Calculating Heat Transfer in Copper Conductors: A Thermodynamics Question

[1Keenan]

Hello,

I have a question for expert in termodynamics.

Let's assume I have a copper conductor made of two big disks connected with a very thin wire.
Let's imagine I have a current pulse starting somewhere in the wire, and assume the current density is high, like 100A/mm^2.
This current will heat the wire locally but I guess the heathing will be distributed in the whole conductor.
How can I calculate the time required for heath to be distributed in the whole conductor?

To be more general: how long it take 1°C of thermal jump to flow in a copper conductor?
Is there any formula that answer my question?

Thank you

 

[Nidum]

Two possibilities :

(a) The current pulse blows the wire and then nothing else happens at all .
(b) The current pulse does not blow the wire and then nothing else happens at all .

Not a very meaningful problem .

You would find it more instructive to look at the heat flow in a plain round bar which is intensely heated over a local area for a short period of time . You could probably analyse this situation mathematically without too much difficulty .

 

[Chestermiller]

Hello,

I have a question for expert in termodynamics.

Let's assume I have a copper conductor made of two big disks connected with a very thin wire.
Let's imagine I have a current pulse starting somewhere in the wire, and assume the current density is high, like 100A/mm^2.
This current will heat the wire locally but I guess the heathing will be distributed in the whole conductor.
How can I calculate the time required for heath to be distributed in the whole conductor?

To be more general: how long it take 1°C of thermal jump to flow in a copper conductor?
Is there any formula that answer my question?

Thank you

Are you talking about a small region of high temperature in one portion of the wire, while the rest of the wire is still cool, or are you talking about the entire wire being hot to begin with? Are you thinking of the two big disks as constant temperature reservoirs?

 

[1Keenan]

Hi,
If the current doesn't blow the wire there will be a thermal jump. Why you said nothing happen?

Let's assume the bar problem: how to analyze it?

Thanks

 

[1Keenan]

Are you talking about a small region of high temperature in one portion of the wire, while the rest of the wire is still cool, or are you talking about the entire wire being hot to begin with? Are you thinking of the two big disks as constant temperature reservoirs?

Dear Chestermiller,
Sorry, I just saw your post.
Im talking about the entire wire bieng hot starting from a small region and yes, the big disk as a constant reservoir of temperature.

Can you help me?

 

[Chestermiller]

Dear Chestermiller,
Sorry, I just saw your post.
Im talking about the entire wire bieng hot starting from a small region and yes, the big disk as a constant reservoir of temperature.

Can you help me?

You have a hot conductive rod that is cooled at both ends by a constant temperature reservoir, correct?

 

[1Keenan]

In principle this is what I have, but current pulse is very short (nanosec) and peak is very high (kAmp)

I thinnk the wire (1 turn coil) explodes mainly because of the magnetic field and heating is not relevat because of the disk.
I made analytical calculations and simulatios for magnetic field, Lorentz forces and deformation and heathing.
Also some experimental data seam to confirm that, but I would like to have your opinion, as you get the point

 

[Nidum]

You have a hot conductive rod that is cooled at both ends by a constant temperature reservoir, correct?

A very thin wire is specified in the original problem .

 

[1Keenan]

A very thin wire is specified in the original problem .

Yes it is a thin wire what I have, sorry, but I imagine it works as the rod with two disks as temperature reservoir

 

[Chestermiller]

Yes it is a thin wire what I have, sorry, but I imagine it works as the rod with two disks as temperature reservoir

Are you considering the heat lost out radially from the wire or do you want to assume that it is insulated thermally?

 

[1Keenan]

It is in vacuum, so thermally insulated

 

[Chestermiller]

It is in vacuum, so thermally insulated

It can still radiate, but I'm assuming it is physically insulated thermally.

So heat can be conducted only along the length of the wire. The heat transfer within the wire is controlled by the 1D transient heat conduction equation: $$\frac{\partial T}{\partial t}=\alpha \frac{\partial ^2 T}{\partial x^2}$$ where $\alpha$ is the thermal diffusivity of the wire material and x is distance along the wire. The boundary conditions at the disks are: $T=T_0$ at $x=\pm \frac{L}{2}$ where L is the length of the wire and the origin is at its center. The initial condition on the wire temperature is $T=T_1$ at t = 0.

If you don't know how to solve these equations yourself, the solution for the temperature as a function of time and location is given in Carslaw and Jaeger, Conduction of Heat in Solids. It is also given in Transport Phenomena by Bird, Stewart, and Lightfoot. Bird et al also present a graph of the average temperature as a function of the dimensionless time $\frac{\alpha t}{L^2}$.

 

[1Keenan]

Thank you very much.

YEs, it can still radiate in vacuum, but process if very low efficient, as far as I know.

I admit I'm no able to solve the equation, but the books you suggest are already a good reference. I'll have a look.

Anyway the equation you give me is the one used I am my simulation code, I have to set alpha for copper and inizial temperature for the conductor and for the disks.
So I guess it is solve in the code more efficiently than what I could do.

Now, according my simuation results and also calculating the joule heating analitically I have a thermal jump of the order of 10000 K if I consider the wire itself, if I consider the additional mask of the disks the thermal jump is of the order of 600K.
I'm pretty sure that the disks work as temperature reservoir and the coil explode because the strong magnetic field (Lorentz force is 10^6N).

Do you think my results are reasonable? I mean not in terms of number, but from the conceptual point of view.

Thank you very much for your help

 

[Nidum]

I'm not at all convinced . What are the actual construction details of this set up ? A clear sketch and some dimensions would help us a lot to understand the physics of this problem .

 

[1Keenan]

http://imgur.com/al1XxLW [\IMG]

Hello, here is a sketch of the setup.
wire is 0.025mm squared section with total length of about 3mm. in this 3mm you can see there is a 300° loop, which form a 0.25mm diameter coil.
Disks have 3.5mm diameter and one of them has 1mm diameter hole

 

[Nidum]

This seems to be a very different problem to the one first described .

Never mind - the problem as given now is probably more interesting .

So the current pulse is definitely going to be powerful enough to cause the wire link to fuse and/or explode .

Your question then is effectively how much of the heat generated in the wire by the pulse gets transferred to the discs ?

Intuitively I would say not much .

To actually get a numerical answer though is going to be extremely difficult .

During the application of the current pulse the wire is going to be sequentially solid , molten and vaporised .

Heat transfer is going to take place during all these phases and with the additional complication of a continually changing configuration geometry as well .

May be easier to use limit state methods rather than detailed calculations to get initial answers .

The complete system could possibly be analysed using some advanced computer modelling software but it would require a lot of work .

 

[1Keenan]

Complete modelling using FEM software has been done.
Expansion due to magnetic forces macth quite good with measured expansion during the experiment. This is why I'm quite confident that the heat transferred to the discs is not negligible.
Anyway I would like to have some complete theory on this aspect, to have a solid explanation. I think a simplifyed model proposed at the beginning is more or less ok and also Chestermiller formulation sounds to be appropriate.
I already have numerical solution, I would like to understand if the concept of the two discs as reservior of temperature works from the conceptual point of view or not.
You say "intuitivly not much", can you explain me why?

 

[Nidum]

I don't think that me explaining anything further would serve any useful purpose . .

 

[1Keenan]

thanks...