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 non-ideal 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

2~~V

a

bV

RTP

At small specific volumes, the attractive term is significant.

At very small specific volumes, the molecules begin to touch which causes the pressure to rise sharply.

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.

c

c

c

c

P

RTb

P

TRa

8

1,

64

27 22

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.

)~()

~(~~

bVbbVV

a

bV

RTP

Peng-Robinson

• The a and b parameters are related empirically to the critical properties:

c

c

P

TRa

22

45724.0c

c

P

RTb 07780.0

Peng-Robinson

• The parameter is temperature dependent and also depends on another tabulated,chemical specific, parameter called the “acentric factor”

211 rTS

cr T

TT

226992.054226.137464.0 Sacentric factor

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.

Specific Volume (V)

PressureP

Isotherm above Tc

Cubic EOS roots

Specific Volume (V)

PressureP

Isotherm below Tc

T1

Three roots (3 V’s are predicted by equation)

Below Tc

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 barCompare 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 2Using 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…

...~~~~ 432

V

DRT

V

CRT

V

BRT

V

RTP

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: