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Non-tabular approaches to calculating properties of real gases

Non-tabular approaches to calculating properties of real gases. The critical state. At the critical state (T c, P c ), properties of saturated liquid and saturated vapor are identical if a gas can be liquefied at constant T by application of pressure, T · T c .

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Non-tabular approaches to calculating properties of real gases

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  1. Non-tabular approaches to calculating properties of real gases

  2. The critical state • At the critical state (Tc, Pc), properties of saturated liquid and saturated vapor are identical • if a gas can be liquefied at constant T by application of pressure, T·Tc. • if a gas can be liquefied at constant P by reduction of T, then P·Pc. • the vapor phase is indistinguishable from liquid phase

  3. Properties of the critical isotherm • The SLL and SVL intersect on a P-v diagram to form a maxima at the critical point. • On a P-v diagram, the critical isotherm has a horizontal point of inflexion.

  4. Departures from ideal gas and the compressibility factor • For an ideal gas • One way of quantifying departure from ideal gas behavior to evaluate the “compressibility factor” (Z) for a true gas: • Both Z<1 and Z>1 is possible for true gases

  5. The critical state and ideal gas behavior • At the critical state, the gas is about to liquefy, and has a small specific volume. is very large  Z factor can depart significantly from 1. Whether a gas follows ideal gas is closely related to how far its state (P,T) departs from the critical state (Pc, ,Tc).

  6. Critical properties of a few engineering fluids • Water/steam (power plants): • CP: 374o C, 22 MPa • BP: 100o C, 100 kPa (1 atm) • R134a or 1,1,1,2-Tetrafluoroethane(refrigerant): • CP: 101o C, 4 MPa • BP: -26o C, 100 kPa (1 atm) • Nitrogen/air (everyday, cryogenics): • CP: -147o C, 3.4 MPa • BP: -196o C, 100 kPa (1 atm)

  7. Principle of corresponding states (van der Waal, 1880) • Reduced temperature: Tr=T/Tcr • Reduced pressure: Pr=P/Pcr • Compressibility factor: • Principle of corresponding states: All fluids when compared at the same Tr and Pr have the same Z and all deviate from the ideal gas behavior to about the same degree.

  8. Generalized compressibility chart 1949 Fits experimental data for various gases

  9. Use of pseudo-reduced specificvolume to calculate p(v,T), T(v,p)using GCC Z

  10. Nelson-Obert generalized compressibility chart 1954 Based on curve- fitting experimental data

  11. Equations of state

  12. Some desirable characteristics of equations of state • Adjustments to ideal gas behavior shoujd have a molecular basis (consistency with kinetic theory and statistical mechanics). • Pressure increase leads to compression at constant temperature • Critical isotherm has a horizontal point of inflection: • Compressibility factor (esp. at critical state consistent with experiments on real gases.)

  13. Some equation of states • Two-parameter equations of state • Virial equation of states Z=1+A(T)/v+B(T)/v2+…. (coefficients can be determined from statistical mechanics) • Multi-parameter equations of state with empirically determined coefficients: • Beattie-Bridgeman • Benedict-Webb-Rubin Equation of State Often based on theory

  14. Two-parameter equations of states • Examples: • Van der waals • Dieterici • Redlich Kwong • Parameters (a, b) can be evaluated from critical point data using • Van der Waals:

  15. Critical compressibility of real gases

  16. First law in differential form, thermodynamic definition of specific heats

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