Mathematical modeling and control of a nonisothermal continuous stirred tank reactor, cstr

Project Abstract

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

Project Overview

<p> INTRODUCTION<br>1.0 MATHEMATICAL MODELING AND CONTROL<br>This research work is based on the characterization of a processing system, non-isothermal<br>continuous stirred tank reactor (CSTR) and its behaviour using a set of fundamental<br>dependent quantities (mass, energy, and momentum) whose values describe the natural state<br>of the system, and the modeling of a set of equations in the dependent variables which<br>describe how the natural state of the system changes with time.<br>The study is carried out by the use of mathematical models which are built based on the<br>knowledge of the constitutive equations namely; transport rate equations, equations of state,<br>chemical and phase equilibrium, kinetic rate equations, and dead-time. These were used in<br>characterizing the conservation balances on mass, energy, and momentum. This is done with<br>particular interest on a non-isothermal CSTR reactor for simple irreversible, exothermic<br>reactions in the same phase. Every physical and chemical phenomenon applied, and the<br>balance equations developed were all from the macroscopic viewpoint so as to moderate the<br>size and complexity of the emerging models.<br>Since the fundamental variables cannot be measured conveniently and directly, other<br>variables which can be measured conveniently, and when grouped appropriately determine<br>the values of the fundamental variables, were selected. Thus mass, energy, and momentum<br>can be characterized by variables (state variables) such as density, concentration,<br>temperature, pressure and flow rate which define the state of the system. The equations that<br>relate the state variables (dependent variables) to the various independent variables are<br>derived from application of the conservation principle on the fundamental quantities and are<br>called state equations.<br>The study was considered for a single, irreversible, unimolecular first order reaction,<br>bimolecular second order reaction, and the empirical nth order reaction occurring<br>exothermically in the same phase. The developed models were solved simultaneously by<br>numerical and analytical approach employed for solving non-linear differential equations.<br>The dynamic responses of the system were analyzed and the steady state and the dynamic<br>feed forward controllers design equations were derived. Such other information that may be<br>very necessary for the proper understanding of our mathematical models was also treated.<br>2<br>1.1 AIMS/ OBJECTIVES OF THIS STUDY<br>This work is aimed at the formulation of a mathematical representation of the physical<br>and chemical phenomena (temperature, density, pressure, concentration, flow rate, etc).<br>taking place in a non isothermal continuous stirred tank reactor, CSTR which handles a<br>simple, irreversible first or second order exothermic reaction in same phase. The<br>mathematical model is to describe the variations in the volume of the system, concentrations<br>of the reactants, temperature of the system and the temperature of the coolant over time.<br>Other objectives that this model is called on to satisfy or perform are to ensure the<br>stability in the operation of the chemical reactor, and to suppress the influence of external<br>disturbances on the reactor. The purpose at this stage is to translate all the important<br>phenomena occurring in the physical and chemical processes into quantitative mathematical<br>equations. The models give the understanding of what really make the process “tick”, enable<br>one get to the core of the system to see clearly the cause-and-effect relationships between the<br>variables. The work also gives a physical application of the solution of the models obtained.<br>1.2 SIGNIFICANCE OF THE STUDY<br>Mathematical models can be useful in all area of life, and as in chemical engineering,<br>it is useful in all phases of chemical engineering, from research and development to plant<br>operations, and even in business and economic studies. Most often the physical equipment of<br>chemical process we want to design and control has not been constructed. Consequently, we<br>cannot experiment to determine how the process reacts to various inputs and therefore we<br>cannot deign them and their appropriate control system. But even if the process equipment is<br>available for experimentation, the procedure is usually very costly.<br>Therefore, we need a simple description of how the process reacts of various inputs,<br>and this is what the mathematical models can provide to the process and control engineer.<br>Uses Of This Mathematical Model Are As Follows.<br>(1) Research and development<br>(2) Design of chemical processing equipment and their control.<br>(3) Plant operation and optimization is cheaper, safer, and faster done on a mathematical<br>model then experimentally on an operating unit.<br>3<br>1.3 SCOPE OF THE STUDY<br>The investigation reported in this project bothers on the formulation of a mathematical<br>representation of the physical and chemical phenomena occurring in the state system;<br>non-isothermal CSTR from a microscopic view point. In the work, the principles of<br>conservation of fundamental quantities (mass, energy, and momentum) were applied<br>using already well-developed constitutive models. We did not go into driving a reaction<br>mechanism, kinetic rate equations, transport rate equations, equations of stage, chemical<br>and phase equilibrium equations but the well developed models from a number of<br>postulated mechanisms were used to correlate the constitutive fundamental quantities.<br>The verified mathematical models for the temperature and concentration-dependent<br>terms of the rate equation for first order, second order, and nth order reactions were used<br>and not developed in this investigation Levenspiel (1972). The models were derived for a<br>unimolecular first order, bimolecular second order and nth order reactions respectively.<br>The emerged models were simultaneously solved by analytical and numerical approach<br>employed for solving nonlinear differential equations, and the comparison of the solutions<br>and the subsequent analysis were properly carried out. We considered the dynamic<br>responses of the system, developed its transfer functions (inputs-output interaction), and<br>subsequently the steady state and dynamic feed forward controllers design equations.<br>The research is particular to a non isothermal continuous stirred tank reactor, CSTR<br>that handles a simple, irreversible exothermic reaction in liquid phase to enable us predict<br>the rate mechanisms, rate equations and reaction occurring in it. The result obtained may<br>be extrapolated to cover a simple, irreversible endothermic reaction in same phase to a<br>good accuracy.<br>1.4 LIMITATIONS OF THE STUDY<br>There are series of difficulties that were encountered in an effort to develop a<br>meaningful and realistic mathematical description of this chemical process – a nonisothermal<br>continuous stirred tank reactor, CSTR. Serious difficulties occurred due to<br>incomplete knowledge of the physical and chemical phenomena taking place in the<br>reactor. Even an acceptable degree of knowledge is at times very difficult. How to<br>account for the effect of alteration of the value of the overall heat transfer co-efficient<br>caused by scaling, fouling etc during the operation of the reactor became a limitation.<br>Also we have considered only the first, second, and empirical nth order kinetics to<br>describe the reaction rate.<br>4<br>Again, imprecisely known parameters become an impediment. The dead time is a<br>critical parameter whose value is usually imprecisely known, varying and can lead to<br>serious stability problems. The availability of accurate value for the parameters of a<br>model is indispensable for any quantitative analysis of the process behaviour.<br>Unfortunately they are not always possible. Parameters such as densities r , heat of<br>reaction (- DHr), pre-exponential constant Ko, activation energy E, and overall heat<br>transfer coefficient U, of the jacketed reactor do not remain constant over long periods of<br>time but are in general functions of concentrations CA, CB, and CP, and the temperate To.<br>Hence, for effective modeling we need not only accurate values but also some<br>quantitative description of how the parametric values changes with time. How to decide<br>that this dependence is weak (as to use constant values) or strong (in which case the<br>modeling becomes very complicated) imposes a limitation. Determination of the values of<br>these parameters is difficult.<br>Also, the size and complexity of the model induces some problems. An effort to<br>develop as accurate and precise a model as possible, its size and complexity increase<br>significantly and exceed manageable levels, beyond which the model loses its value and<br>became less attractive.<br>1.5 DEFINITION OF TERMS<br>Fi and Fo = volumetric flow rates of the system’s inlet and outlet streams.<br>r i and r o=densities of the materials in the inlet and outlet streams.<br>Fci and Fco = volumetric flow rates of the coolant in the inlet and outlet streams.<br>r = Density of the material in the system.<br>nA and nB=number of moles of component A and B in the system.<br>np= number of moles of component P in the system.<br>CAi, CBi and CPi =molar concentration (moles/ volume) of A, B, and P in the<br>Inlet streams respectively.<br>CAo, CBo, and CPo = molar concentration (moles/ volume) of A, B, and P in the outlet<br>streams respectively<br>rA, rB, and rP = reaction rate per unit volume of components<br>A, B and P in the system.<br>hi and ho = specific enthalpy (enthalpy per unit mass) of the feed and<br>Outlet streams.<br>UE,KE,PE= internal, kinetic and potential energies of the system, <br></p>