T₁Here's a handy trick for calculating the effect of radiation compared with convective  cooling. It's especially useful for quick analyses in systems where you're not sure about what kind of role radiation heat transfer is playing in the overall picture.

The idea is to find a quantity that has units of heat transfer coefficient, so that it's easy to compare to the convective heat transfer coefficient. This idea comes from a linearization of the radiative heat transfer law for the heat flow between two bodies,

q = s e A (T₁⁴-T₂⁴)

In this basic law, s is the  Stefan-Boltzmann constant (5.7x10^-8 W/m²K⁴) and e is the gray-body emissivity of the object whose absolute temperature is T₁. The object has "full view" of a blackbody environment at absolute temperature T₂. A good application for this law is a system enclosure with average surface temperature T₁ that is completely surrounded by an ambient at temperature T₂. T₁ and T₂ should be of  similar magnitude on the absolute temperature scale. In other words, don't try this for molten glass, arc lamps, fires, or anything else with  large temperature differences. (In those cases, the approximation doesn't apply, and anyway, radiation will probably be the dominant heat transfer mode so you should go to the trouble of treating it more carefully.)

The linearized form of the equation looks like this:

q = h_r A (T₁-T₂₎

If you work out the  algebra according to rules you haven't much thought about since high school, you'll see that a good approximation to the radiative "heat  transfer coefficient" quantity is

h_r ~ 4 s e T_m³

where T_m is the arithmetic mean of T₁ and T₂.

Let's say you have a closed box with electronics inside - for example, a tabletop video-conferencing system. Your Electrical Engineer colleagues have told you that it will dissipate 20 or 30 W - 40 W, worst case (I'm making this up, obviously!), and now they want you to package it. Vents would be a bad idea, because there's usually coffee available to spill on the table and into the box. The general form factor that Marketing wants is,  let's say, 15x20 cm and about 4 cm high. The box material probably has a relatively high emissivity if it's going to look good - either painted or unpainted plastic, or possibly painted metal. So a good guess at the emissivity would be 0.8. The ambient temperature is probably somewhere  between 20 and 35°C (T2=273+27=300 K). For customer perception's sake, the box temperature will probably have to be 50°C or less (T1=273+50=323 K). The general question is, will this work, and what is the power dissipation limit? More specifically, what are the relative effects of radiation and convection?

Both natural convection and radiation are at work here in parallel paths, so we will  want to look at the radiative "heat transfer coefficient" and compare it to an estimate of the natural convection coefficient. I'm going to guess  for such a small temperature difference that h_NC~ 5  W/m²K (feel free to do the calculation for better accuracy). The radiative version works out to be 5.5 W/m²K, or about the same magnitude.

This example illustrates that the effect of radiation can be comparable to natural  convection, even in cases where you might not think it important because of the modest temperature differences. Furthermore, having a quick way to calculate effects of potential changes in size or surface emissivity is valuable for making design decisions on the spot (and defending to  Marketing and Industrial Design why the box has to be made bigger than they'd like!).