OpenSpecEngineering

Finds the compressibility factor Z and the molar volume of a pure gas or liquid from the Peng–Robinson equation of state. Pick a substance to fill in its critical constants, or enter your own. Process and pipeline engineers use Z to correct the ideal gas law for density, line pack and flow metering at high pressure.

Equations

Z = P·v / (R·T)

Z³ − (1 − B)Z² + (A − 3B² − 2B)Z − (AB − B² − B³) = 0

A = a·P/(R·T)²     B = b·P/(R·T)

a = 0.45724·R²T_c²/P_c·α

b = 0.07780·R·T_c/P_c

α = [1 + κ(1 − √T_r)]²

κ = 0.37464 + 1.54226ω − 0.26992ω²

These are the Peng–Robinson equations [1], [2]. P is the absolute pressure, T the absolute temperature, v the molar volume and R = 8.314 J/(mol·K). T_c and P_c are the critical temperature and pressure, T_r = T/T_c and P_r = P/P_c are the reduced values, and ω is the acentric factor. The substance list takes T_c, P_c and ω from Smith, Van Ness and Abbott [3].

Reading the roots

The cubic has one or three real roots. Above T_c there is one, for the single fluid phase. Below T_c there can be three: the largest is the vapor root and the smallest the liquid root. The page shows both, but it doesn’t say which phase is stable. Near the vapor pressure, compare with a vapor pressure value or a fugacity check. When only one root exists below T_c, the page says whether it is liquid-like or vapor-like.

Limits and common mistakes

• Use absolute pressure, not gauge.
• Pure substances only. A mixture needs mixing rules for a and b.
• Z is good to a few percent for gases. Liquid molar volumes from Peng–Robinson are often several percent off, so use a liquid density correlation for liquids.

Worked example

A pipeline carries methane at 10 MPa absolute and 25 °C. How far from ideal is the gas, and what is its molar volume?

1. Enter Temperature 25 °C and Pressure 10000 kPa.
2. Choose Substance Methane. T_c fills in as −82.55 °C (190.6 K), P_c as 4599 kPa and ω as 0.012. Press Solve. The Solution shows: Reduced temperature Tr: 1.56427
   Reduced pressure Pr: 2.17439
   PR parameter A: 0.330168
   PR parameter B: 0.108144
   Z (only real root): 0.829939
   Molar volume: 205.738 cm³/mol
   Root: Above the critical temperature — a single fluid phase

The ideal gas law gives R·T/P = 8.314 × 298.15/10⁷ = 247.9 cm³/mol, so the gas is about 20 % denser than ideal (1/Z = 1.205). Check: 0.829939 × 247.9 = 205.7 cm³/mol. Solving the Peng–Robinson pressure equation for v directly gives the same volume.