Quote:
Originally posted by Scott England
-d(a(du/dx))/dx+(d^2(b*(du^2/dx^2)/dx^2))+co*u+c1*(du/dt)+c2*(du^2/dt^2)=f(x,t)
this equation represents unsteady heat transfer in a fin
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Actually, I think the c1*(du/dt) term is irrelevant. c1 will always be zero because the derivative is taken twice (with no intermediary steps in between) of the governing law (which is a function of t^2 only, not t), thus integration is guaranteed that the constant must equal zero. For comparison, consider an example of a sinusoid that is translated vertically in the y direction:
f(t) = sin(t) + 2
Then:
f'(t) = cos(t)
f''(t) = -sin(t)
Integrating twice yields:
f''(t) = sin(t) + C1t + C2
However, we are guaranteed that the sinusoid is translated, and should have no t term, thus C1 MUST be zero. With your above equation it's the same idea. c1 must be zero.
Seriously, putting aside my lame and incorrect argument above that has no validity at all, heat transfer equations (specifically fin analysis) are without any doubt in my mind the absolute worst equations I have ever come across. I'm actually impressed you found an equation for fins that is so simple... probably because it's in it's most generic form. Anyway, if you really like I can post the most obscene equation I can find in my heat transfer book. I remember it being very long, and containing many variables that took other equations to define what those variables mean, and so on and so on several times.
- Patrick
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