🎓 Lesson 12
D5
Tie-Line Construction and Lever Rule Applications
Tie-line construction is a graphical method used to find how a mixture splits into two liquid phases during extraction, like separating oil from water using a solvent.
🎯 Learning Objectives
- ✓ Construct tie-lines on ternary phase diagrams using equilibrium data
- ✓ Apply the lever rule to calculate mass fractions of raffinate and extract phases
- ✓ Analyze the effect of solvent choice on tie-line slope and extraction efficiency
- ✓ Design a single-stage extractor based on tie-line and lever rule calculations
📖 Why This Matters
In hydrometallurgical processing—such as gold recovery via cyanidation or copper solvent extraction—precise separation of valuable metals from impurities depends on predicting how components partition between two immiscible liquids. Tie-line construction and the lever rule are the foundational graphical tools that allow engineers to size equipment, optimize solvent flow rates, and avoid costly trial-and-error in plant commissioning.
📘 Core Principles
Liquid–liquid equilibrium (LLE) for three-component systems (e.g., metal-bearing aqueous phase + organic diluent + extractant) is represented on an equilateral triangular (ternary) diagram. Each vertex represents a pure component; any point inside represents a mixture composition. The binodal curve encloses the two-phase region; within it, every overall composition splits into two equilibrium phases connected by a tie-line. The direction and length of tie-lines reflect relative affinities (e.g., hydrophobicity of solute), while their density indicates miscibility gaps. The lever rule then quantifies phase amounts based on the inverse ratio of distances along the tie-line—rooted in mass conservation.
📐 Lever Rule for Phase Mass Distribution
The lever rule uses geometry on the tie-line to compute the relative masses of the raffinate (R) and extract (E) phases. Given an overall mixture point M lying on the tie-line between R and E, the mass ratio is inversely proportional to the segment lengths: mass_R / mass_E = length(M→E) / length(M→R). This follows directly from total mass and component mass balances.
💡 Worked Example
Problem: On a ternary diagram for CuSO₄–H₂O–LIX984 system, an overall feed composition M plots 3.2 cm from raffinate point R and 6.8 cm from extract point E along the tie-line. Total feed mass = 100 kg. Calculate masses of raffinate and extract phases.
1.
Step 1: Identify distances — d_ME = 6.8 cm, d_MR = 3.2 cm
2.
Step 2: Apply lever rule — mass_R / mass_E = d_ME / d_MR = 6.8 / 3.2 = 2.125
3.
Step 3: Let mass_E = x → mass_R = 2.125x; solve 2.125x + x = 100 → x = 100 / 3.125 = 32.0 kg
4.
Step 4: Therefore, mass_E = 32.0 kg, mass_R = 68.0 kg
Answer:
The extract phase mass is 32.0 kg and raffinate is 68.0 kg, consistent with typical industrial single-stage Cu SX where aqueous raffinate dominates volume.
🏗️ Real-World Application
At the Tenke Fungurume Mine (DRC), copper solvent extraction uses LIX84 in kerosene to recover Cu²⁺ from leach solution (≈2 g/L Cu, 180 g/L H₂SO₄, balance water). Ternary diagrams constructed from experimental LLE data show strongly sloped tie-lines indicating high Cu affinity for the organic phase. Engineers used tie-line endpoints and lever rule calculations to size mixer-settlers—ensuring >98% Cu transfer per stage while maintaining aqueous:organic phase ratio (A:O) at 2.5:1, validated against pilot-plant assays.
📋 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
📋 Food-Grade Citric Acid Purification via Liquid-Liquid Extraction
High viscosity broth, emulsion formation with tertiary amines, difficult phase separation