Conductor Load-Temp-Resistance Correlation

[CKent]

Does anyone know the correlation between conductor load current and would be resulting temperature? I am actually more interested in the approximate resistance of a particular conductor at a particular load current. From T&D handbook, the temperature effect on dc resistance of a conductor is given by (Rt2/Rt1) = (M+t2)/(M+t1) wherein Rt1 & Rt2 are the resistances at temperatures t1 & t2, respectively, while M is the contstant of the conductor mat'l, inferred absolute zero temp. From these relationship, given the dc resistance of a conductor at a particular temp, we can calculate the resulting resistance at any temp. Then using the relationship between ac and dc resistances, considering the skin effect constant, ac resistance could then be known. However, on normal loadflow simulation, it is the load current of a particular line which could be determined. Hence, I need to know what would be the resulting temperature of a conductor at a particular load current? Are there direct linearity such as that in temp-resistance relation? Any articles publish that deals with this?

 

[electricpete]

People involved in infrared monitoring often search for a correlation between load and temperature for tracking resistive hotspots under varying load conditions.

It is widely reported that temperature rise varies as load^n where n is in the range of 1.5 to 2.0. If you have "Electrical Equipment Maintenance" by A.S. Gill, you will see they use an exponent around 1.8. I have seen an article on-line that referred to a similar number with experimental results.

Here is my derivation.
Look at the electrical circuit. Heat generated is q=I^2*Re.

Look at the thermal circuit. Heat dissipated is
q = dT * Rth

Assume Re is constant for simplicity although we know it varies slightly 0.4%/C

Under the simplest linear model of heat transfer, Rth is constant. dT~q/Rth = I^2*Re/Rth

This simple model would predict temperature rise with load^2.

That linear model corresponds to conduction. But there are also convection and radiation going on. I propose that the dominant mode of heat transfer to ambient is convection.

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Assume thermal resistance is primarily convective and any conductive thermal resistance or radiative resistance in the problem is negligible in comparison.
dT ~ q / Hfilm(dT)

where dT is temp rise, q is heat generated by resistance connection (watts) and Hfilm(dT) is a convective heat transfer parameter varying with dT discussed more at the end of this document

Substitute q = I^2*Relect
dT ~ I^2*Relect / Hfilm(dT)

Neglect small changes in contact resistance with temperature => assume Relect is constant and remove it from the proportionality.
dT ~ I^2 / Hfilm(dT)

Substitute Hfilm ~ dT^m (with the exponent m in the range 0.25 for laminar natural convection flow to 0.33 for turbulent natural convection flow per CRC handbook).
dT ~ I^2 / dT^m

Multiply each side by dT^m
dT^(1+m) ~ I^2

Raise each side to the power 1/(1+m)
dT ~ I^2 ^ (1/[1+m])

If m = 0.25 = 1/4 (laminar)
dT ~ I^2 ^ (1/[1+0.25]) = I^2^(4/5) = I^[2*4/5]
dT ~ I^ 1.6

If m = 0.33 = 1/3 (turbulent)
dT ~ I^2 ^ (1/[1+1/3]) = I^2 ^ 3/4 = I^2*3/4
dT ~ I^1.5

Conclusion dT varies between I^1.5 and I^1.6

I think it is fair to neglect conduction as an important part of heat transfer to ambient in most situations but not quite fair to neglect radiation. I have done some quick approximations to estimate an equivalent power-law fit to the radiation curve for temperature in the range 50C to 100C. If memory serves me right it is not too far away from the results above.







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[jghrist]

If you are concerned with bare overhead conductor, use IEEE Std 738-1993, IEEE standard for calculating the current-temperature relationship of bare overhead conductors.

If you are concerned with insulated cable, you need to use the same type of equations that you would use for cable ampacity. I would recommend Rating of Electric Power Cables, George J. Anders, McGraw-Hill/IEEE Press,1997.

For a load flow, I would just use an average temperature rather than try to explicitely include the relationship in the load calculations. The variation of resistance with load will not be enough to significantly affect the load flow calculations.