Temp change of copper wire w.r.t. time for a constant applied voltage

[AkhilP]

Homework Statement

How do we find the temperature change of the copper coil with respect to time if we supply constant voltage to it. Assume air core coil.
Given values:
Dia of wire=0.14mm
Length of wire=26m
Number of turns to coil=1092
Resistance of wire=35ohms
Tambient= 23deg
Voltage applied=16V

Relevant Equations

V=I*R

Q=m.c.DeltaT, is one approach which could possibly give the relation between Power vs Time. Is it correct ?

 

[Chestermiller]

Homework Statement: How do we find the temperature change of the copper coil with respect to time if we supply constant voltage to it. Assume air core coil.
Given values:
Dia of wire=0.14mm
Length of wire=26m
Number of turns to coil=1092
Resistance of wire=35ohms
Tambient= 23deg
Voltage applied=16V
Relevant Equations: V=I*R

Q=m.c.DeltaT, is one approach which could possibly give the relation between Power vs Time. Is it correct ?

Not if the wire is losing heat to the ambient surroundings.

 

[AkhilP]

Not if the wire is losing heat to the ambient surroundings.

Thankyou @Chestermiller
How to proceed in such case ? do you know any simulation tool which does it ?

 

[Steve4Physics]

Hi @AkhilP. Welcome to PF. If you haven’t already done so, take a look at the forum’s guidelines here: https://www.physicsforums.com/threads/homework-help-guidelines-for-students-and-helpers.635513/

As you will see, you are expected show what effort you have already made yourself, e.g. what you found out from a Google search such as ‘temperature vs. time for wire’.

Can I note that the current through your coil will be (16/35=) 0.46A. This exceeds the recommended max. current for this size of wire by a factor of 10 - see the table here: https://metersuk.co.uk/american-wire-gauge-awg-cable-conductor-sizes/

Your problem is pretty difficult. Can you tell us why you are asking? Is it a homework question or for some practical application?

 

[AkhilP]

Thankyou @Steve4Physics for noticing the max current limitation with respect to the wire gauge. Yes, it is a practical homework, trying to figure out if we can develop a mathematical relation between the time vs temperature for constant power applied.

Also, mentioned above have tried to theoretically relate using the formula Q=m.c.DeltaT, but it does not match with practical experiment conducted. Formula shows there is linear relation between Time and temperature, but practically it isn't the case. Hence, trying to find out what's the void?

 

[BvU]

Hence, trying to find out what's the void?

@Chestermiller pointed that out in post #2

Not easy to quantify

$\$

 

[erobz]

For steady state the basic idea is power lost by the resistor must equal the power convected and perhaps radiated to the surroundings. As the others note difficulty of that depends on the how much stuff you want to account for. But there is a base model that is not terribly difficult(I think). We just have to see the how much stuff we can ignore.

 

[Steve4Physics]

Yes, it is a practical homework, trying to figure out if we can develop a mathematical relation between the time vs temperature for constant power applied.

Is this intended as a simple homework that you are meant to do in an hour or so? Or is it a project to be tackled over a longer timescale?

Also, mentioned above have tried to theoretically relate using the formula Q=m.c.DeltaT, but it does not match with practical experiment conducted. Formula shows there is linear relation between Time and temperature, but practically it isn't the case. Hence, trying to find out what's the void?

As others have stated, the coil is losing energy to the environment while energy is being supplied. This must be taken into account.

Have you tried my (not very subtle) hint to do a Google search for ‘temperature vs. time for wire’?

You need to do research/background-reading. And (to get support here) you need to provide evidence that you’ve done it and what you found. Those are the requirements here to get guidance/feedback.

This is potentially a very hard (IMO) problem. And the data are incomplete - for example there will be different results if the coil is horizontal or vertical (due to different rates of convection).

A good answer will involve setting-up and solving a differential equation. Have you covered differential equations?

 

[kuruman]

Not if the wire is losing heat to the ambient surroundings.

For a straight wire one could perhaps assume that the input electrical power is radiation power out and use $$P=A~\epsilon~\sigma ~T^4$$ where $A$ is the cylindrical area of the wire, $\epsilon$ the emissivity of copper and $\sigma$ the Stefan-Boltzmann constant. However, the geometry here is not a straight wire. We have a coil with N = 1092 having diameter $d=\dfrac{26~(\text {m})}{\pi N}\approx 7.5~\text{mm}$ and length $L=0.14~\text{mm}*1092\approx 15~\text{cm}.$ Perhaps the coil can be modeled as a radiating solid copper cylinder and use the Stefan-Boltzmann law.

OP is looking for the transient temperature rise when constant voltage is applied to the coil, initially at room temperature. An additional complication is the copper resistivity change as the temperature rises which implies that the current drawn at constant voltage and hence the input power are variable. Aye yai yai.

 

[erobz]

The OP says the diameter of the wire is 0.14 mm. Thats more like thread. Its only about 7 Watts being dissipated, but that is tiny wire, with tiny surface area. So, the temp could get pretty high? If the temp gets high, you have to start accounting for radiation, and the resistivity change as @kuruman mentions as opposed to convection (when the temps are reasonably low). I'm saying we can model this with a level of difficulty found in the first chapter of my heat transfer text to start (I know because I found it there!). Granted, if radiation becomes dominant it is already a numerical integration.

 

[DaveE]

It sounds more like an incandescent lamp than a wire or coil to me. I don't think it can really be answered as HW without guidance from the instructor about what can be ignored and what model is applicable.

 

[Chestermiller]

Can you please provide a sketch showing the shape and dimensions of the coil?

 

[AkhilP]

Thankyou all for providing a direction to go ahead. will try some methods or simulations and see if it can be predicted as close as possible to the experimental results. @Steve4Physics Yes, it is a homework, practical project which has high complexity.