# What about the natural convection heat transfer coefficient (*h*) at low Delta-T?

In a system I am modeling for personal research, I have a sphere (S) submerged in air (A), initially at the same temperature Ts = Ta.

Then the temperature of air is slowly increased (or decreased), by something like 0.5 °C/h. In these conditions the Delta-T = Ts-Ta is necessarily low.

But, would h be so low that the main heat transfer mode is not convection anymore?

UPDATE: It is a sphere of say 20 mm radius, made of steel, and exposed to ambient air - which means the temperature of air will change with the diurnal cycle; 0.5 °C/h is a rough estimate based on the observed diurnal excursion in a certain locality.

In light of the informative comments & answers below, I would then like to ask how is the conductive heat transfer coefficient defined for this case.

• If there is heat transfer from the sphere to the air by natural convection (not forced convection), the surface temperature over the sphere will vary, and so will the temperature of the air around the sphere. So "slowly increasing the air temperature by 0.5C/h" is a very poorly defined concept! Also, whether the main heat transfer mode is convection or conduction (it's unlikely to be radiation, for such a small temperature difference) depends on the thermal properties of the sphere and the air, and also on the size of the sphere. – alephzero Jul 10 '18 at 14:47
• It is a sphere of say 20 mm radius, made of steel, and exposed to ambient air - which means the temperature of air will change with the diurnal cycle; 0.5 °C/h is a rough estimate based on the observed diurnal excursion in a certain locality. If the main heat transfer mode is conduction, how is it practically implemented in this case? – Fabio Capezzuoli Jul 11 '18 at 4:05

## 1 Answer

We can't say without knowing more information about the system. Luckily, we have a concept for evaluating this problem: the Rayleigh number. As outlined in the reference, if the number is low, convection is unimportant relative to conduction. If its high, convection is dominant.

I suspect finding correlations for the Grashof number (to plug into the Raleigh number equation) in your application will be difficult. If you have specific questions about that though, please update the question with more details. Otherwise, I suspect it will be easier to use this concept to do some rough estimates to determine how wide the range of possible outcomes is.

Update about conduction: If conduction is dominant, you unfortunately can't treat the air as a bulk material and solve with nodes, but rather you'd need to solve for the temperature profile/gradient. You'd need to solve Fourier's Law of Conduction, either explicitly or numerically. Different equations time!