A thermodynamic Lyapunov Approach to the Stability Analysis of a Nonlinear Irreversible Process Having Multiplicity
Keywords:
dynamical systems, multiplicity, Lyapunov function, stability, control, thermodynamics
Abstract
Following the second law of thermodynamics, the entropy is always created in irreversible processes such as reacting systems, etc. Under certain operating conditions, the reaction system can be operated with multiple steady states (also called the steady state multiplicity behavior). This behavior is considered for the illustration of the stability analysis of all possible steady states by Lyapunov methods using thermodynamics. More precisely, a novel symmetric storage function (or Lyapunov function candidate) is proposed on the basis of the so-called (non-symmetric) thermodynamic availability function. The acid-catalyzed hydration of 2-3-epoxy-1-propanol to glycerol subject to steady state multiplicity is used for further technical developments. The results are discussed with the inclusions of the simulations.
References
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31. Nguyen, C. T., Dang, Q. D., Mai , T. P, and Hoang, N. H. (2016b). Stability analysis of 2,3-epoxy-1-propanol to glycerol having multiplicity behavior: A thermodynamic Lyapunov approach. Proc. The 23rd Regional Symposium on Chemical Engineering (RSCE). Vung Tau, Vietnam. pp. 482–487.
32. Ramírez, H., Maschke, B., and Sbarbaro, D. (2013). Irreversible port-Hamiltonian systems: A general formulation of irreversible processes with application to the CSTR, Chem. Eng. Sci., 89:223– 234.
33. Rodrigues, D., Srinivasan, S., Billeter, J., and Bonvin, D. (2015). Variant and invariant states for chemical reaction systems, Computers and Chemical Engineering, 73:23–33.
34. Rehmus, P., Zimmermann, E. C., and Ross, J. (1983). The periodically forces conversion of 2-3-epoxy-1-propanol to glycerine: A theoretical analysis, J. Chem. Phys, 78:7241–7251.
35. Ruszkowski, M., Garcia-Osorio, V., and Ydstie, B. E. (2005). Passivity based control of transport reaction systems, AIChE Journal, 51: 3147–3166.
36. Sandler, S.I. (1999). Chemical and engineering thermodynamics, 3rd edition, Wiley and Sons.
37. Tarbell, J. M. (1977). A thermodynamic Lyapunov function for the near equilibrium CSTR, Chem. Eng. Sci., 32:1471–1476.
38. Viel, F., Jadot, F., and Bastin, G. (1997). Global stabilization of exothermic chemical reactors under input constraints, Automatica, 33(8):1437– 1448.
39. Vleeschhouwer, P. H. M., Vermeulen D. P., and Fortuin J. M. H. (1988). Transient behaviour of a chemically reacting system in a CSTR, AIChE Journal, 34:1736–1739.
40. Vleeschhouwer, P. H. M., and Fortuin, J. M. H. (1990). Theory and experiments concerning the stability of a reacting system in a CSTR, AIChE Journal, 36:961–965.
41. Ydstie, B. E., and Alonso, A. A. (1997). Process systems and passivity via the Clausius-Planck inequality, Syst. & Contr. Let., 30(5):253–264.
2. Alvarez, J., Alvarez-Ramírez, J., Espinosa-Perez, G., and Schaum, A. (2011). Energy shaping plus damping injection control for a class of chemical reactors, Chem. Eng. Sci., 66(23):6280– 6286.
3. Alvarez-Ramírez, J., and Morales, A. (2000). PI control of continuously stirred tank reactors: Stability and performance, Chem. Eng. Sci., 55(22):5497–5507.
4. Antonelli, R., and Astolfi, A. (2003). Continuous stirred tank reactors: Easy to stabilise?, Automatica, 39:1817– 1827.
5. Bayer, F., Bürger, M., Guay, M., and Allgöwer, F. (2011). On state- constrained control of a CSTR. Proc. The 18th IFAC World Congress, Milano, Italia. pp. 6079–6084. [Conference paper]
6. Bruns, D. D., and Bailey, J. E. (1975). Process operation near an unstable steady state using nonlinear feedback control, Chem. Eng. Sci., 30, 755–762.
7. Callen, H. B. (1985). Thermodynamics and an introduction to thermostatics, 2 nd edition, John Wiley & Sons, New York.
8. Dammers, W. R., and Tels, M. (1974). Thermodynamic stability and entropy production in adiabatic stirred flow reactors, Chem. Eng. Sci., 29(1):83–90.
9. De Groot, S. R., and Mazur, P. (1962). Non-equilibrium thermodynamics, 1st edition, Dover Pub. Inc., Amsterdam.
10. Eberard, D., Maschke, B., and Van Der Schaft, A. (2007). An extension of pseudo-Hamiltonian systems to the thermodynamic space: Towards a geometry of non-equilibrium thermodynamics, Reports on Mathematical Physics, 60(2):175–198.
11. Ederer, M., Gilles, E. D., and Sawodny, O. (2011). The Glansdorff-Prigogine stability criterion for biochemical reaction networks, Automatica, 47:1097–1104.
12. Farschman, C. A., Viswanath, K. P., and Ydstie B. E. (1998). Process systems and inventory control, AIChE Journal, 44(8):1841–1857.
13. Favache, A., and Dochain, D. (2010). Power-shaping of reaction systems: The CSTR case study, Automatica, 46(11):1877–1883.
14. García-Sandoval, J. P., González- Álvarez, V., and Calderón, C. (2015). Stability analysis and passivity properties for a class of chemical reactors: Internal entropy production approach, Computers and Chemical Engineering, 75 :184–195.
15. García-Sandoval, J. P., Hudon, N., Dochain, D., and González-Álvarez, V. (2016). Stability analysis and passivity properties of a class of thermodynamic processes: An internal entropy production approach, Chem. Eng. Sci., 139: 261–272.
16. Georgakis, C. (1986). On the use of extensive variables in process dynamics and control, Chem. Eng. Sci., 41(6):1471–1484.
17. Hangos, K. M., Bokor, J., and Szederkényi, G. (2001). Hamiltonian view on process systems, AIChE Journal, 47(8):1819–1831.
18. Heemskerk, A. H., Dammers, W. R., and Fortuin, J. M. H. (1980). Limit cycles measured in a liquid-phase reaction system, Chem. Eng. Sci., 32:439–445.
19. Hoang, H., Couenne, F., Jallut, C., and Le Gorrec, Y. (2009). Thermodynamic approach for Lyapunov based control. Proc. The 7th IFAC-ADCHEM, Turkey, pp. 367–372.
20. Hoang, H., Couenne, F., Jallut, C., and Le Gorrec, Y. (2011). The Port Hamiltonian approach to modeling and control of Continuous Stirred Tank Reactors, J. Proc. Control, 21(10):1449– 1458.
21. Hoang, H., Couenne, F., Jallut, C., and Le Gorrec, Y. (2012). Lyapunov-based control of non isothermal continuous stirred tank reactors using irreversible thermodynamics, J. Proc. Control, 22(2):412–422.
22. Hoang, H., and Dochain, D. (2013a). On an evolution criterion of homogeneous multi-component mixtures with chemical transformation, Syst. & Contr. Let., 62(2):170–177.
23. Hoang, H., and Dochain, D. (2013b). A thermodynamic approach to the passive boundary control of tubular reactors. Proc. The 9th IFAC-NOLCOS, Toulouse, France. pp. 383–388.
24. Hoang, H., Couenne, F., Jallut, C., and Le Gorrec, Y. (2013a). Thermodynamics-based stability analysis and its use for nonlinear stabilization of CSTR, Computers & Chemical Engineering, 58(11):156–177.
25. Hoang, H., Du, J., and Ydstie, B. E. (2013b). On the passivity of inventory control in the Port Hamiltonian framework. Proc. The American Control Conference, Washington, DC, USA. pp.1642–1647.
26. Hudon, N., Höffner, K., and Guay, M. (2008). Equivalence to dissipative Hamiltonian realization. Proc. The 47th IEEE-CDC, Cancun, Mexico. pp. 3163– 3168.
27. Keenan, J. H. (1951). Avaibility and irreversibility in thermodynamics, British Journal of Applied Physics, 2:183–192.
28. Khalil, H. K. (2002). Nonlinear systems, 3 rd edition, Prentice Hall, Upper Saddle River, NJ.
29. Luyben, W. L. (1990). Process modeling, simulation and control for chemical engineers, 2nd edition, McGraw-Hill, Singapore.
30. Nguyen, C. T., Nguyen, Q. L., and Hoang, N. H. (2016a). A revisit on dissipation and its relation to irreversible processes. Proc. The 1st International Workshop on the Development of Renewable Energy for the Mekong Delta (DREMD-1). Can Tho, Vietnam. pp. 128–138.
31. Nguyen, C. T., Dang, Q. D., Mai , T. P, and Hoang, N. H. (2016b). Stability analysis of 2,3-epoxy-1-propanol to glycerol having multiplicity behavior: A thermodynamic Lyapunov approach. Proc. The 23rd Regional Symposium on Chemical Engineering (RSCE). Vung Tau, Vietnam. pp. 482–487.
32. Ramírez, H., Maschke, B., and Sbarbaro, D. (2013). Irreversible port-Hamiltonian systems: A general formulation of irreversible processes with application to the CSTR, Chem. Eng. Sci., 89:223– 234.
33. Rodrigues, D., Srinivasan, S., Billeter, J., and Bonvin, D. (2015). Variant and invariant states for chemical reaction systems, Computers and Chemical Engineering, 73:23–33.
34. Rehmus, P., Zimmermann, E. C., and Ross, J. (1983). The periodically forces conversion of 2-3-epoxy-1-propanol to glycerine: A theoretical analysis, J. Chem. Phys, 78:7241–7251.
35. Ruszkowski, M., Garcia-Osorio, V., and Ydstie, B. E. (2005). Passivity based control of transport reaction systems, AIChE Journal, 51: 3147–3166.
36. Sandler, S.I. (1999). Chemical and engineering thermodynamics, 3rd edition, Wiley and Sons.
37. Tarbell, J. M. (1977). A thermodynamic Lyapunov function for the near equilibrium CSTR, Chem. Eng. Sci., 32:1471–1476.
38. Viel, F., Jadot, F., and Bastin, G. (1997). Global stabilization of exothermic chemical reactors under input constraints, Automatica, 33(8):1437– 1448.
39. Vleeschhouwer, P. H. M., Vermeulen D. P., and Fortuin J. M. H. (1988). Transient behaviour of a chemically reacting system in a CSTR, AIChE Journal, 34:1736–1739.
40. Vleeschhouwer, P. H. M., and Fortuin, J. M. H. (1990). Theory and experiments concerning the stability of a reacting system in a CSTR, AIChE Journal, 36:961–965.
41. Ydstie, B. E., and Alonso, A. A. (1997). Process systems and passivity via the Clausius-Planck inequality, Syst. & Contr. Let., 30(5):253–264.
Published
2017-06-30
How to Cite
Nguyen, T. C., & Hoang, H. N. (2017). A thermodynamic Lyapunov Approach to the Stability Analysis of a Nonlinear Irreversible Process Having Multiplicity. ASEAN Journal of Chemical Engineering, 17(1), 8-21. Retrieved from https://dev.journal.ugm.ac.id/v3/AJChE/article/view/8920
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