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A solid cylinder of radius $R$ made of a 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 the combined system are maintained at two different temperatures. Then 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
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The two ends of the combined system are maintained at two different temperatures. Then 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
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Verified Answer
The correct answer is:
$\frac{K_{1}+3 K_{2}}{4}$
As there is no loss of heat, and the system is in steady state.
$\begin{array}{llrl}
& \therefore & Q_{\text {total }} & =Q_{1}+Q_{2} \\
\Rightarrow & & \frac{K\left(4 R^{2}\right) d \theta}{d x} & =\frac{K_{1} R^{2} d \theta}{d x}+\frac{K_{2}\left(3 R^{2}\right) d \theta}{d x} \\
\Rightarrow & & 4 K & =K_{1}+3 K_{2} \\
\Rightarrow & & K & =\frac{K_{1}+3 K_{2}}{4}
\end{array}$
$\begin{array}{llrl}
& \therefore & Q_{\text {total }} & =Q_{1}+Q_{2} \\
\Rightarrow & & \frac{K\left(4 R^{2}\right) d \theta}{d x} & =\frac{K_{1} R^{2} d \theta}{d x}+\frac{K_{2}\left(3 R^{2}\right) d \theta}{d x} \\
\Rightarrow & & 4 K & =K_{1}+3 K_{2} \\
\Rightarrow & & K & =\frac{K_{1}+3 K_{2}}{4}
\end{array}$
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