Chapter 4
Transient Heat Conduction
Review Problems
4105
Two large steel plates are stuck together because of the freezing of the water between the two
plates. Hot air is blown over the exposed surface of the plate on the top to melt the ice. The length of
time the hot air should be blown is to be determined.
Assumptions
1
Heat conduction in the plates is onedimensional since the plate is large relative to its
thickness and there is thermal symmetry about the center plane.
3
The thermal properties of the steel
plates are constant.
4
The heat transfer coefficient is constant and uniform over the entire surface.
5
The
Fourier number is
τ
> 0.2 so that the oneterm approximate solutions (or the transient temperature
charts) are applicable (this assumption will be verified).
Properties
The thermal properties of steel plates are given to be
k
= 43 W/m.
°
C and
α
= 1.17
×
10
5
m
2
/s
Analysis
The characteristic length of the plates and the Biot number are
1
.
0
019
.
0
)
C
W/m.
43
(
)
m
02
.
0
)(
C
.
W/m
40
(
m
02
.
0
2
<
=
°
°
=
=
=
=
=
k
hL
Bi
L
A
V
L
c
s
c
Since Bi < 0.1 , the lumped system analysis is applicable. Therefore,
min
8.0
s
482
=
=
→
=



→
=


=
°
×
°
=
=
=


∞
∞
t
e
e
T
T
T
t
T
L
C
h
V
C
hA
b
bt
i
c
p
p
s
)t
s
000544
.
0
(
1

3
6
2
1
50
15
50
0
)
(
s
000544
.
0
m)
(0.02
)
C
.
J/m
10
675
.
3
(
C
.
W/m
40
ρ
ρ
where
C
.
J/m
10
675
.
3
/s
m
10
17
.
1
C
W/m.
43
3
6
2
5
°
×
=
×
°
=
α
=
ρ

k
C
p
Alternative solution:
This problem can also be solved using
the transient chart Fig. 413a,
2
.
0
15
769
.
0
50
15
50
0
6
.
52
019
.
0
1
1
2
=
=
=



=


=
=
∞
∞
o
i
o
r
t
T
T
T
T
Bi
α
τ
Then,
s
513
=
×
=
α
τ
=

/s)
m
10
17
.
1
(
m)
02
.
0
)(
15
(
2
5
2
2
o
r
t
The difference is due to the reading error of the chart.
484
Steel plates
T
i
= 15
°
C
Hot gases
T
∞
= 50
°
C
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Chapter 4
Transient Heat Conduction
4106
A curing kiln is heated by injecting steam into it and raising its inner surface temperature to a
specified value. It is to be determined whether the temperature at the outer surfaces of the kiln changes
during the curing period.
Assumptions
1
The temperature in the wall is affected by the thermal conditions at inner surfaces only
and the convection heat transfer coefficient inside is very large. Therefore, the wall can be considered to
be a semiinfinite medium with a specified surface temperature of 45
°
C.
2
The thermal properties of the
concrete wall are constant.
Properties
The thermal properties of the concrete wall are given to be
k
= 0.9 W/m.
°
C and
α
= 0.23
×
10
5
m
2
/s.
Analysis
We determine the temperature at a depth of
x
= 0.3 m in 3 h using the analytical solution,
=


t
x
erfc
T
T
T
t
x
T
i
s
i
α
2
)
,
(
Substituting,
C
9.1
°
=
=
=
×
×
=



)
,
(
1782
.
0
)
952
.
0
(
)
s/h
3600
h
3
)(
/s
m
10
23
.
0
(
2
m
3
.
0
5
42
2
)
,
(
2
5
t
x
T
erfc
erfc
t
x
T
which is greater than the initial temperature of 2
°
C. Therefore, heat will propagate through the 0.3 m
thick wall in 3 h, and thus it may be desirable to insulate the outer surface of the wall to save energy.
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 Spring '08
 BENARD
 Heat Transfer, TI, Thermal conductivity, transient heat conduction

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