Applying Newton's cooling equation to a pipe cooling

[scootaash]

Hi. I'm trying to apply Newton's cooling equation (if relevent) to the following system. We produce plastic piping, extruded at about 200 'C and cooled via water sprays to 35 'C. We are trying to calculate how fast we can run the pipe through the cooling sprays. It's been a very long time since I did any of this so any help would be appreciated! I'm not even sure if the equation is useful in this instance.


Newton's cooling equation is :

dT/dt = k(T-M) where T is the object temp, t time and M the outside temperature.

So

T = C(e^kt) + M
Where C is the difference between the start and ambient temperature.

I can measure this for certain thicknesses and sizes of pipe, but the point is to be able to predict the final temperature for any size and thickness of pipe, given the same cooling. I have no idea how I would extend to do this!

The main problem is that only the external surface is cooled, so upon leaving the cooling tank the surface heats up again as heat is conducted to the surface.

Any suggestions for how to proceed would be greatly appreciated!

 

[Chestermiller]

This requires solution to the heat conduction equation $$\rho C_p V\frac{\partial T}{\partial z}=k\frac{\partial ^2T}{\partial r^2}$$neglecting the curvature of the cylinder wall, with v representing the axial velocity of the pipe moving through the spray. The boundary condition is $$-k\frac{\partial T}{\partial r}=h(T-T_w)$$ at r = R. Here, h is the heat transfer coefficient between the outside of the pipe and the cooling water. This would have to be determined from some scouting experiments. The distance z of the water spray region would be that required for the calculated average temperature of the pipe to reach the desired level.