Transient Conductive Heat Transfer, Radial System

[DoubleEngineer]

Hello,
I am a recent graduate in the field of mech eng. and I have come across a problem while trying to model the temperature distribution and heat transfer rate of a system. I will try my best to explain the situation as well as I can.

The system is fairly basic, I have a hollow concentric multilayed cylnder. It is supported horizontally along its center axis to allow for rotation (the w is roughly 1 rev per 3 mins, so pretty slow spin). The system is in a vacuum of 60 Torr and goes from Tinitial=atm to Tfinal=350C. The Heating element is around the cylinder so I assume uniformed temperature at the Tsurface.

I know I can use 1-D radial equations to solve this, and have been able to do so at steady state conditions, but realistically the best approach is for transient conditions.

I believe my Boundary Conditions are BC1=Initial Conditions, BC2= convection h around cylinder for Tsurface, and possibly a third BC. This is mainly where I believe I need help solving this.

My Heat Transfer textbook covers Finite-Difference Methods and transient conditions but during the class FDM was not covered and only a very little amount of transient problems were covered, that I can remember.

So as I refresh and read over these texts, I was wondering if anyone could point me in the right direction.

I have fully modeled my system on NX6 but when it comes to the analysis phase the program crashes, so until I can work around that problem I need to form a different approach to solving this.

Attached is a file of equations that I believe I need in order to solve this problem. Any advice to approach and/or equations/modeling would be greatly appreciated.

Thanks and Best Regards,
Devin

 

[corus]

Your 3rd equation is wrong and your final equation is wrong too. In addition you're using an explicit scheme which tends to be unstable. Try and use an unconditionally stable implicit scheme. It just means changing your dT/dt to one that includes T at time i and i-1 instead of i+1 and i. For the last equation check your units to see if they're compatible on either side. You should find they're not. I'm sure that you should have a h^2 term where h = delta R.

There should be examples of this kind of problem in text books that use the stable Crank-Nicholson scheme for example.

http://en.wikipedia.org/wiki/Finite_difference_method

Tata

 

[davefitz]

Corus is correct. Also, some other rules of thumb for ensuring a stable solution:
a) after the first solution is found, re-run the program at 1/2 the original time step. If the results of the 2nd solution is close to the first run, then your time step is in a stable range.
b) the time step chosen ( for convective HT problems) is proportional to the grid size and inversely proportional to the biot number ( check that - its been 32 yrs since I took that course).


Some useful texts that explicitly define the numerical approach to our specific problem are:

"analytical methods in conduction heat transfer" by glen myers, and the numerical methods ( focusing on minimizing storage space, a problem in the 1970's no longer relevant today) are outlined in "the finite element method in partial differential equations" by Mitchell + Wait.

 

[zekeman]

"I know I can use 1-D radial equations to solve this, and have been able to do so at steady state conditions, but realistically the best approach is for transient conditions."

Please explain why a steady state solution is not warranted?

A truly transient problem involves heating for times less than it takes for steady state, like minutes. If it is hours then you use SS which is vastly easier to calculate.

 

[corus]

With a rotating system you probably need a transient solution in order to obtain a quasi-steady state solution. There is a way of obtaining this quasi steady state solution but I forget how I derived it now. As for steady state taking only minutes. Well, that depends on the thermal inertia of the system. Material that has a very low conductivity may take days before approaching a steady state condition.

Tata

 

[DoubleEngineer]

The reason I am using transient and not SS is because the numbers I have calculated dont seem 100% accurate. I want to check these with calculations derived using the finite difference method. All posts have been very helpful, if I have any more questions or "hit a wall" I'll be sure to visit this page again. Thanks.

"The microwave oven is the consolation prize in our struggle to understand physics."~Jason Love
"Science is a wonderful thing if one does not have to earn one's living at it."~Albert Einstein

 

[DoubleEngineer]

okay hopefully the last time, the attached seems to make since.

the first page is almost straight out of my heat transfer book, i just changed the scheme around to be only in 1-D.

the second page is from wiki sites.

i believe the first page to be more accurate of an approach.

"The microwave oven is the consolation prize in our struggle to understand physics."~Jason Love
"Science is a wonderful thing if one does not have to earn one's living at it."~Albert Einstein

 

[sailoday28]

Hi
Why have you assumed the rotational velocity to be negligible>