Non-ideal Equations of State
Review Real Fluids
• Gases do not always obey the ideal gas
law.
– At modest temperatures but high pressures,
the molecules get close enough together that
intermolecular attractive forces become
significant.
• Two things can happen –
– At low temperatures the gas can turn into a liquid
– At higher temperatures the gas stays a gas but behaves
a lot like a liquid this state is called a supercritical fluid
Review
• The temperature at which it becomes
impossible to ever form a liquid regardless
of the pressure is called the critical
temperature. (Tc)
• The pressure at which there is just last
both vapor and liquid is called the critical
pressure. (Pc)
Use of Tc and Pc
• The critical temperature and pressure are
key parameters for calculating the
relationship between P, V, and T for nonideal fluids using empirical EOS’s.
Empirical Equations
• Several empirical “cubic” equations have
been invented to relate P to V and T for
non-ideal gases.
– van der Waals
– Redlich Kwong
– Peng Robinson
– Redlich Kwong Suave
van der Waals
RT
a
P ~
 ~2
V b V
At very small specific
volumes, the
molecules begin to
touch which causes
the pressure to rise
sharply.
At small specific
volumes, the
attractive term is
significant.
One of the earliest and simplest
van der Waals
• The values of a and b are different for
different chemicals, but they are related in
the same way to each chemical’s Tc and
Pc. Critical properties are tabulated.
2
27 R 2Tc
1 RTc
a
,b 
64 Pc
8 Pc
van der Waals EOS
Peng-Robinson
• The vdW equation is Ok but other
empirical EOS’s are more accurate (but
more complicated) One that has a nice
balance of accuracy vs complexity is
the Peng-Robinson EOS.
RT
a
P ~
 ~ ~
~
V  b V (V  b)  b(V  b)
Peng-Robinson
• The a and b parameters are related
empirically to the critical properties:
R 2Tc2
a  0.45724
Pc
RTc
b  0.07780
Pc
Peng-Robinson
• The  parameter is temperature
dependent and also depends on another
tabulated,chemical specific, parameter
called the “acentric factor”
 
  1  S 1  Tr

2
acentric factor
S  0.37464  1.54226  0.26992 2
Tr 
T
Tc
Peng-Robinson
• It is usually a good idea to program the
more complex equations into a
spreadsheet or Maple.
• Because of the way the equation is
written, finding the volume when T and P
are given or finding the temperature when
P and V are given requires trial and error
calculations (root finding)
Beware Multiple Roots
• When the object is to find V with T and P known, then it
is possible to get 3 answers (roots) that all satisfy the
equation. This will only happen for T below the
critical temperature.
• The smallest value is a volume that corresponds to
the liquid at that T and P
• The largest value is a volume that corresponds to the
vapor (most accurate).
• The middle value has no physical meaning (just a
mathematical artifact). In trial and error programs like
Solver, one must achieve the desired root by an initial
guess that is close to desired root.
Cubic EOS roots
Pressure
P
Isotherm above Tc
Specific Volume (V)
Below Tc
Three roots (3 V’s are
predicted by equation)
Pressure
P
Isotherm below Tc
T1
Specific Volume (V)
Root Evaluation
• Below Tc – care must be taken to make sure that
the right root is obtained
– There is one root near the ideal gas law (The large
volume)
• In Excel, make the first guess the ideal gas law – program
will find the “gas root”
– There is one root near b
• This is the liquid root and is hard to get
– There is one root near 3xb
• This is a physically meaningless root (the middle one)
Example with P-R EOS
Problem 1.
Find the specific volume of propane gas at 1000 psia and 260 C
using the PR equation of state.
 = 0.152; Tc = 369.8 K; Pc= 42.48 bar
Compare this value to the value obtained from the ideal gas law.
• Connection
Since the temp is above Tc,
this is the only root.
Peng-Robinson
• Connection
Problem 2
Using the PR equation of state compute how much methane one
could put into a 100,000 m3 storage tank so that the pressure
would not exceed 20 atm at 25 C?
Assuming this value is accurate, compute the % error one would
get if she used the ideal gas law instead of the PR equation.
Using the Excel program, find all three roots for
HFC134a at 10 bar, 60 C.
Vv
Vl
Vmiddle
Virial Equations
• For sophisticated calculations fitting
equations with more adjustable
parameters are used. These are called
virial equations. Some equations (like
those for water) might have 20 or more
adjustible constants…
RT BRT CRT DRT
P  ~  ~ 2  ~ 3  ~ 4  ...
V
V
V
V
Summary
• EOS are more accurate representations of
fluid PVT relationships than the simple
ideal gas law.
– Cubic equations of state have a good balance between simplicity
and accuracy.
– The other main type of empirical equation is a “virial” equation
that attempts to fit the PVT behavior with a long series of
“adjustment” terms: