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December 2005
library  >  Application Notes  >  Ake Malhammar

A Bessel Function Solution for the Temperature Distribution on Convection Cooled Plates with Discrete Heat Sources


introduction


there are many ways to calculate the temperature distribution on convection cooled plates. this article presents yet another one. it is fundamentally based on a modified bessel function solution to the temperature field that surrounds a circular heat source on an infinite surface. this fundamental solution is then step by step adapted to describe the temperature field on a rectangular plate with rectangular heat sources.

the method is a bit particular in the sense that it does not appear to be attractive at a first glance. bessel functions are known to be heavy to calculate and the circular approach may also seem awkward. these disadvantages can, however, be effectively bypassed and make the method fast, very fast. 




solution path
 
figure 1- the 3 steps needed to model a rectangular heat source on a rectangular plate.

the solution path


figure 1 shows the basic path. it starts with a solution for a circular heat source on an infinitely large plate. the next step is to model a rectangular source as an array of circular sources and the last step is to shape the rectangular plate with a mirror process. finding the temperature in a specific point on the plate is therefore a matter of adding the contributions from all these sources.



equations
 
figure 2 - temperature field solution for a circular heat source on an infinitely large plate cooled by convection. double sided cooling is assumed.



the temperature field solution for the first step is shown in figure 2. it holds several bessel functions and could, therefore, appear to be both complicated and heavy to calculate. it should nevertheless be noted that the functions c1 and c0 only depend on the radius of the heat source, r0 . if the purpose is to calculate the entire temperature field on the plate, they only need to be evaluated once. the functions that do need to be re-evaluated repeatedly are consequently only k0(kr) for external points and i0(kr) for internal points.




approximation error
 
figure 3 - the circular heat source approximation error increases with the side ratio but decreases rapidly with the distance from the source.

subdivision


to analyse the best way to approximate a rectangular heat source with an array of circular heat source is quite complicated. the problem is strongly non-linear, which makes a complete analysis almost impossible. in an applied approach one can therefore only focus on the most important matters. the general tendency is that the approximation error decreases with the size of the rectangle and increases with its side ratio. for side ratios up to about 5 the maximum error is found just outside the longest side of the rectangle and it decreases rapidly for more distant points, figure 3.

one issue that always is brought up in this context is whether the approximating circle should have the same area or the same circumference as the rectangle. this analysis is based on the same area. for typical pcb applications there are however indications that a weighted average between these two alternatives is better. the proposed solution can, therefore, be improved but the potential gain is not radical.    




subdivisions
 
figure 4- the circular heat source approximation error can be decreased by subdivision.



given these general tendencies it is evident that the approximation error always can be decreased by subdivisions, figure 4. it is also evident that the subdivision process should strive to produce sub rectangles with side ratios as near 1.0 as possible. an empirical correlation for the relative error, which accounts both for the side ratio and the size effect, is included in figure 4. for overview applications it is usually possible to accept a 10% error. the corresponding maximum heat source size, for a typical multi-layer pcb cooled by forced convection, can then be estimated to be of the order 20x30 mm. if the error level instead is set to 5% the maximum size decreases to 7x10 mm. the number of subdivision needed to simulate a typical component on a pcb is therefore in many cases zero and in any case quite limited!

the mirroring process


the purpose of the mirroring process is to model a rectangular plate with adiabatic sides, figure 1. it is a well know method but it is only effective if the image count is reasonably low. the fact that the temperature difference in a point decreases very rapidly with the distance from its heat source is very helpful in this particular case. the product of this distance, r, and the k-value is the critical parameter. the value of this parameter, at which the temperature contribution can be neglected, depends on the size of the heat source. a reasonably good criteria for typical pcb cases kr>6. 

a typical forced convection cooled multi-layer pcb has a k-value on the 30 1/m level. using the rule above the corresponding critical r-value can be determined to 200 mm. a 200x200 mm pcb can therefore be modelled with 1 - 2 mirror images at each side, which indeed is a limited number. other applications can be less favourable and if the mirror count becomes too large an alternative calculation method should be considered.

another issue of some importance is whether the heat sources on the mirror images need to be subdivided or not. this matter is not easy to address. a superficial analysis indicates that subdivisions only are needed for components that are near the plate edges and for extreme cases of size or side ratio. a simple strategy would be to always subdivide on the nearest layer of mirror images and only for extreme cases on the other ones.


 

calculation speed

 


the basic equation shown in figure 2 essentially holds two terms. one that depends on the approximation radius and one that depends on the distance to the centre of the heat source. the former only has to be calculated once but the latter must be recalculated for every calculation point. if the purpose is to map the temperature distribution on a pcb this needs to be done on a grid of the order 20x20. if there further are 50 components that each need to be mirrored 25 times, it would all in all imply evaluating a bessel function some 500 thousand times. that would be quite a heavy task even for a modern computer.
 
a solution out of this dilemma is to do what engineers did before the computer age, to use tables. to keep a table with some 1000 values is not a difficult task for a modern computer and such a table can in addition easily be stored in the cash memory, which further speeds up the process.

the proposed calculation procedure therefore consists of some initial procedures that are both complicated and heavy but only has to be done once.  once that basic structure is in place it is easy to calculate a temperature contribution from a heat source. all that is needed is to determine the distance to the its centre, interpolate in a bessel function in a table and multiply the result with a value that is characteristic for the heat source. these operations are all very simple and can consequently be done fast. 



pcb example
 
figure 5- comparison of a finite element calculation and a bessel function approximation.



some application aspects


the procedure is ideal for creating temperature overviews on pcbs. a particularly attractive possibility is to get a fast result when the position of a component is changed. how the super position principle can be used for this purpose is explained in the reference article. the procedure is actually so fast that the major part of the waiting time usually is consumed by the screen graphics.

there are nevertheless also limitations. the heat that is dissipated directly from the components to the air is always a problematic issue. compensations, with various degrees of sophistication, can be used to handle this impact but they will never be perfect. the method therefore works best for cases in which the component-to-air dissipation is much smaller than the part dissipated by the pcb. designs that have heat sinks or pcbs with a very low thermal conductivity are therefore better addressed with other methods. another problem is that the uniform heat source assumption does not always comply well with the actual conditions. devices that contact the pcb around their circumference are best modelled as a combination of positive and negative heat sources. modelling that will however always slow down the calculation speed.


despite these difficulties it is often possible to attain results that are surprisingly good. figure 5 shows a comparison with a finite element method. it is evident that the method cannot replace more accurate methods but it can indeed give a designer a good overview of the impact of different layout alternatives.

it could be of some interest to compare the calculation speed of the proposed method with that of the fourier series method in the reference article. for a 10x10 mm heat source on a 200x200 mm pcb the bessel function method uses the sum of some 25 terms whereas the fourier series method uses the sum of some 900 much more complicated terms. the speed difference is coarsely a factor 40!  this advantage will, however, decline for larger heat sources and at sizes of the order 60x60 mm it is reversed. the two methods are, therefore, complementary. the bessel function solution is favourable for small heat sources, the fourier series solution for large heat sources. both methods can of coarse be combined.






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 web site for more thermal information and software tools he has developed, please visit http://akemalhammar.fr/ - see more at: https://www.coolingzone.com/library.php?read=534#sthash.y3rcxrow.dpuf

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

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