Solve Thermodynamics Question on Electric Wire with Plastic Cover

[nickbone59]

Hey, have been working on this question for about 2 weeks now and can't crack it, I found a formula on the net which gives me a value of just over 30C, i am not confident i have the correct value though. It just seems too low. I anyone could point me in the right direction with this it would be greatly appreciated.

Q1)

A high conductivity electric wire, diameter 3 mm and length 10m, is tightly wrapped with a plastic cover of thickness 1mm and thermal conductivity 0.15W/(mK). Electrical measurements indicate that as a current of 10A passes through the wire, there is a voltage drop 6V along the wire. The insulated wire is exposed to ambient conditions at a temperature of 30C with a surface convective heat transfer coefficient of 12W/(m^2K).
Neglecting any temperature variation across the wire cross-section,

i)calculate the temperature at the interface of the wire and the plastic cover under steady state operation.

ii)measurements indicate that the rate of heat loss from the wire increases rather than decreases as the insulation thickness increases. an the measurements be correct. Explain.

Homework Equations

The Attempt at a Solution

Heat generation in a resistive wire

Per unit volume : I^2R/Pi r0^2 L

= 100x0.442/pi x1.5^(2)x10^-3 x 10
=44.2/70.685x10^-6
=625446.7909


Surface Temperature:

To = T infinity + (g ro^2)/4k
= 303 + 62544.7909 x ((3x10^-3)^2/4x0.15)
= 303.938k
= 30.94c

 

[mgb_phys]

Power loss in the wire is P = IV = 60w/10m = 6W/m, this is the heat into the insulator. You can work out the surface area of 1m of wire, and so the heat loss from the outer cover.
The temperature of the outer and the conductivity of the insulator is known - so you know the temperature difference through the insulator.

Part 2 - what happens to the temperature of anything as the insulation is increased? What happens to the resistance of a wire when the temperature increases? What does changing the resistance of the wire do to the power?

 

[nickbone59]

thank you

thanks that's a great help, hopefully i can suss it out now, thanks again.

 

[nickbone59]

actually i am still a bit unsure on how i calculate the temperature difference through the insulator from the temperature of the outer and the conductivity.

 

[mgb_phys]

The conductivity of the insulator is 0.15W/(mK) with a thickness of 1mm.
So for every 150W ( ie 0.15/0.001) of power transferred through the insulation will give a temperature difference of 1K. In the stable state the power into the wire must equal the heat flow through the insulation.

 

[nickbone59]

i still don't understand? what your saying just doesn't make sense? Does the surface convective heat transfer coefficient not come into it?