A cylinder of radius r and of thermal conductivity $k_{1}$ is surrounded by a cylindrical shell of inner radius r and outer radius 2r made of a material of thermal conductivity $k_{2}$ . The effective thermal conductivity of the system is

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Solution

The correct option is D

Since the two cylinders are in parallel, the equivalent thermal resistance R of the combination is given by

$\frac{1}{R} = \frac{1}{R_{1}} + \frac{1}{R_{2}}$

or $\frac{k A}{l} = \frac{k_{1} A_{1}}{l_{1}} + \frac{k_{2} A_{2}}{l_{2}}$ (1)

$N o w A = π (2 r)^{2} = 4 π r^{2}, A_{1} = π r^{2}, A_{2} = π [(2 r)^{2} - r^{2}] = 3 π r^{2} a n d l_{1} = l_{2} = l . U s i n g t h e s e i n (1), w e g e t$

$4 k = k_{1} + 3 k_{2} o r k = \frac{1}{4} (k_{1} + 3 k_{2})$

Hence the correct choice is (d).

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A 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 2R made of material of thermal conductivity $K_{2}$ . 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

A cylinder of radius R made of a material of thermal conductivity $K_{1}$ is surrounded by a cylindrical shell of inner radiusRand outer radius 2Rmade of material of thermal conductivity $K_{2}$ . 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