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Heat Sinks and Reynolds Analogy |
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Home page..................: Frigus Primore home page
Back ground article.....:
Convection, friction and Reynolds Analogy
Calculator....................:
Forced convection heat sink
Figure 1
The upper heat dissipation limit for a heat sink is determined
by the maximum possible temperature rise of the air and the
mass flow rate.
Introduction
Heat sink design is a subject that frequently is discussed.
The background is well known to anyone. Great progress has
also been made over the last years, particularly on the
software side. Almost all problems can therefore nowadays be
solved with modern numerical CFD tools.
There are nevertheless also disadvantages with numerical
methods. When compared with analytical methods the difference
is best explained with an analogy: Numerical methods create
single images but analytical methods make the movie. Things
are not quite that polarised in real life but it is definitely
true that analytical methods are superior for creating good
overviews. This is why they still are important.
This article is about an analytical approach to heat sink
design that uses Reynolds analogy. It ends with a definition
of a characteristic parameter, here called the heat sink
analogy number that can be used to compare heat sinks of
different size and type. The background takes a few pages to
develop but the end result is worth the effort. It basically
follows the standard scheme for all theoretical derivations.
The starting point is an over simplified imaginary case,
which step by step is expanded with various complications.
The author has used the heat sink analogy number concept for
a couple of years. All advantages with it will be advocated
below but there is at least also one disadvantage. It is
difficult, if not impossible, to adopt it to heat sinks with
bypass flow.
This theory has, as far as the author knows, never been
published in any wide spread publication. There are
subsequently no design tools, with one exception, that use
the heat sink analogy number. Those who are interested in
applying it for heat sinks with rectangular fins can however
do so on the web site given in the beginning of this article.
The heat dissipation limit
The heat flow that is dissipated from a heat sink can
basically be calculated in two different ways. As heat
flow rejected from the heat sink or as heat flow absorbed
by the airflow. For the purpose of this article it is the
latter that is of interest. The corresponding equation is
quite simple and states that the heat flow absorbed is
proportional to the mass flow and the temperature increase
of the air, figure 1. If the airflow is regarded as a
constant, it is quite apparent that it is the maximum
possible air temperature increase that sets the upper limit
for the heat that can be dissipated from a heat sink.
As always when discussing general principles it is convenient
to reduce the parameter count of the model as much as
possible. It is therefore, as a first step, assumed that
the thermal conductivity of the fin material is infinite. The
fin efficiency is subsequently always 100%, regardless of the
fin thickness. Another assumption made is that the bottom
plate is isothermal.
Given these assumptions it is always possible to virtually
imagine a heat sink in which the outlet air temperature
almost equals the bottom plate temperature. This would of
coarse imply a very large heat transfer surface but since the
fins can be made negligibly thin, they can also be made
innumerably many and still operate at 100% efficiency. This
conclusion is valid regardless of the total fin cross section
and regardless of the fin shape, whether rectangular, circular
or any other shape.
Figure 2
The mechanical power needed to overcome friction is proportional
to the square of the velocity.
Mechanical power losses and fin shape
The heat that is dissipated from a heat sink does not come free.
It is always associated with a loss of mechanical energy,
(except possibly for natural convection). Reynolds analogy can
be used to estimate the losses on the local level, figure 2. The
equation, which in this case is used in it most simple form, is
basically the same as for flow around single objects. The
difference is that the local velocity, which for single objects
is defined as the velocity outside the boundary layer, is an
ambiguous property in a heat sink. For rectangular fins it can
always be defined as the average velocity in the flow channels.
For pin fins it is a much more difficult matter. The equation
given in figure 2 should therefore rather be regarded as a
reflection of a general tendency than as an exact representation
of a well known phenomenon.
Reynolds analogy predicts that the mechanical power that is
needed to overcome friction is proportional to the square of
the velocity. A high velocity is therefore always associated
with a heavy prise, regardless of the fin shape. If the flow
channels have a uniform cross section it is fairly easy to
evaluate the all over consequences of this fact. If they have
a non-uniform cross section, such as is the case when they are
formed by the space between a bundle of cylinders, it is much
more complicated. The obvious reason for this is of coarse that
the velocity field is quite complex. One can however, as a first
level approach, assume that the velocity of interest varies
cyclically. The general rule that the mean square velocity always
is higher, or equal to, the mean linear velocity can then be
applied. This somewhat half-intuitive deduction indicates that
all velocity variations tend to increase the mechanical power
losses and therefore are unfavourable.
For the discussion that follows it is convenient to compare a
rectangular fin arrangement and a circular pin fin arrangement
with the same total fin cross section, figure 2. The comparison
is difficult to perform for the general case but quite easy
for the two special cases, very dense and very spares fin
spacing.
When the fin density increases the pin fin arrangement will
eventually reach a state where the fins touch each other. The
flow channels in the corresponding rectangular arrangement will
however still remain open. At this state it is apparent that the
rectangular arrangement is the best. For the reverse condition,
very sparse fin density, the flow conditions approach those for
single objects for which it also is known that the rectangular
profile is the best.
To show that the same conclusion also is valid for all conditions
in between these two extremes is more difficult. It can be shown
for some trivial cases, for example when the heat flux is
uniform. It can also be numerically shown for a large range of
simplified assumptions. So far the author has however not been
able to produce a 100% proof and his file with failed attempts
and half proofs is at this time quite thick. To get any further
it is therefore necessary to operate on the "beyond
reasonable doubt" level.
Figure 3
The fin volume can always be redistributed in such a way that
a pin and a rectangular fin arrangement have the same thermal
resistance.
It was shown above that all heat sinks, independently of the
fin type, potentially can operate in the interval between zero
and the upper heat dissipation limit, if the fin efficiency is
100%. It is therefore always possible to take the material in
a pin fin arrangement and redistribute it as a rectangular
arrangement without changing the heat dissipation, figure 3.
As also was deduced above, it is however the rectangular
arrangement that will require the smallest input of mechanical
power.
This conclusion is based on the assumption that the conductivity
of the fin material is infinite. Since the two cases compared have
the same total fin cross section the temperature losses in the
fins must however be equal. The conclusion that the rectangular
arrangement is the best therefore still holds also when the fin
efficiency is lower than 100%.
The same comparison can be made for any kind of fin shape and
for any kind of arrangement, for example staggered fins. The
result will always be the same. The performance of all through
rectangular fins can not be overridden.
Limitations
The conclusion made has a large validity but there are
nevertheless some exceptions. It is obvious that it only can be
applied if the airflow is parallel to the bottom plate and
perpendicular to the front. Heat sinks that have a mini-fan
mounted on the top, such as those found in most PCs, are
therefore definitely outside the scope of this article.
Another limitation is that Reynolds analogy only is valid for
friction losses. In- and outlet losses are therefore not
accounted for. These losses are in most cases small but can, if
the fins are short and thick, increase the pressure drop
significantly. To decrease the inlet losses by rounding the fin
tops could therefore possibly increase the all over performance.
The losses in the outlet side have a different character. It is
therefore not at all apparent that profile changes, such as
making the fins slightly drop shaped, would be beneficial.
A matter, which not is a limitation but still is important to
address, is the difference between economical performance and
physical performance. The former criterion is always used for
actual heat sink design. Heat sinks for non critical applications
can therefore be allowed to have a mediocre physical performance.
There are even cases for which the manufacturer has sacrificed
performance to make space for a logo. For critical applications
it is nevertheless imperative that the gap between the best and
the actual physical performance not is too large. In all cases,
it is also important to have access to tools that can show the
degree of the compromise that must be made.
Figure 4
The deviation from the perfect analogy is by tradition handled
with an analogy number.
The analogy number for a heat sink
Reynolds analogy is easy to formulate on the local level. For
single surfaces, channels and heat sinks it is more complex. One
problem is that of selecting a relevant reference velocity. Another
problem is that non-friction losses cause derivations. The
complications caused by these two problems is by tradition
handled with an analogy number, figure 4. This parameter can be
looked at as an efficiency term although its maximum value
sometimes is higher and sometimes lower than 100%.
The temperature difference definition is an additional problem.
For external flow around single surfaces there is but one
alternative, which is the surface to incoming air temperature
difference. In channels and heat sinks it is more complicated
since the temperature difference varies along the flow path.
Experience has shown that the mean logarithmic temperature
difference is the best choice for these cases.
For channel flow it is natural to select the average channel
velocity as reference velocity. Given this and using the mean
logarithmic temperature difference, the analogy number for long
and narrow rectangular flow channels vary in the range 0.6 - 0.9.
The lower value is valid for the side ratio 1.0 and the higher
value is valid for the almost parallel plate case. (This is the
main tendency for typical heat sinks. A more elaborate expose of
the subject would reveal that there are other situations to
consider but that the analogy number essentially will remain in
the given interval).
Figure 5
Reference velocity definition for a heat sink.
To select the reference velocity definition for heat sinks is a
slightly more intricate matter. It must also be noted that the
analogy number increases with square of the velocity used. A
small difference in the definition can therefore have a large
impact on the analogy number. The definition selected should in
addition be independent of the fin type and make the resulting
analogy number reflect the all over performance of the heat sink.
Given these requirements it is apparently impossible to use any
internal velocity.
The velocity that most frequently is used in heat transfer theory,
is the linear average velocity. It is defined as the ratio of a
volumetric flow and a cross section area. There is no reason to
deviate from this convention here. The flow used must obviously be
the flow that penetrates the heat sink. The issue of selecting the
reference velocity is therefore an issue of selecting an
appropriate cross section. The main problem here is caused by the
two outmost fins, (or fin rows in the pin fin case). The cross
section can be defined with or without double sided cooling for
these, figure 5. There is not much choice than to accepted them
both and let the application decide which one of them that should
be used. A consequence of this is that an analogy number
specification for a heat sink always should be accompanied by an
indication of the velocity definition used.
For a heat sink with rectangular fins it is evident that the fin
front velocity, as defined above, always is lower that the
channel velocity. The analogy number for the heat sink is
therefore always lower than the analogy number for its flow
channels, see the equations in figure 4. The former will in
addition also include the impact of the fin efficiency. Taken
these two factors into account one can conclude that heat sink
analogy number always must be considerably lower than the channel
analogy number.
Figure 6
The heat sink analogy number can be calculated from data that
normally is available for the heat sink.
Figure 6 shows an alternative way to formulate the definition of
the heat sink analogy number. The purpose of this equation is to
make it easier to evaluate data that usually is available. The
only difficulty in this respect is the mean logarithmic
temperature difference, which requires a bit of elaboration to
calculate.
It should also be noted that this equation is quite general. It
not only makes it possible to compare heat sinks of different
size but also heat sinks of different type.
Figure 7
Impact of fin thickness on the analogy number.
Figure 7 shows the analogy number as a function of the fin
thickness for a heat sink with rectangular fins. The values
are, as expected, zero in both ends of the scale. This effect
is caused by low fin efficiency when the fins are thin and by high
internal velocity when the fins are thick. The maximum
analogy number is found between these two extremes. It is on the
0.4 level, which is rather typical for heat sinks of this
kind.
It could be tempting to over interpret the diagram in figure 7.
What it actually reveals is the best choice of fin thickness for
the particular fin count assumed. It could however be that the
thermal resistance for this particular parameter set is far from
the data needed. It could also be that another fin count would be
more favourable. The analogy number is therefore not in itself a
way to optimise a heat sink but there are advantages in using it
as a part of an optimisation process.
A great advantage with the heat sink analogy number concept is
that it puts the heat sink on a reference scale. To check its
value whenever a heat sink is used is therefore a down to earth
everyday typical application. If it is on the 0.2 level or lower,
there are good grounds to assume that the heat sink could be better
designed.
Conclusions
The upper heat dissipation limit for a heat sink depends on the
airflow and the maximum possible air temperature increase. It is
independent of the fin arrangement.
All heat sinks, regardless of their fin type, has the potential to
dissipate heat between zero and the upper heat dissipation limit
if the fin efficiency is 100%. When arrangements with the same
total fin cross section are compared it is however always the
all through rectangular fin arrangement that results in the
smallest mechanical power losses. The latter conclusion is also
valid for fins with a finite conductivity.
It is possible to define a heat sink analogy number. This number
can be used to compare the performance of heat sinks regardless
of their size and fin arrangement.
Ake Malhammar
