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Chapter 2
Heat Conduction Equation
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.
Properties
The thermal conductivity is given to be
k
= 7.2 Btu/h
⋅
ft
⋅
°F.
Analysis
(
a
) Noting that heat transfer is 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
()
[(
1
1
)
]
T
22
160
== °
F
(
b
)
Integrating the differential equation once with respect to
r
gives
Steam
250
°
F
h
=1.25
L
= 15 ft
T
=160
°
F
C
=
1
r
dT
dr
Dividing both sides of the equation above by
r
to bring it to a readily integrable form and then integrating,
dT
dr
C
r
=
1
C r C
ln
=+
12
where
C
and
C
1
2
are arbitrary constants.
Applying the boundary conditions give
r
=
r
1
:
−
+
∞
k
C
r
C
r C
1
1
11 2
l
n
)
]
2
C
2
r
=
r
2
:
T
l
n
21
2
2
=
Solving for
C
simultaneously gives
1
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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, Conductivity, Mass, Heat

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