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 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.
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.
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 nonuniform 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 halfintuitive 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 minifan 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 nonfriction 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.
about ake malhammer
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 transferengineering 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 frontend 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/.
