Chapter 2
Heat Conduction Equation
2133
A cylindrical shell with variable conductivity is
subjected to specified temperatures on both sides. The rate of
heat transfer through the shell is to be determined.
Assumptions
1
Heat transfer is given to be steady and one
dimensional.
2
Thermal conductivity varies quadratically.
3
There is no heat generation.
Properties
The thermal conductivity is given to be
.
kT
k
T
()
(
)
=+
0
2
1
β
Analysis
When the variation of thermal conductivity with
temperature
k
(
T
) is known, the average value of the thermal
conductivity in the temperature range between
TT
is
determined from
12
and
r
2
T
2
r
r
1
T
1
k
(
T
)
⎥
⎦
⎤
⎢
⎣
⎡
+
+
+
=
−
⎥
⎦
⎤
⎢
⎣
⎡
−
+
−
=
−
⎟
⎠
⎞
⎜
⎝
⎛
+
=
−
+
=
−
=
∫
∫
2
1
2
1
2
2
0
1
2
3
1
3
2
1
2
0
1
2
3
0
1
2
2
0
1
2
ave
3
1
3
3
)
1
(
)
(
2
1
2
1
2
1
T
T
T
T
k
T
T
T
T
T
T
k
T
T
T
T
k
T
T
dT
T
k
T
T
dT
T
k
k
T
T
T
T
T
T
This relation is based on the requirement that the rate of heat transfer through a medium with constant
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 Fall '10
 Dr.DanielArenas
 Thermodynamics, Conductivity, Mass, Heat, Heat Transfer, 2 K, 4k, 0.016 M, 10 7 W

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