Q: 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 temperature. These 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) K_{1} + K_{2}

(b) K_{1}K_{2}/(K_{1} + K_{2})

(c) (K_{1} + 3K_{2}) /4

(d) (3K_{1} + K_{2}) /4

Ans: (c)

Sol: Here , thermal resistances are in parallel

$\large \frac{1}{R} = \frac{1}{R_1} + \frac{1}{R_2} $

$\large \frac{K(4\pi R^2)}{l} = \frac{K_1(\pi R^2)}{l} + \frac{K_2(3\pi R^2)}{l} $

$\large K = \frac{K_1 + 3K_2}{4} $