A chemically reacting mixture is stored in a thin-walled spherical container of radius r l = 200 mm, and the exothermic reaction generates heat at a uniform, but temperature-dependent volumetric rate of q = q_o exp (– A/T_o), where qo = 5000 W/m³, A = 75 K, and T_o is the mixture temperature in Kelvin’s. The vessel is enclosed by an insulating material of outer radius r z, thermal conductivity k, and emissivity ε. The outer surface of the insulation experiences convection heat transfer and net radiation exchange with the adjoining air and large surroundings, respectively.

(a) Write the steady-state form of the heat diffusion equation for the insulation. Verify that this equation is satisfied by the temperature distribution Sketch the temperature distribution, T(r), labeling key features.

(b) Applying Fourier's law, show that the rate of heat transfer by conduction through the insulation may be expressed as

Applying an energy balance to a control surface about the container, obtain an alternative expression for q_r' expressing your result in terms of q and r₁.

(c) Applying an energy balance to a control surface placed around the outer surface of the insulation, obtain an expression from which Ts.2 may be determined as a function of q, r₁ h, T_∞, ε, and Tsur.

(d) The process engineer wishes to maintain a reactor temperature of To = T(r₁,) = 95°C under conditions for which k = 0.05 W 1m . K, r₂ = 208 mm, h = 5 W/m² ∙ K, ε =0.9, T_∞ = 25°C, and T_sur = 35°C. What is the actual reactor temperature and the outer surface temperature of the insulation, T_s.2?

(e) Compute and plot the variation of T_s.2 with r₂ for 201 ≤ r₂ ≤ 210 mm. The engineer is concerned about potential burn injuries to personnel who may come into contact with the exposed surface of the insulation. Is increasing the insulation thickness a practical solution to maintaining T_s.2 ≤ 45°C? What other parameter could be varied to reduce T_s.2?

-T Chemical reaction, (T,)- 11 Ambient air Insulation, k, e