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Relationship between power dissipation in wires and temperature

  1. Sep 25, 2012 #1

    I am wondering if anyone knows any correlations or equations that can give me the temperature of a wire due to power losses.

    I understand how to calculate the power loss in the wire, based on wire resistance and current (P=I^2*R) but how do I turn this into a heat transfer problem and find a temperature? In other words, how do I know what that power dissipation will heat the wire too? Are their a set of equations that relate temperature, power dissipation and geometry of the wire?

  2. jcsd
  3. Sep 25, 2012 #2
    Yes, there is a way of calculating this. You need to be able to determine the relationship between the temperature of the wire, and the heat transfer rate to the surroundings. Do do this, you need to make use of the disciplines of fluid mechanics and heat transfer. For a wire suspended in air, there will be an air flow set up around the wire consisting of natural convection. The natural convection is the result of buoyancy created in the air surrounding the wire by thermal expansion from the increased temperatures. You also need a differential energy balance model which determines the effects of the flow and conductive heat transfer within the air on the temperature distribution. The analysis is simplified if you have experimental data to fill in some of the blanks. You may be able to find the results of natural convection experiments from a horizontal cylinder to the surrounding air. You can use such information, together with dimensional analysis to determine the heat transfer coefficient from your wire to the surrounding air. The heat transfer coefficient is defined as the heat flux per unit area of cylinder divided by the temperature difference between the cylinder and the air. Check out key wording such as "natural convection heat transfer from a horizontal cylinder." Check out books on heat transfer, and engineering handbooks such as Perry's Chemical Engineers' Handbook, and Marks' Mechanical Engineers' Handbook.
  4. Sep 25, 2012 #3
    Thanks, I knew it was mostly a heat transfer problem, but was hoping someone had developed a one equation correlation for such an application. Also, I read something that made a lot of sense today. Once the power being dissipated by the wire equals the heat transfer away from the wire (from all 3 modes - radiation, convection, conduction) I will reach a steady state and can determine the temperature of the wire.

    Problem is more complex than I would like, probably will require some numerical solving, but none the less I have the right steps.

  5. Sep 27, 2012 #4
    Just wanted to update this thread if anyone was following it and wanted help. I created a numerical solution for this problem using a simultaneous equations solver. I also used correlations from a Heat Transfer text, by Nellis and Klein.

    At first the energy balance I used was, Power Dissipation (ohmic losses in wire) = Natural Convection Heat Transfer. I can ignore forced convection as the tests were all done in controlled areas.

    I was finding my theoretical results were not matching test results, I was calculating temperatures much too high. The key was I needed to include radiation losses.

    So now the energy balance is, Power Dissipation (ohmic losses) = Natural Convection Heat Transfer + Radiative Losses. Using this balance, my results are very close to the empirical data collected from wire gauges from 0 to 40. I plotted my theoretical results and my empirical data and the curves match up very closely.

    The way I did the radiation calculations, is I assumed the wire had a view factor of 1 to all of it's surroundings, and the emissive power of that was calculated from room temperature. It would be like using Earth to space (a very small entity surrounded by a large entity of uniform temperature). It would be too hard of course to account for everything around the wire in the room, but the wire is in a large enough area where it can be considered like space, except space in this case is 72 degrees F. If the wire was gigantic, like the size of a person, then more radiation calculations would have to be taken into account, but for these purposes, the equations worked out very well.

    I also was hoping, if anyone saw any issues with my rational and thought process to chime in. I am sure I missed some things, but overall I am very pleased with my results.

  6. Sep 27, 2012 #5
    This should be in the electrical engineering section, PM a moderator and ask to have it moved.

    Not equations, but performance tables.

    Look up the guides to the wiring regulations for your country. They will probably have published tables for temp rise in cables for differing environments such as conduit, plaster, exposed etc.
  7. Sep 27, 2012 #6
    Correct, but I would like to perform calculations, as I am working with some different materials, such as high resistive wires for heating. There doesn't seem to be tables that identify several different types of wire material, and also, now I can adjust about any parameter and see how it will affect my heating.

    I am doing some research on new materials that are not widespread as well, so that makes it more difficult to find experimental data.
  8. Sep 27, 2012 #7
    Nevertheless your wiring will have to satisfy the regulations, not calculated formulae.
  9. Sep 27, 2012 #8
    I am not using the wiring for transfer current to a load. I actually want to produce heating in the wire. I have heated these wires to over 2000 degrees F, I'm sure that wouldn't be code for any electrical application lol. The wire is live, but it is used for ohmic dissipation, not electrical energy.

    I will however have to provide current to this high resistive wire, so in order to bring the current there I will have to meet those current requirements on lower resistance wires, as those will be insulated and should not be heated.
  10. Sep 27, 2012 #9
    The wiring regulations (in the UK at least) covers use as well as distribution.

    If you are developing heating elements then the effect of ceramic or other formers needs to be considered.

    I am simply offering help here, but you are following a well trodden path. I still recommend moving this to the EE section where there are many experienced in the subject.
  11. Sep 27, 2012 #10
    Ok thanks for the support, I will see how it can be moved, who do I PM exactly, any moderator? Either way, I pretty much was able to accomplish what I set out. I was just hoping others could comment on the calculations, and not whether it makes sense to work them out in the first place or not.
  12. Sep 27, 2012 #11
    Mike Berk (Berkman) is a really nice guy and the main engineering mod.

    Alternatively click on the 'report button' and type your request into the box that opens and any mod will deal with it.

    (They don't bite on Thursdays)
    Last edited: Sep 27, 2012
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