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Chapter 2
Heat Conduction Equation
289
Heat is generated uniformly in a spherical radioactive material with specified surface temperature.
The mathematical formulation, the variation of temperature in the sphere, and the center temperature are to
be determined for steady onedimensional heat transfer.
Assumptions
1
Heat transfer is steady since there is no indication of any changes with time.
2
Heat transfer
is onedimensional since there is thermal symmetry about the mid point.
3
Thermal conductivity is
constant.
4
Heat generation is uniform.
Properties
The thermal conductivity is given to be
k
= 15 W/m
⋅
°C.
Analysis
(
a
) Noting that heat transfer is steady and onedimensional in the radial
r
direction, the
mathematical formulation of this problem can be expressed as
r
o
0
T
s
=80°C
k
g
r
constant
with
0
1
2
2
=
=
+
⎟
⎠
⎞
⎜
⎝
⎛
g
k
g
dr
dT
r
dr
d
r
&
&
and
80
°
C
(specified surface temperature)
Tr
T
s
()
0
==
dT
dr
0
0
=
(thermal symmetry about the mid point)
(
b
) Multiplying both sides of the differential equation by
r
2
and rearranging gives
2
2
r
k
g
dr
dT
r
dr
d
&
−
=
⎟
⎠
⎞
⎜
⎝
⎛
Integrating with respect to
r
gives
r
dT
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This note was uploaded on 01/14/2012 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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