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Residual Property Derivation Workbook (Step-by-Step PR/SRK)

The Residual Property Derivation Workbook (Step-by-Step PR/SRK) is a pedagogical and computational resource that guides users through the systematic derivation of residual thermodynamic properties—such as residual enthalpy, entropy, and Gibbs free energy—using the Peng–Robinson (PR) and Soave–Redlich–Kwong (SRK) cubic equations of state. It emphasizes analytical integration of departure function expressions, leveraging partial derivatives of pressure with respect to temperature and volume. Designed for graduate-level thermodynamics and process simulation courses, it bridges theoretical EOS fundamentals with practical property estimation.

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📖 Overview

Residual properties quantify the deviation of real fluid behavior from ideal-gas behavior at the same temperature, pressure, and composition. The workbook provides a rigorous, step-by-step framework for deriving these properties from cubic equations of state (CEOS), specifically PR and SRK, which are widely used in chemical process design and simulation due to their accuracy for nonpolar and moderately polar compounds. Each derivation begins with the fundamental residual property definition—e.g., H^R = H − H^ig—and applies Maxwell relations and the thermodynamic network to express departures in terms of integrals involving (∂P/∂T)_V and (∂P/∂V)_T. The workbook explicitly demonstrates variable substitution (e.g., using reduced or dimensionless forms), integration limits (from infinite volume to molar volume V), and handling of temperature-dependent attraction parameters (α(T) for PR/SRK). It also addresses numerical robustness—such as root selection for saturated phases, cubic root discrimination, and handling of metastable states—making it both a teaching tool and a reference for implementing custom property packages in equation-of-state-based simulators like Aspen Plus or open-source alternatives (e.g., Thermo, CoolProp).

📑 Key Components

1 Analytical derivation templates for H^R, S^R, G^R, and V^R

2 PR and SRK parameterization workflows (a, b, α(T), mixing rules)

3 Stepwise integration procedures with Jacobian and chain-rule expansions

🎯 Applications

✓ Thermodynamic property estimation for phase equilibrium calculations (VLE, LLE)
✓ Initialization and convergence aids in rigorous distillation and flash unit simulations
✓ Development and validation of custom EOS-based property methods in process simulators

📐 Key Formulas

Residual Enthalpy (H^R)

H^R = RT^2 \int_{\infty}^{V} \left( \frac{\partial P}{\partial T} \right)_V \frac{dV}{T^2}

Integral form derived from the fundamental residual property relation; evaluated analytically using PR/SRK pressure explicit in V.

Residual Entropy (S^R)

S^R = -R \int_{\infty}^{V} \left( \frac{\partial P}{\partial T} \right)_V dV - R \ln\left(\frac{PV}{RT}\right)

Combines internal energy and Helmholtz departure contributions; accounts for compressibility via the ideal-gas reference.

PR Attraction Parameter (a)

a = 0.45724 \frac{R^2 T_c^2}{P_c} \alpha(T), \quad \alpha(T) = [1 + \kappa(1 - \sqrt{T/T_c})]^2

Temperature-dependent cohesion term in Peng–Robinson EOS; κ depends on acentric factor ω.

🔗 Related Concepts

Departure Functions Cubic Equations of State Thermodynamic Property Estimation

📚 References

Introduction to Chemical Engineering Thermodynamics (8th Ed.), Chapter 6 & 10

The Properties of Gases and Liquids (5th Ed.) – Residual Property Tables & Derivations

NIST Chemistry WebBook: EOS Implementation Guidelines

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