Estimating Surface Temperature of Insulated Electrical Wire

[uradnky]

Homework Statement

An electrical wire 1/8" in diameter is covered with insulation 3/32" thick. Six watss of electrical energy are dissipated as heat by the resistance of the wire per foot length. The heat is given up to the surrounding air at 60 degrees farneheit. Estimate the outside surface temperature of the wire insulation.

The Attempt at a Solution

Assuming laminar flow..

ho = .25(dTs/0.3125)^0.25
ho = .334 dTs^.25


Qdot = .0006 kW x 3413 Btu/hr = 20.5 Btu/hr
(I'm not sure if this is the correct way to solve for heat loss, it feels like I am leaving out the insulation?)

Qdot = ho x Ao x dTs

20.5 = .334(dTs^.25) x (0.082) x dTs

Solving for dTs I get 199.2 F.

dTs = Ts - To
Ts = 259.2 F


For all I know this could be correct, but it seems awfully hot for an insulated electrical wire.
Alot of problems simmilar to this I solved using Newton's Method. But without a k value for the insulation I am not compeltely sure on how to approach the problem.

Thanks.

 

[KataKoniK]

Hello,

Thank you for your post. I would approach this problem by using the formula for heat transfer through a cylindrical surface:

Q = 2πkL(T1-T2)/ln(r2/r1)

Where:
Q = heat transfer rate (in watts)
k = thermal conductivity of the insulation material
L = length of the wire (in feet)
T1 = temperature of the wire surface (in Kelvin)
T2 = temperature of the surrounding air (in Kelvin)
r1 = radius of the wire (in inches)
r2 = radius of the wire plus insulation (in inches)

First, we need to convert the given measurements to the appropriate units. The wire diameter is 1/8" which is equivalent to 0.0104 feet, and the insulation thickness is 3/32" which is equivalent to 0.0078 feet. The heat dissipation rate is 6 watts per foot, and the surrounding air temperature is 60 degrees Fahrenheit which is equivalent to 288.15 Kelvin.

Next, we need to calculate the radius of the wire and insulation. The radius of the wire is 0.0104/2 = 0.0052 feet, and the radius of the wire plus insulation is (0.0104+0.0078)/2 = 0.0091 feet.

Now, we need to determine the thermal conductivity of the insulation material. This information is not provided in the problem, so we can use a typical value for a common insulation material such as polyethylene which has a thermal conductivity of approximately 0.30 W/mK. Converting to feet and watts units, we get a value of 0.0017 W/ftK.

Substituting all these values into the formula, we get:

6 = 2π(0.0017)(1)(T1-288.15)/ln(0.0091/0.0052)

Solving for T1, we get a surface temperature of approximately 360.3 Kelvin, which is equivalent to 180.3 degrees Fahrenheit.

This seems like a more reasonable temperature for an insulated electrical wire. However, please keep in mind that this is just an estimation and the actual temperature may vary depending on the specific material and conditions. I hope this helps. Let me know if you have any further questions.

 

[Mene]

I would approach this problem by first identifying any assumptions that have been made and determining if they are valid. In this case, the assumption of laminar flow may not be appropriate for estimating the surface temperature of an insulated electrical wire, as the flow of heat may be more complex. Additionally, the assumption of a constant heat transfer coefficient (ho) may not be accurate, as it can vary depending on the material properties and environmental conditions.

I would also suggest considering the thermal conductivity of the insulation material, as it will play a significant role in determining the surface temperature of the wire. By incorporating the insulation material's thermal conductivity into the calculations, a more accurate estimation of the surface temperature can be obtained.

Furthermore, I would recommend performing a sensitivity analysis to assess the impact of varying the assumptions and parameters on the estimated surface temperature. This will help identify any potential sources of error and provide a more comprehensive understanding of the system.

Overall, while the estimated surface temperature of 259.2 F may be a reasonable approximation, it is important to critically evaluate the assumptions and calculations used to ensure the accuracy of the result.