ChE 413 Thermo1 PVT

7/24/2019 ChE 413 Thermo1 PVT

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CHE 413 CHE THERMO1

Volumetric properties of pure fluids



PVT behavior of pure substances



In single phase region,f(P,V,T) = 0

If V = V(T,P), then

Volume expansivity

P T

V V dV dT dP

T P

∂ ∂ = +

∂ ∂

1

P

V

V T β

∂ ≡

∂ 1 V ∂

Combining all equations,

For incompressible fluids,

For small changes in T & P, β & κ can be constant

T V P−

∂

dV dT dP

V

β κ = −

0 β κ = =

( ) ( )22 1 2 1

1

lnV

T T P P

V

β κ = − − −



Virial equations of state For real gases

Virial expansions

2

...PV a bP cP= + + +

( )' ' 2 ' 31 ...PV a B P C P D P= + + + +

Z: compressibility factor B’,B: 2nd virial coefficients; C’,C: 3rd virial coefficients

' ' 2 ' 31 ... Z B P C P D P RT

≡ = + + + +

2 31 ...

B C D Z

V V V = + + + +

' B B

RT

=

( )

2'

2

C BC

RT

−=

( )

3'

3

3 2 D BC B D

RT

− +=



Ideal gas

B/V, C/V2 arise due to molecular interactions; if no

interactions, virial expansion reduces to Z = 1 or PV = RT

Ideal gas definition

Equation of state

PV = RT

Internal energy

U = U(T)



Implied relations for ideal gas

Cv is a function of temperature only

Enthalpy

( ) ( )V V

V

dU T U C C T T dT

∂ ≡ = = ∂

Cp is a function of temperature only

Relation b/w Cp and Cv

P V

dH dU C R C R

dT dT = = + = +

( )( )P P

P

dH T H C C T

T dT

∂ ≡ = =

∂



Implied relations for ideal gas For any change of state of an ideal gas

V U C dT ∆ = ∫V

dU C dT =

P H C dT ∆ = ∫P

dH C dT =



Isochoric process Isobaric process

V U C dT ∆ = ∫P

H C dT ∆ = ∫V U C dT ∆ = ∫P

H C dT ∆ = ∫V Q C dT = ∫

0W =

General restrictions

Equations are valid for ideal gas

The process is mechanically reversible.

The system is closed.

PQ C dT =

( )2 1W P V V = − −



Reversible Reversible

Isothermal process Adiabatic process

V W U C T = ∆ = ∆

0U H ∆ = ∆ =

2 1V P= − = −

0Q =

1TV constant γ − =( )1

TP constant γ γ −

=

PV constant γ

=

P

V

C

C γ ≡2 1

1 2

ln lnV P

Q nRT nRT V P

= =

1 2V P P H C T ∆ = ∆



Applications of the virial equations

' ' 2 ' 31 ...PV

Z B P C P D P

RT

= = + + + +

2 31 ...

PV B C D Z

RT V V V = = + + + +

For engineering purposes, their use is practical onlywhen convergence is very rapid

gas at low pressure pressure <50 bar

1PV B Z RT V

= = + 21PV B C Z

RT V V = = + +

1PV BP

Z

RT RT

= = +



Cubic equations of state Van der Waals equation (J.D. van der Waals)

Generic cubic equation of state

2

RT a

P V b V = −

−

2 227

64

c

c

R T

a P=

8

c

c

RT

b P=

a T RT

Redlich/Kwong equation (Otto Redlich & JNS Kwong)

( )1 2

RT a

P V b T V V b= −− +

2 2.50.42748c

c

R T

a P=

0.08664c

c

RT b

P

=

( ) ( )V b V b V bε σ = −

− + +



Generalized correlations for gases Multiplying RK equation by V/RT

1.5

1

1 1

a h

Z h bRT h

= −

− + b b bP

hV ZRT P ZRT

≡ = =

Eliminating a & b

Solution is iterative. Initial value of Z = 1. Get h. Get new

Z, get new value of h, and so on until convergence

1.5

1 4.9340

1 1r

h Z

h T h

= −

− +

0.08664 r

r

Ph ZT

≡

r

c

T T

T ≡

r

c

PPP

≡



Two-parameter theorem of corresponding states

All fluids, when compared at the same Tr & Pr, have

approx. the same Z, and all deviate from ideal-gasbehavior to about the same degree

Acentric factor, ( )0.7

1.0 log sat

r T

Pω =

≡ − −

Three parameter theorem of corresponding states All fluids having the same value of ω, when compared

at the same Tr & Pr, have the same value of Z, and all

deviate from ideal gas behavior to about the samedegree.



Pitzer correlations for compressibility factor Lee/Kesler correlation

0 1 Z Z Z ω = +



Pitzer correlations for virial coefficient Generalized virial coefficient equation (valid only for

low P)

Pitzer proposed the correlation:

Combining the two equations:

1 1 c r

c r

BP P BP Z

RT RT T

= + = +

0 1c

c

BP B B

RT ω = +

Comparing with the Lee/Kesler correlation

0 11 r r

r r

Z B BT T ω = + +

0 01 r

r

P Z B

T = + 1 1 r

r

P Z B

T =

0

1.6

0.4220.083

r

B

T

= − 1

4.2

0.1720.139

r

B

T

= −

ChE 413 Thermo1 PVT

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