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Q.

A cylinder of radius R is surrounded by a cylindrical shell of inner radius R and outer radius 2R. The thermal conductivity of the material of the inner cylinder is $K_{1}$ and that of the outer cylinder is $K_{2}$ . Assuming no loss of heat, the effective thermal conductivity of the system for heat flowing along the length of the cylinder is:

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a

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

b

$K_{1} + K_{2}$

c

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

d

$\frac{2 K_{1} + 3 K_{2}}{5}$

answer is D.

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Detailed Solution

Inner and outer cylinder are in parallel.
So, $\frac{1}{R} = \frac{1}{R_{1}} + \frac{1}{R_{2}}$
$\begin{array}{l} \frac{K A}{l} = \frac{K_{1} A_{1}}{l} + \frac{K_{2} A_{2}}{l} \\ K = \frac{K_{1} A_{1} + K_{2} A_{2}}{A_{1} + A_{2}} \\ = \frac{K_{1} π R^{2} + K_{2} [π (2 R)^{2} - π R^{2}]}{π (2 R)^{2}} \\ = \frac{K_{1} (π R^{2}) + K_{2} (3 π R^{2})}{4 π R^{2}} = \frac{K_{1} + 3 K_{2}}{4} \end{array}$

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A cylinder of radius R is surrounded by a cylindrical shell of inner radius R and outer radius 2R. The thermal conductivity of the material of the inner cylinder is K1 and that of the outer cylinder is K2 . Assuming no loss of heat, the effective thermal conductivity of the system for heat flowing along the length of the cylinder is: