Temp Rise of an Insulated Copper Wire

[madddog]

This sounds pretty simple to you heat transfer qurus but to someone who doesn't do this on a regular basis, HELP.

I need the equations to determine the temperature rise of an insulated #10AWG copper wire carrying 11.23 amps.

I am designing an umbilical cable that feeds power to two 150 HP electric pumps and need to determine the heat rise per foot to design a reel to carry the umbilical. The heat rise will determing the minimum number of layers I can live with.

Thanks

 

[JKEngineer]

This is how I would approach it, but no guarantees!

It is not significant if the wire is insulated IF you are setting up a number of layers. The heat release (as power or energy/time) is (I^2)*R where I is the amperage and R is the resistivity of the wire. Should be in the handbooks. You then have a power or heat/time per length of wire (assuming you use the R as ohms per length). The temperature rise then becomes a function of how you arrange the insulated wire, NOT the rise of a freely exposed piece. The details of the reel, length, sides, etc. should be reflected in the estimate at this point, at least in a rough manner.

This last step could be approached using either simple heat transfer equations, I think, or certainly using FEA, as on my website for some very different examples.
Jack M. Kleinfeld, P.E. Kleinfeld Technical Services, Inc.
Infrared Thermography, Finite Element Analysis, Process Engineering

http://www.KleinfeldTechnical.com

 

[poetix99]

Just to amplify on what JKEngr said: the (equilibrium) temperature rise isn't a characteristic of the wire, voltage and current alone; you must model the entire geometry to calculate the temperature rise.

Again, as per JKE, the heat generated per unit length is what you need to start with. You must then model the geometry of the umbilical spool. The insulation DOES matter, at least insofar as it determines how densely packed the wires will be and how much power loss as heat is put into the spool per unit volume. (And that the insulation rating of the wire is probably the practical temperature limit for your problem.)

Some simplifications might be possible. For example, if the spool diameter is much much larger than the insulated wire diameter (seems likely), then you could model a cross-section of the spool as a two-dimensional problem.

 

[JKEngineer]

I was, perhaps, not completely clear. As poetix99 points out, the insulation becomes important in the assembly, which is what I meant to imply by "function of how you arrange the insulated wire". Since the beahvior of a single piece of wire is fairly unimportant (for this problem, unless there is only one layer wrapped on the reel), the insulation effects do not become significant until you create your assembled mass on the reel.

HTH

Jack M. Kleinfeld, P.E. Kleinfeld Technical Services, Inc.
Infrared Thermography, Finite Element Analysis, Process Engineering

http://www.KleinfeldTechnical.com

 

[madddog]

Thanks for all the great input and poetix99 you are correct but I have to begin by defining the heat rise contributed by the insulated copper wire first before I can start calculating the heat flow through the different layers of inslulation.

JEngineer you are also correct but can you lead me to that handbook.

Once again I need the equation that calculates the temperature rise of the insulated copper wire with 11.23Amps flowing thought it. It is with this calculation that I can start calculating the heat transfer through the other layers of inslation and all of this together will determine the number of layers on a reel.

 

[JKEngineer]

I looked up the resistivity of 10AWG wire. At 77F it is 1.02ohms/1000ft. At 149F it is 1.18. (Thomas Glover, Pocket Ref, second edition, ISBN 1-885071-00-0).

Having answered your direct question, let me reiterate by reacting to your proposed procedure. NO!!! The heat generated by a single piece of wire is important, but the "heat rise", by which I take it you mean the temperature rise, is irrelevant, unimportant, and meaningless unless you define the local conditions around the wire and unless the geometry of the final usage is similar to a single wire. It may be a significant calculation if it tells you that you cannot put a group of wires together because even one overheats. Since the same reference reports a single wire amperage rating at 86F ambient and a 140F TW insulation as 40 amps, and a triple wire cable rating as 30 amps (other conditions the same)this does not appear to be the case.

As poetix and I have said, you need to look at the proposed assembly. You will need to increase the wraps from 1 to n until you reach the temperature limit imposed by insulation, safety, the reel, or something else, repeating the model at each step.

At the risk of being red flagged, if you need more substantial help, or would like to have this done for you, contact me.
Jack M. Kleinfeld, P.E. Kleinfeld Technical Services, Inc.
Infrared Thermography, Finite Element Analysis, Process Engineering

http://www.KleinfeldTechnical.com

 

[daSapient]

The general d.e. (for constant area, and long wire) that you are looking for looks as follows:

(A/rho) *V'' - (A/rho^2)*V'*T'* (drho/dt) = 0 The current constancy equation.
V'=dV/dx
rho=rho0(1+alpha (T-T0)), alpha can be suitably obtained from Handbooks.
This d.e. can be readily solved.

Then dq/dx= (A/rho)* (V')^2, where V' is dV/dx.

This is fairly good model for your case in my humble opinion. As you might have guessed, I am just a student and mostly like to play with d.e.s. If you need further clarificaitions or feel that the eq is incorrect, do tell me.

 

[abeltio]

madddog,
please refer to:
Heat Transfer by Max Jakob (sixth printing, march 1958)
QUOTE
Article 10-3. Heat conduction in electrical coils at uniform development of heat.
An important practical use in which heat conduction must be considered in connection with uniform heat production is that of an electrical coil in which heat is generated by the current in the wire... [from here on it is a nigtmare of equations]...

Article 10-4. Heat conduction in electrical coils at non-uniform development of heat.
10-4.e. adaptation of the results to real coils

Article 10-5. Determination of the apparent thermal conductivity of electrical coils
UNQUOTE

I think those articles are what you are looking for.
HTH
Saludos
a.