🎓 Lesson 3
D2
Calculating Diffusion Fluxes Using Fick’s First Law
Diffusion flux is how much of a substance moves through a material per second due to a concentration difference — like how quickly smoke spreads from a campfire into clean air.
🎯 Learning Objectives
- ✓ Calculate diffusion flux across a rock matrix given concentration gradient and diffusion coefficient
- ✓ Analyze how porosity and tortuosity affect effective diffusion coefficients in fractured ore bodies
- ✓ Explain the physical meaning of each term in Fick’s First Law using dimensional analysis
- ✓ Apply Fick’s First Law to estimate contaminant migration rates in mine backfill or leach pad liners
📖 Why This Matters
In mining, understanding diffusion is critical for predicting cyanide transport in heap leaching, acid migration in waste rock piles, and radon gas exhalation from uranium tailings. Underestimating diffusion flux can lead to environmental non-compliance, groundwater contamination, or inefficient metal recovery — making Fick’s First Law not just theoretical, but a regulatory and operational necessity.
📘 Core Principles
Diffusion arises from random molecular motion (Brownian motion) driving net movement from high to low concentration. Fick’s First Law formalizes this as a linear response: flux magnitude depends on how steeply concentration changes in space (the gradient) and how easily molecules move through the medium (diffusivity). In porous geological media, the 'effective' diffusion coefficient accounts for reduced cross-sectional area (porosity, φ) and winding pathways (tortuosity, τ), yielding D_eff = D₀·φ/τ. Steady-state assumption means concentration at any point doesn’t change over time — essential for long-term seepage modeling in containment systems.
📐 Key Calculation
Fick’s First Law expresses diffusion flux J (mass per unit area per unit time) as proportional to the concentration gradient dC/dx. The negative sign indicates flux direction opposes increasing concentration. In mining applications, it's applied one-dimensionally across liners, backfill, or weathered rock zones.
💡 Worked Example
Problem: A copper leach pad liner has a 2.5 mm thick HDPE geomembrane. Cyanide concentration drops from 120 mg/L on the leach solution side to 0.8 mg/L on the groundwater side. The diffusion coefficient of CN⁻ in HDPE is 1.2 × 10⁻¹² m²/s. Calculate the steady-state diffusion flux.
1.
Step 1: Convert thickness to meters: 2.5 mm = 0.0025 m; compute concentration gradient: (0.8 − 120) mg/L / 0.0025 m = −47,680 mg/(L·m)
2.
Step 2: Convert gradient to SI units: 1 mg/L = 1 g/m³ = 1 kg/m³ × 10⁻³ → −47,680 mg/(L·m) = −47.68 kg/(m⁴) [since 1 mg/L = 10⁻³ kg/m³]
3.
Step 3: Apply Fick’s Law: J = −D·dC/dx = −(1.2 × 10⁻¹² m²/s) × (−47.68 kg/m⁴) = 5.72 × 10⁻¹¹ kg/(m²·s)
4.
Step 4: Convert to practical unit: 5.72 × 10⁻¹¹ kg/(m²·s) × (3600 s/h) × (24 h/day) × (10⁶ mg/kg) ≈ 0.0049 mg/(m²·day)
Answer:
The diffusion flux is 5.72 × 10⁻¹¹ kg/(m²·s), equivalent to ~0.0049 mg/(m²·day) — well below the EPA-recommended limit of 0.1 mg/(m²·day) for liner systems.
🏗️ Real-World Application
At the Goldstrike Mine (Nevada), engineers used Fick’s First Law to model cyanide breakthrough through compacted clay liners beneath heap leach pads. By measuring in-situ concentration gradients across liner cores and calibrating D_eff using tracer tests, they predicted 50-year cumulative loading and confirmed liner integrity met NRC 10 CFR Part 40 Appendix A requirements — avoiding $12M in potential remediation costs.
📋 Case Connection
📋 CO₂ Capture from Flue Gas using Amine Absorption
Low CO₂ partial pressure (~0.15 bar); amine degradation and solvent carryover