Chapter 2
Heat Conduction Equation
2125
A steam pipe is subjected to convection on both the inner and outer surfaces.
The mathematical
formulation of the problem and expressions for the variation of temperature in the pipe and on the outer
surface temperature are to be obtained for steady onedimensional heat transfer.
Assumptions
1
Heat conduction is steady and onedimensional since the pipe is long relative to its
thickness, and there is thermal symmetry about the center line.
2
Thermal conductivity is constant.
3
There
is no heat generation in the pipe.
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
0
=
⎟
⎠
⎞
⎜
⎝
⎛
dr
dT
r
dr
d
and
−=
−
k
dT r
dr
hT Tr
ii
()
[(
1
1
)
]
−=−
k
dT r
dr
hTr
T
o
[()
]
2
2
o
(
b
) Integrating the differential equation once with respect to
r
gives
r
dT
d
C
=
1
r
2
r
r
1
T
i
h
i
T
o
h
o
r
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, ln r1

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