2.3 Text Problem 2.52 in 7th Ed, Problem 2.41 in 8th Ed. A chemically reacting mixture is stored in a thin-walled spherical container of radius r = 200 mm, and the exother- mic reaction generates heat at a uniform, but temperature- dependent volumetric rate of g = 9, exp(-A/T.), where 9. = 5000 W/m, A = 75 K, and T, is the mixture temper- ature in kelvins. The vessel is enclosed by an insulating material of outer radius r2, thermal conductivity k, and emissivity e. The outer surface of the insulation experi- ences 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 equa- tion is satisfied by the temperature distribution 1 - (rr) T(r) = (1,1-1,2) - (rr) (C) Applying an energy balance to a control surface placed around the outer surface of the insulation, obtain an expression from which 7.2 may be deter- mined as a function of q, r, h, T., E, and Tour (d) The process engineer wishes to maintain a reactor temperature of T, = T(r) = 95C under conditions for which k = 0.05 W/m.K, r = 208 mm, h = 5 W/m.K, E=0.9, T = 25C, and Tsur = 35C. What is the actual reactor temperature and the outer surface temperature T.2 of the insulation? (e) Compute and plot the variation of T.2 with r for 201 Srs 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 Tx2 = 45C? What other parameter could be varied to reduce T 2? [ Tsur 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 4k(T31-T2) 9,= (17r) -(1/r) Chemical reaction, g(T)- 1 1 Ambient air Tih Insulation, ke Applying an energy balance to a control surface about the container, obtain an alternative expression for 9,, expressing your result in terms of g and ri