6.0 context and direction6.1 exothermic chemical reaction in a stirred tank reactor6.2 dynamic model of the reactor 6.3 we encounter nonlinear equations6.4 making linear approximations with Taylor series6.5 linear approximation to the material balance6.6 similarly approximating the energy balance6.7 deriving transfer functions by Laplace transform and block diagram6.8 multiple steady-state operating conditions6.9 response of system to step disturbance6.10 do the linearized models describe instability?6.11 step 1 - specify a control objective for the process6.12 step 2 - assign variables in the dynamic system6.13 step 3 - PID (proportional-integral-derivative) control 6.14 step 4 - choose set points and limits6.15 adjusting heat transfer with a valve6.16 control valve mechanism6.17 control valve failure mode6.18 positive closed loop gain and the sense of the controller 6.19 ambiguity!6.20 closed loop transfer function6.21 closed-loop behavior - laplace transform solution6.22 Bode criterion for closed loop stability6.23 tuning the controller6.24 conclusion6.25 referencesSpring 2006 Process Dynamics, Operations, and Control 10.450 Lesson 6: Exothermic Tank Reactor 6.0 context and direction A tank reactor with an exothermic reaction requires a more elaborate system model, because its outputs are temperature and composition. Furthermore, the model is nonlinear, which forces us to make a linear approximation to solve it. We will add the derivative mode to our PI controller to increase both stability and responsiveness. The closed loop will show how automatic control can stabilize an inherently unstable process. DYNAMIC SYSTEM BEHAVIOR 6.1 exothermic chemical reaction in a stirred tank reactor A second-order dimerization reaction occurs in an overflow stirred tank reactor. The reactor is equipped with a heat transfer surface (perhaps jacket, coils, or bayonet) that contains a flow of cooling water. We wish to know how the outlet composition and temperature may vary with time. 6.2 dynamic model of the reactor With two output variables, we face two balances, as well as several supporting relationships. The mole balance on the reactant A (AAAiArVFCFCdtdCV −−−=) (6.2-1) requires a second-order kinetic rate expression for the rate of disappearance of A, including Arrhenius temperature dependence. 2ARTEo2AAACekkCdtdNV1r−==⎟⎠⎞⎜⎝⎛−=− (6.2-2) The energy balance must account for the reaction and heat transfer. ()QrVH)TT(CF)TT(CFdtdTCVARrefprefipp−−Δ−−ρ−−ρ=ρ (6.2-3) Once again, we will regard physical properties as independent of temperature. Enthalpies are defined with respect to an arbitrary thermodynamic reference temperature. For an exothermic reaction, the heat of reaction ΔHR will be a negative quantity, and will thus tend to raise the reactor temperature T. The rate of heat transfer Q depends on the logarithmic temperature difference cocicociooTTTTln)TT()TT(AUQ−−−−−= (6.2-4) revised 2005 Mar 30 1Spring 2006 Process Dynamics, Operations, and Control 10.450 Lesson 6: Exothermic Tank Reactor in which the well-mixed tank temperature T is uniform and the coolant temperature Tc varies from inlet to outlet. We will presume that the coolant supply temperature Tci is quite stable and thus not consider it as a disturbance. The overall heat transfer coefficient depends on the film coefficients on the inner and outer surfaces of the heat transfer barrier; we will neglect any conduction resistance in the barrier itself. 1iiooohAAh1U−⎟⎟⎠⎞⎜⎜⎝⎛+= (6.2-5) The outer film coefficient ho depends on the rate of stirring in the tank, as well as the variation of physical properties with temperature. With constant physical properties, there is no reason for ho to vary. Inner coefficient hi depends on the flow of coolant. Invoking typical internal-flow behavior, we write mnciiPrRekDh= (6.2-6) If we write (6.2-6) at a reference condition, we can express the flow dependence of hi as ncrciriFFhh⎟⎟⎠⎞⎜⎜⎝⎛= (6.2-7) For flow in tubes, n is often about 0.8. The main structure of the model is given by the balances (6.2-1) and (6.2-3). These relate the outlet temperature and composition to their inlet values. Supplementary equations are needed to describe the reaction kinetics and heat transfer. We see that the two balances will be coupled through the temperature dependence of the reaction rate parameter k in (6.2-2). Heat transfer is described by equipment performance equation (6.2-4) and the empirical relationship (6.2-7) that describes convective heat transfer in conduits. These latter equations show how the coolant flow Fc influences the reactor outlet temperature T. Even so, we are not finished, because we have not yet accounted for the outlet coolant temperature Tco in (6.2-4). Therefore, we must write an energy balance on the coolant. Q)TT(CF)TT(CFdtTdCVrefcopcccrefcipccccpccc+−ρ−−ρ=ρ (6.2-8) revised 2005 Mar 30 2Spring 2006 Process Dynamics, Operations, and Control 10.450 Lesson 6: Exothermic Tank Reactor where ‹T c› is the average coolant temperature in the coolant volume. ∫=cVcccdVTV1T (6.2-9) To proceed with (6.2-8), we must express the average temperature in terms of the inlet and outlet temperatures. This would be an entertaining diversion, but for the primary purposes of Lesson 6 we will assume that heat exchanger outlet temperature adjusts much more quickly than does the tank temperature T, so that we can neglect the accumulation term in (6.2-8) and write it as Q)TT(CFcicopccc=−ρ (6.2-10) Justifying this assumption would involve comparing the characteristic times of (6.2-3) and (6.2-8). We have implicitly made a similar assumption already in (6.2-4), in which we have said that the rate of heat transfer depends on the instantaneous values of the inlet and outlet temperatures according to a relationship that was derived for the steady state. Taken together, the equations of this section describe how the outlet temperature and composition vary in time due to disturbances in inlet temperature, inlet composition, and coolant flow rate. The coolant outlet temperature is an intermediate variable in the system. 6.3 we encounter nonlinear equations The equations in Section 6.2 are nonlinear: the outlet composition is squared, the temperature is an argument in exponential and logarithmic functions, the coolant flow is raised to a power.