Calculate wire temperature

Question

I have 200 mm long, 0.3 mm diameter nichrome wire. I am heating this wire with 5V AC power supply and current approximately 1.4 A. I need to calculate the exact temperature of the nichrome wire?

Answer 1

Presumably, you want to predict the temperature at the wire surface. The problem you face is that it will vary drastically with conditions, e.g., is the wire supported in free air, straight or coiled, or in contact with materials that will conduct energy away from the wire, and so on.

There are various approximations for some common configurations that can yield a number that may or may not be close to the actual temperature in a practical situation. All depend on the same quantity, the energy input, which is easy to calculate. All make approximations of the various mechanisms that exist for carrying that energy away from the source, i.e., your piece of wire.

In most practical situations it's easier to think of the nichrome element as an energy source, and allow for ample safety margin in the temperature tolerance of the materials used in construction. You should be able to find examples of situations similar to yours on line; noting what element configurations worked in these will offer valuable insight to what you can expect in your design.

It is, of course, possible to analyze your exact situation much more closely, with hard numbers predicted, but you still need to verify the predictions with careful measurement. A good example is here:

http://dspace.library.uu.nl/bitstream/handle/1874/32718/sdarticle.pdf?sequence=2

Depending on what you want to accomplish, it's likely to be easier to run some experiments and dial in the results that fit your design goals.

Answer 2

If you insist on calculating it instead of measuring it, it is not impossible but way more troublesome than measuring it.

Let's assume the surrounding temperature to be \$T_0\$. $$\dot E_i - \dot E_o = mc_p(T-T_0)$$ $$VIcos\theta - \sigma\epsilon A(T^4-T_0^4) -hA(T-T_0) = mc_p(T-T_0)$$ Assuming the circuit is pure resistive, i.e. \$\cos \theta= 1 \$ . $$VI - \sigma\epsilon A(T^4-T_0^4) -\pi hdl(T-T_0) = mc_p(T-T_0)$$

This looks like a homework problem nevertheless, and the calculation above is just theoretical.

Answer 3

You can't because important conditions aren't specified. You know you'll be putting 7 W into 200 mm of wire, or 35 Watts per meter. The temperature will be that power times the thermal resistance from the wire to ambient.

However, that's the big missing piece you don't know. This will greatly depend on air flow, which you haven't specified. In addition to that, the wire will be hot enough to create its own air flow in still air, and that will depend on the orientation of the wire, which is yet another thing you haven't specified.

Probably the best way to find the temperature over different conditions is to measure it. You can calculate the resistance as a function of temperature from the resistivity of nichrome, which is something you should be able to look up. Create a setup that regulates the power going into the wire and measured voltage and current. From that you can compute resistance, from which you can compute temperature. Now measure that temperature over whatever range of conditions you care about. Important variables will be air speed, orientation, humidity, and pressure.

Look up something called a hot wire anemometer. These work on exactly this principle to measure air speed.