Two spherical concentric shells of radii r and 3r are kept at temperatures `T_1` and `T_2` as shown in the adjoining figure. If the space between the shells is filled with a liquid of thermal conductivity K, what will be the radial flow of heat through the liquid?

Explanation

Let us consider a concentric shell of radius R and thickness dR wherer ` r lt= R lt= 3r` .
Radial flow of heat through this shell of thickness dR is
`H =KA (dT)/(dR) = - K (4pi R^2) (dT)/(dR)`
`rArr = (HdR)/(R^2) = - 4pi KdT`
Integrating both sides from r to 3r , we get
`H int_(r)^(3r) R^(-2) dR = - 4pi int_(T_1)^(T_2) dT`
`rArr H [ (R^(-1) )/(-1) ]_(r)^(3r) = - 4pi K [T]_(T_1)^(T_2)`
`H [ - 1/R]_(r)^(3r) = - 4pi K [T]_(T_1)^(T_2)`
`H [ 1/r - (1)/(3r) ] = 4pi K (T_1 -T_2)`
` rArr (2H)/(3r) = 4pi K (T_1 -T_2)`
` rArr H = 6pi Kr (T_1 - T_2)`