Generalized Correlations for Gases (Lee-Kesler)

By: Santosh Koirala Manoj Joshi

Lauren BillingsCory Klemashevich

The Generalized Correlation for Gases

• The generalized Pitzer’s correlation is a three-parameter corresponding states method for estimating thermodynamic properties of pure, nonpolar fluids . For the compressibility factor Z, it takes the form

Z = Z0 + ω Z1

where,

Z0 = Compressibility factor for fluids of nearly spherical molecules,

ω = Pitzer's acentric factor, and

Z1 = Corrects for nonspherical intermolecular forces.

• Appendix E provides values of Z0 and Z1 (Lee-Kessler correlations), from which Z can be calculated and, hence, the molar volume can be computed.

Virial equationsThe Virial equation (up to the second Virial coefficient) provides an approximation of Z and the equation is:

c

c

RT

BPB ˆ

2.41

6.10

10

172.0139.0

422.0083.0

ˆ

r

r

TB

TB

BBB

RT

BPZ 1

r

r

T

PBZ

ˆ1

Cont……

The virial equation (up to the Third coefficient) also provides an approximation of Z. Such virial equation is:

2

22

2

2

ˆˆ

1

ˆ

ˆ

1

ZT

PC

ZT

PBZ

TR

CPC

RT

BPB

V

C

V

BZ

r

r

r

r

c

c

c

c

5.107.21

5.100

10

00242.05539.002676.0

00313.02432.001407.0

ˆ

rr

rr

TTC

TTC

CCC

2.41

6.10

10

172.0139.0

422.0083.0

ˆ

r

r

TB

TB

BBB

Example Problem

Determine the molar volume of n-butane at 510 K and 25 bar by each of the following:

a) The ideal-gas equation, b) The generalized compressibility-factor correlation, c) Generalized correlation for using eq. 3.61, d) Equation 3.68 the third virial coefficient equation.

a

B

131.696,125

)510)(14.83( molcmP

RTV

A. By the Ideal Gas equation:

B. From the values of Tc and Pc given in Table B.1 of App. B

13

10

7.480,125

)510)(14.83)(873(.

873.0

molcmP

ZRTV

ZZZ

2.41

6.10

10

172.0139.0

422.0083.0

ˆ

ˆ1

ˆ

1

r

r

r

r

c

c

TB

TB

BBB

T

PBZ

RT

BPB

RT

BPZ

P 25barT 510KPc 37.96barTc 425.1KPr 0.658587987Tr 1.199717713ω 0.2

B0 -0.232344991B1 0.058943546

-0.220556282

Z 0.878925087

R 83.14cm3bar/molKV 1490.706167cm3/mol

Here, Excel was used to calculate the volume using second virial coefficient equation.

C. Using the Second Virial Equation

B

P 25barT 510KPc 37.96barTc 425.1K

Pr 0.658587987 Z (Guess)Z

(Calculated)Tr 1.199717713 1 0.889316ω 0.2 0.889316 0.876994

0.876994 0.875453B0 -0.232344991 0.875453 0.875258B1 0.058943546 0.875258 0.875233

-0.220556282 0.875233 0.875230.87523 0.875229

C0 0.03312865 0.875229 0.875229C1 0.006760336 0.875229 0.875229

0.034480717 0.875229 0.8752290.875229 0.8752290.875229 0.875229

R 83.14cm3bar/molK 0.875229 0.875229V 1484.438039cm3/mol 0.875229 0.875229

0.875229 0.8752290.875229 0.875229

5.107.21

5.100

10

2.41

6.10

10

2

22

2

2

00242.05539.002676.0

00313.02432.001407.0

ˆ

172.0139.0

422.0083.0

ˆ

ˆˆ

1

ˆ

ˆ

1

rr

rr

r

r

r

r

r

r

c

c

c

c

TTC

TTC

CCC

TB

TB

BBB

ZT

PC

ZT

PBZ

TR

CPC

RT

BPB

V

C

V

BZ

This is the third coefficient virial equation for the same problem. Again Excel was used to obtain the solution. The solution is again very close to the value obtained by the Lee-Kesler method.

D. Using Second and Third Virial coefficient

CC

C

Where the Virial Equation Applies

• The second coefficient virial equation works at low pressures where Z is a linear function. It is used when an approximation of a non ideal gas is needed, but at non extreme temperatures and pressures.

• The third virial coefficient equation provides another correction to the virial equation.

This graph shows the difference obtained for the Z0 value for the Lee/Kesler correlation vs. the virial coefficient equation.