🎓 Lesson 25
D5
Mass Transfer Fundamentals Quiz
Mass transfer is how substances like gases, liquids, or dissolved solids move from one place to another due to differences in concentration — like sugar dissolving and spreading evenly in a cup of tea.
🎯 Learning Objectives
- ✓ Calculate molar flux using Fick’s first law for steady-state diffusion in gases and liquids
- ✓ Analyze concentration profiles across interfaces in gas–liquid systems using two-film theory
- ✓ Design minimum solvent flow rate for a given absorption task using equilibrium and operating line analysis
- ✓ Apply overall mass transfer coefficients (K_G, K_L) to size packed absorption columns
- ✓ Explain the physical significance of dimensionless numbers (Sh, Sc, Re) in predicting mass transfer rates
📖 Why This Matters
In mining and blasting engineering, mass transfer governs critical downstream operations: leaching of gold from crushed ore (cyanide diffusion into particle pores), acid mist scrubbing in ventilation systems, dust suppression via water film formation, and VOC emissions control from explosives storage. Misjudging mass transfer rates leads to inefficient reagent use, environmental noncompliance, or unsafe airborne contaminant buildup — directly impacting safety, cost, and sustainability.
📘 Core Principles
Mass transfer begins with molecular diffusion (Fick’s laws), where species migrate down concentration gradients. In real equipment, bulk fluid motion (convection) dominates transport, but resistance persists at phase boundaries — described by two-film theory. The overall process is quantified using mass transfer coefficients (k_G, k_L) and equilibrium relationships (e.g., Henry’s law). Dimensionless numbers link lab-scale correlations to field-scale equipment: Sherwood (Sh) represents total mass transfer, Schmidt (Sc) reflects fluid diffusivity vs. viscosity, and Reynolds (Re) characterizes flow regime. Understanding these layers — molecular → interfacial → system-level — enables rational design rather than empirical trial-and-error.
📐 Fick’s First Law of Diffusion
Fick’s first law defines the diffusive molar flux of a species A in a stagnant medium under steady-state conditions. It is foundational for estimating diffusion-controlled rates in leaching, membrane separation, and pore-scale transport in rock matrices.
💡 Worked Example
Problem: Calculate the molar flux of cyanide ion (CN⁻) diffusing through a 2-mm-thick stagnant aqueous film separating a high-concentration leach solution (C_A1 = 0.05 mol/m³) and a low-concentration raffinate (C_A2 = 0.002 mol/m³). The binary diffusion coefficient D_AB = 1.8 × 10⁻⁹ m²/s at 25°C.
1.
Step 1: Identify knowns — C_A1 = 0.05 mol/m³, C_A2 = 0.002 mol/m³, Δx = 0.002 m, D_AB = 1.8 × 10⁻⁹ m²/s
2.
Step 2: Apply Fick’s first law: N_A = −D_AB × (dC_A/dx) ≈ −D_AB × (C_A2 − C_A1)/Δx
3.
Step 3: Compute: N_A = −(1.8 × 10⁻⁹) × (0.002 − 0.05)/0.002 = −(1.8 × 10⁻⁹) × (−24.9) = 4.48 × 10⁻⁸ mol/(m²·s)
Answer:
The molar flux is 4.48 × 10⁻⁸ mol/(m²·s), which falls within the typical range for aqueous ionic diffusion (10⁻⁹ to 10⁻⁷ mol/(m²·s)).
🏗️ Real-World Application
At the Porgera Gold Mine (Papua New Guinea), heap leaching efficiency dropped after rain infiltration increased solution viscosity and reduced O₂ diffusion into deeper ore zones. Engineers applied two-film theory to quantify the liquid-phase mass transfer resistance (k_L) and redesigned drip emitters to increase turbulence — raising k_L by 40% and restoring cyanide oxidation kinetics. Post-implementation, gold recovery improved from 68% to 79% over six months, validating model-based intervention.
✏️ Student Exercise
A venturi scrubber treats blast fumes containing 120 ppmv NO₂. The gas enters at 85°C and 1 atm, flowing at 2.5 m³/s. Liquid water flows counter-currently at 0.8 kg/s. Using Henry’s constant H = 1.1 × 10⁵ atm·m³/mol (at 85°C) and an overall gas-phase mass transfer coefficient K_G = 0.025 mol/(m²·s·atm), estimate the required interfacial area (A) to achieve 92% removal. Assume equilibrium is governed by p_A = H·x_A and dilute conditions apply.
🔧 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
📋 CO₂ Capture from Flue Gas using Amine Absorption
Low CO₂ partial pressure (~0.15 bar); amine degradation and solvent carryover