I am working on a program aimed at finding optimal parameters (voltages and feed rates) for a hot wire foam cutter. One starting problem that has to be solved before even attempting to evaluate heat transfer into the foam is finding the steady state temperature for the hot wire at a given supply voltage. I tried to approach it as follows:
The resistive wire gains heat by Joule heating, for a DC supply I use this formula:
Pgain = Vsupply ^ 2 / Rwire
and loses heat by convection
Pconv = heatTransferCoeff * surfaceArea * (Twire - Tambient)
Prad = emissivity * surfaceArea * stefanBoltzmannConst * (Twire ^ 4 - Tambient ^ 4)
, while conduction is not modeled due to small contact area between the wire and mounts.
At a thermal equilibrium the power gained by resistive heating will be equal to the power lost by convection and radiation, so the equation for the dissipated power could be solved for temperatures from ambient to the melting point for wire material, and the number matching the power drawn from the supply will indicate the equilibrium temperature.
However, while the emissivity for a given material could be obtained from various handbooks, I found out that the heat transfer coefficient is temperature dependent itself. To calculate it for a horizontal cylinder with external flow, which is the model for the hot wire, I need the temperature of the cylinder surface. As such, I cannot calculate the wire temperature... without knowing the wire temperature?
I tried avoiding this catch-22 by setting the heat transfer coefficient to 8 like in this calculator, but my results look way off: the calculated temperature for a 0.15 mm thick, 0,5 m long nichrome wire at 24V is in excess of 750 ℃, which should make it oxidize, glow dull red and burn or at least quickly vaporize the styrofoam, and that does not happen when I am using my cutter with such parameters - the wire remains shiny and cuts the foam with a small kerf.
Could anyone please point me down a correct path? I cannot proceed with modeling the foam cutting without the wire temperature, so any help would be really appreciated.