Laura Jenkins writes:
> If I calculate the resistance of conductors using R = k*l/A, wouldn't
> the value of k differ depending on the amount of current through each
> conductor (due to the heating effect). Or for small differences in length,
> let's say 5% or less, is the difference in k insignificant?
> Is there an oscillating effect?
While I mentioned that I couldn't prove that there isn't an oscillating effect, I'm pretty certain that you won't have one in any normal situation, and I _think_ that one could prove that you wouldn't get one at all.
You are exactly right, you need to determine the resistance of the conductors using R=k*l/A, and conductors at different temperatures have different values of k. So to answer your question, you need to figure out what the difference in k will be.
The following
short page on temperature coefficeint of resistance will help here. Very approximately, the value of k changes by about 0.4% per degree C. This means that a 12C difference in wire temperature will have about as much effect on the balance of current flow as a 5% difference in length. Thus I would say that the difference in k is potentially significant.
You now need to take the question further, and ask if there will be a temperature difference, and ask how much it will be. Remember that _both_ wires are carrying current, so both wires will heat up, and for short differences in length, the current will be pretty darn close to the same, so the temperature rise will be pretty darn close to the same, and the temperature difference pretty small. You also need to consider the environment that the wires are in. If they are well cooled, then the overall temperature rise will be pretty small, and the temperature difference smaller still. If they are well insulated but together (say two parallel conduits close together in thermal insulation) then there will be lots of temperature rise, but again the temperatures will be pretty similar. If you had two conduits separated by space, each buried in thermal insulation, then you could potentially get a large temperature difference, and the difference in k would be significant.
There is a very useful approximation for figuring out how hot something will get. It is called the 'lumped resistance' model, and basically you assume that conduction of heat acts just like conduction of electricity. You just use Ohms law, but interpret it differently: E=I * R, where E is the _temperature_ difference in degrees C, I is the _heat flow_ in watts, and R is the _thermal resistance_. You then calculate the heat being generated at the source, divide by the thermal resistance, and this gives you E, the temperature difference (or temperature rise). The higher the thermal resistance, the greater the temperature rise for the heat to be carried away.
I spent some time working backward from table B.310.16 (Ampacities of ...Conductors...In a raceway, cable or Earth), and figure that a resistance of about 4.5C/watt is about right for 4/0AWG wire (3 conductors in a cable). Each conductor would be dissipating about 4.3 watts per foot at the allowable current of 260A, and there are 3 conductors for a total of 12W per foot, and the allowed temperature rise is 60C. However I am not certain of the thermal conductivity number; I've just been playing with the tables and making some assumptions. I a resistance that I calculated at 90C from tables 8 and 9 in chapter 9, and these values don't quite agree with values from other sources. But this should give you an idea of a reasonable value for the thermal resistance to expect with cables.
This should give you an idea of what sort of temperature difference you might find in the situations that you are describing. 4/0 copper at full ampacity is carrying 260A (presuming 90C conductors, with 90C terminations, in a 30C ambient, everything else good for this use, blah blah blah), and dissipating 260^2 * 0.0000633 = 4.3 watts per foot. With three conductors in a raceway and a thermal resistance of 4.5C per watt, you get a temperature rise of 4.3*3 * 4.5 = 58 degrees. Now add a 5% current imbalance. You have 254A in one set of conductors and 266A in the other. The temperature rise for the low current set is 254^2*0.0000633 * 3 * 4.5= 55.1C, and for the high current set 266^2 * 0.0000633 * 3 * 4.5 = 60.4C (Note in this calculation I _presumed_ that k was the same in order to calculate the power dissipated in the wires. But the temperature is clearly different, so k _must_ be different...but this is a second order effect, and I've already mucked with this calculation for too long on a Sunday morning; you are welcome to refine the calculation...)
This 5C difference in temperature corresponds to about a 2% difference in length...so my rough estimate is that given this level of thermal resistance, the compensating effect of temperature change in coefficient of resistance would mean that a 5% difference in length would result in a 3-4% difference in current being carried by the two sets of conductors.
-Jon
