Thermal Facts and Fairytales
by Ross Wilcoxon
Long time readers of Electronics Cooling will undoubtedly recall one of our former editors: Clemens Lasance. One of Clemens’ many contributions to the magazine was the creation of the original “Thermal Facts and Fairy Tales” column, which provided a forum for him to discuss a variety of issues related to the use and misuse of information related to the topic of electronics cooling and thermal management. I have known Clemens for many years, and I feel confident that when I describe him as “willing to say what he thinks,” it is unlikely to provoke an argument from other people who know him. One topic that Clemens has always been willing to express an opinion on is the use of empirical correlations in engineering analysis.
In a number of Electronics Cooling® articles, Clemens outlined his many concerns with the use of empirical correlations. One of these concerns included the fact that, even with the simple geometry of a flat plate with natural convection, correlations developed by different researchers for the same geometry can vary by 100% . He was also pointed out that the incorrect use of non-dimensional parameters can lead to correlations that do not exist. For example, if a given configuration is tested with a wide range of the critical length scale (such as diameter), the Nusselt number will increase with Reynolds number even if the convection coefficient is constant . Another concern was with the use of correlations developed for simple geometries, such as flow channels, for estimating the performance of more complex systems such as flat plate heat sinks. The pressure drop of a heat sink estimated using different correlation approaches was as much as twice the value determined with experiments . The general conclusions from [1-3] can be summarized as “geometrically and physically complex phenomena cannot be described by simple equations” .
An additional concern with using correlations is the fact that they are typically only valid over a specific range of conditions, such as Reynolds number, length scales, etc. Even if one is a fervent believer that empirical correlations are accurate, it is still critical that one understands the conditions under which a correlation was developed, and ensures that the correlation is considered to be appropriate for the specific case in which it is being used.
While Clemens’ concerns with correlations are certainly valid, I didn’t always take them as seriously as I possibly should have. I felt that most the more widely accepted correlations were probably reasonably accurate and most engineers are aware enough to be sure that a given correlation is appropriate for their application. I was recently reminded that this isn’t always the case when I came across a paper that dealt with from a near-nanoscale structure. The paper used a fairly widely known correlation  to estimate the free convection from a very small cylinder that had a diameter of a few microns. The results reported in the paper had what seemed to be quite high heat transfer rates given that it was for a case of natural convection. Upon further review of the details of the analysis, it turned out that the conditions under which the analysis was done were below the range for which the correlation was considered to be appropriate. The very small diameter led to an extremely small Rayleigh number that was at least two orders of magnitude below the data from which the correlation was developed. This led to a heat transfer, in free convection, that was more than 5,000 W/m2 K.
A close review of reference  reveals a few interesting details. First, the log-log charts that compile dimensionless data appear to have errors in the labels for the y-axis. While the correlation appears to be correct despite the errors in the plots, this is a reminder that empirical correlations should be verified to ensure that an error did not find its way into the analysis. Another issue of note in the original paper is an explicit recognition that the correlation is not valid for Rayleigh numbers (Ra) less than 10-6 , which is much larger than the situation calculated for a micron-scale cylinder. Using empirical correlations can always be somewhat dangerous — particularly when they are used outside the range of conditions for which they were developed.
When I first began writing this article, I was fairly certain that I was writing about an excellent example of an empirical correlation being badly misused. The micron-scale cylinder in the paper that I found had a Ra on the order of 10-10 , which was four orders of magnitude outside the appropriate range for the correlation. However, after spending far more time trying to interpret reference  than I ever intended (primarily because of the errors in the scales used in the plots), it appears that there are data for extremely small Ra values and the error in using the correlation for the micron-scale may closer to a factor of 2, rather than the orders of magnitude that I first thought.
While the concerns about correlations discussed in references [1-3] are certainly valid, I still believe that empirical correlations can be useful for generating preliminary assessments of a system, as long as their limitations are recognized, and the application is sufficiently similar to the test conditions that were used to generate the correlation. This becomes more important, and challenging, as we move into applications with world of micro- and nano-scale geometries. These tiny geometries can lead us astray by introducing errors because we are extrapolating them to conditions that are not relevant. Or, they can possibly lead us to significant heat transfer improvements (5kW/m2 K in free convection!) if they are still correct at those geometries and we can determine ways to exploit them.
- https://www.electronics-cooling.com/2010/04/thermal-facts-and-fairy-tales-most-of-us-live-neither-in-wind-tunnels-nor-in-the-world-of-nusselt/ (accessed 8/4/19)
- https://www.electronics-cooling.com/2011/09/thermal-facts-and-fairytales-does-your-correlation-have-an-imposed-slope/ (accessed 8/4/19)
- https://www.electronics-cooling.com/2013/12/heat-sink-correlations-design/ (accessed 8/4/19)
- S. W. Churchill and H.S Chu, “Correlating Equations for Laminar and Turbulent Free Convection from a Horizontal Cylinder”, Int. J. Heat Mass Transfer, Vol 18, pp. 1049-1053, 1975