4-121
4-122
4-158 The temperature at the edge of a steel block after 10 minutes of cooling is to be determined.
Assumptions 1 Two-dimensional heat conduction in x and y directions. 2 Thermal properties are constant. 3 Convection heat
transfer coefficient is constant. 4 Heat transfer by radiation is negligible.
Properties The properties of steel are (
= 7832 kg/m3, cp = 434 J/kg ∙ K, k = 63.9 W/m ∙ K, and
= 18.8 10−6 m2/s.
Analysis For a quarter-infinite medium, at the edge of the steel block (x = y = 0), we have
2
infsemiinfsemiinfsemi )],0([),0(),0(),0,0( tttt −−− ==
where
−
αth
th
TtT
ierfcexperfc(0)
),0(
2
4-123
4-159 A heated large iron slab is placed on a concrete floor; (a) the surface temperature and (b) the temperature of the
concrete floor at the depth of 25 mm are to be determined.
Assumptions 1 The iron slab and concrete floor are treated as semi-infinite solids. 2 Thermal properties are constant. 3 Heat
transfer by radiation is negligible. 4 Contact resistance is negligible.
Properties The properties of iron slab are given to be
= 7870 kg/m3, cp = 447 J/kg ∙ K, and k = 80.2 W/m ∙ K; the properties
of concrete floor are given to be
= 1600 kg/m3, cp = 840 J/kg ∙ K, and k = 0.79 W/m ∙ K.
Analysis (a) For contact of two semi-infinite solids, the surface temperature is
(b) For semi-infinite solid with specified surface temperature, we have
−
),(
,
iB
x
TtxT
4-124
4-160 A hot dog is to be cooked by dropping it into boiling water. The time of cooking is to be determined.
Assumptions 1 Heat conduction in the hot dog is two-dimensional, and thus the temperature varies in both the axial x– and
the radial r– directions. 2 The thermal properties of the hot dog are constant. 4 The heat transfer coefficient is constant and
uniform over the entire surface. 5 The Fourier number is > 0.2 so that the one-term approximate solutions are applicable
(this assumption will be verified).
Properties The thermal properties of the hot dog are given to be k = 0.76 W/m.C, = 980 kg/m3, cp = 3.9 kJ/kg.C, and =
210-7 m2/s.
Analysis This hot dog can physically be formed by the intersection of an infinite plane wall of thickness 2L = 12 cm, and a
long cylinder of radius ro = D/2 = 1 cm. The Biot numbers and corresponding constants are first determined to be
37.47
)C W/m.76.0(
)m 06.0)(C. W/m600( 2
=
== k
hL
Bi
2726.1 and 5380.1 11 ==⎯→⎯ A
895.7
)C W/m.76.0(
)m 01.0)(C. W/m600( 2
=
== k
hr
Bi o
5514.1 and 1249.2 11 ==⎯→⎯ A
Noting that
2
/Lt
=
and assuming > 0.2 in all dimensions and thus the one-term approximate solution for transient heat
conduction is applicable, the product solution for this problem can be written as
2105.0
)01.0(
)102(
)1249.2(exp)5514.1(
)06.0(
)102(
)5380.1(exp)2726.1(
1005
10080
),0(),0(),0,0(
2
7
2
2
7
2
11
2
1
2
1
=
−
−=
−
−
==
−
−
−−
t
t
eAeAttt cylwallblock
which gives
min 4.1 s 244 = =t
Therefore, it will take about 4.1 min for the hot dog to cook. Note that
s) /s)(244m 102(
27
−
t
Water
100C
2 cm Hot dog Ti = 5C
4-125
4-161 The engine block of a car is allowed to cool in atmospheric air. The temperatures at the center of the top surface and at
the corner after a specified period of cooling are to be determined.
Assumptions 1 Heat conduction in the block is three-dimensional, and thus the temperature varies in all three directions. 2
The thermal properties of the block are constant. 3 The heat transfer coefficient is constant and uniform over the entire
surface. 4 The Fourier number is > 0.2 so that the one-term approximate solutions are applicable (this assumption will be
verified).
Properties The thermal properties of cast iron are given to be k = 52 W/m.C and = 1.710-5 m2/s.
Analysis This rectangular block can physically be formed by the intersection of two infinite plane walls of thickness 2L = 40
cm (call planes A and B) and an infinite plane wall of thickness 2L = 80 cm (call plane C). We measure x from the center of
the block.
(a) The Biot number is calculated for each of the plane wall to be
)m 2.0)(C. W/m6( 2
hL
4-126
4-162 A man is found dead in a room. The time passed since his death is to be estimated.
Assumptions 1 Heat conduction in the body is two-dimensional, and thus the temperature varies in both radial r– and x–
directions. 2 The thermal properties of the body are constant. 3 The heat transfer coefficient is constant and uniform over the
entire surface. 4 The human body is modeled as a cylinder. 5 The Fourier number is > 0.2 so that the one-term
approximate solutions are applicable (this assumption will be verified).
Properties The thermal properties of body are given to be k = 0.62 W/m.C and = 0.1510-6 m2/s.
Analysis A short cylinder can be formed by the intersection of a long cylinder of radius D/2 = 14 cm and a plane wall of
thickness 2L = 180 cm. We measure x from the midplane. The temperature of the body is specified at a point that is at the
center of the plane wall but at the surface of the cylinder. The Biot numbers and the corresponding constants are first
determined to be
06.13
)C W/m.62.0(
)m 90.0)(C. W/m9( 2
wall =
== k
hL
Bi
2644.1 and 4495.1 11 ==⎯→⎯ A
03.2
)C W/m.62.0(
)m 14.0)(C. W/m9( 2
cyl =
== k
hr
Bi o
3408.1 and 6052.1 11 ==⎯→⎯ A
Noting that
2
/Lt
=
for the plane wall and
2
/o
rt
=
for
cylinder and J0(1.6052)=0.4524 from Table 4-3, and assuming
that > 0.2 in all dimensions so that the one-term approximate
solution for transient heat conduction is applicable, the product
solution method can be written for this problem as
hours 7.11==⎯→⎯
−
=
−
−
=
−
−
−−
s 600,25
)4524.0(
)14.0(
)1015.0(
)6052.1(exp)3408.1(
)90.0(
)1015.0(
)/()(
1236
1223
),(),0(),,0(
2
6
2
2
6
2
01011
cyl0wall0
2
1
2
1
t
t
t
rrJeAeA
trttr block
D0 = 28 cm
z
Human body
Ti = 36C
r
Air
T = 12C
2L=180 cm
4-127
Fundamentals of Engineering (FE) Exam Problems
4-163 The Biot number can be thought of as the ratio of
(a) the conduction thermal resistance to the convective thermal resistance
(b) the convective thermal resistance to the conduction thermal resistance
(c) the thermal energy storage capacity to the conduction thermal resistance
(d) the thermal energy storage capacity to the convection thermal resistance
(e) None of the above
4-164 Lumped system analysis of transient heat conduction situations is valid when the Biot number is
(a) very small (b) approximately one (c) very large
(d) any real number (e) cannot say unless the Fourier number is also known.
4-165 Polyvinylchloride automotive body panels (k = 0.092 W/mK, cp = 1.05 kJ/kgK,
= 1714 kg/m3), 1-mm thick, emerge
from an injection molder at 120oC. They need to be cooled to 40oC by exposing both sides of the panels to 20oC air before
they can be handled. If the convective heat transfer coefficient is 15 W/m2K and radiation is not considered, the time that the
panels must be exposed to air before they can be handled is
(a) 0.8 min (b) 1.6 min (c) 2.4 min (d) 3.1 min (e) 5.6 min
4-128
4-129
4-168 A 10-cm-inner diameter, 30-cm long can filled with water initially at 25ºC is put into a household refrigerator at 3ºC.
The heat transfer coefficient on the surface of the can is 14 W/m2ºC. Assuming that the temperature of the water remains
uniform during the cooling process, the time it takes for the water temperature to drop to 5ºC is
(a) 0.55 h (b) 1.17 h (c) 2.09 h (d) 3.60 h (e) 4.97 h
t_hour=t*Convert(s, h)
4-130
7-169 A 6-cm-diameter 13–cm-high canned drink ( = 977 kg/m3, k = 0.607 W/mC, cp = 4180 J/kgC) initially at 25°C is
to be cooled to 5C by dropping it into iced water at 0C. Total surface area and volume of the drink are As = 301.6 cm2 and
V
= 367.6 cm3. If the heat transfer coefficient is 120 W/m2C, determine how long it will take for the drink to cool to 5C.
Assume the can is agitated in water and thus the temperature of the drink changes uniformly with time.
(a) 1.5 min (b) 8.7 min (c) 11.1 min (d) 26.6 min (e) 6.7 min
4-131