Peng-Robinson Equation of State

Overview

The Peng-Robinson equation of state (PR76) was published in 1976 and quickly became the most widely used cubic EoS in the petroleum industry. It provides improved liquid density predictions compared to the Soave-Redlich-Kwong (SRK) equation while maintaining computational simplicity.

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The PR Equation

Pure Component Form

$$P = \frac{RT}{V_m - b} - \frac{a(T)}{V_m(V_m + b) + b(V_m - b)}$$

Where:

• $P$ = pressure
• $T$ = temperature
• $V_m$ = molar volume
• $R$ = universal gas constant

Parameters

Co-volume parameter:

$$b = 0.07780 \frac{RT_c}{P_c}$$

Attraction parameter at critical temperature:

$$a_c = 0.45724 \frac{R^2 T_c^2}{P_c}$$

Temperature-dependent attraction:

$$a(T) = a_c \cdot \alpha(T)$$

Alpha Function (Soave-type)

$$\alpha(T) = \left[1 + m\left(1 - \sqrt{T_r}\right)\right]^2$$

Where $T_r = T/T_c$ is the reduced temperature, and:

$$m = 0.37464 + 1.54226\omega - 0.26992\omega^2$$

For heavier components ($\omega > 0.49$), the Graboski-Daubert modification is recommended:

$$m = 0.379642 + 1.48503\omega - 0.164423\omega^2 + 0.016666\omega^3$$

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Cubic Form

Rewriting in terms of the compressibility factor $Z = PV_m/RT$:

$$Z^3 - (1-B)Z^2 + (A - 3B^2 - 2B)Z - (AB - B^2 - B^3) = 0$$

Where:

$$A = \frac{aP}{R^2T^2}, \quad B = \frac{bP}{RT}$$

This cubic equation has either one or three real roots:

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Mixing Rules

Van der Waals One-Fluid Mixing Rules

For mixtures of $N$ components with mole fractions $z_i$:

Attraction parameter:

$$a_{mix} = \sum_i \sum_j z_i z_j a_{ij}$$

$$a_{ij} = \sqrt{a_i a_j}(1 - k_{ij})$$

Co-volume parameter:

$$b_{mix} = \sum_i z_i b_i$$

Binary Interaction Parameters

The $k_{ij}$ values correct for non-ideal mixing. They are symmetric ($k_{ij} = k_{ji}$) and $k_{ii} = 0$.

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Derived Properties

Fugacity Coefficient

For component $i$ in a mixture:

$$\ln \hat{\phi}_i = \frac{b_i}{b_{mix}}(Z-1) - \ln(Z-B) - \frac{A}{2\sqrt{2}B}\left(\frac{2\sum_j z_j a_{ij}}{a_{mix}} - \frac{b_i}{b_{mix}}\right)\ln\frac{Z + (1+\sqrt{2})B}{Z + (1-\sqrt{2})B}$$

Molar Volume

$$V_m = \frac{ZRT}{P}$$

Density

$$\rho = \frac{M_w}{V_m}$$

Where $M_w$ is the mixture molecular weight:

$$M_w = \sum_i z_i M_{w,i}$$

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Volume Translation

The PR equation systematically underestimates liquid densities by 3-5%. The Peneloux volume translation corrects this:

$$V_m^{corrected} = V_m^{EoS} - c$$

Where $c$ is a component-specific shift parameter. Volume translation improves density without affecting phase equilibrium calculations (fugacity ratios are unchanged).

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Applicability and Limitations

Strengths

• Best liquid density among two-parameter cubic EoS
• Accurate vapor pressure predictions for hydrocarbons
• Well-suited for petroleum reservoir fluids
• Robust convergence properties

Limitations

Valid Ranges

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Related Topics

• EoS Overview — When to use EoS vs. correlations
• Flash Calculations — Phase equilibrium using PR
• Phase Envelope — Bubble/dew point curves
• C7+ Characterization — Heavy fraction modeling

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References

1. Peng, D.Y. and Robinson, D.B. (1976). "A New Two-Constant Equation of State." Industrial & Engineering Chemistry Fundamentals, 15(1), 59-64.

2. Robinson, D.B. and Peng, D.Y. (1978). "The Characterization of the Heptanes and Heavier Fractions for the GPA Peng-Robinson Programs." GPA Research Report RR-28.

3. Peneloux, A., Rauzy, E., and Freze, R. (1982). "A Consistent Correction for Redlich-Kwong-Soave Volumes." Fluid Phase Equilibria, 8(1), 7-23.

4. Graboski, M.S. and Daubert, T.E. (1978). "A Modified Soave Equation of State for Phase Equilibrium Calculations." Industrial & Engineering Chemistry Process Design and Development, 17(4), 443-448.

5. Whitson, C.H. and Brule, M.R. (2000). Phase Behavior. SPE Monograph Vol. 20.