- Get link
- X
- Other Apps
The purpose of this article is to show how to evaluate the Weisz–Prater criterion correctly, using consistent units and defensible transport property estimates, so you can quickly determine whether internal pore diffusion limits observed kinetics on a porous catalyst pellet.
1. Why the Weisz–Prater criterion matters in heterogeneous catalysis kinetics.
Measured reaction rates over porous catalysts can be lower than intrinsic rates when reactants cannot diffuse fast enough through pores to supply the active sites in the pellet interior.
If internal diffusion is significant, fitted kinetic parameters become “apparent” and lose predictive value across particle size, temperature, and flow conditions.
The Weisz–Prater criterion is a fast diagnostic that uses observable quantities to test whether intraparticle diffusion limitations are negligible for a chosen reactant and rate expression.
2. Definition of the Weisz–Prater criterion and decision thresholds.
The Weisz–Prater criterion is commonly written as a dimensionless number for a limiting reactant A in a porous catalyst pellet.
N_WP = (r_obs * R_p^2) / (C_s * D_eff).In this expression, r_obs is the observed reaction rate per catalyst pellet volume based on external measurements, R_p is the catalyst particle radius, C_s is the concentration of A at the pellet external surface, and D_eff is the effective diffusivity of A in the porous pellet.
When N_WP is sufficiently small, concentration gradients inside the pellet are small and the effectiveness factor is close to 1.
| Interpretation of N_WP. | Practical meaning for porous catalyst kinetics. | Typical action. |
|---|---|---|
| N_WP ≤ 0.3. | Internal diffusion limitation is usually negligible for many common cases and reaction orders. | Proceed with intrinsic kinetics fitting, while still checking external mass transfer and heat effects. |
| 0.3 < N_WP < 1. | Internal diffusion may be present and can bias kinetic parameters depending on reaction order and pellet structure. | Reduce pellet size, improve diffusivity, or use effectiveness factor models during fitting. |
| N_WP ≥ 1. | Internal diffusion is likely significant and observed rate is strongly transport-affected. | Do not treat data as intrinsic kinetics without transport correction and additional verification. |
Note : The threshold values above are widely used screening rules, but the safest interpretation is comparative, meaning N_WP should decrease approximately with R_p^2 if internal diffusion is the cause, when other conditions are held constant.
3. Required inputs and unit consistency checklist.
The Weisz–Prater criterion is simple, but errors are common because each term must be defined on a consistent basis.
3.1 Symbol definitions and recommended units.
| Symbol. | Meaning. | Recommended unit set. | Common pitfalls. |
|---|---|---|---|
| r_obs. | Observed rate per catalyst pellet volume for species A consumption. | mol·m−3cat·s−1. | Using rate per mass without converting to pellet volume. |
| R_p. | Pellet radius, for the characteristic diffusion length. | m. | Using diameter instead of radius, or using sieve size without conversion. |
| C_s. | Concentration of A at the external pellet surface. | mol·m−3. | Using bulk concentration when external mass transfer is not negligible. |
| D_eff. | Effective diffusivity of A inside pellet pores, including structure effects. | m2·s−1. | Using bulk molecular diffusivity without porosity and tortuosity corrections. |
3.2 Converting observed rate to the correct basis.
Many reactors report rate as mol·kg−1cat·s−1, but N_WP requires mol·m−3cat·s−1.
r_obs,vol = r_obs,mass * rho_p.Here rho_p is the pellet apparent density on a pellet volume basis, meaning mass of pellet divided by total pellet volume including pores, in kg·m−3pellet.
Note : Do not use skeletal density unless you explicitly convert to pellet volume, because the diffusion length R_p refers to the pellet geometric volume, not just the solid framework.
4. Estimating the effective diffusivity D_eff in porous catalyst pellets.
D_eff is the parameter that most often controls the conclusion, so it must be estimated in a way consistent with the operating phase and pore regime.
4.1 Structure correction using porosity and tortuosity.
A common engineering form relates D_eff to a pore-space diffusivity D_pore through pellet porosity ε and tortuosity τ.
D_eff = (epsilon / tau) * D_pore.Porosity is dimensionless, and tortuosity is dimensionless, so D_eff has the same units as D_pore.
4.2 Gas-phase pores and the Bosanquet combination.
In gas-phase catalytic reactions, diffusion in pores can be influenced by both molecular diffusion and Knudsen diffusion, depending on pore size and mean free path.
A commonly used combination is the Bosanquet form.
1 / D_pore = 1 / D_m + 1 / D_K.D_m is the molecular diffusivity of A in the gas mixture, and D_K is the Knudsen diffusivity for A in the pores.
For a cylindrical-pore approximation, Knudsen diffusivity can be expressed as.
D_K = (2/3) * r_pore * sqrt(8 * R * T / (pi * M_A)).Here r_pore is pore radius, R is the gas constant, T is temperature, and M_A is molar mass of A.
Note : If pore size distribution is broad, using a single r_pore can under- or over-estimate D_K, so sensitivity analysis across plausible pore radii is recommended when N_WP is near the decision boundary.
4.3 Liquid-filled pores and diffusion in liquids.
In liquid-phase catalysis, Knudsen diffusion is usually not applicable, and D_pore is typically close to the bulk liquid diffusivity of A in the solvent, then reduced by porosity and tortuosity.
Because liquid diffusivities can be orders of magnitude smaller than gas diffusivities, liquid-phase N_WP often signals internal diffusion limitations at comparatively smaller pellet sizes.
4.4 When surface diffusion or adsorption effects matter.
Some systems show additional transport contributions from surface diffusion of adsorbed species, which can increase the apparent D_pore relative to bulk-only models.
If you suspect strong adsorption and the screening result is borderline, consider verifying with particle-size tests rather than relying on a single D_eff estimate.
5. Step-by-step workflow to evaluate the Weisz–Prater criterion.
This workflow is designed to be implemented as a calculation checklist or automated in a spreadsheet or Python script.
5.1 Step 1. Select the limiting reactant and define r_obs consistently.
Choose a reactant A that appears in the rate expression and is plausibly diffusion-limiting inside pores, often the species with the steepest concentration drop or lowest diffusivity.
Compute r_obs for consumption of A, and convert to a pellet volume basis.
5.2 Step 2. Determine pellet radius R_p and geometry assumptions.
Use the true particle radius for the pellets used in the kinetic experiment.
If pellets are non-spherical, use an equivalent radius based on volume-to-surface relationships consistently across experiments.
5.3 Step 3. Estimate C_s at the pellet surface.
If external mass transfer is negligible, C_s can be approximated by the bulk concentration C_b.
If external mass transfer may be significant, C_s is lower than C_b for reactant consumption, and using C_b will under-predict N_WP and hide internal diffusion problems.
In many practical screening studies, you first ensure external mass transfer is minimized by sufficient mixing or high flow, then set C_s approximately equal to C_b.
Note : For reliable screening, do not change pellet size and stirring speed simultaneously, because external mass transfer changes can mask the R_p^2 scaling expected from internal diffusion.
5.4 Step 4. Compute D_eff using an appropriate pore diffusion model.
Pick D_pore as molecular-only, Knudsen-only, or Bosanquet-combined depending on phase and pore regime, then apply the structure factor (epsilon / tau).
5.5 Step 5. Calculate N_WP and interpret with sensitivity analysis.
Compute N_WP and compare to the screening thresholds.
If N_WP is near 0.3 to 1, recalculate using high and low plausible D_eff values to see if your conclusion changes.
6. Fully worked example calculation with consistent units.
This example uses illustrative values to demonstrate the workflow and unit handling, and it is not intended to represent any specific catalyst system.
6.1 Given data for a porous catalyst pellet.
| Quantity. | Value. | Unit. | Comment. |
|---|---|---|---|
| Observed rate per mass. | 0.020. | mol·kg−1cat·s−1. | Consumption rate of reactant A. |
| Pellet apparent density rho_p. | 1200. | kg·m−3pellet. | Mass divided by geometric pellet volume. |
| Pellet diameter. | 1.0e−3. | m. | So radius R_p = 5.0e−4 m. |
| Surface concentration C_s. | 2.0. | mol·m−3. | Assumed close to bulk under strong mixing. |
| Effective diffusivity D_eff. | 1.0e−6. | m2·s−1. | From pore diffusion estimate and structure factor. |
6.2 Convert the rate to a pellet volume basis.
r_obs,vol = (0.020 mol/kg/s) * (1200 kg/m^3) = 24 mol/m^3/s.6.3 Compute the Weisz–Prater number.
N_WP = (r_obs,vol * R_p^2) / (C_s * D_eff) = (24 * (5.0e-4)^2) / (2.0 * 1.0e-6) = (24 * 2.5e-7) / (2.0e-6) = 6.0e-6 / 2.0e-6 = 3.0.This result indicates a strong likelihood of internal diffusion limitation under the stated conditions.
In practice, you would reduce pellet size, increase temperature carefully while checking heat effects, or use an effectiveness factor model to correct kinetics.
Note : The fastest experimental verification is a particle-size test where r_obs is measured at multiple pellet radii under otherwise identical conditions, because internal diffusion effects scale strongly with R_p^2 in screening diagnostics.
7. Linking the Weisz–Prater criterion to effectiveness factor and the Thiele modulus.
For common pellet geometries, internal diffusion and reaction can also be represented using the Thiele modulus and an effectiveness factor η.
The Thiele modulus compares intrinsic reaction rate strength to diffusion, and η describes the fraction of intrinsic activity that is realized when concentration gradients exist.
N_WP is especially useful because it can be formed directly from observed rate data without first estimating intrinsic kinetic constants, while still providing a transport screening metric.
When N_WP is small, η is close to 1 and the observed rate approximates the intrinsic rate.
8. Practical troubleshooting when results look inconsistent.
8.1 If N_WP predicts no diffusion limitation but pellet-size dependence is observed.
Re-check whether the measured rate truly reflects reaction kinetics rather than external mass transfer, axial dispersion, or changing wetting in liquid-phase systems.
Re-check the basis of r_obs and whether rho_p corresponds to pellet volume or skeletal volume.
Re-check whether D_eff was underestimated by ignoring multicomponent diffusion or overestimated by using bulk gas diffusion without Knudsen limitation.
8.2 If N_WP predicts strong diffusion limitation but rate does not change with pellet size.
Confirm that pellets are truly of different radii rather than different size distributions with overlapping means.
Confirm that internal porosity and pore structure are comparable across pellet sizes, because manufacturing changes can alter ε and τ.
Confirm that chemistry is not changing with pellet size due to heat removal differences, especially for exothermic reactions.
8.3 Minimal reporting checklist for publication-quality kinetics.
| Item. | What to report. | Why it matters for diffusion limitation claims. |
|---|---|---|
| Pellet size distribution. | Mean, range, and how size was measured. | Diffusion diagnostics depend on the correct characteristic radius. |
| Pellet density basis. | Apparent pellet density used for rate conversion. | Incorrect density basis can shift N_WP by large factors. |
| Transport property assumptions. | D_m, D_K or liquid diffusivity, ε, τ, and how chosen. | D_eff dominates uncertainty in N_WP conclusions. |
| Surface concentration assumption. | Why C_s ≈ C_b is justified, or how C_s was estimated. | Using bulk concentration can hide combined external and internal limitations. |
9. Simple implementation template for a Weisz–Prater criterion calculator.
The following calculation template can be adapted into a porous catalyst diffusion limitation calculator, a lab worksheet, or a data pipeline for kinetic screening.
# Inputs must be in consistent SI units. # r_obs_mass: mol/kg_cat/s. # rho_pellet: kg/m^3_pellet. # R_p: m. # C_s: mol/m^3. # D_eff: m^2/s.
r_obs_vol = r_obs_mass * rho_pellet
N_WP = (r_obs_vol * (R_p ** 2)) / (C_s * D_eff)
Screening interpretation.
if N_WP <= 0.3:
verdict = "Internal diffusion limitation is likely negligible."
elif N_WP < 1.0:
verdict = "Internal diffusion may be present and needs verification."
else:
verdict = "Internal diffusion limitation is likely significant."For robust decisions, repeat the calculation with upper and lower bounds for D_eff and C_s, then report the resulting N_WP interval.
FAQ
Which concentration should be used for C_s in the Weisz–Prater criterion.
C_s is the reactant concentration at the external surface of the catalyst pellet.
If external mass transfer resistance is negligible under your test conditions, C_s can be approximated by the bulk concentration.
If external mass transfer is not negligible, C_s is lower than the bulk concentration for a consumed reactant, and you should estimate C_s using an external mass transfer model or experimental verification at higher mixing or flow.
What rate should be used for r_obs in the Weisz–Prater criterion.
Use the observed rate derived directly from reactor performance without correcting for internal diffusion, because the purpose is to test whether the observed rate is already diffusion-limited.
Ensure the rate is expressed per pellet volume, typically by converting from a mass basis using the apparent pellet density.
How do I choose D_eff for gas-phase porous catalyst pellets.
Start with a molecular diffusivity estimate for the gas mixture, then evaluate whether Knudsen diffusion could be comparable using pore size information.
If both contribute, use the Bosanquet combination to obtain a pore-space diffusivity, then apply the porosity and tortuosity correction to get D_eff.
What should I do if N_WP is borderline.
Perform a particle-size test at fixed temperature, composition, and mixing or flow conditions.
If internal diffusion is controlling, the observed rate trends should be consistent with increased transport resistance at larger pellet sizes, while other artifacts remain controlled.
In addition, run a sensitivity analysis on D_eff and C_s to assess whether your conclusion is robust to reasonable parameter uncertainty.