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Chapter 4
Transient Heat Conduction
438
A long cylindrical shaft at 400
°
C is allowed to cool slowly. The center temperature and the heat
transfer per unit length of the cylinder are to be determined.
Assumptions
1
Heat conduction in the shaft is onedimensional since it is long and it has thermal symmetry
about the center line.
2
The thermal properties of the shaft 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 stainless steel 304 at room temperature are given to be
k
= 14.9 W/m.
°
C,
ρ
=
7900 kg/m
3
,
C
p
= 477 J/kg.
°
C,
α
= 3.95
×
10
6
m
2
/s
Analysis
First the Biot number is calculated to be
705
.
0
)
C
W/m.
9
.
14
(
)
m
175
.
0
)(
C
.
W/m
60
(
2
=
°
°
=
=
k
hr
Bi
o
Steel shaft
T
i
= 400
°
C
Air
T
∞
= 150
°
C
The constants
λ
1
and
A
1
corresponding to this
Biot number are, from Table 41,
11
10935
11558
==
..
and
A
The Fourier number is
1548
.
0
m)
175
.
0
(
s)
60
/s)(20
m
10
95
.
3
(
2
2
6
2
=
×
×
=
=
−
L
t
α
τ
which is very close to the value of 0.2. Therefore, the oneterm approximate solution (or the transient
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This note was uploaded on 01/19/2012 for the course PHY 4803 taught by Professor Dr.danielarenas during the Fall '10 term at UNF.
 Fall '10
 Dr.DanielArenas
 Thermodynamics, Mass, Heat

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