# Through which medium is most of the heat transferred down (downwards) a glass tube filled with water: the glass walls or the water column?

If a much hotter object is attached to the tube from the outside, and the heat in question is transferred to the part of the tube that is below the source of heat, so there is no convection. The conductivity of glass is a bit higher than that of water (0.8 to 0.6), but the area of water perpendicular to the flow of heat is way larger than that of the glass, so I think that almost all of the heat will be transferred through the water column. Is my reasoning correct? Is Fourier's law applicable in this case?

• Remember an experiment as a kid at school, a test tube had water with an ice cube weighted with metal mesh so it was at the bottom of the test tube, then the top was heated in the bunsen - water boiled 4” from water frozen... Commented Dec 31, 2020 at 13:26

This is a comment. I do not yet have a "reputation of 50" and this is currently the only way I may leave a thought.

I would say that it depends on the thickness of the glass. A micro-thin tube would likely heat the water fast without much conduction to the lower regions, while I could imagine a very thick tube would allow heat to conduct down its exterior more rapidly, and include a gradiant loss to the water through the thickness.

1. I think that the assumption that "there is no convection" is invalid. Since the heat is coming from the sides, which will very efficiently set up a nice donut vortex (using liberal terminology!) Therefore, Fourier cannot apply.

2. With the symmetry around the Z axis, this is a 2-D problem. Applying the Fourier formula in 1-D would be invalid again.

Therefore, I would guess that Fourier's Law is not applicable.

That's only a hypothesis. But it's easily tested. Set up your test tube and leave it to settle. Then gently add a drop of colour. Again leave to settle but wait until the colour spreads to the level where the heat is coming in, so you'll be able to obseverve the effects. Heat it up and behold your swirling tube of chaos.

Of course I could be completely wrong! :-)

• If I've understood the setup correctly, the conduction solution has the vertical temperature gradient the same in the water and in the glass, and no horizontal temperature gradients, so why would heat come from the sides? Commented Oct 30, 2021 at 9:22
• At time = 0, the heat is applied around the outside. This will set up a temperature gradient in x,y. The heating effects are neither perfectly uniform or instantaneous. But as I said, were arguing hypotheticals. If you want to know the actual answer, use science and do an experiment. Commented Oct 30, 2021 at 20:06
• I think we're interpreting "the outside" to mean different surfaces, and therefore would do different experiments and get different results ;-). Commented Oct 30, 2021 at 20:25

Whether glass or water "wins" depends on both the thickness of the glass (as Jim Clark correctly described) and the Rayleigh number of the water in the column.

Wikipedia does a good job of providing a qualitative definition of the Rayleigh number: (time scale for thermal transport via diffusion)/(time scale for thermal transport via convection)

So if the Rayleigh number is large, the time scale for conduction is large and the time scale for convection is small and the system will transfer most of it's heat through convection. The water will "win."

If the Rayleigh number is small, the time scale for conduction is small and the time scale for convection is large (and in practice no convection will occur for small enough values) so heat transfer will mostly happen through the glass so long as it is thick enough relative to the water.

Rayleigh number increases with the temperature difference and the cube of the characteristic length of the system (which would be the diameter of the tube in this system, I believe). So unless the tube is quite thin or the temperature difference is extremely small you should expect water to "win". However, for a slight temperature difference or a thin (think capillary) tube, glass will "win."

https://en.wikipedia.org/wiki/Rayleigh_number