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.
–
–
 P 

 0
 v Tc
 2P 
 2  0
 v Tc
Departures from ideal gas and the
compressibility factor
Pv
1
RT
• For an ideal gas
• One way of quantifying departure from
ideal gas behavior to evaluate the
“compressibility factor” (Z) for a true gas:
Pv
v
Z

RT videal
• Both Z<1 and Z>1 is possible for true
gases
The critical state and ideal gas
behavior
• At the critical state, the gas is about to
liquefy, and has a small specific volume.
videal  vtable
100%
vtable
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).
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
Fits
experimental
data for
various gases
Use of pseudo-reduced specific
volume to calculate p(v,T), T(v,p)
using GCC
Z
Nelson-Obert generalized
compressibility chart
1954
Based
on curvefitting
experimental
data
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
 P 
constant temperature  v   0
• Critical isotherm has a horizontal point of
 P
 P 
 0, 


inflection:  v   v   0,
• Compressibility factor (esp. at critical state
consistent with experiments on real gases.)
T
2
2
Tc
Tc
Some equation of states
Often
• Two-parameter equations of state
based
on theory
• 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
Two-parameter equations of states
• Examples:
– Van der waals
– Dieterici
– Redlich Kwong
P  RT / (v  b)  a / v 2
P
RT
a 

exp  

vb
 RTv 
P
RT
a

v b
T v (v  b )
• Parameters (a, b) can be evaluated from critical
point data using  Pv   0,  vP   0,




• Van der Waals:
2
2
Tc
Tc
27 R 2Tc2
RTc
a
; b
; Zc  0.375
64 Pc
8 pc
Critical compressibility of real gases
First law in differential form,
thermodynamic definition of
specific heats