Hi all,

 

I am trying to mathematically proof that one can neglect
convection and radiation when two cubes (paraffin-based, 1x1x1cm) are
in direct contact on one side whilst the other 5 sides are perfectly
insulated.

Any ideas on how to calculate the Nusselt number here?

 

How do you justify that radiation can be neglected? Can convection and radiation actually be
neglected?

 

How would the mode of heat transfer be changed if the cubes were not perfectly insulated, but instead were surrounded by air?

It would be fantastic if anyone could give me any hint! Many warm wishes and thanks so much in advance!

If they are in contact there is no convection. You should be able to calculate the radiation heat transfer and justify whether it can be ignored.

If the cubes are paraffin then they can't be very hot.

Unless their surroundings are very cold or very hot then there won't be much heat transfer by radiation.

But there "surroundings" are eachother since all other faces are insulated.

(I'm assuming that the "insulation" blocks radiation too, but that's not guaranteed. It could be a vacuum.)

So, unless one of the cubes starts very cold the radiation will probably be small compared to conduction.

Well, the 5 sides that are perfectly insulated will actually (try to) radiate, but the (theoretical) perfect insulation will reflect 100% of the radiation straight back to where it came from. The effect is that there is no (net) radiation. This is just how we define "perfect insulation".

 

Also, when two solids are in contact, heat is normally "conducted". At an atomic scale, photons may still be emitted, but they don't get very far, and just crash into the next atom. So, you could argue that what we call "conduction" is actually the sum of both the actual conduction of heat (caused by atoms bumping into each other) and some radiation. We cannot really measure it... so we just call it conduction.

 

Convection can be ruled out easily... after all, what would move? In your thought-experiment, all you have are 2 solid blocks of paraffin. Solids don't move, or we don't call them solids.

 

If you have a thought-experiment where you have only solids, then the only method of heat transfer is conduction, for the simple reason that "conduction" is what we call heat transfer in a solid.

 

If the two cubes are surrounded by air, then you would have to take into account radiation and convection for sure. The transfer between the blocks wouldn't change much initially, as conduction is typically much larger than convection or radiation... but on a longer time scale, they would have to be included.

Also, when two solids are in contact, heat is normally "conducted". At an atomic scale, photons may still be emitted, but they don't get very far, and just crash into the next atom. So, you could argue that what we call "conduction" is actually the sum of both the actual conduction of heat (caused by atoms bumping into each other) and some radiation. We cannot really measure it... so we just call it conduction.

 

The wikipedia article on thermal conduction confirms that and puts it well,

 

In condensed matter, such as a solid or liquid, the distinction between conduction and radiative transfer of heat is clear in physical concept, but it is often not phenomenologically clear, unless the material is semi-transparent. In a gas the distinction is both conceptually and phenomenologically clear.

 

Since the material is paraffin-based it probably would be semi-transparent, and so like wiki says, more phenomenologically sensible.

Edited April 2, 201312 yr by Iggy