2. (10 points) Consider an adiabatic (perfectly insulated). CA(t), F, Ti(t) exothermic, perfectly mixed chemical reactor where the reaction A+B C takes place as illustrated in the figure. We want to understand the consequences of operating A+B=0 an exothermic reactor with no active heat removal. (insulated) That is, how does the reactor perform when the cooling system fails? CA(t), F, T(t) Use the following definitions: p. Co = density and heat capacity of reactants & product (assumed same & constant for all streams), Kg/m' and KJ/[Kg-K), respectively F = volumetric flow rate of inlet & outlet streams, a constant, m/sec T.(t) = inlet temperature, OK; T:(0) = Tis T(t) = temperature in reactor, K; T(O)=T. AH, = heat of reaction. Joules/mole (assumed constant) V = volume of liquid in reactor, m To simplify the problem, assume that the kinetics for the reaction are zeroth-order but depend upon temperature T(t) according to ra(t) = ko exp[-E RT()] where: r(t) = rate of reaction, (moles A reacting) (m? - sec) ko = frequency factor = 1 (moles) (m? - sec) E = activation energy, 500 KJ/Kmole R=ideal gas constant, J/(mole - K) Note: "Zeroth-order means that r(t) does not depend upon concentration. = = - (a) Derive the energy balance model for this reactor, linearize it, and put it into deviation variable form and derive the expressions for the open loop or process time constant Tp and the gain factor Ky related to the inlet temperature forcing function in deviation variable form, TP(t). Show that the linear ODE in deviation variable form is: (draw boxes around your final answers) Tp [T(t)]' + T'(t) = K, T:'(t) where: PVCp Tp FpCp+AHVC1 and C is a constant resulting from the linearization of the reaction rate term. Show that: C1 = ko ets * * where u. = E/R. (b) In every system we have analyzed until now, the time constant Tp was positive. As shown in Figure 13.1 in PPC, a system with a positive value of the time constant Tp exhibits a stable response when disturbed, i.e. the process variable y(t) does not blow up to infinity for t > 0. However, when Tp is negative, an unstable response occurs - the reactor temperature increases exponentially with time, sometimes called a "reactor runaway". From the above equation for Tp, derive an expression for the combination of parameters that results in a negative Tp and hence unstable reactor operation. Explain the physical significance of this expression in 1-2 sentences. Using the values of the parameters given below, show that the onset of reactor instability occurs when the flow rate F 0. However, when Tp is negative, an unstable response occurs - the reactor temperature increases exponentially with time, sometimes called a "reactor runaway". From the above equation for Tp, derive an expression for the combination of parameters that results in a negative Tp and hence unstable reactor operation. Explain the physical significance of this expression in 1-2 sentences. Using the values of the parameters given below, show that the onset of reactor instability occurs when the flow rate F