🎓 Lesson 2
D2
Understanding Molecular Diffusion: Fick’s Laws in Steady & Unsteady State
Molecular diffusion is how molecules spread out from areas where they’re crowded to areas where they’re sparse—like ink spreading in still water.
🎯 Learning Objectives
- ✓ Calculate diffusion flux using Fick’s First Law for steady-state systems
- ✓ Apply Fick’s Second Law to analyze transient concentration profiles in porous rock or blast fumes
- ✓ Explain the physical significance of diffusion coefficient values across common mining-related gases and solutes
- ✓ Design sampling intervals for monitoring post-blast gas dispersion based on unsteady-state diffusion time scales
- ✓ Analyze how temperature and porosity affect effective diffusion coefficients in fractured rock media
📖 Why This Matters
In underground mining and blasting operations, toxic gases (e.g., NO₂, CO) and explosive residues diffuse through fractured rock, ventilation ducts, and soil. Understanding *how fast* and *how far* these species spread—via molecular diffusion—is critical for designing safe evacuation timelines, predicting fume clearance, evaluating containment of leachate plumes from heap leaching, and modeling solvent transport in in-situ recovery. Ignoring diffusion dynamics can lead to underestimating exposure risks or overdesigning ventilation systems—both costly and dangerous.
📘 Core Principles
Diffusion arises from Brownian motion: molecules move randomly but net flow occurs down a concentration gradient. Fick’s First Law (1855) states that the diffusive flux J is proportional to the negative concentration gradient (J = −D·dC/dx), valid only when concentration doesn’t change with time (steady state). Fick’s Second Law (∂C/∂t = D·∂²C/∂x²) extends this to dynamic systems—essential for modeling gas breakthrough after blasting or contaminant migration in tailings. In porous media like blasted rock, the *effective diffusion coefficient* (D_eff) accounts for tortuosity and saturation, often reduced 3–10× versus free-air D. Transient solutions use error functions (for semi-infinite domains) or series expansions (for finite slabs)—key tools for predicting arrival times of hazardous species.
📐 Fick’s First and Second Laws
Fick’s First Law gives instantaneous flux magnitude; Fick’s Second Law predicts concentration evolution. For planar diffusion into a semi-infinite medium with constant surface concentration, the analytical solution uses the error function: C(x,t) = C_s · [1 − erf(x / (2√(Dt)))]. This is widely used to estimate time-to-peak exposure at a given distance from a gas source.
💡 Worked Example
Problem: After a blast, nitrogen dioxide (NO₂) is released at the face with surface concentration C_s = 10 ppm. Estimate time required for NO₂ to reach 1 ppm at x = 2 m into fractured sandstone (D_eff = 1.2 × 10⁻⁶ m²/s).
1.
Step 1: Use normalized concentration ratio: C/C_s = 1/10 = 0.1 → so 1 − erf(z) = 0.1 → erf(z) = 0.9.
2.
Step 2: From error function tables or inverse erf: z ≈ 1.163. Then solve z = x/(2√(Dt)) → t = x²/(4·D·z²).
3.
Step 3: Plug in: t = (2)² / (4 × 1.2×10⁻⁶ × 1.163²) = 4 / (4 × 1.2×10⁻⁶ × 1.353) ≈ 4 / (6.494×10⁻⁶) ≈ 615,800 s ≈ 7.1 days.
Answer:
The model predicts ~7.1 days for NO₂ to reach 1 ppm at 2 m depth—highlighting why ventilation (not just diffusion) dominates short-term hazard control.
🏗️ Real-World Application
At the Escondida copper mine (Chile), post-blast NO₂ monitoring in stopes revealed delayed concentration peaks at 3–5 m behind the blast face—beyond immediate ventilation influence. Engineers used Fick’s Second Law with site-calibrated D_eff (1.4 × 10⁻⁶ m²/s in oxidized porphyry) to validate sensor placement and adjust ‘safe re-entry’ protocols. Model-predicted breakthrough times matched field data within ±12%, enabling reduction of mandatory waiting periods from 24 to 18 hours—without compromising safety—by confirming diffusion-limited transport dominates beyond the first 10 m.