[Thermodynamics] Temperature gradient around a warm sphere.

[Dafe]

Hi,

Say I have a sphere of radius r that has a constant surface temperature of T_s.
The sphere is surrounded by air at a constant temperature T_amb.
I am interested in the temperature gradient surrounding the sphere.

From the little I know, I think i have to look at the natural convection and radiation.
I can calculate the heat transfer by using Newtons cooling law and the law for radiation,
but I do not know how to calculate the temperature gradient.

Could someone please point me in the right direction?

Much appreciated!

 

[Mapes]

For temperatures less than hundreds of °C and air as a surrounding medium, convection generally dominates over radiation. It seems like you're interested in free convection (no forced flow) around a sphere; you can find empirical equations for this case in handbooks (possibly on the Internet) and in the heat transfer textbook Fundamentals of Heat and Mass Transfer by Incropera and DeWitt, which includes references to literature reviews of the problem.

 

[Dafe]

Hi Mapes,

yes, I am interested in the free convection around a sphere and other geometries.
When you say that I can find empirical equations, does that mean that there are no analytical ways in solving this?

Thank you.

 

[Mapes]

I could be wrong, but I doubt it. Free/natural convection is a notorious fluid mechanics problem where few analytical solutions exist. Researchers make empirical connections by comparing various nondimensional ratios (e.g., the Rayleigh and Prandtl numbers, which you'll need to calculate for this problem).

 

[Dafe]

I assume that a solution could be approximated by using numerical methods like finite element.
Perhaps Ansys can solve this somewhat easily, will have a look.
Thanks

 

[Mapes]

Sure, a numerical approach is ideal. Just make sure your software can accommodate bulk fluid flow. That's the mechanism that's removing thermal energy from the sphere.