– 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.
INTRODUCTION
This research work is based on the characterization of a processing system, non-isothermal continuous stirred tank reactor (CSTR) and its behavior using a set of fundamental dependent quantities (mass, energy,
and momentum) whose values describe the natural state of the system, and the modeling of a set of equations in the dependent variables which describe how the natural state of the system changes with time.
The study is carried out by the use of mathematical models which are built based on the knowledge of the constitutive equations namely; transport rate equations, equations of state, chemical and phase equilibrium, kinetic rate equations, and dead-time.
These were used in characterizing the conservation balances on mass, energy, and momentum. This is done with particular interest on a non-isothermal CSTR reactor for simple irreversible, exothermic reactions in the same phase.
Every physical and chemical phenomenon applied, and the balance equations developed were all from the macroscopic viewpoint so as to moderate the size and complexity of the emerging models.
Since the fundamental variables cannot be measured conveniently and directly, other variables which can be measured conveniently, and when grouped appropriately determine the values of the fundamental variables, were selected.
Thus mass, energy, and momentum can be characterized by variables (state variables) such as density, concentration, temperature, pressure and flow rate which define the state of the system.
The equations that relate the state variables (dependent variables) to the various independent variables are derived from application of the conservation principle on the fundamental quantities and are called state equations.
REFERENCES
Abowei, M. F. N., and Ogoni, H. A. Development of Design Equations for CSTR Reverse Liquid-phase Reactions, International Journal of Modeling, Simulation, and Control, C Vol, 20, NO. 4, PP 31-43 , (1990).
Andrews, J. G., and Maclone, R. R., Mathematical Modeling, Butterworths, 1980.
Aris, R., The Optimal Design of Chemical Reactors, Academic Press, New York, 1961.
Aris, R., Elementary Chemical Reactors Analysis, Prentice-Hall Inc, Englewood Cliffs, N.J., 1969.
Aris, R., Mathematical Modeling Techniques, Pitman, London, 1978.
Arrhenius, S.Z., Physik Chem. 4, 226 (1889).
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