A cylinder of radius R is surrounded by a cylindrical shell of...

Let ${q}_{1}$ and ${q}_{2}$ be the rate of flow of heat through inner part from outer part respecting net flow of heat


$q = {q}_{1} + {q}_{2}$

$= \dfrac{{K}_{1}\pi{R}^{2}}{l} . \DeltaT + \dfrac{{K}_{2} . \pi\left[{\left(2R\right)}^{2} - {R}^{2}\right]}{l} . \DeltaT$

$= \dfrac{\pi{R}^{2}}{l}\DeltaT\left({K}_{1} + 3{K}_{2}\right)$

If the cylinder is replaced by a single material of thermal conductivity $K$ then

$q = \dfrac{K . {\left(2R\right)}^{2}}{l} . \DeltaT$

On comparison,

$K = \dfrac{{K}_{1} + 3{K}_{2}}{4}$.