determine the heat loss from a spherical shell

Question

The thermal conductivity of a hollow sphere of inside radius (R₁) and outside radius (R₀) is given by

Where, T_i = Inner surface temperature

               T₀ = Outside surface temperature.

Prove that

1. The heat flow rate is given by

Also determine the heat loss from a spherical shell whose D₁ = 2.5 m and covered with 30 cm of insulation. The thermal conductivity of insulation is 0.3 W/mK and 0.2 W/mK at inner and outer surface temperatures of 150⁰C and –15⁰C respectively.