Simulation Of Robust Stabilization Of A Chemical Reactor

The paper presents simulation experiments on the continuous stirred tank reactor for hydrolysis of propylene oxide to propylene glycol. The reactor is exothermic one. There are two parameters with only approximately known values in the reactor. These parameters are the reaction rate constant and the reaction enthalpy. The simplified mathematical model of the reactor consists of four nonlinear ordinary differential equations. The steady state analysis shows the reactor has multiple steady states and the open-loop analysis confirms that the reactor is open-loop unstable around one of these steady states. Then the possibility to stabilize the reactor using static output feedback PI and PID controllers is studied. Because of the presence of uncertainty in the continuous stirred tank reactor, the robust static output feedback is designed. Simulations are used for testing the stabilizability of the reactor around its open-loop unstable steady state. INTRODUCTION Continuous stirred tank reactors (CSTRs) are often used plants in chemical industry and especially exothermic CSTRs are very interesting systems from the control viewpoint because of their potential safety problems and the possibility of exotic behaviour such as multiple steady states, see e.g. (Molnár et al., 2002), (Pedersen and Jørgensen, 1999). Furthermore, operation of chemical reactors is corrupted by many different uncertainties. Some of them arise from varying or not exactly known parameters, as e.g. reaction rate constants, reaction enthalpies and heat transfer coefficients (Antonelli and Astolfi, 2003). The other control problems are due to the high sensitivity of the state and output variables to input changes and process nonlinearites (Alvarez-Ramirez and Femat, 1999). Operating points of reactors change in other cases. In addition, the dynamic characteristics may exhibit a varying sign of the gain in various operating points. All these problems can cause poor performance or even instability of closed-loop control systems. Conventional control strategies, which are often used for reactor control design, can fail for such complicated systems and their effective control requires application some of advanced methods, as e. g. adaptive control (Vojtesek and Dostal, 2008), predictive control (Figueroa et al., 2007), robust control (Alvarez-Ramirez and Femat, 1999), (Gerhard et al., 2004), (Bakošová et al., 2005), (Tlacuahuac et al., 2005) and others. Robust control has grown as one of the most important areas in modern control design since works by (Doyle and Stein, 1981), (Zames and Francis, 1983) and many others. One of the solved problems is also the problem of robust static output feedback control (RSOFC), which has been till now an important open question in control engineering, see e.g. (Syrmos et al., 1997), (Antonelli and Astolfi, 2003). Recently, it has been shown that an extremely wide array of robust controller design problems can be reduced to the linear matrix inequalities (LMIs) problem. Especially, the LMIs in semi-definite programming attract a big interest because of their ability to describe non-trivial control design problems integrating various specifications such as robustness, structural and performance constraints, as well as their suitability for efficient numerical processing through various available solvers, see e.g. (Boyd et al., 1994) and references therein. From the system theory viewpoint, CSTRs belong to a class of nonlinear lumped parameter systems. Their mathematical models are described by sets of nonlinear ordinary differential equations (ODEs). The methods of modelling and simulation of such processes are described e. g. in (Ingham et al., 1994). The models are often used for a preliminary analysis of the steady-state, open-loop and closed-loop behaviour of chemical reactors. The paper presents simulation experiments with the CSTR for hydrolysis of propylene oxide to propylene glycol. The reactor is exothermic one. There are two parameters with only approximately known values in the reactor. The steady state analysis shows that the reactor has multiple steady states and the open-loop analysis confirms that the reactor is open-loop unstable around one of these steady states. Then the possibility to stabilize the reactor using robust static output feedback PI and PID controllers is studied by simulations. Proceedings 23rd European Conference on Modelling and Simulation ©ECMS Javier Otamendi, Andrzej Bargiela, José Luis Montes, Luis Miguel Doncel Pedrera (Editors) ISBN: 978-0-9553018-8-9 / ISBN: 978-0-9553018-9-6 (CD) MODEL OF THE CSTR Hydrolysis of propylene oxide to propylene glycol in a continuous stirred tank reactor (Molnár et al., 2002) was chosen as a controlled process. The reaction is as follows C3H6O+H2O −→ C3H8O2 (1) The reaction is of the first order with respect to propylene oxide as a key component. The dependence of the reaction rate constant on the temperature is described by the Arrhenius equation k = k∞e − Ea RTr (2) where k∞ is the pre-exponential factor, Ea is the activation energy, R is the universal gas constant and Tr is the temperature of the reaction mixture. Assuming ideal mixing in the reactor, the constant reaction volume and the same volumetric flow rates of the inlet and outlet streams, the mass balance for any component j in the system is Vr dcj dt = qr (cj0 − cj) + νjrVr , (3) where Vr is the reaction volume, c is the molar concentration, q is the volumetric flow rate, ν is the stoichiometric coefficient, r is the molar rate of the chemical reaction and subscripts denote j the component, r the reation mixture, 0 the feed. It is assumed further that the specific heat capacities, densities and volumetric flow rates do not depend on temperature and composition, and also the heat of mixing and the mixing volume can be neglected. The simplified enthalpy balance of the reaction mixture used as a standard at reactor design (Ingham et al., 1994) is VrρrCPr dTr dt = qrρrCPr (Tr0 − Tr) (4) −UA (Tr − Tc) + rVr(−∆rH) and the simplified enthalpy balance of the cooling medium is VcρcCPc dTc dt = qcρcCPc (Tc0 − Tc) + UA (Tr − Tc) (5) where T is the temperature, ρ is the density, CP is the specific heat capacity, (−∆rH) is the reaction enthalpy, U is the overall heat transfer coefficient and A is the heat exchange area. The subscripts denote 0 the feed, c the cooling medium and r the reaction mixture. The values of constant parameters and steady-state inputs of the CSTR are summarized in Table 1. Model uncertainties of the over described reactor follow from the fact that there are two physical parameters in this reactor, the reaction enthalpy and the preexponential factor, which values are known within following intervals (Table 2). The nominal values of these parameters are mean values of the intervals and they are Table 1: Constant parameters and steady-state inputs of the CSTR Variable Value Unit Vr 2.407 m 3 Vc 2 m 3 ρr 947.19 kg m −3 ρc 998 kg m −3 CPr 3.7187 kJ kg K CPc 4.182 kJ kg K AU 120 kJ minK Ea/R 10183 K qr 0.072 m min qc 0.6307 m min cC3H6O,0 0.0824 kmol m −3 cC3H8O2,0 0 kmol m −3 Tr0 299.05 K Tc0 288.15 K used for deriving of the nominal model of the CSTR. The minimal and maximal values of the intervals are used for obtaining models, which create the vertex systems (7). Table 2: Uncertain parameters of the CSTR Parameter (−∆rH) k∞ Unit kJ kmol min Minimal Value −5.28× 10 2.4067× 10 Maximal Value −5.64× 10 3.2467× 10 STEADY-STATE AND OPEN-LOOP ANALYSIS The steady-state model of the CSTR in the form of a set of nonlinear algebraic equations (AEs) is obtained from the dynamicmodel (3) – (5) equating the derivative terms to zero. MATLAB function fsolve can be used for solving of the set of nonlinear AEs. The steady state behaviour of the chemical reactor with nominal values and also with 4 combinations of minimal and maximal values of 2 uncertain parameters was studied at first. It can be stated the reactor has always three steady states, two of them are stable and one is unstable. The situation is shown in Figure 1, where the curve QGEN (the curve is marked with * for the nominal model) represents the heat generated by the reaction and the line QOUT is the heat withdrawn from the reactor. The steady states of the reactor are points, where the curves and the line intersect. The steady states are stable if the slope of the cooling line is higher than the slope of the heat generated curve. This condition is satisfied for the nominal model at Tr = 296.7 K and Tr = 377.5 K, and is not satisfied at Tr = 343.1 K. From the viewpoint of safety operation or in the case when the unstable steady-state coincides with the point that yields the maximum reaction rate at a prescribed temperature, it can be necessary to control a CSTR about its open-loop unstable steady-state, see e. g. (Pedersen and Jørgensen, 1999), (Antonelli and Astolfi, 2003), 280 300 320 340 360 380 400 −1 0 1 2 3 4 x 10 4 T r [K] Q G E N ,Q O U T [ k J /m in ] Figure 1: Multiple steady states of the CSTR (González and Alvarez, 2006). In this context, the open-loop behaviour of the reactor in the surroundings of its unstable steady state at Tr = 343.1Kwas studied at first. The initial temperature of the reaction mixture was chosen Tr(0) = 346.95K. Simulation results obtained for the nominal model (curve with *) and 4 vertex systems are shown in Figures 2. They confirm that without feedback control, the temperature of the reaction mixture in the CSTR converges to the values characteristic either for the upper or the lower stable steady states. 0 20 40 60 80 100 280 300 320 340 360 380 400