🎓 Lesson 7
D4
Ternary Phase Diagrams and Tie-Line Construction
A ternary phase diagram is a triangular graph that shows how three liquids mix or separate into layers at different compositions and temperatures.
🎯 Learning Objectives
- ✓ Interpret composition points on a ternary diagram using triangular coordinates
- ✓ Construct and apply tie-lines to determine coexisting phase compositions in a partially miscible three-component system
- ✓ Calculate mass fractions of extract and raffinate phases using the lever rule on a ternary diagram
- ✓ Analyze how changes in solvent-to-feed ratio affect phase split and extraction efficiency
- ✓ Design a single-stage liquid–liquid extraction process for a metal-bearing aqueous feed using ternary equilibrium data
📖 Why This Matters
In mining and metallurgy, liquid–liquid extraction (LLE) is critical for recovering high-purity metals like copper, uranium, and cobalt from leach solutions. Ternary diagrams—especially for water–organic solvent–metal solute systems—guide engineers in selecting optimal solvent ratios, predicting phase separation, and avoiding emulsion formation or third-phase issues. Misreading tie-lines can lead to incomplete extraction, solvent carryover, or failed plant commissioning—costing millions in rework and downtime.
📘 Core Principles
Ternary diagrams use an equilateral triangle where each vertex represents 100% of one component (e.g., water, kerosene-based extractant, metal ion). Composition is read via parallel lines to edges (Gibbs triangle rules). In partially miscible systems (e.g., water–D2EHPA–kerosene), a two-phase region appears as a lens-shaped area bounded by a binodal curve; inside it, tie-lines connect equilibrium compositions of coexisting raffinate (aqueous-rich) and extract (organic-rich) phases. The lever rule quantifies phase amounts: mass of raffinate / mass of extract = length of tie-line segment opposite extract / length opposite raffinate. Temperature and pH strongly shift binodal boundaries—hence real-world diagrams are often built at 25°C and controlled pH.
📐 Lever Rule for Phase Mass Balance
The lever rule converts geometric distances on a ternary diagram into mass ratios of coexisting phases. It applies only along a tie-line and assumes negligible density differences between phases. Used after identifying raffinate (R) and extract (E) compositions and locating the overall mixture point (M) on the same tie-line.
Lever Rule (Mass Ratio)
m_R / m_E = |M−E| / |M−R|Computes mass ratio of raffinate (R) to extract (E) phases given overall mixture point M on a tie-line.
Variables:
| Symbol | Name | Unit | Description |
|---|---|---|---|
| m_R | Mass of raffinate phase | kg | Mass of the aqueous-rich phase after equilibrium contact |
| m_E | Mass of extract phase | kg | Mass of the organic-rich phase after equilibrium contact |
| M | Overall mixture composition | wt% | Composition (in wt%) of combined feed + solvent before phase separation |
| |M−E| | Distance from M to E | cm or arbitrary units | Length of line segment between mixture point M and extract phase point E on diagram |
Typical Ranges:
Copper SX circuits: 0.3 – 2.5
Uranium SX with D2EHPA: 0.6 – 1.8