7.3 Generalized correlations for gases and liquids

Generalized correlations help predict fluid behavior across various conditions. They use reduced properties and the principle of corresponding states to compare different substances. These tools are crucial for understanding how gases and liquids deviate from ideal behavior.

The compressibility factor and virial equation of state are key concepts in these correlations. They account for real gas behavior and intermolecular forces, allowing engineers to make accurate predictions about fluid properties in diverse applications.

Corresponding States and Reduced Properties

Principle of Corresponding States and Reduced Properties

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• Corresponding states principle suggests that fluids at the same reduced state have similar deviation from ideal gas behavior
• Reduced properties are obtained by normalizing thermodynamic properties (temperature, pressure, volume) with respect to critical point values
  • Reduced temperature: $T_R = T/T_c$
  • Reduced pressure: $P_R = P/P_c$
  • Reduced volume: $V_R = V/V_c$
• Critical properties (temperature $T_c$, pressure $P_c$, volume $V_c$) characterize the critical point, where liquid and vapor phases become indistinguishable

Acentric Factor and Fluid Behavior

• Acentric factor $\omega$ is a measure of the deviation of a fluid's behavior from that of a simple fluid (spherical molecules)
  • Simple fluids (argon, krypton) have $\omega \approx 0$
  • Complex fluids (hydrocarbons, refrigerants) have higher values of $\omega$
• Acentric factor is defined as: $\omega = -\log_{10}(P_R^{sat}) - 1$ at $T_R = 0.7$, where $P_R^{sat}$ is the reduced vapor pressure
• Incorporating acentric factor improves the accuracy of corresponding states correlations for a wide range of fluids

Generalized Compressibility Charts

Compressibility Factor and Charts

• Compressibility factor $Z$ relates the actual behavior of a gas to that of an ideal gas: $Z = \frac{PV}{nRT}$
  • For an ideal gas, $Z = 1$
  • Deviations from ideality result in $Z \neq 1$
• Generalized compressibility charts (Nelson-Obert, Brown) plot $Z$ as a function of reduced temperature and pressure
  • Different charts are used for different values of acentric factor

Lee-Kesler and Pitzer Correlations

• Lee-Kesler correlation is a widely used corresponding states method for estimating compressibility factor and other properties
  • Based on a three-parameter corresponding states principle (reduced temperature, reduced pressure, acentric factor)
  • Provides accurate results for a wide range of fluids and conditions
• Pitzer correlations are another set of corresponding states correlations for estimating fluid properties
  • Pitzer's two-parameter corresponding states principle uses reduced temperature and pressure
  • Pitzer's three-parameter corresponding states principle adds the acentric factor for improved accuracy

Virial Equation of State

Virial Equation and Compressibility Factor

• Virial equation of state is a power series expansion that relates pressure to molar volume and temperature: $Z = 1 + \frac{B(T)}{V_m} + \frac{C(T)}{V_m^2} + ...$
  • $B(T)$ is the second virial coefficient, $C(T)$ is the third virial coefficient, and so on
  • Virial coefficients depend on temperature and account for intermolecular interactions
• Truncating the virial equation after the second term yields: $Z = 1 + \frac{B(T)}{V_m}$
  • Suitable for low-density gases
  • Second virial coefficient can be estimated using Pitzer-type correlations or Tsonopoulos correlation
• Virial equation provides a more accurate description of gas behavior compared to the ideal gas equation, especially at higher pressures