2D Finite Element Transient Heat Problem

[mc112]

Hi everyone,

I'm currently working on coding a finite element heat problem in Matlab that is as follows:

I have a cross section of a beam made of an isotropic material (concrete) that is placed in a furnace, and the outside temperature is raised at a constant rate. I am interested in graphically displaying the temperature distribution at specific times to show how fast the interior of the cross section heats up.

I have been able to make a Matlab model that works for boundary conditions in which the outside nodes of the cross section are held at a specific temperature, but I am not familiar enough with the heat problem to know what "force" vector I should characterize the outside temperature change as.

The book that I am using (The Finite Element Method for Engineers by Huebner, talks about specified surface heating as a boundary condition, and characterizes this quantity as [Rq]. However, I am unsure if this is the proper quantity for my problem, as I am haven't taken thermodynamics and have only dealt with steady state heat problems in diff eq up to this point. Should I be dealing with heat flux? I've scoured Google,but have been unable to find any other paper that talks about specified surface heating. Conceptually, I am just unsure of what raising the ambient temperature around the cross section is classified as.

Thanks!

 

[Mapes]

Hi mc112, welcome to PF. If you assume that the surface temperature of the beam equals the air temperature in the furnace (i.e., that convection is very efficient, probably a good first assumption to make), then you have a time-dependent temperature boundary condition. This is sometimes called a time-dependent Dirichlet condition. (This is different from a heat flux boundary condition, aka a Newman condition.) With finite elements, you would set the temperature of each surface node equal to the furnace temperature at each time step. Does this help?

 

[minger]

In a more general sense, when creating a heat-conduction FEA program, you're essentially doing a heat balance. That is, heat in - heat out = heat gained. You could simply set the outer nodes to a fixed temperature, however it is only marginally more difficult to introduce a heat convection boundary condition. In this case, its rather easy to specify a ambient temperature and heat convection coefficient.

Your boundary condition will then calculate the heat in based on the boundary condition temperature and the current nodal temperature. This is essentially the heat flux boundary condition Mapes is referring to, however, as opposed to specifying the heat flux, you're calculating it based on external conditions.

...however, you may not know a good convection coefficient and simply specifying the temp may be good enough.

 

[mc112]

Thanks for the input. I was able to model the cross section assuming that I know the temperature of the exterior surfaces as Mapes suggested. Unfortunately, now I have data for the material that could complicate things... The specific heat and the thermal conductivity of the concrete vary with temperature ( I have an excel file for each containing the data). I was initially calculating my K and C matrices outside the time integration loop. Now that these are not constants, is there a way to integrate this into my program?

 

[Mapes]

Sounds like you'll have to recalculate these properties during iteration.