Question

A cylinder $A$ of a material of thermal conductivity $K_1$ is surrounded by a cylindrical shell of inner radius equal to the cylinder.
The shell is made of a material of thermal conductivity $K_2$. The two ends of the combined system are maintained at two different temperatures. If the effective thermal conductivity of the system is $\frac{4K_1+5K_2}{9}$, ratio of outer radius of the shell to the radius of the cylinder is (Answer upto two digit after the decimal point)

Answer 1

Radius of cylinder = $r$
Outer radius of shell = $nr$
$R_{eq}=\frac{R_1{R}_2}{R_1+R_2}=\frac{\frac{1}{K_1\pi r^2}\times \frac{1}{K_2(n^2-1)\pi r^2}}{\frac{1}{K_1\pi r^2}+\frac{1}{K_2(n^2-1)\pi r^2}}$
But $R_{eq}=\frac{1}{K_{eff}\pi n^2{R}^2}$
On Solving, $K_{eff}=\frac{[K_1+K_2(n^2-1\left)\right]}{n^2}$
But $K_{eff}=\frac{4K_1+5K_2}{9}$(Given)

Comparing both, $n=\frac{3}{2}=1.5$