Chapter 2
Heat Conduction Equation
Variable Thermal Conductivity
294C
During steady onedimensional heat conduction in a plane wall, long cylinder, and sphere with
constant thermal conductivity and no heat generation, the temperature in only the
plane wall
will vary
linearly.
295C
The thermal conductivity of a medium, in general, varies with temperature.
296C
During steady onedimensional heat conduction in a plane wall in which the thermal conductivity
varies linearly, the error involved in heat transfer calculation by assuming constant thermal conductivity at
the average temperature is (
a
)
none
.
297C
No, the temperature variation in a plain wall will not be linear when the thermal conductivity
varies with temperature.
298C
Yes, when the thermal conductivity of a medium varies linearly with temperature, the average
thermal conductivity is always equivalent to the conductivity value at the average temperature.
299
A plate with variable conductivity is subjected to specified
temperatures on both sides. The rate of heat transfer through the plate is
to be determined.
Assumptions
1
Heat transfer is given to be steady and onedimensional.
2
Thermal conductivity varies quadratically.
3
There is no heat
generation.
Properties
The thermal conductivity is given to be
k T
k
T
(
)
(
)
0
2
1
.
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
T
T
1
2
and
can be determined from
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
average thermal conductivity
k
ave
equals the rate of heat transfer through the same medium with variable
conductivity
k
(
T
). Then the rate of heat conduction through the plate can be determined
to be
L
T
T
A
T
T
T
T
k
L
T
T
A
k
Q
2
1
2
1
2
1
2
2
0
2
1
ave
3
1
Discussion
We would obtain the same result if we substituted the given
k
(
T
) relation into the second part
of Eq. 276, and performed the indicated integration.
258
T
2
x
k
(
T
)
L
T
1
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Chapter 2
Heat Conduction Equation
2100
A cylindrical 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.
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 Spring '08
 BENARD
 Heat Transfer

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