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Spread angels, Part 1 |
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| Calculator..........: | Thermal Resistance for a Bottleneck |
Introduction
Thermal conduction problems can nowadays be solved with numerical methods. It is a great help but there are limitations. Numerical methods are both complicated and slow and are therefore difficult to use for overview calculations. An alternative is analytical methods but except for the most elementary cases, they also have disadvantages.
The only way out of this difficulty is to use approximations. There are quite a few available for thermal conduction but for the bottleneck problem, which is very common in electronics, there is little. Spread angles can be used but many engineers are hesitant. It is quite understandable because the spread angle can vary considerably from case to case. The 45 degree rule is an average rule of thumb. It is helpful for many cases but as all approximations it has its limitations, see for example: The 45° Heat Spreading Angle - An Urban Legend? There is also a 20 degree rule around. It is a conservative adaptation of the 45 degree rule. Regrettably, also this rule can result in too optimistic predictions.
The intention of this article is to present a set of simple approximations for bottleneck problems based on the spread angle concept. It can not be done with a fixed angle approach, so to reach the objective it is necessary to model the spread angle as a function of the geometrical context.
Figure 1
The 3 basic cases.
Basics
The temperature on a heat source and its corresponding thermal resistance, can be characterised in three different ways: Maximum, average and isothermal, figure 1. Which of them that is of interest depends on the case. The maximum value would probably be the best choice for chip level calculations. The average value is typically used for point contacts. If the bodies in contact have a large difference in thermal conductivity, the isothermal case is an alternative.
Figure 2
Definition of average surface and thermal resistance.
The thermal resistance for a rod with a uniform cross section can be described with a simple and straight forward equation. The same basic formulation can be used for a cut pyramid provided that the referenced cross section is an average of the two end sections, figure 2. The square root definition is the most common. It is exact for a section of a spherical shell but only approximate for a cut pyramid. This does not matter for the following however, because the error will be handled by a proper choice of spread angle.
Figure 3
Cut pyramid with a spread angle.
Figure 3 shows the spread angle approach applied to a large plate with a small quadratic heat source on one side and isothermal conditions on the other. Using the square root rule for the average cross section, results in a fairly simple equation for the thermal resistance. For small heat sources it simplifies to an expression that is independent of the plate height. This effect is explained by the fact that temperature gradients around small heat sources tend to be extremely sharp, see Chip Level Dynamics, figure 3.
Figure 4
Spread angle for a small quadratic heat source based on theory.
An analytical approach for small heat sources results in the same conclusion, figure 4. Each of the basic cases therefore also corresponds to a specific spread angle. The average value is actually 45.4 degrees! It can also be noted that, if the thermal resistance for the average temperature case is normalised to 100%, the level of the isothermal and the maximum temperature cases are set to 92% and 118% respectively. If a 20% error can be accepted, the 45 degree rule is therefore always relevant for small heat sources. As revealed in the referenced document, it can in addition it be shown that there is little difference between circular and quadratic heat sources provided that the square root of the surface is used as characteristic length.
Figure 5
Heat flux channel approach.
Bottlenecks with quadratic cross sections
Many conduction problems involve arrays of heat sources or bodies that are limited sidewise. The heat flux channel approach could be a good approach for these cases, figure 5. The following the discussion is valid for quadratic cross sections, (also for circular sections if they are converted to the equivalent quadrates). The idea is to model the flow channel as a cut pyramid on top of rod, or for the case the height is small, just a cut pyramid.
The spread angle can be found by extraction from analytical or numerical results. For small sources they must necessarily converge to the values given in figure 4. For large sources, d/W~1, the value really does not matter. It is sufficient to have an idea which way it takes. The problem of finding good approximations is therefore a matter curve fitting between these two extremes.
Figure 6
Spread angle for the maximum temperature case.
Figure 6 shows spread angles for the maximum temperature case. For small heat sources they converge towards the expected value. For large heat sources they converge towards zero. Compared with the isothermal case, for which the spread angle approximately is a constant, this is a distinct difference. A remarkable matter is that the break point between the two calculation cases cut pyramid only and cut pyramid + rod, does not show up as a break point in the curves. Although difficult to explain, it is a notion that the spread angle approach somehow reflects basic physics.
The impact of the channel height ceases when the bottlenecks become fully developed. The shift is gradual but for applied purposes the limit can be set to H/W>1.
Figure 7
Empirical correlations for the spread angle and quadratic cross sections.
Figure 7 shows suggested approximations. They are all simple enough to be implemented in spread sheets. The error for the associated thermal resistance is below 10%. This error should not be confused with the angle prediction error, which for large d/W values can be considerable. An example of this is that there is a substantial discrepancy around H/W=0.1 for the average temperature case. As stated, this is not a matter of concern. The total thermal resistance prediction error around that value is still <10%.
It could be argued that other approximation approaches can achieve the same thing. That is undoubtedly true but it seems as if the spread angle approach results in the simplest formulations. Another advantage is that spread angle values can be intuitively understood, which substantially facilitates debugging of calculation procedures.
Figure 8
Accuracy examples for the maximum temperature case.
Figure 9
Accuracy examples for the average temperature case.
Figure 10
Accuracy examples for the isothermal case.
Figure 8 -10 shows examples of the calculation accuracy. The errors are in all cases below 10%. In view of the fact that it for most applied cases is difficult to clearly distinguish between the three basic temperature cases, it is an acceptable result. The effect that the curves do not clearly converge towards unity for small sources is caused by the fact that also theoretical solutions have prediction difficulties in this region.
As mentioned above the bottleneck problem is not particular sensitive to the cross section shape as long as it is reasonably roto-symmetric. Non-quadrates can therefore be simulated by quadrates of the same surface. It is nonetheless difficult to know how far this tendency can be stretched. The equal surface principle is implemented in the referenced calculator. A test on a few arbitrary rectangular cases reveals that side ratios up to 2:1 can be used without too much penalty. It should however be remained that the spread angle for narrow line sources is as high as ~60 degrees, which calls for caution when using the equal surface principle.