Chapter 2
Heat Conduction Equation
263
A spherical container is subjected to specified temperature on the inner surface and convection on the
outer surface. The mathematical formulation, the variation of temperature, and the rate of heat transfer are
to be determined for steady onedimensional heat transfer.
Assumptions
1
Heat conduction is steady and onedimensional since there is no change with time and there
is thermal symmetry about the midpoint.
2
Thermal conductivity is constant.
3
There is no heat generation.
Properties
The thermal conductivity is given to be
k
= 30 W/m
⋅
°C.
Analysis
(
a
) Noting that heat transfer is onedimensional in the radial
r
direction, the mathematical
formulation of this problem can be expressed as
0
2
=
⎟
⎠
⎞
⎜
⎝
⎛
dr
dT
r
dr
d
and
Tr
T
()
11
0
==
°
C
r
1
r
2
T
1
k
T
∞
h
−=
−
∞
k
dT r
dr
hTr
T
[( )
]
2
2
(
b
)
Integrating the differential equation once with respect to
r
gives
r
dT
dr
C
2
1
=
Dividing both sides of the equation above by
r
to bring it to a readily integrable form and then integrating,
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 Fall '10
 Dr.DanielArenas
 Thermodynamics, Convection, Mass, Heat, Heat Transfer, dr r, dr ⎠

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