Convection question

[Roy Easto]

I have currently made a computer model of Pluto to see if I could get
deep oceans to form from radioactive heating. And I can. I put in a
formula for transfer of heat by convection and the formula was clearly
wrong.

Does anyone know of such a formula or codes used by scientists used to
compute rates of convection?

Roy

[John Murrell]

Hello Roy,

I guess you would need to use computational fluid dynamics to try to simulate the dynamic conditions. However in the static condition the amount of heat generated at the bottom of the ocean must equal the amount lost at the top otherwise the ocean would heat up or cool down and not be static. I would have thought that the temperature drop would just be due to the increase in volume of the 'cone' of liquid from the bottom to the top - the head cannot go sideways in a radially symmetrical model as the adjacent liquid will be at the same temperature.

My thought would be that the ratio of the top to bottom temperatures will be the same as the ratio of the top to bottom areas of the liquid ocean in a simple model. As long as the liquid is incompressible and it's thermal capacity does not change with pressure. Something in my distance past says this is the same as 'boiler plate theory' which calculates the temperature difference between the inside and outside of the walls of a steam boiler but I may be wrong - it was probably about 50 years ago I did that at college !

In reality there may be hotspots due to convection plumes but that would be an difficult CFD problem bearing in mind little is known of the core conditions, the thermal and electrical conductivity of the ocean and any impact of rotation and magnetic fields.

Regards

John

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[John Murrell]

Hi Roy,

Not sure if my logic for the temperature difference was quite clear but I will try to explain it.

Let the bottom of the ocean be at radius 1. If we take a small section of ocean size x by y by Delta R for the purpose of this argument let x, y & Delta R all = 1 m so the volume equals 1 cu m.

Then let the top of the ocean be at radius 2 r . Then Delta R will still equal 1 m but X and Y will both have increase to 2 m. As a result the amount of heat will now be going through a volume of 4 cu m. If the specific heat is constant with pressure the temperature of the water ( in Kelvin) will have to drop to 1/4 of that at the bottom of the ocean as the same amount of heat energy is now distributed in 4 times the volume.

The assumptions are the water is not compressible, the specific heat of water does not change with pressure and the expansion of water is not significant over the difference in temperature.

I don't think the transport mechanism be it convection or conduction is relevant as the system is in a steady state. I think I must have assumed that transport by (thermal) radiation is not significant either.

Hope this helps

[wool...@googlemail.com]

If you find a good resource I could do with a link, I am looking to evaluate the convective cooling from a small very hot metal droplet at work.....

Cheers

Peter

[John Murrell]

Hi Peter,

My first thoughts are that when it is 'very hot' radiation cooling would dominate being proportional to T^4. If this drop is moving i.e. falling or otherwise in motion in a gas I doubt if there would be time for any substantial convection to build up - I would think the heated gas near the surface would be replaced very quickly by cool gas due to the motion. As such it is more a problem of conduction between the surface and the gas though the physics of that being quite complicated due to the boundary layer that would be destroyed or at least influenced by the local heating. Being an engineer I think I would abandon the theory and measure the result - a high speed infra-red camera should be able to track the surface temperature of the drop reasonably well though tracking it so you can use a reasonable magnification may be a challenge.

Good Luck !

[wool...@googlemail.com]

I have two analytical guesses assuming the gas flow is due to falling. I am going to ask a colleague to do a multi physics FE model as another guess. Thermal camera/pyrometers don't like small curved and shiny subjects.... Measuring "the temperature" is going to be interesting!

Thanks

Peter

[Roy Easto]

Thanks for the replies. It does seem tricky and I'll carry on searching.

I started my model from cold with the rocky core at 100K and the
surrounding ice at 0K. Clearly wrong but a good starting point for
working with the model. As the planet heats up the T^4 radiation from
the surface seems to pin the surface to the same temperature as
surrounding space. Once my ocean forms I need a formula for the power
fow. For conduction I have:

def conductive_power_flow(temperature_1, temperature_2, area, thickness,
conductivity):
return (temperature_1-temperature_2) * area * conductivity / thickness

An example of a render of the system is attached (done with a free open
source package called Povray). I thought I'd create a movie too.

Regards,

Roy