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Chapter 2
Heat Conduction Equation
286
A long resistance heater wire is subjected to convection at its outer surface. The surface temperature
of the wire is to be determined using the applicable relations directly and by solving the applicable
differential equation.
Assumptions
1
Heat transfer is steady since there is no indication of any change with time.
2
Heat transfer
is onedimensional since there is thermal symmetry about the center line and no change in the axial
direction.
3
Thermal conductivity is constant.
4
Heat generation in the wire is uniform.
Properties
The thermal conductivity is given to be
k
= 15.1 W/m
⋅
°C.
T
∞
h
r
k
g
T
∞
h
r
o
0
Analysis
(
a
) The heat generation per unit volume of the wire is
&
&&
(.
.
g
Q
V
Q
rL
gen
gen
o
==
=
=
×
wire
2
3
W
m) (6 m)
W/m
π
2
8
2000
0 001
1061 10
The surface temperature of the wire is then (Eq. 268)
TT
gr
h
s
o
=+
=
°
+
×
°
=°
∞
&
)
(
2
30
1061 10
0 001
2 140
8
C
m)
W/m . C)
3
2
409 C
(
b
) The mathematical formulation of this problem can be expressed as
0
1
=
+
⎟
⎠
⎞
⎜
⎝
⎛
k
g
dr
dT
r
dr
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 Fall '10
 Dr.DanielArenas
 Thermodynamics, Resistance, Convection, Mass, Heat

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