🎓 Lesson 5
D3
Estimating Mass Transfer Coefficients from Dimensionless Numbers
It's a way to predict how fast a substance (like a gas or chemical) moves between two phases—such as air and water—using easy-to-calculate, unitless numbers instead of complex lab experiments.
🎯 Learning Objectives
- ✓ Calculate the Sherwood number from given Reynolds and Schmidt numbers using established correlations
- ✓ Apply dimensionless correlations to estimate mass transfer coefficients for gas–liquid and solid–fluid systems in mining-related contexts (e.g., leaching, ventilation, dust suppression)
- ✓ Analyze the impact of flow regime (laminar vs. turbulent) and fluid properties on mass transfer efficiency
- ✓ Explain the physical meaning of Re, Sc, and Sh and their interdependence in interphase transport
- ✓ Design a preliminary leaching column or scrubber by selecting appropriate operating conditions to achieve target mass transfer rates
📖 Why This Matters
In mining operations, mass transfer governs critical processes: cyanide leaching of gold ores, acid mist absorption in ventilation scrubbers, and dust capture in wet suppression systems. Guessing transfer rates leads to underdesigned equipment (poor recovery, unsafe air quality) or overspending on oversized units. Dimensionless numbers let engineers reliably scale lab data to field systems—no trial-and-error, no costly pilot plants. This skill turns theoretical transport into predictable, compliant, and economical process design.
📘 Core Principles
Interphase mass transfer depends on three physical regimes: molecular diffusion (governed by concentration gradients), convective transport (driven by fluid motion), and interfacial resistance. Dimensionless numbers collapse these influences into scalable ratios: Reynolds (Re) compares inertial to viscous forces—indicating flow regime; Schmidt (Sc) compares momentum diffusivity (viscosity) to mass diffusivity—revealing how easily molecules move relative to fluid motion; Sherwood (Sh) directly relates the observed mass transfer coefficient to molecular diffusion—acting as the dimensionless 'gain' of the system. Correlations like Sh = 0.023 Re^0.8 Sc^0.33 (for turbulent pipe flow) encode decades of experimental validation and allow rapid estimation across geometries—from heap leach pads to duct-mounted scrubbers.
📐 Key Correlation: Turbulent Flow in Pipes & Packed Beds
The most widely applied correlation for turbulent flow is the Colburn analogy-based power-law: Sh = a · Re^b · Sc^c. Constants (a, b, c) depend on geometry and flow regime. For turbulent flow in pipes or packed beds (common in leachate distribution or gas cleaning), the classic form is Sh = 0.023 Re^0.8 Sc^0.33. This assumes fully developed turbulent flow (Re > 10,000), moderate Sc (0.6–1000), and smooth surfaces—conditions often met in mining hydrometallurgical and ventilation systems.
💡 Worked Example
Problem: Estimate the liquid-phase mass transfer coefficient k_L for cyanide solution flowing through a 0.3 m diameter leachate distribution pipe at 1.2 m/s. Solution properties: ρ = 1020 kg/m³, μ = 1.1 × 10⁻³ Pa·s, D_AB = 1.8 × 10⁻⁹ m²/s.
1.
Step 1: Calculate Reynolds number: Re = ρ·v·D/μ = (1020)(1.2)(0.3)/(1.1×10⁻³) ≈ 335,455 → turbulent flow confirmed.
2.
Step 2: Calculate Schmidt number: Sc = μ/(ρ·D_AB) = (1.1×10⁻³)/[(1020)(1.8×10⁻⁹)] ≈ 599.
3.
Step 3: Apply correlation: Sh = 0.023 × (335455)^0.8 × (599)^0.33 ≈ 0.023 × 27,120 × 8.42 ≈ 5,250.
4.
Step 4: Solve for k_L: k_L = (Sh × D_AB)/D = (5250 × 1.8×10⁻⁹)/0.3 ≈ 3.15×10⁻⁵ m/s.
Answer:
The estimated liquid-phase mass transfer coefficient is 3.15×10⁻⁵ m/s, consistent with typical values for turbulent aqueous leach solutions (10⁻⁶ – 10⁻⁴ m/s).
🏗️ Real-World Application
At the Goldstrike Mine (Nevada), engineers redesigned the carbon-in-pulp (CIP) circuit after observing low gold adsorption rates. By calculating Sh for pulp flow in agitated tanks using measured Re and Sc, they identified laminar boundary layer limitations near activated carbon particles. They increased agitation energy (raising Re) and added surfactant to reduce interfacial resistance—boosting k_L by 40% and increasing overall recovery from 89% to 94%. This avoided capital expense of new tanks and was validated via tracer studies aligned with Sh–Re–Sc predictions.