Hidden Problems in Heat Transfer

I get a pretty large number of trade magazine, but one that I am always excited to see is "Electronics Cooling". As you would expect, the magazine is largely oriented towards heat transfer in electronics applications, a problem that I am rarely concerned with, but there is at least one article per issue that addresses heat transfer in general, and often in very enlightening ways. This month is a terrific example with the article "Does Your Correlation Have an Imposed Slope?".

This article (combined with the editorial at the front of the issue really take engineers to task for relying excessively on those darling dimensionless numbers. For those not familiar with dimensionless numbers, these are grouping of variables that are dimensionless. The most commonly used one is the Reynolds number which is ρVL/μ, where ρ is the density of the fluid, V is the velocity of the fluid, L is an appropriate length from the test setup and μ is the viscosity of the fluid. When running fluid mechanics test, you don't need to explore the impact of all these variables over their full ranges, you only need to explore the Reynolds number of its full range and you're covered for all situations. Quite a time saver, eh? You can see why engineers love them so much.

The dimensionless numbers in heat transfer are numerous and I won't get into any of them today. Their large number would be expected since there are 4 modes of heat transfer (conduction, forced convection, natural convection and radiation). As the article points out, the numbers by themselves are not a problem. The issue arises when multiple dimensionless numbers are used in heat transfer correlations, and when the same parameters are used in multiple numbers. One example given is when the Nusselt number, Nu, is correlated to the Rayleigh number, Ra, as

Nu = C Raⁿ

Looking at the parameters that make up these numbers, you find:

q L/(ΔT k) = C [g Βρc_pΔT L³ / k ν ]ⁿ

where ΔT (temperature difference), L (length) and k (thermal conductivity) are on both sides of the equation. This then leads to situations were random numbers can show correlations and the article gives a specific example of this.

Correlating dimensionless numbers with each other leads in other situations to an increase in the errors associated with the correlations, something that I remember well from my undergraduate days. I dug out my old heat transfer book and sure enough, I see correlations with 2, 3 or even 4 dimensionless numbers in the same equation. I am quite sure that my professor never warned us of these issues, despite their existence first being described back in 1963!

"Electronics Cooling" is free - you can get a subscription at the bottom of the first page I linked to above. If you work with heat transfer (and unless you work exclusively all day long with room temperature equipment and chemicals, you do work with heat transfer), do yourself a favor and get a subscription.