Thermodynamic Equilibria and Extrema: Analysis of Attainability Regions and Partial Equilibrium

293 Pages·2006·2.98 MB·English

by  Alexander N. Gorban, Boris M. Kaganovich, Sergey P. Filippov, Alexandre V. Keiko, Vitaly A. Shamansky, Igor A. Shirkalin

Thermodynamic Equilibria and Extrema
Alexander N. Gorban Boris M. Kaganovich
Sergey P. Filippov Alexandre V. Keiko
Vitaly A. Shamansky Igor A. Shirkalin
Thermodynamic Equilibria
and Extrema
Analysis of Attainability Regions and
Partial Equilibria
Translated by Marina V. Ozerova,
Valentina P. Yermakova, and Alexandre V. Keiko
Alexander N. Gorban Boris M. Kaganovich
Department of Mathematics Laboratory for Thermodynamics
Mathematical Modelling Centre Melentiev Energy Systems Institute
University of Leicester Irkutsk 664033
Leicester LE1 7RH Russia
UK
and Alexandre V. Keiko
Institute of Computational Modelling Vitaly A. Shamansky
Russian Academy of Sciences Igor A. Shirkalin
Krasnoyarsk 660036 Melentiev Energy Systems Institute
Russia Irkutsk 664033
Russia
Sergey P. Filippov
Energy Research Institute
Moscow
Russia
Library of Congress Control Number: 2006922411
ISBN-10: 0-387-28575-X
ISBN-13: 978-0387-28575-7
Printed on acid-free paper.
Translated from the Russian, by Alexander N. Gorban, Boris M. Kaganovich, and Sergey P. Filippov,
published by “Nauka” Publishers, Novosibirsk, Russia, ⃝C 2001.
⃝C 2006 Springer Science+Business Media, LLC
All rights reserved. This work may not be translated or copied in whole or in part without the written
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In memory of a remarkable personality,
physicist-chemist, and historian of science,
Lev Solomonovich Polak
Preface
The authors are very glad to see the publication of Thermodynamic Equilibria and
Extrema in English and would like to express their gratitude to everybody who
contributed to this end.
The book is devoted to the analysis of attainability regions and partial equilibria
in physicochemical and other systems. This analysis employs the extreme models
of classical equilibrium thermodynamics. Consideration is given to the problem of
choosing, from the set of equilibrium states belonging to the attainability regions,
that equilibrium corresponding to the extreme values of a property of interest to a
researcher. For example, one might desire to maximize the concentration of target
products of a chemical reaction. The problem of coordinating thermodynamics
and kinetics is very important in the analysis presented.
At a glance, it may seem that the objects of study in thermodynamics (the science
of equilibria) and kinetics (the science of motion toward equilibrium) coincide only
in the case of complete and final equilibrium. In reality, joint application of ther-
modynamics and kinetic models gives a clearer understanding of the regularities
of the kinetics involved.
Relativity of the notions of rest and motion was already firmly established
in mechanics when the principles of equilibrium were formulated by Galilei,
D’Alembert, and Lagrange. Historically, the theories of motion and equilibrium
states are related. It is precisely the study of gas kinetics that led Clausius and
Boltzmann to the main principles of thermodynamics. The systematic analysis of
these principles in the classic book by Gibbs, On the Equilibrium of Heterogeneous
Substances [54], demonstrated the feasibility of substituting the models of rest for
the models of motion when studying various physicochemical processes. The clas-
sics of thermodynamics, Gibbs [54], Planck [139], Einstein [43], and Sommerfeld
[158], showed that, in passing from descriptions of processes to descriptions of
equilibrium states, it is possible to use the notion of partial equilibrium (they
used different terminology) as well as complete equilibrium. L.D. Landau and
E.M. Lifshitz in [125] emphasized the importance of studying partial (incomplete)
equilibria in chemical systems where reactions often do not reach the end.
The regions of thermodynamic attainability and possible effects on the path of
physicochemical systems toward final equilibrium were thoroughly analyzed in the
vii
viii Preface
1980s by V.I. Bykov, A.N. Gorban, and G.S. Yablonsky [58, 59, 60]. The essence
of the problem was most clearly revealed in the book by A.N. Gorban, Equilibrium
Encircling (Equations of Chemical Kinetics and Their Thermodynamic Analysis)
[58]. This volume used models of closed system equilibria to describe all of the
following: macroscopic kinetics and thermodynamics; thermodynamic analysis of
chemical and biological system relaxation toward equilibrium; and nonstationary
and nonequilibrium processes, including those in open systems.
The problems arising in kinetics are interpreted on the basis of Lyapunov func-
tions, Markov random processes, topology, and graph theory. A geometrical tech-
nique was developed to pass from the search for the Lyapunov function extremum
on the material balance polyhedron to the search for extremum on the graph—a
thermodynamic tree.
Using the principles formulated in [58], B.M. Kaganovich, S.P. Filippov, and
E.G. Antsiferov [82, 83] constructed and studied thermodynamic models and com-
putational algorithms that would find, for a given function, points where extreme
values will occur in the attainability region. The most detailed discussion of these
models is given in the book Equilibrium Thermodynamics and Mathematical Pro-
gramming [181]. Unlike Equilibrium Encircling, in [81] consideration was given
not to the equations of motion but to possible states; that is, the conventional ther-
modynamic approach was applied. This approach was extended to the analysis of
a number of processes in the fields of themal energy, chemical technology, and
nature.
The current volume expounds the basic principles of both Equilibrium Encir-
cling and Equilibrium Thermodynamics, and synthesizes the ideas of these books.
Twenty years worth of work on the thermodynamic analysis of kinetics of macro-
scopic systems is summarized in this book, and areas for further study are outlined.
There are twice as many authors for this English edition as there were in the
Russian edition. The authors of the Russian edition were A.N. Gorban, B.M.
Kaganovich, and S.P. Filippov. The findings of the “new” authors were heavily
used in the Russian text of the present book. These authors contributed enormously
to the preparation of the English edition. In particular, they helped to eliminate
many inaccuracies in the original text.
The authors owe much to many discussions they held with a remarkable physi-
cist, chemist, and historian of natural science, L.S. Polak. The successful perfor-
mance of many of the studies in this book is due to these conversations. Professor
Polak immediately understood and approved the basic mathematical model of
extreme intermediate states (MEIS) applied by the authors, including versions of
this method intended for analysis of hydraulic and chemical circuits. The remarks
of L.S. Polak on the authors’ interpretation of the history of the development
equilibrium principles were extremely valuable.
The main MEIS versions were also discussed with L.I. Rosonoer, who assisted
the authors in constructing the model of systems with variable extents of reaction
completeness.
E.G. Antsiferov created the first algorithms for calculation of partial equilibria
that correspond to extreme concentrations of given substances [7]. Further
Preface ix
development of these algorithms was based on his idea of a two-stage search for the
extreme state of a thermodynamic system: stage one being initial calculation of the
optimal level of thermodynamic function, and stage two the further search for lo-
cation of the extreme point on the surface of this level. E.G. Antsiferov contributed
greatly to the analysis of mathematical features of the problems considered in this
book and, in particular, to the study of the convexity of thermodynamic functions.
A.P. Merenkov and S.V. Sumarokov helped greatly in the first work on thermo-
dynamic analysis of multi-loop hydraulic systems, substantiation of the extremality
criteria in hydraulic circuit theory, and creation of heterogeneous circuits theory.
The authors believe it is their duty to pay tribute to the memory of V.Ya. Khasilev,
the founder of hydraulic circuit theory, whose ideas were interpreted in terms of
thermodynamics.
The authors also acknowledge the support of the Russian Foundation for Basic
Research (project numbers 05-02-16626 and 05-08-01316).
Alexander N. Gorban
Boris M. Kaganovich
Sergey P. Filippov
Alexandre V. Keiko
Vitaly A. Shamansky
Igor A. Shirkalin
Contents
Preface ................................................................................... vii
Introduction ............................................................................ 1
I.1. Subject of Research ......................................................... 1
I.2. To the Use of Equilibrium Principle ..................................... 4
I.3. Modeling of Open and Closed Systems ................................. 5
I.4. Ideal and Nonideal Systems ............................................... 7
I.5. Modeling of Homogeneous and Heterogeneous Systems............ 8
I.6. Almost Almighty Thermodynamics ..................................... 11
I.7. Problem of Getting Maximum Knowledge from
Available Information....................................................... 14
I.8. Types of Descriptions: Stationary (Where Do We Stay?),
Dynamic (How Do We Run?), Geometrical
(Where Do We Run?)....................................................... 17
I.9. “The Field of Battle”: Balance Polyhedrons ........................... 18
I.10. Roughness and Reliability of Thermodynamics ....................... 19
I.11. Thermodynamically Admissible Paths .................................. 20
I.12. Thermodynamic Functions ................................................ 22
I.13. A Thermodynamic Tree and Space of Admissible Paths............. 24
I.14. From Admissibility to Feasibility ........................................ 25
I.15. Constraints Imposed by the Reaction Mechanism..................... 26
I.16. Constraints on Exchange................................................... 28
I.17. Constraints on Parameters ................................................. 29
I.18. Constraints on the Regions of Process Running ....................... 30
I.19. Stability and Sensitivity .................................................... 31
I.20. The Art of the Possible: Idealized Models of Real Systems......... 33
I.21. The Art of the Possible: Methods for Calculation of Estimates..... 35
I.22. Models of Extreme Concentrations ...................................... 37
I.23. Thermodynamics of Combustion......................................... 39
I.24. Thermodynamics of the Atmosphere .................................... 41
I.25. Thermodynamic Modeling on Graphs................................... 43
xi
xii Contents
1. Principles of Equilibrium and Extremality in Mechanics
and Thermodynamics ............................................................ 47
1.1. Principles of Equilibrium and Extremality in Mechanics ............. 47
1.2. Principles of Equilibrium and Extremality
in Thermodynamics .......................................................... 50
1.3. Thermodynamics and Models of Motion ................................ 56
1.4. Partial Thermodynamic Equilibria ........................................ 66
1.5. A Thermodynamic Analysis of the
Chemical Kinetics Equations............................................... 72
2. Extreme Thermodynamic Models in Terms of
Mathematical Programming ................................................... 102
2.1. Brief Information from Mathematical Programming .................. 102
2.2. The Model of Extreme Intermediate States (MEIS) ................... 109
2.3. Description of Different Types of Thermodynamic Systems......... 121
2.4. Mathematical Features of the Extreme
Thermodynamic Models .................................................... 132
2.5. Convex Analysis of the Thermodynamics Problems .................. 141
3. Thermodynamic Modeling on Graphs....................................... 152
3.1. Problem Statement and History............................................ 152
3.2. Thermodynamic Tree ........................................................ 155
3.3. Thermodynamic Interpretations of
Hydraulic Circuit Theory ................................................... 159
3.4. Thermodynamic Interpretations of Hydraulic Circuit
Theory: Heterogeneous Circuits ........................................... 171
4. Methods and Algorithms of Searching for
Thermodynamic Equilibria..................................................... 189
4.1. E.G. Antsiferov’s General Two-Stage Technique
of Searching for Extreme Concentrations................................ 189
4.2. Optimization of the Initial Composition of Reagents
in a Chemical System by the Simplex Embedding Method .......... 191
4.3. Calculations of Complete and Partial Equilibria
by the Affine Scaling Method .............................................. 194
4.4. Construction of Algorithms Using
the Thermodynamic Tree Idea ............................................. 200
4.5. Analysis of Feasibility and Stability of Partial Equilibria............. 208
5. Application of Extreme Models................................................ 213
5.1. Thermodynamics of Atmosphere.......................................... 213
5.2. Thermodynamics of Combustion.......................................... 224
5.3. Fuel Processing .............................................................. 244