Chapter 3 Thermodynamics Properties

of Fluids

Chemical Engineering Thermodynamics

3.1 Property Relations for Homogeneous Phases

3.2 Residual Properties

3.3 Residual Properties by Equations of State

Chapter Outline

SYSTEM

Process design and analysis

PROPERTY CALCULATION

IN OUT

To find volume, V, we are completely prepared, whether it is ideal gas, non-ideal gas or liquid.

But for other properties such as H, S, U, we have no idea what is the pressure and volume functions especially for non-ideal gas.

As such, we need to develop just such relations so that calculations can be carried out.

We have numerically calculated the main thermodynamics properties such as P, V, T.Now we are going to calculate enthalpy, H and entropy, S from PVT and heat-capacity data using 1st and 2nd law property relations.

And develop generalized correlations which provide estimates of property values in the absence of experimental information.

3.1 Property Relations forHomogeneous Phases

Primary thermodynamic properties are given as: P, V, T, U and S.

Other thermodynamic properties arise only by definition using the relation of these primary properties:Enthalpy: H ≡ U + PVHelmholtz Energy: A ≡ U – TSGibbs Energy: G ≡ H – TS

PdVTdSdU

VdPTdSdH

SdTPdVdA

SdTVdPdG

Fundamental property relations which are general equations for a homogenous fluid of constant composition:

VS S

P

V

T

PS S

V

P

T

TV V

S

T

P

TP P

S

T

V

They are all useful for evaluation ofthermodynamic properties from experimental data.

The thermodynamic properties can be expressed by equation for each of these equations which called as Maxwell’s equations:

Heat capacity at constant pressure

PP

CT

H

Other expression for heat capacity

PP T

ST

T

H

Hence, combine these two equation gives:

T

C

T

S P

P

Enthalpy and Entropy as Functions of T and P

From Maxwell’s equations…

TP P

S

T

V

From fundamental property relations,the corresponding derivative for enthalpy is

VP

ST

P

H

TT

As a result, heat capacity at constant pressure:

PT T

VTV

P

H

The functional relations chosen here for H and S are: H = H ( T , P ) and S = S ( T, P ) Whence,

dPT

VTVdTCdH

PP

dPT

V

T

dTCdS

PP

These are general equations relating the properties of homogeneous fluids of constant composition to constant temperature and pressure.

The pressure dependence of the internal energy is obtained by:

VP

VP

P

H

P

U

TTT

Replaced by previous equation gives:

TPT P

VP

T

VT

P

U

Internal Energy as a Function of P

Internal energy is given by equation U = H - PV

TP

H

The Ideal-Gas State

dTCdH igP

ig P

dPR

T

dTCdS ig

Pig

where superscript “ig” denotes an ideal-gas value.

Alternative Forms for Liquids

VP

S

T

VT

P

H

T

1 VTPP

U

T

At critical point the volume itself is verysmall, as are ß and κ. Thus at most conditions pressure has little effect on the properties of liquid. Hence for incompressible fluid:

VdPTdTCdH P 1

(Example 6.1)

VdPTdT

CdSP

Internal Energy and Entropy as Functions of T and V

Temperature and volume often serve as more convenient independent variables than do temperature and pressure.

These are general equations relating the internal energy, U and entropy, S of homogeneous fluids of constant composition to temperature and volume.

dVPT

PTdTCdU

VV

dVT

P

T

dTCdS

VV

VT

P

dVPTdTCdU V

dVT

dTCdS V

A change of state a constant volume becomes:

Then the previous equations above become:

(Example 6.2)

A fundamental property relation, follows fromthe mathematical identity:

dTRT

GdG

RTRT

Gd

2

1

After algebraic reduction:

dTRT

HdP

RT

V

RT

Gd

2

The Gibbs Energy as a Generating Function

When G/RT is known as a function of T and P, V/RT and H / RT follow by simple differentiation.

TP

RTG

RT

V

PT

RTGT

RT

H

The remaining properties are given by defining equations. In particular,

RT

G

RT

H

R

S

RT

PV

RT

H

RT

U

The Gibbs energy when given as a function of T and P serves as a generating function for the other thermodynamic properties, and represents complete property information .

where; GR = Residual Gibbs energy G = Actual Gibbs energy Gig = Ideal-gas values of Gibbs energy

For residual volume:

3.2 Residual Properties

igR GGG

PRT

VVVV igR

Since V = ZRT/ P,

1 ZP

RTV R

The definition for generic residual property is:

where M is molar value of any extensive thermodynamic property eg. V , U , H , S , or G.

igR MMM

dTRT

HdP

RT

V

RT

Gd

RRR

2

Fundamental property relation for residual properties applies to fluids of constant composition.

T

RR

P

RTG

RT

V

P

RR

T

RTGT

RT

H

Useful restricted forms are:

P

RR

dPRT

V

RT

G0

constTdPRT

V

RT

Gd

RR

Thus the residual Gibbs energy serves as a generating function for the other residual properties and here a direct link with experiment does exist.

Integration from zero pressure to arbitrary pressure P yields:

where at the lower limit GR/RT is equal to zero because the zero-pressure state is an ideal-gas state.

TP

dPZ

RTG P

R

constant 10

Differentiation with respect to temperature in accord to

give:

P

P

R

P

dP

T

ZT

RT

H0

P

RR

T

RTGT

RT

H

Defining equation for Gibbs energy is:

RT

G

RT

H

R

S RRR

The residual entropy is therefore:

The residual entropy is found:

P P

P

R

P

dPZ

P

dP

T

ZT

R

S0 0

1

RRR TSHG

The compressibility factor is defined asZ = PV/RT; values of Z and (∂Z/∂T)p

therefore come from experimental PVT data and the two integrals are evaluated by numerical or graphical method.

Alternatively the two integrals are evaluated analytically when Z is expressed as function of T and P by volume- explicit equation of state .

Thus, given PVT data or an appropriate equation of state, HR and SR can be evaluated and all other residual properties.

H = Hig + HR S = Sig + SR

Thus H and S follow from corresponding ideal-gas and residual properties by simple addition. Since:

T

T

igP

igig dTCHH0

0

and

0

0ln

0 PP

RTdT

CSS TT

ig

P

igig

Substitution into the preceding equations gives:

RT

T

igP

ig HdTCHH 0

0

RTT

ig

P

ig SPP

RTdT

CSS 0

0ln

0

(Example 6.3)

These two equation also may expressed alternatively to include the mean heat capacities:

R

H

igP

ig HTTCHH 00

R

S

ig

P

ig SPP

RTT

CSS 00

0lnln

Residual properties have validity for both gases and liquids.

Since the equations of thermodynamics which derive from the first and second laws do not permit calculation of absolute values for enthalpy and entropy and since in practice only relative values are needed the reference-state conditions T0 and P0

are selected for convenience and values are assigned to igH0 and igS0 arbitrarily.

Compressibility factor, Z is given:

Then,RT

BP

RT

G R

3.3 Residual Properties by Equations of State

RTBP

Z 1

By Equation,

2

,

1

T

B

dT

dB

TR

PT

T

RTGT

RT

H

xP

RR

or

dT

dB

T

B

R

P

RT

H R

Substitution of both equations,

dT

dB

R

P

RT

S R

Equation PV = ZRT can be written in the alternative form,

Differentiation at constant T gives:

RTZPV

Combination both gives:

Z

dZd

P

dP

dZZdRTdP

Upon substitution for dP /P ,

ZZd

ZRT

G R

ln110

10

Z

dTZ

TRTH R

00 1lnd

Zd

TZ

TZRS R

21 CBZ Using the 3-term virial equation:

ZCBRT

G R

ln2

32 2

2

2

1 dT

dC

T

C

dT

dB

T

BT

RT

H R

2

2

1ln

dT

dC

T

C

dT

dB

T

BTZ

RT

S R

The End