CANKAYA UNIVERSITY
FACULTY OF ENGINEERING AND ARCHITECTURE
MECHANICAL ENGINEERING DEPARTMENT
ME 211 THERMODYNAMICS I
CHAPTER 3 EXAMPLES
SOLUTIONS
1) The average atmospheric pressure in Denver (elevation=1610 m) is 83.4 kPa.
Determine the temperature at which water in an uncovered pan will boil in
Denver.
?TTbar834.0p
bar834.0Pa10
bar1Pa1083.4 kPa 83.4 p
sat
5
3
From Table A-3:
C71.96Tbar9.0p
C50.93Tbar8.0p
o
o
C59.94z5.93T
0914.1z
5.9371.96
z
8.09.0
8.0834.0
o
0.8
0.834
0.9
z
96.71 93.5
p
T
2) A rigid tank with a volume of 2.5 m3 contains 5 kg of saturated liquid-vapor
mixture of water at 75oC. Now the water is slowly heated. Determine the quality
x at the initial state. Also, determine the temperature at which the liquid in the
tank is completely vaporized.
v = constant
kg/m5.0kg5
m5.2
m
Vv 3
3
1
T1 = 75oC
From Table A-2 @ T1 = 75oC:
vf = 1.025910-3
m3/kg
vg = 4.131 m3/kg
121.0xvxvv 1fg1f1
@ State 2:
v2 = v1 = 0.5 m3/kg
x2 = 1
/kgm5.0v1 x
/kgm 0.5 v v 3
g
2
3
12
From Table A-2 @ vg = 0.5 m3/kg:
C7.140TC140C150
C140T
kg/m5089.0kg/m3928.0
kg/m5089.0kg/m5.0 o
2oo
o
2
33
33
1
2
75oC
T2
T vapor
liquid
v
3) Consider a two phase mixture at 100oC with x3 = 0.9. Determine the specific
volume.
From Table A-2 @100oC:
kg/m673.1v
kg/m100435.1v
3
g
33
f
kg/m506.1v
kg/m)100435.1673.1(9.0kg/m100435.1v
)vv(xvv
3
3
3333
3
fgf3
Note: Once the quality x, is known, it can be applied to calculate v, h or s in the same
manner as above.
4) Refrigerant 134a with a quality of 0.4 and a temperature of 12oC is contained in
a rigid tank that has a volume of 0.17 m3. Find the mass of liquid present.
gfg mmm,xmm,v
Vm
fgf xvvv
From Table A-10 @12oC:
kg/m046.0v
kg/m000797.0v
3
g
3
f
kg/m0189.0v
kg/m)000797.0046.0(4.0kg/m000797.0vxvvv
3
1
33
1fgf1
Total mass kg9kg/m0189.0
m17.0
v
Vm
3
3
1
kg6.3)9)(4.0(xmmg
kg4.5kg6.3kg9mmm gf
vapor
liquid
T1 = 12oC
x1 = 0.4
V = 0.17 m3
5) Consider Refrigerant-22 at 12oC. It is specific internal energy 144.58 kJ/kg.
Determine phase description and enthalpy.
From Table A-7 @12oC:
kg/kJ38.230u
kg/kJ77.58u
g
f
So uf < u < ug saturated liquid vapor mixture.
5.0kg/kJ)77.5838.230(
kg/kJ)77.5858.144(
)uu(
)uu(x
fg
f
kg/kJ67.156h
kg/kJ)35.5999.253(5.0kg/kJ35.59xhhh fgf
6) Determine the temperature of water at a state of p = 0.5 MPa and h = 2890
kJ/kg.
P = 0.5 MPa = 500 kPa = 5bar
h = 2890 kJ/kg
From Table A-3 for p = 5 bar
h > hg so superheated vapor
From Table A-4 @ p = 5 bar:
C38.216z200T
37.16z
200240
z
4.28559.2939
4.28552890
o
h
2855.4
2890
2939.9
z
240 200 T
T (oC)
200
h (kJ/kg)
2855.4
240 2939.9
5
hg (kJ/kg) hf (kJ/kg) p (bar)
640.23 2748.7
7) Determine the missing properties and the phase descriptions in the following
table for water:
T, oC p, kPa h, kJ/kg x Phase description
(a) 200 0.7
(b) 140 1800
(c) 950 0.0
(d) 80 500
(e) 800 3161.7
(a)
C2.120TT7.0x
bar2kPa200po
sat
kg/kJ2046h)9.2201(7.07.504xhhh fgf saturated mixture
(b)
kPa3.361bar613.3pkg/kJ1800h
C140T o
564.0x
)kg/kJ7.2144(xkg/kJ13.589kg/kJ1800xhhh fgf
saturated mixture
(c)
0x
bar5.9kPa950psaturated liquid
p = 950 kPa = 0.95 MPa = 9.5 bar T = 177.65oC
h = hf = 752.82 kJ/kg
(d)
C9.151Tbar5pC80T
bar5MPa5.0kPa500po
sato
satTT compressed liquid
kg/kJ91.334hhC80T@f o
(e)
kg/kJ7.3161h
bar8MPa8.0kPa800p
g
gf
hh
kg/kJ1.2769h,kg/kJ11.721hbar8p
superheated vapor
C06.350Tkg/kJ7.3161h
bar8po
8) A frictionless piston-cylinder device initially contains 200 L of saturated liquid
refrigerant R-134a. The piston is free to move, and its mass is such that it
maintains a pressure of 800 kPa on the refrigerant. The refrigerant is now heated
until its temperature rises to 50 oC. Calculate the work done during this process.
9) A piston-cylinder device contains 50 kg of water at 150 kPa and 25 oC. The
cross-sectional area of the piston is 0.1 m2. Heat is now transferred to the water,
causing part of it to evaporate and expand. When the volume reaches 0.2 m3, the
piston reaches a linear spring whose spring constant is 100kN/m. More heat is
transferred to the water until the piston rises 20 cm more. Determine (a) the
final pressure and temperature and (b) the work done during this process. Also,
show the process on p-v diagram.
10) The radiator of a steam heating system has a volume of 20 L and is filled with
superheated vapor at 300 kPa and 250°C. At this moment both the inlet and
exit valves to the radiator are closed. Determine the amount of heat that will
be transferred to the room when the steam pressure drops to 100 kPa. Also,
show the process on a p-v diagram with respect to saturation lines.
11) Two kilograms of saturated liquid water at 50 kPa are heated slowly at constant
pressure. During the process 5876 kJ of heat are added. Find the final
temperature.
p1 = p2 = 50 kPa
m = 2 kg
State 1:
kg/kJ5.340hh0x
bar5.0kPa50pf1
1
1
Q – W = U2 – U1
112212
2
1
VpVp)VV(ppdVW
kg/kJ32785.3402
5876h
m
Qh
)hh(mHHUUVpVp)UU(WQ
12
121212112212
State 2:
kg/kJ3278h
bar5.0kPa50p
2
2
kg/kJ9.2645h,kg/kJ5.340hbar5.0p@ gf2
g2 hh superheated vapor
From Table A-4 T2 = 400oC
H2O
Q = 5876 kJ
12) A vessel having a volume of 5 m3 contains 0.05 m
3 of saturated liquid water and
4.95 m3 of saturated water vapor at 0.1 MPa. Heat is transferred until the vessel
is filled with saturated vapor. Determine the heat transfer for this process.
0PEKEUUWQ 121212
State 1:
kg/m694.1v,kg/m001043.0vbar1MPa1.0p 3
1g
3
1f1
kg/kJ1.2506u,kg/kJ36.417u 1g1f
kg94.47kg/m001043.0
m05.0
v
Vm
3
3
1f
f1f
kg92.2kg/m694.1
m95.4
v
Vm
3
3
1g
g
1g
kJ27326)kg/kJ1.2506)(kg92.2()kg/kJ36.417)(kg94.47()um()um(U 1gg1ff1
State 2:
kg92.2kg94.47mmm 1g1f
kg/m09831.0kg86.50
m5
m
Vv 3
3
2
kg/m09831.0v
1x
3
2
2We need to do interpolation
vg (m3/kg) p (bar)
0.09963 20
0.09831 ?
0.07998 25
vapor
liquid
H2O
)pressurefinal(bar3.20p
ug (kJ/kg) p (bar)
2600.3 20
? 20.3
2603.1 25
kg/kJ5.2600u 2
kJ132261)kg/kJ5.2600)(kg86.50(umU 222
kJ104935UUQ 1212
13) A cylinder fitted with a piston has a volume of 0.1 m3 and contains 0.5 kg of
steam at 0.4 MPa. Heat is transferred to the steam until the temperature is 300oC
while the pressure remains constant. Determine the work and heat transfer for
this process.
)vv(pm)VV(pdVppdVW 1212
2
1
2
1
12
)hh(mQ
)vpu()vpu(m)vv(mp)uu(mWUUQ
1212
1112221212121212
State 1:
MPa4.0pandkg/m2.05.0
1.0
m
Vv 1
311
From Table A-3 @ p1 = 0.4 Mpa = 4 bar:
vf = 0.001084 m3/kg, vg = 0.4625 m
3/kg
mixturevaporliquidvvv g1f
4311.0x
kg/m)001084.04625.0(xkg/m001084.0kg/m2.0vxvv
1
3
1
33
fg1f1
kg/kJ7.1524h
kg/kJ)74.6046.2738)(4311.0(kg/kJ74.604hhxhh
1
1fg1f1
State 2:
1
p
V
p2 = p1 = 0.4 MPa
p2 = 0.4 Mpa = 4 bar
T2 = 300oC
p2 = 0.4 Mpa = 4 bar sat
o
sat TT,C6.143T superheated vapor
Using Table A-4:
p = 3 bar
T (oC) h (kJ/kg)
280 3028.6
300 ?
320 3110.1
kg/kJ35.3069z6.3028h75.40z)6.30281.3110(
z
280320
280300
p = 5 bar
T (oC) h (kJ/kg)
280 3022.9
300 ?
320 3105.6
T
280
300
320
z
3105.6 3022.9 h
T
280
300
320
z
3110.1 3028.6 h
kg/kJ25.3064z9.3022h35.41z)9.30226.3105(
z
280320
280300
T = 300 oC
p (bar) h (kJ/kg)
3 3069.35
4 ?
5 3064.25
kg/kJ8.3066z25.3064h55.2z)25.306435.3069(
z
35
342
We do similar interpolation for v2 = 0.6548 m3/kg
kJ96.90m)2.06548.0)(kPa400)(kg5.0()vv(mpW
kJ1.771kg/kJ)7.15248.3066(kg5.0)hh(mQ
3
1212
1212
p
3
4
5
z
3069.35 3064.25 h
14) A frictionless piston is used to provide a constant pressure of 400 kPa in a
cylinder containing steam originally at 200oC with a volume of 2 m
3. Calculate
the final temperature if 3500 kJ of heat added.
State 1:
3
1
o
1
5
1
m2V
C200T
bar4Pa104kPa400p
From Table A-3:
p2 = 4 bar sat
o
sat TT,C6.143T superheated vapor
Using Table A-4 for C200Tandbar4p o
11 :
kg/kJ2860h,kg/kJ2647u,kg/m5342.0v 11
3
1
)VV(kPa400)VV(pdVppdVW 1212
2
1
2
1
12
kg744.3kg/m5342.0
m2
v
Vm
3
3
1
1
)uu(m)VV(kPa400kJ3500
)uu(m)UU(WQ
1212
1212
kg/kJ)2647u)(kg744.3(m)2v744.3(kPa400kJ3500 2
3
2
This problem requires trial error procedure:
1) Guess a value of v2
2) Calculate u2 from above equation
3) If this value checks with u2 from steam tables then the guess is correct one.
Steam
Q = 3500 kJ
A) 1) Let v2 = 1 m3/kg
2) kg/kJ)2647u)(kg744.3(m2)1)(744.3(kPa400kJ3500 2
3
kg/kJ3395u 2
3) From Table A-4 for:
kg/kJ2.3300ukg/m1v
bar4p23
2
2
These two u values are not the same. So revise value of v.
B) 1) Let v2 = 1.06 m3/kg
2) kg/kJ)2647u)(kg744.3(m2)06.1)(744.3(kPa400kJ3500 2
3
kg/kJ3372u 2
3) From Table A-4 for:
kg/kJ3372ukg/m06.1v
bar4p23
2
2
We now accept that u2 values are close enough. So temperature is T2 640oC.
Second Method:
)UU(WQ 12
)VV(pdVppdVW 12
2
1
2
1
)hh(mHHQ)UU()VV(p)UU(WQ 1212121212
kg/kJ3795kg/kJ2860kg744.3
kJ3500h
m
Qh 12
From Table A-4 for:
C641Tkg/kJ3795h
bar4po
2
2
2
15) Determine the enthalpy change h of nitrogen, in kJ/kg, as it is heated from 600
to 1000 K, using (a) the empirical specific heat equation as a function of
temperature (Table A-21), (b) the cP value at the average temperature (Table A-
20), and (c) the cP value at room temperature at 300 K.
a) 32
p
32pTTTK.kmol/kJ314.8cTTT
R
c
From Table A-21:
963 10632.0,10324.2,10208.1,675.3
449336
223
T
T
432
T
T
p
600100010632.04
1600100010324.2
3
1
600100010208.12
16001000675.3
K.kmol/kJ314.8h
T4
1T
3
1T
2
1TK.kmol/kJ314.8dTch
2
1
2
1
kmol
kJ12544h
kg
kJ8.447
kmol/kg01.28
kmol/kJ12544
M
hh
b) From Table A-20 at Tavg = 800 K, cp,avg = 1.121 kJ/kg.K
kg
kJ4.448hK)6001000(
K.kg
kJ121.1)TT(ch 12avg,p
c) Taking the cp at room temperature from Table A-20:
cp = 1.039 kJ/kg.K
kg
kJ6.415hK)6001000(
K.kg
kJ039.1)TT(ch 12p
16) A rigid tank has a volume of 400 cm3 contains air initially at 22
oC and 100 kPa.
A paddle wheel stirs the gas until the temperature is 428oC. During the process,
600 J of heat are transferred from air to surroundings. Calculate the work done
by the paddlewheel two ways:
(a) assuming constant specific heat
(b) assuming variable specific heat
(a) cv = constant
)TT(mcU
WQU
12v
K498C2252
42822T o
avg
From Table A-20:
K.kg/J742K.kg/kJ742.0cv
TR
pVMm
RT
pVmmRTpV
kg0004725.0
)K295)(K.kmol/kJ314.8(
cm10/m)cm400)(kPa100)(kmol/kg97.28(m
3633
J3.742)TT(mcQUQW 12v
(b) variable cv with temperature:
)TcTc(mWQ)TcTc(mU
WQU
11v22v11v22v
From Table A-20:
air
At T1 = 295 K cv1 = 0.718 kJ/kg.K = 718 J/kg.K
T2 = 701 K cv2 = 1.098 kJ/kg.K = 1098 J/kg.K
TR
pVMm
RT
pVmmRTpV
kg0004725.0
)K295)(K.kmol/kJ314.8(
cm10/m)cm400)(kPa100)(kmol/kg97.28(m
3633
J080.100)K295)(K.kg/J718)(kg0004725.0(UTmcU
J682.363)K701)(K.kg/J1098)(kg0004725.0(UTmcU
UUWQ
111v1
222v2
12
J602.863)080.100682.363(J600UUQW 12
17) A mass of 15 kg of air in a piston-cylinder device is heated from 25 to 77°C by
passing current through a resistance heater inside the cylinder. The pressure
inside the cylinder is held constant at 300 kPa during the process, and a heat
loss of 60 kJ occurs. Determine the electric energy supplied, in kWh.
18) Helium (He) gas initially at 2 bar and 200 K undergoes a polytropic process
with n = k to a final pressure of 14 bar. Determine work and heat transfer for the
process, each in kJ/kg of helium. Assume ideal gas behavior.
U W -Q
k1
)TT(mR
k1
)vpvp(m
v
dVmpdVW 121122
V
V
k
V
V
2
1
2
1
To find T2 use:
k
1k
1212 )p/p(TT
667.15.1
5.2
c
ck
v
p
K575667.1
1667.1
2 )1/14(200T
kJ7.1167667.11
)200575)(003.4/314.8(
m
W
m
W)TT(c
m
Wuu
m
QW)uu(mQ 12v1212
Rcc vp
1k
Rc
c
R1k
c
R1
c
c
v
vvv
p
0k1
)TT(R)TT(
1k
R
m
Q 1212
2
1 V
p
He
14
2
PVk = c
19) Determine the specific volume of superheated water vapor at 1.6 MPa and 225 oC, using (a) the ideal-gas equation, (b) the generalized compressibility chart,
and (c) the steam tables. Also determine the error involved in the first two cases.
K.kg/kJ4615.0kmol/kg02.18
K.kmol/kJ314.8
M
RR
MPa09.22bar9.220p
K3.647T
c
c
a) ideal gas law:
kg
m14364.0
kPa1600
)K498)(K.kg/kJ4615.0(
p
RTv
3
b) compressibility chart, see Figure A-1, Figure A-2, Figure A-3:
935.0Z
769.0K3.647
K498
T
TT
072.0MPa09.22
MPa6.1
p
pp
c
R
c
R
%710014364.0
14364.01343.0error
kg/m13430.0)14364.0(935.0p
RTZv 3
c) using superheated table:
13287.0vC225T
bar16MPa6.1p
o
20)
21 )
21) Air is confined to one side of a rigid container divided by a partition, as shown in
Figure below. The other side is initially evacuated. The air is initially at =5 bar , and
=500 K, and . When the partition is removed, the air expands to fill the
entire chamber. Measurements show that and . Assuming the air
behaves as an ideal gas, determine (a) the final temperature, in K, and (b) the heat
transfer, kJ.
solution