Introduction

Non-confined flow is, in this context, another way to express that a heat sink has an airflow bypass. some of the incoming air, therefore, takes a detour around the heat sink, which always results in a performance loss. the majority of heat sinks used for cooling of electronics are probably of this type. Much of the physics is the same as for the non-bypass case but there is the additional difficulty that the bypass flow phenomenon must be modeled. the major steps in the optimising process are nevertheless the same: create an overview, refine the solution and verify the result.

This article is about the overview part. An overview must by its nature cover a large number of design variations. The calculation time is, therefore, always a critical factor, which in this case greatly favours an analytical approach rather than a numerical one.

Some analytical approaches to the problem have been presented. one example can be found in: "a simple method to estimate heat sink airflow bypass". This method works well for small bypasses but not for large ones. For large bypasses there is not much to find in the literature except some empirical correlations. The problem with these is that their generality always can be questioned. The correlation presented in this article has the ambition to fill some of that void. It has a semi-empirical approach which is believed to reduce the generality problem considerably.

figure 1- some bypass definitions.

What is bypass?

The simplest way to define a bypass is as a ratio of the cross sections for the flow channel and the heat sink. It can however be questioned if this type of specification not is too simple. Looking at the problem from a general perspective it is certainty true that it matters whether the bypass section is located above or at the side of the heat sink. from the applied perspective things are however not that certain. a result of this discussion is that there are several ways to define bypass, figure 1.

In this context it should also be pointed out that the size of the heat sink cross section is important. huge heat sinks will only sense bypass impacts at their most peripheral parts whereas this not is the case for small heat sinks. For the typical heat sink sizes that are used for individual component cooling it nevertheless seems as if this scale effect is small.

figure 2- the performance of a heat sink declines when the bypass is increased and levels out at a cross section ratio of about 20.

Another issue is how important bypass is. there are no simple answers to this question except that it is important enough to be considered. performance declines on the 50% level have been reported but for optimised heat sinks it is usually not that much. figure 2 shows the principle tendency. Experience values indicate that the decline levels out when the cross section ratio has reach a value of about 20.

The discussion of how the performance of a heat sink varies with the bypass cross sections is interesting but often not that constructive from the applied point of view. It is rare to find cases that are as clear cut as the one shown in figure 1. The reason is that the airflow in an enclosure mostly has a large number of alternative flow paths. a sub rack filled with pcbs is a typical example. Even if the bypass for a heat sink on an individual pcb can be clearly defined there are many other competing flow paths. In most applied cases one is therefore forced to make the assumption that the bypass cross section is large. This is of coarse a conservative approach but it will on the other hand provide a risk margin.

 

figure 3- if the bypass is large the problem can be approached by treating the heat sink as a porous body.

A simple theory for large bypasses

Figure 3 shows an outline of a theory for large bypasses. The basic idea is that bypass flow is an ejection phenomenon that provides a driving force for the flow through the heat sink. The equilibrium is subsequently reached when the pressure loss in the heat sink matches the pressure loss in the bypass.

There are two velocity definitions involved in the theory. one is the incoming velocity, w₀. The other one is the front velocity for the heat sink, w_f. It is defined as the flow that penetrates the heat sink over its external cross section.

The ratio w_f/w₀ is an important parameter. If it is zero it signifies that the heat sink is a solid body and consequently has no air penetration. If it is close to 1.0 it signifies that the heat sink is extremely porous. To formulate a theory for the bypass phenomenon is therefore a matter of finding a rule for how the bypass pressure loss performs between these two extremes. Realising this, The mathematical formulation of the problem is fairly obvious. It should consist of a correlation for the solid body case combined with a compensation function that depends on the w _f/w ₀ratio .

The pressure loss for a solid body is a function of the drag coefficient, c _d, and the dynamic pressure for the incoming flow. The compensation function is a more difficult case. To use a linear approach is obviously too simple. A second degree function offers more possibilities and this is also what is recommended, figure 3.

 

figure 4- examples of comparisons with measured data.

The associated calculation procedure consists in assuming a value for the w_f/w ₀ ratio that successively is adjusted until the pressure losses for the heat sink and the bypass match. Once the air penetration is known the final step is to calculate the thermal properties for the heat sink. This can basically be made in the same way as for a heat sink without bypass. Although this method is fairly simple it seems to fit measured data rather well. Figure 4 shows two examples. The results are not always that good but the method seems to be able to deliver data with an acceptable accuracy for the applied level.

figure 5- each air velocity has its optimum pitch. the circles mark the discrete pitch choices that are possible.

The optimum pitch problem

A heat sink with dense fins has a large heat transfer surface but also high internal pressure losses. A large portion of the incoming air will therefore bypass the heat sink, which deteriorates its performance. For sparse fins this tendency is reversed and for this case it is the small heat transfer surface that causes the performance decline. The optimum pitch is found between these two extremes.

The incoming air velocity is also an important factor. The pressure loss in the heat sink is essentially caused by laminar flow and is therefore proportional to the air velocity while the bypass pressure loss is proportional to the square of the velocity. an increase in the air velocity will as a result increase the bypass pressure loss more than the heat sink pressure loss and this surplus can be used to increase the fin density. The optimum fin pitch is as a result pushed towards a lower value.

Figure 5 shows an example of these tendencies. The circles in the diagram mark the discrete pitches that are possible. The basic model for these simulations is a heat sink with a 4.5 mm fin pitch. The diagram reveals that it is optimised for velocities around 1 m/s. used at any other velocity it will not deliver peak performance and it can be estimated that the potential loss at 4 m/s is about 20%. This discrepancy is by no means radical but it could make a large difference if the application is critical.

figure 6- considerable performance gains can often be made by optimising the fin thickness.

The optimum fin thickness problem

Figure 6 shows the same basic heat sink as in figure 5 but with the fin thickness as parameter. Thick fins increase the pressure losses while thin fins deteriorate the fin efficiency. The optimum thickness is found as a compromise between these two tendencies. The basic heat sink model for this case had a thin thickness of 1.65 mm. If it could be reduced to 0.1 mm the calculations indicate a 30% gain. This is once more not a radical gain but it could certainly make the difference needed for some cases.

 
figure 7- The flow inside a heat sink is never unidirectional. some of the flow always escapes through the heat sink top. this limits the validity of all one-dimensional approaches to regions where this impact is negligible.

Limitations

The flow inside a heat sink is never unidirectional. some of the flow always escapes through the top of the heat sink, figure 7.  This limits the validity of all one-dimensional approaches to regions where this impact is negligible.

There are many parameters that can amplify this type of air escape. The fin- to fin distance, the velocity and the heat sink length are all important. what is encouraging however, is that the phenomenon only seems to be important when the pitch is fairly below the optimum pitch. This problem region is marked in the left diagram in figure 7. A consequence of this is that it sometimes can be difficult to get good measurement fits for low air velocities.

The problem here is that diagrams showing the thermal resistance as function of the air velocity often cover a large velocity range around the value for which the heat sink has been optimised. The pitch can therefore be considerably below its optimum value for the lowest velocities used. It is this impact that explains why the marked region in the right diagram in figure 7 often is problematic.

figure 8- a couple of helpful correlations.

A couple of helpful correlations

Figure 8 shows a couple of simple but helpful correlations. The first one is a way to fast estimate the optimum fin- to fin distance. It is based on the idea that the optimum distance is found when the boundary layer displacement thickness has grown to half the channel width. Since the actual velocity between the fins usually not is known one is however forced to use the incoming velocity as parameter. The correlation should therefore be regarded as a rule of thumb.

Any higher degree of accuracy can consequently not be expected but it reflects the general tendencies quite well. It can typically be used for quick check ups on the air velocity for which a heat sink has been optimised or as a first level approximation for the fin- to fin distance.

The other correlation is very helpful for interpolation and even extrapolation, of thermal resistance data. The exponent value, -0.65, is typical for heat sinks with large bypasses and it is surprisingly invariant.


Application experiences

The author and many of his former collaborators have used the outlined calculation procedure for more than 10 years. It is particularly useful for two purposes, front-end design and checking up on commercial heat sinks.

The front-end part of the design process is very hectic. A key issue for being an attractive discussion partner in this phase is therefore the ability to deliver responses with such swiftness that the discussion never cools. The diagrams shown in this article only takes a few minutes to create. This soon becomes known in a company environment. The calls for thermal expertise participation in innovation and pre-study groups will subsequently be frequent which efficiently challenge the tendency that thermal design is considered too late in design projects.

A thermal designer is often faced with the choice of creating a custom design heat sink or selecting one form a manufacturer's catalogue. The latter is of coarse both less expensive and less time consuming. In most cases it is therefore also the default alternative. This may however not be a completely satisfactory solution for someone with a critical mind.

There could be a better heat sink somewhere else and the performance compromise, relative to a fully optimised heat sink, is not known. Overviews such as shown in figures 5 and 6 are very helpful in this situation. It is having access to this kind of information that can make a thermal designer sleep well at night.


conclusions

A bypass is always associated with a performance loss.

A porous body approach can be used to develop a reasonably successful bypass flow theory.

A heat sink with a bypass is always optimised for one particular air velocity. the potential loss when used at another velocity is usually moderate.

The fin thickness is a critical parameter and significant gains can often be made if it is optimised.

Overviews of heat sink performance are very appreciated in the front-end design process on the condition that they are created fast enough to keep the discussions alive.

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.