5. Consider a constant volume, continuous stirred tank reactor in which the following irreversible reaction sequence occurs: A + B, B C, 2A + D. The reaction rate constants are denoted ki (first reaction), k2 (second reaction) and k3 (third reaction). Let CA, CB, Cc, and Cp denote the molar concentrations of the four reaction species. The reaction rates per unit volume have the form: T1 ki CA, r2 = k2Cb, r3 = k3C. Let Fi and F denote the volumetric flow rates of the inlet and outlet streams, respectively. The reactor is fed with pure component A at a molar concentration Cai. Assume that the reactor operates isothermally. Consider the following model parameter values (given without units for simplicity): k = 5/6, k2 = 5/3, k3 = 1/6, CAi 10, and F/V = 4/7. The following model equations are derived by writing the necessary material balances: - - dCA dt dCB dt FC (CA - C) kiCa- k3C = fi(Ca) Cp+ CB + k1CA - k2Cb = f2(CA,CB) = = dr = (a) Show that there is a single physically meaningful steady state by solving the nonlinea algebraic equations fi(Ca)= 0 and f2(CA,CB) = 0. (b) Define xi(t) = CA(t) - CA and xz(t) = CB(t) - B. Linearize the nonlinear model equations in Part (a) about the steady state in Part 1 to generate a linear ODE model for det = AX. (c) Find the eigenvalues and eigenvectors of the A matrix in Part (b). (d) Consider the initial condition CA(0) = 10 and CB(0) = 0. Use the general solution form r(t) = Cir(1) edt +C2.0(2)etzt and the initial condition z(0) to find the solution X(t) for the linear ODE model in Part (b). Use the definition of x(t) to find the solutions Ca(t) and CB(t). (e) Sketch the qualitative forms of the solutions Ca(t) and CB(t). = = =