Effectiveness Factor Calculator

The Effectiveness Factor Calculator computes intraparticle effectiveness factors from Thiele modulus, reaction order and pellet geometry for porous catalysts.

What Is a Effectiveness Factor Calculator?

An effectiveness factor calculator estimates how much of your catalyst’s potential reaction rate is realized inside a porous solid. In porous pellets, concentration drops from the surface to the core because reactants diffuse inward and are consumed by reaction. This internal gradient causes part of the catalyst to work below its best conditions.

The effectiveness factor, η, is the ratio of the observed reaction rate in the porous solid to the rate you would get if the entire solid were at the surface concentration. If η is near 1, most of the solid is active at high concentration. If η is small, diffusion limits dominate, and the core contributes less.

By combining measured or assumed kinetics with pellet geometry and effective diffusivity, the calculator produces η and related diagnostics. These include the Thiele modulus, φ, and the Weisz–Prater criterion, which help flag when internal mass transfer is limiting.

How to Use Effectiveness Factor (Step by Step)

Before using the tool, gather a few basics: geometry (slab, cylinder, or sphere), pellet size, effective diffusivity, reaction order, and rate constant. It also helps to know the reactant concentration at the pellet surface and the observed rate per pellet volume or mass.

• Choose the pellet geometry that best matches your catalyst or support.
• Enter pellet size (e.g., radius or half-thickness) and porosity-related data for effective diffusivity.
• Specify the reaction rate model (e.g., first-order: r = k C, or second-order: r = k C²).
• Provide the surface concentration from bulk concentration and any external mass transfer data, if known.
• Input the observed rate, if available, to compare with model predictions.

After you compute, review η and φ together. A small η with a large φ suggests strong internal diffusion limits. If the Weisz–Prater number is also high, consider changing pellet size, pore structure, or operating conditions.

Equations Used by the Effectiveness Factor Calculator

The calculator uses standard porous catalyst theory. It links the Thiele modulus, φ, to the effectiveness factor, η, through geometry-specific formulas for first-order reactions, and it supports numerical solutions for other orders.

• Definition of effectiveness factor: η = (observed rate per pellet volume) / (intrinsic rate at surface conditions).
• Thiele modulus, first-order: φ = R_p sqrt(k / D_eff) for a sphere with radius R_p (similar forms for other shapes).
• Effectiveness factor, slab, first-order: η = tanh(φ) / φ.
• Effectiveness factor, sphere, first-order: η = (3 / φ²) [φ coth(φ) − 1].
• Weisz–Prater criterion: C_WP = (R_obs R_p²) / (C_A,s D_eff). If C_WP ≪ 1, internal diffusion is likely negligible.

For non-first-order kinetics (e.g., r = k Cⁿ), φ depends on concentration and reaction order. The calculator uses iterative or numerical methods to solve the underlying diffusion–reaction equations. It reports η consistently, so you can compare cases even when the math gets more complex.

Inputs, Assumptions & Parameters

To compute η with confidence, you need key inputs that describe geometry, transport, and kinetics. The tool assumes steady state, isothermal behavior, and uniform pellet properties unless you specify otherwise.

• Pellet characteristic size (R_p for spheres, slab half-thickness, or cylinder radius), in meters.
• Effective diffusivity, D_eff, in m²/s, accounting for porosity and tortuosity.
• Kinetic model and rate constant, k, with units tied to reaction order (e.g., 1/s for first order; m³/(mol·s) for second order).
• Surface concentration, C_A,s, in mol/m³, from bulk concentration and external mass transfer, if known.
• Observed reaction rate, R_obs, in mol/(m³·s) or per mass basis, for diagnostic checks and η by definition.

Reasonable ranges: R_p from tens of micrometers to a few millimeters; D_eff from 10⁻¹¹ to 10⁻⁸ m²/s for liquids and up to 10⁻⁵ m²/s for gases in large pores; concentrations from millimolar to molar; and rate constants consistent with your stoichiometry and temperature. Edge cases include very small pellets (φ → 0, η → 1) and very large pellets (φ ≫ 1, η ≪ 1). If you have strong heat effects, or concentration-dependent D_eff, results will be approximate.

Step-by-Step: Use the Effectiveness Factor Calculator

Here’s a concise overview before we dive into the key points:

1. Select the pellet geometry that matches your system.
2. Enter pellet size and effective diffusivity.
3. Choose the kinetic model and provide the rate constant and reaction order.
4. Input surface concentration, or let the tool estimate it from bulk and a film coefficient.
5. Optional: enter the observed rate to compute η by definition and cross-check model results.
6. Click Calculate to solve for φ, η, and diagnostic criteria.

These points provide quick orientation—use them alongside the full explanations in this page.

Worked Examples

Case 1: First-order reaction in a spherical catalyst pellet. Geometry: sphere with radius R_p = 1.0 × 10⁻³ m. Effective diffusivity D_eff = 1.0 × 10⁻⁹ m²/s. Kinetics: r = k C_A with k = 0.20 s⁻¹. Surface concentration C_A,s = 1.0 mol/m³. Thiele modulus φ = R_p sqrt(k / D_eff) = 1.0 × 10⁻³ × sqrt(0.20 / 1.0 × 10⁻⁹) ≈ 14.1. Effectiveness factor for a sphere: η = (3/φ²) [φ coth(φ) − 1]. For large φ, coth(φ) ≈ 1, so η ≈ (3/199) × (14.1 − 1) ≈ 0.20. The intrinsic rate at the surface is r_surface = k C_A,s = 0.20 mol/(m³·s). The effective volume-averaged rate becomes r_effective = η × r_surface ≈ 0.040 mol/(m³·s). What this means: only about 20% of the catalyst volume contributes effectively, so a smaller pellet or higher D_eff could improve use of the interior.

Case 2: First-order reaction in a planar slab (half-thickness L = 5.0 × 10⁻⁴ m). Effective diffusivity D_eff = 5.0 × 10⁻¹⁰ m²/s. Choose a rate constant k so that φ = L sqrt(k / D_eff) = 0.50. Solving for k: k = φ² D_eff / L² = 0.25 × 5.0 × 10⁻¹⁰ / (2.5 × 10⁻⁷) ≈ 5.0 × 10⁻⁴ s⁻¹. With C_A,s = 2.0 mol/m³, r_surface = k C_A,s = 1.0 × 10⁻³ mol/(m³·s). The slab effectiveness factor is η = tanh(φ)/φ = tanh(0.50)/0.50 ≈ 0.462/0.50 ≈ 0.92. The effective rate is r_effective ≈ 0.92 × 1.0 × 10⁻³ = 9.2 × 10⁻⁴ mol/(m³·s). What this means: internal diffusion is mild; most of the slab volume operates near surface conditions, so scale-up risk from diffusion is low here.

Accuracy & Limitations

The tool follows standard diffusion–reaction theory for porous solids. Accuracy depends on how well your inputs reflect the real catalyst and fluid. Internal gradients, heat effects, and non-ideal pore networks can shift results.

• Isothermal assumption can break for exothermic reactions; hot spots change local rates.
• D_eff varies with porosity, tortuosity, and concentration; a single value may oversimplify.
• Reaction order may change with coverage or inhibitors; use kinetic data near your conditions.
• External mass transfer resistance changes C_A,s; ignoring it can inflate η.
• Pellet geometry is idealized; real shapes and size distributions add error.

Use the outputs for screening and sensitivity. Validate with lab measurements when possible. For complex stoichiometry or multi-component diffusion, consider a detailed transport model.

Units Reference

Correct units keep the Thiele modulus and effectiveness factor consistent. Concentration, pellet size, and diffusivity must align so rates remain in molar units tied to volume or mass. Always match the rate constant’s units to reaction order and concentration basis.

Read the table left to right to set input units. If your data use different units, convert before entry. Keep concentration basis consistent across kinetics, stoichiometry, and rate measurements.

Common Issues & Fixes

Most problems trace back to mismatched units or inconsistent kinetic models. Check that k and concentration units match the reaction order. Confirm the pellet size is expressed in meters and not millimeters by mistake.

• If η > 1 appears, recheck units, the rate definition, and whether R_obs is per pellet volume.
• If φ is unrealistically large, verify D_eff and pellet size conversions.
• When C_WP is huge, ensure R_obs and C_A,s are on the same basis and geometry is correct.

If external mass transfer is significant, the surface concentration is below the bulk value. Use a film coefficient or Sherwood correlation to estimate C_A,s, or measure it when possible.

FAQ about Effectiveness Factor Calculator

What does an effectiveness factor near 1 mean?

It means internal diffusion resistance is small. Concentration is almost uniform inside the pellet, so the catalyst volume is well used.

Can the calculator handle non-first-order kinetics?

Yes. For n ≠ 1, it uses numerical methods to solve the diffusion–reaction equation and returns η based on your rate law.

How does pellet size affect η?

Bigger pellets increase φ, which lowers η when diffusion limits are present. Smaller pellets often raise η but may increase pressure drop.

Do I need the observed rate to compute η?

No. You can compute η from φ using kinetics and transport properties. However, the observed rate lets you cross-check theory against data.

Effectiveness Factor Terms & Definitions

Effectiveness factor (η)

The ratio of the actual reaction rate in a porous catalyst to the rate if the entire pellet were at the surface concentration.

Thiele modulus (φ)

A dimensionless number comparing reaction rate to diffusion rate inside a pellet; larger φ suggests stronger internal diffusion limits.

Weisz–Prater criterion

A check using observed rate, pellet size, diffusivity, and surface concentration to judge if internal diffusion limitations are present.

Effective diffusivity (D_eff)

The diffusion coefficient inside pores adjusted for porosity and tortuosity; lower than bulk fluid diffusivity.

Reaction order (n)

The power of concentration in the rate law (r ∝ Cⁿ), tied to stoichiometry and mechanism for the limiting step.

Rate constant (k)

The proportionality factor in the rate law; its units depend on n and concentration basis, such as 1/s or m³/(mol·s).

Surface concentration (C_A,s)

The reactant concentration at the pellet surface; can be below bulk due to an external mass transfer film.

Moles and concentration basis

The accounting of amount of substance, typically in mol and mol/m³, used with stoichiometry to define consistent rates and balances.

References

Here’s a concise overview before we dive into the key points:

• Thiele modulus overview on Wikipedia
• Effectiveness factor article on Wikipedia
• NIST Chemistry WebBook for properties and constants
• LibreTexts: Chemical Reactor Analysis and Design
• Lecture notes on heterogeneous catalysis and diffusion (PDF)

These points provide quick orientation—use them alongside the full explanations in this page.