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 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

• A. $K_{1} + K_{2}$
• B. $\frac{K_{1}K_{2}}{K_{1} + K_{2}}$
• C. $\frac{K_{1} + 3K_{2}}{4}$
• D. $\frac{3K_{1} + K_{2}}{4}$

Answer 1

Answer: (C)

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

Answer 2

We can consider this arrangement as a parallel combination of two materials having different thermal conductivities K₁ and K₂

For parallel combination $K = \frac{K_{1}A_{1} + K_{2}A_{2}}{A_{1} + A_{2}}$

A₁ = Area of cross-section of internal cylinder = πR²,

A₂ = Area of cross-section of outer cylinder = π (2R)² – π(R)² = 3πR²

∴ $K = \frac{K_{1}.\pi R^{2} + K_{2}3\pi R^{2}}{\pi R^{2} + 3\pi R^{2}} = \frac{K_{1} + 3K_{2}}{4}$