🎓 Lesson 13
D5
Computing Hᴿ, Sᴿ, and Cpᴿ from Cubic EOS
Residual properties tell us how much a real gas differs from an ideal gas in terms of enthalpy, entropy, and heat capacity — and cubic equations of state help us calculate those differences.
🎯 Learning Objectives
- ✓ Calculate Hᴿ, Sᴿ, and Cpᴿ using the Peng–Robinson equation of state for hydrocarbon mixtures at high pressure
- ✓ Analyze the sensitivity of residual properties to critical parameters (Tc, Pc, ω) using departure function charts
- ✓ Explain why Cpᴿ is zero for ideal gases but nonzero for real fluids under isobaric conditions
- ✓ Apply residual property calculations to design accurate energy balances in high-pressure separation units
📖 Why This Matters
In mining and explosives engineering, accurate thermodynamic modeling of detonation products (e.g., CO₂, H₂O, N₂, CO) at extreme pressures (>100 bar) and temperatures (>3000 K) is essential for predicting blast energy distribution and shock wave propagation. Ideal-gas assumptions fail catastrophically here — residual properties bridge that gap. Without Hᴿ and Sᴿ, engineers misestimate adiabatic flame temperatures and detonation velocities, leading to unsafe or inefficient blast designs.
📘 Core Principles
Residual properties arise because real fluids deviate from ideality due to attractive/repulsive forces and molecular volume. Cubic EOS (e.g., PR, SRK) model compressibility factor Z = PV/RT as a cubic function in V, enabling closed-form derivatives needed for residual property integrals. Hᴿ and Sᴿ derive from integrals involving (∂P/∂T)ᵥ and (∂Z/∂T)ₚ; Cpᴿ follows from differentiation of Hᴿ with respect to temperature at constant pressure. The Peng–Robinson EOS is preferred in mining applications due to its accuracy near critical regions and for polar/nonpolar mixtures common in post-detonation gases.
📐 Key Calculation
The residual enthalpy Hᴿ is computed using the integral form derived from the Peng–Robinson EOS: Hᴿ = RT_c ∫₀ᴾ [Z − 1 − T(∂Z/∂T)ₚ] dP/P. For practical use, dimensionless reduced forms and generalized correlations simplify evaluation. The PR-based departure functions are implemented in process simulators (Aspen HYSYS, ChemCAD) but require manual verification for novel explosive gas mixtures.
💡 Worked Example
Problem: Calculate Hᴿ for pure methane at T = 350 K, P = 40 bar using Peng–Robinson EOS. Given: Tc = 190.6 K, Pc = 45.99 bar, ω = 0.011, R = 8.314 J/mol·K.
1.
Step 1: Compute reduced Tᵣ = 350/190.6 = 1.836 and Pᵣ = 40/45.99 = 0.869.
2.
Step 2: Evaluate PR parameters: α = [1 + 0.37464 + 1.54226ω − 0.26992ω²]² = 1.002; a = 0.45724 R²T_c²/(P_c) × α = 2.292 Pa·m⁶/mol²; b = 0.07780 RT_c/P_c = 2.97×10⁻⁵ m³/mol.
3.
Step 3: Solve cubic for Z → Z = 0.812; then compute (∂Z/∂T)ₚ numerically or via analytic derivative → −0.00142 K⁻¹.
4.
Step 4: Apply Hᴿ/RT_c = Z−1 − ln(Z−B) − A/(2√2 B) × [(2a/bT_c + 1/T_c)(Z−B) − (A/BT_c)(B/(Z+B))], yielding Hᴿ = −1240 J/mol.
Answer:
The result is −1240 J/mol, which falls within the typical range of −2500 to +500 J/mol for light hydrocarbons at moderate supercritical conditions.
🏗️ Real-World Application
At BHP’s Olympic Dam copper-uranium mine, blast gas composition (mainly CO₂, N₂, H₂O vapor) was modeled during underground confined blasts. Residual properties computed via PR EOS corrected ideal-gas adiabatic flame temperature predictions by +185 K — matching pyrometer measurements within 2%. This adjustment directly improved the design of ventilation-after-blast (VAB) duration and gas scrubbing system sizing.