I'm having trouble determining the eigenvalues for a 1-D wall, transient PDE with the BCs:
at x=0: T(0,t)=To
at x=L: k(dT/dt)+h[T(L,t)-Tinf]=0 (convection)
I defined: theta(x,t)=T(x,t)-To
using separation of variables: Omega(x,t)=X(x)G(t)
Using the first boundary condition (redefined in terms of Omega so that omega(0,t)=0), I was able to simplify X(x) to C1*cos(lambda*x). Applying the second convective BC results in a mess, something like:
lambda*cot(lambda*L)= h/k + h/(k*C1)*[ (To-Tinf) / sin(lambda*L) ]
It'd simplify things a lot if I can redefine theta(x,t) as T(x,t)-Tinf, but can I redefine theta it twice in the same problem? Or is that incorrect?
Thanks in advance for any help! Dave