created at December 5, 2022

# Conductor Heating

Consider having a piece of conductor (wire) with length l and diameter d. It has no isolation and is stretched between two fixed points. Air is flowing around. After we apply a voltage to the conductor, current I = U/R starts flowing, and the conductor starts heating. It continues to heat until reaches some stable temperature. The question is, what will be the temperature in the end?

The wire like that could be used to cut a foam plastic for example.

The active power is produced inside the wire:

$P = R\cdot I^2$

Also, the part of that energy contributes the heating (increasing wire temperature T), another part dissipates into environment. Just after plugging the voltage, the wire temperature is the same as environment temperature T0. In the first seconds a lot of energy is wasted to heat the wire, wire temperature increases. The greater wire temperature is, the larger amount of heat is leaking outside through the wire surface. After some time occurrs the state, when the constant temperature is established and all produced power dissipates into environment.

Let's write the energy balance of the process. In the short amount of time the heat energy is applied to the wire:

$dQ = P\cdot dt$

which is then used to heat the wire inside:

$dQ_1 = c\cdot m\cdot dT$

and partially dissipates into environment:

$dQ_2 = \alpha \cdot A \cdot \left ( T - T_0 \right )$

Here m - wire mass [kg], c - specific heat capacity [J/(kg·K)], dT - elementary wire temperature growth, A - wire outer surface [m2], α - heat transfer coefficient [W/(m2·K)], T0 - environment temperature.

While most of these parameters are easy to find for your case, the heat transfer coefficient requires special attention. The heat transfer coefficient describes heat transfer intensity between the wire metallic surface and the environment matter (air, water or whatever). It depends on surface material, surface quality, environment etc.

For that study case I was interested in using nichrome wire, which is surrounded by air having room temperature and zero velocity. I've found a value α = 46 W/(m2·K).

In the end the energy balance is following:

$P\cdot dt = c\cdot m\cdot dT + \alpha \cdot A \cdot \left ( T - T_0 \right )\cdot dt$

After dividing by dt we see the linear first order differential equation, where the wire temperature T is a desired function of time. Solving that equation will give us the process of heating the wire. To solve it numerically, leave the T derivative alone on the left side:

$\frac{dT}{dt} = \frac{P - \alpha\cdot A\cdot \left (T - T_0 \right )}{c\cdot m}$

Then use Dysolve app to solve the case. For nichrome wire having length 1 m, diameter 0.4 mm, and 12V voltage applied, the heating process looks like this:

Here the heating starts from the room's temperature and ends in approximately 40 seconds, reaching the constant temperature around 300°C.