A dual extremum principle in thermodynamics
Phase equilibria of multicomponent mixtures are considered and a reinterpretation of the Gibbs tangent plane stability criterion is proposed via Lagrangian duality. The starting point is the natural primal problem of minimizing the Gibbs free energy subject to material balance. The stable phase spli...
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| Published in | AIChE journal Vol. 53; no. 8; pp. 2131 - 2147 |
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| Main Authors | , |
| Format | Journal Article |
| Language | English |
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01.08.2007
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| Abstract | Phase equilibria of multicomponent mixtures are considered and a reinterpretation of the Gibbs tangent plane stability criterion is proposed via Lagrangian duality. The starting point is the natural primal problem of minimizing the Gibbs free energy subject to material balance. The stable phase split is the solution of the corresponding dual problem, providing a necessary and sufficient dual extremum principle. Only in the absence of duality gap is the physical phase split also the solution of the primal problem. The only requirements are continuity of the Gibbs free energy and the trivial requirement that each species is present in the overall composition. The number of phases is permitted to be infinite, and does not need to be known a priori. No assumption is made on the presence of all species in all phases. Case studies are presented based on the NRTL and UNIQUAC activity coefficient model. © 2007 American Institute of Chemical Engineers AIChE J, 2007 |
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| AbstractList | Phase equilibria of multicomponent mixtures are considered and a reinterpretation of the Gibbs tangent plane stability criterion is proposed via Lagrangian duality. The starting point is the natural primal problem of minimizing the Gibbs free energy subject to material balance. The stable phase split is the solution of the corresponding dual problem, providing a necessary and sufficient dual extremum principle. Only in the absence of duality gap is the physical phase split also the solution of the primal problem. The only requirements are continuity of the Gibbs free energy and the trivial requirement that each species is present in the overall composition. The number of phases is permitted to be infinite, and does not need to be known a priori. No assumption is made on the presence of all species in all phases. Case studies are presented based on the NRTL and UNIQUAC activity coefficient model. Phase equilibria of multicomponent mixtures are considered and a reinterpretation of the Gibbs tangent plane stability criterion is proposed via Lagrangian duality. The starting point is the natural primal problem of minimizing the Gibbs free energy subject to material balance. The stable phase split is the solution of the corresponding dual problem, providing a necessary and sufficient dual extremum principle. Only in the absence of duality gap is the physical phase split also the solution of the primal problem. The only requirements are continuity of the Gibbs free energy and the trivial requirement that each species is present in the overall composition. The number of phases is permitted to be infinite, and does not need to be known a priori. No assumption is made on the presence of all species in all phases. Case studies are presented based on the NRTL and UNIQUAC activity coefficient model. © 2007 American Institute of Chemical Engineers AIChE J, 2007 Phase equilibria of multicomponent mixtures are considered and a reinterpretation of the Gibbs tangent plane stability criterion is proposed via Lagrangian duality. The starting point is the natural primal problem of minimizing the Gibbs free energy subject to material balance. The stable phase split is the solution of the corresponding dual problem, providing a necessary and sufficient dual extremum principle. Only in the absence of duality gap is the physical phase split also the solution of the primal problem. The only requirements are continuity of the Gibbs free energy and the trivial requirement that each species is present in the overall composition. The number of phases is permitted to be infinite, and does not need to be known a priori. No assumption is made on the presence of all species in all phases. Case studies are presented based on the NRTL and UNIQUAC activity coefficient model. [PUBLICATION ABSTRACT] |
| Author | Mitsos, Alexander Barton, Paul I. |
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| Cites_doi | 10.1007/BF00251538 10.1021/ie00044a020 10.2118/9806-PA 10.1088/0305-4470/12/9/022 10.1002/aic.690410715 10.1007/BF00374979 10.1016/0098-1354(90)80020-C 10.1007/BF01096454 10.1016/0098-1354(90)80021-3 10.1002/aic.690390421 10.1016/0378-3812(82)85002-4 10.1007/BF02070248 10.1007/BF00939667 10.1016/0378-3812(82)85001-2 10.1007/BF00934096 10.1007/BF00281498 10.1002/nav.3800240307 10.1007/BF00280534 10.1007/BF01097059 |
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| Keywords | Gibbs free energy Lagrangian Multicomponent mixture Activity coefficient Gibbs tangent plane Stability criterion Thermodynamic properties Phase equilibrium Modeling duality Material balance extremum principle |
| Language | English |
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| Notes | ark:/67375/WNG-B22C0V90-Z ArticleID:AIC11230 Disclaimer: This report was prepared as an account of work sponsored by an agency of the United States Government. Neither the United States Government nor any agency thereof, nor any of their employees, makes any warranty, express or implied, or assumes any legal liability or responsibility for the accuracy, completeness, or usefulness of any information, apparatus, product, or process disclosed, or represents that its use would not infringe privately owned rights. Reference herein to any specific commercial product, process, or service by trade name, trademark, manufacturer, or otherwise does not necessarily constitute or imply its endorsement, recommendation, or favoring by the United States Government or any agency thereof. The views and opinions of authors expressed herein do not necessarily state or reflect those of the United States Government or any agency thereof. Department of Energy Nuclear Energy Research Initiative - No. DE-FC07-06ID14751 istex:806E6E7955AC70E10C3AE4EF386B97A184723634 Disclaimer This report was prepared as an account of work sponsored by an agency of the United States Government. Neither the United States Government nor any agency thereof, nor any of their employees, makes any warranty, express or implied, or assumes any legal liability or responsibility for the accuracy, completeness, or usefulness of any information, apparatus, product, or process disclosed, or represents that its use would not infringe privately owned rights. Reference herein to any specific commercial product, process, or service by trade name, trademark, manufacturer, or otherwise does not necessarily constitute or imply its endorsement, recommendation, or favoring by the United States Government or any agency thereof. The views and opinions of authors expressed herein do not necessarily state or reflect those of the United States Government or any agency thereof. SourceType-Scholarly Journals-1 ObjectType-Feature-1 content type line 14 ObjectType-Article-2 content type line 23 |
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| References | Mcdonald CM, Floudas CA. Decomposition based and branch-and-bound global optimization approaches for the phase-equilibrium problem. J Global Optim. 1994; 5: 205-251. McDonald C, Floudas CA. Global optimization and analysis for the Gibbs free-energy function using the UNIFAC, Wilson, and ASOG equations. Ind Eng Chem Res. 1995; 34: 1674-1687. Brooke A, Kendrick D, Meeraus A. GAMS: A User's Guide. Redwood City, CA: The Scientific Press, 1988. Smith JV, Missen RW, Smith WR. General optimality criteria for multiphase multireaction chemical-equilibrium. AIChE J. 1993; 39: 707-710. Bertsekas DP. Nonlinear Programming, 2nd ed. Belmont, MA: Athena Scientific, 1999. Tawarmalani M, Sahinidis NV. Convexification and Global Optimization in Continuous and Mixed-Integer Nonlinear Programming. Nonconvex Optimization and its Applications. Boston: Kluwer Academic, 2002. Visweswaran V, Floudas CA. A global optimization algorithm (GOP) for certain classes of nonconvex NLPs. 2. Application of theory and test problems. Comput Chem Eng. 1990; 14: 1419-1434. Falk JE, Hoffman K. A nonconvex max-min problem. Naval Res Logistics. 1977; 24: 441-450. Feinberg M. On Gibbs' phase rule. Arch Rational Mech Anal. 1979; 70: 219-234. Sieniutycz S, Farkas H, editors. Variational and Extremum Principles in Macroscopic Systems. Oxford: Elsevier, 2005. Michelsen ML. The isothermal flash problem. 1. Stability. Fluid Phase Equilibria. 1982; 9: 1-19. Baker LE, Pierce AC, Luks KD. Gibbs energy analysis of phase equilibria. Soc Petrol Eng J. 1982; 22: 731-742. Mcdonald CM, Floudas CA. Global optimization for the phase-stability problem. AIChE J. 1995; 41: 1798-1814. Fosdick R, Patiño J. On the Gibbsian thermostatics of mixtures. Arch Rational Mech Anal. 1986; 93: 203-221. Bertsimas D, Tsitsiklis JN. Introduction to Linear Optimization. Belmont, MA: Athena Scientific, 1997. Tester JW, Modell M. Thermodynamics and Its Applications, 3rd ed. New Jersey: Prentice Hall, 1997. Prentice Hall International Series in the Physical and Chemical Engineering Sciences. Floudas CA, Visweswaran V. A Global optimization algorithm (GOP) for certain classes of nonconvex NLPs. 1. Theory. Comput Chem Eng. 1990; 14: 1397-1417. Mistura I. Solutions of the Gibbs-Duhem equation. J Phys A Math Gen. 1979; 12: 1547-1562. Maranas CD, Floudas CA. Finding all solutions of nonlinearly constrained systems of equations. J Global Optim. 1995; 7: 143-182. Floudas CA, Visweswaran V. Primal-relaxed dual global optimization approach. J Optim Theory Appl. 1993; 78: 187-225. Noll W. On certain convex sets of measures and on phases of reacting mixtures. Arch Rational Mech Anal. 1970; 38: 1-12. Michelsen ML. The isothermal flash problem. 2. Phase-split calculation. Fluid Phase Equilibria. 1982; 9: 21-40. Gutierrez G. Gibbs' phase rule revisited. Theor Math Phys. 1996; 108: 1222-1224. Bank B, Guddat J, Klatte D, Kummer B, Tammer K. Non-Linear Parametric Optimization. Stuttgart: Birkhäuser Verlag, 1983. Blankenship JW, Falk JE. Infinitely constrained optimization problems. J Optim Theory Appl. 1976; 19: 261-281. Rabier PJ, Griewank A. Generic aspects of convexification with applications to thermodynamic equilibrium. Arch Rational Mech Anal. 1992; 118: 349-397. 1995; 41 1993; 78 1979; 12 1986; 93 1993; 39 1990; 14 1995; 34 1982; 22 1982; 9 1997 1992; 118 2005 1983 1977; 24 2002 1970; 38 1996; 108 1994; 5 1979; 70 1999 1976; 19 1995; 7 1988 e_1_2_7_2_2 e_1_2_7_9_2 e_1_2_7_8_2 e_1_2_7_7_2 e_1_2_7_6_2 e_1_2_7_19_2 e_1_2_7_18_2 Bertsimas D (e_1_2_7_4_2) 1997 e_1_2_7_16_2 e_1_2_7_15_2 e_1_2_7_14_2 e_1_2_7_13_2 e_1_2_7_12_2 e_1_2_7_11_2 e_1_2_7_10_2 Tawarmalani M (e_1_2_7_17_2) 2002 e_1_2_7_26_2 e_1_2_7_27_2 Bank B (e_1_2_7_28_2) 1983 Brooke A (e_1_2_7_23_2) 1988 Bertsekas DP (e_1_2_7_3_2) 1999 e_1_2_7_24_2 e_1_2_7_22_2 e_1_2_7_21_2 e_1_2_7_20_2 Tester JW (e_1_2_7_5_2) 1997 Sieniutycz S (e_1_2_7_25_2) 2005 |
| References_xml | – reference: Brooke A, Kendrick D, Meeraus A. GAMS: A User's Guide. Redwood City, CA: The Scientific Press, 1988. – reference: Blankenship JW, Falk JE. Infinitely constrained optimization problems. J Optim Theory Appl. 1976; 19: 261-281. – reference: Gutierrez G. Gibbs' phase rule revisited. Theor Math Phys. 1996; 108: 1222-1224. – reference: Baker LE, Pierce AC, Luks KD. Gibbs energy analysis of phase equilibria. Soc Petrol Eng J. 1982; 22: 731-742. – reference: Visweswaran V, Floudas CA. A global optimization algorithm (GOP) for certain classes of nonconvex NLPs. 2. Application of theory and test problems. Comput Chem Eng. 1990; 14: 1419-1434. – reference: Tester JW, Modell M. Thermodynamics and Its Applications, 3rd ed. New Jersey: Prentice Hall, 1997. Prentice Hall International Series in the Physical and Chemical Engineering Sciences. – reference: Fosdick R, Patiño J. On the Gibbsian thermostatics of mixtures. Arch Rational Mech Anal. 1986; 93: 203-221. – reference: Mistura I. Solutions of the Gibbs-Duhem equation. J Phys A Math Gen. 1979; 12: 1547-1562. – reference: Feinberg M. On Gibbs' phase rule. Arch Rational Mech Anal. 1979; 70: 219-234. – reference: Floudas CA, Visweswaran V. Primal-relaxed dual global optimization approach. J Optim Theory Appl. 1993; 78: 187-225. – reference: Sieniutycz S, Farkas H, editors. Variational and Extremum Principles in Macroscopic Systems. Oxford: Elsevier, 2005. – reference: Tawarmalani M, Sahinidis NV. Convexification and Global Optimization in Continuous and Mixed-Integer Nonlinear Programming. Nonconvex Optimization and its Applications. Boston: Kluwer Academic, 2002. – reference: Noll W. On certain convex sets of measures and on phases of reacting mixtures. Arch Rational Mech Anal. 1970; 38: 1-12. – reference: Michelsen ML. The isothermal flash problem. 2. Phase-split calculation. Fluid Phase Equilibria. 1982; 9: 21-40. – reference: Smith JV, Missen RW, Smith WR. General optimality criteria for multiphase multireaction chemical-equilibrium. AIChE J. 1993; 39: 707-710. – reference: Rabier PJ, Griewank A. Generic aspects of convexification with applications to thermodynamic equilibrium. Arch Rational Mech Anal. 1992; 118: 349-397. – reference: Mcdonald CM, Floudas CA. Decomposition based and branch-and-bound global optimization approaches for the phase-equilibrium problem. J Global Optim. 1994; 5: 205-251. – reference: Bank B, Guddat J, Klatte D, Kummer B, Tammer K. Non-Linear Parametric Optimization. Stuttgart: Birkhäuser Verlag, 1983. – reference: McDonald C, Floudas CA. Global optimization and analysis for the Gibbs free-energy function using the UNIFAC, Wilson, and ASOG equations. Ind Eng Chem Res. 1995; 34: 1674-1687. – reference: Falk JE, Hoffman K. A nonconvex max-min problem. Naval Res Logistics. 1977; 24: 441-450. – reference: Bertsekas DP. Nonlinear Programming, 2nd ed. Belmont, MA: Athena Scientific, 1999. – reference: Michelsen ML. The isothermal flash problem. 1. Stability. Fluid Phase Equilibria. 1982; 9: 1-19. – reference: Maranas CD, Floudas CA. Finding all solutions of nonlinearly constrained systems of equations. J Global Optim. 1995; 7: 143-182. – reference: Bertsimas D, Tsitsiklis JN. Introduction to Linear Optimization. Belmont, MA: Athena Scientific, 1997. – reference: Mcdonald CM, Floudas CA. Global optimization for the phase-stability problem. AIChE J. 1995; 41: 1798-1814. – reference: Floudas CA, Visweswaran V. A Global optimization algorithm (GOP) for certain classes of nonconvex NLPs. 1. Theory. Comput Chem Eng. 1990; 14: 1397-1417. – year: 1983 – volume: 14 start-page: 1397 year: 1990 end-page: 1417 article-title: A Global optimization algorithm (GOP) for certain classes of nonconvex NLPs. 1. Theory publication-title: Comput Chem Eng – volume: 34 start-page: 1674 year: 1995 end-page: 1687 article-title: Global optimization and analysis for the Gibbs free‐energy function using the UNIFAC, Wilson, and ASOG equations publication-title: Ind Eng Chem Res – year: 2005 – volume: 41 start-page: 1798 year: 1995 end-page: 1814 article-title: Global optimization for the phase‐stability problem publication-title: AIChE J – volume: 38 start-page: 1 year: 1970 end-page: 12 article-title: On certain convex sets of measures and on phases of reacting mixtures publication-title: Arch Rational Mech Anal – volume: 14 start-page: 1419 year: 1990 end-page: 1434 article-title: A global optimization algorithm (GOP) for certain classes of nonconvex NLPs. 2. Application of theory and test problems publication-title: Comput Chem Eng – volume: 70 start-page: 219 year: 1979 end-page: 234 article-title: On Gibbs' phase rule publication-title: Arch Rational Mech Anal – volume: 93 start-page: 203 year: 1986 end-page: 221 article-title: On the Gibbsian thermostatics of mixtures publication-title: Arch Rational Mech Anal – volume: 39 start-page: 707 year: 1993 end-page: 710 article-title: General optimality criteria for multiphase multireaction chemical‐equilibrium publication-title: AIChE J – volume: 19 start-page: 261 year: 1976 end-page: 281 article-title: Infinitely constrained optimization problems publication-title: J Optim Theory Appl – volume: 9 start-page: 1 year: 1982 end-page: 19 article-title: The isothermal flash problem. 1. Stability publication-title: Fluid Phase Equilibria – volume: 5 start-page: 205 year: 1994 end-page: 251 article-title: Decomposition based and branch‐and‐bound global optimization approaches for the phase‐equilibrium problem publication-title: J Global Optim – volume: 118 start-page: 349 year: 1992 end-page: 397 article-title: Generic aspects of convexification with applications to thermodynamic equilibrium publication-title: Arch Rational Mech Anal – year: 2002 – year: 1988 – volume: 12 start-page: 1547 year: 1979 end-page: 1562 article-title: Solutions of the Gibbs‐Duhem equation publication-title: J Phys A Math Gen – year: 1997 – volume: 9 start-page: 21 year: 1982 end-page: 40 article-title: The isothermal flash problem. 2. Phase‐split calculation publication-title: Fluid Phase Equilibria – volume: 22 start-page: 731 year: 1982 end-page: 742 article-title: Gibbs energy analysis of phase equilibria publication-title: Soc Petrol Eng J – volume: 108 start-page: 1222 year: 1996 end-page: 1224 article-title: Gibbs' phase rule revisited publication-title: Theor Math Phys – volume: 24 start-page: 441 year: 1977 end-page: 450 article-title: A nonconvex max‐min problem publication-title: Naval Res Logistics – volume: 7 start-page: 143 year: 1995 end-page: 182 article-title: Finding all solutions of nonlinearly constrained systems of equations publication-title: J Global Optim – volume: 78 start-page: 187 year: 1993 end-page: 225 article-title: Primal‐relaxed dual global optimization approach publication-title: J Optim Theory Appl – year: 1999 – ident: e_1_2_7_7_2 doi: 10.1007/BF00251538 – ident: e_1_2_7_19_2 doi: 10.1021/ie00044a020 – ident: e_1_2_7_2_2 doi: 10.2118/9806-PA – ident: e_1_2_7_8_2 doi: 10.1088/0305-4470/12/9/022 – ident: e_1_2_7_13_2 doi: 10.1002/aic.690410715 – ident: e_1_2_7_9_2 doi: 10.1007/BF00374979 – ident: e_1_2_7_20_2 doi: 10.1016/0098-1354(90)80020-C – volume-title: Variational and Extremum Principles in Macroscopic Systems year: 2005 ident: e_1_2_7_25_2 – volume-title: Convexification and Global Optimization in Continuous and Mixed‐Integer Nonlinear Programming. 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| SubjectTerms | Applied sciences Chemical engineering duality Energy Exact sciences and technology extremum principle Gibbs tangent plane Lagrangian Material balance phase equilibrium Thermodynamics |
| Title | A dual extremum principle in thermodynamics |
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