Problem 5) Temperature profile in a cylindrical nuclear fuel rod Learning Objective: Deriving a temperature profile in radial coordinates starting from the differential equation for energy transfer A nuclear fuel rod

generates heat internally and the heat is transferred to the surrounding heat transfer fluid kept at temperature T... The cylindrical nuclear fuel rod in this problem is generating heat according to the following function: q=qmax [¹-6] where à is the heat generation rate per unit volume (we have also used the notation à"" in class) as a function of position r amax is the maximum heat generation rate in the rod (at the center), and R is the radius of the rod. This shows that the heat generation is the highest in the center where radiation is the highest and declines toward the surface. 1 In this problem, the end goal is to derive the steady-state temperature profile in the rod as a function of the fluid temperature. Answer the following questions which lead you to necessary equation and boundary conditions needed to derive this profile. Note: you are not expected to be able to do derivations using vector calculus, but you are expected to recognize the meaning of the terms as used in class and expand them using the attached tables./na) Simplify the general differential equation for energy transfer for a solid cylinder, assuming that the cylinder has R>>L and that there is no angular dependence. Note briefly but explicitly your reasoning for eliminating any term. Do not assume steady state or that physical properties are constant at this step. pC₂ (37+ v · VT) = ▼ · kVT + q + ¢ at b) Simplify the equation in a) further by assuming steady state and constant physical properties. c) Using the answer to part b, write the governing differential equation that can be integrated to derive the steady-state radial temperature profile in the fuel rod in terms of qmax. d) Write appropriate boundary condition(s) for the problem. e) Integrate the differential equation to obtain the temperature profile in the nuclear fuel rod. You can leave the equation in terms of the correct number of integration constants. f) Evaluate the integration constants in part e) to obtain the radial temperature distribution in the fuel rod. (this step was given as extra credit on the exam since it takes a while and is math, not heat transfer concept).