Calculating steady-state temperature for electrically heated wire

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.

• What is the ideal operational temperature of the wire? Sep 13, 2022 at 11:11
• Conduction will matter if the purpose of the wire is to provide heat to something with which it is in contact, such as plastic it is melting. Also, during cutting, there won't be much convection. Will these two effects cancel out? I think that may depend upon how fast you are cutting through the foam . Faster implies more conduction losses to the cooler foam. Slower means the loss of convection may actually cause the temperature to rise. Sep 13, 2022 at 11:23
• Possibly, the solution might be to monitor applied voltage and current and calculate the wire resistance. The wire resistance will indicate the temperature of the wire. So, use a control loop to keep the resistance around where you want it to be for the right temperature. Sep 13, 2022 at 11:55
• I think this is an application where calculation is impractical due to variables in material thickness, density, thermal conductivity, feed rate, ambient temperature, wind, etc. Might be more practical to stop heating and measure wire resistance (and thus temperature) for 1µs every 1ms. The temperature delta will tell you if the wire is heating or cooling, thus power can be varied (and temperature regulated) automatically for any workpiece and ambient conditions. Sep 13, 2022 at 12:04
• I don't understand your problem. If you don't have a better equation, just make an initial guess at the the temperature and iterate to a final solution within whatever bounds you pick. Worse guess -> more iterations but you'll still get there. It's going to be a very rough estimate anyway. Sep 13, 2022 at 12:17