A cylinder of radius R made of a material of thermal conductivity `K_1` is surrounded by a cylinderical 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 effictive thermal conductivity of the system is

Options

A. `K_1+K_2`
B. `K_1K_2//(K_1+K_2)`
C. `(K_1+3K_2)//4`
D. `(3K_1+3K_2)//4`

Answer

C

Explanation

(c) Total transfer of heat per second through the composite
=Heat transfer per second from material with thermal
conductivity `K_1`+Heat transfer per second from
material with thermal conductivity `K_2.`

`(KADeltaT)/l=(K_1A_1DeltaT)/l+(K_2A_2DeltaT)/l`
or, Kpi(2R)^2=K_1piR^2+K_2pi[(2R)^2-R^2]`
or, `K=(K_1+3K_2)/4`