chapter 10 vapor and combined power cyclessv.20file.org/up1/472_9.pdf · carnot vapor cycle 10-1c...
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PROPRIETARY MATERIAL. © 2008 The McGraw-Hill Companies, Inc. Limited distribution permitted only to teachers and educators for course preparation. If you are a student using this Manual, you are using it without permission.
10-1
Chapter 10 VAPOR AND COMBINED POWER CYCLES
Carnot Vapor Cycle
10-1C Because excessive moisture in steam causes erosion on the turbine blades. The highest moisture content allowed is about 10%.
10-2C The Carnot cycle is not a realistic model for steam power plants because (1) limiting the heat transfer processes to two-phase systems to maintain isothermal conditions severely limits the maximum temperature that can be used in the cycle, (2) the turbine will have to handle steam with a high moisture content which causes erosion, and (3) it is not practical to design a compressor that will handle two phases.
10-3E A steady-flow Carnot engine with water as the working fluid operates at specified conditions. The thermal efficiency, the quality at the end of the heat rejection process, and the net work output are to be determined.
Assumptions 1 Steady operating conditions exist. 2 Kinetic and potential energy changes are negligible.
Analysis (a) We note that
and
19.3%=−=−=η
=°===°==
R1.833R0.67211
R0.672F0.212R1.833F1.373
Cth,
psia14.7@sat
psia180@sat
H
L
L
H
TT
TTTT
(b) Noting that s4 = s1 = sf @ 180 psia = 0.53274 Btu/lbm·R,
0.153=−
=−
=44441.1
31215.053274.044
fg
f
sss
x
(c) The enthalpies before and after the heat addition process are
( )( ) Btu/lbm2.111216.85190.014.346Btu/lbm14.346
22
psia180@1
=+=+===
fgf
f
hxhhhh
Thus,
and, ( )( ) Btu/lbm148.1===
=−=−=
Btu/lbm0.7661934.0
Btu/lbm0.76614.3462.1112
inthnet
12in
qw
hhq
η
14.7 psia
180 psia
3
2
4
1
s
T
qin
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10-2
10-4 A steady-flow Carnot engine with water as the working fluid operates at specified conditions. The thermal efficiency, the amount of heat rejected, and the net work output are to be determined.
Assumptions 1 Steady operating conditions exist. 2 Kinetic and potential energy changes are negligible.
Analysis (a) Noting that TH = 250°C = 523 K and TL = Tsat @ 20 kPa = 60.06°C = 333.1 K, the thermal efficiency becomes
36.3%==−=−= 3632.0K523K333.1
11Cth,H
L
TT
η
(b) The heat supplied during this cycle is simply the enthalpy of vaporization,
Thus,
( ) kJ/kg1092.3=⎟⎟⎠
⎞⎜⎜⎝
⎛===
==
kJ/kg3.1715K523K333.1
kJ/kg3.1715
inout
250@in
qTT
hq
H
LL
Cfg o
(c) The net work output of this cycle is
( )( ) kJ/kg623.0 kJ/kg3.17153632.0inthnet === qw η
10-5 A steady-flow Carnot engine with water as the working fluid operates at specified conditions. The thermal efficiency, the amount of heat rejected, and the net work output are to be determined.
Assumptions 1 Steady operating conditions exist. 2 Kinetic and potential energy changes are negligible.
Analysis (a) Noting that TH = 250°C = 523 K and TL = Tsat @ 10 kPa = 45.81°C = 318.8 K, the thermal efficiency becomes
39.04%=−=−=K523K318.811Cth,
H
L
TT
η
(b) The heat supplied during this cycle is simply the enthalpy of vaporization,
Thus,
( ) kJ/kg1045.6=⎟⎟⎠
⎞⎜⎜⎝
⎛===
== °
kJ/kg3.1715K523K318.8
kJ/kg3.1715
inout
C250@in
qTT
hq
H
LL
fg
(c) The net work output of this cycle is
( )( ) kJ/kg669.7=== kJ/kg3.17153904.0inthnet qw η
250°C
s
T
20 kPa3
2
4
1qin
qout
250°C
s
T
10 kPa 3
2
4
1qin
qout
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10-3
10-6 A steady-flow Carnot engine with water as the working fluid operates at specified conditions. The thermal efficiency, the pressure at the turbine inlet, and the net work output are to be determined.
Assumptions 1 Steady operating conditions exist. 2 Kinetic and potential energy changes are negligible.
Analysis (a) The thermal efficiency is determined from
η th, C60 273 K350 273 K
= − = −++
=1 1T
TL
H
46.5%
(b) Note that
s2 = s3 = sf + x3sfg
= 0.8313 + 0.891 × 7.0769 = 7.1368 kJ/kg·K
Thus,
MPa1.40≅⎭⎬⎫
⋅=°=
22
2
KkJ/kg1368.7C350
PsT
(Table A-6)
(c) The net work can be determined by calculating the enclosed area on the T-s diagram,
Thus,
( )( )
( )( ) ( )( ) kJ/kg1623=−−=−−==
⋅=+=+=
5390.11368.760350Area
KkJ/kg5390.10769.71.08313.0
43net
44
ssTTw
sxss
LH
fgf
3
2
4
1
60°Cs
T
350°C
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10-4
The Simple Rankine Cycle
10-7C The four processes that make up the simple ideal cycle are (1) Isentropic compression in a pump, (2) P = constant heat addition in a boiler, (3) Isentropic expansion in a turbine, and (4) P = constant heat rejection in a condenser.
10-8C Heat rejected decreases; everything else increases.
10-9C Heat rejected decreases; everything else increases.
10-10C The pump work remains the same, the moisture content decreases, everything else increases.
10-11C The actual vapor power cycles differ from the idealized ones in that the actual cycles involve friction and pressure drops in various components and the piping, and heat loss to the surrounding medium from these components and piping.
10-12C The boiler exit pressure will be (a) lower than the boiler inlet pressure in actual cycles, and (b) the same as the boiler inlet pressure in ideal cycles.
10-13C We would reject this proposal because wturb = h1 - h2 - qout, and any heat loss from the steam will adversely affect the turbine work output.
10-14C Yes, because the saturation temperature of steam at 10 kPa is 45.81°C, which is much higher than the temperature of the cooling water.
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10-5
10-15E A simple ideal Rankine cycle with water as the working fluid operates between the specified pressure limits. The rates of heat addition and rejection, and the thermal efficiency of the cycle are to be determined.
Assumptions 1 Steady operating conditions exist. 2 Kinetic and potential energy changes are negligible.
Analysis From the steam tables (Tables A-4E, A-5E, and A-6E),
Btu/lbm52.13950.102.138Btu/lbm50.1
ftpsia5.404Btu1psia)6500)(/lbmft01645.0(
)(
/lbmft01645.0
Btu/lbm02.138
inp,12
33
121inp,
3psia6 @1
psia6@1
=+=+==
⎟⎟⎠
⎞⎜⎜⎝
⎛
⋅−=
−=
==
==
whh
PPw
hh
f
f
v
vv
Btu/lbm4.1120)88.995)(9864.0(02.138
9864.058155.1
24739.08075.1
psia6
RBtu/lbm8075.1Btu/lbm0.1630
F1200
psia500
44
44
34
4
3
3
3
3
=+=+=
=−
=−
=
⎭⎬⎫
==
⋅==
⎭⎬⎫
°==
fgf
fg
f
hxhhs
ssx
ssP
sh
TP
Knowing the power output from the turbine the mass flow rate of steam in the cycle is determined from
lbm/s9300.0kJ1
Btu0.94782/lbm1120.4)Btu(1630.0
kJ/s500)(43
outT,43outT, =⎟
⎠⎞
⎜⎝⎛
−=
−=⎯→⎯−=
hhW
mhhmW&
&&&
The rates of heat addition and rejection are
Btu/s913.6Btu/s1386
=−=−=
=−=−=
Btu/lbm)02.1380.4lbm/s)(1129300.0()(
Btu/lbm)52.1390.0lbm/s)(1639300.0()(
14out
23in
hhmQ
hhmQ&&
&&
0.341
and the thermal efficiency of the cycle is
=−=−=1386
6.91311in
outth Q
Q&
&η
qin
qout
6 psia 1
3
2
4
500 psia
s
T
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10-6
10-16 A simple ideal Rankine cycle with water as the working fluid operates between the specified pressure limits. The maximum thermal efficiency of the cycle for a given quality at the turbine exit is to be determined.
Assumptions 1 Steady operating conditions exist. 2 Kinetic and potential energy changes are negligible.
Analysis For maximum thermal efficiency, the quality at state 4 would be at its minimum of 85% (most closely approaches the Carnot cycle), and the properties at state 4 would be (Table A-5)
KkJ/kg7440.6)8234.6)(85.0(9441.0kJ/kg3.2274)3.2335)(85.0(27.289
85.0kPa30
44
44
4
4
⋅=+=+==+=+=
⎭⎬⎫
==
fgf
fgf
sxsshxhh
xP
Since the expansion in the turbine is isentropic,
kJ/kg5.3115 KkJ/kg7440.6
kPa30003
43
3 =⎭⎬⎫
⋅===
hss
P
Other properties are obtained as follows (Tables A-4, A-5, and A-6),
kJ/kg31.29204.327.289kJ/kg04.3 mkPa1
kJ1kPa)303000)(/kgm001022.0()(
/kgm001022.0
kJ/kg27.289
inp,12
33
121inp,
3kPa30@1
kPa30@1
=+=+==
⎟⎠
⎞⎜⎝
⎛
⋅−=
−=
==
==
whh
PPw
hh
f
f
v
vv
Thus,
kJ/kg0.198527.2893.2274kJ/kg2823.231.2925.3115
14out
23in
=−=−==−=−=
hhqhhq
and the thermal efficiency of the cycle is
0.297=−=−=2.28230.198511
in
outth q
qη
qin
qout
30 kPa 1
3
2
4
3 MPa
s
T
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10-7
10-17 A simple ideal Rankine cycle with water as the working fluid operates between the specified pressure limits. The power produced by the turbine and consumed by the pump are to be determined.
Assumptions 1 Steady operating conditions exist. 2 Kinetic and potential energy changes are negligible.
Analysis From the steam tables (Tables A-4, A-5, and A-6),
kJ/kg47.25505.442.251kJ/kg05.4 mkPa1
kJ1kPa)204000)(/kgm001017.0()(
/kgm001017.0
kJ/kg42.251
inp,12
33
121inp,
3kPa20@1
kPa20@1
=+=+==
⎟⎠
⎞⎜⎝
⎛
⋅−=
−=
==
==
whh
PPw
hh
f
f
v
vv
kJ/kg7.2513)5.2357)(9596.0(42.251
9596.00752.7
8320.06214.7
kPa20
KkJ/kg6214.7kJ/kg3.3906
C700kPa4000
44
44
34
4
3
3
3
3
=+=+=
=−
=−
=
⎭⎬⎫
==
⋅==
⎭⎬⎫
°==
fgf
fg
f
hxhhs
ssx
ssP
sh
TP
The power produced by the turbine and consumed by the pump are
kW203
kW69,630
===
=−=−=
kJ/kg)kg/s)(4.0550(
kJ/kg)7.2513.3kg/s)(390650()(
inP,inP,
43outT,
wmW
hhmW
&&
&&
qin
qout
20 kPa 1
3
2
4
4 MPa
s
T
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10-8
10-18E A simple ideal Rankine cycle with water as the working fluid operates between the specified pressure limits. The turbine inlet temperature and the thermal efficiency of the cycle are to be determined.
Assumptions 1 Steady operating conditions exist. 2 Kinetic and potential energy changes are negligible.
Analysis From the steam tables (Tables A-4E, A-5E, and A-6E),
Btu/lbm76.13758.718.130Btu/lbm58.7
ftpsia5.404Btu1psia)52500)(/lbmft01641.0(
)(
/lbmft01641.0
Btu/lbm18.130
inp,12
33
121inp,
3psia5 @1
psia5@1
=+=+==
⎟⎟⎠
⎞⎜⎜⎝
⎛
⋅−=
−=
==
==
whh
PPw
hh
f
f
v
vv
RBtu/lbm52203.1)60894.1)(80.0(23488.0Btu/lbm58.930)5.1000)(80.0(18.130
80.0psia5
44
44
4
4
⋅=+=+==+=+=
⎭⎬⎫
==
fgf
fgf
sxsshxhh
xP
F989.2°==
⎭⎬⎫
⋅===
3
3
43
3 Btu/lbm8.1450
RBtu/lbm52203.1psia2500
Th
ssP
Thus,
Btu/lbm4.80018.13058.930Btu/lbm1313.076.1378.1450
14out
23in
=−=−==−=−=
hhqhhq
The thermal efficiency of the cycle is
0.390=−=−=0.13134.80011
in
outth q
qη
qin
qout
5 psia 1
3
2
4
2500 psia
s
T
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10-9
10-19 A simple ideal Rankine cycle with water as the working fluid operates between the specified pressure limits. The power produced by the turbine, the heat added in the boiler, and the thermal efficiency of the cycle are to be determined.
Assumptions 1 Steady operating conditions exist. 2 Kinetic and potential energy changes are negligible.
Analysis From the steam tables (Tables A-4, A-5, and A-6),
kJ/kg05.43354.1551.417kJ/kg54.15 mkPa1
kJ1kPa)100000,15)(/kgm001043.0()(
/kgm001043.0
kJ/kg51.417
inp,12
33
121inp,
3kPa100@1
kPa100@1
=+=+==
⎟⎠
⎞⎜⎝
⎛
⋅−=
−=
==
==
whh
PPw
hh
f
f
v
vv
kJ/kg5.1911)5.2257)(6618.0(51.417
6618.00562.6
3028.13108.5
kPa100
KkJ/kg3108.5kJ/kg8.2610
1
kPa000,15
44
44
34
4
3
3
3
3
=+=+=
=−
=−
=
⎭⎬⎫
==
⋅==
⎭⎬⎫
==
fgf
fg
f
hxhhs
ssx
ssP
sh
xP
Thus,
kJ/kg0.149451.4175.191105.4338.26105.19118.2610
14out
23in
43outT,
=−=−==−=−=
=−=−=
hhqhhqhhw
kJ/kg2177.8kJ/kg699.3
The thermal efficiency of the cycle is
0.314=−=−=8.21770.149411
in
outth q
qη
qin
qout
100 kPa 1
3 2
4
15 MPa
s
T
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10-10
10-20 A simple Rankine cycle with water as the working fluid operates between the specified pressure limits. The isentropic efficiency of the turbine, and the thermal efficiency of the cycle are to be determined.
Assumptions 1 Steady operating conditions exist. 2 Kinetic and potential energy changes are negligible.
Analysis From the steam tables (Tables A-4, A-5, and A-6),
kJ/kg05.43354.1551.417kJ/kg54.15 mkPa1
kJ1kPa)100000,15)(/kgm001043.0()(
/kgm001043.0
kJ/kg51.417
inp,12
33
121inp,
3kPa100@1
kPa100@1
=+=+==
⎟⎠
⎞⎜⎝
⎛
⋅−=
−=
==
==
whh
PPw
hh
f
f
v
vv
kJ/kg8.1997)5.2257)(70.0(51.417 70.0
kPa100
kJ/kg5.1911)5.2257)(6618.0(51.417
6618.00562.6
3028.13108.5
kPa100
KkJ/kg3108.5kJ/kg8.2610
1
kPa000,15
444
4
44
44
34
4
3
3
3
3
=+=+=⎭⎬⎫
==
=+=+=
=−
=−
=
⎭⎬⎫
==
⋅==
⎭⎬⎫
==
fgf
fgsfs
fg
fs
hxhhxP
hxhhs
ssx
ssP
sh
xP
The isentropic efficiency of the turbine is
0.877=−−
=−−
=5.19118.26108.19978.2610
43
43T
shhhh
η
Thus,
kJ/kg3.158051.4178.1997kJ/kg2177.805.4338.2610
14out
23in
=−=−==−=−=
hhqhhq
The thermal efficiency of the cycle is
0.274=−=−=8.21773.158011
in
outth q
qη
qin
qout
100 kPa 1
3 2
4s
15 MPa
s
T
4
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10-11
10-21E A simple steam Rankine cycle operates between the specified pressure limits. The mass flow rate, the power produced by the turbine, the rate of heat addition, and the thermal efficiency of the cycle are to be determined.
Assumptions 1 Steady operating conditions exist. 2 Kinetic and potential energy changes are negligible.
Analysis From the steam tables (Tables A-4E, A-5E, and A-6E),
Btu/lbm18.7746.772.69Btu/lbm46.7
ftpsia5.404Btu1psia)12500)(/lbmft01614.0(
)(
/lbmft01614.0
Btu/lbm72.69
inp,12
33
121inp,
3psia6 @1
psia1 @1
=+=+==
⎟⎟⎠
⎞⎜⎜⎝
⎛
⋅−=
−=
==
==
whh
PPw
hh
f
f
v
vv
Btu/lbm70.787)7.1035)(6932.0(72.69
6932.084495.1
13262.04116.1
psia1
RBtu/lbm4116.1Btu/lbm0.1302
F800psia2500
44
44
34
4
3
3
3
3
=+=+=
=−
=−
=
⎭⎬⎫
==
⋅==
⎭⎬⎫
°==
fgsfs
fg
fs
hxhhs
ssx
ssP
sh
TP
kJ/kg13.839)70.7870.1302)(90.0(0.1302)( 4s3T3443
43T =−−=−−=⎯→⎯
−−
= hhhhhhhh
sηη
Thus,
Btu/lbm39.45541.7698.1224Btu/lbm41.76972.6913.839Btu/lbm8.122418.771302.0
outinnet
14out
23in
=−=−==−=−==−=−=
qqwhhqhhq
The mass flow rate of steam in the cycle is determined from
lbm/s2.081=⎟⎠⎞
⎜⎝⎛==⎯→⎯=
kJ1Btu0.94782
Btu/lbm455.39kJ/s1000
net
netnetnet w
WmwmW
&&&&
The power output from the turbine and the rate of heat addition are
Btu/s2549
kW1016
===
=⎟⎠⎞
⎜⎝⎛−=−=
Btu/lbm)4.8lbm/s)(122081.2(
Btu0.94782kJ1Btu/lbm)13.8392.0lbm/s)(130081.2()(
inin
43outT,
qmQ
hhmW
&&
&&
and the thermal efficiency of the cycle is
0.3718=⎟⎠⎞
⎜⎝⎛==
kJ1Btu0.94782
Btu/s2549kJ/s1000
in
netth Q
W&
&η
qin
qout
1 psia 1
3
2
4
2500 psia
s
T
4s
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10-12
10-22E A simple steam Rankine cycle operates between the specified pressure limits. The mass flow rate, the power produced by the turbine, the rate of heat addition, and the thermal efficiency of the cycle are to be determined.
Assumptions 1 Steady operating conditions exist. 2 Kinetic and potential energy changes are negligible.
Analysis From the steam tables (Tables A-4E, A-5E, and A-6E),
Btu/lbm18.7746.772.69Btu/lbm46.7
ftpsia5.404Btu1psia)12500)(/lbmft01614.0(
)(
/lbmft01614.0
Btu/lbm72.69
inp,12
33
121inp,
3psia6 @1
psia1 @1
=+=+==
⎟⎟⎠
⎞⎜⎜⎝
⎛
⋅−=
−=
==
==
whh
PPw
hh
f
f
v
vv
Btu/lbm70.787)7.1035)(6932.0(72.69
6932.084495.1
13262.04116.1
psia1
RBtu/lbm4116.1Btu/lbm0.1302
F800psia2500
44
44
34
4
3
3
3
3
=+=+=
=−
=−
=
⎭⎬⎫
==
⋅==
⎭⎬⎫
°==
fgsfs
fg
fs
hxhhs
ssx
ssP
sh
TP
kJ/kg13.839)70.7870.1302)(90.0(0.1302)( 4s3T3443
43T =−−=−−=⎯→⎯
−−
= hhhhhhhh
sηη
The mass flow rate of steam in the cycle is determined from
lbm/s2.048kJ1
Btu0.94782Btu/lbm839.13)(1302.0
kJ/s1000)(43
net43net =⎟
⎠⎞
⎜⎝⎛
−=
−=⎯→⎯−=
hhW
mhhmW&
&&&
The rate of heat addition is
Btu/s2508Btu0.94782
kJ1Btu/lbm)18.772.0lbm/s)(130048.2()( 23in =⎟⎠⎞
⎜⎝⎛−=−= hhmQ &&
and the thermal efficiency of the cycle is
0.3779kJ1
Btu0.94782Btu/s2508kJ/s1000
in
netth =⎟
⎠⎞
⎜⎝⎛==
QW&
&η
The thermal efficiency in the previous problem was determined to be 0.3718. The error in the thermal efficiency caused by neglecting the pump work is then
1.64%=×−
= 1003718.0
3718.03779.0Error
qin
qout
1 psia 1
3
2
4
2500 psia
s
T
4s
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PROPRIETARY MATERIAL. © 2008 The McGraw-Hill Companies, Inc. Limited distribution permitted only to teachers and educators for course preparation. If you are a student using this Manual, you are using it without permission.
10-13
10-23 A 300-MW coal-fired steam power plant operates on a simple ideal Rankine cycle between the specified pressure limits. The overall plant efficiency and the required rate of the coal supply are to be determined.
Assumptions 1 Steady operating conditions exist. 2 Kinetic and potential energy changes are negligible.
Analysis (a) From the steam tables (Tables A-4, A-5, and A-6),
( )( )( )
kJ/kg.0327707.596.271kJ/kg5.07
mkPa1kJ1
kPa255000/kgm00102.0
/kgm000102.0
kJ/kg96.271
in,12
33
121in,
3kPa25@1
kPa25@1
=+=+==
⎟⎟⎠
⎞⎜⎜⎝
⎛
⋅−=
−=
==
==
p
p
f
f
whh
PPw
hh
v
vv
( )( ) kJ/kg2.22765.23458545.096.271
8545.09370.6
8932.08210.6kPa25
KkJ/kg8210.6kJ/kg2.3317
C450MPa5
44
44
34
4
3
3
3
3
=+=+=
=−
=−
=⎭⎬⎫
==
⋅==
⎭⎬⎫
°==
fgf
fg
f
hxhh
sss
xss
P
sh
TP
The thermal efficiency is determined from
kJ/kg2.200496.2712.2276kJ/kg2.304003.2772.3317
14out
23in=−=−==−=−=
hhqhhq
and
Thus, ( )( )( ) 24.5%==××=
=−=−=
96.075.03407.0
3407.02.30402.200411
gencombthoverall
in
outth
ηηηη
ηqq
(b) Then the required rate of coal supply becomes
and
tons/h150.3=⎟ =⎟⎠
⎞⎜⎜⎝
⎛==
===
tons/s 04174.0kg 1000
ton1kJ/kg29,300
kJ/s 1,222,992
kJ/s 992,222,12453.0
kJ/s 300,000
coal
incoal
overall
netin
CQ
m
WQ
&&
&&
η
Qin
Qout
25 kPa 1
3
2
4
5 MPa
s
T
·
·
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PROPRIETARY MATERIAL. © 2008 The McGraw-Hill Companies, Inc. Limited distribution permitted only to teachers and educators for course preparation. If you are a student using this Manual, you are using it without permission.
10-14
10-24 A solar-pond power plant that operates on a simple ideal Rankine cycle with refrigerant-134a as the working fluid is considered. The thermal efficiency of the cycle and the power output of the plant are to be determined.
Assumptions 1 Steady operating conditions exist. 2 Kinetic and potential energy changes are negligible.
Analysis (a) From the refrigerant tables (Tables A-11, A-12, and A-13),
( )
( )( )
kJ/kg40.8958.082.88
kJ/kg58.0mkPa1
kJ1kPa7001400/kgm 0008331.0
/kgm 0008331.0
kJ/kg 82.88
in,12
33
121in,
3MPa7.0@1
MPa7.0 @1
=+=+=
=
⎟⎟⎠
⎞⎜⎜⎝
⎛
⋅−=
−=
==
==
p
p
f
f
whh
PPw
hh
v
vv
( )( ) kJ/kg.2026221.1769839.082.88
9839.058763.0
33230.09105.0MPa7.0
KkJ/kg9105.0
kJ/kg12.276
vaporsat.MPa4.1
44
44
34
4
MPa4.1 @3
MPa4.1 @33
=+=+=
=−
=−
=⎭⎬⎫
==
⋅==
==
⎭⎬⎫=
fgf
fg
f
g
g
hxhh
sss
xss
P
ss
hhP
Thus ,
kJ/kg34.1338.17372.186kJ/kg38.17382.8820.262kJ/kg72.18640.8912.276
outinnet
14out
23in
=−=−==−=−==−=−=
qqwhhqhhq
and
7.1%===kJ/kg186.72
kJ/kg13.34
in
netth q
wη
(b) ( )( ) kW40.02=== kJ/kg13.34kg/s3netnet wmW &&
qin
qout 0.7 MPa
1
3
2
4
1.4 MPa
s
T
R-134a
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PROPRIETARY MATERIAL. © 2008 The McGraw-Hill Companies, Inc. Limited distribution permitted only to teachers and educators for course preparation. If you are a student using this Manual, you are using it without permission.
10-15
10-25 A steam power plant operates on a simple ideal Rankine cycle between the specified pressure limits. The thermal efficiency of the cycle, the mass flow rate of the steam, and the temperature rise of the cooling water are to be determined.
Assumptions 1 Steady operating conditions exist. 2 Kinetic and potential energy changes are negligible.
Analysis (a) From the steam tables (Tables A-4, A-5, and A-6),
( )
( )( )
kJ/kg 198.8706.781.191
kJ/kg7.06mkPa1
kJ1kPa107,000/kgm 0.00101
/kgm 00101.0
kJ/kg.81191
in,12
33
121in,
3kPa10@1
kPa10@1
=+=+=
=
⎟⎟⎠
⎞⎜⎜⎝
⎛
⋅−=
−=
==
==
p
p
f
f
whh
PPw
hh
v
vv
( )( ) kJ/kg3.62151.23928201.081.191
8201.04996.7
6492.08000.6kPa10
KkJ/kg8000.6kJ/kg411.43
C500MPa7
44
44
34
4
3
3
3
3
=+=+=
=−
=−
=⎭⎬⎫
==
⋅==
⎭⎬⎫
°==
fgf
fg
f
hxhh
sss
xss
P
sh
TP
Thus,
kJ/kg7.12508.19615.3212kJ/kg8.196181.1916.2153kJ/kg5.321287.1984.3411
outinnet
14out
23in
=−=−==−=−=
=−=−=
qqwhhqhhq
and
38.9%===kJ/kg3212.5kJ/kg1250.7
in
netth q
wη
(b) skg036 /.kJ/kg1250.7kJ/s45,000
net
net ===wWm&
&
(c) The rate of heat rejection to the cooling water and its temperature rise are
( )( )
( )( ) C8.4°=°⋅
==Δ
===
CkJ/kg4.18kg/s2000kJ/s70,586
)(
kJ/s ,58670kJ/kg1961.8kg/s35.98
watercooling
outwatercooling
outout
cmQ
T
qmQ
&
&
&&
qin
qout
10 kPa1
3
2
4
7 MPa
s
T
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PROPRIETARY MATERIAL. © 2008 The McGraw-Hill Companies, Inc. Limited distribution permitted only to teachers and educators for course preparation. If you are a student using this Manual, you are using it without permission.
10-16
10-26 A steam power plant operates on a simple nonideal Rankine cycle between the specified pressure limits. The thermal efficiency of the cycle, the mass flow rate of the steam, and the temperature rise of the cooling water are to be determined.
Assumptions 1 Steady operating conditions exist. 2 Kinetic and potential energy changes are negligible.
Analysis (a) From the steam tables (Tables A-4, A-5, and A-6),
( )
( )( ) ( )
kJ/kg9.921911.881.191kJ/kg8.11
0.87/mkPa1
kJ1kPa107,000/kgm 00101.0
/
/kgm 00101.0
kJ/kg91.811
in,12
33
121in,
3kPa10@1
kPa10@1
=+=+==
⎟⎟⎠
⎞⎜⎜⎝
⎛
⋅−=
−=
==
==
p
pp
f
f
whh
PPw
hh
ηv
vv
( )( )
( )( )( ) kJ/kg1.23176.21534.341187.04.3411
kJ/kg6.21531.2392820.081.191
8201.04996.7
6492.08000.6kPa10
KkJ/kg8000.6kJ/kg.43411
C500MPa7
433443
43
44
44
34
4
3
3
3
3
=−−=−−=⎯→⎯
−−
=
=+=+=
=−
=−
=⎭⎬⎫
==
⋅==
⎭⎬⎫
°==
sTs
T
fgfs
fg
f
hhhhhhhh
hxhh
sss
xss
P
sh
TP
ηη
Thus,
kJ/kg2.10863.21255.3211kJ/kg3.212581.1911.2317kJ/kg5.321192.1994.3411
outinnet
14out
23in
=−=−==−=−=
=−=−=
qqwhhqhhq
and
33.8%===kJ/kg3211.5kJ/kg1086.2
in
netth q
wη
(b) skg4341 /.kJ/kg1086.2kJ/s45,000
net
net ===wWm&
&
(c) The rate of heat rejection to the cooling water and its temperature rise are
( )( )
( )( ) C10.5°=°⋅
==Δ
===
CkJ/kg 4.18kg/s 2000kJ/s 88,051
)(
kJ/s,05188kJ/kg2125.3kg/s41.43
watercooling
outwatercooling
outout
cmQ
T
qmQ
&
&
&&
10 kPa 1
3
2
4
7 MPa
s
T
4
2qin
qout
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PROPRIETARY MATERIAL. © 2008 The McGraw-Hill Companies, Inc. Limited distribution permitted only to teachers and educators for course preparation. If you are a student using this Manual, you are using it without permission.
10-17
10-27 The net work outputs and the thermal efficiencies for a Carnot cycle and a simple ideal Rankine cycle are to be determined.
Assumptions 1 Steady operating conditions exist. 2 Kinetic and potential energy changes are negligible.
Analysis (a) Rankine cycle analysis: From the steam tables (Tables A-4, A-5, and A-6),
( )( )( )
kJ/kg57.26115.1042.251kJ/kg15.10
mkPa1kJ1kPa20000,10/kgm001017.0
/kgm001017.0
kJ/kg42.251
in,12
33
121in,
3kPa20@1
kPa20@1
=+=+==
⎟⎟⎠
⎞⎜⎜⎝
⎛
⋅−=
−=
==
==
p
p
f
f
whh
PPw
hh
v
vv
( )( )kJ/kg3.1845
5.23576761.042.251
6761.00752.7
8320.06159.5kPa20
KkJ/kg6159.5kJ/kg5.2725
1MPa10
44
44
34
4
3
3
3
3
=
+=+=
=−
=−
=⎭⎬⎫
==
⋅==
⎭⎬⎫
==
fgf
fg
f
hxhh
sss
xss
P
sh
xP
kJ/kg869.9=−=−==−=−==−=−=
0.15949.2463kJ/kg0.159442.2513.1845kJ/kg9.246357.2615.2725
outinnet
14out
23in
qqwhhqhhq
0.353=−=−=9.24630.159411
in
outth q
qη
(b) Carnot Cycle analysis:
kJ/kg9.1093)5.2357)(3574.0(42.251
3574.00752.7
8320.03603.3kPa20
KkJ/kg3603.3kJ/kg8.1407
0C0.311
C0.311kJ/kg5.2725
1MPa10
11
11
21
1
2
2
2
32
3
3
3
3
=+=
+=
=−
=−
=
⎭⎬⎫
==
⋅==
⎭⎬⎫
=°==
°==
⎭⎬⎫
==
fgf
fg
f
hxhhs
ssx
ssP
sh
xTT
Th
xP
kJ/kg565.4=−=−==−=−==−=−=
3.7527.1317kJ/kg4.7519.10933.1845
kJ/kg7.13178.14075.2725
outinnet
14out
23in
qqwhhqhhq
0.430=−=−=7.13174.75111
in
outth q
qη
T
1
2 3
4
Carnot cycle
s
T
1
2
3
4
Rankine cycle
s
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10-18
10-28 A single-flash geothermal power plant uses hot geothermal water at 230ºC as the heat source. The mass flow rate of steam through the turbine, the isentropic efficiency of the turbine, the power output from the turbine, and the thermal efficiency of the plant are to be determined.
Assumptions 1 Steady operating conditions exist. 2 Kinetic and potential energy changes are negligible.
Analysis (a) We use properties of water for geothermal water (Tables A-4 through A-6)
1661.02108
09.64014.990
kJ/kg14.990kPa500
kJ/kg14.9900
C230
22
12
2
11
1
=
−=
−=
⎭⎬⎫
===
=⎭⎬⎫
=°=
fg
f
hhh
xhh
P
hxT
The mass flow rate of steam through the turbine is
kg/s38.20=
==
kg/s)230)(1661.0(123 mxm &&
(b) Turbine:
kJ/kg7.2344)1.2392)(90.0(81.19190.0kPa10
kJ/kg3.2160kPa10
KkJ/kg8207.6kJ/kg1.2748
1kPa500
444
4
434
4
3
3
3
3
=+=+=⎭⎬⎫
==
=⎭⎬⎫
==
⋅==
⎭⎬⎫
==
fgf
s
hxhhxP
hss
P
sh
xP
0.686=−−
=−−
=3.21601.27487.23441.2748
43
43
sT hh
hhη
(c) The power output from the turbine is
kW15,410=−=−= kJ/kg)7.23448.1kJ/kg)(27438.20()( 433outT, hhmW &&
(d) We use saturated liquid state at the standard temperature for dead state enthalpy
kJ/kg83.1040
C250
0
0 =⎭⎬⎫
=°=
hxT
kW622,203kJ/kg)83.104.14kJ/kg)(990230()( 011in =−=−= hhmE &&
7.6%==== 0.0757622,203
410,15
in
outT,th E
W&
&η
production well
reinjection well
separator
steam turbine
1
condenser 2
3
4
5 6
Flash chamber
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PROPRIETARY MATERIAL. © 2008 The McGraw-Hill Companies, Inc. Limited distribution permitted only to teachers and educators for course preparation. If you are a student using this Manual, you are using it without permission.
10-19
10-29 A double-flash geothermal power plant uses hot geothermal water at 230ºC as the heat source. The temperature of the steam at the exit of the second flash chamber, the power produced from the second turbine, and the thermal efficiency of the plant are to be determined.
Assumptions 1 Steady operating conditions exist. 2 Kinetic and potential energy changes are negligible.
Analysis (a) We use properties of water for geothermal water (Tables A-4 through A-6)
1661.0kJ/kg14.990
kPa500
kJ/kg14.9900
C230
212
2
11
1
=⎭⎬⎫
===
=⎭⎬⎫
=°=
xhh
P
hxT
kg/s80.1911661.0230kg/s38.20kg/s)230)(1661.0(
316
123
=−=−====
mmmmxm
&&&
&&
kJ/kg7.234490.0kPa10
kJ/kg1.27481
kPa500
44
4
33
3
=⎭⎬⎫
==
=⎭⎬⎫
==
hxP
hxP
kJ/kg1.26931
kPa150
0777.0kPa150
kJ/kg09.6400
kPa500
88
8
7
7
67
7
66
6
=⎭⎬⎫
==
=°=
⎭⎬⎫
==
=⎭⎬⎫
==
hxP
xT
hhP
hxP
C111.35
(b) The mass flow rate at the lower stage of the turbine is
kg/s.9014kg/s)80.191)(0777.0(678 === mxm &&
The power outputs from the high and low pressure stages of the turbine are
kW15,410kJ/kg)7.23448.1kJ/kg)(27438.20()( 433outT1, =−=−= hhmW &&
kW5191=−=−= kJ/kg)7.23443.1kJ/kg)(269.9014()( 488outT2, hhmW &&
(c) We use saturated liquid state at the standard temperature for the dead state enthalpy
kJ/kg83.1040
C250
0
0 =⎭⎬⎫
=°=
hxT
kW621,203kJ/kg)83.10414kg/s)(990.230()( 011in =−=−= hhmE &&
10.1%==+
== 0.101621,203
5193410,15
in
outT,th E
W&
&η
productionwell
reinjection well
separator
steam turbine
1
condenser2
3
4
5
6
Flash chamber
7
8
9
separator
Flash chamber
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10-20
10-30 A combined flash-binary geothermal power plant uses hot geothermal water at 230ºC as the heat source. The mass flow rate of isobutane in the binary cycle, the net power outputs from the steam turbine and the binary cycle, and the thermal efficiencies for the binary cycle and the combined plant are to be determined.
Assumptions 1 Steady operating conditions exist. 2 Kinetic and potential energy changes are negligible.
Analysis (a) We use properties of water for geothermal water (Tables A-4 through A-6)
1661.0kJ/kg14.990
kPa500
kJ/kg14.9900
C230
212
2
11
1
=⎭⎬⎫
===
=⎭⎬⎫
=°=
xhh
P
hxT
kg/s80.19120.38230kg/s38.20kg/s)230)(1661.0(
316
123
=−=−====
mmmmxm
&&&
&&
kJ/kg7.234490.0kPa10
kJ/kg1.27481
kPa500
44
4
33
3
=⎭⎬⎫
==
=⎭⎬⎫
==
hxP
hxP
kJ/kg04.3770
C90
kJ/kg09.6400
kPa500
77
7
66
6
=⎭⎬⎫
=°=
=⎭⎬⎫
==
hxT
hxP
The isobutane properties are obtained from EES:
/kgm001839.0kJ/kg83.270
0kPa400
kJ/kg01.691C80kPa400
kJ/kg05.755C145kPa3250
310
10
10
10
99
9
88
8
==
⎭⎬⎫
==
=⎭⎬⎫
°==
=⎭⎬⎫
°==
v
hxP
hTP
hTP
( )( )( )
kJ/kg65.27682.583.270kJ/kg.82.5
90.0/mkPa1
kJ1kPa4003250/kgm001819.0
/
in,1011
33
101110in,
=+=+==
⎟⎟⎠
⎞⎜⎜⎝
⎛
⋅−=
−=
p
pp
whh
PPw ηv
An energy balance on the heat exchanger gives
kg/s105.46=⎯→⎯=
−=−
isoiso
118iso766
kg276.65)kJ/-(755.05kg377.04)kJ/-09kg/s)(640.81.191(
)()(
mm
hhmhhm
&&
&&
(b) The power outputs from the steam turbine and the binary cycle are
kW15,410=−=−= kJ/kg)7.23448.1kJ/kg)(27438.19()( 433steamT, hhmW &&
steam turbine
production well
reinjection well
isobutaneturbine
heat exchanger
pump
BINARY CYCLE
separator
air-cooledcondenser
condenser
flash chamber
1
2
3
4
56
7
8
91
1
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10-21
kW6139=−=−=
=−=−=
)kJ/kg82.5)(kg/s46.105(6753
kW6753kJ/kg)01.691.05kJ/kg)(755.46105()(
,isoisoT,binarynet,
98isoT,
inp
iso
wmWW
hhmW
&&&
&&
(c) The thermal efficiencies of the binary cycle and the combined plant are
kW454,50kJ/kg)65.276.05kJ/kg)(755.46105()( 118isobinaryin, =−=−= hhmQ &&
12.2%==== 0.122454,50
6139
binaryin,
binarynet,binaryth, Q
W&
&η
kJ/kg83.1040
C250
0
0 =⎭⎬⎫
=°=
hxT
kW622,203kJ/kg)83.104.14kJ/kg)(990230()( 011in =−=−= hhmE &&
10.6%==+
=+
= 0.106622,203
6139410,15
in
binarynet,steamT,plantth, E
WW&
&&η
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10-22
The Reheat Rankine Cycle
10-31C The pump work remains the same, the moisture content decreases, everything else increases.
10-32C The T-s diagram shows two reheat cases for the reheat Rankine cycle similar to the one shown in Figure 10-11. In the first case there is expansion through the high-pressure turbine from 6000 kPa to 4000 kPa between states 1 and 2 with reheat at 4000 kPa to state 3 and finally expansion in the low-pressure turbine to state 4. In the second case there is expansion through the high-pressure turbine from 6000 kPa to 500 kPa between states 1 and 5 with reheat at 500 kPa to state 6 and finally expansion in the low-pressure turbine to state 7. Increasing the pressure for reheating increases the average temperature for heat addition makes the energy of the steam more available for doing work, see the reheat process 2 to 3 versus the reheat process 5 to 6. Increasing the reheat pressure will increase the cycle efficiency. However, as the reheating pressure increases, the amount of condensation increases during the expansion process in the low-pressure turbine, state 4 versus state 7. An optimal pressure for reheating generally allows for the moisture content of the steam at the low-pressure turbine exit to be in the range of 10 to 15% and this corresponds to quality in the range of 85 to 90%.
10-33C The thermal efficiency of the simple ideal Rankine cycle will probably be higher since the average temperature at which heat is added will be higher in this case.
0 20 40 60 80 100 120 140 160 180200
300
400
500
600
700
800
900
s [kJ/kmol-K]
T [K
]
6000 kPa 4000 kPa
500 kPa
20 kPa 0.2 0.4 0.6 0.8
SteamIAPWS
1
2
3
4
5
6
7
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PROPRIETARY MATERIAL. © 2008 The McGraw-Hill Companies, Inc. Limited distribution permitted only to teachers and educators for course preparation. If you are a student using this Manual, you are using it without permission.
10-23
10-34 [Also solved by EES on enclosed CD] A steam power plant that operates on the ideal reheat Rankine cycle is considered. The turbine work output and the thermal efficiency of the cycle are to be determined.
Assumptions 1 Steady operating conditions exist. 2 Kinetic and potential energy changes are negligible.
Analysis From the steam tables (Tables A-4, A-5, and A-6),
( )( )( )
kJ/kg.5425912.842.251
kJ/kg8.12mkPa1
kJ1kPa208000/kgm 001017.0
/kgm 700101.0
kJ/kg42.251
in,12
33
121in,
3kPa20@1
kPa20@1
=+=+=
=
⎟⎟⎠
⎞⎜⎜⎝
⎛
⋅−=
−=
==
==
p
p
f
f
whh
PPw
hh
v
vv
( )( ) kJ/kg2.23855.23579051.042.251
9051.00752.7
8320.02359.7kPa20
KkJ/kg2359.7kJ/kg2.3457
C500MPa3
kJ/kg1.3105MPa3
KkJ/kg7266.6kJ/kg5.3399
C500MPa8
66
66
56
6
5
5
5
5
434
4
3
3
3
3
=+=+=
=−
=−
=
⎭⎬⎫
==
⋅==
⎭⎬⎫
°==
=⎭⎬⎫
==
⋅==
⎭⎬⎫
°==
fgf
fg
f
hxhh
sss
x
ssP
sh
TP
hss
P
sh
TP
The turbine work output and the thermal efficiency are determined from
and ( ) ( )
( ) ( ) kJ/kg0.34921.31052.345754.2595.3399
2.23852.34571.31055.3399
4523in
6543outT,
=−+−=−+−=
=−+−=−+−=
hhhhq
hhhhw kJ/kg 1366.4
Thus,
38.9%===
=−=−=
kJ/kg3492.5kJ/kg1358.3
kJ/kg3.135812.84.1366
in
netth
in,,net
qw
www poutT
η
1
5
2
6 s
T
3
4 8 MPa
20 kPa
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10-24
10-35 EES Problem 10-34 is reconsidered. The problem is to be solved by the diagram window data entry feature of EES by including the effects of the turbine and pump efficiencies and reheat on the steam quality at the low-pressure turbine exit Also, the T-s diagram is to be plotted.
Analysis The problem is solved using EES, and the solution is given below.
"Input Data - from diagram window" {P[6] = 20 [kPa] P[3] = 8000 [kPa] T[3] = 500 [C] P[4] = 3000 [kPa] T[5] = 500 [C] Eta_t = 100/100 "Turbine isentropic efficiency" Eta_p = 100/100 "Pump isentropic efficiency"}
"Pump analysis" function x6$(x6) "this function returns a string to indicate the state of steam at point 6"
x6$='' if (x6>1) then x6$='(superheated)' if (x6<0) then x6$='(subcooled)'
end
Fluid$='Steam_IAPWS'
P[1] = P[6] P[2]=P[3] x[1]=0 "Sat'd liquid" h[1]=enthalpy(Fluid$,P=P[1],x=x[1]) v[1]=volume(Fluid$,P=P[1],x=x[1]) s[1]=entropy(Fluid$,P=P[1],x=x[1]) T[1]=temperature(Fluid$,P=P[1],x=x[1]) W_p_s=v[1]*(P[2]-P[1])"SSSF isentropic pump work assuming constant specific volume" W_p=W_p_s/Eta_p h[2]=h[1]+W_p "SSSF First Law for the pump" v[2]=volume(Fluid$,P=P[2],h=h[2]) s[2]=entropy(Fluid$,P=P[2],h=h[2]) T[2]=temperature(Fluid$,P=P[2],h=h[2]) "High Pressure Turbine analysis" h[3]=enthalpy(Fluid$,T=T[3],P=P[3]) s[3]=entropy(Fluid$,T=T[3],P=P[3]) v[3]=volume(Fluid$,T=T[3],P=P[3]) s_s[4]=s[3] hs[4]=enthalpy(Fluid$,s=s_s[4],P=P[4]) Ts[4]=temperature(Fluid$,s=s_s[4],P=P[4]) Eta_t=(h[3]-h[4])/(h[3]-hs[4])"Definition of turbine efficiency" T[4]=temperature(Fluid$,P=P[4],h=h[4]) s[4]=entropy(Fluid$,T=T[4],P=P[4]) v[4]=volume(Fluid$,s=s[4],P=P[4]) h[3] =W_t_hp+h[4]"SSSF First Law for the high pressure turbine" "Low Pressure Turbine analysis" P[5]=P[4] s[5]=entropy(Fluid$,T=T[5],P=P[5]) h[5]=enthalpy(Fluid$,T=T[5],P=P[5]) s_s[6]=s[5] hs[6]=enthalpy(Fluid$,s=s_s[6],P=P[6])
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PROPRIETARY MATERIAL. © 2008 The McGraw-Hill Companies, Inc. Limited distribution permitted only to teachers and educators for course preparation. If you are a student using this Manual, you are using it without permission.
10-25
Ts[6]=temperature(Fluid$,s=s_s[6],P=P[6]) vs[6]=volume(Fluid$,s=s_s[6],P=P[6]) Eta_t=(h[5]-h[6])/(h[5]-hs[6])"Definition of turbine efficiency" h[5]=W_t_lp+h[6]"SSSF First Law for the low pressure turbine" x[6]=QUALITY(Fluid$,h=h[6],P=P[6]) "Boiler analysis" Q_in + h[2]+h[4]=h[3]+h[5]"SSSF First Law for the Boiler" "Condenser analysis" h[6]=Q_out+h[1]"SSSF First Law for the Condenser" T[6]=temperature(Fluid$,h=h[6],P=P[6]) s[6]=entropy(Fluid$,h=h[6],P=P[6]) x6s$=x6$(x[6])
"Cycle Statistics" W_net=W_t_hp+W_t_lp-W_p Eff=W_net/Q_in
0.0 1.1 2.2 3.3 4.4 5.5 6.6 7.7 8.8 9.9 11.00
100
200
300
400
500
600
700
s [kJ/kg-K ]
T [C
]
8000 kPa
3000 kPa
20 kPa
3
4
5
6
Ideal R ankine cycle w ith reheat
1,2
SOLUTION Eff=0.389 Eta_p=1 Eta_t=1 Fluid$='Steam_IAPWS' h[1]=251.4 [kJ/kg] h[2]=259.5 [kJ/kg] h[3]=3400 [kJ/kg] h[4]=3105 [kJ/kg] h[5]=3457 [kJ/kg] h[6]=2385 [kJ/kg] hs[4]=3105 [kJ/kg] hs[6]=2385 [kJ/kg] P[1]=20 [kPa] P[2]=8000 [kPa] P[3]=8000 [kPa] P[4]=3000 [kPa] P[5]=3000 [kPa] P[6]=20 [kPa] Q_in=3493 [kJ/kg] Q_out=2134 [kJ/kg] s[1]=0.832 [kJ/kg-K] s[2]=0.8321 [kJ/kg-K] s[3]=6.727 [kJ/kg-K] s[4]=6.727 [kJ/kg-K] s[5]=7.236 [kJ/kg-K] s[6]=7.236 [kJ/kg-K] s_s[4]=6.727 [kJ/kg-K] s_s[6]=7.236 [kJ/kg-K] T[1]=60.06 [C] T[2]=60.4 [C] T[3]=500 [C] T[4]=345.2 [C] T[5]=500 [C] T[6]=60.06 [C] Ts[4]=345.2 [C] Ts[6]=60.06 [C] v[1]=0.001017 [m^3/kg] v[2]=0.001014 [m^3/kg] v[3]=0.04177 [m^3/kg] v[4]=0.08968 [m^3/kg] vs[6]=6.922 [m^3/kg] W_net=1359 [kJ/kg] W_p=8.117 [kJ/kg] W_p_s=8.117 [kJ/kg] W_t_hp=294.8 [kJ/kg] W_t_lp=1072 [kJ/kg] x6s$='' x[1]=0 x[6]=0.9051
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PROPRIETARY MATERIAL. © 2008 The McGraw-Hill Companies, Inc. Limited distribution permitted only to teachers and educators for course preparation. If you are a student using this Manual, you are using it without permission.
10-26
10-36E An ideal reheat steam Rankine cycle produces 5000 kW power. The rates of heat addition and rejection, and the thermal efficiency of the cycle are to be determined.
Assumptions 1 Steady operating conditions exist. 2 Kinetic and potential energy changes are negligible.
Analysis From the steam tables (Tables A-4E, A-5E, and A-6E or EES),
Btu/lbm06.16381.125.161Btu/lbm81.1
ftpsia5.404Btu1psia)10600)(/lbmft01659.0(
)(
/lbmft01659.0
Btu/lbm25.161
inp,12
33
121inp,
3psia10@1
psia10@1
=+=+==
⎟⎟⎠
⎞⎜⎜⎝
⎛
⋅−=
−=
==
==
whh
PPw
hh
f
f
v
vv
Btu/lbm5.1187)33.843)(9865.0(46.355
9865.000219.1
54379.05325.1
psia200
RBtu/lbm5325.1Btu/lbm9.1289
F600psia600
44
44
34
4
3
3
3
3
=+=+=
=−
=−
=
⎭⎬⎫
==
⋅==
⎭⎬⎫
°==
fgf
fg
f
hxhhs
ssx
ssP
sh
TP
Btu/lbm0.1071)82.981)(9266.0(25.161
9266.050391.1
28362.06771.1
psia10
RBtu/lbm6771.1Btu/lbm3.1322
F600psia200
66
46
56
6
5
5
5
5
=+=+=
=−
=−
=
⎭⎬⎫
==
⋅==
⎭⎬⎫
°==
fgf
fg
f
hxhhs
ssx
ssP
sh
TP
Thus,
Btu/lbm0.3528.9097.1261Btu/lbm7.90925.1610.1071
Btu/lbm7.12615.11873.132206.1631289.9)()(
outinnet
16out
4523in
=−=−==−=−=
=−+−=−+−=
qqwhhq
hhhhq
The mass flow rate of steam in the cycle is determined from
lbm/s47.13kJ1
Btu0.94782Btu/lbm352.0
kJ/s5000
net
netnetnet =⎟
⎠⎞
⎜⎝⎛==⎯→⎯=
wW
mwmW&
&&&
The rates of heat addition and rejection are
Btu/s12,250Btu/s16,995
===
===
Btu/lbm).7lbm/s)(90947.13(
Btu/lbm)1.7lbm/s)(12647.13(
outout
inin
qmQ
qmQ&&
&&
and the thermal efficiency of the cycle is
0.2790=⎟⎠⎞
⎜⎝⎛==
kJ1Btu0.94782
Btu/s16,990kJ/s5000
in
netth Q
W&
&η
1
5
2
6
s
T
3
4
600 psia
10 psia
200 psia
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PROPRIETARY MATERIAL. © 2008 The McGraw-Hill Companies, Inc. Limited distribution permitted only to teachers and educators for course preparation. If you are a student using this Manual, you are using it without permission.
10-27
10-37E An ideal reheat steam Rankine cycle produces 5000 kW power. The rates of heat addition and rejection, and the thermal efficiency of the cycle are to be determined for a reheat pressure of 100 psia.
Assumptions 1 Steady operating conditions exist. 2 Kinetic and potential energy changes are negligible.
Analysis From the steam tables (Tables A-4E, A-5E, and A-6E or EES),
Btu/lbm06.16381.125.161Btu/lbm81.1
ftpsia5.404Btu1psia)10600)(/lbmft01659.0(
)(
/lbmft01659.0
Btu/lbm25.161
inp,12
33
121inp,
3psia6 @1
psia10@1
=+=+==
⎟⎟⎠
⎞⎜⎜⎝
⎛
⋅−=
−=
==
==
whh
PPw
hh
f
f
v
vv
Btu/lbm9.1131)99.888)(9374.0(51.298
9374.012888.1
47427.05325.1
psia100
RBtu/lbm5325.1Btu/lbm9.1289
F600psia600
44
44
34
4
3
3
3
3
=+=+=
=−
=−
=
⎭⎬⎫
==
⋅==
⎭⎬⎫
°==
fgf
fg
f
hxhhs
ssx
ssP
sh
TP
Btu/lbm2.1124)82.981)(9808.0(25.161
9808.050391.1
28362.07586.1
psia10
RBtu/lbm7586.1Btu/lbm4.1329
F600
psia100
66
66
56
6
5
5
5
5
=+=+=
=−
=−
=
⎭⎬⎫
==
⋅==
⎭⎬⎫
°==
fgf
fg
f
hxhhs
ssx
ssP
sh
TP
Thus,
Btu/lbm5.3619.9624.1324Btu/lbm9.96225.1612.1124
Btu/lbm4.13249.11314.132907.1631289.9)()(
outinnet
16out
4523in
=−=−==−=−=
=−+−=−+−=
qqwhhq
hhhhq
The mass flow rate of steam in the cycle is determined from
lbm/s11.13kJ1
Btu0.94782Btu/lbm361.5
kJ/s5000
net
netnetnet =⎟
⎠⎞
⎜⎝⎛==⎯→⎯=
wW
mwmW&
&&&
The rates of heat addition and rejection are
Btu/s12,620Btu/s17,360
===
===
Btu/lbm).9lbm/s)(96211.13(
Btu/lbm)4.4lbm/s)(13211.13(
outout
inin
qmQ
qmQ&&
&&
and the thermal efficiency of the cycle is
0.2729=⎟⎠⎞
⎜⎝⎛==
kJ1Btu0.94782
Btu/s17,360kJ/s5000
in
netth Q
W&
&η
Discussion The thermal efficiency for 200 psia reheat pressure was determined in the previous problem to be 0.2790. Thus, operating the reheater at 100 psia causes a slight decrease in the thermal efficiency.
1
5
2
6
s
T
3
4
600 psia
10 psia
100 psia
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10-28
10-38 An ideal reheat Rankine with water as the working fluid is considered. The temperatures at the inlet of both turbines, and the thermal efficiency of the cycle are to be determined.
Assumptions 1 Steady operating conditions exist. 2 Kinetic and potential energy changes are negligible.
Analysis From the steam tables (Tables A-4, A-5, and A-6),
kJ/kg84.19503.481.191kJ/kg03.4 mkPa1
kJ1kPa)104000)(/kgm001010.0()(
/kgm001010.0
kJ/kg81.191
inp,12
33
121inp,
3kPa10@1
kPa10@1
=+=+==
⎟⎠
⎞⎜⎝
⎛
⋅−=
−=
==
==
whh
PPw
hh
f
f
v
vv
C292.2°==
⎭⎬⎫
==
⋅=+=+==+=+=
⎭⎬⎫
==
3
3
43
3
44
44
4
4
kJ/kg4.2939
kPa4000
KkJ/kg3247.6)9603.4)(90.0(8604.1kJ/kg3.2537)0.2108)(90.0(09.640
90.0
kPa500
Th
ssP
sxsshxhh
xP
fgf
fgf
C282.9°==
⎭⎬⎫
==
⋅=+=+==+=+=
⎭⎬⎫
==
5
5
65
5
66
66
6
6
kJ/kg2.3029
kPa500
KkJ/kg3989.7)4996.7)(90.0(6492.0kJ/kg7.2344)1.2392)(90.0(81.191
90.0kPa10
Th
ssP
sxsshxhh
xP
fgf
fgf
Thus,
kJ/kg9.215281.1917.2344kJ/kg4.32353.25372.302984.1954.2939)()(
16out
4523in
=−=−==−+−=−+−=
hhqhhhhq
and
0.335=−=−=4.32359.215211
in
outth q
qη
1
5
2
6
s
T
3
4
4 MPa
10 kPa
500 kPa
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PROPRIETARY MATERIAL. © 2008 The McGraw-Hill Companies, Inc. Limited distribution permitted only to teachers and educators for course preparation. If you are a student using this Manual, you are using it without permission.
10-29
10-39 An ideal reheat Rankine cycle with water as the working fluid is considered. The thermal efficiency of the cycle is to be determined.
Assumptions 1 Steady operating conditions exist. 2 Kinetic and potential energy changes are negligible.
Analysis From the steam tables (Tables A-4, A-5, and A-6 or EES),
kJ/kg51.35897.1754.340kJ/kg97.17 mkPa1
kJ1kPa)5017500)(/kgm001030.0()(
/kgm001030.0
kJ/kg54.340
inp,12
33
121inp,
3kPa50@1
kPa50@1
=+=+==
⎟⎠
⎞⎜⎝
⎛
⋅−=
−=
==
==
whh
PPw
hh
f
f
v
vv
kJ/kg5.2841 kPa2000
KkJ/kg4266.6kJ/kg6.3423
C055
kPa500,17
434
4
3
3
3
3
=⎭⎬⎫
==
⋅==
⎭⎬⎫
°==
hss
P
sh
TP
kJ/kg9.2352)7.2304)(8732.0(54.340
8732.05019.6
0912.17684.6
kPa50
KkJ/kg7684.6kJ/kg2.3024
C300kPa2000
66
66
56
6
5
5
5
5
=+=+=
=−
=−
=
⎭⎬⎫
==
⋅==
⎭⎬⎫
°==
fgf
fg
f
hxhhs
ssx
ssP
sh
TP
Thus,
kJ/kg4.201254.3409.2352kJ/kg8.32475.28412.302451.3586.3423)()(
16out
4523in
=−=−==−+−=−+−=
hhqhhhhq
and
0.380=−=−=8.32474.201211
in
outth q
qη
1
5
2
6
s
T
3
4
17.5MPa
50 kPa
2 MPa
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PROPRIETARY MATERIAL. © 2008 The McGraw-Hill Companies, Inc. Limited distribution permitted only to teachers and educators for course preparation. If you are a student using this Manual, you are using it without permission.
10-30
10-40 An ideal reheat Rankine cycle with water as the working fluid is considered. The thermal efficiency of the cycle is to be determined.
Assumptions 1 Steady operating conditions exist. 2 Kinetic and potential energy changes are negligible.
Analysis From the steam tables (Tables A-4, A-5, and A-6 or EES),
kJ/kg52.35897.1754.340kJ/kg97.17 mkPa1
kJ1kPa)5017500)(/kgm001030.0()(
/kgm001030.0
kJ/kg54.340
inp,12
33
121inp,
3kPa50@1
kPa50@1
=+=+==
⎟⎠
⎞⎜⎝
⎛
⋅−=
−=
==
==
whh
PPw
hh
f
f
v
vv
kJ/kg5.2841 kPa2000
KkJ/kg4266.6kJ/kg6.3423
C055
kPa500,17
434
4
3
3
3
3
=⎭⎬⎫
==
⋅==
⎭⎬⎫
°==
hss
P
sh
TP
kJ/kg0.2638)7.2304)(9968.0(54.340
9968.05019.6
0912.15725.7
kPa50
KkJ/kg5725.7kJ/kg0.3579
C550kPa2000
66
66
56
6
5
5
5
5
=+=+=
=−
=−
=
⎭⎬⎫
==
⋅==
⎭⎬⎫
°==
fgf
fg
f
hxhhs
ssx
ssP
sh
TP
Thus,
kJ/kg4.229754.3400.2638kJ/kg6.38025.28410.357952.3586.3423)()(
16out
4523in
=−=−==−+−=−+−=
hhqhhhhq
and
0.396=−=−=6.38024.229711
in
outth q
qη
The thermal efficiency was determined to be 0.380 when the temperature at the inlet of low-pressure turbine was 300°C. When this temperature is increased to 550°C, the thermal efficiency becomes 0.396. This corresponding to a percentage increase of 4.2% in thermal efficiency.
1
5
2
6
s
T
3
4
17.5MP
50 kPa
2 MPa
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10-31
10-41 A steam power plant that operates on an ideal reheat Rankine cycle between the specified pressure limits is considered. The pressure at which reheating takes place, the total rate of heat input in the boiler, and the thermal efficiency of the cycle are to be determined.
Assumptions 1 Steady operating conditions exist. 2 Kinetic and potential energy changes are negligible.
Analysis (a) From the steam tables (Tables A-4, A-5, and A-6),
( )( )( )
kJ/kg95.20614.1581.191
kJ/kg.1415mkPa1
kJ1kPa 10000,15/kgm 00101.0
/kgm 00101.0
kJ/kg81.191
in,12
33
121in,
3kPa10@sat1
kPa10@sat1
=+=+=
=
⎟⎟⎠
⎞⎜⎜⎝
⎛
⋅−=
−=
==
==
p
p
whh
PPw
hh
v
vv
( )( )( )( )
( )
kJ/kg2.2817MPa15.2
kJ/kg61.3466pressurereheat theC500
KkJ/kg3988.74996.790.06492.0
kJ/kg7.23441.239290.081.191kPa10
KkJ/kg3480.6kJ/kg.83310
C500MPa15
434
4
5
5
65
5
66
66
56
6
3
3
3
3
=⎭⎬⎫
==
==
⎭⎬⎫
=°=
⋅=+=+=
=+=+=
⎭⎬⎫
==
⋅==
⎭⎬⎫
°==
hss
P
hP
ssT
sxss
hxhh
ssP
sh
TP
fgf
fgf
kPa2150
(b) The rate of heat supply is
( ) ( )[ ]( )( ) kW45,039=−+−=
−+−=kJ/kg2.281761.346695.2068.3310kg/s12
4523in hhhhmQ &&
(c) The thermal efficiency is determined from
Thus, ( ) ( )( )
42.6%=−=−=
=−=−=
kJ/s45,039kJ/s25,834
11
kJ/s,83525kJ/kg81.1917.2344kJ/s12
in
outth
16out
hhmQ
&
&
&&
η
1
5
2
6 s
T
3
4 15
10 kPa
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10-32
10-42 A steam power plant that operates on a reheat Rankine cycle is considered. The condenser pressure, the net power output, and the thermal efficiency are to be determined. Assumptions 1 Steady operating conditions exist. 2 Kinetic and potential energy changes are negligible. Analysis (a) From the steam tables (Tables A-4, A-5, and A-6),
( )( )( )
( )( )( )
kJ/kg3.30271.29482.335885.02.3358
?
95.0?
KkJ/kg2815.7kJ/kg2.3358
C450MPa2
kJ/kg3.30271.29485.347685.05.3476
kJ/kg1.2948MPa2
KkJ/kg6317.6kJ/kg 5.3476
C550MPa5.12
655665
65
656
6
66
6
5
5
5
5
4334
43
43
434
4
3
3
3
3
=−−=
−−=⎯→⎯−−
=
=⎭⎬⎫
==
=⎭⎬⎫
==
⋅==
⎭⎬⎫
°==
=−−=
−−=→
−−
=
=⎭⎬⎫
==
⋅==
⎭⎬⎫
°==
sTs
T
s
sT
sT
ss
hhhhhhhh
hss
P
hxP
sh
TP
hhhh
hhhh
hss
P
sh
TP
ηη
η
η
The pressure at state 6 may be determined by a trial-error approach from the steam tables or by using EES from the above equations:
P6 = 9.73 kPa, h6 = 2463.3 kJ/kg, (b) Then,
( )( )( ) ( )
kJ/kg 59.20302.1457.189
kJ/kg 14.020.90/
mkPa 1kJ1kPa73.912,500/kgm0.00101
/
/kgm001010.0
kJ/kg57.189
in,12
33
121in,
3kPa10@1
kPa73.9@1
=+=+=
=
⎟⎟⎠
⎞⎜⎜⎝
⎛
⋅−=
−=
==
==
p
pp
f
f
whh
PPw
hh
ηv
vv
Cycle analysis: ( ) ( )
kW10,242==−=
=−=−=
=−+−=−+−=
kg2273.7)kJ/-.8kg/s)(36037.7()(
kJ/kg7.227357.1893.3027
kJ/kg8.36033.24632.33583.30275.3476
outinnet
16out
4523in
qqmW
hhq
hhhhq
&&
(c) The thermal efficiency is
36.9%==−=−= 369.0kJ/kg3603.8kJ/kg2273.7
11in
outth q
qη
1
5
2s
6ss
T
3
4s12.5 MPa
P = ? 6
4
2
3
6
1 2
Turbine Boiler
Condenser Pump
5
4
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10-33
Regenerative Rankine Cycle
10-43C Moisture content remains the same, everything else decreases.
10-44C This is a smart idea because we waste little work potential but we save a lot from the heat input. The extracted steam has little work potential left, and most of its energy would be part of the heat rejected anyway. Therefore, by regeneration, we utilize a considerable amount of heat by sacrificing little work output.
10-45C In open feedwater heaters, the two fluids actually mix, but in closed feedwater heaters there is no mixing.
10-46C Both cycles would have the same efficiency.
10-47C To have the same thermal efficiency as the Carnot cycle, the cycle must receive and reject heat isothermally. Thus the liquid should be brought to the saturated liquid state at the boiler pressure isothermally, and the steam must be a saturated vapor at the turbine inlet. This will require an infinite number of heat exchangers (feedwater heaters), as shown on the T-s diagram.
Boiler exit
s
TBoiler inlet
qin
qout
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10-34
10-48 Feedwater is heated by steam in a feedwater heater of a regenerative The required mass flow rate of the steam is to be determined.
Assumptions 1 This is a steady-flow process since there is no change with time. 2 Kinetic and potential energy changes are negligible. 3 There are no work interactions. 4 The device is adiabatic and thus heat transfer is negligible.
Properties From the steam tables (Tables A-4 through A-6 or EES),
h1 ≅ hf @ 70°C = 293.07 kJ/kg
kJ/kg7.2789 C160kPa200
22
2 =⎭⎬⎫
°==
hTP
h3 = hf @ 200 kPa = 504.71 kJ/kg
Analysis We take the mixing chamber as the system, which is a control volume since mass crosses the boundary. The mass and energy balances for this steady-flow system can be expressed in the rate form as
Mass balance:
321
outin
(steady) 0systemoutin 0
mmmmm
mmm
&&&
&&
&&&
=+=
=Δ=−
Energy balance:
)(0)peke(since
0
3212211
332211
outin
energiesetc.potential, kinetic,internal,in changeofRate
(steady) 0system
massand work,heat,by nsferenergy tranet ofRate
outin
hmmhmhmWQhmhmhm
EE
EEE
&&&&
&&&&&
&&
444 34444 21&
43421&&
+=+≅Δ≅Δ===+
=
=Δ=−
Solving for 2m& , and substituting gives
kg/s0.926=−−
=−−
=kJ/kg)7.278971.504(kJ/kg)71.50407.293(kg/s)10(
23
3112 hh
hhmm &&
Water 70°C
200 kPa 10 kg/s
Steam 200 kPa 160°C
1
23
Water 200 kPa sat. liq.
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10-35
10-49E In a regenerative Rankine cycle, the closed feedwater heater with a pump as shown in the figure is arranged so that the water at state 5 is mixed with the water at state 2 to form a feedwater which is a saturated liquid. The mass flow rate of bleed steam required to operate this unit is to be determined.
Assumptions 1 This is a steady-flow process since there is no change with time. 2 Kinetic and potential energy changes are negligible. 3 There are no work interactions. 4 The device is adiabatic and thus heat transfer is negligible.
Properties From the steam tables (Tables A-4E through A-6E),
Btu/lbm46.355 0
psia200
Btu/lbm0.1218 F400
psia160
Btu/lbm73.321 F350psia200
psia200@66
6
33
3
F350@11
1
==⎭⎬⎫
==
=⎭⎬⎫
°==
=≅⎭⎬⎫
°==
°
f
f
hhxP
hTP
hhTP
Analysis We take the entire unit as the system, which is a control volume since mass crosses the boundary. The energy balance for this steady-flow system can be expressed in the rate form as
631inP,33311
66inP,33311
outin
energiesetc.potential, kinetic,internal,in changeofRate
(steady) 0system
massand work,heat,by nsferenergy tranet ofRate
outin
)(
0
hmmwmhmhmhmwmhmhm
EE
EEE
&&&&&
&&&&
&&
444 34444 21&
43421&&
+=++
=++=
=Δ=−
Solving this for 3m& ,
lbm/s0.0782=+−
−=
+−−
=1344.046.3550.1218
73.32146.355lbm/s)2()( inP,63
1613 whh
hhmm &&
where
Btu/lbm1344.0ftpsia5.404
Btu1psia)160200)(/lbmft01815.0(
)()(
33
45psia160@454inP,
⎟ =⎟⎠
⎞⎜⎜⎝
⎛
⋅−=
−=−= PPPPw fvv
1 23
45
6
Feedwater200 psia 350°F 2 lbm/s
Steam 160 psia 400°F
200 psiasat. liq.
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10-36
10-50E The closed feedwater heater of a regenerative Rankine cycle is to heat feedwater to a saturated liquid. The required mass flow rate of bleed steam is to be determined.
Assumptions 1 This is a steady-flow process since there is no change with time. 2 Kinetic and potential energy changes are negligible. 3 There are no work interactions. 4 Heat loss from the device to the surroundings is negligible and thus heat transfer from the hot fluid is equal to the heat transfer to the cold fluid.
Properties From the steam tables (Tables A-4E through A-6E),
Btu/lbm94.393 0
psia300
Btu/lbm9.1257 F500psia300
Btu/lbm13.424 0
psia400
Btu/lbm88.342 F370psia400
psia300@44
4
33
3
psia400@22
2
F370@11
1
==⎭⎬⎫
==
=⎭⎬⎫
°==
==⎭⎬⎫
==
=≅⎭⎬⎫
°==
°
f
f
f
hhxP
hTP
hhxP
hhTP
Analysis We take the heat exchanger as the system, which is a control volume. The mass and energy balances for this steady-flow system can be expressed in the rate form as
Mass balance (for each fluid stream):
sfw mmmmmmmmmmm &&&&&&&&&&& ====→=→=Δ=− 4321outin(steady) 0
systemoutin and0
Energy balance (for the heat exchanger):
0)peke (since
0
44223311
outin
energiesetc.potential, kinetic,internal,in changeofRate
(steady) 0system
massand work,heat,by nsferenergy tranet ofRate
outin
≅Δ≅Δ==+=+
=
=Δ=−
WQhmhmhmhm
EE
EEE
&&&&&&
&&
444 344 21&
43421&&
Combining the two,
)()( 4312 hhmhhm sfw −=− &&
Solving for sm& :
fws mhhhh
m &&43
12
−−
=
Substituting,
lbm/s0.0940=−−
= )lbm/s1(94.3939.125788.34213.424
sm&
Steam 300 psia400°F
Feedwater400 psia 370°F
300 psia Sat. liq.
400 psia Sat. liq.
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10-37
10-51 The closed feedwater heater of a regenerative Rankine cycle is to heat feedwater to a saturated liquid. The required mass flow rate of bleed steam is to be determined.
Assumptions 1 This is a steady-flow process since there is no change with time. 2 Kinetic and potential energy changes are negligible. 3 There are no work interactions. 4 Heat loss from the device to the surroundings is negligible and thus heat transfer from the hot fluid is equal to the heat transfer to the cold fluid.
Properties From the steam tables (Tables A-4 through A-6),
kJ/kg3.1008 0
kPa3000kJ/kg7.2623)9.1794)(9.0(3.1008
90.0
kPa3000
kJ/kg5.1061 C245kPa4000
kJ/kg26.852 C200kPa4000
kPa3000@44
4
333
3
C245@22
2
C200@11
1
==⎭⎬⎫
==
=+=
+=⎭⎬⎫
==
=≅⎭⎬⎫
°==
=≅⎭⎬⎫
°==
°
°
f
fgf
f
f
hhxP
hxhhxP
hhTP
hhTP
Analysis We take the heat exchanger as the system, which is a control volume. The mass and energy balances for this steady-flow system can be expressed in the rate form as
Mass balance (for each fluid stream):
sfw mmmmmmmmmmm &&&&&&&&&&& ====→=→=Δ=− 4321outin(steady) 0
systemoutin and0
Energy balance (for the heat exchanger):
0)peke (since
0
44223311
outin
energiesetc.potential, kinetic,internal,in changeofRate
(steady) 0system
massand work,heat,by nsferenergy tranet ofRate
outin
≅Δ≅Δ==+=+
=
=Δ=−
WQhmhmhmhm
EE
EEE
&&&&&&
&&
444 344 21&
43421&&
Combining the two,
)()( 4312 hhmhhm sfw −=− &&
Solving for sm& :
fws mhhhh
m &&43
12
−−
=
Substituting,
kg/s0.777=−−
= )kg/s6(3.10087.2623
26.8525.1061sm&
Steam 3 MPa x = 0.90
Feedwater4 MPa 200°C
3 MPa Sat. liq.
4 MPa 245°C
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10-38
10-52 A steam power plant operates on an ideal regenerative Rankine cycle with two open feedwater heaters. The net power output of the power plant and the thermal efficiency of the cycle are to be determined.
Assumptions 1 Steady operating conditions exist. 2 Kinetic and potential energy changes are negligible.
Analysis
(a) From the steam tables (Tables A-4, A-5, and A-6),
( ) ( )( )kJ/kg95.13720.075.137
kJ/kg02.0mkPa1
kJ1kPa 5200/kgm 0.001005
/kgm 001005.0
kJ/kg75.137
in,12
33
121in,
3kPa5 @1
kPa5 @1
=+=+=
⎟ =⎟⎠
⎞⎜⎜⎝
⎛
⋅−=−=
==
==
pI
pI
f
f
whh
PPw
hh
v
vv
( ) ( )( )
( ) ( )( )kJ/kg10.35
mkPa1kJ1
kPa60010,000/kgm 0.001101
/kgm 001101.0
kJ/kg38.670
liquidsat.MPa6.0
kJ/kg13.50542.071.504kJ/kg0.42
mkPa1kJ1
kPa200600/kgm 0.001061
/kgm 001061.0
kJ/kg71.504
liquidsat.MPa2.0
33
565in,
3MPa6.0 @5
MPa6.0 @55
in,34
33
343in,
3MPa2.0@3
MPa2.0 @33
=
⎟⎟⎠
⎞⎜⎜⎝
⎛
⋅−=−=
==
==
⎭⎬⎫=
=+=+==
⎟⎟⎠
⎞⎜⎜⎝
⎛
⋅−=−=
==
==
⎭⎬⎫=
PPw
hhP
whh
PPw
hhP
pIII
f
f
pII
pII
f
f
v
vv
v
vv
kJ/kg8.2821MPa6.0
KkJ/kg9045.6kJ/kg8.3625
C600MPa10
kJ/kg73.68035.1038.670
878
8
7
7
7
7
in,56
=⎭⎬⎫
==
⋅==
⎭⎬⎫
°==
=+=+=
hss
P
sh
TP
whh pIII
1
2
10
s
T
7
9
0.6 MPa
5 kPa
0.2 MPa
1 - y - z
4y
3
6
5 8
1 - y
10 MPa
7
8
9
10
1 2
3 4
5
6
Turbine Boiler
Condenserfwh I fwh II
P I P II P III
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10-39
( )( ) kJ/kg7.26186.22019602.071.504
9602.05968.5
5302.19045.6MPa2.0
99
99
79
9
=+=+=
=−
=−
=
⎭⎬⎫
==
fgf
fg
f
hxhh
sss
x
ssP
( )( ) kJ/kg0.21050.24238119.075.137
8119.09176.7
4762.09045.6kPa5
1010
1010
710
10
=+=+=
=−
=−
=
⎭⎬⎫
==
fgf
fg
f
hxhh
sss
x
ssP
The fraction of steam extracted is determined from the steady-flow energy balance equation applied to the feedwater heaters. Noting that 0ΔpeΔke ≅≅≅≅WQ && ,
FWH-2:
( ) ( )548554488
outin
(steady)0outin
11
0
hhyyhhmhmhmhmhm
EE
EEE
eeii
system
=−+⎯→⎯=+⎯→⎯=
=
=Δ=−
∑∑ &&&&&
&&
&&&
where y is the fraction of steam extracted from the turbine ( = & / &m m8 5 ). Solving for y,
07133.013.5058.282113.50538.670
48
45 =−−
=−−
=hhhhy
FWH-1:
( ) ( ) 329332299 11 hyhzyzhhmhmhmhmhm eeii −=−−+⎯→⎯=+⎯→⎯=∑∑ &&&&&
where z is the fraction of steam extracted from the turbine ( = & / &m m9 5 ) at the second stage. Solving for z,
( ) ( ) 1373.007136.0195.1377.261895.13771.5041
29
23 =−−−
=−−−
= yhhhh
z
Then,
( )( ) ( )( )kJ/kg2.13888.15560.2945
kJ/kg 1556.8137.752105.01373.007133.011kJ/kg0.294573.6808.3625
outinnet
110out
67in
=−=−==−−−=−−−=
=−=−=
qqwhhzyq
hhq
and
( )( ) MW30.5≅=== kW540,30kJ/kg1388.2kg/s22netnet wmW &&
(b) 47.1%=−=−=kJ/kg 2945.0kJ/kg 1556.8
11in
outth q
qη
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10-40
10-53 [Also solved by EES on enclosed CD] A steam power plant operates on an ideal regenerative Rankine cycle with two feedwater heaters, one closed and one open. The mass flow rate of steam through the boiler for a net power output of 250 MW and the thermal efficiency of the cycle are to be determined.
Assumptions 1 Steady operating conditions exist. 2 Kinetic and potential energy changes are negligible.
Analysis (a) From the steam tables (Tables A-4, A-5, and A-6),
( )( )( )
kJ/kg 10.19229.081.191kJ/kg 29.0
mkPa1kJ1
kPa 10300/kgm 0.00101
/kgm 00101.0
kJ/kg81.191
in,12
33
121in,
3kPa10@1
kPa10@1
=+=+==
⎟⎟⎠
⎞⎜⎜⎝
⎛
⋅−=
−=
==
==
pI
pI
f
f
whh
PPw
hh
v
vv
( )
( )( )
( )( ) kJ/kg0.27555.20479935.087.720
9935.06160.4
0457.26317.6MPa8.0
KkJ/kg6317.6kJ/kg5.3476
C550MPa5.12
kJ/kg727.83 MPa 12.5 ,
C4.170
/kgm 001115.0
kJ/kg87.720
liquidsat.MPa8.0
kJ/kg52.57409.1343.561kJ/kg13.09
mkPa1kJ1
kPa30012,500/kgm 0.001073
/kgm 001073.0
kJ/kg43.561
liquidsat.MPa3.0
99
99
89
9
8
8
8
8
5556
MPa0.8@sat6
3MP8.0 @6
MPa8.0@766
in,34
33
343in,
3MPa3.0@3
MPa3.0 @33
=+=+=
=−
=−
=
⎭⎬⎫
==
⋅==
⎭⎬⎫
°==
=→==
°==
==
===
⎭⎬⎫=
=+=+==
⎟⎟⎠
⎞⎜⎜⎝
⎛
⋅−=
−===
==
⎭⎬⎫=
fgf
fg
f
af
f
pII
pII
f
f
hxhh
sss
xss
P
sh
TP
hPTT
TT
hhhP
whh
PPw
hhP
vv
v
vv
( )( )
( )( ) kJ/kg0.21001.23927977.081.191
7977.04996.7
6492.06317.kPa10
kJ/kg5.25785.21639323.043.561
9323.03200.5
6717.16317.6MPa3.0
1111
1111
811
11
1010
1010
810
10
=+=+=
=−6
=−
=
⎭⎬⎫
==
=+=+=
=−
=−
=
⎭⎬⎫
==
fgf
fg
f
fgf
fg
f
hxhh
sss
x
ssP
hxhh
sss
x
ssP
The fraction of steam extracted is determined from the steady-flow energy balance equation applied to the feedwater heaters. Noting that 0ΔpeΔke ≅≅≅≅WQ && ,
1
2
11 s
T
8
10
0.8 MPa
10 kPa
0.3 MPa
1 - y - z
4 y3
65
9
z
12.5 MPa
7
8
9
10
11
1 2
7
4
6
5
Turbine Boiler
CondenserOpenfwh
P I
P IIClosed
fwh
3
y
z1-y-z
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10-41
( ) ( ) ( ) ( )4569455699
outin
(steady)0systemoutin 0
hhhhyhhmhhmhmhm
EE
EEE
eeii −=−⎯→⎯−=−⎯→⎯=
=
=Δ=−
∑∑ &&&&
&&
&&&
where y is the fraction of steam extracted from the turbine ( 510 / mm &&= ). Solving for y,
0753.087.7200.275552.57483.727
69
45 =−−
=−−
=hhhhy
For the open FWH,
( ) ( ) 310273310102277
outin
(steady)0systemoutin
11
0
hzhhzyyhhmhmhmhmhmhm
EE
EEE
eeii =+−−+⎯→⎯=++⎯→⎯=
=
=Δ=−
∑∑ &&&&&&
&&
&&&
where z is the fraction of steam extracted from the turbine ( = & / &m m9 5 ) at the second stage. Solving for z,
( ) ( ) ( )( ) 1381.010.1925.2578
10.19287.7200753.010.19243.561
210
2723 =−
−−−=
−−−−
=hh
hhyhhz
Then,
( )( ) ( )( )kJ/kg12491.15001.2749
kJ/kg1.150081.1910.21001381.00753.011kJ/kg1.274936.7275.3476
outinnet
111out
58in
=−=−==−−−=−−−=
=−=−=
qqwhhzyq
hhq
and
kg/s200.2===kJ/kg1249
kJ/s250,000
net
net
wWm&
&
(b) 45.4%=−=−=kJ/kg2749.1kJ/kg1500.1
11in
outth q
qη
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10-42
10-54 EES Problem 10-53 is reconsidered. The effects of turbine and pump efficiencies on the mass flow rate and thermal efficiency are to be investigated. Also, the T-s diagram is to be plotted.
Analysis The problem is solved using EES, and the solution is given below.
"Input Data" P[8] = 12500 [kPa] T[8] = 550 [C] P[9] = 800 [kPa] "P_cfwh=300 [kPa]" P[10] = P_cfwh P_cond=10 [kPa] P[11] = P_cond W_dot_net=250 [MW]*Convert(MW, kW) Eta_turb= 100/100 "Turbine isentropic efficiency" Eta_turb_hp = Eta_turb "Turbine isentropic efficiency for high pressure stages" Eta_turb_ip = Eta_turb "Turbine isentropic efficiency for intermediate pressure stages" Eta_turb_lp = Eta_turb "Turbine isentropic efficiency for low pressure stages" Eta_pump = 100/100 "Pump isentropic efficiency"
"Condenser exit pump or Pump 1 analysis"
Fluid$='Steam_IAPWS' P[1] = P[11] P[2]=P[10] h[1]=enthalpy(Fluid$,P=P[1],x=0) {Sat'd liquid} v1=volume(Fluid$,P=P[1],x=0) s[1]=entropy(Fluid$,P=P[1],x=0) T[1]=temperature(Fluid$,P=P[1],x=0) w_pump1_s=v1*(P[2]-P[1])"SSSF isentropic pump work assuming constant specific volume" w_pump1=w_pump1_s/Eta_pump "Definition of pump efficiency" h[1]+w_pump1= h[2] "Steady-flow conservation of energy" s[2]=entropy(Fluid$,P=P[2],h=h[2]) T[2]=temperature(Fluid$,P=P[2],h=h[2])
"Open Feedwater Heater analysis" z*h[10] + y*h[7] + (1-y-z)*h[2] = 1*h[3] "Steady-flow conservation of energy" h[3]=enthalpy(Fluid$,P=P[3],x=0) T[3]=temperature(Fluid$,P=P[3],x=0) "Condensate leaves heater as sat. liquid at P[3]" s[3]=entropy(Fluid$,P=P[3],x=0)
"Boiler condensate pump or Pump 2 analysis" P[5]=P[8] P[4] = P[5] P[3]=P[10] v3=volume(Fluid$,P=P[3],x=0) w_pump2_s=v3*(P[4]-P[3])"SSSF isentropic pump work assuming constant specific volume" w_pump2=w_pump2_s/Eta_pump "Definition of pump efficiency" h[3]+w_pump2= h[4] "Steady-flow conservation of energy" s[4]=entropy(Fluid$,P=P[4],h=h[4]) T[4]=temperature(Fluid$,P=P[4],h=h[4])
"Closed Feedwater Heater analysis" P[6]=P[9] y*h[9] + 1*h[4] = 1*h[5] + y*h[6] "Steady-flow conservation of energy"
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10-43
h[5]=enthalpy(Fluid$,P=P[6],x=0) "h[5] = h(T[5], P[5]) where T[5]=Tsat at P[9]" T[5]=temperature(Fluid$,P=P[5],h=h[5]) "Condensate leaves heater as sat. liquid at P[6]" s[5]=entropy(Fluid$,P=P[6],h=h[5]) h[6]=enthalpy(Fluid$,P=P[6],x=0) T[6]=temperature(Fluid$,P=P[6],x=0) "Condensate leaves heater as sat. liquid at P[6]" s[6]=entropy(Fluid$,P=P[6],x=0)
"Trap analysis" P[7] = P[10] y*h[6] = y*h[7] "Steady-flow conservation of energy for the trap operating as a throttle" T[7]=temperature(Fluid$,P=P[7],h=h[7]) s[7]=entropy(Fluid$,P=P[7],h=h[7])
"Boiler analysis" q_in + h[5]=h[8]"SSSF conservation of energy for the Boiler" h[8]=enthalpy(Fluid$, T=T[8], P=P[8]) s[8]=entropy(Fluid$, T=T[8], P=P[8])
"Turbine analysis" ss[9]=s[8] hs[9]=enthalpy(Fluid$,s=ss[9],P=P[9]) Ts[9]=temperature(Fluid$,s=ss[9],P=P[9]) h[9]=h[8]-Eta_turb_hp*(h[8]-hs[9])"Definition of turbine efficiency for high pressure stages" T[9]=temperature(Fluid$,P=P[9],h=h[9]) s[9]=entropy(Fluid$,P=P[9],h=h[9]) ss[10]=s[8] hs[10]=enthalpy(Fluid$,s=ss[10],P=P[10]) Ts[10]=temperature(Fluid$,s=ss[10],P=P[10]) h[10]=h[9]-Eta_turb_ip*(h[9]-hs[10])"Definition of turbine efficiency for Intermediate pressure stages" T[10]=temperature(Fluid$,P=P[10],h=h[10]) s[10]=entropy(Fluid$,P=P[10],h=h[10]) ss[11]=s[8] hs[11]=enthalpy(Fluid$,s=ss[11],P=P[11]) Ts[11]=temperature(Fluid$,s=ss[11],P=P[11]) h[11]=h[10]-Eta_turb_lp*(h[10]-hs[11])"Definition of turbine efficiency for low pressure stages" T[11]=temperature(Fluid$,P=P[11],h=h[11]) s[11]=entropy(Fluid$,P=P[11],h=h[11]) h[8] =y*h[9] + z*h[10] + (1-y-z)*h[11] + w_turb "SSSF conservation of energy for turbine"
"Condenser analysis" (1-y-z)*h[11]=q_out+(1-y-z)*h[1]"SSSF First Law for the Condenser"
"Cycle Statistics" w_net=w_turb - ((1-y-z)*w_pump1+ w_pump2) Eta_th=w_net/q_in W_dot_net = m_dot * w_net
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10-44
ηturb ηturb ηth m [kg/s] 0.7 0.7 0.3916 231.6
0.75 0.75 0.4045 224.3 0.8 0.8 0.4161 218
0.85 0.85 0.4267 212.6 0.9 0.9 0.4363 207.9
0.95 0.95 0.4452 203.8 1 1 0.4535 200.1
0.7 0.75 0.8 0.85 0.9 0.95 1
0.39
0.4
0.41
0.42
0.43
0.44
0.45
0.46
200
205
210
215
220
225
230
235
ηturb
ηth
m [
kg/s
]
ηturb =ηpump
0 2 4 6 8 10 120
100
200
300
400
500
600
s [kJ/kg-K]
T [C
]
12500 kPa
800 kPa
300 kPa
10 kPa
Steam
1,2
3,4
5,6
7
8
910
11
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10-45
10-55 An ideal regenerative Rankine cycle with a closed feedwater heater is considered. The work produced by the turbine, the work consumed by the pumps, and the heat added in the boiler are to be determined.
Assumptions 1 Steady operating conditions exist. 2 Kinetic and potential energy changes are negligible.
Analysis From the steam tables (Tables A-4, A-5, and A-6),
kJ/kg45.25403.342.251kJ/kg03.3 mkPa1
kJ1kPa)203000)(/kgm001017.0()(
/kgm001017.0
kJ/kg42.251
inp,12
33
121inp,
3kPa20@1
kPa20@1
=+=+==
⎟⎠
⎞⎜⎝
⎛
⋅−=
−=
==
==
whh
PPw
hh
f
f
v
vv
kJ/kg7.2221)5.2357)(8357.0(42.251
8357.00752.7
8320.07450.6
kPa20
kJ/kg9.2851 kPa1000
KkJ/kg7450.6kJ/kg1.3116
C350kPa3000
66
66
46
6
545
5
4
4
4
4
=+=+=
=−
=−
=
⎭⎬⎫
==
=⎭⎬⎫
==
⋅==
⎭⎬⎫
°==
fgf
fg
f
hxhhs
ssx
ssP
hss
P
sh
TP
For an ideal closed feedwater heater, the feedwater is heated to the exit temperature of the extracted steam, which ideally leaves the heater as a saturated liquid at the extraction pressure.
kJ/kg53.763 C9.209
kPa3000
C9.179kJ/kg51.762
0
kPa1000
373
3
7
7
7
7
=⎭⎬⎫
°===
°==
⎭⎬⎫
==
hTT
P
Th
xP
An energy balance on the heat exchanger gives the fraction of steam extracted from the turbine ( 45 / mm &&= ) for closed feedwater heater:
1
qin
2
6
s
T
4
5
3 MPa
20 kPa
1 MPa
1-y 7
y 3
qout
56
12 3
4 Turbine
Boiler
Condenser
Closed fwh Pump
7
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10-46
7325
77332255
11 yhhhyhhmhmhmhm
hmhm eeii
+=++=+
=∑∑&&&&
&&
Rearranging,
2437.051.7629.285145.25453.763
75
23 =−−
=−−
=hhhh
y
Then,
kJ/kg2353
kJ/kg3.03kJ/kg740.9
=−=−=
=
=−−+−=−−+−=
53.7631.3116
)7.22219.2851)(2437.01(9.28511.3116))(1(
34in
inP,
6554outT,
hhqw
hhyhhw
Also,
kJ/kg8.73703.39.740inP,outT,net =−=−= www
3136.02353
8.737
in
netth ===
qw
η
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PROPRIETARY MATERIAL. © 2008 The McGraw-Hill Companies, Inc. Limited distribution permitted only to teachers and educators for course preparation. If you are a student using this Manual, you are using it without permission.
10-47
10-56 An ideal regenerative Rankine cycle with a closed feedwater heater is considered. The change in thermal efficiency when the steam serving the closed feedwater heater is extracted at 600 kPa rather than 1000 kPa is to be determined.
Assumptions 1 Steady operating conditions exist. 2 Kinetic and potential energy changes are negligible.
Analysis From the steam tables (Tables A-4, A-5, and A-6 or EES),
kJ/kg45.25403.342.251kJ/kg03.3 mkPa1
kJ1kPa)203000)(/kgm001017.0()(
/kgm001017.0
kJ/kg42.251
inp,12
33
121inp,
3kPa20@1
kPa20@1
=+=+==
⎟⎠
⎞⎜⎝
⎛
⋅−=
−=
==
==
whh
PPw
hh
f
f
v
vv
kJ/kg7.2221)5.2357)(8357.0(42.251
8357.00752.7
8320.07450.6
kPa20
kJ/kg0.2750)8.2085)(9970.0(38.670
9970.08285.4
9308.17450.6
kPa600
KkJ/kg7450.6kJ/kg1.3116
C350kPa3000
66
66
46
6
55
55
45
5
4
4
4
4
=+=+=
=−
=−
=
⎭⎬⎫
==
=+=+=
=−
=−
=
⎭⎬⎫
==
⋅==
⎭⎬⎫
°==
fgf
fg
f
fgf
fg
f
hxhhs
ssx
ssP
hxhhs
ssx
ssP
sh
TP
For an ideal closed feedwater heater, the feedwater is heated to the exit temperature of the extracted steam, which ideally leaves the heater as a saturated liquid at the extraction pressure.
kJ/kg79.671 C8.158
kPa3000
C8.158kJ/kg38.670
0
kPa600
373
3
7
7
7
7
=⎭⎬⎫
°===
°==
⎭⎬⎫
==
hTT
P
Th
xP
1
qin
2
6
s
T
4
5
3 MPa
20 kPa
0.6 MPa
1-y 7
y 3
qout
56
12 3
4 Turbine
Boiler
Condenser
Closed fwh Pump
7
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10-48
An energy balance on the heat exchanger gives the fraction of steam extracted from the turbine ( 45 / mm &&= ) for closed feedwater heater:
7325
77332255
11 yhhhyhhmhmhmhm
hmhm eeii
+=++=+
=∑∑&&&&
&&
Rearranging,
2007.038.6700.275045.25479.671
75
23 =−−
=−−
=hhhh
y
Then,
kJ/kg244479.6711.3116
kJ/kg3.03kJ/kg788.4)7.22210.2750)(2007.01(0.27501.3116))(1(
34in
inP,
6554outT,
=−=−=
=
=−−+−=−−+−=
hhqw
hhyhhw
Also,
kJ/kg4.78503.34.788inP,outT,net =−=−= www
0.3213===2444
4.785
in
netth q
wη
When the steam serving the closed feedwater heater is extracted at 600 kPa rather than 1000 kPa, the thermal efficiency increases from 0.3136 to 0.3213. This is an increase of 2.5%.
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10-49
10-57 EES The optimum bleed pressure for the open feedwater heater that maximizes the thermal efficiency of the cycle is to be determined by EES.
Analysis The EES program used to solve this problem as well as the solutions are given below.
"Given" P[4]=3000 [kPa] T[4]=350 [C] P[5]=600 [kPa] P[6]=20 [kPa]
P[3]=P[4] P[2]=P[3] P[7]=P[5] P[1]=P[6]
"Analysis" Fluid$='steam_iapws'
"pump I" x[1]=0 h[1]=enthalpy(Fluid$, P=P[1], x=x[1]) v[1]=volume(Fluid$, P=P[1], x=x[1]) w_p_in=v[1]*(P[2]-P[1]) h[2]=h[1]+w_p_in
"turbine" h[4]=enthalpy(Fluid$, P=P[4], T=T[4]) s[4]=entropy(Fluid$, P=P[4], T=T[4]) s[5]=s[4] h[5]=enthalpy(Fluid$, P=P[5], s=s[5]) T[5]=temperature(Fluid$, P=P[5], s=s[5]) x[5]=quality(Fluid$, P=P[5], s=s[5]) s[6]=s[4] h[6]=enthalpy(Fluid$, P=P[6], s=s[6]) x[6]=quality(Fluid$, P=P[6], s=s[6])
"closed feedwater heater" x[7]=0 h[7]=enthalpy(Fluid$, P=P[7], x=x[7]) T[7]=temperature(Fluid$, P=P[7], x=x[7]) T[3]=T[7] h[3]=enthalpy(Fluid$, P=P[3], T=T[3]) y=(h[3]-h[2])/(h[5]-h[7]) "y=m_dot_5/m_dot_4"
"cycle" q_in=h[4]-h[3] w_T_out=h[4]-h[5]+(1-y)*(h[5]-h[6]) w_net=w_T_out-w_p_in Eta_th=w_net/q_in
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10-50
P 6 [kPa] ηth 100 0.32380 110 0.32424 120 0.32460 130 0.32490 140 0.32514 150 0.32534 160 0.32550 170 0.32563 180 0.32573 190 0.32580 200 0.32585 210 0.32588 220 0.32590 230 0.32589 240 0.32588 250 0.32585 260 0.32581 270 0.32576 280 0.32570 290 0.32563
100 140 180 220 260 3000.3238
0.324
0.3243
0.3245
0.3248
0.325
0.3253
0.3255
0.3258
0.326
Bleed pressure [kPa]
ηth
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10-51
10-58 A regenerative Rankine cycle with a closed feedwater heater is considered. The thermal efficiency is to be determined.
Assumptions 1 Steady operating conditions exist. 2 Kinetic and potential energy changes are negligible.
Analysis From the steam tables (Tables A-4, A-5, and A-6 or EES),
kJ/kg45.25403.342.251kJ/kg03.3 mkPa1
kJ1kPa)203000)(/kgm001017.0()(
/kgm001017.0
kJ/kg42.251
inp,12
33
121inp,
3kPa20@1
kPa20@1
=+=+==
⎟⎠
⎞⎜⎝
⎛
⋅−=
−=
==
==
whh
PPw
hh
f
f
v
vv
kJ/kg7.2221)5.2357)(8357.0(42.251
8357.00752.7
8320.07450.6
kPa20
kJ/kg9.2851 kPa1000
KkJ/kg7450.6kJ/kg1.3116
C350kPa3000
66
66
46
6
545
5
4
4
4
4
=+=+=
=−
=−
=
⎭⎬⎫
==
=⎭⎬⎫
==
⋅==
⎭⎬⎫
°==
fgsfs
fg
fss
s
ss
hxhhs
ssx
ssP
hss
P
sh
TP
kJ/kg3.2878)9.28511.3116)(90.0(1.3116)( 5s4T4554
54T =−−=−−=⎯→⎯
−−
= hhhhhhhh
sηη
kJ/kg1.2311)7.22211.3116)(90.0(1.3116)( 6s4T4664
64T =−−=−−=⎯→⎯
−−
= hhhhhhhh
sηη
For an ideal closed feedwater heater, the feedwater is heated to the exit temperature of the extracted steam, which ideally leaves the heater as a saturated liquid at the extraction pressure.
56
12 3
4 Turbine
Boiler
Condenser
Closed fwh Pump
7 1
qin
2
6
s
T
4
5s
3 MPa
20 kPa
1 MPa
1-y 7
y 3
qout
5
6s
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10-52
kJ/kg53.763 C9.209
kPa3000
C9.179kJ/kg51.762
0
kPa1000
373
3
7
7
7
7
=⎭⎬⎫
°===
°==
⎭⎬⎫
==
hTT
P
Th
xP
An energy balance on the heat exchanger gives the fraction of steam extracted from the turbine ( 45 / mm &&= ) for closed feedwater heater:
7325
77332255
11 yhhhyhhmhmhmhm
hmhm eeii
+=++=+
=∑∑&&&&
&&
Rearranging,
2406.051.7623.287845.25453.763
75
23 =−−
=−−
=hhhh
y
Then,
kJ/kg235353.7631.3116
kJ/kg3.03kJ/kg668.5)1.23113.2878)(2406.01(3.28781.3116))(1(
34in
inP,
6554outT,
=−=−=
=
=−−+−=−−+−=
hhqw
hhyhhw
Also,
kJ/kg5.66503.35.668inP,outT,net =−=−= www
0.2829===2353
5.665
in
netth q
wη
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PROPRIETARY MATERIAL. © 2008 The McGraw-Hill Companies, Inc. Limited distribution permitted only to teachers and educators for course preparation. If you are a student using this Manual, you are using it without permission.
10-53
10-59 A regenerative Rankine cycle with a closed feedwater heater is considered. The thermal efficiency is to be determined.
Assumptions 1 Steady operating conditions exist. 2 Kinetic and potential energy changes are negligible.
Analysis From the steam tables (Tables A-4, A-5, and A-6 or EES),
When the liquid enters the pump 10°C cooler than a saturated liquid at the condenser pressure, the enthalpies become
/kgm001012.0kJ/kg34.209
C501006.6010
kPa203
C50@1
C50@1
kPa20@sat 1
1
=≅
=≅
⎭⎬⎫
°≅−=−==
°
°
f
fhhTT
Pvv
kJ/kg02.3 mkPa1kJ1kPa)203000)(/kgm001012.0(
)(
33
121inp,
=⎟⎠
⎞⎜⎝
⎛
⋅−=
−= PPw v
kJ/kg36.21202.334.209inp,12 =+=+= whh
kJ/kg7.2221)5.2357)(8357.0(42.251
8357.00752.7
8320.07450.6
kPa20
kJ/kg9.2851 kPa1000
KkJ/kg7450.6kJ/kg1.3116
C350kPa3000
66
66
46
6
545
5
4
4
4
4
=+=+=
=−
=−
=
⎭⎬⎫
==
=⎭⎬⎫
==
⋅==
⎭⎬⎫
°==
fgsfs
fg
fss
s
ss
hxhhs
ssx
ssP
hss
P
sh
TP
kJ/kg3.2878)9.28511.3116)(90.0(1.3116)( 5s4T4554
54T =−−=−−=⎯→⎯
−−
= hhhhhhhh
sηη
kJ/kg1.2311)7.22211.3116)(90.0(1.3116)( 6s4T4664
64T =−−=−−=⎯→⎯
−−
= hhhhhhhh
sηη
For an ideal closed feedwater heater, the feedwater is heated to the exit temperature of the extracted steam, which ideally leaves the heater as a saturated liquid at the extraction pressure.
56
12 3
4 Turbine
Boiler
Condenser
Closed fwh Pump
7 1
qin
2
6
s
T
4
5s
3 MPa
20 kPa
1 MPa
1-y 7
y 3
qout
5
6s
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10-54
kJ/kg53.763 C9.209
kPa3000
C9.179kJ/kg51.762
0
kPa1000
373
3
7
7
7
7
=⎭⎬⎫
°===
°==
⎭⎬⎫
==
hTT
P
Th
xP
An energy balance on the heat exchanger gives the fraction of steam extracted from the turbine ( 45 / mm &&= ) for closed feedwater heater:
7325
77332255
11 yhhhyhhmhmhmhm
hmhm eeii
+=++=+
=∑∑&&&&
&&
Rearranging,
2605.051.7623.287836.21253.763
75
23 =−−
=−−
=hhhh
y
Then,
kJ/kg235353.7631.3116
kJ/kg3.03kJ/kg657.2)1.23113.2878)(2605.01(3.28781.3116))(1(
34in
inP,
6554outT,
=−=−=
=
=−−+−=−−+−=
hhqw
hhyhhw
Also,
kJ/kg2.65403.32.657inP,outT,net =−=−= www
0.2781===2353
2.654
in
netth q
wη
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PROPRIETARY MATERIAL. © 2008 The McGraw-Hill Companies, Inc. Limited distribution permitted only to teachers and educators for course preparation. If you are a student using this Manual, you are using it without permission.
10-55
10-60 EES The effect of pressure drop and non-isentropic turbine on the rate of heat input is to be determined for a given power plant.
Analysis The EES program used to solve this problem as well as the solutions are given below.
"Given" P[3]=3000 [kPa] DELTAP_boiler=10 [kPa] P[4]=P[3]-DELTAP_boiler T[4]=350 [C] P[5]=1000 [kPa] P[6]=20 [kPa] eta_T=0.90
P[2]=P[3] P[7]=P[5] P[1]=P[6]
"Analysis" Fluid$='steam_iapws' "(a)" "pump I" x[1]=0 h[1]=enthalpy(Fluid$, P=P[1], x=x[1]) v[1]=volume(Fluid$, P=P[1], x=x[1]) w_p_in=v[1]*(P[2]-P[1]) h[2]=h[1]+w_p_in "turbine" h[4]=enthalpy(Fluid$, P=P[4], T=T[4]) s[4]=entropy(Fluid$, P=P[4], T=T[4]) s[5]=s[4] h_s[5]=enthalpy(Fluid$, P=P[5], s=s[5]) T[5]=temperature(Fluid$, P=P[5], s=s[5]) x_s[5]=quality(Fluid$, P=P[5], s=s[5]) s[6]=s[4] h_s[6]=enthalpy(Fluid$, P=P[6], s=s[6]) x_s[6]=quality(Fluid$, P=P[6], s=s[6])
h[5]=h[4]-eta_T*(h[4]-h_s[5]) h[6]=h[4]-eta_T*(h[4]-h_s[6]) x[5]=quality(Fluid$, P=P[5], h=h[5]) x[6]=quality(Fluid$, P=P[6], h=h[6])
"closed feedwater heater" x[7]=0 h[7]=enthalpy(Fluid$, P=P[7], x=x[7]) T[7]=temperature(Fluid$, P=P[7], x=x[7]) T[3]=T[7] h[3]=enthalpy(Fluid$, P=P[3], T=T[3]) y=(h[3]-h[2])/(h[5]-h[7]) "y=m_dot_5/m_dot_4"
"cycle" q_in=h[4]-h[3] w_T_out=h[4]-h[5]+(1-y)*(h[5]-h[6]) w_net=w_T_out-w_p_in Eta_th=w_net/q_in
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10-56
Solution with 10 kPa pressure drop in the boiler:
DELTAP_boiler=10 [kPa] eta_T=0.9 Eta_th=0.2827 Fluid$='steam_iapws' P[3]=3000 [kPa] P[4]=2990 [kPa] q_in=2352.8 [kJ/kg] w_net=665.1 [kJ/kg] w_p_in=3.031 [m^3-kPa/kg] w_T_out=668.1 [kJ/kg] y=0.2405
Solution without any pressure drop in the boiler:
DELTAP_boiler=0 [kPa] eta_T=1 Eta_th=0.3136 Fluid$='steam_iapws' P[3]=3000 [kPa] P[4]=3000 [kPa] q_in=2352.5 [kJ/kg] w_net=737.8 [kJ/kg] w_p_in=3.031 [m^3-kPa/kg] w_T_out=740.9 [kJ/kg] y=0.2437
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10-57
10-61E A steam power plant operates on an ideal reheat-regenerative Rankine cycle with one reheater and two open feedwater heaters. The mass flow rate of steam through the boiler, the net power output of the plant, and the thermal efficiency of the cycle are to be determined. Assumptions 1 Steady operating conditions exist. 2 Kinetic and potential energy changes are negligible. Analysis
(a) From the steam tables (Tables A-4E, A-5E, and A-6E),
( )
( )( )
Btu/lbm84.6912.072.69Btu/lbm0.12
ftpsia 5.4039Btu1
psia140/lbmft0.01614
/lbmft 01614.0
Btu/lbm72.69
in,12
33
121in,
3psia1 @1
psia1 @1
=+=+==
⎟⎟⎠
⎞⎜⎜⎝
⎛
⋅−=
−=
==
==
pI
pI
f
f
whh
PPw
hh
v
vv
( )( )( )
( )( )( )
Btu/lbm41.38031.409.376Btu/lbm4.31
ftpsia 5.4039Btu1
psia2501500/lbmft 0.01865
/lbmft 01865.0Btu/lbm09.376
liquidsat.psia250
Btu/lbm81.23667.014.236
Btu/lbm0.67ftpsia 5.4039
Btu1psia40250/lbmft 0.01715
/lbmft 01715.0Btu/lbm14.236
liquidsat.psia40
in,56
33
565in,
3psia250@5
psia250@55
i,34
33
343in,
3psia40@3
psia40@33
=+=+==
⎟⎟⎠
⎞⎜⎜⎝
⎛
⋅−=
−=
==
==
⎭⎬⎫=
=+=+=
=
⎟⎟⎠
⎞⎜⎜⎝
⎛
⋅−=
−=
==
==
⎭⎬⎫=
pIII
pIII
f
f
npII
pII
f
f
whh
PPw
hhP
whh
PPw
hhP
v
vv
v
vv
Btu/lbm5.1308psia250
RBtu/lbm6402.1Btu/lbm5.1550
F1100psia1500
878
8
7
7
7
7
=⎭⎬⎫
==
⋅==
⎭⎬⎫
°==
hss
P
sh
TP
1
2
12
s
T
7
11250 psia
1 psia
40 psia 1 - y - z
4y
3
6
58 1 - y
1500psia
z 10
9 140 psia
9
7
8
1 2
3
6
Low-P Turbine Boiler
CondenseOpen fwh I
P I P II
y
1-y
Open fwh II
4
10
z
11 121-y-z
P III 5
High-P Turbine
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10-58
( )( )Btu/lbm4.1052
7.10359488.072.69
9488.084495.1
13262.08832.1
psia1
Btu/lbm0.1356psia40
RBtu/lbm8832.1Btu/lbm3.1531
F1000psia140
Btu/lbm8.1248psia140
1212
1212
1012
12
111011
11
10
10
10
10
979
9
=
+=+=
=−
=−
=
⎭⎬⎫
==
=⎭⎬⎫
==
⋅==
⎭⎬⎫
°==
=⎭⎬⎫
==
fgf
fg
f
hxhh
sss
x
ssP
hss
P
sh
TP
hss
P
The fraction of steam extracted is determined from the steady-flow energy balance equation applied to the feedwater heaters. Noting that 0ΔpeΔke ≅≅≅≅WQ && ,
FWH-2:
( ) ( )548554488
outin
(steady)0systemoutin
11
0
hhyyhhmhmhmhmhm
EE
EEE
eeii =−+⎯→⎯=+⎯→⎯=
=
=Δ=−
∑∑ &&&&&
&&
&&&
where y is the fraction of steam extracted from the turbine ( = & / &m m8 5 ). Solving for y,
1300.081.2365.130881.23609.376
48
45 =−−
=−−
=hhhh
y
FWH-1
( ) ( ) 321133221111
outin
(steady)0systemoutin
11
0
hyhzyzhhmhmhmhmhm
EE
EEE
eeii −=−−+⎯→⎯=+⎯→⎯=
=
=Δ=−
∑∑ &&&&&
&&
&&&
where z is the fraction of steam extracted from the turbine ( = & / &m m9 5 ) at the second stage. Solving for z,
( ) ( ) 1125.01300.0184.690.135684.6914.2361
211
23 =−−−
=−−−
= yhhhh
z
Then,
( )( ) ( )( )
( )( ) ( )( )Btu/lbm4.6714.7448.1415
Btu/lbm744.469.721052.41125.01300.011Btu/lbm8.14158.12483.15311300.0141.3805.15501
outinnet
112out
91067in
=−=−==−−−=−−−=
=−−+−=−−+−=
qqwhhzyq
hhyhhq
and
lbm/s282.5=×
==Btu/lbm 1415.8
Btu/s104 5
in
in
m&
&
(b) ( )( ) MW200.1⎟ =⎟⎠
⎞⎜⎜⎝
⎛==
Btu1kJ1.055
Btu/lbm671.4lbm/s282.5netnet wmW &&
(c) 47.4%=−=−=Btu/lbm1415.8Btu/lbm744.4
11in
outth q
qη
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10-59
10-62 A steam power plant that operates on an ideal regenerative Rankine cycle with a closed feedwater heater is considered. The temperature of the steam at the inlet of the closed feedwater heater, the mass flow rate of the steam extracted from the turbine for the closed feedwater heater, the net power output, and the thermal efficiency are to be determined.
Assumptions 1 Steady operating conditions exist. 2 Kinetic and potential energy changes are negligible.
Analysis (a) From the steam tables (Tables A-4, A-5, and A-6),
( )
kJ/kg85.265
43.1442.251
kJ/kg.431488.01)kPa2012,500)(/kgm 0.001017(
/
/kgm 001017.0
kJ/kg42.251
in,12
3
121in,
3kPa20@1
kPa20@1
=
+=
+=
=
−=
−=
==
==
pI
ppI
f
f
whh
PPw
hh
ηv
vv
/kgm001127.0
kJ/kg51.762
liquidsat.MPa1
3MPa1 @3
MPa1 @33
==
==
⎭⎬⎫=
f
fhhPvv
( )
kJ/kg25.77773.1451.762kJ/kg73.14
88.0/)kPa001012,500)(/kgm001127.0(
/
in,311
3
3113in,
=+=+==
−=
−=
pII
ppII
whh
PPw ηv
Also, h4 = h10 = h11 = 777.25 kJ/kg since the two fluid streams which are being mixed have the same enthalpy.
( )( )( ) kJ/kg5.32206.31855.347688.05.3476
kJ/kg6.3185MPa5
KkJ/kg6317.6kJ/kg5.3476
C550MPa5.12
655665
65
656
6
5
5
5
5
=−−=−−=⎯→⎯
−−
=
=⎭⎬⎫
==
⋅==
⎭⎬⎫
°==
sTs
T
s
hhhhhhhh
hss
P
sh
TP
ηη
( )( )( )
C328°=⎭⎬⎫
==
=−−=−−=⎯→⎯
−−
=
=⎭⎬⎫
==
⋅==
⎭⎬⎫
°==
88
8
877887
87
878
8
7
7
7
7
kJ/kg1.3111MPa1
kJ/kg1.31111.30519.355088.09.3550
kJ/kg1.3051MPa1
KkJ/kg1238.7kJ/kg9.3550
C550MPa5
ThP
hhhhhhhh
hss
P
sh
TP
sTs
T
s
ηη
6
5
8
1 2
3
4
High-P turbine
Boiler
Cond. Closed
fwh
PI PII
y
7 1-y
MixingCham.
11
10 9
Low-P turbine
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10-60
( )( )( ) kJ/kg2.24929.23479.355088.09.3550
kJ/kg9.2347kPa20
977997
97
979
9
=−−=−−=⎯→⎯
−−
=
=⎭⎬⎫
==
sTs
T
s
hhhhhhhh
hss
P
ηη
The fraction of steam extracted from the low pressure turbine for closed feedwater heater is determined from the steady-flow energy balance equation applied to the feedwater heater. Noting that & &Q W ke pe≅ ≅ ≅ ≅Δ Δ 0 ,
( )( ) ( )1788.0)51.7621.3111()85.26525.777)(1(
1 38210
=⎯→⎯−=−−
−=−−
yyy
hhyhhy
The corresponding mass flow rate is
kg/s4.29=== kg/s)24)(1788.0(58 mym &&
(c) Then,
( )( ) ( )( ) kJ/kg1.184042.2512.24921788.011kJ/kg7.30295.32209.355025.7775.3476
19out
6745in
=−−=−−==−+−=−+−=
hhyqhhhhq
and
kW28,550=−=−= kJ/kg)1.18407.3029)(kg/s24()( outinnet qqmW &&
(b) The thermal efficiency is determined from
39.3%==−=−= 393.0kJ/kg3029.7kJ/kg1840.1
11in
outth q
qη
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10-61
Second-Law Analysis of Vapor Power Cycles
10-63C In the simple ideal Rankine cycle, irreversibilities occur during heat addition and heat rejection processes in the boiler and the condenser, respectively, and both are due to temperature difference. Therefore, the irreversibilities can be decreased and thus the 2nd law efficiency can be increased by minimizing the temperature differences during heat transfer in the boiler and the condenser. One way of doing that is regeneration.
10-64 The exergy destruction associated with the heat rejection process in Prob. 10-25 is to be determined for the specified source and sink temperatures. The exergy of the steam at the boiler exit is also to be determined.
Assumptions 1 Steady operating conditions exist. 2 Kinetic and potential energy changes are negligible.
Analysis From Problem 10-25,
kJ/kg8.1961kJ/kg4.3411
KkJ/kg8000.6
KkJ/kg6492.0
out
3
43
kPa10@21
==
⋅==
⋅===
qh
ss
sss f
The exergy destruction associated with the heat rejection process is
( ) kJ/kg178.0⎟ =⎟⎠
⎞⎜⎜⎝
⎛+−⎟ =⎟
⎠
⎞⎜⎜⎝
⎛+−=
K290kJ/kg1961.8
8000.66492.0K29041,41041destroyed,
R
R
Tq
ssTx
The exergy of the steam at the boiler exit is simply the flow exergy,
( ) ( )( ) ( )03003
03
023
030033 2ssThhqzssThh
−−−=++−−−=
Vψ
where
( )( ) KkJ/kg2533.0
kJ/kg95.71
K290@kPa100,K290@0
K290@kPa100,K290@0
⋅=≅==≅=
f
f
ssshhh
Thus,
( ) ( )( ) kJ/kg 1440.9=⋅−−−=ψ KkJ/kg2532.0800.6K290kJ/kg95.714.34113
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10-62
10-65E The exergy destructions associated with each of the processes of the Rankine cycle described in Prob. 10-15E are to be determined for the specified source and sink temperatures.
Assumptions 1 Steady operating conditions exist. 2 Kinetic and potential energy changes are negligible.
Analysis From Problem 10-15E,
Btu/lbm38.98202.1384.1120Btu/lbm5.149052.1390.1630
RBtu/lbm8075.1
RBtu/lbm24739.0
24out
23in
43
psia6 @21
=−=−==−=−=
⋅==
⋅===
hhqhhq
ss
sss f
⎟
The exergy destruction during a process of a stream from an inlet state to exit state is given by
⎟⎠
⎞⎜⎜⎝
⎛+−−==
sink
out
source
in0gen0dest T
qT
qssTsTx ie
Application of this equation for each process of the cycle gives
Btu/lbm202.3
Btu/lbm399.8
=⎟⎠⎞
⎜⎝⎛ +−⎟ =⎟
⎠
⎞⎜⎜⎝
⎛+−=
=⎟⎠⎞
⎜⎝⎛ −−⎟ =⎟
⎠
⎞⎜⎜⎝
⎛−−=
R500Btu/lbm38.9828075.124739.0)R500(
R1960Btu/lbm5.149024739.08075.1)R500(
sink
out41041destroyed,
source
in23023destroyed,
Tq
ssTx
Tq
ssTx
Processes 1-2 and 3-4 are isentropic, and thus
00
=
=
34destroyed,
12destroyed,
x
x
qin
qout
6 psia1
3
2
4
500 psia
s
T
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10-63
10-66 The exergy destructions associated with each of the processes of the Rankine cycle described in Prob. 10-17 are to be determined for the specified source and sink temperatures.
Assumptions 1 Steady operating conditions exist. 2 Kinetic and potential energy changes are negligible.
Analysis From Problem 10-17,
kJ/kg3.226242.2517.2513kJ/kg8.365047.2553.3906
KkJ/kg6214.7
KkJ/kg8320.0
14out
23in
43
kPa20@21
=−=−==−=−=
⋅==
⋅===
hhqhhq
ss
sss f
⎟
The exergy destruction during a process of a stream from an inlet state to exit state is given by
⎟⎠
⎞⎜⎜⎝
⎛+−−==
sink
out
source
in0gen0dest T
qT
qssTsTx ie
Application of this equation for each process of the cycle gives
kJ/kg307.0
kJ/kg927.6
=⎟⎠
⎞⎜⎝
⎛ +−⎟ =⎟⎠
⎞⎜⎜⎝
⎛+−=
=⎟⎠
⎞⎜⎝
⎛ −−⎟ =⎟⎠
⎞⎜⎜⎝
⎛−−=
K288kJ/kg3.2262
6214.78320.0)K288(
K1023kJ/kg8.3650
8320.06214.7)K288(
sink
out41041destroyed,
source
in23023destroyed,
Tq
ssTx
Tq
ssTx
Processes 1-2 and 3-4 are isentropic, and thus
00
=
=
34destroyed,
12destroyed,
x
x
qin
qout
20 kPa1
3
2
4
4 MPa
s
T
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10-64
10-67E The exergy destructions associated with each of the processes of the ideal reheat Rankine cycle described in Prob. 10-36E are to be determined for the specified source and sink temperatures.
Assumptions 1 Steady operating conditions exist. 2 Kinetic and potential energy changes are negligible.
Analysis From Problem 10-36E,
Btu/lbm7.909Btu/lbm8.1345.11873.1322
Btu/lbm8.112606.1639.1289RBtu/lbm6771.1RBtu/lbm5325.1
RBtu/lbm28362.0
out
455-4 in,
233-in,2
65
43
psia10@21
=
=−=−=
=−=−=
⋅==⋅==
⋅===
qhhqhhq
ssss
sss f
⎟
The exergy destruction during a process of a stream from an inlet state to exit state is given by
⎟⎠
⎞⎜⎜⎝
⎛+−−==
sink
out
source
in0gen0dest T
qT
qssTsTx ie
Application of this equation for each process of the cycle gives
Btu/lbm161.4
Btu/lbm15.2
Btu/lbm149.0
=⎟⎠⎞
⎜⎝⎛ +−⎟ =⎟
⎠
⎞⎜⎜⎝
⎛+−=
=⎟⎠⎞
⎜⎝⎛ −−⎟ =⎟
⎠
⎞⎜⎜⎝
⎛−−=
=⎟⎠⎞
⎜⎝⎛ −−⎟ =⎟
⎠
⎞⎜⎜⎝
⎛−−=
R537Btu/lbm7.9096771.128362.0)R537(
R1160Btu/lbm8.1345325.16771.1)R537(
R1160Btu/lbm8.112628362.05325.1)R537(
sink
out61061destroyed,
source
5-4 in,45045destroyed,
source
3-2 in,23023destroyed,
Tq
ssTx
Tq
ssTx
Tq
ssTx
Processes 1-2, 3-4, and 5-6 are isentropic, and thus,
000
=
=
=
56destroyed,
34destroyed,
12destroyed,
x
x
x
1
5
2
6
s
T
3
4
600 psia
10 psia
200 psia
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PROPRIETARY MATERIAL. © 2008 The McGraw-Hill Companies, Inc. Limited distribution permitted only to teachers and educators for course preparation. If you are a student using this Manual, you are using it without permission.
10-65
10-68 The exergy destructions associated with each of the processes of the reheat Rankine cycle described in Prob. 10-34 are to be determined for the specified source and sink temperatures.
Assumptions 1 Steady operating conditions exist. 2 Kinetic and potential energy changes are negligible.
Analysis From Problem 10-34,
kJ/kg8.213342.2512.2385
kJ/kg1.3521.31052.3457
kJ/kg0.314054.2595.3399KkJ/kg2359.7KkJ/kg7266.6
KkJ/kg8320.0
16out
in,45
in,23
65
43
kPa20@21
=−=−=
=−=
=−=⋅==⋅==
⋅===
hhq
q
qssss
sss f
Processes 1-2, 3-4, and 5-6 are isentropic. Thus, i12 = i34 = i56 = 0. Also,
( )
( )
( ) kJ/kg212.6
kJ/kg94.1
kJ/kg1245.0
=⎟⎠⎞
⎜⎝⎛ +−⎟ =⎟
⎠
⎞⎜⎜⎝
⎛+−=
⎟ =⎟⎠
⎞⎜⎜⎝
⎛ −+−⎟ =⎟
⎠
⎞⎜⎜⎝
⎛+−=
=⎟⎠⎞
⎜⎝⎛ −
+−⎟ =⎟⎠
⎞⎜⎜⎝
⎛+−=
K300kJ/kg2133.82359.78320.0K300
K1800kJ/kg352.57266.62359.7K300
K1800kJ/kg3140.08320.07266.6K300
61,61061destroyed,
45,45045destroyed,
23,23023destroyed,
R
R
R
R
R
R
Tq
ssTx
Tq
ssTx
Tq
ssTx
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10-66
10-69 EES Problem 10-68 is reconsidered. The problem is to be solved by the diagram window data entry feature of EES by including the effects of the turbine and pump efficiencies. Also, the T-s diagram is to be plotted.
Analysis The problem is solved using EES, and the solution is given below.
function x6$(x6) "this function returns a string to indicate the state of steam at point 6" x6$='' if (x6>1) then x6$='(superheated)' if (x6<0) then x6$='(subcooled)'
end "Input Data - from diagram window" {P[6] = 20 [kPa] P[3] = 8000 [kPa] T[3] = 500 [C] P[4] = 3000 [kPa] T[5] = 500 [C] Eta_t = 100/100 "Turbine isentropic efficiency" Eta_p = 100/100 "Pump isentropic efficiency"} "Data for the irreversibility calculations:" T_o = 300 [K] T_R_L = 300 [K] T_R_H = 1800 [K] "Pump analysis" Fluid$='Steam_IAPWS' P[1] = P[6] P[2]=P[3] x[1]=0 "Sat'd liquid" h[1]=enthalpy(Fluid$,P=P[1],x=x[1]) v[1]=volume(Fluid$,P=P[1],x=x[1]) s[1]=entropy(Fluid$,P=P[1],x=x[1]) T[1]=temperature(Fluid$,P=P[1],x=x[1]) W_p_s=v[1]*(P[2]-P[1])"SSSF isentropic pump work assuming constant specific volume" W_p=W_p_s/Eta_p h[2]=h[1]+W_p "SSSF First Law for the pump" v[2]=volume(Fluid$,P=P[2],h=h[2]) s[2]=entropy(Fluid$,P=P[2],h=h[2]) T[2]=temperature(Fluid$,P=P[2],h=h[2]) "High Pressure Turbine analysis" h[3]=enthalpy(Fluid$,T=T[3],P=P[3]) s[3]=entropy(Fluid$,T=T[3],P=P[3]) v[3]=volume(Fluid$,T=T[3],P=P[3]) s_s[4]=s[3] hs[4]=enthalpy(Fluid$,s=s_s[4],P=P[4]) Ts[4]=temperature(Fluid$,s=s_s[4],P=P[4]) Eta_t=(h[3]-h[4])/(h[3]-hs[4])"Definition of turbine efficiency" T[4]=temperature(Fluid$,P=P[4],h=h[4]) s[4]=entropy(Fluid$,T=T[4],P=P[4]) v[4]=volume(Fluid$,s=s[4],P=P[4]) h[3] =W_t_hp+h[4]"SSSF First Law for the high pressure turbine" "Low Pressure Turbine analysis" P[5]=P[4] s[5]=entropy(Fluid$,T=T[5],P=P[5]) h[5]=enthalpy(Fluid$,T=T[5],P=P[5]) s_s[6]=s[5]
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10-67
hs[6]=enthalpy(Fluid$,s=s_s[6],P=P[6]) Ts[6]=temperature(Fluid$,s=s_s[6],P=P[6]) vs[6]=volume(Fluid$,s=s_s[6],P=P[6]) Eta_t=(h[5]-h[6])/(h[5]-hs[6])"Definition of turbine efficiency" h[5]=W_t_lp+h[6]"SSSF First Law for the low pressure turbine" x[6]=QUALITY(Fluid$,h=h[6],P=P[6]) "Boiler analysis" Q_in + h[2]+h[4]=h[3]+h[5]"SSSF First Law for the Boiler" "Condenser analysis" h[6]=Q_out+h[1]"SSSF First Law for the Condenser" T[6]=temperature(Fluid$,h=h[6],P=P[6]) s[6]=entropy(Fluid$,h=h[6],P=P[6]) x6s$=x6$(x[6]) "Cycle Statistics" W_net=W_t_hp+W_t_lp-W_p Eff=W_net/Q_in "The irreversibilities (or exergy destruction) for each of the processes are:" q_R_23 = - (h[3] - h[2]) "Heat transfer for the high temperature reservoir to process 2-3" i_23 = T_o*(s[3] -s[2] + q_R_23/T_R_H) q_R_45 = - (h[5] - h[4]) "Heat transfer for the high temperature reservoir to process 4-5" i_45 = T_o*(s[5] -s[4] + q_R_45/T_R_H) q_R_61 = (h[6] - h[1]) "Heat transfer to the low temperature reservoir in process 6-1" i_61 = T_o*(s[1] -s[6] + q_R_61/T_R_L) i_34 = T_o*(s[4] -s[3]) i_56 = T_o*(s[6] -s[5]) i_12 = T_o*(s[2] -s[1])
0 .0 1 .1 2 .2 3 .3 4 .4 5 .5 6 .6 7 .7 8 .8 9 .9 11.00
100
200
300
400
500
600
700
s [kJ /kg -K ]
T [C
]
8 0 0 0 kP a
30 0 0 kP a
2 0 kP a
3
4
5
6
Id ea l R an k in e cyc le w ith reh eat
1 ,2
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10-68
SOLUTION
Eff=0.389 Eta_p=1 Eta_t=1 Fluid$='Steam_IAPWS' h[1]=251.4 [kJ/kg] h[2]=259.5 [kJ/kg] h[3]=3400 [kJ/kg] h[4]=3105 [kJ/kg] h[5]=3457 [kJ/kg] h[6]=2385 [kJ/kg] hs[4]=3105 [kJ/kg] hs[6]=2385 [kJ/kg] i_12=0.012 [kJ/kg] i_23=1245.038 [kJ/kg] i_34=-0.000 [kJ/kg] i_45=94.028 [kJ/kg] i_56=0.000 [kJ/kg] i_61=212.659 [kJ/kg] P[1]=20 [kPa] P[2]=8000 [kPa] P[3]=8000 [kPa] P[4]=3000 [kPa] P[5]=3000 [kPa] P[6]=20 [kPa] Q_in=3493 [kJ/kg] Q_out=2134 [kJ/kg] q_R_23=-3140 [kJ/kg] q_R_45=-352.5 [kJ/kg] q_R_61=2134 [kJ/kg]
s[1]=0.832 [kJ/kg-K] s[2]=0.8321 [kJ/kg-K] s[3]=6.727 [kJ/kg-K] s[4]=6.727 [kJ/kg-K] s[5]=7.236 [kJ/kg-K] s[6]=7.236 [kJ/kg-K] s_s[4]=6.727 [kJ/kg-K] s_s[6]=7.236 [kJ/kg-K] T[1]=60.06 [C] T[2]=60.4 [C] T[3]=500 [C] T[4]=345.2 [C] T[5]=500 [C] T[6]=60.06 [C] Ts[4]=345.2 [C] Ts[6]=60.06 [C] T_o=300 [K] T_R_H=1800 [K] T_R_L=300 [K] v[1]=0.001017 [m^3/kg] v[2]=0.001014 [m^3/kg] v[3]=0.04177 [m^3/kg] v[4]=0.08968 [m^3/kg] vs[6]=6.922 [m^3/kg] W_net=1359 [kJ/kg] W_p=8.117 [kJ/kg] W_p_s=8.117 [kJ/kg] W_t_hp=294.8 [kJ/kg] W_t_lp=1072 [kJ/kg] x6s$='' x[1]=0 x[6]=0.9051
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69
10-70 A single-flash geothermal power plant uses hot geothermal water at 230ºC as the heat source. The power output from the turbine, the thermal efficiency of the plant, the exergy of the geothermal liquid at the exit of the flash chamber, and the exergy destructions and exergy efficiencies for the flash chamber, the turbine, and the entire plant are to be determined.
Assumptions 1 Steady operating conditions exist. 2 Kinetic and potential energy changes are negligible.
Analysis (a) We use properties of water for geothermal water (Tables A-4, A-5, and A-6)
kJ/kg.K6841.21661.0
kJ/kg14.990kPa500
kJ/kg.K6100.2kJ/kg14.990
0C230
2
2
12
2
1
1
1
1
==
⎭⎬⎫
===
==
⎭⎬⎫
=°=
sx
hhP
sh
xT
kg/s38.19
kg/s)230)(1661.0(123
=== mxm &&
KkJ/kg7739.7kJ/kg3.2464
95.0kPa10
KkJ/kg8207.6kJ/kg1.2748
1kPa500
4
4
4
4
3
3
3
3
⋅==
⎭⎬⎫
==
⋅==
⎭⎬⎫
==
sh
xP
sh
xP
KkJ/kg8604.1
kJ/kg09.6400
kPa500
6
6
6
6
⋅==
⎭⎬⎫
==
sh
xP
kg/s81.19119.38230316 =−=−= mmm &&&
The power output from the turbine is
kW10,842=−=−= kJ/kg)3.24648.1kJ/kg)(27438.19()( 433T hhmW &&
We use saturated liquid state at the standard temperature for dead state properties
kJ/kg3672.0kJ/kg83.104
0C25
0
0
0
0
==
⎭⎬⎫
=°=
sh
xT
kW622,203kJ/kg)83.104.14kJ/kg)(990230()( 011in =−=−= hhmE &&
5.3%==== 0.0532622,203
842,10
in
outT,th E
W&
&η
(b) The specific exergies at various states are
kJ/kg53.216kJ/kg.K)3672.0K)(2.6100(298kJ/kg)83.104(990.14)( 010011 =−−−=−−−= ssThhψ
kJ/kg44.194kJ/kg.K)3672.0K)(2.6841(298kJ/kg)83.104(990.14)( 020022 =−−−=−−−= ssThhψ
kJ/kg10.719kJ/kg.K)3672.0K)(6.8207(298kJ/kg)83.104(2748.1)( 030033 =−−−=−−−= ssThhψ
kJ/kg05.151kJ/kg.K)3672.0K)(7.7739(298kJ/kg)83.104(2464.3)( 040044 =−−−=−−−= ssThhψ
kJ/kg97.89kJ/kg.K)3672.0K)(1.8604(298kJ/kg)83.104(640.09)( 060066 =−−−=−−−= ssThhψ
productionwell
reinjection well
separator
steam turbine
1
condenser2
3
4
5 6
Flash chamber
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70
The exergy of geothermal water at state 6 is
kW17,257=== kJ/kg)7kg/s)(89.9.81191(666 ψmX &&
(c) Flash chamber:
kW5080=−=−= kJ/kg)44.19453kg/s)(216.230()( 211FCdest, ψψmX &&
89.8%==== 0.89853.21644.194
1
2FCII, ψ
ψη
(d) Turbine:
kW10,854=−=−−= kW10,842-kJ/kg)05.15110kg/s)(719.19.38()( T433Tdest, WmX &&& ψψ
50.0%==−
=−
= 0.500)kJ/kg05.15110kg/s)(719.19.38(
kW842,10)( 433
TTII, ψψ
ηm
W&
&
(e) Plant:
kW802,49kJ/kg)53kg/s)(216.230(11Plantin, === ψmX &&
kW38,960=−=−= 842,10802,49TPlantin,Plantdest, WXX &&&
21.8%==== 0.2177kW802,49kW842,10
Plantin,
TPlantII, X
W&
&η
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71
Cogeneration
10-71C The utilization factor of a cogeneration plant is the ratio of the energy utilized for a useful purpose to the total energy supplied. It could be unity for a plant that does not produce any power.
10-72C No. A cogeneration plant may involve throttling, friction, and heat transfer through a finite temperature difference, and still have a utilization factor of unity.
10-73C Yes, if the cycle involves no irreversibilities such as throttling, friction, and heat transfer through a finite temperature difference.
10-74C Cogeneration is the production of more than one useful form of energy from the same energy source. Regeneration is the transfer of heat from the working fluid at some stage to the working fluid at some other stage.
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72
10-75 A cogeneration plant is to generate power and process heat. Part of the steam extracted from the turbine at a relatively high pressure is used for process heating. The net power produced and the utilization factor of the plant are to be determined. Assumptions 1 Steady operating conditions exist. 2 Kinetic and potential energy changes are negligible. Analysis From the steam tables (Tables A-4, A-5, and A-6),
( )( )( )
kJ/kg38.670
kJ/kg41.19260.081.191kJ/kg0.60
mkPa1kJ1
kPa10600/kgm 0.00101
/kgm 00101.0kJ/kg81.191
MPa0.6@3
inpI,12
33
121inpI,
3kPa10@1
kPa10@1
==
=+=+==
⎟⎟⎠
⎞⎜⎜⎝
⎛
⋅−=
−=
==
==
f
f
f
hh
whh
PPw
hh
v
vv
Mixing chamber:
332244
outin(steady)0
systemoutin 0
hmhmhmhmhm
EEEEE
eeii &&&&&
&&&&&
+=⎯→⎯=
=⎯→⎯=Δ=−
∑∑or, ( )( ) ( )( )
( )( )( )
kJ/kg47.31857.690.311kJ/kg6.57
mkPa1kJ1kPa6007000/kgm 0.001026
/kgm 001026.0
kJ/kg90.31130
38.67050.741.19250.22
inII,45
33
454inII,
3kJ/kg90.311@4
4
33224
=+=+==
⎟⎟⎠
⎞⎜⎜⎝
⎛
⋅−=
−=
=≅
=+
=+
=
=
p
p
hf
whh
PPw
mhmhmh
f
v
vv&
&&
KkJ/kg8000.6kJ/kg4.3411
C500MPa7
6
6
6
6⋅=
=
⎭⎬⎫
°==
sh
TP
( )( ) kJ/kg6.21531.23928201.081.191
8201.04996.7
6492.08000.6kPa10
kJ/kg6.2774MPa6.0
88
88
68
8
767
7
=+=+=
=−
=−
=
⎭⎬⎫
==
=⎭⎬⎫
==
fgf
fg
f
hxhhs
ssx
ssP
hss
P
Then, ( ) ( )
( )( ) ( )( )( )( ) ( )( )
kW 32,866=−=−=
=+=+=
=−+−=−+−=
6.210077,33
kW210.6kJ/kg6.57kg/s30kJ/kg0.60kg/s22.5
kW077,33kJ/kg6.21536.2774kg/s22.5kJ/kg6.27744.3411kg/s30
inp,outT,net
inpII,4inpI,1inp,
878766outT,
WWW
wmwmW
hhmhhmW
&&&
&&&
&&&
Also, ( ) ( )( ) kW782,15kJ/kg38.6706.2774kg/s7.5377process =−=−= hhmQ &&
( ) ( )( ) kW788,9247.3184.3411kg/s30565in =−=−= hhmQ &&
and 52.4%=+
=+
=788,92
782,15866,32
in
processnet
Q
QWu &
&&ε
1
2
8
s
T
6
7
7 MPa
10 kPa
0.6 MPa 4
5
3
Qout ·
Qin ·
Qproces·
6
8
1
5
Turbine Boiler
Condenser
Processheater
P I P II4 2
3
7
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73
10-76E A large food-processing plant requires steam at a relatively high pressure, which is extracted from the turbine of a cogeneration plant. The rate of heat transfer to the boiler and the power output of the cogeneration plant are to be determined. Assumptions 1 Steady operating conditions exist. 2 Kinetic and potential energy changes are negligible. Analysis (a) From the steam tables (Tables A-4E, A-5E, and A-6E),
( )
Btu/lbm13.282
Btu/lbm29.9427.002.94Btu/lbm0.27
ftpsia 5.4039Btu1
psia2)80)(/lbmft 0.01623(86.01
/
/lbmft 01623.0Btu/lbm02.94
psia80@3
inpI,12
3
3
121inpI,
3psia2@1
psia2@1
==
=+=+==
⎟⎟⎠
⎞⎜⎜⎝
⎛
⋅×
−=
−=
==
==
f
p
f
f
hh
whh
PPw
hh
ηv
vv
Mixing chamber: & & &
& &
& & & & &
E E E
E E
m h m h m h m h m hi i e e
in out system (steady)
in out
− = =
=
= ⎯→⎯ = +∑ ∑
Δ 0
4 4 2 2 3 3
0
or, ( )( ) ( )( )
( )( )( ) ( )
Btu/lbm72.17229.343.169Btu/lbm3.29
86.0/ftpsia 5.4039
Btu1psia801000/lbmft 0.01664
/
/lbmft 01664.0
Btu/lbm43.1695
13.282229.943
in,45
33
454inpII,
3Btu/lbm43.169@4
4
33224
=+=+==
⎟⎟⎠
⎞⎜⎜⎝
⎛
⋅−=
−=
=≅
=+
=+
=
=
pII
p
hf
whh
PPw
mhmhm
h
f
ηv
vv
&
&&
RBtu/lbm6535.1Btu/lbm2.1506
F1000psia1000
6
6
6
6⋅=
=
⎭⎬⎫
°==
sh
TP
( )( ) Btu/lbm98.9597.10218475.002.94
8475.074444.1
17499.06535.1psia2
Btu/lbm0.1209psia80
88
88
68
8
767
7
=+=+=
=−
=−
=
⎭⎬⎫
==
=⎭⎬⎫
==
fgsfs
fg
fss
s
s
ss
s
hxhhs
ssx
ssP
hss
P
Then, ( ) ( )( ) Btu/s6667=−=−= Btu/lbm72.1722.1506lbm/s5565in hhmQ &&
(b) ( ) ( )[ ]( ) ( )( ) ( )( )[ ]
kW2026==−+−=
−+−==
Btu/s1921Btu/lbm959.981209.0lbm/s3Btu/lbm1209.01506.2lbm/s586.0
878766,outT, sssTsTT hhmhhmWW &&&& ηη
1
2
8
s
T
6
7s
1000psia
2 psia
80 psia 4
5
3
Qin ·
Qprocess ·
7
8s
6
8
1
5
Turbine Boiler
Condenser
Processheater
P I P II4 2
3
7
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74
10-77 A cogeneration plant has two modes of operation. In the first mode, all the steam leaving the turbine at a relatively high pressure is routed to the process heater. In the second mode, 60 percent of the steam is routed to the process heater and remaining is expanded to the condenser pressure. The power produced and the rate at which process heat is supplied in the first mode, and the power produced and the rate of process heat supplied in the second mode are to be determined. Assumptions 1 Steady operating conditions exist. 2 Kinetic and potential energy changes are negligible. Analysis (a) From the steam tables (Tables A-4, A-5, and A-6),
( )( )( )
( )( )( )
kJ/kg47.65038.1009.640kJ/kg10.38
mkPa1kJ1
kPa50010,000/kgm 0.001093
/kgm 001093.0kJ/kg 09.640
kJ/kg57.26115.1042.251kJ/kg10.15
mkPa1kJ1
kPa0210,000/kgm 0.001017
/kgm 001017.0kJ/kg42.251
inpII,34
33
343inpII,
3MPa5.0@3
MPa5.0@3
inpI,12
33
121inpI,
3kPa20@1
kPa20@1
=+=+==
⎟⎟⎠
⎞⎜⎜⎝
⎛
⋅−=
−=
==
==
=+=+==
⎟⎟⎠
⎞⎜⎜⎝
⎛
⋅−=
−=
==
==
whh
PPw
hh
whh
PPw
hh
f
f
f
f
v
vv
v
vv
Mixing chamber: & & & & &
& & & & &
E E E E E
m h m h m h m h m hi i e e
in out system (steady)
in out − = = → =
= ⎯→⎯ = +∑ ∑Δ 0
5 5 2 2 4 4
0
or, ( )( ) ( )( ) kJ/kg91.4945
47.650357.2612
5
44225 =
+=
+=
mhmhm
h&
&&
KkJ/kg4219.6kJ/kg4.3242
C450MPa10
6
6
6
6⋅=
=
⎭⎬⎫
°==
sh
TP
( )( )
( )( ) kJ/kg0.21145.23577901.042.251
7901.00752.7
8320.04219.6kPa20
kJ/kg6.25780.21089196.009.640
9196.09603.4
8604.14219.6MPa5.0
88
88
68
8
77
77
67
7
=+=+=
=−
=−
=
⎭⎬⎫
==
=+=+=
=−
=−
=
⎭⎬⎫
==
fgf
fg
f
fgf
fg
f
hxhhs
ssx
ssP
hxhhs
ssx
ssP
When the entire steam is routed to the process heater, ( ) ( )( )( ) ( )( ) kW 9693
kW 3319
=−=−=
=−=−=
kJ/kg09.6406.2578kg/s5
kJ/kg6.25784.3242kg/s5
377process
766outT,
hhmQ
hhmW
&&
&&
(b) When only 60% of the steam is routed to the process heater, ( ) ( )
( )( ) ( )( )( ) ( )( ) kW 5816
kW4248
=−=−=
=−+−=−+−=
kJ/kg09.6406.2578kg/s3
kJ/kg0.21146.2578kg/s2kJ/kg6.25784.3242kg/s5
377process
878766outT,
hhmQ
hhmhhmW
&&
&&&
1
2
8 s
T6
7
5
4 3
6
8
1
5
Turbine Boiler
Condens.
Process heater
PI PII
4
2
3
7
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75
10-78 A cogeneration plant modified with regeneration is to generate power and process heat. The mass flow rate of steam through the boiler for a net power output of 15 MW is to be determined. Assumptions 1 Steady operating conditions exist. 2 Kinetic and potential energy changes are negligible.
Analysis
From the steam tables (Tables A-4, A-5, and A-6),
( )( )( )
( )( )( )
kJ/kg73.61007.666.604kJ/kg07.6
mkPa1kJ1kPa4006000/kgm 0.001084
/kgm001084.0
kJ/kg66.604
kJ/kg20.19239.081.191kJ/kg0.39
mkPa1kJ1kPa10400/kgm 0.00101
/kgm 00101.0kJ/kg81.191
inpII,45
33
454inpII,
3MPa4.0@4
MPa4.0@943
inpI,12
33
121inpI,
3kPa10@1
kPa10@1
=+=+==
⎟⎟⎠
⎞⎜⎜⎝
⎛
⋅−=
−=
==
====
=+=+==
⎟⎟⎠
⎞⎜⎜⎝
⎛
⋅−=
−=
==
==
whh
PPw
hhhh
whh
PPw
hh
f
f
f
f
v
vv
v
vv
( )( )
( )( ) kJ/kg7.21281.23928097.081.191
8097.04996.7
6492.07219.6kPa10
kJ/kg7.26654.21339661.066.604
9661.01191.5
7765.17219.6MPa4.0
KkJ/kg7219.6kJ/kg9.3302
C450MPa6
88
88
68
8
77
77
67
7
6
6
6
6
=+=+=
=−
=−
=
⎭⎬⎫
==
=+=+=
=−
=−
=
⎭⎬⎫
==
⋅==
⎭⎬⎫
°==
fgf
fg
f
fgf
fg
f
hxhhs
ssx
ssP
hxhhs
ssx
ssP
sh
TP
Then, per kg of steam flowing through the boiler, we have
( ) ( )( ) ( )( )
kJ/kg852.0kJ/kg7.21287.26654.0kJ/kg7.26659.3302
4.0 8776outT,
=−+−=
−+−= hhhhw
( )( ) ( )
kJ/kg8.84523.60.852
kJ/kg6.23kJ/kg6.07kJ/kg0.394.0
4.0
inp,outT,net
inpII,inpI,inp,
=−=−=
=+=
+=
www
www
Thus,
kg/s17.73===kJ/kg845.8
kJ/s15,000
net
net
wWm&
&
1
2
8
s
T
6
7
6 MPa
10 kPa
0.4 MPa
5
3,4,9
6
8
1
5
Turbine Boiler
Condenser
Processheater
PI PII
9
2
3
7
4
fwh
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76
10-79 EES Problem 10-78 is reconsidered. The effect of the extraction pressure for removing steam from the turbine to be used for the process heater and open feedwater heater on the required mass flow rate is to be investigated.
Analysis The problem is solved using EES, and the solution is given below.
"Input Data" y = 0.6 "fraction of steam extracted from turbine for feedwater heater and process heater" P[6] = 6000 [kPa] T[6] = 450 [C] P_extract=400 [kPa] P[7] = P_extract P_cond=10 [kPa] P[8] = P_cond W_dot_net=15 [MW]*Convert(MW, kW) Eta_turb= 100/100 "Turbine isentropic efficiency" Eta_pump = 100/100 "Pump isentropic efficiency" P[1] = P[8] P[2]=P[7] P[3]=P[7] P[4] = P[7] P[5]=P[6] P[9] = P[7]
"Condenser exit pump or Pump 1 analysis" Fluid$='Steam_IAPWS'
h[1]=enthalpy(Fluid$,P=P[1],x=0) {Sat'd liquid} v1=volume(Fluid$,P=P[1],x=0) s[1]=entropy(Fluid$,P=P[1],x=0) T[1]=temperature(Fluid$,P=P[1],x=0) w_pump1_s=v1*(P[2]-P[1])"SSSF isentropic pump work assuming constant specific volume" w_pump1=w_pump1_s/Eta_pump "Definition of pump efficiency" h[1]+w_pump1= h[2] "Steady-flow conservation of energy" s[2]=entropy(Fluid$,P=P[2],h=h[2]) T[2]=temperature(Fluid$,P=P[2],h=h[2])
"Open Feedwater Heater analysis:" z*h[7] + (1- y)*h[2] = (1- y + z)*h[3] "Steady-flow conservation of energy" h[3]=enthalpy(Fluid$,P=P[3],x=0) T[3]=temperature(Fluid$,P=P[3],x=0) "Condensate leaves heater as sat. liquid at P[3]" s[3]=entropy(Fluid$,P=P[3],x=0)
"Process heater analysis:" (y - z)*h[7] = q_process + (y - z)*h[9] "Steady-flow conservation of energy" Q_dot_process = m_dot*(y - z)*q_process"[kW]" h[9]=enthalpy(Fluid$,P=P[9],x=0) T[9]=temperature(Fluid$,P=P[9],x=0) "Condensate leaves heater as sat. liquid at P[3]" s[9]=entropy(Fluid$,P=P[9],x=0)
"Mixing chamber at 3, 4, and 9:" (y-z)*h[9] + (1-y+z)*h[3] = 1*h[4] "Steady-flow conservation of energy" T[4]=temperature(Fluid$,P=P[4],h=h[4]) "Condensate leaves heater as sat. liquid at P[3]" s[4]=entropy(Fluid$,P=P[4],h=h[4])
"Boiler condensate pump or Pump 2 analysis"
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77
v4=volume(Fluid$,P=P[4],x=0) w_pump2_s=v4*(P[5]-P[4])"SSSF isentropic pump work assuming constant specific volume" w_pump2=w_pump2_s/Eta_pump "Definition of pump efficiency" h[4]+w_pump2= h[5] "Steady-flow conservation of energy" s[5]=entropy(Fluid$,P=P[5],h=h[5]) T[5]=temperature(Fluid$,P=P[5],h=h[5])
"Boiler analysis" q_in + h[5]=h[6]"SSSF conservation of energy for the Boiler" h[6]=enthalpy(Fluid$, T=T[6], P=P[6]) s[6]=entropy(Fluid$, T=T[6], P=P[6])
"Turbine analysis" ss[7]=s[6] hs[7]=enthalpy(Fluid$,s=ss[7],P=P[7]) Ts[7]=temperature(Fluid$,s=ss[7],P=P[7]) h[7]=h[6]-Eta_turb*(h[6]-hs[7])"Definition of turbine efficiency for high pressure stages" T[7]=temperature(Fluid$,P=P[7],h=h[7]) s[7]=entropy(Fluid$,P=P[7],h=h[7]) ss[8]=s[7] hs[8]=enthalpy(Fluid$,s=ss[8],P=P[8]) Ts[8]=temperature(Fluid$,s=ss[8],P=P[8]) h[8]=h[7]-Eta_turb*(h[7]-hs[8])"Definition of turbine efficiency for low pressure stages" T[8]=temperature(Fluid$,P=P[8],h=h[8]) s[8]=entropy(Fluid$,P=P[8],h=h[8]) h[6] =y*h[7] + (1- y)*h[8] + w_turb "SSSF conservation of energy for turbine"
"Condenser analysis" (1- y)*h[8]=q_out+(1- y)*h[1]"SSSF First Law for the Condenser"
"Cycle Statistics" w_net=w_turb - ((1- y)*w_pump1+ w_pump2) Eta_th=w_net/q_in W_dot_net = m_dot * w_net
Pextract [kPa]
ηth m [kg/s]
Qprocess [kW]
100 0.3413 15.26 9508 200 0.3284 16.36 9696 300 0.3203 17.12 9806 400 0.3142 17.74 9882 500 0.3092 18.26 9939 600 0.305 18.72 9984
0 2 4 6 8 10 120
100
200
300
400
500
600
700
s [kJ/kg-K]
T [°
C]
6000 kPa
400 kPa
10 kPa
Steam
1
2 3,4,9
5
6
7
8
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78
100 200 300 400 500 60015
15.5
16
16.5
17
17.5
18
18.5
19
Pextract [kPa]
m [
kg/s
]
100 200 300 400 500 6009500
9600
9700
9800
9900
10000
Pextract [kPa]
Qpr
oces
s[k
W]
100 200 300 400 500 6000.305
0.31
0.315
0.32
0.325
0.33
0.335
0.34
0.345
Pextract [kPa]
ηth
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79
10-80E A cogeneration plant is to generate power while meeting the process steam requirements for a certain industrial application. The net power produced, the rate of process heat supply, and the utilization factor of this plant are to be determined.
Assumptions 1 Steady operating conditions exist. 2 Kinetic and potential energy changes are negligible.
Analysis
(a) From the steam tables (Tables A-4E, A-5E, and A-6E),
Btu/lbm5.1229psia120
RBtu/lbm6348.1Btu/lbm0.1408
F800psia600
Btu/lbm49.208
737
7
6543
753
3
3
3
12
F240@1
=⎭⎬⎫
==
===
⋅====
⎭⎬⎫
°==
≅=≅ °
hss
P
hhhh
sssh
TP
hhhh f
( )( )( )
kW2260==−=
−=
Btu/s2142Btu/lbm5.12290.1408lbm/s12
755net hhmW &&
(b)
( )( ) ( )( ) ( )( )Btu/s19,450=
−+=−−+=
−= ∑∑
49.208185.1229120.14086117766
process
hmhmhm
hmhmQ eeii
&&&
&&&
( )( ) ( )( ) ( )( )Btu/s19,450−=
−−=
−−=−= ∑∑5.1229120.1408649.20818
776611process hmhmhmhmhmQ iiee &&&&&&
(c) εu = 1 since all the energy is utilized.
1
s
T
6
7
600 psia
120 psia
2
3,4,
6
1
2
TurbineBoiler
Process heater
P
4 5 3
7
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80
10-81 A cogeneration plant is to generate power and process heat. Part of the steam extracted from the turbine at a relatively high pressure is used for process heating. The mass flow rate of steam that must be supplied by the boiler, the net power produced, and the utilization factor of the plant are to be determined.
Assumptions 1 Steady operating conditions exist. 2 Kinetic and potential energy changes are negligible.
Analysis From the steam tables (Tables A-4, A-5, and A-6),
( )( )( )
kJ/kg38.670
kJ/kg40.192596.081.191kJ/kg596.0
mkPa1kJ1
kPa01600/kgm 0.00101
/kgm 00101.0kJ/kg81.191
MPa6.0@3
inpI,12
33
121inpI,
3kPa01 @1
kPa10@1
==
=+=+==
⎟⎟⎠
⎞⎜⎜⎝
⎛
⋅−=
−=
==
==
f
f
f
hh
whh
PPw
hh
v
vv
Mixing chamber:
kJ/kg90.311)1()kJ/kg))(192.4075.0(kJ/kg))(670.3825.0( 44
442233
=⎯→⎯=+
=+
hh
hmhmhm &&&
( )( )( )
kJ/kg47.318563.690.311kJ/kg.5636
mkPa1kJ1kPa0067000/kgm 0.001026
/kgm 001026.0
inII,45
33
454inII,
3kJ/kg90.311@4
=+=+==
⎟⎟⎠
⎞⎜⎜⎝
⎛
⋅−=
−=
=≅ =
p
p
hf
whh
PPw
f
v
vv
KkJ/kg8000.6kJ/kg4.3411
C500MPa7
6
6
6
6⋅=
=
⎭⎬⎫
°==
sh
TP
kJ/kg9.2773MPa6.0
767
7 =⎭⎬⎫
==
hss
P
6
8
1
5
Turbine Boiler
Condenser
Process heater
P I P II 4 2
3
7
1
2
8
s
T
6
7
7 MPa
10 kPa
0.6 MPa 4
5
3
Qout·
Qin ·
Qprocess ·
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81
kJ/kg6.2153kPa10
868
8 =⎭⎬⎫
==
hss
P
( )( )
kg/s4.088kJ/kg38.6709.2773kJ/s8600
7
7
377process
=−=
−=
mm
hhmQ
&
&
&&
This is one-fourth of the mass flowing through the boiler. Thus, the mass flow rate of steam that must be supplied by the boiler becomes
kg/s16.35=== kg/s)4.088(44 76 mm &&
(b) Cycle analysis:
( ) ( )( )( ) ( )( )
( )( ) ( )( )kW17,919=−=−=
=+=+=
=−+−=
−+−=
115033,18
kW6.114kJ/kg6.563kg/s16.35kJ/kg0.596kg/s4.088-16.35
kW033,18kJ/kg6.21534.3411kg/s4.088-16.35kJ/kg9.27734.3411kg/s088.4
inp,outT,net
inpII,4inpI,1inp,
868767outT,
WWW
wmwmW
hhmhhmW
&&&
&&&
&&&
(c) Then,
and
( ) ( )( )
52.4%==+
=+
=
=−=−=
524.0581,50
8600919,17
kW581,5046.3184.3411kg/s16.35
in
processnet
565in
QQW
hhmQ
u &
&&
&&
ε
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82
Combined Gas-Vapor Power Cycles
10-82C The energy source of the steam is the waste energy of the exhausted combustion gases.
10-83C Because the combined gas-steam cycle takes advantage of the desirable characteristics of the gas cycle at high temperature, and those of steam cycle at low temperature, and combines them. The result is a cycle that is more efficient than either cycle executed operated alone.
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83
10-84 [Also solved by EES on enclosed CD] A 450-MW combined gas-steam power plant is considered. The topping cycle is a gas-turbine cycle and the bottoming cycle is an ideal Rankine cycle with an open feedwater heater. The mass flow rate of air to steam, the required rate of heat input in the combustion chamber, and the thermal efficiency of the combined cycle are to be determined.
Assumptions 1 Steady operating conditions exist. 2 Kinetic and potential energy changes are negligible. 3Air is an ideal gas with variable specific heats.
Analysis (a) The analysis of gas cycle yields (Table A-17)
( )( )
( )
kJ/kg02462 K460
kJ/kg873518325450141
5450kJ/kg421515K1400
kJ/kg56354019386114
3861kJ/kg19300K300
1212
1110
11
1010
98
9
88
1011
10
89
8
.hT
.h..PPP
P
.P.hT
.h..PPP
P
.P.hT
rr
r
rr
r
=⎯→⎯=
=⎯→⎯=⎟⎠⎞
⎜⎝⎛==
==⎯→⎯=
=⎯→⎯===
==⎯→⎯=
From the steam tables (Tables A-4, A-5, A-6),
( )( )( )
kJ/kg01.25259.042.251kJ/kg0.59
mkPa1kJ1
kPa20600/kgm 0.001017
/kgm 001017.0kJ/kg42.251
inpI,12
33
121inpI,
3kPa20@1
kPa20@1
=+=+==
⎟⎟⎠
⎞⎜⎜⎝
⎛
⋅−=
−=
====
whh
PPw
hh
f
f
v
vv
( )( )( )
kJ/kg53.67815.838.670kJ/kg8.15
mkPa1kJ1kPa6008,000/kgm0.001101
/kgm 001101.0kJ/kg38.670
inpI,34
33
343inpII,
3MPa6.0@3
MPa6.0 @3
=+=+==
⎟⎟⎠
⎞⎜⎜⎝
⎛
⋅−=
−=
====
whh
PPw
hh
f
f
v
vv
( )( )
( )( ) kJ/kg2.20955.23577821.042.251
7821.00752.7
8320.03658.6kPa20
kJ/kg1.25868.20859185.038.670
9185.08285.4
9308.13658.6MPa6.0
KkJ/kg3658.6kJ/kg4.3139
C400MPa8
77
77
57
7
66
66
56
6
5
5
5
5
=+=+=
=−
=−
=
⎭⎬⎫
==
=+=+=
=−
=−
=
⎭⎬⎫
==
⋅==
⎭⎬⎫
°==
fgf
fg
f
fgf
fg
f
hxhhs
ssx
ssP
hxhhs
ssx
ssP
sh
TP
Noting that 0ΔpeΔke ≅≅≅≅WQ && for the heat exchanger, the steady-flow energy balance equation yields
1
2
7 s
T
5
6
8 MPa
20 kPa 0.6 MPa
4
9
3
Qout·
Qin ·
8
12
11
10
GAS CYCLE
STEAM CYCLE
1400 K
400°C
460 K
300 K
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84
( ) ( )
steamkgair /kg8.9902.46280.73553.6784.3139
0
1211
45air
1211air45
outin
(steady)0systemoutin
=−−
=−−
=
−=−⎯→⎯=
=
=Δ=−
∑∑
hhhh
mm
hhmhhmhmhm
EE
EEE
s
seeii
&
&
&&&&
&&
&&&
(b) Noting that & &Q W ke pe≅ ≅ ≅ ≅Δ Δ 0 for the open FWH, the steady-flow energy balance equation yields
( ) ( ) 326336622
outin
(steady)0systemoutin
11
0
hhyyhhmhmhmhmhm
EE
EEE
eeii =−+⎯→⎯=+⎯→⎯=
=
=Δ=−
∑∑ &&&&&
&&
&&&
Thus,
( )
( )( )( )( ) kJ/kg23.9562.20951.25861792.011.25864.3139
1
extracted steam offraction the 1792.001.2521.258601.25238.670
7665
26
23
=−−+−=−−+−=
=−−
=−−
=
hhyhhw
hhhhy
T
( )( )( )
( ) ( )( ) kJ/kg3.44419.3005.6358.73542.1515
kJ/kg56.94815.859.01792.0123.9561
891110in,gasnet,
,,in,steamnet,
=−−−=−−−=−=
=−−−=−−−=−=
hhhhwww
wwywwww
CT
IIpIpTpT
The net work output per unit mass of gas is
( ) kJ/kg8.54956.9483.444 99.81
steamnet,99.81
gasnet,net =+=+= www
and
( ) ( )( ) kW 720,215=−=−=
===
kJ/kg5.63542.1515kg/s818.5
kg/s7.188kJ/kg 549.7
kJ/s 450,000
910airin
net
netair
hhmQ
wWm
&&
&&
(c) 62.5%===ηkW720,215kW450,000
in
net
QW
th &
&
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85
10-85 EES Problem 10-84 is reconsidered. The effect of the gas cycle pressure ratio on the ratio of gas flow rate to steam flow rate and cycle thermal efficiency is to be investigated.
Analysis The problem is solved using EES, and the solution is given below.
"Input data" T[8] = 300 [K] "Gas compressor inlet" P[8] = 14.7 [kPa] "Assumed air inlet pressure" "Pratio = 14" "Pressure ratio for gas compressor" T[10] = 1400 [K] "Gas turbine inlet" T[12] = 460 [K] "Gas exit temperature from Gas-to-steam heat exchanger " P[12] = P[8] "Assumed air exit pressure" W_dot_net=450 [MW] Eta_comp = 1.0 Eta_gas_turb = 1.0 Eta_pump = 1.0 Eta_steam_turb = 1.0 P[5] = 8000 [kPa] "Steam turbine inlet" T[5] =(400+273) "[K]" "Steam turbine inlet" P[6] = 600 [kPa] "Extraction pressure for steam open feedwater heater" P[7] = 20 [kPa] "Steam condenser pressure"
"GAS POWER CYCLE ANALYSIS"
"Gas Compressor anaysis" s[8]=ENTROPY(Air,T=T[8],P=P[8]) ss9=s[8] "For the ideal case the entropies are constant across the compressor" P[9] = Pratio*P[8] Ts9=temperature(Air,s=ss9,P=P[9])"Ts9 is the isentropic value of T[9] at compressor exit" Eta_comp = w_gas_comp_isen/w_gas_comp "compressor adiabatic efficiency, w_comp > w_comp_isen" h[8] + w_gas_comp_isen =hs9"SSSF conservation of energy for the isentropic compressor, assuming: adiabatic, ke=pe=0 per unit gas mass flow rate in kg/s" h[8]=ENTHALPY(Air,T=T[8]) hs9=ENTHALPY(Air,T=Ts9) h[8] + w_gas_comp = h[9]"SSSF conservation of energy for the actual compressor, assuming: adiabatic, ke=pe=0" T[9]=temperature(Air,h=h[9]) s[9]=ENTROPY(Air,T=T[9],P=P[9])
"Gas Cycle External heat exchanger analysis" h[9] + q_in = h[10]"SSSF conservation of energy for the external heat exchanger, assuming W=0, ke=pe=0" h[10]=ENTHALPY(Air,T=T[10]) P[10]=P[9] "Assume process 9-10 is SSSF constant pressure" Q_dot_in"MW"*1000"kW/MW"=m_dot_gas*q_in
"Gas Turbine analysis" s[10]=ENTROPY(Air,T=T[10],P=P[10]) ss11=s[10] "For the ideal case the entropies are constant across the turbine" P[11] = P[10] /Pratio Ts11=temperature(Air,s=ss11,P=P[11])"Ts11 is the isentropic value of T[11] at gas turbine exit" Eta_gas_turb = w_gas_turb /w_gas_turb_isen "gas turbine adiabatic efficiency, w_gas_turb_isen > w_gas_turb"
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86
h[10] = w_gas_turb_isen + hs11"SSSF conservation of energy for the isentropic gas turbine, assuming: adiabatic, ke=pe=0" hs11=ENTHALPY(Air,T=Ts11) h[10] = w_gas_turb + h[11]"SSSF conservation of energy for the actual gas turbine, assuming: adiabatic, ke=pe=0" T[11]=temperature(Air,h=h[11]) s[11]=ENTROPY(Air,T=T[11],P=P[11])
"Gas-to-Steam Heat Exchanger" "SSSF conservation of energy for the gas-to-steam heat exchanger, assuming: adiabatic, W=0, ke=pe=0" m_dot_gas*h[11] + m_dot_steam*h[4] = m_dot_gas*h[12] + m_dot_steam*h[5] h[12]=ENTHALPY(Air, T=T[12]) s[12]=ENTROPY(Air,T=T[12],P=P[12])
"STEAM CYCLE ANALYSIS" "Steam Condenser exit pump or Pump 1 analysis" Fluid$='Steam_IAPWS' P[1] = P[7] P[2]=P[6] h[1]=enthalpy(Fluid$,P=P[1],x=0) {Saturated liquid} v1=volume(Fluid$,P=P[1],x=0) s[1]=entropy(Fluid$,P=P[1],x=0) T[1]=temperature(Fluid$,P=P[1],x=0) w_pump1_s=v1*(P[2]-P[1])"SSSF isentropic pump work assuming constant specific volume" w_pump1=w_pump1_s/Eta_pump "Definition of pump efficiency" h[1]+w_pump1= h[2] "Steady-flow conservation of energy" s[2]=entropy(Fluid$,P=P[2],h=h[2]) T[2]=temperature(Fluid$,P=P[2],h=h[2]) "Open Feedwater Heater analysis" y*h[6] + (1-y)*h[2] = 1*h[3] "Steady-flow conservation of energy" P[3]=P[6] h[3]=enthalpy(Fluid$,P=P[3],x=0) "Condensate leaves heater as sat. liquid at P[3]" T[3]=temperature(Fluid$,P=P[3],x=0) s[3]=entropy(Fluid$,P=P[3],x=0) "Boiler condensate pump or Pump 2 analysis" P[4] = P[5] v3=volume(Fluid$,P=P[3],x=0) w_pump2_s=v3*(P[4]-P[3])"SSSF isentropic pump work assuming constant specific volume" w_pump2=w_pump2_s/Eta_pump "Definition of pump efficiency" h[3]+w_pump2= h[4] "Steady-flow conservation of energy" s[4]=entropy(Fluid$,P=P[4],h=h[4]) T[4]=temperature(Fluid$,P=P[4],h=h[4]) w_steam_pumps = (1-y)*w_pump1+ w_pump2 "Total steam pump work input/ mass steam" "Steam Turbine analysis" h[5]=enthalpy(Fluid$,T=T[5],P=P[5]) s[5]=entropy(Fluid$,P=P[5],T=T[5]) ss6=s[5] hs6=enthalpy(Fluid$,s=ss6,P=P[6]) Ts6=temperature(Fluid$,s=ss6,P=P[6]) h[6]=h[5]-Eta_steam_turb*(h[5]-hs6)"Definition of steam turbine efficiency" T[6]=temperature(Fluid$,P=P[6],h=h[6]) s[6]=entropy(Fluid$,P=P[6],h=h[6]) ss7=s[5] hs7=enthalpy(Fluid$,s=ss7,P=P[7]) Ts7=temperature(Fluid$,s=ss7,P=P[7])
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87
h[7]=h[5]-Eta_steam_turb*(h[5]-hs7)"Definition of steam turbine efficiency" T[7]=temperature(Fluid$,P=P[7],h=h[7]) s[7]=entropy(Fluid$,P=P[7],h=h[7]) "SSSF conservation of energy for the steam turbine: adiabatic, neglect ke and pe" h[5] = w_steam_turb + y*h[6] +(1-y)*h[7] "Steam Condenser analysis" (1-y)*h[7]=q_out+(1-y)*h[1]"SSSF conservation of energy for the Condenser per unit mass" Q_dot_out*Convert(MW, kW)=m_dot_steam*q_out "Cycle Statistics" MassRatio_gastosteam =m_dot_gas/m_dot_steam W_dot_net*Convert(MW, kW)=m_dot_gas*(w_gas_turb-w_gas_comp)+ m_dot_steam*(w_steam_turb - w_steam_pumps)"definition of the net cycle work" Eta_th=W_dot_net/Q_dot_in*Convert(, %) "Cycle thermal efficiency, in percent" Bwr=(m_dot_gas*w_gas_comp + m_dot_steam*w_steam_pumps)/(m_dot_gas*w_gas_turb + m_dot_steam*w_steam_turb) "Back work ratio" W_dot_net_steam = m_dot_steam*(w_steam_turb - w_steam_pumps) W_dot_net_gas = m_dot_gas*(w_gas_turb - w_gas_comp) NetWorkRatio_gastosteam = W_dot_net_gas/W_dot_net_steam
Pratio MassRatio gastosteam
Wnetgas [kW]
Wnetsteam [kW]
ηth [%]
NetWorkRatio gastosteam
10 7.108 342944 107056 59.92 3.203 11 7.574 349014 100986 60.65 3.456 12 8.043 354353 95647 61.29 3.705 13 8.519 359110 90890 61.86 3.951 14 9.001 363394 86606 62.37 4.196 15 9.492 367285 82715 62.83 4.44 16 9.993 370849 79151 63.24 4.685 17 10.51 374135 75865 63.62 4.932 18 11.03 377182 72818 63.97 5.18 19 11.57 380024 69976 64.28 5.431 20 12.12 382687 67313 64.57 5.685
0.0 1.1 2.2 3.3 4.4 5.5 6.6 7.7 8.8 9.9 11.0200
300
400
500
600
700
800
900
1000
1100
1200
1300
1400
1500
1600
s [kJ/kg-K]
T [K
]
8000 kPa
600 kPa
20 kPa
Com bined Gas and Steam Pow er Cycle
8
9
10
11
12
1,23,4
5
6
7
Steam CycleSteam Cycle
Gas CycleGas Cycle
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88
5 9 13 17 21 2555
57.2
59.4
61.6
63.8
66
P ratio
ηth
[%]
5 9 14 18 232.0
2.5
3.0
3.5
4.0
4.5
5.0
5.5
6.0
6.5
Pratio
Net
Wor
kRat
ioga
stos
team
W dot,gas / Wdot,steam vs Gas Pressure Ratio
5 9 14 18 235.0
6.0
7.0
8.0
9.0
10.0
11.0
12.0
13.0
14.0
Pratio
Mas
sRat
ioga
stos
team
Ratio of Gas Flow Rate to Steam Flow Rate vs Gas Pressure Ratio
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89
10-86 A combined gas-steam power cycle uses a simple gas turbine for the topping cycle and simple Rankine cycle for the bottoming cycle. The mass flow rate of air for a specified power output is to be determined.
Assumptions 1 Steady operating conditions exist. 2 The air-standard assumptions are applicable fo Brayton cycle. 3 Kinetic and potential energy changes are negligible. 4 Air is an ideal gas with constant specific heats.
Properties The properties of air at room temperature are cp = 1.005 kJ/kg·K and k = 1.4 (Table A-2a).
Analysis Working around the topping cycle gives the following results:
K8.530K)(8)293( 0.4/1.4/)1(
5
656 ==⎟⎟
⎠⎜
⎞⎜⎝
⎛=
− kk
s PP
TT
K8.57285.0
2938.530293
)()(
5656
56
56
56
56
=−
+=
−+=⎯→⎯
−
−=
−−
=
C
s
p
spsC
TTTT
TTcTTc
hhhh
η
η
K0.75881K)1373(
0.4/1.4/)1(
7
878 =⎟
⎠⎞
⎜⎝⎛=⎟⎟
⎠⎜
⎞⎜⎝
⎛=
− kk
s PP
TT
K5.819)0.7581373)(90.0(1373
)()()(
877887
87
87
87
=−−=
−−=⎯→⎯−
−=
−−
= sTsp
p
sT TTTT
TTcTTc
hhhh
ηη
K548.6C6.275kPa6000@sat 9 =°== TT
Fixing the states around the bottom steam cycle yields (Tables A-4, A-5, A-6):
kJ/kg5.25708.642.251kJ/kg08.6 mkPa1
kJ1kPa)206000)(/kgm001017.0()(
/kgm001017.0
kJ/kg42.251
inp,12
33
121inp,
3kPa20@1
kPa20@1
=+=+==
⎟⎠
⎞⎜⎝
⎛
⋅−=
−=
==
==
whh
PPw
hh
f
f
v
vv
kJ/kg8.2035 kPa20
KkJ/kg1871.6kJ/kg6.2953
C320kPa6000
434
4
3
3
3
3
=⎭⎬⎫
==
⋅==
⎭⎬⎫
°==
shss
P
sh
TP
kJ/kg6.2127)8.20356.2953)(90.0(6.2953
)( 433443
43
=−−=
−−=⎯→⎯−−
= sTs
T hhhhhhhh
ηη
1
2
4ss
T
3
6 MPa
20 kPa
6s
Qout·
Qin ·
5
9
8s
7
GAS CYCLE
STEAMCYCLE
1373 K
293 K
320°C
8 6
4
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90
The net work outputs from each cycle are
kJ/kg2.275K)2937.5725.8191373)(KkJ/kg1.005(
)()( 5687
inC,outT,cyclegasnet,
=+−−⋅=
−−−=
−=
TTcTTc
www
pp
kJ/kg9.81908.6)6.21276.2953(
)( inP,43
inP,outT,cyclesteamnet,
=−−=
−−=
−=
whh
www
An energy balance on the heat exchanger gives
aap
wwpa mm-hh
TTcm-hhmTTcm &&&&& 1010.0
5.2576.2953)6.5485.819)(005.1()(
)()(23
982398 =
−−
=−
=⎯→⎯=−
That is, 1 kg of exhaust gases can heat only 0.1010 kg of water. Then, the mass flow rate of air is
kg/s279.3=×+×
==airkJ/kg)9.8191010.02.2751(
kJ/s000,100
net
net
wW
ma
&&
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91
10-87 A combined gas-steam power cycle uses a simple gas turbine for the topping cycle and simple Rankine cycle for the bottoming cycle. The mass flow rate of air for a specified power output is to be determined.
Assumptions 1 Steady operating conditions exist. 2 The air-standard assumptions are applicable fo Brayton cycle. 3 Kinetic and potential energy changes are negligible. 4 Air is an ideal gas with constant specific heats.
Properties The properties of air at room temperature are cp = 1.005 kJ/kg·K and k = 1.4 (Table A-2a).
Analysis With an ideal regenerator, the temperature of the air at the compressor exit will be heated to the to the temperature at the turbine exit. Representing this state by “6a”
K5.81986 == TT a
The rate of heat addition in the cycle is
kW370,155K)5.8191373(C)kJ/kg005.1(kg/s)3.279(
)( 67in
=−°⋅=
−= apa TTcmQ &&
The thermal efficiency of the cycle is then
0.6436===kW370,155kW000,100
in
netth Q
W&
&η
Without the regenerator, the rate of heat addition and the thermal efficiency are
kW640,224K)7.5721373(C)kJ/kg005.1(kg/s)3.279()( 67in =−°⋅=−= TTcmQ pa&&
0.4452===kW640,224kW000,100
in
netth Q
W&
&η
The change in the thermal efficiency due to using the ideal regenerator is
0.1984=−=Δ 4452.06436.0thη
1
2
4ss
T
3
6 MPa
20 kPa
6s
Qout·
Qin·
5
9
8s
7
GAS CYCLE
STEAM CYCLE
1373 K
293 K
320°C
8 6
4
6a
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92
10-88 The component of the combined cycle with the largest exergy destruction of the component of the combined cycle in Prob. 10-86 is to be determined.
Assumptions 1 Steady operating conditions exist. 2 Kinetic and potential energy changes are negligible.
Analysis From Problem 10-86,
kg/s21.28)3.279(1010.01010.0kJ/kg2.1876kJ/kg1.2696
kJ/kg3.804)(KkJ/kg4627.6KkJ/kg1871.6
KkJ/kg8320.0K293
K5.819
K1373
14out
23in,23
67in,67
4
3
kPa20@21
sink
8cyclesteamsource,
cyclegassource,
====−=
=−=
=−=⋅=⋅=
⋅====
==
=
aw
p
f
mmhhqhhq
TTcqss
sssT
TT
T
&&
( ) kW2278)1871.64627.6)(K293(kg/s)21.28(
process)c(isentropi 0
34034destroyed,
12destroyed,
=−=−=
=
ssTmX
X
w&&
&
kW8665K293kJ/kg2.18761871.68320.0)K293(kg/s)21.28(
sink
out41041destroyed,
=⎟⎠
⎞⎜⎝
⎛ +−=
⎟⎟⎠
⎞⎜⎜⎝
⎛+−=
Tq
ssTmX w&&
kW11260
)8320.01871.6)(293)(21.28(5.8196.548ln)005.1()293)(3.279(
)(ln 2308
90230890exchangerheat destroyed,
=
−+⎥⎦⎤
⎢⎣⎡=
−⎟ +⎟⎠
⎞⎜⎜⎝
⎛=Δ+Δ= ssTm
TT
cTmsTmsTmX apawa &&&&&
kW6280)8ln()287.0(293
572.7(1.005)ln)293)(3.279(lnln5
6
5
6056destroyed, =⎥⎦
⎤⎢⎣⎡ −⎟ =⎟
⎠
⎞⎜⎜⎝
⎛−=
PP
RTT
cTmX pa&&
kW23,970=⎥⎦⎤
⎢⎣⎡ −⎟ =⎟
⎠
⎞⎜⎜⎝
⎛−=
13733.804
572.71373(1.005)ln)293)(3.279(ln
source
in
6
7067destroyed, T
qTT
cTmX pa&&
kW639681ln)287.0(
1373819.5(1.005)ln)293)(3.279(lnln
7
8
7
8078destroyed, =⎥
⎦
⎤⎢⎣
⎡⎟⎠⎞
⎜⎝⎛−⎟ =⎟
⎠
⎞⎜⎜⎝
⎛−=
PP
RTT
cTmX pa&&
The largest exergy destruction occurs during the heat addition process in the combustor of the gas cycle.
1
2
4s s
T
3
6 MPa
20 kPa
6s
Qout·
Qin·
5
9
8s
7
GAS CYCLE
STEAM CYCLE
1373 K
293 K
320°C
8 6
4
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93
10-89 A 450-MW combined gas-steam power plant is considered. The topping cycle is a gas-turbine cycle and the bottoming cycle is a nonideal Rankine cycle with an open feedwater heater. The mass flow rate of air to steam, the required rate of heat input in the combustion chamber, and the thermal efficiency of the combined cycle are to be determined.
Assumptions 1 Steady operating conditions exist. 2 Kinetic and potential energy changes are negligible. 3Air is an ideal gas with variable specific heats.
Analysis (a) Using the properties of air from Table A-17, the analysis of gas cycle yields
( )( )
( )( ) ( )
( )
( )( )( )
kJ/kg02462 K460
kJ/kg844.958735421515860421515
kJ/kg873518325450141
5450kJ/kg421515 K1400
kJ/kg709.182019300563519300
kJ/kg56354019386114
3861kJ/kg19300 K300
1212
111010111110
1110
1110
11
1010
898989
89
98
9
88
1011
10
89
8
.hT
....hhhh
hhhh
.h..PPPP
.P.hT
./.../hhhh
hhhh
.h..PPP
P
.P.hT
sTs
T
srr
r
Css
C
srr
r
=⎯→⎯=
=−−=
−η−=⎯→⎯−−
=η
=⎯→⎯=⎟⎠⎞
⎜⎝⎛==
==⎯→⎯=
=−+=η−+=⎯→⎯
−−
=η
=⎯→⎯===
==⎯→⎯=
From the steam tables (Tables A-4, A-5, and A-6),
( )( )( )
kJ/kg01.25259.042.251kJ/kg0.59
mkPa1kJ1
kPa20600/kgm 0.001017
/kgm 001017.0kJ/kg42.251
inpI,12
33
121inpI,
3kPa20@1
kPa20@1
=+=+==
⎟⎟⎠
⎞⎜⎜⎝
⎛
⋅−=
−=
====
whh
PPw
hh
f
f
v
vv
( )( )( )
kJ/kg52.67815.838.670kJ/kg8.15
mkPa1kJ1
kPa6008,000/kgm0.001101
/kgm 001101.0kJ/kg38.670
inpI,34
33
343inpII,
3MPa6.0@3
MPa6.0 @3
=+=+==
⎟⎟⎠
⎞⎜⎜⎝
⎛
⋅−=
−=
====
whh
PPw
hh
f
f
v
vv
KkJ/kg3658.6kJ/kg4.3139
C400MPa8
5
5
5
5⋅=
=
⎭⎬⎫
°==
sh
TP
1
2
7s s
T
5
6s
4
9s
3
Qout·
Qin·
8
12
11s
10
GAS CYCLE
STEAM CYCLE
9
11
7
6
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94
( )( )
( ) ( )( )
( )( )
( ) ( )( ) kJ/kg3.22411.20954.313986.04.3139
kJ/kg1.20955.23577820.042.251
7820.00752.7
8320.03658.6kPa20
kJ/kg3.26639.25854.313986.04.3139
kJ/kg9.25858.20859184.038.670
9184.08285.4
9308.13658.6MPa6.0
755775
75
77
77
57
7
655665
65
66
66
56
6
=−−=−−=⎯→⎯−−
=
=+=+=
=−
=−
=
⎭⎬⎫
==
=−−=−−=⎯→⎯−−
=
=+=+=
=−
=−
=
⎭⎬⎫
==
sTs
T
fgfs
fg
fs
sTs
T
fgsfs
fg
fss
s
hhhhhhhh
hxhhs
ssx
ssP
hhhhhhhh
hxhhs
ssx
ssP
ηη
ηη
Noting that 0ΔpeΔke ≅≅≅≅WQ && for the heat exchanger, the steady-flow energy balance equation yields
( ) ( )
steamkgair /kg 6.425=−−
=−−
=
−=−⎯→⎯=
=
=Δ=−
∑∑
02.46295.84452.6784.3139
0
1211
45air
1211air45
outin
(steady)0systemoutin
hhhh
mm
hhmhhmhmhm
EE
EEE
s
seeii
&
&
&&&&
&&
&&&
(b) Noting that 0ΔpeΔke ≅≅≅≅WQ && for the open FWH, the steady-flow energy balance equation yields
( ) ( ) 326336622
outin(steady)0
systemoutin
11
0
hhyyhhmhmhmhmhm
EEEEE
eeii =−+⎯→⎯=+⎯→⎯=
=→=Δ=−
∑∑ &&&&&
&&&&&
Thus,
( )
( )( )[ ]( ) ( )( )[ ] kJ/kg5.8243.22413.26631735.013.26634.313986.0
1
extracted steam offraction the 1735.001.2523.266301.25238.670
7665
26
23
=−−+−=−−+−=
=−−
=−−
=
hhyhhw
hhhh
y
TT η
( )( )( )
( ) ( )( ) kJ/kg56.26119.3001.70995.84442.1515
kJ/kg9.81515.859.01735.015.8241
891110in,gasnet,
IIp,Ip,inp,steamnet,
=−−−=−−−=−=
=−−−=−−−=−=
hhhhwww
wwywwww
CT
TT
The net work output per unit mass of gas is
( ) kJ/kg55.3889.815425.6156.261
425.61
steamnet,gasnet,net =+=+= www
kg/s2.1158kJ/kg388.55
kJ/s450,000
net
netair ===
wW
m&
&
and ( ) ( )( ) kW933,850=−=−= kJ/kg1.70942.1515kg/s1158.2910airin hhmQ &&
(c) 48.2%===kW933,850kW450,000
in
net
QW
th &
&η
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95
10-90 EES Problem 10-89 is reconsidered. The effect of the gas cycle pressure ratio on the ratio of gas flow rate to steam flow rate and cycle thermal efficiency is to be investigated.
Analysis The problem is solved using EES, and the solution is given below.
"Input data" T[8] = 300 [K] "Gas compressor inlet" P[8] = 14.7 [kPa] "Assumed air inlet pressure" "Pratio = 14" "Pressure ratio for gas compressor" T[10] = 1400 [K] "Gas turbine inlet" T[12] = 460 [K] "Gas exit temperature from Gas-to-steam heat exchanger " P[12] = P[8] "Assumed air exit pressure" W_dot_net=450 [MW] Eta_comp = 0.82 Eta_gas_turb = 0.86 Eta_pump = 1.0 Eta_steam_turb = 0.86 P[5] = 8000 [kPa] "Steam turbine inlet" T[5] =(400+273) "K" "Steam turbine inlet" P[6] = 600 [kPa] "Extraction pressure for steam open feedwater heater" P[7] = 20 [kPa] "Steam condenser pressure"
"GAS POWER CYCLE ANALYSIS"
"Gas Compressor anaysis" s[8]=ENTROPY(Air,T=T[8],P=P[8]) ss9=s[8] "For the ideal case the entropies are constant across the compressor" P[9] = Pratio*P[8] Ts9=temperature(Air,s=ss9,P=P[9])"Ts9 is the isentropic value of T[9] at compressor exit" Eta_comp = w_gas_comp_isen/w_gas_comp "compressor adiabatic efficiency, w_comp > w_comp_isen" h[8] + w_gas_comp_isen =hs9"SSSF conservation of energy for the isentropic compressor, assuming: adiabatic, ke=pe=0 per unit gas mass flow rate in kg/s" h[8]=ENTHALPY(Air,T=T[8]) hs9=ENTHALPY(Air,T=Ts9) h[8] + w_gas_comp = h[9]"SSSF conservation of energy for the actual compressor, assuming: adiabatic, ke=pe=0" T[9]=temperature(Air,h=h[9]) s[9]=ENTROPY(Air,T=T[9],P=P[9])
"Gas Cycle External heat exchanger analysis" h[9] + q_in = h[10]"SSSF conservation of energy for the external heat exchanger, assuming W=0, ke=pe=0" h[10]=ENTHALPY(Air,T=T[10]) P[10]=P[9] "Assume process 9-10 is SSSF constant pressure" Q_dot_in"MW"*1000"kW/MW"=m_dot_gas*q_in
"Gas Turbine analysis" s[10]=ENTROPY(Air,T=T[10],P=P[10]) ss11=s[10] "For the ideal case the entropies are constant across the turbine" P[11] = P[10] /Pratio Ts11=temperature(Air,s=ss11,P=P[11])"Ts11 is the isentropic value of T[11] at gas turbine exit" Eta_gas_turb = w_gas_turb /w_gas_turb_isen "gas turbine adiabatic efficiency, w_gas_turb_isen > w_gas_turb"
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96
h[10] = w_gas_turb_isen + hs11"SSSF conservation of energy for the isentropic gas turbine, assuming: adiabatic, ke=pe=0" hs11=ENTHALPY(Air,T=Ts11) h[10] = w_gas_turb + h[11]"SSSF conservation of energy for the actual gas turbine, assuming: adiabatic, ke=pe=0" T[11]=temperature(Air,h=h[11]) s[11]=ENTROPY(Air,T=T[11],P=P[11])
"Gas-to-Steam Heat Exchanger" "SSSF conservation of energy for the gas-to-steam heat exchanger, assuming: adiabatic, W=0, ke=pe=0" m_dot_gas*h[11] + m_dot_steam*h[4] = m_dot_gas*h[12] + m_dot_steam*h[5] h[12]=ENTHALPY(Air, T=T[12]) s[12]=ENTROPY(Air,T=T[12],P=P[12])
"STEAM CYCLE ANALYSIS" "Steam Condenser exit pump or Pump 1 analysis" Fluid$='Steam_IAPWS' P[1] = P[7] P[2]=P[6] h[1]=enthalpy(Fluid$,P=P[1],x=0) {Saturated liquid} v1=volume(Fluid$,P=P[1],x=0) s[1]=entropy(Fluid$,P=P[1],x=0) T[1]=temperature(Fluid$,P=P[1],x=0) w_pump1_s=v1*(P[2]-P[1])"SSSF isentropic pump work assuming constant specific volume" w_pump1=w_pump1_s/Eta_pump "Definition of pump efficiency" h[1]+w_pump1= h[2] "Steady-flow conservation of energy" s[2]=entropy(Fluid$,P=P[2],h=h[2]) T[2]=temperature(Fluid$,P=P[2],h=h[2]) "Open Feedwater Heater analysis" y*h[6] + (1-y)*h[2] = 1*h[3] "Steady-flow conservation of energy" P[3]=P[6] h[3]=enthalpy(Fluid$,P=P[3],x=0) "Condensate leaves heater as sat. liquid at P[3]" T[3]=temperature(Fluid$,P=P[3],x=0) s[3]=entropy(Fluid$,P=P[3],x=0) "Boiler condensate pump or Pump 2 analysis" P[4] = P[5] v3=volume(Fluid$,P=P[3],x=0) w_pump2_s=v3*(P[4]-P[3])"SSSF isentropic pump work assuming constant specific volume" w_pump2=w_pump2_s/Eta_pump "Definition of pump efficiency" h[3]+w_pump2= h[4] "Steady-flow conservation of energy" s[4]=entropy(Fluid$,P=P[4],h=h[4]) T[4]=temperature(Fluid$,P=P[4],h=h[4]) w_steam_pumps = (1-y)*w_pump1+ w_pump2 "Total steam pump work input/ mass steam" "Steam Turbine analysis" h[5]=enthalpy(Fluid$,T=T[5],P=P[5]) s[5]=entropy(Fluid$,P=P[5],T=T[5]) ss6=s[5] hs6=enthalpy(Fluid$,s=ss6,P=P[6]) Ts6=temperature(Fluid$,s=ss6,P=P[6]) h[6]=h[5]-Eta_steam_turb*(h[5]-hs6)"Definition of steam turbine efficiency" T[6]=temperature(Fluid$,P=P[6],h=h[6]) s[6]=entropy(Fluid$,P=P[6],h=h[6]) ss7=s[5] hs7=enthalpy(Fluid$,s=ss7,P=P[7]) Ts7=temperature(Fluid$,s=ss7,P=P[7])
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97
h[7]=h[5]-Eta_steam_turb*(h[5]-hs7)"Definition of steam turbine efficiency" T[7]=temperature(Fluid$,P=P[7],h=h[7]) s[7]=entropy(Fluid$,P=P[7],h=h[7]) "SSSF conservation of energy for the steam turbine: adiabatic, neglect ke and pe" h[5] = w_steam_turb + y*h[6] +(1-y)*h[7] "Steam Condenser analysis" (1-y)*h[7]=q_out+(1-y)*h[1]"SSSF conservation of energy for the Condenser per unit mass" Q_dot_out*Convert(MW, kW)=m_dot_steam*q_out "Cycle Statistics" MassRatio_gastosteam =m_dot_gas/m_dot_steam W_dot_net*Convert(MW, kW)=m_dot_gas*(w_gas_turb-w_gas_comp)+ m_dot_steam*(w_steam_turb - w_steam_pumps)"definition of the net cycle work" Eta_th=W_dot_net/Q_dot_in*Convert(, %) "Cycle thermal efficiency, in percent" Bwr=(m_dot_gas*w_gas_comp + m_dot_steam*w_steam_pumps)/(m_dot_gas*w_gas_turb + m_dot_steam*w_steam_turb) "Back work ratio" W_dot_net_steam = m_dot_steam*(w_steam_turb - w_steam_pumps) W_dot_net_gas = m_dot_gas*(w_gas_turb - w_gas_comp) NetWorkRatio_gastosteam = W_dot_net_gas/W_dot_net_steam
Pratio MassRatio gastosteam
Wnetgas [kW]
Wnetsteam [kW]
ηth [%]
NetWorkRatio gastosteam
6 4.463 262595 187405 45.29 1.401 8 5.024 279178 170822 46.66 1.634
10 5.528 289639 160361 47.42 1.806 12 5.994 296760 153240 47.82 1.937 14 6.433 301809 148191 47.99 2.037 15 6.644 303780 146220 48.01 2.078 16 6.851 305457 144543 47.99 2.113 18 7.253 308093 141907 47.87 2.171 20 7.642 309960 140040 47.64 2.213 22 8.021 311216 138784 47.34 2.242
0.0 1.1 2.2 3.3 4.4 5.5 6.6 7.7 8.8 9.9 11.0200
300
400
500
600
700
800
900
1000
1100
1200
1300
1400
1500
1600
s [kJ/kg-K]
T [K
]
8000 kPa
600 kPa
20 kPa
Combined Gas and Steam Power Cycle
8
9
10
11
12
1,23,4
5
6
7
Steam CycleSteam Cycle
Gas CycleGas Cycle
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98
5 9 12 16 19 2345.0
45.5
46.0
46.5
47.0
47.5
48.0
48.5
Pratio
ηth
[%]
Cycle Thermal Efficiency vs Gas Cycle Pressure Ratio
5 9 12 16 19 231.4
1.5
1.6
1.7
1.8
1.9
2.0
2.1
2.2
2.3
Pratio
Net
Wor
kRat
ioga
stos
team
W dot,gas / W dot,steam vs Gas Pressure Ratio
5 9 12 16 19 234.0
4.5
5.0
5.5
6.0
6.5
7.0
7.5
8.0
8.5
Pratio
Mas
sRat
ioga
stos
team
Ratio of Gas Flow Rate to Steam Flow Rate vs Gas Pressure Ratio
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99
10-91 A combined gas-steam power plant is considered. The topping cycle is a gas-turbine cycle and the bottoming cycle is a nonideal reheat Rankine cycle. The moisture percentage at the exit of the low-pressure turbine, the steam temperature at the inlet of the high-pressure turbine, and the thermal efficiency of the combined cycle are to be determined.
Assumptions 1 Steady operating conditions exist. 2 Kinetic and potential energy changes are negligible. 3Air is an ideal gas with variable specific heats.
Analysis (a) We obtain the air properties from EES. The analysis of gas cycle is as follows
( )( ) ( )
( )( )( )
kJ/kg62.475C200
kJ/kg98.87179.7638.130480.08.1304
kJ/kg79.763kPa100
kJ/kg6456.6kPa700C950
kJ/kg8.1304C950
kJ/kg21.55780.0/16.29047.50316.290
/
kJ/kg47.503kPa700
kJ/kg6648.5kPa100
C15kJ/kg50.288C15
1111
109910109
109
10910
10
99
9
99
787878
78
878
8
77
7
77
=⎯→⎯°=
=−−=
−−=⎯→⎯−−
=
=⎭⎬⎫
==
=⎭⎬⎫
=°=
=⎯→⎯°=
=−+=
−+=⎯→⎯−−
=
=⎭⎬⎫
==
=⎭⎬⎫
=°=
=⎯→⎯°=
hT
hhhhhhhh
hss
P
sPT
hT
hhhhhhhh
hss
P
sPT
hT
sTs
T
s
Css
C
s
ηη
ηη
From the steam tables (Tables A-4, A-5, and A-6 or from EES),
( )( )( )
kJ/kg37.19965.781.191kJ/kg.567
80.0/mkPa1
kJ1kPa106000/kgm 0.00101
/
/kgm 00101.0kJ/kg81.191
inpI,12
33
121inpI,
3kPa10@1
kPa10@1
=+=+==
⎟⎟⎠
⎞⎜⎜⎝
⎛
⋅−=
−=
====
whh
PPw
hh
p
f
f
ηv
vv
( )( ) kJ/kg4.23661.23929091.081.191
9091.04996.7
6492.04670.7kPa10
KkJ/kg4670.7kJ/kg5.3264
C400MPa1
66
66
56
6
5
5
5
5
=+=+=
=−
=−
=
⎭⎬⎫
==
⋅==
⎭⎬⎫
°==
fgsfs
fg
fss
s hxhhs
ssx
ssP
sh
TP
1
2
6s s
T
3
6 MPa
10 kPa
8s
Qout·
Qin ·
7
11
10s
9
GAS CYCLE
STEAM CYCLE
950°C
15°C
5
4
1 MPa
10
8
4s
6
3
6
1 2
Steam turbine
Gas turbine
Condenser pump
5
4
7
8 9
10
11
Combustion chamber
Compressor
Heat exchanger
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100
( )( )( )
1.6%==−=−=
=⎭⎬⎫
==
=−−=
−−=⎯→⎯−−
=
0158.09842.011PercentageMoisture
9842.0kJ/kg5.2546kPa10
kJ/kg0.25464.23665.326480.05.3264
6
66
6
655665
65
x
xhP
hhhhhhhh
sTs
T ηη
(b) Noting that 0ΔpeΔke ≅≅≅≅WQ && for the heat exchanger, the steady-flow energy balance equation yields
( ) ( ) ( )[ ] kJ/kg0.2965)62.47598.871)(10()5.3264()37.1995.3346()15.1( 44
1110air4523
outin
=⎯→⎯−=−+−
−=−+−
=
=
∑∑
hh
hhmhhmhhm
hmhm
EE
ss
eeii
&&&
&&
&&
Also,
( )sTs
T
ss
hhhhhhhh
hssP
sh
TP
433443
43
434
4
3
3
3
3 MPa1 ?MPa6
−−=⎯→⎯−−
=
=⎭⎬⎫
==
==
⎭⎬⎫
==
ηη
The temperature at the inlet of the high-pressure turbine may be obtained by a trial-error approach or using EES from the above relations. The answer is T3 = 468.0ºC. Then, the enthalpy at state 3 becomes: h3 = 3346.5 kJ/kg
(c) ( ) ( )( ) kW4328kJ/kg98.8718.1304kg/s10109airgasT, =−=−= hhmW &&
( ) ( )( ) kW2687kJ/kg50.28821.557kg/s1078airgasC, =−=−= hhmW &&
kW164126874328gasC,gasT,gasnet, =−=−= WWW &&&
( ) ( )( ) kW1265kJ/kg0.25465.32640.29655.3346kg/s1.156543ssteamT, =−+−=−+−= hhhhmW &&
( )( ) kW7.8kJ/kg564.7kg/s1.15ssteamP, === pumpwmW &&
kW12567.81265steamP,steamT,steamnet, =−=−= WWW &&&
kW2897=+=+= 12561641steamnet,gasnet,plantnet, WWW &&&
(d) ( ) ( )( ) kW7476kJ/kg21.5578.1304kg/s1089airin =−=−= hhmQ &&
38.8%==== 0.388kW7476kW2897
in
plantnet,th Q
W&
&η
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101
Special Topic: Binary Vapor Cycles
10-92C Binary power cycle is a cycle which is actually a combination of two cycles; one in the high temperature region, and the other in the low temperature region. Its purpose is to increase thermal efficiency.
10-93C Consider the heat exchanger of a binary power cycle. The working fluid of the topping cycle (cycle A) enters the heat exchanger at state 1 and leaves at state 2. The working fluid of the bottoming cycle (cycle B) enters at state 3 and leaves at state 4. Neglecting any changes in kinetic and potential energies, and assuming the heat exchanger is well-insulated, the steady-flow energy balance relation yields
( ) ( )43123142
outin
(steady)0systemoutin 0
hhmhhmorhmhmhmhm
hmhm
EE
EEE
BABABA
iiee
−=−+=+
=
=
=Δ=−
∑∑&&&&&&
&&
&&
&&&
Thus,
&
&
mm
hh h
hA
B=
−−
3 4
2 1
10-94C Steam is not an ideal fluid for vapor power cycles because its critical temperature is low, its saturation dome resembles an inverted V, and its condenser pressure is too low.
10-95C Because mercury has a high critical temperature, relatively low critical pressure, but a very low condenser pressure. It is also toxic, expensive, and has a low enthalpy of vaporization.
10-96C In binary vapor power cycles, both cycles are vapor cycles. In the combined gas-steam power cycle, one of the cycles is a gas cycle.
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102
Review Problems
10-97 It is to be demonstrated that the thermal efficiency of a combined gas-steam power plant ηcc can be expressed as η η η η ηcc g s g s= + − where η g g in=W Q/ and ηs s g,out=W Q/ are the thermal efficiencies of the gas and steam cycles, respectively, and the efficiency of a combined cycle is to be obtained.
Analysis The thermal efficiencies of gas, steam, and combined cycles can be expressed as
η
η
η
cctotal
in
out
in
gg
in
g,out
in
ss
g,out
out
g,out
= = −
= = −
= = −
W
Q
Q
Q
W
Q
Q
Q
W
Q
Q
Q
1
1
1
where Qin is the heat supplied to the gas cycle, where Qout is the heat rejected by the steam cycle, and where Qg,out is the heat rejected from the gas cycle and supplied to the steam cycle.
Using the relations above, the expression η η η ηg s g s+ − can be expressed as
cc
η
ηηηη
=
−=
−++−−+−=
⎟⎟⎠
⎞⎜⎜⎝⎟⎛−⎟
⎠
⎞⎜⎜⎝
⎛−⎟ −⎟
⎠
⎞⎜⎜⎝
⎛−⎟ +⎟
⎠
⎞⎜⎜⎝
⎛−=−+
in
out
in
out
outg,
out
in
outg,
outg,
out
in
outg,
outg,
out
in
outg,
outg,
out
in
outg,sgsg
1
111
1111
Therefore, the proof is complete. Using the relation above, the thermal efficiency of the given combined cycle is determined to be
η η η η ηcc g s g s= + − = + − × =0 4 0 30 0 40 0 30. . . . 0.58
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103
10-98 The thermal efficiency of a combined gas-steam power plant ηcc can be expressed in terms of the thermal efficiencies of the gas and the steam turbine cycles as η η η η ηcc g s g s= + − . It is to be shown that
the value of ηcc is greater than either of η ηg s or .
Analysis By factoring out terms, the relation η η η η ηcc g s g s= + − can be expressed as
η η η η η η η η η
η
cc g s g s g s g
Positive since <1
g
g
= + − = + − >( )11 24 34
or η η η η η η η η η
η
cc g s g s s g s
Positive since <1
s
s
= + − = + − >( )11 24 34
Thus we conclude that the combined cycle is more efficient than either of the gas turbine or steam turbine cycles alone.
10-99 A steam power plant operating on the ideal Rankine cycle with reheating is considered. The reheat pressures of the cycle are to be determined for the cases of single and double reheat.
Assumptions 1 Steady operating conditions exist. 2 Kinetic and potential energy changes are negligible.
Analysis (a) Single Reheat: From the steam tables (Tables A-4, A-5, and A-6),
( )( )( )( )
kPa2780=⎭⎬⎫
=°=
⋅=+=+==+=+=
⎭⎬⎫
==
565
5
66
66
6
6
C600
KkJ/kg5488.74996.792.06492.0kJ/kg5.23921.239292.081.191
92.0kPa10
Pss
T
sxsshxhh
xP
fgf
fgf
(b) Double Reheat :
C600 and
KkJ/kg3637.6C600MPa25
5
5
34
4
33
3
°==
==
⋅=⎭⎬⎫
°==
TPP
ssPP
sTP
xx
Any pressure Px selected between the limits of 25 MPa and 2.78 MPa will satisfy the requirements, and can be used for the double reheat pressure.
1
5
2
6
s
T 3
425
MPa
10 kPa
600°CSINGLE
1
5
2
8
s
T3
4 25
MPa
10 kPa
600°CDOUBLE 7
6
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104
10-100E A geothermal power plant operating on the simple Rankine cycle using an organic fluid as the working fluid is considered. The exit temperature of the geothermal water from the vaporizer, the rate of heat rejection from the working fluid in the condenser, the mass flow rate of geothermal water at the preheater, and the thermal efficiency of the Level I cycle of this plant are to be determined.
Assumptions 1 Steady operating conditions exist. 2 Kinetic and potential energy changes are negligible.
Analysis (a) The exit temperature of geothermal water from the vaporizer is determined from the steady-flow energy balance on the geothermal water (brine),
( )( )( )( )
F267.4°=°−°⋅=−
−=
2
2
12brinebrine
F325FBtu/lbm1.03lbm/h 384,286Btu/h000,790,22T
T
TTcmQ p&&
(b) The rate of heat rejection from the working fluid to the air in the condenser is determined from the steady-flow energy balance on air,
( )( )( )( )
MBtu/h29.7=°−°⋅=
−=
F555.84FBtu/lbm0.24lbm/h 4,195,10089airair TTcmQ p&&
(c) The mass flow rate of geothermal water at the preheater is determined from the steady-flow energy balance on the geothermal water,
( )( )( )
lbm/h 187,120=
°−°⋅=−
−=
geo
geo
inoutgeogeo
F8.2110.154FBtu/lbm1.03Btu/h000,140,11
m
m
TTcmQ p
&
&
&&
(d) The rate of heat input is
and
& & & , , , ,
, ,
&
Q Q Q
W
in vaporizer reheater
net
Btu / h
kW
= + = +
=
= − =
22 790 000 11140 000
33 930 000
1271 200 1071
Then,
10.8%⎟ =⎟⎠
⎞⎜⎜⎝
⎛==
kWh1Btu3412.14
Btu/h 33,930,000kW 1071
in
netth Q
W&
&η
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105
10-101 A steam power plant operating on an ideal Rankine cycle with two stages of reheat is considered. The thermal efficiency of the cycle and the mass flow rate of the steam are to be determined.
Assumptions 1 Steady operating conditions exist. 2 Kinetic and potential energy changes are negligible.
Analysis (a) From the steam tables (Tables A-4, A-5, and A-6),
( )
( )( )
kJ/kg82.15207.1575.137kJ/kg15.07
mkPa1kJ1kPa515,000/kgm 0.001005
/kgm 001005.0kJ/kg75.137
inp,12
33
121inp,
3kPa5 @1
kPa5@1
=+=+==
⎟⎟⎠
⎞⎜⎜⎝
⎛
⋅−=
−=
====
whh
PPw
hh
f
f
v
vv
( )( ) kJ/kg9.23670.24239204.075.137
9204.09176.7
4762.07642.7kPa5
KkJ/kg7642.7kJ/kg1.3479
C500MPa1
kJ/kg3.2971MPa1
KkJ/kg9781.6kJ/kg7.3434
C500MPa5
kJ/kg4.3007MPa5
KkJ/kg3480.6kJ/kg8.3310
C500MPa15
88
88
78
8
7
7
7
7
656
6
5
5
5
5
434
4
3
3
3
3
=+=+=
=−
=−
=
⎭⎬⎫
==
⋅==
⎭⎬⎫
°==
=⎭⎬⎫
==
⋅==
⎭⎬⎫
°==
=⎭⎬⎫
==
⋅==
⎭⎬⎫
°==
fgf
fg
f
hxhhs
ssx
ssP
sh
TP
hssP
sh
TP
hssP
sh
TP
Then, ( ) ( ) ( )
kJ/kg9.18622.22301.4093kJ/kg2.223075.1379.2367
kJ/kg1.40933.29711.34794.30077.343482.1528.3310
outinnet
18out
674523in
=−=−==−=−=
=−+−+−=−+−+−=
qqwhhq
hhhhhhq
Thus,
45.5%===kJ/kg4093.1kJ/kg1862.9
in
netth q
wη
(b) kg/s64.4===kJ/kg1862.9
kJ/s120,000
net
net
wWm&
&
1
5
2
8 s
T
3
4 15 MPa
5 kPa
7
6
5 MPa 1 MPa
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106
10-102 A simple ideal Rankine cycle with water as the working fluid operates between the specified pressure limits. The thermal efficiency of the cycle is to be compared when it is operated so that the liquid enters the pump as a saturated liquid against that when the liquid enters as a subcooled liquid.
determined power produced by the turbine and consumed by the pump are to be determined.
Assumptions 1 Steady operating conditions exist. 2 Kinetic and potential energy changes are negligible.
Analysis From the steam tables (Tables A-4, A-5, and A-6),
kJ/kg67.34613.654.340kJ/kg13.6 mkPa1
kJ1kPa)506000)(/kgm001030.0()(
/kgm001030.0
kJ/kg54.340
inp,12
33
121inp,
3kPa20@1
kPa50@1
=+=+==
⎟⎠
⎞⎜⎝
⎛
⋅−=
−=
==
==
whh
PPw
hh
f
f
v
vv
kJ/kg0.2495)7.2304)(9348.0(54.340
9348.05019.6
0912.11693.7
kPa50
KkJ/kg1693.7kJ/kg8.3658
C600kPa6000
44
44
34
4
3
3
3
3
=+=+=
=−
=−
=
⎭⎬⎫
==
⋅==
⎭⎬⎫
°==
fgf
fg
f
hxhhs
ssx
ssP
sh
TP
Thus,
kJ/kg5.215454.3400.2495kJ/kg1.331267.3468.3658
14out
23in
=−=−==−=−=
hhqhhq
0.3495
and the thermal efficiency of the cycle is
=−=−=1.33125.215411
in
outth q
qη
When the liquid enters the pump 11.3°C cooler than a saturated liquid at the condenser pressure, the enthalpies become
/kgm001023.0kJ/kg07.293
C703.113.813.11
kPa503
C70@1
C70@1
kPa50@sat 1
1
=≅
=≅
⎭⎬⎫
°=−=−==
°
°
f
fhhTT
Pvv
kJ/kg09.6 mkPa1kJ1kPa)506000)(/kgm001023.0(
)(
33
121inp,
=⎟⎠
⎞⎜⎝
⎛
⋅−=
−= PPw v
kJ/kg16.29909.607.293inp,12 =+=+= whh
Then,
kJ/kg9.220109.2930.2495kJ/kg6.335916.2998.3658
14out
23in
=−=−==−=−=
hhqhhq
0.3446=−=−=6.33599.220111
in
outth q
qη
The thermal efficiency slightly decreases as a result of subcooling at the pump inlet.
qin
qout
50 kPa 1
3
2
4
6 MPa
s
T
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107
10-103 An 150-MW steam power plant operating on a regenerative Rankine cycle with an open feedwater heater is considered. The mass flow rate of steam through the boiler, the thermal efficiency of the cycle, and the irreversibility associated with the regeneration process are to be determined.
Assumptions 1 Steady operating conditions exist. 2 Kinetic and potential energy changes are negligible.
Analysis
(a) From the steam tables (Tables A-4, A-5, and A-6),
( )( )( ) ( )
/kgm 001093.0kJ/kg09.640
liquidsat.MPa5.0
kJ/kg33.19252.081.191kJ/kg0.52
95.0/mkPa1
kJ1kPa10500/kgm 0.00101
/
/kgm 00101.0kJ/kg81.191
3MPa5.0 @3
MPa5.0 @33
inpI,12
33
121inpI,
3kPa10@1
kPa10@1
====
⎭⎬⎫=
=+=+==
⎟⎟⎠
⎞⎜⎜⎝
⎛
⋅−=
η−=
==
==
f
f
p
f
f
hhP
whh
PPw
hh
vv
v
vv
( )( )( ) ( )
kJ/kg02.65193.1009.640kJ/kg10.93
95.0/mkPa1
kJ1kPa50010,000/kgm 0.001093
/
inpII,34
33
343inpII,
=+=+==
⎟⎟⎠
⎞⎜⎜⎝
⎛
⋅−=
η−=
whh
PPw pv
( )( )kJ/kg1.2654
0.21089554.009.640
9554.09603.4
8604.15995.6
MPa5.0
KkJ/kg5995.6kJ/kg1.3375
C500MPa10
66
66
56
6
5
5
5
5
=+=+=
=−
=−
=
⎭⎬⎫
==
⋅==
⎭⎬⎫
°==
fgsfs
fg
fss
s
s hxhhs
ssx
ssP
sh
TP
1
qin
2
7s
s
T
5
6s
10 MPa
10 kPa
0.5 MPa
1-y
4
y 3
qout7
6 y
5
7
1 2
3
4
Turbine Boiler
CondenserOpen fwh
P I P II
6 1-y
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108
( )( )( )
kJ/kg3.27981.26541.337580.01.3375
655665
65
=−−=
−−=⎯→⎯−−
= sTs
T hhhhhhhh
ηη
( )( )kJ/kg7.2089
1.23927934.081.191
7934.04996.7
6492.05995.6
kPa1077
77
57
7
=+=+=
=−
=−
=
⎭⎬⎫
==
fgsfs
fg
fss
s
s hxhhs
ssx
ssP
( )( )( )
kJ/kg8.23467.20891.337580.01.3375
755775
75
=−−=
−−=⎯→⎯−−
= sTs
T hhhhhhhh
ηη
The fraction of steam extracted is determined from the steady-flow energy balance equation applied to the feedwater heaters. Noting that & &Q W ke pe≅ ≅ ≅ ≅Δ Δ 0 ,
( ) ( )326332266
outin
(steady)0systemoutin
11
0
hhyyhhmhmhmhmhm
EE
EEE
eeii =−+⎯→⎯=+⎯→⎯=
=
=Δ=−
∑∑ &&&&&
&&
&&&
where y is the fraction of steam extracted from the turbine ( = & / &m m6 3 ). Solving for y,
1718.033.1923.279833.19209.640
26
23 =−−
=−−
=hhhh
y
Then, ( )( ) ( )( )
kJ/kg4.9397.17841.2724kJ/kg7.178481.1918.23461718.011
kJ/kg1.272402.6511.3375
outinnet
17out
45in
=−=−==−−=−−=
=−=−=
qqwhhyq
hhq
and
skg7159 /.kJ/kg939.4
kJ/s150,000
net
net ===wW
m&
&
34.5%
(b) The thermal efficiency is determined from
=−=−=kJ/kg2724.1kJ/kg1784.7
11in
outth q
qη
Also,
KkJ/kg6492.0KkJ/kg8604.1
KkJ/kg9453.6kJ/kg3.2798
MPa5.0
kPa10@12
MPa5.0@3
66
6
⋅===⋅==
⋅=⎭⎬⎫
==
f
f
sssss
shP
Then the irreversibility (or exergy destruction) associated with this regeneration process is
( )[ ]
( ) ( )( ) ( )( )[ ] kJ/kg39.25=−−−=
−−−⎟ =⎟⎠
⎞⎜⎜⎝
⎛+−== ∑∑
6492.01718.019453.61718.08604.1K303
1 2630
0surr
0gen0regen syyssTT
qsmsmTsTiL
iiee
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109
10-104 An 150-MW steam power plant operating on an ideal regenerative Rankine cycle with an open feedwater heater is considered. The mass flow rate of steam through the boiler, the thermal efficiency of the cycle, and the irreversibility associated with the regeneration process are to be determined. Assumptions 1 Steady operating conditions exist. 2 Kinetic and potential energy changes are negligible.
Analysis
(a) From the steam tables (Tables A-4, A-5, and A-6),
( )( )( )
/kgm 001093.0kJ/kg09.640
liquidsat.MPa5.0
kJ/kg30.19250.081.191
kJ/kg0.50mkPa1
kJ1kPa10500/kgm 0.00101
/kgm 00101.0kJ/k81.191
3MPa5.0@3
MPa5.0 @33
inpI,12
33
121inpI,
3kPa10@1
kPa10@1
====
⎭⎬⎫=
=+=+=
⎟ =⎟⎠
⎞⎜⎜⎝
⎛
⋅−=
−=
==
==
f
f
f
f
hhP
whh
PPw
ghh
vv
v
vv
( )( )( )
( )( )
( )( ) kJ/kg7.20891.23927934.081.191
7934.04996.7
6492.05995.6kPa10
kJ/kg1.26540.21089554.009.640
9554.09603.4
8604.15995.6MPa5.0
KkJ/kg5995.6kJ/kg1.3375
C500MPa10
kJ/kg47.65038.1009.640
kJ/kg.3810mkPa1
kJ1kPa50010,000/kgm 0.001093
77
77
57
7
66
66
56
6
5
5
5
5
inpII,34
33
343inpII,
=+=+=
=−
=−
=
⎭⎬⎫
==
=+=+=
=−
=−
=
⎭⎬⎫
==
⋅==
⎭⎬⎫
°==
=+=+=
⎟ =⎟⎠
⎞⎜⎜⎝
⎛
⋅−=
−=
fgf
fg
f
fgf
fg
f
hxhhs
ssx
ssP
hxhhs
ssx
ssP
sh
TP
whh
PPw v
The fraction of steam extracted is determined from the steady-flow energy equation applied to the feedwater heaters. Noting that 0ΔpeΔke ≅≅≅≅WQ && ,
1
qin
2
7
s
T
5
6
10 MPa
10 kPa
0.5 MPa
1-y
4
y 3
qout
y
5
7
1 2
3
4
Turbine Boiler
CondenserOpen fwh
P I P II
6 1-y
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110
( ) ( )326332266
outin(steady)0
systemoutin
11
0
hhyyhhmhmhmhmhm
EEEEE
eeii =−+⎯→⎯=+⎯→⎯=
=→=Δ=−
∑∑ &&&&&
&&&&&
where y is the fraction of steam extracted from the turbine ( = & / &m m6 3 ). Solving for y,
1819.031.1921.265431.19209.640
26
23 =−−
=−−
=hhhh
y
Then, ( )( ) ( )( )
kJ/kg0.11727.15526.2724kJ/kg7.155281.1917.20891819.011
kJ/kg6.272447.6501.3375
outinnet
17out
45in
=−=−==−−=−−=
=−=−=
qqwhhyq
hhq
and skg0128 /.kJ/kg1171.9
kJ/s150,000
net
net ===wWm&
&
(b) The thermal efficiency is determined from
%0.43=−=−=kJ/kg2724.7kJ/kg1552.7
11in
outth q
qη
Also,
KkJ/kg6492.0
KkJ/kg8604.1KkJ/kg5995.6
kPa10@12
MPa5.0@3
56
⋅===
⋅==⋅==
f
f
sss
ssss
Then the irreversibility (or exergy destruction) associated with this regeneration process is
( )[ ]
( ) ( )( ) ( )( )[ ] kJ/kg39.0=−−−=
−−−⎟ =⎟⎠
⎞⎜⎜⎝
⎛+−== ∑∑
6492.01819.015995.61819.08604.1K303
1 2630
0surr
0gen0regen syyssTT
qsmsmTsTiL
iiee
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111
10-105 An ideal reheat-regenerative Rankine cycle with one open feedwater heater is considered. The fraction of steam extracted for regeneration and the thermal efficiency of the cycle are to be determined. Assumptions 1 Steady operating conditions exist. 2 Kinetic and potential energy changes are negligible. Analysis (a) From the steam tables (Tables A-4, A-5, and A-6),
( )( )( )
/kgm 001101.0kJ/k38.670
liquidsat.MPa6.0
kJ/kg53.22659.094.225kJ/kg0.59
mkPa1kJ1
kPa15600/kgm 0.001014
/kgm 001014.0kJ/kg94.225
3MPa6.0@3
MPa6.0 @33
inpI,12
33
121inpI,
3kPa15@1
kPa15@1
====
⎭⎬⎫=
=+=+==
⎟⎟⎠
⎞⎜⎜⎝
⎛
⋅−=
−=
==
==
f
f
f
f
ghhP
whh
PPw
hh
vv
v
vv
( )( )( )
( )( ) kJ/kg8.25183.23729665.094.225
9665.02522.7
7549.07642.7kPa15
kJ/kg2.3310MPa6.0
KkJ/kg7642.7kJ/kg1.3479
C500MPa0.1
kJ/kg8.2783MPa0.1
KkJ/kg5995.6kJ/kg1.3375
C500MPa10
kJ/kg73.68035.1038.670kJ/kg10.35
mkPa1kJ1kPa60010,000/kgm 0.001101
99
99
79
9
878
8
7
7
7
7
656
6
5
5
5
5
inpII,34
33
343inpII,
=+=+=
=−
=−
=
⎭⎬⎫
==
=⎭⎬⎫
==
⋅==
⎭⎬⎫
°==
=⎭⎬⎫
==
⋅==
⎭⎬⎫
°==
=+=+==
⎟⎟⎠
⎞⎜⎜⎝
⎛
⋅−=
−=
fgf
fg
f
hxhhs
ssx
ssP
hssP
sh
TP
hssP
sh
TP
whh
PPw v
The fraction of steam extracted is determined from the steady-flow energy balance equation applied to the feedwater heaters. Noting that 0ΔpeΔke ≅≅≅≅WQ && ,
( ) ( )328332288
outin(steady)0
systemoutin
11
0
hhyyhhmhmhmhmhm
EEEEE
eeii =−+⎯→⎯=+⎯→⎯=
=→=Δ=−
∑∑ &&&&&
&&&&&
where y is the fraction of steam extracted from the turbine ( = & / &m m8 3 ). Solving for y,
0.144=−−
=−−
=53.2262.331053.22638.670
28
23
hhhhy
(b) The thermal efficiency is determined from ( ) ( ) ( ) ( )( )( ) ( )( ) kJ/kg7.196294.2258.25181440.011
kJ/kg7.33898.27831.347973.6801.3375
19out
6745in
=−−=−−==−+−=−+−=
hhyqhhhhq
and 42.1%=−=−=kJ/kg3389.7kJ/kg1962.7
11in
outth q
qη
1
7
2
9 s
T5
6 10 MPa
15 kPa
0.6 MPa
1 MPa
8 4
3
y
5
9
1 2
3
4
Turbine Boiler
Condens.Openfwh
P I P II
8 1-y
6
7
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112
10-106 A nonideal reheat-regenerative Rankine cycle with one open feedwater heater is considered. The fraction of steam extracted for regeneration and the thermal efficiency of the cycle are to be determined. Assumptions 1 Steady operating conditions exist. 2 Kinetic and potential energy changes are negligible.
Analysis
(a) From the steam tables (Tables A-4, A-5, and A-6),
( )( )( )
/kgm001101.0kJ/kg38.670
liquidsat.MPa6.0
kJ/kg54.22659.094.225kJ/kg0.59
mkPa1kJ1
kPa15600/kgm0.001014
/kgm001014.0kJ/kg94.225
3MPa6.0@3
MPa6.0 @33
in,12
33
121in,
3kPa15@1
kPa15@1
====
⎭⎬⎫=
=+=+==
⎟⎟⎠
⎞⎜⎜⎝
⎛
⋅−=
−=
==
==
f
f
pI
pI
f
f
hhP
whh
PPw
hh
vv
v
vv
( )( )( )
kJ/kg8.2783MPa0.1
KkJ/kg5995.6kJ/kg1.3375
C500MPa10
kJ/kg73.68035.1038.670kJ/kg10.35
mkPa1kJ1kPa60010,000/kgm 0.001101
656
6
5
5
5
5
inpII,34
33
343inpII,
=⎭⎬⎫
==
⋅==
⎭⎬⎫
°==
=+=+==
⎟⎟⎠
⎞⎜⎜⎝
⎛
⋅−=
−=
ss
s hssP
sh
TP
whh
PPw v
( )( )( )
kJ/kg4.28788.27831.337584.01.3375
655665
65
=−−=
−η−=⎯→⎯−−
=η sTs
T hhhhhhhh
1
7
2
9 s
T5
6 8 4
3
6s 8s
9s
1-y
y y
5
9
1 2
3
4
Turbine Boiler
CondenserOpen fwh
P IP II
8 1-y
6
7
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113
( ) ( )( )kJ/kg2.3337
2.33101.347984.01.3479
kJ/kg2.3310MPa6.0
KkJ/kg7642.7kJ/kg1.3479
C500MPa0.1
877887
87
878
8
7
7
7
7
=−−=−η−=⎯→⎯
−−
=η
=⎭⎬⎫
==
⋅==
⎭⎬⎫
°==
sTs
T
ss
s
hhhhhhhh
hssP
sh
TP
( )( )
( ) ( )( )kJ/kg5.2672
8.25181.347984.01.3479
kJ/kg8.25183.23729665.094.225
9665.02522.7
7549.07642.7kPa15
977997
97
99
99
79
9
=−−=−−=⎯→⎯
−−
=
=+=+=
=−
=−
=
⎭⎬⎫
==
sTs
T
fgsfs
fg
fss
s
s
hhhhhhhh
hxhhs
ssx
ssP
ηη
The fraction of steam extracted is determined from the steady-flow energy balance equation applied to the feedwater heaters. Noting that 0ΔpeΔke ≅≅≅≅WQ && ,
( ) ( )328332288
outin
(steady)0systemoutin
11
0
hhyyhhmhmhmhmhm
EE
EEE
eeii =−+⎯→⎯=+⎯→⎯=
=
=Δ=−
∑∑ &&&&&
&&
&&&
where y is the fraction of steam extracted from the turbine ( = & / &m m8 3 ). Solving for y,
0.1427=−−
=−−
=53.2263.333553.22638.670
28
23
hhhhy
(b) The thermal efficiency is determined from
( ) ( )( ) ( )( )( ) ( )( ) kJ/kg2.209794.2255.26721427.011
kJ/kg1.32954.28781.347973.6801.3375
19out
6745in
=−−=−−==−+−=
−+−=
hhyq
hhhhq
and
36.4%=−=−=kJ/kg3295.1kJ/kg2097.2
11in
outth q
qη
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114
10-107 A steam power plant operating on the ideal reheat-regenerative Rankine cycle with three feedwater heaters is considered. Various items for this system per unit of mass flow rate through the boiler are to be determined.
Assumptions 1 Steady operating conditions exist. 2 Kinetic and potential energy changes are negligible.
Analysis The compression processes in the pumps and the expansion processes in the turbines are isentropic. Also, the state of water at the inlet of pumps is saturated liquid. Then, from the steam tables (Tables A-4, A-5, and A-6),
kJ/kg2.1216kJ/kg8.1213kJ/kg8.1052kJ/kg7.1049kJ/kg05.421kJ/kg51.417kJ/kg90.191kJ/kg81.191
10
9
6
5
4
3
2
1
========
hhhhhhhh
kJ/kg5.2547kJ/kg6.2974kJ/kg5.3275kJ/kg4.2470kJ/kg8.2931kJ/kg8.3062kJ/kg4.3139
19
18
17
16
15
14
13
=======
hhhhhhh
For an ideal closed feedwater heater, the feedwater is heated to the exit temperature of the extracted steam, which ideally leaves the heater as a saturated liquid at the extraction pressure. Then,
kJ/kg1.1213 C6.275
kPa6000
kJ/kg0.1050 C6.242
kPa3500
11911
11
757
7
=⎭⎬⎫
°===
=⎭⎬⎫
°===
hTT
P
hTT
P
Enthalpies at other states and the fractions of steam extracted from the turbines can be determined from mass and energy balances on cycle components as follows:
Mass Balances:
znmzyx=+=++ 1
16
13
1
Low-P Turbine
Boiler
Condenser
14
2
19
15
High-P Turbine
34 5 6
7
8
9 10 11
12
17
18Closed FWH I
ClosedFWH II
OpenFWH P I P II
P III P IV
xy
z
m
n
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115
Open feedwater heater:
3218 zhnhmh =+
Closed feedwater heater-II:
57154 yhzhyhzh +=+
Closed feedwater heater-I:
911148 )()( xhhzyxhhzy ++=++
Mixing chamber after closed feedwater heater II:
867 )( hzyyhzh +=+
Mixing chamber after closed feedwater heater I:
121110 1)( hhzyxh =++
Substituting the values and solving the above equations simultaneously using EES, we obtain
0.63320.05586
0.23030.08072
=======
nmzyx
hh
6890.0
kJ/kg3.1213kJ/kg7.1050
12
8
Note that these values may also be obtained by a hand solution by using the equations above with some rearrangements and substitutions. Other results of the cycle are
0.3986
kJ/kg1492kJ/kg2481
kJ/kg477.8kJ/kg514.9
=−=−=
=−==−+−=
=−+−=
=−+−+−=
2481149211
)()(
)()()()()(
in
outth
119out
16171213in
19171817LPout,T,
161315131413HPout,T,
hhnqhhzhhq
hhnhhmwhhzhhyhhxw
η
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116
10-108 EES The optimum bleed pressure for the open feedwater heater that maximizes the thermal efficiency of the cycle is to be determined using EES.
Analysis The EES program used to solve this problem as well as the solutions are given below.
"Given" P_boiler=8000 [kPa] P_cfwh1=6000 [kPa] P_cfwh2=3500 [kPa] P_reheat=300 [kPa] P_ofwh=100 [kPa] P_condenser=10 [kPa] T_turbine=400 [C]
"Analysis" Fluid$='steam_iapws'
"turbines" h[13]=enthalpy(Fluid$, P=P_boiler, T=T_turbine) s[13]=entropy(Fluid$, P=P_boiler, T=T_turbine) h[14]=enthalpy(Fluid$, P=P_cfwh1, s=s[13]) h[15]=enthalpy(Fluid$, P=P_cfwh2, s=s[13]) h[16]=enthalpy(Fluid$, P=P_reheat, s=s[13]) h[17]=enthalpy(Fluid$, P=P_reheat, T=T_turbine) s[17]=entropy(Fluid$, P=P_reheat, T=T_turbine) h[18]=enthalpy(Fluid$, P=P_ofwh, s=s[17]) h[19]=enthalpy(Fluid$, P=P_condenser, s=s[17])
"pump I" h[1]=enthalpy(Fluid$, P=P_condenser, x=0) v[1]=volume(Fluid$, P=P_condenser, x=0) w_pI_in=v[1]*(P_ofwh-P_condenser) h[2]=h[1]+w_pI_in
"pump II" h[3]=enthalpy(Fluid$, P=P_ofwh, x=0) v[3]=volume(Fluid$, P=P_ofwh, x=0) w_pII_in=v[3]*(P_cfwh2-P_ofwh) h[4]=h[3]+w_pII_in
"pump III" h[5]=enthalpy(Fluid$, P=P_cfwh2, x=0) T[5]=temperature(Fluid$, P=P_cfwh2, x=0) v[5]=volume(Fluid$, P=P_cfwh2, x=0) w_pIII_in=v[5]*(P_cfwh1-P_cfwh2) h[6]=h[5]+w_pIII_in
"pump IV" h[9]=enthalpy(Fluid$, P=P_cfwh1, x=0) T[9]=temperature(Fluid$, P=P_cfwh1, x=0) v[9]=volume(Fluid$, P=P_cfwh1, x=0) w_p4_in=v[5]*(P_boiler-P_cfwh1) h[10]=h[9]+w_p4_in
"Mass balances" x+y+z=1
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117
m+n=z "Open feedwater heater" m*h[18]+n*h[2]=z*h[3] "closed feedwater heater 2" T[7]=T[5] h[7]=enthalpy(Fluid$, P=P_cfwh1, T=T[7]) z*h[4]+y*h[15]=z*h[7]+y*h[5] "closed feedwater heater 1" T[11]=T[9] h[11]=enthalpy(Fluid$, P=P_boiler, T=T[11]) (y+z)*h[8]+x*h[14]=(y+z)*h[11]+x*h[9] "Mixing chamber after closed feedwater heater 2" z*h[7]+y*h[6]=(y+z)*h[8] "Mixing chamber after closed feedwater heater 1" x*h[10]+(y+z)*h[11]=1*h[12]
"cycle" w_T_out_high=x*(h[13]-h[14])+y*(h[13]-h[15])+z*(h[13]-h[16]) w_T_out_low=m*(h[17]-h[18])+n*(h[17]-h[19]) q_in=h[13]-h[12]+z*(h[17]-h[16]) q_out=n*(h[19]-h[1]) Eta_th=1-q_out/q_in
P open fwh
[kPa] ηth
10 0.388371 20 0.392729 30 0.394888 40 0.396199 50 0.397068 60 0.397671 70 0.398099 80 0.398406 90 0.398624
100 0.398774 110 0.398872 120 0.398930 130 0.398954 140 0.398952 150 0.398927 160 0.398883 170 0.398825 180 0.398752 190 0.398669 200 0.398576
0 50 100 150 200 250 3000.388
0.39
0.392
0.394
0.396
0.398
0.4
Pofwh [kPa]
ηth
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118
10-109E A combined gas-steam power cycle uses a simple gas turbine for the topping cycle and simple Rankine cycle for the bottoming cycle. The thermal efficiency of the cycle is to be determined.
Assumptions 1 Steady operating conditions exist. 2 The air-standard assumptions are applicable for Brayton cycle. 3 Kinetic and potential energy changes are negligible. 4 Air is an ideal gas with constant specific heats.
Properties The properties of air at room temperature are cp = 0.240 Btu/lbm·R and k = 1.4 (Table A-2Ea).
Analysis Working around the topping cycle gives the following results:
R1043R)(10)540( 0.4/1.4/)1(
5
656 ==⎟⎟
⎠⎜
⎞⎜⎝
⎛=
− kk
s PP
TT
R109990.0
5401043540
)()(
5656
56
56
56
56
=−
+=
−+=⎯→⎯
−
−=
−−
=
C
s
p
spsC
TTTT
TTcTTc
hhhh
η
η
R1326101R)2560(
0.4/1.4/)1(
7
878 =⎟
⎠⎞
⎜⎝⎛=⎟⎟
⎠⎜
⎞⎜⎝
⎛=
− kk
s PP
TT
R1449)13262560)(90.0(2560
)()()(
877887
87
87
87
=−−=
−−=⎯→⎯−
−=
−−
= sTsp
p
sT TTTT
TTcTTc
hhhh
ηη
R102850R3.97850psia800@sat 9 =+=+= TT
Fixing the states around the bottom steam cycle yields (Tables A-4E, A-5E, A-6E):
Btu/lbm59.13241.218.130Btu/lbm41.2
ftpsia5.404Btu1psia)5800)(/lbmft01641.0(
)(
/lbmft01641.0
Btu/lbm18.130
inp,12
33
121inp,
3psia5 @1
psia5@1
=+=+==
⎟⎟⎠
⎞⎜⎜⎝
⎛
⋅−=
−=
==
==
whh
PPw
hh
f
f
v
vv
Btu/lbm6.908 psia5
RBtu/lbm4866.1Btu/lbm9.1270
F600
psia800
434
4
3
3
3
3
=⎭⎬⎫
==
⋅==
⎭⎬⎫
°==
shss
P
sh
TP
Btu/lbm7.926)6.9089.1270)(95.0(9.1270
)( 433443
43
=−−=
−−=⎯→⎯−−
= sTs
T hhhhhhhh
ηη
1
2
4ss
T
3
800 psia
5 psia
6s
Qout·
Qin·
5
9
8s
7
GAS CYCLE
STEAM CYCLE
2560 R
540 R
600°F
8 6
4
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119
The net work outputs from each cycle are
Btu/lbm5.132R)540109914492560)(RBtu/lbm240.0(
)()( 5687
inC,outT,cyclegasnet,
=+−−⋅=
−−−=
−=
TTcTTc
www
pp
Btu/lbm8.34141.2)7.9269.1270(
)( inP,43
inP,outT,cyclesteamnet,
=−−=
−−=
−=
whh
www
An energy balance on the heat exchanger gives
aap
wwpa mm-hh
TTcm-hhmTTcm &&&&& 08876.0
59.1329.1270)10281449)(240.0()(
)()(23
982398 =
−−
=−
=⎯→⎯=−
That is, 1 lbm of exhaust gases can heat only 0.08876 lbm of water. Then the heat input, the heat output and the thermal efficiency are
Btu/lbm8.187Btu/lbm)18.1307.926(08876.0R)5401028)(RBtu/lbm240.0(1
)()(
Btu/lbm6.350R)10992560)(RBtu/lbm240.0()(
1419out
67in
=−×+−⋅×=
−+−=
=−⋅=−=
hhmm
TTcmm
q
TTcmm
q
a
wp
a
a
pa
a
&
&
&
&
&
&
0.4643=−=−=6.3508.18711
in
outth q
qη
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120
10-110E A combined gas-steam power cycle uses a simple gas turbine for the topping cycle and simple Rankine cycle for the bottoming cycle. The thermal efficiency of the cycle is to be determined.
Assumptions 1 Steady operating conditions exist. 2 The air-standard assumptions are applicable fo Brayton cycle. 3 Kinetic and potential energy changes are negligible. 4 Air is an ideal gas with constant specific heats.
Properties The properties of air at room temperature are cp = 0.240 Btu/lbm·R and k = 1.4 (Table A-2Ea).
Analysis Working around the topping cycle gives the following results:
R1043R)(10)540( 0.4/1.4/)1(
5
656 ==⎟⎟
⎠⎜
⎞⎜⎝
⎛=
− kk
s PP
TT
R109990.0
5401043540
)()(
5656
56
56
56
56
=−
+=
−+=⎯→⎯
−
−=
−−
=
C
s
p
spsC
TTTT
TTcTTc
hhhh
η
η
R1326101R)2560(
0.4/1.4/)1(
7
878 =⎟
⎠⎞
⎜⎝⎛=⎟⎟
⎠⎜
⎞⎜⎝
⎛=
− kk
s PP
TT
R1449)13262560)(90.0(2560
)()()(
877887
87
87
87
=−−=
−−=⎯→⎯−
−=
−−
= sTsp
p
sT TTTT
TTcTTc
hhhh
ηη
R102850R3.97850psia800@sat 9 =+=+= TT
Fixing the states around the bottom steam cycle yields (Tables A-4E, A-5E, A-6E):
Btu/lbm7.16343.225.161Btu/lbm43.2
ftpsia5.404Btu1psia)10800)(/lbmft01659.0(
)(
/lbmft01659.0
Btu/lbm25.161
inp,12
33
121inp,
3psia10@1
psia10@1
=+=+==
⎟⎟⎠
⎞⎜⎜⎝
⎛
⋅−=
−=
==
==
whh
PPw
hh
f
f
v
vv
Btu/lbm6.946 psia10
RBtu/lbm4866.1Btu/lbm9.1270
F600
psia800
434
4
3
3
3
3
=⎭⎬⎫
==
⋅==
⎭⎬⎫
°==
shss
P
sh
TP
1
2
4ss
T
3
800 psia
10 psia
6s
Qout·
Qin ·
5
9
8s
7
GAS CYCLE
STEAM CYCLE
2560 R
540 R
600°F
8 6
4
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121
Btu/lbm8.962)6.9469.1270)(95.0(9.1270
)( 433443
43
=−−=
−−=⎯→⎯−−
= sTs
T hhhhhhhh
ηη
The net work outputs from each cycle are
Btu/lbm5.132R)540109914492560)(RBtu/lbm240.0(
)()( 5687
inC,outT,cyclegasnet,
=+−−⋅=
−−−=
−=
TTcTTc
www
pp
Btu/lbm7.30543.2)8.9629.1270(
)( inP,43
inP,outT,cyclesteamnet,
=−−=
−−=
−=
whh
www
An energy balance on the heat exchanger gives
aap
wwpa mmhh
TTcmhhmTTcm &&&&& 09126.0
7.1639.1270)10281449)(240.0()(
)()(23
982398 =
−−
=−
−=⎯→⎯−=−
That is, 1 lbm of exhaust gases can heat only 0.09126 lbm of water. Then the heat input, the heat output and the thermal efficiency are
Btu/lbm3.190Btu/lbm)25.1618.962(09126.0R)5401028)(RBtu/lbm240.0(1
)()(
Btu/lbm6.350R)10992560)(RBtu/lbm240.0()(
1419out
67in
=−×+−⋅×=
−+−=
=−⋅=−=
hhmm
TTcmm
q
TTcmm
q
a
wp
a
a
pa
a
&
&
&
&
&
&
0.4573=−=−=6.3503.19011
in
outth q
qη
When the condenser pressure is increased from 5 psia to 10 psia, the thermal efficiency is decreased from 0.4643 to 0.4573.
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122
10-111E A combined gas-steam power cycle uses a simple gas turbine for the topping cycle and simple Rankine cycle for the bottoming cycle. The cycle supplies a specified rate of heat to the buildings during winter. The mass flow rate of air and the net power output from the cycle are to be determined.
Assumptions 1 Steady operating conditions exist. 2 The air-standard assumptions are applicable to Brayton cycle. 3 Kinetic and potential energy changes are negligible. 4 Air is an ideal gas with constant specific heats.
Properties The properties of air at room temperature are cp = 0.240 Btu/lbm·R and k = 1.4 (Table A-2Ea).
Analysis The mass flow rate of water is
lbm/h2495Btu/lbm161.25)-(962.8
Btu/h102 6
14
buildings =×
=−
=hh
Qmw
&&
The mass flow rate of air is then
lbm/h27,340===0.09126
249509126.0
aw
mm
&&
The power outputs from each cycle are
kW1062Btu/h3412.14
kW1R)540109914492560)(RBtu/lbm240.0(lbm/h)340,27(
)()(
)(
5687
inC,outT,cyclegasnet,
=
⎟⎠⎞
⎜⎝⎛+−−⋅=
−−−=
−=
TTcTTc
wwmW
pp
a&&
kW224Btu/h3412.14
kW1)43.28.9629.1270(lbm/h)2495(
)(
)(
inP,43
inP,outT,cyclesteamnet,
=
⎟⎠⎞
⎜⎝⎛−−=
−−=
−=
whhm
wwmW
a
a
&
&&
The net electricity production by this cycle is then
kW1286=+= 2241062netW&
1
2
4s s
T
3
800 psia
10 psia
6s
Qout·
Qin ·
5
9
8s
7
GAS CYCLE
STEAMCYCLE
2560 R
540 R
600°F
8 6
4
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123
10-112 A combined gas-steam power plant is considered. The topping cycle is an ideal gas-turbine cycle and the bottoming cycle is an ideal reheat Rankine cycle. The mass flow rate of air in the gas-turbine cycle, the rate of total heat input, and the thermal efficiency of the combined cycle are to be determined.
Assumptions 1 Steady operating conditions exist. 2 Kinetic and potential energy changes are negligible. 3Air is an ideal gas with variable specific heats.
Analysis (a) The analysis of gas cycle yields
( )( )
( )
kJ/kg63523 K520
kJ/kg35860356545081
5450kJ/kg421515 K1400
kJ/kg125268499231118
23111kJ/kg16290 K290
1111
109
10
99
87
8
77
910
9
78
7
.hT
.h..PPP
P
.P.hT
.h..PPP
P
.P.hT
rr
r
rr
r
=⎯→⎯=
=⎯→⎯=⎟⎠⎞
⎜⎝⎛==
==⎯→⎯=
=⎯→⎯===
==⎯→⎯=
From the steam tables (Tables A-4, A-5, and A-6),
( ) ( )( )
gwhh
PPw
hh
f
f
kJ/k95.20614.1581.191
kJ/kg15.14mkPa1
kJ1kPa1015,000/kgm 0.00101
/kgm 00101.0
kJ/kg81.191
inpI,12
33
121inpI,
3kPa10@1
kPa10@1
=+=+=
⎟ =⎟⎠
⎞⎜⎜⎝
⎛
⋅−=−=
==
==
v
vv
( )( )
( )( ) kJ/kg8.22921.23928783.081.191
8783.04996.7
6492.02355.7kPa10
KkJ/kg2359.7kJ/kg2.3457
C500MPa3
kJ/kg7.27819.17949880.03.1008
9880.05402.3
6454.21434.6MPa3
KkJ/kg1434.6kJ/kg9.3157
C450MPa15
66
66
56
6
5
5
5
5
44
44
34
4
3
3
3
3
=+=+=
=−
=−
=
⎭⎬⎫
==
⋅==
⎭⎬⎫
°==
=+=+=
=−
=−
=
⎭⎬⎫
==
⋅==
⎭⎬⎫
°==
fgf
fg
f
fgf
fg
f
hxhhs
ssx
ssP
sh
TP
hxhhs
ssx
ssP
sh
TP
Noting that 0ΔpeΔke ≅≅≅≅WQ && for the heat exchanger, the steady-flow energy balance equation yields
( ) ( )
( ) kg/s262.9=−−
=−−
=
−=−⎯→⎯=⎯→⎯= ∑∑kg/s30
63.52335.86095.2069.3157
1110
23air
1110air23outin
s
seeii
mhhhhm
hhmhhmhmhmEE
&&
&&&&&&
1
26
s
T
3
15 MPa
10 kPa
8
Qout·
Qin·
7
11
10
9
GAS CYCLE
STEAM CYCLE
1400 K
290 K
5
4
3 MPa
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124
(b) ( ) ( )( )( ) ( )( )
kW 102.80 5×≅=
−+−=−+−=+=
kW280,352kJ/kg7.27812.3457kg/s30kJ/kg12.52642.1515kg/s262.9
45reheat89airreheatairin hhmhhmQQQ &&&&&
(c) ( ) ( )( )( ) ( )( )
55.6%
,
=−=−=
=−+−=
−+−=+=
kW280,352kW124,409
11
kW409124kJ/kg81.1918.2292kg/s30kJ/kg16.29063.523kg/s262.9
in
outth
16711airsteamout,airout,out
hhmhhmQQQ s
&
&
&&&&&
η
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125
10-113 A combined gas-steam power plant is considered. The topping cycle is a gas-turbine cycle and the bottoming cycle is a nonideal reheat Rankine cycle. The mass flow rate of air in the gas-turbine cycle, the rate of total heat input, and the thermal efficiency of the combined cycle are to be determined.
Assumptions 1 Steady operating conditions exist. 2 Kinetic and potential energy changes are negligible. 3Air is an ideal gas with variable specific heats.
Analysis (a) The analysis of gas cycle yields (Table A-17)
( )( )
( )( ) ( )
( )
( )( )( )
kJ/kg63.523K520
kJ/kg4.95835.86042.151585.042.1515
kJ/kg35.8603.565.45081
5.450kJ/kg42.1515K1400
kJ/kg1.58580.0/16.29012.52616.290
/
kJ/kg12.526849.92311.18
2311.1kJ/kg16.290K290
1111
109910109
109
109
10
99
787878
78
87
8
77
910
9
78
7
=⎯→⎯=
=−−=
−η−=⎯→⎯−−
=η
=⎯→⎯=⎟⎠⎞
⎜⎝⎛==
==⎯→⎯=
=−+=
η−+=⎯→⎯−−
=η
=⎯→⎯===
==⎯→⎯=
hT
hhhhhhhh
hPP
PP
PhT
hhhhhhhh
hPPPP
PhT
sTs
T
srs
r
r
Css
C
srs
r
r
s
s
From the steam tables (Tables A-4, A-5, and A-6),
( )( )( )
kJ/kg95.20614.1581.191kJ/kg15.14
mkPa1kJ1kPa1015,000/kgm 0.00101
/kgm 00101.0kJ/kg81.191
inpI,12
33
121inpI,
3kPa10@1
kPa10@1
=+=+==
⎟⎟⎠
⎞⎜⎜⎝
⎛
⋅−=
−=
====
whh
PPw
hh
f
f
v
vv
( )( )
( )( )( )
kJ/kg1.28387.27819.315785.09.3157
kJ/kg7.27819.17949879.03.1008
9880.05402.3
6454.21434.6MPa3
KkJ/kg1428.6kJ/kg9.3157
C450MPa15
433443
43
44
44
34
4
3
3
3
3
=−−=
−η−=⎯→⎯−−
=η
=+=+=
=−
=−
=
⎭⎬⎫
==
⋅==
⎭⎬⎫
°==
sTs
T
fgsfs
fg
fss
s
hhhhhhhh
hxhhs
ssx
ssP
sh
TP
1
26s
s
T
3
15 MPa
10 kPa
8
Qout·
Qin·
7
11
10s
9
GAS CYCLE
STEAM CYCLE
1400 K
290 K
5
4
3 MPa
1
8
4
6
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126
( )( )
( )( )( )
kJ/kg5.24678.22922.345785.02.3457
kJ/kg8.22921.23928782.081.191
8783.04996.7
6492.02359.7kPa10
KkJ/kg2359.7kJ/kg2.3457
C500MPa3
655665
65
66
66
56
6
5
5
5
5
=−−=
−η−=⎯→⎯−−
=η
=+=+=
=−
=−
=
⎭⎬⎫
==
⋅==
⎭⎬⎫
°==
sTs
T
fgsfs
fg
fss
s
hhhhhhhh
hxhhs
ssx
ssP
sh
TP
Noting that 0ΔpeΔke ≅≅≅≅WQ && for the heat exchanger, the steady-flow energy balance equation yields
( ) ( )
( ) kg/s203.6=−−
=−−
=
−=−⎯→⎯=
=
=Δ=−
∑∑kg/s30
63.5234.95895.2069.3157
0
1110
23air
1110air23
outin
(steady)0systemoutin
s
seeii
mhhhhm
hhmhhmhmhm
EE
EEE
&&
&&&&
&&
&&&
(b) ( ) ( )( )( ) ( )( )
kW207,986=−+−=
−+−=+=kJ/kg1.28382.3457kg/s30kJ/kg1.58542.1515kg/s203.6
45reheat89airreheatairin hhmhhmQQQ &&&&&
(c) ( ) ( )( )( ) ( )( )
44.3%=−=−=
=−+−=
−+−=+=
kW207,986kW115,805
11
kW115,805kJ/kg81.1915.2467kg/s30kJ/kg16.29063.523kg/s203.6
in
outth
16711airsteamout,airout,out
hhmhhmQQQ s
&
&
&&&&&
η
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127
10-114 It is to be shown that the exergy destruction associated with a simple ideal Rankine cycle can be expressed as ( )thinqx ηη −= Carnotth,destroyed , where ηth is efficiency of the Rankine cycle and ηth, Carnot is the efficiency of the Carnot cycle operating between the same temperature limits.
Analysis The exergy destruction associated with a cycle is given on a unit mass basis as
∑=R
R
Tq
Tx 0destroyed
where the direction of qin is determined with respect to the reservoir (positive if to the reservoir and negative if from the reservoir). For a cycle that involves heat transfer only with a source at TH and a sink at T0, the irreversibility becomes
( ) ( )[ ] ( )thCthCthth
HHH
qqTT
qqqq
TTq
Tq
TqTx
ηηηη −=−−−=
⎟⎟⎠
⎞⎜⎜⎝
⎛−=−⎟ =⎟
⎠
⎞⎜⎜⎝
⎛−=
,in,in
0
in
outinin
0out
in
0
out0destroyed
11
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128
10-115 A cogeneration plant is to produce power and process heat. There are two turbines in the cycle: a high-pressure turbine and a low-pressure turbine. The temperature, pressure, and mass flow rate of steam at the inlet of high-pressure turbine are to be determined.
Assumptions 1 Steady operating conditions exist. 2 Kinetic and potential energy changes are negligible.
Analysis From the steam tables (Tables A-4, A-5, and A-6),
( )( )
( )( )( )
kJ/kg1.23446.20479.278860.09.2788
kJ/kg6.20471.23927758.081.191
7758.04996.7
6492.04675.6
kPa10
KkJ/kg4675.6kJ/kg9.2788
vaporsat.MPa4.1
544554
54
55
45
45
5
MPa4.1 @4
MPa4.1 @44
=−−=
−−=⎯→⎯−−
=
=+=+=
=−
=−
=
⎭⎬⎫
==
⋅====
⎭⎬⎫=
sTs
T
fgsfs
fg
fss
s
g
g
hhhhhhhh
hxhhs
ssx
ssP
sshhP
ηη
and
kg/min9.107kg/s799.1kJ/kg444.8kJ/s800
kJ/kg8.4441.23449.2788
lowturb,
IIturb,turblow
54lowturb,
====
=−=−=
wW
m
hhw&
&
Therefore ,
kJ/kg0.28439.278815.54
kJ/kg54.15kg/s18.47kJ/s1000
=kg/min11081081000
4highturb,3
43turbhigh,
,turbhighturb,
total
=+=+=
−====
=+=
hwh
hhmW
w
m
I
&
&
& kg/s18.47
( )( ) ( )
( )( ) KkJ/kg4289.61840.49908.02835.2
9908.09.1958
96.8298.2770MPa4.1
kJ/kg8.277075.0/9.27880.28430.2843
/
44
44
34
4
433443
43
⋅=+=+=
=−
=−
=
⎭⎬⎫
==
=−−=
−−=⎯→⎯−−
=
fgsfs
fg
fss
s
s
Tss
T
sxssh
hhx
ssP
hhhhhhhh
ηη
Then from the tables or the software, the turbine inlet temperature and pressure becomes
C227.5MPa2
°==
⎭⎬⎫
⋅==
3
3
3
3KkJ/kg4289.6
kJ/kg0.2843TP
sh
1
2
5s
T
5
44
3
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129
10-116 A cogeneration plant is to generate power and process heat. Part of the steam extracted from the turbine at a relatively high pressure is used for process heating. The rate of process heat, the net power produced, and the utilization factor of the plant are to be determined.
Assumptions 1 Steady operating conditions exist. 2 Kinetic and potential energy changes are negligible.
Analysis From the steam tables (Tables A-4, A-5, and A-6),
( )( )( )
kJ/kg47.908
kJ/kg71.25329.242.251kJ/kg.292
88.0/mkPa1
kJ1kPa022000/kgm 0.001017
/
/kgm 001017.0kJ/kg42.251
MPa2@3
inpI,12
33
121inpI,
3kPa02@1
kPa20@1
==
=+=+==
⎟⎟⎠
⎞⎜⎜⎝
⎛
⋅−=
−=
==
==
f
p
f
f
hh
whh
PPw
hh
ηv
vv
Mixing chamber:
kJ/kg81.491kg/s)11()kJ/kg)71kg/s)(253.411(kJ/kg)47kg/s)(908.4( 44
442233
=⎯→⎯=−+
=+
hh
hmhmhm &&&
( )( )( )
kJ/kg02.49921.781.491kJ/kg.217
88.0/mkPa1
kJ1kPa00208000/kgm 0.001058
/
/kgm 001058.0
inII,45
33
454inII,
3kJ/kg81.491@4
=+=+==
⎟⎟⎠
⎞⎜⎜⎝
⎛
⋅−=
−=
=≅ =
p
pp
hf
whh
PPw
f
ηv
vv
KkJ/kg7266.6kJ/kg5.3399
C500MPa8
6
6
6
6⋅=
=
⎭⎬⎫
°==
sh
TP
kJ/kg4.3000MPa2
767
7 =⎭⎬⎫
==
shss
P
6
8
1
5
Turbine Boiler
Conden.Process heater
P IP II 4 2
3
7
1
2
8s 8 s
T
6
7
8 MPa
20 kPa
2 MPa 4
5
3
Qout·
Qin ·
Qprocess·
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130
( ) ( )( ) kJ/kg3.30484.30005.339988.05.3399766776
76 =−−=−−=⎯→⎯−−
= sTs
T hhhhhhhh
ηη
kJ/kg5.2215kPa20
868
8 =⎭⎬⎫
==
shss
P
( ) ( )( ) kJ/kg6.23575.22155.339988.05.3399866886
86 =−−=−−=⎯→⎯−−
= sTs
T hhhhhhhh
ηη
Then,
( ) ( )( ) kW8559=−=−= kJ/kg47.9083.3048kg/s4377process hhmQ &&
(b) Cycle analysis:
( ) ( )( )( ) ( )( )
( )( ) ( )( )
kW8603=−=−=
=+=+=
=−+−=
−+−=
958698
kW95kJ/kg7.21kg/s11kJ/kg2.29kg/s7
kW8698kJ/kg6.23575.3399kg/s7kJ/kg3.30485.3399kg/s4
inp,outT,net
inpII,4inpI,1inp,
868767outT,
WWW
wmwmW
hhmhhmW
&&&
&&&
&&&
(c) Then,
and ( ) ( )( )
53.8%==+
=+
=
=−=−=
538.0905,31
85598603
kW905,3102.4995.3399kg/s11
in
processnet
565in
Q
QW
hhmQ
u &
&&
&&
ε
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131
10-117 A Rankine steam cycle modified for reheat, a closed feedwater heater, and an open feedwater heater is considered. The T-s diagram for the ideal cycle is to be sketched. The net power output of the cycle and the minimum flow rate of the cooling water required are to be determined.
Assumptions 1 Steady operating conditions exist. 2 Kinetic and potential energy changes are negligible.
Analysis (b) Using the data from the problem statement, the enthalpies at various states are
( )( )( )
( )( )( )
kJ/kg1.83414.4830kJ/kg14.4
mkPa1kJ1
kPa14005000/kgm 0.00115
/kgm 00115.0kJ/kg830
kJ/kg8.25241.14.251kJ/kg41.1
mkPa1kJ1
kPa201400/kgm 0.00102
/kgm 00102.0kJ/kg4.251
inpII,45
33
451inpII,
3kPa1400@4
kPa1400@4
inpI,12
33
121inpI,
3kPa20@1
kPa20@1
=+=+==
⎟⎟⎠
⎞⎜⎜⎝
⎛
⋅−=
−===
===+=+=
=
⎟⎟⎠
⎞⎜⎜⎝
⎛
⋅−=
−=
==
==
whh
PPw
hhwhh
PPw
hh
f
f
f
f
v
vv
v
vv
Also,
operation) valve(throttle kJ/kg532
1213
kPa245@123
hhhhh f
====
An energy balance on the open feedwater heater gives
)(1)1( 437 hhyyh =−+
where y is the fraction of steam extracted from the high-pressure turbine. Solving for y,
103905323400
532830
37
34 .hhhh
y =−−
=−−
=
An energy balance on the closed feedwater heater gives
123210 )1()1( zhhyhyzh +−=−+
where z is the fraction of steam extracted from the low-pressure turbine. Solving for z,
0954205323154
)8.252532)(1039.01())1(
1210
23 .hh
hhyz =
−−−
=−
−(−=
The heat input in the boiler is
kJ/kg3367)33493692)(1039.01()1.8343894())(1()( 8956in
=−−+−=−−+−= hhyhhq
The work output from the turbines is
1
7
2
9
s
T
5
6
245 kPa
20 kPa
1.4 MPa8
4
3
10
11
12
13
1.2 MPa
5 MPa
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132
kJ/kg1449)2620)(09542.01039.01()3154)(09542.0()3692)(1039.01(
)3349)(1039.01()3400)(1039.0(3894)1()1()1( 11109876outT,
=−−−−−+
−−−=
−−−−−+−−−= hzyzhhyhyyhhw
The net work output from the cycle is
kJ/kg144414.4)41.1)(1039.01(1449
)1( inPII,inPI,outT,net
=−−−=
−−−= wwyww
The net power output is
MW144.4====
kW144,400kJ/kg)kg/s)(1444100(netnet wmW &&
(c) The heat rejected from the condenser is
kJ/kg1923)4.251)(1039.01()532)(09542.0()2620)(09542.01039.01(
)1()1( 11311out
=−−+−−=
−−+−−= hyzhhzyq
The mass flow rate of cooling water will be minimum when the cooling water exit temperature is a maximum. That is,
C1.60kPa20@12, °=== satw TTT
Then an energy balance on the condenser gives
kg/s1311=−⋅
=−
=
−=
K)25K)(60.1kJ/kg18.4(kJ/kg)kg/s)(1923100(
)(
)(
w,1w,2wp,
out
w,1w,2wp,out
TTcqm
m
TTcmqm
w
w
&&
&&
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133
10-118 A Rankine steam cycle modified for reheat and three closed feedwater heaters is considered. The T-s diagram for the ideal cycle is to be sketched. The net power output of the cycle and the flow rate of the cooling water required are to be determined.
Assumptions 1 Steady operating conditions exist. 2 Kinetic and potential energy changes are negligible.
Analysis (b) Using the data from the problem statement, the enthalpies at various states are
( )( )( )
( )( )( )
kJ/kg4.56604.54.561kJ/kg04.5
mkPa1kJ1
kPa3005000/kgm 0.001073
/kgm 001073.0kJ/kg4.561
kJ/kg8.19604.58.191kJ/kg04.5
mkPa1kJ1
kPa105000/kgm 0.001010
/kgm 001010.0kJ/kg8.191
inpII,1617
33
161716inpII,
3kPa300@16
kPa300@16
inpI,12
33
121inpI,
3kPa10@1
kPa10@1
=+=+==
⎟⎟⎠
⎞⎜⎜⎝
⎛
⋅−=
−===
===+=+=
=
⎟⎟⎠
⎞⎜⎜⎝
⎛
⋅−=
−=
==
==
whh
PPw
hhwhh
PPw
hh
f
f
f
f
v
vv
v
vv
Also,
operation) valve(throttle operation) valve(throttle
kJ/kg7.747
kJ/kg4.561
kJ/kg4.384
1819
1415
kPa925@145
kPa300@164
kPa75@183
hhhh
hhh
hhh
hhh
f
f
f
==
===
===
===
Energy balances on three closed feedwater heaters give
155410 )1()1( yhhzyhzyyh +−−=−−+
16415311 )()1()1( hzyhzyyhhzyzh ++−−=+−−+
183212 )1()1( whhzyhzywh +−−=−−+
The enthalpies are known, and thus there are three unknowns (y, z, w) and three equations. Solving these equations using EES, we obtain
07063.005863.006335.0
===
wzy
The enthalpy at state 6 may be determined from an energy balance on mixing chamber:
kJ/kg6.725)4.566)(05863.006335.0()7.747)(05863.006335.01()()1( 1756
=++−−=++−−= hzyhzyh
The heat input in the boiler is
kJ/kg3246)36153687()6.7253900()()( 8967in
=−+−=−+−= hhhhq
1
8
2
9
s
T
5
7
11
3
10
12 18
13
4 1615
1417
19
6
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134
The work output from the turbines is
kJ/kg1448)2408)(07063.005863.006335.01()2716)(07063.0(
)3011)(05863.0()3330)(06335.0(368736153900)1( 13121110987outT,
=−−−−−
−−+−=
−−−−−−−+−= hwzywhzhyhhhhw
The net work output from the cycle is
kJ/kg1443)04.5)(05863.006335.0()04.5)(05863.006335.01(1448
)()1( inPII,inPI,outT,net
=+−−−−=
+−−−−= wzywzyww
The net power output is
MW144.3====
kW144,300kJ/kg)kg/s)(1443100(netnet wmW &&
(c) The heat rejected from the condenser is
kJ/kg1803)8.191)(05863.006335.01()4.384)(07063.0()2408)(07063.005863.006335.01(
)1()1( 11913out
=−−−+−−−=
−−−+−−−= hzywhhwzyq
Then an energy balance on the condenser gives
kg/s4313=⋅
=−
=
−=
K)K)(10kJ/kg18.4(kJ/kg)kg/s)(1803100(
)(
)(
w,1w,2wp,
out
w,1w,2wp,out
TTcqm
m
TTcmqm
w
w
&&
&&
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135
10-119 EES The effect of the condenser pressure on the performance a simple ideal Rankine cycle is to be investigated.
Analysis The problem is solved using EES, and the solution is given below.
function x4$(x4) "this function returns a string to indicate the state of steam at point 4" x4$='' if (x4>1) then x4$='(superheated)' if (x4<0) then x4$='(compressed)'
end
P[3] = 5000 [kPa] T[3] = 500 [C] "P[4] = 5 [kPa]" Eta_t = 1.0 "Turbine isentropic efficiency" Eta_p = 1.0 "Pump isentropic efficiency"
"Pump analysis" Fluid$='Steam_IAPWS' P[1] = P[4] P[2]=P[3] x[1]=0 "Sat'd liquid" h[1]=enthalpy(Fluid$,P=P[1],x=x[1]) v[1]=volume(Fluid$,P=P[1],x=x[1]) s[1]=entropy(Fluid$,P=P[1],x=x[1]) T[1]=temperature(Fluid$,P=P[1],x=x[1]) W_p_s=v[1]*(P[2]-P[1])"SSSF isentropic pump work assuming constant specific volume" W_p=W_p_s/Eta_p h[2]=h[1]+W_p "SSSF First Law for the pump" s[2]=entropy(Fluid$,P=P[2],h=h[2]) T[2]=temperature(Fluid$,P=P[2],h=h[2])
"Turbine analysis" h[3]=enthalpy(Fluid$,T=T[3],P=P[3]) s[3]=entropy(Fluid$,T=T[3],P=P[3]) s_s[4]=s[3] hs[4]=enthalpy(Fluid$,s=s_s[4],P=P[4]) Ts[4]=temperature(Fluid$,s=s_s[4],P=P[4]) Eta_t=(h[3]-h[4])/(h[3]-hs[4])"Definition of turbine efficiency" T[4]=temperature(Fluid$,P=P[4],h=h[4]) s[4]=entropy(Fluid$,h=h[4],P=P[4]) x[4]=quality(Fluid$,h=h[4],P=P[4]) h[3] =W_t+h[4]"SSSF First Law for the turbine" x4s$=x4$(x[4])
"Boiler analysis" Q_in + h[2]=h[3]"SSSF First Law for the Boiler"
"Condenser analysis" h[4]=Q_out+h[1]"SSSF First Law for the Condenser"
"Cycle Statistics" W_net=W_t-W_p Eta_th=W_net/Q_in
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136
ηth P4 [kPa]
Wnet [kJ/kg]
x4 Qin [kJ/kg]
Qout [kJ/kg]
0.3956 5 1302 0.8212 3292 1990 0.3646 15 1168 0.8581 3204 2036 0.3484 25 1100 0.8772 3158 2057 0.3371 35 1054 0.8905 3125 2072 0.3283 45 1018 0.9009 3100 2082 0.321 55 988.3 0.9096 3079 2091
0.3147 65 963.2 0.917 3061 2098 0.3092 75 941.5 0.9235 3045 2104 0.3042 85 922.1 0.9293 3031 2109 0.2976 100 896.5 0.9371 3012 2116
0 2 4 6 8 10 120
100
200
300
400
500
600
700
s [kJ/kg-K]
T [°
C]
5000 kPa
5 kPa
Steam
1
3
4
2
0 20 40 60 80 100
800
900
1000
1100
1200
1300
1400
P[4] [kPa]
Wne
t[k
J/kg
]
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137
0 20 40 60 80 1000.28
0.3
0.32
0.34
0.36
0.38
0.4
P[4] [kPa]
ηth
0 20 40 60 80 1000.82
0.84
0.86
0.88
0.9
0.92
0.94
P[4] [kPa]
x[4]
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10-120 EES The effect of superheating the steam on the performance a simple ideal Rankine cycle is to be investigated.
Analysis The problem is solved using EES, and the solution is given below.
function x4$(x4) "this function returns a string to indicate the state of steam at point 4" x4$='' if (x4>1) then x4$='(superheated)' if (x4<0) then x4$='(compressed)'
end
P[3] = 3000 [kPa] {T[3] = 600 [C]} P[4] = 10 [kPa] Eta_t = 1.0 "Turbine isentropic efficiency" Eta_p = 1.0 "Pump isentropic efficiency"
"Pump analysis" Fluid$='Steam_IAPWS' P[1] = P[4] P[2]=P[3] x[1]=0 "Sat'd liquid" h[1]=enthalpy(Fluid$,P=P[1],x=x[1]) v[1]=volume(Fluid$,P=P[1],x=x[1]) s[1]=entropy(Fluid$,P=P[1],x=x[1]) T[1]=temperature(Fluid$,P=P[1],x=x[1]) W_p_s=v[1]*(P[2]-P[1])"SSSF isentropic pump work assuming constant specific volume" W_p=W_p_s/Eta_p h[2]=h[1]+W_p "SSSF First Law for the pump" s[2]=entropy(Fluid$,P=P[2],h=h[2]) T[2]=temperature(Fluid$,P=P[2],h=h[2])
"Turbine analysis" h[3]=enthalpy(Fluid$,T=T[3],P=P[3]) s[3]=entropy(Fluid$,T=T[3],P=P[3]) s_s[4]=s[3] hs[4]=enthalpy(Fluid$,s=s_s[4],P=P[4]) Ts[4]=temperature(Fluid$,s=s_s[4],P=P[4]) Eta_t=(h[3]-h[4])/(h[3]-hs[4])"Definition of turbine efficiency" T[4]=temperature(Fluid$,P=P[4],h=h[4]) s[4]=entropy(Fluid$,h=h[4],P=P[4]) x[4]=quality(Fluid$,h=h[4],P=P[4]) h[3] =W_t+h[4]"SSSF First Law for the turbine" x4s$=x4$(x[4])
"Boiler analysis" Q_in + h[2]=h[3]"SSSF First Law for the Boiler"
"Condenser analysis" h[4]=Q_out+h[1]"SSSF First Law for the Condenser"
"Cycle Statistics" W_net=W_t-W_p Eta_th=W_net/Q_in
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T3 [C]
ηth Wnet [kJ/kg]
x4
250 0.3241 862.8 0.752 344.4 0.3338 970.6 0.81 438.9 0.3466 1083 0.8536533.3 0.3614 1206 0.8909627.8 0.3774 1340 0.9244722.2 0.3939 1485 0.955 816.7 0.4106 1639 0.9835911.1 0.4272 1803 100 1006 0.4424 1970 100 1100 0.456 2139 100
200 300 400 500 600 700 800 900 1000 11000.32
0.34
0.36
0.38
0.4
0.42
0.44
0.46
T[3] [C]
ηth
200 300 400 500 600 700 800 900 1000 1100750
1050
1350
1650
1950
2250
T[3] [C]
Wne
t[k
J/kg
]
0 2 4 6 8 10 120
100
200
300
400
500
600
700
s [kJ/kg-K]T
[°C
]
3000 kPa
10 kPa
Steam
1
2
3
4
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140
10-121 EES The effect of number of reheat stages on the performance an ideal Rankine cycle is to be investigated.
Analysis The problem is solved using EES, and the solution is given below.
function x6$(x6) "this function returns a string to indicate the state of steam at point 6" x6$='' if (x6>1) then x6$='(superheated)' if (x6<0) then x6$='(subcooled)'
end
Procedure Reheat(P[3],T[3],T[5],h[4],NoRHStages,Pratio,Eta_t:Q_in_reheat,W_t_lp,h6) P3=P[3] T5=T[5] h4=h[4] Q_in_reheat =0 W_t_lp = 0 R_P=(1/Pratio)^(1/(NoRHStages+1))
imax:=NoRHStages - 1 i:=0 REPEAT i:=i+1
P4 = P3*R_P
P5=P4 P6=P5*R_P
Fluid$='Steam_IAPWS' s5=entropy(Fluid$,T=T5,P=P5) h5=enthalpy(Fluid$,T=T5,P=P5) s_s6=s5 hs6=enthalpy(Fluid$,s=s_s6,P=P6) Ts6=temperature(Fluid$,s=s_s6,P=P6) vs6=volume(Fluid$,s=s_s6,P=P6) "Eta_t=(h5-h6)/(h5-hs6)""Definition of turbine efficiency" h6=h5-Eta_t*(h5-hs6) W_t_lp=W_t_lp+h5-h6"SSSF First Law for the low pressure turbine" x6=QUALITY(Fluid$,h=h6,P=P6) Q_in_reheat =Q_in_reheat + (h5 - h4) P3=P4
UNTIL (i>imax)
END
"NoRHStages = 2" P[6] = 10"kPa" P[3] = 15000"kPa" P_extract = P[6] "Select a lower limit on the reheat pressure" T[3] = 500"C" T[5] = 500"C" Eta_t = 1.0 "Turbine isentropic efficiency" Eta_p = 1.0 "Pump isentropic efficiency"
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141
Pratio = P[3]/P_extract P[4] = P[3]*(1/Pratio)^(1/(NoRHStages+1))"kPa"
Fluid$='Steam_IAPWS'
"Pump analysis" P[1] = P[6] P[2]=P[3] x[1]=0 "Sat'd liquid" h[1]=enthalpy(Fluid$,P=P[1],x=x[1]) v[1]=volume(Fluid$,P=P[1],x=x[1]) s[1]=entropy(Fluid$,P=P[1],x=x[1]) T[1]=temperature(Fluid$,P=P[1],x=x[1]) W_p_s=v[1]*(P[2]-P[1])"SSSF isentropic pump work assuming constant specific volume" W_p=W_p_s/Eta_p h[2]=h[1]+W_p "SSSF First Law for the pump" v[2]=volume(Fluid$,P=P[2],h=h[2]) s[2]=entropy(Fluid$,P=P[2],h=h[2]) T[2]=temperature(Fluid$,P=P[2],h=h[2])
"High Pressure Turbine analysis" h[3]=enthalpy(Fluid$,T=T[3],P=P[3]) s[3]=entropy(Fluid$,T=T[3],P=P[3]) v[3]=volume(Fluid$,T=T[3],P=P[3]) s_s[4]=s[3] hs[4]=enthalpy(Fluid$,s=s_s[4],P=P[4]) Ts[4]=temperature(Fluid$,s=s_s[4],P=P[4]) Eta_t=(h[3]-h[4])/(h[3]-hs[4])"Definition of turbine efficiency" T[4]=temperature(Fluid$,P=P[4],h=h[4]) s[4]=entropy(Fluid$,h=h[4],P=P[4]) v[4]=volume(Fluid$,s=s[4],P=P[4]) h[3] =W_t_hp+h[4]"SSSF First Law for the high pressure turbine"
"Low Pressure Turbine analysis" Call Reheat(P[3],T[3],T[5],h[4],NoRHStages,Pratio,Eta_t:Q_in_reheat,W_t_lp,h6) h[6]=h6
{P[5]=P[4] s[5]=entropy(Fluid$,T=T[5],P=P[5]) h[5]=enthalpy(Fluid$,T=T[5],P=P[5]) s_s[6]=s[5] hs[6]=enthalpy(Fluid$,s=s_s[6],P=P[6]) Ts[6]=temperature(Fluid$,s=s_s[6],P=P[6]) vs[6]=volume(Fluid$,s=s_s[6],P=P[6]) Eta_t=(h[5]-h[6])/(h[5]-hs[6])"Definition of turbine efficiency" h[5]=W_t_lp+h[6]"SSSF First Law for the low pressure turbine" x[6]=QUALITY(Fluid$,h=h[6],P=P[6]) W_t_lp_total = NoRHStages*W_t_lp Q_in_reheat = NoRHStages*(h[5] - h[4])}
"Boiler analysis" Q_in_boiler + h[2]=h[3]"SSSF First Law for the Boiler" Q_in = Q_in_boiler+Q_in_reheat "Condenser analysis" h[6]=Q_out+h[1]"SSSF First Law for the Condenser" T[6]=temperature(Fluid$,h=h[6],P=P[6]) s[6]=entropy(Fluid$,h=h[6],P=P[6])
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x[6]=QUALITY(Fluid$,h=h[6],P=P[6]) x6s$=x6$(x[6]) "Cycle Statistics" W_net=W_t_hp+W_t_lp - W_p Eta_th=W_net/Q_in
ηth NoRH Stages
Qin [kJ/kg]
Wnet [kJ/kg]
0.4097 1 4085 1674 0.4122 2 4628 1908 0.4085 3 5020 2051 0.4018 4 5333 2143 0.3941 5 5600 2207 0.386 6 5838 2253
0.3779 7 6058 2289 0.3699 8 6264 2317 0.3621 9 6461 2340 0.3546 10 6651 2358
1 2 3 4 5 6 7 8 9 100.35
0.36
0.37
0.38
0.39
0.4
0.41
0.42
NoRHStages
ηth
1 2 3 4 5 6 7 8 9 104000
4500
5000
5500
6000
6500
7000
NoRHStages
Qin
[kJ/
kg]
1 2 3 4 5 6 7 8 9 101600
1700
1800
1900
2000
2100
2200
2300
2400
NoRHStages
Wne
t[k
J/kg
]
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143
10-122 EES The effect of number of regeneration stages on the performance an ideal regenerative Rankine cycle with one open feedwater heater is to be investigated.
Analysis The problem is solved using EES, and the solution is given below.
Procedure Reheat(NoFwh,T[5],P[5],P_cond,Eta_turb,Eta_pump:q_in,w_net)
Fluid$='Steam_IAPWS' Tcond = temperature(Fluid$,P=P_cond,x=0) Tboiler = temperature(Fluid$,P=P[5],x=0) P[7] = P_cond s[5]=entropy(Fluid$, T=T[5], P=P[5]) h[5]=enthalpy(Fluid$, T=T[5], P=P[5]) h[1]=enthalpy(Fluid$, P=P[7],x=0)
P4[1] = P[5] "NOTICE THIS IS P4[i] WITH i = 1"
DELTAT_cond_boiler = Tboiler - Tcond
If NoFWH = 0 Then
"the following are h7, h2, w_net, and q_in for zero feedwater heaters, NoFWH = 0" h7=enthalpy(Fluid$, s=s[5],P=P[7]) h2=h[1]+volume(Fluid$, P=P[7],x=0)*(P[5] - P[7])/Eta_pump w_net = Eta_turb*(h[5]-h7)-(h2-h[1]) q_in = h[5] - h2
else
i=0 REPEAT i=i+1 "The following maintains the same temperature difference between any two regeneration stages." T_FWH[i] = (NoFWH +1 - i)*DELTAT_cond_boiler/(NoFWH + 1)+Tcond"[C]" P_extract[i] = pressure(Fluid$,T=T_FWH[i],x=0)"[kPa]" P3[i]=P_extract[i] P6[i]=P_extract[i] If i > 1 then P4[i] = P6[i - 1]
UNTIL i=NoFWH
P4[NoFWH+1]=P6[NoFWH] h4[NoFWH+1]=h[1]+volume(Fluid$, P=P[7],x=0)*(P4[NoFWH+1] - P[7])/Eta_pump
i=0 REPEAT i=i+1
"Boiler condensate pump or the Pumps 2 between feedwater heaters analysis" h3[i]=enthalpy(Fluid$,P=P3[i],x=0) v3[i]=volume(Fluid$,P=P3[i],x=0) w_pump2_s=v3[i]*(P4[i]-P3[i])"SSSF isentropic pump work assuming constant specific volume" w_pump2[i]=w_pump2_s/Eta_pump "Definition of pump efficiency" h4[i]= w_pump2[i] +h3[i] "Steady-flow conservation of energy" s4[i]=entropy(Fluid$,P=P4[i],h=h4[i]) T4[i]=temperature(Fluid$,P=P4[i],h=h4[i])
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144
Until i = NoFWH i=0 REPEAT i=i+1 "Open Feedwater Heater analysis:" {h2[i] = h6[i]} s5[i] = s[5] ss6[i]=s5[i] hs6[i]=enthalpy(Fluid$,s=ss6[i],P=P6[i]) Ts6[i]=temperature(Fluid$,s=ss6[i],P=P6[i]) h6[i]=h[5]-Eta_turb*(h[5]-hs6[i])"Definition of turbine efficiency for high pressure stages" If i=1 then y[1]=(h3[1] - h4[2])/(h6[1] - h4[2]) "Steady-flow conservation of energy for the FWH" If i > 1 then js = i -1 j = 0 sumyj = 0 REPEAT j = j+1 sumyj = sumyj + y[ j ] UNTIL j = js y[i] =(1- sumyj)*(h3[i] - h4[i+1])/(h6[i] - h4[i+1])
ENDIF T3[i]=temperature(Fluid$,P=P3[i],x=0) "Condensate leaves heater as sat. liquid at P[3]" s3[i]=entropy(Fluid$,P=P3[i],x=0)
"Turbine analysis" T6[i]=temperature(Fluid$,P=P6[i],h=h6[i]) s6[i]=entropy(Fluid$,P=P6[i],h=h6[i]) yh6[i] = y[i]*h6[i] UNTIL i=NoFWH ss[7]=s6[i] hs[7]=enthalpy(Fluid$,s=ss[7],P=P[7]) Ts[7]=temperature(Fluid$,s=ss[7],P=P[7]) h[7]=h6[i]-Eta_turb*(h6[i]-hs[7])"Definition of turbine efficiency for low pressure stages" T[7]=temperature(Fluid$,P=P[7],h=h[7]) s[7]=entropy(Fluid$,P=P[7],h=h[7])
sumyi = 0 sumyh6i = 0 wp2i = W_pump2[1] i=0 REPEAT i=i+1 sumyi = sumyi + y[i] sumyh6i = sumyh6i + yh6[i] If NoFWH > 1 then wp2i = wp2i + (1- sumyi)*W_pump2[i] UNTIL i = NoFWH
"Condenser Pump---Pump_1 Analysis:" P[2] = P6 [ NoFWH] P[1] = P_cond h[1]=enthalpy(Fluid$,P=P[1],x=0) {Sat'd liquid} v1=volume(Fluid$,P=P[1],x=0) s[1]=entropy(Fluid$,P=P[1],x=0) T[1]=temperature(Fluid$,P=P[1],x=0) w_pump1_s=v1*(P[2]-P[1])"SSSF isentropic pump work assuming constant specific volume"
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w_pump1=w_pump1_s/Eta_pump "Definition of pump efficiency" h[2]=w_pump1+ h[1] "Steady-flow conservation of energy" s[2]=entropy(Fluid$,P=P[2],h=h[2]) T[2]=temperature(Fluid$,P=P[2],h=h[2])
"Boiler analysis" q_in = h[5] - h4[1]"SSSF conservation of energy for the Boiler" w_turb = h[5] - sumyh6i - (1- sumyi)*h[7] "SSSF conservation of energy for turbine"
"Condenser analysis" q_out=(1- sumyi)*(h[7] - h[1])"SSSF First Law for the Condenser"
"Cycle Statistics" w_net=w_turb - ((1- sumyi)*w_pump1+ wp2i)
endif END
"Input Data" NoFWH = 2 P[5] = 15000 [kPa] T[5] = 600 [C] P_cond=5 [kPa] Eta_turb= 1.0 "Turbine isentropic efficiency" Eta_pump = 1.0 "Pump isentropic efficiency" P[1] = P_cond P[4] = P[5]
"Condenser exit pump or Pump 1 analysis" Call Reheat(NoFwh,T[5],P[5],P_cond,Eta_turb,Eta_pump:q_in,w_net) Eta_th=w_net/q_in
No FWH
ηth wnet [kJ/kg]
qin [kJ/kg]
0 0.4466 1532 3430 1 0.4806 1332 2771 2 0.4902 1243 2536 3 0.4983 1202 2411 4 0.5036 1175 2333 5 0.5073 1157 2280 6 0.5101 1143 2240 7 0.5123 1132 2210 8 0.5141 1124 2186 9 0.5155 1117 2167
10 0.5167 1111 2151 0 2 4 6 8 10 120
100
200
300
400
500
600
700
s [kJ/kg-K]
T [°
C]
6000 kPa
400 kPa
10 kPa
Steam
1
2 3
4
5
6
7
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0 2 4 6 8 100.44
0.45
0.46
0.47
0.48
0.49
0.5
0.51
0.52
NoFwh
ηth
0 2 4 6 8 101100
1150
1200
1250
1300
1350
1400
1450
1500
1550
NoFwh
wne
t[k
J/kg
]
0 2 4 6 8 102000
2200
2400
2600
2800
3000
3200
3400
3600
NoFwh
q in
[kJ/
kg]
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147
Fundamentals of Engineering (FE) Exam Problems
10-123 Consider a steady-flow Carnot cycle with water as the working fluid executed under the saturation dome between the pressure limits of 8 MPa and 20 kPa. Water changes from saturated liquid to saturated vapor during the heat addition process. The net work output of this cycle is
(a) 494 kJ/kg (b) 975 kJ/kg (c) 596 kJ/kg (d) 845 kJ/kg (e) 1148 kJ/kg
Answer (c) 596 kJ/kg
Solution Solved by EES Software. Solutions can be verified by copying-and-pasting the following lines on a blank EES screen. (Similar problems and their solutions can be obtained easily by modifying numerical values).
P1=8000 "kPa" P2=20 "kPa" h_fg=ENTHALPY(Steam_IAPWS,x=1,P=P1)-ENTHALPY(Steam_IAPWS,x=0,P=P1) T1=TEMPERATURE(Steam_IAPWS,x=0,P=P1)+273 T2=TEMPERATURE(Steam_IAPWS,x=0,P=P2)+273 q_in=h_fg Eta_Carnot=1-T2/T1 w_net=Eta_Carnot*q_in
"Some Wrong Solutions with Common Mistakes:" W1_work = Eta1*q_in; Eta1=T2/T1 "Taking Carnot efficiency to be T2/T1" W2_work = Eta2*q_in; Eta2=1-(T2-273)/(T1-273) "Using C instead of K" W3_work = Eta_Carnot*ENTHALPY(Steam_IAPWS,x=1,P=P1) "Using h_g instead of h_fg" W4_work = Eta_Carnot*q2; q2=ENTHALPY(Steam_IAPWS,x=1,P=P2)-ENTHALPY(Steam_IAPWS,x=0,P=P2) "Using h_fg at P2"
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10-124 A simple ideal Rankine cycle operates between the pressure limits of 10 kPa and 3 MPa, with a turbine inlet temperature of 600°C. Disregarding the pump work, the cycle efficiency is
(a) 24% (b) 37% (c) 52% (d) 63% (e) 71%
Answer (b) 37%
Solution Solved by EES Software. Solutions can be verified by copying-and-pasting the following lines on a blank EES screen. (Similar problems and their solutions can be obtained easily by modifying numerical values).
P1=10 "kPa" P2=3000 "kPa" P3=P2 P4=P1 T3=600 "C" s4=s3 h1=ENTHALPY(Steam_IAPWS,x=0,P=P1) v1=VOLUME(Steam_IAPWS,x=0,P=P1) w_pump=v1*(P2-P1) "kJ/kg" h2=h1+w_pump h3=ENTHALPY(Steam_IAPWS,T=T3,P=P3) s3=ENTROPY(Steam_IAPWS,T=T3,P=P3) h4=ENTHALPY(Steam_IAPWS,s=s4,P=P4) q_in=h3-h2 q_out=h4-h1 Eta_th=1-q_out/q_in
"Some Wrong Solutions with Common Mistakes:" W1_Eff = q_out/q_in "Using wrong relation" W2_Eff = 1-(h44-h1)/(h3-h2); h44 = ENTHALPY(Steam_IAPWS,x=1,P=P4) "Using h_g for h4" W3_Eff = 1-(T1+273)/(T3+273); T1=TEMPERATURE(Steam_IAPWS,x=0,P=P1) "Using Carnot efficiency" W4_Eff = (h3-h4)/q_in "Disregarding pump work"
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10-125 A simple ideal Rankine cycle operates between the pressure limits of 10 kPa and 5 MPa, with a turbine inlet temperature of 600°C. The mass fraction of steam that condenses at the turbine exit is
(a) 6% (b) 9% (c) 12% (d) 15% (e) 18%
Answer (c) 12%
Solution Solved by EES Software. Solutions can be verified by copying-and-pasting the following lines on a blank EES screen. (Similar problems and their solutions can be obtained easily by modifying numerical values).
P1=10 "kPa" P2=5000 "kPa" P3=P2 P4=P1 T3=600 "C" s4=s3 h3=ENTHALPY(Steam_IAPWS,T=T3,P=P3) s3=ENTROPY(Steam_IAPWS,T=T3,P=P3) h4=ENTHALPY(Steam_IAPWS,s=s4,P=P4) x4=QUALITY(Steam_IAPWS,s=s4,P=P4) moisture=1-x4
"Some Wrong Solutions with Common Mistakes:" W1_moisture = x4 "Taking quality as moisture" W2_moisture = 0 "Assuming superheated vapor"
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150
10-126 A steam power plant operates on the simple ideal Rankine cycle between the pressure limits of 10 kPa and 10 MPa, with a turbine inlet temperature of 600°C. The rate of heat transfer in the boiler is 800 kJ/s. Disregarding the pump work, the power output of this plant is
(a) 243 kW (b) 284 kW (c) 508 kW (d) 335 kW (e) 800 kW
Answer (d) 335 kW
Solution Solved by EES Software. Solutions can be verified by copying-and-pasting the following lines on a blank EES screen. (Similar problems and their solutions can be obtained easily by modifying numerical values).
P1=10 "kPa" P2=10000 "kPa" P3=P2 P4=P1 T3=600 "C" s4=s3 Q_rate=800 "kJ/s" m=Q_rate/q_in h1=ENTHALPY(Steam_IAPWS,x=0,P=P1) h2=h1 "pump work is neglected" "v1=VOLUME(Steam_IAPWS,x=0,P=P1) w_pump=v1*(P2-P1) h2=h1+w_pump" h3=ENTHALPY(Steam_IAPWS,T=T3,P=P3) s3=ENTROPY(Steam_IAPWS,T=T3,P=P3) h4=ENTHALPY(Steam_IAPWS,s=s4,P=P4) q_in=h3-h2 W_turb=m*(h3-h4)
"Some Wrong Solutions with Common Mistakes:" W1_power = Q_rate "Assuming all heat is converted to power" W3_power = Q_rate*Carnot; Carnot = 1-(T1+273)/(T3+273); T1=TEMPERATURE(Steam_IAPWS,x=0,P=P1) "Using Carnot efficiency" W4_power = m*(h3-h44); h44 = ENTHALPY(Steam_IAPWS,x=1,P=P4) "Taking h4=h_g"
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151
10-127 Consider a combined gas-steam power plant. Water for the steam cycle is heated in a well-insulated heat exchanger by the exhaust gases that enter at 800 K at a rate of 60 kg/s and leave at 400 K. Water enters the heat exchanger at 200°C and 8 MPa and leaves at 350°C and 8 MPa. If the exhaust gases are treated as air with constant specific heats at room temperature, the mass flow rate of water through the heat exchanger becomes
(a) 11 kg/s (b) 24 kg/s (c) 46 kg/s (d) 53 kg/s (e) 60 kg/s
Answer (a) 11 kg/s
Solution Solved by EES Software. Solutions can be verified by copying-and-pasting the following lines on a blank EES screen. (Similar problems and their solutions can be obtained easily by modifying numerical values).
m_gas=60 "kg/s" Cp=1.005 "kJ/kg.K" T3=800 "K" T4=400 "K" Q_gas=m_gas*Cp*(T3-T4) P1=8000 "kPa" T1=200 "C" P2=8000 "kPa" T2=350 "C" h1=ENTHALPY(Steam_IAPWS,T=T1,P=P1) h2=ENTHALPY(Steam_IAPWS,T=T2,P=P2) Q_steam=m_steam*(h2-h1) Q_gas=Q_steam
"Some Wrong Solutions with Common Mistakes:" m_gas*Cp*(T3 -T4)=W1_msteam*4.18*(T2-T1) "Assuming no evaporation of liquid water" m_gas*Cv*(T3 -T4)=W2_msteam*(h2-h1); Cv=0.718 "Using Cv for air instead of Cp" W3_msteam = m_gas "Taking the mass flow rates of two fluids to be equal" m_gas*Cp*(T3 -T4)=W4_msteam*(h2-h11); h11=ENTHALPY(Steam_IAPWS,x=0,P=P1) "Taking h1=hf@P1"
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PROPRIETARY MATERIAL. © 2008 The McGraw-Hill Companies, Inc. Limited distribution permitted only to teachers and educators for course preparation. If you are a student using this Manual, you are using it without permission.
152
10-128 An ideal reheat Rankine cycle operates between the pressure limits of 10 kPa and 8 MPa, with reheat occurring at 4 MPa. The temperature of steam at the inlets of both turbines is 500°C, and the enthalpy of steam is 3185 kJ/kg at the exit of the high-pressure turbine, and 2247 kJ/kg at the exit of the low-pressure turbine. Disregarding the pump work, the cycle efficiency is
(a) 29% (b) 32% (c) 36% (d) 41% (e) 49%
Answer (d) 41%
Solution Solved by EES Software. Solutions can be verified by copying-and-pasting the following lines on a blank EES screen. (Similar problems and their solutions can be obtained easily by modifying numerical values).
P1=10 "kPa" P2=8000 "kPa" P3=P2 P4=4000 "kPa" P5=P4 P6=P1 T3=500 "C" T5=500 "C" s4=s3 s6=s5 h1=ENTHALPY(Steam_IAPWS,x=0,P=P1) h2=h1 h44=3185 "kJ/kg - for checking given data" h66=2247 "kJ/kg - for checking given data" h3=ENTHALPY(Steam_IAPWS,T=T3,P=P3) s3=ENTROPY(Steam_IAPWS,T=T3,P=P3) h4=ENTHALPY(Steam_IAPWS,s=s4,P=P4) h5=ENTHALPY(Steam_IAPWS,T=T5,P=P5) s5=ENTROPY(Steam_IAPWS,T=T5,P=P5) h6=ENTHALPY(Steam_IAPWS,s=s6,P=P6) q_in=(h3-h2)+(h5-h4) q_out=h6-h1 Eta_th=1-q_out/q_in
"Some Wrong Solutions with Common Mistakes:" W1_Eff = q_out/q_in "Using wrong relation" W2_Eff = 1-q_out/(h3-h2) "Disregarding heat input during reheat" W3_Eff = 1-(T1+273)/(T3+273); T1=TEMPERATURE(Steam_IAPWS,x=0,P=P1) "Using Carnot efficiency" W4_Eff = 1-q_out/(h5-h2) "Using wrong relation for q_in"
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PROPRIETARY MATERIAL. © 2008 The McGraw-Hill Companies, Inc. Limited distribution permitted only to teachers and educators for course preparation. If you are a student using this Manual, you are using it without permission.
153
10-129 Pressurized feedwater in a steam power plant is to be heated in an ideal open feedwater heater that operates at a pressure of 0.5 MPa with steam extracted from the turbine. If the enthalpy of feedwater is 252 kJ/kg and the enthalpy of extracted steam is 2665 kJ/kg, the mass fraction of steam extracted from the turbine is
(a) 4% (b) 10% (c) 16% (d) 27% (e) 12%
Answer (c) 16%
Solution Solved by EES Software. Solutions can be verified by copying-and-pasting the following lines on a blank EES screen. (Similar problems and their solutions can be obtained easily by modifying numerical values).
h_feed=252 "kJ/kg" h_extracted=2665 "kJ/kg" P3=500 "kPa" h3=ENTHALPY(Steam_IAPWS,x=0,P=P3) "Energy balance on the FWH" h3=x_ext*h_extracted+(1-x_ext)*h_feed
"Some Wrong Solutions with Common Mistakes:" W1_ext = h_feed/h_extracted "Using wrong relation" W2_ext = h3/(h_extracted-h_feed) "Using wrong relation" W3_ext = h_feed/(h_extracted-h_feed) "Using wrong relation"
10-130 Consider a steam power plant that operates on the regenerative Rankine cycle with one open feedwater heater. The enthalpy of the steam is 3374 kJ/kg at the turbine inlet, 2797 kJ/kg at the location of bleeding, and 2346 kJ/kg at the turbine exit. The net power output of the plant is 120 MW, and the fraction of steam bled off the turbine for regeneration is 0.172. If the pump work is negligible, the mass flow rate of steam at the turbine inlet is
(a) 117 kg/s (b) 126 kg/s (c) 219 kg/s (d) 288 kg/s (e) 679 kg/s
Answer (b) 126 kg/s
Solution Solved by EES Software. Solutions can be verified by copying-and-pasting the following lines on a blank EES screen. (Similar problems and their solutions can be obtained easily by modifying numerical values).
h_in=3374 "kJ/kg" h_out=2346 "kJ/kg" h_extracted=2797 "kJ/kg" Wnet_out=120000 "kW" x_bleed=0.172 w_turb=(h_in-h_extracted)+(1-x_bleed)*(h_extracted-h_out) m=Wnet_out/w_turb "Some Wrong Solutions with Common Mistakes:" W1_mass = Wnet_out/(h_in-h_out) "Disregarding extraction of steam" W2_mass = Wnet_out/(x_bleed*(h_in-h_out)) "Assuming steam is extracted at trubine inlet" W3_mass = Wnet_out/(h_in-h_out-x_bleed*h_extracted) "Using wrong relation"
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PROPRIETARY MATERIAL. © 2008 The McGraw-Hill Companies, Inc. Limited distribution permitted only to teachers and educators for course preparation. If you are a student using this Manual, you are using it without permission.
154
10-131 Consider a simple ideal Rankine cycle. If the condenser pressure is lowered while keeping turbine inlet state the same, (select the correct statement)
(a) the turbine work output will decrease. (b) the amount of heat rejected will decrease. (c) the cycle efficiency will decrease. (d) the moisture content at turbine exit will decrease. (e) the pump work input will decrease.
Answer (b) the amount of heat rejected will decrease.
10-132 Consider a simple ideal Rankine cycle with fixed boiler and condenser pressures. If the steam is superheated to a higher temperature, (select the correct statement)
(a) the turbine work output will decrease. (b) the amount of heat rejected will decrease. (c) the cycle efficiency will decrease. (d) the moisture content at turbine exit will decrease. (e) the amount of heat input will decrease.
Answer (d) the moisture content at turbine exit will decrease.
10-133 Consider a simple ideal Rankine cycle with fixed boiler and condenser pressures . If the cycle is modified with reheating, (select the correct statement)
(a) the turbine work output will decrease. (b) the amount of heat rejected will decrease. (c) the pump work input will decrease. (d) the moisture content at turbine exit will decrease. (e) the amount of heat input will decrease.
Answer (d) the moisture content at turbine exit will decrease.
10-134 Consider a simple ideal Rankine cycle with fixed boiler and condenser pressures . If the cycle is modified with regeneration that involves one open feed water heater, (select the correct statement per unit mass of steam flowing through the boiler)
(a) the turbine work output will decrease. (b) the amount of heat rejected will increase. (c) the cycle thermal efficiency will decrease. (d) the quality of steam at turbine exit will decrease. (e) the amount of heat input will increase.
Answer (a) the turbine work output will decrease.
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155
10-135 Consider a cogeneration power plant modified with regeneration. Steam enters the turbine at 6 MPa and 450°C at a rate of 20 kg/s and expands to a pressure of 0.4 MPa. At this pressure, 60% of the steam is extracted from the turbine, and the remainder expands to a pressure of 10 kPa. Part of the extracted steam is used to heat feedwater in an open feedwater heater. The rest of the extracted steam is used for process heating and leaves the process heater as a saturated liquid at 0.4 MPa. It is subsequently mixed with the feedwater leaving the feedwater heater, and the mixture is pumped to the boiler pressure. The steam in the condenser is cooled and condensed by the cooling water from a nearby river, which enters the adiabatic condenser at a rate of 463 kg/s.
1. The total power output of the turbine is
(a) 17.0 MW (b) 8.4 MW (c) 12.2 MW (d) 20.0 MW (e) 3.4 MW
Answer (a) 17.0 MW
2. The temperature rise of the cooling water from the river in the condenser is
(a) 8.0°C (b) 5.2°C (c) 9.6°C (d) 12.9°C (e) 16.2°C
Answer (a) 8.0°C
3. The mass flow rate of steam through the process heater is
(a) 1.6 kg/s (b) 3.8 kg/s (c) 5.2 kg/s (d) 7.6 kg/s (e) 10.4 kg/s
Answer (e) 10.4 kg/s
4. The rate of heat supply from the process heater per unit mass of steam passing through it is
(a) 246 kJ/kg (b) 893 kJ/kg (c) 1344 kJ/kg (d) 1891 kJ/kg (e) 2060 kJ/kg
Answer (e) 2060 kJ/kg
6
11
1
5
Turbine Boiler
Condenser
Process heater
P IP II
9
2
3
8
4 fwh
107
h1 = 191.81 h2 = 192.20 h3 = h4 = h9 = 604.66 h5 = 610.73 h6 = 3302.9 h7 = h8 = h10 = 2665.6 h11 = 2128.8
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156
5. The rate of heat transfer to the steam in the boiler is
(a) 26.0 MJ/s (b) 53.8 MJ/s (c) 39.5 MJ/s (d) 62.8 MJ/s (e) 125.4 MJ/s
Answer (b) 53.8 MJ/s
Solution Solved by EES Software. Solutions can be verified by copying-and-pasting the following lines on a blank EES screen. (Similar problems and their solutions can be obtained easily by modifying numerical values).
Note: The solution given below also evaluates all enthalpies given on the figure.
P1=10 "kPa" P11=P1 P2=400 "kPa" P3=P2; P4=P2; P7=P2; P8=P2; P9=P2; P10=P2 P5=6000 "kPa" P6=P5 T6=450 "C" m_total=20 "kg/s" m7=0.6*m_total m_cond=0.4*m_total C=4.18 "kJ/kg.K" m_cooling=463 "kg/s" s7=s6 s11=s6 h1=ENTHALPY(Steam_IAPWS,x=0,P=P1) v1=VOLUME(Steam_IAPWS,x=0,P=P1) w_pump=v1*(P2-P1) h2=h1+w_pump h3=ENTHALPY(Steam_IAPWS,x=0,P=P3) h4=h3; h9=h3 v4=VOLUME(Steam_IAPWS,x=0,P=P4) w_pump2=v4*(P5-P4) h5=h4+w_pump2 h6=ENTHALPY(Steam_IAPWS,T=T6,P=P6) s6=ENTROPY(Steam_IAPWS,T=T6,P=P6) h7=ENTHALPY(Steam_IAPWS,s=s7,P=P7) h8=h7; h10=h7 h11=ENTHALPY(Steam_IAPWS,s=s11,P=P11) W_turb=m_total*(h6-h7)+m_cond*(h7-h11) m_cooling*C*T_rise=m_cond*(h11-h1) m_cond*h2+m_feed*h10=(m_cond+m_feed)*h3 m_process=m7-m_feed q_process=h8-h9 Q_in=m_total*(h6-h5)
10-136 ··· 10-143 Design and Essay Problems
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