# Homework Help: Heat diffusion equation solutions for semi-infinite slab

1. Jan 12, 2010

### Hipp0

1. The problem statement, all variables and given/known data

http://img42.imageshack.us/img42/1082/clipboard01lx.jpg [Broken]

2. Relevant equations

(see solution)

3. The attempt at a solution

I literary just spent 5 hours trying to apply those boundary conditions, trying exponentials, sines, cosines, hyperbolic function etc... I always get complex numbers in the final solution :( but it's not physical to get complex numbers there :(
Note: Aw and Bw are just constants (w is an index).

http://img189.imageshack.us/img189/4475/94889151.jpg [Broken]
Any ideas where I went wrong? General solution seems fine, maybe I'm misunderstanding the boundary conditions? And in my final answer I tried expanding $$\sqrt{i}=\frac{1}{\sqrt{2}}+\frac{1}{\sqrt{2}}i$$ and then using cos(A+B) formula and writing the result using hyperbolic sines and cosines, but it's still complex :(
Thanks

Last edited by a moderator: May 4, 2017
2. Jan 13, 2010

### CFDFEAGURU

Hello,

Perhaps a substitution for cos(wt) (allow w = omega)

So now you have q = q(0) cos(wt) at the boundary, where q(0) is the amplitude of the heat flux and omega is the circular frequency.

Substitute q(0)R(e^iwt) where R = the real part for the original equation of q(0) = cos(wt).

I worked this problem out as it is given in Transport Phenomena by Bird Stewart and Lightfoot and they suggested this substitution. (Actually it is given as an example, but quite a few steps are missing.)

If you have already done that and I have overlooked it in your somewhat hard to follow layout above, I apologize.

I will have to dig my notes out on this one later this evening to really provide you with some help.

Thanks
Matt

Last edited: Jan 13, 2010