Diffusion in catalyst pellets, Thiele modulus and effectiveness factor
Pore diffusion in catalyst pellets: effective diffusivity, Thiele modulus, effectiveness factor for slabs and spheres, strong-diffusion falsified kinetics and the Weisz-Prater test, with worked pellet-size calculations.
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Why it matters
Catalyst pellets are porous so that they offer huge internal surface areas, but reactants must diffuse into the pores to reach that surface. If diffusion is slow compared with reaction, the pellet interior is starved and much of the expensive catalyst does nothing. The Thiele modulus and effectiveness factor quantify this and decide pellet size, shape and the location of active metal.
Key ideas
Effective diffusivity. Diffusion in pores is slower than in open fluid: De = D·εp/τp, where εp is pellet porosity and τp the tortuosity (typically 2–6). In very fine pores, or at low pressure, molecules hit walls more often than each other (Knudsen diffusion), and D is replaced by the Knudsen diffusivity or a combined value. Take τp and pore data from your data book or measurements.
Concentration profile. For a first-order reaction in a slab of half-thickness L with surface concentration CAs, a balance on a thin slice gives De·d²CA/dx² = k·CA. The solution is CA/CAs = cosh(φ·x/L)/cosh φ, with x measured from the centre. When φ is small the profile is flat; when φ is large the reactant is used up near the surface and the centre sees almost nothing.
Thiele modulus. φ = L·√(k/De) for a slab; for a sphere of radius R, φ = R·√(k/De) (some texts use the generalised length L = Vp/Sp = R/3 so that all shapes collapse onto one curve: MT = (Vp/Sp)·√(k/De)). It is the ratio of reaction rate to diffusion rate. k here is the first-order rate constant per unit pellet volume (s⁻¹).
Effectiveness factor. η = actual rate in the pellet / rate if the whole pellet were at CAs.
- η → 1 when φ is small (say φ < 0.4 for a slab): no diffusion limitation.
- η → 1/φ (slab) or 3/φ (sphere with φ = R√(k/De)) when φ is large: strong pore diffusion. In the strong-diffusion regime the rate is proportional to 1/(pellet size), so halving the pellet nearly doubles the rate per unit catalyst volume.
Consequences of strong pore diffusion.
- Apparent activation energy ≈ E/2, because η ∝ 1/√k.
- Apparent order n_app = (n + 1)/2 — a second-order reaction appears 1.5 order.
- Selectivity of series reactions suffers: the intermediate has a long path to escape.
Diagnosing from data — Weisz–Prater criterion. Using only observable quantities, C_WP = (−r'A,obs)·ρc·R²/(De·CAs) = η·φ². If C_WP ≪ 1, internal diffusion is negligible; if C_WP ≫ 1, it is severe. A simpler laboratory test: measure the rate with two pellet sizes; if the rate per unit mass does not change, there is no pore-diffusion limitation.
Design responses. Smaller pellets (at the cost of pressure drop), larger pores, egg-shell catalysts with the active metal near the surface, or shapes with thin walls (rings, trilobes).
Overall rate. With external film resistance as well (first order): −r''A = CAb/[1/kc + 1/(η·k·Vp/Sp)] on an external-area basis; or on a bed basis 1/k_o = 1/(kc·a) + 1/(η·k). Internal and external resistances are in series.
Formulas
De = D·εp/τp
- D: molecular (or Knudsen) diffusivity (m²/s); εp: porosity; τp: tortuosity.
φ = L·√(k/De) (slab, half-thickness L); η = tanh(φ)/φ
φ = R·√(k/De) (sphere, radius R); η = (3/φ²)·(φ·coth φ − 1)
- k: first-order rate constant per pellet volume (s⁻¹); L, R in m.
MT = (Vp/Sp)·√(k/De) (generalised modulus; sphere Vp/Sp = R/3)
η ≈ 1/φ (slab, φ > 3); η ≈ 3/φ (sphere, φ large)
−rA,obs = η·k·CAs (per pellet volume, first order)
C_WP = (−r'A,obs)·ρc·R²/(De·CAs) = η·φ²
- ρc: pellet density (kg/m³); r'A,obs: observed rate per catalyst mass (mol/kg·s).
E_app ≈ E/2, n_app = (n + 1)/2 (strong pore diffusion)
Worked examples
Example 1 (standard, slab). A flat catalyst plate of half-thickness 1.5 mm, sealed on its edges, carries a first-order reaction with k = 4 s⁻¹ and De = 1.0 × 10⁻⁶ m²/s. Find φ and η.
- √(k/De) = √(4/10⁻⁶) = 2000 m⁻¹.
- φ = L·√(k/De) = 1.5 × 10⁻³ × 2000 = 3.0.
- η = tanh(3.0)/3.0 = 0.9951/3.0 = 0.332.
φ = 3.0; η = 0.332 — two-thirds of the catalyst volume is wasted.
Example 2 (GATE level, sphere and pellet size). Spherical pellets of radius 3 mm carry a first-order reaction with k = 10 s⁻¹ and De = 1.0 × 10⁻⁶ m²/s. Find η, and the gain from using 1.5 mm pellets.
- φ = R·√(k/De) = 3 × 10⁻³ × √(10⁷) = 3 × 10⁻³ × 3162 = 9.49.
- η = (3/φ²)·(φ·coth φ − 1) = (3/90.0)·(9.49 × 1.000 − 1) = 0.0333 × 8.49 = 0.283.
- R = 1.5 mm: φ = 4.74; coth 4.74 = 1.0002; η = (3/22.5)·(4.74 − 1) = 0.1333 × 3.74 = 0.499.
- Rate per unit catalyst volume rises by 0.499/0.283 = 1.76.
η = 0.283 (3 mm) and 0.499 (1.5 mm) — halving the pellet size raises the rate by about 1.8 times, approaching the factor 2 expected for strong pore diffusion.
Common mistakes
- Mixing Thiele-modulus definitions (R vs R/3) with the wrong η formula.
- Using molecular diffusivity instead of effective diffusivity.
- Using a rate constant per mass of catalyst without multiplying by pellet density to get k per pellet volume.
- Concluding that raising temperature improves η — k rises faster than De, so φ increases and η falls.
- Forgetting the external film when the overall rate is asked.
For GATE CH
Typical questions: compute φ and η for a slab or sphere, use the asymptote η ≈ 1/φ or 3/φ, compare rates for two pellet sizes, apparent activation energy or order under strong pore diffusion, and the Weisz–Prater test. Practise tanh and coth values and keep the definitions consistent.
Quick check
- What is η for a slab with φ = 0.1?
- Under strong pore diffusion, what is the apparent activation energy if the true value is 120 kJ/mol?
- What apparent order does a second-order reaction show under strong pore diffusion?
- If pellet radius is halved in the strong-diffusion regime, by what factor does η change?
Answers: 1. η ≈ 1 (tanh 0.1/0.1 = 0.997); 2. about 60 kJ/mol; 3. 1.5; 4. it roughly doubles.
Interview questions
All Chemical Reaction Engineering interview questionsTry answering each one aloud before you open it.
1.What is the Thiele modulus and how is it used in chemical reaction engineering?Concept
The Thiele modulus is a dimensionless number that characterizes the relationship between reaction rate and diffusion rate within a porous catalyst pellet. It is used to determine whether a reaction is diffusion-limited or reaction-limited. A high Thiele modulus indicates that diffusion is the limiting step, while a low Thiele modulus suggests that the reaction rate is the limiting factor.
2.Explain the concept of the effectiveness factor in the context of catalyst pellets.Concept
The effectiveness factor is a dimensionless number that measures the efficiency of a catalyst pellet in facilitating a chemical reaction. It is defined as the ratio of the actual reaction rate within the pellet to the reaction rate if the entire pellet were at the surface concentration. An effectiveness factor of 1 indicates no diffusion limitations, while a value less than 1 indicates that diffusion is affecting the reaction rate.
3.How does diffusion in catalyst pellets affect the overall reaction rate?Concept
Diffusion in catalyst pellets can significantly affect the overall reaction rate. If the diffusion rate is slower than the reaction rate, it can lead to concentration gradients within the pellet, causing the reaction to be diffusion-limited. This means that the reaction rate is controlled by how fast reactants can diffuse into the pellet and products can diffuse out.
4.Why is it important to consider both the Thiele modulus and the effectiveness factor when designing a catalytic reactor?Application
Considering both the Thiele modulus and the effectiveness factor is crucial because they provide insights into the interplay between reaction kinetics and mass transfer limitations. The Thiele modulus helps identify whether the reaction is diffusion-limited or reaction-limited, while the effectiveness factor quantifies the extent of diffusion limitations. Together, they guide the design and optimization of catalyst pellets and reactors to ensure efficient operation.
5.What happens if the Thiele modulus is very high in a catalytic reaction?Application
If the Thiele modulus is very high, it indicates that the reaction is diffusion-limited. This means that the reactants cannot diffuse into the catalyst pellet fast enough to sustain the reaction at the surface rate. As a result, the effectiveness factor will be low, and the overall reaction rate will be significantly reduced compared to the rate at the surface concentration.
6.How can the effectiveness factor be improved in a diffusion-limited reaction?Application
The effectiveness factor rises as the Thiele modulus falls, so shorten the diffusion path or speed up diffusion: use smaller pellets or thin-walled shapes (rings, trilobes), make the pores larger or the pellet more porous to raise De, or put the active metal in a thin outer shell (egg-shell catalyst). Raising temperature does not help, because the rate constant grows much faster than the diffusivity, so φ increases and η falls. Smaller pellets increase bed pressure drop, which sets a practical lower limit.
7.Calculate the Thiele modulus for a first-order reaction with a rate constant k = 0.1 s⁻¹, effective diffusivity De = 1.0 × 10⁻⁵ m²/s, and pellet radius R = 0.01 m.Numerical
The Thiele modulus (φ) for a first-order reaction is calculated using the formula: φ = R * sqrt(k / De). Substituting the given values: φ = 0.01 m * sqrt(0.1 s⁻¹ / 1.0 × 10⁻⁵ m²/s) = 0.01 m * sqrt(10000 s⁻¹) = 0.01 m * 100 s⁻¹ = 1.0. Therefore, the Thiele modulus is 1.0.
8.For a catalyst pellet with an effectiveness factor of 0.5, what does this imply about the reaction and diffusion rates?Application
An effectiveness factor of 0.5 implies that the actual reaction rate within the catalyst pellet is half of what it would be if the entire pellet were at the surface concentration. This indicates that diffusion limitations are present, and the reaction rate is being affected by the rate at which reactants can diffuse into the pellet and products can diffuse out.
9.Explain how temperature affects the Thiele modulus and the effectiveness factor.Application
φ = L·√(k/De). The rate constant k rises exponentially with temperature (Arrhenius), while the effective diffusivity rises only weakly (roughly T^1.5 to T^1.75 for molecular diffusion, T^0.5 for Knudsen). So φ increases and η decreases as temperature rises; the reaction moves into the pore-diffusion regime. In that regime the observed rate still increases, but the apparent activation energy is only about half the true value.
10.If a catalyst pellet is too large, how does it affect the Thiele modulus and the effectiveness factor?Application
If a catalyst pellet is too large, it increases the Thiele modulus because the diffusion path length is longer, making the reaction more likely to be diffusion-limited. This can lead to a lower effectiveness factor, as the concentration gradients within the pellet become more pronounced, reducing the overall reaction rate compared to the surface rate.
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