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Copyright ©2020 McGraw-Hill Education. All rights reserved. No reproduction or distribution without the prior written consent of McGraw-Hill
Education.
4-1
Solutions Manual for
Heat and Mass Transfer: Fundamentals & Applications
6th Edition
Yunus A. Çengel, Afshin J. Ghajar
McGraw-Hill Education, 2020
Chapter 4
TRANSIENT HEAT CONDUCTION
PROPRIETARY AND CONFIDENTIAL
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4-2
Lumped System Analysis
4-1C Biot number represents the ratio of conduction resistance within the body to convection resistance at the surface of the
4-2C In heat transfer analysis, some bodies are observed to behave like a "lump" whose entire body temperature remains
essentially uniform at all times during a heat transfer process. The temperature of such bodies can be taken to be a function of
4-3C The lumped system analysis is more likely to be applicable in air than in water since the convection heat transfer
coefficient and thus the Biot number is much smaller in air.
4-5C The lumped system analysis is more likely to be applicable to slender bodies than the well-rounded bodies since the
characteristic length (ratio of volume to surface area) and thus the Biot number is much smaller for slender bodies.
4-6C The lumped system analysis is more likely to be applicable for the body cooled naturally since the Biot number is
4-7C The lumped system analysis is more likely to be applicable for the body allowed to cool in the air since the Biot number
4-8C The temperature drop of the potato during the second minute will be less than 4C since the temperature of a body
4-9C The temperature rise of the potato during the second minute will be less than 5C since the temperature of a body
4-10C The heat transfer is proportional to the surface area. Two half pieces of the roast have a much larger surface area than
4-3
4-4
4-5
4-15 An ASTM A203 B steel plate is exposed to cryogenic fluid at T∞ = −50°C and h = 50 W/m2·K. The plate has an
initial temperature of 20°C. Determine the duration that the plate can be in contact with the cold fluid before it reaches the
minimum suitable temperature of −30°C (ASME B31.3-2014, Table A-1M).
Assumptions1 Heat transfer at the plate edges is assumed negligible. 2 Thermal properties are constant. 3 Radiation effects
are negligible. 4 The convection heat transfer coefficient is constant.
Properties The thermal properties given are cp = 470 J/kg·K, k =
52 W/m·K, and ρ = 7900 kg/m3.
Analysis The characteristic length of the plate is
4-6
4-16 An iron whose base plate is made of an aluminum alloy is turned on. The time for the plate temperature to reach 140C
and whether it is realistic to assume the plate temperature to be uniform at all times are to be determined.
Assumptions 1 85 percent of the heat generated in the resistance wires is transferred to the plate. 2 The thermal properties of
the plate are constant. 3 The heat transfer coefficient is constant and uniform over the entire surface.
Properties The density, specific heat, and thermal diffusivity of the aluminum alloy plate are given to be = 2770 kg/m3, cp =
875 kJ/kg.C, and = 7.310-5 m2/s. The thermal conductivity of the plate can be determined from k =
cp = 177 W/m.C
(or it can be read from Table A-3).
Analysis The mass of the iron's base plate is
kg 4155.0)m 03.0)(m 005.0)(kg/m 2770( 23 ==== LAm
V
Noting that only 85 percent of the heat generated is transferred to the plate, the rate of heat
transfer to the iron's base plate is
Air
22C
Iron
4-7
4-8
Tf
[C]
time
[s]
30
40
50
60
70
80
90
100
110
120
130
140
150
160
170
180
190
200
4.286
9.67
15.08
20.52
25.99
31.49
37.02
42.58
48.17
53.79
59.44
65.12
70.84
76.58
82.35
88.16
94
99.87
20 40 60 80 100 120 140 160 180 200
0
20
40
60
80
100
Tf [C]
time [s]
4-9
4-10
4-19 Stainless steel strip is heat treated as it moves through a furnace. The temperature of the strip exiting the furnace is to be
determined.
Assumptions 1 The thermal properties are constant. 2 The heat transfer coefficient is uniform over the entire surface. 3
Radiation effects are negligible. 4 The Biot number is Bi < 0.1 so that the lumped system analysis is applicable (this
assumption will be verified).
Properties The properties of stainless steel are given as k = 21
W/mK, ρ = 8000 kg/m3, and cp = 570 J/kgK.
Analysis The characteristic length and the Biot number of the
stainless steel strip
mm 5
2===== L
LA
V
4-11
4-12
4-21 Stainless steel plates are heat treated as they move through a furnace. The effect of the plate velocity on the plate
temperature at the furnace exit is to be determined.
Assumptions 1 The thermal properties are constant. 2 The heat transfer coefficient is uniform over the entire surface. 3
Radiation effects are negligible. 4 The Biot number is Bi < 0.1 so that the lumped system analysis is applicable (this
assumption will be verified).
Properties The properties of stainless steel are given as k = 21 W/mK, ρ = 8000 kg/m3, and cp = 570 J/kgK.
Analysis The characteristic length and the Biot number of the plate are
cm 2
2===== L
LA
V
)m 101)(K W/m150(
22
−
hLc
4-13
4-14
4-15
4-24 A long copper rod is cooled to a specified temperature. The cooling time is to be determined.
Assumptions 1 The thermal properties of the geometry are constant. 2 The heat transfer coefficient is constant and uniform
over the entire surface.
Properties The properties of copper are k = 401 W/mºC, ρ = 8933 kg/m3, and cp = 0.385 kJ/kgºC (Table A-3).
Analysis For cylinder, the characteristic length and the Biot number are
m 005.0
4
m 02.0
4
)4/(
2
surface
=====
D
DL
LD
A
L
c
V
D = 2 cm
4-16
4-17
4-27 ASTM B98 bolts are exposed to hot steam at T∞ = 200°C and h = 50 W/m2·K. The bolts have an initial temperature
of 20°C. The duration that the bolts can be in the hot steam before they reach the maximum use temperature (ASME B31.3-
2014, Table A-2M) is to be determined.
Assumptions1 The bolts are cylindrical. 2 Thermal properties are constant. 3 Radiation effects are negligible. 4 The
convection heat transfer coefficient is constant. 5 Heat transfer at the two end surfaces of the cylinder is assumed negligible.
Properties The thermal properties given are cp = 377 J/kg·K, k = 36 W/m·K, and ρ = 8550 kg/m3.
Analysis The characteristic length of the bolt is
𝐿𝑐=V
𝐴𝑠=𝜋𝐷2𝐿 4
⁄
𝜋𝐷𝐿 =𝐷
4=0.003 m
4=0.00075 m
The Biot number becomes
4-18
4-19
4-29 Milk in a thin-walled glass container is to be warmed up by placing it into a large pan filled with hot water. The
warming time of the milk is to be determined.
Assumptions 1 The glass container is cylindrical in shape with a radius of
r0 = 3 cm. 2 The thermal properties of the milk are taken to be the same as
those of water. 3 Thermal properties of the milk are constant at room
temperature. 4 The heat transfer coefficient is constant and uniform over
the entire surface. 5 The Biot number in this case is large (much larger
than 0.1). However, the lumped system analysis is still applicable since
the milk is stirred constantly, so that its temperature remains uniform at
all times.
Properties The thermal conductivity, density, and specific heat of the
milk at 20C are k = 0.598 W/m.C, = 998 kg/m3, and cp = 4.182
kJ/kg.C (Table A-9).
Analysis The characteristic length and Biot number for the glass of milk are
m 01050.0
m) 03.0(2+m) m)(0.07 03.0(2
m) 07.0(m) 03.0(
22
2
2
2
2
==
+
==
rLr
Lr
A
L
oo
o
s
c
V
Water
70C
Milk
3C
4-20
