3-141
3-142
3-183E The U-value of a wall for 7.5 mph winds outside are given. The U–value of the wall for the case of 15 mph winds
outside is to be determined.
Assumptions 1 Steady operating conditions exist. 2 Heat transfer through the wall is one-dimensional. 3 Thermal properties
of the wall and the heat transfer coefficients are constant except the one at the outer surface.
Properties The R-values at the outer surface of a wall for summer (7.5
mph winds) and winter (15 mph winds) conditions are given in Table 3–8
to be
Ro, 7.5 mph =Ro, summer = 0.25 h.ft2.F/Btu
and Ro, 15 mph = Ro, winter = 0.17 h.ft2.F/Btu
AnalysisTheR-value of the wall at 7.5 mph winds (summer) is
Inside
Inside
Outside
7.5 mph
WALL
3-143
3-144
Review Problems
3-186 Two persons are wearing different clothes made of different materials with different surface areas. The fractions of
heat lost from each person’s body by perspiration are to be determined.
Assumptions 1 Heat transfer is steady. 2 Heat transfer is one-dimensional. 3 Thermal conductivities are constant. 4 Heat
transfer by radiation is accounted for in the heat transfer coefficient. 5 The human body is assumed to be cylindrical in shape
for heat transfer purposes.
Properties The thermal conductivities of the leather and synthetic fabric are given to be k = 0.159 W/m°C and k = 0.13
W/m°C, respectively.
Analysis The surface area of each body is first determined from
2
1
m 6675.0m)/2 m)(1.7 25.0(2/
===
DLA
C/W 10930.009988.000942.0
)m 6675.0(C). W/m15(
convleathertotal
22
=+=+=
RRR
hA
The total sensible heat transfer is the sum of heat transferred through the clothes and the skin
C)3032(
W3.18
C/W10930.0
C)3032(
21
total
21
clothes
−
−
=
−
=
−
=
TT
R
TT
Q
3-145
Ri
Ralum
T1
T2
Ro
3-187 Cold conditioned air is flowing inside a duct of square cross-section. The maximum length of the duct for a specified
temperature increase in the duct is to be determined.
Assumptions 1 Heat transfer is steady. 2 Heat transfer is one-dimensional. 3 Thermal conductivities are constant. 4 Steady
one-dimensional heat conduction relations can be used due to small thickness of the duct wall. 5 When calculating the
conduction thermal resistance of aluminum, the average of inner and outer surface areas will be used.
Properties The thermal conductivity of aluminum is given to be 237 W/m°C. The specific heat of air at the given
temperature is cp = 1006 J/kg°C (Table A-15).
Analysis The inner and the outer surface areas of the duct per unit length and the individual thermal resistances are
2
22
2
1
1
m 0.1m) 1(m) 25.0(44
m 88.0m) 1(m) 22.0(44
===
===
LaA
LaA
C/W 00007.0
m 015.0
C/W01515.0
)m 88.0(C). W/m75(
11
2
alum
22
1
i
=
==
=
==
L
R
Ah
R
3-146
3-188 Hot water is flowing through a 15-m section of a cast iron pipe. The pipe is exposed to cold air and surfaces in the
basement, and it experiences a 3°C-temperature drop. The combined convection and radiation heat transfer coefficient at the
outer surface of the pipe is to be determined.
Assumptions 1 Heat transfer is steady since there is no indication of any significant change with time. 2 Heat transfer is one–
dimensional since there is thermal symmetry about the centerline and no significant variation in the axial direction. 3
Thermal properties are constant.
Properties The thermal conductivity of cast iron is given to be k = 52 W/m°C.
Analysis Using water properties at room temperature, the mass flow rate of water and rate of heat
transfer from the water are determined to be
 
W13,296=C)6770)(CJ/kg. kg/s)(4180 06.1(
kg/s 06.1m4/(0.03)m/s) )(1.5kg/m 1000(223
−==
====
TcmQ
VAm
p
cc
V
The thermal resistances for convection in the pipe and
the pipe itself are
)5.1/75.1ln(
2
)/ln(
12
pipe
=
kL
rr
R
Rconv ,i
Rpipe
Rcombined ,o
T1
T2
3-147
3-189 The plumbing system of a house involves some section of a plastic pipe exposed to the ambient air. The pipe is
initially filled with stationary water at 0C. It is to be determined if the water in the pipe will completely freeze during a cold
night.
Assumptions 1 Heat transfer is transient, but can be treated as steady since the water temperature remains constant during
freezing. 2 Heat transfer is one-dimensional since there is thermal symmetry about the centerline and no variation in the axial
direction. 3 Thermal properties of water are constant. 4 The water in the pipe is stationary, and its initial temperature is 0C.
5 The convection resistance inside the pipe is negligible so that the inner surface temperature of the pipe is 0C.
Properties The thermal conductivity of the pipe is given to be k = 0.16 W/m°C. The density and latent heat of fusion of
water at 0C are = 1000 kg/m3 and hif = 333.7 kJ/kg (Table A-9).
Analysis We assume the inner surface of the pipe to be at 0C at all times. The thermal resistances involved and the rate of
heat transfer are
11
C/W 3627.0
)m 5.0(C) W/m.16.0(2
)1/2.1ln(
2
)/ln(
12
pipe
=
==
kL
rr
R
Tair = -5C
3-148
3-190 A nuclear fuel rod is encased in a concentric hollow ceramic cylinder, which created an air gap between the rod and the
hollow cylinder. The surface temperature of the fuel rod is to be determined.
Assumptions1 Heat conduction is steady and one-dimensional. 2 Thermal properties are constant. 3 Heat generation in the
fuel rod is uniform. 4 Heat transfer by radiation is negligible.
Properties The thermal conductivity of ceramic is given to be 0.07 W/m ∙ °C.
Analysis The combined thermal resistance between the nuclear fuel rod surface and the outer surface of the ceramic cylinder
is
Lk
DD
LhDLhD
RRRR
2
)/ln(
11 23
21
cyl cond,cyl conv,rod conv,combined
++=
++=
or
)035.0/110.0ln(
1
1
2
)/ln(
11
23
21
combined
++=
k
DD
hDhD
LR
3-149
3-150
3-192 A spherical vessel is used to store a fluid. The thermal resistances, the rate of heat transfer, and the temperature
difference across the insulation layer are to be determined.
Assumptions 1 Steady operating conditions exist. 2 Heat transfer is one-dimensional.
PropertiesThe thermal conductivity of the insulation is given to be 0.20 W/mK.
Analysis(a) The thermal resistances are
K/W 103.31
K/W 108.56
K/W 108.84 4
3
22
3
21
12
22
m) 10.3()K W/m10(
11
)K W/m2.0(m) m)(1.55 (1.54
m)5.155.1(
4
m) 3()K W/m40(
11
−
−
−
=
==
=
−
=
−
=
=
==
oo
o
ins
ii
i
Ah
R
krr
rr
R
Ah
R
(b) The rate of heat transfer is
W1725=
++
−
=
++
=
K/W )1031.3108.5610(8.84
K )022(
3–3–4–
oinsi RRR
T
Q
(c) The temperature difference across the insulation layer is
ins
ins
ins T
T
R
T
K/W 108.56
3-151
3-152
3-194 A wall is constructed of two large steel plates separated by 1-cm thick steel bars placed 99 cm apart. The remaining
space between the steel plates is filled with fiberglass insulation. The rate of heat transfer through the wall is to be
determined, and it is to be assessed if the steel bars between the plates can be ignored in heat transfer analysis since they
occupy only 1 percent of the heat transfer surface area.
Assumptions 1 Heat transfer is steady since there is no indication of change with time. 2 Heat transfer through the wall can
be approximated to be one-dimensional. 3 Thermal conductivities are constant. 4 The surfaces of the wall are maintained at
constant temperatures.
Properties The thermal conductivities are given to be k = 15 W/m°C for steel plates and k = 0.035 W/m°C for fiberglass
insulation.
Analysis We consider 1 m high and 1 m wide portion of the wall which is representative of entire wall.
Thermal resistance network and individual resistances are
C/W 1926.100053.01915.100053.0
C/W 1915.1
3492.6
1
4667.1
1111
C/W 3492.6
)m 99.0(C) W/m.035.0(
m 22.0
C/W 4667.1
)m 01.0(C) W/m.15(
m 22.0
C/W 00053.0
)m 1(C) W/m.15(
m 008.0
4eqv1total
32eqv
2
insulation3
2
steel2
2
steel41
=++=++=
=⎯→⎯+=+=
=
===
=
===
=
====
RRRR
R
RRR
kA
L
RR
kA
L
RR
kA
L
RRR
eqv
The rate of heat transfer per m2 surface area of the wall is
W45.18
C/W 1.1926
C 22
total
=
=
=R
T
Q
The total rate of heat transfer through the entire wall is then determined to be
99 cm
1 cm
0.8 cm 22 cm 0.8 cm
T1
R1
R2
R3
R4
T2
3-153
3-195 A typical section of a building wall is considered. The temperature on the interior brick surface is to be determined.
Assumptions1 Steady operating conditions exist.
Properties The thermal conductivities are given to be k23b = 50 W/mK, k23a = 0.03 W/mK, k12 = 0.5 W/mK, k34 = 1.0
W/mK.
AnalysisWe consider 1 m2 of wall area. The thermal resistances are
C/Wm 645.2
0.005)C)(0.6 W/m03.0(
m 6.0
m) 08.0(
)(
C/Wm 02.0
C) W/m5.0(
m 01.0
2
23a
a
2323
2
12
12
12
=
+
=
+
=
=
==
LLk
L
tR
k
t
R
ba
a
3-154
3-196 A square cross-section bar consists of a copper layer and an epoxy layer. The rates of heat transfer in different
directions are to be determined.
Assumptions 1 Steady operating conditions exist. 2 Heat transfer is one–
dimensional.
Properties The thermal conductivities of copper and epoxy are given to be 380
and 0.4 W/mK, respectively.
Analysis (a) Noting that the resistances in this case are in parallel, the heat
transfer from front to back is
m )01.002.0)(K W/m4.0(
m )01.002.0)(K W/m380(
1
22
1
+
=
−
−
EpCu L
kA
L
kA
R
12 cm
1 cm
1 cm
2 cm
copper
epoxy
3-155
3-156
mLmkhmL
xLmmkhxLm
TT
TxT
bsinh)/(cosh
)(sinh)/()(cosh
)(
+
−+−
−
−
The values for the temperature variations for parts (a) to (d) are tabulated in the following table:
L, m
T(x), °C
Part (a)
Part (b)
Part (c)
Part (d)
0
350
350
350
350
0.005
318
326
328
325
0.010
290
305
308
304
0.015
264
288
292
285
0.020
241
272
279
270
0.025
220
260
268
256
0.030
201
250
259
246
0.035
184
242
253
237
0.040
169
237
250
231
0.045
155
233
249
227
0.050
142
232
250
224
The temperature variations for parts (a) to (d) are plotted in the following figure:
x, m
0.00 0.01 0.02 0.03 0.04 0.05
T, °C
100
150
200
250
300
350
Infinitely long fin
Adiabatic fin tip
Fin with tip temperature of 250 °C
Convection from the fin tip
Discussion The differences in the temperature variations show that applying the proper boundary condition is very important
in order to perform the analysis correctly.
3-157
3-198 Ten rectangular aluminum fins are placed on the outside surface of an electronic device. The rate of heat loss from the
electronic device to the surrounding air and the fin effectiveness are to be determined.
Assumptions 1 Steady operating conditions exist. 2 The temperature along the fins varies in one direction only (normal to the
plate). 3 The heat transfer coefficient is constant and uniform over the entire fin surface. 4 The thermal properties of the fins
are constant. 5 The heat transfer coefficient accounts for the effect of radiation from the fins.
Properties The thermal conductivity of the aluminum fin is given to be k = 203 W/mK.
Analysis The fin efficiency is to be determined using Fig. 3-43 in the text.
97.0218.0
)004.0)(022.0)(203(
80
)2/004.0020.0()tkL/(h)2/tL()kA/(hL fin
2/3
c
2/3
p
2/3
c=⎯→⎯=+=+==
The rate of heat loss can be determined as follows
==
=+=
Q
Q
Q
m 004.0)004.0100.0(10A
m 0416.0)020.0004.0100.0020.0(102A
fin
fin
fin
2
base
2
fin
3-158
3-199 Using Table 3-3, the efficiency, heat transfer rate, and effectiveness of a straight triangular fin are to be determined.
Assumptions1 Heat conduction is steady and one-dimensional. 2 Thermal properties are constant. 3 Heat transfer by radiation
is negligible.
Properties The thermal conductivity of the fin is given as 236 W/m ∙ °C.
AnalysisFrom Table 3-3, for straight triangular fins, we have
)C W/m25(22 2
h
3-159
3-200 Aluminum pin fins of parabolic profile with blunt tips are attached to a plane surrface. The heat transfer rate from a
single fin and the increase in the heat transfer as a result of attaching fins are to be determined.
Assumptions1 Heat conduction is steady and one-dimensional. 2 Thermal properties are constant. 3 Heat transfer by radiation
is negligible.
Properties The thermal conductivity of the fin is given as 230 W/m ∙ °C.
AnalysisFrom Table 3-3, for pin fins of parabolic profile (blunt tip), we have
3497.0)m 025.0(
)m 004.0)(C W/m230(
)C W/m45(44 2
=
== L
kD
h
mL
2/3
2
2
4
2/3
2
2
4
fin
11
m 025.0
16
)m 004.0(
1116
−
+
=
−
+
=
L
D
A
L = 25mm
D = 4mm
k=230W/mC
h = 45W/m2C
3-160