Parabolic Temperature Profile Through A Pipe Wall

[BrianDamage]

Hey Folks,

Was wondering if anyone knew of any good textbooks or journals which could help me with some theory involving parabolic heat transfer in a pipe wall. I've tried looking through a few textbooks but get a bit lost.

Thanks

 

[MrMyers]

What is your application?

Apart from during a short heat-cool-heat transience, I struggle to understand how else one would achieve it.

 

[sailoday28]

Parabolic heat transfer? Do you mean parabolic velocity distribution- which would occur for Newtonian laminar flow in a pipe?

 

[BrianDamage]

Thanks for the responses, sorry I was far too brief, I'll go into more detail.

I'm looking into a forming process for pipes called Induction Bending. It's where you heat a small section (20-40mm) of pipe (in this case super duplex stainless steel) to around 1100C using an induction coil around the outside of the pipe which puts the material into the super plastic region and is able to be bent.

After doing an experiment with a thick-walled pipe and some thermocouples it turns out that the inside wall of the pipe is approximately 1150C, the centre of the wall is 1200C and the outside wall is 1100C.

This is the main reason for me too look into conduction and the theory of a parabolic temperature profile. I'm not really looking at convection or radiation.

Hope this is clearer.

Thanks

 

[IRstuff]

Given your description, why would you say that you're not looking at convection or radiation? There are only 3 way the heat can leave the region, conduction, convection, and radiation. Since the temperature is concave down through the thickness, only conduction is eliminated.

One can certainly argue that the heat being generated by the induction coil is leaving from both sides of the wall, and is therefore showing a lower temperature than the middle of the wall.

TTFN

FAQ731-376

 

[hacksaw]

with eddy current heating (a function of your coil) tends to be on the outside and decays with penetration depth. For the outside to be cooler than the mid-wall temperature suggest cooling. at those temperatures radiation is a dominant player. There is a lot of literature on induction heating dating from the 1930's but you'll have to reseach it

 

[sailoday28]

For heat generation (induction heating) in a conducting solid with constant thermal conductivity
q'''/k +d^2T/dx^2 =0 wwhere q''' is heat generated per unit volume. Integration with constant q''' will give parabolic profile.

Regards

 

[hacksaw]

well to the extent the generation term is constant that's correct, but depending on the freq and the wall thickness, the heating tends to decay with depth.

I am actually more concerned about the particulars of measuring the mid-plane temperature

 

[sailoday28]

hacksaw (Mechanical)
I was trying to answer the original question, however
With q""", the heat generation per unit volume a function of spaceual coordinate and time, I would use an approximate model of one dimension and time

If thermal diffusivity is constant, solve
1/alpha*dT/d time= q'''/k+ d^2T/dx^2----were d means partial derivative.

Regards