Pressurized water at Tm,j = 200°C is pumped at In = 2 kg/s from a power plant to a nearby industrial user through a thin-walled, round pipe of inside diameter D = 1 m. The pipe is covered with a layer of insulation of thickness t = 0.15 m and thermal conductivity k = 0.05 Wm ∙ K. The pipe, which is of length L = 500 m, is exposed to a cross flow of air at T∞ = -10°C and V = 4 m/s. Obtain a differential equation that could be used to solve for the variation of the mixed mean temperature of the water Tm(x) with the axial coordinate. As a first approximation, the internal flow may be assumed to be fully developed throughout the pipe. Express your results in terms of In, V, T ∞, D, t, k, and appropriate water (w) and air (a) properties. Evaluate the heat loss per unit length of the pipe at the inlet. What is the mean temperature of the water at the outlet?