Thermal Stress of Insulated Wire

[jwilson3]

IEEE Std 142-2007 "Grounding....Power Systems" has two equations [ 2.3a & 2.3b] that help define the thermal boundary for wire insulation protection based on I^2t, the initial wire temp, final temp, and wire xsect area. The equation 2.3a is I^2t/A = 0.0297ln[(Max Temp + 234)/(Initial Temp + 235)]. Does anyone know where this equation came from and does it match the original?

I can't get the math to correlate with the example in Part 2.7.4.4 of 142-2007.

 

[PHovnanian]

I suspect that its a curve fit to some measured or modeled behavior of a section of wire with a certain thermal mass and resistivity (which depend on the conductor type), insulation thermal mass and conductivity and environment (bundled or single, in conduit or free air, etc). Either the test conditions or the the numerical model involve quite a few parameters. So some example cases were tested/plotted and a best fit was made to the data points to produce an easy to use cookbook formula.

 

[electricpete]

I haven't looked at the red book, but I think it is a simple adiabatic (no heat transfer out) calculation. Was discussed and derived here:

http://www.eng-tips.com/viewthread.cfm?qid=238191

=====================================
(2B)+(2B)' ?

 

[cuky2000]

The origin of this equation come from considering that heat generated by the conductor during short circuit is absorbed by cable insulation.
_{NOTE: The cable damage curve used for protective device coordination is based in this principle.}
The equation presented in the IEEE standard has couple of error: a) A should A^2
b) Ln should be log10.

I^2t/A² = 0.0297log₁₀[(Max Temp + 234)/(Initial Temp + 234)].

After those changes, the math should correlate with the example.

 

[electricpete]

The origin of this equation come from considering that heat generated by the conductor during short circuit is absorbed by cable insulation

I assume you have a typo. The equation assumes the heat is absorbed by the conductor.

=====================================
(2B)+(2B)' ?

 

[electricpete]

...more specifically it is the specific heat capacity of the conductor which is relevant for the calculation.

=====================================
(2B)+(2B)' ?

 

[electricpete]

By the way, it was a useful correction from cuky. That will save you some time.

=====================================
(2B)+(2B)' ?

 

[jwilson3]

cuky2000 mentions that 'ln', in my equation above, should be 'log10' and he is correct.

In IEEE 142-2007, it's shown as 'in'; I slipped the 'ln' in.

IEEE does a good job on their standards don't they.

 

[cuky2000]

Hi Pete,

I believe that the heat is generated by passing a current through the metallic conductor resistance (Q= I^2R.t). This equation assume that the heat transfer thought the insulation to the external ambient is negligible (adiabatic) since happens in a short time. This will increase the temperature of the insulation from the initial operating point to higher temperature, hopefully below the insulation limits

_{NOTE: This approach is conservative in the safe side in a range of 5% and 10%.}

 

[electricpete]

I agree with everything in your last post. It is an adiabatic problem... which means there is no heat transfer out of the system.... so the I^2*R*t energy must result in an increase in the stored thermal energy of the system. The relevant stored thermal energy is associated with the conductor thermal capacitance only... the insulation thermal capacitance is negligible in comparison and is appropriately neglected by IEEE in deriving their equation.

In summary:
Resistive Losses From Electrical System => Stored Thermal Energy
I^2*R*t => C*deltaT
where t is time and deltaT is temperature change and
C is thermal capacity of the conductor only

If R didn't vary with temperature, we could simply solve for deltaT =I^2*R*t/C instead of needing the more complicated form of the IEEE equation.

Probably all of the above is familiar to you, but when you said: "The origin of this equation come from considering that heat generated by the conductor during short circuit is absorbed by cable insulation", it sounds like a contradiction to me, since insulation thermal capacity is neglected... the conductor is what "absorbs" the thermal energy. If the bolded portion was intended to convey that there is assumed to be no heat transfer out of the system, then I agree.

=====================================
(2B)+(2B)' ?

 

[cuky2000]

Pete,

I like your rational and detail analysis. Beside some initial semantic issues, I am glad that we are all agreeing.

As a "Grand Finale", here is a graph courtesy of Okonite to address cable allowable thermal stresses during short circuit