The method I have given earlier is known as perturbation significance analysis.
Originally Posted by ryaron
What are the ratios of the conductivities ?
i.e. k1/k2 ?
Here's a quick answer:
make two dimensionless parameters epsilon=k1/k2 and eta characteristic length tube/characteristic length plate, make sure they are << 1.
Denote them as epsilon and eta.
Now expand both expression you've written in powers of epsilon and eta.
The one with the higher order O(epsilon^n),