🎓 Lesson 3
D2
Fick’s Laws & Stefan-Maxwell Equations Applied to Separation
Fick’s Laws describe how substances like gases or dissolved chemicals spread out from areas where they’re concentrated to areas where they’re less concentrated, like how perfume spreads across a room.
🎯 Learning Objectives
- ✓ Calculate steady-state diffusive flux using Fick’s First Law for binary gas or liquid systems
- ✓ Analyze transient concentration profiles using Fick’s Second Law with appropriate boundary conditions
- ✓ Explain the physical significance of Stefan-Maxwell coupling terms in multicomponent diffusion
- ✓ Apply Stefan-Maxwell equations to design a countercurrent gas–liquid absorber for acid gas removal in metallurgical off-gas treatment
- ✓ Compare predictions from Fickian vs. Stefan-Maxwell models for a ternary electrolyte solution in heap leaching
📖 Why This Matters
In mining and metallurgy, diffusion governs critical separation processes: oxygen transport into sulfide ore piles during bioleaching, cyanide diffusion into gold-bearing ore in vat leaching, and volatile metal chloride transport in chlorination circuits. Misapplying Fick’s Laws — e.g., assuming binary behavior in multicomponent leach solutions — leads to underdesigned reactors, poor metal recovery, or hazardous gas accumulation. Understanding when and how to upgrade from Fick to Stefan-Maxwell is not academic: it’s the difference between 85% and 94% Cu recovery in a pressure leach circuit.
📘 Core Principles
Start with Fick’s First Law: diffusion is driven solely by concentration gradients in dilute, ideal, binary systems. Then introduce limitations — real leach solutions contain H⁺, Cl⁻, Cu²⁺, Fe³⁺, SO₄²⁻, and O₂ simultaneously; their mutual interactions invalidate simple linear proportionality. This motivates the Stefan-Maxwell framework, where each species’ flux depends on *all* chemical potential gradients via a matrix of binary diffusivities (Ãᵢⱼ). The theory emphasizes thermodynamic consistency: fluxes must satisfy both mass conservation and the requirement that net force on each species vanishes at steady state. Finally, link to practical approximations — the ‘pseudo-binary’ approach used in industry software (e.g., OLI Stream Analyzer) retains Stefan-Maxwell rigor while enabling tractable computation.
📐 Key Calculation
Fick’s First Law quantifies steady-state diffusion; Stefan-Maxwell equations resolve coupled fluxes in multicomponent mixtures. Use Fick’s Law for preliminary sizing; switch to Stefan-Maxwell when >2 major dissolved species coexist at >0.1 mol/kg or when ionic strength exceeds 1 M — common in chloride leaching of PGM ores.
💡 Worked Example
Problem: A cyanide solution (0.005 M NaCN) diffuses through a 2 mm stagnant boundary layer into a gold ore particle. The bulk CN⁻ concentration is 0.005 M; at the mineral surface it drops to 1×10⁻⁶ M due to reaction. The binary diffusivity of CN⁻ in water at 25°C is 1.6×10⁻⁹ m²/s. Calculate the diffusive flux of CN⁻.
1.
Step 1: Identify knowns — C₁ = 0.005 mol/m³, C₂ = 0.000001 mol/m³, Δx = 0.002 m, D_AB = 1.6×10⁻⁹ m²/s
2.
Step 2: Apply Fick’s First Law: J_A = −D_AB × (dC_A/dx) ≈ −D_AB × (C₂ − C₁)/Δx
3.
Step 3: Compute: J_A = −(1.6×10⁻⁹) × (1×10⁻⁶ − 0.005)/0.002 = −(1.6×10⁻⁹) × (−2.4995) ≈ 4.0×10⁻⁹ mol/(m²·s)
Answer:
The diffusive flux is 4.0×10⁻⁹ mol/(m²·s), well within typical cyanide diffusion-limited leach rates of 10⁻⁹–10⁻⁸ mol/(m²·s).
🏗️ Real-World Application
At the Kupol Gold Mine (Russia), heap leaching performance stalled at ~72% Au recovery despite optimized irrigation. Speciation modeling revealed high Ca²⁺ and SO₄²⁻ concentrations (>0.8 M) suppressed CN⁻ activity and induced strong Stefan-Maxwell coupling — CN⁻ flux was reduced 3.2× relative to Fickian prediction due to counter-diffusion drag from CaSO₄ ion pairs. Engineers recalibrated the leach pad irrigation rate and added pH-controlled CaO dosing to reduce sulfate precipitation, restoring CN⁻ effective diffusivity and lifting recovery to 89%. This case is documented in the 2021 SME Annual Meeting Proceedings (Paper 11B-07).