First slide

Conduction

Question

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

Moderate

A

$K_{1} + K_{2}$
B

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

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

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

Solution

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

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