# What is the practical application of Rayleigh number to heat sinks?

I understand that there is such a metric as the Rayleigh number which governs convective cooling in a medium. Assume for example, a fist sized heatsink like that below in free air:-

Does the Rayleigh number mean that convection might not start between the fins? So in effect, the finned heatsink might as well be a large solid lump? And insofar as thermal resistance goes, the thermal conductivity would only depend on the gross overall dimensions (height x width x length) and not the actual finned area?

• Isn’t the Rayleigh an indicator of the convection flow : laminar or turbulent... – Solar Mike Jun 15 '18 at 21:42
• ^this. There will certainly be convection, And even if there were not, the larger surface area would make the finned heatsink considerably more efficient than an equivalently sized block – Jonathan R Swift Jun 15 '18 at 22:49
• @JonathanRSwift Anecdotal evidence suggests not (in free air)... – Paul Uszak Jun 15 '18 at 22:51
• I think it applies to fluids. – paparazzo Jun 15 '18 at 23:40

The Reynolds number is irrelevant in these situations because it is very small and does not deal with buoyancy and natural convection. There is instead a similitude parameter called the Grashof number that deals with balances between buoyant forces and viscuous forces which is relevant here.

• And Grashof applies to gasses? So it might say that my heat sink (above) is absolutely no different that an equivalent block of solid metal? That's the angle I was going for with the question. – Paul Uszak Sep 14 '18 at 11:55
• Grashof applies to gasses. it will tell you things like what temperature difference is required to set up a convection cell in air. – niels nielsen Sep 14 '18 at 17:07

Let's assume for a moment as per your hypothesis the convection doesn't start and the sink acts as a solid heated from bottom face and in contact with air at 5 other faces.

Heat will infuse this cube and propagate in gradients contoured similar to top half of an onion, core hottest and spherical layers gradually less warm to the surface which is least warm..

Because the air at the core is warmer it will rise like a cumulus cloud and leave a low pressure in zone in the base of this column. This low pressure zone will suck in air from around the base of the heat sink to fill in the vacuum.

Convection will stablish by circulating the colder air in between the teeth of the heat sink and getting it warmer and going up through the teeth and opening up in the shape of a canopy.

• Re. last para. Are you sure? I don't think that it's so simple. Does it not depend entirely on the spacing of the teeth? Compare efficacy of teeth gapped at 0.1mm vs a gap of 10mm. Viscosity effects (Rayleigh number(?)) will prevent all convection in the former case won't they? There'd be no sucking... – Paul Uszak Jun 16 '18 at 13:06
• You are guaranteed to get some convection if there is any heat transfer at all. The question is whether the convective heat transfer is bigger or smaller than the conductive, and the Rayleigh number is a measure of that. This might help: file.scirp.org/pdf/JECTC_2014082913121184.pdf – alephzero Jun 17 '18 at 21:23