Thermal simulation of conduction between metal and air

I am building a (2D) thermal simulation and the situation can be simplified as follow : A room filled with air at a uniform temperature (say 20 celsius) and on the floor, a plate of copper at a constant temperature (say 0 celsius). For my simulation, I separated my 2D room into a grid of 4 by 3 blocks :

X X X

X X X

X X X

X C X

Where X is air and C is the cold metal plate. I already have coded the simulation of conduction in the air using Fourier's law and I now want to simulate the heat exchange between the copper block and it's surroundings. I was thinking of using Newton's law of convection but I can't find a value for h, the heat transfer coefficient corresponding to a border between air and copper. I understand that this coefficient will depend on the surface of contact (which we will name S), on the airflow which we will suppose to be non existent (no wind, and convection current are negligable in front of the conduction happening in air) and maybe other variables I didn't think of. Therefore my question is the following :

Is there a formula giving the value of the heat transfer coefficient (h) in this situation ? If not is there a value that was determined in a situation similar to mine ?

1 Answer

If wind is absent, we have only natural convection. For simplicity, symmetry is often assumed in terms of the temperature difference; that is, in symmetry with heat convectively billowing up from a hot plate, warmer air convectively sinks toward a cold plate in a reversed manner, with equivalent governing equations.

So we're looking for the coefficient of natural convection for a horizontal plate. The outgoing heat flux $$q^{\prime\prime}$$ (in watts per square meter, say) is

$$q^{\prime\prime}=h\Delta T,$$

where $$h$$ is the coefficient and $$\Delta T$$ is the temperature of the plate relative to that of the surrounding fluid. (For a cold plate, both side of the equation are negative.)

("Plate" implies a side length much longer than the thickness and the boundary layer thickness of the fluid flow; if the plate isn't wide, more accurate results might be obtained by calling it a "strip" or "cube" and searching for the corresponding relations.)

Convection is often too complex to model from fluid-motion equations such as the Navier–Stokes equations. Instead, we rely on empirical relations obtained from experiment and simulation.

We might search online for natural convection coefficient "horizontal plate" or consult handbooks or texts such as Incropera & DeWitt's Fundamentals of Heat and Mass Transport. We find the relation

$$\overline{\text{Nu}_L}=0.54\text{Ra}_L^{1/4}\;(10^4\,\text{≲}\text{Ra}_L\,\text{≲}\,10^7,\;\text{Pr}\,\text{≳}\,0.7),$$

where the Nusselt number $$\text{Nu}_L$$ is defined here as

$$\text{Nu}_L\equiv\frac{hL}{k}$$

with plate area–perimeter ratio $$L$$ and fluid conductivity $$k$$; the Rayleigh number $$\text{Ra}_L$$ is

$$\text{Ra}_L\equiv\frac{g\beta\Delta TL^3}{\alpha\nu},$$

with gravitational acceleration $$g$$, fluid thermal expansion coefficient $$\beta$$, fluid thermal diffusivity $$\alpha$$, and fluid kinematic viscosity $$\nu$$; and the Prandtl number $$\text{Pr}$$ is

$$\text{Pr}\equiv \frac{\nu}{\alpha}.$$

This may look complex enough, but it's mostly material and geometric properties. The Prandtl number for air is about 0.7, so that condition can usually be considered satisfied. There are many online calculators for the Rayleigh number, for example, that can be used for a quick check of scenarios.

Ignoring the temperature dependence of the material properties, we see that the convection coefficient scales, over the applicable range, with $$\Delta T^{1/4}$$, which is fairly insensitive to $$\Delta T$$ when $$\Delta T$$ is large. It may even be possible to use a single average value. For this case, I'd expect somewhere within $$1\text{–}10\,\text{W}\,\text{m}^{-2}\,\text{K}^{-1}$$.

Does this get at what you're looking for?