2017-Sustainable Industrial Processing Summit
SIPS 2017 Volume 2. Dodds Intl. Symp. / Energy Production
| Editors: | Kongoli F, Buhl A, Turna T, Mauntz M, Williams W, Rubinstein J, Fuhr PL, Morales-Rodriguez M |
| Publisher: | Flogen Star OUTREACH |
| Publication Year: | 2017 |
| Pages: | 306 pages |
| ISBN: | 978-1-987820-63-8 |
| ISSN: | 2291-1227 (Metals and Materials Processing in a Clean Environment Series) |
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Thermodynamic Stability of Irreversible Processes Based on Lyapunov Function Analysis
Anil A. Bhalekar1; Vijay Tangde1;1RTM NAGPUR UNIVERSITY, Nagpur, India;
Type of Paper: Plenary
Id Paper: 145
Topic: 17
Abstract:
In the preceding paper [1] we have formulated a theory of thermodynamic stability which is an extension to irreversible processes of the Gibbs-Duhem theory of the stability of equilibrium states. This theory involves the concept of virtual displacement in the reverse direction on the real trajectory. In this paper the comprehensive thermodynamic theory of stability of irreversible processes (CTTSIP) has been presented that is based on the celebrated Lyapunov's second method of stability of motion in which we have defined the thermodynamic Lyapunov function using the rate of entropy production both on the perturbed and unperturbed trajectories. From the sign definiteness of the thermodynamic Lyapunov function and the behaviour of its time rate of change it gets established that all thermodynamically describable irreversible processes are thermodynamically stable and out of them the processes under the condition of constancy of U,V; H,p; T,V; T,p; etc. get established as of thermodynamically asymptotic stability and are expected to be of exponentially asymptotic stability too.
Keywords:
Energy; Materials; Sustainability;References:
[1] A. A. Bhalekar and B. Andresen, “Thermodynamic stability of irreversible processes. A Gibbs-Duhem type theory and the fourth law of thermodynamics,” 2017, SIPS 2017 Proceedings, (Accepted).[2] H. A. Bumstead and R. G. V. Name, eds., The Scientific Papers of J. Willard Gibbs, vol. I. Thermodynamics, 1906, Longmas, Green and Company, London and Bombay.
[3] F. G. Donnan and A. Haas, eds., A Commentary on the Scientific Writings of J. Willard Gibbs, vol. I, 1936, Yale University Press, London, New Haven.
[4] P. Needham, Commentary on the Principles of Thermodynamics by Pierre Duhem, 2011, Springer, Dordrecht, New York.
[5] P. Glansdorff and I. Prigogine, Thermodynamic Theory of Structure, Stability and Fluctuations, 1971, Wiley, New York.
[6] I. G. Malkin, “Theory of stability of motion,” in ACE-tr-3352 Physics and Mathematics, 1952, US Atomic Energy Commission, Washington, New York, Moscow, Leningrad.
[7] N. G. Chetayev, The Stability of Motion. 1961, Pergamon Press, Oxford, M. Nadler, Transl.
[8] I. Z. Shtokalo, Linear Differential Equations with Variable Coefficients. Criteria of Stability and Instability of Their Solutions, 1961, Hindustan Publishing, Delhi, India.
[9] I. G. Malkin, Stability and Dynamic Systems, 1962, American Mathematical Society, Providence, Rhode Island.
[10] W. Hahn, Theory and Applications of Lyapunov’s Direct Method. 1963, Prentice-Hall, Englewood Cliffs, New Jersey.
[11] J. P. LaSalle and S. Lefschetz, eds., Nonlinear Differential Equations and Nonlinear Mechanics. 1963, Academic Press, New York.
[12] L. Elsgolts, Differential Equations and the Calculus of Variations, 1970, Mir Publications, Moscow, G. Yankuvsky, Transl.
[13] H. H. E. Leipholz, Stability Theory: An Introduction to the Stability of Dynamic Systems and Rigid Bodies, 1987, B. G. Teubner/John Wiley, Stuttgart/Chichester.
[14] J. P. LaSalle and S. Lefschetz, Stability by Liapunov’s Direct Method with Applications, 1961, Academic Press, New York.
[15] D. A. Sànchez, Ordinary Differential Equations and Stability Theory. An Introduction, 1979, Dover, New York.
[16] I. Barkana, “Can stability analysis be really simplified? (revisiting Lyapunov, Barbalat, LaSalle and all that),” in AIP Conf. Proc., no. 1798, pp. 020017–1–18, July 2017, ICNPAA 2016 World Congress, La Rochelle University, France, July 5 - 10, 2016, American Institute of Physics Publishing, New York.
[17] A. A. Bhalekar, “On a comprehensive thermodynamic theory of stability of irreversible processes: A brief introduction,” Far East J. Appl. Math., 5 (2001), 199 – 210.
[18] A. A. Bhalekar, “Comprehensive thermodynamic theory of stability of irreversible processes (CTTSIP). I. The details of a new theory based on Lyapunov’s direct method of stability of motion and the second law of thermodynamics,” Far East J. Appl. Math., 5 (2001), 381 – 396.
[19] A. A. Bhalekar, “Comprehensive thermodynamic theory of stability of irreversible processes (CTTSIP). II. A study of thermodynamic stability of equilibrium and nonequilibrium stationary states,” Far East J. Appl. Math., 5 (2001), 397 – 416.
[20] A. A. Bhalekar, “The generalized set-up of comprehensive thermodynamic theory of stability of irreversible processes (CTTSIP) and a few illustrative applications,” J. Indian Chem. Soc., 81 (2004), 1119 – 1126.
[21] V. M. Tangde, S. G. Rawat, and A. A. Bhalekar, “Comprehensive thermodynamic theory of stability of irreversible processes (CTTSIP): The set-up for autonomous systems and application,” Int. J. Eng. Tech. Res., 3 (2015), 182 – 191.
[22] J. K. Hale and J. P. LaSalle, eds., Differential Equations and Dynamical Systems. September 1967, Academic Press, New York. Reference 9, pp. 291-297.
A. A. Bhalekar, “On the generalized phenomenological irreversible thermodynamic theory (GPITT),” J. Math. Chem., 5(2) (1990), 187 – 196.
[23] A. A. Bhalekar, “Measure of dissipation in the framework of generalized phenomenological irreversible thermodynamic theory (GPITT),” Proc. Int. Symp. on Efficiency, Costs, Optimization and Simulation of Energy Systems (ECOS’92), Zaragoza, Spain, June 15-18, 1992, A. Valero and G. Tsatsaronis, eds.; Empresa Nacional de Electricidad, Spain and Amer. Soc. Mech. Eng., pp. 121 – 128.
[24] A. A. Bhalekar, “Universal inaccessibility principle,” Pramana - J. Phys., 50(4) (1998), 281 – 294.
[25] A. A. Bhalekar, “On the generalized zeroth law of thermodynamics,” Indian J. Phys., 74B( 2) (2000) 153 – 157.
[26] A. A. Bhalekar, “On the time dependent entropy vis-à-vis Clausius’ inequality: Some of the aspects pertaining to the global and the local levels of thermodynamic considerations,” Asian J. Chem., 12(2) (2000), 417 – 427.
[27] A. A. Bhalekar, “On the irreversible thermodynamic framework for closed systems consisting of chemically reactive components,” Asian J. Chem., 12(2) (2000), 433 – 444.
[28] A. A. Bhalekar and B. Andresen, “On the nonequilibrium thermodynamic roots of the additional facets of chemical interaction,” in Recent Advances in Thermodynamic Research Including Nonequilibrium Thermodynamics, Proceedings of 3rd National Conference on Thermodynamics of Chemical and Biological Systems (NCTCBS-2008), G. S. Natarajan, A. A. Bhalekar, S. S. Dhondge, and H. D. Juneja, eds., (Department of Chemistry), 2008, pp. 53 – 62, R. T. M. Nagpur University, Nagpur, Funded by CSIR, New Delhi, October 16-17.
[29] A. A. Bhalekar, “The dictates and the range of applicability of the laws of thermodynamics for developing an irreversible thermodynamical framework,” Indian J. Phys., 76B (2002), 715 – 721.
[30] A. A. Bhalekar, “Irreversible thermodynamic framework using compatible equations from thermodynamics and fluid dynamics. A second route to generalized phenomenological irreversible thermodynamic theory (GPITT),” Bull. Cal. Math. Soc., 94(2) (2002), 209 – 224.
[31] A. A. Bhalekar, “The universe of operations of thermodynamics vis-à-vis Boltzmann integro-differential equation,” Indian J. Phys., 77B (2003), 391 – 397.
[32] A. A. Bhalekar, “Thermodynamic insight of irreversibility,” Proceedings of the Third International Conference on Lie-Admissible Treatment of Irreversible Processes (ICLATIP - 3), Dhulikhel, Kavre, Nepal, 2011, C. Corda, ed., pp. 135 – 162, R. M. Santilli Foundation, USA and Kathmandu University.
[33] A, A. Bhalekar and B. Andresen, “A comprehensive formulation of generalized phenomenological irreversible thermodynamic theory (GPITT),” (to appear).
[34] S. R. De Groot and P. Mazur, Non-Equilibrium Thermodynamics, 1962, North Holland, Amsterdam.
[35] D. Jou, J. Casas-Vázquez, and G. Lebon, Extended Irreversible Thermodynamics, , 1996, second ed., Springer-Verlag, Berlin.
[36] G. Lebon, D. Jou and J. Casas-Vázquez, Understanding of Non-equilibrium Thermodynamics. Foundations, Applications and Frontiers, 2008, Springer, Berlin.
[37] A, A. Bhalekar and B. Andresen, “Local thermodynamic equilibrium (LTE) revisited and the fifth law of thermodynamics,” (to appear).
[38] K. J. Laidler, Chemical Kinetics, 1967, Tata McGraw-Hill, New Delhi.
[39] S. G. Rawat, V. M. Tangde, C. S. Burande, and A. A. Bhalekar, “Thermodynamic stability of multiple steady state in enzymatic reaction using Lyapunov function analysis,” in Recent Advances in Thermodynamic Research Including Nonequilibrium Thermodynamics, Proceedings of 3rd National Conference on Thermodynamics of Chemical and Biological Systems (NCTCBS-2008) (G. S. Natarajan, A. A. Bhalekar, S. S. Dhondge, and H. D. Juneja, eds.), (Department of Chemistry), 2008, pp. 69 – 74, R. T. M. Nagpur University, Nagpur, Funded by CSIR, New Delhi, October 16-17.
[40] C. S. Burande and A. A. Bhalekar, “A study of thermodynamic stability of stress relaxation processes in visco-elastic fluids within the framework of comprehensive thermodynamic theory of stability of irreversible processes (CTTSIP),” Rom. J. Phys., 47 (7-8) (2002), 701 – 707.
[41] C. S. Burande and A. A. Bhalekar, “Thermodynamic stability of some elementary chemical reactions investigated within the framework of comprehensive thermodynamic theory of stability of irreversible processes (CTTSIP),” J. Indian Chem. Soc., 80 (5) (2003), 583 – 596.
[42] C. S. Burande and A. A. Bhalekar, “Thermodynamic stability of enzyme catalytic reactions by Lyapunov function analysis,” Int. J. Chem. Sci., 2 (4) (2004), 495 – 518.
[43] A. A. Bhalekar and C. S. Burande, “A study of thermodynamic stability of deformation in visco-elastic fluids by Lyapunov function analysis,” J. Non-Equilib. Thermodyn., 30 (2005), 53 – 65.
[44] C. S. Burande and A. A. Bhalekar, “Thermodynamic stability of elementary chemical reactions proceeding at finite rates revisited using Lyapunov function analysis,” Energy, 30 (2005), 897 – 913.
[45] V. M. Tangde, A. A. Bhalekar, and B. Venkataramani, “Thermodynamic stability of sulfur dioxide oxidation by Lyapunov function analysis against temperature perturbation,” Phys. Scr., 75 (2007), 460 – 466.
[46] V. M. Tangde, A. A. Bhalekar, and B. Venkataramani, “Lyapunov function analysis of the thermodynamic stability of ammonia synthesis,” Far East J. Appl. Math., 30 (2008), 297 – 313.
[47] V. M. Tangde, A. A. Bhalekar, and B. Venkataramani, “A study of thermodynamic stability by Lyapunov function analysis of some elementary chemical reactions against sufficiently small temperature perturbation,” Bull. Cal. Math. Soc., 100(1) (2008), 47 – 66.
[48] V. M. Tangde, S. G. Rawat, and A. A. Bhalekar, “A theoretical study of biological Lotka-Volterra Ecological model using CTTSIP,” Int. J. Knowledge Engg., 3 (1) (2012), 91 – 94.
[49] H. A. Antosiewicz, “A survey of Liapunov’s second method,”In “Contributions to the theory of Nonlinear Oscillations (S. Lefschetz, ed.), Vol. IV, Princeton Univ. Press, Princeton, New Jersey, 1958, pp. 141 - 166.