# Heatsink Problem

1. Nov 10, 2015

### Physics_Kid

i need some help understanding this not-so-simple heatsink model.

lets say i have a length of copper wire, say 8ft long, and i put tubular insulation around the wire for just a section of it, say a 4ft length and it is co-centered on the wire (2ft on each side is in ambient air), for giggles the insulation R value is 100.

now, i pass amps through the wire, from the wire properties i can calculate the heat being generated by the wire, hence the exothermic density (flux) is the same everywhere. as example, i know its watts/mm^2.

i know copper has low R value, yet i also know that the wire is exothermic everywhere, so help me understand the answer to my Q.

my question is, does the exposed copper outside of the insulation have significant impact of the temp of center of wire?

2. Nov 10, 2015

### Staff: Mentor

I'm no expert in heat flow, but I can offer a couple thoughts and a question for you.

You could run an experiment with an insulated section of wire and an uninsulated section of wire separately, to determine the Thermal Resistance from the wire to the air (in Degrees C per Watt). Obviously the insulated wire will have a much higher Thermal Resistance, and hence will get hotter for the same current than the uninsulated wire.

So, in your setup with the middle section of wire insulated, you know that the center section will get hotter. Your question asks how much effective heat sinking will the uninsulated wire sections provide, and how much will that help to lower the temperature of that center section (and raise the temperature of the uninsulated sections). What is it about this setup would mitigate the heat sinking effect of the uninsulated sections of wire and keep them from being effective as heat sinks?

3. Nov 10, 2015

### Physics_Kid

i think you understand my Q, but let me clarify. in my setup the ends are clamped to the power source, using heavy metal clamps. the wire sticks out of the insulation some so that it can be connected. so there is some convection mode there outside of the insulation. i am trying to understand a couple of things. does the non-insulated sections act as a heatsink in a significant manner that it affects the temp in the middle? and if so, how long would the insulated section need to be so that the heatsink affects would be insignificant on the temp in the middle of the wire? my goal is to test the temp of the wire vs amps, but need to do so without heatsink affects changing the temp in the center. its not trivial to me because the wire itself is exothermic everywhere on the wire.

4. Nov 10, 2015

### Staff: Mentor

5. Nov 11, 2015

### Staff: Mentor

The temperature in the insulated section would not be affected significantly at distances more than a couple of insulation diameters in from the ends. But if a more precise answer is required, the system in the region near the ends of the insulated section can be modelled pretty easily.

Chet

6. Nov 11, 2015

### Physics_Kid

hi Chet,
what is "insulation diameter" if the wire is laying in the middle of a 2ft x2ft x 4"thick insulation sandwich (sorry, i changed physicals of insulation)? from center the closest "out" for heat would be two inches on each side (lowest R value section of the sandwich), so is "insulation dia" = 4" and the heatsink mode would be insignificant beyond about 8" into the insulation from the end, but up to ~8" from end the heatsink mode will impact wire temp?

also noted, the R value of copper is magnitudes less than R value of the insulation.

Last edited: Nov 11, 2015
7. Nov 11, 2015

### Staff: Mentor

Sorry. I'm having trouble picturing the geometry. I thought it was a single wire with annular insulation around it, stripped for part of the wire length.

Chet

8. Nov 11, 2015

### Physics_Kid

Hi Chet,
the physicals are simple.

12awg bare copper that is 4ft long, a 2x2ft x 2" foam board (two of them) sandwiches the wire co-centered, thus 12" of wire sticks out each side of the foam board sandwich. taking temp measurements of wire at center as amps through the wire varies. the ends of the wire are clamped to a variable power source, etc.

9. Nov 11, 2015

### Staff: Mentor

What is the context of this question? That is an unusual setup. What is the reason for this question? It sounds like a specific situation, not a general heat flow question...

10. Nov 12, 2015

### Staff: Mentor

Are you also worried about the heat loss through the two 2' x 4" rectangular areas at the ends? Do you need to know the answer in advance, or, can you first run the system and then, based on the temperature measurement at the center, use that to estimate the heat sink effect of the wires at the ends? If it is the latter, I can tell you how to make the estimate. If the former, there is a way of getting an upper bound to the temperature effect at the center, but it might be over-conservative.

Chet

11. Nov 12, 2015

### Physics_Kid

i can answer the last two posts here.

post #9, i already stated the purpose, testing wire temps vs amps for a wire that passes through insulation, in this case a 2x2 x4" foam board insulation sandwich, that is my experiment. simple.

post #10, i am not worried about heat loss via the insulation sandwich, just trying to understand how if the ends of the wire are acting as a heatsink (which they are) does this impact the temp of wire in center in a significant way? i could run the system with the ends insulated and non-insulated to see if there's a diff, but i wanted to get something in form of basic equation that would suggest your notion of not more than X # of dia, etc.

12. Nov 12, 2015

### Staff: Mentor

The answer to your question about the heat sink is intimately related to the heat transfer through the insulation sandwich. The temperature disturbance caused by the bare wire at the edge is dissipated through the insulation sandwich. Note, for example that, if the insulation were perfect, the temperature at the center would rise to a huge value, since the only way that heat could be removed would be by conduction along the wire (to the edges).

Chet

13. Nov 12, 2015

### Physics_Kid

so, i already know a few things.

1) the R value of the insulation is magnitudes bigger than the R value of the copper, so up front we know some heat from inside the insulation will want to escape using conductive mode out down the wire to the outside where there is a "heatsink" clamped at the very end. some heat will also escape through the insulation.
2) equilibrium temp is governed +Q(total) + -Q(insulation) + -Q(copper) = 0
simply put, the exothermic wire is generating heat, and that heat can escape via several paths via two modes, doing so until the temp in center reached equilibrium.

now, here's where it is not trivial for me. there is a "heatsink" at the very ends of the wire, copper has a very low R value, the wire is exothermic (constant flux) everywhere, how does the heatsink at the very ends impact the temp at the center.

as a hypo, lets change the dimensions of the insulation sandwich, lets say its 20ft x 20ft x 10"thick, and the wire is 30ft long co-centered in the sandwich (5ft sticking out each end). same amps, same physical "heatsink" on the ends. we still agree that the copper will conduct heat out to the ends, but is this conductive mode significant such that it affects the temp of the wire in the center? the monkey wrench for me is, the wire is exothermic everywhere, so its not as simple as looking at this as if just the center of the wire is exothermic.

i can partially answer my own Q. we know it affects temp because some heat moves down and out the copper, but to what magnitude relevant to -Q(insulation)? a 50/50 split would certainly mean big significance, but a 92/8 split means much smaller significance.

14. Nov 12, 2015

### Staff: Mentor

Let's try to get a handle on this by using a heat transfer model. Instead of a 2' x 2' x 4" sheet, suppose the wire is inside a 4" diameter annulus of insulation, and that the combined wire and insulation is 2' long. To start with, lets assume that the heat sinks at the ends of the wire do not exist, so that the two ends of the wire are considered insulated. In this case, the flow of heat out of the wire and through the insulation will be radial. We assume that the thermal conductivity of the insulation is k, and the heat transfer coefficient from the outer surface of the insulation to the room is h. We call r0 the 2" outer radius of the insulation, and ri the radius of the wire (roughly 0.04" for 12 gauge). The rate of heat generation in the wire per unit length of wire is Q. The radial heat conduction equation within the insulation for this situation is given by:

$$2\pi rk\frac{dT}{dr}=-Q$$
where r is the distance from the cylindrical centerline of the wire.

At the surface of the wire, the boundary condition is:$$2\pi r_0h(T_s-T_0)=Q$$
where Ts is the temperature at the outer surface of the insulation and T0 is the ambient room temperature.

OK so far?

Last edited: Nov 12, 2015
15. Nov 12, 2015

### Physics_Kid

sure, makes sense, we can swap out Q for I^2R, so for 12awg(1inch) its 0.1325x10^-3 * I^2

now need to add the "heatsink" mode that is connected at the very end of the wire. lets assume that the conductive path of the copper that is outside of the insulation has zero heat loss along that path until it reaches the heatsink, so like water flow in a pipe, except in this model the main pipe has a bunch of smaller feeder radial pipes feeding the main pipe along the path (because the wire is exothermic everywhere).

Last edited: Nov 12, 2015
16. Nov 12, 2015

### Staff: Mentor

Whoa pardner. Hold yer horses. Before we start looking at the heat sink case, I want to first get the solution for the no-heat-sink case. Why? If we can't solve that, we certainly won't be able to do the heat sink case. Plus, I want to get some results under our belts that we can look over and apply to the heat sink case. Please be patient.

If Tw is the temperature of the wire at its surface r = ri, are you able to solve the equations I wrote to obtain Tw and Ts in terms of Q, ri, ro, h, and k?

Chet

17. Nov 12, 2015

### Physics_Kid

so you said "at the surface of the wire" for this eq, did you mean to say at the surface of the insulation, the boundary between insulation and ambient air?

not sure what i am solving for, are we solving for Tw to start with?

and isnt Ts considered to be temp of ambient air when calculating the heat gradient across an insulator of value R ? and from that we know the gradient is a function of Δt

18. Nov 12, 2015

### Staff: Mentor

That equation applies to the outer surface of the insulation, as I said in my post.
Were are trying to solve the differential equation and boundary conditions for (a) the temperature as a function of radial position (b) Tw and (c) Ts.
No. There is typically a convective heat transfer resistance between the insulation surface and the ambient room air.

I wanted to give you a chance to solve the equations yourself, but if you are uncomfortable and uncertain about that, I can do it. What's your preferrance?

Chet

19. Nov 12, 2015

### Physics_Kid

sure, ok, please show me an example.

we can use Qwire = 0.159watt = 0.159J/s = 572.4J/hr = 0.5724kJ/hr = 0.5425btu/hr
this is 10amps on 1ft of 12awg copper wire

but, i think you are driving at a answer to dT/dr so that we can find the temp of rwire, am i correct?

Last edited: Nov 12, 2015
20. Nov 12, 2015

### Physics_Kid

i found this from a UK site

where d1 is conductor diameter and d2 is insulation diameter