UNIT 10 ENTROPY - PART I
Unit Structure
10.011 Overview
10.1 Objectives
10.2 Introduction Entro!"
10.2.1 T#e $%&usius Ine'u&%it"
10.2.2 Entro!"
10.2.( T#e Incre&se o) Entro!" Princi!%e
10.( Entro!" Re%&tions
10.(.1 Entro!" $#&n*e o) Pure Subst&nces
10.(.2 Entro!" $#&n*e o) Ide&% +&ses
10.(.( Isentro!ic Process o) Ide&% +&ses
10.,2 Tutori&% S#eet
10.- Su..&r"
10./ Answers to Activities &nd Tutori&% S#eet
10.0O0ER0IE1
In Units 8 and 9, the second law of thermodynamics was introduced and its application to
cycles and cyclic devices was seen.
In this unit, the second law will be applied to processes and a new property called entropy
will be defined. A discussion of the Clausius Inequality which forms the basis of the
definition of entropy will first be done. The increase in entropy principle based on the
Clausius inequality will be introduced.
The entropy chanes that ta!e place durin processes for pure substances, incompressible
substances and ideal ases will be then discussed. Also, a special class of idealised
process, called isentropic processes, will be e"amined.
Unit 10
#
10.1 2EARNIN+ O34E$TI0ES
$y the end of this unit, you will be able to do the followin%&
#. 'istinuish between the application of (
nd
law to cycles and processes.
(. Understand the new property, entropy.
). Calculate entropy chanes for ideal ases.
*. Identify isentropic processes.
+. ,ead entropy values from tables for pure substances.
10.2 INTRO5U$TION ENTROPY
-ntropy is an abstract property of a system. It can be viewed as a measure of molecular
disorder or molecular randomness durin a process. Unli!e enery, entropy is a non&
conserved property and there is no such thin as the conservation of entropy principle.
-ntropy is best understood and appreciated by studyin its uses in commonly
encountered enineerin processes.
The concept of entropy may even be applied to human beins. -fficient people are those
who lead low&entropy lives. It ta!es minimum enery for them to find anythin they
need. 'isoranised people, however, lead hih entropy lives and are hihly inefficient.
It ta!es them hours to find somethin they need in a hurry instead of seconds.
10.2.1 T#e $%&usius Ine'u&%it"
$efore definin entropy, let us see how the Clausius inequality forms the basis for the
definition of entropy.
The Clausius Inequality is e"pressed as
6T#e c"c%ic inte*r&% o)
T
.
is &%w&"s %ess t#&n or e'u&% to 7ero89 that is,
/
T
Q
.
Unit 10
(
This inequality is valid for all cycles, reversible or irreversible. The validity of the
Clausius Inequality is demonstrated by considerin a system connected to a thermal
enery reservoir at an absolute temperature T throuh a reversible cyclic device.
0or reversible processes,
/
T
.
and irreversible processes,
/ <
T
Q
10.2.2 Entro!"
To develop a relation for the definition of entropy, let us e"amine the quantity
T
.
more closely. 1e have a quantity whose cyclic interal is 2ero for reversible processes
and less than 2ero for irreversible processes. 3et us consider a cycle consistin of two
internally reversible processes A and $ and another cycle consistin of two internally
reversible processes A and C.
1ritin the Clausius Inequality for the reversible processes, we have
B A rev
T
Q
T
Q
T
Q
,
_
+
,
_
,
_
#
(
(
#
for cycle A&$
C A rev
T
Q
T
Q
T
Q
,
_
+
,
_
,
_
#
(
(
#
for cycle A&C
And subtractin the two processes, we will have
Unit 10
)
A
3
$
1
2
:i*ure 10.1; Reversib%e Processes
C B
T
Q
T
Q
,
_
,
_
#
(
#
(
4 constant for all reversible processes between # and (.
Therefore, the quantity
,
_
T
Q
depends only on states # and ( and not on paths followed.
It is a point function !nown as -5T,678. A point function is a function which does not
depend on the path followed but on initial and final states whereas a path function
depends on the path followed.
Chane in entropy is denoted by 9 where
9 4
rev
T
Q
ds
int,
(
#
(
#
,
_
9 depends on initial and final states. The entropy chane 9, is the same between two
specified states, no matter what path, reversible or irreversible is followed durin a
process.
5ote % 1e have defined the chane in entropy instead of entropy itself. As for enery, the
entropy of a substance is assined a 2ero value at an arbitrarily selected reference state.
Units of entropy, 9 4
K
J
K
kJ
,
Units of specific entropy, s 4
K g
kJ
K kg
kJ
.
,
.
Activit" 1
A heat enine receives :// !; of heat from a hih temperature source at #/// < durin a
cycle. It converts #+/ !; of this heat to net wor! and re=ects the remainin *+/ !; to a
low temperature sin! at )// <. 'etermine if this heat enine violates the second law of
thermodynamics on the basis of >a? the Clausius Inequality and >b? the Carnot 7rinciple.
Unit 10
*
10.2.( T#e Incre&se o) Entro!" Princi!%e
The Clausius inequality lin!ed with the definition of entropy lead to an inequality !nown
as the Increase of -ntropy 7rinciple which states that
@T#e Entro!" o) &n Iso%&ted s"ste. durin* & !rocess &%w&"s incre&ses or in t#e
%i.itin* c&se o) & reversib%e !rocess9 re.&ins const&nt.A
A system and its surroundins can be viewed as subsystems of an I963AT-' 989T-B.
That is, -5T,678 CCA5D- 60 A5 I963AT-' 989T-B 9UB 60 -5T,678
CCA5D-9 60 TC- 989T-B A5' IT9 9U,,6U5'I5D9.
0 S S
sur sys Isolated
+ S
Total -ntropy applicable to both closed and open systems
Chane or
-ntropy Deneration
The increase in -ntropy principle states that the T6TA3 -5T,678 CCA5D-
A996CIAT-' 1ITC A 7,6C-99 must be positive or 2ero.
0or reversible processes, 9
total
4 9
sys
E 9
sur
4 /
Irreversible processes, 9
total
4 9
sys
E 9
sur
F/
5ote that the quantity 9
sys
can be less than 2ero but the sum 9
sys
E 9
sur
is always
positive.
Unit 10
+
'
<
>
PRO$ESSES I<POSSI32E 0
PRO$ESSES RE0ERSI32E 0
PRO$ESSES 2E IRRE0ERSI3 0
S
tot&%
0or e"ample, let us evaluate 9
total
when an amount of heat, . is transferred to
surroundins from a system at a temperature T
C
.
The temperature of surroundins is denoted by T
sur
9
sys
4
C
T
.
9
sur
4
sur
T
.
9
total
4
'
H sur
T T
Q
# #
Activit" 2
A frictionless piston and cylinder device contains saturated liquid water at (// !7a
pressure. 5ow, *+/ !; of heat is transferred to water from a source at +//
/
C and part of
the liquid vaporises at constant pressure.
'etermine the total entropy chane for this process, in !;G<. Is this process reversible,
irreversible or impossibleH
10.( ENTROPY RE2ATIONS
Two relations !nown as entropy relations are very important in thermodynamics to
evaluate chanes in entropy for both closed and open systems.
0rom the definition of entropyI
Tds q
rev
Unit 10
:
=
T
sur
:i*ure 10.2
T
C
0rom the #
st
law of thermodynamics applied to reversible closed systems,
Pdv du Tds
du Pdv Tds
du q
rev rev
+
E'u&tion 10.1
This relation is also !nown as the Dibbs -quation.
0rom the definition of enthalpy, h 4 u E 7v
$y differentiation, we obtain
dh = du + Pdv + vdP E'u&tion 10.2
0rom Dibbs -quation #/.#, Tds = du + Pdv E'u&tion 10.(
,eplacin #/.) into #/.(,
dh = Tds + vdP
Tds = dh vdP E'u&tion 10.,
These two relations >#/.#? and >#/.*?, !nown as entropy relations are very important
since they relate entropy chanes of a system to the chanes in other properties such as
enthalpy, pressure, volume and internal enery. Therefore, if variations in these
properties are !nown , chanes in entropy can be calculated.
The two equations >#/.#? and >#/.*? are valid for both reversible and irreversible systems
and for closed and open systems, and can also be e"pressed in terms of the entropy
chane as follows%&
'
+
T
vd7
T
du
ds
T
7dv
T
du
ds
Unit 10
J
The entropy chane durin a process can be determined by interatin either of these
equations between the initial and final states.
10.(.1 Entro!" $#&n*e o) Pure Subst&nces
The determination of entropy relations for pure substances is quite complicated and it is
preferable to use values of entropy from tables.
The readin of values of entropy is done in the same way as for other parameters for
different reions in the 7&K&T diaram of pure substances, that is
>#? Compressed liquidGsuperheated vapour. Kalues of entropy are directly
read from tables.
>(? 9aturated liquidGvapour mi"ture. The formula 9 4 9
f
Ex.9
f
is used to
calculate the averae value of the entropy of the mi"ture based on entropy
values for saturated liquid, saturated vapour and the quality.
>)? Compressed 3iquid ,eion
As for other properties, in the absence of compressed liquid data, revert to
the saturated liquid data and use s4s
f
at the correspondin temperature.
Unit 10
8
Compressed
liquid
9uperheated
vapour
9aturated
liquidGvapour
mi"ture
P
0
:i*ure 10.(; P-0 5i&*r&. )or Pure Subst&nces
Activit" (
A riid tan! contains + ! of ,efrierant #( >,
#(
? initially at (/
/
C and #*/ <7a. The
refrierant is cooled while bein stirred until its pressure drops to #// <7a. 'etermine
the entropy chane of the refrierant of the refrierant durin the process.
10.(.2 ENTROPY $>AN+E O: I5EA2 +ASES
There are basically two equations >#/.# and #/.*? to evaluate entropy chane of ideal
ases which are derived from the two entropy relations.
>a? 0rom Dibbs -quation,
T
PdV
T
dU
dS or
T
Pdv
T
du
ds + +
0or an ideal as, dU 4 C
v
dT where C
v
4 Ceat Capacity at Constant Kolume e"pressed as
>!;G<?
T
PdV
T
dT
C dS
v
+
0rom equation ).(, 7K 4 n,
u
T
,eplacin in the d9 relation above, we et
V
dV
nR
T
dT
C dS
u v
+
#
(
#
(
K
K
#
T
T
# n nR n C S
u v
+
E'u&tion 10.-
Units of
K
kJ
S
>b? 0rom the second entropy relation
T
VdP
T
dH
dS or
T
vdP
T
dh
ds
0or an ideal asI dH 4 C
p
dT where C
p
4 Ceat Capacity at Constant 7ressure >!;G<?.
Unit 10
9
T
VdP
T
dT C
dS
p
P
dP
nR
T
dT
C S
u p
#
(
#
(
# #
P
P
n nR
T
T
n C S
u p
E'u&tion 10./
5ote% An appro"imation has been done to evaluate
T
dT
C
v
and
T
dT
C
p
for the
determination of the entropy chane in the above equations. It has been assumed that C
v
and C
p
are independent of temperature. C
v
and C
p
are in fact dependent on temperature
and if we want to be precise, we have to ta!e into account the variation of C
p
and C
v
with
temperature and another method called the -LACT method has to be used. 0or the
purpose of this course, we will assume that C
p
and C
v
can be ta!en to be independent of
temperature so that we can evaluate the interals directly.
Activit" ,
Air at #+
/
C and #.(+ bars occupies /.(m
)
. The air is heated at constant volume until the
pressure is *.( bars and then cooled at constant pressure bac! to the oriinal temperature.
Calculate the net heat flow to or from the air and the net entropy chane usin the
followin data%
c
v
4 /.J#) !;G!.< c
p
4 #.// !;G!.<
,
u
4 8.)# !;G!mole.< Bolecular weiht of air 4 (9 !G!mole
10.(.( Isentro!ic Processes o) Ide&% +&ses
Isentropic processes are processes where the entropy is !ept constant.
0or an isentropic process, d9 4 ds 4 /
9ince, for a reversible process,
Tds Q
I therefore,
/ Q
Unit 10
#/
,eplacin
/ Q
in the #
st
law of thermodynamics,
dT C PdV
dU Q
v
1e !now from equation ).(, that
V
T nR
p
u
K PV V P V
V P
V P
V
V
V P
V P
n
V
V
V P
V P
T
T
T
T
V
V
n
T
dT
V
dV
C
nR
C
C
C
C
nR C C
T
dT
V
dV
C
nR
T
dT
C
V
dV
R n
dT C
V
TdV
nR
v
u
v
p
v
p
u v p
v
u
v u
v u
,
_
,
_
,
_
,
_
( ( # #
( (
# #
#
#
(
( (
# #
#
#
(
( (
# #
(
#
(
#
#
#
(
7
# ln
therefore , , * . ) eq 0rom
ln # #
# # Usin
$ut
E'u&tion 10.?
The relation
K PV
holds for isentropic processes of ideal ases and is applicable
when C
v
is assumed to be independent of temperature.
Usin equation ).( for ideal ases, alternative forms of this relation are
K TV
K P T
#
#
Unit 10
##
10., TUTORIA2 S>EET
# # ! of steam at J bars entropy :.+ !;G!.< is heated reversibly at constant
pressure until the temperature is (+/
/
C. Calculate the heat supplied and show on
a T&s diaram the area which represent the heat flow.
( A riid cylinder of volume /./(+ m
)
contains steam at 8/ bars and )+/
/
C. The
cylinder is cooled until the pressure is +/ bars. Calculate the state of the steam
after coolin and the amount of heat re=ected by the steam.
) 'ry saturated steam at #// bars e"pands isothermally and reversibly to a pressure
of #/ bars. Calculate the heat supplied and the wor! done per ! of steam durin
the process.
* /./) m
)
of nitroen >molar mass (8 !G!mol? contained in a cylinder behind a
piston is initially at #./+ bars and #+
/
C. The as is compressed isothermally
and reversibly until the pressure is *.( bars. Calculate the chane of entropy,
the heat flow and the wor! done. Assume 5itroen to act as a perfect as
+ 9team at #// bars, )J+
/
C e"pands isentropically in a cylinder behind a piston to a
pressure of #/ bars. Calculate the wor! done per ! of steam.
: In a steam enine, the steam at the beinnin of the e"pansion process is at J
bars, dryness fraction /.9+ and the e"pansion follows the law 7v
#.#
4 constant,
down to a pressure of /.)* bars. Calculate the chane of entropy per ! of steam
durin the process.
10.-SU<<ARY
Unit 10
#(
The second law of thermodynamics leads to the definition of a new property called
entropy, which is a quantitative measure of microscopic disorder for a system.
The definition of entropy is based on the Clausius Inequality, iven by
( ) !;G< /
T
Q
where the equality holds for internally or totally reversible processes and the inequality
for irreversible processes.
-ntropy is defined as
rev
T
Q
ds
int,
,
_
and has units >!;G<?
The entropy chane durin a process is obtained by interatin this relation
( ) !;G<
int,
(
#
# (
rev
T
Q
S S S
,
_
The Clausius inequality for irreversible pressures combined with the definition of entropy
yields an inequality which is !nown as the increase of entropy principle, whereby
0 S S
0
sur sys
isolated
+
S
Thus the total entropy chane durin a process is positive >for actual processes? or 2ero
>for reversible processes?. The entropy of a system or its surroundins may decrease
durin a process, but the sum of these two >system and surroundin? can never decrease.
-ntropy is a property, and it can be e"pressed in terms of more familiar properties such as
entropy throuh the Tds relations, e"pressed as
Unit 10
#)
Tds = du + Pdv
Tds = dh vdp
The successful use of these Tds relations also !nown as entropy relations depends on the
availability of property relations and can be used for ideal ases as follows%
0or ideal ases,
#
(
#
(
# (
#
(
#
(
# (
# #
# #
P
P
n nR
T
T
n C S S
V
V
n nR
T
T
n C S S
u p
u v
+
Cowever, for pure substances, entropy values are used directly from tables, where 949
(
&
9
#
and 9
#
and 9
(
are read from tables.
Isentropic processes are processes where the chane in entropy is equal to 2ero.
The ideal isentropic processes can be represented by 7K
4 cst where 4 ratio of specific
heat capacities at constant pressure and volume respectively.
In the ne"t unit, you will see the application of entropy to enineerin processes and the
use of property diarams.
Unit 10
#*
10./ ANS1ERS TO A$TI0ITIES AN5 TUTORIA2 S>EET
Activit" 1
>a?
!
!
H
H
T
Q
T
Q
T
Q
>since heat is supplied at cst temperature,
T
C
and re=ected at cst temperature, T
3
.
4
< G !; 9 . /
)//
*+/
#///
://
+
9ince the cyclic interal of
T
Q
is neative, this cycle satisfies the Clausius Inequality.
>b? 3et us compare efficiencies of Carnot enine and that of the present heat enine.
n
th
of Carnot 4
"
T
T
H
!
J/
#///
)//
# #
n
th
of heat enine 4
"
Q
Q
H
!
(+
://
*+/
# #
Unit 10
#+
T> @ 1000 A
:i*ure 10.,
T2 @ (00 A
,-0 B4
/00 B4 B4B4
1@1-0 B4
9ince n
th
of the heat enine C- M n
th
of Carnot heat enine, the heat enine is in
compliance with the Carnot principle.
Activit" 2
At (// <7a , T
sat
4 #(/.()
/
C.
9
Total
4 9
sys
E 9
sur
9
1ater
4
( )
!;G< #*+/ . #
(J) #(/
*+/
+
+
-ntropy chane of surroundin air, which serves as source
( )
( )
kJ#K $%&'()
$%&*(+ , +%+-&$ S S S
kJ#K % S
kJ Q Q
sur ./0er 1 To0/
sur
s2s sur
+
+
+8(# /
(J) +//
*+/
*+/
9ince 9
total
F /, then process is irreversible.
Activit" (
9tate # % 7
#
4 #*/ <7a
T
#
4 (/
/
C
9tate ( % 7
(
4 #// <7a
T
(
4 H
At 7
#
4 #*/ <pa, T
#
sat 4 &(#.9#
/
C, therefore state # is in the superheated reion.
Unit 10
#:
1&ter 100
0
$
,-0 B4
R
12
< ; - B*
:i*ure 10.-
:i*ure 10./
!;G!.! /.8/)+ s #*/ 7
G m #)9J . / K (/ T
# #
)
#
/
#
'
kP/
kg C
9ince this is a riid tan!, the specific volume remains the same at /.#)9J m
)
G!.
At 7
(
4 #// <7a ,
f
4 /.///:J m
)
!
4 /.#: m
)
G!
1e are in the saturated liquid vapour reion with x 4 /.8J)
Cence, 9
(
4 9
f
E x9
f
4 /.:)# !;G!.!
And -ntropy chane, 9 4 m>9
(
N 9
#
? 4 + O >/.:)# N /.8/)+?
4 &/.8:) !;G<
Activit" ,
T
#
4 #+
/
C 7
(
4 *.( bars 7
)
4 *.( bars
7
#
4 #.(+ bars
vo1
3s0
K
(
4 /./( m
)
pres
3s0
T
)
4 #+
/
C
K
#
4 /./( m
)
T
(
4 H K
)
4
,
_
,
_
,
_
,
_
(
4
u p
+
(
u
+
(
v
P
P
+n nR
T
T
+n 5C S
V
V
+n nR
T
T
+n 5C S
(
)
(
#
'etermination of states%
kg
T R
PV6
5
K
P
P T
T
u
)/ . /
J . 9:J
#
( #
(
#&(%14/
Unit 10
#J
( ) kJ T 5C U Q
v
)9 . #*+ (88 J . 9:J J#) . / ) . /
( )
( ) kJ Q
kJ
8 . ++ )9 . (// )9 . #*+ . .
)9 . (// 9:J.J & (88 #.// /.) .
T mC C Kd7 & C 7dK U . U 1 & . % ) (
() #( #)
p
+
+
Chane in -ntropy%
.< /.)*+!;G! &
. G (## . #
9:J.J
(88
# // . # 9
!;G!.< 8:J . /
(88
9:J.J
J#) . /
() #(
()
#(
+
,
_
,
_
S S S
K kg kJ n
+n S
0o0/1
Tutori&% S#eet
#.
!;G! (8) Q
(.
kg kJ Q
x
G # . +##
J+ . /
).
!;G! :)J.*
!;G! #( . 9/*
Q
*.
!;G! &##8.*
!;G! ##8.* . G *) . *##
kgK J ds
+.
kg kJ G * . )9/
:. 9 4 /.(9 !;G!.<
Unit 10
#8
