introduction
optimum heat sink design has grown to become a hot subject. the evident reason, of course, is that the heat densities in electronics steadily are growing. the days when almost any heat sink could be used with a satisfactory result are therefore gone. things have become more complicated.
the major problem when optimising a heat sink is that there are quite a few parameters that can be changed. the process must therefore by necessity span over a large collection of parameter sets from which the best alternatives can be picked out. there are basically three steps in this process.
the first is to create an overview; the second is to refine the analyses and the third is to verify the result. all three steps are important but it is obvious that if the overview is missing or incomplete, large difficulties will follow.
heat sinks can basically operate under two distinctly different flow conditions, confined and nonconfined flow. this article is about optimisations for the confined case. the nonconfined case will be brought up in a coming issue. figure 1 the best compromise between fin pitch and fin thickness is the combination that results in the highest analogy number.
fin pitch and fin thickness for a given air velocity
the purpose of this type of optimisation is to find the best combination of fin pitch and fin thickness for a given incoming air velocity. the most elementary case is a heat sink with an isothermal bottom plate and rectangular fins. figure 1 shows an example. the thermal performance target is in this case set to 15 c/w. there are several pitch and thickness combinations that can meet this requirement. although they are thermally equal they do not have the flow properties. the best combination must evidently be the one that has the lowest pressure loss, or differently expressed, the highest analogy number.
the example shown is somewhat simplified in the respect that the fin pitch has been treated as a continuous parameter whereas it actually only can take discrete values. this has however been made intentionally and with the purpose of making the diagram more intuitively understandable. for applied designs it is better draw the data as points that each represent an actual possibility.
the tendencies in figure 1 are not difficult to understand. thin fins have low efficiencies and must therefore compensate this handicap with a larger total convection surface, which increases the pressure losses. thick fins need a smaller total convection surface but they also create higher internal air velocities that increase the pressure losses. the best combination is somewhere in between these two extremes.
it should also be noted that the differences between the alternatives are quite small. this reveals a tendency which is very typical for all heat sink optimizations; the curves are always flat arcs. this fact has two important consequences. the first is that the optimum point often is difficult to find. the second is that design parameters can be allowed to deviate considerably from their optimum value without causing too much penalty.
fin shape
it is relatively easy to compare heat sinks that have the same outer measures but different fin shapes. that type of evaluation is however merely a comparison of heat sinks a, b and c. a fair evaluation of the fin shape impact must by necessity be based on the principle of comparing the best with the best and that is a much more difficult task.
besides the outer measures there are several internal parameters that must be optimised. for rectangular fins it is the pitch, the thickness and various staggered arrangements. for circular pin fins it is the diameter, the pitch in two directions and the array pattern. such comparisons might be possible to perform numerically but to do them experimentally is, if not impossible, at least next to it.
there is fortunately an analytical approach to the fin shape problem. it is valid for all cases when the air flows from front to back, which covers the vast majority of cases in confined flow. it can be found in the article "heat sinks and reynolds analogy". the theory basically provides a proof as for why rectangular nonstaggered fins always perform better than any other fin shape. this fact is very important because it means that an optimised heat sink with rectangular fins always can serve as a reference point that marks the best possible option.
figure 2  the flow through a heat sink is determined by a system curve and a fan curve.
figure 3  thermal performance for various combinations of pitch and fin thickness for a heat sink placed in a fan system.
fin pitch and fin thickness for a given fan curve
the fan curve and the system curve determine the airflow in a fan system, figure 2. the fan curve is a function of the fan used and its voltage feed. the system curve is a function of the pressure loss in the heat sink plus the pressure losses in the channel work. the point where these two curves cross gives the equilibrium.
a heat sink that is placed in a fan system must obviously have an impact on the system curve. if its pressure losses are high it will sense a lower air velocity than if its pressure losses are low. this impact is however to some extent counterbalanced by the fact that high pressure losses also tend to increase the thermal performance. a typical result of these impacts is shown in figure 3. the tendency for large fin pitches is roughly the same as that for the given air velocity case, figure 1, but for small pitches the curves dive, which is it is radically different. each parameter set now has its maximum and it is the highest of these that defines the best choice.
it should be noted that there is no radical difference between the alternatives shown. this again underlines the fact that optimum curves are flat arcs. it should also be noted that figure 3 only shows the main tendency. actual applications are not that clearcut. the fin pitch can only take discrete values. secondary problems such as air leaks, speed controlled fans and air filter pressure loss must also be considered.
another important matter is that each curve in figure 3, although it appears to be continuos, actually is a line train composed of 40 single points. the entire diagram therefore comprises 160 calculations. it is probably possible to get away with a third of that number and still get an acceptable overview. it must nevertheless be stressed that all heat sink optimisation procedures are based on a large number of calculations. the time that each calculation requires is therefore a critical factor. if it is on the minute level the chances are that the diagram in figure 3 never would have been created. if it is one the second level or below, it is a much easier task. figure 4  the bottom plate thickness is a critical parameter when the heat source is discrete.
bottom plate thickness for a discrete heat source
another optimisation case appears when the bottom plate no longer is isothermal, figure 4. these cases emerge with a growing frequency, which reflects that extreme hot spot problems are becoming increasingly common.
as always it is important to quantify the problem. a straightforward way to do this is to define a thermal efficiency for the bottom plate. it follows the basic principle for any fin efficiency definition, that is, as a ratio between a virtual isothermal case and an actual case, figure 4. the order of this efficiency can vary considerably. for heat sinks that only are slightly wider than the heat source it is near 100% but for extreme cases it can be as low as 30%. values on that level are difficult to accept for a thermal designer. this is also why various heat pipe and vapour chamber solutions are seeing an increased interest.
a straightforward way to reduce the temperature losses in a bottom plate is to make it thicker. this can sometimes be done without too much difficulty but in most cases it must be made at the expense of the fin height. an example of the latter is shown in figure 4. the result is again a curve with a weak optimum. this method is a bit brutal in the sense that the bottom plate really only needs to be thickened near the heat source. it is nevertheless widely used because heat sinks with flat bottom plates are simple to manufacture.
figure 5  a triangular fin profile can result in some gains if the fin efficiency is low.
fin profile
for idealised conditions it can be shown that the best fin profile is a profile with slightly concave sides. it can also be shown that the difference between this profile and the triangular profile is small. in applications it is not that simple. all nonrectangular profiles tend to distort the flow channel cross section and thereby also the distribution of the local heat transfer coefficient. the idealised conditions for which the referenced theory is applicable are therefore no longer valid.
the fin profile problem is quite complex. before launching a big optimisation project it is therefore essential that the potential gains are estimated. figure 5 shows a comparison of the fin efficiency for rectangular and triangular profiles having the same volume and working under theoretically ideal conditions. it basically reveals that a triangular profile does not generate any important gains unless the fin efficiency is low. because the flow conditions in a heat sink not are ideal, it should in addition be considered that these gains are the best that can be hoped for.
nonrectangular fin profiles are some times used in forced convection applications. in the majority of cases it is not for thermal performance reasons but because they have been manufactured by extrusion. figure 6  comparison of the fin efficiency for a circular fins. both fin profiles have the same volume and but the nonrectangular fin is 4 times thicker at the heat source.
figure 7  to optimise a bottom plate profile is a very difficult task.
bottom plate profile
when the bottom plate efficiency is low it can be advantageous to give it a thickness profile. if the total height is limited this is an extraordinary difficult problem because the resulting variations of the fin height requires that each fin is optimised individually. if further the bottom plate thickness varies in the flow direction there are flow phenomena such as separation to consider.
as with any difficult problem it is important to estimate the potential gains before getting involved into details. even this is not easy and the result of the proposed method is truly approximate. the first step is to regard the bottom plate as a cooling fin. because rectangular fins are quit difficult to deal with, the second step is to make a circular approximation. a first coarse indication can therefore be given by comparisons such as in figure 6.
manufacturing difficulties often makes a total thickness optimisation impossible. in most cases one therefore has to settle for a variation in one direction only, figure 7. this reduces the potential gain with roughly a factor 2. the height of the fins above the heat source must in addition be reduced and since they are the ones that have the highest temperature, there will be further losses.
a reasonably realistic estimation is that about 30% of the gain shown in figure 6 is what can be hoped for. a heat sink with a uniform bottom plate thickness that operates at 30% efficiency would thus have a potential improvement factor of 1.2. this value is far from radical but it agrees well with the author's experience. if the application is critical it can nevertheless make the difference.
to make a bottom plate thickness optimisation is both a heavy and a frustrating task. there are two main reasons. the first is that there are quite a few measures that can be altered. the second is that the optimum is weak which makes it very difficult to determine how near the optimum point a specific parameter set is. the difference between two alternatives can often be as low as 0.1 c although both of them are quite far from the optimum point and there is a potential to gain of several degrees.
some help in this dilemma can be given by the fact that heat transfer theory predicts that the best profile is the one that results in a temperature that decreases linearly with the distance from the heat source. the bumplike temperature profile that results when a bottom plate with uniform thickness has a discrete heat source, figure 4, therefore indicates an optimised profile should have a thickness that declines very fast outside the heat source. in short, the best profile should be highly concave. conclusions
optimisation curves always resemble flat arcs. large deviations from the optimum parameter values can therefore often be made without much penalty.
the number of cases that needs to be covered to create a good overview often exceeds 100. the calculation time is therefore a critical factor.
cases with uniform bottom plate and fin thickness are relatively easy to optimise.
all optimisation processes that involve thickness variations, of the fins or of the bottom plate, are extremely time consuming. before getting involved in their details it is recommended that some time be spent to estimate the gains that possibly can be made.
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/.
