Consider a thin concentric shell of thickness (dr) and of radius (r) and let the temperature of inner and outer surfaces of this shell be T and (T – dT) respectively. The radial rate of flow of heat through this elementary shell will be
dQ
dt
=
KA[(T−dT)−T]
dr
=
−KAdT
dr
=−4πKr2
dT
dr
(∵A=4πr2) Since the area of the surface through which heat will flow is not constant. Integrating both sides between the limits of radii and temperatures of the two shells, we get (