Thermodynamics of wire calculation

[TLewis]

I have been over and over this and I need some help.
I have a copper wire (.0019 inch diameter) running in a wire preheater. I heat the wire at 24 volts AC, 800Hz and pull 192 amps. The wire is approximately 1 meter in length and is moving at 2500 ft/sec. The ambient temp. is 73 deg. F.
What is the temperature of the wire?
I know the "black body radiation law" applies here but this is what I am after;

What is an approximating equation I can use to determine the wire temp if I know the following:
Diameter of the wire
Voltage applied
Current measured
Frequency of the source voltage
Emissity of the copper
Wire speed
Wire length 1 meter
Ambient temperature
I am collecting all of the above in a PLC application and want to approximate the exiting wire temp. What about just in a steady-state condition?
I know, this is pretty long hair, but I know there are real engineers/ physics majors that know the laws of thermodynamics and might be able to give me some direction.

thank you

 

[marcusl]

Is this a trick question? There is no wire! Air pressure and turbulence at more than Mach 2 blow away your long little wisp of a wire (half the diameter of human hair) into dust. Even if that didn't happen, application of 192A would instantly vaporize the wire.

 

[L62]

you have a 1.9 mil diameter wire with 192 Amps flowing through it??

 

[TLewis]

Please forgive my typo. That's 2500 ft/min. This application is called an induction preheater. It is used in the wire manufacturing industry to preheat the 24-26 gauge wire to about 330 deg. F before going into an extruder guider and die to extrud FEP (Floro-ethelene polymers) onto it for insulation. The wire is preheated to gain better adhesion of the outside insulation before it is cooled in a water bath. The wire doesn,t burn up because of the speed it is traveling at between the insulated sheaves inducing the current. Sorry guys this is a real life application. Cut apart a CAT5E computer network ing cable (solid wire of course) if it is marked 'Plenum' wire it is a FEP design, 24 -26 gauge. Sorry for the confusion. Any suggestions?

 

[L62]

...What is an approximating equation I can use to determine the wire temp if I know the...

..here's one possible way to simplify the analysis down to get an approximate equation:

(lots of simplifying assumptions here)

Assume a steady state condition, as you have said.. Also assume that the temperature is uniform throughout the wire. Start with the energy balance on the wire:
energy generated + energy in = energy out. (1)

Now define the individual terms in (1):

Energy generated in the wire can be approximated as = I^2 * R (you know the current through the wire, and you know the resistance of the wire though this is also temperature-dependent so you can look up the temperature-dependence of resistivity for the copper wire)

Assume Energy In = 0, i.e. assume no heat is entering the wire from outside sources (neglecting heating from friction of the wire against the electrical contact surfaces for example...maybe this shouldn't be neglected...I don't know, I don't have a good intuition for your application)

Energy out = heat lost from the wire to the surroundings. This is further broken down into:
(a) Convective losses. The wire is moving at a certain speed. How about instead assuming the wire is stationary and the air is moving over the wire at that speed. So use this to calculate the forced convection coefficient. There are all kinds of formulations for the convection coefficients based on geometry and boundary conditions, you would have to look these up in a heat transfer book... Then heat loss by convection = h * A * (Tinf - Twire)
[where h is that convection coefficient, A is the wire surface area which you can get from the length and diameter, Tinf = temperature of the ambient]...to calculate h you will also need to know the properties of the surrounding fluid (which I assume is air) but which you can look up...

(b) radiative losses. Assume blackbody radiation, then radiative loss can be approximate as sigma*epsilon*A* (Tinf^4 - Twire^4)
where sigma = stefan-Boltzmann constant; epsilon = the wire's emissivity at that temperature, A = wire surface area

(c) Conductive losses: assume 1-D conduction to the electrical contacts or support structures that is holding the wire up. (note that this assumes good thermal contact so we can ignore contact resistance) You would need to know the approximate temperature of those structures. Then conduction losses = k*A*(Tstruc - Twire)/L
where k = the thermal conductivity which you can look up, A = the cross sectional area normal to the direction of the heat flux, L is the conduction path lengthSo define (or rather approximate) these individual terms and put them into Equation (1) and you will have a single equation - a fourth-order polynomial in Twire. But I don't know how accurate this will be relative to how accurate you need it to be