Catalyst Design: Thiele Modulus & Effectiveness Factor

simulator intermediate ~10 min
Loading simulation...
Effectiveness: 72% — Thiele modulus phi=1.8

A catalyst particle of 2mm radius with 300 m2/g surface area and 10nm pores at 550K gives Thiele modulus 1.8 and effectiveness factor 0.72. About 28% of the catalyst interior is underutilized due to diffusion limitations.

Formula

phi = R * sqrt(k_s / D_eff)
eta = (3/phi) * (phi*coth(phi) - 1) / phi^2
D_eff = D_AB * epsilon_p / tau

Catalysis: Accelerating Chemistry

Catalysts are substances that increase reaction rates without being consumed, and they underpin roughly 90% of all chemical manufacturing processes. Heterogeneous catalysts - solid materials that catalyze reactions of gas or liquid phase reactants - are especially important because they are easily separated from products. From the platinum in your car's catalytic converter to the iron in Haber-Bosch ammonia synthesis, catalyst design determines both economic viability and environmental impact.

The Thiele Modulus: Reaction vs. Diffusion

The central question in heterogeneous catalysis is: can reactants reach the catalyst's interior fast enough to keep up with the surface reaction? The Thiele modulus phi answers this by comparing reaction rate to diffusion rate. When phi is small, reaction is slow relative to diffusion and the entire particle is utilized. When phi is large, diffusion cannot keep up, and only a thin shell near the surface participates. This simulation visualizes the concentration profile inside the particle, showing exactly where the reaction occurs.

Surface Area and Pore Architecture

A gram of modern catalyst can have a surface area exceeding a football field, thanks to intricate pore networks. BET surface area, measured by nitrogen adsorption, quantifies this. But raw surface area is not enough - the pores must be large enough for reactant molecules to enter and products to exit. This creates a fundamental tradeoff: smaller pores give more area but slower diffusion. The most advanced catalysts use hierarchical pore structures spanning nano to micro scales.

Engineering the Active Site

At the molecular level, catalysis occurs at specific active sites where reactant molecules adsorb, react, and desorb. The turnover frequency (TOF) - reactions per site per second - measures intrinsic catalytic activity. Catalyst design aims to maximize the number of active sites (high dispersion), ensure they are accessible (good pore structure), and maintain their activity over time (resistance to sintering and poisoning).

FAQ

What is the Thiele modulus?

The Thiele modulus phi is a dimensionless number that compares the rate of surface reaction to the rate of diffusion within a catalyst particle. phi = R * sqrt(k/D_eff), where R is particle radius, k is rate constant, and D_eff is effective diffusivity. Large phi means diffusion-limited; small phi means kinetics-limited.

What is the effectiveness factor?

The effectiveness factor eta is the ratio of actual reaction rate to the rate that would occur if the entire particle interior were at surface conditions. For a sphere: eta = (1/phi)*(3*phi*coth(3*phi) - 1)/(3*phi^2). Eta ranges from 1 (no diffusion limitation) to near 0 (severe limitation).

How does pore size affect catalysis?

Pore size controls the effective diffusivity of reactants within the catalyst. Micropores (<2nm) provide high surface area but restrict diffusion; mesopores (2-50nm) balance area and transport; macropores (>50nm) allow fast diffusion but have low area. Hierarchical pore structures combine the benefits of multiple scales.

What is BET surface area?

BET (Brunauer-Emmett-Teller) surface area is the total surface area per gram of catalyst, measured by nitrogen adsorption. High surface area (hundreds of m2/g) provides more active sites. Activated carbon can exceed 1500 m2/g; zeolites typically range from 300-900 m2/g.

Sources

Embed

<iframe src="https://homo-deus.com/lab/chemical-engineering/catalyst-design/embed" width="100%" height="400" frameborder="0"></iframe>
View source on GitHub