Question

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 a 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) K_1 + K_2 (b) K_1K_2//(K_1+K_2) (c )(K_1 + 3K_2)//4 (d) (3K_1 + K_2)//4.

Answer 1

The effective thermal conductivity of the system is:

• 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 two cylinders are in parallel, so equivalent thermal resistance is

              (1/R) = (1/R₁) + (1/R₂)

• But $R = \frac{KL}{A}$

                 where R = Thermal resistance

                             K = Thermal conductivity

                             L = Length

                             A = Area

• Combining above two equations, we get : (KL/A) = (K₁L₁/A₁) + (K₂L₂/A₂)
• But L = L₁ = L₂ ,

              A₁ = $\pi R^2$ ,

              A₂ = $\pi (2R)^2 -\pi (R)^2 = 3\pi R^2$,

              A = $\pi (2R)^2 = 4\pi R^2$

• Putting these values, we get : K = ( K₁ + 3K₂ )/4
• Therefore, Option C is correct.