Chapter 2
Heat Conduction Equation
2101
A spherical shell with variable conductivity is subjected to specified temperatures on both sides.
The variation of temperature and the rate of heat transfer through the shell are to be determined.
Assumptions
1
Heat transfer is given to be steady and onedimensional.
2
Thermal conductivity varies
linearly.
3
There is no heat generation.
Properties
The thermal conductivity is given to be
kT
k
T
()
(
)
=
+
0
1
β
.
Solution
(
a
)
The rate of heat transfer through the shell is expressed as
&
Qk
r
r
TT
rr
sphere
ave
=
−
−
4
12
21
π
where
r
1
is the inner radius,
r
2
is the outer radius, and
⎟
⎠
⎞
⎜
⎝
⎛
+
+
=
=
2
1
)
(
1
2
0
ave
ave
T
T
k
T
k
k
r
1
r
2
T
1
k
(
T
)
r
T
2
is the average thermal conductivity.
(
b
) To determine the temperature distribution in the shell, we begin with the Fourier’s law of heat
conduction expressed as
&
T
A
dT
dr
=−
where the rate of conduction heat transfer
is constant and the heat conduction area
A
= 4
π
r
&
Q
2
is variable.
Separating the variables in the above equation and integrating from
r
=
r
1
where
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 Fall '10
 Dr.DanielArenas
 Thermodynamics, Conductivity, Mass, Heat, Heat Transfer, Thermal conductivity, r2 − r1, kave

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