The chief engineer at a university that is constructing a large number of new student dormitories decides to install a counter flow concentric tube heat exchanger on each of the dormitory shower drains. The thin-walled copper drains are of diameter D_i = 50 mm. Wastewater from the shower enters the heat exchanger at T_h,i = 38?C while fresh water enters the dormitory at T_c,i = 10?C. The wastewater flows down the vertical wall of the drain in a thin, falling film, providing h_h = 10,000 W/m² ? K.

(a) If the annular gap is d = 10 mm, the heat exchanger length is L = 1 m, and the water flow rate is m = 10 kg/min, determine the heat transfer rate and the outlet temperature of the warmed fresh water.

(b) If a helical spring is installed in the annular gap so the fresh water is forced to follow a spiral path from the inlet to the fresh water outlet, resulting in he = 9050 W/m² ? K, determine the heat transfer rate and the outlet temperature of the fresh water.

(c) Based on the result for part (b), calculate the daily savings if 15,000 students each take a 10-minute shower per day and the cost of water heating is $0.07/kW ? h.

Thi, m, Hot waste water Warm fresh water D- Copper tube Annulus Falling film Cold fresh water Cool waste water