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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 specific volume to calculate p(v,T), T(v,p) using GCC
Z Source:

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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