A spherical vessel used as a reactor for producing pharmaceuticals has a 5-mm-thick stainless steel wall (k = 17 W/m ∙ K) and an inner diameter of D _j = 1.0 m. During production, the vessel is filled with reactants for which p = 1100 kg/m³ and c = 2400 J/kg ∙ K. while exothermic reactions release energy at a volumetric rate of q = 10 4 W/m³. As first approximations, the reactants may be assumed to be well stirred and the thermal capacitance of the vessel may be neglected.

(a) The exterior surface of the vessel is exposed to ambient air (T_∞, = 25°C) for which a convection coefficient of h = 6 W/m² ∙ K may be assumed. If the initial temperature of the reactants is 25°C, what is the temperature of the reactants after five hours of process time? What is the corresponding temperature at the outer surface of the vessel?

(b) Explore the effect of varying the convection coefficient on transient thermal conditions within the reactor.