The outer surface of a hollow sphere of radius r₂ is subjected to a uniform heat flux q₂. The inner surface at r₁ is held at a constant temperature T_∞.1.

(a) Develop an expression for the temperature distribution T(r) in the sphere wall in terms of q₂, T_s.1, r₁, r₂, and the thermal conductivity of the wall material k.

(b) If the inner and outer tube radii are r₁ = 50 mm and r₂ = 100 mm. what heat flux q₂ is required to maintain the outer surface at T_s.2 = 50°C, while the inner surface is at T_S.1 = 20°C? The thermal conductivity of the wall material is k = 10 W/m ∙ K.