1. A long cylindrical rod (Fig. 1) of radius R = 2 cm and length L = 40 cm is made of a material that generates energy from within. The conductivity of the material is k = 15 W/m K. The temperature profile inside the rod can be assumed to vary only in the radial direction and can be represented as T(r) [K]. As shown in the figure, the rod can be represented in cylindrical coordinates (x-r). One end of the cylinder can be assumed to be at x = 0 and the other end at x = L. It was found that the governing equation that governs the heat transfer in the rod can be written as dT dr ( + e = 0, where e is a constant and r is measured in meter. x=0 X-L Figure 1: Long cylinder (a) Integrate the equation to find the functional form of temperature T(r) in terms of R, e, L, and arbitrary constants of integrations. (b) What is the unit of the quantity e? (8 pt) (4 pt) (2 pt) (4 pt) = -k] What is the conductive heat flux in the rod in x direction at x = 4/2 and r = R/2? (d) What is the conductive heat flux in the rod in r direction at x = 4/2 and r = R/2? [Hint: Conductive heat flux in radial direction (i.e., in cylindrical coordinates) is q