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Standard XIII

Physics

Conduction Law

A cylinder of...

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 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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