Chapter 11, Problem 11.57P

The chief engineer at a university that is constructing alarge number of new student dormitories decides toinstall a counterflow concentric tube heat exchangeron each of the dormitory shower drains. The thin- walled copper drains are of diameter $D_i=50\text{ mm}$ .Wastewater from the shower enters the heat exchangerat $T_{h,i}=38°C$ while fresh water enters the dormitory at $T_{c,i}=10°C$ .The wastewater flows down the verticalwall of the drain in a thin, falling fun, providing $h_h=10,000{\text{ W/m}}^2\cdot \text{K}$ .

(a) If the annular gap is $d=10\text{ mm}$ , the heatexchanger length is $L=1\text{ m}$ , and the water flowrate is $\stackrel{\dot{}}{m}=10\text{ kg/min}$ , determine the heat transferrate and the outlet temperature of the warmedfresh water.
(b) If a helical spring is installed in the annular gap sothe fresh water b forced to follow a spiral path
from the inlet to the fresh water outlet, resulting in $h_c=9050{\text{ W/m}}^2\cdot \text{K}$ , determine the heat transfer
rate and the outlet temperature of the fresh water.
(c) Based on the result for part (b), calculate the dailysavings if $15,000$ students each take a 10-minute
shower per day and the cost of water heating is $\$0.0\text{7/kW}\cdot \text{h}$ .