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Advanced Thermodynamics

Thermodynamic Properties of Fluids

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Property relations for homogeneous phases

First law for a closed system dWdQnUd +=)(

a special reversible process

revrev dWdQnUd +=)()(nVPddWrev = )(nSTddQrev=

)()()( nVPdnSTdnUd =

Only properties of system are involved:

It can be applied to any process in a closed system (not necessarily

reversible processes)

The change occurs between e!uilibrium states

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PdVTdSdU =

SandUTVP """"The primary thermodynamic properties:

PVUH +The enthalpy:

TSUA The #elmholt$ energy:

TSHG The %ibbs energy:

For one mol of homogeneous fluid of constant composition:

)()()( nVPdnSTdnUd =

VdPTdSdH +=

SdTPdVdA =

SdTVdPdG = &a'wells e!uations

VS S

P

V

T

=

PS S

V

P

T

=

TV V

S

T

P

=

TP P

S

T

V

=

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nthalpy" entropy and internal energy change calculationsf (P,T)

dP

P

HdT

T

HdH

TP

+

= dP

P

SdT

T

SdS

TP

+

=

VdPTdSdH +=P

P

CT

H=

VP

ST

P

H

TT

+

=

dPVP

STdTCdH

T

P

+

+=

TP P

S

T

V

=

dPT

V

TVdTCdHP

P

+=

T

C

T

S P

P

=

dPT

V

T

dT

CdSP

P

=PP T

ST

T

H

=

TP P

S

T

V

=

PVUH +

VP

VP

P

U

P

H

TTT

+

+

=

TTT P

V

PP

V

TP

U

=

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*etermine the enthalpy and entropy changes of li!uid water for a

change of state from + bar and ,-./ to +000 bar and -0./ The data

for water are given

#+and 1+at + bar" ,-./#,and 1,at + bar" -0./ #2and 12at +000 bar" -0./

dPT

VTVdTCdH

P

P

+=

dPT

V

T

dTCdS

P

P

=

( ) )(+)( +,,+, PPVTTTCH P +=

( )VdPTdTCdH P += +

V

T

V

P

=

volume e'pansivity

)(ln +,+

, PPVT

TCS P =

( )molJH 32400+0

)++000)(,04+5()+-2,2)(+0-+2(+

)+-,65+-2,2(2+07-

8

=

+=

( )molJS 3+2-+0

)++000)(,04+5(+0-+2

+-,65

+-2,2

ln2+07-

8

=

=

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Internal energy and entropy change calculationsf (V,T)

dVVUdT

TUdU

TV

+

=

dVVSdT

TSdS

TV

+

=

V

V

CT

U=

dVPV

STdTCdU

T

V

+=

VT T

P

V

S

=

dVPT

PTdTCdU

V

V

+=

T

C

T

S V

V

=

dV

T

P

T

dTCdS

V

V

+=

PV

ST

V

U

TT

=

VV T

ST

T

U

=

VT T

P

V

S

=

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SdTVdPdG =

%ibbs energy G = G (P,T)

dTRT

GdG

RTRT

Gd

,

+

Thermodynamic property of great potential utility

SdTVdPdG =

dTRTHdP

RTV

RTGd

,

TSHG

dTTRTGdP

PRTG

RTGd

PT

+ =)3()3(

TP

RTG

RT

V

=)3(

PTRTGT

RTH

= )3(

G/RT = g (P,T)

The %ibbs energy serves as a generating function

for the other thermodynamic properties

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9esidual properties

The definition for the generic residual property:

;MandMgare the actual and ideal" #" 1" or %

The residual %ibbs energy serves as a generatingfunction for the other residual properties:

gR MMM

dTRT

HdP

RT

V

RT

Gd

RRR

,=

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dTRT

HdP

RT

V

RT

Gd

RRR

,=

const TdP

RT

V

RT

Gd

RR

=

= P

RR

dPRT

V

RT

G

0

P

RT

P

!RTVVV gR ==

=P

R

P

dP!

RT

G

0

)+(PT

RTGTRTH =

)3(

=P

P

R

P

dP

T

!T

RT

H

0

=

PP

P

R

P

dP!

P

dP

T

!T

R

S

00)+(

RT

G

RT

H

R

S RRR=

const T

const T

? @ P=39T: e'perimental measurement %iven P=T data or an appropriate e!uation

of state" we can evaluate #9and 19and hence all other residual properties

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{ }

{ }

( ){ }

( ){ }

++=

++=

++=

++=+=

P

P

g

P

PH

g

P

g

P

P

g

P

P

T

T

g

P

gRg

P

dP

T

!RTTT"C#ATTMCPHRH

P

dP

T

!RTTTCH

P

dP

T

!RT"C#ATT$CPHRH

P

dP

T

!RTdTCHHHH

0

,

000

0

,

00

0

,

00

0

,

0

)"""A"(

)"""A"(

0

+

+=

+

+=

+

+=

+

+=+=

PP

P

g

PP

PS

g

P

g

PP

P

g

PP

P

T

T

g

P

gRg

P

dP

!P

dP

T

!

TRP

P

RT

T

"C#ATTMCPSRS

P

dP!

P

dP

T

!TR

P

PR

T

TCS

P

dP

!P

dP

T

!

TRP

P

R"C#ATT$CPSRS

P

dP!

P

dP

T

!TR

P

PR

T

dTCSSSS

0000

00

0000

0

000

00

000

0

)+(lnln)"""A"(

)+(lnln

)+(ln)"""A"(

)+(ln0

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/alculate the enthalpy and entropy of saturated isobutane vapor at 280

B from the following information: (+) compressibility

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9esidual properties by e!uation

of state If ? @ f (P"T):

The calculation of residual properties for gases

and vapors through use of the virial e!uations and

cubic e!uation of state

=

P

P

R

P

dP

T

!T

RT

H

0

=

PP

P

R

P

dP!

P

dP

T

!T

R

S

00

)+(

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>se of the virial e!uation of state

RT

#P! =+

=P

R

P

dP!

RT

G

0

)+(two

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,+ C#! +=

=P

R

PdP!

RTG

0

)+(

Three

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>se of the cubic e!uation of state

The generic cubic e!uation of state:

))((

)(

'V'V

Ta

'V

RTP

++

=

)+)(+(+

+

''

'(

'!

++

=

'RT

Ta(

)(

($'d

! =

)+ln()+(

0

$dT

d(d

T

!=

0

T)on*+

''

'd$

++

0

)+)(+(

)(

($Td

Td

!RT

H

r

r

R

+= +ln)(ln

+

( ) ($Td

Td!

R

S

r

r

R

+= +

ln

)(lnln

RT

'P

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Find values for the residual enthalpy #9and the residual entropy 19

for n

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Two

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GG = dGdG =SdTVdPdG =

dTSdPVdTSdPV *a+*a+

=

V

S

VV

SS

dT

dP*a+

=

=

VdPTdSdH += STH =The latent heat of phase transition

VT

H

dT

dP*a+

=The /lapeyron e!uation

Ideal gas" and =l =g

*a+

v

P

RTVV ==

)3+(

ln

Td

PdRH

*a+lv =

The /lausius3/lapeyron e!uation

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VT

H

dT

dP*a+

=

The /lapeyron e!uation: a vital connection between the properties of different phases

)(TfP=

T

#AP*a+ =ln For the entire temperature range from the triple point

to the critical point

CT

#AP*a+

+=ln The Antoine e!uation" for a specific temperature range

+++=

+ln

82-+ "C#AP*a+r

The Cagner e!uation" over a wide

temperature range

rT=+

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Two

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Thermodynamic diagrams and tables

A thermodynamic diagram represents the temperature"

pressure" volume" enthalpy" and entropy of a substance on

a single plot /ommon diagrams are:

; T1 diagram

; P# diagram (ln P vs #)

; #1 diagram (&ollier diagram)

In many instances" thermodynamic properties are reported

in tables The steam tables are the most thoroughcompilation of properties for a single material

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Fig 8,

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Fig 82

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Fig 84

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1uperheated steam originally at P+and T+e'pands through a no$$le to an

e'haust pressure P, Assuming the process is reversible and adiabatic" determine

the downstream state of the steam and G# for the following conditions:

(a) P+@ +000 HPa" T+@ ,-0./" and P,@ ,00 HPa

(b) P+@ +-0 psia" T+@ -00.F" and P,@ -0 psia

1ince the process is both reversible and adiabatic" the entropy change

of the steam is $ero(a) From the steam stable and the use of interpolation"

At P+@ +000 HPa" T+@ ,-0./:#+@ ,64,6 H3Hg" 1+@ 86,-, H3HgB

At P,@ ,00 HPa"

1,@ 1+@ 86,-, H3HgB 1saturated vapor"

the final state is in the two

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(a) From the steam stable and the use of interpolation"

At P+@ +-0 psia" T+@ -00.F:

#+@ +,742 Jtu3lbm" 1+@ +880, Jtu3lbm9

At P,@ -0 psia"

1,@ 1

+@ +880, Jtu3lbmB K 1

saturated vapor"

the final state is in the superheat region:.T ,5,52,=

l'm

#+/HHH 066+, ==

l'm#+/H 32++7-,=

A 2 H i H f li id i ilib i i h

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A +- m2tanH contains -00 Hg of li!uid water in e!uilibrium with pure water

vapor" which fills the remainder of the tanH The temperature and pressure are

+00./" and +0+22 HPa From a water line at a constant temperature of 70./ and

a constant pressure somewhat above +0+22 HPa" 7-0 Hg of li!uid is bled into

the tanH If the temperature and pressure in the tanH are not to change as a resultof the process" how much energy as heat must be transferred to the tanHL

mHQd+

mUd )v += )(

nergy balance:d+

dmHQ

d+

mUd )v)v += )(

vv mHmUQ = )(

PVUH +=

vvv mHPmVmHQ = )()(

vvv mHHmHmQ = )()( ++,,

At the end of the process" the tanH still contains saturated li!uid and

saturated vapor in e!uilibrium at +00./ and +0+22 HPa

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vvv mHHmHmQ = )()( ++,,Cl(/d*a+/ra+ed

-g

-JH 70M0,62=

Cl(/d*a+/ra+ed-g

-JHl)v

+00M+4+6=

Cva0or*a+/ra+ed-g

-JH

v

)v

+00M0,878=

From the steam table" the specific volumes of saturated li!uid and saturated

vapor at +00./ are 000+044 and +872 m23Hg " respectively

-JHmHmHm vvll)v ,++8+8)0,878(872+

)00+0440)(-00(-+)+4+6(-00)( ++++++ =

+=+=

vl mmm ,,, 7-0872+

)00+0440)(-00(-+-00 +=+

+= lv mm ,, 00+0440872+-+ +=

-JHmHmHm vvll)v -,44-5)( ,,,,,, =+=

-JmHHmHmQ vvv 6206,)0,62)(7-0(,++8+8-,44-5)()( ++,, ===

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rPPP= rTTT=

= P

P

R

P

dP

T

!T

RT

H

0

=

PP

P

R

P

dP

!P

dP

T

!

TR

S

00 )+(

= r

r

P

r

r

Pr

r

R

P

dP

T

!T

RT

H

0

,

= rr

r

P

r

rP

r

r

Pr

r

R

P

dP

!P

dP

T

!

TR

S

00 )+(

+0 !!! +=

=

r

r

r

r

P

r

r

Pr

r

P

r

r

Pr

r

)

R

PdP

T!T

PdP

T!T

RTH

0

+

,

0

0

,

= rr

r

rr

r

P

r

rP

r

r

Pr

r

P

r

rP

r

r

Pr

r

R

P

dP!

P

dP

T

!T

P

dP!

P

dP

T

!T

R

S

0

+

0

+

0

0

0

0

)+()+(

( ) ( ))""(

+0

1M&GAPRTRHR#RT

H

RT

H

RT

H

R

R

R

=

+=

( ) ( ))""(

+0

1M&GAPRTRSR#R

S

R

S

R

S RRR

=

+= Table - N +,

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RT

T

g

P

g HdTCHH ,0,,

0 ++=

RT

T

g

P

g HdTCHH +0++

0 ++=

RRT

T

g

P HHdTCH +,,

+

+=

RT

T

g

P

g SP

PR

T

dTCSS ,

0

,0,

,

0

ln ++=

RT

T

g

P

g SP

PR

T

dTCSS

+0

+

0+

+

0

ln

++=

RRT

T

g

P SSP

PR

T

dTCS +,

+

,,

+

ln +=

RR

H

g

P HHTTCH +,+, )( +=

RR

S

g

P SSP

PCS +,

+

,ln +=

ti t = > # d 1 f + b t t ,00./ d 70 b if # d 1

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stimate =" >" #" and 1 for +

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Fig 87

#

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1tep (a)

T

#AP*a+ =ln

6,880+22+ln

#A=

4,04240ln

#A=

For @ ,72+-B"

Psat@ +,77+ bar

67666200

)0+2+(ln06,+=

=

nr

)

n

lv

n

T

P

RT

HThe latent heat at normal boiling point:

The latent heat at ,72+- B:

250

+

+

=

nr

r

lv

n

lv

T

T

H

H

mol

JHlv ,+5+0=

STH =%mol

JSlv

= 5476

1tep (b)8-00=rT 02+80=rP( ) ( )

065-0)""(

+0

==

+= 1M&GAPRTRHR#

RT

H

RT

H

RT

H

R

R

R

( ) ( )+0820)""(

+0

==

+= 1M&GAPRTRSR#

R

S

R

S

R

S RRR

mol

JHR 244+ =

%mol

JSR

= 550+

1tep (c)

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1tep (c)

mol

J&&$CPHHg ,0-84)00"85276"28202+"687+A+-472"+-,72(2+45 ==

%mol

J

&&$CPSSg

=

=

+5,,,77++

70ln2+45)00"85276"28202+"687+A+-472"+-,72(2+45

1tep (d) +,7+=rT 72++=rP

( ) ( )420,

+0

=

+=)

R

)

R

)

R

RT

H

RT

H

RT

H

( ) ( )70-+

+0

=

+=

R

S

R

S

R

S RRR

mol

JH

R

545-, =

%mol

JSR

= +5+4,

molJHH 24,22545-,0-84)244(,+5+0 =+==

%mol

JSS

=+== 7,55+5+4+5,,)550(5476

Total

mol

J

PVHU 2,,+5+03)5,57)(70(24,22 ===

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%as mi'tures

1imple linear mi'ing rules

2

)P) T2T

)P) P2P

stimate =" #9" and 19for an e!uimolar mi'ture of carbon dio'ide and propane

at 4-0B and +40 bar

%T2T

)P) 227)5286)(-0(),204)(-0( =+=

'arP2P

)P) +--5)454,)(-0()5272)(-0( =+= 4+,=0rP

22-+=0rT 86700 =!

,0-0+ =!

+550)+-,0)(-0(),,40)(-0( =+=

2 7280+0 =+= !!!

mol

)m

P

!RTV

2

7+68== ( ) ( ) 78,++0

=

+=

0

R

0

R

0

R

RT

H

RT

H

RT

H

( ) ( )0,6+

+0

=

+=

R

S

R

S

R

S RRR

lJH R 34627 %lJS R 3-85