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

Options

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)`

Answer

C

Explanation

For the inner and outer cylinders, thermal resistances are `R_(1) = (L)/(K_(1) (pi R^(2))) and R_(2) = (L)/(K_(2) (4pi R^(2) - piR^(2))) = (L)/(K_(2) (3pi R^(2)))`
For an equivalent conductor of length L, radius 2R and thermal conductivity K, thermal resistance `R= (L)/(K(4pi R^(2)))`
As the inner and outer cylinders are effectively in parallel between the same temperature difference,
`:. (1)/(R ) = (1)/(R_(1)) + (1)/(R_(2)) or (K(4pi R^(2)))/(L) = (K_1)(pi R^(2)))/(L) + (K_(2) (3pi R^(2)))/(L) or K = (K_(1) + 3K_(2))/(4)`