The fin shape issue

1
The fin shape issue
1
There are heat sinks with all kind of fin shapes
on the market. Which one is best?
2
The theoretical limit
2
The heat flow absorbed by the air pushed through
a heat sink is given by The maximum
possible heat dissipation, for fixed flow, is
determined by the maximum air temperature
increase, DTmax. This limit is independent of the
fin shape and it is approached when the fin
density is high.
3
Heat dissipation and mechanical power losses
3
The heat dissipated by a heat sink is associated
with a mechanical power loss. Some kind of
pumping action is therefore always needed, (also
for natural convection). The theoretical
dissipation limit is independent of the fin
shape. This is not the case for the mechanical
power loss. The optimum fin shape is the shape
that requires the least mechanical power input
for a given heat dissipation.
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Reynolds analogy
4
Reynolds analogy correlates the mechanical power
losses and the heat dissipation. The following
formulation is valid for small surface
element Note that the mechanical power
losses are strongly dependent on the velocity.
The lower is the velocity, the smaller are the
losses. At very low velocities the losses are so
small that the buoyancy forces are able to
provide sufficient pumping power, (natural
convection).
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The analogy number
5
Typical designs
Reynolds analogy is an idealization. Most devices
do not live up to it. It is therefore common
practice to introduce a compensation factor to
account for this discrepancy For typical
designs the mechanical power losses are only a
fraction of the heat dissipated.
6
The analogy number for a cylinder
6
The analogy number for flat plates is ? 1.27. The
flow conditions for cylinders are more
complicated. For low Re-conditions the tangential
velocity has a peak that is twice the approaching
velocity. If the approaching velocity, w0, is
chosen as the reference velocity, the theoretical
value for the analogy number for cylinders and
low velocity is ? 0.63. Higher air velocities
create vortex streets travelling downstream.
These vortexes consume mechanical energy but they
contribute little to heat transfer. The analogy
number therefore has a tendency to decrease
drastically when the velocity is increased.
7
The heat sink analogy number
7
The value of the analogy number for a heat sink
with rectangular fins depends on the
definition. The local analogy number based on the
velocity between the fins, wsp , is slightly
velocity dependent and has values in the range
0.6 - 0.8. The local definition is not sensitive
to the fin thickness and is therefore less
convenient for comparisons. It is better to base
the analogy number definition on the upstream
velocity, w0. It is also convenient to use the
average logarithmic temperature
difference It should be noted that this
definition can be used for any kind of heat sink,
regardless of the fin shape. It is therefore
ideal for comparisons.
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The importance of porosity
8
If the local analogy number has the value 0.7,
the mechanical power need to push the air through
a heat sink with rectangular fins is
approximately This equation is based on the
slot velocity, wsp, and not the upstream
velocity, w0. A small reformulation results
in The corresponding heat sink analogy number
is The heat sink analogy number is very
sensitive to the porosity of the heat sink. Low
porosity generates high internal air velocities
that dramatically decrease the analogy number.
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Optimization curve
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The image above shows calculated values for the
heat sink analogy number when the fin thickness
is varied. All impacts such as fin efficiency,
exit and entrance losses have been accounted for.
The analogy number is low when the porosity is
low. This effect is caused by high internal
velocities. At the other end of the curve, the
analogy number is low because the fin efficiency
declines.
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Heat sink comparisons
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A comparison
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Compare a heat sink with a circular pin fins and
a heat sink with a rectangular straight fins. If
the fin volume is conserved, this can be done in
several ways. The straight fins could be made
thin and dense or they could be made thick and
sparse. If the thermal conductivity is infinite,
the heat dissipation could be anything from
almost zero up to the dissipation limit. It must
therefore always be possible to find an
arrangement that exactly matches the heat
dissipation for the pin fins. If the thermal
conductivity of the fins is finite, this
conclusion is still valid because both
arrangements have the same fin cross section and
therefore also the same conduction losses. No
fin shape has any advantage over any other fin
shape for the two qualities heat dissipation and
weight.
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The winner
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The heat transfer surfaces in the rectangular
arrangement senses an almost uniform air
velocity. In the circular pin fin arrangement
however, the corresponding velocity is far from
uniform. The mechanical power that is required to
push the air along the surfaces is strongly
dependent on the velocity. The circular fin
arrangement will therefore always be penalized by
its velocity peaks and the result is that it will
consume more mechanical power.
You can not beat the rectangular fin arrangement!
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Conclusion
13
The analogy number for heat sinks makes it
possible to compare the mechanical power losses
with the heat dissipated, regardless of the fin
arrangement. The rectangular fin arrangement is
the best technical solution.
Note 1This conclusion is only valid when the air
enters the heat sink from the front.
Note 2All applied thermal design should be
targeted to produce economically optimized
products. The best technical solution is
therefore not always the best economical
solution.
A more elaborated approach can be found
inhttp//www.frigprim.com/downloads/Reynolds_ana
logy_heatsinks.PDF