Conduction - Heat Equation - Units Don't Add Up

[chrissimpson]

Conduction - Heat Equation - Units Don't Add Up!

Hi there

I have what I think/hope is a simple question:

I've been working on heat inputs and outputs in inertia friction welds and have managed to produce a net power term (W) as a function of time.

I now want to use that in the heat equation to come up with heating rates, temperature profiles and final cooling rates (these can then be related back to the final weld microstructure).

So, the equation I wish to use is:

ut-αuxx=(α/κ)*f(x,t) - a non-homogeneous pde

The problem I'm having with this is that my heat input term is in watts or watts/m^2, not watts/m^3. This is because my heat input is coming from one surface into a long/infinite length bar. When heat generation is in watts/m^3 you get consistent units of K/s throughout the equation. I don't!

Any ideas?

Cheers

Chris

 

[chrissimpson]

I think that I may have made a mistake with my formulation of the problem; I actually have a non-homogeneous boundary condition, so:

ut=αuxx

with Neumann boundary conditions of:

ux(0,t)=Q/kA - with both sides having units of K/m

ux(L,t)=0


Does this seem like the correct way of looking at the problem?

Cheers

Chris