1. (10 points) A component balance on a nonisothermal stirred reactor with second-order kinetics and constant flow rate F, volume V, and inlet concentration Cao yields the nonlinear ODE: - F dCa(t) dt - Cao - Ca()] kem [Cu(t] where: CAO) = CA: (moles A/volume), the initial exit stream concentration of A a=E/R CAO (moles A/volume) = the inlet stream concentration of A, a constant Showing all steps, linearize this equation around the initial conditions Ca; and T., and derive the equations for the open loop time constant Tp and gain factor Kp. Show that the linear ODE in deviation variable form is: (draw boxes around your final answers) To CP(t)' +C.P(t) = K, TP(t) 1 -kai where: Tp = Bueno and = F ka2+ ka2 F/ where the constants a, and a, come from linearizing the component balance equation such that: -CE -CC a1 = els () * cs and az = 2eTs (CAS) Hints: You are not required to derive the initial component balance equation but you should know how to derive it if the problem were posed on a test. Assume that the fluid properties - density, heat capacity - are constant and that the flow rate F and hence the reacting fluid volume V are constant. Terms involving Ca(t) - the process variable dependent variable - appear on the left-hand side of the linearized equation while the function T(t) appears as a forcing function on the right-hand side