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A Fourier Series Solution |
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Convection cooled plate with discrete heat sources
Introduction
Various types of finite methods are often used to solve combined
conduction and convection problems. These methods are both flexible
and accurate but they are also complex to program and relatively
slow. At times it can there-fore be advantageous to use an analytical
approach. The list of pros and cons for this category of methods is
very much a mirror of those for the finite methods. They are easy
to program, usually fast but not very flexible.
This article is about a Fourier series solution to the problem of
calculating the temperature distribution on a convection cooled plate
with discrete heat sources. It is by no means the only way to tackle
with this problem. At times it is nevertheless the most convenient
one.
The solution has been known in the heat transfer community for
several years. As far as the author knows its origin is to find at
the University of Poitiers, France. The purpose of this article
is therefore not to explain the how it is derived but to discuss
some of its application aspects.
Figure 1
The 2-D solution is a Fourier series in which the coefficients are
given by another Fourier series.
The solution
The 2-D solution is shown in figure 1. It is basically a Fourier
series in which the coefficients are given by another Fourier
series. Double sided cooling has been assumed in this particular
case. This can however easily be changed by dividing the heat
transfer coefficient with a factor 2, (or by suppressing number
2 in the fourth equation). It should also be noted that the
solution is the simplest in a family of solutions that originates
from a 3-D solution for a plate with axis dependent thermal
conductivities. Since their application aspects are similar the
discussion that follows is relevant for all members of this
family.
The main advantage with the solution is its simplicity. It can
programmatically be fitted into a few procedures, which greatly
facilitates code writing. The main disadvantage is that its
application window is limited to simple cases. It can sometimes
also be slow. As this article will reveal there are nonetheless
several tricks that can counterbalance these drawbacks.
Figure 2
Fourier series can be used to model square wave functions. The fit
is strongly dependent on the number of terms included.
To show the total derivation of the solution would require too much
space but the basic principle is to apply two perpendicular square
functions to model the heat source flux. These functions take the
value 1 above the heat source and the value 0 elsewhere. A standard
approach for describing such functions is to use Fourier series.
These series can be made to model just about any function but their
drawback is that the fit only is perfect if an infinite number of
terms are included. One must therefore always settle for an
approximation, figure 2.
To determine the term count needed in a Fourier series is always
problematic. The terms do have a tendency to decrease as the series
is expanded but not in an orderly and regular manner. It is
therefore difficult to use the value of last term as criteria for
halting the summation. For the square wave function there is
fortunately another possibility. Each term in the modelling
series can be looked at as cycloid function with a wavelength
that successively decreases. It is evident that a good fit only
can be achieved unless the last wavelength is smaller than the
width of the square wave. The rule of thumb is to break the
summation when the width is covered by 1.5 wavelengths. The
equations for N and M in figure 1 reflect this rule and for the
particular case in figure 2 it yields N=15.
Figure 3
The accuracy impact of the term count is most important in the
centre of the heat source. At some distance it is therefore
possible to use half the value recommended in figure 1.
The term count has a large impact on the calculation time. It is
therefore highly desirable to reduce it as much as possible.
Figure 3 shows the impact on the temperature level for the same
example as in figure 2. The accuracy seems to reach saturation
for N>15, which confirms that the equations given for M and N
in figure 1 are reasonable. When the maximum summation index is
reduced to N=8 there is some impact inside the heat source but
hardly any on its outside. It therefore seems possible to reduce
the recommended values for M and N with a factor 2 for points
that are at some distance from the heat source. This is important
because it reduces the calculation time with a factor 4.
Figure 4
Superposition can be used to simulate the impact of several heat
sources.
Superposition
The super position principle can always be used if the sum of the
solutions to an equation also is a solution to that equation. All
conduction problems in which the material conductivity is an invariant
belong to this category. The superposition principle can therefore be
used for almost all conduction problems in thermal design but with a
possible exception for half conductors such as silicone. The solution
discussed is actually a large scale application of this principle.
Superposition can naturally also be used to simulate the combined
impact of several heat sources on a plate, figure 4. This opens up the
possibility for fast studies on how a change of a heat source property
in a multi-heat source environment impacts the total temperature
field. A procedure that manages this has five steps:
1. Calculate the temperature field for the heat source.
2. Subtract it from the total field.
3. Change the properties of the heat source.
4. Calculate the new temperature field for the heat source.
5. Add it to the total field.
The advantage with this procedure is that the temperature field only
has to be recalculated for one heat source at a time. If there are
50 of them this can save a lot of calculation time. This technique
can for example be used for rapid layout studies on PCBs. The solution
discussed is however still is a bit too slow to make this fully
enjoyable when the PCBs are large.
Figure 5
Various heat source shapes can be created by combining heat sources.
The superposition principle can also be used to simulate various heat
source shapes. Figure 5 shows a few examples. It should nevertheless
be noted that using overlapping, although mathematically elegant,
always implies the risk that the two calculation errors combine
unfavourably.
Execution speed
The execution speed is highly dependent on the term count. When it is
low, such as for the example in figure 3, this is not a concern. For
small heat sources on large plates the situation is however radically
different. A typical 200x200 mm PCB with a 10x10 mm heat source would
for example require 60 terms in each series which adds up to 3600
terms when the two nested series are combined. If the ambition is to
create an overview of the temperature field, calculations must further
be made on a grid of the order 20x20. All in all this results in 1.4
million terms. If there in addition are some 50 components, it is easy
to realise that the task is heavy.
There are nevertheless a couple of measures that can be taken to speed
up the process. To reduce the term count with a factor 2 for points
outside the heat source has already been mentioned. Another
possibility is to save the terms in the second summation, f(n,y), in
a list and reuse them for all points that have the same y-value.
These two measures boost the calculation speed considerably but for
large densely packed PCBs it is still difficult to get down to
enjoyable execution times.
The situation is much better for heat sinks. They are usually smaller
than PCBs and the heat source count is in most cases limited to less
than 10. Overviews can therefore be created in a matter of a few
seconds, which is a key issue for successful front-end thermal design.
Those who not have tired this method can get an idea of how it works
on the web site given at the be-ginning of this article. The full
potential of the method can however not be exploited unless the fluid
temperature increase problem is reasonably well managed.
Figure 6
The basic procedure calculates the temperature distribution as if the
fluid temperature was a constant. An array of large heaters that cover
the entire plate can be used to compensate for this error.
Fluid temperature increase compensation
One problem with the Fourier series solution is that it only is valid
for uniform fluid temperatures and heat transfer coefficients. This
limitation always causes an error. It is often negligible for small
heat sinks but for large ones cooled by small flow rates, it can be
considerable.
Figure 6 shows a case with a high fluid temperature increase. For this
case it is obvious the cooling is much more effective at the inlet
than at the outlet. This effect is partly caused by a variation of
the local temperature difference and partly by a variation of the
local heat transfer coefficient.
Figure 7
Different array arrangements can be used for the compensation heat
sources but the sum of their heat dissipations must always be zero.
One way to rectify for this discrepancy is to use compensation heat
sources. They are typically only kept alive during the calculation
process and they should cover the entire plate. Various array
arrangements are possible but the simplest is to use a single inline
layout, figure 7. Another restriction is that the sum of their heat
dissipations always must equal zero.
A first level approach, which for many cases is sufficient, is to
initially assume that the plate is isothermal, figure 6. It is a
simple task to calculate the corresponding fluid temperature
profile. The deviation between the average fluid temperature and the
actual fluid temperature determines the strengths of the compensation
sources and this is done with a very simple equation. The calculation
is performed with the average heat transfer coefficient as parameter
and the average fluid temperature as an offset value.
The first level approach basically simulates the fluid temperature
profile as if the plate had been isothermal but it is possible to go
one step further. The second level approach consists in calculating
the average temperature for the compensation sources and use these as
a bases for a better approximation of the temperature profile. The
basic principle is the same as above but with the complication that
the plate now has to be looked at as composed of several isothermal
segments. Another difference is that it for this case can make sense
to use a two dimensional layout array.
It should also be noted that the compensation sources are large
sources that calculate rapidly. Compensating for the fluid
temperature increase is therefore not a process that considerably
charges the all over execution time.
Figure 8
Comparison of the temperature distribution in a 210x260 mm heat sink
bottom plate, with and without fluid temperature compensation.
Figure 8 shows an example of the impact of an air temperature
increase compensation for rather extreme case. The heat sources are
in this case located near the inlet. Their temperature would
consequently be calculated too high if no fluid temperature
compensation was made. The result difference is on the 20% level.
It is not radical but it considerably expands the usefulness of the
solution.
Conclusions
The main advantage with the method discussed is that it simple to
program.
The calculation speed can be a problem when the heat source is much
smaller than the plate.
The superposition principle can be use to rapidly examine how the
properties of an individual source in a multi-source environment
impacts the total temperature field.
Compensation heat sources can be used to include the impact of the
temperature change of the cooling fluid.
Ake Malhammar