Advertisement

Nonisothermal Stirred-Tank Reactor with Irreversible Exothermic Reaction AB: 2. Nonlinear Phenomena

  • M. Pérez-Polo
  • P. Albertos
Part of the Lecture Notes in Control and Information Sciences book series (LNCIS, volume 361)

Abstract

In this chapter, the non-linear dynamics of a nonisothermal CSTR where a simple first order reaction takes place is considered. From the mathematical model of the reactor without any control system, and by using dimensionless variables, it has been corroborated that an external periodic disturbance, either inlet stream temperature and coolant flow rate, can lead to chaotic dynamics. The chaotic behavior is analyzed from the sensitivity to initial conditions, the Lyapunov exponents and the power spectrum of reactant concentration. Another interesting case is the one researched from the self-oscillating regime, showing that a periodic variation of coolant flow rate can also produce chaotic behavior. Finally, steady-state, self-oscillating and chaotic behavior with two PI controllers have been investigated. From different parameters of the PI controllers, it has been verified that a new self-oscillating regime can appear. In this case, the saturation in a control valve gives a Shilnikov type homoclinic orbit, which implies chaotic dynamics. The existence of a new set of strange attractors with PI control has been analyzed from the initial conditions sensitivity. An Appendix to show a computationally simple form to implement the calculation of Lyapunov exponents is also presented.

Keywords

Equilibrium Point Lyapunov Exponent Chaotic Dynamic Control Valve Homoclinic Orbit 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. [1]
    P. Albertos and M. Pérez Polo. Selected Topics in Dynamics and Control of Chemical and Biochemical Processes, chapter Nonisothermal stirred-tank reactor with irreversible exothermic reaction A → B. 1.Modelling and local control. LNCIS. Springer-Verlag, 2005 (in this volume).Google Scholar
  2. [2]
    J. Alvarez-Ramírez, R. Femat, and J. González-Trejo. Robust control of a class of uncertain first-order systems with least prior knowledge. Chem. Eng. Sci., 53(15):2701–2710, 1998.CrossRefGoogle Scholar
  3. [3]
    J. Alvarez-Ramirez, J. Suarez, and R. Femat. Robust stabilization of temperature in continuous-stirred tank reactors. Chem. Eng. Sci., 52(14):2223–2230, 1997.CrossRefGoogle Scholar
  4. [4]
    A.A. Andronov, A. Vitt, and S.E. Khaikin. Theory of Oscillations. Pergamon Press, Oxford, 1966.Google Scholar
  5. [5]
    G. Benettin, L. Galgani, and J-M. Strelcyn. Lyapunov characteristic exponents for smooth dynamical systems and for Hamiltonian systems: a method for computing all of them. Meccanica, 15:9–20, 1980.zbMATHCrossRefGoogle Scholar
  6. [6]
    G. Benettin, L. Galgani, and J-M. Strelcyn. Lyapunov characteristic exponents for smooth dynamical systems and for Hamiltonian systems: a method for computing all of them. Meccanica, 15:21–30, 1980.CrossRefGoogle Scholar
  7. [7]
    M.L. Cartwright and J.E. Littlewood. On nonlinear differential equations of the second order, I. The equation: ÿ + k(ly 2).y + y = bλcos(λt + a), k large. Lond. Math. Soc, 20:180–189, 1945.zbMATHCrossRefMathSciNetGoogle Scholar
  8. [8]
    M. Dolnik, A.S. Banks, and I.R. Epstein. Oscillatory chemical reaction in a CSTR with feedback control of flow rate. Journal Phys. Chem. A, 101:5148–5154, 1997.CrossRefGoogle Scholar
  9. [9]
    R. Femat. Chaos in a class of reacting systems induced by robust asymptotic feedback. Physica D, 136:193–204, 2000.zbMATHCrossRefMathSciNetGoogle Scholar
  10. [10]
    P. Glendinning. Stability, instability and chaos: an introduction to the theory of nonlinear differential equations. Cambridge University Press, 1994.Google Scholar
  11. [11]
    J. Guckenheimer and P. Holmes. Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields. Springer-Verlag, New York, 1983.zbMATHGoogle Scholar
  12. [12]
    E. Hopf. Abzweigung einer periodischen Lösung von einer stationären Lösung eines Differetiasystems. Ver Math.-Phys. Kl. Sachs, Acad Wiss, Leipzig 94:1–22, 1942.Google Scholar
  13. [13]
    Z. Kubickova, M. Kubicek, and M. Marek. Feed-batch operation of stirred reactors. Chem. Eng. Sci., 42(2):327–333, 1987.CrossRefGoogle Scholar
  14. [14]
    M.J. Kurtz, G. Yan Zhu, and M.A. Henson. Constrained output feedback control of a multivariable polymerization reactor. IEEE Trans. Contr. Syst. Technol., 8:87–97, 2000.CrossRefGoogle Scholar
  15. [15]
    A.J. Lichtenberg and M.A. Lieberman. Regular and Chaotic Dynamics. Springer-Verlag, New York, 2nd. edition, 1992.zbMATHGoogle Scholar
  16. [16]
    W.L. Luyben. Process Modeling, Simulation and Control for Chemical Engineering. McGraw-Hill, New York, 2nd. edition, 1990.Google Scholar
  17. [17]
    J.C. Mankin and J.L. Hudson. Oscillatory and chaotic behaviour of a forced exothermic chemical reaction. Chem. Eng. Sci., 39(12):1807–1814, 1984.CrossRefGoogle Scholar
  18. [18]
    J.E. Marsden and M. McCraken. The Hopf Bifurcation and its Applications. Springer, New York, 1976.zbMATHGoogle Scholar
  19. [19]
    E. Ott. Chaos in Dynamical Systems. Cambridge University Press, 2002.Google Scholar
  20. [20]
    L. Pellegrini and G. Biardi. Chaotic behaviour of a controlled CSTR. Comput. Chem. Eng., 14:1237–1247, 1990.CrossRefGoogle Scholar
  21. [21]
    M. Pérez and P. Albertos. Self-oscillating and chaotic behaviour of a PI-controlled CSTR with control valve saturation. J. Process Control, 14:51–57, 2004.CrossRefGoogle Scholar
  22. [22]
    M. Pérez, R. Font, and M.A. Montava. Regular self-oscillating and chaotic dynamics of a continuos stirred tank reactor. Comput. Chem. Eng., 26:889–901, 2002.CrossRefGoogle Scholar
  23. [23]
    J.B. Planeaux and K.F. Jensen. Bifurcation phenomena in CSTR dynamics: A system with extraneous thermal capacitance. Chem. Eng. Sci., 41(6): 1497–1523, 1986.CrossRefGoogle Scholar
  24. [24]
    R. Seydel. Practical Bifurcation and Stability Analysis From Equilibrium to Chaos. Springer-Verlag, New York, 2nd. edition, 1994.zbMATHGoogle Scholar
  25. [25]
    L.P. Shilnikov. A case of the existence of a demensurable set of periodic motions. Sov. Math. Dokl., 6:163–166, 1965.Google Scholar
  26. [26]
    L.P. Shilnikov, A.L. Shilnikov, P.V. Turaev, and O.L. Chua. Part II. World Scientific, chapter Methods of qualitative theory in nonlinear dynamics. Series in Nonlinear Science, 2001.Google Scholar
  27. [27]
    M. Soroush. Nonlinear state-observer design with applications to reactors. Chem. Eng. Sci., 52(3):387–404, 1987.CrossRefMathSciNetGoogle Scholar
  28. [28]
    G. Stephanopoulos. Chemical Process Control: An introduction to theory and practice. Prentice Hall, New Jersey, 1984.Google Scholar
  29. [29]
    F. Teymour. Dynamics of semibatch polymerization reactors: I. Theoretical analysis. A.I.Ch.E. Journal, 43(1):145–156, 1997.Google Scholar
  30. [30]
    F. Teymour and W.H. Ray. The dynamic behavior of continuous solution polymerization reactors-IV. Dynamic stability and bifurcation analysis of an experimental reactor. Chem. Eng. Sci., 44(9):1967–1982, 1989.CrossRefGoogle Scholar
  31. [31]
    K. Tomita. Periodically forced nonlinear oscillators. A.V. Holden, Princenton Univ. Press.Google Scholar
  32. [32]
    Y. Uppal, W.H. Ray, and A.B. Poore. On the dynamic behavior of continuous stirred tank reactors. Chem. Eng. Sci., 29:967–985, 1974.CrossRefGoogle Scholar
  33. [33]
    D.A. Vaganov, N.G.V. Samoilenko, and V.G. Abranov. Periodic regimes of continuous stirred tank reactors. Chem. Eng. Sci., 33:1133–1140, 1978.Google Scholar
  34. [34]
    S. Wiggins. Global Bifurcations and Chaos. Springer, New York, 1988.zbMATHGoogle Scholar
  35. [35]
    S. Wiggins. Introduction to Applied Nonlinear Dynamical System and Chaos. Springer, New York, 1990.Google Scholar
  36. [36]
    A.M. Zhabotinsky and A.B. Rovinskii. Self-Organization, Autowaves and Structures Far from Equilibrium. Springer, Berlin, 1984.Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2007

Authors and Affiliations

  • M. Pérez-Polo
    • 1
  • P. Albertos
    • 2
  1. 1.Department of PhysicsSystem Engineering and Signal Theory, EPSUSA
  2. 2.Department of Systems Engineering and ControlETSIIUSA

Personalised recommendations