. η → 1 for φ ≪ 1 and η ≈ 1/φ for φ ≫ 1.

As a rule of thumb: φ < 0.4 is reaction-limited (η ≈ 1, the internal concentration is nearly uniform), 0.4 ≤ φ ≤ 4 is a mixed regime where diffusion cannot be ignored, and φ > 4 is strongly diffusion-limited (η ≈ 1/φ, the reaction runs only in a thin shell near the surface). In the diffusion-limited regime, most of the expensive catalyst sits idle. This tool computes φ and η and automatically reports which regime you are in.

Raising η means lowering φ, so the options are: (1) make the pellet smaller to shorten the characteristic length L_c, (2) enlarge the pores to increase the effective diffusivity De, or (3) slow the reaction (lower activity or run at a lower temperature). In practice, highly active catalysts are the most prone to diffusion limitation, so engineers deliberately use small-diameter pellets or thin shell catalysts (active component coated only on the outer shell) to shorten the internal diffusion path.

Real-World Applications

Oil refining and petrochemical processes: Catalytic reforming, hydrodesulfurization and fluid catalytic cracking (FCC) — every major refinery process uses porous catalyst pellets. These reactions are fast, so φ tends to be large, and the catalyst is supplied as small-diameter shaped bodies (a few mm or less) or as shell catalysts to suppress the loss of effectiveness from internal diffusion. φ and η are used routinely as design metrics by catalyst manufacturers.

Ammonia and methanol synthesis: Even in large-scale processes that pack enormous amounts of catalyst into fixed-bed reactors, intra-pellet diffusion cannot be ignored. Making the pellet smaller raises η, but it also increases the pressure drop across the packed bed and therefore the compressor power. The trade-off between η (catalyst utilisation) and pressure drop is the heart of sizing the pellet.

Automotive exhaust catalysts: In three-way catalysts and diesel oxidation catalysts, the precious metals (Pt, Pd, Rh) are dispersed in a thin catalyst layer (washcoat) on the honeycomb walls. The layer is kept thin precisely to make L_c small, lower φ and use the limited precious metal effectively. Cold-start low-temperature activity is also closely tied to internal diffusion.

Reactor simulation and scale-up: In fixed-bed reactor design, the apparent reaction rate r_obs = η·k·Cₛ is built into the reactor model. Applying laboratory data taken on small particles (η ≈ 1) directly to industrial-size large particles (η < 1) leads to a failed scale-up. Correcting for internal diffusion with φ and η is a prerequisite for a reliable scale-up.

Common Misconceptions and Pitfalls

The most common one is assuming that a reaction rate measured in the laboratory holds unchanged in an industrial reactor. Laboratories often use finely crushed catalyst, in which case φ is essentially zero, η ≈ 1, and the "true reaction rate" (the intrinsic activity) is measured. But on a large industrial pellet φ becomes large and η < 1, so the apparent rate drops by a factor of η. Scaling up laboratory data without this correction can give a reactor that delivers a fraction of the expected performance. If the rate changes when you change the particle size, that is a clear sign of internal diffusion limitation.

Next, confusing the effective diffusivity De with the molecular diffusivity. De is the effective diffusion coefficient inside the catalyst pores, and it is far smaller than the free-space molecular diffusivity. It is discounted by the porosity (the void fraction), discounted further by the tortuosity (how winding the pores are), and when pores are small the Knudsen-diffusion effect adds in too. It is not unusual for De to be around one-tenth of the free-space value. Using the molecular diffusivity for De underestimates φ and misses the diffusion limitation.

Finally, do not assume that η = tanh(φ)/φ is exact in every case. This formula is exact only for an isothermal, first-order reaction in a slab. By adopting the generalised Thiele modulus (using L_c = Vₚ/Sₚ as the characteristic length), this tool gives a good approximation for spheres and cylinders too, but it is still an approximation. For an exothermic reaction with a temperature gradient inside the pellet (non-isothermal), η can even exceed 1. And if the reaction order is not first order, or the external mass-transfer resistance is large, a different correction is required. φ and η are above all a powerful first guide to "whether a reaction is diffusion-limited"; the final design should be verified with a detailed model that includes non-isothermal effects and reaction order.

Thiele Modulus & Catalyst Effectiveness Factor Simulator

What is the Thiele Modulus and the Effectiveness Factor?

Frequently Asked Questions

Real-World Applications

Common Misconceptions and Pitfalls

How to Use

Worked Example

Practical Notes