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Chapter 2
Heat Conduction Equation
Assumptions
1
Heat conduction is steady and onedimensional since there is no change with time and
there is thermal symmetry about the mid point.
2
Thermal conductivity is constant.
3
There is no heat
generation in the container.
Properties
The thermal conductivity is given to be
k
= 1.5 W/m
⋅
°C. The specific heat of water at the
average temperature of (100+20)/2 = 60
°
C is 4.185 kJ/kg
⋅°
C (Table A9).
Analysis
(
a
) Noting that the 90% of the 500 W generated by the strip heater is transferred to the container,
the heat flux through the outer surface is determined to be
2
2
2
2
2
W/m
0
.
213
m)
(0.41
4
W
500
90
.
0
4
=
×
=
=
=
ππ
r
Q
A
Q
q
s
s
s
&
&
&
Noting that heat transfer is onedimensional in the radial
r
direction and heat flux is in the negative
r
direction, the mathematical formulation of this problem can be expressed as
0
2
=
⎟
⎠
⎞
⎜
⎝
⎛
dr
dT
r
dr
d
and
Tr
T
()
11
100
== °
C
k
dT r
dr
q
s
&
2
=
(
b
)
Integrating the differential equation once with respect to
r
gives
r
dT
dr
C
2
1
=
r
1
r
2
T
1
k
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 Fall '10
 Dr.DanielArenas
 Thermodynamics, Conductivity, Mass, Heat

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