Heat transfer/cooling problem (multilayers)

[spudsquad]

i am a sophomore and just started my first co-op and the boss left me with this problem to solve that seems to be slightly beyond my knowledge so far. The whole problem is too long but here is the part i am stuck on.

I have a sample material that is being made and fed out of a machine, so it is infinitely long. The sample looks like this:

it comes out at a temperature of 450F, and into a tank that is 65 degrees F.
What i need to find is how long before the fiberglass is 300F.

I have looked up specific heats, thermal conductivities, and all the dimensions.

I am having trouble even just finding the when the PVC will be 300F, i have read for hours online for ways to do this and can't seem to find a similar problem. This is my first post on the forum and would really appreciate any tips on the method to solve this, or if i even have enough information.
thanks.

attempts:
- i tried Newtons law of cooling and the conduction equation. the main problem i keep having is that all the formulas are dependent on difference in temperature and the temperature for both the acrylic and pvc are constantly changing.

 

[Integral]

Newtons law cooling should be the way to go, but it is a differential equation, and needs to be solved accordingly.

Have you had a course in differential equations?

If not, then you should approach your boss or one of the engineers, and ask about borrowing a intro Differential equation text, such as Boyce and DiPrima. They may be willing to get you started.



A common example problem is that of controlling temperature in a house with multilayed insulation. This should be of interest to you.

 

[spudsquad]

thanks for the reply.
i had a differential equations class and know how to solve a simple two body Newton cooling problem. But the problem that i have run into is that the cooling between air and the acrylic depend on difference in temperature, but the temperature of the acrylic is losing energy to the air but in turn will then gain energy from the pvc which is still at a higher temperature. Also i do not have another time and temperature to solve for k.

 

[Integral]

That is why I referred you to an example of a wall with multilayers of insulation, you need to set up a statement of the heat equation in each region with approbriate boundary condtions at the layer interfaces. Use Newtons Law of cooling as the heat loss boundary condition with the water.

 

[AlephZero]

A quick search on Google suggests the PVC would be molten at 450F and solidified at 300F.

If that's true, your problem is WAY more complicated than solving the diffusion equation for heat conduction in a solid. I think the only practical way to go would be use a computer model.

 

[Integral]

Since apparently there is a void in the center I think it is safe to assume that the PVC is solid or very nearly, or it would not hold the shape. Naturally any phase changes will complicate things a lot.

 

[AlephZero]

Since apparently there is a void in the center I think it is safe to assume that the PVC is solid or very nearly, or it would not hold the shape. Naturally any phase changes will complicate things a lot.

http://www.matweb.com/reference/deflection-temperature.asp says the MP of acrylic is 130C (266F)

http://www.dynalabcorp.com/technical_info_pvc.asp the MP of PVC is 80C (176F).

Something must be wrong somewhere.

 

[spudsquad]

i've dropped the problem, but i appreciate the advice.
Clearly he didn't think the problem through before he gave me the numbers, or severely over estimated the abilities of a second year student.