Question

A cylinder of radius $R$ made of material of thermal conductivity $K_{1}$ is surrounded by a cylindrical shell of inner radius $R$ and outer radius $3 R$ made of a material of thermal conductivity $K_{2}$. The two ends of the combined system are maintained at two different temperatures. What is the effective thermal conductivity of the system ?

• A. $K_{1}+K_{2}$
• B. $\frac{K_{1}+8 K_{2}}{9}$
• C. $\frac{K_{1} K_{2}}{K_{1}+K_{2}}$
• D. $\frac{8 K_{1}+K_{2}}{9}$

Answer 1

Answer: (B)

As we use
$k_{e q}=\frac{K_{1} A_{1}+K_{2} A_{2}}{A_{1}+A_{2}}$
$\Rightarrow k_{e q}=\frac{K_{1}\left[\pi R^{2}\right]+K_{2}\left[\pi\left(9 R^{2}\right)-\pi R^{2}\right]}{\pi R^{2}+\pi\left(9 R^{2}-R^{2}\right)}$
$\Rightarrow k_{e q}=\frac{K_{1}+8 K_{2}}{9}$