ME152Slides CB Ch6

7/23/2019 ME152Slides CB Ch6

1/36

ME 152 1

Entropy

Cengel & Boles,

Chapter 6


2/36

ME 152 2

Entropy

From the 1st Carnot principle:

this is valid or t!o thermal reservoirs

"

11

1,,

LL

HH

H

H

L

L

H

L

H

L

H

L

H

L

H

L

H

outnet

T

Q

T

Q

T

Q

T

Q

T

T

Q

Q

T

T

Q

Q

T

T

Q

W

revthth


3/36

ME 152 #

The Clausius Inequality For a s$stem %ndergoing a c$cle and

comm%nicating !ith N thermal

reservoirs, it can e sho!n that

'his can e %rther generali(ed into

the Cla%si%s )ne*%alit$:

!here Q is the heat transer at a

partic%lar location along the s$stemo%ndar$ d%ring a portion o the c$cleand Tis the asol%te temperat%re at thatlocation+ 'his is called a c$clic integral+

=

N

i i

i

T

Q

1

"

"TQ


4/36

ME 152

The Clausius

Inequality, cont.

) the c$cle is internall$ reversile,

then

What does this mean? Consider:

=

"

revintT

Q

%nctions-dependent.pathare,/

","

-propertiesare,,/

",","

QWQW

PTV

dPdTdV

===


5/36

ME 152 5

New Property: Entropy

'hereore, the *%antit$ Q/T is a

propert$ o the s$stem in dierential

orm !hen the integration isperormed along an internall$

reversile path+ 'his propert$ is

0no!n as entropy /S-:

==

=

2

1revint

12

revint

or,

TQSSS

T

QdS


6/36

ME 152 6

New Property: Entropy

'his integral deines a ne! propert$ o the

s$stem called entropy /S-:

entrop$ is an etensive propert$ !ith %nits o034 speciic entrop$ is deined $s = S/m,

!ith %nits o 030g.4

the integration !ill onl$ $ield entrop$ change

!hen carried o%t along an int+ rev+ path

li0e enthalp$, entrop$ is a convenient and %se%lpropert$ that has een introd%ced !itho%t

ph$sical motivation its %tilit$ !ill e discovered

as !e learn more ao%t its characteristics

revint

2

112

= T

QSS


7/36

ME 152

Entropy Change and

Heat Transfer

7%ppose !e have a closed s$stem

%ndergoing an internall$ reversile

process !ith heat transer

i heat is added /Q8"-, then S28S1or

entrop$ increases

i heat is removed /Q9"-, then S29S1

or entrop$ decreases

i s$stem is adiaatic /Q"-, then

S2S1or entrop$ is constant /isentropic-

rev

int

2

112

= TQ

SS


8/36

ME 152 ;

Entropy Change and

Heat Transfer, cont.

Entrop$ e*%ation can e rearranged:

!hen temperat%re is plotted againstentrop$, the area %nder a process path is

e*%al to the heat transer !hen the

process is internall$ reversile+

( )

==

=2

1

2

1rev

int

rev

int

or TdsmTdSQ

TdSQ


9/36

ME 152

in determining an isentropic process rom the

ideal gas tales

const1

2

1

2

=

=

sr

r

P

P

P

P


24/36

ME 152 2

!elati*e #olue, vr

From the ideal gas la!,

'he *%antit$ T/Pris also ta%lated in

the ideal gas tales and is 0no!n as therelative vol%me, vr i the speciic

vol%me ratio is 0no!n, then vris %se%l

in inding the isentropic process $

setting

-/-/

33

1

2

11

22

1

1

2

2

1

2

T#T#

PTPT

'TP

P'T

vv

r

r ==

=

const1

2

1

2

=

=

sr

r

v

v

v

v


25/36

ME 152 25

Constant %pecific Heats&(ppro'. Isentropic (nalysis)

ecall:

!ith ) = (p/(v and (p= (v% ' & theollo!ing relations res%lt:

note thatPv) constant or isentropicprocesses th%s, these processes arepo*ytropic!here n = )

"lnln

"lnln

1

2

1

2,12

1

2

1

2

,12

=

=

+

P

P'

T

T(ss

v

v

'T

T

(ss

avp

avv

)))

)

v

v

P

P

v

v

T

T

P

P

T

T

=

=

=

2

1

1

2

1

2

1

1

2

1

1

2

1

2 ,,


26/36

ME 152 26

!e*ersi$le %teady+low

-or

ecall relation et!een heat transer

and entrop$ or an int+ rev+ process:

( )

=

==

=

2

112

2

1

2

1rev

2

1revint or,

vdPhh

vdPdh

Tds+

TdSQ


27/36

ME 152 2


-or, cont.

From C energ$ alance,

or a reversile process,

( )

( ) -/or

-/

21

2

2

2

121

21

12

2

1

2

22112

,,-hh+.

,,-hh.+

+++=

++=

VV

VV

( ) -/21

2

2

2

12

1

21

2

112rev

,,-

hhvdPhh.

+

++= VV

( ) -/ 212

2

2

121

2

1,,-vdP ++= VV


28/36

ME 152 2;


-or, cont.

ote that i .rev ", !e have the simple

orm o theernou**i e+uation

For t%rines, compressors, and p%mps!ith negligile 4E, GE eects:

( )-/

21

2

2

2

12

12

1rev ,,-vdP.

++= VV

=2

1rev vdP.


29/36

ME 152 2

Documents

ME152Slides CB Ch6