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Solution Manual
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9-1
Solutions Manual for
Thermodynamics: An Engineering Approach Seventh Edition in SI Units
Yunus A. Cengel, Michael A. Boles McGraw-Hill, 2011
Chapter 9 GAS POWER CYCLES
PROPRIETARY AND CONFIDENTIAL This Manual is the proprietary property of The McGraw-Hill Companies, Inc. (“McGraw-Hill”) and protected by copyright and other state and federal laws. By opening and using this Manual the user agrees to the following restrictions, and if the recipient does not agree to these restrictions, the Manual should be promptly returned unopened to McGraw-Hill: This Manual is being provided only to authorized professors and instructors for use in preparing for the classes using the affiliated textbook. No other use or distribution of this Manual is permitted. This Manual may not be sold and may not be distributed to or used by any student or other third party. No part of this Manual may be reproduced, displayed or distributed in any form or by any means, electronic or otherwise, without the prior written permission of McGraw-Hill.
PROPRIETARY MATERIALpreparation. If you are a student using this Manual, you are using it without permission.
. © 2011 The McGraw-Hill Companies, Inc. Limited distribution permitted only to teachers and educators for course
9-2
Actual and Ideal Cycles, Carnot cycle, Air-Standard Assumptions, Reciprocating Engines
9-1C It represents the net work on both diagrams.
9-2C The air standard assumptions are: (1) the working fluid is air which behaves as an ideal gas, (2) all the processes are internally reversible, (3) the combustion process is replaced by the heat addition process, and (4) the exhaust process is replaced by the heat rejection process which returns the working fluid to its original state.
9-4C The cold air standard assumptions involves the additional assumption that air can be treated as an ideal gas with constant specific heats at room temperature.
9-5C The clearance volume is the minimum volume formed in the cylinder whereas the displacement volume is the volume displaced by the piston as the piston moves between the top dead center and the bottom dead center.
9-6C It is the ratio of the maximum to minimum volumes in the cylinder.
9-6C The MEP is the fictitious pressure which, if acted on the piston during the entire power stroke, would produce the same amount of net work as that produced during the actual cycle.
9-7C Yes.
9-8C Assuming no accumulation of carbon deposits on the piston face, the compression ratio will remain the same (otherwise it will increase). The mean effective pressure, on the other hand, will decrease as a car gets older as a result of wear and tear.
9-9C The SI and CI engines differ from each other in the way combustion is initiated; by a spark in SI engines, and by compressing the air above the self-ignition temperature of the fuel in CI engines.
9-10C Stroke is the distance between the TDC and the BDC, bore is the diameter of the cylinder, TDC is the position of the piston when it forms the smallest volume in the cylinder, and clearance volume is the minimum volume formed in the cylinder.
PROPRIETARY MATERIALpreparation. If you are a student using this Manual, you are using it without permission.
. © 2011 The McGraw-Hill Companies, Inc. Limited distribution permitted only to teachers and educators for course
9-3
9-11 The maximum possible thermal efficiency of a gas power cycle with specified reservoirs is to be determined.
Analysis The maximum efficiency this cycle can have is
64.0%==+
+−=−= 1
T0.640
K )273(500K )273(51Carnotth,
H
L
Tη
specified processes. The cycle is to be
2 Kinetic and potential energy changes are negligible. 3 Air is
erties The properties of air are given as R = 0.287 kPa·m3/kg·K, c = 1.005 kJ/kg·K, cv = 0.718 kJ/kg·K, and k =
the cycle are shown in the figures.
(b) Process 1-2: Isentropic com
TTmcw in −= v
9-12 An air-standard cycle executed in a piston-cylinder system is composed of three sketcehed on the P-v and T-s diagrams and the back work ratio are to be determined.
Assumptions 1 The air-standard assumptions are applicable. an ideal gas with constant specific heats.
Prop p1.4.
Analysis (a) The P-v and T-s diagrams of
pression
)( 12,21−
11
s
T
3 2
1
v
P
32
1
1−
−⎞⎛
kv
2
112 =⎟⎟
⎠⎜⎜⎝
= krTTTv
Process 2-3: Constant pressure heat addition
2,3 TTmRPPdw out −=−== ∫− VVv
rk ratio is
3
)()( 23232 2
The back wo
)()(
23
12
,32
,21
TTmRTTmc
ww
rout
inbw −
−==
−
− v
Noting that
PROPRIETARY MATERIAL. © 2011 The McGraw-Hill Companies, Inc. Limited distribution permitted only to teachers and educators for course
1
thus,and and =−=c
kccR pp v −
=k
Rcc
vv
From ideal gas relation,
rTT
===2
1
2
3
2
3
vv
vv
Substituting these into back work relation,
( )
( )0.256=
−−
−=
−−
−=
−
⎟⎠⎞
⎜⎝⎛ −
−=
−−
−=
−
−−
1661
14.11
11
11
1
11
11
)1/()/1(1
1
4.0
11
23
21
2
2
rr
krr
k
TTTT
TT
RkRr
kk
bw
preparation. If you are a student using this Manual, you are using it without permission.
9-4
9-13 The three processes of an air-standard cycle are described. The cycle is to be shown on the P-v and T-s diagrams, and the back work ratio and the thermal efficiency are to be determined.
Assumptions 1 The air-standard assumptions are applicable. 2 Kinetic and potential energy changes are negligible. 3 Air is an ideal gas with constant specific heats.
Properties The properties of air are given as R = 0.287 kJ/kg.K, cp = 1.005 kJ/kg.K, cv = 0.718 kJ/kg·K, and
k = 1.4.
Analysis (a) The P-v and T-s diagrams of the cycle are shown in the figures.
(b) During process 1-3, we have
PROPRIETARY MATERIAL. © 2011 The McGraw-Hill Companies, Inc. Limited distribution permitted only to teachers and educators for course
=−⋅−=
−−=− TTRP VV
uring process 2-3, we have
,13 −=−= ∫− Pdw in v
kJ/kg 516.600)K21K)(300kJ/kg 287.0(
)()( 31311
1
3
s
T
32
1
v
P
3
2
1
D
kJ/kg 8.1172n7K)(2100)KlkJ/kg 287.0(
7ln7
lnln2
2
2
33
2
3
2,32 == ∫∫− Pdw out v
=⋅=
=== RTRTRTdRTV
V
V
Vv
V
The back work ratio is then
0.440===−
−
kJ/kg 8.1172kJ/kg 6.516
,32
,13
out
inbw w
wr
Hprocess 1-3,
eat input is determined from an energy balance on the cycle during
kJ/kg 1172.8300)K)(2100kJ/kg 718.0()( ,3213
,3231
31,32,31
+−⋅=
+−=
+∆=−
∆=−
−
−−
−−−
outv
out
outin
wTTcwu
uwq
he net work output is
,31− inq
k 2465= J/kg
T
kJ/kg 2.6566.5168.1172,13,32 =−=−= −− inoutnet www
(c) The thermal efficiency is then
26.6%==== 266.0kJ 2465kJ 656.2
in
netth q
wη
preparation. If you are a student using this Manual, you are using it without permission.
9-5
9-14 The three processes of an ideal gas power cycle are described. The cycle is to be shown on the P-v and T-s diagrams, and the maximum temperature, expansion and compression works, and thermal efficiency are to be determined.
Assumptions 1 Kinetic and potential energy changes are negligible. 2 The ideal gas has constant specific heats.
Properties The properties of ideal gas are given as R = 0.3 kJ/kg.K, cp = 0.9 kJ/kg.K, cv = 0.6 kJ/kg·K, and
k = 1.5.
Analysis (a) The P-v and T-s diagrams of the cycle are shown in the figures.
(b) The maximum temperature is determined from
PROPRIETARY MATERIAL. © 2011 The McGraw-Hill Companies, Inc. Limited distribution permitted only to teachers and educators for course
K 734.8=+==⎟⎟⎠
⎜⎜⎝
== 12
112max 27(rTTTT
V
⎞ −−−
11.511
K)(6) 273kk
V
) An energy balance during process 2-3 gives
voutin TTTTcuwq 232332,32,32 since 0)(−−− ==−=∆=−
Then, the work of compression is
s
T
32
1
v
P
3
2
1
⎛
(c
ouin wq ,32,32 −− = t
kJ/kg 395.0=⋅=
=== ∫K)ln6 K)(734.8kJ/kg 3.0(
lnln 22
32
22
rRTRTdRTPdV
Vv
Vv
(d) The work during isentropic compression is determined from an energy
−=∆= −
300))( 1221,2 TTcuw vin
) Net work output is
== ∫−−
33
,32,32 wq outin
balance during process 1-2:
−1
kJ/kg 260.9=
⋅= K)(734.8kJ/kg 6.0( −
(e
kJ/kg 1.1349.2600.395,21,32 =−=−= −− inoutnet www
The thermal efficiency is then
33.9%==== 339.0kJ 395.0kJ 134.1
in
netth q
wη
preparation. If you are a student using this Manual, you are using it without permission.
9-6
9-15 The four processes of an air-standard cycle are described. The cycle is to be shown on P-v and T-s diagrams, and the net work output and the thermal efficiency are to be determined.
Assumptions 1 The air-standard assumptions are applicable. 2 Kinetic and potential energy changes are negligible. 3 Air is an ideal gas with variable specific heats.
Properties The properties of air are given in Table A-17.
Analysis (b) The properties of air at various states are
v
P
1
2
4
3qin
qout
s
T
1
24
3qin
qout
( )
( )
( ) kJ/kg 71.73979.32601.9kPa 1835.6
kPa 100
kPa 6.1835kPa 600K 490.3K 1500
9.601kJ/kg 41.1205
K 1500
K 3.490kJ/kg 352.29
841.71.3068kPa 100kPa 600
3068.1kJ/kg .17295
K 2951 ⎯→⎯=T
43
4
22
323
33
2
2
1
2
1
34
3
12
1
=⎯→⎯===
=
==
⎯→⎯=
==
⎯→⎯===
==
hPPP
P
TT
Pu
T
Tu
PPP
P
Ph
rr
r
rr
r
From energy balances,
(c) Then the thermal efficiency becomes
32233 ==⎯→⎯= PT
PPP vv
T
kJ/kg 408.6=−=−=
=−=−=
=−=−=
5.4441.853
kJ/kg .544417.29571.739
kJ/kg 1.85329.35241.1205
outinoutnet,
14out
23in
qqw
hhq
uuq
47.9%==== 479.0kJ/kg 853.1kJ/kg 408.6
in
outnet,th q
wη
PROPRIETARY MATERIALpreparation. If you are a student using this Manual, you are using it without permission.
. © 2011 The McGraw-Hill Companies, Inc. Limited distribution permitted only to teachers and educators for course
9-7
9-16 Problem 9-15 is reconsidered. The effect of the maximum temperature of the cycle on the net work output and thermal efficiency is to be investigated. Also, T-s and P-v diagrams for the cycle are to be plotted.
Analysis Using EES, the problem is solved as follows:
"Input Data" T[1]=295 [K] P[1]=100 [kPa] P[2] = 600 [kPa] T[3]=1500 [K] P[4] = 100 [kPa] "Process 1-2 is isentropic compression" s[1]=entropy(air,T=T[1],P=P[1]) s[2]=s[1] T[2]=temperature(air, s=s[2], P=P[2]) P[2]*v[2]/T[2]=P[1]*v[1]/T[1] P[1]*v[1]=R*T[1] R=0.287 [kJ/kg-K] "Conservation of energy for process 1 to 2" q_12 -w_12 = DELTAu_12 q_12 =0"isentropic process" DELTAu_12=intenergy(air,T=T[2])-intenergy(air,T=T[1]) "Process 2-3 is constant volume heat addition" s[3]=entropy(air, T=T[3], P=P[3]) {P[3]*v[3]/T[3]=P[2]*v[2]/T[2]} P[3]*v[3]=R*T[3] v[3]=v[2] "Conservation of energy for process 2 to 3" q_23 -w_23 = DELTAu_23 w_23 =0"constant volume process" DELTAu_23=intenergy(air,T=T[3])-intenergy(air,T=T[2]) "Process 3-4 is isentropic expansion" s[4]=entropy(air,T=T[4],P=P[4]) s[4]=s[3] P[4]*v[4]/T[4]=P[3]*v[3]/T[3] {P[4]*v[4]=0.287*T[4]} "Conservation of energy for process 3 to 4" q_34 -w_34 = DELTAu_34 q_34 =0"isentropic process" DELTAu_34=intenergy(air,T=T[4])-intenergy(air,T=T[3]) "Process 4-1 is constant pressure heat rejection" {P[4]*v[4]/T[4]=P[1]*v[1]/T[1]} "Conservation of energy for process 4 to 1" q_41 -w_41 = DELTAu_41 w_41 =P[1]*(v[1]-v[4]) "constant pressure process" DELTAu_41=intenergy(air,T=T[1])-intenergy(air,T=T[4]) q_in_total=q_23 w_net = w_12+w_23+w_34+w_41 Eta_th=w_net/q_in_total*100 "Thermal efficiency, in percent"
PROPRIETARY MATERIALpreparation. If you are a student using this Manual, you are using it without permission.
. © 2011 The McGraw-Hill Companies, Inc. Limited distribution permitted only to teachers and educators for course
9-8
T3 [K]
ηth qin,total [kJ/kg]
Wnet [kJ/kg]
1500 1600 1700 1800 1900 2000 2100 2200 2300 2400 2500
47.91 48.31 48.68 49.03 49.35 49.66 49.95 50.22 50.48 50.72 50.95
852.9 945.7 1040 1134 1229 1325 1422 1519 1617 1715 1813
408.6 456.9 506.1 556
606.7 658.1 710.5 763
816.1 869.8 924
1500 1700 1900 2100 2300 250047.5
48
48.5
49
49.5
50
50.5
51
T[3] [K]
ηth
1500 1700 1900 2100 2300 2500800
1020
1240
1460
1680
1900
T[3
g]
] [K]
q in,
tota
l [k
J/k
PROPRIETARY MATERIALpreparation. If you are a student using this Manual, you are using it without permission.
. © 2011 The McGraw-Hill Companies, Inc. Limited distribution permitted only to teachers and educators for course
9-9
1500 1700 1900 2100 2300 2500400
500
600
700
800
900
1000
T[3] [K]
wne
t [k
J/kg
]
5.0 5.5 6.0 6.5 7.0 7.5 8.0 8.50
500
1000
1500
2000
s [kJ/kg-K]
T [K
]
100 kPa
600 kPa
Air
1
2
3
4
10-2 10-1 100 101 102101
102
103
104
v [m3/kg]
P [k
Pa]
295 K 1500 K
Air
1
2
3
4
PROPRIETARY MATERIALpreparation. If you are a student using this Manual, you are using it without permission.
. © 2011 The McGraw-Hill Companies, Inc. Limited distribution permitted only to teachers and educators for course
9-10
9-17 The three processes of an air-standard cycle are described. The cycle is to be shown on P-v and T-s diagrams, and the heat rejected and the thermal efficiency are to be determined.
Assumptions 1 The air-standard assumptions are applicable. 2 Kinetic and potential energy changes are negligible. 3 Air is an ideal gas with constant specific heats.
Properties The properties of air at room temperature are cp = 1.005 kJ/kg.K, cv = 0.718 kJ/kg·K, and k = 1.4 (Table A-2).
Analysis (b) The temperature at state 2 and the heat input are
PROPRIETARY MATERIAL. © 2011 The McGraw-Hill Companies, Inc. Limited distribution permitted only to teachers and educators for course
( )( ) K 579.2
kPa 100kPa 1000
K 3000.4/1.4/1
1
212 =⎟⎟
⎠
⎞⎜⎜⎝
⎛=⎟⎟
⎠
⎞⎜⎜⎝
⎛=
− kk
PP
TT
( ) ( )( )( )( )2.579KkJ/kg 1.005kg 0.004kJ 2.76 3 ⎯→⎯−⋅= T K 12663
2323in
=
−=−=
T
TTmchhmQ p
Process 3-1 is a straight line on the P-v diagram, thus the w31 is simply the area under the process curve,
( )
( )
kJ/kg 7.273=
KkJ/kg 0.287kPa 1000K 1266
kPa 100K 300
2kPa 1001000
22area
3
3
1
11331
1331
⋅⎟⎟⎠
⎞⎜⎜⎝
⎛−⎟
⎠
⎞⎜⎝
⎛ +=
⎟⎟⎠
⎞⎜⎜⎝
⎛−
+=−
+==
PRT
PRTPPPP
w vv
Energy bal nce for process 3-1 gives
−−−=
−=−−→
K1266-300KkJ/kg 0.718273.7kg 0.004)(
)(
31out31,31out31,out31,
31out31,out31,systemoutin
TTcwmTTmcmwQ
uumWQ
vv
) The thermal efficiency is then
a
( )[ ]( ) ( )( )[ ] kJ 1.679=⋅+−=
−+−
v
P
32
1
qin
qout
s
T
32
1 qout
qin
⎯⎯∆=− EEE
=
(c
39.2%=−=−=kJ 2.76kJ 1.679
11in
outth Q
Qη
preparation. If you are a student using this Manual, you are using it without permission.
9-11
9-18 A Carnot cycle with the specified temperature limits is considered. The net work output per cycle is to be determined.
Assumptions Air is an ideal gas with constant specific heats.
Properties The properties of air at room temperature are cp = 1.005 kJ/kg.K, cv = 0.718 kJ/kg·K, R = 0.287 kJ/kg.K, and k = 1.4 (Table A-2).
Analysis The minimum pressure in the cycle is P3 and the maximum pressure is P1. Then,
PROPRIETARY MATERIAL. © 2011 The McGraw-Hill Companies, Inc. Limited distribution permitted only to teachers and educators for course
or
( )
( )( ) kPa 1888
K 300K 1100
kPa 201.4/0.41/
3
232
/1
3
2
3
2⎟⎟⎠
⎜⎜⎝
=PT
=⎟⎟⎠
⎞⎜⎜⎝
⎛=⎟⎟
⎠
⎞⎜⎜⎝
⎛=
⎞⎛
−
−
kk
kk
TT
PP
PT
t input is determined from
hen,
s
T
3
2qin
qout4
11100
300 The hea
( )
( ) ( )( )( )
( )( ) kJ 63.8===
==−=−=
=⋅=−=
⋅=⋅−=−=−
kJ 87.730.7273
.7%727273.0K 1100K 300
11
kJ 73.87KkJ/kg 0.1329K 1100kg 0.6
KkJ/kg 0.1329kPa 3000kPa 1888lnKkJ/kg 0.287lnln
inthoutnet,
th
12in
1
20
1
212
QW
TT
ssmTQ
PP
RTT
css
H
L
H
p
η
η
T
preparation. If you are a student using this Manual, you are using it without permission.
9-12
9-19 A Carnot cycle executed in a closed system with air as the working fluid is considered. The minimum pressure in the cycle, the heat rejection from the cycle, the thermal efficiency of the cycle, and the second-law efficiency of an actual cycle operating between the same temperature limits are to be determined.
Assumptions Air is an ideal gas with constant specific heats.
Properties The properties of air at room temperatures are R = 0.287 kJ/kg.K and k = 1.4 (Table A-2).
Analysis (a) The minimum temperature is determined from
( )( ) ( )( ) K 350K750KkJ/kg 0.25kJ/kg 10012net =⎯→⎯−⋅=⎯→⎯−−= LLLH TTTTssw
The pressure at state 4 is determined from
or
( )
( )
kPa 1.110K 350K 750
kPa 800 4
1.4/0.4
4
1/
4
141
/1
4
1
4
1
=⎯→⎯⎟⎟⎠
⎞⎜⎜⎝
⎛=
⎟⎟⎠
⎞⎜⎜⎝
⎛=
⎟⎟⎠
⎞⎜⎜⎝
⎛=
−
−
PP
TT
PP
PP
TT
kk
kk
s
T
3
2qin
4
1750 K
wnet=100 kJ/kg
qout
The minimum pressure in the cycle is determined from
( ) kPa 46.1=⎯→⎯⋅−=⋅−
−=∆−=∆ 3412 lncss p
33
3
40
3
4
kPa 110.1lnKkJ/kg 0.287KkJ/kg 25.0
ln
PP
PP
RTT
(b) The heat rejection from the cycle is
kgkJ/ 87.5==∆= kJ/kg.K K)(0.25 350(12out sTq L )
) The thermal efficiency is determined from
(c
PROPRIETARY MATERIAL. © 2011 The McGraw-Hill Companies, Inc. Limited distribution permitted only to teachers and educators for course
0.533=−=−=K 350
11 LTη
K 750HT
(d) The power output for the Carnot cycle is
Then, the second-law efficiency of the actual cycle becomes
th
kW 9000kJ/kg) kg/s)(100 90(netCarnot === wmW &&
0.578===kW 9000kW 5200
Carnot
actualII W
W&
&η
preparation. If you are a student using this Manual, you are using it without permission.
9-13
9-20 An ideal gas Carnot cycle with air as the working fluid is considered. The maximum temperature of the low-temperature energy reservoir, the cycle's thermal efficiency, and the amount of heat that must be supplied per cycle are to be determined.
Assumptions 1 The air-standard assumptions are applicable. 2 Kinetic and potential energy changes are negligible. 3 Air is an ideal gas with constant specific heats.
Properties The properties of air at room temperature are cp = 1.005 kJ/kg.K, cv = 0.718 kJ/kg·K, and k = 1.4 (Table A-2a).
Analysis The temperature of the low-temperature reservoir can be found by applying the isentropic expansion process relation
K 481.1=⎟⎠⎞
⎜⎝⎛+=⎟⎟
⎠
⎞⎜⎜⎝
⎛=
−− 14.11
1
221 12
1K) 2731027(k
TTv
v
s
T
3
2qin
qout4
11300 K Since the Carnot engine is completely reversible, its efficiency is
0.630=+
−=−=K 273)(1027
K 481.111Carnotth,HTLT
η
The work output per cycle is
kJ/cycle 20min 1cycle/min 1500net
⎠⎝n&s 60kJ/s 500net =⎟
⎞⎜⎛==
WW
&
ccording the definition of the cycle efficiency,
A to
kJ/cycle 31.75===⎯→⎯=0.63kJ/cycle 20
Carnotth,
netin
in
netCarnotth, η
ηW
QQW
PROPRIETARY MATERIALpreparation. If you are a student using this Manual, you are using it without permission.
. © 2011 The McGraw-Hill Companies, Inc. Limited distribution permitted only to teachers and educators for course
9-14
9-21 An air-standard cycle executed in a piston-cylinder system is composed of three specified processes. The cycle is to be sketcehed on the P-v and T-s diagrams; the heat and work interactions and the thermal efficiency of the cycle are to be determined; and an expression for thermal efficiency as functions of compression ratio and specific heat ratio is to be obtained.
Assumptions 1 The air-standard assumptions are applicable. 2 Kinetic and potential energy changes are negligible. 3 Air is an ideal gas with constant specific heats.
Properties The properties of air are given as R = 0.3 kJ/kg·K and cv = 0.3 kJ/kg·K.
Analysis (a) The P-v and T-s diagrams of the cycle are shown in the figures.
(b) Noting that
1.429
7.00.1
K kJ/kg 0.13.07.0
===
⋅=+=+=
v
v
cc
k
Rcc
p
p
s
T
32
1
v
P
32
1Process 1-2: Isentropic compression
K 4.584)5)(K 293( 429.011
1
2
112 ===⎟⎟
⎠
⎞⎜⎜⎝
⎛= −
−k
k
rTTTvv
kJ/kg 204.0=−⋅=−=−1 K )2934.584)(KkJ/kg 7.0()( 12in2, TTcw v
rom ideal gas relation,
0=q −21
F
2922)5)(4.584(3133 =⎯→⎯=== Tr
T v
v
v
v
22=
T
Constant pressure heat addition
−=−== ∫−
K )4.
)()( 23232
3
out3,2 TTmRPPdw VVv
huq
p
out
ant volume heat rejection
2
Process 2-3:
kJ/kg 701.3=−⋅= 5842922)(KkJ/kg 3.0(2
kJ/kg 2338=−⋅=−=
∆=∆+= −−−−
K )4.5842922)(KkJ/kg 1()( 23
3232,32in3,2
TTcw
Process 3-1: Const
=∆= −− (31out1,3 cuq v kJ/kg 1840.3=⋅=− K 293)-K)(2922kJ/kg 7.0()13 TT
(c) Net work is
0=−13w
KkJ/kg 3.4970.2043.701in2,1out3,2net ⋅=−=−= −− www
The thermal efficiency is then
21.3%==== 213.0kJ 2338kJ 497.3
in
netth q
wη
PROPRIETARY MATERIALpreparation. If you are a student using this Manual, you are using it without permission.
. © 2011 The McGraw-Hill Companies, Inc. Limited distribution permitted only to teachers and educators for course
9-15
(d) The expression for the cycle thermal efficiency is obtained as follows:
PROPRIETARY MATERIAL. © 2011 The McGraw-Hill Companies, Inc. Limited distribution permitted only to teachers and educators for course
⎟⎠
⎞⎜⎝
⎛ −−
−⎟⎠⎞
⎜⎝⎛ −=
⎟⎠
⎞⎜⎝
⎛ −−
−=
⎟⎟⎠
⎞⎜⎜⎝
⎛−
−−=
−
⎟⎟⎠
⎞⎜⎜⎝
⎛−
−=
−
−−=
−−−−
=
−==
−
−
−
−
−−
−−
−
−−
1
1
11
1
11
11
111
11
11
11
1
23
1223
in
in2,1out3,2
in
netth
11)1(
111
11)1(
1
1)1(
1
)1(
1
)()(
)()()(
k
kp
kp
kp
kk
v
p
kkp
kv
p
p
v
rrkk
rrkcR
rTT
rkcR
rrTc
rTT
rTc
cR
rTrrTcTrTc
cR
TTcTTcTTR
qww
qw
η
since
111kc
cc
cccR
p
v
p
vp
p−=−=
−=
preparation. If you are a student using this Manual, you are using it without permission.
9-16
Otto Cycle
9-22C For actual four-stroke engines, the rpm is twice the number of thermodynamic cycles; for two-stroke engines, it is equal to the number of thermodynamic cycles.
9-23C The ideal Otto cycle involves external irreversibilities, and thus it has a lower thermal efficiency.
9-24C The four processes that make up the Otto cycle are (1) isentropic compression, (2) v = constant heat addition, (3) isentropic expansion, and (4) v = constant heat rejection.
9-25C They are analyzed as closed system processes because no mass crosses the system boundaries during any of the processes.
9-26C It increases with both of them.
9-27C Because high compression ratios cause engine knock.
9-28C The thermal efficiency will be the highest for argon because it has the highest specific heat ratio, k = 1.667.
9-29C The fuel is injected into the cylinder in both engines, but it is ignited with a spark plug in gasoline engines.
PROPRIETARY MATERIALpreparation. If you are a student using this Manual, you are using it without permission.
. © 2011 The McGraw-Hill Companies, Inc. Limited distribution permitted only to teachers and educators for course
9-17
9-30 An ideal Otto cycle is considered. The thermal efficiency and the rate of heat input are to be determined.
Assumptions 1 The air-standard assumptions are applicable. 2 Kinetic and potential energy changes are negligible. 3 Air is an ideal gas with constant specific heats.
Properties The properties of air at room temperature are cp = 1.005 kJ/kg.K, cv = 0.718 kJ/kg·K, and k = 1.4 (Table A-2a).
Analysis The definition of cycle thermal efficiency reduces to
v
P
4
1
3
2
qin
qout
61.0%==−=−=−− 10.51th kr
6096.01111 11.4η
The rate of heat addition is then
kW 148===0.6096
kW 90
th
netin η
WQ
&&
9-31 An ideal Otto cycle is considered. The thermal efficiency and the rate of heat input are to be determined.
Assumptions 1 The air-standard assumptions are applicable. 2 Kinetic and potential energy changes are negligible. 3 Air is
erties The properties of air at room temperature are cp = 1.005 kJ/kg.K, cv = 0.718 kJ/kg·K, and k = 1.4 (Table A-
nalysis The definition of cycle thermal efficiency reduces to
an ideal gas with constant specific heats.
Prop2a).
A
v
P
4
1
3
2
qin
qout
57.5%==−=−=1111thη
−−5752.0
8.5 11.41kr
he rate of heat addition is then
T
kW 157===0.5752
kW 90
th
netin η
WQ
&&
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. © 2011 The McGraw-Hill Companies, Inc. Limited distribution permitted only to teachers and educators for course
9-18
9-32 The two isentropic processes in an Otto cycle are replaced with polytropic processes. The heat added to and rejected from this cycle, and the cycle’s thermal efficiency are to be determined.
Assumptions 1 The air-standard assumptions are applicable. 2 Kinetic and potential energy changes are negligible. 3 Air is an ideal gas with constant specific heats.
Properties The properties of air at room temperature are R = 0.287 kPa·m3/kg·K, cp = 1.005 kJ/kg·K, cv = 0.718 kJ/kg·K, and k = 1.4 (Table A-2a).
Analysis The temperature at the end of the compression is
v
P
4
1
3
2
PROPRIETARY MATERIAL. © 2011 The McGraw-Hill Companies, Inc. Limited distribution permitted only to teachers and educators for course
K 4.537K)(8) 288(12
12 ==⎟⎟⎠
⎜⎜⎝
= rTTTv
13.111
1 =⎞⎛ −−
−n
nv
And the temperature at the end of the expansion is
K 4.7898
K) 1473(34
34 =⎟⎠
⎜⎝
=⎟⎠
⎜⎝
=⎟⎟⎠
⎜⎜⎝
=r
TTTv
11 13.1113 ⎞⎛⎞⎛⎞⎛ −−− nn
v
he integr of the work expression for the polytropic compression gives
T al
kJ/kg 6.238)18(13.1
K) K)(288kJ/kg 287.0(11
13.11 ⎤⎡ −n
2
1121 =−
−⋅
=⎥⎥
⎦⎢⎢
⎣−⎟⎟
⎠
⎞⎜⎜⎝
⎛−
= −− n
RTw
v
v
Similarly, the work produced during the expansion is
kJ/kg 0.654181
13.1K) K)(1473kJ/kg 287.0(
113
43 ⎢⎢⎜⎜⎛
−−=− n
RT 13.1
4
3 =⎥⎥⎦
⎤
⎢⎢⎣
⎡−⎟
⎠⎞
⎜⎝⎛
−⋅
−=⎥⎥
⎦⎣−⎟⎟
⎠
⎞
⎝
−
wv
v
pplicatio of the first law to each of the four processes gives
84.537)(KkJ/kg 718.0(kJ/kg 6.238)( 122121
1 ⎤⎡ −n
A n
28 kJ/kg 53.59K) =−⋅−=−−= −− TTcwq v
1473)(KkJ/kg 718.0()( 2332 ⋅=−=− TTcq v kJ/kg 8.671K)4.537 =−
kJ/kg 2.163K)4.7891473)(KkJ/kg 718.0(kJ/kg 0.654)( 43434 3 =−⋅−=−−= − TTcwq v −
kJ/kg 718.0()( 1414 ⋅=−=− TTcq v kJ/kg 0.360K)2884.789)(K =−
he head added and rejected from the cycle are
The thermal efficiency of this cycle is then
T
kJ/kg 419.5kJ/kg 835.0
=+=+==+=+=
−−
−−
0.36053.592.1638.671
1421out
4332in
qqqqqq
0.498=−=−=0.8355.41911
in
outth q
qη
preparation. If you are a student using this Manual, you are using it without permission.
9-19
9-33 An ideal Otto cycle is considered. The heat added to and rejected from this cycle, and the cycle’s thermal efficiency are to be determined.
Assumptions 1 The air-standard assumptions are applicable. 2 Kinetic and potential energy changes are negligible. 3 Air is an ideal gas with constant specific heats.
Properties The properties of air at room temperature are R = 0.287 kPa·m3/kg·K, cp = 1.005 kJ/kg·K, cv = 0.718 kJ/kg·K, and k = 1.4 (Table A-2a).
Analysis The temperature at the end of the compression is
v
P
4
1
3
2
PROPRIETARY MATERIAL. © 2011 The McGraw-Hill Companies, Inc. Limited distribution permitted only to teachers and educators for course
K 7.661K)(8) 288(12
12 ==⎟⎟⎠
⎜⎜⎝
= rTTTv
14.111
1 =⎞⎛ −−
−k
kv
and the temperature at the end of the expansion is
K 2.648
K) 1473(34
34 =⎟⎠
⎜⎝
=⎟⎠
⎜⎝
=⎟⎟⎠
⎜⎜⎝
=r
TTTv
111 14.1113 ⎞⎛⎞⎛⎞⎛ −−− kk
v
pplicatio of the first law to the heat addition process gives A n
=−= ()( TTcq kJ/kg 582.5=−⋅ K)7.6611473)(KkJ/kg 718.023in v
ilarly, the heat rejected is
Sim
⋅=−= KkJ/kg 718.0()( TTcq kJ/kg 253.6=− K)2882.641)(14v
a
out
l efficiency of this cycle is then The therm
0.565=−=−=5.5826.25311
in
outth q
qη
preparation. If you are a student using this Manual, you are using it without permission.
9-20
9-34 A six-cylinder, four-stroke, spark-ignition engine operating on the ideal Otto cycle is considered. The power produced by the engine is to be determined.
Assumptions 1 The air-standard assumptions are applicable. 2 Kinetic and potential energy changes are negligible. 3 Air is an ideal gas with constant specific heats.
Properties The properties of air at room temperature are R = 0.287 kPa·m3/kg.K (Table A-1), cp = 1.005 kJ/kg·K, cv = 0.718 kJ/kg·K, and k = 1.4 (Table A-2a).
Analysis From the data specified in the problem statement,
v
P
4
1
3
2
143.7140 1
1
2
1 ===v
v
v
v
.r
Since the compression and expansion processes are isentropic,
( ) K 7.636143.7K) 290( 14.111
1
2
112 ===⎟⎟
⎠
⎞⎜⎜⎝
⎛= −−
−k
k
rTTTvv
K 6.5207.143
K) 1143(34
34 =⎟⎠
⎜⎝
=⎟⎠
⎜⎝
=⎟⎟⎠
⎜⎜⎝
=r
TTTv
11 14.1113 ⎞⎛⎞⎛⎞⎛ −−− kk
v
pplicatio of the first law to the compression and expansion processes gives
=−−−=
A n
)()( 1243net −−−= TTcTTcw vv
kJ/kg 0.198K)2907.636)(kJ/kg·K 718.0(K)6.5201143)(kJ/kg·K 718.0(
When each cylinder is charged with the air-fuel mixture,
PROPRIETARY MATERIAL. © 2011 The McGraw-Hill Companies, Inc. Limited distribution permitted only to teachers and educators for course
/kgm 8761.0kPa 951
1 ==P
vK) 290)(K/kgmkPa 287.0( 3
31 =
⋅⋅RT
he total air mass taken by all 6 cylinders when they are charged is T
kg 004218.0/kgm 8761.0
m)/4 099.0(m) 089.0()6(4/2∆ πV SB3
2
1cyl ==
πv
Nm 1
cyl ==v
N
The net work produced per cycle is
kJ/cycle 8352.0kJ/kg) kg(198.0 (0.004218netnet === mwW
The power produced is determined from
kW 17.4===rev/cycle 2
rev/s) 2500/60kJ/cycle)( (0.8352
rev
netnet N
nWW
&&
since there are two revolutions per cycle in a four-stroke engine.
preparation. If you are a student using this Manual, you are using it without permission.
9-21
9-35 An ideal Otto cycle with air as the working fluid has a compression ratio of 9.5. The highest pressure and temperature in the cycle, the amount of heat transferred, the thermal efficiency, and the mean effective pressure are to be determined.
Assumptions 1 The air-standard assumptions are applicable. 2 Kinetic and potential energy changes are negligible. 3 Air is an ideal gas with constant specific heats.
Properties The properties of air at room temperature are cp = 1.005 kJ/kg·K, cv = 0.718 kJ/kg·K, R = 0.287 kJ/kg·K, and k = 1.4 (Table A-2).
Analysis (a) Process 1-2: isentropic compression.
v
P
4
1
3
2Qin Qout
( )( )
( ) ( ) kPa 2338kPa 100K 308K 757.9
9.5
K 757.99.5K 308
11
2
2
12
1
11
2
22
0.41
2
112
=⎟⎟⎠
⎞⎜⎜⎝
⎛==⎯→⎯=
==⎟⎟⎠
⎞⎜⎜⎝
⎛=
−
PTT
PT
PT
P
TTk
v
vvv
v
v
Process 3-4: isentropic expansion.
PROPRIETARY MATERIAL. © 2011 The McGraw-Hill Companies, Inc. Limited distribution permitted only to teachers and educators for course
( )( )=⎟⎟⎠
⎜⎜⎝
=3
443 9.5K 800TT
vK 1969=
⎞⎛−
0.41k
v
rocess 2-3: v = constant heat addition. P
( ) kPa 6072=⎟⎟⎠
⎞⎜⎜⎝
⎛==⎯→⎯= kPa 2338
K 757.9K 1969
22
33
2
22
3
33
TP
TT
PT
PP vv
( )( )(b) ( )( )
kg10788.6K 308
m 0.0006kPa 100 43
11 −×=PV
K/kgmkPa 0.287 3
1 ⋅⋅==
RTm
( ) ( ) ( )( )( ) kJ 0.590=−⋅×=−=−= − K757.91969KkJ/kg 0.718kg106.788 42323in TTmcuumQ v
(c) Process 4-1: v = constant heat rejection.
( ) ( )( )( ) kJ0.2 40K308800KkJ/kg 0.718kg106.788)( 41414out =−⋅×−=−=−= −TTmcuumQ v
kJ 0.350240.0590.0outinnet =−=−= QQW
59.4%===kJ 0.590kJ 0.350
in
outnet,th Q
Wη
( )( )kPa 652=⎟
⎟⎠
⎞⎜⎜⎝
⎛ ⋅
−=
−=
−=
==
kJmkPa
1/9.51m 0.0006kJ 0.350
)/11(MEP
3
31
outnet,
21
outnet,
max2min
rWW
r
VVV
VVV (d)
preparation. If you are a student using this Manual, you are using it without permission.
9-22
9-36 An Otto cycle with air as the working fluid has a compression ratio of 9.5. The highest pressure and temperature in the cycle, the amount of heat transferred, the thermal efficiency, and the mean effective pressure are to be determined. Assumptions 1 The air-standard assumptions are applicable. 2 Kinetic and potential energy changes are negligible. 3 Air is an ideal gas with constant specific heats.
PROPRIETARY MATERIAL. © 2011 The McGraw-Hill Companies, Inc. Limited distribution permitted only to teachers and educators for course
Properties The properties of air at room temperature are cp = 1.005 kJ/kg·K, cv = 0.718 kJ/kg·K, R = 0.287 kJ/kg·K, and k = 1.4 (Table A-2). Analysis (a) Process 1-2: isentropic compression.
( )( )
( ) ( ) 2338kPa 100K 308K 757.9
9.5
K 757.99.5
11
2
2
12
1
11
2
22
0.41
=⎟⎟⎠
⎞⎜⎜⎝
⎛==⎯→⎯=
=⎞⎛
−
PTT
PT
PT
P
k
v
vvv
v
rocess 3- polytropic expansion.
kPa v
P
4
1
3
2
Qin
Qout
Polytropic
800 K
308 KK 3082
112 =⎟⎟
⎠⎜⎜⎝
= TTv
P 4:
( )( )( )( )
kg10788.6K 308K/kgmkPa 0.287
m 43
3
1
11 −×=⋅⋅
==RT
m
0.0006kPa 100PV
( )( )
( ) ( )( )( )kJ 0.5338
1.351K1759800KkJ/kg 0.287106.788
1
434
34 =−
−⋅×=
−−
=−
nTTmR
W
9.5K 800 35.01
3
443 ==⎟⎟
⎠
⎞⎜⎜⎝
⎛=
−
TTn
K 1759v
v
ring the expansion process (This is not realistic, and probably is due to ssuming constant specific heats at room temperature). ) Process 2-3: v = constant heat addition.
Then energy balance for process 3-4 gives
( )( ) ( )
( )( )( ) kJ 0.0664kJ 0.5338K1759800KkJ/kg 0.718kg106.788 4in34,
out34,34out34,34in34,
34out34,in34,
outin
=+−⋅×=
+−=+−=
−=−
∆=−
−QWTTmcWuumQ
uumWQ
EEE system
v
That is, 0.066 kJ of heat is added to the air dua(b
( ) kPa 5426=⎟⎟⎠
⎞⎜⎜⎝
⎛==⎯→⎯= kPa 2338
K
757.9K 1759
22
33
2
22
3
33 PTT
PT
PT
P vv
3,
=−⋅×=
−=−=−Q
TTmcuumQ v
Therefore,
( ) ( )( )( )( ) kJ 0.4879K757.91759KkJ/kg 0.718kg106.788 4
in23,
2323in2
kJ 0.5543=+=+= 0664.04879.0in34,in23,in QQQ
(c) Process 4-1: v = constant heat rejection.
( ) ( ) ( )( )( ) kJ 0.2398K308800KkJ/kg 0.718kg 106.788 41414out =−⋅×=−=−= −TTmcuumQ v
kJ 0.31452398.05543.0outinoutnet, =−=−= QQW
56.7%===kJ 0.5543kJ 0.3145
in
outnet,th Q
Wη
( )( )kPa 586=⎟
⎟⎠
⎞⎜⎜⎝
⎛ ⋅
−=
−=
−=
==
kJmkPa
1/9.51m 0.0006kJ 0.3145
)/11(MEP
3
31
outnet,
21
outnet,
max2min
rWW
r
VVV
VVV (d)
preparation. If you are a student using this Manual, you are using it without permission.
9-23
9-37 An ideal Otto cycle with air as the working fluid has a compression ratio of 8. The amount of heat transferred to the air during the heat addition process, the thermal efficiency, and the thermal efficiency of a Carnot cycle operating between the same temperature limits are to be determined.
Assumptions 1 The air-standard assumptions are applicable. 2 Kinetic and potential energy changes are negligible. 3 Air is an ideal gas with variable specific heats.
Properties The properties of air are given in Table A-17.
Analysis (a) Process 1-2: isentropic compression.
v
P
4
1
3
2
qin
qout
1340 K
300K
2.621kJ/kg 07.214
K 3001
11 =
=⎯→⎯=
r
uT
v
( ) kJ/kg 22.49165.772.62181
222 r rrr v11
22 =⎯→⎯==== uvv
vv
Process 2-3: v = constant heat addition.
==
⎯→⎯=
22.
247.10kJ/kg 94.1058
K 1340 33
3
uT
rv
) Proces -4: isentropic expansion.
−=−= 49194.105823 uuqin kJ/kg 567.72=
(b s 3
PROPRIETARY MATERIAL. © 2011 The McGraw-Hill Companies, Inc. Limited distribution permitted only to teachers and educators for course
( )( 247.1084 === rvvv
v ) kJ/kg 82.480976.81 43
334=⎯→⎯= urrr v
rocess 4- v = constant heat rejection.
P 1:
kJ/kg 75.26607.21482.48014out =−=−= uuq
53.0%==−=−= 530.0kJ/kg 567.72kJ/kg 266.75
11in
outth q
qη
77.6%==−=−= 776.0K 1340K 300
11Cth,H
L
TT
η (c)
preparation. If you are a student using this Manual, you are using it without permission.
9-24
9-38 An ideal Otto cycle with argon as the working fluid has a compression ratio of 8. The amount of heat transferred to the argon during the heat addition process, the thermal efficiency, and the thermal efficiency of a Carnot cycle operating between the same temperature limits are to be determined.
Assumptions 1 The air-standard assumptions are applicable with argon as the working fluid. 2 Kinetic and potential energy changes are negligible. 3 Argon is an ideal gas with constant specific heats.
Properties The properties of argon are cp = 0.5203 kJ/kg.K, cv = 0.3122 kJ/kg.K, and k = 1.667 (Table A-2).
Analysis (a) Process 1-2: isentropic compression.
v
P
4
1
3
2
qinqout
( )( )8K 3002
12 =⎟⎟⎠
⎜⎜⎝
= TTv
K 12010.6671
1 =⎞⎛
−kv
Process 2-3: v = constant heat addition.
( ) ( )( ) kJ/kg 43.45=−=−=−= k 0.31222323in TTcuuq v K12011340J/kg.K
) Process 3-4: isentropic expansion. (b
( ) K 8.33481K 1340
0.6671−k
4
334 =⎟
⎠⎞
⎜⎝⎛=⎟⎟
⎠
⎞⎜⎜⎝
⎛= TT
v
v
rocess 4- v = constant heat rejection.
P 1:
)( ) ( ( ) kJ/kg .8510K300334.8kJ/kg.K 0.31221414out =−=−=−= TTcuuq v
75.0%==−=−= 750.0kJ/kg 43.45kJ/kg 10.85
11in
outth q
qη
77.6%==−=−= 776.0K 1340K 300
11Cth,H
L
TT
η (c)
PROPRIETARY MATERIALpreparation. If you are a student using this Manual, you are using it without permission.
. © 2011 The McGraw-Hill Companies, Inc. Limited distribution permitted only to teachers and educators for course
9-25
9-39 A gasoline engine operates on an Otto cycle. The compression and expansion processes are modeled as polytropic. The temperature at the end of expansion process, the net work output, the thermal efficiency, the mean effective pressure, the engine speed for a given net power, and the specific fuel consumption are to be determined.
Assumptions 1 The air-standard assumptions are applicable. 2 Kinetic and potential energy changes are negligible. 3 Air is an ideal gas with constant specific heats.
Properties The properties of air at 850 K are cp = 1.110 kJ/kg·K, cv = 0.823 kJ/kg·K, R = 0.287 kJ/kg·K, and k = 1.349 (Table A-2b).
Analysis (a) Process 1-2: polytropic compression
1
Qin
2
3
4
P
V
Qout
( )( )
( )( )11kPa 100 1.3
2
112 ==⎟⎟
⎠⎜⎜⎝
= PPv
kPa 2258
K 5.63611K 310 1-1.31
2
112
⎞⎛
==⎟⎟⎠
⎞⎜⎜⎝
⎛=
−
n
n
TT
v
v
v
kJ/kg 3.3121.31
310)KK)(636.5kJ/kg (0.2871
)( 1212 −=
−−⋅
=−
−=
nTTR
w
Process 2-3: constant volume heat addition
( ) K 2255kPa 2258kPa 8000K 636.5
2
32
PROPRIETARY MATERIAL. © 2011 The McGraw-Hill Companies, Inc. Limited distribution permitted only to teachers and educators for course
3 =⎟⎠⎞
⎜⎝
=⎟⎟⎠
⎜⎜⎝
=P
TT
Process 3-4: polytropic expansion.
⎛⎞⎛ P
( )( )( ) kJ/kg 1332K636.52255KkJ/kg 0.823
2323in
=−⋅=−=−= TTcuuq v
( ) K 1098=⎟⎠⎞
⎜⎝⎛=⎟⎟
⎠⎜⎜⎝
=4
334 11
1K 2255TTv
⎞⎛− 1-1.31n
v
( ) kPa 2.354111kPa 8000
1.3
1
234 =⎟
⎠⎞
⎜⎝⎛=⎟⎟
⎞⎜⎜⎛
=n
PPvv
⎠⎝
kJ/kg 11061.31
2255)KK)(1098kJ/kg (0.2871
)( 3434 =
−−⋅
=−−
=nTTR
w
Process 4-1: constant volume heat rejection.
(b) The net work output and the thermal efficiency are
kJ/kg 794=−=−= 3.31211061234outnet, www
59.6%==== 596.0kJ/kg 1332kJ/kg 794
in
outnet,th q
wη
(c) The mean effective pressure is determined as follows
( )( )
( )( )kPa 982=⎟
⎟⎠
⎞⎜⎜⎝
⎛ ⋅
−=
−=
−=
==
==⋅⋅
==
kJmkPa
1/111/kgm 0.8897kJ/kg 794
)/11(MEP
/kgm 0.8897kPa 100
K 310K/kgmkPa 0.287
3
31
outnet,
21
outnet,
max2min
max3
3
1
11
rww
r
PRT
vvv
vvv
vv
preparation. If you are a student using this Manual, you are using it without permission.
9-26
(d) The clearance volume and the total volume of the engine at the beginning of compression process (state 1) are
33
m 00016.0m 0016.0
11 =⎯→⎯+
=⎯→⎯+
= cc
c
c
dcr VV
V
V
VV
31 0016.000016.0 =+=+= dc VVV m 00176.0
The total m ss contained in the cylinder is a
( )( )kg 0.001978
K 310K/kgmkPa 0.287)m 76kPa)/0.001 100( 3PV
31
11 =⋅⋅
==RT
mt
The engine speed for a net power output of 50 kW is
rev/min 3820=⎟⎠⎞
⎜⎝⎛ 60kJ/s 50W&
⋅ min 1s
cycle)kJ/kg
ote that t wo revolutions in one cycle in four-stroke engines.
cle is
==kg)(794 001978.0(
rev/cycle) 2(2net
net
wmn
t&
N here are t
(e) The mass of fuel burned during one cy
kg 0001164.0kg) 001978.0(
16AF =⎯→⎯−
=⎯→⎯−
== ff
f
f
ft
f
a mm
mm
mmmm
Finally, the specific fuel consumption is
g/kWh 267=⎟⎠⎞
⎜⎝⎛
⎟⎟⎠
⎞⎜⎜⎝
⎛==
kWh 1kJ 3600
kg 1g 1000
kJ/kg) kg)(794 001978.0(kg 0001164.0sfc
netwmm
t
f
PROPRIETARY MATERIALpreparation. If you are a student using this Manual, you are using it without permission.
. © 2011 The McGraw-Hill Companies, Inc. Limited distribution permitted only to teachers and educators for course
9-27
9-40 The expressions for the maximum gas temperature and pressure of an ideal Otto cycle are to be determined when the compression ratio is doubled.
Assumptions 1 The air-standard assumptions are applicable. 2 Kinetic and potential energy changes are negligible. 3 Air is an ideal gas with constant specific heats.
Analysis The temperature at the end of the compression varies with the compression ratio as
11
212 =⎟⎟
⎠⎜⎜⎝
= rTTTv
1
1 −−
⎞⎛ kk
v
T1 is fixed. The temperature rise during the combustion remains constant since the mount of heat addition is fixed. Then, the maximum cycle temperature is given by
he smallest gas specific volume during the cycle is
v
P
4
1
3
2 qout
qinsince a
11in2in3 // −+=+= krTcqTcqT vv
T
r1
3v
v =
When this is combined with the maximum temperature, the maximum pressure is given by
( )11in
13
33 / −+== krTcqRrRTP vvv
9-41 It is to be determined if the polytropic exponent to be used in an Otto cycle model will be greater than or less than the isentropic exponent.
Assumptions 1 The air-standard assumptions are applicable. 2 Kinetic and potential energy changes are negligible. 3 Air is n ideal ga with constant specific heats.
rocess,
=kPv
heat is lost during the expansion of the gas,
where T4s is the temperature that would occur if the expansion were reversible and adiabatic (n=k). This can only occur when
a s
Analysis During a polytropic p
PROPRIETARY MATERIAL. © 2011 The McGraw-Hill Companies, Inc. Limited distribution permitted only to teachers and educators for course
constant
constant/)1( =
=− nn
n
TP
Pv
v
P
4
1
3
2 qout
qinand for an isentropic process,
constant/)1( =− kkTP
constant
If
sTT 44 >
kn ≤
preparation. If you are a student using this Manual, you are using it without permission.
9-28
Diesel Cycle
9-42C A diesel engine differs from the gasoline engine in the way combustion is initiated. In diesel engines combustion is initiated by compressing the air above the self-ignition temperature of the fuel whereas it is initiated by a spark plug in a gasoline engine.
9-43C The Diesel cycle differs from the Otto cycle in the heat addition process only; it takes place at constant volume in the Otto cycle, but at constant pressure in the Diesel cycle.
9-44C The gasoline engine.
9-45C Diesel engines operate at high compression ratios because the diesel engines do not have the engine knock problem.
9-46C Cutoff ratio is the ratio of the cylinder volumes after and before the combustion process. As the cutoff ratio decreases, the efficiency of the diesel cycle increases.
9-47 An ideal diesel cycle has a compression ratio of 20 and a cutoff ratio of 1.3. The maximum temperature of the air and the rate of heat addition are to be determined.
Assumptions 1 The air-standard assumptions are applicable. 2 Kinetic and potential energy changes are negligible. 3 Air is an ideal gas with constant specific heats.
Properties The properties of air at room temperature are cp = 1.005 kJ/kg·K, cv = 0.718 kJ/kg·K, R = 0.287 kJ/kg·K, and k = 1.4 (Table A-2a).
Analysis We begin by using the process types to fix the temperatures of the states.
PROPRIETARY MATERIAL. © 2011 The McGraw-Hill Companies, Inc. Limited distribution permitted only to teachers and educators for course
( ) K 6.95420K) 288( 14.111
1
2
112 ===⎟⎟
⎠
⎞⎜⎜⎝
⎛= −−
−k
k
rTTTv
v
K 1241===⎟⎟⎠
⎜⎜⎝
= K)(1.3) 6.954(22
23 crTTTv
⎞⎛ 3v
ombining e first law as applied to the various processes with the process equations gi
C th ves v
P
4
1
2 3qin
qout
6812.0)13.1(4.1
13.1111 4.1 −−kr20
1)1(
11.41th =
−−=
−−=
−c
ck rkr
η
According to the definition of the thermal efficiency,
1−
kW 367===0.6812
kW 250
th
netin η
WQ
&&
preparation. If you are a student using this Manual, you are using it without permission.
9-29
9-48 An ideal dual cycle has a compression ratio of 14 and cutoff ratio of 1.2. The thermal efficiency, amount of heat added, and the maximum gas pressure and temperature are to be determined.
Assumptions 1 The air-standard assumptions are applicable. 2 Kinetic and potential energy changes are negligible. 3 Air is an ideal gas with constant specific heats.
Properties The properties of air at room temperature are cp = 1.005 kJ/kg·K, cv = 0.718 kJ/kg·K, R = 0.287 kJ/kg·K, and k = 1.4 (Table A-2a).
Analysis The specific volume of the air at the start of the compression is
v
P
4
1
2
3
qout
x
qin/kgm 051.1
kPa 8011 ===
Pv
K) 293)(K/kgmkPa 287.0( 33
1 ⋅⋅RT
olume at the end of the compression is and the specific v
/kgm 07508.014
12 ===
/kgm 051.1 33
rv
he pressu at the end of the compression is
v
T re
( )
PROPRIETARY MATERIAL. © 2011 The McGraw-Hill Companies, Inc. Limited distribution permitted only to teachers and educators for course
kPa 321914kPa) 80( 4.11
2
112 ===⎟⎟
⎠⎜⎜⎝
= krPPPv
temperature at the end of the compression is
⎞⎛k
v
and the maximum pressure is
x kPa 4829==== kPa) 3219)(5.1(23 PrPP p
The
( ) K 0.84214K) 293( 14.111
112 =⎟⎟
⎠
⎞⎜⎜⎝
⎛= −krTTT
v
v1
2== −
−k
and K 1263kPa 3219k 4829K) 0.842(3
2 ⎜⎛=⎟⎟
⎞⎜⎜⎛
=P
TTxPa
2=⎟
⎠⎞
⎝⎠⎝ P
rom the definition of cutoff ratio
==== vvv cxc rr
F
/kgm 09010.0)/kgm 07508.0)(2.1( 332 3
The remaining state temperatures are then
K 1516=⎟⎠⎞
⎜⎝⎛=⎟⎟
⎠
⎞⎜⎜⎝
⎛=
0.075080.09010K) 1263(3
3x
xTTv
v
K 5.5671.051
0.09010K) 1516(14.11
434 ⎜⎜⎝
= TTv
3 =⎟⎠⎞
⎜⎝⎛=⎟⎟
⎠
⎞⎛ −−kv
pplying t e first law and work expression to the heat addition processes gives
The heat rejected is
A h
kJ/kg 556.5=−⋅+−⋅=
−+−=
K)12631516)(KkJ/kg 005.1(K)0.8421263)(KkJ/kg 718.0(
)()( 32in xpx TTcTTcq v
kJ/kg 1.197K)2935.567)(KkJ/kg 718.0()( 14out =−⋅=−= TTcq v
0.646=−=−=kJ/kg 556.5kJ/kg 197.1
11in
outth q
qη Then,
preparation. If you are a student using this Manual, you are using it without permission.
9-30
9-49 An ideal dual cycle has a compression ratio of 14 and cutoff ratio of 1.2. The thermal efficiency, amount of heat added, and the maximum gas pressure and temperature are to be determined.
Assumptions 1 The air-standard assumptions are applicable. 2 Kinetic and potential energy changes are negligible. 3 Air is an ideal gas with constant specific heats.
Properties The properties of air at room temperature are cp = 1.005 kJ/kg·K, cv = 0.718 kJ/kg·K, R = 0.287 kJ/kg·K, and k = 1.4 (Table A-2).
Analysis The specific volume of the air at the start of the compression is
/kgm 9076.0kPa 801
1 ===P
vK) 253)(K/kgmkPa 287.0( 3
31 ⋅⋅RT
olume at the end of the compression is
v
P
4
1
2
3
qout
x
qinand the specific v
/kgm 06483.014
12 ===
/kgm 9076.0 33
rv
he pressu at the end of the compression is
v
T re
( )
PROPRIETARY MATERIAL. © 2011 The McGraw-Hill Companies, Inc. Limited distribution permitted only to teachers and educators for course
kPa 321914kPa) 80( 4.11
2
112 ===⎟⎟
⎠⎜⎜⎝
= krPPPv
temperature at the end of the compression is
⎞⎛k
v
and the maximum pressure is
x kPa 4829==== kPa) 3219)(5.1(23 PrPP p
The
( ) K 1.72714K) 253( 14.111
112 =⎟⎟
⎠
⎞⎜⎜⎝
⎛= −krTTT
v
v1
2== −
−k
and K 1091kPa 3219kP 4829K) 1.727(3
2 ⎜⎛=⎟⎟
⎞⎜⎜⎛
=P
TTxa
2=⎟
⎠⎞
⎝⎠⎝ P
rom the definition of cutoff ratio
==== vvv cxc rr
F
/kgm 07780.0)/kgm 06483.0)(2.1( 332 3
The remaining state temperatures are then
K 1309=⎟⎠⎞
⎜⎝⎛=⎟⎟
⎠
⎞⎜⎜⎝
⎛=
0.064830.07780K) 1091(3
3x
xTTv
v
K 0.4900.9076
0.07780K) 1309(14.11
434 ⎜⎜⎝
= TTv
3 =⎟⎠⎞
⎜⎝⎛=⎟⎟
⎠
⎞⎛ −−kv
pplying t e first law and work expression to the heat addition processes gives
The heat rejected is
A h
kJ/kg 480.4=−⋅+−⋅=
−+−=
K)10911309)(KkJ/kg 005.1(K)1.7271091)(KkJ/kg 718.0(
)()( 32in xpx TTcTTcq v
kJ/kg 2.170K)2530.490)(KkJ/kg 718.0()( 14out =−⋅=−= TTcq v
0.646=−=−=kJ/kg 480.4kJ/kg 170.2
11in
outth q
qη Then,
preparation. If you are a student using this Manual, you are using it without permission.
9-31
9-50 An air-standard Diesel cycle with a compression ratio of 18.2 is considered. The cutoff ratio, the heat rejection per unit mass, and the thermal efficiency are to be determined.
Assumptions 1 The air-standard assumptions are applicable. 2 Kinetic and potential energy changes are negligible. 3 Air is an ideal gas with variable specific heats.
Properties The properties of air are given in Table A-17.
PROPRIETARY MATERIAL. © 2011 The McGraw-Hill Companies, Inc. Limited distribution permitted only to teachers and educators for course
v
P
4
1
2 3 qin
qout
1700 KAnalysis (a) Process 1-2: isentropic compression.
2.621kJ/kg 07.214
K 3001
11 =
=⎯→⎯=
r
uT
v
( )kJ/kg 83.934
K 7.90113.34 2.621
2.181112
====r rrr vv
vv
11
2
22
==
⎯→⎯hTv
rocess 2- P = constant heat addition. P 3:
1.885===⎯→⎯=K 901.7K 1700
2
3
2
3
2
22
3
33
TT
TP
TP
v
vvv
(b) K 001733
=⎯→⎯=T
rv
kJ/kg 27.94583.9341.1880
761.4kJ/kg 1.1880
23
3
=−=−=
=
hhq
h
in
Process 3-4: isentropic expansion.
( ) kJ/kg 91.60297.45761.4885.1
2.18885.1885.1 4
2
4
3
43334
=⎯→⎯===== urrrrr vv
v
vv
v
vv
Process 4-1: v = constant heat rejection.
(c) 58.9%
kJ/kg 388.84
==−=−=
=−=−=
589.0kJ/kg 945.27kJ/kg 388.84
11
07.21491.602
in
outth
14out
uuq
η
preparation. If you are a student using this Manual, you are using it without permission.
9-32
9-51 An air-standard Diesel cycle with a compression ratio of 18.2 is considered. The cutoff ratio, the heat rejection per unit mass, and the thermal efficiency are to be determined.
Assumptions 1 The air-standard assumptions are applicable. 2 Kinetic and potential energy changes are negligible. 3 Air is an ideal gas with constant specific heats.
Properties The properties of air at room temperature are cp = 1.005 kJ/kg.K, cv = 0.718 kJ/kg.K, and k = 1.4 (Table A-2).
Analysis (a) Process 1-2: isentropic compression.
PROPRIETARY MATERIAL. © 2011 The McGraw-Hill Companies, Inc. Limited distribution permitted only to teachers and educators for course
( )( ) K 5.95718.2K 3002
12 =⎟⎟⎠
⎜⎜⎝
= TTv
0.41
1 =⎞⎛
−kv
rocess 2-3: P = constant heat addition. P
1.775===⎯→⎯=K 957.5K 1700
2
3
2
3
2
22
3
33
T TT
TPP
v
vvv
(b) ( ) ( )( ) kJ/kg 2.746K5.9571700kJ/kg.K 1.0052323in =−=−=−= TTchhq p
rocess 3-4: isentropic expansion.
v
P
4
1
2 3qin
P
( ) K 1.67018.21.775K 1700
775.1 23
334 ⎟⎟⎜⎜
⎝
⎛=⎟⎟
⎠
⎞⎜⎜⎝
⎛=
vTTT
v
v
v
0.41
4
1
4=⎟
⎠⎞
⎜⎝⎛=
⎠
⎞−− kk
v = constant heat rejection.
(c)
Process 4-1:
( )( )( )
64.4%
kJ/kg 265.7
==−=−=
=−=−=−=
644.0kJ/kg 746.2kJ/kg 265.7
11
K300670.1kJ/kg.K 0.718
in
outth
1414out
TTcuuq
η
v
preparation. If you are a student using this Manual, you are using it without permission.
9-33
9-52 An ideal diesel engine with air as the working fluid has a compression ratio of 20. The thermal efficiency and the mean effective pressure are to be determined.
Assumptions 1 The air-standard assumptions are applicable. 2 Kinetic and potential energy changes are negligible. 3 Air is an ideal gas with constant specific heats.
Properties The properties of air at room temperature are cp = 1.005 kJ/kg·K, cv = 0.718 kJ/kg·K, R = 0.287 kJ/kg·K, and k = 1.4 (Table A-2).
Analysis (a) Process 1-2: isentropic compression.
v
P
4
1
2 3 qin
qout
( )( ) K 971.120K 293 0.41
2
112 ==⎟⎟
⎠
⎞⎜⎜⎝
⎛=
−k
TTV
V
Process 2-3: P = constant heat addition.
2.265K971.1K2200
2
3
223 TT V3223 ===⎯→⎯=
TTPP VVV
Process 3-4: isentropic expansion.
3
( )
( ) ( )( )( ) ( )( )
63.5%===
=−=−=
=−⋅=−=−=
=−⋅=−=−=
=⎟⎠⎞
⎜⎝⎛=⎟
⎠⎞
⎜⎝⎛=⎟⎟
⎠
⎞⎜⎜⎝
⎛=⎟⎟
⎠
⎞⎜⎜⎝
⎛=
−−−
kJ/kg 1235kJ/kg 784.4
kJ/kg 784.46.4501235
kJ/kg 450.6K293920.6KkJ/kg 0.718
kJ/kg 1235K971.12200KkJ/kg 1.005
K 920.620
2.265K 2200265.2265.2
in
outnet,th
outinoutnet,
1414out
2323in
0.41
3
1
4
23
1
4
334
qw
qqw
TTcuuq
TTchhq
rTTTT
p
kkk
η
v
V
V
V
V
( )( )
( ) ( )( )kPa933
kJmkPa
1/201/kgm 0.885kJ/kg 784.4
/11MEP
/kgm 0.885kPa 95
K 293K/kgmkPa 0.287
3
31
outnet,
21
outnet,
max2min
max3
3
1
11
=⎟⎟⎠
⎞⎜⎜⎝
⎛ ⋅
−=
−=
−=
==
==⋅⋅
==
rww
r
PRT
vvv
vvv
vv (b)
PROPRIETARY MATERIALpreparation. If you are a student using this Manual, you are using it without permission.
. © 2011 The McGraw-Hill Companies, Inc. Limited distribution permitted only to teachers and educators for course
9-34
9-53 A diesel engine with air as the working fluid has a compression ratio of 20. The thermal efficiency and the mean effective pressure are to be determined. Assumptions 1 The air-standard assumptions are applicable. 2 Kinetic and potential energy changes are negligible. 3 Air is an ideal gas with constant specific heats. Properties The properties of air at room temperature are cp = 1.005 kJ/kg·K, cv = 0.718 kJ/kg·K, and k = 1.4 (Table A-2). Analysis (a) Process 1-2: isentropic compression.
v
P
4
1
2 3 qin
qout
Polytropic ( )( ) K 971.120K 293 0.41
2
112 ==⎟⎟
⎠
⎞⎜⎜⎝
⎛=
−k
TTV
V
Process 2-3: P = constant heat addition.
2.265K 971.1K 2200
2
3
223=⎯→⎯=
TT V
PROPRIETARY MATERIAL. © 2011 The McGraw-Hill Companies, Inc. Limited distribution permitted only to teachers and educators for course
3223 ==TTPP VVV
Process 3-4: polytropic expansion.
3
( )
( ) ( )( )( ) ( )( ) kJ/kg 526.3K 2931026KkJ/kg 0.7181414out =−⋅=−=−= TTcuuq v
te that q
kJ/kg 1235K 971.12200KkJ/kg 1.005
K 102620
2.265K 2200r
2.2652.265
2323in
0.351n
3
1n
4
23
1
4
334
=−⋅=−=−=
=⎟⎠⎞
⎜⎝⎛=⎟
⎠⎞
⎜⎝⎛=⎟⎟
⎠
⎞⎜⎜⎝
⎛=⎟⎟
⎠
⎞⎜⎜⎝
⎛=
−−−
TTchhq
TTTT
p
n
V
V
V
V
t some heat is also rejected during the polytropic rocess, w ch is determined from an energy balance on process 3-4:
No oup hi
in this case does not represent the entire heat rejected since
( ) ( )( )
( )( )( )
kJ/kg 120.1K 22001026KkJ/kg 0.718kJ/kg
34out34,in34,34out34,in34,
system
=−⋅+
−+=⎯→⎯−=− TTcwquuwq v
h means that 120.1 kJ/kg of heat is transferred to the combustion gases during the expansion process. This is unrealistic since the gas is at a much higher temperature than the surroundings, and a hot gas loses heat during polytropic
pansion. The cause of this unrealistic result is the constant specific heat assumption. If we were to use u data from the ir table, we would obtain
kJ/kg 9631.351
K 22001026KkJ/kg 0.2871
outin
34out34,
∆=−
=−
−⋅=
−−
=
EEEnTTR
w
963=
whic
exa
( ) kJ/kg 1.128)4.18723.781(96334out34,in34, −=−+=−+= uuwq
whic hh is a eat loss as expected. Then qout becomes kJ/kg 654.43.5261.128out41,out34,out =+=+= qqq
and
47.0%===
=−=−=
kJ/kg 1235kJ/kg 580.6
kJ/kg 580.64.6541235
in
outnet,th
outinoutnet,
qw
qqw
η
( )( )
( ) ( )( )kPa 691=⎟
⎟⎠
⎞⎜⎜⎝
⎛ ⋅
−=
−=
−=
==
==⋅⋅
==
kJmkPa 1
1/201/kgm 0.885kJ/kg 580.6
/11MEP
/kgm 0.885kPa 95
K 293K/kgmkPa 0.287
3
31
outnet,
21
outnet,
max2min
max3
3
1
11
rww
r
PRT
vvv
vvv
vv (b)
preparation. If you are a student using this Manual, you are using it without permission.
9-35
9-54 Problem 9-53 is reconsidered. The effect of the compression ratio on the net work output, mean effective pressure, and thermal efficiency is to be investigated. Also, T-s and P-v diagrams for the cycle are to be plotted.
Analysis Using EES, the problem is solved as follows:
Procedure QTotal(q_12,q_23,q_34,q_41: q_in_total,q_out_total) q_in_total = 0 q_out_total = 0 IF (q_12 > 0) THEN q_in_total = q_12 ELSE q_out_total = - q_12 If q_23 > 0 then q_in_total = q_in_total + q_23 else q_out_total = q_out_total - q_23 If q_34 > 0 then q_in_total = q_in_total + q_34 else q_out_total = q_out_total - q_34 If q_41 > 0 then q_in_total = q_in_total + q_41 else q_out_total = q_out_total - q_41 END "Input Data" T[1]=293 [K] P[1]=95 [kPa] T[3] = 2200 [K] n=1.35 {r_comp = 20} "Process 1-2 is isentropic compression" s[1]=entropy(air,T=T[1],P=P[1]) s[2]=s[1] T[2]=temperature(air, s=s[2], P=P[2]) P[2]*v[2]/T[2]=P[1]*v[1]/T[1] P[1]*v[1]=R*T[1] R=0.287 [kJ/kg-K] V[2] = V[1]/ r_comp "Conservation of energy for process 1 to 2" q_12 - w_12 = DELTAu_12 q_12 =0"isentropic process" DELTAu_12=intenergy(air,T=T[2])-intenergy(air,T=T[1]) "Process 2-3 is constant pressure heat addition" P[3]=P[2] s[3]=entropy(air, T=T[3], P=P[3]) P[3]*v[3]=R*T[3] "Conservation of energy for process 2 to 3" q_23 - w_23 = DELTAu_23 w_23 =P[2]*(V[3] - V[2])"constant pressure process" DELTAu_23=intenergy(air,T=T[3])-intenergy(air,T=T[2]) "Process 3-4 is polytropic expansion" P[3]/P[4] =(V[4]/V[3])^n s[4]=entropy(air,T=T[4],P=P[4]) P[4]*v[4]=R*T[4] "Conservation of energy for process 3 to 4" q_34 - w_34 = DELTAu_34 "q_34 is not 0 for the ploytropic process" DELTAu_34=intenergy(air,T=T[4])-intenergy(air,T=T[3]) P[3]*V[3]^n = Const w_34=(P[4]*V[4]-P[3]*V[3])/(1-n) "Process 4-1 is constant volume heat rejection" V[4] = V[1] "Conservation of energy for process 4 to 1" q_41 - w_41 = DELTAu_41 w_41 =0 "constant volume process" DELTAu_41=intenergy(air,T=T[1])-intenergy(air,T=T[4]) Call QTotal(q_12,q_23,q_34,q_41: q_in_total,q_out_total) w_net = w_12+w_23+w_34+w_41
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. © 2011 The McGraw-Hill Companies, Inc. Limited distribution permitted only to teachers and educators for course
9-36
Eta_th=w_net/q_in_total*100 "Thermal efficiency, in percent" "The mean effective pressure is:" MEP = w_net/(V[1]-V[2])
rcomp ηth MEP [kPa]
wnet [kJ/kg]
14 47.69 970.8 797.9 16 50.14 985 817.4 18 52.16 992.6 829.8 20 53.85 995.4 837.0 22 55.29 994.9 840.6 24 56.54 992 841.5
4.0 4.5 5.0 5.5 6.0 6.5 7.0 7.5200400600800
10001200140016001800200022002400
s [kJ/kg-K]
T [K
]
95
kPa
340
.1 kP
a
592
0 kP
a
0.0
44
0.1
0.8
8 m
3/kg
Air
2
1
3
4
10-2 10-1 100 101 102101
102
103
104
101
102
103
104
v [m3/kg]
P [k
Pa] 293 K
1049 K
2200 K
5.69 6.74 kJ/kg-K
Air
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. © 2011 The McGraw-Hill Companies, Inc. Limited distribution permitted only to teachers and educators for course
9-37
14 16 18 20 22 24790
800
810
820
830
840
850
rcomp
wne
t [k
J/kg
]
14 16 18 20 22 2447
49
51
53
55
57
rcomp
ηth
14 16 18 20 22 24970
975
980
985
990
995
1000
rcomp
MEP
[kP
a]
PROPRIETARY MATERIALpreparation. If you are a student using this Manual, you are using it without permission.
. © 2011 The McGraw-Hill Companies, Inc. Limited distribution permitted only to teachers and educators for course
9-38
9-55 A four-cylinder ideal diesel engine with air as the working fluid has a compression ratio of 22 and a cutoff ratio of 1.8. The power the engine will deliver at 2300 rpm is to be determined.
Assumptions 1 The cold air-standard assumptions are applicable. 2 Kinetic and potential energy changes are negligible. 3 Air is an ideal gas with constant specific heats.
Properties The properties of air at room temperature are cp = 1.005 kJ/kg·K, cv = 0.718 kJ/kg·K, R = 0.287 kJ/kg·K, and k = 1.4 (Table A-2).
Analysis Process 1-2: isentropic compression.
v
P
4
1
2 3 Qin
Qout
( )( ) K 118122K 343 0.41
2
112 ==⎟⎟
⎠
⎞⎜⎜⎝
⎛=
−k
TTV
V
Process 2-3: P = constant heat addition.
( )( ) K 2126K 11811.88.1 222
33
23 TT223 ====⎯→⎯= TTT
PPv
vvv
Process 3-4: isentropic expansion.
3
( )
( ) ( )
( ) ( )( )( )( )
( )( ) kW 48.0=== kJ/rev 1.251rev/s 2300/60outnet,outnet, WnW &&
Discussion No
=−=−=
=−⋅=−=−=
=−⋅=−=−=
=⋅⋅
==
=⎟⎠⎞
⎜⎝⎛=⎟
⎠⎞
⎜⎝⎛=⎟⎟
⎠
⎞⎜⎜⎝
⎛=⎟⎟
⎠
⎞⎜⎜⎝
⎛=
−−−
kJ/rev 251.16198.0871.1
kJ 6198.0K343781KkJ/kg 0.718kg 001971.0
kJ .8711K)11812216)(KkJ/kg 1.005)(kg 001971.0(
kg 001971.0)K 343)(K/kgmkPa 0.287(
)m 0.0020)(kPa 97(
K 78122
8.1K 22162.22.2
outinoutnet,
1414out
2323in
3
3
1
11
4.01
3
1
4
23
1
4
334
QQW
TTmcuumQ
TTmchhmQ
RTP
m
rTTTT
v
p
kkk
V
V
V
V
V
te that for 2-stroke engines, 1 thermodynamic cycle is equivalent to 1 mechanical cycle (and thus revolutions).
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9-39
9-56 A four-cylinder ideal diesel engine with nitrogen as the working fluid has a compression ratio of 22 and a cutoff ratio of 1.8. The power the engine will deliver at 2300 rpm is to be determined.
Assumptions 1 The air-standard assumptions are applicable with nitrogen as the working fluid. 2 Kinetic and potential energy changes are negligible. 3 Nitrogen is an ideal gas with constant specific heats.
Properties The properties of nitrogen at room temperature are cp = 1.039 kJ/kg·K, cv = 0.743 kJ/kg·K, R = 0.2968 kJ/kg·K, and k = 1.4 (Table A-2).
Analysis Process 1-2: isentropic compression.
v
P
4
1
2 3 Qin
Qout
PROPRIETARY MATERIAL. © 2011 The McGraw-Hill Companies, Inc. Limited distribution permitted only to teachers and educators for course
( )( )22K 3432
12 =⎟⎟⎠
⎜⎜⎝
= TTV
K 11810.41
1 =⎞⎛
−kV
Process 2-3: P = constant heat addition.
( )( ) K 2126K 11811.88.1 222
33
2
22
3
33 =⎯→⎯= TTT
=== TTPP
v
vvv
rocess 3- isentropic expansion.
P 4:
( )
( ) ( )
( ) ( )( )( )( )
( )( ) kW 47.9===
=−=−=
=−⋅=−=−=
=−⋅=−=−=
=⋅⋅
==
=⎟⎠⎞
⎜⎝⎛=⎟
⎠⎞
⎜⎝⎛=⎟⎟
⎠
⎞⎜⎜⎝
⎛=⎟⎟
⎠
⎞⎜⎜⎝
⎛=
−−− 4.0111
Tkkk
kJ/rev 1.251rev/s 2300/60
kJ/rev 251.16202.0871.1
kJ 6202.0K343781KkJ/kg 0.743kg 001906.0
kJ .8711K)11812216)(KkJ/kg 1.039)(kg 001906.0(
kg 001906.0)K 343)(K/kgmkPa 0.2968(
)m 0.0020)(kPa 97(
K 78122
8.1K 22162.22.2
outnet,outnet,
outinoutnet,
1414out
2323in
3
3
1
11
34
23
4
334
WnW
QQW
TTmcuumQ
TTmchhmQ
RTP
m
rTTT
v
p
&&
V
V
V
V
V
Discussion Note that for 2-stroke engines, 1 thermodynamic cycle is equivalent to 1 mechanical cycle (and thus revolutions).
preparation. If you are a student using this Manual, you are using it without permission.
9-40
9-57 An ideal dual cycle has a compression ratio of 18 and cutoff ratio of 1.1. The power produced by the cycle is to be determined.
Assumptions 1 The air-standard assumptions are applicable. 2 Kinetic and potential energy changes are negligible. 3 Air is an ideal gas with constant specific heats.
Properties The properties of air at room temperature are cp = 1.005 kJ/kg·K, cv = 0.718 kJ/kg·K, R = 0.287 kJ/kg·K, and k = 1.4 (Table A-2a).
Analysis We begin by fixing the temperatures at all states.
( ) K 7.92418K) 291( 14.111
1
2
112 ===⎟⎟
⎠
⎞⎜⎜⎝
⎛= −−
−k
k
rTTTv
v
v
P
4
1
2
3
qout
x
qin
( ) kPa 514818kPa) 90( 4.11
2
112 ===⎟⎟
⎠
⎞⎜⎜⎝
⎛= k
k
rPPPv
v
x kPa 5663kPa) 5148)(1.1(23 ==== PrPP p
K 1017kPa 5148kPa 5663K) 7.924(
22 =⎟
⎠⎞
⎜⎝⎛=⎟⎟
⎠
⎞⎜⎜⎝
⎛=
PP
TT x
xc
x
K 1119K) 1017)(1.1( === TrT 3
K 8.365181.1K) 1119(
14.11
3
1
4
33 4 =⎟
⎠⎞
⎜⎝⎛=⎟⎟
⎠
⎞⎜⎜⎝
⎛=⎟⎟
⎠
⎞⎜⎜⎝
⎛=
−−− kc
k
rr
TTTv
v
Applying the first law to each of the processes gives
kJ/kg 0.455K)2917.924)(KkJ/kg 718.0()(
PROPRIETARY MATERIAL. © 2011 The McGraw-Hill Companies, Inc. Limited distribution permitted only to teachers and educators for course
1221 =−⋅=−= TTcw − v
kJ/kg 5.102K)10171119)(KkJ/kg 005.1()( 33 =−⋅=−=− xpx TTcq
)( 333 −−= −− xxx TTcqw v kJ/kg 26.29K)10171119)(KkJ/kg 718.0(5.102 =−⋅−=
kJ/kg 8.540K)8.3651119)(KkJ/kg 718.0()( 434 3 =−⋅=−=− TTcw v
The net work of the cycle is
kJ/kg 1.1150.45526.298.54021343net =−+=−+= −−− wwww x
The mass in the device is given by
kg 003233.0K) 291)(K/kgmkPa 287.0(
)m kPa)(0.003 90(3
3
1
11 =⋅⋅
==RTP
mV
The net power produced by this engine is then
kW 24.8=== cycle/s) 0/60kJ/kg)(400 115.1kg/cycle)( 003233.0(netnet nmwW &&
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9-41
9-58 A dual cycle with non-isentropic compression and expansion processes is considered. The power produced by the cycle is to be determined.
Assumptions 1 The air-standard assumptions are applicable. 2 Kinetic and potential energy changes are negligible. 3 Air is an ideal gas with constant specific heats.
Properties The properties of air at room temperature are cp = 1.005 kJ/kg·K, cv = 0.718 kJ/kg·K, R = 0.287 kJ/kg·K, and k = 1.4 (Table A-2a).
Analysis We begin by fixing the temperatures at all states.
PROPRIETARY MATERIAL. © 2011 The McGraw-Hill Companies, Inc. Limited distribution permitted only to teachers and educators for course
( ) K 7.92418K) 291( 14.111
1
2
112 ===⎟⎟
⎠
⎞⎜⎜⎝
⎛= −−
−k
k
s rTTTv
v
K 10370.85
K )291(924.7K) 291(1212
12
12 =−
+=−
+=⎯→⎯−−
=η
ηTT
TTTTTT ss
v
P
4
1
2
3
qout
x
qin
( ) kPa 514818kPa) 90( 4.11
2
112 ===⎟⎟
⎠
⎞⎜⎜⎝
⎛= k
k
rPPPv
v
kPa) 5148)(1.1(23 ==== PrPP px 5663 kPa
K 1141kPa 5148kPa 5663K) 1037(
22 =⎟
⎠⎞
⎜⎝⎛=⎟⎟
⎠
⎞⎜⎜⎝
⎛=
PP
TT xx
K 1255K) 1141)(1.1(3 === xcTrT
K 3.410181.1K) 1255(
14.11
3
1
4
334 =⎟
⎠⎞
⎜⎝⎛=⎟⎟
⎠
⎞⎜⎜⎝
⎛=⎟⎟
⎠
⎞⎜⎜⎝
⎛=
−−− kc
k
s rr
TTTv
v
K 8.494K )3.410(1255)90.0(K) 1255()( 433443
43 =−−=−−=⎯→⎯−−
= ss
TTTTTTTT
ηη
Applying the first law to each of the processes gives
kJ/kg 6.535K)2911037)(KkJ/kg 718.0()( 122 1 =−⋅=−= TTcw v
5 =−⋅
−
00.1()( 33 =−=− xpx TTcq kJ/kg 6.114K)11411255)(KkJ/kg
kJ/kg 75.32K)11411255)(KkJ/kg 718.0(6.114)( 333 =− ⋅−=−−= −− xxx TTcqw v
kJ/kg 8.545K)8.4941255)(KkJ/kg 718.0()( 4343 =−⋅=−=− TTcw v
The net work of the cycle is
kJ/kg 95.426.53575.328.54521343 net =−+=−+= −−− wwww x
The mass in the device is given by
kg 003233.0K) 291)(K/kgmkPa 287.0(
)m kPa)(0.003 90(3
3
1
11 =⋅⋅
==RTP
mV
The net power produced by this engine is then
kW 9.26=== cycle/s) 0/60kJ/kg)(400 42.95kg/cycle)( 003233.0(netnet nmwW &&
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9-42
9-59 An ideal dual cycle has a compression ratio of 15 and cutoff ratio of 1.4. The net work, heat addition, and the thermal efficiency are to be determined.
Assumptions 1 The air-standard assumptions are applicable. 2 Kinetic and potential energy changes are negligible. 3 Air is an ideal gas with constant specific heats.
Properties The properties of air at room temperature are R = 0.287 kPa·m3/kg.K (Table A-1), cp = 1.005 kJ/kg·K, cv = 0.718 kJ/kg·K, and k = 1.4 (Table A-2a).
Analysis Working around the cycle, the germane properties at the various states are
( ) K 4.87715K) 297( 14.111
1
2
112 ===⎟⎟
⎠
⎞⎜⎜⎝
⎛= −−
−k
k
rTTTv
v
v
P
4
1
2
3
qout
x
qin( ) kPa 434315kPa) 98( 4.11
2
112 ===⎟⎟
⎠
⎞⎜⎜⎝
⎛= k
k
rPPPv
v
23 === PrP px kPa 4777kPa) 4343)(1.1( =P
K 1.965kPa 4343kPa 4777K) 4.877(
22 =⎟
⎠⎞
⎜⎝⎛=⎟⎟
⎠
⎞⎜⎜⎝
⎛=
PP
TT xx
K 1351K)(1.4) 1.965(33 ===⎟⎟
⎠⎜⎜⎝
= cxx
x rTTTv
⎞⎛ v
K 2.523151.4K) 1351(
14.11
3
1
4
33 4 =⎟
⎠⎞
⎜⎝⎛=⎟⎟
⎠
⎞⎜⎜⎝
⎛=⎟⎟
⎠
⎞⎜⎜⎝
⎛=
−−− kc
k
rr
TTTv
v
Applying the first law to each of the processes gives
kJ/kg 7.416K)2974.877)(KkJ/kg 718.0()( 1221 =−⋅=−=− TTcw v
kJ/kg 0.63K)4.8771.965)(KkJ/kg 718.0()(
PROPRIETARY MATERIAL. © 2011 The McGraw-Hill Companies, Inc. Limited distribution permitted only to teachers and educators for course
22 =−⋅=−= TTcq
33
− xx v
)( kJ/kg 8.387K)1.9651351)(KkJ/kg 005.1( =−⋅=xp −= TTcq −x
( 333 kJ/kg 8.110K)1.9651351)(KkJ/kg 718.0(kJ/kg 8.387) =−⋅−=
31351)(KkJ/kg 718.0()( 4343
−−= −− xxx TTcqw v
52 kJ/kg 4.594K)2. =−⋅=−=− TTcw v
The net work of the cycle is
kJ/kg 289=−+=−+= −−− 7.4168.1104.59421343 net wwww x
and the net heat addition is
Hence, the thermal efficiency is
kJ/kg 451=+=+= −− 8.3870.6332in xx qqq
0.641===kJ/kg 451kJ/kg 289
in
netth q
wη
preparation. If you are a student using this Manual, you are using it without permission.
9-43
9-60 A six-cylinder compression ignition engine operates on the ideal Diesel cycle. The maximum temperature in the cycle, the cutoff ratio, the net work output per cycle, the thermal efficiency, the mean effective pressure, the net power output, and the specific fuel consumption are to be determined.
Assumptions 1 The air-standard assumptions are applicable. 2 Kinetic and potential energy changes are negligible. 3 Air is an ideal gas with constant specific heats.
Properties The properties of air at 850 K are cp = 1.110 kJ/kg·K, cv = 0.823 kJ/kg·K, R = 0.287 kJ/kg·K, and k = 1.349 (Table A-2b).
Analysis (a) Process 1-2: Isentropic compression
1
Qin
2 3
4
Qout
( )( )
( )( ) kPa 504419kPa 95
K 1.95019K 340
1.349
2
112
1-1.3491
2
112
==⎟⎟⎠
⎞⎜⎜⎝
⎛=
==⎟⎟⎠
⎞⎜⎜⎝
⎛=
−
k
k
PP
TT
v
v
v
v
The clearance volume and the total volume of the engine at the beginning of compression process (state 1) are
3
3
m 0001778.0=cV
m 0045.019
+=⎯→⎯
+=
c
c
c
dcrV
V
V
VV
m 003378.00032.00001778.0 =+=+= dc VVV 31
The total mass contained in the cylinder is
( )( )kg 0.003288
K 340K/kgmkPa 0.287)m 378kPa)(0.003 95(
3
3
1
11 =⋅⋅
==RTP
mV
The mass of fuel burned during one cycle is
kg 0001134.0kg) 003288.0(
28 =⎯→⎯−
=⎯→⎯−
== ff
f
f
f
f
a mm
mm
mmmm
AF
Process 2-3: constant pressure heat addition
PROPRIETARY MATERIAL. © 2011 The McGraw-Hill Companies, Inc. Limited distribution permitted only to teachers and educators for course
kJ 723.48)kJ/kg)(0.9 kg)(42,500 0001134.0(HVin ==Q = cf qm η
003288.0(kJ 723.4)( TTTTmcv
The cutoff ratio is
kg)(0.823 Q K 2244=⎯→⎯−=⎯→⎯−= 3323in K)1.950(kJ/kg.K)
2.362===K 950.1K 2244
2
3
TT
β
(b) 33
12 m 0001778.0
19m 0.003378
===r
VV
23
14
3323 m 0004199.0)m 0001778.0)(362.2(
PP ==
===VV
VV β
preparation. If you are a student using this Manual, you are using it without permission.
9-44
Process 3-4: isentropic expansion.
( )
( ) kPa 9.302m 0.003378m 0.0004199kPa 5044
K 1084m 0.003378m 0.0004199K 2244
1.349
3
3
4
334
1-1.349
3
31
4
334
=⎟⎟⎠
⎞⎜⎜⎝
⎛=⎟⎟
⎠
⎞⎜⎜⎝
⎛=
=⎟⎟⎠
⎞⎜⎜⎝
⎛=⎟⎟
⎠
⎞⎜⎜⎝
⎛=
−
k
k
PP
TT
V
V
V
V
Process 4-1: constant voume heat rejection.
( ) ( )( ) kJ 013.2K3401084KkJ/kg 0.823kg) 003288.0(14out =−⋅=−= TTmcQ v
The net work output and the thermal efficiency are
kJ 2.710=−=−= 013.2723.4outinoutnet, QQW
57.4%==== 5737.0kJ 4.723in
th Qη kJ 2.710outnet,W
(c) The mean effective pressure is determined to be
kPa 847=⎟⎟⎠
⎞⎜⎜⎝
⎛ ⋅
−=
−=
00017.0003378.0(MEP
21 VV kJmkPa
m)78kJ 2.710 3
3outnet,W
) The power for engine speed of 1750 rpm is (d
kW 39.5=⎟⎠⎞⎛ min 1(rev/min) 1750n&
PROPRIETARY MATERIAL. © 2011 The McGraw-Hill Companies, Inc. Limited distribution permitted only to teachers and educators for course
⎜⎝ s 60rev/cycle)
ote that there are two revolutions in one cycle in four-stroke engines
(e) Finally, the specific fuel consumption is
==2(
kJ/cycle) (2.7102netnet WW&
N .
g/kWh 151=⎟⎠⎞
⎜⎝⎛
⎟⎟⎠
⎞⎜⎜⎝
⎛==
kWh 1kJ 3600
kg 1g 1000
kJ/kg 2.710kg 0001134.0sfc
netWm f
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9-45
9-61 An expression for cutoff ratio of an ideal diesel cycle is to be developed.
Assumptions 1 The air-standard assumptions are applicable. 2 Kinetic and potential energy changes are negligible. 3 Air is an ideal gas with constant specific heats.
Analysis Employing the isentropic process equations,
PROPRIETARY MATERIAL. © 2011 The McGraw-Hill Companies, Inc. Limited distribution permitted only to teachers and educators for course
−= krTT
1 TrTrrcTTcq kk −− −=−=
When this is solved for cutoff ratio, the result is
v
P
4
1
2 3 qin
qout
21
1
while the ideal gas law gives
11
23 TrrrTT kcc
−==
When the first law and the closed system work integral is applied to the constant pressure heat addition, the result is
)()( 11
123in cpp
11
in1Trc
qr kp
c −+=
preparation. If you are a student using this Manual, you are using it without permission.
9-46
9-62 An expression for the thermal efficiency of a dual cycle is to be developed and the thermal efficiency for a given case is to be calculated.
Assumptions 1 The air-standard assumptions are applicable. 2 Kinetic and potential energy changes are negligible. 3 Air is an ideal gas with constant specific heats.
Properties The properties of air at room temperature are cp = 1.005 kJ/kg·K, cv = 0.718 kJ/kg·K, R = 0.287 kJ/kg·K, and k = 1.4 (Table A-2)
Analysis The thermal efficiency of a dual cycle may be expressed as
)()( 32in
thxpx TTcTTcq −+−v
By applying the isentropic process relations for ideal gases with constant specific heats to the processes 1-2 and 3-4, as wellas the ideal gas equation of state, the tem
)(11 14out TTcq −
−=−= vη
peratures may be eliminated from the thermal efficiency expression. This yields
the result
⎥⎥⎦
⎤
⎢⎢⎣
⎡
−+−
−−=
− 1)1(111
1thpcp
kcp
k rrkrrr
rη
v
P
4
1
2
3
qout
x
qinwhere
PROPRIETARY MATERIAL. © 2011 The McGraw-Hill Companies, Inc. Limited distribution permitted only to teachers and educators for course
2PpP
r x= and x
cr v
v 3=
hen rc = , we obtain
W rp
⎟⎟
⎠
⎞
⎜⎜
⎝
⎛1 r
−+−
−−=
+
− 1)(
11
2
1
1thppp
kp
k rrrkrη
For the case r = 20 and rp = 2,
0.660=⎟⎟⎠
⎞⎜⎜⎝
⎛
−+−
−−=
+
− 12)22(4.112
2011
2
14.1
14.1thη
preparation. If you are a student using this Manual, you are using it without permission.
9-47
9-63 An expression regarding the thermal efficiency of a dual cycle for a special case is to be obtained.
Assumptions 1 The air-standard assumptions are applicable. 2 Kinetic and potential energy changes are negligible. 3 Air is an ideal gas with constant specific heats.
Analysis The thermal efficiency of a dual cycle may be expressed as
)()( 32in
thxpx TTcTTcq −+−v
By applying the isentropic process relations for ideal gases with constant specific heats to the processes 1-2 and 3-4, as wellas the ideal gas equation of state, the tem
)(11 14out TTcq −
−=−= vη
peratures may be eliminated from the thermal efficiency expression. This yields
the result
⎥⎥⎦
⎤
⎢⎢⎣
⎡
−+−
−−=
− 1)1(111
1thpcp
kcp
k rrkrrr
rη
v
P
4
1
2
3
qout
x
qinwhere
PROPRIETARY MATERIAL. © 2011 The McGraw-Hill Companies, Inc. Limited distribution permitted only to teachers and educators for course
2P
rp and Px=
xcr v
v 3=
hen rc = , we obtain
W rp
⎟⎟
⎠
⎞
⎜⎜
⎝ −+−
−−=
− 1)(
11
21thppp
pk rrrkr
η
Rearrangement of this result gives
⎛ +1 1kr
1th2
1
)1(1)(
1 −+
−=−+−
− k
ppp
kp r
rrrk
rη
preparation. If you are a student using this Manual, you are using it without permission.
9-48
9-64 The five processes of the dual cycle is described. The P-v and T-s diagrams for this cycle is to be sketched. An expression for the cycle thermal efficiency is to be obtained and the limit of the efficiency is to be evaluated for certain cases.
Assumptions 1 The air-standard assumptions are applicable. 2 Kinetic and potential energy changes are negligible. 3 Air is an ideal gas with constant specific heats.
Analysis (a) The P-v and T-s diagrams for this cycle are as shown.
s
T
52
1
4
3
v
P
5
1
2
4
qout
3
qin
(b) Apply first law to the closed system for processes 2-3, 3-4, and 5-1 to show:
PROPRIETARY MATERIAL. © 2011 The McGraw-Hill Companies, Inc. Limited distribution permitted only to teachers and educators for course
3 2 4 3
5 1
v p
out v
q C T T C T T= − + −
The cycle thermal efficiency is given by
( ) ( )in
( )q C T T= −
( )( ) ( )
( )( ) ( )
( )( ) ( )
5 1 1 5 1
3 2 4 3 2 3 2 3 4 3
5 1
34 3
1 1
/ 11 1 1
/ 1 / 1
/ 11
/ 1
voutth
in v p
th
C T T T T Tqq C T T C T T T T T kT T T
T TTT k T TT
η
η
− −= − = − = −
− + − − + −
−= −
−
23 2/ 1T T
T− +
Process 1-2 is isentropic; therefore, 1
12 1
1 2
kkT V r
T V
−
−⎛ ⎞= =⎜ ⎟
⎝ ⎠
Process 2-3 is constant volume; therefore,
3 3 3 3T PV P r= = =
Process 3-4 is constant pressure; therefore,
2 2 2 2pT PV P
3 34 4 4 4
4 3 3 3T T T V cPVPV T V r= ⇒ = =
Process 4-5 is isentropic; therefore, 1 1
5 34 4
k k
c cT r VV V− − 1 1 1
2
4 5 1 1 1
k k kcr V r
T V V V V r
− − −⎛ ⎞ ⎛ ⎞ ⎛ ⎞ ⎛ ⎞ ⎛ ⎞= = = = =⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟⎝ ⎠⎝ ⎠ ⎝ ⎠ ⎝ ⎠⎝ ⎠
Process 5-1 is constant volume; however T5/T1 is found from the following. 1
5 5 34 2 1c
kc k k
c p pr
T T T T T rr r r r r
−−⎛ ⎞= = =⎜ ⎟
T T TT T
1 4 3 2 1 ⎝ ⎠
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9-49
PROPRIETARY MATERIAL. © 2011 The McGraw-Hill Companies, Inc. Limited distribution permitted only to teachers and educators for course
The ratio T3/T1 is found from the following.
3 3 2
1 2 1
1kp
T T TT T T
r r −= =
The efficiency becomes
( ) ( )1 11 1th kp c
kpr r k rr r− −
11
kc pr r
η−
= −− + −
(c) In the limit as rp approaches unity, the cycle thermal efficiency becomes
( ) ( )
( )
11 1
1
1
1
1lim 1 lim
1 1
11lim 11
p p
p
kc p
th kr rp c
kc
th th Dieselrc
kp
k
r rr r k r
rk r
r r
r
η
η η
−→ →
→
−
−
⎧ ⎫−⎪ ⎪= − ⎨ ⎬− + −⎪ ⎪⎩ ⎭
⎡ ⎤−⎢ ⎥= − =−⎢ ⎥⎣ ⎦
(d) In the limit as rc approaches unity, the cycle thermal efficiency becomes
( ) ( ) ( )1 11 1
1
1
1
1 1lim 1 lim 1
1 1
1lim 1
c p
c
kc p p
th k kr rp c
th th Ottor
kp
k
r r rr r k r r rr r
r
η
η η
− −→ →
→
−
−
⎧ ⎫− −⎪ ⎪= − = −⎨ ⎬− + − −⎪ ⎪⎩ ⎭
= − =
1p
⎧ ⎫⎪ ⎪⎨ ⎬⎪ ⎪⎩ ⎭
preparation. If you are a student using this Manual, you are using it without permission.
9-50
Stirling and Ericsson Cycles
9-65C The Stirling cycle.
9-66C The two isentropic processes of the Carnot cycle are replaced by two constant pressure regeneration processes in the Ericsson cycle.
9-67C The efficiencies of the Carnot and the Stirling cycles would be the same, the efficiency of the Otto cycle would be less.
9-68C The efficiencies of the Carnot and the Ericsson cycles would be the same, the efficiency of the Diesel cycle would be less.
9-69 An ideal steady-flow Ericsson engine with air as the working fluid is considered. The maximum pressure in the cycle, the net work output, and the thermal efficiency of the cycle are to be determined.
Assumptions Air is an ideal gas.
Properties The gas constant of air is R = 0.287 kJ/kg.K (Table A-1).
Analysis (a) The entropy change during process 3-4 is
s
T
3
2qin
qout
4
1 1200 K
300 K
KkJ/kg 0.5K 300
kJ/kg 150
0
out34,34 ⋅−=−=−=−
Tq
ss
and
PROPRIETARY MATERIAL. © 2011 The McGraw-Hill Companies, Inc. Limited distribution permitted only to teachers and educators for course
( ) KkJ/kg 0.5kPa 120
PlnKkJ/kg
ln
4
3
40
4
⋅−=⋅
−PPRT
It yields P4 = 685.2 kPa
sible cycles,
ln3
34 =−T
css p
0.287−=
(b) For rever
( ) kJ/kg 600kJ/kg 150K 300in TTq LH
K 1200outin
t ===⎯→⎯= qT
qTq HL
Thus,
(c) The thermal efficiency of this totally reversible cycle is determined from
ou
kJ/kg 450=−=−= 150600outinoutnet, qqw
75.0%=−=−=K1200K300
11thH
L
TT
η
preparation. If you are a student using this Manual, you are using it without permission.
9-51
9-70 An ideal Stirling engine with air as the working fluid operates between the specified temperature and pressure limits. The net work produced per cycle and the thermal efficiency of the cycle are to be determined.
Assumptions Air is an ideal gas with constant specific heats.
Properties The properties of air at room temperature are cp = 1.005 kJ/kg·K, cv = 0.718 kJ/kg·K, R = 0.287 kJ/kg·K, and k = 1.4 (Table A-2a).
Analysis Since the specific volume is constant during process 2-3,
s
T
3
2qin
qout
4
1 800 K
300 K
kPa 7.266K 3003
32⎠⎝T
K 800kPa) 100(2 =⎟⎞
⎜⎛==
TPP
Heat is only added to the system during reversible process 1-2. Then,
kJ/kg .6462)KkJ/kg 0.5782)( K800()( 121in =⋅=−= ssTq
KkJ/kg 5782.0kPa 2000kPa 266.7ln)KkJ/kg 0.287(0
lnln1
20
1
212
⋅=
⎟⎠⎞
⎜⎝⎛⋅−=
−=−PP
RTT
css p
he therm reversible cycle is determined from T al efficiency of this totally
0.625=−=−=K 800K 300
11thH
L
TT
η
Then,
kJ 289.1=== kJ/kg) kg)(462.6 (0.625)(1inthnet mqW η
PROPRIETARY MATERIALpreparation. If you are a student using this Manual, you are using it without permission.
. © 2011 The McGraw-Hill Companies, Inc. Limited distribution permitted only to teachers and educators for course
9-52
9-71 An ideal Stirling engine with air as the working fluid operates between the specified temperature and pressure limits. The power produced and the rate of heat input are to be determined.
Assumptions Air is an ideal gas with constant specific heats.
Properties The properties of air at room temperature are cp = 1.005 kJ/kg·K, cv = 0.718 kJ/kg·K, R = 0.287 kJ/kg·K, and k = 1.4 (Table A-2a).
Analysis Since the specific volume is constant during process 2-3,
s
T
3
2qin
qout
4
1 800 K
300 K
kPa 7.266K 3003
32⎠⎝T
K 800kPa) 100(2 =⎟⎞
⎜⎛==
TPP
Heat is only added to the system during reversible process 1-2. Then,
kJ/kg .6462)KkJ/kg 0.5782)( K800()( 121in =⋅=−= ssTq
KkJ/kg 5782.0kPa 2000kPa 266.7ln)KkJ/kg 0.287(0
lnln1
20
1
212
⋅=
⎟⎠⎞
⎜⎝⎛⋅−=
−=−PP
RTT
css p
he therm reversible cycle is determined from T al efficiency of this totally
0.625K 800K 300
11th =−=−=H
L
TT
η
Then,
kJ 289.1kJ/kg) kg)(462.6 (0.625)(1inth net === mqW η
e is
hile the power produced by the engine is
The rate at which heat is added to this engin
kW 10,020=== cycle/s) 0/60kJ/kg)(130 462.6kg/cycle)( 1(inin nmqQ &&
w
kW 6264=== cycle/s) 1300/60kJ/cycle)( 1.289()netnet nWW &&
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9-53
9-72 An ideal Ericsson cycle operates between the specified temperature limits. The rate of heat addition is to be determined.
Analysis The thermal efficiency of this totally reversible cycle is determined from
s
T
3
2qin
qout
4
1 900 K
280 K
0.6889K 900th
HT
According t
K 28011 =−=−= LT
η
o the general definition of the thermal efficiency, the rate of heat
addition is
kW 726===0.6889
kW 500
th
netin η
WQ
&&
9-73 An ideal Ericsson cycle operates between the specified temperature limits. The power produced by the cycle is to be
hen the cycle is repeated 2000 times er minute. Then the work per cycle is
determined.
Analysis The power output is 500 kW w
s
T
3
2qin
qout
4
1 900 K
280 K
p
kJ/cycle 15cycle/s (2000/60)
kJ/s 500netW&net ===
nW
&
hen the cycle is repeated 3000 times per minute, the power output will be
W
kW 750=== kJ/cycle) 5cycle/s)(1 000/603(netnet WnW &&
PROPRIETARY MATERIALpreparation. If you are a student using this Manual, you are using it without permission.
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9-54
9-74 An ideal Stirling engine with air as the working fluid operates between specified pressure limits. The heat added to the cycle and the net work produced by the cycle are to be determined.
Assumptions Air is an ideal gas with constant specific heats.
Properties The properties of air at room temperature are R = 0.287 kPa·m3/kg.K, cp = 1.005 kJ/kg·K, cv = 0.718 kJ/kg·K, and k = 1.4 (Table A-2a).
Analysis Applying the ideal gas equation to the isothermal process 3-4 gives
s
T
3
2qin
qout
4
1
298 K
kPa 600kPa)(12) 50(4
34 v3 ===
vPP
Since process 4-1 is one of constant volume,
K 1788kPa 6004
41⎠⎝⎟
⎠⎜⎝ P
kPa 3600K) 298(1 =⎟⎞
⎜⎛=⎟
⎞⎜⎛
=P
TT
dapting t work integral to the heat addition process gives A he first law and
kJ/kg 1275=⋅=== − K)ln(12) 1788)(KkJ/kg 0.287(ln1
2121in v
vRTwq
Similarly,
kJ/kg 212.5=⎟⎠⎞
⎜⎝⎛⋅== = wq − 12
1K)ln 298)(KkJ/kg 0.287(ln3
4343t v
vRT
he net work is then
Properties The properties of air at room temperature are R = 0.287 kPa·m /kg.K, cp = 1.005 kJ/kg·K, cv = 0.718 kJ/kg·K, nd k = 1.4 (Table A-2a).
Analysis Applying the ideal gas equation to the isothermal process 3-4 gives
ou
T
kJ/kg 1063=−=−= 5.2121275outinnet qqw
9-75 An ideal Stirling engine with air as the working fluid operates between specified pressure limits. The heat transfer in the regenerator is to be determined.
Assumptions Air is an ideal gas with constant specific heats. 3
a
kPa 600kPa)(12) 50(4
s
T
3
2qin
qout
4
1
298 K
3v3 ===
vPP 4
Since process 4-1 is one of constant volume,
K 1788kPa 600kPa 3600K) 298(
4
141 =⎟
⎠⎞
⎜⎝⎛=⎟⎟
⎠
⎞⎜⎜⎝
⎛=
PP
TT
Application of the first law to process 4-1 gives
kJ/kg 1070=−⋅=−= K)2981788)(KkJ/kg 718.0()( 41regen TTcq v
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. © 2011 The McGraw-Hill Companies, Inc. Limited distribution permitted only to teachers and educators for course
9-55
Ideal and Actual Gas-Turbine (Brayton) Cycles
9-76C They are (1) isentropic compression (in a compressor), (2) P = constant heat addition, (3) isentropic expansion (in a turbine), and (4) P = constant heat rejection.
9-77C For fixed maximum and minimum temperatures, (a) the thermal efficiency increases with pressure ratio, (b) the net work first increases with pressure ratio, reaches a maximum, and then decreases.
9-78C Back work ratio is the ratio of the compressor (or pump) work input to the turbine work output. It is usually between 0.40 and 0.6 for gas turbine engines.
9-79C In gas turbine engines a gas is compressed, and thus the compression work requirements are very large since the steady-flow work is proportional to the specific volume.
9-80C As a result of turbine and compressor inefficiencies, (a) the back work ratio increases, and (b) the thermal efficiency decreases.
PROPRIETARY MATERIALpreparation. If you are a student using this Manual, you are using it without permission.
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9-56
9-81 A simple ideal Brayton cycle with air as the working fluid has a pressure ratio of 10. The air temperature at the compressor exit, the back work ratio, and the thermal efficiency are to be determined.
Assumptions 1 Steady operating conditions exist. 2 The air-standard assumptions are applicable. 3 Kinetic and potential energy changes are negligible. 4 Air is an ideal gas with variable specific heats.
Properties The properties of air are given in Table A-17.
Analysis (a) Noting that process 1-2 is isentropic,
PROPRIETARY MATERIAL. © 2011 The McGraw-Hill Companies, Inc. Limited distribution permitted only to teachers and educators for course
s
T
1
2 4
3qin
qout
1100 K
290 K
2311.1kJ/kg 16.290
K 0291
11 =
=⎯→⎯=
rPh
T
( )( )kJ/kg 06.561
311.122311.1102
2
112
=== PP
P rr2
==
⎯→⎯hTP K 555.6
roces 3-4 is isentropic, and thus
(b) P s
( )
kJ/kg 62.54945.61107.1161
kJ/kg 9.27016.29006.561
kJ/kg 45.61171.161.167101
1.167kJ/kg 07.1161
K 0011
43outT,
12inC,
43
4
3 =h3
34
3
=−=−=
=−=−=
=⎯→⎯=⎟⎠⎞
⎜⎝⎛==
=⎯→⎯=
hhw
hhw
hPPP
P
PT
rr
r
Then the back-work ratio becomes
49.3%==== 493.0kJ/kg 549.62kJ/kg 270.9
outT,
inC,bw w
wr
(c)
46.5%====
=−=−=
=−=−=
465.0kJ/kg 600.01kJ/kg 278.72
kJ/kg 72.2789.27062.549
kJ/kg 01.60006.56107.1161
in
outnet,th
inC,outT,outnet,
23in
qw
www
hhq
η
preparation. If you are a student using this Manual, you are using it without permission.
9-57
9-82 A simple Brayton cycle with air as the working fluid has a pressure ratio of 10. The air temperature at the turbine exit, the net work output, and the thermal efficiency are to be determined.
s
T
1
2s
4s
3qin
qout
1240 K
295 K
2
4
Assumptions 1 Steady operating conditions exist. 2 The air-standard assumptions are applicable. 3 Kinetic and potential energy changes are negligible. 4 Air is an ideal gas with variable specific heats.
Properties The properties of air are given in Table A-17.
Analysis (a) Noting that process 1-2s is isentropic,
3068.1kJ/kg .17295
K 2951
11 =
=⎯→⎯=
rPh
T
( )( )
( )
( )( )( )
kJ/kg 83.04707.70293.132487.093.1324
K 6.689 and kJ/kg .0770223.273.272101
3.272kJ/kg 1324.93
K 1240
kJ/kg .6062683.0
17.29526.57017.295
K 564.9 and kJ/kg 570.2607.133068.110
433443
43
443
4
33
1212
12
12
221
2
34
3
12
=−−=
−−=⎯→⎯−−
=
==⎯→⎯=⎟⎠⎞
⎜⎝⎛==
==
⎯→⎯=
=−
+=
−+=⎯→⎯
−−
=
==⎯→⎯===
sTs
T
ssrr
r
C
ssC
ssrr
hhhhhhhh
ThPPP
P
Ph
T
hhhh
hhhh
ThPPP
P
ηη
ηη
Thus,
T = 764.4 K
(c)
4
(b)
kJ/kg 210.4=−=−=
=−=−=
=−=−=
9.4873.698
kJ/kg 487.917.29504.783
kJ/kg .369860.62693.1324
outinoutnet,
14out
23in
qqw
hhq
hhq
30.1%==== 3013.0kJ/kg 698.3kJ/kg 210.4
in
outnet,th q
wη
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. © 2011 The McGraw-Hill Companies, Inc. Limited distribution permitted only to teachers and educators for course
9-58
9-83 Problem 9-82 is reconsidered. The mass flow rate, pressure ratio, turbine inlet temperature, and the isentropic efficiencies of the turbine and compressor are to be varied and a general solution for the problem by taking advantage of the
be developed.
nalysis Using EES, the problem is solved as follows:
m diagram window"
]
P[1])
compressor, assuming: adiabatic, ke=pe=0"
_in= m_dot*h[3]"SSSF First Law for the heat exchanger, assuming W=0, ke=pe=0"
ot_t + m_dot*h[4] "SSSF First Law for the actual compressor, assuming: adiabatic, ke=pe=0"
ycle work, kW" fficiency"
determined only to produce a T-s plot"
[4]=entropy(air,T=T[4],P=P[4])
diagram window method for supplying data to EES is to
A
"Input data - fro{P_ratio = 10} {T[1] = 295 [K] P[1]= 100 [kPa] T[3] = 1240 [K] m_dot = 20 [kg/s
ta_c = 83/100 EEta_t = 87/100} "Inlet conditions" h[1]=ENTHALPY(Air,T=T[1]) s[1]=ENTROPY(Air,T=T[1],P= "Compressor anaysis" s_s[2]=s[1] "For the ideal case the entropies are constant across the compressor"
sure ratio - to find P[2]" P_ratio=P[2]/P[1]"Definition of presT_s[2]=TEMPERATURE(Air,s=s_s[2],P=P[2]) "T_s[2] is the isentropic value of T[2] at compressor exit" h_s[2]=ENTHALPY(Air,T=T_s[2])
mpressor adiabatic efficiency; Eta_c = W_dot_c_ideal/W_dot_c_actual. " Eta_c =(h_s[2]-h[1])/(h[2]-h[1]) "Com_dot*h[1] +W_dot_c=m_dot*h[2] "SSSF First Law for the actual
lysis" "External heat exchanger anaP[3]=P[2]"process 2-3 is SSSF constant pressure" h[3]=ENTHALPY(Air,T=T[3]) m_dot*h[2] + Q_dot "Turbine analysis" s[3]=ENTROPY(Air,T=T[3],P=P[3]) s_s[4]=s[3] "For the ideal case the entropies are constant across the turbine" P_ratio= P[3] /P[4] T_s[4]=TEMPERATURE(Air,s=s_s[4],P=P[4]) "Ts[4] is the isentropic value of T[4] at turbine exit" h_s[4]=ENTHALPY(Air,T=T_s[4]) "Eta_t = W_dot_t /Wts_dot turbine adiabatic efficiency, Wts_dot > W_dot_t" Eta_t=(h[3]-h[4])/(h[3]-h_s[4]) m_dot*h[3] = W_d "Cycle analysis" W_dot_net=W_dot_t-W_dot_c"Definition oEta=W_dot_net/Q_dot_in"Cycle thermal e
f the net c
Bwr=W_dot_c/W_dot_t "Back work ratio" "The following state points areT[2]=temperature(air,h=h[2]) T[4]=temperature(air,h=h[4]) s[2]=entropy(air,T=T[2],P=P[2]) s
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9-59
Bwr η Pratio Wc
[kW] Wnet [kW]
Wt [kW]
Qin [kW]
0.5229 0.1 2 1818 1659 3477 16587 0.6305 0.1644 4 4033 2364 6396 14373 0.7038 0.1814 6 5543 2333 7876 12862 0.7611 0.1806 8 6723 2110 8833 11682 0.8088 0.1702 10 7705 1822 9527 10700
0.85 0.1533 12 8553 1510 10063 9852 0.8864 0.131 14 9304 1192 10496 9102 0.9192 0.1041 16 9980 877.2 10857 8426 0.9491 0.07272 18 10596 567.9 11164 7809 0.9767 0.03675 20 11165 266.1 11431 7241
2 4 6 8 10 12 14 16 18 200.12
0.16
0.2
0.24
0.28
0.32
0.36
2250
2700
3150
3600
4050
4500
Pratio
η
Wne
t [k
W]
4.5 5.0 5.5 6.0 6.5 7.0 7.5 8.0 8.50
500
1000
1500
s [kJ/kg-K]
T [K
]
100 kPa
1000 kPa
Air
1
2
3
42s 4s
PROPRIETARY MATERIALpreparation. If you are a student using this Manual, you are using it without permission.
. © 2011 The McGraw-Hill Companies, Inc. Limited distribution permitted only to teachers and educators for course
9-60
9-84 A simple Brayton cycle with air as the working fluid has a pressure ratio of 10. The air temperature at the turbine exit, the net work output, and the thermal efficiency are to be determined.
Assumptions 1 Steady operating conditions exist. 2 The air-standard assumptions are applicable. 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-2).
Analysis (a) Using the compressor and turbine efficiency relations, ( )
( )( )
( )( )
( )( )
( )( ) ( )
( )( )K 720=
−−=−−=⎯→⎯
−
−=
−−
=
=−
+=
−+=⎯→⎯
−
−=
−−
=
=⎟⎠⎞
⎜⎝⎛=⎟⎟
⎠
⎞⎜⎜⎝
⎛=
==⎟⎟⎠
⎞⎜⎜⎝
⎛=
−
−
3.642124087.01240
K 625.883.0
2956.569295
K 642.3101K 1240
K 6.56910K 295
433443
43
43
43
1212
12
12
12
12
0.4/1.4/1
3
434
0.4/1.4/1
1
212
sTsp
p
sT
C
s
p
spsC
kk
s
kk
s
TTTTTTcTTc
hhhh
TTTT
TTcTTc
hhhh
PP
TT
PP
TT
ηη
ηη
s
T
1
2s
4s
3qin
qout
1240 K
295 K
2
4
(b) ( ) ( )( )( ) ( )( )
kJ/kg 190.2=−=−=
=−⋅=−=−=
=−⋅=−=−=
1.4273.617
kJ/kg 427.1K295
PROPRIETARY MATERIAL. © 2011 The McGraw-Hill Companies, Inc. Limited distribution permitted only to teachers and educators for course
720KkJ/kg 1.005
kJ/kg 17.36K625.81240KkJ/kg 1.005
outinoutnet,
1414out
2323
qqw
TTchhq
TTchhq
p
p
(c)
in
30.8%==== 3081.0kJ/kg 617.3kJ/kg 190.2
in
outnet,th q
wη
preparation. If you are a student using this Manual, you are using it without permission.
9-61
9-85 A simple ideal Brayton cycle with helium has a pressure ratio of 14. The power output is to be determined.
Assumptions 1 Steady operating conditions exist. 2 The air-standard assumptions are applicable. 3 Kinetic and potential energy changes are negligible. 4 Helium is an ideal gas with constant specific heats.
Properties The properties of helium are cp = 5.1926 kJ/kg·K and k = 1.667 (Table A-2a).
Analysis Using the isentropic relations for an ideal gas,
s
T
1
2
4
3 qin
qout
973 K
288 K
K 9.827K)(14) 288( 70.667/1.66/)1(1
/)1(
1
212T ===⎟⎟
⎠
⎞⎜⎜⎝
⎛= −
−kk
p
kk
rTPP
T
Similarly,
K 5.33814
K) 973(33
34 =⎟⎠
⎜⎝
=⎟⎠
⎜⎝
=⎟⎟⎠
⎜⎜⎝
=pr
TP
TT 11 70.667/1.66/)1(/)1(4 ⎞⎛⎟
⎞⎜⎛⎞⎛
−− kkkkP
t law to the constant-pressure heat addition process 2-3 produces
p
Similarly,
14
Applying he first
q kJ/kg 4.753K)9.827973)(kJ/kg·K .19265()( 23in =−=−= TTc
1926.5()(
PROPRIETARY MATERIAL. © 2011 The McGraw-Hill Companies, Inc. Limited distribution permitted only to teachers and educators for course
kJ/kg 2.262K)2885.338)(kJ/kg·K out =−= TTcq =−p
net work production is then
nd
The
kJ/kg 2.4912.2624.753outinnet =−=−= qqw
a
kW 409=⎟⎠⎞
⎜⎝⎛==
s 60min 1kJ/kg) 2.491(kg/min) 50(netnet wmW &&
preparation. If you are a student using this Manual, you are using it without permission.
9-62
9-86 A simple Brayton cycle with helium has a pressure ratio of 14. The power output is to be determined.
Assumptions 1 Steady operating conditions exist. 2 The air-standard assumptions are applicable. 3 Kinetic and potential energy changes are negligible. 4 Helium is an ideal gas with constant specific heats.
Properties The properties of helium are cp = 5.1926 kJ/kg·K and k = 1.667 (Table A-2a).
Analysis For the compression process,
K 9.827K)(14) 288( 70.667/1.66/)1(1
/)1(
1
212 ===⎟⎟
⎠
⎞⎜⎜⎝
⎛= −
−kk
p
kk
s rTPP
TT
4
s
T
1
2s
3 qin
qout
973 K
288 K
2
K 3.85695.0
2889.827288
)()( 12
1212
12
12
12
=−
+=
−+=⎯→⎯
−
−=
−−
=C
s
p
spsC
TTTT
TTcTTc
hhhh
ηη
For the isentropic expansion process,
PROPRIETARY MATERIAL. © 2011 The McGraw-Hill Companies, Inc. Limited distribution permitted only to teachers and educators for course
K 53⎛
=T .338141K) 973(1
33
44 =⎟
⎠⎞
⎜⎝⎛=⎟
⎟⎠
⎞⎜⎜⎝
⎛=⎟⎟
⎠
⎞⎜⎜⎝ pr
TPP
T
ssure heat addition process 2-3 produces
=−
imilarly,
70.667/1.66/)1(/)1( −− kkkk
Applying the first law to the constant-pre
.19265()( 23in =−= TTcq p kJ/kg 0.606K)3.856973)(kJ/kg·K
S
kJ/kg 2.262K)2885.338)(kJ/kg·K 1926.5()( 14out =−=−= TTcq p
T orhe net w k production is then
and
kJ/kg 8.3432.2620.606outinnet =−=−= qqw
kW 287=⎟⎠⎞
⎜⎝⎛==
s 60min 1kJ/kg) 8.343(kg/min) 50(netnet wmW &&
preparation. If you are a student using this Manual, you are using it without permission.
9-63
9-87 A simple Brayton cycle with air as the working fluid operates between the specified temperature and pressure limits. The effects of non-isentropic compressor and turbine on the back-work ratio is to be compared.
Assumptions 1 Steady operating conditions exist. 2 The air-standard assumptions are applicable. 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 For the compression process,
4s
s
T
1
2s
4
3 qin
qout
873 K
288 K
2
K 8.585K)(12) 288( 0.4/1.4/)1(
1
212 ==⎟⎟
⎠
⎞⎜⎜⎝
⎛=
− kk
s PP
TT
K 9.61890.0
2888.585288
)()( 12
1212
12
12=
−=C chh
η 12
=−
+=
−+=⎯→⎯
−
−−
C
s
p
sps TTTT
TTTTchh
η
For the expansion process,
PROPRIETARY MATERIAL. © 2011 The McGraw-Hill Companies, Inc. Limited distribution permitted only to teachers and educators for course
K 2.429121K) 873(4
34 =⎟⎠⎞
⎜⎝⎛=⎟⎟
⎞⎜⎜⎛
=s PP
TT 0.4/1.4/)1(
3 ⎠⎝
− kk
K 6.473)2.429873)(90.0(873
)()()(
433443
43
43
43
=−−=
−−=⎯→⎯−
−=
−−
= sTsp
p
s T TTTT
TTcTTc
hhhh
ηη
of compressor and turbine are
The isentropic and actual work
kJ/kg 3.299K)2888.585)(KkJ/kg 1.005()( 12Comp, =−⋅=−= TTcW sps
8KkJ/kg 1.005()( 12Comp =⋅=−= TTcW p
4
289.618)( − kJ/kg 6.332K)
kJ/kg 0.446K)2.429873)(KkJ/kg 1.005()( 43Turb, =−⋅=−= sps TTcW
( 3Turb kJ/kg 4.401K)6.473873)(KkJ/kg 1.005() =−⋅=sTW
compressor and isentropic turbine case is
−= p Tc
The back work ratio for 90% efficient
0.7457===kJ/kg 446.0kJ/kg 332.6
Turb,
Compbw
sWW
r
The back work ratio for 90% efficient turbine and isentropic compressor case is
0.7456===kJ/kg 401.4kJ/kg 299.3
Turb
Comp,bw W
Wr s
The two results are almost identical.
preparation. If you are a student using this Manual, you are using it without permission.
9-64
9-88 A gas turbine power plant that operates on the simple Brayton cycle with air as the working fluid has a specified pressure ratio. The required mass flow rate of air is to be determined for two cases.
Assumptions 1 Steady operating conditions exist. 2 The air-standard assumptions are applicable. 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-2).
PROPRIETARY MATERIAL. © 2011 The McGraw-Hill Companies, Inc. Limited distribution permitted only to teachers and educators for course
Analysis (a) Using the isentropic relations,
( )( )( )
( )( )
( ) ( )( )( ) ( )( )
kg/s 352===
=−=−=
=−⋅=−=−=
=−⋅=−=−=
=⎟⎠⎞
⎜⎝⎛=⎟⎟
⎠
⎞⎜⎜⎝
⎛=
==⎟⎟⎠
⎞⎜⎜⎝
⎛=
−
−
kJ/kg 199.1kJ/s 70,000
kJ/kg 99.1175.31184.510
kJ/kg 510.84K491.71000KkJ/kg 1.005
kJ/kg 311.75K300610.2KkJ/kg 1.005
K 491.7121K 1000
K 610.212K 300
outnet,s,
outnet,
inC,s,outT,s,outnet,s,
4343outT,s,
1212inC,s,
0.4/1.4/1
3
434
0.4/1.4/1
1
212
wW
m
www
TTchhw
TTchhw
PP
TT
PP
TT
s
sps
sps
kk
s
kk
s
&&
s
T
1
2s
4s
31000 K
300 K
2
4
(b) The net work output is determined to be
( )( )
kg/s 1037===
=−=
−=−=
kJ/kg 67.5kJ/s 70,000
kJ/kg 67.50.85311.7584.51085.0
/
outnet,a,
outnet,
inC,s,outT,s,inC,a,outT,a,outnet,a,
wW
m
wwwww
a
CT
&&
ηη
preparation. If you are a student using this Manual, you are using it without permission.
9-65
9-89 An actual gas-turbine power plant operates at specified conditions. The fraction of the turbine work output used to drive the compressor and the thermal efficiency are to be determined.
Assumptions 1 Steady operating conditions exist. 2 The air-standard assumptions are applicable. 3 Kinetic and potential energy changes are negligible. 4 Air is an ideal gas with variable specific heats.
Properties The properties of air are given in Table A-17.
Analysis (a) Using the isentropic relations,
PROPRIETARY MATERIAL. © 2011 The McGraw-Hill Companies, Inc. Limited distribution permitted only to teachers and educators for course
T h
T h
1 1
2 2
= ⎯ →⎯ =
= ⎯ →⎯ =
300 K 300.19 kJ / kg
580 K 586.04 kJ / kg
( )
( ) ( )( ) kJ/kg 542.0905.831536.0486.0
kJ/kg 285.8519.30004.586
kJ/kg 05.83973.6711.47471
11.474
kJ/kg1536.0404.586950
7100700
43outT,
12inC,
43
4
323in
1
2
34
3
=−=−=
=−=−=
=⎯→⎯=⎟⎠⎞
⎜⎝⎛==
=→
=+=⎯→⎯−=
===
sT
srr
r
p
hhw
hhw
hPPP
P
P
hhhq
PP
r
η
s
T
1
2s
4s
3950 kJ/kg
580 K
300 K
2
4
Thus,
52.7%===kJ/kg 542.0kJ/kg 285.85
outT,
inC,bw w
wr
27.0%===
=−=−=
kJ/kg 950kJ/kg 256.15
kJ/kg 256.15285.85.0542
in
outnet,th
inC,outT,net.out
qw
www
η
(b)
preparation. If you are a student using this Manual, you are using it without permission.
9-66
9-90 A gas-turbine power plant operates at specified conditions. The fraction of the turbine work output used to drive the compressor and the thermal efficiency are to be determined.
Assumptions 1 Steady operating conditions exist. 2 The air-standard assumptions are applicable. 3 Kinetic and potential energy changes are negligible. 4 Air is an ideal gas with constant specific heats.
PROPRIETARY MATERIAL. © 2011 The McGraw-Hill Companies, Inc. Limited distribution permitted only to teachers and educators for course
Properties The properties of air at room temperature are cp = 1.005 kJ/kg·K and k = 1.4 (Table A-2).
Analysis (a) Using constant specific heats,
( )( ) ( )
( )( )
( ) ( )( )( ) ( ) ( )( )( ) kJ/kg 562.2K874.81525.3KkJ/kg 1.00586.0
kJ/kg 281.4K300580KkJ/kg1.005
K 874.871K 1525.3
K 1525.3KkJ/kg 1.005/kJ/kg 950K 580
/
7100700
4343outT,
1212inC,
0.4/1.4/k1k
3
434s
in232323in
1
2
=−⋅=−=−=
=−⋅=−=−=
=⎟⎠⎞
⎜⎝⎛=⎟⎟
⎠
⎞⎜⎜⎝
⎛=
=⋅+=
+=⎯→⎯−=−=
===
−
spTsT
p
pp
p
TTchhw
TTchhw
PP
TT
cqTTTTchhq
PP
r
ηη
s
T
1
2s
4s
3950 kJ/kg
580 K
300 K
2
4
Thus,
50.1%===kJ/kg 562.2kJ/kg 281.4
outT,
inC,bw w
wr
29.6%===
=−=−=
kJ/kg 950kJ/kg 280.8
kJ/kg 280.8281.42.562
in
outnet,th
inC,outT,outnet,
qw
www
η
(b)
preparation. If you are a student using this Manual, you are using it without permission.
9-67
9-91 An aircraft engine operates as a simple ideal Brayton cycle with air as the working fluid. The pressure ratio and the rate of heat input are given. The net power and the thermal efficiency are to be determined.
Assumptions 1 Steady operating conditions exist. 2 The air-standard assumptions are applicable. 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 For the isentropic compression process,
PROPRIETARY MATERIAL. © 2011 The McGraw-Hill Companies, Inc. Limited distribution permitted only to teachers and educators for course
=== − kkprTT 2 K .1527K)(10) 273( 0.4/1.4/)1(
1
s
T
1
2
4
3 qin
qout273 K
The heat addition is
kJ/kg 500kg/s 1
kW 500inin ===
mQ
q&
&
Applying the first law to the heat addition process,
K 1025
KkJ/kg 1.005kJ/kg 500
)( 23in p
qK 1.527in
23 =⋅
+=+=
−
pcTT
TTq
The temperature at the exit of the turbine is
= c
K 9.5301011 0.4/1.4/)1(
⎞⎛⎞⎛− kk
K) 1025(3 =⎟⎠
⎜⎝
=⎟⎟⎠
⎜⎜⎝
=pr
TT
tic turbine and the compressor produce
4
Applying the first law to the adiaba
1025)(KkJ/kg 1.005()( 43T ⋅=−= TTcw p kJ/kg 6.496K)9.530 =−
=−⋅
The net power produced by the engine is then
inally the thermal efficiency is
1.005()( 12C =−= TTcw p kJ/kg 4.255K)2731.527)(KkJ/kg
kW 241.2=−=−= kJ/kg)4.2556kg/s)(496. 1()( CTnet wwmW &&
F
0.482===kW 500kW 241.2
in
netth Q
W&
&η
preparation. If you are a student using this Manual, you are using it without permission.
9-68
9-92 An aircraft engine operates as a simple ideal Brayton cycle with air as the working fluid. The pressure ratio and the rate of heat input are given. The net power and the thermal efficiency are to be determined.
Assumptions 1 Steady operating conditions exist. 2 The air-standard assumptions are applicable. 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 For the isentropic compression process,
PROPRIETARY MATERIAL. © 2011 The McGraw-Hill Companies, Inc. Limited distribution permitted only to teachers and educators for course
=== − kkprTT 2 K .8591K)(15) 273( 0.4/1.4/)1(
1
s
T
1
2
4
3 qin
qout273 K
The heat addition is
kJ/kg 500kg/s 1
kW 500inin ===
mQ
q&
&
Applying the first law to the heat addition process,
K 1089
KkJ/kg 1.005kJ/kg 500
)( 23in p
qK 8.591in
23 =⋅
+=+=
−
pcTT
TTq
The temperature at the exit of the turbine is
= c
K 3.5021511 0.4/1.4/)1(
⎞⎛⎞⎛− kk
K) 1089(3 =⎟⎠
⎜⎝
=⎟⎟⎠
⎜⎜⎝
=pr
TT
tic turbine and the compressor produce
4
Applying the first law to the adiaba
1089)(KkJ/kg 1.005()( 43T ⋅=−= TTcw p kJ/kg 6.589K)3.502 =−
=−⋅
The net power produced by the engine is then
Finally the thermal efficiency is
1.005()( 12C =−= TTcw p kJ/kg 4.320K)2738.591)(KkJ/kg
kW 269.2=−=−= kJ/kg)4.3206kg/s)(589. 1()( CTnet wwmW &&
0.538===kW 500kW 269.2
in
netth Q
W&
&η
preparation. If you are a student using this Manual, you are using it without permission.
9-69
9-93 A gas-turbine plant operates on the simple Brayton cycle. The net power output, the back work ratio, and the thermal efficiency are to be determined. Assumptions 1 The air-standard assumptions are applicable. 2 Kinetic and potential energy changes are negligible. 3 Air is an ideal gas with variable specific heats. Properties The gas constant of air is R = 0.287 kJ/kg·K (Table A-1). Analysis (a) For this problem, we use the properties from EES software. Remember that for an ideal gas, enthalpy is a function of temperature only whereas entropy is functions of both temperature and pressure.
PROPRIETARY MATERIAL. © 2011 The McGraw-Hill Companies, Inc. Limited distribution permitted only to teachers and educators for course
⋅=⎭⎬⎫
=°=
=⎯→⎯°=
sPT
hT
kPa 20002 =
⎭⎬⎫=
shP
Process 1-2: Compression
KkJ/kg 749.5kPa 100
C40kJ/kg 6.313C40
11
1
11
kJ/kg 7.736kJ/kg.K 749.512 == ss
2
kJ/kg 4.8116.313
6.
1
Combustion chamber
Turbine
23
4
Compress.
100 kPa 40°C
650°C
2 MPa
3137.73685.0 2212
12 =⎯→⎯−
−=⎯→⎯
−−
= hhhh
hh sη
0 4 =⎯→⎯° h
C
Process 3-4: Expansion
654 =T kJ/kg 2.959C
ss hhh 3
3
43T 88.0
−=⎯→⎯
−=η
hh
4
2.959−
. However, using the following lines in EES together with the isentropic J/kg, T3 = 1421ºC, s3 = 6.736 kJ/kg.K. The solution by hand would require a trial-
error approach. h_3=enthalpy(Air, T=T_3)
=T_3, P=P_2) P_1, s=s_3)
The mass flow rate is determined from
hh 43 −
We cannot find the enthalpy at state 3 directlyefficiency relation, we find h3 = 1873 k
s_3=entropy(Air, Th_4s=enthalpy(Air, P=
( )( )kg/s 99.12
K 27340K/kgmkPa 0.287)s/m 0kPa)(700/6 100(
3
3
1
11 =+⋅⋅
==RTP
mV&
&
=−=−=&
WWW &&
(b) The back work ratio is
The net power output is
kW 6464)kJ/kg6.3134kg/s)(811. 99.12()( 12inC, =−=−= hhmW &&
2.959kg/s)(1873 99.12()( 43outT, hhmW & kW 868,11)kJ/kg
net& kW 5404=−=−= 6464868,11inC,outT,
0.545===kW 868,11
kW 6464
outT,
inC,bw W
Wr
&
&
(c) The rate of heat input and the thermal efficiency are
kW 788,13)kJ/kg4.811kg/s)(1873 99.12()( 23in =−=−= hhmQ &&
39.2%==== 392.0kW 788,13
kW 5404
in
net
QW
th &
&η
preparation. If you are a student using this Manual, you are using it without permission.
9-70
9-94 A simple Brayton cycle with air as the working fluid operates between the specified temperature and pressure limits. The cycle is to be sketched on the T-s cycle and the isentropic efficiency of the turbine and the cycle thermal efficiency are to be determined.
Assumptions 1 Steady operating conditions exist. 2 The air-standard assumptions are applicable. 3 Kinetic and potential energy changes are negligible. 4 Air is an ideal gas with constant specific heats.
Properties The properties of air are given as cv = 0.718 kJ/kg·K, cp = 1.005 kJ/kg·K, R = 0.287 kJ/kg·K, k = 1.4.
Analysis (b) For the compression process,
PROPRIETARY MATERIAL. © 2011 The McGraw-Hill Companies, Inc. Limited distribution permitted only to teachers and educators for course
)( 12
=−
−= TTcmW p&
For the turbine during the isentropic process, 4s
s
T
1
2s
4
3 873 K
303 K
2
Comp&
kW 300,60K)30330)(KkJ/kg 1.005(kg/s) 200( ⋅=
K 9.772kPa 803
34⎝⎟
⎠⎜⎝
s P 0kPa 100K) 1400(
0.4/1.4/)1(4 =⎟
⎠⎞
⎜⎛=⎟
⎞⎜⎛
=− kk
PTT
The actual power output from the turbine is
=+=+= WWW &&
the turbine is then
kW 050,126K)9.7721400)(KkJ/kg 1.005(kg/s) 200()( 43sTurb, =−⋅=−= sp TTcmW &&
CompTurbnet −= WWW &&&
Turb& kW 300,120300,60000,60Turbnet
The isentropic efficiency of
95.4%====,126sTurb,
Turb W&η 954.0
kW 050kW 300,120TurbW&
) The rat input is
The thermal efficiency is then
(c e of heat
kW 200,160K)]273330(1400)[(KkJ/kg 1.005(kg/s) 200()( 23in =+−⋅=−= TTcmQ p&&
37.5%==== 375.0kW 200,160kW 000,60
in
netth Q
W&
&η
preparation. If you are a student using this Manual, you are using it without permission.
9-71
9-95 A modified Brayton cycle with air as the working fluid operates at a specified pressure ratio. The T-s diagram is to be sketched and the temperature and pressure at the exit of the high-pressure turbine and the mass flow rate of air are to be determined.
Assumptions 1 Steady operating conditions exist. 2 The air-standard assumptions are applicable. 3 Kinetic and potential energy changes are negligible. 4 Air is an ideal gas with constant specific heats.
Properties The properties of air are given as cv = 0.718 kJ/kg·K, cp = 1.005 kJ/kg·K, R = 0.287 kJ/kg·K, k = 1.4.
Analysis (b) For the compression process,
K 5.494K)(8) 273( 0.4/1.4/)1(
1
212 ==⎟⎟
⎠
⎞⎜⎜⎝
⎛=
− kk
PP
TT
s
T
1
4
5
3
P4
1500 K
273 K
2
P5
P3 = P2
The power input to the compressor is equal to the power output from the high-pressure turbine. Then,
K 1278.5=−+=−+=−=−
−=− )( 12 cmTTcm p &&
=
5.4942731500
)(
2134
4312
43
outTurb, HPinComp,
TTTTTTTT
TT
WW
p
&&
The pressure at this state is
kPa 457.3=⎟⎠⎝⎟
⎠⎜⎝
⎟⎠
⎜⎝ 3
1433 K 1500TTP
(c) The temperature at state 5 is determined f
⎞⎜⎛=⎟
⎞⎜⎛
=⎯→⎯⎟⎞
⎜⎛
=−− 4.0/4.1)1/(
4)1/(
4 K 5.1278kPa) 100(8kkkk
TrPP
TP
rom
4
K 1.828kPa 3.457
kPa 100K) 5.1278(0.4/1.4/)1(
4
545 =⎟
⎠⎞
⎜⎝⎛=⎟⎟
⎠
⎞⎜⎜⎝
⎛=
− kk
PP
TT
The net power is that generated by the low-pressure turbine since the power output from the high-pressure turbine is equal the power input to the compressor. Then,
to
kg/s 441.8=−⋅
=−
=
−=
K)1.8285.1278)(KkJ/kg 1.005(kW 000,200
)(
)(
54
Turb LP
54Turb LP
TTcW
m
TTcmW
p
p
&&
&&
PROPRIETARY MATERIALpreparation. If you are a student using this Manual, you are using it without permission.
. © 2011 The McGraw-Hill Companies, Inc. Limited distribution permitted only to teachers and educators for course
9-72
9-96 A simple Brayton cycle with air as the working fluid operates at a specified pressure ratio and between the specified temperature and pressure limits. The cycle is to be sketched on the T-s cycle and the volume flow rate of the air into the compressor is to be determined. Also, the effect of compressor inlet temperature on the mass flow rate and the net power output are to be investigated.
Assumptions 1 Steady operating conditions exist. 2 The air-standard assumptions are applicable. 3 Kinetic and potential energy changes are negligible. 4 Air is an ideal gas with constant specific heats.
Properties The properties of air are given as cv = 0.718 kJ/kg·K, cp = 1.005 kJ/kg·K, R = 0.287 kJ/kg·K, k = 1.4.
Analysis (b) For the compression process,
PROPRIETARY MATERIAL. © 2011 The McGraw-Hill Companies, Inc. Limited distribution permitted only to teachers and educators for course
( ) K 0.4767K) 273( 0.4/1.4/)1(
11
212 ==⎟⎟
⎠
⎞⎜⎜⎝
⎛=
− kk
s PP
TT
4s
s
T
1
2s
4
3 1500 K
273 K
2
K 8.526
27380.0
2 −=
T2730.476
)()(
2
12
12
12
12
Comp
sComp,Comp
=⎯→⎯−
−−
=−
−==
T
TTTT
TTcmTTcm
W
W s
p
sp
&
&
&
&η
or the expansion process, F
K 3.86071K) 1500(
0.4/1.4/)1(
3
434 =⎟
⎠⎞
⎜⎝⎛=⎟⎟
⎠
⎞⎜⎜⎝
⎛=
− kk
s PP
TT
K 3.924
3.8601500 −1500
90
)(
44
4343Turb
=⎯→⎯−
=
−=
−==
TT
TTTTcmW p&&η
Given the net power, the mass flow rate is determined from
)( 4343sTurb,Turb −− TTTTcmW ssp&&
.0
[ ]
[ ]
[ ]kg/s 7.63
)2738.526()3.9241500()KkJ/kg 1.005(
)( 123
net
=−−−⋅
=
−=
TTTcW
mp
&&
4
kW 000,150
()
)()(
()(
4
1243
1243CompTurbnet
−−
−−−=
−−−=−=
T
TTTTcmW
TTcmTTcmWWW
p
pp
&
&&&&&
The specific volume and the volume flow rate at the inlet of the compressor are
)
net&
/kgm 7835.0kPa 100
K) 273)(KkJ/kg 287.0( 3
1
11 =
⋅==
PRT
v
(c) For a fixed compressor inlet velocity and flow area, when the compressor inlet temperature increases, the specific
volume increases since
3m 342.2=== )/kgm 7835.0)(kg/s 7.436( 311 vV m&&
PRT
=v . When specific volume increases, the mass flow rate decreases since vV&
& =m . Note that
volume flow rate is the same since inlet velocity and flow area are fixed ( ). When mass flow rate decreases, the net power decreases since . Therefore, when the inlet temperature increases, both mass flow rate and
the net power decrease.
AV=V&
)( CompTurbnet wwmW −= &&
preparation. If you are a student using this Manual, you are using it without permission.
9-73
Brayton Cycle with Regeneration
9-97C Regeneration increases the thermal efficiency of a Brayton cycle by capturing some of the waste heat from the exhaust gases and preheating the air before it enters the combustion chamber.
9-98C Yes. At very high compression ratios, the gas temperature at the turbine exit may be lower than the temperature at the compressor exit. Therefore, if these two streams are brought into thermal contact in a regenerator, heat will flow to the exhaust gases instead of from the exhaust gases. As a result, the thermal efficiency will decrease.
9-99C The extent to which a regenerator approaches an ideal regenerator is called the effectiveness ε, and is defined as ε = qregen, act /qregen, max.
9-100C (b) turbine exit.
9-101C The steam injected increases the mass flow rate through the turbine and thus the power output. This, in turn, increases the thermal efficiency since in/ QW=η and W increases while Qin remains constant. Steam can be obtained by utilizing the hot exhaust gases.
PROPRIETARY MATERIALpreparation. If you are a student using this Manual, you are using it without permission.
. © 2011 The McGraw-Hill Companies, Inc. Limited distribution permitted only to teachers and educators for course
9-74
9-102 A Brayton cycle with regeneration produces 150 kW power. The rates of heat addition and rejection are to be determined.
Assumptions 1 The air standard assumptions are applicable. 2 Air is an ideal gas with constant specific heats at room temperature. 3 Kinetic and potential energy changes are negligible.
Properties The properties of air at room temperature are cp = 1.005 kJ/kg.K and k = 1.4 (Table A-2a).
Analysis According to the isentropic process expressions for an ideal gas,
PROPRIETARY MATERIAL. © 2011 The McGraw-Hill Companies, Inc. Limited distribution permitted only to teachers and educators for course
=== − kkprTT 2 K 8.530K)(8) 293( 0.4/1.4/)1(
1
s
T
1
2 5
4 qin1073 K
293 K
3
6
qout
K 3.59281K) 1073(1 0.4/1.4/)1(
45 =⎟⎠⎞
⎜⎝⎛=⎟
⎟⎠
⎞⎜⎜⎝
⎛=
− kk
prTT
When the first law is applied to the heat exchanger, the result is
6523 TTTT −=−
while the regenerator temperature specification gives
=−=−= TT
mult neous solution of these two results gives
K 3.582103.5921053
The si a
K 8.540)8.5303.582(3.592)( 2356 =−−=−−= TTTT
Application of the first law to the turbine and compressor gives
hen,
kJ/kg 244.1K )2938.530)(KkJ/kg 005.1(K )3.5921073)(KkJ/kg 005.1(
)()( 1254net
=−⋅−−⋅=
−−−= TTcTTcw pp
T
kg/s 6145.0kW 150net ===W
m&
& kJ/kg 244.1
he first law to the combustion chamber produces
TTcmQ p&
imilarly,
netw
Applying t
in& kW 303.0=−⋅=−= K)3.5821073)(KkJ/kg 005.1(kg/s) 6145.0()( 34
S
kW 153.0=−⋅=−= K)2938.540)(KkJ/kg 005.1(kg/s) 6145.0()( 16out TTcmQ p&&
preparation. If you are a student using this Manual, you are using it without permission.
9-75
9-103 A Brayton cycle with regeneration produces 150 kW power. The rates of heat addition and rejection are to be determined.
Assumptions 1 The air standard assumptions are applicable. 2 Air is an ideal gas with constant specific heats at room temperature. 3 Kinetic and potential energy changes are negligible.
Properties The properties of air at room temperature are cp = 1.005 kJ/kg.K and k = 1.4 (Table A-2a).
Analysis For the compression and expansion processes we have
K 8.530K)(8) 293( 0.4/1.4/)1(12 === − kk
ps rTT
s
T
1
2 5s
4qin1073 K
293 K
3 6
qout
52s K 3.566
87.02938.530293
)()( 12
1212
12
=−
+=
−+=⎯→⎯
−
−=
C
s
p
spC
TTTT
TTcTTc
ηη
K 3.59281K) 1073(1
45 =⎟⎠⎞
⎜⎝⎛=⎟
⎟⎜⎜=s r
TT 0.4/1.4/)1(
⎠
⎞
⎝
⎛− kk
p
PROPRIETARY MATERIAL. © 2011 The McGraw-Hill Companies, Inc. Limited distribution permitted only to teachers and educators for course
K 9.625=)3.5921073)(93.0(1073
)554 −−=
ssp
hen the first law is applied to the heat exchanger, the result is
hile the regenerator temperature specification gives
multaneous solution of these two results gives
()()(
44554 −−=⎯→⎯
−
−= T
pT TTTT
TTcTTc
ηη
W
6523 TTTT −=−
w
K 9.615109.6251053 =−=−= TT
The si
K 3.576)3.5669.615(9.625)( 2356 =−−=−−= TTTT
Application of the first law to the turbine and compressor gives
=−⋅−−⋅=
−−− TTcTT p
Then,
kJ/kg 7.174K )2933.566)(KkJ/kg 005.1(K )9.6251073)(KkJ/kg 005.1(
)()( 1254net = cw p
kg/s 8586.0kJ/kg 174.7kW 150
net
net ===wW
m&
&
Applying the first law to the combustion chamber produces
Similarly,
kW 394.4=−⋅=−= K)9.6151073)(KkJ/kg 005.1(kg/s) 8586.0()( 34in TTcmQ p&&
kW 244.5=−⋅=−= K)2933.576)(KkJ/kg 005.1(kg/s) 8586.0()( 16out TTcmQ p&&
preparation. If you are a student using this Manual, you are using it without permission.
9-76
9-104 A Brayton cycle with regeneration is considered. The thermal efficiencies of the cycle for parallel-flow and counter-flow arrangements of the regenerator are to be compared.
Assumptions 1 The air standard assumptions are applicable. 2 Air is an ideal gas with constant specific heats at room temperature. 3 Kinetic and potential energy changes are negligible.
Properties The properties of air at room temperature are cp = 1.005 kJ/kg⋅K and k = 1.4 (Table A-2a).
Analysis According to the isentropic process expressions for an ideal gas,
PROPRIETARY MATERIAL. © 2011 The McGraw-Hill Companies, Inc. Limited distribution permitted only to teachers and educators for course
=== − kkprTT 2 K 9.510K)(7) 293( 0.4/1.4/)1(
1
s
T
1
2 5
4qin1000 K
293 K
3
6
qout
K 5.57371K) 1000(1 0.4/1.4/)1(
45 =⎟⎠⎞
⎜⎝⎛=⎟
⎟⎠
⎞⎜⎜⎝
⎛=
− kk
prTT
When the first law is applied to the heat exchanger as originally arranged, the result is
6523 TTTT −=−
while the regenerator temperature specification gives
=−=−= TT
he simul ution of these two results gives
K 5.56765.573653
T taneous sol
5.5675.5732356 +−=+−= TTTT K 9.5169.510 =
The thermal efficiency of the cycle is then
0.482=−
−−=
− 34in TTq−
−=−=5.5671000
293516.9111 16outth
TTqη
or the rearranged version of this cycle,
n energy balance on the eat exchanger gives
he soluti hese two equations is
6 =T
The thermal efficiency of the cycle is then
F
663 −= TT
s
T
1
2 5
4qin1000 K
293 K
3 6
qout
A h
6523 TTTT −=−
T on of t
K 2.5393 =T
K 2.545
0.453=−
−−=
−−
−=−=2.5391000
293545.211134
16
in
outth TT
TTqq
η
preparation. If you are a student using this Manual, you are using it without permission.
9-77
9-105 A car is powered by a gas turbine with a pressure ratio of 4. The thermal efficiency of the car and the mass flow rate of air for a net power output of 70 kW are to be determined.
Assumptions 1 Steady operating conditions exist. 2 Air is an ideal gas with variable specific heats. 3 The ambient air is 300 K and 100 kPa. 4 The effectiveness of the regenerator is 0.9, and the isentropic efficiencies for both the compressor and the turbine are 80%. 5 The combustion gases can be treated as air.
Properties The properties of air at the compressor and turbine inlet temperatures can be obtained from Table A-17.
Analysis The gas turbine cycle with regeneration can be analyzed as follows:
PROPRIETARY MATERIAL. © 2011 The McGraw-Hill Companies, Inc. Limited distribution permitted only to teachers and educators for course
( )( )
( ) kJ/k 76.87350.590.23814
434
=⎯→⎯=⎟⎞
⎜⎛== srr hP
PP g
4
0.238kJ/kg 79.1277
K 1200
kJ/kg 49.446544.5386.14
386.1kJ/kg 19.300
K 300
3
33
21
2
11
3
12
1
⎠⎝
==
⎯→⎯=
=⎯→⎯===
==
⎯→⎯=
r
srr
r
Ph
T
hPPP
P
Ph
and
T
s
T
1
2s4s
3 qin1200 K
300 K
5
42
P
kJ/kg 57.954 76.87379.1277
79.1277 443
212
−− hhh0.80
kJ/kg 07.483 19.300
19.30049.4460.80
443
turb
212
comp
=→−
=→−
=
=→−
−=→
−−
=
hhh
hhhh
hh
s
s
η
η
Then the thermal efficiency of the gas turbine cycle becomes
kJ/kg 35.424)07.48357.954(9.0)( 24regen =−=−= hhq ε
kJ/kg 34.140)19.30007.483()57.95479.1277()()(
kJ/kg 370.37=35.424)07.48379.1277()(
1243inC,outT,outnet,
regen23in
=−−−=−−−=−=
−−=−−=
hhhhwww
qhhq
0.379kJ/kg 370.37kJ/kg 140.34
in
outnet,th %37.9====
qw
η
Finally, the mass flow rate of air through the turbine becomes
kg/s 0.499===kJ/kg 140.34
kW 70
net
netair w
Wm
&&
preparation. If you are a student using this Manual, you are using it without permission.
9-78
9-106 The thermal efficiency and power output of an actual gas turbine are given. The isentropic efficiency of the
1 Air is an ideal gas with variable specific heats. 2 Kinetic and potential energy changes are negligible. 3 The same, and the properties of combustion gases are the same as
e A-17.
Analysis The properties at various states are
turbine and of the compressor, and the thermal efficiency of the gas turbine modified with a regenerator are to be determined.
Assumptionsmass flow rates of air and of the combustion gases are the those of air.
Properties The properties of air are given in Tabl
PROPRIETARY MATERIAL. © 2011 The McGraw-Hill Companies, Inc. Limited distribution permitted only to teachers and educators for course
( )( )
( ) 47.485.7127.14
14
434
=⎯→⎯=⎟⎠⎞
⎜⎝⎛== srr hP
PP kJ/kg 23.825
5.712kJ/kg 1710.0
K 1561C1288
kJ/kg 25.65310.214356.17.14
4356.1kJ/kg .21303
K 303=C30
3
33
21
2
11
3
12
1
==
⎯→⎯=°=
=⎯→⎯===
==
⎯→⎯°=
r
srr
r
P
Ph
T
hPPP
P
Ph
T
s
T
1
2s 4s
3qin1561 K
303 K
5
4
2
The net work output and the heat input per unit mass are
C7.658K 7.931 kJ/kg 55.96821.30334.665 kJ/kg 34.66566.3720.1038netinout =−=−= wqq
kJ/kg 0.67210381710
kJ/kg 0.10380.359
kJ/kg 372.66
kJ/kg 66.372h 1
s 3600kg/h 1,536,000kW 159,000
41out414t
3223in
th
netin
netnet
°==→=+=+=→−=
=−=−=→−=
===
=⎟⎠⎞
⎜⎝⎛==
Thqhhhq
qhhhhq
wq
mW
w
in
η
&
&
ou
Then the compressor and turbine efficiencies become
94.9%
83.8%
==−
−=
−−
=
==−−
=−−
=
949.021.303672
21.30325.653
838.023.825171055.9681710
12
12
43
43
hhhh
hhhh
sC
sT
η
η
When a regenerator is added, the new heat input and the thermal efficiency become
44.1%====
−=−=
−=
441.0kJ/kg 845.2kJ/kg 372.66
kJ/kg 845.2=8.1921038
kJ/kg 192.8=672.0)-.55(0.65)(968=)(
newin,
netnewth,
regeninnewin,
24regen
qw
qqq
hhq
η
ε
Discussion Note a 65% efficient regenerator would increase the thermal efficiency of this gas turbine from 35.9% to 44.1%.
preparation. If you are a student using this Manual, you are using it without permission.
9-79
9-107 Problem 9-106 is reconsidered. A solution that allows different isentropic efficiencies for the compressor and turbine is to be developed and the effect of the isentropic efficiencies on net work done and the heat supplied to the cycle is to be studied. Also, the T-s diagram for the cycle is to be plotted.
Analysis Using EES, the problem is solved as follows:
"Input data" T[3] = 1288 [C] Pratio = 14.7 T[1] = 30 [C] P[1]= 100 [kPa] {T[4]=659 [C]} {W_dot_net=159 [MW] }"We omit the information about the cycle net work" m_dot = 1536000 [kg/h]*Convert(kg/h,kg/s) {Eta_th_noreg=0.359} "We omit the information about the cycle efficiency." Eta_reg = 0.65 Eta_c = 0.84 "Compressor isentropic efficiency" Eta_t = 0.95 "Turbien isentropic efficiency" "Isentropic Compressor anaysis" s[1]=ENTROPY(Air,T=T[1],P=P[1]) s_s[2]=s[1] "For the ideal case the entropies are constant across the compressor" P[2] = Pratio*P[1] s_s[2]=ENTROPY(Air,T=T_s[2],P=P[2]) "T_s[2] is the isentropic value of T[2] at compressor exit" Eta_c = W_dot_compisen/W_dot_comp "compressor adiabatic efficiency, W_dot_comp > W_dot_compisen" "Conservation of energy for the compressor for the isentropic case: E_dot_in - E_dot_out = DELTAE_dot=0 for steady-flow" m_dot*h[1] + W_dot_compisen = m_dot*h_s[2] h[1]=ENTHALPY(Air,T=T[1]) h_s[2]=ENTHALPY(Air,T=T_s[2]) "Actual compressor analysis:" m_dot*h[1] + W_dot_comp = m_dot*h[2] h[2]=ENTHALPY(Air,T=T[2]) s[2]=ENTROPY(Air,T=T[2], P=P[2]) "External heat exchanger analysis" "SSSF First Law for the heat exchanger, assuming W=0, ke=pe=0 E_dot_in - E_dot_out =DELTAE_dot_cv =0 for steady flow" m_dot*h[2] + Q_dot_in_noreg = m_dot*h[3] q_in_noreg=Q_dot_in_noreg/m_dot h[3]=ENTHALPY(Air,T=T[3]) P[3]=P[2]"process 2-3 is SSSF constant pressure" "Turbine analysis" s[3]=ENTROPY(Air,T=T[3],P=P[3]) s_s[4]=s[3] "For the ideal case the entropies are constant across the turbine" P[4] = P[3] /Pratio s_s[4]=ENTROPY(Air,T=T_s[4],P=P[4])"T_s[4] is the isentropic value of T[4] at turbine exit" Eta_t = W_dot_turb /W_dot_turbisen "turbine adiabatic efficiency, W_dot_turbisen > W_dot_turb" "SSSF First Law for the isentropic turbine, assuming: adiabatic, ke=pe=0 E_dot_in -E_dot_out = DELTAE_dot_cv = 0 for steady-flow" m_dot*h[3] = W_dot_turbisen + m_dot*h_s[4] h_s[4]=ENTHALPY(Air,T=T_s[4])
PROPRIETARY MATERIALpreparation. If you are a student using this Manual, you are using it without permission.
. © 2011 The McGraw-Hill Companies, Inc. Limited distribution permitted only to teachers and educators for course
9-80
"Actual Turbine analysis:" m_dot*h[3] = W_dot_turb + m_dot*h[4] h[4]=ENTHALPY(Air,T=T[4]) s[4]=ENTROPY(Air,T=T[4], P=P[4]) "Cycle analysis" "Using the definition of the net cycle work and 1 MW = 1000 kW:" W_dot_net*1000=W_dot_turb-W_dot_comp "kJ/s" Eta_th_noreg=W_dot_net*1000/Q_dot_in_noreg"Cycle thermal efficiency" Bwr=W_dot_comp/W_dot_turb"Back work ratio" "With the regenerator the heat added in the external heat exchanger is" m_dot*h[5] + Q_dot_in_withreg = m_dot*h[3] q_in_withreg=Q_dot_in_withreg/m_dot h[5]=ENTHALPY(Air, T=T[5]) s[5]=ENTROPY(Air,T=T[5], P=P[5]) P[5]=P[2] "The regenerator effectiveness gives h[5] and thus T[5] as:" Eta_reg = (h[5]-h[2])/(h[4]-h[2]) "Energy balance on regenerator gives h[6] and thus T[6] as:" m_dot*h[2] + m_dot*h[4]=m_dot*h[5] + m_dot*h[6] h[6]=ENTHALPY(Air, T=T[6]) s[6]=ENTROPY(Air,T=T[6], P=P[6]) P[6]=P[4] "Cycle thermal efficiency with regenerator" Eta_th_withreg=W_dot_net*1000/Q_dot_in_withreg "The following data is used to complete the Array Table for plotting purposes." s_s[1]=s[1] T_s[1]=T[1] s_s[3]=s[3] T_s[3]=T[3] s_s[5]=ENTROPY(Air,T=T[5],P=P[5]) T_s[5]=T[5] s_s[6]=s[6] T_s[6]=T[6]
PROPRIETARY MATERIALpreparation. If you are a student using this Manual, you are using it without permission.
. © 2011 The McGraw-Hill Companies, Inc. Limited distribution permitted only to teachers and educators for course
9-81
ηt ηc ηth,noreg ηth,withreg Qinnoreg
[kW] Qinwithreg
[kW] Wnet [kW]
0.7 0.75 0.8
0.85 0.9
0.95 1
0.84 0.84 0.84 0.84 0.84 0.84 0.84
0.2044 0.2491 0.2939 0.3386 0.3833 0.4281 0.4728
0.27 0.3169 0.3605 0.4011 0.4389 0.4744 0.5076
422152 422152 422152 422152 422152 422152 422152
319582 331856 344129 356403 368676 380950 393223
86.3 105.2 124.1 142.9 161.8 180.7 199.6
4.5 5.0 5.5 6.0 6.5 7.0 7.5 8.0 8.5-100
100
300
500
700
900
1100
1300
1500
1700
s [kJ/kg-K]
T [C
] 100 kPa
1470 kPa
T-s Diagram for Gas Turbine with Regeneration
1
2s2
5
3
4s
4
6
0.7 0.75 0.8 0.85 0.9 0.95 180
100
120
140
160
180
200
η t
Wne
t [k
W]
0.7 0.75 0.8 0.85 0.9 0.95 1310000
330000
350000
370000
390000
410000
430000
η t
Qin
[kW
]
no regeneration
with regeneration
0.7 0.75 0.8 0.85 0.9 0.95 10.2
0.25
0.3
0.35
0.4
0.45
0.5
0.55
ηt
ηth
no regeneration
with regeneration
PROPRIETARY MATERIALpreparation. If you are a student using this Manual, you are using it without permission.
. © 2011 The McGraw-Hill Companies, Inc. Limited distribution permitted only to teachers and educators for course
9-82
9-108 A Brayton cycle with regeneration using air as the working fluid is considered. The air temperature at the turbine exit, the net work output, and the thermal efficiency are to be determined.
Assumptions 1 The air standard assumptions are applicable. 2 Air is an ideal gas with variable specific heats. 3 Kinetic and potential energy changes are negligible.
PROPRIETARY MATERIAL. © 2011 The McGraw-Hill Companies, Inc. Limited distribution permitted only to teachers and educators for course
Properties The properties of air are given in Table A-17.
Analysis (a) The properties of air at various states are
( )( )
( ) ( ) ( )
( )
( ) ( )( ) kJ/kg 803.14711.801219.250.821219.25
kJ/kg 711.8059.2815.20071
15.200kJ/kg 1219.25
K 1501
kJ/kg 618.260.75/310.24541.2610.243/
kJ/kg 541.2688.105546.17
5546.1kJ/kg 310.24
K 310
433443
43
43
4
33
121212
12
21
2 == PP
P
1
34
3
12
1
=−−=−−=⎯→⎯−−
=
=⎯→⎯=⎟⎠⎞
⎜⎝⎛==
==
⎯→⎯=
=−+=−+=⎯→⎯−−
=
=⎯→⎯=
==
⎯→⎯=
sTs
T
srr
r
Css
C
srr
r
hhhhhhhh
hPPP
P
Ph
hhhhhhhh
hP
Ph
ηη
ηη
Thus,
T4 = 782.8 K
(b)
)
s
T
1
2s4s
3qin1150 K
310 K
5
6
42
1T
T
( ) ( )( ) (
kJ/kg 108.09=−−−=
−−−=−=
24.31026.61814.80325.12191243inC,outT,net hhhhwww
( )( )(
kJ/kg 738.43618.26803.140.6526.618
242524
25
=−+=
−+=⎯→⎯−−
= hhhhhhhh
εε (c) )
Then,
22.5%===
=−=−=
kJ/kg 480.82kJ/kg 108.09
kJ/kg 480.82738.4319.2512
in
netth
53in
qw
hhq
η
preparation. If you are a student using this Manual, you are using it without permission.
9-83
9-109 A stationary gas-turbine power plant operating on an ideal regenerative Brayton cycle with air as the working fluid is considered. The power delivered by this plant is to be determined for two cases.
Assumptions 1 The air standard assumptions are applicable. 2 Air is an ideal gas. 3 Kinetic and potential energy changes are negligible.
Properties When assuming constant specific heats, the properties of air at room temperature are cp = 1.005 kJ/kg.K and k = 1.4 (Table A-2a). When assuming variable specific heats, the properties of air are obtained from Table A-17.
Analysis (a) Assuming constant specific heats,
( )( )( )
PROPRIETARY MATERIAL. © 2011 The McGraw-Hill Companies, Inc. Limited distribution permitted only to teachers and educators for course
( )( )
( )( )
( )( ) kW 39,188===
=−
−−=
−−
−=−
−−=−=
====⎯→⎯=
=⎟⎠⎞
⎜⎝⎛=⎟⎟
⎠
⎞⎜⎜⎝
⎛=
=⎞⎛
−
−
kW 75,0000.5225
5225.02.6071100
2903.5251111
K 525.3andK 607.2%100
K 607.281K 1100
K 525.3
innet
53
16
53
16
in
outth
2645
0.4/1.4/1
3
434
0.4/1.4/1
QW
TTTT
TTcTTc
TTTT
PP
TT
P
T
p
p
kk
kk
&& η
η
ε
) Assuming variable specific heats,
=⎟⎟⎠
⎜⎜⎝
= 8K 2901
212 P
TT
(b
( )( )
( )
( )( ) kW 40,283===
=−−
−=−−
−=−=
====⎯→⎯=
=⎯→⎯=⎟⎠⎞
⎜⎝⎛==
==
⎯→⎯=
=⎯→⎯===
==
⎯→⎯=
kW 75,0000.5371
5371.037.65107.116116.29012.526111
kJ/kg 526.12andkJ/kg 651.37%100
kJ/kg 651.3789.201.16781
1.167kJ/kg 1161.07
K1100
kJ/kg 526.128488.92311.18
2311.1kJ/kg 90.162
K029
innet
53
16
in
outth
2645
43
4
33
21
2
11
34
3
12
1
QW
hhhh
hhhh
hPPP
P
Ph
T
hPPP
P
Ph
T
T
rr
r
rr
r
&& η
η
ε
s
T
1
2 4
375,000 kW1100 K
290 K
5
qout
6
preparation. If you are a student using this Manual, you are using it without permission.
9-84
9-110 A regenerative gas-turbine engine using air as the working fluid is considered. The amount of heat transfer in the regenerator and the thermal efficiency are to be determined.
Assumptions 1 The air standard assumptions are applicable. 2 Air is an ideal gas with variable specific heats. 3 Kinetic and potential energy changes are negligible.
Properties The properties of air are given in Table A-17.
Analysis (a) The properties at various states are
PROPRIETARY MATERIAL. © 2011 The McGraw-Hill Companies, Inc. Limited distribution permitted only to teachers and educators for course
( )
( )( )(
( ) ( )( ) kJ/kg 192.7=−=−==
−−=−−=⎯→⎯
−−
=
=⎯→⎯=⎟⎠⎞
⎜⎝⎛==
==⎯→⎯==⎯→⎯=
=⎯→⎯=
===
84.65974.90080.0kJ/kg .74900
44.83242.151590.042.1515
kJ/kg .4483206.505.45091
.5450kJ/kg 1515.42K 0014
kJ/kg .84659K 065kJ/kg 10.243K 310
9100/900/
24regen
433443
43
43
4
33
22
11
12
34
3
hhq
hhhhhhhh
hPPP
P
PhThThT
PPr
sTs
T
srr
r
)
ε
ηη
(b)
p
( ) ( )( ) ( )( ) ( )
40.0%====
=−−=−−=
=−−−=−−−=−=
400.0kJ/kg 662.88kJ/kg 265.08
kJ/kg .886627.192.846591515.42
kJ/kg .08265.24310659.8474.90042.1515
in
netth
regen23in
1243inC,outT,net
qw
qhhq
hhhhwww
η
s
T
1
2s4s
3 qin1400 K
310 K
5
6
4650 K 2
preparation. If you are a student using this Manual, you are using it without permission.
9-85
9-111 A regenerative gas-turbine engine using air as the working fluid is considered. The amount of heat transfer in the regenerator and the thermal efficiency are to be determined.
Assumptions 1 The air standard assumptions are applicable. 2 Air is an ideal gas with constant specific heats. 3 Kinetic and potential energy changes are negligible.
Properties The properties of air at room temperature are cp = 1.005 kJ/kg.K and k = 1.4 (Table A-2a).
PROPRIETARY MATERIAL. © 2011 The McGraw-Hill Companies, Inc. Limited distribution permitted only to teachers and educators for course
Analysis (a) Using the isentropic relations and turbine efficiency,
( )( )
( )( ) ( )
( )( )
( ) ( ) ( )( )( ) kJ/kg 130.7=−⋅=−=−==
−−=−−=⎯→⎯
−
−=
−−
=
=⎟⎠⎞
⎜⎝⎛=⎟⎟
⎠
⎞⎜⎜⎝
⎛=
===
−
K065812.6KkJ/kg 1.0050.80K .6812
3.747140090.01400
K .374791K 1400
9100/900/
2424regen
433443
43
43
43
4.1/4.0/1
3
434
12
TTchhq
TTTTTTcTTc
hhhh
PP
TT
PPr
p
sTsp
p
sT
kk
s
p
εε
ηη s
T
1
2s 4s
3qin1400 K
310 K
5
6
4650 K 2
( ) ( )( ) ( ) ( )[ ]( ) ( )( )( )
39.9%====
=−−⋅=
−−=−−=
=−−−⋅=
−−−=−=
399.0kJ/kg 623.1kJ/kg 248.7
kJ/kg .1623.7130K0651400KkJ/kg 1.005
kJ/kg .7248K310650.68121400KkJ/kg 1.005
in
netth
regen23regen23in
1243inC,outT,net
qw
qTTcqhhq
TTcTTcwww
p
pp
η
(b)
preparation. If you are a student using this Manual, you are using it without permission.
9-86
9-112 A regenerative gas-turbine engine using air as the working fluid is considered. The amount of heat transfer in the regenerator and the thermal efficiency are to be determined.
Assumptions 1 The air standard assumptions are applicable. 2 Air is an ideal gas with variable specific heats. 3 Kinetic and potential energy changes are negligible.
Properties The properties of air are given in Table A-17.
PROPRIETARY MATERIAL. © 2011 The McGraw-Hill Companies, Inc. Limited distribution permitted only to teachers and educators for course
Analysis (a) The properties at various states are
( )
( )( )(
( ) ( )( ) kJ/kg 168.6=−=−==
−−=−−=⎯→⎯
−−
=
=⎯→⎯=⎟⎠⎞
⎜⎝⎛==
==⎯→⎯==⎯→⎯=
=⎯→⎯=
===
84.65974.90070.0kJ/kg .74900
44.83242.151590.042.1515
kJ/kg .4483206.505.45091
.5450kJ/kg 1515.42K 0014
kJ/kg .84659K 065kJ/kg 10.243K 310
9100/900/
24regen
433443
43
43
4
33
22
11
12
34
3
hhq
hhhhhhhh
hPPP
P
PhThThT
PPr
sTs
T
srr
r
)
ε
ηη
(b)
p
s
T
1
2s4s
3 qin1400 K
310 K
5
6
4650 K 2
( ) ( )( ) ( )( ) ( )
38.6%====
=−−=−−=
=−−−=−−−=−=
386.0kJ/kg 687.18kJ/kg 265.08
kJ/kg 18.6876.168.846591515.42
kJ/kg .08265.24310659.8474.90042.1515
in
netth
regen23in
1243inC,outT,net
qw
qhhq
hhhhwww
η
preparation. If you are a student using this Manual, you are using it without permission.
9-87
9-113 An expression for the thermal efficiency of an ideal Brayton cycle with an ideal regenerator is to be developed.
Assumptions 1 The air standard assumptions are applicable. 2 Air is an ideal gas with constant specific heats at room temperature. 3 Kinetic and potential energy changes are negligible.
Analysis The expressions for the isentropic compression and expansion processes are
PROPRIETARY MATERIAL. © 2011 The McGraw-Hill Companies, Inc. Limited distribution permitted only to teachers and educators for course
−= 2kk
prTT /)1(1
s
T
1
2 4
3qin
5
6qout
kk
prTT
/)1(
341
−
⎟⎟⎠
⎞⎜⎜⎝
⎛=
For an ideal regenerator,
45
TT =
26
TT =
The thermal efficiency of the cycle is
kkp
kkp
kkp
rTT
r
rTT
TTTT
TT
TTTT
TT
TTTT
/)1(
3
1
/)1(
/)1(
3
1
34
12
3
1
35
16
3
1
53
16
in
outth
1
1
11
)/(11)/(
1
)/(11)/(
111
−
−−
−
−=
−
−−=
−−
−=
−−
−=−−
−=−=η
preparation. If you are a student using this Manual, you are using it without permission.
9-88
Brayton Cycle with Intercooling, Reheating, and Regeneration
9-114C As the number of compression and expansion stages are increased and regeneration is employed, the ideal Brayton cycle will approach the Ericsson cycle.
9-115C Because the steady-flow work is proportional to the specific volume of the gas. Intercooling decreases the average specific volume of the gas during compression, and thus the compressor work. Reheating increases the average specific volume of the gas, and thus the turbine work output.
9-116C (a) decrease, (b) decrease, and (c) decrease.
9-117C (a) increase, (b) decrease, and (c) decrease.
9-118C (a) increase, (b) decrease, (c) decrease, and (d) increase.
9-119C (a) increase, (b) decrease, (c) increase, and (d) decrease.
9-120C (c) The Carnot (or Ericsson) cycle efficiency.
PROPRIETARY MATERIALpreparation. If you are a student using this Manual, you are using it without permission.
. © 2011 The McGraw-Hill Companies, Inc. Limited distribution permitted only to teachers and educators for course
9-89
9-121 An ideal gas-turbine cycle with two stages of compression and two stages of expansion is considered. The back work ratio and the thermal efficiency of the cycle are to be determined for the cases of with and without a regenerator.
Assumptions 1 The air standard assumptions are applicable. 2 Air is an ideal gas with variable specific heats. 3 Kinetic and potential energy changes are negligible.
s
T
3
4
1
5qin1200 K
300 K
86
7
10
9
2
Properties The properties of air are given in Table A-17.
Analysis (a) The work inputs to each stage of compressor are identical, so are the work outputs of each stage of the turbine since this is an ideal cycle. Then,
( )( )
( )
( ) ( )( ) ( ) kJ/kg 62.86636.94679.127722
kJ/kg 2.142219.30026.41122
kJ/kg 946.3633.7923831
5
⎞⎛P
238kJ/kg 77.7912
K 2001
kJ/kg 411.26158.4386.13
386.1kJ/kg 300.19
K 300
65outT,
12inC,
865
6
755
421
2
11
56
12
1
=−=−=
=−=−=
==⎯→⎯=⎟⎠
⎜⎝
==
===
⎯→⎯=
==⎯→⎯===
==
⎯→⎯=
hhw
hhw
hhPP
P
Phh
T
hhPPP
P
Ph
T
rr
r
rr
r
Thus,
33.5%===kJ/kg 662.86kJ/kg 222.14
outT,
inC,bw w
wr
( ) ( ) ( ) ( )
36.8%===
=−=−=
=−+−=−+−=
kJ/kg 1197.96kJ/kg 440.72
kJ/kg 440.72222.1486.662
kJ/kg 1197.9636.94679.127726.41179.1277
in
netth
inC,outT,net
6745in
qw
www
hhhhq
η
(b) When a regenerator is used, rbw remains the same. The thermal efficiency in this case becomes
( ) ( )( )
55.3%===
=−=−=
=−=−=
kJ/kg 796.63kJ/kg 440.72
kJ/kg 796.6333.40196.1197
kJ/kg 33.40126.41136.94675.0
in
netth
regenoldin,in
48regen
qw
qqq
hhq
η
ε
PROPRIETARY MATERIALpreparation. If you are a student using this Manual, you are using it without permission.
. © 2011 The McGraw-Hill Companies, Inc. Limited distribution permitted only to teachers and educators for course
9-90
9-122 A gas-turbine cycle with two stages of compression and two stages of expansion is considered. The back work ratio and the thermal efficiency of the cycle are to be determined for the cases of with and without a regenerator.
Assumptions 1 The air standard assumptions are applicable. 2 Air is an ideal gas with variable specific heats. 3 Kinetic and potential energy changes are negligible.
Properties The properties of air are given in Table A-17.
Analysis (a) The work inputs to each stage of compressor are identical, so are the work outputs of each stage of the turbine. Then,
PROPRIETARY MATERIAL. © 2011 The McGraw-Hill Companies, Inc. Limited distribution permitted only to teachers and educators for course
( )( )
( )( ) ( )
( )
( )( )( )
( ) ( )( ) ( ) kJ/kg 3.325813.98679.127722
kJ/kg .4626419.30042.43222
kJ/kg .1398636.94679.127788.079.1277
kJ/kg 946.3633.7923831
238kJ/kg 1277.79K 1200
kJ/kg .42432
/
kJ/kg 411.26158.4386.13
386.1kJ/kg 300.19K 300
65outT,
12inC,
6558665
65
865
6
755
1214212
421
2
11
56
5
12
1
=−=−=
=−=−=
=−−=
−−==⎯→⎯−−
=
==⎯→⎯=⎟⎠⎞
⎜⎝⎛==
===⎯→⎯=
=
−+==⎯→⎯−−
=
=⎯→⎯===
==⎯→⎯=
hhw
hhw
hhhhhhhhh
hhPPP
P
PhhT
hhhhhhhhh
hhPPP
P
PhT
sTs
T
ssrr
r
Css
C
ssrr
r
ηη
ηη
Thus,
s
T
3
4
1
5qin
86
7
1
9
2
86s
84.0/19.30026.41119.30012 −+=
=
45.3%==== 453.0kJ/kg 583.32kJ/kg 264.46
outT,
inC,bw w
wr
4 2s
( ) ( ) ( ) ( )
28.0%====
=−=−=
=−+−=−+−=
280.0kJ/kg 1137.03kJ/kg 318.86
kJ/kg .8631846.26432.583
kJ/kg 1137.03986.131277.79432.421277.79
in
netth
inC,outT,net
6745in
qw
www
hhhhq
η
(b) When a regenerator is used, rbw remains the same. The thermal efficiency in this case becomes
( ) ( )( )
44.2%====
=−=−=
=−=−=
442.0kJ/kg 721.75kJ/kg 318.86
kJ/kg .7572128.41503.1137
kJ/kg 15.28442.43213.98675.0
in
netth
regenoldin,in
48regen
qw
qqq
hhq
η
ε
preparation. If you are a student using this Manual, you are using it without permission.
9-91
9-123 An ideal regenerative gas-turbine cycle with two stages of compression and two stages of expansion is considered. The power produced and consumed by each compression and expansion stage, and the rate of heat rejected are to be determined.
Assumptions 1 The air standard assumptions are applicable. 2 Air is an ideal gas with constant specific heats at room temperature. 3 Kinetic and potential energy changes are negligible.
Properties The properties of air at room temperature are cp = 1.005 kJ/kg⋅K and k = 1.4 (Table A-2a).
Analysis The pressure ratio for each stage is
464.312 ==pr
s
T
3
4
1
6
288 K
97
8
2
5
1
According to the isentropic process expressions for an ideal gas,
K 7.410K)(3.464) 288( 0.4/1.4/)1(142 ==== − kk
prTTT
Since this is an ideal cycle,
7.410504975
PROPRIETARY MATERIAL. © 2011 The McGraw-Hill Companies, Inc. Limited distribution permitted only to teachers and educators for course
K 7.43828 =+
For the isentropic expansion processes,
==== − kkprTTT
he heat i is
=+=== TTTT
K 7.625K)(3.464) 7.438( 0.4/1.4/)1(786
T nput
(2 56in kJ/kg 8.375K )7.438K)(625.7kJ/kg (1.0052) =−⋅=
e mass flow rate is then
−= TTcq p
Th
kg/s .3311kJ/kg 8.375inqkJ/s 500in ===
Qm
&&
pplication of the first law to the expansion process 6-7 gives
K )7.438K)(625.7kJ/kg 5kg/s)(1.00 331.1(
he same amount of power is produced in process 8-9. When e first law is adapted to the compression process 1-2 it becomes
K )288K)(410.7kJ/kg 5kg/s)(1.00 331.1(
Compression process 3-4 uses the same amount of power. The rate of heat rejection from the cycle is
A
−= )( 76out7,-6 TTcmW p&&
kW 250.1=−⋅=
T th
−= )( 1212,in TTcmW p&&
kW 164.1=
−⋅=
kW 328.3=−⋅=
−=
K )288K)(410.7kJ/kg 5kg/s)(1.00 331.1(2
)(2 32out TTcmQ p&&
preparation. If you are a student using this Manual, you are using it without permission.
9-92
9-124 A regenerative gas-turbine cycle with two stages of compression and two stages of expansion is considered. The thermal efficiency of the cycle is to be determined.
Assumptions 1 The air standard assumptions are applicable. 2 Air is an ideal gas with constant specific heats at room temperature. 3 Kinetic and potential energy changes are negligible.
Properties The properties of air at room temperature are cp = 1.005 kJ/kg⋅K and k = 1.4 (Table A-2a).
Analysis The temperatures at various states are obtained as follows
K 9.430K)(4) 290( 0.4/1.4/)1(142 ==== − kk
prTTT
PROPRIETARY MATERIAL. © 2011 The McGraw-Hill Companies, Inc. Limited distribution permitted only to teachers and educators for course
=+=+= TT 5 K 9.450209.430204
s
T
3
4
1
6
290 K
97
8
2 5
10
K 4.749
KkJ/kg 1.005kJ/kg 300
K 9.450
)(
in56
56in
=⋅
+=+=
−=
p
p
cq
TT
TTcq
K 3.50441K) 4.749(1 0.4/1.4/)1(
67 =⎟⎠⎞
⎜⎝⎛=⎟
⎟⎠
⎞⎜⎜⎝
⎛=
− kk
prTT
K 8.802KkJ/kg 1.005
kJ/kg 300K 3.504in78 =
⋅+=+=
pcq
TT
K 2.54041K) 8.802(1 0.4/1.4/)1(
89⎜⎛
= TT =⎟⎠⎞
⎜⎝⎛=⎟
⎟⎠
⎞⎜⎝
− kk
pr
=−=
The heat input is
The heat rejected is
=−+−⋅=
pp
The thermal efficiency of the cycle is then
20910 −= TT K 2.520202.540
kJ/kg 600300300in =+=q
kJ/kg 373.0R )2909.430290K)(520.2kJ/kg (1.005
)()( 32110
out −+−= TTcTTcq
0.378=−=−=600
0.37311in
outth q
qη
preparation. If you are a student using this Manual, you are using it without permission.
9-93
9-125 A regenerative gas-turbine cycle with three stages of compression and three stages of expansion is considered. The thermal efficiency of the cycle is to be determined.
Assumptions 1 The air standard assumptions are applicable. 2 Air is an ideal gas with constant specific heats at room temperature. 3 Kinetic and potential energy changes are negligible.
Properties The properties of air at room temperature are cp = 1.005 kJ/kg⋅K and k = 1.4 (Table A-2a).
Analysis The temperatures at various states are obtained as follows
K 9.430K)(4) 290( 0.4/1.4/)1(1642 ===== − kk
prTTTT
PROPRIETARY MATERIAL. © 2011 The McGraw-Hill Companies, Inc. Limited distribution permitted only to teachers and educators for course
=+=+= TT
s
T
3
4
1290 K
9
8
2
5
1
11
67
12
13
14
7 K 9.450209.430206
K 4.749
KkJ/kg 1.005kJ/kg 300
K 9.450
)(
in78
78in
=⋅
+=+=
−=
p
p
cq
TT
TTcq
K 3.50441K) 4.749(1 0.4/1.4/)1(
89 =⎟⎠⎞
⎜⎝⎛=⎟
⎟⎠
⎞⎜⎜⎝
⎛=
− kk
prTT
K 8.802KkJ/kg 1.005
kJ/kg 300K 3.504in910 =
⋅+=+=
pcq
TT
K 2.54041K) 8.802(1 0.4/1.4/)1(
1011 =⎟⎠⎞
⎜⎝⎛=⎟
⎟⎠
⎞⎜⎜⎝
⎛=
− kk
prTT
K 7.838KkJ/kg 1.005
kJ/kg 300K 2.540in
1112 =⋅
+=+=pc
qTT
K 4.56441K) 7.838(1 0.4/1.4/)1(
1213 ⎜⎜⎝
=r
TT =⎟⎠⎞
⎜⎝⎛=⎟
⎟⎠
⎞⎛− kk
p
The heat input is
he heat r ected is
The thermal efficiency of the cycle is then
K 4.544204.564201314 =−=−= TT
kJ/kg 900300300300in =++=q
T ej
kJ/kg .9538R )2909.4302909.430290K)(544.4kJ/kg (1.005
)()()( 5432114out
=−+−+−⋅=
−+−+−= TTcTTcTTcq ppp
40.1%==−=−= 0.401900
9.53811in
outth q
qη
preparation. If you are a student using this Manual, you are using it without permission.
9-94
9-126 A regenerative gas-turbine cycle with three stages of compression and three stages of expansion is considered. The thermal efficiency of the cycle is to be determined.
Assumptions 1 The air standard assumptions are applicable. 2 Air is an ideal gas with constant specific heats at room temperature. 3 Kinetic and potential energy changes are negligible.
Properties The properties of air at room temperature are cp = 1.005 kJ/kg⋅K and k = 1.4 (Table A-2a).
Analysis Since all compressors share the same compression ratio and begin at the same temperature,
K 9.430K)(4) 290( 0.4/1.4/)1(1642 ===== − kk
prTTTT
PROPRIETARY MATERIAL. © 2011 The McGraw-Hill Companies, Inc. Limited distribution permitted only to teachers and educators for course
−= TT
s
T
3
4
1290 K
9
8
2
5
1
11
67
12
13
14
From the problem statement,
7 6513
The relations for heat input and expansion processes are
p
p cq
TTTTcq in7878in )( +=⎯→⎯−=
kk
prTT
/)1(
891
−
⎟⎟⎠
⎞⎜⎜⎝
⎛=
kk
prT
pcq
TT in910 += , T 1
/)1(
1011
−
⎟⎟⎠
⎞⎜⎜⎝
⎛=
pcq inTT 1112 += ,
kk /)1(1
−
⎟⎞
⎜⎛
prTT 1213 ⎟
⎠⎜⎝
=
The simultaneous solution of above equations using EES software gives the following results
131211
9
=====
TTTT
lance on the regenerator,
61414136
141367
=+=+=⎯→⎯−=
−=−
TTTTT
TTTT
he heat i is
he heat rejected is
The thermal efficiency of the cycle is then
K 7.585 K, 4.870 K, 9.571 K, 8.849K 3.551
10
K, 2.819 K, 7.520 87 == TTT
From an energy ba
)40( 13 −−T K 9.495659.43065
T nput
kJ/kg 900300300300in =++=q
T
kJ/kg 1.490R )2909.4302909.430290K)(495.9kJ/kg (1.005
)()()( 5432114out
=−+−+−⋅=
−+−+−= TTcTTcTTcq ppp
45.5%==−=−= 455.0900
1.49011in
outth q
qη
preparation. If you are a student using this Manual, you are using it without permission.
9-95
Jet-Propulsion Cycles
9-127C The power developed from the thrust of the engine is called the propulsive power. It is equal to thrust times the aircraft velocity.
9-128C The ratio of the propulsive power developed and the rate of heat input is called the propulsive efficiency. It is determined by calculating these two quantities separately, and taking their ratio.
9-129C It reduces the exit velocity, and thus the thrust.
PROPRIETARY MATERIALpreparation. If you are a student using this Manual, you are using it without permission.
. © 2011 The McGraw-Hill Companies, Inc. Limited distribution permitted only to teachers and educators for course
9-96
9-130 A turboprop engine operating on an ideal cycle is considered. The thrust force generated is to be determined.
Assumptions 1 Steady operating conditions exist. 2 The air standard assumptions are applicable. 3 Air is an ideal gas with constant specific heats at room temperature. 4 The turbine work output is equal to the compressor work input.
Properties The properties of air at room temperature are R = 0.287 kPa⋅m3/kg⋅K (Table A-1), cp = 1.005 kJ/kg⋅K and k = 1.4 (Table A-2a).
Analysis Working across the two isentropic processes of the cycle yields
PROPRIETARY MATERIAL. © 2011 The McGraw-Hill Companies, Inc. Limited distribution permitted only to teachers and educators for course
=== − kkprTT 2 K 7.482K)(10) 250( 0.4/1.4/)1(
1
s
T
1
2 4
3qin
5qout
K 0.403101K) 778(1 0.4/1.4/)1(
35 =⎟⎠⎞
⎜⎝⎛=⎟
⎟⎠
⎞⎜⎜⎝
⎛=
− kk
prTT
Since the work produced by expansion 3-4 equals that used by compression 1-2, an energy balance gives
K 3.545)2507.482(778)( 1234 =−−=−−= TTTT
The excess enthalpy generated by expansion 4-5 is used to increase the kinetic energy of the flow through the propeller,
2
)(2
inlet2
exit54
VVmTTcm ppe
−=− &&
ich when solved for the velocity at which the air leaves the propeller gives wh
m/s 1.216
)m/s 180(kJ/kg 1
/sm 1000K)0.4033.545)(KkJ/kg 005.1(2012
)(21
2 ⎥⎤
⎢⎡
+−= VTTcm
V e&
2/12
22
2/
inlet54exit
=
⎥⎥⎦
⎤
⎢⎢⎣
⎡+
⎠⎜⎜⎝
⎛−⋅=
⎥⎦⎢⎣ m pp&
⎟⎟⎞
The mass flow rate through the propeller is
kg/s 0.975/kgm 305.1
m/s 1804m) 3(
4
/kgm 305.1kPa 55
K) 250()mkPa
287.0(RT
3
2
1
12
1
1
33
1
====
=⋅
==
ππvv
v
VDAVm
P
p&
The thrust force generated by this propeller is then
kN 35.2=⎟⎟⎠
⎞⎜⎜⎝
⎛
⋅−=−=
2inletexitm/skg 1000
kN 10)m/s181kg/s)(216. (975.0)( VVmF p&
preparation. If you are a student using this Manual, you are using it without permission.
9-97
9-131 A turboprop engine operating on an ideal cycle is considered. The thrust force generated is to be determined.
Assumptions 1 Steady operating conditions exist. 2 The air standard assumptions are applicable. 3 Air is an ideal gas with constant specific heats at room temperature. 4 The turbine work output is equal to the compressor work input.
Properties The properties of air at room temperature are R = 0.287 kPa⋅m3/kg⋅K (Table A-1), cp = 1.005 kJ/kg⋅K and k = 1.4 (Table A-2a).
Analysis Working across the two isentropic processes of the cycle yields
K 7.482K)(10) 250( 0.4/1.4/)1(12 === − kk
prTT
PROPRIETARY MATERIAL. © 2011 The McGraw-Hill Companies, Inc. Limited distribution permitted only to teachers and educators for course
K 0.40310
K) 778(35 =⎟⎠
⎜⎝
=⎟⎟⎠
⎜⎜⎝
=pr
TT 11 0.4/1.4/)1(⎞⎛⎞⎛
− kk
ince the work produced by expansion 3-4 equals that used by pression 1-2, an energy balance gives
123s
T
1
24
3 qin
5 qout
Scom
7.482(778)( K 3.545)2504 −−=−−= TTTT =
ler is The mass flow rate through the propel
kg/s 0.624
/kgm 305.144 311
====vv
m p&m/s 180m) 4.2(
/kgm 305.1kPa 55
K) 250()mkPa 287.0(
21
21
33
1 =⋅
==
ππ
v
VDAVP
RT
ccording the previous problem,
A to
kg/s 75.4820
kg/s 0.975m&
20=== p
em&
The excess enthalpy generated by expansion 4-5 is used to increase the kinetic energy of the flow through the propeller,
2)(
2inlet
2exit
54VV
mTTcm ppe−
=− &&
which when solved for the velocity at which the air leaves the propeller gives
m/s 0.234
)m/s 180(kJ/kg 1
/sm 1000K)0.4033.545)(KkJ/kg 005.1(kg/s 0.624kg/s 75.48
2
)(2
2/12
22
2/12 ⎤⎡ me&
inlet54it
=
⎥⎥⎦
⎤
⎢⎢⎣
⎡+⎟
⎟⎠
⎞⎜⎜⎝
⎛−⋅=
⎥⎥⎦⎢
⎢⎣
+−= VTTcm
V pp&
The thrust force generated by this propeller is then
ex
kN 33.7=⎟⎟⎠
⎞⎜⎜⎝
⎛
⋅−=−=
2inletexitm/skg 1000
kN 10)m/s180kg/s)(234. (624.0)( VVmF p&
preparation. If you are a student using this Manual, you are using it without permission.
9-98
9-132 A turbofan engine operating on an ideal cycle produces 50,000 N of thrust. The air temperature at the fan outlet needed to produce this thrust is to be determined.
Assumptions 1 Steady operating conditions exist. 2 The air standard assumptions are applicable. 3 Air is an ideal gas with constant specific heats at room temperature. 4 The turbine work output is equal to the compressor work input.
Properties The properties of air at room temperature are R = 0.287 kPa⋅m3/kg⋅K, cp = 1.005 kJ/kg⋅K and k = 1.4 (Table A-2a).
Analysis The total mass flow rate is
kg/s 1.676
/kgm 452.1m/s 200
4m) 5.2(
4
/kgm 452.1kPa 50
K) 253()mkPa 287.0(
3
2
1
12
1
1
33
1
====
=⋅
==
ππvv
v
VDAVm
PRT
&
s
T
1
2 4
3qin
5
qoutNow,
kg/s 51.8488e
kg/s 1.676===
mm&
&
he mass flow rate through the fan is
order to roduce the specified thrust force, the velocity at the fan exit will be
T
kg/s 6.59151.841.676 =−=−= ef mmm &&&
In p
m/s 284.5N 1m/skg 1
kg/s 591.6N 50,000m/s) (200
)( inletexit −= f VVmF &
2
inletexit =⎟⎟⎠
⎞⎜⎜⎝
⎛ ⋅+=+=
fmFVV&
balance on the stream passing through the fan gives
An energy
K 232.6=
⎟⎠
⎞⎜⎝
⎛⋅
−−=
−−=
−=−
22
22
2inlet
2exit
45
2inlet
2exit
54
/sm 1000kJ/kg 1
)KkJ/kg 005.1(2)m/s 200()m/s 5.284(K 253
2
2)(
p
p
cVV
TT
VVTTc
PROPRIETARY MATERIALpreparation. If you are a student using this Manual, you are using it without permission.
. © 2011 The McGraw-Hill Companies, Inc. Limited distribution permitted only to teachers and educators for course
9-99
9-133 A pure jet engine operating on an ideal cycle is considered. The velocity at the nozzle exit and the thrust produced are to be determined. Assumptions 1 Steady operating conditions exist. 2 The air standard assumptions are applicable. 3 Air is an ideal gas with constant specific heats at room temperature. 4 The turbine work output is equal to the compressor work input. Properties The properties of air at room temperature are R = 0.287 kPa⋅m3/kg⋅K, cp = 1.005 kJ/kg⋅K and k = 1.4 (Table A-2a). Analysis (a) We assume the aircraft is stationary and the air is moving towards the aircraft at a velocity of V 1 = 240 m/s. Ideally, the air will leave the diffuser with a negligible velocity (V 2 ≅ 0).
PROPRIETARY MATERIAL. © 2011 The McGraw-Hill Companies, Inc. Limited distribution permitted only to teachers and educators for course
Diffuser:
( )( )
( )( )( )
( ) kPa 88.64K 260K 288.7kPa 45
K 7.288/sm 1000
kJ/kg 1KkJ/kg 1.0052
m/s 240K 2602
2/02
02/2/
1.4/0.41/
1
212
22
221
12
2112
21
022
122
222
11
outin(steady) 0
systemoutinE
=⎟⎟⎠
⎞⎜⎜⎝
⎛=⎟⎟
⎠
⎞⎜⎜⎝
⎛=
=⎟⎟⎠
⎞⎜⎜⎝
⎛
⋅+=+=
−−=
−+−=⎯→⎯+=+
=⎯→⎯∆=−
−kk
p
p
TT
PP
cV
TT
VTTc
VVhhVhVh
EEEE &&&&
pressor:
&
Com( )( ) ( )( )
( )( )(1K 288.73
23 =⎟⎟⎞
⎜⎜⎛
=PP
TT ) K 7.6003
kPa 5.843kPa 64.8813
0.4/1.4/1
2
243
=⎠⎝
====
− kk
p PrPP
Turbine:
( ) ( )54235423outturb,incomp, TTcTTchhhhww pp −=−⎯→⎯−=−⎯→⎯=
or K 0.5187.2887.6008302345 =+−=+−= TTTT Nozzle:
( )( )
( ) 2/02
0
2/2/
K 3.359kPa 843.5
kPa 45K 830
2656
025
26
56
266
255
outin(steady) 0
systemoutin
0.4/1.4/1
4
646
VTTcVV
hh
VhVh
EEEEE
PP
TT
p
kk
+−=⎯→⎯−
+−=
+=+
=⎯→⎯∆=−
=⎟⎟⎠
⎞⎜⎜⎝
⎛=⎟⎟
⎠
⎞⎜⎜⎝
⎛=
−
&&&&&
( )( )( ) m/s 564.8=⎟⎟⎠
⎞⎜⎜⎝
⎛−⋅==
kJ/kg 1/sm 1000K3.359518.0KkJ/kg 1.0052
22
6 exitVV or
The mass flow rate through the engine is
kg/s 0.291
/kgm 658.1m/s 240
4m) 6.1(
4
/kgm 658.1kPa 45
K) 260()mkPa 287.0(
3
2
1
12
1
1
33
1
====
=⋅
==
ππvv
v
VDAVm
PRT
&
The thrust force generated is then
N 94,520=⎟⎟⎠
⎞⎜⎜⎝
⎛
⋅−=−= 2inletexit m/skg 1
N 10)m/s248kg/s)(564. (291.0)( VVmF &
qin
qouts
T
1
2
4
3 5
6
preparation. If you are a student using this Manual, you are using it without permission.
9-100
9-134 A turbojet aircraft flying at an altitude of 9150 m is operating on the ideal jet propulsion cycle. The velocity of exhaust gases, the propulsive power developed, and the rate of fuel consumption are to be determined. Assumptions 1 Steady operating conditions exist. 2 The air standard assumptions are applicable. 3 Air is an ideal gas with constant specific heats at room temperature. 4 Kinetic and potential energies are negligible, except at the diffuser inlet and the nozzle exit. 5 The turbine work output is equal to the compressor work input. Properties The properties of air at room temperature are cp = 1.005 kJ/kg.K and k = 1.4 (Table A-2a). Analysis (a) We assume the aircraft is stationary and the air is moving towards the aircraft at a velocity of V 1 = 320 m/s. Ideally, the air will leave the diffuser with a negligible velocity (V 2 ≅ 0).
PROPRIETARY MATERIAL. © 2011 The McGraw-Hill Companies, Inc. Limited distribution permitted only to teachers and educators for course
Diffuser:
( )( )
( )( )( )
( ) kPa 62.6K 241K 291.9
kPa 32
K 291.9/sm 1000
kJ/kg 1KkJ/kg 1.0052
m/s 320K 241
2
2/02
02/2/
1.4/0.41/
1
212
22
221
12
2112
21
022
122
222
11
outin(steady) 0
systemoutin
=⎟⎟⎠
⎞⎜⎜⎝
⎛=⎟⎟
⎠
⎞⎜⎜⎝
⎛=
=⎟⎟⎠
⎞⎜⎜⎝
⎛
⋅+=+=
−−=
−+−=⎯→⎯+=+
=⎯→⎯∆=−
−kk
p
p
TT
PP
cV
TT
VTTc
VVhhVhVh
EEEEE &&&&
ompressor:
&
s
T
1
2
4
3
Qi
5
6
·
C( )( ) ( )( )
( )( )( ) K 593.712K 291.9
kPa 751.2kPa 62.612
0.4/1.4/1
2
32
3
243
==⎟⎟⎠
⎞⎜⎜⎝
⎛=
====
− kk
p
PP
TT
PrPP
ine:
Turb
( ) ( )54235423outturb,incomp, TTcTTchhhhww pp −=−⎯→⎯−=−⎯→⎯=
K1098.2291.9593.714002345 =+−=+−= TTTT or
Nozzle:
( )( )
( ) 2/02
0
2/2/
K 568.2kPa 751.2
kPa 32K 1400
2656
025
26
56
266
255
outin(steady) 0
systemoutin
0.4/1.4/1
4
646
VTTcVV
hh
VhVh
EEEEE
PP
TT
p
kk
+−=⎯→⎯−
+−=
+=+
=⎯→⎯∆=−
=⎟⎟⎠
⎞⎜⎜⎝
⎛=⎟⎟
⎠
⎞⎜⎜⎝
⎛=
−
&&&&&
( )( )( ) m/s1032 kJ/kg 1
/sm 1000K568.21098.2KkJ/kg 1.0052
22
6 =⎟⎟⎠
⎞⎜⎜⎝
⎛−⋅=V or
( ) ( )( ) ( ) kW 13,670=⎟⎟⎠
⎞⎜⎜⎝
⎛−=−=
22aircraftinletexit/sm 1000
kJ/kg 1m/s 320m/s3201032kg/s 60VVVmW p && (b)
( ) ( ) ( )( )( )
kg/s 1.14===
=−⋅=−=−=
kJ/kg 42,700kJ/s 48,620
HV
kJ/s 48,620K593.71400KkJ/kg 1.005kg/s 60
infuel
3434in
Qm
TTcmhhmQ p
&&
&&& (c)
preparation. If you are a student using this Manual, you are using it without permission.
9-101
9-135 A turbojet aircraft is flying at an altitude of 9150 m. The velocity of exhaust gases, the propulsive power developed, and the rate of fuel consumption are to be determined.
Assumptions 1 Steady operating conditions exist. 2 The air standard assumptions are applicable. 3 Air is an ideal gas with constant specific heats at room temperature. 4 Kinetic and potential energies are negligible, except at the diffuser inlet and the nozzle exit.
Properties The properties of air at room temperature are cp = 1.005 kJ/kg.K and k = 1.4 (Table A-2a).
Analysis (a) For convenience, we assume the aircraft is stationary and the air is moving towards the aircraft at a velocity of V 1 = 320 m/s. Ideally, the air will leave the diffuser with a negligible velocity (V 2 ≅ 0).
Diffuser:
PROPRIETARY MATERIAL. © 2011 The McGraw-Hill Companies, Inc. Limited distribution permitted only to teachers and educators for course
( )
( )( )( )
( )( ) kPa 62.6
K 241K 291.9
kPa 32
K 291.9/sm 1000
kJ/kg 1KkJ/kg 1.0052
m/s 320K 241
2
2/02
0
2/2/
1.4/0.41/
1
212
22
221
12
2112
21
022
12
222
211
outin
(steady) 0systemoutin
=⎟⎟⎠
⎞⎜⎜⎝
⎛=⎟⎟
⎠
⎞⎜⎜⎝
⎛=
=⎟⎟⎠
⎞⎜⎜⎝
⎛
⋅+=+=
−−=
−+−=
+=+
=
∆=−
−kk
p
p
TT
PP
cV
TT
VTTc
VVhh
VhVh
EE
EEE&&
&&&
s
T
1
2
4
3
Qin
5s
6
·
5
Compressor:
( )( ) ( )( )( )
( )( ) K 593.712K 291.9
kPa 751.2kPa 62.612
0.4/1.4/1
2
323
243
==⎟⎟⎠
⎞⎜⎜⎝
⎛=
====
− kk
s
p
PP
TT
PrPP
( )( )
( ) ( ) ( ) K 669.20.80/291.9593.71.929/2323
23
23
23
23
=−+=−+=
−
−=
−−
=
Cs
p
spsC
TTTT
TTcTTc
hhhh
η
η
Turbine:
or, ( ) ( )
K 1022.7291.9669.214002345
54235423outturb,incomp,
=+−=+−=
−=−⎯→⎯−=−⎯→⎯=
TTTT
TTcTTchhhhww pp
( )( )
( ) ( )( )
( ) kPa 197.7K 1400K 956.1
kPa 751.2
K 956.185.0/1022.714001400/1.4/0.41/
4
545
5445
54
54
54
54
=⎟⎟⎠
⎞⎜⎜⎝
⎛=⎟⎟
⎠
⎞⎜⎜⎝
⎛=
=−−=−−=
−
−=
−−
=
−kks
Ts
sp
p
sT
TT
PP
TTTT
TTcTTc
hhhh
η
η
preparation. If you are a student using this Manual, you are using it without permission.
9-102
Nozzle:
( )( )
( ) 2/02
0
2/2/
K 607.8kPa 197.7
kPa 32K 1022.7
2656
025
26
56
266
255
outin
(steady) 0systemoutin
0.4/1.4/1
5
656
VTTc
VVhh
VhVh
EE
EEE
PP
TT
p
kk
+−=
−+−=
+=+
=
∆=−
=⎟⎟⎠
⎞⎜⎜⎝
⎛=⎟⎟
⎠
⎞⎜⎜⎝
⎛=
−
&&
&&&
or,
( )( )( ) m/s 913.2=⎟⎟⎠
⎞⎜⎜⎝
⎛−⋅=
kJ/kg 1/sm 1000
K607.81022.7KkJ/kg 1.005222
6V
( )( )( ) ( )
kW 11,390=⎟⎟⎠
⎞⎜⎜⎝
⎛−=
−=
22
aircraftinletexit
/sm 1000kJ/kg 1
m/s 320m/s320913.2kg/s 60
VVVmW p && (b)
(c) ( ) ( ) ( )( )( )
kg/s 1.03===
=−⋅=−=−=
kJ/kg 42,700kJ/s 44,067
HV
kJ/s 44,067K669.21400KkJ/kg 1.005kg/s 60
infuel
3434in
Qm
TTcmhhmQ p
&&
&&&
PROPRIETARY MATERIALpreparation. If you are a student using this Manual, you are using it without permission.
. © 2011 The McGraw-Hill Companies, Inc. Limited distribution permitted only to teachers and educators for course
9-103
9-136 A turbojet aircraft that has a pressure rate of 9 is stationary on the ground. The force that must be applied on the brakes to hold the plane stationary is to be determined.
Assumptions 1 Steady operating conditions exist. 2 The air standard assumptions are applicable. 3 Air is an ideal gas with variable specific heats. 4 Kinetic and potential energies are negligible, except at the nozzle exit.
Properties The properties of air are given in Table A-17.
Analysis (a) Using variable specific heats for air,
PROPRIETARY MATERIAL. © 2011 The McGraw-Hill Companies, Inc. Limited distribution permitted only to teachers and educators for course
Compressor:
2311.1kJ/kg 0.1629K 029
1
11
==⎯→⎯=
rPhT
( )( )
( )( )
5.568kJ/kg 1611.65.1067.07544
kJ/kg 5.1067kg/s 20
kJ/s 21,350
kJ/s 350,21kJ/kg 42,700kg/s 0.5HV
kJ/kg .0754408.112311.19
3
12
2323in
inin
fuelin
21
2
=⎯→⎯=+=+=⎯→⎯−=
===
==×=
=⎯→⎯===
rin
rr
Pqhhhhq
mQ
q
mQ
hPPP
P
&
&
&&
s
T
1
3
2
qin
4
5
Turbine:
kJ/kg 7.1357.16290.07544.616111234
4312outturb,incomp,
=+−=+−=
−=−⎯→⎯=
hhhh
hhhhww or
Nozzle:
( )
20
2/2/
kJ/kg .5688817.63915.568
024
25
45
255
244
outin
(steady) 0systemoutin
53
535
VVhh
VhVh
EE
EEE
hPP
PP rr
−+−=
+=+
=
∆=−
=⎯→⎯=⎟⎠⎞
⎜⎝⎛=⎟⎟
⎠
⎞⎜⎜⎝
⎛=
&&
&&&
or
( ) ( )( ) m/s .6968kJ/kg 1
/sm 1000kJ/kg.568881357.72222
545 =⎟⎟⎠
⎞⎜⎜⎝
⎛−=−= hhV
( ) ( )( ) N 19,370=⎟⎟⎠
⎞⎜⎜⎝
⎛
⋅−=−
2inletexit m/skg 1N 1m/s0968.6kg/s 20VVm& Brake force = Thrust =
preparation. If you are a student using this Manual, you are using it without permission.
9-104
9-137 Problem 9-136 is reconsidered. The effect of compressor inlet temperature on the force that must be applied to the brakes to hold the plane stationary is to be investigated.
Analysis Using EES, the problem is solved as follows:
P_ratio =9 T_1 = 7 [C] T[1] = T_1+273 "[K]" P[1]= 95 [kPa] P[5]=P[1] Vel[1]=0 [m/s] V_dot[1] = 18.1 [m^3/s] HV_fuel = 42700 [kJ/kg] m_dot_fuel = 0.5 [kg/s] Eta_c = 1.0 Eta_t = 1.0 Eta_N = 1.0 "Inlet conditions" h[1]=ENTHALPY(Air,T=T[1]) s[1]=ENTROPY(Air,T=T[1],P=P[1]) v[1]=volume(Air,T=T[1],P=P[1]) m_dot = V_dot[1]/v[1] "Compressor anaysis" s_s[2]=s[1] "For the ideal case the entropies are constant across the compressor" P_ratio=P[2]/P[1]"Definition of pressure ratio - to find P[2]" T_s[2]=TEMPERATURE(Air,s=s_s[2],P=P[2]) "T_s[2] is the isentropic value of T[2] at compressor exit" h_s[2]=ENTHALPY(Air,T=T_s[2]) Eta_c =(h_s[2]-h[1])/(h[2]-h[1]) "Compressor adiabatic efficiency; Eta_c = W_dot_c_ideal/W_dot_c_actual. " m_dot*h[1] +W_dot_c=m_dot*h[2] "SSSF First Law for the actual compressor, assuming: adiabatic, ke=pe=0" "External heat exchanger analysis" P[3]=P[2]"process 2-3 is SSSF constant pressure" h[3]=ENTHALPY(Air,T=T[3]) Q_dot_in = m_dot_fuel*HV_fuel m_dot*h[2] + Q_dot_in= m_dot*h[3]"SSSF First Law for the heat exchanger, assuming W=0, ke=pe=0" "Turbine analysis" s[3]=ENTROPY(Air,T=T[3],P=P[3]) s_s[4]=s[3] "For the ideal case the entropies are constant across the turbine" {P_ratio= P[3] /P[4]} T_s[4]=TEMPERATURE(Air,h=h_s[4]) "Ts[4] is the isentropic value of T[4] at turbine exit" {h_s[4]=ENTHALPY(Air,T=T_s[4])} "Eta_t = W_dot_t /Wts_dot turbine adiabatic efficiency, Wts_dot > W_dot_t" Eta_t=(h[3]-h[4])/(h[3]-h_s[4]) m_dot*h[3] = W_dot_t + m_dot*h[4] "SSSF First Law for the actual compressor, assuming: adiabatic, ke=pe=0" T[4]=TEMPERATURE(Air,h=h[4]) P[4]=pressure(Air,s=s_s[4],h=h_s[4]) "Cycle analysis" W_dot_net=W_dot_t-W_dot_c"Definition of the net cycle work, kW" W_dot_net = 0 [kW] "Exit nozzle analysis:" s[4]=entropy('air',T=T[4],P=P[4]) s_s[5]=s[4] "For the ideal case the entropies are constant across the nozzle" T_s[5]=TEMPERATURE(Air,s=s_s[5], P=P[5]) "T_s[5] is the isentropic value of T[5] at nozzle exit" h_s[5]=ENTHALPY(Air,T=T_s[5]) Eta_N=(h[4]-h[5])/(h[4]-h_s[5]) m_dot*h[4] = m_dot*(h_s[5] + Vel_s[5]^2/2*convert(m^2/s^2,kJ/kg))
PROPRIETARY MATERIALpreparation. If you are a student using this Manual, you are using it without permission.
. © 2011 The McGraw-Hill Companies, Inc. Limited distribution permitted only to teachers and educators for course
9-105
m_dot*h[4] = m_dot*(h[5] + Vel[5]^2/2*convert(m^2/s^2,kJ/kg)) T[5]=TEMPERATURE(Air,h=h[5]) s[5]=entropy('air',T=T[5],P=P[5]) "Brake Force to hold the aircraft:" Thrust = m_dot*(Vel[5] - Vel[1]) "[N]" BrakeForce = Thrust "[N]" "The following state points are determined only to produce a T-s plot" T[2]=temperature('air',h=h[2]) s[2]=entropy('air',T=T[2],P=P[2])
BrakeForce
[N]
m [kg/s]
T3
[K] T1
[C]
21232 21007 20788 20576 20369 20168 19972 19782 19596 19415 19238
23.68 23.22 22.78 22.35 21.94 21.55 21.17 20.8
20.45 20.1
19.77
1284 1307 1330 1352 1375 1398 1420 1443 1466 1488 1510
-20 -15 -10 -5 0 5
10 15 20 25 30
-20 -10 0 10 20 3019000
19450
19900
20350
20800
21250
T1 [C]
Bra
keFo
rce
[N]
4.5 5.0 5.5 6.0 6.5 7.0 7.5 8.0 8.50
500
1000
1500
0x100
5x102
103
103
s [kJ/kg-K]
T [K
]
Air
855 kPa
95 kPa
1
2s
3
4s
5
PROPRIETARY MATERIALpreparation. If you are a student using this Manual, you are using it without permission.
. © 2011 The McGraw-Hill Companies, Inc. Limited distribution permitted only to teachers and educators for course
9-106
9-138 Air enters a turbojet engine. The thrust produced by this turbojet engine is to be determined.
Assumptions 1 Steady operating conditions exist. 2 The air standard assumptions are applicable. 3 Air is an ideal gas with variable specific heats. 4 Kinetic and potential energies are negligible, except at the diffuser inlet and the nozzle exit.
Properties The properties of air are given in Table A-17.
Analysis We assume the aircraft is stationary and the air is moving towards the aircraft at a velocity of V 1 = 300 m/s. Taking the entire engine as our control volume and writing the steady-flow energy balance yield
PROPRIETARY MATERIAL. © 2011 The McGraw-Hill Companies, Inc. Limited distribution permitted only to teachers and educators for course
J / kg
J / kg
T h
T h
1 1
2 2
280 28013
700 713 27
= ⎯ →⎯ =
= ⎯ →⎯ =
K k
K k
.
.
( ) ( )⎥⎥⎦
⎤
⎢⎢⎣
⎡⎟⎟⎠
⎞⎜⎜⎝
⎛−+−=
⎟⎟⎠
⎞⎜⎜⎝
⎛ −+−=
+=++ 211in ( VhmQ &&
=
∆=−
22
222
21
22
12in
222
outin
(steady) 0systemoutin
/sm 1000kJ/kg 1
2m/s 300
13.28027.713kg/s 16kJ/s 000,15
2
)2/()2/
V
VVhhmQ
Vhm
EE
EEE
&&
&
&&
&&&
It gives
V 2 = 1048 m/s
Thus,
15,000 kJ/s
1
7°C 300 m/s16 kg/s
2
427°C
( ) ( )( ) N 11,968=−=−= m/s3001048kg/s 1612 VVmFp &
preparation. If you are a student using this Manual, you are using it without permission.
9-107
Second-Law Analysis of Gas Power Cycles
9-139 The process with the highest exergy destruction for an ideal Otto cycle described in Prob. 9-33 is to be determined.
Analysis From Prob. 9-33, qin = 582.5 kJ/kg, qout = 253.6 kJ/kg, T1 = 288 K, T2 = 661.7 K, T3 = 1473 K, and T4 = 641.2 K. The exergy destruction during a process of the cycle is
PROPRIETARY MATERIAL. © 2011 The McGraw-Hill Companies, Inc. Limited distribution permitted only to teachers and educators for course
⎟⎠
⎜⎝
+−∆==sinksource
0gen0dest TTsTsTx ⎟
⎞⎜⎛ outin qq
ss of the cycle gives
st,1x (isentropic process)
v
P
4
1
3
2
s
T
1
2
4
3 qin
qout
Application of this equation for each proce
de 0=2-
Kk J/kg5746.00K 7.661K 1473ln)KkJ/kg 718.0(
lnln2
3
2
31423
⋅=+⋅=
+=−=−v
vv R
TT
cssss
kJ/kg 51.59=
⎟⎠
⎞⎜⎝
⎛ −⋅=
⎟⎟⎠
⎞⎜⎝
2303-dest,2 TssT ⎜
⎛−−=
K 1473kJ/kg 582.5KkJ/kg 0.5746K) 288(
source
inqx
(isentropic process)
0=4-dest,3x
kJ/kg 88.12=
⎟⎠
⎞⎜⎝
⎛ +⋅−=
⎟⎟⎠
⎜⎜⎝
+−=sink
out4101-dest,4 T
ssTx⎞⎛
K 288kJ/kg 253.6KkJ/kg 0.5746K) 288(
q
The largest exergy destruction in the cycle occurs during the heat-rejection process.
preparation. If you are a student using this Manual, you are using it without permission.
9-108
9-140 The exergy loss of each process for an ideal dual cycle described in Prob. 9-58 is to be determined.
Analysis From Prob. 9-58, qin,x-3 = 114.6 kJ/kg, T1 = 291 K, T2 = 1037 K, Tx = 1141 K, T3 = 1255 K, and T4 = 494.8 K. Also,
kJ/kg 74.67K)10371141)(KkJ/kg 718.0()( 22,in =−⋅=−=− TTcq xx v
kJ/kg 3.146K)2918.494)(KkJ/kg 718.0()( 14out =−⋅=−= TTcq v
v
P
4
1
2
3
qout
x
qinThe exergy destruction during a process of the cycle is
PROPRIETARY MATERIAL. © 2011 The McGraw-Hill Companies, Inc. Limited distribution permitted only to teachers and educators for course
⎟⎟⎠
⎜⎜⎝
+−∆==sink
out
source
in0gen0dest TT
sTsTx ⎞⎛ qq
Application of this equation for each process of the cycle gives
KkJ/kg 1158.0kPa 90kPa 5148ln)KkJ/kg 287.0(
K 291K 1037ln)KkJ/kg 005.1(
lnln1
2
1
212
⋅=
⋅−⋅=
−=−PP
RTT
css p
kJ/kg 33.7=⋅=−= K)kJ/kg K)(0.1158 291()( 1202-1 dest, ssTx
KkJ/kg 06862.00K 1037K 1141ln)KkJ/kg 718.0(
lnln22
2
⋅=+⋅=
+=−v
vv
xxx R
TT
css
kJ/kg 2.65=⎟⎠
⎞⎜⎝
⎛ −⋅=⎟⎟⎠
⎞⎜⎜⎝
⎛−−=
K 1255kJ/kg 74.67KkJ/kg 0.06862K) 291(
source
-2 in,20-2 dest, T
qssTx x
xx
KkJ/kg 09571.00K 1141K 1255ln)KkJ/kg 005.1(lnln 33
3 ⋅=−⋅=−=−xx
px PP
RTT
css
kJ/kg 1.28=⎟⎠
⎞⎜⎝
⎛ −⋅=⎟⎟⎠
⎞⎜⎜⎝
⎛−−=
K 1255kJ/kg 114.6KkJ/kg 0.09571K) 291(
source
3- in,303- dest, T
qssTx x
xx
KkJ/kg 1339.01.1
18ln)KkJ/kg 287.0(K 1255K 8.494ln)KkJ/kg 718.0(
lnlnlnln3
4
3
4
3
434
⋅=⋅+⋅=
+=+=−crrR
TT
cRTT
css vv v
v
kJ/kg 39.0=⋅=−= K)kJ/kg K)(0.1339 291()( 3404-3 dest, ssTx
KkJ/kg 3811.00K 8.494
K 291ln)KkJ/kg 718.0(lnln4
1
4
141 ⋅−=+⋅=+=−
v
vv R
TT
css
kJ/kg 35.4=⎟⎠
⎞⎜⎝
⎛ +⋅−=⎟⎟⎠
⎞⎜⎜⎝
⎛+−=
K 291kJ/kg 146.3KkJ/kg 0.3811K) 291(
sink
out4101-dest,4 T
qssTx
preparation. If you are a student using this Manual, you are using it without permission.
9-109
9-141 The exergy loss of each process for an air-standard Stirling cycle described in Prob. 9-74 is to be determined.
Analysis From Prob. 9-74, qin = 1275 kJ/kg, qout = 212.5 kJ/kg, T1 = T2 = 1788 K, T3 = T4 = 298 K. The exergy destruction during a process of the cycle is
⎟⎟⎠
⎞⎜⎜⎝
⎛+−∆==
sink
out
source
in0gen0dest T
qT
qsTsTx
PROPRIETARY MATERIAL. © 2011 The McGraw-Hill Companies, Inc. Limited distribution permitted only to teachers and educators for course
Application of this equation for each process of the cycle gives
KkJ/kg 7132.0)12ln()KkJ/kg 287.0(0
lnln1
2
1
212
⋅=⋅+=
+=−v
vv R
TT
css
0kJ/kg 0.034 ≈=⎟⎠
⎞⎜⎝
⎛ −⋅=
⎟⎟⎠
⎞⎜⎜⎝
⎛−−=
K 1788kJ/kg 1275
KkJ/kg 0.7132K) 298(
source
in1202-1 dest, T
qssTx
s
T
3
2 qin
qout
4
1
KkJ/kg 7132.012 ⎠⎝
1ln)KkJ/kg 287.0(0
lnln3
4
3
434
⋅−=⎟⎞
⎜⎛⋅+=
+=−v
vv R
TT
css
0kJ/kg 0.034 ≈−=⎟⎠
⎞⎜⎝
⎛ +⋅−=
⎟⎟⎠
⎞⎛ outq⎜⎜⎝
+−=
K 298kJ/kg 212.5
KkJ/kg 0.7132K) 298(
sink1202-1 dest, T
ssTx
These results are not surprising since Stirling cycle is totally reversible. Exergy destructions are not calculated for processes 2-3 and 4-1 because there is no interaction with the surroundings during these processes to alter the exergy destruction.
preparation. If you are a student using this Manual, you are using it without permission.
9-110
9-142 The exergy destruction associated with each of the processes of the Brayton cycle described in Prob. 9-82 is to be determined.
Analysis From Prob. 9-82, qin = 698.3 kJ/kg, qout = 487.9 kJ/kg, and
PROPRIETARY MATERIAL. © 2011 The McGraw-Hill Companies, Inc. Limited distribution permitted only to teachers and educators for course
Thus,
KkJ/kg 2.66807kJ/kg 783.04
KkJ/kg 3.21751K 1240
KkJ/kg 2.44117kJ/kg .60626
KkJ/kg .685151K 295
44
33
22
11
⋅=⎯→⎯=
⋅=⎯→⎯=
⋅=⎯→⎯=
⋅=⎯→⎯=
o
o
o
o
sh
sT
sh
sT
( )
( ) ( ) ( )( )
( )
( )
( ) ( ) ( )( )
( ) kJ/kg 183.2
kJ/kg 34.53
kJ/kg 105.4
kJ/kg 29.51
=⎟⎟⎠
⎞⎜⎜⎝
⎛+−=
⎟⎟⎠
⎞⎜⎜⎝
⎛+−−=⎟
⎟⎠
⎞⎜⎜⎝
⎛+−==
=⋅−−=
=⎟⎟⎠
⎞⎜⎜⎝
⎛−−=−==
=⎟⎟⎠
⎞⎜⎜⎝
⎛−−=
⎟⎟⎠
⎞⎜⎜⎝
⎛ −+−−=⎟
⎟⎠
⎞⎜⎜⎝
⎛+−==
=⋅−−=
=⎟⎟⎠
⎞⎜⎜⎝
⎛−−=−==
K 310kJ/kg 487.9
2.668071.68515K 310
ln
1/10lnKkJ/kg 0.2873.217512.66807K 310
ln
K 1600kJ/kg 698.32.441173.21751K 310
ln
10lnKkJ/kg 0.2871.685152.44117K 310
ln
out0
4
1410
41,41041gen,041 destroyed,
3
434034034gen,034 destroyed,
in0
2
3230
23,23023gen,023 destroyed,
1
212012012gen,012 destroyed,
LR
R
HR
R
Tq
PP
RssTT
qssTsTx
PP
RssTssTsTx
Tq
PP
RssTT
qssTsTx
PP
RssTssTsTx
oo
oo
oo
oo
preparation. If you are a student using this Manual, you are using it without permission.
9-111
9-143 Exergy analysis is to be used to answer the question in Prob. 9-87.
Analysis From Prob. 9-87, T1 = 288 K, T2s = 585.8 K, T2 = 618.9 K, T3 = 873 K, T4s = 429.2 K, T4 = 473.6 K, rp = 12. The exergy change of a flow stream between an inlet and exit state is given by
)(0 ieie ssThh −−−=∆ψ
4s
s
T
1
2s
4
3 qin
qout
873 K
288 K
2
This is also the expression for reversible work. Application of this equation for isentropic and actual compression processes gives
KkJ/kg 0003998.0
)12ln()KkJ/kg 287.0(K 288K 8.585ln)KkJ/kg 005.1(
lnln1
2
1
212
⋅=
⋅−⋅=
−=−PP
RTT
css sps
kJ/kg 2.299)KkJ/kg 0003998.0)(K 288(K)2888.585)(KkJ/kg 005.1(
)()( 12s01221 rev,
=⋅−−⋅=
−−−=− ssTTTcw sps
KkJ/kg 05564.0)12ln()KkJ/kg 287.0(K 288K 9.618ln)KkJ/kg 005.1(
lnln1
2
1
212
⋅=⋅−⋅=
−=−PP
RTT
css p
)()( 1201221 rev, −−−=− ssTTTcw p
PROPRIETARY MATERIAL. © 2011 The McGraw-Hill Companies, Inc. Limited distribution permitted only to teachers and educators for course
kJ/kg 5.316)KkJ/kg 05564.0)(K 288(K)2889.618)(KkJ/kg 005.1( =⋅−−⋅=
inimum work that must be supplied to the compressor by The irreversibilities therefore increase the m
kJ/kg 17.3=−=−=∆ −− 2.2995.31621 rev,21 rev,Crev, swww
Repeating the calculations for the turbine,
KkJ/kg 0003944.0)12ln()KkJ/kg 287.0(K2.429
K 873ln)KkJ/kg 005.1(
lnln4
3
4
343
⋅=⋅−⋅=
−=−PP
RTT
csss
ps
v, −−−=− ssps ssTTTcw
)()( 4304343 re
kJ/kg 9.445)KkJ/kg 0003944.0)(K 288(K)2.429873)(KkJ/kg 005.1( =⋅−−⋅=
KkJ/kg 09854.0)12ln()KkJ/kg 287.0(K6.473
K 873ln)KkJ/kg 005.1(
lnln4
3
4
343
⋅−=⋅−⋅=
−=−PP
RTT
css p
kJ/kg 8.429)KkJ/kg 09854.0)(K 288(K)6.473873)(KkJ/kg 005.1(
)()( 4304343 rev,
=⋅−−−⋅=
−−−=− ssp ssTTTcw
kJ/kg 16.1=−=−=∆ −− 8.4299.44543 rev,43 rev,Trev, www s
Hence, it is clear that the compressor is a little more sensitive to the irreversibilities than the turbine.
preparation. If you are a student using this Manual, you are using it without permission.
9-112
9-144 The total exergy destruction associated with the Brayton cycle described in Prob. 9-108 and the exergy at the exhaust gases at the turbine exit are to be determined.
Properties The gas constant of air is R = 0.287 kJ/kg·K (Table A-1).
Analysis From Prob. 9-108, qin = 480.82, qout = 372.73 kJ/kg, and
s
T
1
2s 4s
3qin1150 K
310 K
5
6
42
KkJ/kg 08156.2kJ/kg 738.43
KkJ/kg 69407.2kJ/kg 14.803
KkJ/kg 12900.3K 1150
KkJ/kg 42763.2kJ/kg 26.618
KkJ/kg 1.73498K 310
55
44
33
22
11
⋅=⎯→⎯=
⋅=⎯→⎯=
⋅=⎯→⎯=
⋅=⎯→⎯=
⋅=⎯→⎯=
o
o
o
o
o
sh
sh
sT
sh
sT
and, from an energy balance on the heat exchanger,
PROPRIETARY MATERIAL. © 2011 The McGraw-Hill Companies, Inc. Limited distribution permitted only to teachers and educators for course
⋅=⎯→⎯
=−−=⎯→⎯−=−os
hhhhh
Thus,
KkJ/kg 2.52861
kJ/kg 97.682)26.61843.738(14.803
6
66425
( )
( ) ( ) ( )( )
( )
( ) ( ) ( )( )( ) ( )[ ] ( ) ( )[ ]
( )( )
( )
( ) kJ/kg 142.6=⎟⎟⎠
⎜⎜⎝
+−K 290
2.528611.73498K 290
kJ/kg 58.09
kJ/kg 4.37
kJ/kg 35.83
kJ/kg 38.91
⎞⎛=
⎟⎟⎠
⎞⎜⎜⎝
⎛+−−=⎟
⎟⎠
⎞⎜⎜⎝
⎛+−==
=⎟⎟⎠
⎞⎜⎜⎝
⎛−−=
⎟⎟⎠
⎞⎜⎜⎝
⎛−−−=⎟
⎟⎠
⎞⎜⎜⎝
⎛−−==
=−+−=
−+−=−+−==
=⋅−−=
⎟⎟⎠
⎞⎜⎜⎝
⎛−−=−==
=⋅−−=
⎟⎟⎠
⎞⎜⎜⎝
⎛−−=−==
kJ/kg 372.73
ln
K 1500kJ/kg 480.822.608153.12900K 290
ln
2.694072.528612.427632.60815K 290
1/7lnKkJ/kg0.2873.129002.69407K 290
ln
7lnKkJ/kg 0.2871.734982.42763K 290
ln
out0
6
1610
61,61061gen,061 destroyed,
0
5
3530
53,53053gen,053 destroyed,
4625046250regengen,0regen destroyed,
3
434034034gen,034 destroyed,
1
212012012gen,012 destroyed,
LR
R
H
in
R
R
Tq
PP
RssTT
qssTsTx
Tq
PP
RssTT
qssTsTx
ssssTssssTsTx
PP
RssTssTsTx
PP
RssTssTsTx
oo
oo
oooo
oo
oo
oting tha 0 = h@ 290 K = 290.16 kJ/kg, the stream exergy at the exit of the regenerator (state 6) is determined from
N t h
( ) ( ) 06
026
060066 2gzVssThh ++−−−=φ
where
KkJ/kg 0.793631.734982.52861ln0
1
6161606 ⋅=−=−−=−=−
PP
Rssssss oo
Thus,
( )( ) kJ/kg 162.7=⋅−−= KkJ/kg 0.79363K 2900.162997.6826φ
preparation. If you are a student using this Manual, you are using it without permission.
9-113
9-145 Prob. 9-144 is reconsidered. The effect of the cycle pressure on the total irreversibility for the cycle and the exergy of the exhaust gas leaving the regenerator is to be investigated.
Analysis Using EES, the problem is solved as follows:
"Given" T[1]=310 [K] P[1]=100 [kPa] Ratio_P=7 P[2]=Ratio_P*P[1] T[3]=1150 [K] eta_C=0.75 eta_T=0.82 epsilon=0.65 T_H=1500 [K] T0=290 [K] P0=100 [kPa] "Analysis for Problem 9-154" q_in=h[3]-h[5] q_out=h[6]-h[1] h[5]-h[2]=h[4]-h[6] s[2]=entropy(Fluid$, P=P[2], h=h[2]) s[4]=entropy(Fluid$, h=h[4], P=P[4]) s[5]=entropy(Fluid$, h=h[5], P=P[5]) P[5]=P[2] s[6]=entropy(Fluid$, h=h[6], P=P[6]) P[6]=P[1] h[0]=enthalpy(Fluid$, T=T0) s[0]=entropy(Fluid$, T=T0, P=P0) x_destroyed_12=T0*(s[2]-s[1]) x_destroyed_34=T0*(s[4]-s[3]) x_destroyed_regen=T0*(s[5]-s[2]+s[6]-s[4]) x_destroyed_53=T0*(s[3]-s[5]-q_in/T_H) x_destroyed_61=T0*(s[1]-s[6]+q_out/T0) x_total=x_destroyed_12+x_destroyed_34+x_destroyed_regen+x_destroyed_53+x_destroyed_61 x6=h[6]-h[0]-T0*(s[6]-s[0]) "since state 0 and state 1 are identical" "Analysis for Problem 9-116" Fluid$='air' "(a)" h[1]=enthalpy(Fluid$, T=T[1]) s[1]=entropy(Fluid$, T=T[1], P=P[1]) s_s[2]=s[1] "isentropic compression" h_s[2]=enthalpy(Fluid$, P=P[2], s=s_s[2]) eta_C=(h_s[2]-h[1])/(h[2]-h[1]) h[3]=enthalpy(Fluid$, T=T[3]) s[3]=entropy(Fluid$, T=T[3], P=P[3]) P[3]=P[2] s_s[4]=s[3] "isentropic expansion" h_s[4]=enthalpy(Fluid$, P=P[4], s=s_s[4]) P[4]=P[1] eta_T=(h[3]-h[4])/(h[3]-h_s[4]) q_regen=epsilon*(h[4]-h[2]) "(b)" w_C_in=(h[2]-h[1]) w_T_out=h[3]-h[4]
PROPRIETARY MATERIALpreparation. If you are a student using this Manual, you are using it without permission.
. © 2011 The McGraw-Hill Companies, Inc. Limited distribution permitted only to teachers and educators for course
9-114
w_net_out=w_T_out-w_C_in q_in=(h[3]-h[2])-q_regen eta_th=w_net_out/q_in
Ratio_P xtotal
[kJ/kg] x6
[kJ/kg] 6 7 8 9
10 11 12 13 14
270.1 280
289.9 299.5 308.8 317.8 326.6 335.1 343.3
137.2 143.5 149.6 155.5 161.1 166.6 171.9 177.1 182.1
6 7 8 9 10 11 12 13 14130
150
170
190
210
230
250
270
290
310
330
350
RatioP
x [k
J/kg
]
xtotal,dest
x6
PROPRIETARY MATERIALpreparation. If you are a student using this Manual, you are using it without permission.
. © 2011 The McGraw-Hill Companies, Inc. Limited distribution permitted only to teachers and educators for course
9-115
9-146 The exergy loss of each process for a regenerative Brayton cycle with three stages of reheating and intercooling described in Prob. 9-126 is to be determined.
Analysis From Prob. 9-126,
s
T
3
4
1
9
8
2
5
10
11
67
12
13
14
rp = 4, qin,7-8 = qin,9-10 = qin,11-12 = 300 kJ/kg,
qout,14-1 = 206.9 kJ/kg, qout,2-3 = qout,4-5 = 141.6 kJ/kg,
T1 = T3 = T5 = 290 K , T2 = T4 = T6 = 430.9 K
K 9.495 K, 7.585 K, 4.870 K, 9.571 K, 8.849
K 3.551 K, 2.819 K, 7.520
1413
121110
987
=====
===
TTTTT
TTT
The exergy destruction during a process of a stream from an inlet state to exit state is given by
PROPRIETARY MATERIAL. © 2011 The McGraw-Hill Companies, Inc. Limited distribution permitted only to teachers and educators for course
⎟⎠
⎜⎝
+−−==sinksource
0gen0dest TTssTsTx ie ⎟
⎞⎜⎛ outin qq
Application of this equation for each process of the cycle gives
0kJ/kg 0.03 ≈=⎥⎦⎤
⎢⎣⎡ −=
⎟⎟⎠
⎞⎜⎜⎝
⎛−===
)4ln()287.0(290
430.9(1.005)ln)290(
lnln1
2
1
206-5 dest,4-3 dest,2-1 dest, P
PR
TT
cTxxx p
kJ/kg 32.1=⎥⎦⎤
⎢⎣⎡ −−=⎟⎟
⎠
⎞⎜⎜⎝
⎛−−=
4.8703000
520.7819.2(1.005)ln)290(lnln
source
8-in,7
7
8
7
808-7 dest, T
qPP
RTT
cTx p
kJ/kg 26.2=⎥⎦⎤
⎢⎣⎡ −−=⎟⎟
⎠
⎞⎜⎜⎝
⎛−−=
4.8703000
551.3849.8(1.005)ln)290(lnln
source
10-in,9
7
8
9
10010-9 dest, T
qPP
RTT
cTx p
kJ/kg 22.5=⎥⎦⎤
⎢⎣⎡ −−=⎟⎟
⎠
⎞⎜⎜⎝
⎛−−=
4.8703000
571.9870.4(1.005)ln)290(lnln
source
12-in,11
11
12
11
12012-11 dest, T
qPP
RTT
cTx p
0kJ/kg 0.05 ≈−=⎥⎦
⎤⎢⎣
⎡⎟⎠⎞
⎜⎝⎛−=⎟⎟
⎠
⎞⎜⎜⎝
⎛−=
41ln)287.0(
819.2551.3(1.005)ln)290(lnln
8
9
8
9098- dest, P
PR
TT
cTx p
0kJ/kg 0.04 ≈−=⎥⎦
⎤⎢⎣
⎡⎟⎠⎞
⎜⎝⎛−=⎟⎟
⎠
⎞⎜⎜⎝
⎛−=
41ln)287.0(
849.8571.9(1.005)ln)290(lnln
10
11
10
11011-10 dest, P
PR
TT
cTx p
0kJ/kg 0.08 ≈−=⎥⎦
⎤⎢⎣
⎡⎟⎠⎞
⎜⎝⎛−=⎟⎟
⎠
⎞⎜⎜⎝
⎛−=
41ln)287.0(
870.4585.7(1.005)ln)290(lnln
12
13
12
13013-12 dest, P
PR
TT
cTx p
kJ/kg 50.6=⎥⎦⎤
⎢⎣⎡ +−=⎟⎟
⎠
⎞⎜⎜⎝
⎛+−=
2909.2060
495.9290(1.005)ln)290(lnln
sink
1-out,14
14
1
14
101-14 dest, T
qPP
RTT
cTx p
kJ/kg 26.2=⎥⎦⎤
⎢⎣⎡ +−=⎟⎟
⎠
⎞⎜⎜⎝
⎛+−==
2906.1410
430.9290(1.005)ln)290(lnln
sink
3-out,2
2
3
2
305-4 dest,3-2 dest, T
qPP
RTT
cTxx p
kJ/kg 6.66=⎥⎦⎤
⎢⎣⎡ +=
⎟⎟⎠
⎞⎜⎜⎝
⎛+=∆+∆= −−
585.7495.9(1.005)ln
430.9520.7(1.005)ln)290(
lnln)(13
14
6
701413760regendest, T
Tc
TT
cTssTx pp
preparation. If you are a student using this Manual, you are using it without permission.
9-116
9-147 A gas-turbine plant uses diesel fuel and operates on simple Brayton cycle. The isentropic efficiency of the compressor, the net power output, the back work ratio, the thermal efficiency, and the second-law efficiency are to be determined.
PROPRIETARY MATERIAL. © 2011 The McGraw-Hill Companies, Inc. Limited distribution permitted only to teachers and educators for course
Assumptions 1 The air-standard assumptions are applicable. 2 Kinetic and potential energy changes are negligible. 3 Air is an ideal gas with constant specific heats. Properties The properties of air at 500ºC = 773 K are cp = 1.093 kJ/kg·K, cv = 0.806 kJ/kg·K, R = 0.287 kJ/kg·K, and k = 1.357 (Table A-2b). Analysis (a) The isentropic efficiency of the compressor may be determined if we first calculate the exit temperature for the isentropic case
( ) K 6.505kPa 100kPa 700K 303
1)/1.357-(1.357/)1(
1
212 =⎟
⎠⎞
⎜⎝⎛=⎟⎟
⎠
⎞⎜⎜⎝
⎛=
− kk
s PP
TT
1
Combustion chamber
Turbine
23
4
Compress.
100 kPa30°C
Diesel fuel
700 kPa260°C
0.881=−
=−
=K)303533(12 TTCη −− K)3036.505(12 TT s
(b) The total mass flowing through the turbine and the rate of heat input are
kg/s 81.12kg/s 21.0kg/s 6.1260AF
kJ/kg)(0.9 00kg/s)(42,0 21.0(== qmQ η&&
kg/s 6.12kg/s 6.12 =+=+=+=+= aafa
mmmmm
&&&&&
HVin =cf
perature at the exit of combustion chamber is
3323 =⎯→⎯ Tp
The temperature at the turbine exit is determined using isentropic efficiency relation
t
kW 85557)
The tem
)K533kJ/kg.K)( 3kg/s)(1.09 81.12(kJ/s 8555)( −=⎯→⎯−= TTTcmQ && K 1144in
( ) K 7.685kPa 70
K 11443
34 ⎜⎝
=⎟⎠
⎜⎝
=s PTT
0kPa 100 1)/1.357-(1.357/)1(
4 =⎟⎠⎞⎛⎟
⎞⎜⎛
− kkP
K 4.754K)7.6851144(
K)1144(85.0 4
4
43
43 =⎯→⎯−
−=⎯→⎯
−−
= TT
TTTT
sTη
The net power and the back work ratio are
=−=−= TTcmW &&
t WWW
kW 3168)K30333kJ/kg.K)(5 3kg/s)(1.09 6.12()( 12inC, =−=−= TTcmW pa&&
kW 5455)K4.754144kJ/kg.K)(1 3kg/s)(1.09 81.12()( 43outT, p
&&ne& kW 2287=−=−= 31685455inC,outT,
0.581===kW 5455kW 3168inC,
bw WW
r &
&
outT,
(c) The thermal efficiency is
0.267===kW 8555kW 2287
in
netth Q
W&
&η
The second-law efficieny of the cycle is defined as the ratio of actual thermal efficiency to the maximum possible thermal efficiency (Carnot efficiency). The maximum temperature for the cycle can be taken to be the turbine inlet temperature. That is,
735.0K 1144K 30311
3
1max =−=−=
TT
η
0.364===735.0267.0
max
thII η
ηη and
preparation. If you are a student using this Manual, you are using it without permission.
9-117
9-148 A modern compression ignition engine operates on the ideal dual cycle. The maximum temperature in the cycle, the net work output, the thermal efficiency, the mean effective pressure, the net power output, the second-law efficiency of the cycle, and the rate of exergy of the exhaust gases are to be determined.
Assumptions 1 The air-standard assumptions are applicable. 2 Kinetic and potential energy changes are negligible. 3 Air is an ideal gas with constant specific heats.
Properties The properties of air at 1000 K are cp = 1.142 kJ/kg·K, cv = 0.855 kJ/kg·K, R = 0.287 kJ/kg·K, and k = 1.336 (Table A-2b).
Analysis (a) The clearance volume and the total volume of the engine at the beginning of compression process (state 1) are
xcc
c
c
dcr VVVV
V
V
VV===⎯→⎯
+=⎯→⎯
+= 2
33
m 00012.0m 0018.0
16
43
1 m 00192.00018.000012.0 VVVV ==+=+= dc
1
Qin2
3
4
P
V
Qout
xProcess 1-2: Isentropic compression
( )( )
( )( ) kPa 385916kPa 95
K 7.87016K 343
1.336
2
112
1-1.3361
2
112
==⎟⎟⎠
⎞⎜⎜⎝
⎛=
==⎟⎟⎠
⎞⎜⎜⎝
⎛=
−
k
k
PP
TT
v
v
v
v
Process 2-x and x-3: Constant-volume and constant pressure heat addition processes:
K 1692kPa 3859kPa 7500K) 7.870(
22 ===
PP
TT xx
kJ/kg 6.702K)7.8701692(kJ/kg.K) (0.855)( 2-2 =−=−= TTcq xx v
K)1692(kJ/kg.K) (0.855kJ/kg 6.702)( TTTTcqq xpxx
=
K 2308=⎯→⎯−=⎯→⎯−==− 3333-2
(b) 6.7023-2in +=+= − xx qqq kJ/kg 14056.702
3333 m 0001636.0
K 1692K 2308)m 00012.0( ===
xx T
TVV
Process 3-4: isentropic expansion.
( )
( ) kPa 4.279
PROPRIETARY MATERIAL. © 2011 The McGraw-Hill Companies, Inc. Limited distribution permitted only to teachers and educators for course
m 0.001924 ⎝⎠⎝Vm 0.0001636kPa 7500
K 1009m 92m 0.0001636K 2308
1.336
3
33
3
1-1.336
3
313
34
=⎟⎟⎠
⎞⎜⎜⎛
=⎟⎟⎞
⎜⎜⎛
=
=⎟⎟⎠
⎞⎜⎜⎛
=⎟⎟⎞
⎜⎜⎛
=−k
PP
TT
V
V
rocess 4-1: constant voume heat rejection.
0.0014 ⎝⎠⎝k
V
4
P
( ) ( )( ) kJ/kg 3.569K3431009KkJ/kg 0.85514out =−⋅=−= TTcq v
The net work output and the thermal efficiency are
kJ/kg 835.8=−=−= 3.5691405outinoutnet, qqw
59.5%==== 5948.0kJ/kg 1405kJ/kg 835.8
in
outnet,th q
wη
preparation. If you are a student using this Manual, you are using it without permission.
9-118
(c) The mean effective pressure is determined to be
( )( )kg 0.001853
K 343K/kgmkPa 0.287)m 92kPa)(0.001 95(
3
3
1
11 =⋅⋅
==RTP
mV
kPa 860.4=⎟⎟⎠
⎞⎜⎜⎝
⎛ ⋅
−=
−=
0001.000192.0(MEP
21 VV kJmkPa
m)2kJ/kg)kg)(835.8 (0.001853 3
3outnet,mw
d) The power for engine speed of 3500 rpm is (
kW 28.39=⎟⎠⎞
⎜⎝
==60rev/cycle) 2(
kJ/kg) kg)(835.8 001853.0(2netnet mwW& ⎛
s min 1(rev/min) 2200n&
(e) The second-law efficieny of the cycle is defined as the ratio of actual thermal efficiency to the maximum possible thermal ef ency (Carnot efficiency). We take the dead state temperature and pressure to be 25ºC and 100 kPa.
Note that there are two revolutions in one cycle in four-stroke engines.
fici
8709.0K 2308
K )27325(113
0max =
+−=−=
TT
η
and
68.3%==== 683.08709.05948.0η
max
th
ηη
The rate of exergy of the exhaust gases is determined as follows
II
( )
( )( ) kJ/kg 0.285100
4.279ln)kJ/kg.K 287.0(298
1009kJ/kg.K)ln 142.1()298(29810090.855
lnln)(0
4
0
4004040044
=⎥⎦⎤
⎢⎣⎡ −−−=
⎥⎦
⎤⎢⎣
⎡−−−=−−−=
PP
RTT
cTTTcssTuux pv
kW 9.683=⎟⎠⎞
⎜⎝⎛==
s 60min 1
rev/cycle) 2((rev/min) 2200kJ/kg) kg)(285.0 001853.0(
244nmxX&&
PROPRIETARY MATERIALpreparation. If you are a student using this Manual, you are using it without permission.
. © 2011 The McGraw-Hill Companies, Inc. Limited distribution permitted only to teachers and educators for course
9-119
Review Problems
9-149 An Otto cycle with a compression ratio of 7 is considered. The thermal efficiency is to be determined using constant and variable specific heats.
Assumptions 1 The air-standard assumptions are applicable. 2 Kinetic and potential energy changes are negligible.
Properties The properties of air at room temperature are R = 0.287 kPa·m3/kg·K, cp = 1.005 kJ/kg·K, cv = 0.718 kJ/kg·K, and k = 1.4 (Table A-2a).
Analysis (a) Constant specific heats:
PROPRIETARY MATERIAL. © 2011 The McGraw-Hill Companies, Inc. Limited distribution permitted only to teachers and educators for course
54.1%==−=−=−− 7 11.41th kr
0.54081111η
properties from Table A-17)
Process 1-2: isentropic compression.
v
P
4
1
3
2(b) Variable specific heats: (using air
1.688kJ/kg 48.205
K 2881
11 =
=⎯→⎯=
r
uT
v
kJ/kg 62.4473.98)1.688(112 === vvv
v7 222
12 =⎯→⎯= u
r rrr v
rocess 2-3: v = constant heat addition.
2kJ/kg 51.998
23
3
=−=−=
=
uuq
u
P
1K 1273
33 =
⎯→⎯=Trv
kJ/kg .8955062.44751.998045.
in
Process 3-4: isentropic expansion.
kJ/kg 54.47532.84)045.12)(7( 4333
44 =⎯→⎯==== u r rrr vv
v
vv
rocess 4-1: v = constant heat rejection.
P
kJ/kg 06.27048.20554.47514out =−=−= uuq
51.0%==−=−= 0.5098kJ/kg 550.89kJ/kg 270.0611
in
outth q
qη
preparation. If you are a student using this Manual, you are using it without permission.
9-120
9-150 A simple ideal Brayton cycle with air as the working fluid operates between the specified temperature limits. The net work is to be determined using constant and variable specific heats.
Assumptions 1 Steady operating conditions exist. 2 The air-standard assumptions are applicable. 3 Kinetic and potential energy changes are negligible.
Properties The properties of air at room temperature are cp = 1.005 kJ/kg·K and k = 1.4 (Table A-2a).
Analysis (a) Constant specific heats:
K 2.549K)(12) 270( 0.4/1.4/)1(12 === − kk
prTT
s
T
1
2
4
3 qin
qout
800 K
270 K
K 3.393121K) 800(1 0.4/1.4/)1(
34 =⎟⎠⎞
⎜⎝⎛=⎟
⎟⎠
⎞⎜⎜⎝
⎛=
− kk
prTT
)(
)()(
2143
1243
compturbnet
TTTTc
TTcTTc
www
p
pp
(b) Variable specific heats: (using air properties from Table A-17)
kJ/kg 128.1=K)2.5492703.393800)(kJ/kg·K 005.1( −+−=
−+−=
−−−=
−=
9590.0kJ/kg 11.270
K 2701
11 =
=⎯→⎯=
rPh
T
kJ/kg 46.55051.11)9590.0)(12( 211
22 =⎯→⎯=== hP
PP
P rr
kJ/kg 04.406979.3)75.47(121
75.47kJ/kg 95.821
K 800
433
44
3
33
=⎯→⎯=⎟⎠⎞
⎜⎝⎛==
==
⎯→⎯=
hPPP
P
Ph
T
rr
r
kJ/kg 135.6=−−−=
−−−=
−=
)11.27046.550()04.40695.821()()( 1243
compturbnet
hhhh
www
PROPRIETARY MATERIALpreparation. If you are a student using this Manual, you are using it without permission.
. © 2011 The McGraw-Hill Companies, Inc. Limited distribution permitted only to teachers and educators for course
9-121
9-151 A turbocharged four-stroke V-16 diesel engine produces 3500 hp at 1200 rpm. The amount of power produced per cylinder per mechanical and per thermodynamic cycle is to be determined.
Analysis Noting that there are 16 cylinders and each thermodynamic cycle corresponds to 2 mechanical cycles (revolutions), we have
(a)
cycle)mech kJ/cyl 8.16(= hp 1Btu/min 42.41
rev/min) (1200cylinders) (16hp 3500
cycles) mechanical of (No.cylinders) of (No.producedpower Total
mechanical
⋅⋅=
⎟⎟⎠
⎞⎜⎜⎝
⎛=
=
cycle mechBtu/cyl 7.73
w
(b)
cycle) thermkJ/cyl 16.31(= cycle thermBtu/cyl 15.46
hp 1Btu/min 42.41
rev/min) (1200/2cylinders) (16hp 3500
cycles) amic thermodynof (No.cylinders) of (No.producedpower Total
micthermodyna
⋅⋅=
⎟⎟⎠
⎞⎜⎜⎝
⎛=
=w
ified temperature limits is considered. The pressure ratio for
2 The air-standard assumptions are applicable. 3 Kinetic and potential nergy cha ges are negligible. 4 Air is an ideal gas with constant specific heats.
is k =1.4 (Table A-2).
Analysis We treat air as an ideal gas with constant specific heats. Using the isentropic relations, the temperatures at the ompressor and turbine exit can be expressed as
9-152 A simple ideal Brayton cycle operating between the specwhich the compressor and the turbine exit temperature of air are equal is to be determined.
Assumptions 1 Steady operating conditions exist.e n
Properties The specific heat ratio of air
c( )
PROPRIETARY MATERIAL. © 2011 The McGraw-Hill Companies, Inc. Limited distribution permitted only to teachers and educators for course
( )( )
( ) ( ) kk
p
kk
kk
rT
PPTT
/1
3
/1
3
434
/1
1
1−−
−
⎟⎟⎠
⎞⎜⎜⎝
⎛=⎟⎟
⎠
⎞⎜⎜⎝
⎛=
⎠⎝
Setting T2 = T4 and solving for rp gives
TkkprT
PPTT /1
12
12−=⎟⎟
⎞⎜⎜⎛
=
s1
2
4
3 qin
qout
T3
T1( )16.7=⎟⎟
⎠
⎞⎜⎜⎝
⎛=⎟⎟
⎠
⎞⎜⎜⎝
⎛=
− 1.4/0.812/
1
3
K 300K 1500
kk
p TT
r
Therefore, the compressor and turbine exit temperatures will be equal when the compression ratio is 16.7.
preparation. If you are a student using this Manual, you are using it without permission.
9-122
9-153 A four-cylinder spark-ignition engine with a compression ratio of 8 is considered. The amount of heat supplied
ons are applicable. 2 Kinetic and potential energy changes are negligible. 3 Air is
e A-1). The properties of air are given in Table A-17.
Analysis (a) Process 1-2: isentropic compression.
per cylinder, the thermal efficiency, and the rpm for a net power output of 60 kW are to be determined.
Assumptions 1 The air-standard assumptian ideal gas with variable specific heats.
Properties The gas constant of air is R = 0.287 kJ/kg·K (Tabl
PROPRIETARY MATERIAL. © 2011 The McGraw-Hill Companies, Inc. Limited distribution permitted only to teachers and educators for course
( )
kJ/kg .295642 =⎯→⎯ u
50.543.5725.10
11
3.572kJ/kg 221.25K 031
1
2
11
112
1
====
==⎯→⎯=
r
uT
rrr
r
vvv
vv
v
rocess 2- v = constant heat addition. P 3:
( )( )( )( )
( ) ( )( ) kJ 0.5336=−×=−=
×=
=
−
−
kJ/kg.295641775.3kg 104.406
kg 10.4064K 310K
m 0.0004kPa 98
356.2kJ/kg .3
423in
43
3
uumQ
P
r
V
v
rocess 3-4: isentropic expansion.
v
P
4
1
3
2
Qin
Qout
1800 K
=⎯→⎯= 1775K 0021 33 uT
⋅⋅==
/kgmkPa 0.287 31
11
RTm
(b) P
( )( ) 05.76474.24356.25.10 44
334=⎯→⎯==== ur rrr vv
vv kJ/kg
3v
Process 4-1: v = constant heat rejection.
( ) ( )( )
55.2%====
=−=−=
=−×=−=
5517.0kJ 0.5336kJ 0.2944
kJ 0.29442392.05336.0
kJ 0.2392kJ/kg.25221764.05kg 104.406
in
netth
outinnet
-414out
QW
QQW
uumQ
η
rpm 4586=⎟⎟⎠
⎞⎜⎜⎝
⎛×
==min 1
s 60kJ/cycle) 0.2944(4
kJ/s 45rev/cycle) 2(2cylnet,cyl
net
WnW
n&
& (c)
Note that for four-stroke cycles, there are two revolutions per cycle.
preparation. If you are a student using this Manual, you are using it without permission.
9-123
9-154 Problem 9-153 is reconsidered. The effect of the compression ratio net work done and the efficiency of the s for the cycle are to be plotted.
nalysis Using EES, the problem is solved as follows:
K]
ber of cyclinders"
nit mass." ession"
2]) v[1]/T[1]
fer (s=const.) with work input"
T=T[1]) eat addition"
3]) 2]*v[2]/T[2]}
ro for v=const, heat is added"
nergy(air,T=T[2]) ion"
fer (s=const) with work output"
T=T[3])
ro for v=const; heat is rejected"
der is found from the volume of the cylinder:"
et=m*w_net"kJ/cyl"*N_cyl*N_dot"mechanical cycles/min"*1"min"/60"s"*1"thermal cycle"/2"mechanical cycles"
cycle is to be investigated. Also, the T-s and P-v diagram
A
"Input Data" T[1]=(37+273) [P[1]=98 [kPa] T[3]= 2100 [K]
(L, m^3) V_cyl=0.4 [L]*Convertr_v=10.5 "Compression ratio"
45 [kW] W_dot_net =_cyl=4 "numN
v[1]/v[2]=r_v
one per u"The first part of the solution is d1-2 is isentropic compr"Process
s[1]=entropy(air,T=T[1],P=P[1]) s[2]=s[1]
T=T[2], v=v[s[2]=entropy(air, P[2]*v[2]/T[2]=P[1]*P[1]*v[1]=R*T[1] R=0.287 [kJ/kg-K] "Conservation of energy for process 1 to 2: no heat trans w_in = DELTAu_12
)-intenergy(air,DELTAu_12=intenergy(air,T=T[2]e h"Process 2-3 is constant volum
T=T[3], P=P[s[3]=entropy(air, /T[3]=P[{P[3]*v[3]
P[3]*v[3]=R*T[3] v[3]=v[2] "Conservation of energy for process 2 to 3: the work is zeq_in = DELTAu_23
])-inteDELTAu_23=intenergy(air,T=T[33-4 is isentropic expans"Process
s[4]=entropy(air,T=T[4],P=P[4]) s[4]=s[3] P[4]*v[4]/T[4]=P[3]*v[3]/T[3] {P[4]*v[4]=R*T[4]} "Conservation of energy for process 3 to 4: no heat trans - w_out = DELTAu_34
34=intenergy(air,T=T[4])-intenergy(air,DELTAu_"Process 4-1 is constant volume heat rejection" v[4]=v[1] "Conservation of energy for process 2 to 3: the work is ze- q_out = DELTAu_41 DELTAu_41=intenergy(air,T=T[1])-intenergy(air,T=T[4]) w_net = w_out - w_in
/q_in*Convert(, %) "Thermal efficiency, in percent" Eta_th=w_net"The mass contained in each cylinV_cyl=m*v[1] "The net work done per cycle is:" W_dot_n
PROPRIETARY MATERIALpreparation. If you are a student using this Manual, you are using it without permission.
. © 2011 The McGraw-Hill Companies, Inc. Limited distribution permitted only to teachers and educators for course
9-124
rv ηth
[%] wnet
[kJ/kg] 5 6 7 8 9 10 11
42 45.55 48.39 50.74 52.73 54.44 55.94
568.3 601.9 625.7 642.9 655.5 664.6 671.2
10-2 10-1 100 101 102101
102
103
104
v [m3/kg]P
[kPa
]
310 K 2100 K
1
2
3
4
s = const
Air Otto Cycle P-v Diagram
4.0 4.5 5.0 5.5 6.0 6.5 7.0 7.5 8.0 8.50
500
1000
1500
2000
2500
3000
s [kJ/kg-K]
T [K
]
98 kPa
6971 kPa
Air Otto Cycle T-s Diagram
1
2
3
4
v = const
5 6 7 8 9 10 1142
44
46
48
50
52
54
56
rv
ηth
[%
]
5 6 7 8 9 10 11
560
580
600
620
640
660
680
rv
wne
t [k
J/kg
]
PROPRIETARY MATERIALpreparation. If you are a student using this Manual, you are using it without permission.
. © 2011 The McGraw-Hill Companies, Inc. Limited distribution permitted only to teachers and educators for course
9-125
9-155 An ideal gas Carnot cycle with helium as the working fluid is considered. The pressure ratio, compression ratio, and minimum temperature of the energy source are to be determined.
Assumptions 1 Kinetic and potential energy changes are negligible. 2 Helium is an ideal gas with constant specific heats.
Properties The specific heat ratio of helium is k = 1.667 (Table A-2a).
Analysis From the definition of the thermal efficiency of a Carnot heat engine,
s
T
3
2qin
qout4
1TH
288 K
K 576=−+
=−
=⎯→⎯−=0.501
K 273)(151
1Carnotth,
Carnotth, ηη L
HH
L TT
TT
An isentropic process for an ideal gas is one in which Pvk remains constant. Then, the pressure ratio is
PROPRIETARY MATERIAL. © 2011 The McGraw-Hill Companies, Inc. Limited distribution permitted only to teachers and educators for course
5.6=⎟⎠
⎜⎝
=⎟⎟⎠
⎜⎜⎝
=11 K 288TP
5⎞⎛⎞⎛ −− )1667.1/(667.1)1/(2 K 576
kkTP
ased on the process equation, the compression ratio is
2
B
2.83==⎟⎟⎠
⎞⎜⎜⎝
⎛= 667.1/1
/1
1
2
2
1 )65.5(k
PP
v
v
9-156 The compression ratio required for an ideal Otto cycle to produce certain amount of work when consuming a given amount of fuel is to be determined.
Assumptions 1 The air-standard assumptions are applicable. 2 Kinetic and potential energy changes are negligible. 3 Air isan ideal gas with constant specific heats. 4 The combustion efficiency is 100 percent.
cv = 0.718 kJ/kg·K, and k = 1.4 (Table A-
Analysis The heat input to the cycle for 0.043 grams of fuel consumption is 3
HVfuelin =× −
he thermal efficiency is then
Properties The properties of air at room temperature are cp = 1.005 kJ/kg.K,2).
kJ 505.1kJ/kg) kg)(43,000 10035.0(== qmQ
T
6645.0kJ 1.505
kJ 1netth ===
QW
η in
rom the definition of thermal efficiency, we obtain the required compression ratio to be v
P
4
1
3
2 qout
qin
F
15.3=−
=−
=⎯→⎯−=−−− )14.1/(1)1/(1
th1th
)6645.01(1
)1(111
kkr
r ηη
preparation. If you are a student using this Manual, you are using it without permission.
9-126
9-157 An ideal Otto cycle with air as the working fluid with a compression ratio of 9.2 is considered. The amount of heat transferred to the air, the net work output, the thermal efficiency, and the mean effective pressure are to be determined.
Assumptions 1 The air-standard assumptions are applicable. 2 Kinetic and potential energy changes are negligible. 3 Air is an ideal gas with variable specific heats.
Properties The gas constant of air is R = 0.287 kJ/kg·K (Table A-1). The properties of air are given in Table A-17.
Analysis (a) Process 1-2: isentropic compression.
v
P
4
1
3
2
qin
qout
( )
( ) ( ) kPa 2129kPa 98K 300K 708.3
9.2
kJ/kg 9.518K 3.70852.672.621
2.911
2.621kJ/kg 214.07K 300
11
2
2
12
1
11
2
22
2
21
2
11
112
1
=⎟⎟⎠
⎞⎜⎜⎝
⎛==⎯→⎯=
==⎯→⎯====
==⎯→⎯=
PTT
PT
PT
P
uT
r
uT
rrr
r
v
vvv
vvv
vv
v
Process 2-3: v = constant heat addition.
( )( )
kJ/kg609.8 9.5187.1128
593.8kJ/kg 1128.7K 1416.63.70822
23
3222
323 PTT
3223
3
=−=−=
=
=⎯→⎯====⎯→⎯=
uuq
uTTP
TPP
in
rv
vv
(b) Process 3-4: isentropic expansion.
3
( )( ) kJ/kg 487.7506.79593.82.9 43
433
PROPRIETARY MATERIAL. © 2011 The McGraw-Hill Companies, Inc. Limited distribution permitted only to teachers and educators for course
4r =⎯→⎯==== ur rr vv
vv
ess 4-1: v = constant heat rejection.
14t
=−=−==−=−=
qqwuuq
(c)
v
Proc
kJ/kg336.1 7.2738.609kJ/kg 273.707.21475.487
outin
net
ou
55.1%===kJ/kg 609.8kJ/kg 336.1
in
netth q
wη
( )( )
( ) ( )( )kPa429
kJ 1mkPa 1
1/9.21/kgm 0.879kJ/kg 336.1
/11MEP
/kgm 0.879kPa 98
K 300K/kgmkPa 0.287
3
31
net
21
net
max2min
33
1
11max
=⎟⎟⎠
⎞⎜⎜⎝
⎛ ⋅
−=
−=
−=
==
=⋅⋅
===
rww
r
PRT
vvv
vvv
vv (d)
preparation. If you are a student using this Manual, you are using it without permission.
9-127
9-158 An ideal Otto cycle with air as the working fluid with a compression ratio of 9.2 is considered. The amount of heat transferred to the air, the net work output, the thermal efficiency, and the mean effective pressure are to be determined.
Assumptions 1 The air-standard assumptions are applicable. 2 Kinetic and potential energy changes are negligible. 3 Air is an ideal gas with constant specific heats.
PROPRIETARY MATERIAL. © 2011 The McGraw-Hill Companies, Inc. Limited distribution permitted only to teachers and educators for course
Properties The properties of air at room temperature are cp = 1.005 kJ/kg.K, cv = 0.718 kJ/kg·K, and k = 1.4 (Table A-2).
Analysis (a) Process 1-2 is isentropic compression:
( )( )
( ) ( ) kPa 2190kPa 98K 300K 728.8
9.2
K 728.89.2K 300
11
2
2
12
1
11
2
22
0.41
2
112
=⎟⎟⎠
⎞⎜⎜⎝
⎛==⎯→⎯=
==⎟⎟⎠
⎞⎜⎜⎝
⎛=
−
PTT
PT
PT
P
TTk
v
vvv
v
v
v
P
4
1
3
2
qin
qout
Process 2-3: v = constant heat addition.
( )( )
( ) ( )( ) kJ/kg523.3 K728.81457.6KkJ/kg 0.718
K 457.618.72822
2323
222
323 PTT
3223
=−⋅=−=−=
====⎯→⎯=
TTcuuq
TTP
TPP
in v
vv
(b) Process 3-4: isentropic expansion.
3
( ) K 600.09.21K 1457.6
0.413 ⎛⎞⎛
−kv
43 =⎟
⎠⎞
⎜⎝
=⎟⎟⎠
⎜⎜⎝
= TTv
ess 4-1: v = constant heat rejection.
4
Proc
( ) ( )( )
kJ/kg307.9 4.2153.523
kJ/kg 215.4K300600KkJ/kg 0.718
outinnet
1414out
=−=−=
=−⋅=−=−=
qqw
TTcuuq v
(c) 58.8%===kJ/kg 523.3kJ/kg 307.9
in
netth q
wη
( )( )
( ) ( )( )kPa393
kJ 1mkPa 1
1/9.21/kgm 0.879kJ/kg 307.9
/11MEP
/kgm 0.879kPa 98
K 300K/kgmkPa 0.287
3
31
net
21
net
max2min
33
1
11max
=⎟⎟⎠
⎞⎜⎜⎝
⎛ ⋅
−=
−=
−=
==
=⋅⋅
===
rww
r
PRT
vvv
vvv
vv (d)
preparation. If you are a student using this Manual, you are using it without permission.
9-128
9-159 An ideal dual cycle with air as the working fluid with a compression ratio of 12 is considered. The thermal efficiency of the cycle is to be determined.
Assumptions 1 The air-standard assumptions are applicable. 2 Kinetic and potential energy changes are negligible. 3 Air is an ideal gas with constant specific heats.
Properties The properties of air at room temperature are cp = 1.005 kJ/kg.K, cv = 0.718 kJ/kg.K, and k = 1.4 (Table A-2).
Analysis The mass of air is
( )( )( )( )
kg 001726.0K 323K/kgmkPa 0.287
m 101600kPa 1003
3-6
1
11 =⋅⋅
×==
RTP
mV
PROPRIETARY MATERIAL. © 2011 The McGraw-Hill Companies, Inc. Limited distribution permitted only to teachers and educators for course
Process 1-2: isentropic compression.
( )( ) K 2.92814K 3232
12 =⎟⎟⎠
⎜⎜⎝
= TTV
0.41
1 =⎞⎛
−kV
rocess 2-x: v = constant heat addition,
)( ) K 1412K2.928KJ/kg x
22in,2
=⎯→⎯−⋅
−=−=−
x
xxx
TT
TTmcuumQ v
rocess x- P = constant heat addition.
v
P
4
1
2
31.1 kJ
Qout
x
0.6 kJ
P
( )( k 0.718kg 0.001726kJ 0.6 =
( ) ( )
P 3:
( ) ( )( )( )( )
1.449K 1412K 2046
K 2046K1412KkJ/kg 1.005kg 0.001726kJ 1.1
3
3
3
33 ==⎯→⎯=x
cx
xx rTT V
33
33in,3
==
=⎯→⎯−⋅=
−−
x
xpxx
TTPP
TT
TT
VVV
rocess 3- isentropic expansion.
=−= mchhmQ
P 4:
( ) K 8.82514
1.449K 2046449.1449.1 0.41
3
1
4
13
1
4
334 =⎟
⎠⎞
⎜⎝⎛=⎟
⎠⎞
⎜⎝⎛=⎟⎟
⎠
⎞⎜⎜⎝
⎛=⎟⎟
⎠
⎞⎜⎜⎝
⎛=
−−− kkk
rTTTT
VV
VV
Process 4-1: v = constant heat rejection.
( ) ( )
( )( )( )
63.3%==−=−=
=−⋅=−=−=
633.0kJ 1.7
kJ 0.623111
kJ 0.6231K323825.8KkJ/kg 0.718kg 0.001726
in
outth
1414out
TTmcuumQ
η
v
preparation. If you are a student using this Manual, you are using it without permission.
9-129
9-160 An ideal Stirling cycle with air as the working fluid is considered. The maximum pressure in the cycle and the net work output are to be determined.
Assumptions 1 The air-standard assumptions are applicable. 2 Kinetic and potential energy changes are negligible. 3 Air is an ideal gas with constant specific heats.
s
T
3
2
qin = 900 kJ/kg
qout
4
1 1800 K
350 K
Properties The properties of air at room temperature are R = 0.287 kJ/kg.K, cp = 1.005 kJ/kg.K, cv = 0.718 kJ/kg·K, and k = 1.4 (Table A-2).
Analysis (a) The entropy change during process 1-2 is
KkJ/kg 0.5K 1800
kJ/kg 9001212 ⋅===−
HTq
ss
and
( )
( )( ) kPa 5873=⎟⎟⎠
⎞⎜⎜⎝
⎛===⎯→⎯=
=⎯→⎯⋅=⋅⎯→⎯+=− ln1
12 Tcss v
K 350K 800
5.710kPa 200
710.5lnKkJ/kg 0.287KkJ/kg 0.5ln
3
1
1
23
3
1
1
331
1
11
3
33
1
2
1
2
1
20
2
TT
PTT
PPT
PT
P
RT
v
v
v
vvv
v
v
v
v
v
v
(b) The net work output is
1
( ) kJ/kg 725=⎟⎟⎠
⎞⎜⎜⎝
⎛−=⎟⎟
⎠
⎞⎜⎜⎝
⎛−== kJ/kg 900
K 1800K 350
11 ininthnet qTT
qwH
Lη
PROPRIETARY MATERIALpreparation. If you are a student using this Manual, you are using it without permission.
. © 2011 The McGraw-Hill Companies, Inc. Limited distribution permitted only to teachers and educators for course
9-130
9-161 A simple ideal Brayton cycle with air as the working fluid is considered. The changes in the net work output per unit mass and the thermal efficiency are to be determined.
Assumptions 1 The air-standard assumptions are applicable. 2 Kinetic and potential energy changes are negligible. 3 Air is an ideal gas with variable specific heats.
Properties The properties of air are given in Table A-17.
PROPRIETARY MATERIAL. © 2011 The McGraw-Hill Companies, Inc. Limited distribution permitted only to teachers and educators for course
For rp = 6,
s
T
1
2
4
3 qin
qout
2′
3′ Analysis The properties at various states are
T h
P
T h
P
r
r
1 1
3 3
1
3
1 386
330 9
= ⎯ →⎯ ==
= ⎯ →⎯ ==
300 K 300.19 kJ / kg
1300 K 1395.97 kJ / kg
.
.
( )( )
( )
%9.37kJ/kg 894.57kJ/kg 339.46
kJ/kg 339.4611.55557.894kJ/kg 555.1119.3003.855
kJ/kg 894.5740.50197.1395
kJ/kg 855.315.559.33061
kJ/kg 501.40316.8386.16
in
netth
outinnet
outw
14
23in
43
4
21
2
34
12
===
=−=−==−=−=
=−=−=
=⎯→⎯=⎟⎠⎞
⎜⎝⎛==
=⎯→⎯===
qw
qqhhqhhq
hPPP
P
hPPP
P
rr
rr
η
rp = 12,
For
( )( )
( )
%5.48kJ/kg 785.37kJ/kg 380.96
kJ/kg 380.9641.40437.785kJ/kg 404.4119.3006.704
kJ/kg 785.3760.61097.1395
kJ/kg 704.658.279.330121
kJ/kg 610.663.16386.112
in
netth
outinnet
14out
23in
43
4
21
2
34
12
===
=−=−==−=−=
=−=−=
=⎯→⎯=⎟⎠⎞
⎜⎝⎛==
=⎯→⎯===
qw
qqwhhqhhq
hPPP
P
hPPP
P
rr
rr
η
Thus,
(a) ( )increase46.33996.380net kJ/kg 41.5=−=∆w
( )increase%9.37%5.48th 10.6%=−=∆η (b)
preparation. If you are a student using this Manual, you are using it without permission.
9-131
9-162 A simple ideal Brayton cycle with air as the working fluid is considered. The changes in the net work output per unit mass and the thermal efficiency are to be determined.
Assumptions 1 The air-standard assumptions are applicable. 2 Kinetic and potential energy changes are negligible. 3 Air is an ideal gas with constant specific heats.
Properties The properties of air at room temperature are R = 0.287 kJ/kg.K, cp = 1.005 kJ/kg.K, cv = 0.718 kJ/kg·K, and k = 1.4 (Table A-2).
Analysis Processes 1-2 and 3-4 are isentropic. Therefore, For rp = 6,
( )( )( )
( )( )
( )( )( )
( )( )( )
%1.40kJ/kg 803.4kJ/kg 321.9
kJ/kg 321.95.4814.803
kJ/kg 481.5K300779.1KkJ/kg 1.005
kJ/kg 803.4K500.61300KkJ/kg 1.005
K 779.161K 1300
K 500.66K 300
in
netth
outinnet
1414out
2323in
0.4/1.4/1
3
434
0.4/1.4/1
1
212
===
=−=−=
=−⋅=−=−=
=−⋅=−=−=
=⎟⎠⎞
⎜⎝⎛=⎟⎟
⎠
⎞⎜⎜⎝
⎛=
==⎟⎟⎠
⎞⎜⎜⎝
⎛=
−
−
qw
qqw
TTchhq
TTchhq
PP
TT
PP
TT
p
p
kk
kk
η
s
T
1
24
3 qin
qout
2
3′
For rp = 12, ( )
( )( )
( )( )
( )( )( )
( )( )( )
%8.50kJ/kg 693.2kJ/kg 352.3
kJ/kg 352.39.3402.693
kJ/kg 340.9K300639.2KkJ/kg 1.005
kJ/kg 693.2K610.21300KkJ/kg 1.005
K 639.2121K 1300
K 610.212K 300
in
netth
outinnet
1414out
2323in
0.4/1.4/1
3
434
0.4/1.4/1
1
212
===
=−=−=
=−⋅=−=−=
=−⋅=−=−=
=⎟⎠⎞
⎜⎝⎛=⎟⎟
⎠
⎞⎜⎜⎝
⎛=
==⎟⎟⎠
⎞⎜⎜⎝
⎛=
−
−
qw
qqw
TTchhq
TTchhq
PP
TT
PP
TT
p
p
kk
kk
η
Thus,
(a) ( )increase9.3213.352net kJ/kg 30.4=−=∆w
( )increase%1.40%8.50th 10.7%=−=∆η (b)
PROPRIETARY MATERIALpreparation. If you are a student using this Manual, you are using it without permission.
. © 2011 The McGraw-Hill Companies, Inc. Limited distribution permitted only to teachers and educators for course
9-132
9-163 A regenerative gas-turbine engine operating with two stages of compression and two stages of expansion is considered. The back work ratio and the thermal efficiency are to be determined.
Assumptions 1 The air-standard assumptions are applicable. 2 Kinetic and potential energy changes are negligible. 3 Air is an ideal gas with constant specific heats.
PROPRIETARY MATERIAL. © 2011 The McGraw-Hill Companies, Inc. Limited distribution permitted only to teachers and educators for course
Properties The properties of air at room temperature are cp = 1.005 kJ/kg.K, cv = 0.718 kJ/kg·K, and k = 1.4 (Table A-2).
Analysis The work inputs to each stage of compressor are identical, so are the work outputs of each stage of the turbine.
( )( )( )
( )( ) ( )
( ) ( )
( )( )
( )( ) ( )
( )( )
( )( ) ( )
( )( )
( ) ( ) ( )( )
( ) ( ) ( )( ) kJ/kg .5791K10061400KkJ/kg 1.005222
kJ/kg 375.7K300486.9KkJ/kg 1.005222
K .48769.486100675.09.486
K 10061.942140086.01400
K .194241K 1400
K 486.978.0/3008.445300
/
K 445.84K 300
7676outT,
1212inC,
494549
45
49
45
7667976
76
76
76
0.4/1.4/1
6
7679
1212412
12
12
12
0.4/1.4/1
1
2124
=−⋅=−=−=
=−⋅=−=−=
=−+=
−+=⎯→⎯−
−=
−−
=
=−−=
−−==⎯→⎯−
−=
−−
=
=⎟⎠⎞
⎜⎝⎛=⎟⎟
⎠
⎞⎜⎜⎝
⎛==
=−+=
−+==⎯→⎯−
−=
−−
=
==⎟⎟⎠
⎞⎜⎜⎝
⎛==
−
−
TTchhw
TTchhw
TTTTTTcTTc
hhhh
TTTTTTTcTTc
hhhh
PP
TTT
TTTTTTTcTTc
hhhh
PP
TTT
p
p
p
p
sTsp
p
sT
kk
ss
Csp
spsC
kk
ss
εε
ηη
ηη
s
T
3
4
1
6
97
8
5
2
97s
4 2s
Thus,
0.475===kJ/kg 791.5kJ/kg 375.7
outT,
inC,bw w
wr
( ) ( ) ( ) ( )[ ]( ) ( ) ( )[ ]
45.1%====
=−=−=
=−+−⋅=−+−=−+−=
451.0kJ/kg 922.0kJ/kg 415.8
kJ/kg .84157.3755.791
kJ/kg .0922K10061400.48761400KkJ/kg 1.005
in
netth
inC,outT,net
78567856in
qw
www
TTTTchhhhq p
η
preparation. If you are a student using this Manual, you are using it without permission.
9-133
9-164 Problem 9-163 is reconsidered. The effect of the isentropic efficiencies for the compressor and turbine and regenerator effectiveness on net work done and the heat supplied to the cycle is to be investigated. Also, the T-s diagram for
nalysis Using EES, the problem is solved as follows:
[K]
cy" sentropic efficiency"
ies are constant across the compressor"
ssor for the isentropic case: dy-flow"
])
T[2], P=P[2])
ideal case the entropies are constant across the HP compressor"
r for the isentropic case: dy-flow"
])
P[4])
the cycle is to be plotted.
A
"Input data" T[6] = 1400 T[8] = T[6] Pratio = 4 T[1] = 300 [K] P[1]= 100 [kPa] T[3] = T[1] Eta_reg = 0.75 "Regenerator effectiveness
ta_c =0.78 "Compressor isentorpic efficien"
EEta_t =0.86 "Turbien i "LP Compressor:" "Isentropic Compressor anaysis" s[1]=ENTROPY(Air,T=T[1],P=P[1]) s_s[2]=s[1] "For the ideal case the entropP[2] = Pratio*P[1] s_s[2]=ENTROPY(Air,T=T_s[2],P=P[2]) "T_s[2] is the isentropic value of T[2] at compressor exit" Eta_c = w_compisen_LP/w_comp_LP "compressor adiabatic efficiency, W_comp > W_compisen" "Conservation of energy for the LP compree_in - e_out = DELTAe=0 for steah[1] + w_compisen_LP = h_s[2] h[1]=ENTHALPY(Air,T=T[1]) h_s[2]=ENTHALPY(Air,T=T_s[2 "Actual compressor analysis:"h[1] + w_comp_LP = h[2]
[2]=ENTHALPY(Air,T=T[2]) hs[2]=ENTROPY(Air,T= "HP Compressor:" s[3]=ENTROPY(Air,T=T[3],P=P[3]) s_s[4]=s[3] "For theP[4] = Pratio*P[3] P[3] = P[2] s_s[4]=ENTROPY(Air,T=T_s[4],P=P[4]) "T_s[4] is the isentropic value of T[4] at compressor exit" Eta_c = w_compisen_HP/w_comp_HP "compressor adiabatic efficiency, W_comp > W_compisen" "Conservation of energy for the compressoe_in - e_out = DELTAe=0 for steah[3] + w_compisen_HP = h_s[4] h[3]=ENTHALPY(Air,T=T[3]) h_s[4]=ENTHALPY(Air,T=T_s[4 "Actual compressor analysis:"h[3] + w_comp_HP = h[4] h[4]=ENTHALPY(Air,T=T[4]) s[4]=ENTROPY(Air,T=T[4], P=
PROPRIETARY MATERIALpreparation. If you are a student using this Manual, you are using it without permission.
. © 2011 The McGraw-Hill Companies, Inc. Limited distribution permitted only to teachers and educators for course
9-134
PROPRIETARY MATERIAL. © 2011 The McGraw-Hill Companies, Inc. Limited distribution permitted only to teachers and educators for course
ntercooling heat loss:"
eat exchanger, assuming W=0, ke=pe=0 _in - e_out =DELTAe_cv =0 for steady flow"
[6]=ENTHALPY(Air,T=T[6]) s SSSF constant pressure"
r,T=T[6],P=P[6])
_s[7]=ENTROPY(Air,T=T_s[7],P=P[7])"T_s[7] is the isentropic value of T[7] at HP turbine exit"
ic turbine, assuming: adiabatic, ke=pe=0 teady-flow"
s[7] =T_s[7])
[7]=ENTHALPY(Air,T=T[7]) (Air,T=T[7], P=P[7])
[7] + q_in_reheat = h[8] T[8])
e analysis"
r,T=T[8],P=P[8])
_s[9]=ENTROPY(Air,T=T_s[9],P=P[9])"T_s[9] is the isentropic value of T[9] at LP turbine exit" n > w_turb"
ic turbine, assuming: adiabatic, ke=pe=0 teady-flow"
[9] =T_s[9])
[9]=ENTHALPY(Air,T=T[9]) ir,T=T[9], P=P[9])
ta_th_noreg=w_net/(q_in_total_noreg)*Convert(, %) "[%]" "Cycle thermal efficiency" atio"
t added in the external heat exchanger is"
ALPY(Air, T=T[5]) [5]=ENTROPY(Air,T=T[5], P=P[5])
"Ih[2] = q_out_intercool + h[3] "External heat exchanger analysis" "SSSF First Law for the heh[4] + q_in_noreg = h[6] hP[6]=P[4]"process 4-6 i "HP Turbine analysis" s[6]=ENTROPY(Ais_s[7]=s[6] "For the ideal case the entropies are constant across the turbine" P[7] = P[6] /Pratio sEta_t = w_turb_HP /w_turbisen_HP "turbine adiabatic efficiency, w_turbisen > w_turb" "SSSF First Law for the isentrope_in -e_out = DELTAe_cv = 0 for sh[6] = w_turbisen_HP + h_h_s[7]=ENTHALPY(Air,T"Actual Turbine analysis:" h[6] = w_turb_HP + h[7] hs[7]=ENTROPY "Reheat Q_in:" hh[8]=ENTHALPY(Air,T= "HL Turbin P[8]=P[7] s[8]=ENTROPY(Ais_s[9]=s[8] "For the ideal case the entropies are constant across the turbine" P[9] = P[8] /Pratio sEta_t = w_turb_LP /w_turbisen_LP "turbine adiabatic efficiency, w_turbise "SSSF First Law for the isentrope_in -e_out = DELTAe_cv = 0 for sh[8] = w_turbisen_LP + h_sh_s[9]=ENTHALPY(Air,T"Actual Turbine analysis:" h[8] = w_turb_LP + h[9] hs[9]=ENTROPY(A "Cycle analysis" w_net=w_turb_HP+w_turb_LP - w_comp_HP - w_comp_LP q_in_total_noreg=q_in_noreg+q_in_reheat EBwr=(w_comp_HP + w_comp_LP)/(w_turb_HP+w_turb_LP)"Back work r "With the regenerator, the heah[5] + q_in_withreg = h[6] h[5]=ENTHsP[5]=P[4]
preparation. If you are a student using this Manual, you are using it without permission.
9-135
PROPRIETARY MATERIAL. © 2011 The McGraw-Hill Companies, Inc. Limited distribution permitted only to teachers and educators for course
]-h[4]) ives h[10] and thus T[10] as:"
ALPY(Air, T=T[10]) [10]=ENTROPY(Air,T=T[10], P=P[10])
_in_total_withreg=q_in_withreg+q_in_reheat
g data is used to complete the Array Table for plotting purposes."
OPY(Air,T=T[5],P=P[5])
_s[10]=s[10] _s[10]=T[10]
η qi g qin eg
"The regenerator effectiveness gives h[5] and thus T[5] as:" Eta_reg = (h[5]-h[4])/(h[9"Energy balance on regenerator gh[4] + h[9]=h[5] + h[10] h[10]=ENTHsP[10]=P[9] "Cycle thermal efficiency with regenerator" qEta_th_withreg=w_net/(q_in_total_withreg)*Convert(, %) "[%]" "The followins_s[1]=s[1] T_s[1]=T[1] s_s[3]=s[3] T_s[3]=T[3] s_s[5]=ENTRT_s[5]=T[5] s_s[6]=s[6] T_s[6]=T[6] s_s[8]=s[8] T_s[8]=T[8] sT
ηreg ηC ηt ηth,noreg [%]
th,withreg
[%] n,total,nore
[kJ/kg] ,total,withr
[kJ/kg] wnet
[kJ/kg] 0.6
0.65 0.7
0.75 0.8
0.85 0.9
0.95 1
0.78 0.78
0.86 0.86
30.57 30.57
51.89 53.87
1434 1434
845.1 814.1
438.5 438.5
0.78 0.78 0.78 0.78 0.78 0.78 0.78
0.86 0.86 0.86 0.86 0.86 0.86 0.86
30.57 30.57 30.57 30.57 30.57 30.57 30.57
41.29 42.53 43.85 45.25 46.75 48.34 50.06
1434 1434 1434 1434 1434 1434 1434
1062 1031 1000 969.1 938.1 907.1 876.1
438.5 438.5 438.5 438.5 438.5 438.5 438.5
4.5 5.0 5.5 6.0 6.5 7.0 7.5 8.0 8.50
400
800
1200
1600
s [kJ/kg-K]
T [K
]
100 k
Pa
400 k
Pa
160
0 kP
a
Air
1
2
3
4
5
6
7
8
9
10
preparation. If you are a student using this Manual, you are using it without permission.
9-136
0.6 0.65 0.7 0.75 0.8 0.85 0.9 0.95 128
32
36
40
44
48
52
ηreg
ηth
[%
]
No regeneration
With regeneration
0.6 0.65 0.7 0.75 0.8 0.85 0.9 0.95 110
15
20
25
30
35
40
45
50
55
η t
ηth
[%
]
With regeneration
No regeneration
0.6 0.65 0.7 0.75 0.8 0.85 0.9 0.95 1150
200
250
300
350
400
450
500
550
600
η t
wne
t [k
J/kg
]
PROPRIETARY MATERIALpreparation. If you are a student using this Manual, you are using it without permission.
. © 2011 The McGraw-Hill Companies, Inc. Limited distribution permitted only to teachers and educators for course
9-137
0.6 0.65 0.7 0.75 0.8 0.85 0.9 0.95 1
800
900
1000
1100
1200
1300
1400
1500
η t
q in,
tota
l [kJ
/kg]
With regeneration
No regeneration
0.6 0.65 0.7 0.75 0.8 0.85 0.9 0.95 1
25
30
35
40
45
50
55
ηc
ηth
[%
] With regeneration
No regeneration
PROPRIETARY MATERIALpreparation. If you are a student using this Manual, you are using it without permission.
. © 2011 The McGraw-Hill Companies, Inc. Limited distribution permitted only to teachers and educators for course
9-138
9-165 A regenerative gas-turbine engine operating with two stages of compression and two stages of expansion is considered. The back work ratio and the thermal efficiency are to be determined.
Assumptions 1 The air-standard assumptions are applicable. 2 Kinetic and potential energy changes are negligible. 3 Helium is an ideal gas with constant specific heats.
PROPRIETARY MATERIAL. © 2011 The McGraw-Hill Companies, Inc. Limited distribution permitted only to teachers and educators for course
Properties The properties of helium at room temperature are cp = 5.1926 kJ/kg.K and k = 1.667 (Table A-2).
Analysis The work inputs to each stage of compressor are identical, so are the work outputs of each stage of the turbine.
( )( )( )
( )( ) ( )
( ) ( )
( )( )
( )( ) ( )
( )( )
( )( ) ( )
( )( )
( ) ( ) ( )( )
( ) ( ) ( )( ) kJ/kg 5323K.48871400KkJ/kg 5.1926222
kJ/kg 2961K300585.2KkJ/kg 5.1926222
K .88112.5854.88775.02.585
K .48870.804140086.01400
K 0.80441K 1400
K .258578.0/3004.522300
/
K .45224K 300
7676outT,
1212inC,
494549
45
49
45
7667976
76
76
76
70.667/1.66/1
6
7679
1212412
12
12
12
70.667/1.66/1
1
2124
=−⋅=−=−=
=−⋅=−=−=
=−+=
−+=⎯→⎯−
−=
−−
=
=−−=
−−==⎯→⎯−
−=
−−
=
=⎟⎠⎞
⎜⎝⎛=⎟⎟
⎠
⎞⎜⎜⎝
⎛==
=−+=
−+==⎯→⎯−
−=
−−
=
==⎟⎟⎠
⎞⎜⎜⎝
⎛==
−
−
TTchhw
TTchhw
TTTTTTcTTc
hhhh
TTTTTTTcTTc
hhhh
PP
TTT
TTTTTTTcTTc
hhhh
PP
TTT
p
p
p
p
sTsp
p
sT
kk
ss
Csp
spsC
kk
ss
εε
ηη
ηη
s
T
3
4
1
6qin
97
8
5
2
97s
4 2s
Thus,
0.556===kJ/kg 5323kJ/kg 2961
outT,
inC,bw w
wr
( ) ( ) ( ) ( )[ ]( ) ( ) ( )[ ]
41.3%====
=−=−=
=−+−⋅=−+−=−+−=
4133.0kJ/kg 5716kJ/kg 2362
kJ/kg 236229615323
kJ/kg 5716K.48871400.88111400KkJ/kg 5.1926
in
netth
inC,outT,net
78567856in
qw
www
TTTTchhhhq p
η
preparation. If you are a student using this Manual, you are using it without permission.
9-139
9-166 An ideal gas-turbine cycle with one stage of compression and two stages of expansion and regeneration is considered. The thermal efficiency of the cycle as a function of the compressor pressure ratio and the high-pressure turbine to compressor inlet temperature ratio is to be determined, and to be compared with the efficiency of the standard regenerative cycle.
Analysis The T-s diagram of the cycle is as shown in the figure. If the overall pressure ratio of the cycle is rp, which is the pressure ratio across the compressor, then the pressure ratio across each turbine stage in the ideal case becomes √ rp. Using the isentropic relations, the temperatures at the compressor and turbine exit can be expressed as
( )( )( )
( ) ( )( )
( ) ( )( ) ( ) ( ) ( ) kk
pkk
pkk
pkk
p
kk
p
kk
kkp
kk
p
kk
kkp
kk
rTrrTrTr
TPPTT
rTr
TPPTTT
rTPPTTT
2/11
2/1/11
2/12
/1
5
/16
5
2/13
/1
3
/1
3
4347
/11
/1
1
2125
1
1
−−−−
−−
−
−−
−−
===⎟⎟
⎠
⎞
⎜⎜
⎝
⎛=⎟⎟
⎠
⎞⎜⎜⎛
=
=⎟⎟
⎠
⎞
⎜⎜
⎝
⎛=⎟⎟
⎠
⎞⎜⎜⎝
⎛==
=⎟⎟⎠
⎞⎜⎜⎝
⎛==
Then,
65⎝
( ) ( )( )( ) ( )( )12/1 −=−=−= − kkrTcTTchhq
1
1161
2/137373in −=−=−= −
ppp
kkppp rTcTTchhq
6out
and thus ( )( )
( )( )kkpp rTcq 2/1
3inth
111
−−−=−=η
which simplifies to
kkpp rTcq 2/1
1out 1− −
( ) kkpr
TT 2/1
3
1th 1 −−=η
The thermal efficiency of the single stage ideal regenerative cycle is given as
( ) kkpr
TT /1
3
1th 1 −−=η
Therefore, the regenerative cycle with two stages of expansion has a higher thermal efficiency than the standard regenerative cycle with a single stage of expansion for any given value of the pressure ratio rp.
s
T
1
2
7
qin
5
6
4
3
qout
PROPRIETARY MATERIALpreparation. If you are a student using this Manual, you are using it without permission.
. © 2011 The McGraw-Hill Companies, Inc. Limited distribution permitted only to teachers and educators for course
9-140
9-167 A gas-turbine plant operates on the regenerative Brayton cycle with reheating and intercooling. The back work ratio, the net work output, the thermal efficiency, the second-law efficiency, and the exergies at the exits of the combustion chamber and the regenerator are to be determined.
Assumptions 1 The air-standard assumptions are applicable. 2 Kinetic and potential energy changes are negligible. 3 Air is an ideal gas with variable specific heats.
Properties The gas constant of air is R = 0.287 kJ/kg·K.
Analysis (a) For this problem, we use the properties from EES software. Remember that for an ideal gas, enthalpy is a function of temperature only whereas entropy is functions of both temperature and pressure.
Optimum intercooling and reheating pressure is
PROPRIETARY MATERIAL. © 2011 The McGraw-Hill Companies, Inc. Limited distribution permitted only to teachers and educators for course
kPa 4.346)1200)(100(412 === PPP
Process 1-2, 3-4: Compression
KkJ/kg 7054.5kPa 100K 300
kJ/kg 43.300K 300
11
1
11
⋅=⎭⎬⎫
==
=⎯→⎯=
sPT
hT
=⎭⎬⎫
===
shss
P kJ/kg 79.428
kJ/kg.K 7054.5kPa 4.346
212
2
kJ/kg 88.46043.300
43.30079.42880.0 2212
12C =⎯→⎯
−−
=⎯→⎯−−
= hhhh
hh sη
s
T
3
4
1
5 qin
8 6
7
10
9
2
8s 6s
4s 2s
KkJ/kg 5040.5kPa 4.346
K 350kJ/kg 78.350K 350
33
3
33
⋅=⎭⎬⎫
==
=⎯→⎯=
sPT
hT
43
=⎭⎬⎫
== shss
kJ/kg 42.500kJ/kg.K 5040.5
kPa 12004 =P
4
kJ/kg 83.53778.350
78.35042.50080.0 4434
34C =⎯→⎯
−−
=⎯→⎯−−
= hhhh
hh sη
Process 6-7, 8-9: Expansion
=⎭⎬⎫
===
shss
P
KkJ/kg 6514.6kPa 1200K 1400
kJ/kg 9.1514K 1400
66
6
66
⋅=⎭⎬⎫
==
=⎯→⎯=
sPT
hT
kJ/kg 9.1083kJ/kg.K 6514.6
kPa 4.3467
67
7
kJ/kg 1.11709.10839.1514
9.151480.0 7
7
76T =
hη 76 =⎯→⎯
−−
=⎯→⎯−−
hh
hhh
s
8
88
⋅=⎭⎬⎫
==
=⎯→⎯=
sPT
h
96kPa 100
99 =
⎭⎬⎫=
shP
T
KkJ/kg 9196.6kPa 4.346
K 1300kJ/kg 6.1395K 1300
88
kJ/kg 00.996kJ/kg.K 91.689 == ss
preparation. If you are a student using this Manual, you are using it without permission.
9-141
kJ/kg 9.107500.9966.1395
6.139580.0 9
9
98
98T =⎯→⎯
−−
=⎯→⎯−−
= hh
hhhh
sη
C ysycle anal is:
kJ/kg 50.34778.35083.53743.30088.4603412inC, =−+−=−+−= hhhhw
kJ/kg 50.6649.10756.13951.11709.15149876outT, =−+−=−+−= hhhhw
0.523===50.66450.347
outT,
inC,bw w
wr
kJ/kg 317.0=−=−= 50.34750.664inC,outT,net www
Regenerator analysis:
kJ/kg 36.67283.5379.1075
9.107575.0 10
10
49
109regen =⎯→⎯
−−
=⎯→⎯−−
= hh
hhhh
ε
1010
⋅=⎭⎬=
sP
(b) =−=−= hhq
KkJ/kg 5157.6kPa 100
K 36.67210 ⎫=h
kJ/kg 40.94183.53736.6729.1075 5545109regen =⎯→⎯−=−⎯→⎯−=−= hhhhhhq
kJ/kg 54.57340.9419.151456in
0.553===54.5730.317
in
netth q
wη
(c) The second-law efficieny of the cycle is defined as the ratio of actual thermal efficiency to the maximum possible thermal efficiency (Carnot efficiency). The maximum temperature for the cycle can be taken to be the turbine inlet temperature. That is,
786.0K 1400K 30011
6
1max =−=−=
TT
η
and
0.704===786.0553.0
maxηη
η thII
(d) The exergies at the combustion chamber exit and the regenerator exit are
kJ/kg 930.7=−−−=
−−−=kJ/kg.K)7054.56514.6)(K 300(kJ/kg)43.3009.1514(
)( 060066 ssThhx
kJ/kg 128.8=−−−=
−−−=kJ/kg.K)7054.55157.6)(K 300(kJ/kg)43.30036.672(
)( 010001010 ssThhx
PROPRIETARY MATERIALpreparation. If you are a student using this Manual, you are using it without permission.
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9-142
9-168 The thermal efficiency of a two-stage gas turbine with regeneration, reheating and intercooling to that of a three-stage gas turbine is to be compared.
Assumptions 1 The air standard assumptions are applicable. 2 Air is an ideal gas with constant specific heats at room temperature. 3 Kinetic and potential energy changes are negligible.
Properties The properties of air at room temperature are cp = 1.005 kJ/kg⋅K and k = 1.4 (Table A-2a).
Analysis
Two Stages:
PROPRIETARY MATERIAL. © 2011 The McGraw-Hill Companies, Inc. Limited distribution permitted only to teachers and educators for course
The pressure ratio across each stage is
416 ==pr
The temperatures at the end of compression and expansion are
c K
s
T
3
4
1
6 873 K
283 K
97
8
2
5
10 5.420K)(4) 283( 0.4/1.4/)1(min === − kk
prTT
K 5.58741K) 873(1 0.4/1.4/)1(
max =⎟⎠⎞
⎜⎝⎛=⎟
⎟⎠
⎞⎜⎜⎝
⎛=
− kk
pe r
TT
The heat input and heat output are
kJ/kg 9.573K )5.587K)(873kJ/kg 2(1.005)(2 maxin =−⋅=−= ep TTcq
kJ/kg 4.276K )283K)(420.5kJ/kg 2(1.005)(2 minout =−⋅=−= TTcq
e therm of the cycle is then
cp
Th al efficiency
0.518=−=−= 1thη9.573inq4.2761outq
hree Stages:
The pressure ratio across each stage is
==r
The temperatures at the end of compression and expansion are
T
p 520.216 3/1
K 5.368K)(2.520) 283( 0.4/1.4/)1(min === − kk
pc rTT
K 4.6702.520
1K) 873(1/)1(
=⎟⎞
⎜⎛
=− kk
TT0.4/1.4
max =⎟⎠⎞
⎜⎝⎛
⎟⎠
⎜⎝ pr
he heat input and heat output are
kg =−⋅
s
T
3
4
1
9
8
2
5
10
11
6
7
12
13
14
e
T
kJ/ 3(1.005)(3 maxin =−= ep TTcq kJ/kg 8.610K )4.670K)(873
out kJ/kg 8.257K )283K)(368.5kJ/kg (1.0053)(3 min =−⋅=−= TTcq cp
The thermal efficiency of the cycle is then
0.578=−=−=8.6108.25711
in
outth q
qη
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9-143
9-169 A pure jet engine operating on an ideal cycle is considered. The thrust force produced per unit mass flow rate is to be determined.
Assumptions 1 Steady operating conditions exist. 2 The air standard assumptions are applicable. 3 Air is an ideal gas with constant specific heats at room temperature. 4 The turbine work output is equal to the compressor work input.
Properties The properties of air at room temperature are R = 0.287 kPa⋅m3/kg⋅K (Table A-1), cp = 1.005 kJ/kg⋅K and k = 1.4 (Table A-2a).
Analysis Working across the two isentropic processes of the cycle yields
K 6.509K)(9) 272( 0.4/1.4/)1(12 === − kk
prTT
s
T
1
2 4
3qin
5
qout
K 2.34391K 643(35 =⎟
⎟⎠
⎜⎜⎝
=pr
TT )1 0.4/1.4/)1(
=⎟⎠⎞
⎜⎝⎛⎞⎛
− kk
pressor is
The work input to the com
K)(509.6kJ/kg 005.1()( 12C ⋅=−= TTcw p kJ/kg 8.238272)K- =
kJ/kg 005.1(C53
=−−=
p
engine exit is determined from
An energy balance gives the excess enthalpy to be
)( −−=∆ wTTch
kJ/kg 50.62kJ/kg 8.238)K2.343K)(643⋅
The velocity of the air at the
2
2inletexit VV
h−
=∆ 2
earranging,
R
( )
m/s 6.504
)m/s 360(kJ/kg 1
/sm 1000kJ/kg) 50.62(2
22/1
222
2/12inletexit
=
⎥⎥⎦
⎤
⎢⎢⎣
⎡+⎟
⎟⎠
⎞⎜⎜⎝
⎛=
+∆= VhV
The specific impulse is then
m/s 145=−=−= 3606.504inletexit VVmF&
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. © 2011 The McGraw-Hill Companies, Inc. Limited distribution permitted only to teachers and educators for course
9-144
9-170 The electricity and the process heat requirements of a manufacturing facility are to be met by a cogeneration plant consisting of a gas-turbine and a heat exchanger for steam production. The mass flow rate of the air in the cycle, the back work ratio, the thermal efficiency, the rate at which steam is produced in the heat exchanger, and the utilization efficiency of the cogeneration plant are to be determined.
Assumptions 1 The air-standard assumptions are applicable. 2 Kinetic and potential energy changes are negligible. 3 Air is an ideal gas with variable specific heats.
PROPRIETARY MATERIAL. © 2011 The McGraw-Hill Companies, Inc. Limited distribution permitted only to teachers and educators for course
⋅=⎭⎬⎫
=°=
=⎯→⎯°=
sPT
hT
kPa 10002 =
⎭⎬⎫=
shP
1
Combustion chamber
Turbine
23
4
Compress.
100 kPa20°C
450°C1 MPa
325°C 15°C
Sat. vap. 200°C
Heat exchanger
5
Analysis (a) For this problem, we use the properties of air from EES software. Remember that for an ideal gas, enthalpy is a function of temperature only whereas entropy is functions of both temperature and pressure.
Process 1-2: Compression
KkJ/kg 682.5kPa 100
C20kJ/kg 5.293C20
11
1
11
kJ/kg 2.567kJ/kg.K 682.512 == ss
2
kJ/kg 8.6115.293
5.2932.56786.0 2212
12C =⎯→⎯
−−
=⎯→⎯−−
= hhhh
hh sη
Process 3-4: Expansion
kJ/kg 5.738C4504 =⎯→⎯°= hT 4
ss hhh 343T −− h
h
4
3 5.738−
We cann ly. However, using the following lines in EES together with the isentropic c relation, we find h3 = 1262 kJ/kg, T3 = 913.2ºC, s3 = 6.507 kJ/kg.K. The solution by hand would require a trial-
h_3=enthalpy(Air, T=T_3)
5 =→h
The inlet water is compressed liquid at 15ºC and at the saturation pressure of steam at 200ºC (1555 kPa). This is not available in the tables but we can obtain it in EES. The alternative is to use saturated liquid enthalpy at the given temperature.
C15
22
2 =⎭⎬⎫
=°=
=⎫°=
ww hx
T
T
12in, =−=−= hhw
hh 43 88.0 =⎯→⎯−
=η
ot find the enthalpy at state 3 directefficien yerror approach.
s_3=entropy(Air, T=T_3, P=P_2)
h_4s=enthalpy(Air, P=P_1, s=s_3)
Also,
C3255 ⎯⎯°=T kJ/kg 4.605
kJ/kg 27921
C200
kJ/kg 47.64kPa 1555 1
1 ⎭⎬= wh
P1w
The net work output is
kJ/kg 2.3185.2938.611C
kJ/kg 4.5235.738126243outT, =−=−= hhw
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9-145
kJ/kg 2.2052.3184.523inC,outT,net =−=−= www
The mass flow rate of air is
PROPRIETARY MATERIAL. © 2011 The McGraw-Hill Companies, Inc. Limited distribution permitted only to teachers and educators for course
kg/s 7.311===kJ/s 1500netW
m&
& kJ/kg 2.205netwa
(b) The back work ratio is
0.608===2.318inC,
bww
r 4.523outT,w
The rate of heat input and the thermal efficiency are
kW 4753)kJ/kg8.611kg/s)(1262 311.7()( 23in =−=−= hhmQ a&&
31.6%==== 3156.0kW 4753kW 1500
in
netth Q
W&
&η
(c) An energy balance on the heat exchanger gives
(d) The heat supplied to the water in the heat exchanger (process heat) and the utilization efficiency are
kg/s 0.3569=⎯→⎯−=−
−=−
ww
wwwa
mm
hhmhhm
&&
&&
)kJ/kg47.64(2792)kJ/kg4.6055kg/s)(738. 311.7(
)()( 1254
kW 5.973)kJ/kg47.64kg/s)(2792 3569.0()( 12p =−=−= www hhmQ &&
52.0%==+
=+
= 5204.0kW 4753
5.9731500
in
pnet
Q
QWu &
&&ε
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9-146
9-171 A turbojet aircraft flying is considered. The pressure of the gases at the turbine exit, the mass flow rate of the air through the compressor, the velocity of the gases at the nozzle exit, the propulsive power, and the propulsive efficiency of the cycle are to be determined.
Assumptions 1 The air-standard assumptions are applicable. 2 Potential energy changes are negligible. 3 Air is an ideal gas with variable specific heats.
Properties The gas constant of air is R = 0.287 kJ/kg·K (Table A-1).
PROPRIETARY MATERIAL. © 2011 The McGraw-Hill Companies, Inc. Limited distribution permitted only to teachers and educators for course
Analysis (a) For this problem, we use the properties from EES software. Remember that for an ideal gas, enthalpy is a function of temperature only whereas entropy is functions of both temperature and pressure.
Diffuser, Process 1-2:
kJ/kg 23.238C35 11 =⎯→⎯°−= hT
kJ/kg 37.269/sm 1000
kJ/kg 12m/s) (15
/sm 1000kJ/kg 1
2m/s) (900/3.6kJ/kg) 23.238(
22
222
2
222
2
22
2
21
1
=⎯→⎯⎟⎠⎞
⎜⎝⎛+=⎟
⎠⎞
⎜⎝⎛+
+=+
hh
VhVh
s
T
1
2
4
3
qin
5
qout
6
kPa 50kJ/kg 37.2692
=Ph
KkJ/kg 7951.522
⋅=⎭⎬⎫=
s
Compressor, Process 2-3:
=⎭⎬⎫
===
shss
P kJ/kg 19.505
kJ/kg.K 7951.5kPa 450
323
3
kJ/kg 50.55337.269
37.26919.50583.0 3323
23C =⎯→⎯
−−
=⎯→⎯−−
= hhhh
hh sη
urbine, P cess 3-4:
1028.130437.26950.553 555423 =⎯→⎯−=−⎯→⎯−=− hhhhhh
here the ass flow rates through the compressor and the turbine are assumed equal.
T ro
kJ/kg 8.1304C950 =⎯→⎯°= hT 44
.0 kJ/kg 6
w m
kJ/kg 45.9628.1304
6.10208.130483.0 5554
54T =⎯ →⎯
−−
=⎯→⎯−−
= sss
hhhh
hhη
4⎭=P
545 KkJ/kg 7725.6
kgP
ss
(b) The mass flow rate of the air through the compressor is
KkJ/kg 7725.6C9504 ⋅=⎬
⎫°=s
T
kPa 4504
=5 kJ/ 45.962h s kPa 147.4=
⎭⎬⎫
⋅==
kg/s 1.760=−
=−
=kJ/kg )37.26950.553(
kJ/s 500
23
C
hhW
m&
&
(c) Nozzle, Process 5-6:
⎫=h KkJ/kg 8336.6
kPa 4.147 55
⋅=⎭⎬=
sP
kJ/kg 6.10205
preparation. If you are a student using this Manual, you are using it without permission.
9-147
kJ/kg 66.709kJ/kg.K 8336.6
kPa 406
56
6 =⎭⎬⎫
===
shss
P
PROPRIETARY MATERIAL. © 2011 The McGraw-Hill Companies, Inc. Limited distribution permitted only to teachers and educators for course
kJ/kg 52.76266.7096.1020
6.102083.0 6
6
65
65N =⎯→⎯
−−
=⎯→⎯−−
= hh
hhhh
sη
m/s 718.5=⎯→⎯⎟⎠⎞
⎜⎝⎛+=+
+=+
622
26
26
6
25
5
/sm 1000kJ/kg 1
2kJ/kg 52.7620kJ/kg) 6.1020(
22
VV
VhVh
where the velocity at nozzle inlet is assumed zero.
(d) The propulsive power and the propulsive efficiency are
kW 206.1=⎟⎠⎞
⎜⎝⎛−=−= 22116 /sm 1000
kJ/kg 1m/s) 250)(m/s 250m/s 5.718(kg/s) 76.1()( VVVmWp &&
kW 1322kJ/kg)50.5538.1304(kg/s) 76.1()( 34in =−=−= hhmQ &&
0.156===kW 1322kW 1.206
inQ
W pp &
&η
preparation. If you are a student using this Manual, you are using it without permission.
9-148
9-172 The three processes of an air standard cycle are described. The cycle is to be shown on the P-v and T-s diagrams, and the expressions for back work ratio and the thermal efficiency are to be obtained.
Assumptions 1 The air-standard assumptions are applicable. 2 Kinetic and potential energy changes are negligible. 3 Air is an ideal gas with constant specific heats.
Analysis (a) The P-v and T-s diagrams for this cycle are as shown.
(b) The work of compression is found by the first law for process 1-2:
PROPRIETARY MATERIAL. © 2011 The McGraw-Hill Companies, Inc. Limited distribution permitted only to teachers and educators for course
2 1 2 1 2
1 2 1 2 2 1
1 2 2 1
)
v
comp v
q w urocess
w u C T T
w w C T T
−
− −
−
− = ∆
= −∆ = − −
= − = −
ound by
)3 2 3 22
v v R T T− = −
The back work ratio is
( )( )
1− −
1 2 0(q isentropic p− =
The expansion work is f
( )exp 2 3w w Pdv P−= = =∫ (3
( )( )
( ))(
3 1 3 11
3 2
/ 1/ 1
comp v vw C T T T TC TT
− −−
Process 1-2 is isentropic; therefore,
s
T
3 2
1
v
P
32
1
exp 3 2 2w R T T R T T= =
−
1
1 21
1k
T VT V r −= =
2 1
k −⎛ ⎞⎜ ⎟⎝ ⎠
and 2P VP V
= 1
1 2
kkr
⎛ ⎞=⎜ ⎟
⎝ ⎠
Process 2-3 is constant pressure; therefore,
3 3 3 32 2 1
3 2 2 2 2
PV T VPV V rT T T V V
= ⇒ = = =
Process 3-1 is constant volume; therefore,
3 3 3 31 1 2 kPV T PPV P r= ⇒ = = =
The back work ratio becomes (Cv=R/(k-1))
3 1 1 1 1T T T P P
1
1exp 1w k − −
1 1 11
kcomp
k
w rr r
−
−
−=
(c) Apply first law to the closed system for processes 2-3 and 3-1 to show:
3 1out vq C T T= −
The cycle thermal efficiency is given by
( )3 2in pq C T T= −
( )
( )( )
( )( )
3 1 1 3 1
3 2 2 3 2
/ 111 1 1/ 1
voutth
in p
C T T T T Tqq C T T k T T T
η− −
= − = − = −− −
The efficiency becomes
preparation. If you are a student using this Manual, you are using it without permission.
9-149
111th kk r r
η −= −−
1 1 1kr −
ine the value of the back work ratio and efficiency as r goes to unity.
(d) Determ
( )( )
1
1exp
21 1
1 1 11 1exp 1
1exp
1 1 11 1
11 1 1 1 1 1lim lim lim lim1 1 1 1 1
1 1 1 1lim 11 1 1 1
kcomp
k
kk kcomp
k k k kr r rr
comp
r
w rw k r r
w k rr rw k r r k r r k kr k r
w k kw k k k k
−
−
−− −
− − −→ → →→
→
−=
− −
⎧ ⎫21 k −
⎧ ⎫−⎧ ⎫− −⎪ ⎪ ⎪= = =⎨ ⎬ ⎨ ⎬ ⎨− − − − − − −⎪⎬
⎪ ⎪⎩ ⎭⎪ ⎪ ⎩ ⎭⎩ ⎭− −⎧ ⎫ ⎧ ⎫= = =⎨ ⎬ ⎨ ⎬− − + −⎩ ⎭ ⎩ ⎭
( )
1
1
1 1 11 1 1 1
1
1 1 111
1 1 1 1 1 1lim 1 lim 1 lim 1 lim1 1
1 1lim 1 1 01 1
k
th k
k k k
th k k k kr r r r
thr
rk r r
r r krk r r k r r k kr k r
k kk k k k
η
η
η
−
−
− − −→ → → →
→
−= −
−
2k −
⎧ ⎫⎧ ⎫ ⎧ ⎫− − ⎪ ⎪= − = − = −⎨ ⎬ ⎨ ⎬ ⎨− − − − ⎬⎪ ⎪⎩ ⎭ ⎩ ⎭ ⎩ ⎭
⎧ ⎫ ⎧ ⎫= − = − =⎨ ⎬ ⎨ ⎬− +⎩ ⎭ ⎩ ⎭
These results show that if there is no compression (i.e. r = 1), there can be no expansion and no net work will be done even though heat may be added to the system.
PROPRIETARY MATERIALpreparation. If you are a student using this Manual, you are using it without permission.
. © 2011 The McGraw-Hill Companies, Inc. Limited distribution permitted only to teachers and educators for course
9-150
9-173 The three processes of an air standard cycle are described. The cycle is to be shown on the P-v and T-s diagrams, and the expressions for back work ratio and the thermal efficiency are to be obtained.
Assumptions 1 The air-standard assumptions are applicable. 2 Kinetic and potential energy changes are negligible. 3 Air is an ideal gas with constant specific heats.
Analysis (a) The P-v and T-s diagrams for this cycle are as shown.
(b) The work of expansion is found by the first law for process 2-3:
PROPRIETARY MATERIAL. © 2011 The McGraw-Hill Companies, Inc. Limited distribution permitted only to teachers and educators for course
T
)1 3 3 13
Pdv P v v R T T= − = − − = −∫
The back work ratio is
( )( )
2 3 2 3 2 3
2 3
2 3 2 3 3 2
exp 2 3 2 3
0( )
v
v
q w uq isentropic processw u C T
w w C T T
− − −
−
− −
−
− = ∆=
= −∆ = − −
= = −
The compression work is found by
( ) (1
3 1compw w −= −
( )( )
( )( )
( )( )
3 1 1 3 1 3
2 3 2 3
1 / 1 // 1 / 1
comp vw C T T T T T TC T CT R T T
− − −=
−
Process 3-1 is constant pressure; therefore,
3
exp 2 3 3
v v
w R T T R T T= =
− −
v
P
3
2
1
s
T
3
2
1
3 3 1 1 1 1 2
3 1 3 3V V3
1PV PV T V VT T T r
= ⇒ = = =
Process 2-3 is isentropic; therefore, 1k
T V−
⎛ ⎞ 32
3 2
kkVP12 2
3 2
krT V
−= =⎜ ⎟⎝ ⎠
and rP V
⎛ ⎞= =⎜ ⎟
⎝ ⎠
The back work ratio becomes (Cv=R/(k-1))
( ) 1 1
11w − − −
exp
1 111 1
mpk k
k rrkw r r r− −= − =
− −
(c) Apply first law to the closed system for processes 1-2 and 3-1 to show:
2 1in vq C T T= −
The cycle thermal efficiency is given by
co
( )( )3 1out pq C T T= −
( )( )
( )( )
3 1 1 3 1
2 1 1 2 1
/ 11 1 1
/ 1poutqηth
in v
C T T T T Tk
q C T T T T T− −
= − = − = −− −
Process 1-2 is constant volume; therefore,
2 2
T1 1 2 2 2
2 1 1 1 3
krT T P P
= ⇒ = = =
The efficiency becomes
PV PV T P P
preparation. If you are a student using this Manual, you are using it without permission.
9-151
111th k
rkr
η −= −
−
(d) Determine the value of the back work ratio and efficiency as r goes to unity.
( )
( ) ( )
( )
1 1exp
11
1exp
11 1 111 1
1
1lim 1 11
compk k
kr
comp
r
w k rrkw r r r
r kr
wk
w k
− −
−→
→
− − −= − =
− −
−⎭ ⎩ ⎭
⎧ ⎫= − =⎨ ⎬−⎩ ⎭
1 1exp
1 1lim 1 lim 1 limcompkr r
w rk kw r→ →
⎧ ⎫ ⎧ ⎫−= − = −⎨ ⎬ ⎨ ⎬−⎩
11 1 1
1
111
1 1lim 1 lim 1 lim1
1lim 1 0
th k
th k kr r r
thr
rkr
rk kr k
kk
η
η
η
−→ → →
→
r
−= −
−⎧ ⎫ ⎧−
= − = −⎨ ⎬ ⎨−⎩ ⎭ ⎩⎧ ⎫= − =⎨ ⎬⎩ ⎭
⎫⎬⎭
These results show that if there is no compression (i.e. r = 1), there can be no expansion and no net work will be done even though heat may be added to the system.
PROPRIETARY MATERIALpreparation. If you are a student using this Manual, you are using it without permission.
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9-152
9-174 The four processes of an air-standard cycle are described. The cycle is to be shown on the P-v and T-s diagrams; an expression for the cycle thermal efficiency is to be obtained; and the limit of the efficiency as the volume ratio during heat rejection approaches unity is to be evaluated.
Analysis (a) The P-v and T-s diagrams of the cycle are shown in the figures.
(b) Apply first law to the closed system for processes 2-3 and 4-1 to show:
PROPRIETARY MATERIAL. © 2011 The McGraw-Hill Companies, Inc. Limited distribution permitted only to teachers and educators for course
3 2
4 1
v
out p
q C T T= −
The cycle thermal efficiency is given by
( )( )q C T T= −
in
( )( )
( )( )
4 11 1 poutth q C T T
η 1 4 1
3 2 2 3 2
/ 11
/ 1in v
C T T T T Tq kT T T
− −= − = − = −
− −
Process 1-2 is isentropic; therefore,
1
1 21
s
T
42
1
3
v
P3
2
1
4
2 1
1k
k
T VT V r
−
−
⎛ ⎞= =⎜ ⎟
⎝ ⎠
Process 3-4 is isentropic; therefore,
113 4
4 3
kk
eT V rT V
−
−⎛ ⎞= =⎜ ⎟
⎝ ⎠
Process 4-1 is constant pressure; therefore,
4 4 1 1 4 4p
PV PV T V r= ⇒ = = 4 1 1 1T T VT
113 3 4 1
12 4 1 2
1 kk e
e p k
T T rT T r r rT T T T r r
−−
−
⎛ ⎞= = = ⎜ ⎟⎝ ⎠
p
tant volume and V3 = V2, Since process 2-3 is cons
4 4 4 1
3 2 1 2e pV V V V
V V V Vr r r= = = =
1
3
2
kp k
p p
r rT r rT r
−⎛ ⎞
= =⎜ ⎟⎝ ⎠
The efficiency becomes
1
1111
pth k k
p
rk
r rη −
−= −
−
(c) In the limit as rp approaches unity, the cycle thermal efficiency becomes
1 11 1
1 11
11 1lim 1 lim 1 lim1
1 1 1lim 1 1
p p p
p
pth k k k kr r r
p p
th th Ottok kr
rk k
r r r kr
kr k r
η
η η
− −→ → →
− −→
⎧ ⎫ ⎧−⎪ ⎪ ⎪= − = −⎨ ⎬ ⎨−⎪ ⎪ ⎪⎩ ⎭ ⎩⎧ ⎫= − = − =⎨ ⎬⎩ ⎭
11
1−
⎫⎪⎬⎪⎭
preparation. If you are a student using this Manual, you are using it without permission.
9-153
9-175 The four processes of an air-standard cycle are described. The back work ratio and its limit as rp goes to unity are to be determined, and the result is to be compared to the expression for the Otto cycle.
Analysis The work of compression for process 1-2 is found by the first law:
PROPRIETARY MATERIAL. © 2011 The McGraw-Hill Companies, Inc. Limited distribution permitted only to teachers and educators for course
2 1 2 1 2
1 2 1 2 2 1
,1 2 1 2 2 1
v
comp v
q w u
w u C T T
w w C T T
−
− −
− −
− = ∆
= −∆ = − −
= − = −
)4 1 1 4 4 14
Pdv P v v R T T− = − = − − = −∫
The work of expansion for process 3-4 is found by the first law:
3 4 4 3
exp,3 4 3 4 3 4
v
v
q w u
w u C T T
w w C T T
− − −
− −
− −
− = ∆
= −∆ = − −
= − = −
The back work ratio is
( )( )
1− −
1 2 0( )q isentropic process− =
The work of compression for process 4-1 is found by
( ) (1
,4 1compw w− = −
( )( )
3 4 3 4 3 4
3 4 0( )q isentropic process− =
3 4
( ) ( )( )
( ) ( )
( )4 1 2 1
4 1 2 1 1
exp 3 4 3 4 3
/ 1 / 1
1 /comp v v
v
R T T T Tw R T T C T T CTw C T T T T T
− + −− + −
= =− −
Using data from the previous problem and Cv = R/(k-1)
( ) ( )11
exp 31 1
( 1) 1 1
11
kpcomp
k kpr r⎝ ⎠
k r rw Tw T
−
− −
− − + −=
⎛ ⎞−⎜ ⎟⎜ ⎟
( ) ( ) ( )( ) ( )1 11 1
1 1exp 3 3
11 1
11
1exp 3
1
( 1) 1 1 1 0 1lim lim 11 11
1lim 11
p p
p
k kpcomp
r r
kk kp
kcomp
r
k
k r r k rw T Tw T T
rr r
w T rw T
r
− −
→ →
−− −
−
→
−
⎧ ⎫⎧ ⎫⎪ ⎪⎪ ⎪⎪ ⎪− − + − − + −⎪ ⎪ ⎪= =⎨ ⎬ ⎨
⎛ ⎞⎪ ⎪ ⎪ −−⎜ ⎟⎪ ⎪ ⎪⎜ ⎟ ⎩ ⎭⎝ ⎠⎪ ⎪⎩ ⎭⎧ ⎫⎪ ⎪−
= ⎨ ⎬⎪ ⎪−⎩ ⎭
⎪⎬⎪⎪
This result is the same expression for the back work ratio for the Otto cycle.
preparation. If you are a student using this Manual, you are using it without permission.
9-154
9-176 The effects of compression ratio on the net work output and the thermal efficiency of the Otto cycle for given
nalysis Using EES, the problem is solved as follows:
]
ession"
P=P[2]) v[1]/T[1]
rocess 1 to 2"
T=T[1]) eat addition"
2 to 3"
nergy(air,T=T[2]) ion"
rocess 3 to 4"
T=T[3]) n"
to 1"
nergy(air,T=T[1])-intenergy(air,T=T[4])
Eta_th=w_net/q_in_total*Convert(, %) "Thermal efficiency, in percent"
operating conditions is to be investigated.
A
"Input Data" T[1]=300 [K]
] P[1]=100 [kPa[3] = 2000 [KT
r_comp = 12
1-2 is isentropic compr"Process s[1]=entropy(air,T=T[1],P=P[1]) s[2]=s[1]
(air, s=s[2],T[2]=temperatureP[2]*v[2]/T[2]=P[1]*P[1]*v[1]=R*T[1] R=0.287 [kJ/kg-K] V[2] = V[1]/ r_comp "Conservation of energy for pq_12 - w_12 = DELTAu_12 q_12 =0"isentropic process"
12=intenergy(air,T=T[2])-intenergy(air,DELTAu_"Process 2-3 is constant volume hv[3]=v[2] s[3]=entropy(air, T=T[3], P=P[3]) P[3]*v[3]=R*T[3] "Conservation of energy for process q_23 - w_23 = DELTAu_23 w_23 =0"constant volume process"
23=intenergy(air,T=T[3])-inteDELTAu_"Process 3-4 is isentropic expanss[4]=s[3] s[4]=entropy(air,T=T[4],P=P[4]) P[4]*v[4]=R*T[4] "Conservation of energy for pq_34 -w_34 = DELTAu_34 q_34 =0"isentropic process"
=intenergy(air,T=T[4])-intenergy(air,DELTAu_34"Process 4-1 is constant volume heat rejectioV[4] = V[1] "Conservation of energy for process 4q_41 - w_41 = DELTAu_41
_41 =0 "constant volume process" wDELTAu_41=inte q_in_total=q_23 q_out_total = -q_41 w_net = w_12+w_23+w_34+w_41
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9-155
ηth[%]
rcomp wnet[kJ/kg]
45.83 6 567.4 48.67 7 589.3 51.03 8 604.9 53.02 9 616.2 54.74 10 624.3 56.24 11 630 57.57 12 633.8 58.75 13 636.3 59.83 14 637.5 60.8 15 637.9
6 7 8 9 10 11 12 13 14 15560
570
580
590
600
610
620
630
640
rcomp
wne
t [k
J/kg
]
6 7 8 9 10 11 12 13 14 1545
48.5
52
55.5
59
62.5
rcomp
ηth
[%
]
PROPRIETARY MATERIALpreparation. If you are a student using this Manual, you are using it without permission.
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9-156
9-177 The effects of pressure ratio on the net work output and the thermal efficiency of a simple Brayton cycle is to be investigated. The pressure ratios at which the net work output and the thermal efficiency are maximum are to be determined.
Analysis Using EES, the problem is solved as follows:
P_ratio = 8 T[1] = 300 [K] P[1]= 100 [kPa] T[3] = 1800 [K] m_dot = 1 [kg/s] Eta_c = 100/100 Eta_t = 100/100 "Inlet conditions" h[1]=ENTHALPY(Air,T=T[1]) s[1]=ENTROPY(Air,T=T[1],P=P[1]) "Compressor anaysis" s_s[2]=s[1] "For the ideal case the entropies are constant across the compressor" P_ratio=P[2]/P[1]"Definition of pressure ratio - to find P[2]" T_s[2]=TEMPERATURE(Air,s=s_s[2],P=P[2]) "T_s[2] is the isentropic value of T[2] at compressor exit" h_s[2]=ENTHALPY(Air,T=T_s[2]) Eta_c =(h_s[2]-h[1])/(h[2]-h[1]) "Compressor adiabatic efficiency; Eta_c = W_dot_c_ideal/W_dot_c_actual. " m_dot*h[1] +W_dot_c=m_dot*h[2] "SSSF First Law for the actual compressor, assuming: adiabatic, ke=pe=0" "External heat exchanger analysis" P[3]=P[2]"process 2-3 is SSSF constant pressure" h[3]=ENTHALPY(Air,T=T[3]) m_dot*h[2] + Q_dot_in= m_dot*h[3]"SSSF First Law for the heat exchanger, assuming W=0, ke=pe=0" "Turbine analysis" s[3]=ENTROPY(Air,T=T[3],P=P[3]) s_s[4]=s[3] "For the ideal case the entropies are constant across the turbine" P_ratio= P[3] /P[4] T_s[4]=TEMPERATURE(Air,s=s_s[4],P=P[4]) "Ts[4] is the isentropic value of T[4] at turbine exit" h_s[4]=ENTHALPY(Air,T=T_s[4]) "Eta_t = W_dot_t /Wts_dot turbine adiabatic efficiency, Wts_dot > W_dot_t" Eta_t=(h[3]-h[4])/(h[3]-h_s[4]) m_dot*h[3] = W_dot_t + m_dot*h[4] "SSSF First Law for the actual compressor, assuming: adiabatic, ke=pe=0" "Cycle analysis" W_dot_net=W_dot_t-W_dot_c"Definition of the net cycle work, kW" Eta=W_dot_net/Q_dot_in"Cycle thermal efficiency" Bwr=W_dot_c/W_dot_t "Back work ratio" "The following state points are determined only to produce a T-s plot" T[2]=temperature(air,h=h[2]) T[4]=temperature(air,h=h[4]) s[2]=entropy(air,T=T[2],P=P[2]) s[4]=entropy(air,T=T[4],P=P[4])
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9-157
Bwr η Pratio Wc
[kW] Wnet[kW]
Wt[kW]
Qin[kW]
0.254 0.3383 5 175.8 516.3 692.1 1526 0.2665 0.3689 6 201.2 553.7 754.9 1501 0.2776 0.3938 7 223.7 582.2 805.9 1478 0.2876 0.4146 8 244.1 604.5 848.5 1458 0.2968 0.4324 9 262.6 622.4 885 1439 0.3052 0.4478 10 279.7 637 916.7 1422 0.313 0.4615 11 295.7 649 944.7 1406 0.3203 0.4736 12 310.6 659.1 969.6 1392 0.3272 0.4846 13 324.6 667.5 992.1 1378 0.3337 0.4945 14 337.8 674.7 1013 1364 0.3398 0.5036 15 350.4 680.8 1031 1352 0.3457 0.512 16 362.4 685.9 1048 1340 0.3513 0.5197 17 373.9 690.3 1064 1328 0.3567 0.5269 18 384.8 694.1 1079 1317 0.3618 0.5336 19 395.4 697.3 1093 1307 0.3668 0.5399 20 405.5 700 1106 1297 0.3716 0.5458 21 415.3 702.3 1118 1287 0.3762 0.5513 22 424.7 704.3 1129 1277 0.3806 0.5566 23 433.8 705.9 1140 1268 0.385 0.5615 24 442.7 707.2 1150 1259 0.3892 0.5663 25 451.2 708.3 1160 1251 0.3932 0.5707 26 459.6 709.2 1169 1243 0.3972 0.575 27 467.7 709.8 1177 1234 0.401 0.5791 28 475.5 710.3 1186 1227 0.4048 0.583 29 483.2 710.6 1194 1219 0.4084 0.5867 30 490.7 710.7 1201 1211 0.412 0.5903 31 498 710.8 1209 1204 0.4155 0.5937 32 505.1 710.7 1216 1197 0.4189 0.597 33 512.1 710.4 1223 1190 0.4222 0.6002 34 518.9 710.1 1229 1183
5 10 15 20 25 30 35500
525
550
575
600
625
650
675
700
725
0.3
0.35
0.4
0.45
0.5
0.55
0.6
0.65
Pratio
Wne
t [k
W]
η th
PROPRIETARY MATERIALpreparation. If you are a student using this Manual, you are using it without permission.
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9-158
9-178 The effects of pressure ratio on the net work output and the thermal efficiency of a simple Brayton cycle is to be investigated assuming adiabatic efficiencies of 85 percent for both the turbine and the compressor. The pressure ratios at which the net work output and the thermal efficiency are maximum are to be determined.
Analysis Using EES, the problem is solved as follows:
P_ratio = 8 T[1] = 300 [K] P[1]= 100 [kPa] T[3] = 1800 [K] m_dot = 1 [kg/s] Eta_c = 85/100 Eta_t = 85/100 "Inlet conditions" h[1]=ENTHALPY(Air,T=T[1]) s[1]=ENTROPY(Air,T=T[1],P=P[1]) "Compressor anaysis" s_s[2]=s[1] "For the ideal case the entropies are constant across the compressor" P_ratio=P[2]/P[1]"Definition of pressure ratio - to find P[2]" T_s[2]=TEMPERATURE(Air,s=s_s[2],P=P[2]) "T_s[2] is the isentropic value of T[2] at compressor exit" h_s[2]=ENTHALPY(Air,T=T_s[2]) Eta_c =(h_s[2]-h[1])/(h[2]-h[1]) "Compressor adiabatic efficiency; Eta_c = W_dot_c_ideal/W_dot_c_actual. " m_dot*h[1] +W_dot_c=m_dot*h[2] "SSSF First Law for the actual compressor, assuming: adiabatic, ke=pe=0" "External heat exchanger analysis" P[3]=P[2]"process 2-3 is SSSF constant pressure" h[3]=ENTHALPY(Air,T=T[3]) m_dot*h[2] + Q_dot_in= m_dot*h[3]"SSSF First Law for the heat exchanger, assuming W=0, ke=pe=0" "Turbine analysis" s[3]=ENTROPY(Air,T=T[3],P=P[3]) s_s[4]=s[3] "For the ideal case the entropies are constant across the turbine" P_ratio= P[3] /P[4] T_s[4]=TEMPERATURE(Air,s=s_s[4],P=P[4]) "Ts[4] is the isentropic value of T[4] at turbine exit" h_s[4]=ENTHALPY(Air,T=T_s[4]) "Eta_t = W_dot_t /Wts_dot turbine adiabatic efficiency, Wts_dot > W_dot_t" Eta_t=(h[3]-h[4])/(h[3]-h_s[4]) m_dot*h[3] = W_dot_t + m_dot*h[4] "SSSF First Law for the actual compressor, assuming: adiabatic, ke=pe=0" "Cycle analysis" W_dot_net=W_dot_t-W_dot_c"Definition of the net cycle work, kW" Eta=W_dot_net/Q_dot_in"Cycle thermal efficiency" Bwr=W_dot_c/W_dot_t "Back work ratio" "The following state points are determined only to produce a T-s plot" T[2]=temperature(air,h=h[2]) T[4]=temperature(air,h=h[4]) s[2]=entropy(air,T=T[2],P=P[2]) s[4]=entropy(air,T=T[4],P=P[4])
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9-159
Bwr η Pratio Wc
[kW] Wnet[kW]
Wt[kW]
Qin[kW]
0.3515 0.2551 5 206.8 381.5 588.3 1495 0.3689 0.2764 6 236.7 405 641.7 1465 0.3843 0.2931 7 263.2 421.8 685 1439 0.3981 0.3068 8 287.1 434.1 721.3 1415 0.4107 0.3182 9 309 443.3 752.2 1393 0.4224 0.3278 10 329.1 450.1 779.2 1373 0.4332 0.3361 11 347.8 455.1 803 1354 0.4433 0.3432 12 365.4 458.8 824.2 1337 0.4528 0.3495 13 381.9 461.4 843.3 1320 0.4618 0.355 14 397.5 463.2 860.6 1305 0.4704 0.3599 15 412.3 464.2 876.5 1290 0.4785 0.3643 16 426.4 464.7 891.1 1276 0.4862 0.3682 17 439.8 464.7 904.6 1262 0.4937 0.3717 18 452.7 464.4 917.1 1249 0.5008 0.3748 19 465.1 463.6 928.8 1237 0.5077 0.3777 20 477.1 462.6 939.7 1225 0.5143 0.3802 21 488.6 461.4 950 1214 0.5207 0.3825 22 499.7 460 959.6 1202 0.5268 0.3846 23 510.4 458.4 968.8 1192 0.5328 0.3865 24 520.8 456.6 977.4 1181
5 9 13 17 21 25380
390
400
410
420
430
440
450
460
470
0.24
0.26
0.28
0.3
0.32
0.34
0.36
0.38
0.4
Pratio
Wne
t [k
W]
η thPratio for
W net,max
W net η th
PROPRIETARY MATERIALpreparation. If you are a student using this Manual, you are using it without permission.
. © 2011 The McGraw-Hill Companies, Inc. Limited distribution permitted only to teachers and educators for course
9-160
9-179 The effects of pressure ratio, maximum cycle temperature, and compressor and turbine inefficiencies on the net work output per unit mass and the thermal efficiency of a simple Brayton cycle with air as the working fluid is to be investigated. Constant specific heats at room temperature are to be used.
Analysis Using EES, the problem is solved as follows:
Procedure ConstPropResult(T[1],P[1],r_comp,T[3]:Eta_th_ConstProp,Eta_th_easy) "For Air:" C_V = 0.718 [kJ/kg-K] k = 1.4 T2 = T[1]*r_comp^(k-1) P2 = P[1]*r_comp^k q_in_23 = C_V*(T[3]-T2) T4 = T[3]*(1/r_comp)^(k-1) q_out_41 = C_V*(T4-T[1]) Eta_th_ConstProp = (1-q_out_41/q_in_23)*Convert(, %) "[%]" "The Easy Way to calculate the constant property Otto cycle efficiency is:" Eta_th_easy = (1 - 1/r_comp^(k-1))*Convert(, %) "[%]" END "Input Data" T[1]=300 [K] P[1]=100 [kPa] {T[3] = 1000 [K]} r_comp = 12 "Process 1-2 is isentropic compression" s[1]=entropy(air,T=T[1],P=P[1]) s[2]=s[1] T[2]=temperature(air, s=s[2], P=P[2]) P[2]*v[2]/T[2]=P[1]*v[1]/T[1] P[1]*v[1]=R*T[1] R=0.287 [kJ/kg-K] V[2] = V[1]/ r_comp "Conservation of energy for process 1 to 2" q_12 - w_12 = DELTAu_12 q_12 =0"isentropic process" DELTAu_12=intenergy(air,T=T[2])-intenergy(air,T=T[1]) "Process 2-3 is constant volume heat addition" v[3]=v[2] s[3]=entropy(air, T=T[3], P=P[3]) P[3]*v[3]=R*T[3] "Conservation of energy for process 2 to 3" q_23 - w_23 = DELTAu_23 w_23 =0"constant volume process" DELTAu_23=intenergy(air,T=T[3])-intenergy(air,T=T[2]) "Process 3-4 is isentropic expansion" s[4]=s[3] s[4]=entropy(air,T=T[4],P=P[4]) P[4]*v[4]=R*T[4] "Conservation of energy for process 3 to 4" q_34 -w_34 = DELTAu_34 q_34 =0"isentropic process" DELTAu_34=intenergy(air,T=T[4])-intenergy(air,T=T[3]) "Process 4-1 is constant volume heat rejection" V[4] = V[1] "Conservation of energy for process 4 to 1" q_41 - w_41 = DELTAu_41
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9-161
w_41 =0 "constant volume process" DELTAu_41=intenergy(air,T=T[1])-intenergy(air,T=T[4]) q_in_total=q_23 q_out_total = -q_41 w_net = w_12+w_23+w_34+w_41 Eta_th=w_net/q_in_total*Convert(, %) "Thermal efficiency, in percent" Call ConstPropResult(T[1],P[1],r_comp,T[3]:Eta_th_ConstProp,Eta_th_easy) PerCentError = ABS(Eta_th - Eta_th_ConstProp)/Eta_th*Convert(, %) "[%]"
PerCentError [%]
rcomp ηth[%]
ηth,ConstProp[%]
ηth,easy[%]
T3[K]
3.604 12 60.8 62.99 62.99 1000 6.681 12 59.04 62.99 62.99 1500 9.421 12 57.57 62.99 62.99 2000 11.64 12 56.42 62.99 62.99 2500
6 7 8 9 10 11 123.6
3.7
3.8
3.9
4
4.1
4.2
4.3
rcom p
Per
Cen
tErr
or [
%]
P ercent E rror = |η th - η th ,ConstProp | / η th
Tm ax = 1000 K
1000 1200 1400 1600 1800 2000 2200 2400 26004
6.2
8.4
10.6
12.8
15
T[3] [K]
Per
Cen
tErr
or [
%]
rcomp = 6
=12
PROPRIETARY MATERIALpreparation. If you are a student using this Manual, you are using it without permission.
. © 2011 The McGraw-Hill Companies, Inc. Limited distribution permitted only to teachers and educators for course
9-162
9-180 The effects of pressure ratio, maximum cycle temperature, and compressor and turbine efficiencies on the net work output per unit mass and the thermal efficiency of a simple Brayton cycle with air as the working fluid is
.
nalysis Using EES, the problem is solved as follows:
m diagram window"
P[1])
for the actual compressor, assuming: adiabatic, ke=pe=0"
ot_t + m_dot*h[4] "SSSF First Law for the actual compressor, assuming: adiabatic, ke=pe=0"
ycle work, kW"
determined only to produce a T-s plot"
s[4]=entropy(air,T=T[4],P=P[4])
to be investigated. Variable specific heats are to be used
A
"Input data - fro{P_ratio = 8}
{T[1] = 300 [K]P[1]= 100 [kPa] T[3] = 800 [K] m_dot = 1 [kg/s]
ta_c = 75/100 EEta_t = 82/100} "Inlet conditions" [1]=ENTHALPY(Air,T=T[1]) h
s[1]=ENTROPY(Air,T=T[1],P= "Compressor anaysis" s_s[2]=s[1] "For the ideal case the entropies are constant across the compressor"
sure ratio - to find P[2]" P_ratio=P[2]/P[1]"Definition of presT_s[2]=TEMPERATURE(Air,s=s_s[2],P=P[2]) "T_s[2] is the isentropic value of T[2] at compressor exit" h_s[2]=ENTHALPY(Air,T=T_s[2])
mpressor adiabatic efficiency; Eta_c = W_dot_c_ideal/W_dot_c_actual. " Eta_c =(h_s[2]-h[1])/(h[2]-h[1]) "Com_dot*h[1] +W_dot_c=m_dot*h[2] "SSSF First Law
lysis" "External heat exchanger anaP[3]=P[2]"process 2-3 is SSSF constant pressure"
ir,T=T[3]) h[3]=ENTHALPY(Am_dot*h[2] + Q_dot_in= m_dot*h[3]"SSSF First Law for the heat exchanger, assuming W=0, ke=pe=0""Turbine analysis"
,T=T[3],P=P[3]) s[3]=ENTROPY(Airs_s[4]=s[3] "For the ideal case the entropies are constant across the turbine" P_ratio= P[3] /P[4]
=s_s[4],P=P[4]) "Ts[4] is the isentropic value of T[4] at turbine exit" T_s[4]=TEMPERATURE(Air,sh_s[4]=ENTHALPY(Air,T=T_s[4]) "Eta_t = W_dot_t /Wts_dot turbine adiabatic efficiency, Wts_dot > W_dot_t"
ta_t=(h[3]-h[4])/(h[3]-h_s[4]) Em_dot*h[3] = W_d "Cycle analysis"
f the net cW_dot_net=W_dot_t-W_dot_c"Definition oEta=W_dot_net/Q_dot_in"Cycle thermal efficiency"
work ratio" Bwr=W_dot_c/W_dot_t "Back "The following state points are T[2]=temperature('air',h=h[2]) T[4]=temperature('air',h=h[4]) s[2]=entropy(air,T=T[2],P=P[2])
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9-163
Bwr η Pratio Wc
[kW] Wnet[kW]
Wt[kW]
Qin[kW]
0.5229 0.1 2 1818 1659 3477 16587 0.6305 0.1644 4 4033 2364 6396 14373 0.7038 0.1814 6 5543 2333 7876 12862 0.7611 0.1806 8 6723 2110 8833 11682 0.8088 0.1702 10 7705 1822 9527 10700
0.85 0.1533 12 8553 1510 10063 9852 0.8864 0.131 14 9304 1192 10496 9102 0.9192 0.1041 16 9980 877.2 10857 8426 0.9491 0.07272 18 10596 567.9 11164 7809 0.9767 0.03675 20 11165 266.1 11431 7241
5.0 5.5 6.0 6.5 7.0 7.50
500
1000
1500
s [kJ/kg-K]
T [K
]
100 kPa
800 kPa
1
2s
2
3
4
4s
Air Standard Brayton Cycle
Pressure ratio = 8 and Tmax = 1160K
2 4 6 8 10 12 14 16 18 200.00
0.05
0.10
0.15
0.20
0.25
0
500
1000
1500
2000
2500
Pratio
Cyc
le e
ffic
ienc
y,
Wne
t [k
W]
η Wnet
Tmax=1160 K
Note Pratio for maximum work and η
ηc = 0.75
ηt = 0.82
η
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9-164
9-181 The effects of pressure ratio, maximum cycle temperature, and compressor and turbine efficiencies on the net work output per unit mass and the thermal efficiency of a simple Brayton cycle with helium as the working fluid is to be investigated.
Analysis Using EES, the problem is solved as follows:
Function hFunc(WorkFluid$,T,P) "The EES functions teat helium as a real gas; thus, T and P are needed for helium's enthalpy." IF WorkFluid$ = 'Air' then hFunc:=enthalpy(Air,T=T) ELSE hFunc: = enthalpy(Helium,T=T,P=P) endif END Procedure EtaCheck(Eta_th:EtaError$) If Eta_th < 0 then EtaError$ = 'Why are the net work done and efficiency < 0?' Else EtaError$ = '' END "Input data - from diagram window" {P_ratio = 8} {T[1] = 300 [K] P[1]= 100 [kPa] T[3] = 800 [K] m_dot = 1 [kg/s] Eta_c = 0.8 Eta_t = 0.8 WorkFluid$ = 'Helium'} "Inlet conditions" h[1]=hFunc(WorkFluid$,T[1],P[1]) s[1]=ENTROPY(WorkFluid$,T=T[1],P=P[1]) "Compressor anaysis" s_s[2]=s[1] "For the ideal case the entropies are constant across the compressor" P_ratio=P[2]/P[1]"Definition of pressure ratio - to find P[2]" T_s[2]=TEMPERATURE(WorkFluid$,s=s_s[2],P=P[2]) "T_s[2] is the isentropic value of T[2] at compressor exit" h_s[2]=hFunc(WorkFluid$,T_s[2],P[2]) Eta_c =(h_s[2]-h[1])/(h[2]-h[1]) "Compressor adiabatic efficiency; Eta_c = W_dot_c_ideal/W_dot_c_actual. " m_dot*h[1] +W_dot_c=m_dot*h[2] "SSSF First Law for the actual compressor, assuming: adiabatic, ke=pe=0" "External heat exchanger analysis" P[3]=P[2]"process 2-3 is SSSF constant pressure" h[3]=hFunc(WorkFluid$,T[3],P[3]) m_dot*h[2] + Q_dot_in= m_dot*h[3]"SSSF First Law for the heat exchanger, assuming W=0, ke=pe=0" "Turbine analysis" s[3]=ENTROPY(WorkFluid$,T=T[3],P=P[3]) s_s[4]=s[3] "For the ideal case the entropies are constant across the turbine" P_ratio= P[3] /P[4] T_s[4]=TEMPERATURE(WorkFluid$,s=s_s[4],P=P[4]) "Ts[4] is the isentropic value of T[4] at turbine exit" h_s[4]=hFunc(WorkFluid$,T_s[4],P[4]) "Eta_t = W_dot_t /Wts_dot turbine adiabatic efficiency, Wts_dot > W_dot_t" Eta_t=(h[3]-h[4])/(h[3]-h_s[4]) m_dot*h[3] = W_dot_t + m_dot*h[4] "SSSF First Law for the actual compressor, assuming: adiabatic, ke=pe=0" "Cycle analysis" W_dot_net=W_dot_t-W_dot_c"Definition of the net cycle work, kW" Eta_th=W_dot_net/Q_dot_in"Cycle thermal efficiency" Call EtaCheck(Eta_th:EtaError$) Bwr=W_dot_c/W_dot_t "Back work ratio" "The following state points are determined only to produce a T-s plot" T[2]=temperature(air,h=h[2]) T[4]=temperature(air,h=h[4]) s[2]=entropy(air,T=T[2],P=P[2]) s[4]=entropy(air,T=T[4],P=P[4])
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9-165
Bwr η Pratio Wc
[kW] Wnet
[kW] Wt
[kW] Qin
[kW] 0.5229 0.1 2 1818 1659 3477 16587 0.6305 0.1644 4 4033 2364 6396 14373 0.7038 0.1814 6 5543 2333 7876 12862 0.7611 0.1806 8 6723 2110 8833 11682 0.8088 0.1702 10 7705 1822 9527 10700
0.85 0.1533 12 8553 1510 10063 9852 0.8864 0.131 14 9304 1192 10496 9102 0.9192 0.1041 16 9980 877.2 10857 8426 0.9491 0.07272 18 10596 567.9 11164 7809 0.9767 0.03675 20 11165 266.1 11431 7241
5.0 5 .5 6 .0 6 .5 7 .0 7 .50
500
1000
1500
s [kJ /kg -K ]
T [K
]
100 kP a
800 kP a
1
2 s
2
3
4
4 s
B rayto n C yc le
P ressu re ra tio = 8 an d T m ax = 1160K
2 4 6 8 10 12 14 16 18 200.00
0.05
0.10
0.15
0.20
0.25
0
500
1000
1500
2000
2500
Pratio
Cyc
le e
ffici
ency
,
Wne
t [k
W]
ηWnet
Tmax=1160 K
Note Pratio for maximum work and η
η c = 0.75η t = 0.82
η
Brayton Cycle using Airm air = 20 kg/s
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9-166
9-182 The effect of the number of compression and expansion stages on the thermal efficiency of an iregenerative B
deal rayton cycle with multistage compression and expansion and air as the working fluid is to be
nalysis Using EES, the problem is solved as follows:
umber of compression and expansion stages"
cy" iency"
ysis"
r for the isentropic case: dy-flow"
)
the same pressure ssor is the same. The total compressor work is:"
tween turbines is" ges - 1)*C_P*(T_8 - T_7)
to occur until:"
ming: adiabatic, ke=pe=0 steady-flow"
7s)
tages*w_turb
" mp/w_turb "Back work ratio"
investigated.
A
"Input data for air" C_P = 1.005 [kJ/kg-K] k = 1.4 "Nstages is the n
Nstages = 1T_6 = 1200 [K]Pratio = 12 T_1 = 300 [K] P_1= 100 [kPa]
Eta_reg = 1.0 "regenerator effectiveness"entorpic efficienEta_c =1.0 "Compressor is
Eta_t =1.0 "Turbine isentropic efficR_p = Pratio^(1/Nstages)
ressor ana"Isentropic CompT_2s = T_1*R_p^((k-1)/k) P_2 = R_p*P_1 "T_2s is the isentropic value of T_2 at compressor exit"
ta_c = w_compisen/w_comp E"compressor adiabatic efficiency, W_comp > W_compisen"
ompresso"Conservation of energy for the cteae_in - e_out = DELTAe=0 for s
_1 w_compisen = C_P*(T_2s-TActual compressor analysis:""
w_comp = C_P*(T_2 - T_1)
ccur such that T_3 = T_1 and the compressors have"Since intercooling is assumed to oratio, the work input to each comprew_comp_total = Nstages*w_comp "External heat exchanger analysis" "SSSF First Law for the heat exchanger, assuming W=0, ke=pe=0 e_in - e_out =DELTAe_cv =0 for steady flow"
t exchanger + the reheat be"The heat added in the external hea C_P*(T_6 - T_5) +(Nstaq_in_total =
"Reheat is assumedT_8 = T_6 "Turbine analysis" P_7 = P_6 /R_p "T_7s is the isentropic value of T_7 at turbine exit"
) T_7s = T_6*(1/R_p)^((k-1)/k"Turbine adiabatic efficiency, w_turbisen > w_turb" Eta_t = w_turb /w_turbisen
c turbine, assu"SSSF First Law for the isentropi= 0 fore_in -e_out = DELTAe_cv
w_turbisen = C_P*(T_6 - T_"Actual Turbine analysis:"
_turb = C_P*(T_6 - T_7) ww_turb_total = Ns "Cycle analysis"
turb_total-w_comp_total "[kJ/kg]w_net=w_Bwr=w_co
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9-167
P_4=P_2 P_5=P_4 P_6=P_5
PROPRIETARY MATERIAL. © 2011 The McGraw-Hill Companies, Inc. Limited distribution permitted only to teachers and educators for course
rator effectiveness gives T_5 as:"
as:"
ycle thermal efficiency with regenerator"
mperatures, T_6 and T_1 for this problem." ta_th_Ericsson = (1 - T_1/T_6)*Convert(, %) "[%]"
ηth,Ericksson ηth,Regenerative Nstages
T_4 = T_2 "The regeneEta_reg = (T_5 - T_4)/(T_9 - T_4) T_9 = T_7 "Energy balance on regenerator gives T_10T_4 + T_9=T_5 + T_10 "CEta_th_regenerative=w_net/q_in_total*Convert(, %) "[%]" "The efficiency of the Ericsson cycle is the same as the Carnot cycle operating between the same max and min teE
[%] [%] 75 49.15 1 75 64.35 2 75 68.32 3 75 70.14 4 75 72.33 7 75 73.79 15 75 74.05 19 75 74.18 22
0 2 4 6 8 10 12 14 16 18 20 22 2440
50
60
70
80
Nstages
ηth
[%]
Ericsson
Ideal Regenerative Brayton
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9-168
9-183 The effect of the number of compression and expansion stages on the thermal efficiency of an idealregenerative B
rayton cycle with multistage compression and expansion and helium as the working fluid is to be
nalysis Using EES, the problem is solved as follows:
umber of compression and expansion stages"
cy" iency"
ysis"
r for the isentropic case: dy-flow"
)
the same pressure ssor is the same. The total compressor work is:"
ing W=0, ke=pe=0
tween turbines is" ges - 1)*C_P*(T_8 - T_7)
to occur until:"
ming: adiabatic, ke=pe=0 steady-flow"
_7s)
tages*w_turb
investigated.
A
"Input data for Helium" C_P = 5.1926 [kJ/kg-K] k = 1.667 "Nstages is the n
1} {Nstages = T_6 = 1200 [K]Pratio = 12 T_1 = 300 [K] P_1= 100 [kPa]
Eta_reg = 1.0 "regenerator effectiveness"entorpic efficienEta_c =1.0 "Compressor is
Eta_t =1.0 "Turbine isentropic effic_p = Pratio^(1/Nstages) R
"Isentropic Compressor ana T_2s = T_1*R_p^((k-1)/k) P_2 = R_p*P_1 "T_2s is the isentropic value of T_2 at compressor exit" Eta_c = w_compisen/w_comp
> W_compisen" "compressor adiabatic efficiency, W_compompresso"Conservation of energy for the c
teae_in - e_out = DELTAe=0 for s_1 w_compisen = C_P*(T_2s-T
"Actual compressor analysis:" w_comp = C_P*(T_2 - T_1)
ccur such that T_3 = T_1 and the compressors have"Since intercooling is assumed to oratio, the work input to each comprew_comp_total = Nstages*w_comp "External heat exchanger analysis" SSSF First Law for the heat exchanger, assum"
e_in - e_out =DELTAe_cv =0 for steady flow"
t exchanger + the reheat be"The heat added in the external hea C_P*(T_6 - T_5) +(Nstaq_in_total =
"Reheat is assumedT_8 = T_6 "Turbine analysis" P_7 = P_6 /R_p "T_7s is the isentropic value of T_7 at turbine exit"
) T_7s = T_6*(1/R_p)^((k-1)/k"Turbine adiabatic efficiency, w_turbisen > w_turb" Eta_t = w_turb /w_turbisen
c turbine, assu"SSSF First Law for the isentropi_in -e_out = DELTAe_cv = 0 fore
w_turbisen = C_P*(T_6 - T "Actual Turbine analysis:"
_turb = C_P*(T_6 - T_7) ww_turb_total = Ns "Cycle analysis"
_net=w_turb_total-w_comp_total w
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9-169
PROPRIETARY MATERIAL. © 2011 The McGraw-Hill Companies, Inc. Limited distribution permitted only to teachers and educators for course
mp/w_turb "Back work ratio"
rator effectiveness gives T_5 as:"
nergy balance on regenerator gives T_10 as:"
ycle thermal efficiency with regenerator"
mperatures, T_6 and T_1 for this problem." ta_th_Ericsson = (1 - T_1/T_6)*Convert(, %) "[%]"
ηth,Ericksson ηth,Regenerative Nstages
Bwr=w_co P_4=P_2 P_5=P_4 P_6=P_5 T_4 = T_2 "The regeneEta_reg = (T_5 - T_4)/(T_9 - T_4) T_9 = T_7 "ET_4 + T_9=T_5 + T_10 "CEta_th_regenerative=w_net/q_in_total*Convert(, %) "[%]" "The efficiency of the Ericsson cycle is the same as the Carnot cycle operating between the same max and min teE
[%] [%] 75 32.43 1 75 58.9 2 75 65.18 3 75 67.95 4 75 71.18 7 75 73.29 15 75 73.66 19 75 73.84 22
0 2 4 6 8 10 12 14 16 18 20 22 2430
40
50
60
70
80
Nstages
ηth
[%]
EricssonIdeal Regenerative Brayton
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9-170
Fundamentals of Engineering (FE) Exam Problems
9-184 An Otto cycle with air as the working fluid has a compression ratio of 10.4. Under cold air standard conditions, the thermal efficiency of this cycle is
(a) 10% (b) 39% (c) 61% (d) 79% (e) 82%
Answer (c) 61%
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).
r=10.4 k=1.4 Eta_Otto=1-1/r^(k-1) "Some Wrong Solutions with Common Mistakes:" W1_Eta = 1/r "Taking efficiency to be 1/r" W2_Eta = 1/r^(k-1) "Using incorrect relation" W3_Eta = 1-1/r^(k1-1); k1=1.667 "Using wrong k value"
9-185 For specified limits for the maximum and minimum temperatures, the ideal cycle with the lowest thermal efficiency is
(a) Carnot (b) Stirling (c) Ericsson (d) Otto (e) All are the same
Answer (d) Otto
9-186 A Carnot cycle operates between the temperatures limits of 300 K and 2000 K, and produces 600 kW of net power. The rate of entropy change of the working fluid during the heat addition process is
(a) 0 (b) 0.300 kW/K (c) 0.353 kW/K (d) 0.261 kW/K (e) 2.0 kW/K
Answer (c) 0.353 kW/K
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).
TL=300 "K" TH=2000 "K" Wnet=600 "kJ/s" Wnet= (TH-TL)*DS "Some Wrong Solutions with Common Mistakes:" W1_DS = Wnet/TH "Using TH instead of TH-TL" W2_DS = Wnet/TL "Using TL instead of TH-TL" W3_DS = Wnet/(TH+TL) "Using TH+TL instead of TH-TL"
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9-171
9-187 Air in an ideal Diesel cycle is compressed from 2 L to 0.13 L, and then it expands during the constant pressure heat addition process to 0.30 L. Under cold air standard conditions, the thermal efficiency of this cycle is
(a) 41% (b) 59% (c) 66% (d) 70% (e) 78%
Answer (b) 59%
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).
V1=2 "L" V2= 0.13 "L" V3= 0.30 "L" r=V1/V2 rc=V3/V2 k=1.4 Eta_Diesel=1-(1/r^(k-1))*(rc^k-1)/k/(rc-1) "Some Wrong Solutions with Common Mistakes:" W1_Eta = 1-(1/r1^(k-1))*(rc^k-1)/k/(rc-1); r1=V1/V3 "Wrong r value" W2_Eta = 1-Eta_Diesel "Using incorrect relation" W3_Eta = 1-(1/r^(k1-1))*(rc^k1-1)/k1/(rc-1); k1=1.667 "Using wrong k value" W4_Eta = 1-1/r^(k-1) "Using Otto cycle efficiency"
9-188 Helium gas in an ideal Otto cycle is compressed from 20°C and 2.5 L to 0.25 L, and its temperature increases by an additional 700°C during the heat addition process. The temperature of helium before the expansion process is
(a) 1790°C (b) 2060°C (c) 1240°C (d) 620°C (e) 820°C
Answer (a) 1790°C
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).
k=1.667 V1=2.5 V2=0.25 r=V1/V2 T1=20+273 "K" T2=T1*r^(k-1) T3=T2+700-273 "C" "Some Wrong Solutions with Common Mistakes:" W1_T3 =T22+700-273; T22=T1*r^(k1-1); k1=1.4 "Using wrong k value" W2_T3 = T3+273 "Using K instead of C" W3_T3 = T1+700-273 "Disregarding temp rise during compression" W4_T3 = T222+700-273; T222=(T1-273)*r^(k-1) "Using C for T1 instead of K"
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9-172
9-189 In an ideal Otto cycle, air is compressed from 1.20 kg/m3 and 2.2 L to 0.26 L, and the net work output of the cycle is 440 kJ/kg. The mean effective pressure (MEP) for this cycle is
(a) 612 kPa (b) 599 kPa (c) 528 kPa (d) 416 kPa (e) 367 kPa
Answer (b) 599 kPa
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).
rho1=1.20 "kg/m^3" k=1.4 V1=2.2 V2=0.26 m=rho1*V1/1000 "kg" w_net=440 "kJ/kg" Wtotal=m*w_net MEP=Wtotal/((V1-V2)/1000) "Some Wrong Solutions with Common Mistakes:" W1_MEP = w_net/((V1-V2)/1000) "Disregarding mass" W2_MEP = Wtotal/(V1/1000) "Using V1 instead of V1-V2" W3_MEP = (rho1*V2/1000)*w_net/((V1-V2)/1000); "Finding mass using V2 instead of V1" W4_MEP = Wtotal/((V1+V2)/1000) "Adding V1 and V2 instead of subtracting"
9-190 In an ideal Brayton cycle, air is compressed from 95 kPa and 25°C to 1100 kPa. Under cold air standard conditions, the thermal efficiency of this cycle is
(a) 45% (b) 50% (c) 62% (d) 73% (e) 86%
Answer (b) 50%
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=95 "kPa" P2=1100 "kPa" T1=25+273 "K" rp=P2/P1 k=1.4 Eta_Brayton=1-1/rp^((k-1)/k) "Some Wrong Solutions with Common Mistakes:" W1_Eta = 1/rp "Taking efficiency to be 1/rp" W2_Eta = 1/rp^((k-1)/k) "Using incorrect relation" W3_Eta = 1-1/rp^((k1-1)/k1); k1=1.667 "Using wrong k value"
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9-173
9-191 Consider an ideal Brayton cycle executed between the pressure limits of 1200 kPa and 100 kPa and temperature limits of 20°C and 1000°C with argon as the working fluid. The net work output of the cycle is
(a) 68 kJ/kg (b) 93 kJ/kg (c) 158 kJ/kg (d) 186 kJ/kg (e) 310 kJ/kg
Answer (c) 158 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=100 "kPa" P2=1200 "kPa" T1=20+273 "K" T3=1000+273 "K" rp=P2/P1 k=1.667 Cp=0.5203 "kJ/kg.K" Cv=0.3122 "kJ/kg.K" T2=T1*rp^((k-1)/k) q_in=Cp*(T3-T2) Eta_Brayton=1-1/rp^((k-1)/k) w_net=Eta_Brayton*q_in "Some Wrong Solutions with Common Mistakes:" W1_wnet = (1-1/rp^((k-1)/k))*qin1; qin1=Cv*(T3-T2) "Using Cv instead of Cp" W2_wnet = (1-1/rp^((k-1)/k))*qin2; qin2=1.005*(T3-T2) "Using Cp of air instead of argon" W3_wnet = (1-1/rp^((k1-1)/k1))*Cp*(T3-T22); T22=T1*rp^((k1-1)/k1); k1=1.4 "Using k of air instead of argon" W4_wnet = (1-1/rp^((k-1)/k))*Cp*(T3-T222); T222=(T1-273)*rp^((k-1)/k) "Using C for T1 instead of K"
9-192 An ideal Brayton cycle has a net work output of 150 kJ/kg and a backwork ratio of 0.4. If both the turbine and the compressor had an isentropic efficiency of 85%, the net work output of the cycle would be
(a) 74 kJ/kg (b) 95 kJ/kg (c) 109 kJ/kg (d) 128 kJ/kg (e) 177 kJ/kg
Answer (b) 95 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).
wcomp/wturb=0.4 wturb-wcomp=150 "kJ/kg" Eff=0.85 w_net=Eff*wturb-wcomp/Eff "Some Wrong Solutions with Common Mistakes:" W1_wnet = Eff*wturb-wcomp*Eff "Making a mistake in Wnet relation" W2_wnet = (wturb-wcomp)/Eff "Using a wrong relation" W3_wnet = wturb/eff-wcomp*Eff "Using a wrong relation"
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9-174
9-193 In an ideal Brayton cycle, air is compressed from 100 kPa and 25°C to 1 MPa, and then heated to 927°C before entering the turbine. Under cold air standard conditions, the air temperature at the turbine exit is
(a) 349°C (b) 426°C (c) 622°C (d) 733°C (e) 825°C
Answer (a) 349°C
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=100 "kPa" P2=1000 "kPa" T1=25+273 "K" T3=900+273 "K" rp=P2/P1 k=1.4 T4=T3*(1/rp)^((k-1)/k)-273 "Some Wrong Solutions with Common Mistakes:" W1_T4 = T3/rp "Using wrong relation" W2_T4 = (T3-273)/rp "Using wrong relation" W3_T4 = T4+273 "Using K instead of C" W4_T4 = T1+800-273 "Disregarding temp rise during compression"
9-194 In an ideal Brayton cycle with regeneration, argon gas is compressed from 100 kPa and 25°C to 400 kPa, and then heated to 1200°C before entering the turbine. The highest temperature that argon can be heated in the regenerator is
(a) 246°C (b) 846°C (c) 689°C (d) 368°C (e) 573°C
Answer (e) 573°C
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). k=1.667 Cp=0.5203 "kJ/kg.K" P1=100 "kPa" P2=400 "kPa" T1=25+273 "K" T3=1200+273 "K" "The highest temperature that argon can be heated in the regenerator is the turbine exit temperature," rp=P2/P1 T2=T1*rp^((k-1)/k) T4=T3/rp^((k-1)/k)-273 "Some Wrong Solutions with Common Mistakes:" W1_T4 = T3/rp "Using wrong relation" W2_T4 = (T3-273)/rp^((k-1)/k) "Using C instead of K for T3" W3_T4 = T4+273 "Using K instead of C" W4_T4 = T2-273 "Taking compressor exit temp as the answer"
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9-195 In an ideal Brayton cycle with regeneration, air is compressed from 80 kPa and 10°C to 400 kPa and 175°C, is heated to 450°C in the regenerator, and then further heated to 1000°C before entering the turbine. Under cold air standard conditions, the effectiveness of the regenerator is
(a) 33% (b) 44% (c) 62% (d) 77% (e) 89%
Answer (d) 77%
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).
k=1.4 Cp=1.005 "kJ/kg.K" P1=80 "kPa" P2=400 "kPa" T1=10+273 "K" T2=175+273 "K" T3=1000+273 "K" T5=450+273 "K" "The highest temperature that the gas can be heated in the regenerator is the turbine exit temperature," rp=P2/P1 T2check=T1*rp^((k-1)/k) "Checking the given value of T2. It checks." T4=T3/rp^((k-1)/k) Effective=(T5-T2)/(T4-T2) "Some Wrong Solutions with Common Mistakes:" W1_eff = (T5-T2)/(T3-T2) "Using wrong relation" W2_eff = (T5-T2)/(T44-T2); T44=(T3-273)/rp^((k-1)/k) "Using C instead of K for T3" W3_eff = (T5-T2)/(T444-T2); T444=T3/rp "Using wrong relation for T4"
9-196 Consider a gas turbine that has a pressure ratio of 6 and operates on the Brayton cycle with regeneration between the temperature limits of 20°C and 900°C. If the specific heat ratio of the working fluid is 1.3, the highest thermal efficiency this gas turbine can have is
(a) 38% (b) 46% (c) 62% (d) 58% (e) 97%
Answer (c) 62%
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).
k=1.3 rp=6 T1=20+273 "K" T3=900+273 "K"
PROPRIETARY MATERIAL. © 2011 The McGraw-Hill Companies, Inc. Limited distribution permitted only to teachers and educators for course
"Some Wrong Solutions with Common Mistakes:" W1_Eta = 1-((T1-273)/(T3-273))*rp^((k-1)/k) "Using C for temperatures instead of K" W2_Eta = (T1/T3)*rp^((k-1)/k) "Using incorrect relation" W3_Eta = 1-(T1/T3)*rp^((k1-1)/k1); k1=1.4 "Using wrong k value (the one for air)"
Eta_regen=1-(T1/T3)*rp^((k-1)/k)
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PROPRIETARY MATERIAL. © 2011 The McGraw-Hill Companies, Inc. Limited distribution permitted only to teachers and educators for course
9-197 An ideal gas turbine cycle with many stages of compression and expansion and a regenerator of 100 percent effectiveness has an overall pressure ratio of 10. Air enters every stage of compressor at 290 K, and every stage of turbine at 1200 K. The thermal efficiency of this gas-turbine cycle is
(a) 36% (b) 40% (c) 52% (d) 64% (e) 76%
Answer (e) 76%
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).
k=1.4 rp=10 T1=290 "K" T3=1200 "K" Eff=1-T1/T3 "Some Wrong Solutions with Common Mistakes:" W1_Eta = 100 W2_Eta = 1-1/rp^((k-1)/k) "Using incorrect relation" W3_Eta = 1-(T1/T3)*rp^((k-1)/k) "Using wrong relation" W4_Eta = T1/T3 "Using wrong relation"
9-198 Air enters a turbojet engine at 320 m/s at a rate of 30 kg/s, and exits at 650 m/s relative to the aircraft. The thrust developed by the engine is
(a) 5 kN (b) 10 kN (c) 15 kN (d) 20 kN (e) 26 kN
Answer (b) 10 kN
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).
Vel1=320 "m/s" Vel2=650 "m/s" Thrust=m*(Vel2-Vel1)/1000 "kN" m= 30 "kg/s" "Some Wrong Solutions with Common Mistakes:" W1_thrust = (Vel2-Vel1)/1000 "Disregarding mass flow rate" W2_thrust = m*Vel2/1000 "Using incorrect relation"
9-199 ··· 9-207 Design and Essay Problems.
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