Heat Transfer Diff. Equation: Fixed vs Free Extremities

[Jairo]

I know the heat transfer diferential equation, but I only saw it being used when the temperatures of the extremities of the material are fixed. Is there any phormula that aplies for situations where at least one of the extremities is free to change its temperature?

Ex.: the left side of an iron bar with dimentions A,B,C is heated to T1 while the right side starts with T2. Ignoring radiation loss, when the temperature of the right side will be 90% of T1? And if we consider radiation?

If it´s too hard to solve, is there any numerical aproximation? Thanks.

 

[Dale]

Yes there are two typical kinds of boundary conditions. One type defines the temperature on the boundaries the other defines the heat transfer on the boundaries. A common heat-transfer type of boundary is the insulating boundary where no heat is allowed to transfer in or out of the boundary. I think for most general cases that it requires numerical solution to the PDE.

-Dale

 

[jubug]

There is a second order, transient differential equation, which is typically solved with series analysis. Oftentimes, if the heat change is perdioc, one can find an exact solution to the problem. If not, then a numeric solution is in order. You will likely not find this in an undergraduate text, as non-transient heat transfer solutions are not addressed until you take a graduate level heat transfer course. I suggest you look in a college library or a graduate heat transfer text or a graduate diff E book for Fourier series analysis of 2nd order transiet diff equations.