Hot water for an industrial washing operation is produced by recovering heat from the flue gases of a furnace. A cross-flow heat exchanger is used, with the gases passing over the tubes and the water making a single pass through the tubes. The steel tubes (k = 60 W/m ∙ K) have inner and outer diameters of D_i = 15 mm and D_o = 20 mm, while the staggered tube array has longitudinal and transverse pitches of S_T = S_L = 40 mm. The plenum in which the array is installed has a width (corresponding to the tube length) of W = 2 m and a height (normal to the tube axis) of H = 1.2 m. The number of tubes in the transverse plane is there-fore N_T = H/ST = 30. The gas properties may be approximated as those of atmospheric air, and the convection coefficient associated with water flow in the tubes may be approximated as 3000 W/m² ∙ K.

(a) If 50 kg/s of water are to be heated from 290 to 350 K by 40 kg/s of flue gases entering the exchanger at 700 K, what is the gas outlet temperature and how many tube rows N_L are required?

(b) The water outlet temperature may be controlled by varying the gas flow rate and/or inlet temperature. For the value of N L determined in part (a) and the prescribed values of H, W, S_T, m_c, and T_c,i compute and plot T_c,o as a function of m_h over the range 20 < m_h < 40 kg/s for values of T_h,i = 500, 600, and 700 K. Also plot the corresponding variations of T_h,p If T_h,o must not drop below 400 K to prevent condensation of corrosive vapors on the heat exchanger surfaces, are there any constraints on m_h and T_h,i?