A point source of heat of power P is placed at the centre of a thin spherical shell of mean radius R . The material of the shell has thermal conductivity K . Calculate the thickness of the shell if the temperature difference between the outer and inner surfaces of the shell in steady-state is T .

Step-by-step solution

Consider a concentric spherical shell of radius r and thickness d r as shown in diagram. The radial rate of flow of heat through this shell in steady state will be, H = d Q d t = - KA d θ d r [Negative sign is used because, with an increase in r , θ decreases.] Now as for spherical shell A = 4 π r 2 So H = - 4 π r 2 K d θ d r or ∫ a b d r r 2 = - 4 π K H ∫ θ 1 θ 2 d θ which on integration and simplification gives H = d Q d t = K 4 π a b θ 1 - θ