Solve the following problem. For deviation variables, you can use hats. Represent as block diagram. G(s) and Gas) are transfer functions for input and disturbance variables. (a) Figure P4.2 shows a system for heating a continuous flow water kettle using a hot plate. Assuming that the hot plate temperature can be changed instantaneously by adjusting the hot plate rate of heat input and assuming uniform water kettle temperature, show that the following is a reasonable model for the process: dT Pdt = c(1, - T,) + FPC, 7o - FpC where p, C, are, respectively, water density and heat capacity, and C = a constant F = volumetric flowrate in and out of the kettle. PVC (P4.1) F,T F.T. Assuming that the simpler conditions of Problems 4.2(a) apply and that the objective is to control the water temperature T, by adjusting Tz (a) Which of the variables indicated in Eq. (P4.1) is the input variable, and which is the disturbance variable? (b) By defining appropriate deviation variables (in terms of T7", 12*, and T,*, the steady-state values of T1, T2, T, respectively) rewrite Eq. (P4.1) in the form: dy dt = ay + bu + pel (P4.2) Explicitly state a, b, and y in terms of process parameters p, Cp, C, F, and V. (c) Express this differential equation model in the transform-domain transfer function form. What are g(s) and 8d(s) in this case