In open heart surgery under hypothermic conditions, the patient's blood is cooled before the surgery and re-warmed afterward. It is proposed that a concentric tube, counter flow heat exchanger of length 0.5 m be used for this purpose, with the thin-walled inner tube having a diameter of 55 mm. The specific heat of the blood is 3500 J/kg ∙ K.

(a) If water at T_h,i = 60°C and m_h = 0.10 kg/s is used to heat blood entering the exchanger at T_c,i = 18°C and m_c = 0.05 kg/s, what is the temperature of the blood leaving the exchanger? The overall heat transfer coefficient is 500 W/m₂ ∙ K.

(b) The surgeon may wish to control the heat rate q and the outlet temperature T_c,o of the blood by altering the flow rate and/or inlet temperature of the water during the re-warming process. To assist in the development of an appropriate controller for the prescribed values of m_c and T_c,i, compute and plot q and T_c,o as a function of m_h for 0.05 < mh < 0.20 kg/s and values of T_h,i = 50, 60, and 70°C. Since the dominant influence on the overall heat transfer coefficient is associated with the blood flow conditions, the value of U may be assumed to remain at 500 W/m² ∙ K. Should certain operating conditions be excluded?