ME 5243: Final Project

Advanced Thermodynamics, Spring 2013

Thermodynamic Relations for a Cubic Equation of State

Submission by:

Jitin Samuel

Cenk Sarikaya

Equation Number 9 as given in the project statement.

The Patel-Teja Equation of state (Patel,Teja 1982) is given as

( )

( ) ( )

The following parameters are used in the derivations in this project.

R = Molar gas constant

P = Pressure

Pr = Reduced pressure

Pc = Critical pressure

T = Temperature

Tr = Reduced temperature

Tc = Critical temperature

v = Specific volume

vr = Reduced specific volume

vc = Critical volume

= Specific volume of ideal gas

vrf = Reduced volume at liquid state

vrg = Reduced volume at gaseous state

Z = Compressibility factor

Zr = Reduced compressibility factor

Zc = Critical compressibility factor

= Specific enthalpy for real gas

specific internal energy for real gas

Specific entropy for real gas

Speed of sound

a) Constants ( a, b ,c )

.

/

= ( )

( ( ) ( ))

( )

.

/

= .

( ) (( ) ( ))

( ( ) ( )) /

.

/

= 0 = ( )

( ( ) ( ))

( )

.

/

= 0 = .

( ) (( ) ( ))

( ( ) ( )) /

(

)

(

)

(

)

b) Equation of state in reduced form

( )

( ) ( ) (1)

, , (2)

( ) ( )

. ( ) /

(

)

. ( ) / (

) ( (

) )

,

. ( ) /

( )

. ( ) / (

) ( (

) )

. /

( )

(

)

( )

(

)

( )

( )

c) Critical compressibility factor

. /

( )

(

)

( )

0.86 (please see iterative solution)

Using,

= 0.006701

= -0.15739

= -1.03124

d) Equation of saturation curve

( ) ∫

( )∫ (

( )

(

) )

( )∫ (

( )

( )

)

( ), ( ) ( )

( )-

( ) , (

) (

)

(

)

e) Reduced specific volumes of saturated liquid and vapor

( )

( )

( )

( )

( ) ( )

( )( )

( ) ( )

Substitute TR into equation for PR

( )( )

( ) ( )

( )

g) From PR-TR curve, deduce , C

along the saturated curve,

Rln PR

BA

T

At critical point PR=1 TR=1 & ln PR=0

Rln P 0

01

R

BA

T

BA A B

R R

1 1ln P (1 ) P (1 )

R R

A Exp AT T

R

2

1(1 )

R R R

dP AExp A

dT T T

0R 1(1 )

1R cp

dPAExp A Ae A

dT

( )

( )

( ) For = 1

A=0.983

( ) (

)

h) Departure enthalpy

∫ [ (

) ]

(

)

( ) ( )

*

[ (

√ )

]

√ +

Departure entropy:

∫[(

) ]

)

∫[

]

, ( ) -

[ (

)]

[ (

)]

Departure internal energy:

( )

*

[ (

√ )

]

√ +

*

[ (

√ )

]

√ +

i ) (f* - f)/RTc and (g* - g)/RTc

∫

∫ 0

1

( )

∫ 0

1

( )

∫ 0

1

( )

∫

∫ 0

1

( ( ) )

∫ 0

1

( )

∫ 0

1

( ( ) )

( )

( )

, 0.86

( ) ( )

∫ [ ]

∫ * (

( )

( )

)+

( ) ( )( ) )

.

/ .

/

( .

/ .

/ ( ))

( .

/ .

/ ( ) )

j) Speed Of Sound

For an ideal gas the speed of sound is expressed as kRTc . It can be

shown that the speed of sound waves, c, is given by the formula :

s

Pc

Tds = du + Pdv.

21/ (1/ )v dv

2

PTds du d

For simp e compressib e substa ces u u(P ρ)

u udu dp d

p

2

u u PTds dp d d

p

ds=0 divide both side by d and find s

P

2

p

s

u P

P

u

P

cos tanvu c T n t v p v

Pu c C R c c

R

v

p v

c Pu C

c c

1

1

Pu C

k

1 1

1

u

p k

1

1 2

u P

k

( 1)s

P P P Pk k

2

s

P Pc k kPv

Given equation of state

( )

( ) ( )

.

/

=

( )

( )

( ( ) ( ))

21/ /v d dv v

Isothermal speed of sound

2 2

TT

P Pc k v k

v

( )

( )

( ( ) ( ))

Refernce

1. N.C. Patel, A.S. Teja, Chem. Eng. Sci. 37 (1982) 463–473