# Mathematical Modeling and Control of a Nonisothermal Continuous Stirred Tank Reactor, Cstr

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Abstract

Mathematical Models describing the variations in the volume of the system, concentration of reactant (s) yet to react, temperature of the system, and the temperature of the cooling jacket over time in a non-isothermal CSTR that handles a simple, irreversible, first order or second order exothermic reaction in liquid phase were formulated. This work is with a particular reference to the synthesis of propylene from cyclopropane and that of cumene (isopropyl benzene) from benzene and propylene. The models were solved simultaneously by analytical approach rather than the normal numerical approach employed for solving non-linear differential equations. We noticed that the major determinants of the reactants conversion level and the extent of reaction are the feed concentrations, feed temperature and the cooling jacket inlet temperature. The system is found to have a single, locally stable, steady state with periodic (underdamped) behaviors due to the existence of both inherent negative and positive feedback in it. Nonlinear feedforward control equations show that feed flowrate does not have to be changed when feed temperature Ti changes, rather its changes inversely with feed concentration CAi. Again, the cooling -jacket temperature Tc changes linearly with feed temperature Ti and nonlinearly (inversely) with feed concentration CAi. The models were utilized to explore the dynamic response and the controllers design equations of the system. We noticed from the dynamic response that the system is self regulatory. Also feed forward controller is physically realizable and has two (Gc1 and Gc3) lead elements and one gain-only element (Gc2) controller for the control of concentration CAO and CBO, and a lag element (GC1), gain-only element (GC2) and a lead-lag element (GC3) controller for the control of temperature, To.

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