A cylinder of radius R made of a material of thermal conductivity K1 is surrounded by a cylindrical shell of inner radius R and outer radius 2R made of a material of thermal conductivity K2. 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

Qt=Q1t+Q2t Qt=KAθ1−θ2LKot π4R2dθL=K1π4R2dθL+K2π3R2dθL⇒Kof =K1+3K24

A cylinder of radius R made of a material of thermal conductivity K1 is surrounded by a cylindrical shell of inner radius R and outer radius 2R made of a material of thermal conductivity K2. 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

Qt=Q1t+Q2t Qt=KAθ1−θ2LKot π4R2dθL=K1π4R2dθL+K2π3R2dθL⇒Kof =K1+3K24

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A cylinder of radius R made of a material of thermal conductivity K₁ is surrounded by a cylindrical shell of inner radius R and outer radius 2R made of a material of thermal conductivity K₂. 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

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$K_{1} + K_{2}$

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

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

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

answer is B.

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Detailed Solution

$\begin{array}{l} \frac{Q}{t} = \frac{Q_{1}}{t} + \frac{Q_{2}}{t} \frac{Q}{t} = \frac{KA (θ_{1} - θ_{2})}{L} \\ K_{ot} π 4 R^{2} \frac{dθ}{L} = \frac{K_{1} π 4 R^{2} dθ}{L} + \frac{K_{2} π 3 R^{2} dθ}{L} \\ \Rightarrow K_{of} = \frac{K_{1} + 3 K_{2}}{4} \end{array}$