I'm trying to get a formula that I can reuse to help me determine wire size and properties. The goal is to heat a 430' wire between 40-100F using 12 volts and 10 amps of direct current. Everyone is talking about ambient temperature but I'm more interested in the actual wire temperature, properties (copper, zinc, aluminium) and resistance. I'm looking to engineer the wire to fit the purpose not make my system fit the wire (too late).
The resistance of the wire will be 12/10 = 1.2 ohms
R = rho * length / Area (where rho is the resistivity of the material chosen, remembering to convert length etc. to appropriate unit)
then Area = rho * length /1.2 this will give you the cross sectional area. Look up the X sectional area to give a wire gauge.
All this will do is give you a wire of the correct length and resistance for a given material.
The temperature the wire will 'get up to' depends on how quickly energy is lost to the environment. (Power out = power in @ equilibrium temperature).
There is no set formula for the final temperature of the wire because there are so many variables e.g. how well insulated the wire is, how it is wound (large or small coils), in air or vacuum and so on and so on ...
You have two basic options.
Experiment: Get your wire (as calculated above), feed it with 12V and see what happens temperature wise.
Devise a circuit to limit how hot the wire gets (within your desired range) by controlling the amount of power dissipated in the wire. (e.g. - fit a thermostat)
You might look at the link I found here, Current capacity of copper wires in vacuum
I think they also did wires in air.
The big question is how does the wire lose heat? Convective cooling is hard to quantify.
Just looking at your numbers, I would guess that 120 Watts of power is not all that much to raise the temp of a 430' length of wire, Unless it's wound in a tight coil, or very thin...
For the actual wire temperature you need to know three things-
- The power going into the wire per unit length (that's easy to calculate from the current and resistivity). It won't vary much with temperature.
- The ambient temperature (you'll get a rise above ambient temperature that may be a nonlinear function of temperature difference)
- The heat loss per unit length. That may be difficult to calculate. Newtonian heat loss is simple conduction like Ohms law- you'll get so many degrees rise per Watt of power dissipation (per unit length). Convection and radiation are nonlinear and very nonlinear respectively. King's Law (derived in the early 1900s) predicts heat loss from an infinite cylinder in a fluid of low Reynolds number, which fits air fairly well under some conditions.
When it comes right down to it though, you may be best off to build a prototype that is maybe 5' long and test it under typical conditions.