Nusselt for vertical cylinder with free convection

Main Question or Discussion Point

I am trying to find out how fast a vertical cylindrical water column heats due to the ambient temperature. I already calculated the heat transfer due to conduction but I am now interested in the role of natural convection. So to be clear there is a temperature difference between the walls and the fluid inside the cilindrical column (the top and bottom are preferably insulated) which might create due to instabilities convective flows.

Does anybody know (a reference to) a relation between Nusselt and probably the Rayleigh number for this situation?

Answers and Replies

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siddharth
Homework Helper
Gold Member
I think a more significant effect would be the heat loss to the air outside the cylinder via convection.

For this, you'll have to figure out the prandtl number and the grashof number. From that, you'll be able toget the nusselt number and the heat transfer coefficient via an empirical equation. Look up a heat transfer text for the eqn.

I am interested in the response time of a water column for the purpose of using it as a thermometer, so want to know if the average temperature in a water column can 'keep up' with the changing day air temperature.

Do you think the main resistance to heat transfer is in the boundary layer around the cylinder (~20cm diameter, height ~1m) and not in the water in the cylinder itself?

I am not too sure about that, so would like to compare the nusselt numbers for both mechanisms. The convection around (!) a cylinder is easily found in the literature, but for free convection in the flow inside (!) a cylinder the literature is very scarce however. The closest thing I found was that of a vertical rectangular cavity with two opposite sides at different temperatures. This is maybe good as an appriximation, but for my purpose sadly inconclusive.

siddharth
Homework Helper
Gold Member
I think that the convection inside, if it is significant, would depend on the temperature of the walls of the cylinder. Ususally, if the walls are made of metal, this effect shouldn't matter much, as the temperature on the wall along the vertical axis will be more or less uniform, and so, there'd be very less convection on the inside of the cylinder.

I think the correlation you use is the one by Churchill and Chu?

Also, here's a link which has some correlations for convection inside enclosed surfaces. Tell me if you find it useful.

http://batman.mech.ubc.ca/~mech475/MECH 475 web Lecture 6 and lecture 7 free convection.pdf"

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Thanks for the help, I think I indeed used the correlation of Churchill and Chu, I got it off the internet somewhere.

Right now I'm running a full Fluent simulation in the Bousinesq approximation with a 1,5m water column with a few cm thick perspex wall. Hope something usefull comes out of that. Of course I did not take into account the external flow but instead put a temperature difference between the outside of the tube and the water temperature inside. I am really interested in the time needed for the water column to respond to this. If as you say convection doesn't play a large role it should come out that this time is just given by an easily calculated time needed for conduction. However for our purpose we have both the top and bottom of the cylinder insulated and hopefully there is some natural convection to speed op the heat transfer.

But still, it would be nice to have a nusselt relation for this situation so we can play with the paramters easily. It all comes down to whcih processes are important and which aren't. My feeling says the external boundary layer is not much of a resistance; a little more resistance is introduced by the perspex. But I still think the main resistance is within the fluid itself: the time needed for conduction (and hopefully convection to speed this up) to allow the temperature difference to heat up the water.

Can anyone tell me if there is a formula, or natural convection rate for a column of water (that is insulated)
For example; a 40 foot column by 3 foot diam. (with really high "R" value around it) Maybe the top is open to allow heat to escape. What is the natural convection (or thermal siphon) that occurs between the top and bottom. (Bottom temp./Top temp) Am trying to figure out temp. for a passive solar storage device of such size; where panels feed heat to top and push down the water column as heat capacity builds. ( am wondering if I can have cold water at bottom and hot water at top.

Thanks.
Terry

Could anyone write down the equations for the flow of an incompressible fluid, driven by a heat source?
I have to simulate the convective flow of coolant in a TRIGA (pool type) reactor - which is basically a cylinder full of water with the core at the bottom.

Thanks!

Vladimir