Download - Chapter 6 Entropy Ppt
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Chapter 6ENTROPY
Assoc.Prof.Sommai Priprem, PhD.
Department of Mechanical EngineeringKhon Kaen University
Reference: Sonntag R.E., and an !ylen ".#.,Introduction toThermodynamics: $lassical and Statistical, %rdEd., #ohn !iley & Sons, '(('
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.. 2
Introduction
Last chapter: 2ndLaw apply to CYCLE
This chapter: 2ndLaw apply to PROCESS
stlaw deal with Ener!y and itsConser"ation
2ndlaw deal with Entropy# it is notconser"ed
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Topics
Ine%uality o& Clausius Entropy Entropy o& pure su'stances Entropy Chan!e durin! Re"ersi'le Process Two I(portant ther(odyna(ic Relations Principle o& Increase o& Entropy Entropy chan!e o& solids and Li%uids
Entropy chan!e o& ideal !ases Isentropic Process o& Ideal )ases Second Law E&&iciency and Isentropic E&&iciency
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.. *
Ine%uality o& CL+,SI,S
Consider a Re"ersi'le -eat En!ineTheory: .L/ 01 .-.L3 4re"
T-and TL3 constant
0 TQ
HeatEngine
Source, TH
Sink, TL
Wrev
QH
QL
0...
0:....
0
=
==
==
=
T
Qcyclereversibleafor
T
Q
T
Q
T
QTherefore
T
Q
T
Q
T
Q
T
QFrom
QQQ
L
L
H
H
L
L
H
H
L
L
H
H
LH
L
L
H
H
T
Q
T
Q
=
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Now consider an Irre"ersi'le -eat En!ine which operates 'etween the sa(eT-and TLand recei"es the sa(e .-'ut reects .L7 and produce 4irr
Theory: 4irr 84re" 1 .-.7L3 4irrthere&ore .7L / .L
HeatEngine
Source, TH
Sink, TL
Wirr
QH
QL
)1.6(0....
0...
0:....
0
'
'
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ENTROPY: + Property o& a Syste(
C
B
A
1
2
Consider Two Re"ersi'le Cycles +9 and +C
)3.6...(..........
)2.6.........(
............
0;.
0;.
0..
2
112
2
1
2
1
2
1
2
1
2
1
2
1
rev
rev
CB
CA
BA
T
QSS
T
QdSDefined
PROPERTYPATHonnob!saefinalandiniialonde"endsermThis
T
Q
T
Q
T
Q
T
Q
T
QCACycle
T
Q
T
Q
T
QBACycle
T
Qcyclerevfor
=
=
=
+
==
+
==
=
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Notes re!ardin! e%n 6;2
Re"ersi'le Process Irre"ersi'le process entropychan!e
entropy chan!e ! statepoint "# 2
Process #"#!! State 2 entropychan!e $ Process Re"ersi'le %$ Irre"ersi'le Process !&Entropy Property
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The Entropy o& a Pure Su'stance
re&erence "alue : assi!ned s& 4ater s&3 0 at 0;0
oC
Re&ri!erants s&3 0 at *0o
C ?eter(ine "alues the sa(e way as
Other properties1 u# h
See Ta'les @ollier dia!ra(: Ts and hs dia!;
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@ollier dia!ra(
)*s diagram
h*s diagram
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Entropy Chan!e in Re"ersei'le Process
==
===
==
=
3
2
3
232
212
1
2
112
2
112
1
11
)2.6(
TdsQm
#
T
h
T
#Q
mTT
Q
msss
T
QSSe#from
f$
rev
f$
rev
T
s
P
s1
1
s2
s3
2
3
Example: -eatin! sat; li%uid superheated "apor
Process 1-2phase chan!e sat;li% sat;"ap;# T 3constant
Inte!rate and apply st law
% 3 h2h3 h&!
Process 2-3sat; "apor superheated "apor# Tnot constant# need relation 'etween T and . toper&or( inte!ration
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System: R*'+.
nitial state: )'- satrated vapor/ state fi0ed. 1inal state: P+2no3n.
Process: Reversi4le and adia4atic.
Model: R*'+ ta4les.
Analysis:
1irst la3, adia4atic:
'5+6 +7 '8 '3+6 9
'3+6 '*+
Second la3, reversi4le and adia4atic:
s'6 s+
Example 8.1 Consider a cylinder &itted with a piston that containssaturate R2 "apor at 0C; Let this "apor 'e co(pressed in are"ersi'le adia'atic process until the pressure is ;6 @Pa; ?eter(inethe worB per Bilo!ra( o& R2 &or this process;
T
ss1=s
2
1
2
P
P2
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.. 2
Solution
ro( the R2 ta'les#
u3 66;*0 BDB!
s3 s23 0;0* BDB!FP23 ;6 @Pa
There&ore# &ro( the superheat ta'les &or R2#
T23 2;2GC# u23 200;5 BDB!;
w23 uH u2 3 66;*0 200;5 BDB!
3 $*; BDB!
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.. $
Two I(portant Ther(odyna(ics Relations
Consider a internally re"ersi'le CLOSE? syste(1 st Law
. 3 d, J 4
TdS 3 d, JPdK T ds = du + Pdv ;;;;;;;;;
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*
Principle o& Increase o& Entropy
)8.6(0
)7.6(0
)6.6(0
11,
11
;
0
0
0
0
0
+=
+=
>
+=
=
$ss!rro!ndinsysem$eneraion
isolae
$ss!rro!ndinsysemne
$ss!rro!ndinsysemne
$ss!rro!ndinsysem
dSdSdSDefined
dS
dSdSdS
Posiiveal%aysTT
henTTSince
TTQ
T
Q
T
QdSdSdS
T
QdS
T
QdS
Surroundings,temperature = T
S!stem,temperature = T
W
Q
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5
So(e Re(arBs a'out Entropy
Processes will occur only i& StotalM 0
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6
Principle o& Increase o& Entropy
SisolatedM 0
S!en3Stotal3Ssyste(JSsurroundin!sM 0 / 0 Irre"ersi'le process
3 0 Re"ersi'le process
8 0 I(possi'le
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S 3
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>
E!"#P$E 8.2 Suppose that B! o& saturated water "apor at00GC is condensed to a saturated li%uid at 00GC in a constantpressure process 'y heat trans&er to the surroundin! air# which isat 25GC; 4hat is the net increase in entropy o& the syste( plussurroundin!sQ
Solution
or the syste(# &ro( the stea( ta'les#
Ssyste( 3 (s&!3 6;0*>0 3 6;0*>0 BDF
Concernin! the surroundin!s# we ha"e
. to surroundin!s 3 (h&!3 225;0 3 225 BD
Ssurr 3 .To3 2A>;52A>;5 3 ;500 BDF
Snet 3 Ssyste( JSsurr 3 6;0*>0 J ;500
3 ;5220 BDF
)his increase in entropy is in accordance 3ith the principle of the increase
of entropy, and tells s, as does or e0perience, that this process can ta2e place.
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A
Entropy Chan!e o& a Solid or Li%uid
Solid Li%uid
Speci&ic -eat 3 Constant
K "ery s(all hu %
ds 3
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..
20
EXAMPLE 8.3 One kilogra o! li"#i$ %a&er i' (ea&e$ !ro 20)* &o +0)*.
*al#la&e &(e en&ro- (ange, a''#ing on'&an& '-ei!i (ea&, an$ o-are
&(e re'#l& %i&( &(a& !o#n$ %(en #'ing &(e '&ea &a/le'.
System: 4ater;nitial and final states: Fnown;Model: Constant speci&ic heat# "alue at roo( te(perature;
Solutionor constant speci&ic heat#
s2s3 *;>* ln A5> BDB!F
Co(parin! calculation &ro( "alue &ro( ta'le:
s2s3 s&A0C s&20C 3 ;A25 H 0;2A66 3 0;>A5> BDB!F
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2
Entropy Chan!e o& an Ideal )as
)12.6...(ln
)11.6...(,
,
)10.6...(ln
)+.6...(,
2
11
212
2
11
212
=
=
===
+=
+=
==+=
P
PR
T
dTCss
P
dPR
T
dTCdsherefore
PRTvanddTCdh$as&deal
vdPdhTdsfrom
Similarly
vvR
TdTCss
v
dvR
T
dTCdsherefore
vRTPanddTCd!$as&deal
Pdvd!Tdsfrom
"o
"o
"o
vo
vo
vo
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22
)1.6...(lnln)12.6.(
)13.6...(lnln)10.6.(
1
2
1
212
1
2
1
212
PPR
TTCssE#
v
vR
T
TCssE#
"o
vo
=
+=
; To intr!rate E%n;
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2$
E!"#P$E >;* Consider Ea(ple 5;6# in which oy!en is heated&ro( $00 to 500 F; +ssu(e that durin! this process the pressuredropped &ro( 200 to 50 BPa; Calculate the chan!e in entropy perBilo!ra(;
Solution#ethod 1; The (ost accurate answer &or the entropy chan!e# assu(in! ideal!as'eha"ior# would 'e &ro( the ideal!as ta'les# Ta%le ".1&; This result is# usin! E'. (.1(#
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2*
#ethod 3; or constantp at3,Ta'le +;2
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25
EXAMPLE 8. *al#la&e &(e (ange in en&ro- -er kilogra a' air i'
(ea&e$ !ro 300 &o 600 4 %(ile -re''#re $ro-' !ro 00 &o 300 kPa.
A''#e: 1. *on'&an& '-ei!i (ea&. 2. 5aria/le '-ei!i (ea&.
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26
Isentropic Process o& Ideal )ases
(6.17)....................on'&an&
0
0
0)(
1
1
0)(,
)(1
e"n.a'$eal
0
'en&ro-i0,$'-roe'',a$ia/a&ian8or1
,1
!ro
=
=+
=+
=++
=++
+=
=+=+=
=
=
=
=
=
'
v
v
"v
vP
v
"
Pvv
dv'
P
dP
'PdvvdP
PdvvdPPdv
'
PdvvdPPdvR
Cherefore
vdPPdvR
dT
PdvdTCPdvd!Tds
'
'RC
'
RChen
CCRand
C
C'
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..
2
).....(6.1+
...(6.18)..........an$
on'&an&
9:Pe"#a&ionga'i$eal%i&*o/ine
1
2
1
)1(
1
2
1
2
2
1
2
1
1
2
2211
=
=
=
=
====
'''
''
'''
v
v
P
P
T
T
(
(
v
v
P
P
vPvPPv
Isentropic Process o& Ideal )ases
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2>
1
2
1
2
1
2
1
2
r
r
r
r
v
v
v
v
PP
PP
=
=
Ta'le +
&or isentropic process o& ideal !as only
Relati"e Pressure# Pr
Relati"e Kolu(e# "r
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.. 2A
).....(6.21
...(6.20)..........an$
/eoe6.1+
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.. $0
P
E l 8 0 I i'l it i d i
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.. $
Example 8.0In a re"ersi'le process nitro!en is co(pressed in acylinder &ro( 00 BPa# 20GC to 500 BPa; ?urin! the co(pressionprocess the relation 'etween pressure and "olu(e is PK;$ 3constant; Calculate the worB and heat trans&er per Bilo!ra(# andshow this process on P" and T s dia!ra(s;
Syste(: Nitro!en;
Initial state: P# T1 state Bnown;
inal state: P2
Process: Re"ersi'le# polytropic with eponent n 3;$
?ia!ra(: P"# Ts
@odel: Ideal !as# constant speci&ic heat"alue at $00 F;
+nalysis: 9oundary (o"e(ent worB; 43PdKirst law: %3 u2uJw
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Solution
polytropic process P"n3c
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.. $$
Second Law E&&iciency
6.2).........(:$e;i,e',on'#e$%ork
6.2).........(:$e;i,e'-ro$#,e$ %ork
)e!!i,ien,A$ia/a&i,(ore!!i,ien,7'en&ro-i,
6.23).........(P#-=ea&
(6.22)..........engine=ea&
-o''i/le)(a>.7$eal;'.A,al:e!!i,ien,la%2n$
ac!al
isenro"ic
isen
isenro"ic
ac!alisen
rev
ac!al
rev
ac!al
%
%%
%
COPCOPCOP
=
=
=
=
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.. $*
"
ss1=s2s s2
P1
2
1
2s
P2"2
"2s
"1
#a#s
$sentropicProcess
%ctua&Process
T
ss1=s2s s2
P1
2
1
2s
P2T2
T2s
T1
$sentropicProcess
%ctua&Process
7'en&ro-i E!!iien o! Compressors
12
12
1212
,
an$:la%1'&
hh
hhherefore
hh%hh%
%
%
%
%
scom"
ass
a
s
ac!al
isenro"ic
isen
=
==
==
w
'ompressor
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.. $5
T
ss1=s2s s2
P2
22s
1
P1
T2
T1
T2s
$sentropicProcess
%ctua&Process
7'en&ro-i E!!iien o! Turbines
12
12
1212
,
,
an$:la%1'&
hh
hhherefore
hh%hh%
%
%
%
%
s
!rbine
ass
s
a
isenro"ic
ac!al!rbisen
=
==
==
wTur(ine
Ea(ple 6> Stea( enters an adia'atic tur'ine steadily at $
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.. $6
Ea(ple 6 > Stea( enters an adia'atic tur'ine steadily at $@Pa and *00C and lea"es at 50 BPa and 00C; i& the power outputo& the tur'ine is 2 @4 and FE and PE are ne!li!i'le# deter(ine
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.. $
Solution
+ssu(e Ideal !as : P" 3 RT
Isentropic process : P"B3 constant
B 3 CpC"
Ea(ple 6A +ir is co(pressed 'y an adia'atic co(pressor &ro(00 BPa and 2C and to a pressur o& >00 BPa at a stead rate o& 0;2B!s; I& the the adia'atic e&&iciency o& the co(pressor is >0V#deter(ine
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Su((ary
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.. $A
Ine%uality o& CL+,SI,S
Consider a Re"ersi'le -eat En!ineTheory: .L/ 01 .-.L3 4re"
T-and TL3 constant
0 TQ
HeatEngine
Source, TH
Sink, TL
Wrev
QH
QL
0...
0:....
0
=
==
==
=
T
Qcyclereversibleafor
T
Q
T
Q
T
QTherefore
T
Q
T
Q
T
Q
T
QFrom
QQQ
L
L
H
H
L
L
H
H
L
L
H
H
LH
L
L
H
H
T
Q
T
Q
=
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.. *0
Now consider an Irre"ersi'le -eat En!ine which operates 'etween the sa(eT-and TLand recei"es the sa(e .-'ut reects .L7 and produce 4irr
Theory: 4irr 84re" 1 .-.7L3 4irrthere&ore .7L / .L
HeatEngine
Source, TH
Sink, TL
Wirr
QH
QL
)1.6(0....
0...
0:....
0
'
'
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.. *
ENTROPY: + Property o& a Syste(
C
B
A
1
2
Consider Two Re"ersi'le Cycles +9 and +C
)3.6...(..........
)2.6.........(
............
0;.
0;.
0..
2
112
2
1
2
1
2
1
2
1
2
1
2
1
rev
rev
CB
CA
BA
T
QSS
T
QdSDefined
PROPERTYPATHonnob!saefinalandiniialonde"endsermThis
T
Q
T
Q
T
Q
T
Q
T
QCACycle
T
Q
T
Q
T
QBACycle
T
Qcyclerevfor
=
=
=
+
==
+
==
=
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.. *$
Principle o& Increase o& Entropy
)8.6(0
)7.6(0
)6.6(0
11,
11
;
0
0
0
0
0
+=
+=
>
+=
=
$ss!rro!ndinsysem$eneraion
isolae
$ss!rro!ndinsysemne
$ss!rro!ndinsysemne
$ss!rro!ndinsysem
dSdSdSDefined
dS
dSdSdS
Posiiveal%aysTT
henTTSince
TTQ
T
Q
T
QdSdSdS
T
QdS
T
QdS
Surroundings,temperature = T
S!stem,temperature = T
W
Q
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.. **
Entropy Chan!e o& a Solid or Li%uid
Solid Li%uid
Speci&ic -eat 3 Constant
K "ery s(all hu %
ds 3
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.. *5
Entropy Chan!e o& an Ideal )as
)12.6...(ln
)11.6...(,
,
)10.6...(ln
)+.6...(,
2
11
212
2
11
212
=
=
===
+=
+=
==+=
P
PR
T
dTCss
P
dPR
T
dTCdsherefore
PRTvanddTCdh$as&deal
vdPdhTdsfrom
Similarly
vvR
TdTCss
v
dvR
T
dTCdsherefore
vRTPanddTCd!$as&deal
Pdvd!Tdsfrom
"o
"o
"o
vo
vo
vo
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.. *6
).....(6.1+
...(6.18)..........an$
on'&an&
9:Pe"#a&ionga'i$eal%i&*o/ine
1
2
1
)1(
1
2
1
2
2
1
2
1
1
2
2211
=
=
=
=
====
'''
''
'''
v
v
P
P
T
T
(
(
v
v
P
P
vPvPPv
Isentropic Process o& Ideal )ases
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.. *
).....(6.21
...(6.20)..........an$
/eoe6.1+
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.. *>
Second Law E&&iciency
6.2).........(:$e;i,e',on'#e$%ork
6.2).........(:$e;i,e'-ro$#,e$ %ork
)e!!i,ien,A$ia/a&i,(ore!!i,ien,7'en&ro-i,
6.23).........(P#-=ea&
(6.22)..........engine=ea&
-o''i/le)(a>.7$eal;'.A,al:e!!i,ien,la%2n$
ac!al
isenro"ic
isen
isenro"ic
ac!alisen
rev
ac!al
rev
ac!al
%
%%
%
COP
COPCOP
=
=
=
=
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So(e Selected Pro'le(s
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. . 0
>; Consider a Carnotcycle heat en!ine with water as the worBin!&luid; The heat trans&er to the worBin! &luid taBes place at $00GC1 durin! thisprocess the water chan!es &ro( saturated li%uid to saturated "apor; -eat isreected &ro( the worBin! &luid at *0GC; * +n insulated cylinder &itted with a &rictionless piston contains
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. . 1
>;* +n insulated cylinder &itted with a &rictionless piston contains0; B! o& water at 00GC# A0 percent %uality; The piston is (o"ed#co(pressin! the water until it reaches a pressure o& ;2 @Pa; -ow (uchworB is re%uired in this processQ
>;5 + cylinder containin! R22 at 0GC# 00 BPa# has an initial"olu(e o& 20 L; + piston co(presses the R22 in a re"ersi'le#isother(al process until it reaches the saturated "apor point; Calculatethe re%uired worB and heat trans&er needed to acco(plish this chan!e o&state;
>;6 One Bilo!ra( o& water at $00GC epands a!ainst a piston ina cylinder unril is reaches a('ient pressure# 00 BPa# at which point thewater has a %uality o& A0 percent; It (ay 'e assu(ed that theepansion is re"ersi'le and adia'atic;;> + ri!id# insulated "essel contains superheated "apor stea(t > @P $5O C + l th l i d ll i t t
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. . 2
at ;> @Pa# $5OoC ; + "al"e on the "essel is opened# allowin! stea( toescape; It (ay 'e assu(ed that at any instant the stea( re(ainin!inside the "essel has under!one a re"ersi'le# adiWa'atic epansion;?eter(ine the &raction o& stea( that has escaped when the stea(re(ainin! inside the "essel reaches the saturated "apor line;
>;A +n insulated cylinder &itted with a &rictionless piston contains0; B! o& superheated "apor stea(; The stea( epands to a('ientpressure# 00 BPa# at which point the te(perature o& the stea( insidethe cylinder is 50GC; The stea( does 50 BD o& worB a!ainst the pistondurin! this epansion; 4hat were the initial pressure and te(peratureQ
>;2 4e wish to cool a !i"en %uantity o& (aterial rapidly to ate(perature o& lOGC; The process re%uires a heat reection o& 2000 BD;One possi'ility is to i((erse the (aterial in a (iture o& ice and water#allowin! heat trans&er &ro( the (aterial to the ice# which (elts the ice;+nother possi'ility is to cool the (aterial 'y e"aporatin! RW22 at 20GC;The heat trans&er to the R22 chan!es it &ro( a saturated li%uid to a
saturated "apor; + third possi'ility is to e"aporate li%uid nitro!en at0;$BPa pressure;
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605 Nitro!en !as is co(pressed &ro( >0 BPa and2GC to *>0 BPa 'y a OB4 co(pressor; ?eter(ine the (ass&low rate o& nitro!en throu!h the co(pressor# assu(in! the
co(pression process to 'e =a> isentropic;
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. .
60A Stea( enters an adia'atic tur'ine at > @Pa and 500GC witha (ass &low rate o& $ B!s and lea"es at $0 BPa; The adia'atic e&&iciencyo& the tur'ine is 0;A0; Ne!lectin! the Binetic ener!y chan!e o& the stea(#deter(ine 0 (sand lea"es at 50 BPa# 00GC# and *0 (s; I& the power output o& thetur'ine is 5 @4# deter(ine 0 percent# deter(ine
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. .
6* +ir enters an adia'atic co(pressor at 00 BPa and GC at arate o& ;2 ($s# and it eits at 25GC; The co(pressor has anadia'atic e&&iciency o& >* percent; Ne!lectin! the chan!es in Binetic
and potential ener!ies# deter(ine 0 (so I& the
adia'atic e&&iciency o& the co(pressor is >5 percent# deter(ine
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End of Chapter 6