🎓 Lesson 1
D1
Getting Started with Mass Transfer and Separation Processes
Mass transfer is how substances move from one place to another—like sugar dissolving in coffee or water vapor rising from a mine ventilation shaft.
🎯 Learning Objectives
- ✓ Explain the physical mechanisms driving mass transfer (diffusion vs. convection) using Fick’s and Newton’s laws
- ✓ Calculate molecular diffusivity and convective mass transfer coefficients for typical mining-related systems
- ✓ Analyze concentration profiles across interfaces using steady-state diffusion models
- ✓ Apply equilibrium relationships (e.g., Henry’s law, distribution coefficients) to design simple leaching or gas scrubbing units
- ✓ Compare separation process selection criteria (efficiency, energy use, scalability) for ore processing applications
📖 Why This Matters
In mining, mass transfer isn’t abstract—it determines whether cyanide reaches gold particles in a heap leach pad, whether acid penetrates sulfide ore in bioleaching, or whether methane disperses safely in an underground mine. Poor mass transfer leads to incomplete recovery, hazardous gas accumulation, or environmental leakage. Mastering it means designing safer, more efficient, and more sustainable extraction and remediation systems.
📘 Core Principles
Mass transfer begins with gradients: just as heat flows from hot to cold, mass flows from high to low concentration. At the molecular level, diffusion dominates in stagnant or laminar zones (e.g., pore spaces in ore), governed by Fick’s first law. In moving fluids—like air in ventilation ducts or solution in agitated tanks—convection enhances transfer, described by dimensionless numbers (Sh, Re, Sc). Equilibrium relationships define phase boundaries (e.g., how much CO₂ dissolves in water at a given partial pressure), while rate-based models separate thermodynamic feasibility from kinetic reality—critical when scaling lab leach tests to field operations.
📐 Fick’s First Law of Diffusion
This fundamental equation quantifies steady-state molecular diffusion flux. It applies where concentration changes linearly over distance—common in porous media like tailings filters or mineral grain boundaries.
Fick’s First Law (Molecular Diffusion)
N_A = -D_AB \frac{dC_A}{dx}Quantifies molar flux of species A due to concentration gradient in stagnant medium.
Variables:
| Symbol | Name | Unit | Description |
|---|---|---|---|
| N_A | Molar flux of species A | mol/(m²·s) | Rate of A transferred per unit area perpendicular to direction x |
| D_AB | Binary diffusion coefficient | m²/s | Diffusivity of A in B; depends on T, P, and medium properties |
| dC_A/dx | Concentration gradient | mol/m⁴ | Change in concentration of A with distance x |
Typical Ranges:
Ions in aqueous solution (25°C): 1 × 10⁻¹⁰ to 2 × 10⁻⁹ m²/s
Gases in air (25°C): 1 × 10⁻⁵ to 2 × 10⁻⁵ m²/s
Organic vapors in porous ore: 1 × 10⁻⁸ to 5 × 10⁻⁸ m²/s
💡 Worked Example
Problem: A gold-bearing ore sample has a cyanide concentration gradient of −0.8 mol/m⁴ across a 2 mm diffusion path in a pore. The effective diffusivity of CN⁻ in the saturated pore solution is 1.2 × 10⁻⁹ m²/s. Calculate the molar flux of cyanide toward the gold surface.
1.
Step 1: Identify knowns: dC/dx = −0.8 mol/m⁴, D_AB = 1.2 × 10⁻⁹ m²/s
2.
Step 2: Apply Fick’s law: N_A = −D_AB × (dC/dx) = −(1.2 × 10⁻⁹) × (−0.8) = 9.6 × 10⁻¹⁰ mol/(m²·s)
3.
Step 3: Verify units and magnitude: This flux is typical for slow diffusion-controlled leaching—consistent with field observations where >70% of leach time is diffusion-limited.
Answer:
The molar flux is 9.6 × 10⁻¹⁰ mol/(m²·s), which falls within the typical range of 10⁻¹¹ to 10⁻⁹ mol/(m²·s) for aqueous ion diffusion in consolidated ore.
🏗️ Real-World Application
At the Pueblo Viejo gold mine (Dominican Republic), heap leaching efficiency dropped after rainfall saturated the ore堆. Engineers diagnosed reduced oxygen and cyanide mass transfer into deeper layers due to collapsed pore structure and increased liquid holdup. By introducing intermittent irrigation cycles and adding crushed limestone to maintain porosity, they restored convective–diffusive transport—increasing gold recovery by 12% and cutting cycle time by 3 weeks. This case underscores that mass transfer isn’t just chemistry—it’s geometry, flow, and geomechanics.
✏️ Student Exercise
A ventilation shaft carries air containing 0.5% (v/v) methane at 25°C and 1 atm. The shaft wall is lined with activated carbon. Using Henry’s law (H = 1.4 × 10⁵ atm·m³/mol), calculate the equilibrium dissolved methane concentration (mol/m³) at the carbon–air interface. Then estimate the convective mass transfer coefficient (k_c) using the Sherwood number correlation Sh = 0.023 × Re^0.8 × Sc^0.4, assuming Re = 8,500 and Sc = 0.62 (for CH₄ in air). Finally, compute the initial mass transfer flux if the bulk gas-phase concentration is twice the interfacial value.
🔧 Interactive Calculator
🔧 Open Mass Transfer and Separation Processes Calculator📋 Case Connection
📋 Ethanol-Water Separation in Biofuel Plant
High energy demand for azeotropic distillation; poor purity (<92%) in first-pass product