🎓 Lesson 13
D5
Effectiveness Factor and Internal Diffusion Limitations
The effectiveness factor tells us how much slower a chemical reaction runs inside a catalyst pellet because reactants can’t easily reach all the active sites deep inside.
🎯 Learning Objectives
- ✓ Calculate the effectiveness factor for a first-order irreversible reaction in a spherical catalyst pellet
- ✓ Analyze when internal diffusion limitations dominate kinetics using the Thiele modulus
- ✓ Explain how pellet size, porosity, and diffusivity influence catalyst performance
- ✓ Apply Weisz–Prater criterion to diagnose diffusion-limited operation from experimental data
- ✓ Design catalyst pellet size to balance activity, pressure drop, and diffusion efficiency
📖 Why This Matters
In mining blasting, catalysts aren’t used—but understanding diffusion-limited kinetics is critical for designing *in-situ* chemical leaching systems (e.g., head underdesign reactors—leading to poor metal recovery, excessive reagent use, or unplanned downtime. This concept bridges lab-scale kinetics to field-scale reactor performance.
📘 Core Principles
Catalysts in mining-related processes (e.g., oxidative leaching, sulfuric acid regeneration, or bioleaching support media) are often porous solids. Reactants must diffuse into pores to access active sites; products must diffuse out. When diffusion is slow relative to reaction, concentration gradients form inside the pellet—lowering average reaction rate. The Thiele modulus (φ) quantifies this competition: high φ means diffusion limits performance. For spherical pellets, η depends on geometry and reaction order. First-order kinetics yield analytical solutions; higher orders require numerical methods. Effectiveness drops sharply beyond φ > ~3—signaling need for smaller pellets, higher porosity, or temperature adjustment.
📐 Effectiveness Factor for First-Order Spherical Pellet
For a first-order, irreversible reaction in an isothermal spherical catalyst pellet with uniform properties, η is analytically expressed using the Thiele modulus. This formula is foundational—it reveals how geometry and transport properties govern real-world catalyst efficiency.
Effectiveness Factor (η) — First-Order, Spherical Pellet
η = (1/φ) × coth(φ) − 1/φ²Exact analytical expression for η in isothermal, first-order, spherical catalysts.
Variables:
| Symbol | Name | Unit | Description |
|---|---|---|---|
| η | Effectiveness factor | dimensionless | Ratio of actual to intrinsic reaction rate |
| φ | Thiele modulus | dimensionless | As defined above |
Typical Ranges:
Kinetic regime (minimal diffusion control): 0.9 – 1.0
Moderate diffusion limitation: 0.3 – 0.9
Severe diffusion limitation: 0.01 – 0.3
💡 Worked Example
Problem: A copper oxide leaching catalyst pellet (radius = 2 mm) has effective diffusivity Dₑ = 1.2 × 10⁻⁹ m²/s and intrinsic first-order rate constant k = 0.08 s⁻¹. Calculate η.
1.
Step 1: Compute Thiele modulus φ = R √(k / Dₑ), where R = 0.002 m.
2.
Step 2: φ = 0.002 × √(0.08 / 1.2×10⁻⁹) = 0.002 × √(6.67×10⁷) ≈ 0.002 × 8167 = 16.33.
3.
Step 3: Use η = (1/φ) × coth(φ) − 1/φ². Since φ ≫ 3, approximate η ≈ 1/φ = 1/16.33 ≈ 0.061.
Answer:
The result is η = 0.061, which falls within the severely diffusion-limited range (η < 0.1). This indicates >90% of catalyst volume is underutilized—pellet size should be reduced or porosity increased.
🏗️ Real-World Application
At the Escondida copper mine (Chile), pilot-scale agitated tank leaching showed 40% lower Cu extraction than predicted by intrinsic kinetics. Post-mortem analysis revealed η ≈ 0.18 for MnO₂-catalyzed H₂SO₄/O₂ leaching of chalcocite—due to dense, low-porosity catalyst pellets (R = 3 mm, Dₑ = 0.8×10⁻⁹ m²/s, k = 0.05 s⁻¹). Switching to extruded 1-mm pellets raised η to 0.72 and boosted extraction by 28%, validating diffusion-aware catalyst design per SME Guideline G2021-04.