1. The fuel element in a nuclear reactor consists of a cylindrical thorium (k = 60 Wm^-1K^-1) fuel rod of 25 mm diameter that is wrapped in a 2 mm-thick aluminum (k = 205 Wm^-1K^-1) cladding. The fuel rod may be considered to be in perfect contact with the cladding material so that contact resistances may be neglected. Energy generation occurs uniformly in the fuel rod, and the cladding is cooled with 95°C water and a convective heat transfer coefficient of h = 7000 Wm^-2K^-1. The melting points of thorium and aluminum are 2023K and 933K, respectively.

a. Steady-state operation of the fuel rod with a volumetric energy generation rate of 700 MWm^-3 is proposed.

Is the proposed stead-state operation feasible?

b. In the event of a loss of coolant, the heat transfer coefficient h would decrease drastically. Explore the effects of energy generation rate and heat transfer coefficient on the temperatures at the center of the fuel rod and at the fuel rod-cladding interface. Plot these temperatures as functions of volumetric energy generation rate in the fuel rod (over the range 100-1000 MWm^-3) for different values of h (over the range 1-10 MWm^-2K^-1) in order to suggest an envelope of acceptable operating conditions.

2. Consider an electric space heater made of coiled Nichrome wire of 1 mm diameter, with cross flow of 50°C air over the wire to establish a convective heat transfer coefficient of 250 Wm^-2K^-1. The housing that encloses the coiled wire is at T_sur = 50°C. The Nichrome wire has electrical resistivity of ρ_e = 10^-6 m, thermal conductivity of k = 25 Wm^-1K^-1, and surface emissivity of ε = 0.20.

a. Assuming negligible temperature variations within the wire, estimate the maximum allowable electric current to avoid wire temperatures greater than 1200°C.

b. Assess the validity of the assumption of uniform wire temperature in part (a) by computing the centerline temperature corresponding to the maximum rate of heat generation.

c. (Extra credit) What is the corresponding length of wire that may be used in the heater for a maximum available voltage of 110V? What is the power rating of the heater?

3. Plate glass at 600°C is cooled during its manufacture by passing cooled air over its surfaces with a convective heat transfer coefficient of 5 Wm^-2K^-1. To prevent cracking, the temperature gradient must not be allowed to exceed 15 K/mm at any point in the glass during the cooling process. Plate glass has a thermal conductivity of k = 1.4 Wm^-1K^-1 and a surface emissivity of ε = 0.80. Assuming the air temperature to be the same as that of the surrounding walls, what is the lowest air temperature that can initially be used for the cooling process?