A completely enclosed stirred-tank heating process is used to heat an incoming stream whose flow rate varies. T; w--~ Heating coil Figure E2.2 Q The heating rate from this coil and the volume are both constant. (a) Develop a mathematical model (differential and algebraic equations) that describes the exit temperature if heat losses to the ambient occur and if the ambient temperature (Ta) and the incoming stream's temperature (T;) both can vary. (b) Discuss qualitatively what you expect to happen as T; and w increase (or decrease). Justify by reference to your model. Notes: p and CP are constants. U, the overall heat transfer coefficient, is constant. As is the surface area for heat losses to ambient. T; > Ta (inlet temperature is higher than ambient temperature).
Question_2_2.2:A completely enclosed stirred-tank heating process is used to heat an incoming stream whoseflow rate varies.The heating rate from this coil and the volume are both constant. (a) Develop a mathematical model (differential and algebraic equations) that describes the exit temperature if heat losses to the ambient occur and if the ambient temperature (Ta) and the incoming stream's temperature (T ) both ian vary. (b) Discuss qualitatively what you expect to happen as T and w inciease (or decrease). Justify by reference to your model.Notes: and C arepconstants.U, the overall heat transfer coefficient, is constant.A ss the surface area for heat losses to ambient.T i Ta (inlet temperature is higher than ambient temperature).\nAnswerStep 1 of 2(a)The energy balance can be written as, dU int dt = (wH) + Q……….(1) where U is the internal energy of the system, H is the enthalpy per unit mass, w is the mass intflow rate, and Q is the rate of heat transfer to the system.The operator denotes the difference between outlet conditions and inlet conditions of theflowing streams.Consequently, the (wH) term represents the enthalpy of the inlet stream(s) minus theenthalpy of the outlet stream(s).The liquid inlet stream consists of a single component with a mass flow ratand an inlet itemperature T iThe tank contents are agitated and heated using an electrical heater that provides a heating rate,Q.A dynamic model will be developed based on the following assumptions: 1. Perfect mixing; thus, the exit temperature T is also the temperature of the tank contents.\n 2. The inlet and outlet flow rates are equal; thus, w i= w and the liquid holdup V is constant. 3. The density and heat capacity C of the liquid are assumed to be constant. Thus, their temperature dependence is neglected. 4. Heat losses are negligible.After simplifying the general energy balance in equation (1), we can write the followingexpression which gives the desired dynamic model of the stirred-tank heating system. d[V (TTref] C p dt = wC (Tp Ti ref) wC (p T ref UA (Ts T ) + aOn simplifying, dT V C p dt = wC (p ) iwC (T) pUA (T T ) s Q a dT V C p dt = wC (Tp T)i UA (T T )s+ Q a
