Effectiveness Factor Calculator

The Effectiveness Factor Calculator helps engineers and researchers estimate how effectively a catalyst is performing in a porous structure, such as a particle, slab, or cylinder. It compares the actual reaction rate inside the particle, where diffusion might slow down the process, to the reaction rate at the surface where no diffusion limitation exists.

This calculator belongs to the chemical reaction engineering calculator category and is mainly used in industries dealing with catalytic reactions, such as chemical manufacturing, pharmaceuticals, and petrochemicals.

The effectiveness factor (η) indicates whether internal diffusion is limiting the reaction rate. A value of 1 means the catalyst is fully effective (no diffusion limitations), while lower values show increasing resistance due to slow diffusion inside the particle.

formula of Effectiveness Factor Calculator

Formula for Effectiveness Factor (η):

η = (Actual reaction rate with diffusion limitations) / (Reaction rate without diffusion limitations)

Or, mathematically:

η = (integral over particle volume of [reaction rate at local concentration]) / (reaction rate at surface concentration * particle volume)

For a spherical particle and first-order reaction:

η = (3 / φ) * ( (1 / tanh(φ)) - (1 / φ) )

Where:

• η = Effectiveness Factor (dimensionless, 0 to 1)
• φ = Thiele Modulus, calculated as: φ = R * sqrt(k / D_eff)
• R = Radius of the catalyst particle (m or cm)
• k = First-order reaction rate constant (s⁻¹)
• D_eff = Effective diffusivity (m²/s or cm²/s)
• tanh(φ) = Hyperbolic tangent of φ

For Other Geometries:

Slab Geometry (Flat Plate Catalyst):
η = tanh(φ) / φ
φ = L * sqrt(k / D_eff)
L = Half-thickness of slab

Cylindrical Geometry:
η = (2 * I_1(φ)) / (φ * I_0(φ))
I_0 and I_1 = Modified Bessel functions of the first kind, order 0 and 1

For Non-First-Order Reactions:

φ_n = (R / 3) * sqrt( (n + 1) * k * C_s^(n-1) / (2 * D_eff) )
Where:

• C_s = Concentration at surface (mol/m³)
• n = Reaction order

Effectiveness Factor is then solved using numerical methods or charts.

Limiting Cases:

• If φ < 0.3 → η ≈ 1 (no diffusion resistance)
• If φ > 3 → η ≈ 3 / φ (reaction occurs only near surface for spheres)

General Reference Table

Example of Effectiveness Factor Calculator

Given:

• R = 0.002 m (2 mm)
• k = 0.1 s⁻¹
• D_eff = 1 × 10⁻⁹ m²/s

Step 1: Calculate φ
φ = R * sqrt(k / D_eff)
φ = 0.002 * sqrt(0.1 / 1e-9) ≈ 20

Step 2: Use the formula for spherical particles
η = (3 / φ) * ((1 / tanh(φ)) - (1 / φ))
tanh(20) ≈ 1, so
η ≈ (3 / 20) * (1 - 1/20) = 0.15 * 0.95 = 0.1425

So, the catalyst is only 14.25% effective due to internal diffusion resistance.

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