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 therefore 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 the 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 beginning 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.
  
  
  

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about ake malhammar

ake obtained his master of science degree in 1970 at kth, (royal school of technology), stockholm. he then continued his studies and financed them with various heat transfer-engineering activities such as deep freezing of hamburgers, nuclear power plant cooling and teaching. his ph.d. degree was awarded in 1986 with a thesis about frost growth on finned surfaces. since that year and until december 2000 he was employed at ericsson as a heat transfer expert. currently he is establishing himself as an independent consultant.

having one foot in the university world and the other in the industry, ake has dedicated himself to applying heat transfer theory to the requirements of the electronic industry. he has developed and considerably contributed to several front-end design methods, he holds several patents and he is regularly lecturing thermal design for electronics.

to read ake's website for more thermal information and software tools he has developed, visit http://akemalhammar.fr/.