🎓 Lesson 16
D5
Solution-Diffusion Model for RO and Pervaporation
The solution-diffusion model explains how molecules cross a membrane by first dissolving into it and then moving through it like sugar spreading in a gel.
🎯 Learning Objectives
- ✓ Explain the physical basis of the solution-diffusion mechanism using molecular-level reasoning
- ✓ Calculate permeate flux and separation factor for RO and pervaporation systems using the solution-diffusion equations
- ✓ Analyze how feed composition, temperature, and membrane properties affect selectivity and flux
- ✓ Apply the model to compare RO desalination vs. pervaporation for ethanol/water dehydration
- ✓ Design a preliminary membrane stage by estimating required membrane area given target recovery and flux
📖 Why This Matters
Every day, over 100 million cubic meters of freshwater are produced globally via reverse osmosis—most relying on the solution-diffusion model to predict performance. In biorefineries, pervaporation membranes separate bioethanol from fermentation broth with >99.5% purity—enabling energy-efficient alternatives to distillation. Understanding this model isn’t just theoretical: it’s the engineering bedrock for sizing plants, troubleshooting fouling-induced flux decline, and selecting membranes for aggressive feeds like acid mine drainage or solvent-laden leachates—critical challenges in mining and hydrometallurgical operations.
📘 Core Principles
The model rests on two coupled phenomena: thermodynamic partitioning (solution) and kinetic transport (diffusion). First, components partition into the membrane according to their activity coefficients and polymer–solute interactions—governed by Flory–Huggins theory for polymers. Second, dissolved species diffuse at rates determined by molecular size, polarity, and membrane free volume; smaller, more condensable (higher S), and less hindered (higher D) molecules permeate faster. Critically, selectivity (α) arises not from size exclusion (as in UF/MF), but from differences in S/D products—e.g., water swells hydrophilic polyamide (high S_H₂O), while organics diffuse faster in hydrophobic PDMS. Temperature exponentially increases D but may reduce S—hence the trade-off in pervaporation optimization.
📐 Key Calculation
The solution-diffusion model expresses steady-state molar flux (J_i) of component i as proportional to its upstream–downstream chemical potential gradient, approximated by concentration difference across the dense film. For thin-film composite membranes, it simplifies to Fick’s first law with sorption boundary conditions.
💡 Worked Example
Problem: A polyamide RO membrane (thickness δ = 0.2 µm) treats seawater (C_feed = 35,000 mg/L NaCl ≈ 0.6 mol/L) at 25°C and ΔP = 55 bar. Given P_water = 4.2 × 10⁻⁷ m²/s and reflection coefficient σ = 0.95, calculate water flux J_w (L/m²·h) and salt passage SP (%).
1.
Step 1: Convert P_water to L/m²·bar·h: P_water = 4.2 × 10⁻⁷ m²/s × (3600 s/h) × (1000 L/m³) / (1 bar) = 1.512 L/m²·bar·h
2.
Step 2: Apply RO flux equation J_w = A(ΔP − σΔπ); estimate Δπ ≈ iMRT = 2 × 0.6 mol/m³ × 8.314 kPa·m³/mol·K × 298 K ≈ 2970 kPa = 29.7 bar → J_w = 1.512 × (55 − 0.95×29.7) = 1.512 × (55 − 28.2) = 1.512 × 26.8 ≈ 40.5 L/m²·h
3.
Step 3: Salt passage SP = exp(−J_w × δ / P_salt); assume P_salt ≈ 10⁻⁹ m²/s → P_salt (L/m²·bar·h) ≈ 0.0036; then SP = exp(−40.5 × 0.2×10⁻⁶ / 0.0036) = exp(−0.00225) ≈ 0.998 → SP ≈ 0.2%
Answer:
Water flux = 40.5 L/m²·h (within typical SWRO range of 20–45 L/m²·h); salt passage = 0.2% (meets ISO 15527:2018 < 1% requirement).
🏗️ Real-World Application
At the BHP Olympic Dam hydrometallurgical plant (South Australia), a pervaporation system using PDMS–PTFE composite membranes dehydrates spent electrolyte containing 10–15 wt% H₂SO₄ and 5–8 wt% Cu²⁺ prior to electrowinning. The solution-diffusion model guided membrane selection: PDMS offers high water solubility (S_H₂O ≈ 0.04 cm³(STP)/cm³·bar) and low water diffusivity hindrance in acidic media, enabling >99.9% water removal at 80°C with 3× lower specific energy vs. vacuum distillation—reducing corrosion-related maintenance and acid loss by 40% (AusIMM Proc., 2021).
🔧 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