Chapter 17
Steady Heat Conduction
Critical Radius Of Insulation
1783C
In a cylindrical pipe or a spherical shell, the additional insulation increases the conduction
resistance of insulation, but decreases the convection resistance of the surface because of the increase in
the outer surface area. Due to these opposite effects, a critical radius of insulation is defined as the outer
radius that provides maximum rate of heat transfer. For a cylindrical layer, it is defined as
r
k
h
cr
/
where k is the thermal conductivity of insulation and
h
is the external convection heat transfer
coefficient.
1784C
It will decrease.
1785C
Yes, the measurements can be right. If the radius of insulation is less than critical radius of
insulation of the pipe, the rate of heat loss will increase.
1786C
No.
1787C
For a cylindrical pipe, the critical radius of insulation is defined as
r
k
h
cr
/
. On windy days,
the external convection heat transfer coefficient is greater compared to calm days. Therefore critical
radius of insulation will be greater on calm days.
1788
An electric wire is tightly wrapped with a 1mm thick plastic cover. The interface temperature
and the effect of doubling the thickness of the plastic cover on the interface temperature are to be
determined.
Assumptions
1
Heat transfer is steady since there is no indication of any change with time.
2
Heat
transfer is onedimensional since there is thermal symmetry about the centerline and no variation in the
axial direction.
3
Thermal properties are constant.
4
The thermal contact resistance at the interface is
negligible.
5
Heat transfer coefficient accounts for the radiation effects, if any.
Properties
The thermal conductivity of plastic cover is given to be
k
= 0.15 W/m
°C.
Analysis
In steady operation, the rate of heat transfer from the wire is equal to the heat generated within
the wire,
W
80
)
A
10
)(
V
8
(
VI
W
Q
e
The total thermal resistance is
C/W
4051
.
0
0735
.
0
3316
.
0
C/W
0735
.
0
)
m
10
(
C)
W/m.
15
.
0
(
2
)
1
/
2
ln(
2
/
ln(
C/W
3316
.
0
m)]
m)(10
(0.004
C)[
.
W/m
24
(
1
1
plastic
conv
total
1
2
plastic
2
conv
R
R
R
kL
r
r
R
A
h
R
o
o
Then the interface temperature becomes
C
62.4
)
C/W
4051
.
0
)(
W
80
(
C
30
total
1
total
2
1
R
Q
T
T
R
T
T
Q
The critical radius of plastic insulation is
mm
25
.
6
m
00625
.
0
C
.
W/m
24
C
W/m.
15
.
0
2
h
k
r
cr
Doubling the thickness of the plastic cover will increase the outer radius of the wire to 3 mm, which is
less than the critical radius of insulation. Therefore, doubling the thickness of plastic cover will increase
the rate of heat loss and decrease the interface temperature.
17

63
R
conv
T
2
R
plastic
T
1
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Chapter 17
Steady Heat Conduction
1789E
An electrical wire is covered with 0.02in thick plastic insulation. It is to be determined if the
plastic insulation on the wire will increase or decrease heat transfer from the wire.
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 Spring '08
 Chung
 Heat Transfer, Insulation, heat sink, Steady heat conduction, cosh aL

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