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Natural Convection and Inclined Parallel Plates |
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Back ground article....:
A volumetric approach to natural convection
Back ground article....:
Natural Convection and Chimneys
Home page.................:
Frigus Primore home page
Figure 1
The air temperature increases is higher for heat sinks
with vertical fins than heat sinks with inclined fins.
Background
Natural convection has been used to cool electronics for
ages. It is a simple and reliable method. The big drawback,
of coarse, is that the cooling capacity is limited. All
possible means to enhance natural convection are therefore
more than welcome. This article is about one particular
method, inclined parallel plates.
My first encounter with this issue happened when I was
trying to design a cooling system for active electronics
mounted in radio masts. Fans were excluded for reliability
reasons. The only realistic alternative was therefore
natural convection. The equipment was enclosed in metal
boxes that needed to be cooled from one side. Applying a
wide but not very high heat sink could have solved the
problem but horizontal structures are not considered
esthetical in radio masts. A second restraint was
therefore that the heat sink not could be made much
wider than the mast itself.
I was about to give up on the problem when a colleague of
mine came to my rescue. Supported by CFD calculations he
suggested that there were gains to be made if the heat
sink was made quite high and with inclined fins. Even
if the project, for other reasons than thermal, later
was abandoned, this discovery trigged me to find out
more about the issue.
A first overview can be achieved with a very simple
comparison. Figure 1 shows a heat sink with vertical
fins and another one with inclined fins. It is somewhat
difficult to compare their performance on single unit
bases but if they are stacked things get much clearer.
For the vertical arrangement it is apparent that the
top heat sink is less effective than the bottom heat
sink, because it receives heated air from below. For
the inclined arrangement this is not the case. Each
stacked unit will in fact approximately contribute as
much as the units below. My first idea was therefore
that there must be a critical height above which inclined
fins perform better than the vertical fins. That turned
out to be wrong! What I found was that inclined fins
always perform better than vertical fins.
Figure 2
The angular orientation has a rather limited impact on the
cooling of a single plate.
The angle impact
Basic heat transfer theory reveals that the angular
orientation has a rather limited impact on the cooling
of a single plate, figure 2. For parallel plates there
are reports that the angle does not have much impact
either. Some reports even suggest no impact at all if
the inclination angle is smaller than 60 deg. These
observations are of coarse true but it is doubtful if
they also are valid for closely spaced plates. This is
important since the optimum spacing always is found in a
region between single plate characteristics and closely
spaced plates characteristics. See: A Volumetric Approach
to Natural Convection.
Figure 3
The asymmetry of the velocity profile must reasonably be
less important for plates with a small spacing than for
plates with large spacing.
A particular difficulty when dealing with inclined parallel
plates is the asymmetric velocity profile. For single
plates it is apparent that the velocity gradient is
higher on the downward side than on the upward side. This
is also the case for parallel plates separated by a large
distance, figure 3. For closely spaced plates it however
seems reasonable to assume that the viscous forces are so
strong that any important asymmetry not can be formed.
For plates with large spacing it is therefore reasonable
to assume that the describing equations are independent
of the inclination angle and the same as for parallel
vertical plates. For plates with small spacing it is
reasonable to assume that the velocity profile is
symmetric. The describing equations would in that case
also be the same as for vertical plates but with the
difference that gravity is replaced by the gravity
component in the plate direction.
The problem is to find a realistic approach for plates
placed on optimum distance. A straightforward way could
be to use the same method as for vertical plates, which
is to let a smooth function operate on the two extreme
cases small and large plate spacing. The drawback with
this line of attack is that the optimisation equations
become quite complex. Another possibility is to assume
that the velocity profile is symmetric at optimum plate
spacing. Both these methods were tried and the result
difference was found to be minor.
The equations presented here are based on the latter
assumption. Unfortunately there does not seem to exist
any hard laboratory evidence that they are correct.
Using the vocabulary that a theory is a confirmed
hypothesis, what follows is definitely a hypothesis.
The result is however so interesting that it is worth
sharing with others.
Figure 4
Optimum equations.
Optimum equations
Figure 4 shows the optimum equations and table 1 the
corresponding proportionality parameters. The equations
are similar to those for vertical parallel plates but
there are a couple of extra terms that handle the angle
impact. It should also be noted that the optimum plate
spacing is a function of the plate length, which in
this case not is uniform. The plates in the corners
must therefore be given a closer spacing than other
plates.
Table 1
Parameters for the equations in figure 4.
Equation 3 is based on equation 2 but an efficiency
term has been added. The idea behind this term is
that it should take care of all deviations from
ideal conditions such as non-isothermal plates,
non-negligible plate thickness, non-optimum plate
spacing and U-shaped flow channels. The temperature
difference term has in addition been indexed to
underline that the equation is valid for
non-isothermal conditions. The order of the
efficiency that can be achieved can probably vary
considerably. Based on the experience for vertical
plates a course estimate would be 70% - 90%.
Figure 5
Optimum inclination angle and possible gains relative
to vertical plates.
Figure 5 shows the optimum angle and the gains that
can be made relative to vertical plates. The optimum
angle is very small for low H/W ratios and it
asymptotically approaches 45 deg for large H/W
ratios. The heat dissipation relative to vertical
plates is described by the "Kgamma" term in equation 2.
This function is mapped in the lower image and it
shows the somewhat surprising phenomenon that an
inclination angle always is beneficial. The gains
are however very modest for small H/W ratios.
Discussion
The benefits at H/W<1 are obviously too small to
make inclined arrangements worthwhile. For higher
H/W ratios there might however be some attractive
cases. A coarse rule of thumb could be that the
matter should be considered if H/W>4. The potential
gain at this ratio is 50%, which could be a
sufficient incitement to take the extra costs
associated with an inclined arrangement.
It might be possible that a stack of inclined PCBs
could be a solution to some odd problem but the major
application for inclined plates is nevertheless heat
sinks. There are nevertheless several practical
problems associated with such designs. Manufacturing
is one of them. Casting seems to be the only
realistic alternative. Another difficulty is that a
heat sink not is effective unless the base plate
temperature is reasonably uniform. It is apparent
that this can be difficult to arrange for high but
not very wide heat sinks. A chimney arrangement,
that occupies the same volume but where the heat
sink itself is much smaller, could therefore be a
more attractive solution. An additional problem is
that heat sinks with inclined fins not can be placed
side by side.
Even if there are few attractive applications for
electronics there is however one that seems to be
both realistic and beneficial: cooling of high
voltage transformers. These devices often have a
considerable H/W ratio and their cooling fins seem
to be manufactured by casting.
