The axisymmetric annulus is a conducting medium. The annulus occupies ri r ro and 0 x L, where ri is the radius of the inner cylinder, ro is the radius of the outer cylinder, and L is the length. x = 0, x = L, and r = ro surfaces are insulated. r = ri surface is a condensation surface at 400 C (the condensation surface is assumed to be a constant surface temperature boundary). The annulus is initially at 300 C, and the condensation at r = ri is activated for t 0. Consider ri = 12.5 mm, ro = 37.5 mm, and L = 1.2 m. The thermal conductivity is 50 W/(m K), the specific heat is 1.75 kJ/(kg K), and the density is 1500 kg/m3.

a) Find the temperature vs. time at (r,x) = (15 mm, 500 mm), (22.5 mm, 500 mm), and (30 mm, 500 mm) for 0 t 2000 s. Use an analytical method. b) Use explicit and implicit methods to find the temperature vs. time at (22.5 mm, 500 mm) for 0 t 2000 s. Compare it against the solution obtained in part (a)

Please use separation of variables as the analytical solution and partition the solution into steady state and transient components.