➤ Problem 2. Consider the stirred-tank heating system shown in Figure 2. The liquid inlet stream consists of a single component with a mass flow rate w, and an inlet temperature T. The tank contents are agitated and heated using an electrical heater that provides a heating rate, Q. (a) Develop the dynamic models (unsteady-state energy balances for the tank and the heating element) based on following assumption: Perfect mixing; thus, the exit temperature 7 is also the temperature of the tank contents. • The inlet and outlet flow rates are equal; thus ww and the liquid holdup V is constant. The density p and heat capacity C of the liquid are assumed to be constant. Thus their temperature dependence is neglected. Heat losses are negligible. ⚫ Energy transferred instantaneously from the heating element to the contents of the tank. ⚫ Neglect the temperature gradients in the heating element that result from heat conduction and assume that the element has a uniform temperature, Te. This temperature can be interpreted as the average temperature for the heating element. (b) Consider the models of the electrically heated stirred-tank system in section (a): 1. Drive transfer functions relating changes in outlet temperature T to changes in the two input variables; heater Q (assuming no change in inlet temperature), and inlet temperature Ti (for no change in heater input). 2. Show how these transfer functions are simplified when negligible thermal capacitance of the heating element (m.C. →0) is assumed. Wi V Heater Figure 2: Stirred-tank heating with constant holdup, V, related to problem 2.