Answer Happy

3. (Difficult former exam problem) A cylindrical kiln is made by wrapping a thick electrical wire around a hollow cylinder. For simplicity, you can assume the heat is generated uniformly in an annular region around the cylinder and that exterior of the annular region is open to the environment as shown in the following figure.
h, To ri ro agen The exterior environment is characterized by a heat transfer coefficient h and an ambient temperature of T. The heat generation region is characterized by a volumetric heat generation rate, Agen and a thermal conductivity k. The inner region is sealed and you can assume the ends of the kiln are adiabatic (i.e. assume 2D). (a) What is the governing equation that determines the temperature in the heat generation region in simplest form (i.e. neglect all terms that are zero). (b) If the furnace is operating at steady state, what boundary conditions should be applied to the inner and outer surface of the heat generation region. As a small reminder: in radial coordinates doutward-k. (Hint: if it is steady state, how much heat can be going into the kiln?) (c) What is the general form of the temperature solution in the annular region (d) If h = 20 W/(m²K), Tx = 35 C, agen = 80x105 W/m³, k = 100 W/(mK), Ti = 30 cm, and ro= 30.6 cm, determine i. The temperature inside the furnace at steady-state operating condi- tions. ii. The temperature on the outer surface at steady-state operating con- ditions.