Heat Chap04-047 - Chapter 4 Transient Heat Conduction 4-47...

Chapter 4 Transient Heat Conduction
4-47 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 one-dimensional 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
one-term 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. 4-14b 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. 4-14a 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
4-34
Water
94 ° C
Hot dog

Chapter 4 Transient Heat Conduction
4-48 Using the data and the answers given in Prob. 4-43, 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 one-dimensional 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
one-term 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
P4-47 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 4-1,
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 4-2 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