There must be a version of Fourier’s law written in terms of
the relaxation time for temperature varying sinusoidally over
distance in a solid. I expect an initial sine wave in
temperature to remain a sine wave of the same frequency and
phase as time goes by, but also expect the amplitude to decay
exponentially.
Wanting to define a rate as dt(temperature)/temperature and
see the constant in:
rate = constant * diffusivity / wavelength^2,
I whipped up a FEM model and played with the wavelength and
diffusivity. The constant appears to be around 39. What’s
the meaning of 39? Why isn’t it sqrt(pi) or e^e or something
all insightful and crap? Did I do something stupid?
Pardon my ignorance, but as a M.E., is FEM the correct model for what you are after? As far as I have ever seen, FEM finds stresses due to loads. The basis is that all beams are merely very-stiff springs. IIRC, the basic premise of FEM has its roots in the basic physics of springs relating energy to deflection by knowing the spring constant. In structures, a beam’s (or, spring’s) stiffness is known perhaps from a bulk modulus value, or some other known property of the beam’s stiffness. Once this is known, FEM performs an energy analysis to find places of max deflection.
At best, I wager you may be using FEM to study thermal stresses? Just wondering if your initial model is correct… - Jinx
It sounds like you’re modeling the one-dimensional diffusion equation,
T[sub]t[/sub] = d T[sub]xx[/sub]
(using subscripts to denote partial derivatives, and using d for what you’re calling the diffusivity). Following your guess, we try a solution of the form
T(x,t) = T[sub]0[/sub] + T(t) sin(2 π x / L)
(with L the wavelength of the sinusoidal temperature variation, T[sub]0[/sub] the average temperature, and T(t) the time-varying amplitude of the sinusoidal oscillation). Plugging this in gives the results you expect, along with a formula for the constant.
I think I see. Txx means the second derivative of temperature with respect to position. Taking the second derivative of sin(2 Pi distance) brings 2 Pi out twice and becomes negative sine. And -(2Pi)^2 is -39.5 which is what I was getting (I didn’t mention the sign in the OP, thinking it was just which direction I was looking or something trivial like that).
It’s been almost 30 years since I used calculus much - my FEM work is with a canned package. I don’t think I can write this out properly, but I think I see where my constant came from now.
And, hey, Jinx, FEM is a method like regression or FFT - it’s applicable in many fields. Check out pdesolutions.com who makes the nifty tool I use.
Napier is correct on this one. FEM is great for stress analysis, and that’s it’s most famous use. Most FEM packages work great for heat transfer calculations, very simple fluid dynamics solutions, and a host of other things. It’s versatile enough to adapt to many computational applications.
You could even say that CFD is an just a special case of FEM, specifically adapted to use in fluid flows of various types. Both tools are about grid-based, iterative, numerical solutions.