Non-tabular approaches to calculating properties of real

gases

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

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.

– –

0cT

P

v

2

20

cT

P

v

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

1Pv

RT

ideal

Pv v

vZ

RT

The critical state and ideal gas behavior

• At the critical state, the gas is about to liquefy, and has a small specific volume.

100%ideal table

table

v

v

v is very large

Z factor can depart significantlyfrom 1.

Whether a gas follows ideal gas is closelyrelated to how far its state (P,T) departsfrom the critical state (Pc, ,Tc).

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)

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.

Generalized compressibility chart

1949

Fitsexperimentaldata for various gases

Use of pseudo-reduced specificvolume to calculate p(v,T), T(v,p)

using GCC

Z

Nelson-Obert generalized compressibility chart

1954

Basedon curve-fittingexperimentaldata

Equations of state

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.)

0T

P

v

2

20, 0,

c cT T

P P

v v

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

Oftenbasedon theory

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:

2/ ( ) /P RT v b a v

2

20, 0,

c cT T

P P

v v

expa

RTv

RTP

v b

( )

RT a

v b TvP

v b

2 227; ; Z

640.375

8c c

cc c

R T RT

pa b

P

Critical compressibility of real gases

First law in differential form, thermodynamic definition of

specific heats