1. Radioactive waste (k_w = 20 Win^-/Kⁱ) is to be stored in spherical stainless steel (k_s = 16 Wm^-1K^-1) containers. Heat generation occurs uniformly within the radioactive waste at a volumetric rate of 100 kffim^-3, while the outer surface of the container is cooled by 25 °C water with a convective heat transfer coefficient of h = 1000 Wm^-2K¹.

a. Safety considerations require that the maximum steady-state temperature of the system remain below 475 °C. Will a spherical container with inner and outer diameters of 1.0 m and 1.2 m, respectively, satisfy this requirement?

b. Determine the steady-state temperature at the outer surface of the container in part (a).

c. It is proposed to use a spherical container of 1.0 m inner diameter for storing radioactive waste with the same thermal conductivity but twice the volumetric heat generation rate. Safety considerations require that the container wall thickness be no less than 4 cm. If the outside heat transfer coefficient can be increased up to a maximum value of 5000 Wm^-2K¹, determine if the proposed design is feasible. If so, recommend operating and design conditions for h and container wall thickness, respectively.

2. A spherical reactor with diameter of 20 cm is filled with a reacting mixture undergoing an exothermic reaction that generates heat uniformly at a temperature-dependent volumetric rate of Q₀exp(-A/T), where Q₀= 5 kWm^-3, A = 75 K and T is the mixture temperature in K. The reactor is enclosed in an insulation layer (k = 0.05 Wm^-le , £ = 0.9) of 8-mm thickness. The exterior surface of -2 ₁ the insulation layer is exposed to 25 °C air with convective heat transfer coefficient of h = 5 Wm K¹, and also exchanges radiation with the surrounding surfaces that are at 35 °C. The thickness of the reactor wall is negligibly small and the reactant mixture is well stirred so that the reactant mixture and the reactor wall may be considered to be at the same temperature.

a. What are the steady-state temperatures of the reacting mixture and the outer surface of the insulation?

b. You are asked to maintain a steady-state reactor temperature of 95 °C. At the same time, to avoid potential burn injuries to those who may come into contact with the exposed surface of the insulation, we would like to keep the temperature at the outer surface of the insulation below 45 °C. Plot the outer surface temperature and the reactor temperature as a function of insulation thickness (in the range of 0 to 10 mm). Is increasing the insulation thickness a practical solution?

c. What other parameter could be varied to achieve the desired reactor operation? Provide quantitative justification.

3. Aluminum cylindrical-spine fins (k = 205 Wm^-1K^-1) with diameters d = 2, 5, and 10 mm are to be used in a convection environment with h = 40 Wm^-2K^-1. If all fins are of length L = 5 cm, use the result obtained in class for an insulated-tip fin of constant cross-sectional area to answer the following questions.

a. Compare the relative performance of the fins in terms of heat transfer rate.

b. Compare the relative performance of the fins in terms of heat transfer rate per unit mass.