Chapter 4
Transient Heat Conduction
447
A hot dog is dropped into boiling water, and temperature measurements are taken at certain time
intervals. The thermal diffusivity and thermal conductivity of the hot dog and the convection heat
transfer coefficient are to be determined.
Assumptions
1
Heat conduction in the hot dog is onedimensional since it is long and it has thermal
symmetry about the center line.
2
The thermal properties of the hot dog are constant.
3
The heat transfer
coefficient is constant and uniform over the entire surface.
4
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 properties of hot dog available are given to be
ρ
= 980 kg/m
3
and
C
p
= 3900 J/kg.
°
C.
Analysis
(
a
) From Fig. 414b we have
15
.
0
1
1
17
.
0
94
59
94
88
=
=
=
=
=


=


∞
∞
o
o
o
o
o
hr
k
Bi
r
r
r
r
T
T
T
T
The Fourier number is determined from Fig. 414a to be
20
.
0
47
.
0
94
20
94
59
15
.
0
1
2
=
=
=


=


=
=
∞
∞
o
i
o
o
r
t
T
T
T
T
hr
k
Bi
α
τ
The thermal diffusivity of the hot dog is determined to be
/s
m
10
2.017
2
7

×
=
=
=
α
→
=
α
s
120
m)
011
.
0
)(
2
.
0
(
2
.
0
20
.
0
2
2
2
t
r
r
t
o
o
(
b
) The thermal conductivity of the hot dog
is determined from
C
W/m.
0.771
°
=
°
×
=
αρ
=

C)
J/kg.
)(3900
kg/m
/s)(980
m
10
017
.
2
(
3
2
7
p
C
k
(
c
) From part (
a
) we have
15
.
0
1
=
=
o
hr
k
Bi
. Then,
m
0.00165
m)
011
.
0
)(
15
.
0
(
15
.
0
0
=
=
=
r
h
k
Therefore, the heat transfer coefficient is
C
.
W/m
467
2
°
=
°
=
→
=
m
0.00165
C
W/m.
771
.
0
00165
.
0
h
h
k
434
Water
94
°
C
Hot dog
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Chapter 4
Transient Heat Conduction
448
Using the data and the answers given in Prob. 443, the center and the surface temperatures of the
hot dog 4 min after the start of the cooking and the amount of heat transferred to the hot dog are to be
determined.
Assumptions
1
Heat conduction in the hot dog is onedimensional since it is long and it has thermal
symmetry about the center line.
2
The thermal properties of the hot dog are constant.
3
The heat transfer
coefficient is constant and uniform over the entire surface.
4
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 properties of hot dog and the convection heat transfer coefficient are given or obtained in
P447 to be
k
= 0.771 W/m.
°
C,
ρ
= 980 kg/m
3
, C
p
= 3900 J/kg.
°
C,
α
= 2.017
×
10
7
m
2
/s, and
h
= 467 W/
m
2
.
°
C.
Analysis
The Biot number is
66
.
6
)
C
W/m.
771
.
0
(
)
m
011
.
0
)(
C
.
W/m
467
(
2
=
°
°
=
=
k
hr
Bi
o
The constants
λ
1
1
and
A
corresponding to this
Biot number are, from Table 41,
5357
.
1
and
0785
.
2
1
1
=
=
A
λ
The Fourier number is
2
.
0
4001
.
0
m)
011
.
0
(
s/min)
60
min
/s)(4
m
10
017
.
2
(
2
2
7
2
=
×
×
=
=

L
t
α
τ
Then the temperature at the center of the hot dog is determined to be
C
73.8
°
=
→
=


=
=
=


=


∞
∞
0
0
)
4001
.
0
(
)
0785
.
2
(
1
0
,
2727
.
0
94
20
94
2727
.
0
)
5357
.
1
(
2
2
1
T
T
e
e
A
T
T
T
T
i
cyl
o
τ
λ
θ
From Table 42 we read
J
0
=0.2194 corresponding to the constant
λ
1
=2.0785. Then the temperature at
the surface of the hot dog becomes
C
89.6
°
=
→
=


=
=
=




∞
∞
)
,
(
05982
.
0
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 Spring '11
 ENgel
 Heat Transfer, TI, transient heat conduction, T∞

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