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A cylinder A of a material of B thermal conductivity $K_{1}$ is surrounded by a cylindrical shell of inner radius equal to the cylinder. The shell is made of a material of thermal conductivity $K_{2}$ . The two ends of the combined system are maintained at two different temperature. If the effective thermal conductivity of the system is $\frac{4 K_{1} + 5 K_{2}}{9}$ the ratio of outer radius of the shell to the radius of the cylinder.

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2.0

$\frac{5}{2}$

$\frac{5}{3}$

$\frac{3}{2}$

answer is D.

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Detailed Solution

Radius of cylinder = R and outer radius of shell = nR

$R_{eq} = \frac{R_{1} R_{2}}{R_{1} + R_{2}} = \frac{\frac{1}{K_{1} {πR}^{2}} \times \frac{1}{K_{2} (n^{2} - 1) {πR}^{2}}}{\frac{1}{K_{1} {πR}^{2}} + \frac{1}{K_{2} (n^{2} - 1) {πR}^{2}}}$

But $R_{eq} = \frac{1}{{Kπn}^{2} R^{2}}$

On solving, $K = \frac{[K_{1} + K_{2} (n^{2} - 1)]}{n^{2}}$

= $\frac{4 K_{1} + 5 K_{2}}{9} (given)$

= $\frac{\frac{K_{1} + 5 K_{2}}{4}}{\frac{9}{4}}$

n = $\frac{3}{2}$

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