Convection, Friction and Reynolds Analogy

Home page :	Frigus Primore home page
More :	A Method for Comparing Heat Sinks

Introduction

A very characteristic element in thermal design for electronics is the race for increased cooling capacity. Once a new powerful cooling method has been introduced it does not take long until it is fully exploited and must be replaced by yet another even more powerful method.

A decade ago it was usually sufficient to use natural convection. Those days are long gone. A large-scale invasion of fans followed. They calmed things down for a couple of years but engineers soon realised that the noise and the fan power problems not are that easy to overcome. Heat sinks was a way out of that dilemma and the use of these devices has increased dramatically in the last years.




Figure 1
A coarse sketch of the cooling capacity for different generations of air-cooling.

Figure 1 shows a coarse sketch of the capacity for different generations of air-cooling techniques. The dissipation levels can vary from application type to application type but the general tendency is indisputable. The historic movement from one generation to the next has by no way been halted. The current shift is pushing the front line from forced convection with local heat sinks to forced convection with PCB scale heat sinks. Whatever comes after this is difficult to know but it does seem as if air cooling rapidly is approaching the end of its performance potential.

PCB designers often take a deep sigh of relief when a new and radically improved cooling method is introduced. Experienced thermal designers react differently. They know that the temperature margins that initially appear to be large are quite volatile. They also know that using the breathing pause to prepare for coming days is a good strategy. That is, for the days when the gnawing of a degree here and a degree there, again becomes a daily routine.

The introduction of heat sinks was no exception in this respect. They did improve the cooling possibilities radically. The first generation of heat sinks therefore did not need to be particularly performing. As time has passed and the margins have melted,the chase to gain a few degrees here and there has subsequently restarted. The main difference compared with previous times is that there is an additional element to consider, the heat sink.

How heat sinks should be designed and in particular how the fins should be shaped, has been debated for quite a time. At this time conventional rectangular fins seem to be dominant but there are competing ideas. Pin fins with various cross sections have been on the market for some time. Odd designs, such as fins shaped by folded nets, also appear now and then. In addition there are at least two alternative fin arrangements, inline fins and staggered fins.

It is not is easy to navigate in this jungle of ideas. It is nevertheless possible to create a map that can provide some bearing. The profound correlation between convection and friction is the bases for it. The entire subject is too large to be covered in a single piece. This article will therefore only deal with single bodies. The consequences for actual heat sink designs will be covered in a coming issue.


Reynolds analogy

The fact that convection somehow is correlated with friction has been known for a long time. Reynolds is famous for being the first to describe this dependency more precisely. His finding is generally called Reynolds analogy. There are several different ways to formulate it. The most well known form is probably the one that correlates the Stanton number with the friction coefficient. That formulation works well for channel flow but can not be used for flow around objects. This article will therefore use an alternative approach that consists in comparing the heat dissipated with the mechanical power needed to overcome the friction.

The full proof of Reynolds analogy requires quite a deep dive into boundary layer theory. There is not enough space for it here. If one is less stringent it is however possible to get away with simpler means. Two such derivations are presented here. What is striking about both of them is that they are very simple and yet generate surprisingly good results.



Figure 2
A small mass element that absorbs heat from a surface has zero velocity and must later be accelerated to the full velocity of the airflow.

Figure 2 shows how a small mass element absorbs heat on a surface and thereafter is transported away. The velocity at the surface is zero and the velocity in the wake is the velocity of the airflow. A comparison of the heat absorbed and the transport energy required for accelerating the element, results in the underlined equation in figure 2. Although this derivation is extremely simple, the result is almost theoretically correct. The author has no idea why it works so well.



Figure 3
Assuming linear velocity and temperature difference profiles creates an equation that is almost identical with the one in figure 2.

The second derivation is slightly more elaborate and uses a line of attack that is much closer to the full proof methodology. The bases for this derivation is that heat transfer theory predicts that the Pr-number is a measure on how the velocity and the temperature difference boundary layers are related. For low Pr-numbers the latter is thicker than the former, for large Pr-numbers it is the reverse. For Pr-numbers near 1.0 they have the same thickness and in addition also the same profile, figure 3.

Liquids generally have Pr>>1 but one exception from this rule are liquid met-als that have Pr<<1. For most gases, including air, the Pr-number is about 0.7. One could therefore suspect that a derivation based on an assumption of equal thickness for the velocity and the temperature difference layers could be fertile.

Figure 3 shows a simplified derivation in which it has been assumed that both the velocity and the temperature difference profiles are linear. The resulting equations are quite simple and the solution process is straightforward. The final result is similar to the result in the former derivation. The only difference is that a Pr-number appears. The Pr-number was however assumed to be 1.0, which actually makes the two derivations exactly equal.



Figure 4
Analogy equation for a plate and the definition of the analogy number.


The analogy number

A more complete derivation for flat plates results in the top left equation in figure 4. The only difference between this equation and the equations derived above, is the exponent in the Pr-number term.

The analogy has so far only been treated on the local level. For a flat plate it is a matter of a simple integration to make it valid also for the entire plate. As will be shown below it, is not that easy for curved surfaces.

Experimental data shows that the actual mechanical power needed often is much larger than the one predicted the simple analogy. One reason for this is that all bodies, that not are negligibly thin, tend to create vortex streets in their wakes. These vortexes consume mechanical power but contribute very little to the heat transfer. Another phenomenon that creates deviations from the simple theory is variations in the velocity field that surrounds a body. This will be explained more in detail below.

It is convenient to introduce a compensation factor that can handle these discrepancies. It is called the analogy number. The definition is shown at the top right of figure 4. A great advantage with this formulation is that it is general and consequently can be applied both to channel flow and external flow.

The analogy number functions as an efficiency factor but with the difference that its maximum value can exceed 1.0. The analogy number for a flat smooth and thin plate is for example 1.27.



Figure 5
Flow around a cylinder.


The analogy number for a cylinder

The flow pattern around a cylinder is a very complicated. It is reasonably smooth at very low Re-numbers but as the velocity increases the flow be-comes increasingly disturbed, figure 5. It is apparent that it is impossible to use a simple analogy theory to deal with this complexity. For low Re-numbers there might however be a chance. The velocity field can for this case be predicted with potential flow theory. Figure 5 shows the equation for the tangential velocity. It must however be pointed out that the velocity of interest is the velocity just outside the boundary layer. Since the boundary layer thickness increases in the flow direction, the actual shape that need to be simulated is egg-shaped rather than round. The velocity equation given must therefore be regarded as an approximation.

It can be assumed that Reynolds analogy is fulfilled for each local surface element. From an applied point of view it is nevertheless the all-over performance that is of interest. Several steps are therefore needed to create something more useful. A reference velocity must be defined and the local heat dissipation and mechanical power must be integrated over the entire surface. For the velocity definition there is not much other choice than the up stream velocity. The integration is somewhat more intricate.


Figure 6
Integration of the local analogy to an all-over analogy for a cylinder.

The steps in the integration process are shown in figure 6. The problem that appears in this process is that some kind of variation must be assumed for the heat dissipation. For the case treated it can be assumed that it is much smaller than the variation of the velocity and therefore approximately can be regarded as a constant. If this line of approach is followed the result is that the all-over analogy number for a cylinder and at low Re-numbers approximately equals 0.63.


Figure 7
Equations for extracting the analogy number from empirical data.

Considerable simplifications were made to arrive to this result. Its validity can therefore not be regarded as particularly high without some kind empirical support. Figure 7 shows the equations that can be used to extract the analogy number from known correlations for the Nu-number and the drag factor, Cd.


Figure 8
Theoretical result compared with empirical data.

A comparison between available experimental data and the calculations made for the cylinder is shown in figure 8. The agreement is far from perfect but the fact that the analogy number for cylinders is considerably lower than that for flat plates is confirmed.

Figure 8 also shows that the analogy number for a cylinder decreases dramatically when the Re-number is increased. It is essentially the vortex streets in the wake that cause this effect. They consume mechanical power but do not contribute much to the heat transfer. One might however wonder if the chaotic flow in the wake of a cylinder could contribute positively if the several cylinders were placed in a bundle? It is definitely true that the heat transfer coefficient for a cylinder that is hit by a non-stable flow is higher than if it had been hit by a uniform flow. Empirical studies on the debits and credits of bundle arrangements are however unanimous; the analogy number might be higher than for single cylinders but it still remains much lower than that for flat plates.


A speculation

The simple analogy equation shows that the mechanical power which is required to overcome friction not only increases with the heat dissipated but also with the square of the velocity. A body such as a cylinder, for which the velocity is forced to increase to let the flow around, is therefore punished by its peak velocity. This is why its analogy number is lower that for a flat plate.

One might therefore wonder if it is possible to apply the simple rule that between two arrangements, with the same linear average velocity, it is the one with the lowest peak velocity that wins. The author has searched for such a proof, or at least for a very strong indication, for several years. So far without much success. He is definitely convinced that it is the case when the difference in peak velocity is substantial but even that is difficult to prove theoretically.

There is of coarse always the possibility to prove it empirically. Such proofs are by their nature somewhat weak but they are nevertheless better than nothing.