🎓 Lesson 2
D2
Mass Transfer Fundamentals: Diffusion, Convection & Interphase Transport
Mass transfer is how substances like gases, liquids, or dissolved solids move from one place to another — like smoke spreading in air or salt dissolving in water.
🎯 Learning Objectives
- ✓ Calculate molecular diffusion flux using Fick’s first law for gaseous and liquid systems
- ✓ Analyze convective mass transfer coefficients using dimensionless numbers (Sh, Re, Sc)
- ✓ Design interphase mass transfer rates for gas–liquid systems (e.g., CO₂ scrubbing in mine ventilation)
- ✓ Explain the physical significance of boundary layers and resistance-in-series models in multi-phase separations
- ✓ Apply equilibrium relationships (e.g., Henry’s law) to predict driving force in interphase transport
📖 Why This Matters
In mining and metallurgical engineering, mass transfer governs critical operations: cyanide leaching of gold ores, acid mist capture in heap leach pads, VOC removal from underground ventilation air, and SO₂ scrubbing in smelter off-gas treatment. Poor understanding leads to inefficient reagent use, environmental non-compliance, or catastrophic failures—like acid fog formation in bioleach ducts or inadequate H₂S removal in confined spaces. Mastering diffusion, convection, and interphase transport enables engineers to design safer, greener, and more economical separation systems.
📘 Core Principles
Mass transfer occurs via three primary mechanisms: (1) Molecular diffusion — random thermal motion driving net movement from high to low concentration (Fick’s law); (2) Convective mass transfer — enhanced transport due to bulk fluid flow (laminar or turbulent), modeled using mass transfer coefficients (kₗ, k₉); and (3) Interphase transport — simultaneous diffusion across a phase boundary (e.g., gas–liquid), requiring equilibrium thermodynamics (Henry’s law, Raoult’s law) and resistance-in-series analysis. The overall rate is limited by the slowest step — often the stagnant film near the interface. Dimensionless numbers (Sherwood, Reynolds, Schmidt) unify scaling across lab, pilot, and full-scale equipment.
📐 Fick’s First Law of Diffusion
Fick’s first law quantifies steady-state molecular diffusion flux in one dimension. It applies to gases (e.g., O₂ ingress into tailings ponds) and liquids (e.g., CN⁻ diffusion into gold-bearing pyrite grains). Valid when concentration gradient is linear and no bulk flow dominates.
Fick’s First Law (1D)
N_A = -D_{AB} \frac{dC_A}{dx}Molar diffusive flux of species A in binary mixture AB due to concentration gradient.
Variables:
| Symbol | Name | Unit | Description |
|---|---|---|---|
| N_A | Molar flux of species A | mol/m²·s | Rate of A transported per unit area perpendicular to x-direction. |
| D_{AB} | Binary diffusion coefficient | m²/s | Measure of mobility of A in B; depends on temperature, pressure, and molecular size. |
| dC_A/dx | Concentration gradient of A | mol/m⁴ | Change in molar concentration of A per unit distance in x-direction. |
Typical Ranges:
O₂ in water at 25°C: 1.8 – 2.4 × 10⁻⁹ m²/s
CO₂ in air at 25°C: 1.6 × 10⁻⁵ m²/s
💡 Worked Example
Problem: A 10-mm-thick water film separates air containing 21% O₂ (gas phase) from anoxic groundwater. At 25°C, the aqueous O₂ concentration at the air–water interface is 8.3 mg/L, dropping linearly to 0.2 mg/L at the film’s base. The binary diffusion coefficient of O₂ in water is 2.1 × 10⁻⁹ m²/s. Calculate the diffusive O₂ flux (kg/m²·s).
1.
Step 1: Convert concentrations to mol/m³: 8.3 mg/L = 8.3 g/m³ → ÷32 g/mol = 0.259 mol/m³; 0.2 mg/L = 0.00625 mol/m³.
2.
Step 2: Compute concentration gradient: dC/dx = (0.00625 − 0.259) mol/m³ ÷ 0.01 m = −25.275 mol/m⁴.
3.
Step 3: Apply Fick’s law: N_A = −D_AB × dC/dx = −(2.1×10⁻⁹)(−25.275) = 5.31×10⁻⁸ mol/m²·s → ×0.032 kg/mol = 1.70×10⁻⁹ kg/m²·s.
Answer:
The diffusive O₂ flux is 1.70 × 10⁻⁹ kg/m²·s — consistent with typical natural attenuation rates in saturated zones (10⁻¹⁰ to 10⁻⁸ kg/m²·s).
🏗️ Real-World Application
At the Boddington Gold Mine (Western Australia), a packed-bed CO₂ absorber treats 120,000 m³/h of diesel exhaust air from underground haulage to meet WA EPA limit of <5,000 ppm CO₂. Engineers used two-film theory (Higbie penetration model) and Sherwood–Reynolds–Schmidt correlations to size the column and select MEA concentration. Measured kₗ was 4.2 × 10⁻⁵ m/s (liquid-side), and k₉ was 1.8 × 10⁻³ m/s (gas-side), confirming liquid-phase resistance dominated — leading to optimized irrigation rate and packing geometry. Post-commissioning data showed >92% removal efficiency, validating the interphase transport model.