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Unformatted text preview: Spring 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 ( A A Ai A r V FC FC dt dC V = ) (6.2-1) requires a second-order kinetic rate expression for the rate of disappearance of A, including Arrhenius temperature dependence. 2 A RT E o 2 A A A C e k kC dt dN V 1 r = = = (6.2-2) The energy balance must account for the reaction and heat transfer. ( ) Q r V H ) T T ( C F ) T T ( C F dt dT C V A R ref p ref i p p = (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 H R 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 co ci co ci o o T T T T ln ) T T ( ) T T ( A U Q = (6.2-4) revised 2005 Mar 30 1 Spring 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 T c varies from inlet to outlet. We will presume that the coolant supply temperature T ci 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. 1 i i o o o h A A h 1 U + = (6.2-5) The outer film coefficient h o 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 h o to vary. Inner coefficient h i depends on the flow of coolant. Invoking typical internal- flow behavior, we write m n c i i Pr Re k D h = (6.2-6) If we write (6.2-6) at a reference condition, we can express the flow If we write (6....
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- Spring '03