A solid 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. Then 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

Options

A. `K_(1) + K_(2)`
B. `(K_(1)K_(2))/(K_(1) + K_(2))`
C. `(3K_(1) + K_(2))/(4)`
D. `(K_(1) + 3K_(2))/(4)`

Answer

D

Explanation

Area of cross-section of inner cylinder = `piR^(2)`
Area of cross-section of outer shell = `pi (2R)^(2) - piR^(2) = 3piR^(2)`
Rate of heat flow in inner cylinder
`H_(1) = K_(1) piR^(2)(T_(1) - T_(2))/(L)`
Rate of heat flow in outer shell
`H_(2) = K_(2) 3piR^(2) (T_(1) - T_(2))/(L)`
Rate of heat flow in the combine system
`H = K 4piR^(2)(T_(1) - T_(2))/(L)`
At steady state,
`H = H_(1) + H_ (2)`
`:. K4piR^(2)(T_(1) - T_(2))/(L) = (K_(1)piR^(2)(T_(1) - T_(2)))/(L) + (K_(2)3piR^(2)(T_(1) - T_(2)))/(L)`
`4K = K_(1) + 3K_(2)` or `K = (K_(1) + 3K_(2))/(4)`