# When does heat conduction become convection?

Okay, I have a hard time wrapping my head around something, I think, pretty basic.

For a little hobby project of mine I was making a fermentation-box for beer brewing allowing me to accurately control temperature. This is nothing more than a fridge with a heater element added. However, I wanted to model the system (including the control part) to assess the dynamics a bit and, you know,... for fun.

Let's get to the problem at hand:

I first wanted to model the fridge without anything in it, except air. This is like the most basic thermodynamics problem imaginable:

• Ambient temperature $T_a$.
• Fridge is filled with air, with a total mass of $m_{f}$, temperature $T_f$ and specific heat constant $c_p$.
• Contact surface area of the walls of the fridge to the inside is $A$ and the walls have a thickness of $d$.
• Heat transfer from outside to inside and this is somehow dependent on the difference in temperature with the outside via a constant $C$.

Okay, so the total system is described as: $\dfrac{dT_f}{dt} = \dfrac{C}{c_p m_f} (T_a - T_f)$

So, what is $C$?

When looking at the heat transfer from the outside to the inside there are, as I believe, 2 dominant mechanisms: conduction and convection.

My assumption is that if the thickness of the wall is 'thin', whatever that may be, it can be seen as a simple conductor. This means that $C$ is described as: $C = \dfrac{k A}{d}$, with $k$ being the conductive heat transfer coefficient of the material (e.g. styrofoam). This means that the wall does not have any significant thermal capacity, since it is thin.

So, what happens when the wall thickness is 1 meter? Then it definitely has thermal capacity, but how do I model it now? Do I use the average temperature between atmosphere and inside? This seems wrong, because the conduction formula does not seem to hold up anymore according to me, since a part of the heat flux is used to heat up the wall.

So, is it then solely convection? The air heats up the wall, and the wall heats up the insides? Looking at the formula of convective heat transfer it looks like $q = h_c A dT$, where $dT$ is the 'temperature difference between the surface and the bulk of the fluid'. Okay, so from the atmosphere to the wall the fluid is the air and the surface is the outside temperature of the wall. This heat flux will then heat up the wall which will heat up the insides. On the inside the fluid is the air also and the surface the inside temperature of the wall. But are the inside and outside surface temperatures the same? Is there a gradient or not?

Maybe I am making it way to complicated, but my questions boil down to this:

• In this example, with thin walls, is it conduction, convection, or both?
• With thick walls, is it conduction, convection or both?
• There will be a temperature gradient in the wall. However, is it significant?