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The purpose of this article is to explain how to compute the Thiele modulus and the effectiveness factor for a first-order reaction in porous pellets, and how to use them to diagnose and quantify internal diffusion limitations in heterogeneous catalysis.
1. Why the Thiele modulus matters in catalyst pellet design.
In porous catalyst pellets, reactant must diffuse from the pellet surface into internal pores while reacting on internal surface area. When reaction is fast relative to diffusion, the reactant concentration drops toward the pellet center, and a portion of the catalyst volume becomes underutilized. This is called internal mass transfer limitation, and it reduces the observed reaction rate compared with the intrinsic kinetic rate. The Thiele modulus is the standard dimensionless group that compares intrinsic reaction to intraparticle diffusion for a given pellet size and effective diffusivity.
For a first-order reaction, the Thiele modulus has a clean definition and yields closed-form effectiveness factor formulas for common pellet shapes. These results are widely used for catalyst sizing, kinetic parameter interpretation, scale-up, and quick screening calculations in fixed-bed reactor design.
2. Physical model and assumptions for first-order pellets.
The classic pellet model is a steady-state diffusion–reaction balance inside the porous solid. The most common assumptions are isothermal pellet, constant pellet properties, constant effective diffusivity, and a first-order intrinsic rate in the pore fluid concentration. The pellet is treated as a continuum with an effective diffusivity that lumps pore structure effects such as porosity and tortuosity.
For a single reactant A with first-order kinetics, the intrinsic volumetric consumption rate can be written as r_A = k C_A, where k has units of 1/s when the rate is expressed per pellet pore-fluid volume, or can be converted to an equivalent volumetric form depending on the chosen definition of k and the catalyst internal surface area model. The key point is that the governing equation becomes linear in concentration for first-order kinetics, which enables analytic solutions.
Note : Effectiveness factor formulas in this article assume the surface concentration equals the bulk concentration at the pellet boundary, which requires negligible external mass transfer resistance or a sufficiently large mass-transfer Biot number. If external transfer is not negligible, use a coupled film and pellet model as described later in this article.
3. Governing equation and boundary conditions.
Let C(r) be the reactant concentration inside the pellet pores. For constant effective diffusivity D_e and first-order reaction rate k C, the steady-state diffusion–reaction equation in radial coordinates is D_e ∇²C − k C = 0.
Boundary conditions are symmetry at the center and a specified concentration at the pellet surface. Symmetry requires dC/dr = 0 at r = 0 for cylinder and sphere, and dC/dx = 0 at x = 0 for a slab. At the external surface, the simplest boundary condition is C = C_s at r = R, where C_s is the concentration at the pellet surface in the pore fluid.
4. Definition of the Thiele modulus for first-order reaction.
The Thiele modulus is defined as a characteristic pellet length multiplied by the square root of the ratio of intrinsic first-order rate constant to effective diffusivity. For a slab of half-thickness L, φ = L √(k / D_e). For a cylinder of radius R, φ = R √(k / D_e). For a sphere of radius R, φ = R √(k / D_e).
The characteristic length is not arbitrary in these standard closed-form solutions. For the slab solution, the half-thickness L is used because the symmetry plane is at the midplane. For the cylinder and sphere solutions, the radius R is used because diffusion is radial from the surface to the center.
5. Definition of effectiveness factor and what it measures.
The effectiveness factor η is the ratio of the actual overall reaction rate inside the pellet to the rate that would occur if the entire pellet were at the surface concentration. For first-order kinetics, this can be written as η = (1 / V_p) ∫(C / C_s) dV over the pellet volume, where V_p is pellet volume. This definition makes η a volume-average of the normalized concentration for first-order reaction.
The observed pellet consumption rate becomes R_obs = η k C_s V_p for the first-order volumetric form. In reactor models, this η is used to correct intrinsic kinetics so that bed-scale rate expressions reflect intraparticle diffusion limitations.
6. Closed-form effectiveness factor formulas for common pellet shapes.
6.1 Slab pellet effectiveness factor.
For a slab of half-thickness L with φ = L √(k / D_e), the effectiveness factor is η = tanh(φ) / φ.
For small φ, η approaches 1, meaning diffusion is fast and the pellet is nearly uniform in concentration. For large φ, η decreases approximately as 1/φ, meaning the reaction is confined near the surface region.
6.2 Cylindrical pellet effectiveness factor.
For a cylinder of radius R with φ = R √(k / D_e), the effectiveness factor is η = 2 I1(φ) / (φ I0(φ)), where I0 and I1 are modified Bessel functions of the first kind.
For practical work, this expression is typically evaluated using scientific calculators, spreadsheet functions, or standard numerical libraries. For large φ, the cylindrical effectiveness factor also scales approximately as 2/φ, which reflects a shrinking reactive region near the surface.
6.3 Spherical pellet effectiveness factor.
For a sphere of radius R with φ = R √(k / D_e), the effectiveness factor is η = 3(φ coth(φ) − 1) / φ².
This spherical formula is the most commonly used in packed-bed catalyst calculations because many catalysts are approximated as spheres or pellets with a representative spherical radius.
7. Useful limiting approximations for quick checks.
Small Thiele modulus implies weak internal gradients and η close to 1. For a slab, η ≈ 1 − φ²/3 for φ much smaller than 1. For a cylinder, η ≈ 1 − φ²/8 for φ much smaller than 1. For a sphere, η ≈ 1 − φ²/15 for φ much smaller than 1.
Large Thiele modulus implies strong diffusion limitation and η inversely proportional to φ. For a slab, η ≈ 1/φ for φ much larger than 1. For a cylinder, η ≈ 2/φ for φ much larger than 1. For a sphere, η ≈ 3/φ for φ much larger than 1.
| Pellet shape. | Thiele modulus definition. | Effectiveness factor for first-order reaction. | Small-φ approximation. | Large-φ approximation. |
|---|---|---|---|---|
| Slab. | φ = L √(k / D_e). | η = tanh(φ) / φ. | η ≈ 1 − φ²/3. | η ≈ 1/φ. |
| Cylinder. | φ = R √(k / D_e). | η = 2 I1(φ) / (φ I0(φ)). | η ≈ 1 − φ²/8. | η ≈ 2/φ. |
| Sphere. | φ = R √(k / D_e). | η = 3(φ coth(φ) − 1) / φ². | η ≈ 1 − φ²/15. | η ≈ 3/φ. |
8. How to estimate effective diffusivity inside porous pellets.
Effective diffusivity D_e represents diffusion through a porous network and is lower than the molecular diffusivity in free gas or liquid. A common engineering form is D_e = (ε/τ) D_p, where ε is pellet porosity and τ is tortuosity, and D_p is an appropriate pore diffusion coefficient.
In gases, pore diffusion may involve both molecular diffusion and Knudsen diffusion. A frequently used combination is the Bosanquet relation, 1/D_p = 1/D_m + 1/D_K, where D_m is molecular diffusivity and D_K is Knudsen diffusivity. In liquids, Knudsen diffusion is usually negligible and D_p is often approximated by D_m modified by ε/τ.
Knudsen diffusivity is often estimated as D_K = (2/3) r_p √(8 R_g T / (π M_A)), where r_p is a representative pore radius, R_g is the gas constant, T is temperature, and M_A is molar mass of A. This estimate is sensitive to pore size assumptions, so it is typically used as an order-of-magnitude screening tool unless pore-size distribution data are available.
Note : Uncertainty in D_e directly propagates into φ because φ scales with 1/√D_e. If you are fitting kinetics from pellet data, separating intrinsic kinetics and internal diffusion requires independent characterization of pellet porosity, tortuosity, and pore size distribution, or experiments with multiple pellet sizes.
9. Step-by-step workflow to compute η for a first-order pellet.
9.1 Inputs you need.
You need the intrinsic first-order rate constant k at the pellet operating temperature and composition. You need an effective diffusivity D_e for the reactant in the pellet under the same conditions. You need a representative pellet size, which is half-thickness for slabs or radius for cylinders and spheres.
9.2 Computation steps.
Step 1 is compute φ using the appropriate shape definition. Step 2 is compute η from the shape-specific closed-form formula. Step 3 is compute the observed pellet-scale rate using R_obs = η k C_s V_p for a volumetric first-order model, or equivalently apply η as a multiplicative factor in your reactor rate expression. Step 4 is check whether η is close to 1 or far below 1, and decide whether pellet size, pore structure, or operating conditions should be changed to reduce internal diffusion limitation.
# Example calculation for a spherical pellet. # Given. # k = 0.50 1/s. # De = 1.0e-6 m^2/s. # R = 1.5e-3 m. # Cs = 2.0 mol/m^3. import math k = 0.50 De = 1.0e-6 R = 1.5e-3 Cs = 2.0 phi = R * math.sqrt(k / De) # Effectiveness factor for a sphere: eta = 3*(phi*coth(phi) - 1)/phi^2. coth = math.cosh(phi) / math.sinh(phi) eta = 3.0 * (phi * coth - 1.0) / (phi ** 2) phi, eta, eta * k * Cs This workflow yields φ and η, and the final term η k C_s is the effective first-order consumption rate per unit pore-fluid volume at the pellet scale, consistent with the volumetric rate constant definition used in the input.
10. Interpreting results and practical design decisions.
If φ is much smaller than 1, then η is close to 1 and internal diffusion is not rate limiting. In that regime, observed kinetics approximate intrinsic kinetics, and pellet size has little impact on rate. If φ is much larger than 1, then η becomes small, and the reaction is concentrated near the pellet surface, which means the interior catalyst is underutilized.
To increase η under diffusion limitation, you can reduce pellet radius, increase effective diffusivity by increasing porosity or reducing tortuosity, increase pore size when Knudsen diffusion is limiting, or reduce intrinsic rate per pellet volume by lowering temperature if selectivity constraints allow. Because φ scales linearly with pellet size, pellet diameter changes are often the fastest lever for regaining effectiveness, but they can increase pressure drop in packed beds, so reactor hydrodynamics must be checked.
11. When external mass transfer cannot be ignored.
If there is a significant concentration drop across the external gas or liquid film around the pellet, then the pellet surface concentration C_s is lower than the bulk concentration C_b, and using C_s = C_b overestimates the observed rate. A common way to assess this is with a mass-transfer Biot number, Bi_m = k_g R / D_e, where k_g is the external film mass-transfer coefficient and R is pellet radius for a sphere or cylinder.
Large Bi_m implies the surface concentration is close to bulk concentration and external resistance is small. Moderate or small Bi_m implies external resistance matters and a coupled boundary condition is needed, typically −D_e (dC/dr)|_R = k_g (C_b − C_s). In that case, the overall observed rate reduction is influenced by both internal and external transport, and η alone does not capture the full resistance network.
Note : Many field problems blamed on “bad catalyst” are actually combined internal diffusion plus external film limitation caused by low superficial velocity, maldistribution, or unexpected viscosity changes. Always evaluate both pellet-scale and film-scale transport before concluding kinetic deactivation.
12. Common pitfalls in Thiele modulus calculations.
A frequent error is mixing definitions of k. A first-order rate constant reported per catalyst mass or per internal surface area must be converted to a consistent volumetric pellet form before using φ = R √(k / D_e). Another common error is using free-fluid molecular diffusivity instead of effective diffusivity, which can underpredict φ and overpredict η. A third error is using pellet diameter where radius is required, which doubles φ and can materially change η in the transition regime.
It is also important to check that the first-order assumption is valid over the concentration range inside the pellet. If the intrinsic kinetics are not first order, then φ can be generalized, but the closed-form formulas given here are no longer exact and a numerical solution is typically required.
FAQ
What is the physical meaning of the Thiele modulus for a first-order catalyst pellet.
The Thiele modulus is a dimensionless ratio that compares the intrinsic first-order reaction rate scale to the intraparticle diffusion rate scale. A small Thiele modulus means diffusion is fast enough to keep the pellet nearly uniform in concentration. A large Thiele modulus means diffusion is slow relative to reaction and strong concentration gradients develop, reducing catalyst utilization.
How do I choose the correct pellet length scale for the Thiele modulus.
Use the half-thickness for a slab, and use the radius for cylinders and spheres when applying the standard closed-form first-order effectiveness factor formulas. If you have irregular pellets, you can approximate them as spheres with an equivalent radius based on volume-to-surface considerations, but you should treat the result as an engineering approximation and validate with sensitivity checks.
Can I use the same effectiveness factor formulas for nth-order or Langmuir–Hinshelwood kinetics.
The formulas in this article are exact for first-order kinetics with constant effective diffusivity. For other kinetic forms, effectiveness factors can still be defined, but you typically need numerical solution of the diffusion–reaction equation, or you need specialized approximations. The first-order formulas are sometimes used for screening by linearizing kinetics around an operating concentration, but that should be treated as an approximation.
What effectiveness factor should I use in a packed-bed reactor model.
Use the internal effectiveness factor η to correct intrinsic kinetics at the pellet scale, and also evaluate whether external mass transfer causes a surface-to-bulk concentration drop. If external resistance is negligible, you can take C_s equal to the bulk concentration and apply η directly to the intrinsic rate expression. If external resistance matters, you need a film–pellet coupling so that the rate depends on both η and the external mass transfer coefficient.
How can I tell whether internal diffusion is distorting my kinetic parameter estimates.
A standard diagnostic is to repeat experiments with different pellet sizes while keeping external conditions comparable. If the apparent rate constant changes with pellet size, internal diffusion is likely influencing the observed rate. You can then use Thiele modulus and effectiveness factor analysis, together with independent estimates of effective diffusivity, to separate intrinsic kinetics from transport effects.