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- A cylinder of radius R made of material of thermal conductivity K1 is surrounded by a cylindrical shell of inner radius R and outer radius 3 R made of a material of thermal conductivity K2. The two ends of the combined system are maintained at two different temperatures. What is the effective thermal conductivity of the system ?
Q. 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 ?
Thermal Properties of Matter
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Solution:
As we use
$k_{e q}=\frac{K_{1} A_{1}+K_{2} A_{2}}{A_{1}+A_{2}}$
$\Rightarrow k_{e q}=\frac{K_{1}\left[\pi R^{2}\right]+K_{2}\left[\pi\left(9 R^{2}\right)-\pi R^{2}\right]}{\pi R^{2}+\pi\left(9 R^{2}-R^{2}\right)}$
$\Rightarrow k_{e q}=\frac{K_{1}+8 K_{2}}{9}$
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