3. Non-ideal Gases 3.ppt

8/14/2019 3. Non-ideal Gases 3.ppt

1/26

Non-ideal Equations of State


2/26

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 happenAt 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


3/26

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)


4/26

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 EOSs.


5/26

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


6/26

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


7/26

van der Waals

The values of a and b are different for

different chemicals, but they are related in

the same way to each chemicals Tc and

Pc. Critical properties are tabulated.

c

c

c

c

P

RTb

P

TRa

8

1,

64

27 22

van der Waals EOS


8/26

Peng-Robinson

The vdW equation is Ok but other

empirical EOSs are more accurate (but

more complicated) One that has a nice

balance of accuracy vs complexityisthe Peng-Robinson EOS.

)~

()~

(~~

bVbbVV

a

bV

RTP


9/26

Peng-Robinson

The a and b parameters are related

empirically to the critical properties:

c

c

P

TRa

22

45724.0

c

c

P

RTb 07780.0


10/26

Peng-Robinson

The parameter is temperature

dependentand also depends onanother

tabulated,chemical specific, parameter

called the acentric factor

211r

TS

c

r

T

TT

226992.054226.137464.0 S

acentric factor


11/26

Peng-Robinson

It is usually a good idea to programthe

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 errorcalculations (root finding)
http://c/Documents%20and%20Settings/carlson1/My%20Documents/RH%20303-F2002/Lecture%20Materials/Spreadsheets/10-24%20(PR).xlshttp://c/Documents%20and%20Settings/carlson1/My%20Documents/RH%20303-F2002/Lecture%20Materials/Spreadsheets/10-24%20(PR).xls


12/26

Beware Multiple Roots

When the object is to find V with T and P known, then itis possible to get 3 answers (roots) that all satisfy theequation. This will only happen for T below thecritical temperature.

The smallest valueis a volume that corresponds tothe liquidat that T and P

The largest valueis a volume that corresponds to thevapor(most accurate).

The middle valuehas no physical meaning(just amathematical artifact). In trial and error programs likeSolver, one must achieve the desired root by an initialguess that is close to desired root.


13/26

Specific Volume (V)

Pressure

PIsotherm above Tc

Cubic EOS roots


14/26

Specific Volume (V)

Pressure

P

Isotherm below Tc

T1

Three roots (3 Vs are

predicted by equation)

Below Tc


15/26

Root Evaluation

Below Tccare 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 lawprogram

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)


16/26

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
http://c/Documents%20and%20Settings/carlson1/My%20Documents/RH%20303-F2002/Lecture%20Materials/Spreadsheets/10-24%20(PR).xlshttp://c/Documents%20and%20Settings/carlson1/My%20Documents/RH%20303-F2002/Lecture%20Materials/Spreadsheets/10-24%20(PR).xls


17/26

Since the temp is above Tc,this is the only root.


18/26

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.
http://c/Documents%20and%20Settings/carlson1/My%20Documents/RH%20303-F2002/Lecture%20Materials/Data%20Base%20Info/Critical%20properties%20(1).pdfhttp://c/Documents%20and%20Settings/carlson1/My%20Documents/RH%20303-F2002/Lecture%20Materials/Data%20Base%20Info/Critical%20properties%20(1).pdf


19/26


20/26

Using the Excel program, find all three roots for

HFC134a at 10 bar, 60 C.

Vv

Vl

Vmiddle


21/26


22/26


23/26


24/26


25/26

Virial Equations

For sophisticated calculations fitting

equations with more adjustable

parameters are used. These are called

virial equations. Some equations (likethose for water) might have 20 or more

adjustible constants

...~~~~ 432 V

DRT

V

CRT

V

BRT

V

RTP


26/26

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:

3. Non-ideal Gases 3.ppt

Documents

Transcript of 3. Non-ideal Gases 3.ppt