A cylinder of radius r and thermal conductivity $k_{1}$ is placed inside a cylindrical shell of inner and outer radius r & 3r and thermal conductivity $k_{2}$ . Ends of the cylinder are maintained at two different temperatures and curved surface is insulated, then equivalent thermal conductivity of the system is

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$\frac{k_{1} + 9 k_{2}}{9}$

$\frac{k_{1} + 8 k_{2}}{9}$

$\frac{k_{2} + 8 k_{1}}{9}$

$\frac{k_{2} + 9 k_{1}}{8}$

answer is B.

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Detailed Solution

$R = \frac{R_{1} R_{2}}{R_{1} + R_{2}} = \frac{\frac{l}{k_{1} {πr}^{2}} \times \frac{l}{k_{2} 8 {πr}^{2}}}{\frac{l}{k_{1} {πr}^{2}} + \frac{l}{k_{2} 8 {πr}^{2}}} \frac{1}{k_{eq} 9 {πr}^{2}} = \frac{l}{{πr}^{2}} \frac{1}{8 k_{2} + k_{1}} = \frac{1}{k_{1} + 8 k_{2}}$

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