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 a combined systems are maintained at two different temperatures. What is the thermal conductivity of the system?

K_{1} + K_{2}

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

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\frac{K_{1} K_{2}}{K_{1} + K_{2}}

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

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Solution

The correct option is B $\frac{K_{1} + 8 K_{2}}{9}$
Area of inner cylinder $A_{1} = π R^{2}$
Area of outer cylinder $A_{2} = π ((3 R)^{2} - R^{2})$
Let $K$ be thermal conductivity of the equivalent system,
Effective area of equivalent system is $A = π (3 R)^{2}$

Two cylinders are in parallel, therefore equivalent thermal resistance R is given by
$\frac{1}{R} = \frac{1}{R_{1}} + \frac{1}{R_{2}}$
$\frac{K A}{l} = \frac{K_{1} A_{1}}{l} + \frac{K_{2} A_{2}}{l}$
$K π (3 R)^{2} = K_{1} π R^{2} + K_{2} π ((3 R)^{2} - R^{2})$
$9 K = K_{1} + 8 K_{2}$
$K = \frac{K_{1} + 8 K_{2}}{9}$

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A cylinder of radius R made of a material of thermal conductivity $K_{1}$ is surrounded by a cylindrical shell of inner radiusRand outer radius 2Rmade of material of thermal conductivity $K_{2}$ . The two ends of the combined system are maintained at two different temperatures. There is no loss of heat across the cylindrical surface and the system is in steady state. The effective thermal conductivity of the system is