com THERMODYNAMICS OF IRREVERSIBLE PROCESSES www.com Thertnodynatnics of Irreversible Processes BERNARD H. LAVENDA Universita di Napoli Istituto di Fisica Sperimenta/e della Facolta di Scienze M www. Lavenda 1978 Softcover reprint of the hardcover 1st edition 1978 All rights reserved. No part of this publication may be reproduced or transmitted, in any form or by any means, without permission First published 1978 by THE MACMILLAN PRESS L TO London and Basingstoke Associated companies in Delhi Dublin Hong Kong Johannesburg Lagos Melbourne New York Singapore and Tokyo British Library Cataloguing in Publication Data Lavenda, Bernard H Thermodynamics of irreversible processes.7 QC311 ISBN 978-1-349-03256-3 ISBN 978-1-349-03254-9 (eBook) DOI 10.1007/978-1-349-03254-9 This book is sold subject to the standard conditions of the Net Book Agreement.com To Marlene www.com Contents Preface IX Introduction XI Formulations of non-equilibrium thermodynamics xi Quasi-thermodynamic approach to nonlinear thermodynamics xii Equilibrium Thermodynamics 1 1.1 Definitions and conditions of equilibrium 3 1.2 The first law 4 1.3 The second law 5 1.3 Geometry of the Gibbs space 14 1.4 Equilibrium extremum principles 19 1.5 Principle of maximum work 21 2 Classical Non-equilibrium Thermodynamics 25 2.1 The predecessors of Onsager 25 2.2 The statistical basis of Onsager's reciprocal relations 29 2.3 Criticisms of classical non-equilibrium thermodynamics 34 3 'Rational' Thermodynamics 41 3.1 Axioms of rational thermodynamics 42 3.1 Axiom of admissibility 42 3.2 Axiom of determinism 44 3.3 Axiom of equipresence 44 3.4 Axiom of material frame-indifference or objectivity 44 3.2 The formalism of rational thermodynamics 45 3.3 The interpretations of 'dissipation' 49 3.4 Limitations of rational thermodynamics 55 4 'Generalised' Thermodynamics 60 4.1 The development of generalised thermodynamics 62 4.2 Thermodynamic versus kinetic stability criteria 69 www.com Vlll CONTENTS 5 Nonlinear Thermodynamics 76 5.1 The thermodynamic principle of the balance of power 79 5.2 The balance equation of mechanical power in the entropy representation 82 5.3 Properties and forms of the balance equation of mechanical power 88 6 Non-equilibrium Variational Principles 93 6.1 The dynamic Le Chatelier principle 94 6.2 The principle of least dissipation of energy 97 6.3 Gauss's principle in non-equilibrium thermodynamics 101 6.4 Variational principles of nonlinear thermodynamics 103 6.5 Kinetic formulation of thermodynamic variational principles 105 7 Quasi-Thermodynamic Stability Theory 111 7.1 Elements of kinetic stability theory 112 7.2 The complex power method 115 7.3 The significance of the antisymmetric components of the phenomenological coefficient matrices 119 7.4 Stability of non-equilibrium stationary states 120 7.5 The mechanical bases of phenomenological symmetries and antisymmetries 123 7.6 Stability of nonlinear irreversible processes 127 8 Field Thermodynamics 132 8.1 The velocity potential analysis of multistationary state transitions 134 8.2 Thermokinematics of rotational non-equilibrium processes 136 8.3 Thermodynamics of force fields 140 8.4 Elements of the field theory 141 8.5 Thermodynamic principles of the field 144 8.6 Description of fields by thermodynamic variational principles 147 9 Continuum Thermodynamics 150 9.1 The balance equations of the continua 151 9.2 The generalised power equation 153 9.3 Variational equations of the continua 157 9.4 Thermodynamic variational principles of the continua 161 9.5 An illustration of the internal state variable representation 163 9.6 Thermodynamic evolutionary criteria 167 9.7 Thermodynamics of nonlinear dissipative wavetrains 168 Glossary of Principal Symbols 175 Index 179 www.com Preface In view of the large amount of recent work on the thermodynamics of non- equilibrium processes, there is a real need for a book giving a clear exposition of the thermodynamic formalism applicable to nonlinear thermodynamic pro- cesses at a mathematical level, accessible to physicists and to theoretically inclined biologists and chemists.
This book attempts to fill this need by making a definite statement in regard to the present-day status oflinear and nonlinear thermodynamics. For the past two decades, two schools of non-equilibrium thermodynamics have dominated the literature: the school of 'rational' thermodynamics of Coleman and co-workers and the school of'generalised' thermodynamics that is associated with the names of Glansdorff and Prigogine. Although both these schools praise themselves for their all-embracing coverage of the field, I have never seen 'generalised' thermodynamics applied to elastic materials with memory, nor have I seen 'rational' thermodynamics used in the analysis of chemical instabilities. The apparent incompatibility of the theories may bewilder the reader who wants to understand what non-equilibrium thermodynamics is all about.
Moreover, the vastly different usage of concepts and notations has not helped matters. A case in point is the ambiguity in the meanings of'dissipation' and 'irreversibility'. These terms are often regarded as synonymous, and the precise meanings of each left in doubt. Confusion also arises over the meaning of the term 'nonlinear' in thermodynamics.
While everyone knows what a nonlinear differential equation looks like, its usage in thermodynamics is far from being self-evident. In this book I attempt to resolve such types of ambiguity and misconception. After the introductory chapter on equilibrium thermodynamics, which serves to form a common background and as a reference for all future developments, the book is divided into two parts. The first part (chapters 1-4) is a critical analysis of the classical theory of non-equilibrium thermodynamics and its more recent offshoots.
The second part (chapters 5-9) deals with my own interpretation of what a theory of non-equilibrium thermodynamics should include. In the same way that equilibrium thermodynamics offers criteria for its validity so, too, non-equilibrium thermodynamics must provide for similar www.com X PREFACE criteria. My point of view is that these criteria must ultimately come from nonlinear mechanics and kinetic stability theory. This is to say that equilibrium thermodynamics itself does not provide a broad enough basis that will incorporate all types of kinetic process.
This book is formalistic rather than applicative in character. My feeling is that the presentation of a self-consistent and clear-cut thermodynamic formulation will automatically lead to its application. I hope that the interdisciplinary character of the book will allow the reader to draw upon the many interesting analogies that exist among the seemingly diverse branches of macroscopic physics, chemistry and theoretical biology. I would like to express my deep gratitude to Gabriel Stein of the Hebrew University, Jerusalem, whose encouragement and advice had a great deal to do with making this book a reality.
A very special role has been played by my wife Fanny, to whom I am greatly indebted. Lucrino, Italy BERNARD H. LA VENDA www.com Introduction Formulations of non-equilibrium thermodynamics In the first part of this book (chapters 1-4), there is given a brief exposition of classical equilibrium and linear thermodynamics and a critical review of two recent formulations of non-equilibrium thermodynamics. Chapter I briefly presents the classic formulations of equilibrium thermody- namics.
There is a twofold objective: (1) to accentuate the inherent differences between the axiomatic formulation of Caratheodory and the phenomenologi- cal approach of Gibbs, and (2) to evaluate the relative merits of the two equilibrium formulations for the construction of a non-equilibrium theory. The conclusion is reached that although Caratheodory's theory is mathematically more rigorous, it lacks the elements which would make it readily adaptable as a basis for the development of a theory of non-equilibrium thermodynamics. Chapter 2 gives a chronological account of the developments in linear thermodynamics. Onsager's derivation of a class of reciprocal relations in which the flux is the time derivative of an extensive thermodynamic variable, forms the corner-stone of linear thermodynamics.
The advantage of placing Onsager's derivation in its historical original form is that it affords a better grasp of the reciprocal relations and their specificity. An objection is raised concerning the Onsager- Casimir demonstration, in that their interpretation of the principle of microscopic reversibility at equilibrium is apparently incongruous with the non-conservative nature of the phenomenological regression laws. Chapter 3 presents a resume of one school of thought which uses a formal statement of the second law, the so-called Clausius- Duhem inequality, as a restriction on the types of thermodynamic process that can occur in elastic materials. Why this restriction? Since the second law does not, in general, constitute a criterion of stability, this restriction would necessarily exclude various forms of nonlinear thermodynamic processes which are stable kineti- cally.
In the last section of this chapter there is an example of where the restriction on the form of the constitutive relations leads to a contradiction; the results invalidate the Clausius- Duhem inequality. Chapter 4 is a review of the work of still another school of thought, which www.com Xll INTRODUCTION uses the sign criteria of the second variation of the entropy and its time-rate-of- change as criteria of stability in the small. The inability to make a direct connection with the second law, on account of the fact that the first variation of the entropy does not vanish in a non-equilibrium stationary state, makes it necessary to turn to a weaker justification of the proposed stability criteria based on an analogy with a Liapounov function. Notwithstanding the fact that Liapounov's second method is addressed to stability in the large, the analogy is found to be spurious.
One of the two criteria ofLiapounov's second method is satisfied automatically by supposing the system to be in a state of local equilibrium. The criteria of local equilibrium have nothing whatsoever to do with the asymptotic stability properties of kinetic processes. Furthermore, it is shown that the sign criterion of the time-rate-of-change of the second variation of the entropy does not coincide with the necessary and sufficient conditions of stability that are obtained from Liapounov's first method. Quasi-thermodynamic approach to nonlinear thermodynamics In the second part of the book (chapters 5-9), there is undertaken a detailed exposition of an approach to nonlinear thermodynamics which is based on a confluence of thermodynamic and kinetic concepts regarding evolution and stability.
For the major part, the analyses are limited to thermodynamic systems that are found in the immediate neighbourhood of a non-equilibrium stationary state. Only in the last sections of chapters 7 and 9 is the approach extended to include the phenomena of nonlinear periodic processes in space and time that may occur at a finite distance from an unstable non-equilibrium stationary state. The fundamental idea is that the principles of thermodynamics are com- patible with, and can be sharpened by, the precise criteria of kinetic stability analysis. In carrying out the implications of the fundamental idea, it was found necessary to correlate thermodynamic variables, needed to specify the thermo- dynamic state, with mechanical variables that satisfy the same types of differential equation.
This approach is to be regarded as 'quasi- thermodynamic' in character and this is what distinguishes it from the thermodynamic approaches that have been discussed in the first part of the book. In chapter 5, the development of nonlinear thermodynamics is begun, having already appreciated the fact that linear thermodynamic processes evolve to a state of least dissipation of energy or equivalently to a state of minimum entropy production. This implies that the evolution of linear thermodynamic processes can be accounted for by a single thermodynamic potential. The relevant thermodynamic principle states that the entropy production is equal to the energy dissipated.
In nonlinear thermodynamics we are dealing with processes that occur in open systems in which the external forces prevent the system from relaxing to equilibrium. We can no longer expect that the linear thermodynamic principle will be valid or that the evolution of such processes can be accounted for in terms of the properties of a single thermodynamic potential. It is found www.com INTRODUCTION Xlll necessary to derive an extension of the thermodynamic principle which will govern the evolution of nonlinear thermodynamic processes. It is now found that the dissipation function is no longer synonymous to the entropy production but rather that their difference is a measure of the absorbed power.
This is to say, the absorbed power appears as the time-rate-of-change of the entropy less that which is dissipated. The thermodynamic principle of the balance of power forms the basis for the discussion of non-equilibrium variational principles in chapter 6 and the quasi-thermodynamic stability analysis of chapter 7. In chapter 6, there is a fairly complete treatment of the variational principles of linear and nonlinear thermodynamics.