1. fluids 1
TRANSCRIPT
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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