Simulation Analyses of Continuous Stirred Tank Reactor

This paper presents simulation experiments on the continuous stirred tank reactor (CSTR) which is widely used equipment mainly in the chemical industry. The behaviour of these types of systems is usually nonlinear with other negative properties such as a time-delay or a non-minimum phase behaviour and simulation could help us with the understanding of them without making real experiments which could be dangerous, time or cost demanding. The simple iteration method and the RungeKutta’s method were used for solving of a steady-state and dynamics of the system. Used adaptive control is based on the recursive identification of an External Linear Model (ELM) as a representation of the originally nonlinear system. The polynomial approach together with the LQ approach gives sufficient control results although the system has negative control properties. INTRODUCTION The simulation of the system on a computer enroll big boom nowadays when speed and availability of the computer technologies grows rapidly. On the contrary, the purchasing prise and running costs are relatively low. The computer simulation is usually connected with the mathematical model as a result of modeling procedure (Ingham et al. 2000). Material and heat balances are one way how to describe the system and relations between unknown quantities in the mathematical way. These balances are then represented by ordinary or partial differential equations depending on the type of systems. The Continuous Stirred Tank Reactor (CSTR) is typical equipment used in the industry for its good properties from control point of view. The CSTR belongs to the class of lumped parameters systems, a mathematical model of which is described by the set of ordinary differential equations (ODE). The simulation analysis of the system usually consists of steady-state and dynamic analyses (Ingham et al. 2000, Luyben 1989). The simple iteration method and Runge-Kutta’s method (Lyuben 1989) were used in the work for numerical solving of the steady-state and dynamic analyses. These methods are well known, simple and Runge-Kutta’s method is fully implemented in the used mathematical software Matlab. Results from simulation experiments are then used for choosing of the control strategy and designing of the controller. The nonlinearity and negative control properties of the system should be overcome with the use of Adaptive control (Åström 1989). Adaptive approach used in this work is based the choice of an External Linear Model (ELM) parameters of which are recomputed recursively during the control (Bobal et al. 2005). The external delta models (Middleton and Goodwin 2004) were used for parameter estimation. Although delta models belong to the range of discrete models, parameters of these models approaches to their continuous-time counterparts up to some assumptions (Stericker and Sinha 1993). Ordinary recursive least squares method (Fikar and Mikles 1999) was used for parameter estimation during the control. A polynomial approach with one degree-of-freedom (1DOF) configuration used for the controller synthesis has satisfied basic control requirements and connected with the LQ control technique, it fulfills the requirements of stability, asymptotic tracking of the reference signal and compensation of disturbances (Kucera 1993). MODEL OF THE PLANT As it is written above, the chemical process under consideration is the Continuous Stirred Tank Reactor (CSTR). The schematical representation of the CSTR is in Figure 1. We supposed that reactant is perfectly mixed and react to the final product with the concentration cA(t). The heat produced by the reaction is represented by the temperature of the reactant T(t). Furthermore we expect that volume, heat capacities and densities are constant during the control due to simplification. A mathematical model of this system is derived from the material and heat balances of the reactant and cooling. The resulted model is then set of two Ordinary Differential Equations (ODEs) (Gao et al. 2002):