From Equation 3.85, the heat loss is
Hence for copper,
Similarly, for the aluminum alloy and stainless steel, respectively, the heat rates are
q
f
5.6 W and 1.6 W.
2.
Since there is no heat loss from the tip of an infinitely long rod, an estimate of the
validity of this approximation may be made by comparing Equations 3.81 and 3.85. To
a satisfactory approximation, the expressions provide equivalent results if tanh
mL
0.99 or
mL
2.65. Hence a rod may be assumed to be infinitely long if
For copper,
Results for the aluminum alloy and stainless steel are
L
0.13 m and
L
0.04 m,
respectively.
L
2.65
398
W/m K
(
/4)(0.005
m)
2
100
W/m
2
K
(0.005
m)
1/2
0.19
m
L
L
2.65
m
2.65
kA
c
hP
1/2
8.3
W
398
W/m K
4
(0.005
m)
2
1/2
(100
25) C
q
f
100
W/m
2
K
0.005
m
q
f
hPkA
c
b
T
(
°
C)
100
300
250
200
150
100
50
0
20
40
80
60
T
∞
Cu
2024 Al
316 SS
x
(mm)
3.6
Heat Transfer from Extended Surfaces
163

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Comments:
1.
The foregoing results suggest that the fin heat transfer rate may accurately be predicted
from the infinite fin approximation if
mL
2.65. However, if the infinite fin approxi-
mation is to accurately predict the temperature distribution
T
(
x
), a larger value of
mL
would be required. This value may be inferred from Equation 3.84 and the requirement
that the tip temperature be very close to the ﬂuid temperature. Hence, if we require that
(
L
)/
b
exp(
mL
)
0.01, it follows that
mL
4.6, in which case
L
0.33, 0.23,
and 0.07 m for the copper, aluminum alloy, and stainless steel, respectively. These
results are consistent with the distributions plotted in part 1.
2.
This example is solved in the
Advanced
section of
IHT
.
3.6.3
Fin Performance
Recall that fins are used to increase the heat transfer from a surface by increasing the effec-
tive surface area. However, the fin itself represents a conduction resistance to heat transfer
from the original surface. For this reason, there is no assurance that the heat transfer rate
will be increased through the use of fins. An assessment of this matter may be made by
evaluating the
fi
n effectiveness
f
. It is defined as the
ratio of the fi
n heat transfer rate to the
heat transfer rate that would exist without the fi
n.
Therefore
(3.86)
where
A
c
,
b
is the fin cross-sectional area at the base. In any rational design the value of
f
should be as large as possible, and in general, the use of fins may rarely be justified unless
f
2.
Subject to any one of the four tip conditions that have been considered, the effectiveness
for a fin of uniform cross section may be obtained by dividing the appropriate expression for
q
f
in Table 3.4 by
hA
c
,
b
b
. Although the installation of fins will alter the surface convection
coefficient, this effect is commonly neglected. Hence, assuming the convection coefficient
of the finned surface to be equivalent to that of the unfinned base, it follows that, for the infi-
nite fin approximation (Case D), the result is
(3.87)
Several important trends may be inferred from this result. Obviously, fin effectiveness is
enhanced by the choice of a material of high thermal conductivity. Aluminum alloys and
copper come to mind. However, although copper is superior from the standpoint of thermal