thermal expansion of copper solenoid with a rectangular cross section

[eng142857]

Hi,

I'm trying to model the thermal expansion of a copper rectangular cross sectional solenoid - it has N number of rectangular ring turns and can't be modelled as a normal circular ring solenoid. At the moment I've modelled the rectangular cross section area as 4 rods that each linearly expand, and the length dimension (the turns) as also linearly expanding. I know that for the length dimension this is OK to do because I have found a paper supporting that method, but I think modelling the rectangular ring cross section as 4 rods is wrong. I've measured the resistance of the coil at several temperatures and used a resistivity value for copper at these temperatures to give me a rough idea of the total length of the copper coil with respect to temperature and it's not the same value/trend as multiplying the circumference (modelled as linearly expanding) of one ring by the number of turns.

I'd really appreciate some advice because I can't find much literature on the subject! Also, I don't use ANSYS or solidworks so I'd prefer to do the calculations myself but if you think that a software approach would be best then let me know!

Many thanks,
eng142857

 

[Engdoitbetter]

At a first glance, I would say that total wire length is much larger than cross section side length, so thermal expansion may be neglected for the latter. So I think that any change in resistance is due to change in resistivity and length.

We should also discuss the effects of curvature and of the angle between solenoid axis and wire.

It would be fine if you post some pictures

Hope it helps.

Stefano

 

[eng142857]

Hi Stefano, thanks for your reply. I've uploaded a diagram of the coil (a X b X length).

The temperature change the coil will experience is quite large, from around -160C to room temp, so thermal expansion is not insignificant enough to ignore. I need to know how each dimension of the coil (a, b and length) is changing over this temp range, because this will affect the magnitude of magnetic field through the solenoid. Modelling the length dimension as linearly expanding is fine, but I think that my thermal expansion calculations for a and b are incorrect... Also will the effects of curvature be significant??

Many thanks!

 

[Engdoitbetter]

I'm sorry but it seems I totally misunderstood the geometry of the problem... Please don't consider my previous post I beg your pardon.
It is obvious that there isn't any curvature involved.

Now, looking at the picture, I think your calculation should be quite correct: a & b sides should expand as straight rods and the same goes for the length. At this point of analysis I think that only two factors may be missing:
1) accounting for the angle between solenoid axis and the planes of the rectangular rings, but from the drawing the angle seems negligible as it is in a circular solenoid
2) since the temperature change is quite large, the expansion coefficient may depend on temperature; in this case you need to calculate an integral instead of using a simple multiplication i.e.

coeff = cost. --->> DeltaL = coeff * L * DeltaT
coeff = coeff(T) --->> DeltaL = L * int (coeff(t)*dT)

any suggestions?

 

[Engdoitbetter]

I have a doubt regarding my previous post: why do you need to calculate the length again if you have already taken into account the expansion of a & b?

 

[eng142857]

Thanks again.

I have used a paper which has calculated the thermal expansion coefficient of copper at different temperatures and put this into the thermal expansion equation for each temperature.

Sorry, by length I mean the length dimension of the solenoid in the figure (so where the turns are) and not the total length of the unwrapped wire. I found a paper that says you can treat this dimension as linearly expanding, so it's the rectangular ring aspect I'm not quite sure about especially whether it's OK to treat the ring as 4 rods... The magnetic field is dependent on a, b and length so I need to see how each dimension changes with temp.

Also I've heard that copper may not expand linearly and is different to normal metals - have you heard anything about this?

Thanks!

 

[Engdoitbetter]

Regarding thermal expansion of copper I don't really know. What does it mean linearly? All non linearity is usually confined in the thermal expansion coefficient, and, as I said, this deviation from linearity is negligible if the temp. change is small.
Anyway, you can have a look here:

http://www.itl.nist.gov/div898/handbook/pmd/section6/pmd64.htm

As far as the solenoid is concerned, since the length variation of the solenoid wire due to temperature is always the same as that of a straight wire, I believe that you have two opposite ideal situations:
a) the dims of the rect. cross section are constrained: the total length of the coil will change.
b) the coil length doesn't change, but the dims of the rect. section do.
To make a long story short, maybe you're taking into account the change in length two times instead of one.
This is my guess.

 

[eng142857]

Thanks! I think I'm going to model the solenoid with the length constraint and let the dimensions of the rect. cross section expand - is this still OK to treat as 4 wires?

Also just a quick question - in the linear expansion formula:

length_final = length_initial * (1 + alpha*(Temp_final - Temp_initial))

which temperature should you take the alpha (thermal expansion coefficient) value for - at temp final or the temp you start with..?

Thanks

 

[Engdoitbetter]

I think you don't have many more option than using the 4 wires method...

Anyway, regarding coeff. of thermal expansion, apart from using the integral above (which would require to obtain a polynomial describing the variation of alpha), you should proceed like this:

- divide the total variation of temperature in smaller intervals so that exp. coeff. is approximately constant in each of them
- choose the value of coeff. for each interval, I would take an average of start and final.
- multiply the coeff.s with the relevant intervals of temperature, then sum

Hope it helps.

Stefano

 

[Compositepro]

Solving thermal expansion problems is much simpler if you understand the basic concepts. When a homogenous material expands all points within it move apart uniformly. It does not matter the shape or if there is a hole or gap between two points. This is a very simple and basic principle. I've argued with a number of engineers about this point over the years,so it appears to be a principle that is sometimes overlooked.

For every material there is a unique specific volume (or density) at any given temperature. CTE (or alpha) is simply the slope of the line between any two points on this curve.