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 $2 R$ made of a material of thermal conductivity $K_{2}$ . The two ends of combined system are maintained at two different temp. There is no loss of heat across cylindrical surface and system is in steady state. Calculate effective thermal conductivity of system.

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

$K_{1} + K$

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

$\frac{K_{1} + 3 K_{2}}{4}$

answer is A.

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Detailed Solution

$K_{e} = \frac{K_{1} A_{1} + K_{2} A_{2}}{A_{1} + A_{2}}$ (For parallel combination)

$\frac{K_{1} [{πR}^{2}] + K_{2} [π {(2 R)}^{2} - {πR}^{2}]}{4 {πR}^{2}} K_{e} = \frac{K_{1} + 3 K_{2}}{4}$

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