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Chapter 2
Heat Conduction Equation
291
A long homogeneous resistance heater wire with specified surface temperature is used to boil water.
The temperature of the wire 2 mm from the center is to be determined in steady operation.
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
= 8 W/m
⋅
°C.
Analysis
Noting that heat transfer is steady and onedimensional in the radial
r
direction, the mathematical
formulation of this problem can be expressed as
0
1
=
+
⎟
⎠
⎞
⎜
⎝
⎛
k
g
dr
dT
r
dr
d
r
&
and
180
°
C
(specified surface temperature)
Tr
T
s
()
0
==
dT
dr
0
0
=
(thermal symmetry about the centerline)
Multiplying both sides of the differential equation by
r
and rearranging gives
g
180°C
Resistance wire
r
r
o
r
k
g
dr
dT
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 Fall '10
 Dr.DanielArenas
 Thermodynamics, Resistance, Mass, Heat

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