🎓 Lesson 3
D2
Cubic EOS Derivation and Physical Interpretation
A cubic equation of state is a mathematical formula that predicts how gases and liquids behave under different pressures, temperatures, and volumes—like a 'weather forecast' for fluids in industrial processes.
🎯 Learning Objectives
- ✓ Derive the Peng–Robinson cubic EOS from fundamental thermodynamic principles
- ✓ Explain the physical meaning of the a and b parameters in terms of molecular attraction and excluded volume
- ✓ Calculate compressibility factor (Z) and phase-split compositions for a binary hydrocarbon mixture at reservoir conditions
- ✓ Analyze the limitations of cubic EOS in predicting liquid-phase densities and high-pressure phase behavior
- ✓ Apply EOS-based flash calculations to design separator operating conditions in mineral processing gas handling systems
📖 Why This Matters
In mining and metallurgical process simulation—such as modeling acid-gas scrubbing, CO₂ capture from smelter off-gases, or solvent extraction vent streams—accurate fluid property predictions are essential for equipment sizing, safety analysis, and energy optimization. Cubic EOS form the backbone of commercial simulators (Aspen HYSYS, ChemCAD) used daily by process engineers. Misapplying them leads to oversized compressors, failed dew-point control, or unsafe pressure relief design—costing millions in rework and downtime.
📘 Core Principles
Cubic EOS originate from the virial expansion and molecular thermodynamics: they replace the ideal gas assumption with physically interpretable correction terms. The 'a' parameter quantifies intermolecular attraction (energy-driven), while 'b' represents the finite molecular volume (size-driven). Unlike linear or quadratic models, the cubic form allows up to three real roots in Z—enabling coexistence of vapor and liquid phases via Maxwell’s equal-area rule. SRK and PR improve upon van der Waals by introducing temperature-dependent 'a(T)' functions calibrated to vapor pressure data, making them suitable for hydrocarbons, acid gases (CO₂, H₂S), and polar solvents common in hydrometallurgy.
📐 Peng–Robinson Equation of State
The Peng–Robinson (PR) EOS is the industry-standard cubic EOS for process simulation in mineral and metallurgical engineering due to its accuracy for nonpolar and moderately polar mixtures. It expresses pressure as a function of temperature, molar volume, and composition-dependent parameters.
Peng–Robinson EOS
P = \frac{RT}{v - b} - \frac{a(T)}{v(v + b) + b(v - b)}Pressure as a function of molar volume v, temperature T, and composition-dependent parameters a(T) and b.
Variables:
| Symbol | Name | Unit | Description |
|---|---|---|---|
| P | Absolute pressure | Pa | System pressure |
| R | Universal gas constant | J/(mol·K) | 8.314462618 J/(mol·K) |
| T | Absolute temperature | K | System temperature |
| v | Molar volume | m³/mol | Volume per mole of mixture |
| a(T) | Temperature-dependent attraction parameter | Pa·m⁶/mol² | Cohesive energy term scaled by α(T) |
| b | Covolume | m³/mol | Effective molecular excluded volume |
Typical Ranges:
Hydrocarbon processing: a = 10⁴–10⁶ Pa·m⁶/mol²; b = 10⁻⁵–10⁻⁴ m³/mol
CO₂-rich systems: a ≈ 0.23–0.25 Pa·m⁶/mol² at 300 K; b ≈ 2.97×10⁻⁵ m³/mol
💡 Worked Example
Problem: Calculate the compressibility factor Z for pure propane at 350 K and 10 bar using the Peng–Robinson EOS. Critical properties: Tc = 369.8 K, Pc = 42.48 bar, ω = 0.152.
1.
Step 1: Compute critical parameters → Tr = 350 / 369.8 = 0.946; α(Tr) = [1 + 0.37464 + 1.54226ω − 0.26992ω²] × (1 − √Tr)² = [1 + 0.37464 + 1.54226×0.152 − 0.26992×(0.152)²] × (1 − √0.946)² ≈ 0.864
2.
Step 2: Calculate a = 0.45724 × R²Tc² / Pc × α = 0.45724 × (8.314)² × (369.8)² / (4.248×10⁶) × 0.864 ≈ 9.17×10⁵ Pa·m⁶/mol²; b = 0.07780 × RTc / Pc = 0.07780 × 8.314 × 369.8 / (4.248×10⁶) ≈ 5.62×10⁻⁵ m³/mol
3.
Step 3: Solve cubic in Z: Z³ − (1 − B)Z² + (A − 2B − 3B²)Z − AB − B² − B³ = 0, where A = aP/(R²T²) = 0.254, B = bP/(RT) = 0.0327 → Roots: Z₁ ≈ 0.82 (liquid), Z₂ ≈ 0.94 (vapor); select stable root based on fugacity test → Z ≈ 0.94
Answer:
The vapor-phase compressibility factor Z is 0.94, consistent with typical values for propane near critical conditions (Z = 0.75–0.95).
🏗️ Real-World Application
At the Olympic Dam copper-uranium mine (South Australia), Aspen HYSYS simulations using the Peng–Robinson EOS predicted CO₂–H₂O–H₂S phase behavior in the acid-gas removal unit. When feed gas composition shifted unexpectedly (higher H₂S), the original SRK model overpredicted liquid dropout by 22%, causing foaming in the amine contactor. Switching to PR with Wong–Sandler mixing rules and binary interaction parameters (kij) from NIST ThermoData Engine reduced prediction error to <3%, restoring stable operation and avoiding a 72-hour production outage.
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