Q. A cylinder of radius $R$ is surrounded by a cylindrical shell of inner radius $R$ and outer radius $2R$. The thermal conductivity of the material of the inner cylinder is $K_1$ and that of the outer cylinder is $K_2$. Assuming no loss of heat, the effective thermal conductivity of the system for heat flowing along the length of the cylinder is:
Solution:
$K_{eq} = \frac{K_{1}A_{1}+K_{2}A_{2}}{A_{1}+A_{2}}$
$ =\frac{K_{1}\left(\pi R^{2}\right)+K_{2}\left(3\pi R^{2}\right)}{4\pi R^{2}} $
$ = \frac{K_{1} + 3K_{2}}{4} $
